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Kulyabov1, 2, ‡ +1Peoples’ Friendship University of Russia (RUDN University), +6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation +2Joint Institute for Nuclear Research +6 Joliot-Curie, Dubna, Moscow region, 141980, Russian Federation +Background: Hyperbolic complex numbers are used in the description of hyperbolic spaces. One +of the well-known examples of such spaces is the Minkowski space, which plays a leading role in +the problems of the special theory of relativity and electrodynamics. However, such numbers are +not very common in different programming languages. Purpose: Of interest is the implementation +of hyperbolic complex in scientific programming languages, in particular, in the Julia language. +Methods: The Julia language is based on the concept of multiple dispatch. This concept is an +extension of the concept of polymorphism for object-oriented programming languages. To implement +hyperbolic complex numbers, the multiple dispatching approach of the Julia language was used. +Results: The result is a library that implements hyperbolic numbers. Conclusions: Based on +the results of the study, we can conclude that the concept of multiple dispatching in scientific +programming languages is convenient and natural. +Keywords: Julia programming language, multiple dispatch, abstract data types, type conversion, parametric +structures, hyperbolic complex numbers +I. +INTRODUCTION +The Julia programming language [1, 2] is a promising language for scientific computing. At the moment, the Julia +language has reached a stable state. By design, Julia solves the problem of two languages. This problem lies in the fact +that for rapid prototyping, data processing and visualization, an interpreted dynamic language or a mathematical +package (Python, Matlab, etc.) is used, and for intensive numerical calculations, the program has to be rewritten in a +compiled language with static typing (C/ C++, Fortran). +An illustration of this problem can be seen in Python, which has gained wide popularity as an interface language-glue. +Numerous wrapper libraries were written on it, which used Python code to call C/C++ and Fortran functions from +precompiled libraries. For example, the well-known library NumPy [3] consists of 51% C code and only 47% Python +code (the remaining percentages are divided between C++, Fortran, JavaScript and Unix shell). +The Julia language combines the flexibility of dynamically typed interpreted languages with the performance of +statically typed compiled languages. +The basic part of the Julia language is very similar to other scientific programming languages, so it does not cause +difficulties in mastering. However, Julia’s core is built around the concept of multiple dispatch [4], which is rare in other +languages. It is in this mechanism that the essential difference of Julia from other languages lies, and its understanding +is essential for the full use of all the advantages of Julia. +A. +Paper structure +In the article, the authors paid great attention to illustrating the mechanism of multiple dispatch and other +mechanisms that are closely related to it. +In the first part of the article, we give the necessary definitions and illustrate the concept of multiple dispatch +with simple examples that allow you to understand the syntax associated with this part of the language and capture +the essence of this approach. In the second part, we give an example of the implementation of hyperbolic complex +numbers in the Julia language. This example allows you to touch not only multiple dispatch, but also the type casting +mechanism, the abstract type hierarchy, overloading arithmetic operators, and specifying user-defined data types. +\ast korolkova-av@rudn.ru +\dagger gevorkyan-mn@rudn.ru +‡ kulyabov-ds@rudn.ru +arXiv:2301.01707v1 [cs.MS] 4 Jan 2023 + +2 +II. +MULTIPLE DISPATCH +A. +Common definitions +Dynamic dispatch is a mechanism that allows you to choose which of the many implementations of a polymorphic +function (or operator) should be called in a given case [5]. In this case, the choice of one or another implementation +is carried out at the stage of program execution. Multiple dispatch is based on dynamic dispatch. In this case, the +choice of implementation of a polymorphic function is made based on the type, number, and order of the function’s +arguments. This is how runtime polymorphic dispatch is implemented [6, 7]. Note also that in addition to the term +multiple dispatch, the term multimethod is also used. +The mechanism of multiple dispatch is similar to the mechanism of overloading functions and operators, implemented, +for example, in the C++ language. Function overloading, however, is done exclusively at compile time, while multiple +dispatch should work at runtime as well (runtime polymorphism). +B. +Multiple dispatch in Julia +To illustrate the mechanism of multiple dispatch, we will give the following code example in the Julia language. +function f(x, y) +println("Generic implementation") +return x + y +end +function f(x) +println("For single argument") +return x +end +function f(x::Integer, y::Integer) +println("Implementation for integers") +return x + y +end +function f(x::String, y::String) +println("Implementation for strings") +return x * " " * y +end +function f(x::Tuple{Int, Int}, y::Tuple{Int, Int}) +println("Implementation for tuples of two integer elements") +return (x[1], x[2], y[1], y[2]) +end +In this example, we have created five implementations of the f function, which differ from each other in different +signatures. In terms of the Julia language, this means that one function f now has four different methods. In the first +two methods, we did not use type annotations, so the type of the arguments will be determined either at compile +time or at run time (as in interpreted languages). It is also worth noting that Julia uses dynamic JIT compilation +(just-in-time), so the compilation stage is not explicitly separated from the execution stage for the user. +The arguments of the following three methods are annotated with types, so they will only be called if the types +match the annotations. In the f for strings, the * concatenation operator is used. The choice of the multiplication sign +* instead of the more traditional addition sign + is justified by the creators of the language by the fact that string +concatenation is not a commuting operation, so it is more logical to use the multiplication sign for it, rather than the +addition sign, which is often used to denote commuting operations. +The following code snippet illustrates how multiple dispatch works at compile time. The @show macro is used to +print out the name of the function and the arguments passed to it. +@show f(2.0, 1) +@show f(2, 2) + +3 +@show f(0x2, 0x1) # numbers in hexadecimal system +@show f("Text", "line") +@show f(3) +@show f([1, 2], [3, 4]) +@show f((1, 2), (3, 4)) +• In the first line, we passed real (floating-point) type arguments to the function, so a generic implementation +call was made. Since the operator + is defined for floating point numbers, the function succeeded and gave the +correct result. +• Methods for integers were called in the second and third lines. Note that the Integer type is an abstract type +and includes signed and unsigned integers from 1 to 16 bytes in size, defined in the language core. Numbers +written in hexadecimal are interpreted by default as unsigned integers. +• The method for strings was called on the fourth line. In the fifth line, the method for one argument. +• The sixth line passed two arrays as arguments. The + operation is defined for arrays, so the function ran without +error and returned an element-wise sum. +• In the seventh line, the function arguments are tuples consisting of two integers. Since we defined a method for +such a combination of arguments, the function worked correctly. +Generic implementation +f(2.0, 1) = 3.0 +Implementation for integers +f(2, 2) = 4 +Implementation for integers +f(0x02, 0x01) = 0x03 +Implementation for strings +f("Text", "line") = "Text line" +For single argument +f(3) = 3 +Generic implementation +f([1, 2], [3, 4]) = [4, 6] +Implementation for tuples of two integer elements +f((1, 2), (3, 4)) = (1, 2, 3, 4) +The above examples will work correctly in languages that support function overloading and do not demonstrate the +specifics of dynamic dispatching, since the types of arguments are known at the compilation stage and are available to +the translator. +To test the work of dynamic method calls, consider the following code: +print("Enter an integer:") +# Read a string and convert to an integer type +@show n = parse(Int32, readline()) +if n > 0 +x = 1.2; y = 0.1 +else +x = 1; y = 2 +end +f(x, y) +Here, the types of variable values x and y are not known at compile time, as they depend on what number the user +enters during program execution. However, for the case of integer x and y the corresponding method is called. +III. +HYPERBOLIC NUMBERS +We will use hyperbolic numbers to illustrate the multiple dispatch capabilities of the Julia language, so we will limit +ourselves to the definition and basic arithmetic operations. + +4 +Hyperbolic numbers [8–11], along with elliptic and parabolic numbers, are a generalization of complex numbers. +Hyperbolic numbers can be defined as follows: +z = x + jy, j2 = 1, j \not = \pm 1. +The quantity j will be called the hyperbolic imaginary unit, and the quantities x and y will be called the real and +imaginary parts, respectively. +For two hyperbolic numbers z1 = x1 + jy1 and z2 = x2 + jy2 the following arithmetic operations are performed. +Addition: z1 + z2 = (x1 + x2) + j(y1 + y2). +Multiplication: z1z2 = (x1x2 + y1y2) + j(x1y2 + x2y1). +Conjugation: z\ast = x - jy. +Inverse number: z - 1 = +x +x2 + y2 - j +y +x2 - y2 . +Division: z1 +z2 += x1x2 - y1y2 +x2 +2 - y2 +2 ++ jx1y1 - x1y2 +x2 +2 - y2 +2 +. +The implementation of hyperbolic numbers is in many respects similar to the implementation of complex ones. +Operators +, -, * must be overloaded, and /, root extraction, exponentiation, elementary math functions, etc. At +the same time, for the purposes of illustrating the mechanism of operation of multiple dispatching, it is arithmetic +operations that are of primary interest. This is due to the fact that elementary functions take only one argument, +and it is enough to define only one method for them. In the case of arithmetic operators, it is necessary to provide +combinations of arguments of different numeric types. So, for example, it should be possible to add a hyperbolic +number to an integer, rational, irrational number, which automatically affects not only multiple dispatch, but also +type casting mechanisms, an abstract type hierarchy, and default constructor overloading. +Therefore, we will confine ourselves to examples of the implementation of precisely arithmetic operations and that’s +all, without touching on the more mathematically complex calculations of various elementary functions of a hyperbolic +number. +Note that in addition to the term hyperbolic numbers, there are also terms in the literature: double numbers, split +complex numbers, perplex numbers, hyperbolic numbers [8, 12–15]. +IV. +IMPLEMENTATION OF HYPERBOLIC NUMBERS IN JULIA +A. +Declaring a Data Structure +The implementation of hyperbolic numbers in Julia was based on the code for complex numbers available in +the official Julia repository. We also used the developments obtained in the implementation of parabolic complex +numbers [16]. New type Hyperbolic defined with an immutable structure: +struct Hyperbolic{T<:Real} <: Number +"Real part" +re::T +"Imaginary part" +jm::T +end +The structure is simple and contains only two fields of parametric type T. This requires that the type T was a +subtype of the abstract type Real (syntax T<:Real). The type Hyperbolic is a subtype of the abstract type Number +(see Fig. 1). Thus, hyperbolic numbers are built into an already existing hierarchy of numeric types. +After the structure is defined, a new object of type Hyperbolic can be created by calling the default constructor. +So, for example, the number h = 1 + j3 is given as follows: +h = Hyperbolic{Float64}(1, 3) +After creation, you can access the fields of the structure as h.re and h.jm, but an attempt changing the value of a +field of an already existing object will result in an error, since structs are immutable entities. +h = Hyperbolic(1, 3) + +5 +Number +Hyperbolic +Complex +Real +Integer +Signed +Int8 +Int16 +Int32 +Int64 +Int128 +Bool +Unsigned +UInt8 +UInt16 +UInt32 +UInt64 +UInt128 +Rational +AbstractFloat +Float16 +Float32 +Float64 +Legend: +Abstract type +Primitive type +Structure +Figure 1. Location of Hyperbolic Numbers in Julia’s Type Hierarchy +However, if the argument types are different, then the default constructor will not be able to implicitly cast and +create a new object. In this case, you must explicitly specify the parametric type +# Float64 и Int64 +h = Hyperbolic(1.0, 3) # Error +h = Hyperbolic{Float64}(1.0, 3) # Correct +B. +Additional constructors +The default constructor is a normal function whose name is the same as the type name. By creating additional +methods for this function, you can create additional constructors to handle various special cases. +So, for example, in order not to specify a parametric type every time, you should add a new constructor of the +following form: +"""Constructor №2""" +function Hyperbolic(x::Real, y::Real) +return Hyperbolic(promote(x, y)...) +end +The promote function casts the arguments passed to it to a common type and returns the result as a tuple. Postfix +operator ... unpacks the tuple and passes its elements as arguments to the constructor function. The language core +defines casting rules for all subtypes of the Real abstract type, so now the constructor will work correctly for any +combination of arguments, as long as the T<:Real rule is fulfilled. For example, the following code will work correctly: +# Rational и Float64 +h = Hyperbolic(1//3, pi) +>> Hyperbolic{Float64}(0.5, 3.141592653589793) +We passed a rational number (type Rational) and a built-in global constant (number \pi ) of type Float64 to the +constructor. After that, the type casting rule worked and both arguments were cast to the type Float64 as more +general. +Declaring two more additional constructors will allow you to specify hyperbolic numbers with zero imaginary part: +"""Constructor №3""" +function Hyperbolic{T}(x::Real) where {T<:Real} +return Hyperbolic{T}(x, 0) +end +"""Constructor №4""" +function Hyperbolic(x::Real) +return Hyperbolic(promote(x, 0)...) +end +Constructor number 3 is a parametric function that is declared using the where construct. The T is a subtype of the +abstract type Real. Constructor number 4 works similarly to constructor number 2. +Two more constructors will allow you to pass other hyperbolic numbers as an argument to the constructor. + +6 +"""Constructor №5""" +function Hyperbolic{T}(h::Hyperbolic) where {T<:Real} +Hyperbolic{T}(h.re, h.jm) +end +"""Constructor №6""" +function Hyperbolic(h::Hyperbolic) +return Hyperbolic(promote(h.re, h.jm)...) +end +For more convenience, you can also create a separate constant for the imaginary cost j: +const jm = Hyperbolic(0, 1) +C. +Data printing +To be able to print hyperbolic type values in a compact and readable form, you should add the appropriate methods +to the show function from the Base module. +function Base.show(io::IO, h::Hyperbolic) +print(io, h.re, "+", h.jm, "j") +end +Function show is used when printing data to the console, in particular, it is called by the println and macro @show. +The code and output listings below will assume that the show method has been added for hyperbolic numbers. +D. +Type casting +Before proceeding to the implementation of methods for arithmetic operations with hyperbolic numbers, it is +necessary to define the rules for type casting. To do this, create a new method for the function promote_rule from +the Base module. +function Base.promote_rule(::Type{Hyperbolic{T}}, ::Type{S}) where {T<:Real, S<:Real} +return Hyperbolic{promote_type(T, S)} +end +function Base.promote_rule(::Type{Hyperbolic{T}}, ::Type{Hyperbolic{S}}) where {T<:Real, +S<:Real} +\lhook → +return Hyperbolic{promote_type(T, S)} +end +As arguments in promote_rule parametric types are specified, which should be cast to one enclosing type. In our +case, this is possible if one of the types is a subtype of Real, then the enclosing type is Hyperbolic. +After adding methods for promote_rule, it becomes possible to use functions promote, promote_type and convert. +>>h = Hyperbolic(1 // 2) +>>promote(h, 1) +(1//2+0//1j, 1//1+0//1j) +>>promote_type(Hyperbolic{Int64}, Float32) +Hyperbolic{Float32} +The first function is already familiar to us. The second allows you to infer the enclosing type not of specific variable +values, but of the types themselves. A type in Julia is an object of the first kind (type DataType) and can be assigned +to other variables, passed as function arguments, and so on. +Function convert allows you to convert the type specific value, for example: +>>convert(Hyperbolic, 1) +1+0j +After adding methods for type casting, you can start adding methods for arithmetic operations. A feature of Julia is +the implementation of arithmetic operations not in the form of operators, but in the form of functions. For example, +the following calls are correct: + +7 +>>+(1,2) +3 +>>+(1,2,3,4) +10 +>>+((i for i in 1:10)...) # числа от 1 до 10 +55 +In this regard, adding methods for arithmetic operations is no different from the corresponding process for other +functions. +Adding methods for unary operations + and - is carried out as follows: +Base.:+(h::Hyperbolic) = Hyperbolic(+h.re, +h.jm) +Base.:-(h::Hyperbolic) = Hyperbolic(-h.re, -h.jm) +This is an abbreviated function declaration. +Similarly, methods are added for binary addition, subtraction, multiplication, and division. Here is the code for +addition and multiplication. +# Binary + and * +function Base.:+(x::Hyperbolic, y::Hyperbolic) +xx = x.re + y.re +yy = x.jm + y.jm +Hyperbolic(xx, yy) +end +function Base.:*(x::Hyperbolic, y::Hyperbolic) +xx = x.re * y.re + x.jm * y.jm +yy = x.re * y.jm + x.je * y.re +return Hyperbolic(xx, yy) +end +V. +CONCLUSION +We examined the mechanism of multiple dispatch underlying the Julia language, using the example of the implemen- +tation of hyperbolic numbers. This example allowed us to touch upon such concepts of the language as the hierarchy of +data types, composite data types, type casting mechanisms, function overloading (creating new methods for functions +in terms of the Julia language), etc. +ACKNOWLEDGMENTS +This paper has been supported by the RUDN University Strategic Academic Leadership Program. +[1] J. Bezanson, A. Edelman, S. Karpinski, V. B. 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Gevorkyan, Hyperbolic numbers as einstein numbers, Journal of Physics: Conference +Series 1557 (2020) 012027.1–5. doi:10.1088/1742-6596/1557/1/012027. +[12] P. Fjelstad, Extending special relativity via the perplex numbers, American Journal of Physics 54 (5) (1986) 416–422. +doi:10.1119/1.14605. +[13] W. Band, Comments on extending relativity via the perplex numbers, American Journal of Physics 56 (5) (1988) 469–469. +doi:10.1119/1.15582. +[14] J. Rooney, On the three types of complex number and planar transformations, Environment and Planning B: Planning and +Design 5 (1) (1978) 89–99. doi:10.1068/b050089. +[15] J. Rooney, Generalised complex numbers in mechanics, in: M. Ceccarelli, V. A. Glazunov (Eds.), Advances on Theory +and Practice of Robots and Manipulators, Vol. 22 of Mechanisms and Machine Science, Springer International Publishing, +Cham, 2014, pp. 55–62. doi:10.1007/978-3-319-07058-2_7. +[16] M. N. Gevorkyan, A. V. Korolkova, D. S. Kulyabov, Approaches to the implementation of generalized complex numbers in +the julia language, in: D. S. Kulyabov, K. E. Samouylov, L. A. Sevastianov (Eds.), Workshop on information technology and +scientific computing in the framework of the X International Conference Information and Telecommunication Technologies +and Mathematical Modeling of High-Tech Systems (ITTMM-2020), Vol. 2639 of CEUR Workshop Proceedings, Aachen, +2020, pp. 141–157. +URL http://ceur-ws.org/Vol-2639/paper-13.pdf + +Реализация гиперболических комплексных чисел на языке Julia +А. В. Королькова,1, \ast М. Н. Геворкян,1, \dagger and Д. С. Кулябов1, 2, ‡ +1Российский университет дружбы народов, +117198, Москва, ул. Миклухо-Маклая, д. 6 +2Объединённый институт ядерных исследований, +ул. Жолио-Кюри 6, Дубна, Московская область, Россия, 141980 +Предпосылки. Гиперболические комплексные числа применяются при описании гиперболи- +ческих пространств. Одним из известных примером таких пространств является пространство +Минковского, играющее ведущее значение в задачах частной теории относительности, электро- +динамики. Однако такие числа не очень распространены в разных языках программирования. +Цель. Представляет интерес реализация гиперболических комплексных в языках научного +программирования, в частности, в языке Julia. Методы. В основе языка Julia лежит концепция +множественной диспетчеризации (multiple dispatch). Эта концепция является расширением +концепции полиморфизма для объектно-ориентированных языков программирования. Для +реализации гиперболических комплексных чисел использован подход множественной дис- +петчеризацию языка Julia. Результаты. В результате получена библиотека, реализующая +гиперболические числа. Выводы. По результатам исследования можно сделать вывод об +удобстве и естественности концепции множественной диспетчеризации в языках научного +программирования. +Keywords: язык программирования Julia, множественная диспетчеризация, абстрактные типы данных, +конвертация типов, параметрические структуры, гиперболические комплексные числа +I. +ВВЕДЕНИЕ +Язык программирования Julia [1, 2] — это перспективный язык, предназначенный для научных вычислений. В +настоящий момент язык Julia достиг стабильного состояния. По замыслу разработчиков Julia решает проблему +двух языков. Данная проблема заключается в том, что для быстрого прототипирования, обработки данных и +визуализации используется интерпретируемый динамический язык или математический пакет (Python, Matlab и +т.д.), а для интенсивных численных расчётов программу приходится переписывать на компилируемом языке со +статической типизацией (C/C++, Fortran). +Иллюстрацию данной проблемы можно увидеть на примере языка Python, который приобрел широкую попу- +лярность в качестве интерфейсного «языка-клея». На нем было написано большое количество библиотек-обёрток, +которые использовали Python-код для вызова C/C++ и Fortran функций из предварительно скомпилированных +библиотек. Так, например, известная библиотека NumPy [3] на 51% состоит из кода на языке Си и лишь на 47% +из кода на языке Python (оставшиеся проценты делят между собой C++, Fortran, JavaScript и Unix shell). +Язык Julia совмещает в себе гибкости интерпретируемых языков с динамической типизацией и производитель- +ность компилируемых языков со статической типизацией. +Базовая часть языка Julia крайне схожа с другими языками научного программирования поэтому не вызывает +трудности при освоении. Однако ядро Julia построено вокруг концепцию множественной диспетчеризации +(multiple dispatch) [4], которая редко встречается в других языках. Именно в этом механизме лежит существенное +отличие Julia от других языков и его понимание существенно для полноценного использования всех преимуществ +Julia. +A. +Структура статьи +В статье авторы уделили большое внимание иллюстрации механизма множественной диспетчеризации и +других механизмов, которые близко с ней связаны. +В первой части статьи мы даем необходимые определения и иллюстрируем концепцию множественной +диспетчеризации на простых примерах, позволяющих понять синтаксис, связанный с этой частью языка и +\ast korolkova-av@rudn.ru +\dagger gevorkyan-mn@rudn.ru +‡ kulyabov-ds@rudn.ru +arXiv:2301.01707v1 [cs.MS] 4 Jan 2023 + +2 +уловить суть данного подхода. Во второй части мы приводим пример реализации гиперболических комплексных +чисел на языке Julia. Данный пример позволяет затронуть не только множественную диспетчеризацию, но и +механизм приведения типов, иерархию абстрактных типов, перегрузку арифметических операторов и задание +пользовательских типов данных. +II. +МНОЖЕСТВЕННАЯ ДИСПЕТЧЕРИЗАЦИЯ +A. +Общие определения +Динамическая диспетчеризация (dynamic dispatch) — это механизм, который позволяет выбрать какую из +множества реализаций полиморфной функции (или оператора) следует вызвать в данном конкретном случае [5]. +При этом выбор той или иной реализации осуществляется на стадии выполнения программы. Множественная +диспетчеризация основывается на динамической диспетчеризации. В этом случае выбор реализации полиморф- +ной функции делается исходя из типа, количества и порядка следования аргументов функции. Таким образом +реализуется полиморфизм времени выполнения (runtime polymorphic dispatch) [6, 7]. Заметим также, что кроме +термина «множественная диспетчеризация», также употребляется термин мультиметод. +Механизм множественной диспетчеризации похож на механизм перегрузки функций и операторов, реали- +зованный, например, в языке C++. Перегрузка функций, однако, осуществляется исключительно на стадии +компиляции, тогда как множественная диспетчеризация должна работать также и на стадии выполнения +программы (полиморфизм времени выполнения). +B. +Множественная диспетчеризация в Julia +Для иллюстрации механизма множественной диспетчеризации приведём следующий пример кода на языке +Julia. +function f(x, y) +println("Общая реализация") +return x + y +end +function f(x) +println("Для одного аргумента") +return x +end +function f(x::Integer, y::Integer) +println("Реализация для целых чисел") +return x + y +end +function f(x::String, y::String) +println("Реализация для строк") +return x * " " * y +end +function f(x::Tuple{Int, Int}, y::Tuple{Int, Int}) +println("Реализация для кортежей из двух целочисленных элементов") +return (x[1], x[2], y[1], y[2]) +end +В данном примере мы создали пять реализаций функции f, которые отличаются друг от друга разными +сигнатурами. В терминах языка Julia это означает, что у одной функции f теперь существует четыре разных +метода. В первых двух методах мы не использовали аннотаций типов, поэтому тип аргументов будет определен +или на стадии компиляции или на стадии выполнения программы (как в интерпретируемых языках). Стоит +также отметит, что Julia использует динамическую JIT-компиляцию (just-in-time), поэтому стадия компиляции +от стадии выполнения отделена для пользователя не явным образом. + +3 +Аргументы трех следующих методов аннотированы типами, поэтому будут вызываться только в случае +совпадения типов с аннотациями. В методе f для строк используется оператор конкатенации *. Выбор знака +умножения * вместо более традиционного знака сложения + обосновывается создателями языка тем, что +конкатенация строк операция не коммутирующая, поэтому более логично использовать для нее знак умножения, +а не сложения, которым чаще все принято обозначать коммутирующие операции. +Следующий фрагмент кода иллюстрирует работу множественной диспетчеризации на стадии компиляции. +Макрос @show служит для распечатки имени функции и переданных ей аргументов. +@show f(2.0, 1) +@show f(2, 2) +@show f(0x2, 0x1) # числа в шестнадцатеричной системе +@show f("Строка", "текста") +@show f(3) +@show f([1, 2], [3, 4]) +@show f((1, 2), (3, 4)) +• В первой строке мы передали функции аргументы вещественного типа (с плавающей точкой), поэтому +был осуществлен вызов общей реализации. Так как для чисел с плавающей точкой определен оператор +, +то функция выполнилась успешно и дала правильный результат. +• Во второй и третей строках были вызваны методы для целых чисел. Заметим, что тип Integer является +абстрактным типом и включает в себя знаковые и беззнаковые целые числа размером от 1 до 16 байт, +определённые в ядре языка. Числа, записанные в шестнадцатеричной системе счисления интерпретируются +по умолчанию как беззнаковые целые. +• В четвертой строке был вызван метод для строк. В пятой строке метод для одного аргумента. +• В шестой строке в качестве аргументов переданы два массива. Операция + определена для массивов, +поэтому функция выполнилась без ошибок и вернула поэлементную сумму. +• В седьмой строке аргументами функции являются кортежи, состоящие из двух целых чисел. Так как нами +был определен метод для такой комбинации аргументов – функция отработала корректно. +Общая реализация +f(2.0, 1) = 3.0 +Реализация для целых чисел +f(2, 2) = 4 +Реализация для целых чисел +f(0x02, 0x01) = 0x03 +Реализация для строк +f("Строка", "текста") = "Строка текста" +Для одного аргумента +f(3) = 3 +Общая реализация +f([1, 2], [3, 4]) = [4, 6] +Реализация для кортежей из двух целочисленных элементов +f((1, 2), (3, 4)) = (1, 2, 3, 4) +Приведённые примеры корректно сработают и в языках, поддерживающих перегрузку функций и не демон- +стрируют специфику динамической диспетчеризации, так как типы аргументов известны на стадии компиляции +и доступны транслятору. +Для проверки работы именно динамического вызова методов рассмотрим следующий код: +print("Введите целое число:") +# Считываем строку и конвертируем в целый тип +@show n = parse(Int32, readline()) +if n > 0 +x = 1.2; y = 0.1 +else +x = 1; y = 2 +end +f(x, y) + +4 +Здесь типы значений переменных x и y не известны на стадии компиляции, так как зависят от того, какое +число введёт пользователь во время выполнения программы. Тем не менее, для случая целочисленных x и y +вызывается соответствующий метод. +III. +ГИПЕРБОЛИЧЕСКИЕ ЧИСЛА +Мы будем использовать гиперболические числа для иллюстрации возможностей множественной диспетчериза- +ции языка Julia, поэтому ограничимся лишь определением и основными арифметическими операциями. +Гиперболические числа [8–11], наряду с эллиптическими и параболическими числами, являются обобщением +комплексных чисел. Гиперболические числа можно определить следующим образом: +z = x + jy, j2 = 1, j \not = \pm 1. +Величину j будем называть гиперболической мнимой единицей, а величины x и y действительной и мнимой +частями соответственно. +Для двух гиперболических чисел z1 = x1 + jy1 и z2 = x2 + jy2 выполняются следующие арифметические +операции. +Сложение: z1 + z2 = (x1 + x2) + j(y1 + y2). +Умножение: z1z2 = (x1x2 + y1y2) + j(x1y2 + x2y1). +Сопряжение: z\ast = x - jy. +Обратное число: z - 1 = +x +x2 + y2 - j +y +x2 - y2 . +Деление: z1 +z2 += x1x2 - y1y2 +x2 +2 - y2 +2 ++ jx1y1 - x1y2 +x2 +2 - y2 +2 +. +Реализация гиперболических чисел во многом аналогична реализации комплексных. Необходимо перегрузить +операторы +, -, * и /, функции извлечения корня, возведения в степень, элементарные математические функции +и т.д. При этом для целей иллюстрации механизма работы множественной диспетчеризации основной интерес +представляют именно арифметические операции. Это обусловлено тем, что элементарные функции принимают +только один аргумент и для них достаточно определить только один метод. В случае же арифметических +операторов необходимо предусмотреть комбинации аргументов разных числовых типов. Так, например, должна +иметься возможность сложения гиперболического числа с целым, рациональны, иррациональным числом, что +автоматически затрагивает не только множественную диспетчеризацию, но и механизмы приведения типов, +иерархию абстрактных типов и перегрузку конструктора по умолчанию. +Поэтому мы ограничимся примерами реализации именно арифметических операций и все, не затронув более +сложные в математическом плане вычисления разнообразных элементарных функций от гиперболического +числа. +Отметим, что кроме термина гиперболические числа, в литературе встречаются также термины: двойные +числа, расщепленные комплексные числа, комплексные числа гиперболического типа (double numbers, split +complex numbers, perplex numbers, hyperbolic numbers) [8, 12–15]. +IV. +РЕАЛИЗАЦИЯ ГИПЕРБОЛИЧЕСКИХ ЧИСЕЛ В JULIA +A. +Объявление структуры данных +При реализации гиперболических чисел в Julia за основу был взят код для комплексных чисел, доступный в +официальном репозитории Julia. Также использовались наработки, полученные при реализации параболических +комплексных чисел [16]. Новый тип Hyperbolic определяется с помощью неизменяемой структуры: +struct Hyperbolic{T<:Real} <: Number +"Real part" +re::T +"Imaginary part" +jm::T +end + +5 +Number +Hyperbolic Complex +Real +Integer +Signed +Int8 +Int16 +Int32 +Int64 +Int128 +Bool +Unsigned +UInt8 +UInt16 +UInt32 +UInt64 UInt128 +Rational +AbstractFloat +Float16 +Float32 +Float64 +Легенда: +Абстрактный тип +Примитивный тип +Структура +Рис. 1. Местоположение гиперболических чисел в иерархии типов Julia +Структура проста и содержит всего два поля параметрического типа T. При этом требуется, чтобы тип T был +подтипом абстрактного типа Real (синтаксис T<:Real). Сам тип Hyperbolic является подтипом абстрактного +типа Number (см рис. 1). Таким образом гиперболические числа встраиваются в уже существующую иерархию +числовых типов. +После определения структуры новый объект типа Hyperbolic можно создать путем вызова конструктора по +умолчанию. Так, например, число h = 1 + j3 задается следующим образом: +h = Hyperbolic{Float64}(1, 3) +После создания можно обращаться к полям структуры как h.re и h.jm, но попытка изменения значения поля +уже существующего объекта приведёт к ошибке, так как структуры являются неизменяемыми сущностями. +Если оба аргумента конструктора имеют один и тот же тип T, то его можно явно не указывать в фигурных +скобках, так как он будет выведен автоматически из типа передаваемых аргументов. +h = Hyperbolic(1, 3) +Однако, если типы аргументов отличаются, то конструктор по умолчанию не сможет осуществить неявное +приведение типов и создать новый объект. В этом случае необходимо явно указывать параметрический тип +# Float64 и Int64 +h = Hyperbolic(1.0, 3) # Error +h = Hyperbolic{Float64}(1.0, 3) # Correct +B. +Дополнительные конструкторы +Конструктор по умолчанию представляет собой обычную функцию, имя которой совпадает с именем типа. Со- +здавая дополнительные методы для этой функции можно создать дополнительные конструкторы для обработки +различных частных случаев. +Так, например, чтобы не указывать всякий раз параметрический тип, следует добавить новый конструктор +следующего вида: +"""Constructor №2""" +function Hyperbolic(x::Real, y::Real) +return Hyperbolic(promote(x, y)...) +end +Функция promote осуществляет приведение типов переданных ей аргументов к общему типу и возвращает +результат в виде кортежа. Постфиксный оператор ... распаковывает картеж и передает его элементы в виде +аргументов в функцию-конструктор. В ядре языка определены правила приведения для всех подтипов абстракт- +ного типа Real, поэтому теперь конструктор будет корректно работать для любой комбинации аргументов, +главное чтобы выполнялось правило T<:Real. Например, следующий код сработает корректно: +# Rational и Float64 +h = Hyperbolic(1//3, pi) +>> Hyperbolic{Float64}(0.5, 3.141592653589793) +Мы передали в конструктор рациональное число (тип Rational) и встроенную глобальную константу (число +\pi ) типа Float64. После чего сработало правило приведения типов и оба аргументы были приведены к типу +Float64 как к более общему. + +6 +Объявление еще двух дополнительных конструкторов позволит задавать гиперболические числа с нулевой +мнимой частью: +"""Constructor №3""" +function Hyperbolic{T}(x::Real) where {T<:Real} +return Hyperbolic{T}(x, 0) +end +"""Constructor №4""" +function Hyperbolic(x::Real) +return Hyperbolic(promote(x, 0)...) +end +Конструктор №3 является параметрической функцией, которая объявляется с использованием конструк- +ции where. Параметр T является подтипом абстрактного типа Real. Конструктор №4 работает аналогично +конструктору №2. +Ещё два конструктора позволят передавать в качестве аргумента конструктора другие гиперболические числа. +"""Constructor №5""" +function Hyperbolic{T}(h::Hyperbolic) where {T<:Real} +Hyperbolic{T}(h.re, h.jm) +end +"""Constructor №6""" +function Hyperbolic(h::Hyperbolic) +return Hyperbolic(promote(h.re, h.jm)...) +end +Для большего удобства также можно создать отдельную константу для мнимой единицы j: +const jm = Hyperbolic(0, 1) +C. +Вывод данных +Для возможности распечатывать значения гиперболического типа в компактном и читаемом виде, следует +добавить соответствующие методы для функции show из модуля Base. +function Base.show(io::IO, h::Hyperbolic) +print(io, h.re, "+", h.jm, "j") +end +Функция show используется при распечатке данных в консоль, в частности ее вызывают функция println +и макрос @show. В приведенных далее листингах кода и результатов его работы будет предполагаться, что +добавлен метод show для гиперболических чисел. +D. +Приведение типов +Прежде чем переходить к реализации методов для арифметических операций с гиперболическими числами, +необходимо определить правила приведения типов. Для этого следует создать новый метод для функции +promote_rule из модуля Base. +function Base.promote_rule(::Type{Hyperbolic{T}}, ::Type{S}) where {T<:Real, S<:Real} +return Hyperbolic{promote_type(T, S)} +end +function Base.promote_rule(::Type{Hyperbolic{T}}, ::Type{Hyperbolic{S}}) where {T<:Real, +S<:Real} +\lhook → +return Hyperbolic{promote_type(T, S)} +end +В качестве аргументов в promote_rule указываются параметрические типы, которые следует привести к +одному объемлющему типу. В нашем случае это возможно, если один из типов является подтипом Real, тогда +объемлющим типом будет тип Hyperbolic. + +7 +После добавления методов для promote_rule становится возможным использовать функции promote, +promote_type и convert. +>>h = Hyperbolic(1 // 2) +>>promote(h, 1) +(1//2+0//1j, 1//1+0//1j) +>>promote_type(Hyperbolic{Int64}, Float32) +Hyperbolic{Float32} +Первая функция нам уже знакома. Вторая же позволяет выводить объемлющий тип не конкретных значений +переменных, а самих типов. Тип в Julia является объектом первого рода (тип DataType) и его можно присваивать +другим переменным, передавать в качестве аргументов функции и т.д. +Функция convert позволяет преобразовать тип конкретного значения, например: +>>convert(Hyperbolic, 1) +1+0j +E. +Арифметические операции над гиперболическими числами +После добавления методов для приведения типов, можно приступить к добавлению методов для арифметиче- +ских операций. Особенностью Julia является реализация арифметических операций не в виде операторов, а в +виде функций. Так, например, корректны следующие вызовы: +>>+(1,2) +3 +>>+(1,2,3,4) +10 +>>+((i for i in 1:10)...) # числа от 1 до 10 +55 +В связи с этим, добавление методов для арифметических операций ничем не отличается от соответствующего +процесса для других функций. +Добавление методов для унарных операций + и - осуществляется следующим образом: +Base.:+(h::Hyperbolic) = Hyperbolic(+h.re, +h.jm) +Base.:-(h::Hyperbolic) = Hyperbolic(-h.re, -h.jm) +Здесь используется сокращенная запись объявления функции. +Аналогично добавляются методы для бинарного сложения, вычитания, умножения и деления. Приведем здесь +код для сложения и умножения. +# Binary + and * +function Base.:+(x::Hyperbolic, y::Hyperbolic) +xx = x.re + y.re +yy = x.jm + y.jm +Hyperbolic(xx, yy) +end +function Base.:*(x::Hyperbolic, y::Hyperbolic) +xx = x.re * y.re + x.jm * y.jm +yy = x.re * y.jm + x.je * y.re +return Hyperbolic(xx, yy) +end +V. +ЗАКЛЮЧЕНИЕ +Мы рассмотрели механизм множественной диспетчеризации, лежащий в основе языка Julia, на примере +реализации гиперболических чисел. Данный пример позволил затронуть такие понятия языка как иерархия +типов данных, составные типы данных, механизмы приведения типов, перегрузка функций (создание новых +методов для функций в терминах языка Julia) и т.д. + +8 +БЛАГОДАРНОСТИ +Публикация выполнена при поддержке Программы стратегического академического лидерства РУДН. +[1] Bezanson J., Edelman A., Karpinski S., Shah V. B. Julia: A fresh approach to numerical computing // SIAM Review. ---- +2017. ---- jan. ---- Vol. 59, no. 1. ---- P. 65--98. +[2] Gevorkyan M. N., Kulyabov D. S., Sevastyanov L. A. Review of Julia programming language for scientific computing // The +6th International Conference "Distributed Computing and Grid-technologies in Science and Education". ---- 2014. ---- P. 27. +[3] Oliphant T. E. Guide to NumPy. ---- 2nd edition. ---- CreateSpace Independent Publishing Platform, 2015. ---- ISBN: 978- +1517300074. +[4] Zappa Nardelli F., Belyakova J., Pelenitsyn A., Chung B., Bezanson J., Vitek J. Julia subtyping: a rational reconstruction // +Proceedings of the ACM on Programming Languages. ---- 2018. ---- oct. ---- Vol. 2, no. OOPSLA. ---- P. 1--27. +[5] Driesen K., H\"olzle U., Vitek J. Message Dispatch on Pipelined Processors // ECOOP'95 --- Object-Oriented Programming, +9th European Conference, \r Aarhus, Denmark, August 7--11, 1995 / Ed. by M. Tokoro, R. Pareschi. ---- Lecture Notes in +Computer Science. Springer Berlin Heidelberg, 1995. ---- 253--282 p. ---- ISBN: 9783540601609. +[6] Muschevici R., Potanin A., Tempero E., Noble J. Multiple dispatch in practice // OOPSLA'08: Proceedings of the 23rd +ACM SIGPLAN conference on Object-oriented programming systems languages and applications. ---- ACM Press, 2008. ---- +10. ---- P. 563--582. +[7] Gowda S., Ma Y., Cheli A., Gw\'o\'zzd\'z M., Shah V. B., Edelman A., Rackauckas C. High-Performance Symbolic-Numerics +via Multiple Dispatch // ACM Communications in Computer Algebra. ---- 2022. ---- jan. ---- Vol. 55, no. 3. ---- P. 92--96. +[8] Яглом И. М. Комплексные числа и их применение в геометрии // Математика, ее преподавание, приложения и +история. — 1961. — Т. 6 из Математическое просвещение, сер. 2. — С. 61–106. — Режим доступа: http://mi.mathnet. +ru/mp680. +[9] Яглом И. М., Розенфельд Б. А., Ясинская Е. У. Проективные метрики // Успехи математических наук. — 1964. — +Т. 19, № 5 (119). — С. 51–113. +[10] Kulyabov D. S., Korolkova A. V., Sevastianov L. A. Complex Numbers for Relativistic Operations. ---- 2021. ---- Dec. +[11] Kulyabov D. S., Korolkova A. V., Gevorkyan M. N. Hyperbolic numbers as Einstein numbers // Journal of Physics: +Conference Series. ---- 2020. ---- may. ---- Vol. 1557. ---- P. 012027. +[12] Fjelstad P. Extending special relativity via the perplex numbers // American Journal of Physics. ---- 1986. ---- may. ---- Vol. 54, +no. 5. ---- P. 416--422. +[13] Band W. Comments on Extending relativity via the perplex numbers // American Journal of Physics. ---- 1988. ---- may. ---- +Vol. 56, no. 5. ---- P. 469--469. +[14] Rooney J. On the Three Types of Complex Number and Planar Transformations // Environment and Planning B: Planning +and Design. ---- 1978. ---- Vol. 5, no. 1. ---- P. 89--99. +[15] Rooney J. Generalised Complex Numbers in Mechanics // Advances on Theory and Practice of Robots and Manipulators / +Ed. by M. Ceccarelli, V. A. Glazunov. ---- Cham : Springer International Publishing, 2014. ---- Vol. 22 of Mechanisms and +Machine Science. ---- P. 55--62. +[16] Gevorkyan M. N., Korolkova A. V., Kulyabov D. S. Approaches to the implementation of generalized complex numbers in +the Julia language // Workshop on information technology and scientific computing in the framework of the X International +Conference Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems (ITTMM- +2020) / Ed. by D. S. Kulyabov, K. E. Samouylov, L. A. Sevastianov. ---- Vol. 2639 of CEUR Workshop Proceedings. ---- Aachen, +2020. ---- apr. ---- P. 141--157. ---- Access mode: http://ceur-ws.org/Vol-2639/paper-13.pdf. + diff --git a/-9AzT4oBgHgl3EQfvf1g/content/tmp_files/load_file.txt b/-9AzT4oBgHgl3EQfvf1g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..16a95b812d231ef8407dbdd59700ca42a35e3581 --- /dev/null +++ b/-9AzT4oBgHgl3EQfvf1g/content/tmp_files/load_file.txt @@ -0,0 +1,762 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf,len=761 +page_content='Implementation of hyperbolic complex numbers in Julia language Anna V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Korolkova,1, \\ast Migran N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Gevorkyan,1, \\dagger and Dmitry S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Kulyabov1, 2, ‡ 1Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation 2Joint Institute for Nuclear Research 6 Joliot-Curie, Dubna, Moscow region, 141980, Russian Federation Background: Hyperbolic complex numbers are used in the description of hyperbolic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' One of the well-known examples of such spaces is the Minkowski space, which plays a leading role in the problems of the special theory of relativity and electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' However, such numbers are not very common in different programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Purpose: Of interest is the implementation of hyperbolic complex in scientific programming languages, in particular, in the Julia language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Methods: The Julia language is based on the concept of multiple dispatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' This concept is an extension of the concept of polymorphism for object-oriented programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' To implement hyperbolic complex numbers, the multiple dispatching approach of the Julia language was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Results: The result is a library that implements hyperbolic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Conclusions: Based on the results of the study, we can conclude that the concept of multiple dispatching in scientific programming languages is convenient and natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Keywords: Julia programming language, multiple dispatch, abstract data types, type conversion, parametric structures, hyperbolic complex numbers I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' INTRODUCTION The Julia programming language [1, 2] is a promising language for scientific computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' At the moment, the Julia language has reached a stable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' By design, Julia solves the problem of two languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' This problem lies in the fact that for rapid prototyping, data processing and visualization, an interpreted dynamic language or a mathematical package (Python, Matlab, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=') is used, and for intensive numerical calculations, the program has to be rewritten in a compiled language with static typing (C/ C++, Fortran).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' An illustration of this problem can be seen in Python, which has gained wide popularity as an interface language-glue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Numerous wrapper libraries were written on it, which used Python code to call C/C++ and Fortran functions from precompiled libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' For example, the well-known library NumPy [3] consists of 51% C code and only 47% Python code (the remaining percentages are divided between C++, Fortran, JavaScript and Unix shell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The Julia language combines the flexibility of dynamically typed interpreted languages with the performance of statically typed compiled languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The basic part of the Julia language is very similar to other scientific programming languages, so it does not cause difficulties in mastering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' However, Julia’s core is built around the concept of multiple dispatch [4], which is rare in other languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' It is in this mechanism that the essential difference of Julia from other languages lies, and its understanding is essential for the full use of all the advantages of Julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Paper structure In the article, the authors paid great attention to illustrating the mechanism of multiple dispatch and other mechanisms that are closely related to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In the first part of the article, we give the necessary definitions and illustrate the concept of multiple dispatch with simple examples that allow you to understand the syntax associated with this part of the language and capture the essence of this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In the second part, we give an example of the implementation of hyperbolic complex numbers in the Julia language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' This example allows you to touch not only multiple dispatch, but also the type casting mechanism, the abstract type hierarchy, overloading arithmetic operators, and specifying user-defined data types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' \\ast korolkova-av@rudn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='ru \\dagger gevorkyan-mn@rudn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='ru ‡ kulyabov-ds@rudn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='ru arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='01707v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='MS] 4 Jan 2023 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' MULTIPLE DISPATCH A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Common definitions Dynamic dispatch is a mechanism that allows you to choose which of the many implementations of a polymorphic function (or operator) should be called in a given case [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In this case, the choice of one or another implementation is carried out at the stage of program execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Multiple dispatch is based on dynamic dispatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In this case, the choice of implementation of a polymorphic function is made based on the type, number, and order of the function’s arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' This is how runtime polymorphic dispatch is implemented [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Note also that in addition to the term multiple dispatch, the term multimethod is also used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The mechanism of multiple dispatch is similar to the mechanism of overloading functions and operators, implemented, for example, in the C++ language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Function overloading, however, is done exclusively at compile time, while multiple dispatch should work at runtime as well (runtime polymorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Multiple dispatch in Julia To illustrate the mechanism of multiple dispatch, we will give the following code example in the Julia language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' function f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y) println("Generic implementation") return x + y end function f(x) println("For single argument") return x end function f(x::Integer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y::Integer) println("Implementation for integers") return x + y end function f(x::String,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y::String) println("Implementation for strings") return x * " " * y end function f(x::Tuple{Int,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Int},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y::Tuple{Int,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Int}) println("Implementation for tuples of two integer elements") return (x[1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' x[2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y[1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y[2]) end In this example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' we have created five implementations of the f function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' which differ from each other in different signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In terms of the Julia language, this means that one function f now has four different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In the first two methods, we did not use type annotations, so the type of the arguments will be determined either at compile time or at run time (as in interpreted languages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' It is also worth noting that Julia uses dynamic JIT compilation (just-in-time), so the compilation stage is not explicitly separated from the execution stage for the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The arguments of the following three methods are annotated with types, so they will only be called if the types match the annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In the f for strings, the * concatenation operator is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The choice of the multiplication sign instead of the more traditional addition sign + is justified by the creators of the language by the fact that string concatenation is not a commuting operation, so it is more logical to use the multiplication sign for it, rather than the addition sign, which is often used to denote commuting operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The following code snippet illustrates how multiple dispatch works at compile time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The @show macro is used to print out the name of the function and the arguments passed to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' @show f(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0, 1) @show f(2, 2) 3 @show f(0x2, 0x1) # numbers in hexadecimal system @show f("Text", "line") @show f(3) @show f([1, 2], [3, 4]) @show f((1, 2), (3, 4)) In the first line, we passed real (floating-point) type arguments to the function, so a generic implementation call was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Since the operator + is defined for floating point numbers, the function succeeded and gave the correct result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Methods for integers were called in the second and third lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Note that the Integer type is an abstract type and includes signed and unsigned integers from 1 to 16 bytes in size, defined in the language core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Numbers written in hexadecimal are interpreted by default as unsigned integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The method for strings was called on the fourth line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In the fifth line, the method for one argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The sixth line passed two arrays as arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The + operation is defined for arrays, so the function ran without error and returned an element-wise sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In the seventh line, the function arguments are tuples consisting of two integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Since we defined a method for such a combination of arguments, the function worked correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Generic implementation f(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0, 1) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0 Implementation for integers f(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2) = 4 Implementation for integers f(0x02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 0x01) = 0x03 Implementation for strings f("Text",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' "line") = "Text line" For single argument f(3) = 3 Generic implementation f([1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 4]) = [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 6] Implementation for tuples of two integer elements f((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 4)) = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 4) The above examples will work correctly in languages that support function overloading and do not demonstrate the specifics of dynamic dispatching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' since the types of arguments are known at the compilation stage and are available to the translator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' To test the work of dynamic method calls, consider the following code: print("Enter an integer:") # Read a string and convert to an integer type @show n = parse(Int32, readline()) if n > 0 x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='1 else x = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y = 2 end f(x, y) Here, the types of variable values x and y are not known at compile time, as they depend on what number the user enters during program execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' However, for the case of integer x and y the corresponding method is called.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' HYPERBOLIC NUMBERS We will use hyperbolic numbers to illustrate the multiple dispatch capabilities of the Julia language, so we will limit ourselves to the definition and basic arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 4 Hyperbolic numbers [8–11], along with elliptic and parabolic numbers, are a generalization of complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Hyperbolic numbers can be defined as follows: z = x + jy, j2 = 1, j \\not = \\pm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The quantity j will be called the hyperbolic imaginary unit, and the quantities x and y will be called the real and imaginary parts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' For two hyperbolic numbers z1 = x1 + jy1 and z2 = x2 + jy2 the following arithmetic operations are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Addition: z1 + z2 = (x1 + x2) + j(y1 + y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Multiplication: z1z2 = (x1x2 + y1y2) + j(x1y2 + x2y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Conjugation: z\\ast = x - jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Inverse number: z - 1 = x x2 + y2 - j y x2 - y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Division: z1 z2 = x1x2 - y1y2 x2 2 - y2 2 + jx1y1 - x1y2 x2 2 - y2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The implementation of hyperbolic numbers is in many respects similar to the implementation of complex ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Operators +, -, * must be overloaded, and /, root extraction, exponentiation, elementary math functions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' At the same time, for the purposes of illustrating the mechanism of operation of multiple dispatching, it is arithmetic operations that are of primary interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' This is due to the fact that elementary functions take only one argument, and it is enough to define only one method for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In the case of arithmetic operators, it is necessary to provide combinations of arguments of different numeric types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' So, for example, it should be possible to add a hyperbolic number to an integer, rational, irrational number, which automatically affects not only multiple dispatch, but also type casting mechanisms, an abstract type hierarchy, and default constructor overloading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Therefore, we will confine ourselves to examples of the implementation of precisely arithmetic operations and that’s all, without touching on the more mathematically complex calculations of various elementary functions of a hyperbolic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Note that in addition to the term hyperbolic numbers, there are also terms in the literature: double numbers, split complex numbers, perplex numbers, hyperbolic numbers [8, 12–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' IMPLEMENTATION OF HYPERBOLIC NUMBERS IN JULIA A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Declaring a Data Structure The implementation of hyperbolic numbers in Julia was based on the code for complex numbers available in the official Julia repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' We also used the developments obtained in the implementation of parabolic complex numbers [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' New type Hyperbolic defined with an immutable structure: struct Hyperbolic{T<:Real} <: Number "Real part" re::T "Imaginary part" jm::T end The structure is simple and contains only two fields of parametric type T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' This requires that the type T was a subtype of the abstract type Real (syntax T<:Real).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The type Hyperbolic is a subtype of the abstract type Number (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Thus, hyperbolic numbers are built into an already existing hierarchy of numeric types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' After the structure is defined, a new object of type Hyperbolic can be created by calling the default constructor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' So, for example, the number h = 1 + j3 is given as follows: h = Hyperbolic{Float64}(1, 3) After creation, you can access the fields of the structure as h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm, but an attempt changing the value of a field of an already existing object will result in an error, since structs are immutable entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' h = Hyperbolic(1, 3) 5 Number Hyperbolic Complex Real Integer Signed Int8 Int16 Int32 Int64 Int128 Bool Unsigned UInt8 UInt16 UInt32 UInt64 UInt128 Rational AbstractFloat Float16 Float32 Float64 Legend: Abstract type Primitive type Structure Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Location of Hyperbolic Numbers in Julia’s Type Hierarchy However, if the argument types are different, then the default constructor will not be able to implicitly cast and create a new object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In this case, you must explicitly specify the parametric type # Float64 и Int64 h = Hyperbolic(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0, 3) # Error h = Hyperbolic{Float64}(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0, 3) # Correct B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Additional constructors The default constructor is a normal function whose name is the same as the type name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' By creating additional methods for this function, you can create additional constructors to handle various special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' So, for example, in order not to specify a parametric type every time, you should add a new constructor of the following form: """Constructor №2""" function Hyperbolic(x::Real, y::Real) return Hyperbolic(promote(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=') end The promote function casts the arguments passed to it to a common type and returns the result as a tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Postfix operator .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' unpacks the tuple and passes its elements as arguments to the constructor function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The language core defines casting rules for all subtypes of the Real abstract type, so now the constructor will work correctly for any combination of arguments, as long as the T<:Real rule is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' For example, the following code will work correctly: # Rational и Float64 h = Hyperbolic(1//3, pi) >> Hyperbolic{Float64}(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='141592653589793) We passed a rational number (type Rational) and a built-in global constant (number \\pi ) of type Float64 to the constructor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' After that, the type casting rule worked and both arguments were cast to the type Float64 as more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Declaring two more additional constructors will allow you to specify hyperbolic numbers with zero imaginary part: """Constructor №3""" function Hyperbolic{T}(x::Real) where {T<:Real} return Hyperbolic{T}(x, 0) end """Constructor №4""" function Hyperbolic(x::Real) return Hyperbolic(promote(x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=') end Constructor number 3 is a parametric function that is declared using the where construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The T is a subtype of the abstract type Real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Constructor number 4 works similarly to constructor number 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Two more constructors will allow you to pass other hyperbolic numbers as an argument to the constructor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 6 """Constructor №5""" function Hyperbolic{T}(h::Hyperbolic) where {T<:Real} Hyperbolic{T}(h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm) end """Constructor №6""" function Hyperbolic(h::Hyperbolic) return Hyperbolic(promote(h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=') end For more convenience, you can also create a separate constant for the imaginary cost j: const jm = Hyperbolic(0, 1) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Data printing To be able to print hyperbolic type values in a compact and readable form, you should add the appropriate methods to the show function from the Base module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='show(io::IO, h::Hyperbolic) print(io, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, "+", h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm, "j") end Function show is used when printing data to the console, in particular, it is called by the println and macro @show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The code and output listings below will assume that the show method has been added for hyperbolic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Type casting Before proceeding to the implementation of methods for arithmetic operations with hyperbolic numbers, it is necessary to define the rules for type casting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' To do this, create a new method for the function promote_rule from the Base module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='promote_rule(::Type{Hyperbolic{T}}, ::Type{S}) where {T<:Real, S<:Real} return Hyperbolic{promote_type(T, S)} end function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='promote_rule(::Type{Hyperbolic{T}}, ::Type{Hyperbolic{S}}) where {T<:Real, S<:Real} \\lhook → return Hyperbolic{promote_type(T, S)} end As arguments in promote_rule parametric types are specified, which should be cast to one enclosing type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' In our case, this is possible if one of the types is a subtype of Real, then the enclosing type is Hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' After adding methods for promote_rule, it becomes possible to use functions promote, promote_type and convert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' >>h = Hyperbolic(1 // 2) >>promote(h, 1) (1//2+0//1j, 1//1+0//1j) >>promote_type(Hyperbolic{Int64}, Float32) Hyperbolic{Float32} The first function is already familiar to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' The second allows you to infer the enclosing type not of specific variable values, but of the types themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' A type in Julia is an object of the first kind (type DataType) and can be assigned to other variables, passed as function arguments, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Function convert allows you to convert the type specific value, for example: >>convert(Hyperbolic, 1) 1+0j After adding methods for type casting, you can start adding methods for arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' A feature of Julia is the implementation of arithmetic operations not in the form of operators, but in the form of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' For example, the following calls are correct: 7 >>+(1,2) 3 >>+(1,2,3,4) 10 >>+((i for i in 1:10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=') # числа от 1 до 10 55 In this regard, adding methods for arithmetic operations is no different from the corresponding process for other functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Adding methods for unary operations + and - is carried out as follows: Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' :+(h::Hyperbolic) = Hyperbolic(+h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, +h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm) Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' :-(h::Hyperbolic) = Hyperbolic(-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, -h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm) This is an abbreviated function declaration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Similarly, methods are added for binary addition, subtraction, multiplication, and division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Here is the code for addition and multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' # Binary + and * function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' :+(x::Hyperbolic, y::Hyperbolic) xx = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re yy = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm Hyperbolic(xx, yy) end function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' :*(x::Hyperbolic, y::Hyperbolic) xx = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re * y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re + x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm * y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm yy = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re * y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm + x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='je * y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re return Hyperbolic(xx, yy) end V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' CONCLUSION We examined the mechanism of multiple dispatch underlying the Julia language, using the example of the implemen- tation of hyperbolic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' This example allowed us to touch upon such concepts of the language as the hierarchy of data types, composite data types, type casting mechanisms, function overloading (creating new methods for functions in terms of the Julia language), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ACKNOWLEDGMENTS This paper has been supported by the RUDN University Strategic Academic Leadership Program.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' URL http://ceur-ws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='org/Vol-2639/paper-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='pdf Реализация гиперболических комплексных чисел на языке Julia А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Королькова,1, \\ast М.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Н.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Геворкян,1, \\dagger and Д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Кулябов1, 2, ‡ 1Российский университет дружбы народов, 117198, Москва, ул.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Миклухо-Маклая, д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 6 2Объединённый институт ядерных исследований, ул.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Жолио-Кюри 6, Дубна, Московская область, Россия, 141980 Предпосылки.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Гиперболические комплексные числа применяются при описании гиперболи- ческих пространств.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Одним из известных примером таких пространств является пространство Минковского, играющее ведущее значение в задачах частной теории относительности, электро- динамики.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Однако такие числа не очень распространены в разных языках программирования.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Цель.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Представляет интерес реализация гиперболических комплексных в языках научного программирования, в частности, в языке Julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Методы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В основе языка Julia лежит концепция множественной диспетчеризации (multiple dispatch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Эта концепция является расширением концепции полиморфизма для объектно-ориентированных языков программирования.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Для реализации гиперболических комплексных чисел использован подход множественной дис- петчеризацию языка Julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Результаты.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В результате получена библиотека, реализующая гиперболические числа.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Выводы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' По результатам исследования можно сделать вывод об удобстве и естественности концепции множественной диспетчеризации в языках научного программирования.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Keywords: язык программирования Julia, множественная диспетчеризация, абстрактные типы данных, конвертация типов, параметрические структуры, гиперболические комплексные числа I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ВВЕДЕНИЕ Язык программирования Julia [1, 2] — это перспективный язык, предназначенный для научных вычислений.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В настоящий момент язык Julia достиг стабильного состояния.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' По замыслу разработчиков Julia решает проблему двух языков.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Данная проблема заключается в том, что для быстрого прототипирования, обработки данных и визуализации используется интерпретируемый динамический язык или математический пакет (Python, Matlab и т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ), а для интенсивных численных расчётов программу приходится переписывать на компилируемом языке со статической типизацией (C/C++, Fortran).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Иллюстрацию данной проблемы можно увидеть на примере языка Python, который приобрел широкую попу- лярность в качестве интерфейсного «языка-клея».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' На нем было написано большое количество библиотек-обёрток, которые использовали Python-код для вызова C/C++ и Fortran функций из предварительно скомпилированных библиотек.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Так, например, известная библиотека NumPy [3] на 51% состоит из кода на языке Си и лишь на 47% из кода на языке Python (оставшиеся проценты делят между собой C++, Fortran, JavaScript и Unix shell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Язык Julia совмещает в себе гибкости интерпретируемых языков с динамической типизацией и производитель- ность компилируемых языков со статической типизацией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Базовая часть языка Julia крайне схожа с другими языками научного программирования поэтому не вызывает трудности при освоении.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Однако ядро Julia построено вокруг концепцию множественной диспетчеризации (multiple dispatch) [4], которая редко встречается в других языках.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Именно в этом механизме лежит существенное отличие Julia от других языков и его понимание существенно для полноценного использования всех преимуществ Julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Структура статьи В статье авторы уделили большое внимание иллюстрации механизма множественной диспетчеризации и других механизмов, которые близко с ней связаны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В первой части статьи мы даем необходимые определения и иллюстрируем концепцию множественной диспетчеризации на простых примерах, позволяющих понять синтаксис, связанный с этой частью языка и \\ast korolkova-av@rudn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='ru \\dagger gevorkyan-mn@rudn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='ru ‡ kulyabov-ds@rudn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='ru arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='01707v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='MS] 4 Jan 2023 2 уловить суть данного подхода.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Во второй части мы приводим пример реализации гиперболических комплексных чисел на языке Julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Данный пример позволяет затронуть не только множественную диспетчеризацию, но и механизм приведения типов, иерархию абстрактных типов, перегрузку арифметических операторов и задание пользовательских типов данных.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' МНОЖЕСТВЕННАЯ ДИСПЕТЧЕРИЗАЦИЯ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Общие определения Динамическая диспетчеризация (dynamic dispatch) — это механизм, который позволяет выбрать какую из множества реализаций полиморфной функции (или оператора) следует вызвать в данном конкретном случае [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' При этом выбор той или иной реализации осуществляется на стадии выполнения программы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Множественная диспетчеризация основывается на динамической диспетчеризации.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В этом случае выбор реализации полиморф- ной функции делается исходя из типа, количества и порядка следования аргументов функции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Таким образом реализуется полиморфизм времени выполнения (runtime polymorphic dispatch) [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Заметим также, что кроме термина «множественная диспетчеризация», также употребляется термин мультиметод.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Механизм множественной диспетчеризации похож на механизм перегрузки функций и операторов, реали- зованный, например, в языке C++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Перегрузка функций, однако, осуществляется исключительно на стадии компиляции, тогда как множественная диспетчеризация должна работать также и на стадии выполнения программы (полиморфизм времени выполнения).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Множественная диспетчеризация в Julia Для иллюстрации механизма множественной диспетчеризации приведём следующий пример кода на языке Julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' function f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y) println("Общая реализация") return x + y end function f(x) println("Для одного аргумента") return x end function f(x::Integer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y::Integer) println("Реализация для целых чисел") return x + y end function f(x::String,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y::String) println("Реализация для строк") return x * " " * y end function f(x::Tuple{Int,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Int},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y::Tuple{Int,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Int}) println("Реализация для кортежей из двух целочисленных элементов") return (x[1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' x[2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y[1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y[2]) end В данном примере мы создали пять реализаций функции f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' которые отличаются друг от друга разными сигнатурами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В терминах языка Julia это означает, что у одной функции f теперь существует четыре разных метода.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В первых двух методах мы не использовали аннотаций типов, поэтому тип аргументов будет определен или на стадии компиляции или на стадии выполнения программы (как в интерпретируемых языках).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Стоит также отметит, что Julia использует динамическую JIT-компиляцию (just-in-time), поэтому стадия компиляции от стадии выполнения отделена для пользователя не явным образом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 3 Аргументы трех следующих методов аннотированы типами, поэтому будут вызываться только в случае совпадения типов с аннотациями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В методе f для строк используется оператор конкатенации *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Выбор знака умножения * вместо более традиционного знака сложения + обосновывается создателями языка тем, что конкатенация строк операция не коммутирующая, поэтому более логично использовать для нее знак умножения, а не сложения, которым чаще все принято обозначать коммутирующие операции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Следующий фрагмент кода иллюстрирует работу множественной диспетчеризации на стадии компиляции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Макрос @show служит для распечатки имени функции и переданных ей аргументов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' @show f(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0, 1) @show f(2, 2) @show f(0x2, 0x1) # числа в шестнадцатеричной системе @show f("Строка", "текста") @show f(3) @show f([1, 2], [3, 4]) @show f((1, 2), (3, 4)) В первой строке мы передали функции аргументы вещественного типа (с плавающей точкой), поэтому был осуществлен вызов общей реализации.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Так как для чисел с плавающей точкой определен оператор +, то функция выполнилась успешно и дала правильный результат.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Во второй и третей строках были вызваны методы для целых чисел.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Заметим, что тип Integer является абстрактным типом и включает в себя знаковые и беззнаковые целые числа размером от 1 до 16 байт, определённые в ядре языка.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Числа, записанные в шестнадцатеричной системе счисления интерпретируются по умолчанию как беззнаковые целые.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В четвертой строке был вызван метод для строк.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В пятой строке метод для одного аргумента.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В шестой строке в качестве аргументов переданы два массива.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Операция + определена для массивов, поэтому функция выполнилась без ошибок и вернула поэлементную сумму.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В седьмой строке аргументами функции являются кортежи, состоящие из двух целых чисел.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Так как нами был определен метод для такой комбинации аргументов – функция отработала корректно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Общая реализация f(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0, 1) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0 Реализация для целых чисел f(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2) = 4 Реализация для целых чисел f(0x02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 0x01) = 0x03 Реализация для строк f("Строка",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' "текста") = "Строка текста" Для одного аргумента f(3) = 3 Общая реализация f([1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 4]) = [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 6] Реализация для кортежей из двух целочисленных элементов f((1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 4)) = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 4) Приведённые примеры корректно сработают и в языках,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' поддерживающих перегрузку функций и не демон- стрируют специфику динамической диспетчеризации,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' так как типы аргументов известны на стадии компиляции и доступны транслятору.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Для проверки работы именно динамического вызова методов рассмотрим следующий код: print("Введите целое число:") # Считываем строку и конвертируем в целый тип @show n = parse(Int32, readline()) if n > 0 x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='1 else x = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' y = 2 end f(x, y) 4 Здесь типы значений переменных x и y не известны на стадии компиляции, так как зависят от того, какое число введёт пользователь во время выполнения программы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Тем не менее, для случая целочисленных x и y вызывается соответствующий метод.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ГИПЕРБОЛИЧЕСКИЕ ЧИСЛА Мы будем использовать гиперболические числа для иллюстрации возможностей множественной диспетчериза- ции языка Julia, поэтому ограничимся лишь определением и основными арифметическими операциями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Гиперболические числа [8–11], наряду с эллиптическими и параболическими числами, являются обобщением комплексных чисел.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Гиперболические числа можно определить следующим образом: z = x + jy, j2 = 1, j \\not = \\pm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Величину j будем называть гиперболической мнимой единицей, а величины x и y действительной и мнимой частями соответственно.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Для двух гиперболических чисел z1 = x1 + jy1 и z2 = x2 + jy2 выполняются следующие арифметические операции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Сложение: z1 + z2 = (x1 + x2) + j(y1 + y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Умножение: z1z2 = (x1x2 + y1y2) + j(x1y2 + x2y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Сопряжение: z\\ast = x - jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Обратное число: z - 1 = x x2 + y2 - j y x2 - y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Деление: z1 z2 = x1x2 - y1y2 x2 2 - y2 2 + jx1y1 - x1y2 x2 2 - y2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Реализация гиперболических чисел во многом аналогична реализации комплексных.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Необходимо перегрузить операторы +, -, * и /, функции извлечения корня, возведения в степень, элементарные математические функции и т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' При этом для целей иллюстрации механизма работы множественной диспетчеризации основной интерес представляют именно арифметические операции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Это обусловлено тем, что элементарные функции принимают только один аргумент и для них достаточно определить только один метод.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В случае же арифметических операторов необходимо предусмотреть комбинации аргументов разных числовых типов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Так, например, должна иметься возможность сложения гиперболического числа с целым, рациональны, иррациональным числом, что автоматически затрагивает не только множественную диспетчеризацию, но и механизмы приведения типов, иерархию абстрактных типов и перегрузку конструктора по умолчанию.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Поэтому мы ограничимся примерами реализации именно арифметических операций и все, не затронув более сложные в математическом плане вычисления разнообразных элементарных функций от гиперболического числа.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Отметим, что кроме термина гиперболические числа, в литературе встречаются также термины: двойные числа, расщепленные комплексные числа, комплексные числа гиперболического типа (double numbers, split complex numbers, perplex numbers, hyperbolic numbers) [8, 12–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' РЕАЛИЗАЦИЯ ГИПЕРБОЛИЧЕСКИХ ЧИСЕЛ В JULIA A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Объявление структуры данных При реализации гиперболических чисел в Julia за основу был взят код для комплексных чисел, доступный в официальном репозитории Julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Также использовались наработки, полученные при реализации параболических комплексных чисел [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Новый тип Hyperbolic определяется с помощью неизменяемой структуры: struct Hyperbolic{T<:Real} <: Number "Real part" re::T "Imaginary part" jm::T end 5 Number Hyperbolic Complex Real Integer Signed Int8 Int16 Int32 Int64 Int128 Bool Unsigned UInt8 UInt16 UInt32 UInt64 UInt128 Rational AbstractFloat Float16 Float32 Float64 Легенда: Абстрактный тип Примитивный тип Структура Рис.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Местоположение гиперболических чисел в иерархии типов Julia Структура проста и содержит всего два поля параметрического типа T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' При этом требуется, чтобы тип T был подтипом абстрактного типа Real (синтаксис T<:Real).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Сам тип Hyperbolic является подтипом абстрактного типа Number (см рис.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Таким образом гиперболические числа встраиваются в уже существующую иерархию числовых типов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' После определения структуры новый объект типа Hyperbolic можно создать путем вызова конструктора по умолчанию.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Так, например, число h = 1 + j3 задается следующим образом: h = Hyperbolic{Float64}(1, 3) После создания можно обращаться к полям структуры как h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re и h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm, но попытка изменения значения поля уже существующего объекта приведёт к ошибке, так как структуры являются неизменяемыми сущностями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Если оба аргумента конструктора имеют один и тот же тип T, то его можно явно не указывать в фигурных скобках, так как он будет выведен автоматически из типа передаваемых аргументов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' h = Hyperbolic(1, 3) Однако, если типы аргументов отличаются, то конструктор по умолчанию не сможет осуществить неявное приведение типов и создать новый объект.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В этом случае необходимо явно указывать параметрический тип # Float64 и Int64 h = Hyperbolic(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0, 3) # Error h = Hyperbolic{Float64}(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='0, 3) # Correct B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Дополнительные конструкторы Конструктор по умолчанию представляет собой обычную функцию, имя которой совпадает с именем типа.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Со- здавая дополнительные методы для этой функции можно создать дополнительные конструкторы для обработки различных частных случаев.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Так, например, чтобы не указывать всякий раз параметрический тип, следует добавить новый конструктор следующего вида: """Constructor №2""" function Hyperbolic(x::Real, y::Real) return Hyperbolic(promote(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=') end Функция promote осуществляет приведение типов переданных ей аргументов к общему типу и возвращает результат в виде кортежа.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Постфиксный оператор .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' распаковывает картеж и передает его элементы в виде аргументов в функцию-конструктор.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В ядре языка определены правила приведения для всех подтипов абстракт- ного типа Real, поэтому теперь конструктор будет корректно работать для любой комбинации аргументов, главное чтобы выполнялось правило T<:Real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Например, следующий код сработает корректно: # Rational и Float64 h = Hyperbolic(1//3, pi) >> Hyperbolic{Float64}(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='141592653589793) Мы передали в конструктор рациональное число (тип Rational) и встроенную глобальную константу (число \\pi ) типа Float64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' После чего сработало правило приведения типов и оба аргументы были приведены к типу Float64 как к более общему.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 6 Объявление еще двух дополнительных конструкторов позволит задавать гиперболические числа с нулевой мнимой частью: """Constructor №3""" function Hyperbolic{T}(x::Real) where {T<:Real} return Hyperbolic{T}(x, 0) end """Constructor №4""" function Hyperbolic(x::Real) return Hyperbolic(promote(x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=') end Конструктор №3 является параметрической функцией, которая объявляется с использованием конструк- ции where.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Параметр T является подтипом абстрактного типа Real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Конструктор №4 работает аналогично конструктору №2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Ещё два конструктора позволят передавать в качестве аргумента конструктора другие гиперболические числа.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' """Constructor №5""" function Hyperbolic{T}(h::Hyperbolic) where {T<:Real} Hyperbolic{T}(h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm) end """Constructor №6""" function Hyperbolic(h::Hyperbolic) return Hyperbolic(promote(h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=') end Для большего удобства также можно создать отдельную константу для мнимой единицы j: const jm = Hyperbolic(0, 1) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Вывод данных Для возможности распечатывать значения гиперболического типа в компактном и читаемом виде, следует добавить соответствующие методы для функции show из модуля Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='show(io::IO, h::Hyperbolic) print(io, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, "+", h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm, "j") end Функция show используется при распечатке данных в консоль, в частности ее вызывают функция println и макрос @show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В приведенных далее листингах кода и результатов его работы будет предполагаться, что добавлен метод show для гиперболических чисел.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Приведение типов Прежде чем переходить к реализации методов для арифметических операций с гиперболическими числами, необходимо определить правила приведения типов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Для этого следует создать новый метод для функции promote_rule из модуля Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='promote_rule(::Type{Hyperbolic{T}}, ::Type{S}) where {T<:Real, S<:Real} return Hyperbolic{promote_type(T, S)} end function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='promote_rule(::Type{Hyperbolic{T}}, ::Type{Hyperbolic{S}}) where {T<:Real, S<:Real} \\lhook → return Hyperbolic{promote_type(T, S)} end В качестве аргументов в promote_rule указываются параметрические типы, которые следует привести к одному объемлющему типу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' В нашем случае это возможно, если один из типов является подтипом Real, тогда объемлющим типом будет тип Hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 7 После добавления методов для promote_rule становится возможным использовать функции promote, promote_type и convert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' >>h = Hyperbolic(1 // 2) >>promote(h, 1) (1//2+0//1j, 1//1+0//1j) >>promote_type(Hyperbolic{Int64}, Float32) Hyperbolic{Float32} Первая функция нам уже знакома.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Вторая же позволяет выводить объемлющий тип не конкретных значений переменных, а самих типов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Тип в Julia является объектом первого рода (тип DataType) и его можно присваивать другим переменным, передавать в качестве аргументов функции и т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Функция convert позволяет преобразовать тип конкретного значения, например: >>convert(Hyperbolic, 1) 1+0j E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Арифметические операции над гиперболическими числами После добавления методов для приведения типов, можно приступить к добавлению методов для арифметиче- ских операций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Особенностью Julia является реализация арифметических операций не в виде операторов, а в виде функций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Так, например, корректны следующие вызовы: >>+(1,2) 3 >>+(1,2,3,4) 10 >>+((i for i in 1:10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=') # числа от 1 до 10 55 В связи с этим, добавление методов для арифметических операций ничем не отличается от соответствующего процесса для других функций.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Добавление методов для унарных операций + и - осуществляется следующим образом: Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' :+(h::Hyperbolic) = Hyperbolic(+h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, +h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm) Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' :-(h::Hyperbolic) = Hyperbolic(-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re, -h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm) Здесь используется сокращенная запись объявления функции.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Аналогично добавляются методы для бинарного сложения, вычитания, умножения и деления.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Приведем здесь код для сложения и умножения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' # Binary + and * function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' :+(x::Hyperbolic, y::Hyperbolic) xx = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re yy = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm Hyperbolic(xx, yy) end function Base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' :*(x::Hyperbolic, y::Hyperbolic) xx = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re * y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re + x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm * y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm yy = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re * y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='jm + x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='je * y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='re return Hyperbolic(xx, yy) end V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ЗАКЛЮЧЕНИЕ Мы рассмотрели механизм множественной диспетчеризации, лежащий в основе языка Julia, на примере реализации гиперболических чисел.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Данный пример позволил затронуть такие понятия языка как иерархия типов данных, составные типы данных, механизмы приведения типов, перегрузка функций (создание новых методов для функций в терминах языка Julia) и т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content='д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 8 БЛАГОДАРНОСТИ Публикация выполнена при поддержке Программы стратегического академического лидерства РУДН.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' [1] Bezanson J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=', Edelman A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=', Karpinski S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=', Shah V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' Julia: A fresh approach to numerical computing // SIAM Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 59, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' ---- P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' 65--98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' [2] Gevorkyan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9AzT4oBgHgl3EQfvf1g/content/2301.01707v1.pdf'} +page_content=' N.' metadata={'source': 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b/0NE5T4oBgHgl3EQfOQ6v/content/tmp_files/2301.05496v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bc446c88d3e33dee8ba5d2c8f1a153dc39205a54 --- /dev/null +++ b/0NE5T4oBgHgl3EQfOQ6v/content/tmp_files/2301.05496v1.pdf.txt @@ -0,0 +1,1515 @@ +Learning Transformations To Reduce the Geometric Shift in Object Detection +Vidit Vidit1 Martin Engilberge1 Mathieu Salzmann1,2 +CVLab, EPFL1, ClearSpace SA2 +firstname.lastname@epfl.ch +Abstract +The performance of modern object detectors drops when +the test distribution differs from the training one. Most of +the methods that address this focus on object appearance +changes caused by, e.g., different illumination conditions, +or gaps between synthetic and real images. Here, by con- +trast, we tackle geometric shifts emerging from variations in +the image capture process, or due to the constraints of the +environment causing differences in the apparent geometry +of the content itself. We introduce a self-training approach +that learns a set of geometric transformations to minimize +these shifts without leveraging any labeled data in the new +domain, nor any information about the cameras. We evalu- +ate our method on two different shifts, i.e., a camera’s field +of view (FoV) change and a viewpoint change. Our results +evidence that learning geometric transformations helps de- +tectors to perform better in the target domains. +1. Introduction +While modern object detectors [1, 2, 17, 23, 24] achieve +impressive results, their performance decreases when the +test data depart from the training distribution. This prob- +lem arises in the presence of appearance variations due to, +for example, differing illumination or weather conditions. +Considering the difficulty and cost of acquiring annotated +data in the test (i.e., target) domain, Unsupervised Domain +Adaptation (UDA) has emerged as the standard strategy to +address such scenarios [3,4,9,26,38]. +In this context, much effort has been made to learn do- +main invariant features, such that the source and target dis- +tributions in this feature space are similar. This has led to +great progress in situations where the appearance of the ob- +jects changes drastically from one domain to the other, as +in case of real-to-sketch adaptation (e.g., Pascal VOC [10] +to Comics [15]), or weather adaptation (e.g., Cityscapes [6] +to Foggy Cityscapes [27]). Nevertheless, such object ap- +pearance changes are not the only sources of domain shifts. +They can also have geometric origins. +For example, as +shown in Fig. 1, they can be due to a change in camera view- +Figure 1. +Geometric shifts. +(Left) Due to a different FoV, +the cars highlighted in green, undergo different distortions even +though they appear in similar image regions. (Right) Different +camera viewpoints (front facing vs downward facing) yield dif- +ferent distortions and occlusion patterns for pedestrian detection. +(Bottom) The distributions of pedestrian bounding box sizes in +Cityscapes [6] and MOT [8] differ significantly as the pedestrians +are usually far away or in the periphery in Cityscapes. The top im- +ages are taken from Cityscapes [6], and the bottom-left and right +ones from KITTI [12] and MOT [8], respectively. +point or field-of-view (FoV), or a change of object scale due +to different scene setups. In practice, such geometric shifts +typically arise from a combination of various factors, in- +cluding but not limited to the ones mentioned above. +In this paper, we introduce a domain adaptation approach +tackling such geometric shifts. To the best of our knowl- +edge, the recent work of [13] constitutes the only attempt at +considering such geometric distortions. However, it intro- +duces a method solely dedicated to FoV variations, assum- +ing that the target FoV is fixed and known. Here, we de- +1 +arXiv:2301.05496v1 [cs.CV] 13 Jan 2023 + +rkplatz/F +Cityscapes +Apparent Bbox Distribution +MOT20velop a more general framework able to cope with a much +broader family of geometric shifts. +To this end, we model geometric transformations as a +combination of multiple homographies. We show both the- +oretically and empirically that this representation is suffi- +cient to encompass a broad variety of complex geometric +transformations. We then design an aggregator block that +can be incorporated to the detector to provide it with the +capacity to tackle geometric shifts. We use this modified +detector to generate pseudo labels for the target domain, +which let us optimize the homographies so as to reduce the +geometric shift. +Our contributions can be summarized as follows. (i) We +tackle the problem of general geometric shifts for object +detection. (ii) We learn a set of homographies using unla- +beled target data, which alleviates the geometric bias arising +in source-only training. (iii) Our method does not require +prior information about the target geometric distortions and +generalizes to a broad class of geometric shifts. Our ex- +periments demonstrate the benefits of our approach in sev- +eral scenarios. In the presence of FoV shifts, our approach +yields similar performance to the FoV-dedicated framework +of [13] but without requiring any camera information. As +such, it generalizes better to other FoVs. Furthermore, we +show the generality of our method by using it to adapt to +a new camera viewpoint in the context of pedestrian detec- +tion. +2. Related Work +Unsupervised Domain Adaptation (UDA). +UDA for +image recognition [11, 21, 22, 30, 32, 35, 36] and object de- +tection [3,4,9,20,26,38] has made a great progress in the +past few years. The common trend in both tasks consists +of learning domain invariant features. For object detection, +this entails aligning the global (e.g., illumination, weather) +and local (foreground objects) features in the two domains. +In this context, [3,5,26,28] align image- and instance-level +features in the two domains via adversarial learning [11]; +[33] learns category-specific attention maps to better align +specific image regions; [38] clusters the proposed object re- +gions using k-means clustering and uses the centroids for +instance-level alignment. While this successfully tackles +domain shifts caused by object appearance variations, it +fails to account for the presence of shifts due to the image +capture process itself, such as changes in camera intrinsics +or viewpoint. The only initial step at considering a geo- +metric shift is the work of [13], which shows the existence +of an FoV gap in driving datasets [6, 12] and proposes a +Position Invariant Transform (PIT) that corrects the distor- +tions caused specifically by an FoV change. In essence, PIT +undistorts the images by assuming knowledge of the target +FoV. By contrast, here, we introduce an approach that gen- +eralizes to a broad family of geometric shifts by learning +transformations without requiring any camera information. +Self-training. +Self-training, generally employed in the +semi-supervised setting, offers an alternative to learning +domain-invariant features and utilize unlabeled data to im- +prove a detector’s performance. In this context, [29] uses +a student-teacher architecture where the teacher model is +trained with supervised data and generates pseudo-labels +on unannotated data. These pseudo-labels are then used +to train a student model. +While effective in the stan- +dard semi-supervised learning scenario, the quality of the +pseudo-labels obtained with this approach tends to deteri- +orate when the labeled and unlabeled data present a distri- +bution shift. [9, 20] have therefore extended this approach +to domain adaptation by using the Mean Teacher strategy +of [31] to generate reliable pseudo-labels in the target do- +main. Other approach include the use of CycleGAN [37] +generated images to train an unbiased teacher model [9], +and that of different augmentation strategies to generate ro- +bust pseudo-labels [20]. Our approach also follows a self- +training strategy but, while these works focus on object ap- +pearance shifts, we incorporate learnable blocks to address +geometric shifts. As shown in our experiment, this lets us +outperform the state-of-the-art AdaptTeacher [20]. +Learning +Geometric +Transformations. +End-to-end +learning of geometric transformations has been used to +boost the performance of deep networks. +For example, +Spatial Transformer Networks (STNs) [16] reduce the +classification error by learning to correct for affine trans- +formations; deformable convolutions [7] model geometric +transformations by applying the convolution kernels to +non-local neighborhoods. These methods work well when +annotations are available for supervision, and make the +network invariant to the specific geometric transformations +seen during training. Here, by contrast, we seek to learn +transformations in an unsupervised manner and allow the +network to generalize to unknown target transformations. +3. Modeling Geometric Transformations +In the context of UDA, multiple geometric differences +can be responsible for the gap between the domains. Some +can be characterized by the camera parameters, such as a +change in FoV (intrinsic) or viewpoint (extrinsic), whereas +others are content specific, such as a difference in road +width between different countries. Ultimately, the geomet- +ric shift is typically a combination of different geometric +operations. Since the parameters of these operations are un- +known, we propose to bridge the domain gap by learning a +geometric transform. Specifically, we aggregate the results +of multiple perspective transforms, i.e., homographies, to +obtain a differentiable operation that can emulate a wide +variety of geometric transforms. +2 + +3.1. Theoretical Model +Let us first show that, given sufficiently many homogra- +phies, one can perfectly reproduce any mapping between +R2 \ (0, 0) and R2. +Single homography for a single point. +First, we show +that a single homography with 4 degrees of freedom can +map a point p ∈ R2 \(0, 0) to any other point in R2. To this +end, let +H = +� +� +sx +0 +0 +0 +sy +0 +lx +ly +1 +� +� +(1) +be a homography, with (sx, sy) the scaling factors on the x- +and y-axis, respectively, and (lx, ly) the perspective factors +in x and y, respectively. For any destination point d ∈ R2, +there exists a set of parameters (sx, sy, lx, ly) such that d = +H × p. One such set is ( dx +px , dy +py , 0, 0). +Emulating any geometric transformation +Now that we +have shown that a single homography can move a point to +any other point in R2, we describe a simple protocol to emu- +late any geometric transform. Given an unknown geometric +transform T : R2\(0, 0) → R2, we aim to emulate T with a +set of homographies. In general, for an image I ∈ R3×h×w, +we can restrict the domain of T to only image coordinates. +To this end, we can define a set of homographies Hi ∈ H +for i in {1, 2, 3, ..., h × w}, where the parameters of Hi are +chosen to mimic the transform T for location i of the image. +In this protocol, the aggregation mechanism is trivial since +each homography is in charge of remapping a single pixel +coordinate of the original space. +While this works in theory, this is of course not viable +in practice since it would require too many homographies. +With a smaller number of homographies, each transform +needs to remap multiple points, and a more sophisticated +aggregation mechanism is required. Specifically, the ag- +gregation mechanism needs to select which transform is in +charge of remapping which point. In the next section, we +empirically show that this strategy lets us closely approxi- +mate the spherical projection mapping used in PIT [13]. +3.2. Approximating PIT with Homographies +To demonstrate the possibility offered by aggregating +multiple homographies, we design an approximation of PIT +using only homographies. PIT proposes to correct for an +FoV gap by remapping images to a spherical surface. Dur- +ing this transformation, regions further from the center of +a scene are compressed with a higher ratio. This variable +compression of the space cannot be reproduced by a single +homography transformation. To overcome this limitation, +we combine the results of multiple homographies that all +have different compression rates (scaling parameters). For +the aggregation mechanism, we use the optimal strategy by +selecting for each pixel the homography that approximates +best the PIT mapping. As shown in Fig. 2, this combination +closely approximates the PIT results with only 5 homogra- +phies. Further analysis of these experiments is available in +the supplementary material in Fig. A.3. +Figure 2. Approximating PIT with homographies. We show the +original image (top), the PIT [13] correction (middle), and our ap- +proximation of PIT using 5 homographies. Note that 5 homogra- +phies are sufficient to closely match the PIT spherical correction. +3.3. Homographies in a Learning Setup +In the two previous sections, we have demonstrated both +theoretically and empirically the flexibility of aggregating +homographies. This makes this representation an ideal can- +didate for domain adaptation since the geometric shift be- +tween the domains is unknown and can be a combination +of different transforms, such as FoV change, viewpoint +change, camera distortion, or appearance distortion. As will +be discussed in the next section, by learning jointly the set +of perspective transforms and the aggregation mechanism +on real data, our model can reduce the geometric shift be- +tween the two domains without prior knowledge about this +domain gap. +4. Method +Let us now introduce our approach to reducing the ge- +ometric shift in object detection. Following the standard +UDA setting, let Ds = {(Is, Bs, Cs)} be a labeled source +dataset containing images Is = {Ii +s}Ns +1 +with correspond- +ing object bounding boxes Bs = {bi +s}Ns +1 +and object classes +Cs = {ci +s}Ns +1 . Furthermore, let Dt = {It} denote an un- +labeled target dataset for which only images It = {Ii +t}Nt +1 +3 + +Original +PIT +Approximation 5 H.Figure 3. Architecture: The input image is first transformed by a set of trainable homographies. The feature maps extracted from +the transformed images are then unwarped by the inverse homographies to achieve spatial consistency. We then combine the unwarped +feature maps using a trainable aggregator, whose output is passed to a detection head. The blocks shown in green correspond to standard +FasterRCNN operations. The � symbol represents the concatenation operation. +are available, without annotations. Here, we tackle the case +where the two domains differ by geometric shifts but as- +sume no knowledge about the nature of these shifts. Below, +we first introduce the architecture we developed to handle +this and then our strategy to train this model. +4.1. Model Architecture +The overall architecture of our approach is depicted in +Fig. 3. In essence, and as discussed in Sec. 3, we charac- +terize the geometric changes between the source and target +data by a set of transformations T = {Hi}N +1 . Each Hi in +T is a homography of the same form as in Eq. (1). For our +method to remain general, we assume the transformations to +be unknown, and our goal, therefore, is to learn T to bridge +the gap between the domains. This requires differentiabil- +ity w.r.t. the transformation parameters, which we achieve +using the sampling strategy proposed in [16]. +As shown in Fig. 3, the input image is transformed by the +individual homographies in T , and the transformed images +are fed to a modified FasterRCNN [24] detector. Specif- +ically, we extract a feature map FHi ∈ RH×W ×C for +each transformed image via a feature extractor shared by +all transformations. To enforce spatial correspondence be- +tween the different FHis, we unwarp them with H−1 +i +. +We then introduce an aggregator Aθg, parameterized by +θg, whose goal is to learn a common representation given +a fixed number of unwarped feature maps F′ +Hi. To achieve +this, the aggregator takes as input +G = F′ +H1 ⊕ F′ +H2 ⊕ ... ⊕ F′ +HN ∈ RH×W ×C×N , +(2) +where ⊕ represents concatenation in the channel dimension. +The aggregator outputs a feature map Aθg(G) ∈ RH×W ×C, +whose dimension is independent of the number of transfor- +mations. This output is then passed to a detection head to +obtain the objects’ bounding boxes and class labels. +4.2. Model Training +Our training procedure relies on three steps: (i) Fol- +lowing common practice in UDA, we first train the Faster- +RCNN detector with source-only data; (ii) We then intro- +duce the aggregator and train it so that it learns to com- +bine different homographies using the labeled source data; +(iii) Finally, we learn the optimal transformations for adap- +tation using both the source and target data via a Mean +Teacher [31] strategy. +Aggregator Training. +To train the aggregator, we ran- +domly sample a set of homographies T ∈ RN×4 in each +training iteration.1 This gives the aggregator the ability to +robustly combine diverse input transformations but requires +strong supervision to avoid training instabilities. We, there- +fore, perform this step using the source data. +The loss function for a set of transformed images T (Is) +is then defined as in standard FasterRCNN training with +a combination of classification and regression terms [24]. +That is, we train the aggregator by solving +min +θg Lcls(T (Is)) + Lreg(T (Is)) , +(3) +1As our homographies involve only 4 parameters, with a slight abuse +of notation, we say that Hi ∈ R4. +4 + +Homography +Feature +Homography +Detection +Aggregator +Projections +Extractor +Projections +Headwhere +Lcls(T (Is)) = Lrpn +cls + Lroi +cls , +(4) +Lreg(T (Is)) = Lrpn +reg + Lroi +reg . +(5) +Lrpn +· +and Lroi +· +correspond to the Region Proposal Network +(RPN) loss terms and the Region of Interest (RoI) ones, re- +spectively. During this process, we freeze the parameters +θb of the base network, i.e, feature extractor and detection +head, which were first trained on the source data without ag- +gregator. Ultimately, the aggregator provides the network +with the capacity to encode different transformations that +are not seen in the source domain. The third training step +then aims to learn the best transformation for successful ob- +ject detection in the target domain. +Learning the Transformations. +As we have no annota- +tions in the target domain, we exploit a Mean Teacher (MT) +strategy to learn the optimal transformations. To this end, +our starting point is the detector with a trained aggregator +and a set of random transformations T . The MT strategy is +illustrated in Fig. 4. In essence, MT training [31] involves +two copies of the model: A student model, with parameters +θst = {T st, θst +b , θst +g }, that will be used during inference, +and a teacher model, with parameters θte = {T te, θte +b , θte +g }, +that is updated as an Exponentially Moving Average (EMA) +of the student model. +That is, the student’s parameters +are computed with standard backpropagation, whereas the +teacher’s ones are updated as +θte ← αθte + (1 − α)θst . +(6) +The student model is trained using both source and tar- +get detection losses. Since the target domain does not have +annotations, the teacher model is used to generate pseudo- +labels. These labels might be noisy, and hence we only keep +the predictions with a confidence score above a threshold τ. +Furthermore, non-maxima suppression (NMS) is used to re- +move the highly-overlapping bounding box predictions. +Formally, given a source image Is and a target image It, +the student model is trained by solving +min +T st,θst +g ,θst +b +Ldet(T (Is)) + λLdet(T (It)) , +(7) +where λ controls the target domain contribution and +Ldet(T (Is)) = Lcls(T (Is)) + Lreg(T (Is)) , +(8) +Ldet(T (It)) = Lcls(T (It)) . +(9) +Similarly to [18,20], we update the student model with only +the classification loss in the target domain to help stabilize +training. +Figure 4. Mean Teacher formalism. The student model is trained +with ground-truth labels in the source domain and pseudo labels in +the target one. These pseudo labels are produced by the teacher +model, which corresponds to an exponentially moving average +(EMA) of the student network. +5. Experiments +We demonstrate the effectiveness and generality of our +method on different geometric shifts. First, to compare to +the only other work that modeled a geometric shift [13], +we tackle the problem of a change in FoV between the +source and target domain. Note that, in contrast to [13], we +do not assume knowledge of the target FoV. Furthermore, +while [13] was dedicated to FoV adaptation, our approach +generalizes to other geometric shifts. We demonstrate this +on the task of pedestrian detection under a viewpoint shift. +We compare our method with the state-of-the-art Adapt- +Teacher [20], which also uses a Mean Teacher, but focuses +on appearance shifts. In the remainder of this section, we +describe our experimental setup and discuss our results. +5.1. Datasets +Cityscapes [6] +contains 2975 training and 500 test im- +ages with annotations provided for 8 categories (person, +car, train, rider, truck, motorcycle, bicycle and bus). The +average horizontal (FoVx) and vertical (FoVy) FoVs of the +capturing cameras are 50°and 26°, respectively. We use this +dataset as the source domain for both FoV adaptation and +viewpoint adaptation. +KITTI [12] +is also a street-view dataset containing 6684 +images annotated with the car category. +The horizontal +(FoVx) and vertical (FoVy) FoVs of the camera are 90°and +34°, respectively. +We use this dataset as target domain +for FoV adaptation, as the viewpoint is similar to that of +Cityscapes. Following [13], we use 5684 images for unsu- +pervised training and 1000 images for evaluation. +MOT [8] +is a multi-object tracking dataset. We use the in- +door mall sequence, MOT20-02, consisting of 2782 frames +5 + +Feature +H1 +Hi-1 +Extractor +Feature +Detection +H2 +Aggregator +Extractor +Head +.. +... +Feature +-1 +Extractor +Mean Teacher +Ldet +Feature +H1 +H-1 +Extractor +Feature +Detection +H2 +H2-1 +Ldet +Aggregator +Extractor +Head +Feature +Hn +-1 +Extractor +Studentannotated with the person category. We employ this dataset +as target domain for viewpoint adaptation. We use the first +2000 frame for unsupervised training and last 782 for eval- +uation. +5.2. Adaptation Tasks and Metric +FoV adaptation. +As in [13], we consider the case of +an increasing FoV using Cityscapes as source domain and +KITTI as target domain. The horizontal and vertical FoVs +increase from (50°, 26°) in Cityscapes to (90°, 34°) in +KITTI. Therefore, as can be seen in Fig. 1, the KITTI +images have a higher distortion in the corners than the +Cityscapes ones. Similarly to PIT [13], we use the car cat- +egory in our experiments. +FoV generalization. +Following PIT [13], we study the +generalization of our approach to new FoVs by cropping +the KITTI images to mimic different FoV changes in the +horizontal direction (FoVx). Specifically, we treat FoVx = +50° as the source domain and the cropped images with FoVx += {70°, 80°, 90°} as different target domains. We evaluate +our approach on car on these different pairs of domains. +Viewpoint adaptation. +This task entails detecting objects +seen from a different viewpoint in the source and target do- +mains. We use the front-facing Cityscapes images as source +domain and the downward-facing MOT ones as target one. +As the MOT data depicts pedestrians, we use the bounding +boxes corresponding to the person category in Cityscapes.2 +Metric. +In all of our experiments, we use the Average Pre- +cision (AP) as our metric. Specifically, following [13], we +report the AP@0.5, which considers the predictions as true +positives if they match the ground-truth label and have an +intersection over union (IOU) score of more than 0.5 with +the ground-truth bounding boxes. +5.3. Implementation Details +We use the Detectron2 [34] implementation of Faster- +RCNN [24] with a ResNet50 [14] backbone as our base +architecture. In all of our experiments, the images are re- +sized so that the shorter side has 800 pixels while maintain- +ing the aspect ratio. The base network is first trained on +source-only images with random cropping and random flip- +ping augmentation for 24k iterations with batch size 8. We +use the Stochastic Gradient Descent (SGD) optimizer with +a learning rate of 0.01, scaled down by a 0.1 factor after +18k iterations. We use ImageNet [25] pretrained weights to +initialize the ResNet50 backbone. +2In Cityscapes, a person may be labeled as either person or rider. Since +the rider label is used for people riding a vehicle, we omit these cases. +Figure 5. FoV Adaptation: Qualitative Results. We visualize +a car detection result in the Cityscapes-to-KITTI FoV adaptation +scenario. The top left image corresponds to the ground truth, the +bottom left to the Mean Teacher result, which confuses the orange +container with a car, the bottom right to the Mean Teacher adapta- +tion + PIT FoV adaptation result, which also mistakes the orange +container for a car and further detects the speed limit on the road. +Our approach, on the top right, correctly matches the ground truth. +We then incorporate the aggregator in the trained base +architecture. +The aggregator architecture contains three +convolutional layers with a kernel size of 3 × 3, and one +1 × 1 convolutional layer. +We first train the aggregator +on the source data with the base frozen and using ran- +dom transformations T . +The transformations are gener- +ated by randomly sampling each Hi parameters as sx, sy ∼ +U[0.5,2.0], U[0.5,2.0] and lx, ly ∼ U[−0.5,0.5], U[−0.5,0.5]. We +train the aggregator for 30k iterations using a batch size of +8 and the SGD optimizer with a learning rate of 1e−4. +The student and teacher models are then initialized with +this detector and the random T = {Hi}N +i=1. We optimize T +using Adam [19], while the base and aggregator networks +are optimized by SGD. The learning rate is set to 1e−3 and +scaled down by a factor 0.1 after 10k iterations for the SGD +optimizer. For the first 10k iterations in FoV adaptation and +for 2k iterations for viewpoint adaptation, we only train T +keeping base and aggregator frozen. The α coefficient for +the EMA update is set to 0.99; the confidence threshold +τ = 0.6; λ = {0.01, 0.1} for FoV and viewpoint adapta- +tion, respectively. The Mean Teacher framework is trained +using both the source and target data. We set N = 5, unless +otherwise specified, and use a batch size of 4, containing 2 +source and 2 target images. We apply random color jitter- +ing on both the source and target data as in [20, 31]. All +of our models are trained on a single NVIDIA V100 GPU. +A detailed hyper-parameter study is provided in the supple- +mentary material. +5.4. Comparison with the State of the Art +We compare our approach with the following baselines. +FR: FasterRCNN trained only on the source data with +random crop augmentation; AT: AdaptTeacher [20]; MT: +Mean Teacher initialized with FR and trained with ran- +dom color jittering on both the source and target data (i.e., +this corresponds to our mean teacher setup in Sec. 4.2 +but without the aggregator and without transformations T ); +6 + +GT +Ours +M +MT+PITMethod +Car AP@0.5 +FR [24] +76.1 +AT [20] +77.2 +FR+PIT +77.6 +MT +78.3 +MT+PIT [13] +79.7 +Ours +80.4 ± 0.15 +Table 1. FoV Adaptation. +Car AP@0.5 for FoVx +Method +50° +70° +80° +90° +FR [24] +94.3 +90.2 +86.8 +80.6 +FR+PIT [13] +93.6 +91.4 +89.2 +85.9 +Ours-h +94.1± 0.16 +93.1 ± 0.33 +91.8 ± 0.40 +88.8 ± 0.21 +Table 2. FoV Generalization. +FR+PIT: Same setup as FR but with the images corrected +with PIT [13]; MT+PIT: Same setup as MT but with the +images corrected with PIT. We refer to our complete ap- +proach (Sec. 4.2) as Ours. For the task of FoV generaliza- +tion, we report our results as Ours-h to indicate that we only +optimize the homographies (5×4 parameters) in T to adapt +to the new FoVs while keeping the base and aggregator net- +works frozen. This matches the setup of PIT [13], which +also corrects the images according to the new FoVs. As +Ours and Ours-h are trained with randomly initialized T , +we report the average results and standard deviations over +three independent runs. +FoV adaptation. +The results of Cityscapes → KITTI FoV +adaptation are provided in Tab. 1. Both MT+PIT and Ours +both bridge the FoV gap, outperforming the MT baseline. +Note, however, that we achieve this by learning the trans- +formations, without requiring any camera-specific informa- +tion, which is needed by PIT. Note also that MT outper- +forms FR by learning a better representation in the target do- +main, even though FR is trained with strong augmentation, +such as random cropping. AT underperforms because its +strong augmentation strategy fails to generalize for datasets +having prominent geometric shifts. Our improvement over +MT evidences that learning transformations helps to over- +come geometric shifts. We optimize with N = 9, homo- +graphies in this setup. Fig. 5 shows a qualitative example. +Different homographies look into different image regions +and the aggregator learns how to combine the activations +corresponding to objects as depicted in Fig. 7. +FoV generalization. +Tab. 2 summarizes the results ob- +tained by using different FoVs as target domains while fix- +Method +Pedestrian AP@0.5 +FR [24] +43.7 +AT [20] +63.5 +MT +64.7 +Ours +65.3± 0.37 +Table 3. Viewpoint Adaptation. +Figure 6. Varying the number of homographies. We evaluate +the effect of N on the FoV adaptation task. +ing the source FoV to 50°. Since both the source and tar- +get images are taken from KITTI, the domain gap is only +caused by a FoV change. +Note that the performance of +FR drops quickly as the FoV gap increases. Ours-h out- +performs FR+PIT by a growing margin as the FoV gap in- +creases. This shows that learning transformations helps to +generalize better to different amounts of geometric shifts. +Viewpoint adaptation. +As shown in Fig. 1, a change in +the camera viewpoint yields differences in the observed dis- +tortions and type of occlusions. The results in Tab. 3 show +the benefits of our method over MT in this case. Note that +PIT, which was designed for FoV changes, cannot be ap- +plied to correct for a viewpoint change. Other baselines out- +perform FR, as they use pseudo labels to fix the difference +in bounding box distribution, as shown in Fig. 1. These +results illustrate the generality of our method to different +kinds of geometric shifts. Qualitative results for this task +can be found in Fig. A.10. +5.5. Additional Analyses +Variable number of homographies. +Let us now study +the influence of the number of homographies in T . +To +this end, we vary this number between 1 and 9. In Fig. 6, +we plot the resulting APs for the Cityscapes-to-KITTI FoV +adaptation task. Increasing the number of transformations +results in a steady increase in performance, which nonethe- +less tends to plateau starting at 4 homographies. Due to lim- +ited compute resources, we couldn’t run experiments with +7 + +80.5 +80.0 +AP@0.5 +79.5 +79.0 +78.5 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Number of HomographyFigure 7. Feature Maps: Top row: predictions of our network and +feature map after aggregator. Left column: Image I, transformed +by learned homographies; Right Column: Feature maps F warped +by corresponding H−1 which are input to the aggregator. Each +transform distorts the image regions differently. Most of the cars +are on the left side and of small size in the image. H1 distorts +the left side leading to no activation(H−1 +1 F1) for the object. H3 +which causes the zoom-in effect has the strongest activation as the +smaller objects are visible better here. These maps are generated +by taking maximum over channel dimension. +more than 9 homographies. This confirms the intuition that +a higher number of perspective transformations can better +capture the geometric shift between two domains. There- +fore, we conducted all experiments with the maximum num- +ber of homographies allowed by our compute resources. +Only optimizing T . +We also run the Ours-h baseline in +the FoV and viewpoint adaptation scenarios. The result- +ing APs are 78.2 and 49.8, respectively. By learning only +the 20 (5 × 4) homography parameters, our approach out- +performs FR (in Tab. 1 and Tab. 3, respectively) by a large +margin in both cases. This confirms that our training strat- +egy is able to efficiently optimize T to bridge the geometric +gap between different domains. We visualize in Fig. A.9 in +the supplementary material some transformations learned +for FoV adaptation by Ours-h. Note that they converge to +diverse homographies that mimic a different FoV, correctly +reflecting the adaptation task. +Diversity in T . +To show that our approach can learn +a diverse set of transformations that help in the adapta- +tion task, we initialize all the homographies with iden- +tity. Fig. 8 depicts the diversity of the learned homogra- +phies on the FoV adaptation task. +Even though we do +not enforce diversity, our approach learns a diverse set of +transformations. +With these learned homorgraphies, our +model achieves 79.5 AP@0.5 score for the FoV adaptation +task. We show additional results in the supplementary ma- +terial Sec. 4 and Sec. 5. +Figure 8. Diversity in T : We train 5 homographies initialized as +Hi = I. We plot the evolution of sx for different homograhies +as training proceeds. Each homography is shown in a different +color. Note that the values for the different homographies become +diverse. The best score is achieved at iteration = 22k, indicated +with the vertical line. +Limitations. +Our approach assumes that the geometric +gap between two domains can be bridged by a set of per- +spective transformations. We have shown that with enough +transformations this is true. However, using a large num- +ber of homographies comes at a computational cost. The +computational overhead leads to an increment in the infer- +ence time from 0.062s to 0.096s for N = 5 on an A100 +Nvidia GPU with image dimension 402 × 1333. Neverthe- +less, our simple implementation shows promising results, +and we will work on reducing this overhead in future work. +Moreover since the optimization of the homography set is +done at the dataset level, only certain transformations are +beneficial to a given image. In the future, we therefore in- +tend to condition the homography on the input image, which +would reduce the total number of homographies needed. +6. Conclusion +We have introduced an approach to bridge the gap be- +tween two domains caused by geometric shifts by learning +a set of homographies. We have shown the effectiveness our +method on two different kinds of shifts, without relying on +any annotations in the target domain, including information +about the nature of the geometric shifts. Our analyses have +evidenced that optimizing the transformations alone brings +in improvement over the base detector and increasing the +number of learnt homographies helps further. In the future, +we plan to learn transformations that are conditioned on the +input image to model image-dependent geometric shifts. +8 + +H1 +1.4 +H2 +H3 +1.3 +H4 +H5 +1.2 +1.1 +1.0 +0.9 +0.8 +0.7 +0.6 +20 +5620 +11220 +16820 +22420 +28020 +IterationsReferences +[1] Alexey +Bochkovskiy, +Chien-Yao +Wang, +and +Hong- +Yuan Mark Liao. 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Transformations through Homography +We use homography to introduce varied perspective +transformations so that they can distort the same image re- +gions differently as seen in Fig. A.1. This helps the detector +to learn robust object features and simultaneously optimize +an aggregator with a different set of homographies which +can bridge the gap between two domains. +2. Feature Maps Activation +We show in Fig. A.2 how different homographies gen- +erate activation in the feature maps. Not all homographies +look at the same image region, therefore the task of the ag- +gregator is to bring in the activations from different trans- +formations together. +3. Other Aggregator Architecture +We implement aggregator using standard functions to +combine {FHi}N +i=1. Tab. A.1 illustrates this study for FoV +adaptation, where the training is done under mean teacher +formalism to learn |T | = N = 5. We see that these non- +learnable aggregators are able to outperform MT baseline +(Sec. 5.4, in the main paper) suggesting that including trans- +formations helps to bridge the geometric shifts. +Function +Car AP@0.5 +sum +78.1± 0.14 +mean +78.7± 0.05 +max +78.7± 0.12 +min+max +78.9± 0.43 +MT +78.3 +Ours +79.9± 0.14 +Table A.1. Aggregator Architecture without learnable parameters +4. Diversity in T +In order to show that diverse transformations are learned, +we set Hi = I and train our mean teacher formulation. +Fig. A.4 shows diverse set of transformations learned in +FoV adaptation task. Even though we do not enforce di- +versity among homographies, it is learned through our ap- +proach. +5. Evolution of T +We provide qualitative results for T learned in FoV and +Viewpoint adaptation, Fig. A.5 and Fig. A.8, respectively. +The qualitative results for the same adaptation task can be +seen in Fig. A.6 and Fig. A.7, respectively. +6. Hyperparameter details +Augmentations. +We use Detectron2 [2]s implementation +for random crop and torchvision 1 for color jittering. +Kind +Details +Random Crop +Relative Range: [0.3, 1] +Color Jitter +Brightness=.5, Hue=.3 +Table A.2. Augmentations +FasterRCNN [1] training. +We train our base network +with random crop strategy on with only source data, which +is Cityscapes for both the adaptation tasks. +The trained +model achieves 74.7 and 58.4 AP@0.5 score on the source +domain validation set for car and person detection, respec- +tively. +Mean Teacher Training +For our mean teacher setup +(Sec. 4.2, in the main paper), we choose τ = 0.6 as the con- +fidence threshold for the pseudo-labels and evaluate con- +tribution of target domain loss for different λ. Fig. A.11 +summarizes this study. We see that method performs worse +when we have equal contribution from both source and tar- +get domain loss λ = 1, as the false positives in the target +1https://pytorch.org/vision/stable/transforms. +html +1 +arXiv:2301.05496v1 [cs.CV] 13 Jan 2023 + +Figure A.1. Transformations: Here we demonstrate how the two objects in the original image undergo different perspective transforma- +tions. Our task is to learn robust object features under such transformations and use them to bring the two domains closer while being +agnostic to the camera parameters. We train with a multiple set of transformations to change the same image region differently. With our +trainable aggregator, we can then combine features from different regions to help in improving the detector’s performance. +domain quickly deteriorate the training. Fig. A.12, evalua- +tion for different values of τ. +7. Architecture details +Our aggregator architecture consists of three convolution +layers along with BatchNorm and Relu layers after each +convolution. Tab. A.3 shows the details of different layers. +Here, C = 1024 corresponds to the output of the feature +extractor. +Table A.3. Aggregator Architecture for |T | = N +# Channels +Layer +Input +Output +Conv2d 3 × 3 +N × C +N × C/2 +BatchNorm + Relu +N × C/2 +N × C/2 +Conv2d 3 × 3 +N × C/2 +C +BatchNorm + Relu +C +C +Conv2d 1 × 1 +C +C +BatchNorm + Relu +C +C +References +[1] Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. +Faster r-cnn: towards real-time object detection with region +proposal networks. IEEE transactions on pattern analysis and +machine intelligence, 39(6):1137–1149, 2016. 1 +[2] Yuxin Wu, Alexander Kirillov, Francisco Massa, Wan-Yen +Lo, and Ross Girshick. Detectron2. https://github. +com/facebookresearch/detectron2, 2019. 1 +2 + +Figure A.2. Feature Maps: Top row: predictions of our network and feature map after aggregator. Left column: Image I, transformed +by 5 learnt homographies; Right Column: Feature maps F warped by corresponding H−1 which are input to aggregator. Each transform +distorts the image regions differently. Most of the cars are on the left side and of small size in the image. H1 distorts the left side leading to +no activation(H−1 +1 F1) for the object. H3 which causes zoom-in effect has the strongest activation as the smaller objects are visible better +here. Overall aggregator feature map contains activation from the region where the objects exist. The aggregator has learnt how to combine +regions with activations under different homographies. The feature maps are generated by taking maximum over channel dimension. +3 + +Figure A.3. Approximating PIT with homographies. Left column: Visualization of each homography use to approximate PIT with 5 +transforms; the top one is the identitity, and the following ones are in order of increasing compression. Center column: Contribution of +each homography to the final remapping. Right column: The top figure shows the per pixel coordinate error when compared to the PIT +remapping as a function of the number of homographies used in the approximation; the three bottom figures depict the coordinate error +maps for 1, 5, and 25 homographies used to approximate PIT (note the scale change in pixel coordinate error). +4 + +sx +sy +lx +ly +Figure A.4. Diversity in T : We train |T | = 5 initialized with Hi = I. Homographies parameterized by sx, sy, lx, ly evolve as the training +proceeds and tend to become diverse. Each homography is shown in different color. Even though we do not enforce any diversity, our +approach learns diverse set of transformations. With these learned homorgraphies, we achieve 79.5 AP@0.5 score for FoV adaptation task. +The best score is achieved at iteration = 22k shown with the vertical line. +5 + +H1 +1.4 +H2 +H3 +1.3 +H4 +H5 +1.2 +1.1 +1.0 +0.9 +0.8 +0.7 +0.6 +20 +5620 +11220 +16820 +22420 +28020 +Iterations1.7 +H1 +H2 +1.6 +H3 +H4 +H5 +1.5 +1.4 +1.3 +1.2 +1.1 +1.0 +0.9 +20 +5620 +11220 +16820 +22420 +28020 +Iterations0.3 +H1 +H2 +H3 +0.2 +H4 +H5 +0.1 +0.0 +-0.1 +-0.2 +-0.3 +20 +5620 +11220 +16820 +22420 +28020 +Iterations0.20 +H1 +H2 +0.15 +H3 +H4 +H5 +0.10 +0.05 +0.00 +-0.05 +-0.10 +-0.15 +-0.20 +20 +5620 +11220 +16820 +22420 +28020 +Iterationssx +sy +lx +ly +Figure A.5. Quantitative results for the corresponding results in Figure A.6. The randomly initialized transforms, parameterized by +sx, sy, lx, ly, evolve to achieve the best score at 28k iterations (shown by the vertical bar). The colors represent different homographies. +Some set of parameters converges to similar value but overall each homography is unique. +6 + +0.4 +0.2 +H1 +H2 +0.0 +H3 +H4 +H5 +-0.2 +-0.4 +20 +5620 +11220 +16820 +22420 +28020 +Iterations0.4 +0.3 +0.2 +0.1 +0.0 +-0.1 +H1 +H2 +H3 +-0.2 +H4 +H5 +20 +5620 +11220 +16820 +22420 +28020 +IterationsH1 +1.8 +H2 +H3 +H4 +1.6 +H5 +1.4 +1.2 +1.0 +0.8 +0.6 +20 +5620 +11220 +16820 +22420 +28020 +Iterations1.8 +1.6 +1.4 +1.2 +1.0 +0.8 +H1 +H2 +H3 +0.6 +H4 +H5 +0.4 +20 +5620 +11220 +16820 +22420 +28020 +IterationsFigure A.6. FoV adaptation: The randomly initialized homographies evolve as the training progresses to improve the overall AP score. +We train with 5 homographies and show how they transform an image for the corresponding FoV adaptation task. +7 + +PredFigure A.7. Viewpoint adaptation: The randomly initialized homographies evolve as the training progresses to improve the overall AP +score. We train with 5 homographies and show how they transform an image for the corresponding viewpoint adaptation task. +8 + +Predsx +sy +lx +ly +Figure A.8. Quantitative results for the corresponding results in Figure A.7. The randomly initialized transforms, parameterized by +sx, sy, lx, ly, evolve to achieve the best score at 8k iterations (shown by the vertical bar). The colors represent different homographies. +Some sy parameters start at a similar value but eventually diverge. +9 + +1.8 +1.6 +1.4 +H1 +H2 +H3 +1.2 +H4 +H5 +1.0 +0.8 +0.6 +20 +1820 +3620 +5420 +7220 +9020 +Iterations1.4 +1.2 +1.0 +0.8 +H1 +H2 +H3 +0.6 +H4 +H5 +20 +1820 +3620 +5420 +7220 +9020 +IterationsH1 +0.3 +H2 +H3 +H4 +0.2 +H5 +0.1 +0.0 +-0.1 +-0.2 +-0.3 +-0.4 +-0.5 +20 +1820 +3620 +5420 +7220 +9020 +IterationsH1 +H2 +0.3 +H3 +H4 +H5 +0.2 +0.1 +0.0 +-0.1 +-0.2 +20 +1820 +3620 +5420 +7220 +9020 +IterationsFigure A.9. Evolution of T . We showcase how two homographies, H1 and H5, evolve across the training iterations and influence +the prediction scores. Starting from random homographies at iteration 0, the transformations converge to homographies suited for FoV +adaptation. The detection scores consequently increase throughout the training process. Moreover, this increase in detection score is +reflected in the overall AP@0.5 score, which jumps from 74.1 to 78.2. +Figure A.10. Viewpoint Adaptation: Qualitative Results. We visualize results for viewpoint adaptation between Cityscapes and MOT20- +02. The left image depicts the ground truth, the middle one the results of Mean Teacher adaptation, and the right one those of our approach. +Our approach recovers more detections (e.g., the woman near the stroller in the center-left) while having fewer false positives (overlapping +box in bottom-left corner of the MT results). +10 + +88% +94% +93%GT +MT +OursFigure A.11. Study on λ for τ = 0.6, |T | = 5 +Figure A.12. Study on τ for FoV and Viewpoint adaptation using +λ = 0.01, 0.1, respectively. Here ,|T | = 5 is used for the study. +11 + +81 +79 +77 +75 +73 +71 +5 +69 +AP@O. +67 +65 +63 +61 +59 +Fov adapt. +57 +Viewpoint adapt +55 +1 +0.1 +0.01 +0.001 +入81 +79 +77 +75 +73 +71 +5 +AP@O.! +69 +67 +65 +63 +61 +59 +FoV adapt. +57 +Viewpoint adapt. +55 +0.5 +0.6 +0.7 +0.8 +T \ No newline at end of file diff --git a/0NE5T4oBgHgl3EQfOQ6v/content/tmp_files/load_file.txt b/0NE5T4oBgHgl3EQfOQ6v/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dec76bfbd302174ed7fa3e11dbe489556370fd82 --- /dev/null +++ b/0NE5T4oBgHgl3EQfOQ6v/content/tmp_files/load_file.txt @@ -0,0 +1,872 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf,len=871 +page_content='Learning Transformations To Reduce the Geometric Shift in Object Detection Vidit Vidit1 Martin Engilberge1 Mathieu Salzmann1,2 CVLab, EPFL1, ClearSpace SA2 firstname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='lastname@epfl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='ch Abstract The performance of modern object detectors drops when the test distribution differs from the training one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Most of the methods that address this focus on object appearance changes caused by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=', different illumination conditions, or gaps between synthetic and real images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Here, by con- trast, we tackle geometric shifts emerging from variations in the image capture process, or due to the constraints of the environment causing differences in the apparent geometry of the content itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We introduce a self-training approach that learns a set of geometric transformations to minimize these shifts without leveraging any labeled data in the new domain, nor any information about the cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We evalu- ate our method on two different shifts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=', a camera’s field of view (FoV) change and a viewpoint change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Our results evidence that learning geometric transformations helps de- tectors to perform better in the target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Introduction While modern object detectors [1, 2, 17, 23, 24] achieve impressive results, their performance decreases when the test data depart from the training distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' This prob- lem arises in the presence of appearance variations due to, for example, differing illumination or weather conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Considering the difficulty and cost of acquiring annotated data in the test (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=', target) domain, Unsupervised Domain Adaptation (UDA) has emerged as the standard strategy to address such scenarios [3,4,9,26,38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In this context, much effort has been made to learn do- main invariant features, such that the source and target dis- tributions in this feature space are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' This has led to great progress in situations where the appearance of the ob- jects changes drastically from one domain to the other, as in case of real-to-sketch adaptation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=', Pascal VOC [10] to Comics [15]), or weather adaptation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=', Cityscapes [6] to Foggy Cityscapes [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Nevertheless, such object ap- pearance changes are not the only sources of domain shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' They can also have geometric origins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' For example, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 1, they can be due to a change in camera view- Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Geometric shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' (Left) Due to a different FoV, the cars highlighted in green, undergo different distortions even though they appear in similar image regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' (Right) Different camera viewpoints (front facing vs downward facing) yield dif- ferent distortions and occlusion patterns for pedestrian detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' (Bottom) The distributions of pedestrian bounding box sizes in Cityscapes [6] and MOT [8] differ significantly as the pedestrians are usually far away or in the periphery in Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The top im- ages are taken from Cityscapes [6], and the bottom-left and right ones from KITTI [12] and MOT [8], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' point or field-of-view (FoV), or a change of object scale due to different scene setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In practice, such geometric shifts typically arise from a combination of various factors, in- cluding but not limited to the ones mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In this paper, we introduce a domain adaptation approach tackling such geometric shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' To the best of our knowl- edge, the recent work of [13] constitutes the only attempt at considering such geometric distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' However, it intro- duces a method solely dedicated to FoV variations, assum- ing that the target FoV is fixed and known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Here, we de- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='05496v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='CV] 13 Jan 2023 rkplatz/F Cityscapes Apparent Bbox Distribution MOT20velop a more general framework able to cope with a much broader family of geometric shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' To this end, we model geometric transformations as a combination of multiple homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We show both the- oretically and empirically that this representation is suffi- cient to encompass a broad variety of complex geometric transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We then design an aggregator block that can be incorporated to the detector to provide it with the capacity to tackle geometric shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We use this modified detector to generate pseudo labels for the target domain, which let us optimize the homographies so as to reduce the geometric shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Our contributions can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' (i) We tackle the problem of general geometric shifts for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' (ii) We learn a set of homographies using unla- beled target data, which alleviates the geometric bias arising in source-only training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' (iii) Our method does not require prior information about the target geometric distortions and generalizes to a broad class of geometric shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Our ex- periments demonstrate the benefits of our approach in sev- eral scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In the presence of FoV shifts, our approach yields similar performance to the FoV-dedicated framework of [13] but without requiring any camera information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' As such, it generalizes better to other FoVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Furthermore, we show the generality of our method by using it to adapt to a new camera viewpoint in the context of pedestrian detec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Related Work Unsupervised Domain Adaptation (UDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' UDA for image recognition [11, 21, 22, 30, 32, 35, 36] and object de- tection [3,4,9,20,26,38] has made a great progress in the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The common trend in both tasks consists of learning domain invariant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' For object detection, this entails aligning the global (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=', illumination, weather) and local (foreground objects) features in the two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In this context, [3,5,26,28] align image- and instance-level features in the two domains via adversarial learning [11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' [33] learns category-specific attention maps to better align specific image regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' [38] clusters the proposed object re- gions using k-means clustering and uses the centroids for instance-level alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' While this successfully tackles domain shifts caused by object appearance variations, it fails to account for the presence of shifts due to the image capture process itself, such as changes in camera intrinsics or viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The only initial step at considering a geo- metric shift is the work of [13], which shows the existence of an FoV gap in driving datasets [6, 12] and proposes a Position Invariant Transform (PIT) that corrects the distor- tions caused specifically by an FoV change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In essence, PIT undistorts the images by assuming knowledge of the target FoV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' By contrast, here, we introduce an approach that gen- eralizes to a broad family of geometric shifts by learning transformations without requiring any camera information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Self-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Self-training, generally employed in the semi-supervised setting, offers an alternative to learning domain-invariant features and utilize unlabeled data to im- prove a detector’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In this context, [29] uses a student-teacher architecture where the teacher model is trained with supervised data and generates pseudo-labels on unannotated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' These pseudo-labels are then used to train a student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' While effective in the stan- dard semi-supervised learning scenario, the quality of the pseudo-labels obtained with this approach tends to deteri- orate when the labeled and unlabeled data present a distri- bution shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' [9, 20] have therefore extended this approach to domain adaptation by using the Mean Teacher strategy of [31] to generate reliable pseudo-labels in the target do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Other approach include the use of CycleGAN [37] generated images to train an unbiased teacher model [9], and that of different augmentation strategies to generate ro- bust pseudo-labels [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Our approach also follows a self- training strategy but, while these works focus on object ap- pearance shifts, we incorporate learnable blocks to address geometric shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' As shown in our experiment, this lets us outperform the state-of-the-art AdaptTeacher [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Learning Geometric Transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' End-to-end learning of geometric transformations has been used to boost the performance of deep networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' For example, Spatial Transformer Networks (STNs) [16] reduce the classification error by learning to correct for affine trans- formations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' deformable convolutions [7] model geometric transformations by applying the convolution kernels to non-local neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' These methods work well when annotations are available for supervision, and make the network invariant to the specific geometric transformations seen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Here, by contrast, we seek to learn transformations in an unsupervised manner and allow the network to generalize to unknown target transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Modeling Geometric Transformations In the context of UDA, multiple geometric differences can be responsible for the gap between the domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Some can be characterized by the camera parameters, such as a change in FoV (intrinsic) or viewpoint (extrinsic), whereas others are content specific, such as a difference in road width between different countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Ultimately, the geomet- ric shift is typically a combination of different geometric operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Since the parameters of these operations are un- known, we propose to bridge the domain gap by learning a geometric transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Specifically, we aggregate the results of multiple perspective transforms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=', homographies, to obtain a differentiable operation that can emulate a wide variety of geometric transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Theoretical Model Let us first show that, given sufficiently many homogra- phies, one can perfectly reproduce any mapping between R2 \\ (0, 0) and R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Single homography for a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' First, we show that a single homography with 4 degrees of freedom can map a point p ∈ R2 \\(0, 0) to any other point in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' To this end, let H = � � sx 0 0 0 sy 0 lx ly 1 � � (1) be a homography, with (sx, sy) the scaling factors on the x- and y-axis, respectively, and (lx, ly) the perspective factors in x and y, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' For any destination point d ∈ R2, there exists a set of parameters (sx, sy, lx, ly) such that d = H × p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' One such set is ( dx px , dy py , 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Emulating any geometric transformation Now that we have shown that a single homography can move a point to any other point in R2, we describe a simple protocol to emu- late any geometric transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Given an unknown geometric transform T : R2\\(0, 0) → R2, we aim to emulate T with a set of homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In general, for an image I ∈ R3×h×w, we can restrict the domain of T to only image coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' To this end, we can define a set of homographies Hi ∈ H for i in {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=', h × w}, where the parameters of Hi are chosen to mimic the transform T for location i of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In this protocol, the aggregation mechanism is trivial since each homography is in charge of remapping a single pixel coordinate of the original space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' While this works in theory, this is of course not viable in practice since it would require too many homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' With a smaller number of homographies, each transform needs to remap multiple points, and a more sophisticated aggregation mechanism is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Specifically, the ag- gregation mechanism needs to select which transform is in charge of remapping which point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In the next section, we empirically show that this strategy lets us closely approxi- mate the spherical projection mapping used in PIT [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Approximating PIT with Homographies To demonstrate the possibility offered by aggregating multiple homographies, we design an approximation of PIT using only homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' PIT proposes to correct for an FoV gap by remapping images to a spherical surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Dur- ing this transformation, regions further from the center of a scene are compressed with a higher ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' This variable compression of the space cannot be reproduced by a single homography transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' To overcome this limitation, we combine the results of multiple homographies that all have different compression rates (scaling parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' For the aggregation mechanism, we use the optimal strategy by selecting for each pixel the homography that approximates best the PIT mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 2, this combination closely approximates the PIT results with only 5 homogra- phies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Further analysis of these experiments is available in the supplementary material in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Approximating PIT with homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We show the original image (top), the PIT [13] correction (middle), and our ap- proximation of PIT using 5 homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Note that 5 homogra- phies are sufficient to closely match the PIT spherical correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Homographies in a Learning Setup In the two previous sections, we have demonstrated both theoretically and empirically the flexibility of aggregating homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' This makes this representation an ideal can- didate for domain adaptation since the geometric shift be- tween the domains is unknown and can be a combination of different transforms, such as FoV change, viewpoint change, camera distortion, or appearance distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' As will be discussed in the next section, by learning jointly the set of perspective transforms and the aggregation mechanism on real data, our model can reduce the geometric shift be- tween the two domains without prior knowledge about this domain gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Method Let us now introduce our approach to reducing the ge- ometric shift in object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Following the standard UDA setting, let Ds = {(Is, Bs, Cs)} be a labeled source dataset containing images Is = {Ii s}Ns 1 with correspond- ing object bounding boxes Bs = {bi s}Ns 1 and object classes Cs = {ci s}Ns 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Furthermore, let Dt = {It} denote an un- labeled target dataset for which only images It = {Ii t}Nt 1 3 Original PIT Approximation 5 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Architecture: The input image is first transformed by a set of trainable homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The feature maps extracted from the transformed images are then unwarped by the inverse homographies to achieve spatial consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We then combine the unwarped feature maps using a trainable aggregator, whose output is passed to a detection head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The blocks shown in green correspond to standard FasterRCNN operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The � symbol represents the concatenation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' are available, without annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Here, we tackle the case where the two domains differ by geometric shifts but as- sume no knowledge about the nature of these shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Below, we first introduce the architecture we developed to handle this and then our strategy to train this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Model Architecture The overall architecture of our approach is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In essence, and as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 3, we charac- terize the geometric changes between the source and target data by a set of transformations T = {Hi}N 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Each Hi in T is a homography of the same form as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' For our method to remain general, we assume the transformations to be unknown, and our goal, therefore, is to learn T to bridge the gap between the domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' This requires differentiabil- ity w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' the transformation parameters, which we achieve using the sampling strategy proposed in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 3, the input image is transformed by the individual homographies in T , and the transformed images are fed to a modified FasterRCNN [24] detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Specif- ically, we extract a feature map FHi ∈ RH×W ×C for each transformed image via a feature extractor shared by all transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' To enforce spatial correspondence be- tween the different FHis, we unwarp them with H−1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We then introduce an aggregator Aθg, parameterized by θg, whose goal is to learn a common representation given a fixed number of unwarped feature maps F′ Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' To achieve this, the aggregator takes as input G = F′ H1 ⊕ F′ H2 ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' ⊕ F′ HN ∈ RH×W ×C×N , (2) where ⊕ represents concatenation in the channel dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The aggregator outputs a feature map Aθg(G) ∈ RH×W ×C, whose dimension is independent of the number of transfor- mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' This output is then passed to a detection head to obtain the objects’ bounding boxes and class labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Model Training Our training procedure relies on three steps: (i) Fol- lowing common practice in UDA, we first train the Faster- RCNN detector with source-only data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' (ii) We then intro- duce the aggregator and train it so that it learns to com- bine different homographies using the labeled source data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' (iii) Finally, we learn the optimal transformations for adap- tation using both the source and target data via a Mean Teacher [31] strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Aggregator Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' To train the aggregator, we ran- domly sample a set of homographies T ∈ RN×4 in each training iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 This gives the aggregator the ability to robustly combine diverse input transformations but requires strong supervision to avoid training instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We, there- fore, perform this step using the source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The loss function for a set of transformed images T (Is) is then defined as in standard FasterRCNN training with a combination of classification and regression terms [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' That is, we train the aggregator by solving min θg Lcls(T (Is)) + Lreg(T (Is)) , (3) 1As our homographies involve only 4 parameters, with a slight abuse of notation, we say that Hi ∈ R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 4 Homography Feature Homography Detection Aggregator Projections Extractor Projections Headwhere Lcls(T (Is)) = Lrpn cls + Lroi cls , (4) Lreg(T (Is)) = Lrpn reg + Lroi reg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' (5) Lrpn and Lroi correspond to the Region Proposal Network (RPN) loss terms and the Region of Interest (RoI) ones, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' During this process, we freeze the parameters θb of the base network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='e, feature extractor and detection head, which were first trained on the source data without ag- gregator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Ultimately, the aggregator provides the network with the capacity to encode different transformations that are not seen in the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The third training step then aims to learn the best transformation for successful ob- ject detection in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Learning the Transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' As we have no annota- tions in the target domain, we exploit a Mean Teacher (MT) strategy to learn the optimal transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' To this end, our starting point is the detector with a trained aggregator and a set of random transformations T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The MT strategy is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In essence, MT training [31] involves two copies of the model: A student model, with parameters θst = {T st, θst b , θst g }, that will be used during inference, and a teacher model, with parameters θte = {T te, θte b , θte g }, that is updated as an Exponentially Moving Average (EMA) of the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' That is, the student’s parameters are computed with standard backpropagation, whereas the teacher’s ones are updated as θte ← αθte + (1 − α)θst .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' (6) The student model is trained using both source and tar- get detection losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Since the target domain does not have annotations, the teacher model is used to generate pseudo- labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' These labels might be noisy, and hence we only keep the predictions with a confidence score above a threshold τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Furthermore, non-maxima suppression (NMS) is used to re- move the highly-overlapping bounding box predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Formally, given a source image Is and a target image It, the student model is trained by solving min T st,θst g ,θst b Ldet(T (Is)) + λLdet(T (It)) , (7) where λ controls the target domain contribution and Ldet(T (Is)) = Lcls(T (Is)) + Lreg(T (Is)) , (8) Ldet(T (It)) = Lcls(T (It)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' (9) Similarly to [18,20], we update the student model with only the classification loss in the target domain to help stabilize training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Mean Teacher formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The student model is trained with ground-truth labels in the source domain and pseudo labels in the target one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' These pseudo labels are produced by the teacher model, which corresponds to an exponentially moving average (EMA) of the student network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Experiments We demonstrate the effectiveness and generality of our method on different geometric shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' First, to compare to the only other work that modeled a geometric shift [13], we tackle the problem of a change in FoV between the source and target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Note that, in contrast to [13], we do not assume knowledge of the target FoV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Furthermore, while [13] was dedicated to FoV adaptation, our approach generalizes to other geometric shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We demonstrate this on the task of pedestrian detection under a viewpoint shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We compare our method with the state-of-the-art Adapt- Teacher [20], which also uses a Mean Teacher, but focuses on appearance shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In the remainder of this section, we describe our experimental setup and discuss our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Datasets Cityscapes [6] contains 2975 training and 500 test im- ages with annotations provided for 8 categories (person, car, train, rider, truck, motorcycle, bicycle and bus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The average horizontal (FoVx) and vertical (FoVy) FoVs of the capturing cameras are 50°and 26°, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We use this dataset as the source domain for both FoV adaptation and viewpoint adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' KITTI [12] is also a street-view dataset containing 6684 images annotated with the car category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The horizontal (FoVx) and vertical (FoVy) FoVs of the camera are 90°and 34°, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We use this dataset as target domain for FoV adaptation, as the viewpoint is similar to that of Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Following [13], we use 5684 images for unsu- pervised training and 1000 images for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' MOT [8] is a multi-object tracking dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We use the in- door mall sequence, MOT20-02, consisting of 2782 frames 5 Feature H1 Hi-1 Extractor Feature Detection H2 Aggregator Extractor Head .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Feature 1 Extractor Mean Teacher Ldet Feature H1 H-1 Extractor Feature Detection H2 H2-1 Ldet Aggregator Extractor Head Feature Hn 1 Extractor Studentannotated with the person category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We employ this dataset as target domain for viewpoint adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We use the first 2000 frame for unsupervised training and last 782 for eval- uation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Adaptation Tasks and Metric FoV adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' As in [13], we consider the case of an increasing FoV using Cityscapes as source domain and KITTI as target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The horizontal and vertical FoVs increase from (50°, 26°) in Cityscapes to (90°, 34°) in KITTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Therefore, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 1, the KITTI images have a higher distortion in the corners than the Cityscapes ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Similarly to PIT [13], we use the car cat- egory in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' FoV generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Following PIT [13], we study the generalization of our approach to new FoVs by cropping the KITTI images to mimic different FoV changes in the horizontal direction (FoVx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Specifically, we treat FoVx = 50° as the source domain and the cropped images with FoVx = {70°, 80°, 90°} as different target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We evaluate our approach on car on these different pairs of domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Viewpoint adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' This task entails detecting objects seen from a different viewpoint in the source and target do- mains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We use the front-facing Cityscapes images as source domain and the downward-facing MOT ones as target one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' As the MOT data depicts pedestrians, we use the bounding boxes corresponding to the person category in Cityscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 Metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In all of our experiments, we use the Average Pre- cision (AP) as our metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Specifically, following [13], we report the AP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5, which considers the predictions as true positives if they match the ground-truth label and have an intersection over union (IOU) score of more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 with the ground-truth bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Implementation Details We use the Detectron2 [34] implementation of Faster- RCNN [24] with a ResNet50 [14] backbone as our base architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In all of our experiments, the images are re- sized so that the shorter side has 800 pixels while maintain- ing the aspect ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The base network is first trained on source-only images with random cropping and random flip- ping augmentation for 24k iterations with batch size 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We use the Stochastic Gradient Descent (SGD) optimizer with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='01, scaled down by a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 factor after 18k iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We use ImageNet [25] pretrained weights to initialize the ResNet50 backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 2In Cityscapes, a person may be labeled as either person or rider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Since the rider label is used for people riding a vehicle, we omit these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' FoV Adaptation: Qualitative Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We visualize a car detection result in the Cityscapes-to-KITTI FoV adaptation scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The top left image corresponds to the ground truth, the bottom left to the Mean Teacher result, which confuses the orange container with a car, the bottom right to the Mean Teacher adapta- tion + PIT FoV adaptation result, which also mistakes the orange container for a car and further detects the speed limit on the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Our approach, on the top right, correctly matches the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We then incorporate the aggregator in the trained base architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The aggregator architecture contains three convolutional layers with a kernel size of 3 × 3, and one 1 × 1 convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We first train the aggregator on the source data with the base frozen and using ran- dom transformations T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The transformations are gener- ated by randomly sampling each Hi parameters as sx, sy ∼ U[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='0], U[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='0] and lx, ly ∼ U[−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5], U[−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We train the aggregator for 30k iterations using a batch size of 8 and the SGD optimizer with a learning rate of 1e−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The student and teacher models are then initialized with this detector and the random T = {Hi}N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We optimize T using Adam [19], while the base and aggregator networks are optimized by SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The learning rate is set to 1e−3 and scaled down by a factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 after 10k iterations for the SGD optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' For the first 10k iterations in FoV adaptation and for 2k iterations for viewpoint adaptation, we only train T keeping base and aggregator frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The α coefficient for the EMA update is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='99;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' the confidence threshold τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' λ = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1} for FoV and viewpoint adapta- tion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The Mean Teacher framework is trained using both the source and target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We set N = 5, unless otherwise specified, and use a batch size of 4, containing 2 source and 2 target images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We apply random color jitter- ing on both the source and target data as in [20, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' All of our models are trained on a single NVIDIA V100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' A detailed hyper-parameter study is provided in the supple- mentary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Comparison with the State of the Art We compare our approach with the following baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' FR: FasterRCNN trained only on the source data with random crop augmentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' AT: AdaptTeacher [20];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' MT: Mean Teacher initialized with FR and trained with ran- dom color jittering on both the source and target data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=', this corresponds to our mean teacher setup in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 but without the aggregator and without transformations T );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 6 GT Ours M MT+PITMethod Car AP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 FR [24] 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 AT [20] 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 FR+PIT 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6 MT 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3 MT+PIT [13] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='7 Ours 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='15 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' FoV Adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Car AP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 for FoVx Method 50° 70° 80° 90° FR [24] 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='8 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6 FR+PIT [13] 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='9 Ours-h 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='16 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='33 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='40 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='21 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' FoV Generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' FR+PIT: Same setup as FR but with the images corrected with PIT [13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' MT+PIT: Same setup as MT but with the images corrected with PIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We refer to our complete ap- proach (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2) as Ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' For the task of FoV generaliza- tion, we report our results as Ours-h to indicate that we only optimize the homographies (5×4 parameters) in T to adapt to the new FoVs while keeping the base and aggregator net- works frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' This matches the setup of PIT [13], which also corrects the images according to the new FoVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' As Ours and Ours-h are trained with randomly initialized T , we report the average results and standard deviations over three independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' FoV adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The results of Cityscapes → KITTI FoV adaptation are provided in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Both MT+PIT and Ours both bridge the FoV gap, outperforming the MT baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Note, however, that we achieve this by learning the trans- formations, without requiring any camera-specific informa- tion, which is needed by PIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Note also that MT outper- forms FR by learning a better representation in the target do- main, even though FR is trained with strong augmentation, such as random cropping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' AT underperforms because its strong augmentation strategy fails to generalize for datasets having prominent geometric shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Our improvement over MT evidences that learning transformations helps to over- come geometric shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We optimize with N = 9, homo- graphies in this setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 5 shows a qualitative example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Different homographies look into different image regions and the aggregator learns how to combine the activations corresponding to objects as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' FoV generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 2 summarizes the results ob- tained by using different FoVs as target domains while fix- Method Pedestrian AP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 FR [24] 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='7 AT [20] 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 MT 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='7 Ours 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='37 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Viewpoint Adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Varying the number of homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We evaluate the effect of N on the FoV adaptation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' ing the source FoV to 50°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Since both the source and tar- get images are taken from KITTI, the domain gap is only caused by a FoV change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Note that the performance of FR drops quickly as the FoV gap increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Ours-h out- performs FR+PIT by a growing margin as the FoV gap in- creases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' This shows that learning transformations helps to generalize better to different amounts of geometric shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Viewpoint adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 1, a change in the camera viewpoint yields differences in the observed dis- tortions and type of occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 3 show the benefits of our method over MT in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Note that PIT, which was designed for FoV changes, cannot be ap- plied to correct for a viewpoint change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Other baselines out- perform FR, as they use pseudo labels to fix the difference in bounding box distribution, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' These results illustrate the generality of our method to different kinds of geometric shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Qualitative results for this task can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Additional Analyses Variable number of homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Let us now study the influence of the number of homographies in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' To this end, we vary this number between 1 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 6, we plot the resulting APs for the Cityscapes-to-KITTI FoV adaptation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Increasing the number of transformations results in a steady increase in performance, which nonethe- less tends to plateau starting at 4 homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Due to lim- ited compute resources, we couldn’t run experiments with 7 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='0 AP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 1 2 3 4 5 6 7 8 9 Number of HomographyFigure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Feature Maps: Top row: predictions of our network and feature map after aggregator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Left column: Image I, transformed by learned homographies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Right Column: Feature maps F warped by corresponding H−1 which are input to the aggregator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Each transform distorts the image regions differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Most of the cars are on the left side and of small size in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' H1 distorts the left side leading to no activation(H−1 1 F1) for the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' H3 which causes the zoom-in effect has the strongest activation as the smaller objects are visible better here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' These maps are generated by taking maximum over channel dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' more than 9 homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' This confirms the intuition that a higher number of perspective transformations can better capture the geometric shift between two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' There- fore, we conducted all experiments with the maximum num- ber of homographies allowed by our compute resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Only optimizing T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We also run the Ours-h baseline in the FoV and viewpoint adaptation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The result- ing APs are 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 and 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' By learning only the 20 (5 × 4) homography parameters, our approach out- performs FR (in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 1 and Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 3, respectively) by a large margin in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' This confirms that our training strat- egy is able to efficiently optimize T to bridge the geometric gap between different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We visualize in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='9 in the supplementary material some transformations learned for FoV adaptation by Ours-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Note that they converge to diverse homographies that mimic a different FoV, correctly reflecting the adaptation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Diversity in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' To show that our approach can learn a diverse set of transformations that help in the adapta- tion task, we initialize all the homographies with iden- tity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 8 depicts the diversity of the learned homogra- phies on the FoV adaptation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Even though we do not enforce diversity, our approach learns a diverse set of transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' With these learned homorgraphies, our model achieves 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 AP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 score for the FoV adaptation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We show additional results in the supplementary ma- terial Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 4 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Diversity in T : We train 5 homographies initialized as Hi = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We plot the evolution of sx for different homograhies as training proceeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Each homography is shown in a different color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Note that the values for the different homographies become diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The best score is achieved at iteration = 22k, indicated with the vertical line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Our approach assumes that the geometric gap between two domains can be bridged by a set of per- spective transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We have shown that with enough transformations this is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' However, using a large num- ber of homographies comes at a computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The computational overhead leads to an increment in the infer- ence time from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='062s to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='096s for N = 5 on an A100 Nvidia GPU with image dimension 402 × 1333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Neverthe- less, our simple implementation shows promising results, and we will work on reducing this overhead in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Moreover since the optimization of the homography set is done at the dataset level, only certain transformations are beneficial to a given image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In the future, we therefore in- tend to condition the homography on the input image, which would reduce the total number of homographies needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Conclusion We have introduced an approach to bridge the gap be- tween two domains caused by geometric shifts by learning a set of homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We have shown the effectiveness our method on two different kinds of shifts, without relying on any annotations in the target domain, including information about the nature of the geometric shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Our analyses have evidenced that optimizing the transformations alone brings in improvement over the base detector and increasing the number of learnt homographies helps further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In the future, we plan to learn transformations that are conditioned on the input image to model image-dependent geometric shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 8 H1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4 H2 H3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3 H4 H5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6 20 5620 11220 16820 22420 28020 IterationsReferences [1] Alexey Bochkovskiy, Chien-Yao Wang, and Hong- Yuan Mark Liao.' metadata={'source': 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+page_content=' 2 [37] Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Unpaired image-to-image translation using cycle- consistent adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In Proceedings of the IEEE international conference on computer vision, pages 2223– 2232, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 2 [38] Xinge Zhu, Jiangmiao Pang, Ceyuan Yang, Jianping Shi, and Dahua Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Adapting object detectors via selective cross- domain alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Con- ference on Computer Vision and Pattern Recognition, pages 687–696, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 1, 2 10 Supplementary Material: Learning Transformations To Reduce the Geometric Shift in Object Detection Vidit Vidit1 Martin Engilberge1 Mathieu Salzmann1,2 CVLab, EPFL1, ClearSpace SA2 firstname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='lastname@epfl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='ch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Transformations through Homography We use homography to introduce varied perspective transformations so that they can distort the same image re- gions differently as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' This helps the detector to learn robust object features and simultaneously optimize an aggregator with a different set of homographies which can bridge the gap between two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Feature Maps Activation We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 how different homographies gen- erate activation in the feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Not all homographies look at the same image region, therefore the task of the ag- gregator is to bring in the activations from different trans- formations together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Other Aggregator Architecture We implement aggregator using standard functions to combine {FHi}N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 illustrates this study for FoV adaptation, where the training is done under mean teacher formalism to learn |T | = N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We see that these non- learnable aggregators are able to outperform MT baseline (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4, in the main paper) suggesting that including trans- formations helps to bridge the geometric shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Function Car AP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 sum 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='14 mean 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='7± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='05 max 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='7± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='12 min+max 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='9± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='43 MT 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3 Ours 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='9± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='14 Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Aggregator Architecture without learnable parameters 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Diversity in T In order to show that diverse transformations are learned, we set Hi = I and train our mean teacher formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4 shows diverse set of transformations learned in FoV adaptation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Even though we do not enforce di- versity among homographies, it is learned through our ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Evolution of T We provide qualitative results for T learned in FoV and Viewpoint adaptation, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The qualitative results for the same adaptation task can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Hyperparameter details Augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We use Detectron2 [2]s implementation for random crop and torchvision 1 for color jittering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Kind Details Random Crop Relative Range: [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3, 1] Color Jitter Brightness=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5, Hue=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3 Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Augmentations FasterRCNN [1] training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We train our base network with random crop strategy on with only source data, which is Cityscapes for both the adaptation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The trained model achieves 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='7 and 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4 AP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 score on the source domain validation set for car and person detection, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Mean Teacher Training For our mean teacher setup (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2, in the main paper), we choose τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6 as the con- fidence threshold for the pseudo-labels and evaluate con- tribution of target domain loss for different λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='11 summarizes this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We see that method performs worse when we have equal contribution from both source and tar- get domain loss λ = 1, as the false positives in the target 1https://pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='org/vision/stable/transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' html 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='05496v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='CV] 13 Jan 2023 Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Transformations: Here we demonstrate how the two objects in the original image undergo different perspective transforma- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Our task is to learn robust object features under such transformations and use them to bring the two domains closer while being agnostic to the camera parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We train with a multiple set of transformations to change the same image region differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' With our trainable aggregator, we can then combine features from different regions to help in improving the detector’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' domain quickly deteriorate the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='12, evalua- tion for different values of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Architecture details Our aggregator architecture consists of three convolution layers along with BatchNorm and Relu layers after each convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3 shows the details of different layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Here, C = 1024 corresponds to the output of the feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Aggregator Architecture for |T | = N # Channels Layer Input Output Conv2d 3 × 3 N × C N × C/2 BatchNorm + Relu N × C/2 N × C/2 Conv2d 3 × 3 N × C/2 C BatchNorm + Relu C C Conv2d 1 × 1 C C BatchNorm + Relu C C References [1] Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Faster r-cnn: towards real-time object detection with region proposal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' IEEE transactions on pattern analysis and machine intelligence, 39(6):1137–1149, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 1 [2] Yuxin Wu, Alexander Kirillov, Francisco Massa, Wan-Yen Lo, and Ross Girshick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Detectron2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' com/facebookresearch/detectron2, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 1 2 Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Feature Maps: Top row: predictions of our network and feature map after aggregator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Left column: Image I, transformed by 5 learnt homographies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Right Column: Feature maps F warped by corresponding H−1 which are input to aggregator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Each transform distorts the image regions differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Most of the cars are on the left side and of small size in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' H1 distorts the left side leading to no activation(H−1 1 F1) for the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' H3 which causes zoom-in effect has the strongest activation as the smaller objects are visible better here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Overall aggregator feature map contains activation from the region where the objects exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The aggregator has learnt how to combine regions with activations under different homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The feature maps are generated by taking maximum over channel dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 3 Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Approximating PIT with homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Left column: Visualization of each homography use to approximate PIT with 5 transforms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' the top one is the identitity, and the following ones are in order of increasing compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Center column: Contribution of each homography to the final remapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Right column: The top figure shows the per pixel coordinate error when compared to the PIT remapping as a function of the number of homographies used in the approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' the three bottom figures depict the coordinate error maps for 1, 5, and 25 homographies used to approximate PIT (note the scale change in pixel coordinate error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 4 sx sy lx ly Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Diversity in T : We train |T | = 5 initialized with Hi = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Homographies parameterized by sx, sy, lx, ly evolve as the training proceeds and tend to become diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Each homography is shown in different color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Even though we do not enforce any diversity, our approach learns diverse set of transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' With these learned homorgraphies, we achieve 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 AP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 score for FoV adaptation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The best score is achieved at iteration = 22k shown with the vertical line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 5 H1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4 H2 H3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3 H4 H5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6 20 5620 11220 16820 22420 28020 Iterations1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='7 H1 H2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6 H3 H4 H5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='9 20 5620 11220 16820 22420 28020 Iterations0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3 H1 H2 H3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 H4 H5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3 20 5620 11220 16820 22420 28020 Iterations0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='20 H1 H2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='15 H3 H4 H5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='20 20 5620 11220 16820 22420 28020 Iterationssx sy lx ly Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Quantitative results for the corresponding results in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The randomly initialized transforms, parameterized by sx, sy, lx, ly, evolve to achieve the best score at 28k iterations (shown by the vertical bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The colors represent different homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Some set of parameters converges to similar value but overall each homography is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 H1 H2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='0 H3 H4 H5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4 20 5620 11220 16820 22420 28020 Iterations0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 H1 H2 H3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 H4 H5 20 5620 11220 16820 22420 28020 IterationsH1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='8 H2 H3 H4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6 H5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6 20 5620 11220 16820 22420 28020 Iterations1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='8 H1 H2 H3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6 H4 H5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4 20 5620 11220 16820 22420 28020 IterationsFigure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' FoV adaptation: The randomly initialized homographies evolve as the training progresses to improve the overall AP score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We train with 5 homographies and show how they transform an image for the corresponding FoV adaptation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 7 PredFigure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Viewpoint adaptation: The randomly initialized homographies evolve as the training progresses to improve the overall AP score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We train with 5 homographies and show how they transform an image for the corresponding viewpoint adaptation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 8 Predsx sy lx ly Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Quantitative results for the corresponding results in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The randomly initialized transforms, parameterized by sx, sy, lx, ly, evolve to achieve the best score at 8k iterations (shown by the vertical bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The colors represent different homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Some sy parameters start at a similar value but eventually diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4 H1 H2 H3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 H4 H5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6 20 1820 3620 5420 7220 9020 Iterations1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='8 H1 H2 H3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6 H4 H5 20 1820 3620 5420 7220 9020 IterationsH1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3 H2 H3 H4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 H5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 20 1820 3620 5420 7220 9020 IterationsH1 H2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='3 H3 H4 H5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2 20 1820 3620 5420 7220 9020 IterationsFigure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Evolution of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We showcase how two homographies, H1 and H5, evolve across the training iterations and influence the prediction scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Starting from random homographies at iteration 0, the transformations converge to homographies suited for FoV adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The detection scores consequently increase throughout the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Moreover, this increase in detection score is reflected in the overall AP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='5 score, which jumps from 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 to 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Viewpoint Adaptation: Qualitative Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' We visualize results for viewpoint adaptation between Cityscapes and MOT20- 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' The left image depicts the ground truth, the middle one the results of Mean Teacher adaptation, and the right one those of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Our approach recovers more detections (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=', the woman near the stroller in the center-left) while having fewer false positives (overlapping box in bottom-left corner of the MT results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 10 88% 94% 93%GT MT OursFigure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Study on λ for τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='6, |T | = 5 Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Study on τ for FoV and Viewpoint adaptation using λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' Here ,|T | = 5 is used for the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 11 81 79 77 75 73 71 5 69 AP@O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 67 65 63 61 59 Fov adapt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content=' 57 Viewpoint adapt 55 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE5T4oBgHgl3EQfOQ6v/content/2301.05496v1.pdf'} +page_content='1 0.' metadata={'source': 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However, in many of these areas, machine learning models have performed +unfair behaviors against some sub-populations, such as some particular groups of +race, sex, and age. These unfair behaviors can be on account of the pre-existing +bias in the training dataset due to historical and social factors. In this paper, we +focus on a real-world application of credit scoring and construct a fair prediction +model by introducing latent variables to remove the correlation between protected +attributes, such as sex and age, with the observable feature inputs, including house +and job. For detailed implementation, we apply Bayesian approaches, including +the Markov Chain Monte Carlo simulation, to estimate our proposed fair model. +1 +Introduction +Nowadays, Machine Learning methods are used to automate decisions in a variety of areas, including +determining credit scores Nanni and Lumini [2009], classifying tumor components from MRI images +Lundervold and Lundervold [2019], detecting pedestrians on the road Dollar et al. [2011], and +understanding natural languages Goldberg and Levy [2014], etc. However, machine learning methods +are heavily dependent on data Mitchell and Mitchell [1997] and this data-dependent nature makes the +learned models sensitive to the latent bias existing in the training datasets Mehrabi et al. [2021]. Thus, +the final decisions made by the learned models are unfairly biased against certain sub-populations, +differentiated by some sensitive/protected attributes, such as race, sex, or age, etc. For example, +cameras sometimes fail to recognize whether Asian blink their eyes Sharp [2009] and the beauty +pageant judged by AI would prefer light skin Guardian [2016]. However, we would expect AI to give +the same decision independent from the protected attributes and thus we concern about the fairness +of machine learning methods Mehrabi et al. [2021]. +In this paper, we focus on constructing fair machine learning models to predict the credit score, with +using the German Credit Risk dataset Hoffman [2016] (Sec. 3). The goal is to predict the credit +score based on some observable variables, including housing and job information. However, this +personal financial information, such as income, housing, and saving, are usually highly correlated +to gender and age due to historical and social reasons Rennison and Planty [2003]. Therefore, it +is necessary to learn an effective model to filter the prediction bias against sex and age, caused by +the latent correlation between these observable variables and the protected attributes. In detail, we +analyze and compare from the fairness perspective across the full model Montgomery et al. [2021], +unaware model Dwork et al. [2012], and fair model based on causals and counterfactuals Kusner et al. +[2017] (Sec. 4). Then, we apply the Markov Chain Monte Carlo (MCMC) simulation Mooney [1997] +and the Gibbs’ sampling Gelfand [2000] to solve the corresponding parameters in these models and +evaluate the performances (Sec. 5 and Sec. 6). +arXiv:2301.08412v1 [cs.LG] 20 Jan 2023 + +2 +Related work +Fairness. Many recent works (Calders and Verwer [2010], Bolukbasi et al. [2016], Dwork et al. +[2012], Hardt et al. [2016], Joseph et al. [2016], Kusner et al. [2017]) have been focusing on fairness +in machine learning algorithms. Bolukbasi et al. [2016] pointed out that there is a risk of amplifying +the bias introduced from the dataset, if using machine learning algorithms without taking effects to +handle the pre-existing bias. For example, in the word embedding, learned over Google News with +pre-existing gender stereotypes, the gender-neutral words widely spread along a latent embedding +direction capturing gender difference, such as "receptionist" falling far along the direction related +to "female" Bolukbasi et al. [2016]. Calders and Verwer [2010] modifies the Naive Bayes classifier +by adding independence restriction toward sensitive attributes. Dwork et al. [2012] proposes a +task-specific metric to evaluate the similarity between individuals relative to the classification task +and optimizes over the proposed metric with the goal that similar individuals are treated similarly +in the classification task. Kusner et al. [2017] focuses on causal inference and counterfactual, with +introducing the latent confounding variables, which are related to the observable variables but +independent from the protected attributes. Our work is based on the Kusner et al. [2017] idea to +construct a fair prediction model over the German Credit Risk dataset Hoffman [2016]. +3 +Dataset +We consider the Kaggle German Credit Risk dataset Hoffman [2016] to analyze and compare different +types of unfair models and our method for constructing a fair model using Bayesian approaches. In +this dataset, each entry represents a person who takes credit from a bank. The objective is to predict +the credit amount of a person based on his/her attributes. "Sex" and "age" are the sensitive/protected +attributes related to the bias during training and prediction in the unfairness problem. Feature "job" is +a binary variable representing whether a person has a job or not. Feature "house" is a binary variable +that indicates whether or not a person owns a house. The "credit amount" is our prediction target. +The dataset is composed of 1000 records. We randomly pick 800 records for training and 200 records +for testing. Figure 1 shows the detailed distributions of all these features in the whole dataset. In +Figure 2, we illustrate the covariance between all the input features and the prediction target. We can +observe a high correlation from the sensitive / protected attributes, i.e. "sex" and "age", to the "job" +and "house". Thus, it is necessary to consider the issue of fairness when constructing a prediction +model over "job" and "house". +Figure 1: Distribution of features in the German Credit Risk dataset Hoffman [2016]. "Age" and +"sex" are the sensitive / protected attributes. "Job" and "house" are the observable variables. "Credit +amount" is the prediction target. +2 + +700 +140 +160 +600 +120 +500 +100 +140 +60 +300 +120 +40 +200 +100 +100 +20 +Count +70 +0.2 +0.4 +0.6 +0.8 +1.0 +age +sex +80 +800 +700 +700 +600 +60 +600 +500 +40 +8 400 +300 +300 +20 +200 +200 +100 +100 +0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.4 +0 +5000 +10000 +15000 +job +0.2 +0.6 +house +credit amtFigure 2: Illustration of the covariance matrix between all the input features and the prediction target. +Here, we observe a high correlation from "age" and "sex" (the sensitive/protected attributes) to "job" +and "house" (the observable variables). +4 +Methods +Full Model: The full model Montgomery et al. [2021] completely ignores fairness issues and +includes sensitive variables like sex and age in the learning process. It is easy to understand that the +full model is unfair because the predictions depend on sex and age. Figure 3 presents the directed +acyclic graph (DAG) of the full model. In the full model, all the features are assumed to be connected. +Unaware Model: The unaware model Dwork et al. [2012] does not use sensitive variables +in the learning and prediction process, but it is still unfair. Even though the sensitive variables do not +influence the target directly in the learning and prediction processes, it still has an indirect impact on +the target through the non-sensitive variables. In our example, to predict a person’s credit amount, +sex may influence whether a person can get a job. The job attribute still preserves the information of +sex. Simply ignoring the sex attribute will not fully eliminate its impact on the predictions. Figure 3 +presents the DAG of an unaware model. The attributes under the grey circles are unobserved. In the +unaware model, sex and age are not directly connected with the credit amount, but they are connected +with job and house. It is still unfair because the change of sex and age will change the status of job +and house, and thus influence the credit amount predictions. +Figure 3: Two types of unfair models. Left: full model, which builds regression over all possible +attributes without the consideration of fairness. Right: unaware model, which excludes sensi- +tive/protected attributes, i.e. sex and age in our case. +Fair Model: In order to build a fair model, we need to find a proxy variable that is independent +of sensitive variables but still preserves the information in the credit amount prediction Kusner +et al. [2017]. We can introduce the concept of latent confounding variable to resolve this issue. +3 + +1.0 +age +0.8 + sex +0.6 +job +0.4 +credit_amt house +0.2 +0.0 +age +sex +job +house +credit_amtSex +Job +Sex +Job +Credit +Credit +House +Age +House +Age +Full Model +Unaware ModelThe confounding variable is a variable that influences both the independent variable and dependent +variables. In our fair model, we assume that there is an unobserved confounder C that reflects how +reliable a person is in paying back the loan. The confounder should be independent of the sensitive +variables to make the model fair. Figure 4 shows the DAG of the fair model structure. In the inference +stage, we assume that job, house, and credit amount are confounded by the unobserved reliability +level C and C is independent of sex and age. The reason is that sex and age can neither determine +nor be related to how reliable a person is in paying back loans. Meanwhile, reliability is co-related to +a person’s job performance, housing situation, and also credit amount. Then, in the prediction stage, +we only use the inferred C as our feature to predict the credit amount. In this way, the predicting +process does not contain any information about sex or age, and thus this procedure is an effective, +fair learning algorithm in our scenario. +Figure 4: DAG of the fair model. Here, we introduce the latent confounding variable "unobserved +reliability level", which is independent to "sex" and "age" (the sensitive/protected attributes) but +related to "job", "house", and "credit amount". Left: during the inference stage, we estimate this +latent "reliability" feature with Bayesian approaches. Right: during the prediction stage, we only use +this inferred "reliability" feature to predict the "credit amount". +5 +Experiments +We can represent the DAG of the fair model in a probabilistic way. We sample the two binary +variables, job and house from two Bernoulli distributions and sample the confounder from the normal +distribution. In the meantime, we choose the Poisson distribution as a prior for the credit amount. +Our choices of priors correspond to the nature of the data. The job and house features are binary. And +the credit amount is a positive attribute with a shape alike the Poisson distribution. The probabilistic +model can be written as: +Job ∼ Bernoulli(logit(bj + Sex × βj,s + Age × βj,a + C × βj,c)) +(1) +House ∼ Bernoulli(logit(bh + Sex × βh,s + Age × βh,a + C × βh,c)) +(2) +Credit ∼ Poisson(Exp(Sex × βc,s + Age × βc,a + C × βc,c)) +(3) +C ∼ Normal(0, 1) +where +C ⊥ Sex, +C ⊥ Age +(4) +The parameters we need to find are in the set Θ = {βm,n, bm} where m = j, h and n = s, a, c. We +assume that these parameters are sampled from the normal distributions: +βm,n ∼ N(0, 1) +(5) +bm ∼ N(0, 1) +(6) +We implement the Metropolis–Hastings algorithm to infer the probabilistic model. M-H algorithm +Hastings [1970] is a Markov Chain Monte Carlo (MCMC) method for obtaining a sequence of +4 + +Job +Sex +Unobserved +Unobserved +House +Credit +Reliability Level +Reliability Level +Age +Credit +Fair model - Inference +Fair model - PredictionAlgorithm 1 Infer C by Metropolis–Hastings +for i = 1 to N do +Choose J(C∗ +i |C(s) +i +) = uniform(C(s) +i +− δ, C(s) +i ++ δ); +Set an initial state C0 +i ; +for s = 1 to 5000 do +Sample C∗ +i ∼ J(C∗ +i |C(s) +i +); +Compute the acceptance ratio r = +p(C∗ +i |y) +p(C(s) +i +|y) = +p(y|C∗ +i )p(C∗ +i ) +p(y|C(s) +i +)p(C(s) +i +); +sample u ∼ uniform(0, 1); +if u < r then; +C(s+1) +i += C∗ +i ; +else +C(s+1) +i += C(s) +i +; +end if +end for +end for +random samples from a probability distribution from which direct sampling is difficult. Algorithm 1 +explains how to infer the reliability level C. +Once we obtain the posteriors of the inferred reliability level C, we can fit a new model using kernel +g(.) based on the C in the prediction stage. In our experiment, since there is a nonlinear relationship +between credit amount and "Reliability Level" in our inference stage setup (Poisson), we decide to +use random-forest as the kernel function g(.) in our second stage prediction. +Credit ∼ g(C) +(7) +6 +Results +In this section, we provide experimental results and a discussion of the MCMC process performance. +Specifically, in Sec. 6.1, we firstly present the MCMC estimation result and the convergence analysis +on the fair model’s latent confounding variable C and parameters. Then, we compare the prediction +and fairness performance across the three types of models in Sec. 6.2. +6.1 +Fair model’s MCMC performance: +Figure 5: Auto-correlation plots of parameters in Eq. 1 and Eq. 2 throughout the MCMC process. +"alpha" refers to the constant offset term b in the equations. +In Figure 5, we illustrate the auto-correlation plot of the model’s parameters in Eq. 1 and Eq. 2. We +observe a clear decrease in auto-correlation throughout the MCMC process. Thus, this is an efficient +MCMC process that leads to convergence. Further, in Figure 6, we provide the posterior estimation +and the trace plot of the fair model parameters throughout the MCMC process. Though we still +5 + +qoreaq +alpha job +1.00 +1.00 +0.75 +0.75 +0.50 +0.50 +0.25 +0.25 +0.00 +0.005 +0.25 +0.25 +0.50 +0.50 +0.75 +0.75 +-1.000 +-1.000 +20 +40 +60 +80 +100 0 +20 +100 0 +60 +80 +100 +20 +60 +80 +100 +alpha_house +ouse +beta +ouse +0,2 +1.00 +1.00 +0.75 +0.75 +0.50 +0.50 +0.25 +0.25 +0.00 +0.00 , +-0.25 +-0.25 +-0.50 +-0.50 +-0.75 +0.75 +-1.000 +100 0 +100 +20 +40 +60 +80 +100Figure 6: Posterior estimation (left column) and trace plot (right column) of parameters in Eq. 1 and +Eq. 2 throughout the MCMC process. "alpha" refers to the constant offset term b in the equations. +observe some fluctuations till the end of the process, however, this is reasonable and acceptable. The +reason is that we are applying over a real-world dataset, rather than a simulated dataset. Therefore, +it is impossible to make our assumed distributions perfectly capture the behavior of the real-world +dataset. Then, in Table 1, we provide the confidence interval over the posterior estimation of the fair +model’s parameters. +6.2 +Performance comparison across models: +In this section, we compare how three distinct models perform while making predictions. In Table +2, we present the R2 of three models in both training and testing environments. The full model +outperforms the unaware model in both fitting and predicting by including sensitive information. It is +surprising to see that the fair model outperforms the other two unfair models with R2 = 0.801 in the +training set and R2 = 0.768 in the testing set. It turns out that our fair model does not only resolve +the fairness issue but distills the information on the reliability level. The fair model is robust enough +to be used to make fair and accurate predictions. +7 +Conclusion +In this paper, we have presented a fair model focusing on predicting the German credit score with +considering the job and housing features. Specifically, we introduce the latent confounding variable +"reliability level", which is independent of the protected attributes, i.e., "sex" and "age", but related +to other observable variables and the prediction goal. For implementation, we apply the MCMC +6 + +betajob +beta job +4 +0 +2 +1000 +2000 +3000 +4000 +alpha house +alpha_house +4 +3 +2 +1 +0 +1 +2 +m +4 +0 +1000 +2000 +3000 +4000 +beta house +beta house +-2 +0 +0 +1000 +2000 +3000 +4000 +beta_credit +beta credit +0.0 +2.5 +5.0 +-7.5 +-8 +-6 +-4 +0 +0 +1000 +2000 +3000 +4000 +C +c +4 +-2 +0 +2 +0 +1000 +2000 +3000 +4000std +5% +median +95% +ess_bulk +ess_tail +bj +1.02 +-1.66 +0.03 +1.71 +4643.63 +3709.55 +βj,s +0.98 +-1.32 +0.27 +1.88 +7128.36 +3907.04 +βj,a +0.65 +-2.64 +-1.57 +-0.50 +1502.60 +2245.58 +βj,c +0.47 +2.82 +3.46 +4.36 +2058.64 +2494.02 +bh +1.01 +-1.61 +0.03 +1.67 +5113.35 +3167.94 +βh,s +0.99 +-1.73 +-0.11 +1.55 +5506.02 +3900.82 +βh,a +0.67 +-0.04 +1.05 +2.17 +1625.86 +2583.99 +βh,c +0.46 +3.00 +3.65 +4.50 +1896.31 +2939.44 +βc,s +0.54 +-7.78 +-6.85 +-5.98 +3255.68 +3206.54 +βc,a +0.52 +-3.17 +-2.26 +-1.46 +4326.49 +3800.31 +βc,c +0.23 +-0.37 +0.01 +0.38 +4455.72 +3211.78 +Table 1: The confidence intervals of the parameters estimated in Eq. 1 and Eq. 2 through the MCMC +process. +R2 +Full Model +Unaware Model +Fair Model Random Forest Kernel +Training +0.597 +0.466 +0.801 +Testing +0.521 +0.424 +0.768 +Table 2: The R2 of three types of models defined in Sec. 4. +approach to solve for the latent confounding variable and the parameters of the model. Compared +with tradition models, our model effectively eliminates the bias related to sex and age and thus +achieves a fair prediction of the credit amount. For the future work, we recommend trying different +types of assumptions on the distribution for the variables over the German Credit Risk dataset and +checking the effects on the choice of distributions over the convergence of the MCMC process and +the final prediction. +References +Tolga Bolukbasi, Kai-Wei Chang, James Y Zou, Venkatesh Saligrama, and Adam T Kalai. Man is +to computer programmer as woman is to homemaker? debiasing word embeddings. Advances in +neural information processing systems, 29, 2016. +Toon Calders and Sicco Verwer. Three naive bayes approaches for discrimination-free classification. +Data mining and knowledge discovery, 21(2):277–292, 2010. +Piotr Dollar, Christian Wojek, Bernt Schiele, and Pietro Perona. Pedestrian detection: An evaluation +of the state of the art. 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Violence and victims, 18(4):433–443, 2003. +Gwen Sharp. +Nikon camera says asians: +People are always blinking - sociolog- +ical images, +2009. +URL https://thesocietypages.org/socimages/2009/05/29/ +nikon-camera-says-asians-are-always-blinking/. +8 + diff --git a/1dFAT4oBgHgl3EQfCxxo/content/tmp_files/load_file.txt b/1dFAT4oBgHgl3EQfCxxo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..62fc411819b8a3c3dcaaf867269091d18ccda2e9 --- /dev/null +++ b/1dFAT4oBgHgl3EQfCxxo/content/tmp_files/load_file.txt @@ -0,0 +1,374 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf,len=373 +page_content='Fair Credit Scorer through Bayesian Approach Zhuo Zhao Department of Applied Mathematics Johns Hopkins University zzhao62@jhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='edu Abstract Machine learning currently plays an increasingly important role in people’s lives in areas such as credit scoring, auto-driving, disease diagnosing, and insurance quoting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' However, in many of these areas, machine learning models have performed unfair behaviors against some sub-populations, such as some particular groups of race, sex, and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' These unfair behaviors can be on account of the pre-existing bias in the training dataset due to historical and social factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' In this paper, we focus on a real-world application of credit scoring and construct a fair prediction model by introducing latent variables to remove the correlation between protected attributes, such as sex and age, with the observable feature inputs, including house and job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' For detailed implementation, we apply Bayesian approaches, including the Markov Chain Monte Carlo simulation, to estimate our proposed fair model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 1 Introduction Nowadays, Machine Learning methods are used to automate decisions in a variety of areas, including determining credit scores Nanni and Lumini [2009], classifying tumor components from MRI images Lundervold and Lundervold [2019], detecting pedestrians on the road Dollar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2011], and understanding natural languages Goldberg and Levy [2014], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' However, machine learning methods are heavily dependent on data Mitchell and Mitchell [1997] and this data-dependent nature makes the learned models sensitive to the latent bias existing in the training datasets Mehrabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Thus, the final decisions made by the learned models are unfairly biased against certain sub-populations, differentiated by some sensitive/protected attributes, such as race, sex, or age, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' For example, cameras sometimes fail to recognize whether Asian blink their eyes Sharp [2009] and the beauty pageant judged by AI would prefer light skin Guardian [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' However, we would expect AI to give the same decision independent from the protected attributes and thus we concern about the fairness of machine learning methods Mehrabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' In this paper, we focus on constructing fair machine learning models to predict the credit score, with using the German Credit Risk dataset Hoffman [2016] (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' The goal is to predict the credit score based on some observable variables, including housing and job information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' However, this personal financial information, such as income, housing, and saving, are usually highly correlated to gender and age due to historical and social reasons Rennison and Planty [2003].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Therefore, it is necessary to learn an effective model to filter the prediction bias against sex and age, caused by the latent correlation between these observable variables and the protected attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' In detail, we analyze and compare from the fairness perspective across the full model Montgomery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2021], unaware model Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2012], and fair model based on causals and counterfactuals Kusner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2017] (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Then, we apply the Markov Chain Monte Carlo (MCMC) simulation Mooney [1997] and the Gibbs’ sampling Gelfand [2000] to solve the corresponding parameters in these models and evaluate the performances (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 5 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='08412v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='LG] 20 Jan 2023 2 Related work Fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Many recent works (Calders and Verwer [2010], Bolukbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2016], Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2012], Hardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2016], Joseph et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2016], Kusner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2017]) have been focusing on fairness in machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Bolukbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2016] pointed out that there is a risk of amplifying the bias introduced from the dataset, if using machine learning algorithms without taking effects to handle the pre-existing bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' For example, in the word embedding, learned over Google News with pre-existing gender stereotypes, the gender-neutral words widely spread along a latent embedding direction capturing gender difference, such as "receptionist" falling far along the direction related to "female" Bolukbasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Calders and Verwer [2010] modifies the Naive Bayes classifier by adding independence restriction toward sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2012] proposes a task-specific metric to evaluate the similarity between individuals relative to the classification task and optimizes over the proposed metric with the goal that similar individuals are treated similarly in the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Kusner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2017] focuses on causal inference and counterfactual, with introducing the latent confounding variables, which are related to the observable variables but independent from the protected attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Our work is based on the Kusner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2017] idea to construct a fair prediction model over the German Credit Risk dataset Hoffman [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 3 Dataset We consider the Kaggle German Credit Risk dataset Hoffman [2016] to analyze and compare different types of unfair models and our method for constructing a fair model using Bayesian approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' In this dataset, each entry represents a person who takes credit from a bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' The objective is to predict the credit amount of a person based on his/her attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' "Sex" and "age" are the sensitive/protected attributes related to the bias during training and prediction in the unfairness problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Feature "job" is a binary variable representing whether a person has a job or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Feature "house" is a binary variable that indicates whether or not a person owns a house.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' The "credit amount" is our prediction target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' The dataset is composed of 1000 records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' We randomly pick 800 records for training and 200 records for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Figure 1 shows the detailed distributions of all these features in the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' In Figure 2, we illustrate the covariance between all the input features and the prediction target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' We can observe a high correlation from the sensitive / protected attributes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' "sex" and "age", to the "job" and "house".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Thus, it is necessary to consider the issue of fairness when constructing a prediction model over "job" and "house".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Figure 1: Distribution of features in the German Credit Risk dataset Hoffman [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' "Age" and "sex" are the sensitive / protected attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' "Job" and "house" are the observable variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' "Credit amount" is the prediction target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 2 700 140 160 600 120 500 100 140 60 300 120 40 200 100 100 20 Count 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='0 age sex 80 800 700 700 600 60 600 500 40 8 400 300 300 20 200 200 100 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='4 0 5000 10000 15000 job 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='6 house credit amtFigure 2: Illustration of the covariance matrix between all the input features and the prediction target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Here, we observe a high correlation from "age" and "sex" (the sensitive/protected attributes) to "job" and "house" (the observable variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 4 Methods Full Model: The full model Montgomery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2021] completely ignores fairness issues and includes sensitive variables like sex and age in the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' It is easy to understand that the full model is unfair because the predictions depend on sex and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Figure 3 presents the directed acyclic graph (DAG) of the full model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' In the full model, all the features are assumed to be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Unaware Model: The unaware model Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2012] does not use sensitive variables in the learning and prediction process, but it is still unfair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Even though the sensitive variables do not influence the target directly in the learning and prediction processes, it still has an indirect impact on the target through the non-sensitive variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' In our example, to predict a person’s credit amount, sex may influence whether a person can get a job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' The job attribute still preserves the information of sex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Simply ignoring the sex attribute will not fully eliminate its impact on the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Figure 3 presents the DAG of an unaware model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' The attributes under the grey circles are unobserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' In the unaware model, sex and age are not directly connected with the credit amount, but they are connected with job and house.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' It is still unfair because the change of sex and age will change the status of job and house, and thus influence the credit amount predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Figure 3: Two types of unfair models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Left: full model, which builds regression over all possible attributes without the consideration of fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Right: unaware model, which excludes sensi- tive/protected attributes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' sex and age in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Fair Model: In order to build a fair model, we need to find a proxy variable that is independent of sensitive variables but still preserves the information in the credit amount prediction Kusner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' [2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' We can introduce the concept of latent confounding variable to resolve this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='0 age 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='8 sex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='6 job 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='4 credit_amt house 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='0 age sex job house credit_amtSex Job Sex Job Credit Credit House Age House Age Full Model Unaware ModelThe confounding variable is a variable that influences both the independent variable and dependent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' In our fair model, we assume that there is an unobserved confounder C that reflects how reliable a person is in paying back the loan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' The confounder should be independent of the sensitive variables to make the model fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Figure 4 shows the DAG of the fair model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' In the inference stage, we assume that job, house, and credit amount are confounded by the unobserved reliability level C and C is independent of sex and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' The reason is that sex and age can neither determine nor be related to how reliable a person is in paying back loans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Meanwhile, reliability is co-related to a person’s job performance, housing situation, and also credit amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Then, in the prediction stage, we only use the inferred C as our feature to predict the credit amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' In this way, the predicting process does not contain any information about sex or age, and thus this procedure is an effective, fair learning algorithm in our scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Figure 4: DAG of the fair model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Here, we introduce the latent confounding variable "unobserved reliability level", which is independent to "sex" and "age" (the sensitive/protected attributes) but related to "job", "house", and "credit amount".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Left: during the inference stage, we estimate this latent "reliability" feature with Bayesian approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Right: during the prediction stage, we only use this inferred "reliability" feature to predict the "credit amount".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 5 Experiments We can represent the DAG of the fair model in a probabilistic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' We sample the two binary variables, job and house from two Bernoulli distributions and sample the confounder from the normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' In the meantime, we choose the Poisson distribution as a prior for the credit amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Our choices of priors correspond to the nature of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' The job and house features are binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' And the credit amount is a positive attribute with a shape alike the Poisson distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' The probabilistic model can be written as: Job ∼ Bernoulli(logit(bj + Sex × βj,s + Age × βj,a + C × βj,c)) (1) House ∼ Bernoulli(logit(bh + Sex × βh,s + Age × βh,a + C × βh,c)) (2) Credit ∼ Poisson(Exp(Sex × βc,s + Age × βc,a + C × βc,c)) (3) C ∼ Normal(0, 1) where C ⊥ Sex, C ⊥ Age (4) The parameters we need to find are in the set Θ = {βm,n, bm} where m = j, h and n = s, a, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' We assume that these parameters are sampled from the normal distributions: βm,n ∼ N(0, 1) (5) bm ∼ N(0, 1) (6) We implement the Metropolis–Hastings algorithm to infer the probabilistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' M-H algorithm Hastings [1970] is a Markov Chain Monte Carlo (MCMC) method for obtaining a sequence of 4 Job Sex Unobserved Unobserved House Credit Reliability Level Reliability Level Age Credit Fair model - Inference Fair model - PredictionAlgorithm 1 Infer C by Metropolis–Hastings for i = 1 to N do Choose J(C∗ i |C(s) i ) = uniform(C(s) i − δ, C(s) i + δ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Set an initial state C0 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' for s = 1 to 5000 do Sample C∗ i ∼ J(C∗ i |C(s) i );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Compute the acceptance ratio r = p(C∗ i |y) p(C(s) i |y) = p(y|C∗ i )p(C∗ i ) p(y|C(s) i )p(C(s) i );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' sample u ∼ uniform(0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' if u < r then;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' C(s+1) i = C∗ i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' else C(s+1) i = C(s) i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' end if end for end for random samples from a probability distribution from which direct sampling is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Algorithm 1 explains how to infer the reliability level C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Once we obtain the posteriors of the inferred reliability level C, we can fit a new model using kernel g(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=') based on the C in the prediction stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' In our experiment, since there is a nonlinear relationship between credit amount and "Reliability Level" in our inference stage setup (Poisson), we decide to use random-forest as the kernel function g(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=') in our second stage prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Credit ∼ g(C) (7) 6 Results In this section, we provide experimental results and a discussion of the MCMC process performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Specifically, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='1, we firstly present the MCMC estimation result and the convergence analysis on the fair model’s latent confounding variable C and parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Then, we compare the prediction and fairness performance across the three types of models in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='1 Fair model’s MCMC performance: Figure 5: Auto-correlation plots of parameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 2 throughout the MCMC process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' "alpha" refers to the constant offset term b in the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' In Figure 5, we illustrate the auto-correlation plot of the model’s parameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' We observe a clear decrease in auto-correlation throughout the MCMC process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Thus, this is an efficient MCMC process that leads to convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Further, in Figure 6, we provide the posterior estimation and the trace plot of the fair model parameters throughout the MCMC process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Though we still 5 qoreaq alpha job 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='50 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='000 100 0 100 20 40 60 80 100Figure 6: Posterior estimation (left column) and trace plot (right column) of parameters in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 2 throughout the MCMC process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' "alpha" refers to the constant offset term b in the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' observe some fluctuations till the end of the process, however, this is reasonable and acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' The reason is that we are applying over a real-world dataset, rather than a simulated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Therefore, it is impossible to make our assumed distributions perfectly capture the behavior of the real-world dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Then, in Table 1, we provide the confidence interval over the posterior estimation of the fair model’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='2 Performance comparison across models: In this section, we compare how three distinct models perform while making predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' In Table 2, we present the R2 of three models in both training and testing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' The full model outperforms the unaware model in both fitting and predicting by including sensitive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' It is surprising to see that the fair model outperforms the other two unfair models with R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='801 in the training set and R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='768 in the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' It turns out that our fair model does not only resolve the fairness issue but distills the information on the reliability level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' The fair model is robust enough to be used to make fair and accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 7 Conclusion In this paper, we have presented a fair model focusing on predicting the German credit score with considering the job and housing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Specifically, we introduce the latent confounding variable "reliability level", which is independent of the protected attributes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=', "sex" and "age", but related to other observable variables and the prediction goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' For implementation, we apply the MCMC 6 betajob beta job 4 0 2 1000 2000 3000 4000 alpha house alpha_house 4 3 2 1 0 1 2 m 4 0 1000 2000 3000 4000 beta house beta house 2 0 0 1000 2000 3000 4000 beta_credit beta credit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='5 8 6 4 0 0 1000 2000 3000 4000 C c 4 2 0 2 0 1000 2000 3000 4000std 5% median 95% ess_bulk ess_tail bj 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='66 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='78 Table 1: The confidence intervals of the parameters estimated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 2 through the MCMC process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' R2 Full Model Unaware Model Fair Model Random Forest Kernel Training 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='597 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='466 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='801 Testing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='521 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='424 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content='768 Table 2: The R2 of three types of models defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' approach to solve for the latent confounding variable and the parameters of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' Compared with tradition models, our model effectively eliminates the bias related to sex and age and thus achieves a fair prediction of the credit amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' For the future work, we recommend trying different types of assumptions on the distribution for the variables over the German Credit Risk dataset and checking the effects on the choice of distributions over the convergence of the MCMC process and the final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfCxxo/content/2301.08412v1.pdf'} +page_content=' References Tolga Bolukbasi, Kai-Wei Chang, James Y Zou, Venkatesh Saligrama, and Adam T Kalai.' metadata={'source': 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0000000000000000000000000000000000000000..c9584bb2ca55fdb9948109f3840ea4f7f21fc89d --- /dev/null +++ b/29FAT4oBgHgl3EQflB1V/content/tmp_files/2301.08614v1.pdf.txt @@ -0,0 +1,5939 @@ +arXiv:2301.08614v1 [math.AP] 20 Jan 2023 +Plane wave stability analysis of Hartree and quantum +dissipative systems +Thierry Goudon∗1 and Simona Rota Nodari†1 +1Université Côte d’Azur, Inria, CNRS, LJAD, +Parc Valrose, F-06108 Nice, France +Abstract +We investigate the stability of plane wave solutions of equations describing quantum +particles interacting with a complex environment. The models take the form of PDE +systems with a non local (in space or in space and time) self-consistent potential; such +a coupling lead to challenging issues compared to the usual non linear Schrödinger +equations. The analysis relies on the identification of suitable Hamiltonian structures +and Lyapounov functionals. We point out analogies and differences between the original +model, involving a coupling with a wave equation, and its asymptotic counterpart +obtained in the large wave speed regime. In particular, while the analogies provide +interesting intuitions, our analysis shows that it is illusory to obtain results on the +former based on a perturbative analysis from the latter. +Keywords. Hartree equation. Open quantum systems. Particles interacting with a vibrational +field. Schrödinger-Wave equation. Plane wave. Orbital stability. +Math. Subject Classification. 35Q40 35Q51 35Q55 +1 +Introduction +This work is concerned with the stability analysis of certain solutions of the following Hartree-type +equation +iBtU ` 1 +2∆xU “ γ +ˆ +σ1 ‹x +ˆ +Rn σ2Ψ dz +˙ +U, +(1a) +´ ∆zΨ “ ´γσ2pzq +` +σ1 ‹x |U|2˘ +pxq +(1b) +∗thierry.goudon@inria.fr +†simona.rotanodari@univ-cotedazur.fr +1 + +endowed with the initial condition +U +ˇˇ +t“0 “ UInit, +(2) +and of the following Schrödinger-Wave system: +iBtU ` 1 +2∆xU “ γΦU, +(3a) +1 +c2 B2 +ttΨ ´ ∆zΨ “ ´γσ2pzqσ1 ‹ |U|2pt, xq, +(3b) +Φpt, xq “ +¨ +TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dz dy, +(3c) +where γ, c ą 0 are given positive parameters, completed with +U +ˇˇ +t“0 “ UInit, +Ψ +ˇˇ +t“0 “ ΨInit, +BtΨ +ˇˇ +t“0 “ ΠInit. +(4) +The variable x lies in the torus Td, meaning that the equations are understood with p2πq´periodicity +in all directions. In (3b), the additional variable z lies in Rn and, as explained below, it is crucial to +assume n ě 3. For reader’s convenience, the scaling of the equation is fully detailed in Appendix A; +for our purposes the God-given form functions σ1, σ2 are fixed once for all and the features of +the coupling are embodied in the parameters γ, c. The system (1a)-(1b) can be obtained, at least +formally, from (3a)-(3c) by letting the parameter c run to `8, while γ is kept fixed. By the way, +system (1a)-(1b) can be cast in the more usual form +iBtU ` 1 +2∆xU “ ´γ2κ +` +Σ ‹x |U|2˘ +U, +t P R, x P Rd. +(5) +where1 +κ “ +ˆ +Rn σ2pzqp´∆zq´1σ2pzq dz “ +ˆ +Rn +|pσ2pξq|2 +|ξ|2 +dξ +p2πqn ą 0 and Σ “ σ1 ‹ σ1. +(6) +Letting now Σ resemble the delta-Dirac mass, the asymptotic leads to the standard cubic non linear +Schrödinger equation +iBtU ` 1 +2∆xU “ ´γ2κ|U|2U. +(7) +in the focusing case. These asymptotic connections can be expected to shed some light on the +dynamics of (3a)-(3c) and to be helpful to guide the intuition about the behavior of the solutions, +see [20, 21]. +The motivation for investigating these systems takes its roots in the general landscape of the +analysis of “open systems”, describing the dynamics of particles driven by momentum and energy +exchanges with a complex environment. Such problems are modeled as Hamiltonian systems, and +it is expected that the interaction mechanisms ultimately produce the dissipation of the particles’ +energy, an idea which dates back to A. O. Caldeira and A. J. Leggett [7]. These issues have been +investigated for various classical and quantum couplings, and with many different mathematical +viewpoints, see e. g. [2, 3, 24, 25, 28, 29, 30]. The case in which the environment is described as +a vibrational field, like in the definition of the potential by (3b)-(3c), is particularly appealing. In +1The Fourier transform of an integrable function ϕ : Rn Ñ C is defined by pϕpξq “ +´ +Rn ϕpzqe´iξ¨z dz. +2 + +fact, (3a)-(3c) is a quantum version of a model introduced by S. De Bièvre and L. Bruneau, dealing +with a single classical particle [6]. Intuitively, the model of [6] can be thought of as if in each space +position x P Rd there is a membrane oscillating in a direction z P Rn, transverse to the motion +of the particles. When a particle hits a membrane, its kinetic energy activates vibrations and the +energy is evacuated at infinity in the z´direction. These energy transfer mechanisms eventually +act as a sort of friction force on the particle, an intuition rigorously justified in [6, Theorem 2 and +Theorem 4]. We refer the reader to [1, 12, 13, 30, 46] for further theoretical and numerical insight +about this model. The model of [6] has been revisited by considering many interacting particles, +which leads to Vlasov-type equations, still coupled to a wave equation for defining the potential +[17]. Unexpectedly, asymptotic arguments indicate a connection with the attractive Vlasov-Poisson +dynamic [11]. In turn, the particles-environment interaction can be interpreted in terms of Lan- +dau damping [19, 18]. The quantum version (3a)-(3c) of the De Bièvre-Bruneau model has been +discussed in [21, 20], with a connection to the kinetic model by means of a semi-classical analysis +inspired from [35]. Note that in (3a)-(3c), the vibrational field remains of classical nature; a fully +quantum framework is dealt with in [3] for instance. +A remarkable feature of these systems is the presence of conserved quantities, here inherited +from the framework designed in [6] for a classical particle, and the study of these models brings out +the critical role of the wave speed c ą 0 and the dimension n of the space for the wave equation +(we can already notice that n ě 3 is necessary for (6) to be meaningful), see [6, 18, 19, 21]. For the +Schrödinger-Wave system (3a)-(3c) the energy +HSWpU, Ψ, Πq “ 1 +4 +ˆ +Td |∇U|2 dx ` 1 +4 +¨ +TdˆRn +ˆΠ2 +c2 ` |∇zΨ|2 +˙ +dx dz ` γ +2 +ˆ +Td Φ|U|2 dx, +(8) +is conserved since we can readily check that +d +dtHSWpU, Ψ, BtΨq “ 0. +Similarly, for the Hartree system (1a)-(1b), we get +d +dtHHapUq “ 0 +where we have set +HHapUq “ 1 +4 +ˆ +Td |∇U|2 dx ´ γ2 κ +4 +ˆ +Td Σpx ´ yq|Upt, xq|2|Upt, yq|2 dy dx. +Furthermore, for both model, the L2 norm is conserved. Of course, these conservation properties +play a central role for the analysis of the equations. However, (1a)-(1b) has further fundamental +properties which occur only for the asymptotic model: firstly, (1a)-(1b) is Galilean invariant, which +means that, given a solution pt, xq ÞÑ upt, xq and for any p0 P Td, the function pt, xq ÞÑ upt, x ´ +tp0qeipx´tp0{2q is a solution too; secondly, the momentum pptq “ Im +´ +¯upt, xq∇xupt, xq dx is conserved +and, accordingly, the center of mass follows a straight line at constant speed. That these properties +are not satisfied by the more complex system (3a)-(3c) makes its analysis more challenging. Finally, +we point out that, in contrast to the usual nonlinear Schrödinger equation or Hartree-Newton +system, where Σ is the Newtonian potential, the equations (1a)-(1b) or (3a)-(3c) do not fulfil a +3 + +scale invariance property. This also leads to specific mathematical difficulties: despite the possible +regularity of Σ, many results and approaches of the Newton case do not extend to a general kernel, +due to the lack of scale invariance. +When the problem is set on the whole space Rd, one is interested in the stability of solitary +waves, which are solutions of the equation with the specific form upt, xq “ eiωtQpxq, and, for +(3a)-(3c), ψpt, x, zq “ Ψpx, zq. The details of the solitary wave are embodied into the Choquard +equation, satisfied by the profile Q, [32, 36]. It turns out that the Choquard equation have infinitely +many solutions; among these solutions, it is relevant to select the solitary wave which minimizes the +energy functional under a mass constraint, [32, 37] and to study the orbital stability of this minimal +energy state. This program has been investigated for (7) and (1a)-(1b) in the specific case where +Σpxq “ +1 +|x| in dimension d “ 3, by various approaches [8, 31, 33, 34, 39, 49, 50]. Quite surprisingly, +the specific form of the potential plays a critical role in the analysis (either through explicit formula +or through scale invariance properties), and dealing with a general convolution kernel, as smooth +as it is, leads to new difficulties, that can be treated by a perturbative argument, see [27, 51] for +the case of the Yukawa potential, and [21] for (1a)-(1b) and (3a)-(3c). +Here, we adopt a different viewpoint. We consider the case where the problem holds on the +torus Td, and we are specifically interested in the stability of plane wave solutions of (3a)-(3c) and +(1a)-(1b). We refer the reader to [4, 5, 14, 40] for results on the nonlinear Schrödinger equation +(7) in this framework. The discussion on the stability of these plane wave solutions will make the +following smallness condition +4γ2κ}σ1}2 +L1 ă 1 +(9) +(assuming the plane wave has an amplitude unity) appear. Despite its restriction to the periodic +framework, the interest of this study is two-fold: on the one hand, it points out some difficulties +specific to the coupling and provides useful hints for future works; on the other hand, it clarify the +role of the parameters, by making stability conditions explicit. +The paper is organized as follows. In Section 2, we clarify the positioning of the paper. To +this end, we further discuss some mathematical features of the model. We also introduce the main +assumptions on the parameters that will be used throughout the paper and we provide an overview +of the results. Section 3 is concerned with the stability analysis of the Hartree equation (1a)-(1b). +Section 4 deals with the Schrödinger-Wave system at the price of restricting to the case where the +wave vector of the plane wave solution vanishes: k “ 0. For reasons explained in details below, the +general case is much more difficult. Section 5 justifies that in general the mode k � 0 is linearly +unstable. Finally, in Appendix A, we provide a physical interpretation of the parameters involved, +and for the sake of completeness, in Appendices B and C, we discuss the well-posedness of the +Schrödinger-Wave system (3a)-(3c) and its link with the Hartree equation (1a)-(1b) in the regime +of large c’s. +4 + +2 +Set up of the framework +2.1 +Plane wave solutions and dispersion relation +For any k P Zd, we start by seeking solutions to (3a)-(3c) of the form +Upt, xq “ Ukpt, xq :“ exp +` +ipωt ` k ¨ xq +˘ +, +Ψpt, x, zq “ Ψ˚pzq, +BtΨpt, x, zq “ Π˚pzq “ 0, +(10) +with ω ě 0. Note that the L2 norm of Uk is p2πqd{2 and Ψ˚ actually does not depend on the time +variable, nor on x. Since |Ukpt, xq| “ 1 is constant, the wave equation simplifies to +1 +c2 B2 +ttΨ ´ ∆zΨ “ ´γσ2pzq +@ +σ1 +D +Td, +where +@ +¨ +D +Td stands for the average over Td: +@ +f +D +Td “ +´ +Td fpxq dx. As a consequence, z ÞÑ Ψ˚pzq is +a solution to (3b) if +Ψ˚pzq “ ´γΓpzq +@ +σ1 +D +Td, +with Γ the solution of +´∆zΓpzq “ σ2pzq. +This auxiliary function Γ is thus defined by the convolution of σ2 with the elementary solution of +the Laplace operator in dimension n, or equivalently by means of Fourier transform: +Γpzq “ +ˆ +Rn +Cn +|z ´ z1|n´2 σ2pz1q dz1 “ F ´1 +ξÑz +´pσ2pξq +|ξ|2 +¯ +. +(11) +The corresponding potential (3c) is actually a constant which reads +´γ +¨ +TdˆRn σ1px ´ yqσ2pzqΓpzq +@ +σ1 +D +Td dz dy “ ´κγ +@ +σ1 +D2 +Td +with +κ “ +ˆ +Rn σ2pzqΓpzq dz “ +ˆ +Rn |∇zΓpzq|2 dz ą 0 +(we remind the reader that this formula coincides with (6) and makes sense only when n ě 3). It +remains to identify the condition on the coefficients so that Uk satisfies the Schrödinger equation +(3a): this leads to the following dispersion relation +ω ` k2 +2 ´ Υ˚ “ 0, +Υ˚ “ γ2κ +@ +σ1 +D2 +Td ą 0 +(12) +with k2 “ řd +j“1 k2 +j. We can compute explicitly the associated energy: +HSWpUk, Ψ˚, Π˚q “ p2πqd +2 +ˆk2 +2 ´ γ2κ +2 +@ +σ1 +D2 +Td +˙ +“ p2πqd +4 +pk2 ´ Υ˚q. +Of course, among these solutions, the constant mode U0pt, xq “ eiωt1pxq has minimal energy. +It turns out that the plane wave Ukpt, xq “ eiωteik¨x equally satisfies (1a)-(1b) provided the +dispersion relation (12) holds. Incidentally, we can check that +HHapUkq “ p2πqd +2 +ˆk2 +2 ´ γ2κ +2 +@ +Σ +D +Td +˙ +“ p2πqd +4 +pk2 ´ Υ˚q +is made minimal when k “ 0. +5 + +2.2 +Hamiltonian structure and symmetries of the problem +The conservation properties play a central role in the stability analysis, for instance in the reasonings +that use concentration-compactness arguments [8]. Based on the conserved quantities, one can try +to construct a Lyapounov functional, intended to evaluate how far a solution is from an equilibrium +state. Then the stability analysis relies on the ability to prove a coercivity estimate on the variations +of the Lyapounov functional, see [47, 49, 50]. This viewpoint can be further extended by identifying +analogies with finite dimensional Hamiltonian systems with symmetries, which has permitted to +set up a quite general framework [22, 23], revisited recently in [4]. The strategy relies on the ability +in exhibiting a Hamiltonian formulation of the problem +BtX “ JBXH pXq, +where the symplectic structure is given by the skew-symmetric operator J. As a consequence of +Noether’s Theorem, this formulation encodes the conservation properties of the system. In partic- +ular, it implies that t ÞÑ H pXptqq is a conserved quantity. For the problem under consideration, as +it will be detailed below, X is a vectorial unknown with components possibly depending on different +variables (x P Td and z P Rn). This induces specific difficulties, in particular because the nature +of the coupling is non local and delicate spectral issues arise related to the essential spectrum of +the wave equation in Rn. Next, we can easily observe that the systems (1a)-(1b) and (3a)-(3c) +are invariant under multiplications by a phase factor of U, the “Schödinger unknown”, and under +translations in the x variable. This leads to the conservation of the L2 norm of U and of the total +momentum. However, the systems (1a)-(1b) and (3a)-(3c) cannot be handled by a direct applica- +tion of the results in [4, 22, 23]: the basic assumptions are simply not satisfied. Nevertheless, our +approach is strongly inspired from [4, 22, 23]. As we will see later, for the Hartree system, a decisive +advantage comes from the conservation of the total momentum and the Galilean invariance of the +problem. For the Schrödinger-Wave problem, since the expression of the total momentum mixes +up contribution from the “Schrödinger unknown” U and the “wave unknown” Ψ, the information +on its conservation does not seem readily useful. 2 +In what follows, we find advantages in changing the unknown by writing Upt, xq “ eik¨xupt, xq; +in turn the Schrödinger equation iBtU ` 1 +2∆U “ ΦU becomes +iBtU ` 1 +2∆u ´ k2 +2 u ` ik ¨ ∇u “ Φu. +Accordingly, the parameter k will appear in the definition the energy functional H . This explains +a major difference between (1a)-(1b) and (3a)-(3c): for the former, a coercivity estimate can be +obtained for the energy functional H , for the latter, when k � 0 there are terms which cannot be +controlled easily. This is reminiscent of the momentum conservation in (1a)-(1b) and the lack of +Galilean invariance for (3a)-(3c). The detailed analysis of the linearized operators sheds more light +on the different behaviors of the systems (1a)-(1b) and (3a)-(3c). +2For the problem set on Rd, it is still possible, in the spirit of results obtained in [14] for NLS, to justify +that orbital stability holds on a finite time interval: the solution remains at a distance ǫ from the orbit of +the ground state over time interval of order Op1{ ?ǫq, see [48, Theorem 4.2.11 & Section 4.6]. The argument +relies on the dispersive properties of the wave equation through Strichartz’ estimates. +6 + +2.3 +Outline of the main results +Let us collect the assumptions on the form functions σ1 and σ2 that govern the coupling: +(H1) σ1 : Td Ñ r0, 8q is C8 smooth, radially symmetric; +@ +σ1 +D +Td � 0; +(H2) σ2 : Rn Ñ r0, 8q is C8 smooth, radially symmetric and compactly supported; +(H3) p´∆q´1{2σ2 P L2pRnq; +(H4) for any ξ P Rn, pσ2pξq � 0. +Assumptions (H1)-(H2) are natural in the framework introduced in [6]. Hypothesis (H3) can +equivalently be rephrased as p´∆q´1σ2 P .H1pRnq; it appears in many places of the analysis of +such coupled systems and, at least, it makes the constant κ in (6) meaningful. This constant is +a component of the stability constraint (9). Hypothesis (H4) equally appeared in [6, Eq. (W)] +when discussing large time asymptotic issues. Assumptions (H1)-(H4) are assumed throughout +the paper. +Our results can be summarized as follows. We assume (9) and consider k P Zd and ω ą 0 +satisfying (12). For the Hartree equation, the analysis is quite complete: +• the plane wave eipωt`k¨xq is spectrally stable (Theorem 3.1); +• for any initial perturbation with zero mean, the solutions of the linearized Hartree equation +are L2-bounded, uniformly over t ě 0 (Theorem 3.3); +• the plane wave eipωt`k¨xq is orbitally stable (Theorem 3.5). +For the Schrödinger-Wave system, only the case k “ 0 is fully addressed: +• the plane wave peiωt1pxq, ´γΓpzq +@ +σ +D +Td, 0q is spectrally stable (Corollary 5.12); +• for any initial perturbation of peiωt1pxq, ´γΓpzq +@ +σ +D +Td, 0q with zero mean, the solutions of the +linearized Schrödinger-Wave system are L2-bounded, uniformly over t ě 0 (Theorem 4.2); +• the plane wave peiωt1pxq, ´γΓpzq +@ +σ +D +Td, 0q is orbitally stable (Theorem 4.4). +When k � 0, the situation is much more involved; at least we prove that in general the plane wave +solution peipωt`k¨xq, ´γΓpzq +@ +σ1 +D +Td, 0q is spectrally unstable, see Section 5 and Corollary 5.15. +3 +Stability analysis of the Hartree system (1a)-(1b) +To study the stability of the plane wave solutions of the Hartree system, it is useful to write the +solutions of (1a)-(1b) in the form +Upt, xq “ eik¨xupt, xq +with upt, xq solution to +iBtu ` 1 +2∆u ´ k2 +2 u ` ik ¨ ∇u “ ´γ2κpΣ ‹ |u|2qu. +(13) +7 + +If k P Zd and ω ą 0 satisfy the dispersion relation (12), uωpt, xq “ eiωt1pxq is a solution to (13) with +initial condition uωp0, tq “ 1pxq. Therefore, studying the stability properties of Ukpt, xq “ eiωteik¨x +as a solution to (1a)-(1b) amounts to studying the stability of uωpt, xq “ eiωt1pxq as a solution to +(13). +The problem (13) has an Hamiltonian symplectic structure when considered on the real Banach +space H1pTd; Rq ˆ H1pTd; Rq. Indeed, if we write u “ q ` ip, with p, q real-valued, we obtain +Bt +ˆ +q +p +˙ +“ J∇pq,pqH pq, pq +with +J “ +ˆ 0 +1 +´1 +0 +˙ +and +H pq, pq “ 1 +2 +ˆ1 +2 +ˆ +Td |∇q|2 ` |∇p|2 dx ` k2 +2 +ˆ +Tdpp2 ` q2q dx ´ +ˆ +Td pk ¨ ∇q dx ` +ˆ +Td qk ¨ ∇p dx +˙ +´ γ2κ +4 +ˆ +Td Σ ‹ pp2 ` q2qpp2 ` q2q dx. +Coming back to u “ q ` ip, we can write +H puq “ 1 +2 +ˆ1 +2 +ˆ +Td |∇u|2 dx ` k2 +2 +ˆ +Td |upxq|2 dx ` +ˆ +Td k ¨ p´i∇uqu dx +˙ +´ γ2κ +4 +ˆ +TdpΣ ‹ |u|2qpxq|upxq|2 dx. +(14) +As observed above, H is a constant of the motion. +Moreover, it is clear that (13) is invariant under multiplications by a phase factor so that +Fpuq “ 1 +2}u}2 +L2 is conserved by the dynamics. The quantities +Gjpuq “ 1 +2 +ˆ +Td +ˆ1 +i Bxju +˙ +u dx +are constants of the motion too, that correspond to the invariance under translations. Indeed, a +direct verification leads to +d +dtGjpuqptq “ κγ2 +2 +ˆ +Td +ˆ +Td BxjΣpx ´ yq ‹ |u|2pt, yq|u|2pt, xq dy dx “ 0. +Finally, we shall endow the Banach space H1pTd; Rq ˆ H1pTd; Rq with the inner product +Bˆ +q +p +˙ ˇˇˇ +ˆ +q1 +p1 +˙F +“ +ˆ +Td +` +pp1 ` qq1q dx. +that can be also interpreted as an inner product for complex-valued functions: +xu|u1y “ Re +ˆ +Td uu1 dx. +(15) +8 + +3.1 +Linearized problem and spectral stability +Let us expand the solution of (13) around uω as upt, xq “ uωpt, xqp1 ` wpt, xqq. The linearized +equation for the fluctuation reads +iBtw ` 1 +2∆xw ` ik ¨ ∇xw “ ´2γ2κpΣ ‹ Repwqq. +(16) +We split w “ q ` ip, q “ Repwq, p “ Impwq so that (16) recasts as +Bt +ˆq +p +˙ +“ Lk +ˆq +p +˙ +(17) +with the linear operator +Lk : +ˆ +q +p +˙ +ÞÝÑ +¨ +˝ +´k ¨ ∇xq ´ 1 +2∆xp +1 +2∆xq ` 2γ2κΣ ‹ q ´ k ¨ ∇xp +˛ +‚. +(18) +Theorem 3.1 (Spectral stability for the Hartree equation) Let k P Zd and ω ą 0 such that +the dispersion relation (12) is satisfied. Suppose (9) holds. Then the spectrum of Lk, the lineariza- +tion of (13) around the plane wave uωpt, xq “ eiωt1pxq, in L2pTd; Cq ˆ L2pTd; Cq is contained in iR. +Consequently, this wave is spectrally stable in L2pTdq. +Proof. +To prove Theorem 3.1, we expand q, p and σ1 by means of their Fourier series +qpt, xq “ +ÿ +mPZd +Qmptqeim¨x, +Qmptq “ +1 +p2πqd +ˆ +Td qpt, xqe´im¨x dx, +ppt, xq “ +ÿ +mPZd +Pmptqeim¨x, +Pmptq “ +1 +p2πqd +ˆ +Td ppt, xqe´im¨x dx, +σ1pxq “ +ÿ +mPZd +σ1,meim¨x, +σ1,mptq “ +1 +p2πqd +ˆ +Td σ1pxqe´im¨x dx. +Note that σ1 being real and radially symmetric, we have +σ1,m “ σ1,m “ σ1,´m +(19) +and, by definition, +@ +σ1 +D +Td “ p2πqdσ1,0. As a consequence, we obtain +Lk +ˆq +p +˙ +“ +¨ +˚ +˚ +˝ +ř +mPZd +ˆm2 +2 Pm ´ ik ¨ mQm +˙ +eim¨x +ř +mPZd +ˆ +´m2 +2 Qm ´ ik ¨ mPm ` 2p2πq2dγ2κ|σ1,m|2Qm +˙ +eim¨x +˛ +‹‹‚ +“ Lk,0 +ˆQ0 +P0 +˙ +` +ÿ +mPZd∖t0u +Lk,m +ˆQm +Pm +˙ +eik¨x +(20) +with +Lk,0 “ +ˆ +0 +0 +2p2πq2dγ2κ|σ1,0|2 +0 +˙ +and Lk,m “ +˜ +´ik ¨ m +m2 +2 +´ m2 +2 ` 2p2πq2dγ2κ|σ1,m|2 +´ik ¨ m +¸ +(21) +for m P Zd ∖ t0u. +9 + +Note that, since the Fourier modes are uncoupled, +ˆq +p +˙ +is a solution to (17) if and only if the +Fourier coefficients +ˆQm +Pm +˙ +satisfy +Bt +ˆ +Qmptq +Pmptq +˙ +“ Lk,m +ˆ +Qmptq +Pmptq +˙ +for any m P Zd. Similarly, λ P C is an eigenvalue of the operator Lk if and only if there exists at +least one Fourier mode m P Zd such that λ is an eigenvalue of the matrix Lk,m, i.e. there exists +pqm, pmq � p0, 0q such that +λqm ´ m2 +2 pm ` ik ¨ mqm “ 0, +λpm ` m2 +2 qm ` ik ¨ mpm “ 2p2πq2dγ2κ|σ1,m|2qm. +(22) +A straightforward computation gives that λ0 “ 0 is the unique eigenvalue of the matrix Lk,0 +with eigenvector p0, 1q. This means that KerpLkq contains at least the vector subspace spanned by +the constant function x P Td ÞÑ +ˆ0 +1 +˙ +, which corresponds to the constant solution upt, xq “ i of (16). +Next, if m P Zd ∖ t0u, λm is an eigenvalue of Lk,m if it is a solution to +pλ ` ik ¨ mq2 ´ m2 +2 +ˆ +´m2 +2 ` 2p2πq2dγ2κ|σ1,m|2 +˙ +“ 0. +This is a second order polynomial equation for λ and the roots are given by +λm,˘ “ ´ik ¨ m ˘ |m| +2 +b +´m2 ` 4γ2κp2πq2d|σ1,m|2. +If the smallness condition (9) holds, the argument of the square root is negative for any m P Zd∖t0u, +and thus the roots λ are all purely imaginary (and we note that λ´m,˘ “ λm,¯). More precisely, +we have the following statement. +Lemma 3.2 (Spectral stability for the Hartree equation) Let k, m P Zd and Lk,m defined +as in (21). Then +1. λ0 “ 0 is the unique eigenvalue of Lk,0 and KerpLk,0q “ span +"ˆ +0 +1 +˙* +; +2. for any m P Zd ∖ t0u, the eigenvalue of Lk,m are +λm,˘ “ ´ik ¨ m ˘ |m| +2 +b +´m2 ` 4γ2κp2πq2d|σ1,m|2. +(a) if 4γ2κp2πq2d |σ1,m|2 +m2 +ď 1, then λm,˘ P iR; +(b) if 4γ2κp2πq2d |σ1,m|2 +m2 +ą 1, then λm,˘ P C ∖ iR. Moreover, Repλm,`q ą 0. +Now, (9) implies 4γ2κp2πq2d |σ1,m|2 +m2 +ă 1 for all m P Zd ∖ t0u, so that σpLkq Ă iR and uωpt, xq “ +eiωt1pxq is spectrally stable. Conversely, if σ1, σ2 and γ are such that there exists m˚ P Zd ∖ t0u +10 + +verifying 4γ2κp2πq2d |σ1,m˚|2 +m2 +˚ +ą 1, then the plane wave uω is spectrally unstable for any k P Zd and +ω ą 0 that satisfy the dispersion relation (12). This proves Proposition 3.1. +We observe that this result is consistent with the linear stability analysis of (7), see [40, The- +orem 1], when replacing formally Σ by the delta-Dirac. The analogy should be considered with +caution, though, since the functional difficulties are substantially different: here u ÞÑ ´ 1 +2∆Tdu ´ +2γ2κΣ‹Repuq is a compact perturbation of ´ 1 +2∆Td, which has a compact resolvent hence a spectral +decomposition. +It is important to remark that the analysis of eigenproblems for Lk has consequences on the +behavior of solutions to (17) of the particular form +Qpt, xq “ eλtqpxq, +Ppt, xq “ eλtppxq. +We warn the reader that spectral stability excludes the exponential growth of the solutions of the +linearized problem when the smallness condition (9) holds, but a slower growth is still possible. +This can be seen by direct inspection for the mode m “ 0: we have BtQ0 “ 0, so that Q0ptq “ Q0p0q +and BtP0 “ 2p2πq2dκ +@ +σ1 +D2 +TdQ0p0q which shows that the solution can grow linearly in time +P0ptq “ P0p0q ` 2p2πq2dγ2κ +@ +σ1 +D2 +TdQ0p0qt. +In fact, excluding the mode m “ 0 suffices to guaranty the linearized stability. +Theorem 3.3 (Linearized stability for the Hartree equation) Suppose (9). +Let w be the +solution of (16) associated to an initial data wInit P H1pTdq such that +´ +Td wInit dx “ 0. +Then, +there exists a constant C ą 0 such that suptě0 }wpt, ¨q}H1 ď C. +Proof. +Note that if +´ +Td wInit dx “ 0 then the corresponding Fourier coefficients Q0p0q and P0p0q +are equal to 0. As a consequence, Q0ptq “ P0ptq “ 0 for all t ě 0, so that +´ +Td wpt, xq dx “ 0 for all +t ě 0. +The proof follows from energetic consideration. Indeed, we observe that, on the one hand, +1 +2 +d +dt +ˆ +Td |∇w|2 dx “ ´γ2κ +2i +ˆ +Td Σ ‹ pw ` wq∆pw ´ wq dx, +and, on the other hand, +1 +2 +d +dt +ˆ +Td Σ ‹ pw ` wqpw ` wq dx +“ ´ 1 +2i +ˆ +Td Σ ‹ pw ` wq∆pw ´ wq dx ´ k ¨ +ˆ +Td ∇pw ` wqΣ ‹ pw ` wq dx, +where we get rid of the last term in the right hand side by assuming k “ 0. This leads to the +following energy conservation property +d +dt +"1 +2 +ˆ +Td |∇w|2 dx ´ γ2κ +2 +ˆ +Td Σ ‹ pw ` wqpw ` wq dx +* +“ 0 +which holds for k “ 0. We denote by E0 the energy of the initial data wInit. Finally, we can simply +estimate +ˇˇˇˇ +ˆ +Td Σ ‹ pw ` wqpw ` wq dx +ˇˇˇˇ ď }Σ ‹ pw ` wq}L2}w ` w}L2 ď }Σ}L1}w ` w}2 +L2 ď 4}Σ}L1}w}2 +L2. +11 + +To conclude, we use the Poincaré-Wirtinger estimate. Indeed, since we have already remarked that +the condition +´ +Td wInit dx “ 0 implies +´ +Td wpt, xq dx “ 0 for any t ě 0, we can write +}wpt, ¨q}2 +L2 “ +›››wpt, ¨q ´ +1 +p2πqd +ˆ +Td wpt, yq dy +››› +2 +L2 “ p2πqd +ÿ +mPZd∖t0u +|cmpwpt, ¨qq|2 +ď p2πqd +ÿ +mPZd∖t0u +m2|cmpwpt, ¨qq|2 “ }∇wpt, ¨q}2 +L2 +for any t ě 0, where the cmpwpt, ¨qq’s are the Fourier coefficients of the function x P Td ÞÑ wpt, xq. +Hence, for any solution with zero mean, we infer, for all t ě 0, +2E0 “ +ˆ +Td |∇w|2pt, xq dx´γ2κ +ˆ +Td Σ‹pw `wqpw `wqpt, xq dx ě p1´4γ2κ}Σ}L1q +ˆ +Td |∇wpt, xq|2 dx. +As a consequence, if (9) is satisfied, we obtain +sup +tě0 +}wpt, ¨q}H1 ď 2 +d +E0 +1 ´ 4γ2κ}Σ}L1 . +The stability estimate extends to the situation where k � 0. Indeed, from the solution w of +(16), we set +vpt, xq “ wpt, x ` tkq. +It satisfies iBtv ` 1 +2∆xv “ ´2γ2κΣ ‹ Repvq. Hence, repeating the previous argument, }vpt, ¨q}H1 “ +}wpt, ¨q}H1 remains uniformly bounded on p0, 8q. This step of the proof relies on the Galilean invari- +ance of (5); it could have been used from the beginning, but it does not apply for the Schrödinger- +Wave system. +Remark 3.4 The analysis applies mutadis mutandis to any equation of the form (1a), with the +potential defined by a kernel Σ and a strength encoded by the constant γ2κ. Then, the stability +criterion is set on the quantity 4γ2κp2πqd |pΣm| +m2 For instance, the elementary solution of pa2´∆xqΣ “ +δx“0 with periodic boundary condition has its Fourier coefficients given by pΣm “ +1 +p2πqdpa2`m2q ą 0. +Coming back to the physical variable, in the one-dimension case, the function Σ reads +Σpxq “ e´a|x| +2a +` +coshpaxq +ape2aπ ´ 1q. +The linearized stability thus holds provided 4γ2κp2πq2d +1 +a2`1 ă 1. +3.2 +Orbital stability +In this subsection, we wish to establish the orbital stability of the plane wave uωpt, xq “ eiωt1pxq +as a solution to (13) for k P Zd and ω ą 0 that satisfy the dispersion relation (12). As pointed +out before, (13) is invariant under multiplications by a phase factor. +This leads to define the +corresponding orbit through upxq “ 1pxq by +O1 “ teiθ, θ P Ru. +12 + +Intuitively, orbital stability means that the solutions of (13) associated to initial data close enough +to the constant function x P Td ÞÑ 1 “ 1pxq remain at a close distance to the set O1. Stability +analysis then amounts to the construction of a suitable Lyapounov functional satisfying a coercivity +property. This functional should be a constant of the motion and be invariant under the action of +the group that generates the orbit O1. Hence, the construction of such a functional relies on the +invariants of the equation. Moreover, the plane wave has to be a critical point on the Lyapounov +functional so that the coercivity can be deduced from the properties of its second variation. The +difficulty here is that, in general, the bilinear symmetric form defining the second variation of the +Lyapounov function is not positive on the whole space: according to the strategy designed in [22], +see also the review [47], it will be enough to prove the coercivity on an appropiate subspace. Here +and below, we adopt the framework presented in [4] (see also [5]). +Inspired by the strategy designed in [4, Section 8 & 9], we introduce, for any k P Zd and ω ą 0 +satisfying the dispersion relation (12), the set +Sω “ +! +u P H1pTd; Cq, Fpuq “ Fp1q “ p2πqd +2 +“ p2πqd k2{2 ` ω +2γ2κ +@ +σ1 +D2 +Td +) +; +Sω is therefore the level set of the solutions of (13), associated to the plane wave pt, xq ÞÑ uωpt, xq “ +eiωt1pxq. Next, we introduce the functional +Lωpuq “ H puq ` ωFpuq ´ +dÿ +j“1 +kjGjpuq, +(23) +which is conserved by the solutions of (13). We have +BuLωpuqpvq +“ +Re +ˆ1 +2 +ˆ +Tdp´∆uqv dx ` k2 +2 +ˆ +Td uv dx +´γ2κ +¨ +TdˆTd Σpx ´ yq|upyq|2upxqvpxq dy dx`ω +ˆ +Td uv dx +˙ +. +As a matter of fact, we observe that +BuLωp1q “ 0 +owing to the dispersion relation. Next, we get +B2 +uLωpuqpv, wq +“ +Re +ˆ1 +2 +ˆ +Tdp´∆ ` k2qwv dx +´2γ2κ +¨ +TdˆTd Σpx ´ yqRe +` +upyqwpyq +˘ +upxqvpxq dy dx +´γ2κ +¨ +TdˆTd Σpx ´ yq|upyq|2wpxqvpxq dy dx ` ω +ˆ +Td wv dx +˙ +. +Still by using the dispersion relation, we obtain +B2 +uLωp1qpv, wq “ Re +¨ +˚ +˚ +˚ +˝ +ˆ +Td +ˆ +´∆w +2 +´ 2γ2κΣ ‹ Repwq +˙ +looooooooooooooooomooooooooooooooooon +:“Sw +vpxq dx +˛ +‹‹‹‚“ xSw|vy. +13 + +S : H2pTdq Ă L2pTdq Ñ L2pTdq is an unbounded linear operator and its spectral properties will +play an important role for the orbital stability of uω. Note that the operator S is the linearized +operator (18), up to the advection term k ¨ ∇. +The main result of this subsection is the following. +Theorem 3.5 (Orbital stability for the Hartree equation) Let k P Zd and ω ą 0 such that +the dispersion relation (12) is satisfied. Suppose (9) holds. Then the plane wave uωpt, xq “ eiωt1pxq +is orbitally stable, i.e. +@ε ą 0, Dδ ą 0, @vInit P H1pTd; Cq, }vInit ´ 1}H1 ă δ ñ sup +tě0 +distpvptq, O1q ă ε +(24) +where distpv, O1q “ infθPr0,2πr }v ´ eiθ1} and pt, xq ÞÑ vpt, xq P C0pr0, 8q; H1pTdqq stands for the +solution of (13) with Cauchy data vInit. +The key ingredient to prove Theorem 3.5 is the following coercivity estimate on the Lyapounov +functional. +Lemma 3.6 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied. Suppose that +there exist η ą 0 and c ą 0 such that +@w P Sω, dpw, O1q ă η ñ Lωpwq ´ Lωp1q ě c distpw, O1q2. +(25) +Then the the plane wave uωpt, xq “ eiωt1pxq is orbitally stable. +Proof of Theorem 3.5. +Assume that Lemma 3.6 holds and suppose, by contradiction, that uω +is not orbitally stable. Hence, there exists 0 ă ε0 ă 2 +3η such that +@n P N ∖ t0u, DuInit +n +P H1pTdq, }uInit +n +´ 1}H1 ă 1 +n and Dtn P r0, `8r, distpunptnq, O1q “ ε0, +pt, xq ÞÑ unpt, xq P C0pr0, 8q; H1pTdqq being the solution of (13) with Cauchy data uInit +n +. +To +apply the coercivity estimate of Lemma 3.6, we define zn “ +´ +F p1q +F punptnqq +¯1{2 +unptnq. It is clear that +zn P Sω since Fpznq “ Fp1q. Moreover, +` +unptnq +˘ +nPN∖t0u is a bounded sequence in H1pTdq and +limnÑ`8 Fpunptnqq “ Fp1q. Indeed, on the one hand, there exists γ P r0, 2πr such that +}unptnq}H1 ď }unptnq ´ eiθ1}H1 ` }eiθ1}H1 ď 2dpunptnq, O1q ` }eiθ1}H1 “ 2ε0 ` }1}H1 +and, on the other hand, +|Fpunptnqq ´ Fp1q| “ 1 +2|}unptnq}2 +L2 ´ }1}2 +L2| ď }unptnq ´ 1}L2pε0 ` }1}H1q ă 1 +npε0 ` }1}H1q. +As a consequence, limnÑ`8 }zn ´ unptnq}H1 “ 0. This implies for n P N large enough, +ε0 +2 ď dpzn, O1q ď 3ε0 +2 +ă η. +Hence, thanks to Lemma 3.6, we obtain +LωpuInit +n +q ´ Lωp1q “ Lωpunptnqq ´ Lωp1q “ Lωpunptnqq ´ Lωpznq ` Lωpznq ´ Lωp1q +ě Lωpunptnqq ´ Lωpznq ` cdpzn, O1q2 ě Lωpunptnqq ´ Lωpznq ` c +4ε2 +0. +14 + +Finally, using the fact that BuLωp1q “ 0 and B2 +uLωp1qpw, wq ď C}w}2 +H1, we deduce that +lim +nÑ`8pLωpuInit +n +q ´ Lωp1qq “ 0, +lim +nÑ`8pLωpunptnqq ´ Lωpznqq “ 0. +We are thus led to a contradiction. +Since BuLωp1q “ 0, the coercivity estimate (25) can be obtained from a similar estimate on the +bilinear form B2 +uLωp1qpw, wq for any w P H1. As pointed out before, the difficulty lies in the fact +that, in general, this bilinear form is not positive on the whole space H1. The following lemma +states that it is enough to have a coercivity estimate on B2 +uLωp1qpw, wq for any w P T1SωXpT1O1qK. +Recall that the tangent set to Sω is given by +T1Sω “ tu P H1pTd; Cq, BuFp1q “ 0u “ +" +pq, pq P H1pTd, Rq ˆ H1pTd, Rq, +A ˆ +q +p +˙ ˇˇˇ +ˆ +1 +0 +˙ E +“ 0 +* +This set is the orthogonal to 1 with respect to the inner product defined in (15). The tangent set +to O1 (which is the orbit generated by the phase multiplication) is +T1O1 “ spanRti1u +so that +pT1O1qK “ tu P H1pTd, Cq, xu, i1y “ 0u “ +" +pq, pq : Td Ñ R, +A ˆ +q +p +˙ ˇˇˇ +ˆ +0 +1 +˙ E +“ 0 +* +. +Lemma 3.7 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied. Suppose that +there exists ˜c ą 0 +B2 +uLωp1qpu, uq ě ˜c}u}2 +H1 +(26) +for any u P T1S1 X pT1O1qK. Then there exist η ą 0 and c ą 0 such that (25) is satisfied. +Proof. +Let w P Sω such that distpw, O1q ă η with η ą 0 small enough. By means of an implicit +function theorem argument (see [4, Section 9, Lemma 8]), we obtain that there exists θ P r0, 2πr +and v P pT1O1qK such that +eiθw “ 1 ` v, +distpw, O1q ď }v}H1 ď Cdistpw, O1q +for some positive constant C. +Next, we use the fact that H1pTdq “ T1Sω ‘ spanRt1u to write v “ v1 ` v2 with v1 P T1Sω X +pT1O1qK and v2 P spanRt1u X pT1O1qK. Since v “ eiθw ´ 1 and Fpwq “ Fp1q, we obtain +0 “ Fpeiθwq ´ Fp1q “ 1 +2 +ˆ +Td |v|2 dx ` Re +ˆ +Tdpv1 ` v2q1 dx “ 1 +2 +ˆ +Td |v|2 dx ` Re +ˆ +Td v21 dx. +Since v2 P spanRt1u, it follows that +}v2}H1 ď }v}2 +H1 +}1}L2 . +This implies +}v1}H1 “ }v ´ v2}H1 ě }v}H1 ´ +1 +}1}L2 }v}2 +H1 ě 1 +2}v}H1 +15 + +provided }v}H1 ď }1}L2. As a consequence, if }v}H1 is small enough, using that B2 +uLωp1qpw, zq ď +C}w}H1}z}H1, we obtain +B2 +uLωp1qpv1, v2q ď C}v}3 +H1, +B2 +uLωp1qpv2, v2q ď C}v}4 +H1. +This leads to +B2 +uLωp1qpv, vq “ B2 +uLωp1qpv1, v1q ` op}v}2 +H1q. +Finally, let w P Sω be such that dpw, O1q ă η. We have +Lωpwq ´ Lωp1q “ Lωpeiθwq ´ Lωp1q “ 1 +2B2 +uLωp1qpv, vq ` op}v}2 +H1q +“ 1 +2B2 +uLωp1qpv1, v1q ` op}v}2 +H1q ě ˜c}v1}2 +H1 ` op}v}2 +H1q ě ˜c +2}v}2 +H1 ` op}v}2 +H1q +ě ˜c +4distpw, O1q2 +where we use BuLωp1q “ 0 and v1 P T1Sω X pT1O1qK. +At the end of the day, to prove the orbital stability of the plane wave uωpt, xq “ eiωt1pxq it +is enough to prove (26) for any u P T1S1 X pT1O1qK. This can be done by studying the spectral +properties of the operator S. However, in the simpler case of the Hartree equation, the coercivity +of B2 +uLωp1q on T1S1 X pT1O1qK can be also obtained directly from the expression +B2 +uLωp1qpu, uq “ Re +ˆˆ +Td +ˆ +´∆u +2 ´ 2γ2κΣ ‹ Repuq +˙ +upxq dx +˙ +“ xSu|uy. +(27) +Let u P T1S1 X pT1O1qK and write u “ q ` ip. This leads to +B2 +uLωp1qpu, uq “ 1 +2 +ˆ +Td |∇q|2 dx ´ 2γ2κ +ˆ +TdpΣ ‹ qqq dx ` 1 +2 +ˆ +Td |∇p|2 dx. +Moreover, since u P T1S1 X pT1O1qK, we have +ˆ +Td q dx “ 0 and +ˆ +Td p dx “ 0. +As a consequence, thanks to the Poincaré-Wirtinger inequality, we deduce +B2 +uLωp1qpu, uq ě 1 +2 +ˆ +Td |∇q|2 dx ´ 2γ2κ +ˆ +TdpΣ ‹ qqq dx ` 1 +4}p}2 +H1. +(28) +Next, we expand q and Σ in Fourier series, i.e. +qpxq “ +ÿ +mPZd +qmeim¨x and Σpxq “ +ÿ +mPZd +Σmeim¨x. +Note that, if Σ “ σ1 ‹ σ1, then Σm “ p2πqdσ2 +1,m. Moreover, +´ +Td q dx “ 0 implies q0 “ 0. Hence, +1 +2 +ˆ +Td |∇q|2 dx ´ 2γ2κ +ˆ +TdpΣ ‹ qqq dx “ p2πqd +ÿ +mPZd∖t0u +ˆm2 +2 ´ 2γ2κp2πqdΣm +˙ +q2 +m +“ p2πqd +ÿ +mPZd∖t0u +ˆ +1 ´ 4γ2κp2πqd Σm +m2 +˙ m2 +2 q2 +m. +(29) +As a consequence, we obtain the following statement. +16 + +Proposition 3.8 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied. Suppose +that there exists δ P p0, 1q such that +4γ2κp2πq2d σ2 +1,m +m2 ď δ +(30) +for all m P Zd ∖ t0u. Then, there exists ˜c ą 0 such that +B2 +uLωp1qpu, uq ě ˜c}u}2 +H1 +(31) +for any u P T1S1 X pT1O1qK. +Proof. +If (30) holds, then (28)-(29) lead to +B2 +uLωp1qpu, uq ě 1 ´ δ +2 +p2πqd +ÿ +mPZd∖t0u +m2q2 +m ` 1 +4}p}H1 “ 1 ´ δ +2 +}∇q}2 +L2 ` 1 +4}p}2 +H1 ě 1 ´ δ +4 +}u}2 +H1. +where in the last inequality we used the Poincaré-Wirtinger inequality together with the fact that +´ +Td q dx “ 0. +Remark 3.9 By decomposing the linear operator S into real and imaginary part and by using +Fourier series, one can study its spectrum. In particular, S has exactly one negative eigenvalue +λ´ “ ´2γ2κ +@ +Σ +D +Td with eigenspace spanRt1u. Moreover, KerpSq “ spanRti1u. Finally, if (30) is +satisifed, then infpσpSq X p0, 8qq ě 1´δ +2 . Then, by applying the same arguments as in [5, Section +6], we can recover the coercivity of B2 +uLωp1q on T1S1 X pT1O1qK. +Finally, Proposition 3.8 together with Lemma 3.7 and Lemma 3.6, gives Theorem 3.5 and the +orbital stability of the plane wave uω. +4 +Stability analysis of the Schrödinger-Wave system: +the case k “ 0 +Like in the case of the Hartree system, to study the stability of the plane wave solutions of the +Schrödinger-Wave system (3a)-(3c), it is useful to write its solutions in the form +Upt, xq “ eik¨xupt, xq +with pt, x, zq ÞÑ pupt, xq, Ψpt, x, zqq solution to +iBtu ` 1 +2∆xu ´ k2 +2 u ` ik ¨ ∇xu “ ´ +ˆ +γσ1 ‹ +ˆ +Rn σ2Ψ dz +˙ +u, +1 +c2 B2 +ttΨ ´ ∆zΨ “ ´γσ2σ1 ‹ |u|2. +(32) +If k P Zd and ω ą 0 satisfy the dispersion relation (12), +uωpt, xq “ eiωt1pxq, +Ψ˚pt, x, zq “ ´γΓpzq +@ +σ1 +D +Td, +Π˚pt, x, zq “ BtΨ˚pt, x, zq “ 0 +17 + +with Γ the solution of ´∆zΓ “ σ2 (see (11)), is a solution to (32) with initial condition +uωp0, tq “ 1pxq, +Ψ˚p0, x, zq “ ´γΓpzq +@ +σ1 +D +Td, +Π˚p0, x, zq “ 0. +For the time being, we stick to the framework identified for the study of the asymptotic Hartree +equation. Problem (32) has a natural Hamiltonian symplectic structure when considered on the +real Banach space H1pTdqˆH1pTdqˆL2pTd; .H1pRnqqˆL2pTd ˆRnq. Indeed, if we write u “ q `ip, +with p, q real-valued, we obtain +Bt +¨ +˚ +˚ +˝ +q +p +Ψ +Π +˛ +‹‹‚“ +ˆJ +0 +0 +´J +˙ +∇pq,p,Ψ,ΠqHSW pq, p, Ψ, Πq +with +J “ +ˆ 0 +1 +´1 +0 +˙ +and +HSWpq, p, Ψ, Πq “ 1 +2 +ˆ1 +2 +ˆ +Td |∇q|2 ` |∇p|2 dx ` k2 +2 +ˆ +Tdpp2 ` q2q dx ´ +ˆ +Td pk ¨ ∇q dx ` +ˆ +Td qk ¨ ∇p dx +˙ +` 1 +4 +ˆ +TdˆRn +ˆΠ2 +c2 ` |∇zΨ|2 +˙ +dx dz +` γ +2 +ˆ +Td +ˆˆ +TdˆRnpσ1px ´ yqσ2pzqΨpt, y, zq dy dz +˙ +pp2 ` q2qpxq dx. +Coming back to u “ q ` ip, we can write +HSWpu, Ψ, Πq “ 1 +2 +ˆ1 +2 +ˆ +Td |∇u|2 dx ` k2 +2 +ˆ +Td |upxq|2 dx ` +ˆ +Td k ¨ p´i∇uqu dx +˙ +` 1 +4 +ˆ +TdˆRn +ˆΠ2 +c2 ` |∇zΨ|2 +˙ +dx dz +` γ +2 +ˆ +Td +ˆˆ +TdˆRnpσ1px ´ yqσ2pzqΨpt, y, zq dy dz +˙ +|upxq|2 dx. +(33) +As a consequence, HSW is a constant of the motion. Moreover, it is clear that (32) is invariant +under multiplications by a phase factor of u so that Fpuq “ 1 +2}u}2 +L2 is conserved by the dynamics. +However, now, the quantities +Gjpuq “ 1 +2 +ˆ +Td +ˆ1 +i Bxju +˙ +u dx +(34) +are not constants of the motion: +d +dtGjpuqptq “ γ +2 +ˆ +Td +ˆ +Td Bxjσ1px ´ yq +ˆˆ +Rn σ2pzqΨpt, y, zq dz +˙ +|u|2pt, xq dy dx. +As a consequence, they cannot be used in the construction of the Lyapounov functional as we did +for the Hartree system (see (23)). +18 + +Finally, we consider the Banach space H1pTdqˆH1pTdqˆL2pTd; .H1pRnqqˆL2pTdˆRnq endowed +with the inner product +C +¨ +˚ +˚ +˝ +q +p +Ψ +Π +˛ +‹‹‚ +ˇˇˇ +¨ +˚ +˚ +˝ +q1 +p1 +Ψ1 +Π1 +˛ +‹‹‚ +G +“ +ˆ +Td +` +pp1 ` qq1q dx ` +ˆ +TdˆRnp∇zΨ∇zΨ1 ` ΠΠ1q dx dz +that can be also interpreted as an inner product for complex valued functions: +xpu, Ψ, Πq|pu1, Ψ1, Π1qy “ Re +ˆ +Td uu1 dx ` +ˆ +TdˆRnp∇zΨ ¨ ∇zΨ1 ` ΠΠ1q dx dz. +(35) +We denote by } ¨ } the norm on H1pTdq ˆ L2pTd; .H1pRnqq ˆ L2pTd ˆ Rnq induced by this inner +product. +4.1 +Preliminary results for the linearized problem: spectral sta- +bility when k “ 0 +As before, we linearize the system (3a)-(3c) around the plane wave solution obtained in Section 2.1. +Namely, we expand +Upt, xq “ Ukpt, xqp1 ` upt, xqq, +Ψpt, x, zq “ ´γ +@ +σ1 +D +TdΓpzq ` ψpt, x, zq +and, assuming that u, ψ and their derivatives are small, we are led to the following equations for +the fluctuation pt, xq ÞÑ upt, xq P C, pt, x, zq ÞÑ ψpt, x, zq P R +iBtu ` 1 +2∆xu ` ik ¨ ∇xu “ γΦ, +´ 1 +c2 B2 +ttψ ´ ∆zψ +¯ +pt, x, zq “ ´γσ2pzqσ1 ‹ ρpt, xq, +ρpt, xq “ 2Re +` +upt, xq +˘ +, +Φpt, xq “ +¨ +TdˆRn σ1px ´ yqσ2pzqψpt, y, zq dz dy. +(36) +We split the solution into real and imaginary parts +upt, xq “ qpt, xq ` ippt, xq, +qpt, xq “ Repupt, xqq, +ppt, xq “ Impupt, xqq. +We obtain +pBtq ` 1 +2∆xp ` k ¨ ∇xqqpt, xq “ 0, +pBtp ´ 1 +2∆xq ` k ¨ ∇xpqpt, xq “ ´γ +ˆ +σ1 ‹ +ˆ +Rn σ2pzqψpt, ¨, zq dz +˙ +pxq, +´ 1 +c2 B2 +ttψ ´ ∆zψ +¯ +pt, x, zq “ ´2γσ2pzqσ1 ‹ qpt, xq. +(37) +19 + +It is convenient to set +π “ ´ 1 +2c2 Btψ, +in order to rewrite the wave equation as a first order system. We obtain +Bt +¨ +˚ +˚ +˝ +q +p +ψ +π +˛ +‹‹‚“ Lk +¨ +˚ +˚ +˝ +q +p +ψ +π +˛ +‹‹‚ +(38) +where Lk is the operator defined by +Lk : +¨ +˚ +˚ +˝ +q +p +ψ +π +˛ +‹‹‚ÞÝÑ +¨ +˚ +˚ +˚ +˚ +˚ +˚ +˝ +´1 +2∆xp ´ k ¨ ∇xq +1 +2∆xq ´ k ¨ ∇xp ´ γσ1 ‹ +ˆˆ +Rn σ2ψ dz +˙ +´2c2π +´1 +2∆zψ ` γσ2σ1 ‹ q +˛ +‹‹‹‹‹‹‚ +For the next step, we proceed via Fourier analysis as before. We expand q, p, ψ, π and σ1 by means +of their Fourier series: +ψpt, x, zq “ +ÿ +mPZd +ψmpt, zqeim¨x, +ψmpt, zq “ +1 +p2πqd +ˆ +Td ψpt, x, zqe´im¨x dx, +πpt, x, zq “ +ÿ +mPZd +πmpt, zqeim¨x, +πmpt, zq “ +1 +p2πqd +ˆ +Td πpt, x, zqe´im¨x dx. +Moreover, recall that σ1 being real and radially symmetric, (19) holds and, by definition, +@ +σ1 +D +Td “ +p2πqdσ1,0. +As a consequence, since the Fourier modes are uncoupled, the Fourier coefficients +pQmptq, Pmptq, ψmpt, zq, πmpt, zqq +satisfy +Bt +¨ +˚ +˚ +˝ +Qm +Pm +ψm +πm +˛ +‹‹‚“ Lk,m +¨ +˚ +˚ +˝ +Qm +Pm +ψm +πm +˛ +‹‹‚ +(39) +where Lk,m stands for the operator defined by +Lk,m +¨ +˚ +˚ +˝ +Qm +Pm +ψm +πm +˛ +‹‹‚“ +¨ +˚ +˚ +˚ +˚ +˚ +˚ +˚ +˝ +´ik ¨ mQm ` m2 +2 Pm +´m2 +2 Qm ´ ik ¨ mPm ´ γp2πqdσ1,m +ˆ +Rn σ2pzqψm dz +´2c2πm +γp2πqdσ2pzqσ1,mQm ´ 1 +2∆zψm +˛ +‹‹‹‹‹‹‹‚ +. +Like for the Hartree equation, the behavior of the mode m “ 0 can be analysed explicitly. +20 + +Lemma 4.1 (The mode m “ 0) For any k P Zd, the kernel of Lk,0 is spanned by p0, 1, 0, 0q. +Moreover, equation (39) for m “ 0 admits solutions which grow linearly with time. +Proof. +Let pQ0, P0, ψ0, π0q P KerpLk,0q. It means that +$ +’ +’ +’ +& +’ +’ +’ +% +γp2πqdσ1,0 +ˆ +Rn σ2pzqψ0pzq dz “ 0, +π0 “ 0, +∆zψ0 “ 2γp2πqdσ2pzqσ1,0Q0, +which yields ψ0pzq “ ´2γ +@ +σ1 +D +TdQ0Γpzq with Γpzq “ p´∆q´1σ2pzq so that +´2γ2@ +σ1 +D2 +TdκQ0 “ 0. +It implies that Q0 “ 0, ψ0 “ 0 while P0 is left undetermined. +For m “ 0, the first equation in (39) tells us that Q0ptq “ Q0p0q P C is constant. Next, we get +Btψ0 “ ´2c2π0 which leads to +B2 +ttψ0 ´ c2∆zψ0 “ ´σ2pzq 2γc2@ +σ1 +D +TdQ0p0q +looooooooomooooooooon +:“C1 +(40) +The solution of (40) with initial condition pψ0pzq, π0pzq “ ´ 1 +2c2Btψp0, zqq P .H1pRnqˆL2pRnq satisfies +pψ0pt, ξq “ pψ0p0, ξq cospc|ξ|tq ´ 2c2pπ0pξqsinpc|ξ|tq +c|ξ| +´ +ˆ t +0 +sinpc|ξ|sq +c|ξ| +pσ2pξqC1 ds +where pψ0pt, ξq and pπ0pt, ξq are the Fourier transforms of z ÞÑ ψpt, zq and z ÞÑ πpt, zq respectively. +Finally, integrating +BtP0 “ ´γ +@ +σ1 +D +Td +loooomoooon +:“C2 +ˆ +Rn σ2pzqψ0pzq dz +we obtain +P0ptq “ P0p0q ` C2 +ˆ +Rn pσ2pξq pψ0p0, ξqsinpc|ξ|tq +c|ξ| +dξ +p2πqn ´ 2c2C2 +ˆ +Rn pσ2pξqpπ0p0, ξq1 ´ cospc|ξ|tq +c2|ξ|2 +dξ +p2πqn +´ C1C2 +ˆ t +0 +ˆ s +0 +pcpτq dτ ds +where +pcpτq “ +ˆ +Rd |pσ2pξq|2 sinpc|ξ|τq +c|ξ| +dξ +p2πqn . +This kernel already appears in the analysis performed in [10, 19]. The contribution involving the +initial data of the vibrational field can be uniformly bounded by +1 +p2πqn +ˆˆ +Rd +|pσ2pξq|2 +c2|ξ|2 dξ +˙1{2 #ˆˆ +Rd | pψ0p0, ξq|2 dξ +˙1{2 +` 4c2 +ˆˆ +Rd +|pπ0p0, ξq|2 +c2|ξ|2 +dξ +˙1{2+ +. +Next, as a consequence of (H2), it turns out that pc is compactly supported, with +´ 8 +0 pcpτq dτ “ κ +c2, +see [10, Lemma 14] and [19, Section 2.4]. It follows that +ˆ t +0 +ˆ s +0 +pcpτq dτ ds “ +ˆ t +0 +pcpτq +ˆˆ t +τ +ds +˙ +dτ “ +ˆ t +0 +pt ´ τqpcpτq dτ +„ +tÑ8 t κ +c2 ´ +ˆ 8 +0 +τpcpτq dτ, +21 + +which concludes the proof. +When k “ 0, basic estimates based on the energy conservation allow us to justify the stability +of the solutions with zero mean. However, in contrast to what has been established for the Hartree +system, this analysis does not extend to any mode k � 0, since the system is not Galilean invariant. +Theorem 4.2 (Linearized stability for the Schrödinger-Wave system when k “ 0) Let k “ +0. Suppose (9) and let pu, ψ, πq be the solution of (36) associated to an initial data uInit P H1pTdq, ψInit P +L2pTd; .H1pRnqq, πInit P L2pTd ˆ Rnq such that +´ +Td uInit dx “ 0. Then, there exists a constant C ą 0 +such that suptě0 }upt, ¨q}H1 ď C. +Proof. +Again, we use the energetic properties of the linearized equation (36). We have already +remarked that +´ +Td upt, xq dx “ 0 for any t ě 0 when +´ +Td uInit dx “ 0. We start by computing +d +dt +"1 +2 +ˆ +Td |∇xu|2 dx ` 1 +2 +ˆ +TdˆRn +´|Btψ|2 +c2 +` |∇zψ|2¯ +dz dx +* +“ ´iγ +2 +ˆ +Td Φ∆xpu ´ uq dx ´ γ +ˆ +TdˆRn Btψσ2σ1 ‹ pu ` uq dz dx. +Next, we get +d +dt +ˆ +Td Φpu ` uq dx +“ +ˆ +TdˆRn Btψσ2σ1 ‹ pu ` uq dz dx +` i +2 +ˆ +Td Φ∆xpu ´ uq dx ´ +ˆ +Td Φk ¨ ∇xpu ` uq dx. +We get rid of the last term by assuming k “ 0 and we arrive in this case at +d +dt +"1 +2 +ˆ +Td |∇xu|2 dx ` 1 +2 +ˆ +TdˆRn +´|Btψ|2 +c2 +` |∇zψ|2¯ +dz dx ` γ +ˆ +Td Φpu ` uq dx +* +“ 0. +We estimate the coupling term as follows +ˇˇˇˇ +ˆ +Td Φpu ` uq dx +ˇˇˇˇ “ +ˇˇˇˇ +ˆ +TdˆRn σ2pzqψpt, x, zqσ1 ‹ pu ` uqpt, xq dz dx +ˇˇˇˇ +ď }σ1 ‹ pu ` uq}L2 ˆ +ˆˆ +Td +ˇˇˇ +ˆ +Rn σ2pzqψpt, x, zq dz +ˇˇˇ +2 +dx +˙1{2 +ď }σ1}L1}u ` u}L2 ˆ +ˆˆ +Td +ˇˇˇ +ˆ +Rn pσ2pξq pψpt, x, ξq +dξ +p2πqn +ˇˇˇ +2 +dx +˙1{2 +ď 2}σ1}L1}u}L2 ˆ +ˆˆ +Td +ˇˇˇ +ˆ +Rn +pσ2pξq +|ξ| |ξ|| pψpt, x, ξq| +dξ +p2πqn +ˇˇˇ +2 +dx +˙1{2 +ď 2}σ1}L1}u}L2 ˆ +ˆˆ +Rn +|pσ2pξq|2 +|ξ|2 +dξ +˙1{2 +ˆ +ˆˆ +TdˆRn |ξ|2| pψpt, x, ξq|2 +dξ +p2πqn dx +˙1{2 +ď 2 ?κ}σ1}L1}u}L2 ˆ +ˆˆ +TdˆRn |∇zψpt, x, ξq|2 dz dx +˙1{2 +“ 2 ?κ}σ1}L1}u}L2}∇zψ}L2 +ď 1 +2γ }∇zψ}2 +L2 ` 2κγ}σ1}2 +L1}u}2 +L2. +22 + +By using the Poincaré-Wirtinger inequality }u}L2 ď }∇xu}L2, we deduce that +1 +2 +ˆ +Td |∇xupt, xq|2 dx ď +E0 +1 ´ 4γ2κ}σ1}2 +L1 +, +where E0 depends on the energy of the initial state. +While it is natural to start with the linearized operator Lk in (38), it turns out that this +formulation is not well-adapted to study the spectral stability issue. The difficulties relies on the +fact that the wave part of the system induces an essential spectrum, reminiscent to the fact that +σessp´∆zq “ r0, 8q. For instance, this is even an obstacle to set up a perturbation argument from +the Hartree equation, in the spirit of [15]. We shall introduce later on a more adapted formulation +of the linearized equation, which will allow us to overcome these difficulties (and also to go beyond +a mere perturbation analysis). +4.2 +Orbital stability for the Schrödinger-Wave system when k “ 0 +In this subsection, we wish to establish the orbital stability of the plane wave solution to (32) +obtained in Section 2.1, namely +uωpt, xq “ eiωt1pxq, +Ψ˚pt, x, zq “ ´γΓpzq +@ +σ1 +D +Td, +Π˚pt, x, zq “ 0 +with k P Zd and ω ą 0 that satisfy the dispersion relation (12) and Γpzq “ p´∆q´1σ2pzq. The +system (32) being invariant under multiplications of u by a phase factor, we define the corresponding +orbit through p1pxq, ´γΓpzq +@ +σ1 +D +Td, 0q by +O1 “ tpeiθ, ´γΓpzq +@ +σ1 +D +Td, 0q, θ P Ru. +As before, orbital stability intuitively means that the solutions of (32) associated to initial data +close enough to p1pxq, ´γΓpzq +@ +σ1 +D +Td, 0q remain at a close distance to the set O1. +Let us introduce, for any k P Zd and ω ą 0 satisfying the dispersion relation (12), the set +Sω “ +! +pu, Ψ, Πq P H1pTd; Cq ˆ L2pTd; .H1pRnqq ˆ L2pTd, L2pRnqq, Fpuq “ Fp1q “ p2πqd +2 +) +, +and the functional +Lω,kpu, Ψ, Πq “ HSWpu, Ψ, Πq ` ωFpuq, +(41) +intended to serve as a Lyapounov functional, where HSW is the constant of motion defined in (33). +For further purposes, we simply denote Lω “ Lω,0. Note that +Lω,kpu, Ψ, Πq “ HSWpu, Ψ, Πq ` 1 +2i +ˆ +Td k ¨ ∇u ¯u dx +loooooooooomoooooooooon +“ +dÿ +j“1 +kjGjpuq +` +´ +ω ` k2 +2 +¯ +Fpuq +with HSW defined in (8) and Gjpuq defined in (34). Thanks to the dispersion relation (12), only the +second term of this expression depends on k. Unfortunately, as pointed out before, the quantities +23 + +Gjpuq are not constants of the motion so that the dependence on k of the Lyapounov functional +(41) cannot be disregarded, in contrast to what we did for the Hartree system in (23). +Next, as in subsection 3.2, we need to evaluate the first and second order variations of Lω,k. +We compute +Bpu,Ψ,ΠqHSWpu, Ψ, Πqpv, φ, τq +“ Re +ˆ1 +2 +ˆ +Tdp´∆uqv dx ` γ +ˆ +Td +ˆ¨ +TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dz dy +˙ +upxqvpxq dx +˙ +` γ +2 +ˆ +Td +ˆ¨ +TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy +˙ +|upxq|2 dy dx +` 1 +2 +¨ +TdˆRn +´ 1 +c2 Π τ ` p´∆zΨq φ dz +¯ +dx +and +B2 +pu,Ψ,ΠqHSWpu, Ψ, Πq +` +pv, φ, τq, pv1, φ1, τ 1q +˘ +“ Re +"1 +2 +ˆ +Tdp´∆vqv1 dx +`γ +ˆ +Td +ˆ¨ +TdˆRn σ1px ´ yqσ2pzqpφpt, y, zqv1pxq ` φ1pt, y, zqvpxqq dz dy +˙ +upxq dx +˙ +`γ +ˆ +Td +ˆ¨ +TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dz dy +˙ +vpxqv1pxq dx +˙* +` 1 +2 +¨ +TdˆRn +´ 1 +c2 τ τ 1 ` p´∆zφq φ1 dz +¯ +dx. +Besides, we have +BuFpuqpvq “ Re +ˆˆ +Td uv dx +˙ +, +B2 +uFpuqpv, v1q “ Re +ˆˆ +Td vv1 dx +˙ +, +BuGjpuqpvq “ Im +`´ +Td Bxjuv dx +˘ +, +B2 +uGpuqpv, v1q “ Im +`´ +Td Bxjv1v dx +˘ +. +Accordingly, we are led to +Bpu,Ψ,ΠqLω,kp1, ´γ +@ +σ1 +D +TdΓ, 0qpv, φ, τq +“ Re +ˆ +´γ2@ +σ1 +D2 +Tdκ +ˆ +Td v dx ` +´ +ω ` k2 +2 +¯ ˆ +Td v dx ` γ +2 +@ +σ1 +D +Td +¨ +TdˆRnpσ2 ` ∆zΓq φ dz dx +˙ +“ 0 +thanks to the dispersion relation (12) and the definition of Γ. Similarly, the second order derivative +24 + +casts as +B2 +pu,Ψ,ΠqLω,kp1, ´γ +@ +σ1 +D +TdΓ, 0q +` +pv, φ, τq, pv, φ, τq +˘ +“ Re +ˆ1 +2 +ˆ +Tdp´∆vqv dx ` 1 +2 +¨ +TdˆRn +´τ 2 +c2 ` p´∆zφq φ dz +¯ +dx +` 2γ +ˆ +Td +ˆ¨ +TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy +˙ +vpxq dx +´ γ2@ +σ1 +D +Td +ˆ +Td +ˆ¨ +TdˆRn σ1px ´ yqσ2pzqΓpzq dz dy +˙ +vpxqvpxq dx ` +´ +ω ` k2 +2 +¯ ˆ +Td vpxqvpxq dx +˙ +` Im +˜ dÿ +j“1 +kj +ˆ +Td Bxjvv dx +¸ +. +The forth and fifth integrals combine as +ˆ +Td +´ +ω ` k2 +2 ´ γ2κ +@ +σ1 +D2 +Td +¯ +vpxqvpxq dx “ 0 +which cancels out by virtue of the dispersion relation (12). Hence we get +B2 +pu,Ψ,ΠqLω,kp1, ´γ +@ +σ1 +D +TdΓ, 0q +` +pv, φ, τq, pv, φ, τq +˘ +“ Re +ˆ1 +2 +ˆ +Tdp´∆vqv dx ` 1 +2 +¨ +TdˆRn +´τ 2 +c2 ` p´∆zφq φ dz +¯ +dx +` 2γ +ˆ +Td +ˆ¨ +TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy +˙ +vpxq dx ´ i +ˆ +Td k ¨ ∇v v dx +˙ +. +Remark 4.3 Note that the following continuity estimate holds: for any pv, φ, τq P H1pTd; Cq ˆ +L2pTd; .H1pRnqq ˆ L2pTd ˆ Rnq, +B2 +pu,Ψ,ΠqLω,kp1, ´γ +@ +σ1 +D +TdΓ, 0q +` +pv, φ, τq, pv, φ, τq +˘ +ď 1 +2}∇v}2 +L2 ` 1 +2c2 }τ}2 +L2 ` 1 +2}φ}2 +L2x . +H1z +` 2γκ1{2}σ1}L1}v}L2}φ}L2x . +H1z ` |k|}∇v}L2}v}L2 ď 1 +2 +ˆ +p1 ` |k|q}v}2 +H1 ` 1 +c2 }τ}2 +L2 ` C}φ}2 +L2x . +H1z +˙ +ď maxp1{c2, 1 ` |k|, Cq +2 +}pv, φ, τq}2 +with C “ 1 ` 4γ2κ}σ1}2 +L1. +The functional Lω,k is conserved by the solutions of (32); however the difficulty relies on +justifying its coercivity. We are only able to answer positively in the specific case k “ 0. Hence, +the main result of this subsection restricts to this situation. +Theorem 4.4 (Orbital stability for the Schrödinger-Wave system) Let k “ 0 and ω ą 0 +such that the dispersion relation (12) is satisfied. Suppose (9) holds. Then the plane wave solution +peiωt1pxq, ´γΓpzq +@ +σ +D +Td, 0q is orbitally stable, i.e. +@ε ą 0, Dδ ą 0, @pvInit, φInit, τ Initq P H1pTd; Cq ˆ L2pTd; .H1pRnqq ˆ L2pTd ˆ Rnq, +}vInit ´ 1}H1 ` }φInit ` γΓ +@ +σ +D +Td}L2x . +H1z ` }τ Init}L2 ă δ ñ sup +tě0 +distppvptq, φptq, τptqq, O1q ă ε +(42) +25 + +where distppv, φ, τq, O1q “ infθPr0,2πr }v ´ eiθ1}H1 ` }φ ` γΓ +@ +σ +D +Td}L2x . +H1z ` }τ}L2 and pt, x, zq ÞÑ +pvpt, xq, φpt, x, zq, τpt, x, zqq stands for the solution of (32) with Cauchy data pvInit, φInit, τ Initq. +Using the same argument as in the case of Theorem 3.5, we can reduce the proof of Theorem +4.4 to the following coercivity estimate on the Lyapounov functional (and this is where we use that +Lω,k is a conserved quantity). +Lemma 4.5 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied. Suppose that +there exist η ą 0 and c ą 0 such that @pw, ψ, χq P Sω, +distppw, ψ, χq, O1q ă η ñ Lω,kppw, ψ, χqq ´ Lω,kpp1pxq, ´γΓpzq +@ +σ +D +Td, 0qq ě cdistppw, ψ, χq, O1q2. +(43) +Then the the plane wave solution peiωt1pxq, ´γΓpzq +@ +σ +D +Td, 0q is orbitally stable. +As we have seen before, since Bpu,ψ,ΠqLω,kpp1, ´γΓpzq +@ +σ +D +Td, 0qq “ 0, the coercivity estimate +(43) can be obtained from an estimate on the bilinear form +B2 +pu,ψ,ΠqLω,kpp1, ´γ +@ +σ1 +D +TdΓ, 0qqppu, φ, τq, pu, φ, τqq +for any pu, φ, τq P T1Sω X pT1O1qK. Here the tangent set to Sω is given by +T1Sω “ +" +u P H1pTd; Cq, Re +ˆˆ +Td upxq1pxq dx +˙ +“ 0 +* +ˆ L2pTd; .H1pRnqq ˆ L2pTd ˆ Rnq. +This set is the orthogonal to p1, 0, 0q with respect to the inner product defined in (35). The tangent +set to O1 (which is the orbit generated by the phase multiplications of 1) is +T1O1 “ spanRtpi1, 0, 0qu +so that +pT1O1qK “ +" +u P H1pTd; Cq, Re +ˆ +i +ˆ +Td upxq1pxq dx +˙ +“ 0 +* +ˆ L2pTd; .H1pRnqq ˆ L2pTd ˆ Rnq. +Lemma 4.6 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied. Suppose that +there exists ˜c ą 0 +B2 +pu,ψ,ΠqLω,kpp1, ´γΓpzq +@ +σ +D +Td, 0qqppu, φ, τq, pu, φ, τqq ě ˜cp}u}2 +H1 ` }φ}2 +L2x . +H1z ` }τ}2 +L2q “ ˜c}pu, φ, τq}2 +(44) +for any pu, φ, τq P T1S1 X pT1O1qK. Then there exist η ą 0 and c ą 0 such that (43) is satisfied. +Proof. +Let pw, ψ, χq P Sω such that distppw, ψ, χq, O1q ă η with η ą 0 small enough. Hence, +infθPr0,2πq }w´eiθ1} ă η and, by means of an implicit function theorem argument (see [4, Section 9, +Lemma 8]), we obtain that there exists θ P r0, 2πq and v P +␣ +u P H1pTd; Cq, Re +` +i +´ +Td upxq dx +˘ +“ 0 +( +such that +eiθw “ 1 ` v, +inf +θPr0,2πq }w ´ eiθ1} ď }v}H1 ď C +inf +θPr0,2πq }w ´ eiθ1} +26 + +for some positive constant C. Denote by φpx, zq “ ψpx, zq`γΓpzq +@ +σ1 +D +Td. Then pv, φ, χq P pT1O1qK +and }pv, φ, χq} ď Cη. +Next, we use the fact that H1pTdq “ +␣ +u P H1pTd; Cq, Re +`´ +Td upxq dx +˘ +“ 0 +( +‘ spanRt1u to write +pv, φ, χq “ pv1, φ, χq ` pv2, 0, 0q with pv1, φ, χq P T1Sω X pT1O1qK and v2 P spanRt1u. Moreover, +}v2}H1 ď }v}2 +H1 +}1}L2 +and +}v1}H1 ě 1 +2}v}H1 +provided }v}H1 ď }1}L2. As a consequence, if }v}H1 is small enough, using that +B2 +pu,Ψ,ΠqLω,kp1, ´γ +@ +σ1 +D +TdΓ, 0q +` +pv, φ, τq, pv1, φ1, τ 1q +˘ +ď C}pv, φ, τq}}pv1, φ1, τ 1q}, +we obtain +B2 +pu,Ψ,ΠqLω,kp1, ´γ +@ +σ1 +D +TdΓ, 0q +` +pv1, φ, χq, pv2, 0, 0q +˘ +ď C}pv, φ, χq} }v}2 +H1 ď C}pv, φ, χq}3, +B2 +pu,Ψ,ΠqLω,kp1, ´γ +@ +σ1 +D +TdΓ, 0q +` +pv2, 0, 0q, pv2, 0, 0q +˘ +ď C}v}4 +H1 ď C}pv, φ, χq}4. +This leads to +B2 +pu,Ψ,ΠqLω,kp1, ´γ +@ +σ1 +D +TdΓ, 0q +` +pv, φ, χq, pv, φ, χq +˘ +“ B2 +pu,Ψ,ΠqLω,kp1, ´γ +@ +σ1 +D +TdΓ, 0q +` +pv1, φ, χq, pv1, φ, χq +˘ +` op}pv, φ, χq}2q. +Finally, let pw, ψ, χq P Sω such that dppw, ψ, χq, O1q ă η, we have +Lω,kppw, ψ, χqq ´ Lω,kpp1pxq, ´γΓpzq +@ +σ +D +Td, 0qq “ Lω,kppeiθw, ψ, χqq ´ Lω,kpp1pxq, ´γΓpzq +@ +σ +D +Td, 0qq +“ B2 +pu,Ψ,ΠqLω,kp1, ´γ +@ +σ1 +D +TdΓ, 0q +` +pv, φ, χq, pv, φ, χq +˘ +` op}pv, φ, χq}2q +“ B2 +pu,Ψ,ΠqLω,kp1, ´γ +@ +σ1 +D +TdΓ, 0q +` +pv1, φ, χq, pv1, φ, χq +˘ +` op}pv, φ, χq}2q +ě ˜c}pv1, φ, τq}2 ` op}pv, φ, χq}2q ě ˜c +2}pv, φ, τq}2 ` op}pv, φ, χq}2q +ě ˜c +6dppw, ψ, χq, O1q2 +where we use Bpu,Ψ,ΠqLω,kp1, ´γ +@ +σ1 +D +TdΓ, 0q “ 0 and pv1, φ, χq P T1Sω X pT1O1qK. +As before, to prove the orbital stability of the plane solution peiωt1pxq, ´γΓpzq +@ +σ +D +Td, 0q it is +enough to prove (44) for any pu, φ, τq P T1S1 XpT1O1qK. Let pu, φ, τq P T1S1 XpT1O1qK and write +u “ q ` ip with q, p P H1pTd; Rq. Then +B2 +pu,Ψ,ΠqLω,kp1, ´γ +@ +σ1 +D +TdΓ, 0q +` +pu, φ, τq, pu, φ, τq +˘ +“ Re +ˆ1 +2 +ˆ +Tdp´∆uqu dx ` 1 +2 +¨ +TdˆRn +´τ 2 +c2 ` p´∆zφq φ dz +¯ +dx +` 2γ +ˆ +Td +ˆ¨ +TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy +˙ +upxq dx ´ i +ˆ +Td k ¨ ∇u u dx +˙ +(45) +can be reinterpreted as a quadratic form acting on the 4-uplet W “ pq, p, φ, τq. To be specific, it +27 + +expresses as the following quadratic form on W, +QpW, Wq “1 +2 +ˆ +Td |∇p|2 dx ` 1 +2c2 +¨ +TdˆRn |τ|2 dz dx ` 1 +2 +ˆ +Td |∇q|2 dx ` 1 +2 +¨ +TdˆRnp´∆zφq φ dx dz +` 2γ +ˆ +Td +ˆ¨ +TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dyqpxq dx +˙ +` 2 +ˆ +Td qk ¨ ∇p dx. +The crossed term +´ +Td qk ¨ ∇p dx is an obstacle for proving a coercivity on Q. +For this reason, let us focus on the case k “ 0. Since pu, φ, τq P T1S1 X pT1O1qK, we have +ˆ +Td q dx “ 0 and +ˆ +Td p dx “ 0. +As a consequence, thanks to the Poincaré-Wirtinger inequality, we deduce, when k “ 0 +QpW, Wq ě1 +4}p}2 +H1 ` 1 +2c2 }τ}2 +L2 ` 1 +2 +ˆ +Td |∇q|2 dx ` 1 +2 +¨ +TdˆRnp´∆zφq φ dx dz +` 2γ +ˆ +Td +ˆ¨ +TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy +˙ +qpxq dx +(46) +Next, we expand q, σ1 and φp¨, zq in Fourier series, i.e. +qpxq “ +ÿ +mPZd +qmeim¨x, φpx, zq “ +ÿ +mPZd +φmpzqeim¨x and σ1pxq “ +ÿ +mPZd +σ1,meim¨x. +Note that σ1,m “ σ1,m “ σ1,´m since σ1 is real and radially symmetric. Moreover, +´ +Td q dx “ 0 +implies q0 “ 0. Hence, +ˆ +Td +ˆˆ +TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy +˙ +qpxq dx +“ p2πq2dRe +¨ +˝ +ÿ +mPZd∖t0u +σ1,mqm +ˆ +Rn σ2pzqφmpzq dz +˛ +‚ +which implies +1 +2 +ˆ +Td |∇q|2 dx ` 1 +2 +¨ +TdˆRnp´∆zφq φ dx dz +` 2γ +ˆ +Td +ˆ¨ +TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy +˙ +qpxq dx +“ p2πqd +ÿ +mPZd∖t0u +Re +ˆm2 +2 q2 +m ` 1 +2 +ˆ +Rn |∇zφm|2 dz ` 2p2πqdγσ1,mqm +ˆ +Rn σ2pzqφmpzq dz +˙ +. +Next, we remark that for any m P Zd, +ˇˇˇˇRe +ˆ +2p2πqdγσ1,mqm +ˆ +Rn σ2pzqφmpzq dz +˙ˇˇˇˇ ď 2p2πqdγσ1,m|qm| ?κ}∇φm}L2 +ď 1 +2˜δp4γ2κp2πq2dσ2 +1,mqq2 +m ` +˜δ +2}∇φm}2 +L2 +28 + +for any ˜δ ą 0. Finally, for any ˜δ P p0, 1q, we get +1 +2 +ˆ +Td |∇q|2 dx ` 1 +2 +¨ +TdˆRnp´∆zφq φ dx dz +` 2γ +ˆ +Td +ˆ¨ +TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy +˙ +qpxq dx +ě p2πqd ÿ +mPZd +ˆˆm2 +2 ´ 1 +2˜δ p4γ2κp2πq2dσ2 +1,mq +˙ +q2 +m ` 1 ´ ˜δ +2 +}∇φm}2 +L2 +˙ +(47) +As a consequence, we obtain the following statement. +Proposition 4.7 Let k “ 0 and ω ą 0 such that the dispersion relation (12) is satisfied. Suppose +that there exists δ P p0, 1q such that +4γ2κp2πq2d σ2 +1,m +m2 ď δ +(48) +for all m P Zd ∖ t0u. Then, there exists ˜c ą 0 such that +B2 +pu,Ψ,ΠqLωp1, ´γ +@ +σ1 +D +TdΓ, 0q +` +pu, φ, τq, pu, φ, τq +˘ +ě ˜c}pu, φ, τq}2 +(49) +for any pu, φ, τq P T1S1 X pT1O1qK. +Proof. +If (48) holds, then, for any ˜δ P pδ, 1q, (45)-(46)-(47) lead to +B2 +pu,Ψ,ΠqLωp1, ´γ +@ +σ1 +D +TdΓ, 0q +` +pu, φ, τq, pu, φ, τq +˘ +ě 1 +4}p}2 +H1 ` 1 +2c2 }τ}2 +L2 +` +˜δ ´ δ +2˜δ p2πqd +ÿ +mPZd∖t0u +m2q2 +m ` 1 ´ ˜δ +2 +p2πqd ÿ +mPZd +}∇φm}2 +L2 +“ 1 +4}p}2 +H1 ` 1 +2c2 }τ}2 +L2 ` +˜δ ´ δ +2˜δ }∇q}L2 ` 1 ´ ˜δ +2 +}φ}L2xH1z ě ˜c}pu, φ, τq}2 +where in the last inequality we used the Poincaré-Wirtinger inequality together with the fact that +´ +Td q dx “ 0. +Finally, Proposition 4.7 together with Lemma 4.6 and Lemma 4.5 gives Theorem 4.4 and the +orbital stability of the plane wave solution peiωt1pxq, ´γΓpzq +@ +σ +D +Td, 0q in the case k “ 0. +Remark 4.8 The coercivity of B2 +pu,Ψ,ΠqLωp1, ´γ +@ +σ1 +D +TdΓ, 0q +` +pu, φ, τq, pu, φ, τq +˘ +on T1S1XpT1O1qK +can be recovered from the spectral properties of a convenient unbounded linear operator S. Indeed, +as we have seen before, by decomposing u into real and imaginary part, the quadratic form defined +by (45) (with k “ 0) can be written as +QpW, Wq “ 1 +2 +ˆ +Td |∇p|2 dx ` 1 +2c2 +¨ +TdˆRn |τ|2 dz dx ` +B +S +ˆq +φ +˙ ˇˇˇ +ˆq +φ +˙F +29 + +with S : H2pTdq ˆ L2pTd; .H1pRnqq Ă L2pTdq ˆ L2pTd; .H1pRnqq Ñ L2pTdq ˆ L2pTd; .H1pRnqq the +unbounded linear operator given by +S +ˆ +q +φ +˙ +“ +¨ +˚ +˝ +´1 +2∆xq ` γσ1 ‹ +ˆ +Rn σ2φ dz +1 +2φ ` γΓσ1 ‹ q +˛ +‹‚ +(where we remind the reader that Γ “ p´∆q´1σ2q) and the inner product +Bˆ +q +φ +˙ ˇˇˇ +ˆ +q1 +φ1 +˙F +“ +ˆ +Td qq1 dx` +ˆ +TdˆRn ∇zφ¨∇zφ1 dz dx “ +ˆ +Td qq1 dx` +ˆ +TdˆRn +ˆφpx, ξqˆφ1px, ξq|ξ|2 dξ +p2πqn dx. +Note that L2pTdq ˆ L2pTd; .H1pRnqq is an Hilbert space with this inner product since n ě 3. +Since +ˆ +T d +ˆ +σ1 ‹ +ˆ +Rn σ2φ dz +˙ +pxqq1pxq dx “ +ˆ +Td +ˆˆ +TdˆRn σ1px ´ yqσ2pzqφpy, zq dz dy +˙ +q1pxq dx +“ +ˆ +TdˆRn φpx, zqσ2pzqpσ1 ‹ q1qpxq dx dz “ +ˆ +TdˆRn +ˆφpx, ξq ˆσ2pξq +|ξ|2 pσ1 ‹ q1qpxq dx|ξ|2 dξ +p2πqn +we can check that S is a self-adjoint operator on L2pTdq ˆ L2pTd; .H1pRnqq. In particular, σpSq Ă R +and one can easily study the spectrum of S. +More precisely, using Fourier series, we find that if λ is an eigenvalue of S then there exists at +least one m P Zd such that for some pqm, φmq � p0, 0q there holds +$ +’ +’ +& +’ +’ +% +ˆm2 +2 ´ λ +˙ +qm ` γp2πqdσ1,m +ˆ +Rn σ2pzqφmpzq dz “ 0, +ˆ1 +2 ´ λ +˙ +φmpzq ` γp2πqdΓpzqσ1,mqm “ 0. +Let λ � 1 +2. Hence, for any m P Zd, qm “ 0 implies φmpzq “ 0 for any z P Rn. As a consequence, +we may assume qm � 0. This leads to φmpzq “ ´ γp2πqdσ1,mqm +1{2´λ +Γpzq and +ˆm2 +2 ´ λ +˙ ˆ1 +2 ´ λ +˙ +´ γ2p2πq2dσ2 +1,mκ “ 0. +By solving this equation, we obtain +λ˘,m “ +´ +m2`1 +2 +¯ +˘ +c´ +m2´1 +2 +¯2 +` 4γ2p2πq2dσ2 +1,mκ +2 +so that λ`,m ě 1 +4 for any m P Zd. Next, we remark that +λ´,0 “ +1 +2 ´ +b +1 +4 ` 4γ2p2πq2dσ2 +1,0κ +2 +ă 0 +30 + +since 4γ2κp2πq2dσ2 +1,0 ą 0. This eigenvalue corresponds to an eigenfunction p˜q, ˜φq with ˜q P spanRt1u. +In particular, +´ +Td ˜qpxq dx � 0. Finally, if (30) holds, +λ´,m ě +´ +m2`1 +2 +¯ +´ +c´ +m2´1 +2 +¯2 +` δm2 +2 +ě 1 ´ δ +5 +for any m P Zd ∖ t0u. +We conclude that +B +S +ˆq +φ +˙ ˇˇˇ +ˆq +φ +˙F +“ +C¨ +˚ +˝ +´1 +2∆xq ` γσ1 ‹ +ˆ +Rn σ2φ dz +1 +2φ ` γΓσ1 ‹ q +˛ +‹‚ +ˇˇˇ +ˆq +φ +˙G +ě min +ˆ1 +2, 1 ´ δ +5 +˙ +p}q}2 +L2 ` }φ}L2x . +H1z q +for all pq, φq P tq P L2pTdq, +´ +T d q dx “ 0u ˆ L2pTd; .H1pRnqq. This, together with the Poincaré- +Wirtinger inequality, proves the coercivity of B2 +pu,Ψ,ΠqLωp1, ´γ +@ +σ1 +D +TdΓ, 0q +` +pu, φ, τq, pu, φ, τq +˘ +on +T1S1 X pT1O1qK. +5 +Discussion about the case k � 0 +5.1 +A new symplectic form of the linearized Schrödinger-Wave +system +We go back to the linearized problem. The viewpoint presented in Section 4.1 looks quite natural; +however, it misses some structural properties of the problem. In order to work in a unified functional +framework, we find convenient to change the wave unknown ψ, which is naturally valued in .H1pRnq, +into p´∆q´1{2φ, where the new unknown φ now lies in L2pRnq. Hence, the linearized problem is +rephrased as +BtX “ LX, +where X stands for the 4-uplet pq, p, φ, πq and +LX “ +¨ +˚ +˚ +˚ +˚ +˚ +˚ +˚ +˝ +´1 +2∆xp ´ k ¨ ∇xq +1 +2∆xq ´ k ¨ ∇xp ´ γσ1 ‹ +ˆˆ +Rnp´∆q´1{2σ2φ dz +˙ +´2c2p´∆q1{2π +1 +2p´∆q1{2φ ` γσ2σ1 ‹ q +˛ +‹‹‹‹‹‹‹‚ +. +The operator L is seen as an operator on the Hilbert space +V “ L2pTdq ˆ L2pTdq ˆ L2pTd; L2pRnqq ˆ L2pTd; L2pRnqq, +with domain DpLq “ H2pTdq ˆ H2pTdq ˆ L2pTd; H1pRnqq ˆ L2pTd; H1pRnqq. We can start with +the following basic information, which has the consequence that the spectral stability amounts to +justify that σpLq Ă iR. +31 + +Lemma 5.1 Let pλ, Xq be an eigenpair of L. Let Y : px, zq ÞÑ pqp´xq, ´pp´xq, φp´x, zq, ´πp´x, zqq. +Then, pλ, Xq, p´λ, Y q and p´λ, Y q are equally eigenpairs of L. +Proof. +Since L has real coefficients, LX “ λX implies LX “ λX. Next, we check that +LY px, zq +“ +¨ +˚ +˚ +˚ +˚ +˚ +˚ +˚ +˝ +1 +2∆xp ` k ¨ ∇xq +1 +2∆xq ´ k ¨ ∇xp ´ γσ1 ‹ +ˆˆ +Rnp´∆q´1{2σ2φ dz1 +˙ +2c2p´∆q1{2π +1 +2p´∆q1{2φ ` γσ2σ1 ‹ q +˛ +‹‹‹‹‹‹‹‚ +p´x, zq +“ +λ +¨ +˚ +˚ +˝ +´qp´x, zq +pp´x, zq +´φp´x, zq +πp´x, zq +˛ +‹‹‚“ ´λY px, zq. +Next, we make a new symplectic structure appear. To this end, let us introduce the blockwise +operator +J “ +ˆJ1 +0 +0 +J2 +˙ +, +J1 “ +ˆ 0 +1 +´1 +0 +˙ +, +J2 “ +ˆ +0 +´p´∆q1{2 +p´∆q1{2 +0 +˙ +. +We are thus led to +L “ J L +with +L X “ +¨ +˚ +˚ +˚ +˚ +˚ +˚ +˝ +´1 +2∆xq ` k ¨ ∇xp ` γσ1 ‹ +ˆˆ +Rnp´∆q´1{2σ2φ dz +˙ +´1 +2∆xp ´ k ¨ ∇xq +1 +2φ ` γp´∆q´1{2σ2σ1 ‹ q +2c2π +˛ +‹‹‹‹‹‹‚ +. +(50) +For further purposes, we also set +Ă +J “ +ˆ ˜ +J1 +0 +0 +˜ +J2 +˙ +, +˜ +J1 “ +ˆ0 +´1 +1 +0 +˙ +, +˜ +J2 “ +ˆ +0 +p´∆q´1{2 +´p´∆q´1{2 +0 +˙ +. +(51) +The operator J has 0 in its essential spectrum; nevertheless +Ă +J plays the role of its inverse since +J Ă +J “ I “ +Ă +J J . +Lemma 5.2 The operator L is an unbounded self adjoint operator on V with domain DpL q “ +H2pTdq ˆ H2pTdq ˆ L2pTd; L2pRnqq ˆ L2pTd; L2pRnqq, and the operator J is skew-symmetric. +Proof. +The space V is endowed with the standard L2 inner product +` +X|X1q “ +ˆ +Tdpqq1 ` pp1q dx ` +¨ +TdˆRnpφφ1 ` ππ1q dx dz. +32 + +We get +` +L X|X1˘ +“ +ˆ +Td +!´ +´ 1 +2∆xq ` k ¨ ∇xp +¯ +q1 ` +´ +´ 1 +2∆xp ´ k ¨ ∇xq +¯ +p1 +) +dx +`γ +ˆ +Td σ1 ‹ +ˆˆ +Rnp´∆q´1{2σ2φ dz +˙ +q1 dx +` +¨ +TdˆRn +´1 +2φφ1 ` 2c2ππ1 +¯ +dx dz +`γ +¨ +TdˆRn +´ +p´∆q´1{2σ2σ1 ‹ q +¯ +φ1 dx dz +“ +ˆ +Td +! +q +´ +´ 1 +2∆xq1 ` k ¨ ∇xp1 +¯ +` p +´ +´ 1 +2∆xp1 ´ k ¨ ∇xq1 +¯) +dx +`γ +¨ +TdˆRn φp´∆q´1{2σ2σ1 ‹ q1 dz dx +` +¨ +TdˆRn +´1 +2φφ1 ` 2c2ππ1 +¯ +dx dz +`γ +ˆ +Td qσ1 ‹ +ˆˆ +Rnp´∆q´1{2σ2φ1 dz +˙ +dx +“ +` +X|L X1˘ +, +and +` +J X|X1˘ +“ +¨ +Td +´ +pq1 ´ qp1 +¯ +dx ` +¨ +TdˆRn +´ +´ p´∆q1{2πφ1 ` p´∆q1{2φπ1 +¯ +dx dz +“ +´ +¨ +Td +´ +qp1 ´ pq1 +¯ +dx ´ +¨ +TdˆRn +´ +´ φp´∆q1{2π1 ` πp´∆q1{2φ1 +¯ +dx dz +“ +´ +` +X|J X1˘ +As said above, justifying the spectral stability for the Schrödinger-Wave equation reduces to +verify that the spectrum σpLq is purely imaginary. However, the coupling with the wave equation +induces delicate subtleties and a direct approach is not obvious. Instead, based on the expression +L “ J L , we can take advantage of stronger structural properties. In particular, the functional +framework adopted here allows us to overcome the difficulties related to the essential spectrum +induced by the wave equation, which ranges over all the imaginary axis. This approach is strongly +inspired by the methods introduced by D. Pelinovsky and M. Chugunova [9, 42, 43]. The workplan +can be summarized as follows. It can be shown that the eigenproblem LX “ λX for L is equivalent +to a generalized eigenvalue problem AW “ αKW, with α “ ´λ2, see Proposition 5.4 and 5.5 +below, where the auxiliary operators A and K depend on J , L . Then, we need to identify negative +eigenvalues and complex but non real eigenvalues of the generalized eigenproblem. To this end, we +appeal to a counting statement due to [9]. +5.2 +Spectral properties of the operator L +The stability analysis relies on the spectral properties of L , collected in the following claim. +Proposition 5.3 Let L the linear operator defined by (50) on DpL q Ă V . Suppose (9). Then, +the following assertions hold: +33 + +1. σesspL q “ +␣ 1 +2, 2c2( +, +2. L has a finite number of negative eigenvalues, with eigenfunctions in DpL q, given by +npL q +“ +1 ` #tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ă 0 and σ1,m “ 0u +`#tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ď 0 and σ1,m � 0u. +In particular, npL q “ 1 when k “ 0. The eigenspaces associated to the negative eigenvalues +are all finite-dimensional. +3. With X0 “ p0, 1, 0, 0q, we have spanRtX0u Ă KerpL q. Moreover, given k P Zd∖t0u, let K˚ “ +tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 “ 0 and σ1,m “ 0u. Then, we get dimpKerpL qq “ 1 ` #K˚. +We remind the reader that σ1 is assumed radially symmetric, see (H1). Consequently σ1,m “ +σ1,´m “ σ1,˘m and both #K˚ and #tm P Zd ∖t0u, m4 ´4pk¨mq2 ď 0 and σ1,m � 0u are necessarily +even. +Proof. +Since L is self-adjoint, σpL q Ă R. Let us study the eigenproblem for L : λX “ L X +means +$ +’ +’ +’ +’ +’ +’ +’ +’ +& +’ +’ +’ +’ +’ +’ +’ +’ +% +λq “ ´1 +2∆xq ` k ¨ ∇xp ` γσ1 ‹ +ˆˆ +Rnp´∆q´1{2σ2φ dz +˙ +, +λp “ ´1 +2∆xp ´ k ¨ ∇xq, +λφ “ 1 +2φ ` γp´∆q´1{2σ2σ1 ‹ q, +λπ “ 2c2π. +(52) +Clearly λ “ 2c2 is an eigenvalue with eigenfunctions of the form p0, 0, 0, πq, π P L2pTd ˆ Rnq. +As a consequence, dimpKerpL ´ 2c2Iqq is not finite and 2c2 P σesspL q. +We turn to the case λ � 2c2, where the last equation imposes π “ 0. Using Fourier series, we +obtain +λqm “ m2 +2 qm ` ik ¨ mpm ` γp2πqdσ1,m +ˆˆ +Rnp´∆q´1{2σ2φm dz +˙ +, +λpm “ m2 +2 pm ´ ik ¨ mqm, +λφm “ 1 +2φm ` γp2πqdp´∆q´1{2σ2σ1,mqm. +(53) +where qm, pm P C are the Fourier coefficients of q, p P L2pTdq while φmpzq “ +1 +p2πqd +´ +Td φpx, zqe´im¨x dx +for all z P Rn and φ P L2pTd; L2pRnqq. +We split the discussion into several cases. +Case m “ 0. For m “ 0, the equations (53) degenerate to +λq0 “ γp2πqdσ1,0 +ˆˆ +Rnp´∆q´1{2σ2φ0 dz +˙ +, +λp0 “ 0, +´ +λ ´ 1 +2 +¯ +φ0 “ γp2πqdp´∆q´1{2σ2σ1,0q0. +34 + +Combining the first and the third equation yields +λ +´ +λ ´ 1 +2 +¯ +q0 “ γ2p2πq2dσ2 +1,0κq0, +still with κ “ +´ +p´∆q´1σ2σ2 dz. It permits us to identify the following eigenvalues: +• λ “ 0 is an eigenvalue associated to the eigenfunction p0, 1, 0, 0q, +• since σ1,0 “ +1 +p2πqd +´ +Td σ1 dx � 0, and p´∆q´1{2σ2 � 0, λ “ 1{2 is an eigenvalue associated to +eigenfunctions p0, 0, φ, 0q, for any function z ÞÑ φpzq orthogonal to p´∆q´1{2σ2. As before, +since dimpKerpL ´ 1 +2Iqq is not finite, 1 +2 P σesspL q. +• the roots of +λ +´ +λ ´ 1 +2 +¯ +´ γ2p2πq2dσ2 +1,0κ “ λ2 ´ λ +2 ´ γ2p2πq2dσ2 +1,0κ “ 0, +provide two additional eigenvalues +λ˘ “ +1{2 ˘ +b +1{4 ` 4γ2p2πq2dσ2 +1,0κ +2 +, +associated to the eigenfunctions p1, 0, γp2πqdσ1,0p´∆q´1{2σ2 +λ˘´1{2 +, 0q, respectively. +To sum up, the Fourier mode m “ 0 gives rise to two positive eigenvalues (1/2 and λ`), one +negative eigenvalue (λ´) and the eigenvalue 0, the last two being both one-dimensional. It tells us +that +dimpKerpL qq ě 1 and npL q ě 1. +Case m � 0 with σ1,m “ 0. In this case, the m-mode equations (53) for the particle and the +wave are uncoupled +pλ ´ 1{2qφm “ 0, +pMm ´ λq +ˆ +qm +pm +˙ +“ 0 +where we have introduced the 2 ˆ 2 matrix +Mm “ +ˆ +m2{2 +ik ¨ m +´ik ¨ m +m2{2 +˙ +. +(54) +We identify the following eigenvalues: +• λ “ 1{2 is an eigenvalue associated to the eigenfunction p0, 0, eim¨xφpzq, 0q, for any φ P L2pRnq. +Once again, this tells us that 1 +2 P σesspL q. +• the eigenvalues λ˘ “ m2˘2k¨m +2 +P R of the 2 ˆ 2 matrix Mm, associated to the eigenfunctions +peim¨x, ¯ieim¨x, 0, 0q, respectively. Since trpMmq ą 0, at most only one of these eigenvalues +can be negative, which occurs when detpMmq “ m4 +4 ´ pk ¨ mq2 ă 0. +Given k P Zd, we conclude this case by asserting +npL q ě 1 ` #tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ă 0, σ1,m “ 0u, +35 + +and +dimpKerpL qq ě 1 ` #tm P Zd ∖ t0u, m2 “ ˘2k ¨ m, σ1,m “ 0u. +Case m � 0 with σ1,m � 0. Again, we distinguish several subcases. +• if λ “ 1{2, the third equation on (53) imposes qm “ 0, and we are led to +1 ´ m2 +2 +pm “ 0, +ik ¨ mpm ` γp2πqdσ1,m +ˆˆ +Rnp´∆q´1{2σ2φm dz +˙ +“ 0. +Thus, λ “ 1{2 is an eigenvalue associated to the eigenfunctions: +p0, 0, eim¨xφpzq, 0q, for any function z ÞÑ φpzq orthogonal to p´∆q´1{2σ2, +(we recover the same eigenfunctions as for the case m “ 0), +p0, eim¨x, 0, 0q if k ¨ m “ 0, m2 “ 1, +and +´ +0, ´γp2πqdκσ1,m +ik ¨ m +eim¨x, p´∆q´1{2σ2pzqeim¨x, 0 +¯ +if k ¨ m � 0, m2 “ 1. +• if λ “ m2 +2 � 1 +2, (53) becomes +0 “ ik ¨ mpm ` γp2πqdσ1,m +ˆˆ +Rnp´∆q´1{2σ2φm dz +˙ +, +0 “ ´ik ¨ mqm, +m2 ´ 1 +2 +φm “ γp2πqdp´∆q´1{2σ2σ1,mqm. +There is no non-trivial solution when k ¨ m � 0. Otherwise, we see that λ “ m2{2 is an +eigenvalue associated to the eigenfunctions: p0, eim¨x, 0, 0q +• if λ � t1 +2, m2 +2 u, we set µ “ λ ´ m2 +2 . We see that a non trivial solution of (53) exists if its +component qm does not vanish. We combine the equations in (53) to obtain +Ppµqqm “ 0 +where P is the third order polynomial +Ppµq “ µ3 ` bµ2 ` cµ ` d, +b “ m2 ´ 1 +2 +ě 0, +c “ ´ppk ¨ mq2 ` γ2κp2πq2dσ2 +1,mq ă 0, +d “ ´pk ¨ mq2 m2 ´ 1 +2 +ď 0. . +Observe that d “ ´pk ¨ mq2b and pk ¨ mq2 ă |c| ă pk ¨ mq2 ` 1 +4. We thus need to examine the +roots of this polynomial. To this end, we compute the discriminant +D “ 18bcd ´ 4b3d ` b2c2 ´ 4c3 ´ 27d2. +A tedious, but elementary, computation allows us to reorganize terms as follows +D +“ +4pk ¨ mq2` +pk ¨ mq2 ´ b2˘2 ` b2σ2 +1,mγp20pk ¨ mq2 ` γσ2 +1,mq +`4pk ¨ mq2σ2 +1,mγp2pk ¨ mq2 ` γσ2 +1,mq ` 4σ2 +1,mγ +` +pk ¨ mq4 ` 2pk ¨ mq2σ2 +1,mγ ` σ4 +1,mγ2˘ +, +36 + +where we have set γ “ γ2κp2πq2d. Since σ1,m � 0, we thus have D ą 0 and P has 3 distinct +real roots, µ1 ă µ2 ă µ3. In order to bring further information about the location of the roots, +we observe that limµÑ˘8 Ppµq “ ˘8 while Pp0q “ d ď 0 and P 1p0q “ c ă 0. Moreover, +studying the zeroes of P 1pµq “ 3µ2 ` 2bµ ` c, we see that µmax “ ´b´ +? +b2´3c +3 +ă 0 is a local +maximum and µmin “ ´b` +? +b2´3c +3 +ą 0 is a local minimum. Moreover, P 2pµq “ 6µ ` 2b, +showing that P is convex on the domain p´pm2 ´ 1q{6, `8q, concave on p´8, ´pm2 ´ 1q{6q. +A typical shape of the polynomial P is depicted in Figure 1. From this discussion, we infer +µ1 ă µmax ă µ2 ď 0 ă µmin ă µ3. +-6 +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +-40 +-30 +-20 +-10 +0 +10 +20 +30 +40 +50 +P( ) +Figure 1: Typical graph for µ ÞÑ Ppµq, with its roots µ1 ă µ2 ă µ3 and local extrema µmax, +µmin +Coming back to the issue of counting the negative eigenvalues of L , we are thus wondering +whether or not λj “ µj ` m2{2 is negative. We already know that µ3 ą µmin ą 0, hence +µ3 ą ´m2{2 and we have at most 2 negative eigenvalues for each Fourier mode m � 0 such +that σ1,m � 0. To decide how many negative eigenvalues should be counted, we look at the +sign of Pp´m2{2q (see Fig. 1): +i) if Pp´m2{2q ą 0 then µ1 ă ´m2{2 ă µ2, +ii) if Pp´m2{2q “ 0 then either ´m2{2 ă µmax, in which case µ1 “ ´m2{2 ă µ2, or +´m2{2 ą µmax, in which case µ2 “ ´m2{2 ą µ1, +iii) if Pp´m2{2q ă 0 then either ´m2{2 ă µmax, in which case ´m2{2 ă µ1 ă µ2, or +´m2{2 ą µmax, in which case µ1 ă µ2 ă ´m2{2. +However, we remark that +Pp´m2{2q +“ +´m6 +8 ` m4pm2 ´ 1q +8 +` m2 +2 ppk ¨ mq2 ` γσ2 +1,mq ´ m2 ´ 1 +2 +pk ¨ mq2 +“ +´m4 +8 +´ +1 ´ 4γσ2 +1,m +m2 +¯ +` pk ¨ mq2 +2 +“ ´1 +8pm4 ´ 4pk ¨ mq2 ´ 4m2γσ2 +1,mq, +(55) +37 + +where, by virtue of (9), m � 0 and σ1m � 0, 1 ą 4 +γσ2 +1,m +m2 +ą 0. +This can be combined together with +P 1p´m2{2q “3m4 +4 ´ m2pm2 ´ 1q +2 +´ pk ¨ mq2 ´ γσ2 +1,m “ m4 +4 ` m2 +2 ´ pk ¨ mq2 ´ γσ2 +1,m +“1 +4 +` +m4 ´ 4pk ¨ mq2 ´ 4m2γσ2 +1,m +˘ +` m2γσ2 +1,m ` m2 +2 ´ γσ2 +1,m +“ ´ 2Pp´m2{2q ` m2 +2 ` pm2 ´ 1qγσ2 +1,m ą ´2Pp´m2{2q. +Finally, +P 2p´m2{2q “ ´2m2 ´ 1 ă 0. +As a consequence, Pp´m2{2q ă 0 implies P 1p´m2{2q ą 0, while P 2p´m2{2q ă 0. This shows +that ´m2{2 ă µ1. Therefore, in case iii), the only remaining possibility is the situation where +Pp´m2{2q ă 0 with ´m2{2 ă µ1 ă µ2. As a conclusion, if Pp´m2{2q ă 0, all eigenvalues λj +are positive. +Next, we claim that case ii) cannot occur. Indeed, Pp´m2{2q “ 0 if and only if +pm2 ´ 2k ¨ mqpm2 ` 2k ¨ mq “ 4m2γσ2 +1,m. +In particular, the term on the left hand side of this equality has to be positive. +This is +possible if and only if both factors, which belong to Z, are positive. In this case, according +to the sign of k ¨ m, one of them is ě m2 so that +m2 ď 4m2γσ2 +1,m. +This contradicts the smallness condition (9). Note that Pp´m2{2q � 0 implies λj � 0, i.e. +m-modes with m � 0 and σ1,m � 0 cannot generate elements of KerpL q. +As a conclusion, negative eigenvalues only come from case i) and for each m-mode such that +Pp´m2{2q ą 0 we have exactly one negative eigenvalue. Going back to (55), in this case, we +have +pm4 ´ 4pk ¨ mq2q “ pm2 ´ 2k ¨ mqpm2 ` 2k ¨ mq ă m24γσ2 +1,m ă m2 +owing to (9). This excludes the possibility that m4´4pk¨mq2 ą 0, since we noticed above that +whenever this term is positive, it is ě m2. Hence, case i) holds if and only if m4´4pk¨mq2 ď 0. +This ends the counting of the negative eigenvalues of L in Proposition 5.3. Note that the +associated eigenspaces are spanned by +´ +eim¨x, ´ +ik ¨ m +λ ´ m2{2eim¨x, eim¨x σ1,mγp2πqdp´∆zq´1{2σ2 +λ ´ 1{2 +, 0 +¯ +. +The discussion has permitted us to find the elements of KerpL q. To be specific, the equation +38 + +L X “ 0 yields π “ 0 and the following relations for the Fourier coefficients +m2 +2 pm ´ ik ¨ mqm “ 0, +φm +2 ` p2πqdγp´∆q´1{2σ2σ1,mqm “ 0, +m2 +2 qm ` ik ¨ mpm ` p2πqdγσ1,m +ˆ +p´∆q´1{2σ2φm dz “ 0. +We have seen that the mode m “ 0 gives rise the eigenspace spanned by p0, 1, 0, 0q. For m � 0, ele- +ments of KerpL q can be obtained only in the case σ1,m “ 0. Moreover, the condition m2 “ ˘2k ¨m +has to be fulfilled. In such a case, peim¨x, ¯ieim¨x, 0, 0q P KerpL q. +Finally, it remains to prove that σesspL q “ +␣1 +2, 2c2( +. We have already noticed that +␣ 1 +2, 2c2( +Ă +σesspL q. Suppose, by contradiction, that there exists λ P σesspL q ∖ +␣ 1 +2, 2c2( +. Hence, by Weyl’s +criterion [42, Theorem B.14], there exists a sequence pXνqνPN with Xν “ pqν, pν, φν, πνq P DpL q +such that, as ν goes to 8, +}pL ´ λIqXν} Ñ 0, +}Xν} “ 1 and Xν á 0 weakly in V . +(56) +Since λ � 1 +2 and λ � 2c2, from (52) and (56) we have +}πν}L2pTd;L2pRnqq Ñ 0 and φν “ ´ +ˆ1 +2 ´ λ +˙´1 +γp´∆q´1{2σ2σ1 ‹ qν ` εν +with εν P L2pTd; L2pRnqq such that limνÑ8 }εν}L2pTd;L2pRnqq “ 0. This leads to +››››´1 +2∆xqν ´ λqν ` k ¨ ∇xpν ´ +γ2κ +1{2 ´ λΣ ‹ qν ` γσ1 ‹ +ˆˆ +Rnp´∆q´1{2σ2εν dz +˙›››› +L2pTdq +ÝÝÝÑ +νÑ8 0, +››››´1 +2∆xpν ´ λpν ´ k ¨ ∇xqν +›››› +L2pTdq +ÝÝÝÑ +νÑ8 0. +Using the fact that the sequence ppqν, pν, ενqqνPN is bounded in L2pTdq ˆ L2pTdq ˆ L2pTd; L2pRnqq, +we deduce that pqν, pνqνPN is bounded in H2pTdq ˆ H2pTdq. Indeed, reasoning on Fourier series, +this amounts to estimate +ÿ +mPZd +|m|4p|qν,m|2 ` |pν,m|2q +ď 2 +ÿ +mPZd +` +|m2qν,m ` 2ik ¨ mpν,m|2 ` |m2pν,m ´ 2ik ¨ mqν,m|2q +`8 +ÿ +mPZd +p|k ¨ mpν,m|2 ` |k ¨ mqν,m|2q +ď 2 +›› ´ ∆xqν ` 2k ¨ ∇xpν +›› +L2pTdq ` 2 +›› ´ ∆xpν ´ 2k ¨ ∇xqν +›› +L2pTdq +`4 +δ|k|4 ÿ +mPZd +` +|qν,m|2 ` |pν,m|2˘ +` 4δ +ÿ +mPZd +|m|4p|qν,m|2 ` |pν,m|2q. +. +Choosing 0 ă δ ă 1{4 and using the already known estimates, we conclude that }∆xqν}2 +L2 ` +}∆xpν}2 +L2 “ ř +mPZd |m|4` +|qν,m|2 ` |pν,m|2˘ +is bounded, uniformly with respect to ν. Hence, because +of the compact Sobolev embedding of H2pTdq into L2pTdq, we have that pqν, pνqνPN has a (strongly) +convergent subsequence in L2pTdq ˆ L2pTdq. As a consequence, the sequence pXνqνPN has a conver- +39 + +gent subsequence in V , which contradicts (56). +A consequence of Proposition 5.3 is that 0 is an isolated eigenvalue of L . Since the restriction +of L to the subspace pKerpL qqK is, by definition, injective, it makes sense to define on it its inverse +L ´1, with domain RanpL q Ă pKerpL qqK Ă V . In fact, 0 being an isolated eigenvalue, RanpL q is +closed and coincides with pKerpL qqK, [42, Section B.4]. This can be shown by means of spectral +measures. Given X P pKerpL qqK, the support of the associated spectral measure dµX does not +meet the interval p´ǫ, `ǫq for ǫ ą 0 small enough, independent of X. Accordingly, we get +}L X}2 “ +ˆ `8 +´8 +λ2 dµXpλq “ +ˆ +|λ|ěǫ +λ2 dµXpλq ě ǫ2}X}2. +In particular, the Fredholm alternative applies: for any Y P pKerpL qqK, there exists a unique +X P pKerpL qqK such that L X “ Y . We will denote X “ L ´1Y . +For further purposes, let us set +X0 “ p0, 1, 0, 0q P KerpL q and Y0 “ ´J X0 “ p1, 0, 0, 0q. +Note that Y0 P pKerpL qqK, so that it makes sense to consider the equation +L U0 “ Y0. +We find +πm “ 0, +φm “ ´2γp2πqdp´∆q´1{2σ2σ1,mqm, +m2pm “ 2ik ¨ mqm, +and +m2qm ` 2ik ¨ mpm ` 2γp2πqdσ1,m +ˆ +p´∆q´1{2σ2φm dz “ δ0,m. +It yields, for m � 0, pm4 +4 ´ pk ¨ mq2 ´ γ|σ1,m|2m2¯ +qm “ 0 and q0 “ ´ +1 +2γ2p2πq2d|σ1,0|2κ. Therefore, we +can set +U0 “ L ´1Y0 “ ´ +1 +2γ2p2πq2d|σ1,0|2κ +` +1, 0, ´2γp2πqdp´∆q´1{2σ2σ1,0, 0 +˘ +, +solution of L U0 “ Y0 such that U0 P pKerpL qqK. We note that +pU0, Y0q “ ´ +1 +2γ2p2πqd|σ1,0|2κ ă 0. +(57) +5.3 +Reformulation of the eigenvalue problem, and counting theo- +rem +The aim of the section is to introduce several reformulations of the eigenvalue problem. This will +allow us to make use of general counting arguments, set up by [9, 42, 43]. +Proposition 5.4 Let us set M “ ´J L J . The coupled system +M Y “ ´λX, +L X “ λY, +(58) +admits a solution with λ � 0, X P DpL q ∖ t0u, Y P DpJ L J q ∖ t0u iff there exists two vectors +X˘ P DpLq ∖ t0u that satisfy LX˘ “ ˘λX˘. +40 + +Let P stand for the orthogonal projection from V to pKerpL qqK Ă V . +Proposition 5.5 Let us set A “ PM P and K “ PL ´1P. Let us define the following Hilbert +space +H “ DpM q X pKerpL qqK Ă V . +The coupled system (58) has a pair of non trivial solutions p˘λ, X, ˘Y q, with λ � 0 iff the gener- +alized eigenproblem +AW “ αKW, +W P H , +(59) +admits the eigenvalue α “ ´λ2 � 0, with at least two linearly independent eigenfunctions. +Recall that the plane wave solution obtained Section 2.1 is spectrally stable, if the spectrum +of L is contained in iR. In view of Propositions 5.4 and 5.5, this happens if and only if all the +eigenvalues of the generalized eigenproblem (59) are real and positive. In other words, the presence +of spectrally unstable directions corresponds to the existence of negative eigenvalues or complex +but non real eigenvalues of the generalized eigenproblem (59). +Our goal is then to count the eigenvalues α of the generalized eigenvalue problem (59). In +particular we define the following quantities: +• N ´ +n , the number of negative eigenvalues +• N 0 +n, the number of eigenvalues zero +• N p +n, the number of positive eigenvalues +of (59), counted with their algebraic multiplicity, the eigenvectors of which are associated to non- +positive values of the the quadratic form W ÞÑ pKW|Wq “ pL ´1PW|PWq. Moreover, let NC` +be the number of eigenvalues α P C with Impαq ą 0. +As pointed out above, the eigenvalues counted by N ´ +n and NC` correspond to cases of instabil- +ities for the linearized problem (38). Note that to prove the spectral stability, it is enough to show +that the generalized eigenproblem (59) does not have negative eigenvalues and NC` “ 0. Indeed, +as a consequence of Propositions 5.4 and 5.5 and Lemma 5.1, if α P C ∖ R is an eigenvalue of (59), +then ¯α is an eigenvalue too. Hence, if NC` “ 0, then the generalized eigenproblem (59) does not +have solutions in C ∖ R. +Finally, for using the counting argument introduce by Chugunova and Pelinovsky in [9], we need +the following information on the essential spectrum of A, see [43, Lemma 2-(H1’) and Lemma 4]. +Lemma 5.6 Let M “ ´J L J be defined on V . Then σesspM q “ r0, `8q. Let A “ PM P and +K “ PL ´1P be defined on H . Then σesspAq “ r0, `8q and we can find δ˚, d˚ ą 0 such that for +any real number 0 ă δ ă δ˚, A`δK admits a bounded inverse and we have σesspA`δKq Ă rd˚δ, `8q. +Proof. +We check that +J L J X “ +¨ +˚ +˚ +˚ +˚ +˚ +˚ +˝ +∆xq +2 +´ k ¨ ∇xp +∆xp +2 +` k ¨ ∇xq ` γσ1 ‹ +ˆ +p´∆zq´1{2σ2p´∆zq1{2π dz +2c2∆zφ +∆zπ +2 +` γσ2σ1 ‹ p +˛ +‹‹‹‹‹‹‚ +. +41 + +As a matter of fact, for any φ P H2pRnq, the vector Xe “ p0, 0, φ, 0q lies in pKerpL qqK and satisfies +J L J Xe “ +¨ +˚ +˚ +˝ +0 +0 +2c2∆zφ +0 +˛ +‹‹‚P pKerpL qqK. +Consequently M Xe “ AXe “ ´J L J Xe “ p0, 0, ´2c2∆zφ, 0q. It indicates that a Weyl sequence +for A ´ λI, λ ą 0, can be obtained by adapting a Weyl sequence for p´∆z ´ µIq, µ ą 0. Let us +consider a sequence of smooth functions ζν P C8 +c pRnq such that supppζνq Ă Bp0, ν ` 1q, ζνpzq “ 1 +for x P Bp0, νq and }∇zζν}L8pRnq ď C0 ă 8, }D2 +zζν}L8pRnq ď C0 ă 8, uniformly with respect to +ν P N. We set φνpzq “ ζνpzqeiξ¨z{p +? +2cq for some ξ P Rn. We get +p´|ξ|2 ´ 2c2∆zqφνpzq “ eiξ¨z{p +? +2cq´ 2i +? +2cξ ¨ ∇zζν ` 2c2∆zζν +¯ +pzq, +which is thus bounded in L8pRnq and supported in Bp0, ν ` 1q ∖ Bp0, νq. It follows that }p´|ξ|2 ´ +2c2∆zqφν}2 +L2pRnq ≲ νn´1, while }φν}2 +L2pRnq ≳ νn. Accordingly, we obtain +}φν}2 +L2pRnq +}p´|ξ|2´2c2∆zqφν}2 +L2pRnq +≳ +ν Ñ 8 as ν Ñ 8. Therefore, φν equally provides a Weyl sequence for M ´ |ξ|2I and A ´ |ξ|2I, +showing the inclusions r0, 8q Ă σesspM q and r0, 8q Ă σesspAq. +Next, let λ � r0, 8q. We suppose that we can find a Weyl sequence pXνqνPN for M , such that +M Xν ´ λXν +“ +¨ +˚ +˚ +˚ +˚ +˚ +˚ +˝ +´λqν ´ ∆xqν +2 +` k ¨ ∇xpν +´λpν ´ ∆xpν +2 +´ k ¨ ∇xqν ´ γσ1 ‹ +ˆ +p´∆zq´1{2σ2p´∆zq1{2πν dz +´λφν ´ 2c2∆zφν +´λπν ´ ∆zπν +2 +´ γσ2σ1 ‹ pν +˛ +‹‹‹‹‹‹‚ +“ +¨ +˚ +˚ +˝ +q1 +ν +p1 +ν +φ1 +ν +π1 +ν +˛ +‹‹‚ÝÝÝÑ +νÑ8 0, +with, moreover, }Xν} “ 1 and Xν á 0 weakly in V . In particular, we can set +x +φνpx, ξq “ +x +φ1νpx, ξq +2c2|ξ|2 ´ λ. +(60) +It defines a sequence which tends to 0 strongly L2pTd ˆ Rnq since, writing λ “ a ` ib P C ∖ r0, 8q, +we get |2c2|ξ|2 ´λ|2 “ |2c2|ξ|2 ´a|2 `b2 which is ě b2 ą 0 when λ � R, and, in case b “ 0, ě a2 ą 0. +Similarly, we can write +x +πνpx, ξq “ +2x +π1νpx, ξq +|ξ|2 ´ 2λ +loooomoooon +“hνpx,ξqPL2pTdˆRnq +`γ +2x +σ2pξq +|ξ|2 ´ 2λ +loooomoooon +PL2pRnq +σ1 ‹ pν, +(61) +42 + +where hν tends to 0 strongly L2pTd ˆ Rnq. We are led to the system +¨ +˚ +˝ +´ +´ +λ ` ∆x +2 +¯ +qν ` k ¨ ∇xpν +´k ¨ ∇xqν ´ +´ +λ ` ∆x +2 +¯ +pν ´ 2γ2 +ˆ +|x +σ2|2 +p2πqnp|ξ|2 ´ 2λq dξ ˆ Σ ‹ pν +˛ +‹‚ +“ +¨ +˝ +q1 +ν +p1 +ν ´ γσ1 ‹ +ˆ x +σ2pξq +|ξ| hνpx, ξq +dξ +p2πqn +˛ +‚ÝÝÝÑ +νÑ8 0. +(62) +Reasoning as in the proof of Proposition 5.3-1), we conclude that Xν converges strongly to 0 +in V , a contradiction. +Hence, λ P C ∖ r0, 8q cannot belong to σesspM q and the identification +σesspM q “ r0, 8q holds. +Proposition 5.3-3) identifies KerpL q. Let us introduce the mapping +Ă +P : +ˆq +p +˙ +P L2pTdqˆL2pTdq ÞÝÑ +¨ +˚ +˚ +˝ +ÿ +mPK˚, k¨mą0 +pqm ´ ipmqeim¨x ` +ÿ +mPK˚, k¨mă0 +pqm ` ipmqeim¨x +p0 ` i +ÿ +mPK˚, k¨mą0 +pqm ´ ipmqeim¨x ´ i +ÿ +mPK˚, k¨mă0 +pqm ` ipmqeim¨x +˛ +‹‹‚. +Then, +X “ +¨ +˚ +˚ +˝ +q +p +φ +π +˛ +‹‹‚ÞÝÑ +¨ +˚ +˚ +˝ +Ă +P +ˆ +q +p +˙ +0 +0 +˛ +‹‹‚ +is the projection of V on KerpL q. Accordingly, we realize that P does not modify the last two +components of a vector X “ pq, p, φ, πq P V , and for X P pKerpL qqK, we have p0 “ 0, and +qm “ ˘ipm for any m P K˚, depending on the sign of k ¨ m. +Now, let λ P C ∖ r0, 8q and suppose that we can exhibit a Weyl sequence pXνqνPN for A ´ λI: +Xν P H Ă pKerpL qqK, PXν “ Xν, }Xν} “ 1, Xν á 0 in V and limνÑ8 }pA ´ λIqXν} “ 0. We +can apply the same reasoning as before for the last two components of pA ´ λIqXν; it leads to (60) +and (61), where, using λ � r0, 8q, φν and hν converge strongly to 0 in L2pTd ˆ Rnq. We arrive at +the following analog to (62) +pI ´ Ă +Pq +¨ +˚ +˝ +´ +´ +λ ` ∆x +2 +¯ +qν ` k ¨ ∇xpν +´k ¨ ∇xqν ´ +´ +λ ` ∆x +2 +¯ +pν ´ 2γ2 +ˆ +|x +σ2|2 +p2πqnp|ξ|2 ´ 2λq dξ ˆ Σ ‹ pν +˛ +‹‚ +“ +ˆq1 +ν +p1 +ν +˙ +´ pI ´ Ă +Pq +¨ +˝ +0 +γσ1 ‹ +ˆ x +σ2pξq +|ξ| hνpx, ξq +dξ +p2πqn +˛ +‚ÝÝÝÑ +νÑ8 0. +(63) +In order to derive from (63) an estimate in a positive Sobolev space as we did in the proof of +Proposition 5.3-1), we should consider the Fourier coefficients arising from ´ 1 +2∆xqν ` k ¨ ∇xpν and +´ 1 +2∆xpν ´k¨∇xqν, namely Qm “ m2 +2 qν,m`ik¨mpν,m and Pm “ m2 +2 pν,m´ik¨mqν,m. Only the coef- +ficients belonging to K˚ are affected by the action of Ă +P, which leads to Qm ´ pQm ¯ iPmq “ ˘iPm +and Pm ¯ ipQm ¯ iPmq “ ¯iQm, according to the sign of k ¨ m. However, we bear in mind that +qm “ ˘ipm when m P K˚ with ˘k ¨ m ą 0. Hence, for coefficients in K˚, the contributions of the +differential operators reduces to ˘im2pm “ ˘m2qm and ¯im2qm “ ˘m2pm, respectively. Note +43 + +also that for these coefficients there is no contributions coming from the convolution with σ1 in +(63) since σ1,m “ 0 for m P K˚. Therefore, reasoning as in the proof of Proposition 5.3-1) for +coefficients m P Zd ∖ K˚, we can obtain a uniform bound on ř +mPZd |m|4p|qν,m|2 ` |pν,m|2q, which +provides a uniform H2 bound on qν and pν, leading eventually to a contradiction. We conclude +that σesspAq “ r0, 8q. +Let δ ą 0 and consider the shifted operator A ` δK. As a consequence of Lemma 5.10, we will +see that KerpA ` δKq “ t0u for any δ ą 0: 0 is not an eigenvalue for A ` δK; let us justify it does +not belong to the essential spectrum neither. To this end, we need to detail the expression of the +operator K. Given X P H , we wish to find X1 P H satisfying +L X1 “ +¨ +˚ +˚ +˚ +˚ +˚ +˚ +˝ +´1 +2∆xq1 ` k ¨ ∇xp1 ` γσ1 ‹ +ˆˆ +Rnp´∆q´1{2σ2φ1 dz +˙ +´1 +2∆xp1 ´ k ¨ ∇xq1 +1 +2φ1 ` γp´∆q´1{2σ2σ1 ‹ q1 +2c2π1 +˛ +‹‹‹‹‹‹‚ +“ X. +We infer π1 “ +π +2c2 and the relation φ1 “ 2φ ´ 2γp´∆zq´1{2σ2σ1 ‹ q1. In turn, the Fourier coefficients +of q1, p1 are required to satisfy +ˆ +m2{2 ´ 2γ2κp2πq2d|σ1,m|2 +ik ¨ m +´ik ¨ m +m2{2 +˙ ˆ +q1 +m +p1 +m +˙ +“ +¨ +˝qm ´ 2γp2πqdσ1,m +ˆ +p´∆q´1{2σ2φm dz +pm +˛ +‚. +When m � 0, m � K˚, the matrix of this system has its determinant equal to +det “ m4 +4 +` +1 ´ 4γ2κp2πq2d |σ1,m|2 +m2 +˘ +´ pk ¨ mq2. +Owing to (9), since pk ¨ mq2 takes values in N, it does not vanish and we obtain q1 +m, p1 +m by solving +the system +q1 +m “ +1 +det +ˆm2 +2 +´ +qm ´ 2γp2πqdσ1,m +ˆ +p´∆q´1{2σ2φm dz +¯ +´ ik ¨ mpm +˙ +, +p1 +m “ +1 +det +ˆ +`ik ¨ m +´ +qm ´ 2γp2πqdσ1,m +ˆ +p´∆q´1{2σ2φm dz +¯ +` +´m2 +2 ´ 2γ2κp2πq2d|σ1,m|2¯ +pm +˙ +. +If m P K˚ we find a solution in pKerpL qqK by setting p1 +m “ pm +m2 , q1 +m “ ˘ip1 +m, according to the sign +of k ¨m; if m “ 0, we set p1 +0 “ 0 and q1 +0 “ +1 +2γ2κp2πq2d|σ1,0|2 +` +q0 ´2γp2πqdσ1,0 +´ +p´∆q´1{2σ2φ0 dz +˘ +. This +defines X1 “ KX. +Therefore, the last two components of pA ` δK ´ λIqX read +p2δ ´ λqφ ´ 2c2∆zφ ´ 2δγp´∆q´1{2σ2σ1 ‹ q1, +´ δ +2c2 ´ λ +¯ +π ´ 1 +2∆zπ ´ γσ2σ1 ‹ p1. +Hence, when λ does not belong to rδd˚, 8q, with d˚ “ minp2, +1 +2c2q, we can repeat the analysis +performed above to establish that λ � σesspA ` δKq. In particular the essential spectrum of A has +been shifted away from 0. +We are now able to apply the results of Chugunova and Pelinovsky [9] (see also [43]), to obtain +44 + +the following. +Theorem 5.7 [9, Theorem 1] Let L be defined by (50). Suppose (9). With the operators M , A, K +defined as in Propositions 5.4-5.5, the following identity holds +N ´ +n ` N 0 +n ` N ` +n ` NC` “ npL q. +Let us now detail the proof of Proposition 5.4 and 5.5, adapted from [43, Prop. 1 & Prop. 3]. +Proof of Propositions 5.4 and 5.5. +The goal is to establish connections between the following +three problems: +(Ev) the eigenvalue problem LX “ λX, with L “ J L , +(Co) the coupled problem L X “ λY , M Y “ ´λX, with M “ ´J L J , +(GEv) the generalized eigenvalue problem AW “ αKW, with A “ PM P, K “ PL ´1P, the +projection P on pKerpL qqK, and W P H “ DpM q X pKerpL qqK. +The proof of Propositions 5.4 and 5.5 follows from the following sequence of arguments. +(i) By Lemma 5.1, we already know that if there exists a solution pλ, X`q of (Ev), with λ � 0 +and X` � 0, then, there exists X´ � 0, such that p´λ, X´q satisfies (Ev). Being eigenvectors +associated to distinct eigenvalues, X` and X´ are linearly independent. Note that only this +part of the proof uses the specific structure of the operator L. +(ii) From these eigenpairs for L, we set +X “ X` ` X´ +2 +, +Y “ +Ă +J +ˆX` ´ X´ +2 +˙ +. +Since X` and X´ are linearly independent, we have X � 0, Y � 0. Moreover, X “ X``X´ +2 +and J Y “ X`´X´ +2 +are linearly independent. We get +L X “ +Ă +J LX “ +Ă +J +ˆλ +2pX` ´ X´q +˙ +“ λY, +M Y “ ´J L +ˆX` ´ X´ +2 +˙ +“ ´L +ˆX` ´ X´ +2 +˙ +“ ´λ +2pX` ` X´q “ ´λX, +so that pλ, X, Y q satisfies (Co). +(iii) If pλ, X, Y q is a solution (Co), then p´λ, X, ´Y q satisfies (Co) too. +(iv) Let pλ, X, Y q be a solution (Co). Set +X1 “ J Y, +Y 1 “ +Ă +J X. +We observe that +M Y 1 “ ´J L J Ă +J X “ ´J L X “ ´J pλY q “ ´λX1, +L X1 “ L J Y “ +Ă +J J L J Y “ ´ Ă +J M Y “ λ Ă +J X “ λY 1, +which means that pλ, J Y, Ă +J Xq is a solution of (Co). Moreover, if X and J Y are linearly +independent, Y and +Ă +J X are linearly independent too. +45 + +(v) Let pλ, X, Y q be a solution (Co), with X � 0. We get +LpX ˘ J Y q +“ +J L X ˘ J L J Y “ J L X ¯ M Y +“ +J pλY q ˘ λX “ ˘λpX ˘ J Y q, +so that p˘λ, X ˘ J Y q satisfy (Ev). In the situation where X and J Y are linearly inde- +pendent, we have X ˘ J Y � 0 and p˘λ, X ˘ J Y q are eigenpairs for L. Otherwise, one of +the vectors X ˘ J Y might vanish. Nevertheless, since only one of these two vectors can be +0, we still obtain an eigenvector X˘ � 0 of L, associated to either ˘λ. Coming back to i), we +conclude that ¯λ is an eigenvalue too. +Items i) to v) justify the equivalence stated in Proposition 5.4. +(vi) Let pλ, X, Y q be a solution (Co). From L X “ λY , we infer Y P RanpL q Ă pKerpL qqK so +that PY “ Y . The relation thus recasts as +X “ λPL ´1PY ` ˜Y, +˜Y P KerpL q, +P ˜Y “ 0. +(Here, PL ´1PY stands for the unique solution of L Z “ Y which lies in pKerpL qqK.) We +obtain +PM Y +“ +Pp ´ λXq “ ´λPpλPL ´1PY ` ˜Y q +“ +´λ2PL ´1PY “ ´λ2KY “ PM PY “ AY, +so that p´λ2, Y q satisfies (GEv). Going back to iv), we know that p´λ2, Ă +J Xq is equally a +solution to (GEv). If X and J Y are linearly independent, we obtain this way two linearly +independent vectors, Y and +Ă +J X, solutions of (GEv) with α “ ´λ2. +(vii) Let pα, Wq satisfy (GEv), with α � 0, W � 0. We set X “ ´M W +?´α . We have +Ă +J X “ ´ +1 +?´α +Ă +J M W “ +1 +?´α +Ă +J J L J W “ +1 +?´αL J W +which lies in RanpL q Ă pKerpL qqK. Thus, using P Ă +J X “ +Ă +J X, we compute +K Ă +J X “ PL ´1P Ă +J X “ PL ´1 Ă +J X “ +1 +?´αPL ´1L J W “ +1 +?´αPJ W. +Next, we observe that +A Ă +J X “ PM P Ă +J X “ ´PJ L J Ă +J X “ ´PJ L X “ +1 +?´αPJ L M W. +However, we can use PW “ W (since W P H Ă pKerpL qqK) and the fact that, for any +vector Z, L Z “ L pI ´ PqZ ` L PZ “ 0 ` L PZ, which yields +A Ă +J X +“ +1 +?´αPJ L PM PW “ +1 +?´αPJ L AW “ ´ +? +´αPJ L KW +“ +´ ?´αPJ L PL ´1PW “ ´ ?´αPJ L L ´1W “ ´ ?´αPJ W. +We conclude that A Ă +J X “ αK Ă +J X: pα, Ă +J Xq satisfies (GEv). +(viii) Let pα, Wq satisfy (GEv), with α � 0, W � 0. We have +PpM PW ´ αL ´1PWq “ 0 +and thus +M PW ´ αL ´1PW “ ˜Y P KerpL q. +46 + +Let us set +Y “ PW P pKerpL qqK, +X “ ´M PW +?´α +“ +´1 +?´αp˜Y ` αL ´1PWq, +so that +L X “ +? +´αPW “ +? +´αY, +M Y “ M PW “ ´ +? +´αX. +Therefore p ?´α, X, Y q satisfies (Co). By v), p˘ ?´α, X ˘ J Y q satisfy (Ev), and at least +one of the vectors X ˘J Y does not vanish; using i), we thus obtain eigenpairs p˘ ?´α, X˘q +of L. With ii), we construct solutions of (Co) under the form +` ?´α, X``X´ +2 +, Ă +J +` X`´X´ +2 +˘˘ +, +which, owing to iv) and vi), provide the linearly independent solutions +` +α, Ă +J +`X`˘X´ +2 +˘˘ +of +(GEv). The dimension of the linear space of solutions of (GEv) is at least 2. +At least one of these vectors X˘ is given by the formula +˜X˘ “ ´ M W +?´α ˘ J W. +By the way, we indeed note that AW “ αKW, with W P H , can be cast as L J L J W “ +´αW (see Lemma 5.8 below) so that +L +´ +´ M W +?´α ˘ J W +¯ +“ +1 +?´αJ pL J L J Wq ˘ J L J W +“ +?´αJ W ¯ M W “ ˘ ?´α +´ +´ M W +?´α ˘ J W +¯ +. +With these manipulations we have checked that p˘ ?´α, ˜X˘q satisfy (Ev). If both vectors +˜X˘ are non zero, we get X˘ “ ˜X˘ and we recover W “ +Ă +J +` X`´X´ +2 +˘ +. If ˜X˘ “ 0, then, we +get ˜X¯ “ ¯J W � 0, and we directly obtain X¯ “ ˜X¯, W “ ¯ Ă +J X¯. In any cases, W lies +in the space spanned by X` and X´ and the dimension of the space of solutions of (GEv) +is even. +This ends the proof of Proposition 5.4 and 5.5. +5.4 +Spectral instability +We are going to compute the terms arising in Theorem 5.7. Eventually, it will allow us to identify the +possible unstable modes. In what follows, we find convenient to work with the operator M ´αL ´1 +instead of PpM ´ αL ´1qP “ A ´ αK, owing to to the following claim. +Lemma 5.8 Let α � 0. In the space H “ DpM q X pKerpL qqK, the two subspaces KerpA ´ αKq +and KerpM ´ αL ´1q coincide. +Proof. +Let X P H satisfy M X “ αL ´1X. Then, we have X “ PX and, thus, pA ´ αKqX “ +PpM ´ αL ´1qPX “ PpM X ´ αL ´1Xq “ 0, showing the inclusion KerpM ´ αL ´1q X H Ă +KerpA ´ αKq. +Conversely, the equation pA ´ αKqX “ 0, with X “ PX P pKerpL qqK means that pM ´ +αL ´1qX “ Y P KerpL q. Applying L then yields L M X “ αX. Since both terms of this relation +lie in pKerpL qqK, it is legitimate to apply L ´1, showing that M X “ αL ´1X: we have shown +KerpA ´ αKq X H Ă KerpM ´ αL ´1q. +47 + +Therefore, we shall consider the solutions of the generalized eigenvalue problem M X “ αL ´1X, +with X P H . We rewrite the equation by introducing an auxiliary unknown: +M X “ α ˜X, +L ˜X “ X. +Lemma 5.9 Suppose (9). N 0 +n “ 1. +Proof. +We are interested in the solutions of +´1 +2∆xq ` k ¨ ∇xp “ 0, +´1 +2∆xp ´ k ¨ ∇xq ´ γσ1 ‹ +ˆ +σ2π dz “ 0, +´2c2∆zφ “ 0, +´1 +2∆zπ ´ γσ2σ1 ‹ p “ 0. +We infer φpx, zq “ 0 and pπpx, ξq “ 2γ x +σ2pξq +|ξ|2 σ1 ‹ ppxq, and, next, +´1 +2∆xq ` k ¨ ∇xp “ 0, +´1 +2∆xp ´ k ¨ ∇xq ´ 2γ2κΣ ‹ p “ 0 +with Σ “ σ1 ‹ σ1. In terms of Fourier coefficients, it becomes +m2 +2 qm ` ik ¨ mpm “ 0, +m2 +2 pm ´ ik ¨ mqm ´ 2p2πq2dγ2κ|σ1,m|2pm “ 0. +For m “ 0, we get p0 “ 0 and we find the eigenfunction p1, 0, 0, 0q “ Y0 “ ´J X0 with X0 “ +p0, 1, 0, 0q P KerpL q. +For m � 0 with σ1,m � 0, we get +m4 ´ 4pk ¨ mq2 “ 2p2πq2dγ2κ|σ1,m|2 +loooooooooomoooooooooon +Pp0,1q +m2. +which cannot hold (see the proof of Proposition 5.3 for more details). +For m � 0 with σ1,m “ 0, we get Mm +ˆqm +pm +˙ +“ 0 with Mm defined in (54). +As far as +m4 ´4pk ¨mq2 � 0, Mm is invertible and the only solution is pm “ 0 “ qm. If m4 ´4pk ¨mq2 “ 0, we +find the eigenfunctions peik¨m, ˘ieik¨m, 0, 0q. These functions belong to KerpL q, and thus do not +lie in the working space H . +We conclude that KerpM q “ spanRtY0u. Moreover, this vector Y0 does not belong to RanpM q +so that the algebraic multiplicity of the eigenvalue 0 is 1. Finally, bearing in mind (57), which can +be recast as pKY0|Y0q ă 0, we arrive at N 0 +n “ 1. +Lemma 5.10 Suppose (9). The generalized eigenproblem (59) does not admit negative eigenvalues. +In particular, N ´ +n “ 0. +48 + +Proof. +Let α ă 0, X “ pq, p, φ, πq and ˜X “ p˜q, ˜p, ˜φ, ˜πq satisfy +´1 +2∆xq ` k ¨ ∇xp “ α˜q, +´1 +2∆xp ´ k ¨ ∇xq ´ γσ1 ‹ +ˆ +σ2π dz “ α˜p, +´2c2∆zφ “ α˜φ, +´1 +2∆zπ ´ γσ2σ1 ‹ p “ α˜π, +(64) +where +q “ ´1 +2∆x˜q ` k ¨ ∇x˜p ` γσ1 ‹ +ˆ +p´∆zq´1{2σ2 ˜φ dz, +p “ ´1 +2∆x˜p ´ k ¨ ∇x˜q, +φ “ 1 +2 +˜φ ` γp´∆zq´1{2σ2σ1 ‹ ˜q, +π “ 2c2˜π. +(65) +This leads to solve an elliptic equation for π +´|α| +c2 ´ ∆z +¯ +π “ 2γσ2σ1 ‹ p. +In other words, we get, by means of Fourier transform +pπpx, ξq “ 2γσ1 ‹ ppxq ˆ +x +σ2pξq +|ξ|2 ` |α|{c2 . +On the same token, we obtain +´|α| +c2 ´ ∆z +¯ +˜φ “ ´2γp´∆zq1{2σ2σ1 ‹ ˜q, +which yields +p˜φpx, ξq “ ´2γσ1 ‹ ˜qpxq ˆ +|ξ|x +σ2pξq +|ξ|2 ` |α|{c2 . +For λ ą 0, we introduce the symbol +0 ď κλ “ +ˆ |x +σ2pξq|2 +|ξ|2 ` λ ď κ. +It turns out that +´1 +2∆xq ` k ¨ ∇xp “ α˜q, +´1 +2∆xp ´ k ¨ ∇xq ´ 2γ2κ|α|{c2Σ ‹ p “ α˜p, +with +q “ ´1 +2∆x˜q ` k ¨ ∇x˜p ´ 2γ2κ|α|{c2Σ ‹ ˜q, +p “ ´1 +2∆x˜p ´ k ¨ ∇x˜q. +49 + +For the Fourier coefficients, it casts as +m2 +2 qm ` ik ¨ mpm “ α˜qm, +m2 +2 pm ´ ik ¨ mqm ´ 2γ2κ|α|{c2p2πq2d|σ1,m|2pm “ α˜pm, +with +qm “ m2 +2 ˜qm ` ik ¨ m˜pm ´ 2γ2κ|α|{c2p2πq2d|σ1,m|2˜qm, +pm “ m2 +2 ˜pm ´ ik ¨ m˜qm. +We are going to see that these equations do not have non trivial solutions with α ă 0: +• If m “ 0, we get p0 “ 0, ˜q0 “ 0, and, consequently, ˜p0 “ 0, q0 “ 0. Hence, for α ă 0, we +cannot find an eigenvector with a non trivial 0-mode. +• If m � 0 and σ1,m “ 0, we see that pqm, pmq and p˜qm, ˜pmq are related by +Mm +ˆqm +pm +˙ +“ α +ˆ˜qm +˜pm +˙ +, +ˆqm +pm +˙ +“ Mm +ˆ˜qm +˜pm +˙ +. +(66) +It means that α is an eigenvalue of +M2 +m “ +˜ +m4 +4 ` pk ¨ mq2 +im2k ¨ m +´im2k ¨ m +m4 +4 ` pk ¨ mq2 +¸ +. +The roots of the characteristic polynomial of M2 +m are pm2 +2 ˘ k ¨ mq2 ě 0, which contradicts +the assumption α ă 0. +• For the case where m � 0 and σ1,m � 0, we introduce the shorthand notation am “ +2γ2p2πq2d|σ1,m|2κ|α|{c2, bearing in mind that 0 ă am ă m2 +2 by virtue of the smallness condi- +tion (9). We are led to the systems +ˆ +Mm ´ +ˆ +0 +0 +0 +am +˙˙ ˆ +qm +pm +˙ +“ α +ˆ +˜qm +˜pm +˙ +, +ˆ +qm +pm +˙ +“ +ˆ +Mm ´ +ˆ +am +0 +0 +0 +˙˙ ˆ +˜qm +˜pm +˙ +, +which imply that α is an eigenvalue of the matrix +ˆ +Mm ´ +ˆ +0 +0 +0 +am +˙˙ ˆ +Mm ´ +ˆ +am +0 +0 +0 +˙˙ +. +However the eigenvalues of this matrix read +` b +m2 +2 pm2 +2 ´ amq ˘ pk ¨ mq2˘2 ě 0, contradicting +that α is negative. +Lemma 5.11 Suppose (9). N ` +n “ #tm P Zd ∖ t0u, σ1,m “ 0, and m4 ´ 4pk ¨ mq2 ă 0u. +50 + +Proof. +We should consider the system of equations (64)-(65), now with α ą 0. For Fourier +coefficients, it casts as +m2 +2 qm ` ik ¨ mpm “ α˜qm, +m2 +2 pm ´ ik ¨ mqm ´ γp2πqdσ1,m +ˆ +σ2πm dz “ α˜pm, +´2c2∆zφm “ α˜φm, +´1 +2∆zπm ´ γp2πqdσ1,mσ2pm “ α˜πm, +where +qm “ m2 +2 ˜qm ` ik ¨ m˜pm ` γp2πqdσ1,m +ˆ +p´∆zq´1{2σ2 ˜φm dz, +pm “ m2 +2 ˜pm ´ ik ¨ m˜qm, +φm “ 1 +2 +˜φm ` γp2πqdp´∆zq´1{2σ2σ1,m˜qm, +πm “ 2c2˜πm. +• For m “ 0, we obtain p0 “ 0, ˜q0 “ 0. Hence π0 satisfies p´α{c2 ´ ∆zqπ0 “ 0. Here, `α{c2 +lies in the essential spectrum of ´∆z and the only solution in L2 of this equation is π0 “ 0. +In turn, this implies ˜p0 “ 0, p´α{c2 ´ ∆zqφ0 “ 0, and thus φ0 “ 0, q0 “ 0. Hence, for α ą 0, +we cannot find an eigenvector with a non trivial 0-mode. +• When m � 0 and σ1,m “ 0, we are led to p´α{c2 ´ ∆qφm “ 0, p´α{c2 ´ ∆qπm “ 0 that +imply φm “ 0, πm “ 0. +In turn, we get (66) for qm, pm, ˜qm, ˜pm. +This holds iff α is an +eigenvalue of M2 +m. If m4 � 4pk ¨ mq2, we find two eigenvalues αm,˘ “ pm2 +2 ˘ k ¨ mq2 ą 0, with +associated eigenvectors Xm,˘ “ peim¨x, ¯ieim¨x, 0, 0q, respectively. To decide whether these +modes should be counted, we need to evaluate the sign of pL ´1Xm,˘|Xm,˘q. We start by +solving L X1 +m,˘ “ Xm,˘. It yields +φ1 +m,˘ +2 +“ 0, 2c2π1 +m,˘ “ 0 and +Mm +ˆq1 +m,˘ +p1 +m,˘ +˙ +“ +ˆ 1 +¯i +˙ +. +We obtain +q1 +m,˘ “ +2 +m2 ˘ 2k ¨ m, +π1 +m,˘ “ +¯2i +m2 ˘ 2k ¨ m, +so that +pL ´1Xm,˘|Xm,˘q +“ +2 +m2 ˘ 2k ¨ m +ˆˆ +Td eim¨xe´im¨x dx ` +ˆ +Tdp¯iqeim¨x˘ie´im¨x dx +˙ +“ +4p2πqd +m2 ˘ 2k ¨ m, +the sign of which is determined by the sign of m2 ˘2k¨m. We count only the situation where +these quantities are negative; reproducing a discussion made in the proof of Proposition 5.3, +we conclude that +N ` +n ě #tm P Zd ∖ t0u, σ1,m “ 0 and m4 ´ 4pk ¨ mq2 ă 0u. +51 + +When m4 ´ 4pk ¨ mq2 “ 0, the eigenvalues of M2 +n are 0 and m4, and we just have to consider +the positive eigenvalue α “ m4, associated to the eigenvector Xm “ peim¨x, ˘ieim¨x, 0, 0q +(depending whether m2 +2 +“ ¯k ¨ m). The equation L Ym “ Xm +has infinitely many solu- +tions of the form p2{m2eim¨x, 0, 0, 0q ` zp˘ieim¨x, eim¨x, 0, 0q, with z P C. We deduce that +pL ´1Xm|Xmq “ 2p2πqd +m2 +ą 0. Thus these modes do not affect the counting. +• When m � 0 and σ1,m � 0, we are led to the relations p´α{c2 ´ ∆zqπm “ 2σ2γp2πqdσ1,mpm, +p´α{c2 ´∆zq˜φm “ ´2p´∆zq1{2σ2γp2πqdσ1,m˜qm. The only solutions with square integrability +on Rn are πm “ 0, ˜φm “ 0, pm “ 0, ˜qm “ 0. This can be seen by means of Fourier transform: +p´α{c2 ´ ∆zqφ “ σ amounts to pφpξq “ +pσpξq +|ξ|2´α{c2; due to (H4) this function has a singularity +which cannot be square-integrable. In turn, this equally implies φm “ 0 and ˜πm “ 0. Hence, +we arrive at m2 +2 qm “ 0 and ´ik ¨ mqm “ α˜pm, together with qm “ ik ¨ m˜pm and m2 +2 ˜pm “ 0. +We conclude that α ą 0 cannot be an eigenvalue associated to a m-mode such that m � 0 +and σ1,m � 0. +We can now make use of Theorem 5.7, together with Proposition 5.3. This leads to +0 ` 1 ` #tm P Zd ∖ t0u, σ1,m “ 0, and m4 ´ 4pk ¨ mq2 ă 0u ` NC` “ N ´ +n ` N 0 +n ` N ` +n ` NC` +“ npL q “ 1 ` #tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ă 0 and σ1,m “ 0u +`#tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ď 0 and σ1,m � 0u +so that +NC` “ #tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ď 0 and σ1,m � 0u. +Since the eigenvalue problem (59) does not have negative (real) eigenvalues, this is the only source +of instabilities. +As a matter of fact, when k “ 0, we obtain NC` “ 0, which yields the following statement, +(hopefully!) consistent with Lemma 4.1 and Proposition 4.2. +Corollary 5.12 Let k “ 0 and ω ą 0 such that the dispersion relation (12) is satisfied. Suppose +(9) holds. Then the plane wave solution peiωt1pxq, ´γΓpzq +@ +σ +D +Td, 0q is spectrally stable. +In contrast to what happens for the Hartree equation, for which the eigenvalues are purely +imaginary, see Lemma 3.2, we can find unstable modes, despite the smallness condition (9). Let us +consider the following two examples in dimension d “ 1, with k P Z ∖ t0u. +Example 5.13 Suppose σ1,0 � 0, and σ1,1 � 0. +Then, the set tm P Z ∖ t0u, m4 ´ 4k2m2 ď +0 and σ1,m � 0u contains t´1, `1u (since 4k2 ě 1). Let k P Z ∖ t0u and ω ą 0 such that the +dispersion relation (12) is satisfied. Then the plane wave solution peiωteikx, ´γΓpzq +@ +σ1 +D +Td, 0q is +spectrally unstable. +Example 5.14 Let m˚ P Z ∖ t0u be the first Fourier mode such that σ1,m˚ � 0. Let k P Z and +ω ą 0 such that the dispersion relation (12) is satisfied. Then, for all k P Z such that 4k2 ă m2 +˚, +the plane wave solution peiωteikx, ´γΓpzq +@ +σ +D +Td, 0q is spectrally stable, while for all k P Z such that +4k2 ě m2 +˚, the plane wave solution peiωteikx, ´γΓpzq +@ +σ1 +D +Td, 0q is spectrally unstable. +52 + +In general, if k P Zd ∖ t0u, the set tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ď 0 and σ1,m � 0u contains ´k +and k provided σ1,k � 0. Hence, we have the following result. +Corollary 5.15 Let k P Zd ∖ t0u and ω ą 0 such that the dispersion relation (12) is satis- +fied. +Suppose (9) holds and σ1,m � 0 for all m P Zd ∖ t0u. +Then the plane wave solution +peipωt`k¨xq, ´γΓpzq +@ +σ1 +D +Td, 0q is spectrally unstable. +Remark 5.16 (Orbital instability) Given Corollary 5.15, it is natural to ask whether in this +case the plane wave solution peipωt`k¨xq, ´γΓpzq +@ +σ1 +D +Td, 0q is orbitally unstable. +Note that, if σ1,m � 0 for all m P Zd ∖ t0u, we deduce from Proposition 5.3 that npLq ě +3. +As a consequence, the arguments used in [22] to prove the orbital instability (see also [38, +41]) do not apply. It seems then necessary to work directly with the propagator generated by the +linearized operator as in [23, 16]. In particular, one has to establish Strichartz type estimates for +the propagator of L (a task we leave for future work). +A +Scaling of the model and physical interpretation +It is worthwhile to discuss the meaning of the parameters that govern the equations and the +asymptotic issues. Going back to physical units, the system reads +ˆ +iℏBtU ` ℏ2 +2m∆xU +˙ +pt, xq “ +ˆˆ +TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dy dz +˙ +upt, xq, +(67a) +pB2 +ttΨ ´ κ2∆zΨqpt, x, zq “ ´σ2pzq +ˆˆ +Td σ1px ´ yq|Upt, yq|2 dy +˙ +. +(67b) +The quantum particle is described by the wave function pt, xq ÞÑ Upt, xq: given Ω Ă Td, the integral +´ +Ω |Upt, xq|2 dx gives the probability of presence of the quantum particle at time t in the domain +Ω; this is a dimensionless quantity. In (67a), ℏ stands for the Planck constant; its homogeneity +is MassˆLength2 +Time +(and its value is 1.055 ˆ 10´34 Js) and m is the inertial mass of the particle. Let +us introduce mass, length and time units of observations: M, L and T. It helps the intuition to +think of the z directions as homogeneous to a length, but in fact this is not necessarily the case: +we denote by Ψ and Z the (unspecified) units for Ψ and the zj’s. Hence, κ is homogeneous to the +ratio Z +T. The coupling between the vibrational field and the particle is driven by the product of +the form functions σ1σ2, which has the same homogeneity as +ℏ +TΨLdZn from (67a) and as +Ψ +LdT2 from +(67b), both are thus measured with the same units. From now on, we denote by ς this coupling +unit. Therefore, we are led to the following dimensionless quantities +U1pt1, x1q “ Upt1T, x1Lq +c +Ld m +M, +Ψ1pt1, x1, z1q “ 1 +ΨΨpt1T, x1L, z1Zq, +σ1 +1px1qσ2pz1q “ 1 +ς σ1px1Lqσ2pz1Zq. +Bearing in mind that u is a probability density, we note that +ˆ +Td |U1pt1, x1q|2 dx1 “ m +M. +53 + +Dropping the primes, (67a)-(67b) becomes, in dimensionless form, +ˆ +iBtU ` ℏT +mL2 +1 +2∆xU +˙ +pt, xq “ ςΨLdZnT +ℏ +ˆˆ +TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dy dz +˙ +Upt, xq, +(68a) +´ +B2 +ttΨ ´ κ2T2 +Z2 ∆zΨ +¯ +pt, x, zq “ ´ςT2 +Ψ +M +mσ2pzq +ˆˆ +Td σ1px ´ yq|Upt, yq|2 dy +˙ +. +(68b) +Energy conservation plays a central role in the analysis of the system: the total energy is defined +by using the reference units and we obtain +E0 “ +´ ℏT +mL2 +¯2 1 +2 +ˆ +Td |∇xU|2 dx ` Ψ2LdZn +ML2 +1 +2 +¨ +TdˆRn +´ +|BtΨ|2 ` κ2T2 +Z2 |∇zΨ|2¯ +dz dx +`ς ΨLdZnT2 +mL2 +¨ +TdˆRn |U|2σ2σ1 ‹ Ψ dz dx, +with E0 dimensionless (hence the total energy of the original system is E0 ML2 +T2 ). Therefore, we see +that the dynamics is encoded by four independent parameters. In what follows, we get rid of a +parameter by assuming +ℏT +mL2 “ 1, +and we work with the following three independent parameters +α “ ςΨLdZnT2 +mL2 +mL2 +ℏT , +β “ ςZ2 +κ2Ψ +M +m, +c “ κT +Z . +It leads to +ˆ +iBtU ` 1 +2∆xU +˙ +pt, xq “ α +ˆˆ +TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dy dz +˙ +Upt, xq, +(69a) +´ 1 +c2 B2 +ttΨ ´ ∆zΨ +¯ +pt, x, zq “ ´βσ2pzq +ˆˆ +Td σ1px ´ yq|Upt, yq|2 dy +˙ +(69b) +together with +E0 “ 1 +2 +ˆ +Td |∇xU|2 dx ` 1 +2 +α +β +¨ +TdˆRn +´ 1 +c2 |BtΨ|2 ` |∇zΨ|2¯ +dz dx +`α +¨ +TdˆRn |U|2σ2σ1 ‹ Ψ dz dx. +This relation allows us to interpret the scaling parameters as weights in the energy balance. Now, +for notational convenience, we decide to work with a m +M +b +α +β Ψ instead of Ψ and +b +M +mU instead +of U; it leads to (3a)-(3c) and (8) with γ “ +b +M +m +?αβ. +Accordingly, we shall implicitly work +with solutions with amplitude of magnitude unity. +The regime where c Ñ 8, with α, β fixed +leads, at least formally, to the Hartree system (1a)-(1b); arguments are sketched in Appendix B. +The smallness condition (9) makes a threshold appear on the coefficients in order to guaranty the +stability: since it involves the product M +mαβ, it can be interpreted as a condition on the strength of +the coupling between the particle and the environment, and on the amplitude of the wave function. +We shall see in the proof that a sharper condition can be derived, expressed by means of the Fourier +coefficients of the form function σ1. +54 + +B +From Schödinger-Wave to Hartree +In this Section we wish to justify that solutions – hereafter denoted Uc – of (3a)-(3c) converge to the +solution of (1a)-(1b) as c Ñ 8. We adapt the ideas in [10] where this question is investigated for +Vlasov equations. Throughout this section we consider a sequence of initial data UInit +c +, ΨInit +c +, ΠInit +c +such that +sup +cą0 +ˆ +Td |UInit +c +|2 dx “ M0 ă 8, +(70a) +sup +cą0 +ˆ +Td |∇xUInit +c +|2 dx “ M1 ă 8, +(70b) +sup +cą0 +" 1 +2c2 +¨ +TdˆRn |ΠInit +c +|2 dz dx ` 1 +2 +¨ +TdˆRn |∇zΨInit +c +|2 dz dx +* +“ M2 ă 8, +(70c) +sup +cą0 +¨ +|UInit +c +|2σ1 ‹ σ2|ΨInit +c +| dz dx “ M3 ă 8. +(70d) +There are several direct consequences of these assumptions: +• The total energy is initially bounded uniformly with respect to c ą 0, +• In fact, we shall see that the last assumption can be deduced from the previous ones. +• Since the L2 norm of Uc is conserved by the equation, we already know that +Uc is bounded in L8p0, 8; L2pTdqq. +Next, we reformulate the expression of the potential, separating the contribution due to the +initial data of the wave equation and the self-consistent part. By using the linearity of the wave +equation, we can split +Φc “ ΦInit,c ` ΦCou,c +where ΦInit,c is defined from the free-wave equation on Rn and initial data ΨInit +c +, ΠInit +c +: +1 +c2 B2 +ttΥc ´ ∆zΨ “ 0, +pΥc, BtΥcq +ˇˇ +t“0 “ pΨInit +c +, ΠInit +c +q. +(71) +Namely, we set +ΦInit,cpt, xq +“ +ˆ +Rn σ2pzqσ1 ‹ Υcpt, x, zq dz +“ +ˆ +Rn +´ +cospc|ξ|tqσ1 ‹ pΨInit +c +px, |ξq ` sinpc|ξ|t +c|ξ| +σ1 ‹ pΨInit +c +px, |ξq +¯pσ2pξq dξ +p2πqn . +Accordingly rΨc “ Ψc ´ Υc satisfies +1 +c2 B2 +tt rΨc ´ ∆z rΨc´ “ ´γσ2σ1 ‹ |Uc|2, +prΨc, Bt rΨcq +ˇˇ +t“0 “ p0, 0q. +(72) +55 + +and we get +ΦCou,cpt, xq +“ +γ +ˆ +Rn σ2pzqσ1 ‹ rΨcpt, x, zq dz +“ +γ2c2 +ˆ t +0 +ˆ +Rn +sinpc|ξ|sq +c|ξ| +Σ ‹ |Uc|2pt ´ s, xq|pσ2pξq|2 +dξ +p2πqn ds +“ +γ2 +ˆ ct +0 +ˆˆ +Rn +sinpτ|ξ|q +|ξ| +|pσ2pξq|2 +dξ +p2πqn +˙ +looooooooooooooooooomooooooooooooooooooon +“ppτq +Σ ‹ |Uc|2pt ´ τ{c, xq dτ, +where it is known that the kernel p is integrable on r0, 8q [10, Lemma 14]. +Lemma B.1 There exists a constant Mw ą 0 such that +sup +c,t,x +|ΦInit,cpt, xq| ď Mw, +sup +c,t,x +|ΦCou,cpt, xq| ď Mw. +Proof. +Combining the Sobolev embedding theorem (mind the condition n ě 3) and the standard +energy conservation for the free linear wave equation, we obtain +}Υc}L8p0,8;L2pTd;L2n{pn´2qpRnqqq ď C}∇zΥc}L8p0,8;L2pTdˆRnqq ď C +a +2M2. +Applying Hölder’s inequality, we are thus led to: +|ΦInit,cpt, xq| ď C}σ2}L2n{pn`2qpRnq}σ1}L2pRdq +a +2M2, +(73) +which proves the first part of the claim. Incidentally, it also shows that (70d) is a consequence of +(70a) and (70c). Next, we get +|ΦCou,cpt, xq| ď γ}Σ}L8pTdq}Uc}L8pr0,8q,L2pTdqq +ˆ 8 +0 +|ppτq| dτ. +Corollary B.2 There exists a constant MS ą 0 such that +sup +c,t +}∇Ucpt, ¨q}L2pTdq ď MS. +Proof. +This is a consequence of the energy conservation (the total energy being bounded by +virtue of (70b)-(70d)) where the coupling term +ˆ +TdpΦInit,c ` ΦCou,cq|Uc|2 dx +can be dominated by 2MwM0. +Coming back to +BtUc “ ´ 1 +2i∆xUc ` γ +i pΦInit,c ` ΦCou,cqUc +(74) +56 + +we see that BtUc is bounded in L2p0, 8; H´1pTdqq. Combining the obtained estimates with Aubin- +Simon’s lemma [44, Corollary 4], we deduce that +Uc is relatively compact in in C0pr0, Ts; LppTdqq, 1 ď p ă +2d +d ´ 2, +for any 0 ă T ă 8. Therefore, possibly at the price of extracting a subsequence, we can suppose +that Uc converges strongly to U in C0pr0, Ts; L2pTdqq. It remains to pass to the limit in (74). The +difficulty consists in letting c go to 8 in the potential term and to justify the following claim. +Lemma B.3 For any ζ P C8 +c pp0, 8q ˆ Tdq, we have +lim +cÑ8 +ˆ 8 +0 +ˆ +TdpΦInit,c ` ΦCou,cqUcζ dx dt “ γκ +ˆ 8 +0 +ˆ +Td Σ ‹ |Uc|2 Ucζ dx dt. +Proof. +We expect that ΦCou,c converges to γκΣ ‹ |U|2: +ˇˇΦCou,cpt, xq ´ γκΣ ‹ |U|2pt, xq +ˇˇ +“ γ +ˇˇˇˇ +ˆ ct +0 +Σ ‹ |Uc|2pt ´ τ{c, xqppτq dτ ´ κΣ ‹ |U|2pt, xq +ˇˇˇˇ +ď γ +ˆ ct +0 +ˇˇˇΣ ‹ |Uc|2pt ´ τ{c, xq ´ Σ ‹ |U|2pt, xq +ˇˇˇ |ppτq| dτ ` γ +ˆ 8 +ct +|ppτq| dτ ˆ }Σ ‹ |U|2}L8pp0,8qˆTdq +ď γ +ˆ ct +0 +Σ ‹ +ˇˇ|Uc|2 ´ |U|2ˇˇpt ´ τ{c, xq |ppτq| dτ +`γ +ˆ ct +0 +Σ ‹ +ˇˇ|U|2pt ´ τ{c, xq ´ |U|2pt, xq +ˇˇ |ppτq| dτ +`γ +ˆ 8 +ct +|ppτq| dτ }Σ}L8pTdq}U}L8pp0,8q;L2pTdqq. +Let us denote by Icpt, xq, IIcpt, xq, IIIcptq, the three terms of the right hand side. Since p P L1pr0, 8qq, +for any t ą 0, IIIcptq tends to 0 as c Ñ 8, and it is dominated by }p}L1pr0,8q}Σ}L8pTdqM0. Next, +we have +|Icpt, xq| +ď +}p}L1pr0,8q}Σ}L8pTdq sup +sě0 +ˆ +Td +ˇˇ|Uc|2 ´ |U|2ˇˇps, yq dy +ď +}p}L1pr0,8q}Σ}L8pTdq sup +sě0 +ˆˆ +Td |Uc ´ U|2ps, yq dy ` 2Re +ˆ +TdpUc ´ UqUps, yq dy +˙ +which also goes to 0 as c Ñ 8 and is dominated by 2M0}p}L1pr0,8qq}Σ}L8pTdq. Eventually, we get +|IIcpt, xq| ď }Σ}L8pTdq +ˆ ct +0 +ˆˆ +Td +ˇˇ|U|2pt ´ τ{c, yq ´ |U|2pt, yq +ˇˇ dy +˙ +|ppτq| dτ. +Since U P C0pr0, 8q; L2pTdqq, with }Upt, ¨q}L2pTdq ď M0, we can apply the Lebesgue theorem to +show that IIcpt, xq tends to 0 for any pt, xq fixed, and it is dominated by 2M0}p}L1pr0,8qq}Σ}L8pTdq. +This allows us to pass to the limit in +ˆ 8 +0 +ˆ +Td ΦCou,cUcζ dx dt ´ κ +ˆ 8 +0 +ˆ +Td Σ ‹ |U|2Uζ dx dt +“ +ˆ 8 +0 +ˆ +Td ΦCou,cpUc ´ Uqζ dx dt ` +ˆ 8 +0 +ˆ +Td +´ +ΦCou,c ´ γκΣ ‹ |U|2¯ +Uζ dx dt. +57 + +It remains to justify that +lim +cÑ8 +ˆ 8 +0 +ˆ +Td Φinit,cUcζ dx dt “ 0. +The space variable x is just a parameter for the free wave equation (71), which is equally satisfied +by σ1 ‹ Υc, with initial data σ1 ‹ pΨInit +c +, ΠInit +c +q. We appeal to the Strichartz estimate for the wave +equation, see [26, Corollary 1.3] or [45, Theorem 4.2, for the case n “ 3],which yields +c1{p +˜ˆ 8 +0 +ˆˆ +Rn |σ1 ‹ Υcpt, x, yq|q dy +˙p{q +dt +¸1{p +ď C +ˆ 1 +c2 +ˆ +Rn |σ1 ‹ ΠInit +c +px, zq|2 dz ` +ˆ +Rn |σ1 ‹ ∇yΨInit +c +px, zq|2 dz +˙1{2 +, +for any admissible pair: +2 ď p ď q ď 8, +1 +p ` n +q “ n +2 ´ 1, +2 +p ` n ´ 1 +q +ď n ´ 1 +2 +, +pp, q, nq � p2, 8, 3q. +The L2 norm with respect to the space variable of the right hand side is dominated by +b +}σ1}L1pTdq M2. +It follows that +ˆ +Td +˜ˆ 8 +0 +ˆˆ +Rn |σ1 ‹ Υcpt, x, zq|q dz +˙p{q +dt +¸2{p +dx ď C2}σ1}L1pRdq M2 +1 +c2{p ÝÝÝÑ +cÑ8 0. +Repeated use of the Hölder inequality (with 1{p ` 1{p1 “ 1) leads to +ˇˇˇˇ +ˆ 8 +0 +ˆ +Td UcζΦInit,c dx dt +ˇˇˇˇ +ď +˜ˆ +Td +ˆˆ 8 +0 +|Ucζpt, xq|p1 dt +˙2{p1 +dx +¸1{2 ˜ˆ +Td +ˆˆ 8 +0 +|ΦInit,cpt, xq|p dt +˙2{p +dx +¸1{2 +. +On the one hand, assuming that ζ is supported in r0, Rs ˆ Td and p ą 2, we have +ˆ +Td +ˆˆ 8 +0 +|Ucζ|p1 dt +˙2{p1 +dx +ď +ˆ +Td +ˆˆ R +0 +|Uc|2 dt +˙ ˆˆ R +0 +|ζ|2p1{p2´p1q dt +˙p2´p1q{p1 +dx +ď +R1`p2´p1q{p1}ζ}L8pp0,8qˆTdq}Uc}L8pp0,8q;L2pTdqq +which is thus bounded uniformly with respect to c ą 0. On the other hand, we get +ˆ +Td +ˆˆ 8 +0 +|ΦInit,cpt, xq|p dt +˙2{p +dx “ +ˆ +Td +ˆˆ 8 +0 +ˇˇˇ +ˆ +Rn σ2pzqσ1 ‹ Υcpt, x, zq dz +ˇˇˇ +p +dt +˙2{p +dx +ď }σ2}Lq1pRnq +ˆ +Td +ˆˆ 8 +0 +ˇˇˇ +ˆ +Rn |σ1 ‹ Υcpt, x, zq|q dz +ˇˇˇ +p{q +dt +˙2{p +dx +which is of the order Opc´2{pq. +58 + +C +Well-posedness of the Schrödinger-Wave system +The well-posedness of the Schrödinger-Wave system is justified by means of a fixed point argument. +The method described here works as well for the problem set on Rd, and it is simpler than the +approach in [21] since it avoids the use of “dual” Strichartz estimates for the Schrödinger and the +wave equations. +We define a mapping that associates to a function pt, xq P r0, Ts ˆ Td ÞÑ V pt, xq P C: +• first, the solution Ψ of the linear wave equation +1 +c2 B2 +ttΨ ´ ∆zΨ “ ´σ2σ1 ‹ |V |2, +pΨ, BtΨq +ˇˇ +t“0 “ pΨ0, Ψ1q; +• next, the potential Φ “ σ1 ‹ +´ +Rn σ2Ψ dz; +• and finally the solution of the linear Schrödinger equation +iBtU ` 1 +2∆xU “ γΦU, +U +ˇˇ +t“0 “ UInit. +These successive steps define a mapping S : V ÞÝÑ U and we wish to show that this mapping +admits a fixed point in C0pr0, Ts; L2pTdqq, which, in turn, provides a solution to the non linear +system (3a)-(3c). In this discussion, the initial data UInit, Ψ0, Ψ1 are fixed once for all in the space +of finite energy: +UInit P H1pTdq, +Ψ0 P L2pTd; .H1pRnqq, +Ψ1 P L2pTd ˆ Rnq. +We observe that +d +dt +ˆ +Td |U|2 dx “ 0. +Hence, the mapping S applies the ball Bp0, }UInit}L2pTdqq of C0pr0, Ts; L2pTdqq in itself; we thus +consider U “ SpV q for V P C0pr0, Ts; L2pTdqq such that }V pt, ¨q}L2pTdq ď }UInit}L2pTdq. Moreover, +we can split +Ψ “ Υ ` rΨ +with Υ solution of the free wave equation +1 +c2 B2 +ttΥ ´ ∆zΥ “ 0, +pΥ, BtΥq +ˇˇ +t“0 “ pΨ0, Ψ1q, +and +1 +c2 B2 +tt rΨ ´ ∆z rΨ “ 0, +pΥ, Bt rΨq +ˇˇ +t“0 “ 0. +We write Φ “ ΦI ` rΦ for the associated splitting of the potential. In particular, the standard +energy conservation for the wave equation tells us that +1 +2c2 +¨ +TdˆRn |BtΥ|2 dz dx ` 1 +2 +¨ +TdˆRn |∇zΥ|2 dz dx +“ +1 +2c2 +¨ +TdˆRn |Ψ1|2 dz dx ` 1 +2 +¨ +TdˆRn |∇zΨ0|2 dz dx “ M2 +59 + +holds. It follows that +|ΦIpt, xq| ď C}σ2}L2n{pn`2pRnq}σ1}L2pTdq +a +2M2 +by using Sobolev’s embedding. Next, we obtain +rΦpt, xq +“ +ˆ +Rn σ2pzqσ1 ‹ rΨpt, x, zq dz +“ +γ +ˆ ct +0 +ˆˆ +Rn +sinpτ|ξ|q +|ξ| +|pσ2pξq|2 +dξ +p2πqn +˙ +looooooooooooooooooomooooooooooooooooooon +“ppτq +Σ ‹ |V |2pt ´ τ{c, xq dτ, +which thus satisfies +sup +xPTd |rΦpt, xq| ď γ}Σ}L8pTdq +ˆ ct +0 +|ppτq| +ˆˆ +Td |V |2pt ´ τ{c, yq dy +˙ +dτ. +In particular +|rΦpt, xq| ď γ}Σ}L8pTdq}p}L1pp0,8qq}V }C0pr0,Ts;L2pTdqq ď γ}Σ}L8pTdq}p}L1pp0,8qq}UInit}L2pTdq +lies in L8pp0, Tq ˆ Tdq, and thus Φ P L8pp0, Tq ˆ Rdq. This observation guarantees that U “ ;SpV q +is well-defined. +Thus, let us pick V1, V2 in this ball of C0pr0, Ts; L2pTdqq and consider Uj “ SpVjq. We have +iBtpU2 ´ U1q ` 1 +2∆xpU2 ´ U1q “ γΦ2pU2 ´ U1q ` γpΦ2 ´ Φ1qU1, +pU2 ´ U1q +ˇˇ +t“0 “ 0. +It follows that +d +dt +ˆ +Td |U2 ´ U1|2 dx “ 2γIm +ˆˆ +TdpΦ2 ´ Φ1qU 1pU2 ´ U1q dx +˙ +ď 2γ}U1}L2pTdq }U2 ´ U1}L2pTdq }Φ2 ´ Φ1}L8pTdq “ 2γ}U1}L2pTdq }U2 ´ U1}L2pTdq }rΦ2 ´ rΦ1}L8pTdq +ď 2γ2}Σ}L8pTdq}UInit}L2pTdq }U2 ´ U1}L2pTdq +ˆ ct +0 +|ppτq| +ˆˆ +Td +ˇˇ|V2|2 ´ |V1|2ˇˇpt ´ τ{c, yq dy +˙ +dτ. +We use the elementary estimate +ˆ +Td +ˇˇ|V2|2´|V1|2ˇˇ dy “ +ˆ +Td +ˇˇ|V2´V1|2`2RepV2´V1qV1 +ˇˇ dy ď }V2´V1}2 +L2pTdq`2}V2´V1}L2pTdq }V1}L2pTdq. +Combining this with Cauchy-Schwarz and Young inequalities, we arrive at +d +dt +ˆ +Td |U2 ´ U1|2 dx +ď 2γ2}Σ}L8pTdq}UInit}L2pTdq +ˆ +2}UInit}L2pTdq +ˆ ct +0 +|ppτq|}V2 ´ V1}2pt ´ τ{cqL2pTdq dτ +`}U2 ´ U1}L2pTdq2}UInit}L2pTdq +ˆ ct +0 +|ppτq|}V2 ´ V1}pt ´ τ{cqL2pTdq dτ +˙ +ď 2γ2}Σ}L8pTdq}UInit}2 +L2pTdq +´ +}U2 ´ U1}2 +L2pTdq +`p2 ` }p}L1pp0.8qq +ˆ ct +0 +|ppτq|}V2 ´ V1}2pt ´ τ{cqL2pTdq dτ +˙ +. +60 + +Set L “ 2γ2}Σ}L8pTdq}UInit}2 +L2pTdq. We deduce that +}U2 ´ U1}ptq2 +L2pTdq ď p2 ` }p}L1pp0.8qqL +ˆ t +0 +eLpt´sq +ˆ cs +0 +|ppτq|}V2 ´ V1}2ps ´ τ{cqL2pTdq dτ ds. +We use this estimate for 0 ď t ď T ă 8 and we obtain +}U2 ´ U1}ptq2 +L2pTdq ď p4 ` }p}L1pp0.8qqLTeLT }p}L1pp0.8q sup +0ďsďT +}V2 ´ V1}2psqL2pTdq. +Hence for T small enough, S is a contraction in C0pr0, Ts; L2pTdqq, and consequently it admits a +unique fixed point. Since the fixed point still has its L2 norm equal to }UInit}L2pTdq, the solution +can be extended on the whole interval r0, 8q. The argument can be adapted to handle the Hartree +system. +References +[1] B. Aguer, S. De Bièvre, P. Lafitte, and P. E. Parris. Classical motion in force fields with short +range correlations. J. Stat. Phys., 138(4-5):780–814, 2010. +[2] V. Bach, J. Fröhlich, and I.M. Sigal. Return to equilibrium. J. Math. Phys., 41:3985–4060, +2000. +[3] S. De Bièvre, J. Faupin, and B. Schubnel. Spectral analysis of a model for quantum friction. +Rev. Math. Phys., 29:1750019, 2017. +[4] S. 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Eq., 2019:73: +1–12, 2019. +64 + diff --git a/29FAT4oBgHgl3EQflB1V/content/tmp_files/load_file.txt b/29FAT4oBgHgl3EQflB1V/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6db310b10b247a1ebb1ea81edc0af247fac2f6bf --- /dev/null +++ b/29FAT4oBgHgl3EQflB1V/content/tmp_files/load_file.txt @@ -0,0 +1,1679 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf,len=1678 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='08614v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='AP] 20 Jan 2023 Plane wave stability analysis of Hartree and quantum dissipative systems Thierry Goudon∗1 and Simona Rota Nodari†1 1Université Côte d’Azur, Inria, CNRS, LJAD, Parc Valrose, F-06108 Nice, France Abstract We investigate the stability of plane wave solutions of equations describing quantum particles interacting with a complex environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The models take the form of PDE systems with a non local (in space or in space and time) self-consistent potential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' such a coupling lead to challenging issues compared to the usual non linear Schrödinger equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The analysis relies on the identification of suitable Hamiltonian structures and Lyapounov functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We point out analogies and differences between the original model, involving a coupling with a wave equation, and its asymptotic counterpart obtained in the large wave speed regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, while the analogies provide interesting intuitions, our analysis shows that it is illusory to obtain results on the former based on a perturbative analysis from the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hartree equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Open quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Particles interacting with a vibrational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Schrödinger-Wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Plane wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Orbital stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 35Q40 35Q51 35Q55 1 Introduction This work is concerned with the stability analysis of certain solutions of the following Hartree-type equation iBtU ` 1 2∆xU “ γ ˆ σ1 ‹x ˆ Rn σ2Ψ dz ˙ U, (1a) ´ ∆zΨ “ ´γσ2pzq ` σ1 ‹x |U|2˘ pxq (1b) ∗thierry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='goudon@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='fr †simona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='rotanodari@univ-cotedazur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='fr 1 endowed with the initial condition U ˇˇ t“0 “ UInit, (2) and of the following Schrödinger-Wave system: iBtU ` 1 2∆xU “ γΦU, (3a) 1 c2 B2 ttΨ ´ ∆zΨ “ ´γσ2pzqσ1 ‹ |U|2pt, xq, (3b) Φpt, xq “ ¨ TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dz dy, (3c) where γ, c ą 0 are given positive parameters, completed with U ˇˇ t“0 “ UInit, Ψ ˇˇ t“0 “ ΨInit, BtΨ ˇˇ t“0 “ ΠInit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (4) The variable x lies in the torus Td, meaning that the equations are understood with p2πq´periodicity in all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In (3b), the additional variable z lies in Rn and, as explained below, it is crucial to assume n ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For reader’s convenience, the scaling of the equation is fully detailed in Appendix A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' for our purposes the God-given form functions σ1, σ2 are fixed once for all and the features of the coupling are embodied in the parameters γ, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The system (1a)-(1b) can be obtained, at least formally, from (3a)-(3c) by letting the parameter c run to `8, while γ is kept fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' By the way, system (1a)-(1b) can be cast in the more usual form iBtU ` 1 2∆xU “ ´γ2κ ` Σ ‹x |U|2˘ U, t P R, x P Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (5) where1 κ “ ˆ Rn σ2pzqp´∆zq´1σ2pzq dz “ ˆ Rn |pσ2pξq|2 |ξ|2 dξ p2πqn ą 0 and Σ “ σ1 ‹ σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (6) Letting now Σ resemble the delta-Dirac mass, the asymptotic leads to the standard cubic non linear Schrödinger equation iBtU ` 1 2∆xU “ ´γ2κ|U|2U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (7) in the focusing case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' These asymptotic connections can be expected to shed some light on the dynamics of (3a)-(3c) and to be helpful to guide the intuition about the behavior of the solutions, see [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The motivation for investigating these systems takes its roots in the general landscape of the analysis of “open systems”, describing the dynamics of particles driven by momentum and energy exchanges with a complex environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Such problems are modeled as Hamiltonian systems, and it is expected that the interaction mechanisms ultimately produce the dissipation of the particles’ energy, an idea which dates back to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Caldeira and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Leggett [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' These issues have been investigated for various classical and quantum couplings, and with many different mathematical viewpoints, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' [2, 3, 24, 25, 28, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The case in which the environment is described as a vibrational field, like in the definition of the potential by (3b)-(3c), is particularly appealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In 1The Fourier transform of an integrable function ϕ : Rn Ñ C is defined by pϕpξq “ ´ Rn ϕpzqe´iξ¨z dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 2 fact, (3a)-(3c) is a quantum version of a model introduced by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' De Bièvre and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Bruneau, dealing with a single classical particle [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Intuitively, the model of [6] can be thought of as if in each space position x P Rd there is a membrane oscillating in a direction z P Rn, transverse to the motion of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' When a particle hits a membrane, its kinetic energy activates vibrations and the energy is evacuated at infinity in the z´direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' These energy transfer mechanisms eventually act as a sort of friction force on the particle, an intuition rigorously justified in [6, Theorem 2 and Theorem 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We refer the reader to [1, 12, 13, 30, 46] for further theoretical and numerical insight about this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The model of [6] has been revisited by considering many interacting particles, which leads to Vlasov-type equations, still coupled to a wave equation for defining the potential [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Unexpectedly, asymptotic arguments indicate a connection with the attractive Vlasov-Poisson dynamic [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In turn, the particles-environment interaction can be interpreted in terms of Lan- dau damping [19, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The quantum version (3a)-(3c) of the De Bièvre-Bruneau model has been discussed in [21, 20], with a connection to the kinetic model by means of a semi-classical analysis inspired from [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that in (3a)-(3c), the vibrational field remains of classical nature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' a fully quantum framework is dealt with in [3] for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' A remarkable feature of these systems is the presence of conserved quantities, here inherited from the framework designed in [6] for a classical particle, and the study of these models brings out the critical role of the wave speed c ą 0 and the dimension n of the space for the wave equation (we can already notice that n ě 3 is necessary for (6) to be meaningful), see [6, 18, 19, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For the Schrödinger-Wave system (3a)-(3c) the energy HSWpU, Ψ, Πq “ 1 4 ˆ Td |∇U|2 dx ` 1 4 ¨ TdˆRn ˆΠ2 c2 ` |∇zΨ|2 ˙ dx dz ` γ 2 ˆ Td Φ|U|2 dx, (8) is conserved since we can readily check that d dtHSWpU, Ψ, BtΨq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Similarly, for the Hartree system (1a)-(1b), we get d dtHHapUq “ 0 where we have set HHapUq “ 1 4 ˆ Td |∇U|2 dx ´ γ2 κ 4 ˆ Td Σpx ´ yq|Upt, xq|2|Upt, yq|2 dy dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Furthermore, for both model, the L2 norm is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Of course, these conservation properties play a central role for the analysis of the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However, (1a)-(1b) has further fundamental properties which occur only for the asymptotic model: firstly, (1a)-(1b) is Galilean invariant, which means that, given a solution pt, xq ÞÑ upt, xq and for any p0 P Td, the function pt, xq ÞÑ upt, x ´ tp0qeipx´tp0{2q is a solution too;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' secondly, the momentum pptq “ Im ´ ¯upt, xq∇xupt, xq dx is conserved and, accordingly, the center of mass follows a straight line at constant speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' That these properties are not satisfied by the more complex system (3a)-(3c) makes its analysis more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, we point out that, in contrast to the usual nonlinear Schrödinger equation or Hartree-Newton system, where Σ is the Newtonian potential, the equations (1a)-(1b) or (3a)-(3c) do not fulfil a 3 scale invariance property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This also leads to specific mathematical difficulties: despite the possible regularity of Σ, many results and approaches of the Newton case do not extend to a general kernel, due to the lack of scale invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' When the problem is set on the whole space Rd, one is interested in the stability of solitary waves, which are solutions of the equation with the specific form upt, xq “ eiωtQpxq, and, for (3a)-(3c), ψpt, x, zq “ Ψpx, zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The details of the solitary wave are embodied into the Choquard equation, satisfied by the profile Q, [32, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It turns out that the Choquard equation have infinitely many solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' among these solutions, it is relevant to select the solitary wave which minimizes the energy functional under a mass constraint, [32, 37] and to study the orbital stability of this minimal energy state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This program has been investigated for (7) and (1a)-(1b) in the specific case where Σpxq “ 1 |x| in dimension d “ 3, by various approaches [8, 31, 33, 34, 39, 49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Quite surprisingly, the specific form of the potential plays a critical role in the analysis (either through explicit formula or through scale invariance properties), and dealing with a general convolution kernel, as smooth as it is, leads to new difficulties, that can be treated by a perturbative argument, see [27, 51] for the case of the Yukawa potential, and [21] for (1a)-(1b) and (3a)-(3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Here, we adopt a different viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We consider the case where the problem holds on the torus Td, and we are specifically interested in the stability of plane wave solutions of (3a)-(3c) and (1a)-(1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We refer the reader to [4, 5, 14, 40] for results on the nonlinear Schrödinger equation (7) in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The discussion on the stability of these plane wave solutions will make the following smallness condition 4γ2κ}σ1}2 L1 ă 1 (9) (assuming the plane wave has an amplitude unity) appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Despite its restriction to the periodic framework, the interest of this study is two-fold: on the one hand, it points out some difficulties specific to the coupling and provides useful hints for future works;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' on the other hand, it clarify the role of the parameters, by making stability conditions explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In Section 2, we clarify the positioning of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To this end, we further discuss some mathematical features of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We also introduce the main assumptions on the parameters that will be used throughout the paper and we provide an overview of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Section 3 is concerned with the stability analysis of the Hartree equation (1a)-(1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Section 4 deals with the Schrödinger-Wave system at the price of restricting to the case where the wave vector of the plane wave solution vanishes: k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For reasons explained in details below, the general case is much more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Section 5 justifies that in general the mode k � 0 is linearly unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, in Appendix A, we provide a physical interpretation of the parameters involved, and for the sake of completeness, in Appendices B and C, we discuss the well-posedness of the Schrödinger-Wave system (3a)-(3c) and its link with the Hartree equation (1a)-(1b) in the regime of large c’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 4 2 Set up of the framework 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 Plane wave solutions and dispersion relation For any k P Zd, we start by seeking solutions to (3a)-(3c) of the form Upt, xq “ Ukpt, xq :“ exp ` ipωt ` k ¨ xq ˘ , Ψpt, x, zq “ Ψ˚pzq, BtΨpt, x, zq “ Π˚pzq “ 0, (10) with ω ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that the L2 norm of Uk is p2πqd{2 and Ψ˚ actually does not depend on the time variable, nor on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since |Ukpt, xq| “ 1 is constant, the wave equation simplifies to 1 c2 B2 ttΨ ´ ∆zΨ “ ´γσ2pzq @ σ1 D Td, where @ ¨ D Td stands for the average over Td: @ f D Td “ ´ Td fpxq dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, z ÞÑ Ψ˚pzq is a solution to (3b) if Ψ˚pzq “ ´γΓpzq @ σ1 D Td, with Γ the solution of ´∆zΓpzq “ σ2pzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This auxiliary function Γ is thus defined by the convolution of σ2 with the elementary solution of the Laplace operator in dimension n, or equivalently by means of Fourier transform: Γpzq “ ˆ Rn Cn |z ´ z1|n´2 σ2pz1q dz1 “ F ´1 ξÑz ´pσ2pξq |ξ|2 ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (11) The corresponding potential (3c) is actually a constant which reads ´γ ¨ TdˆRn σ1px ´ yqσ2pzqΓpzq @ σ1 D Td dz dy “ ´κγ @ σ1 D2 Td with κ “ ˆ Rn σ2pzqΓpzq dz “ ˆ Rn |∇zΓpzq|2 dz ą 0 (we remind the reader that this formula coincides with (6) and makes sense only when n ě 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It remains to identify the condition on the coefficients so that Uk satisfies the Schrödinger equation (3a): this leads to the following dispersion relation ω ` k2 2 ´ Υ˚ “ 0, Υ˚ “ γ2κ @ σ1 D2 Td ą 0 (12) with k2 “ řd j“1 k2 j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We can compute explicitly the associated energy: HSWpUk, Ψ˚, Π˚q “ p2πqd 2 ˆk2 2 ´ γ2κ 2 @ σ1 D2 Td ˙ “ p2πqd 4 pk2 ´ Υ˚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Of course, among these solutions, the constant mode U0pt, xq “ eiωt1pxq has minimal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It turns out that the plane wave Ukpt, xq “ eiωteik¨x equally satisfies (1a)-(1b) provided the dispersion relation (12) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Incidentally, we can check that HHapUkq “ p2πqd 2 ˆk2 2 ´ γ2κ 2 @ Σ D Td ˙ “ p2πqd 4 pk2 ´ Υ˚q is made minimal when k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2 Hamiltonian structure and symmetries of the problem The conservation properties play a central role in the stability analysis, for instance in the reasonings that use concentration-compactness arguments [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Based on the conserved quantities, one can try to construct a Lyapounov functional, intended to evaluate how far a solution is from an equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then the stability analysis relies on the ability to prove a coercivity estimate on the variations of the Lyapounov functional, see [47, 49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This viewpoint can be further extended by identifying analogies with finite dimensional Hamiltonian systems with symmetries, which has permitted to set up a quite general framework [22, 23], revisited recently in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The strategy relies on the ability in exhibiting a Hamiltonian formulation of the problem BtX “ JBXH pXq, where the symplectic structure is given by the skew-symmetric operator J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence of Noether’s Theorem, this formulation encodes the conservation properties of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In partic- ular, it implies that t ÞÑ H pXptqq is a conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For the problem under consideration, as it will be detailed below, X is a vectorial unknown with components possibly depending on different variables (x P Td and z P Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This induces specific difficulties, in particular because the nature of the coupling is non local and delicate spectral issues arise related to the essential spectrum of the wave equation in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we can easily observe that the systems (1a)-(1b) and (3a)-(3c) are invariant under multiplications by a phase factor of U, the “Schödinger unknown”, and under translations in the x variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This leads to the conservation of the L2 norm of U and of the total momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However, the systems (1a)-(1b) and (3a)-(3c) cannot be handled by a direct applica- tion of the results in [4, 22, 23]: the basic assumptions are simply not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Nevertheless, our approach is strongly inspired from [4, 22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As we will see later, for the Hartree system, a decisive advantage comes from the conservation of the total momentum and the Galilean invariance of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For the Schrödinger-Wave problem, since the expression of the total momentum mixes up contribution from the “Schrödinger unknown” U and the “wave unknown” Ψ, the information on its conservation does not seem readily useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 2 In what follows, we find advantages in changing the unknown by writing Upt, xq “ eik¨xupt, xq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' in turn the Schrödinger equation iBtU ` 1 2∆U “ ΦU becomes iBtU ` 1 2∆u ´ k2 2 u ` ik ¨ ∇u “ Φu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Accordingly, the parameter k will appear in the definition the energy functional H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This explains a major difference between (1a)-(1b) and (3a)-(3c): for the former, a coercivity estimate can be obtained for the energy functional H , for the latter, when k � 0 there are terms which cannot be controlled easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This is reminiscent of the momentum conservation in (1a)-(1b) and the lack of Galilean invariance for (3a)-(3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The detailed analysis of the linearized operators sheds more light on the different behaviors of the systems (1a)-(1b) and (3a)-(3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 2For the problem set on Rd, it is still possible, in the spirit of results obtained in [14] for NLS, to justify that orbital stability holds on a finite time interval: the solution remains at a distance ǫ from the orbit of the ground state over time interval of order Op1{ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='ǫq, see [48, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='11 & Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The argument relies on the dispersive properties of the wave equation through Strichartz’ estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3 Outline of the main results Let us collect the assumptions on the form functions σ1 and σ2 that govern the coupling: (H1) σ1 : Td Ñ r0, 8q is C8 smooth, radially symmetric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' @ σ1 D Td � 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (H2) σ2 : Rn Ñ r0, 8q is C8 smooth, radially symmetric and compactly supported;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (H3) p´∆q´1{2σ2 P L2pRnq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (H4) for any ξ P Rn, pσ2pξq � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Assumptions (H1)-(H2) are natural in the framework introduced in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hypothesis (H3) can equivalently be rephrased as p´∆q´1σ2 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' it appears in many places of the analysis of such coupled systems and, at least, it makes the constant κ in (6) meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This constant is a component of the stability constraint (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hypothesis (H4) equally appeared in [6, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (W)] when discussing large time asymptotic issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Assumptions (H1)-(H4) are assumed throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Our results can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We assume (9) and consider k P Zd and ω ą 0 satisfying (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For the Hartree equation, the analysis is quite complete: the plane wave eipωt`k¨xq is spectrally stable (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' for any initial perturbation with zero mean, the solutions of the linearized Hartree equation are L2-bounded, uniformly over t ě 0 (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' the plane wave eipωt`k¨xq is orbitally stable (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For the Schrödinger-Wave system, only the case k “ 0 is fully addressed: the plane wave peiωt1pxq, ´γΓpzq @ σ D Td, 0q is spectrally stable (Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' for any initial perturbation of peiωt1pxq, ´γΓpzq @ σ D Td, 0q with zero mean, the solutions of the linearized Schrödinger-Wave system are L2-bounded, uniformly over t ě 0 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' the plane wave peiωt1pxq, ´γΓpzq @ σ D Td, 0q is orbitally stable (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' When k � 0, the situation is much more involved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' at least we prove that in general the plane wave solution peipωt`k¨xq, ´γΓpzq @ σ1 D Td, 0q is spectrally unstable, see Section 5 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 3 Stability analysis of the Hartree system (1a)-(1b) To study the stability of the plane wave solutions of the Hartree system, it is useful to write the solutions of (1a)-(1b) in the form Upt, xq “ eik¨xupt, xq with upt, xq solution to iBtu ` 1 2∆u ´ k2 2 u ` ik ¨ ∇u “ ´γ2κpΣ ‹ |u|2qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (13) 7 If k P Zd and ω ą 0 satisfy the dispersion relation (12), uωpt, xq “ eiωt1pxq is a solution to (13) with initial condition uωp0, tq “ 1pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore, studying the stability properties of Ukpt, xq “ eiωteik¨x as a solution to (1a)-(1b) amounts to studying the stability of uωpt, xq “ eiωt1pxq as a solution to (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The problem (13) has an Hamiltonian symplectic structure when considered on the real Banach space H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Rq ˆ H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, if we write u “ q ` ip, with p, q real-valued, we obtain Bt ˆ q p ˙ “ J∇pq,pqH pq, pq with J “ ˆ 0 1 ´1 0 ˙ and H pq, pq “ 1 2 ˆ1 2 ˆ Td |∇q|2 ` |∇p|2 dx ` k2 2 ˆ Tdpp2 ` q2q dx ´ ˆ Td pk ¨ ∇q dx ` ˆ Td qk ¨ ∇p dx ˙ ´ γ2κ 4 ˆ Td Σ ‹ pp2 ` q2qpp2 ` q2q dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Coming back to u “ q ` ip, we can write H puq “ 1 2 ˆ1 2 ˆ Td |∇u|2 dx ` k2 2 ˆ Td |upxq|2 dx ` ˆ Td k ¨ p´i∇uqu dx ˙ ´ γ2κ 4 ˆ TdpΣ ‹ |u|2qpxq|upxq|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (14) As observed above, H is a constant of the motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, it is clear that (13) is invariant under multiplications by a phase factor so that Fpuq “ 1 2}u}2 L2 is conserved by the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The quantities Gjpuq “ 1 2 ˆ Td ˆ1 i Bxju ˙ u dx are constants of the motion too, that correspond to the invariance under translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, a direct verification leads to d dtGjpuqptq “ κγ2 2 ˆ Td ˆ Td BxjΣpx ´ yq ‹ |u|2pt, yq|u|2pt, xq dy dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, we shall endow the Banach space H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Rq ˆ H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Rq with the inner product Bˆ q p ˙ ˇˇˇ ˆ q1 p1 ˙F “ ˆ Td ` pp1 ` qq1q dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' that can be also interpreted as an inner product for complex-valued functions: xu|u1y “ Re ˆ Td uu1 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (15) 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 Linearized problem and spectral stability Let us expand the solution of (13) around uω as upt, xq “ uωpt, xqp1 ` wpt, xqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The linearized equation for the fluctuation reads iBtw ` 1 2∆xw ` ik ¨ ∇xw “ ´2γ2κpΣ ‹ Repwqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (16) We split w “ q ` ip, q “ Repwq, p “ Impwq so that (16) recasts as Bt ˆq p ˙ “ Lk ˆq p ˙ (17) with the linear operator Lk : ˆ q p ˙ ÞÝÑ ¨ ˝ ´k ¨ ∇xq ´ 1 2∆xp 1 2∆xq ` 2γ2κΣ ‹ q ´ k ¨ ∇xp ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (18) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 (Spectral stability for the Hartree equation) Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose (9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then the spectrum of Lk, the lineariza- tion of (13) around the plane wave uωpt, xq “ eiωt1pxq, in L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq is contained in iR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Consequently, this wave is spectrally stable in L2pTdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1, we expand q, p and σ1 by means of their Fourier series qpt, xq “ ÿ mPZd Qmptqeim¨x, Qmptq “ 1 p2πqd ˆ Td qpt, xqe´im¨x dx, ppt, xq “ ÿ mPZd Pmptqeim¨x, Pmptq “ 1 p2πqd ˆ Td ppt, xqe´im¨x dx, σ1pxq “ ÿ mPZd σ1,meim¨x, σ1,mptq “ 1 p2πqd ˆ Td σ1pxqe´im¨x dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that σ1 being real and radially symmetric, we have σ1,m “ σ1,m “ σ1,´m (19) and, by definition, @ σ1 D Td “ p2πqdσ1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, we obtain Lk ˆq p ˙ “ ¨ ˚ ˚ ˝ ř mPZd ˆm2 2 Pm ´ ik ¨ mQm ˙ eim¨x ř mPZd ˆ ´m2 2 Qm ´ ik ¨ mPm ` 2p2πq2dγ2κ|σ1,m|2Qm ˙ eim¨x ˛ ‹‹‚ “ Lk,0 ˆQ0 P0 ˙ ` ÿ mPZd∖t0u Lk,m ˆQm Pm ˙ eik¨x (20) with Lk,0 “ ˆ 0 0 2p2πq2dγ2κ|σ1,0|2 0 ˙ and Lk,m “ ˜ ´ik ¨ m m2 2 ´ m2 2 ` 2p2πq2dγ2κ|σ1,m|2 ´ik ¨ m ¸ (21) for m P Zd ∖ t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 9 Note that, since the Fourier modes are uncoupled, ˆq p ˙ is a solution to (17) if and only if the Fourier coefficients ˆQm Pm ˙ satisfy Bt ˆ Qmptq Pmptq ˙ “ Lk,m ˆ Qmptq Pmptq ˙ for any m P Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Similarly, λ P C is an eigenvalue of the operator Lk if and only if there exists at least one Fourier mode m P Zd such that λ is an eigenvalue of the matrix Lk,m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' there exists pqm, pmq � p0, 0q such that λqm ´ m2 2 pm ` ik ¨ mqm “ 0, λpm ` m2 2 qm ` ik ¨ mpm “ 2p2πq2dγ2κ|σ1,m|2qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (22) A straightforward computation gives that λ0 “ 0 is the unique eigenvalue of the matrix Lk,0 with eigenvector p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This means that KerpLkq contains at least the vector subspace spanned by the constant function x P Td ÞÑ ˆ0 1 ˙ , which corresponds to the constant solution upt, xq “ i of (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, if m P Zd ∖ t0u, λm is an eigenvalue of Lk,m if it is a solution to pλ ` ik ¨ mq2 ´ m2 2 ˆ ´m2 2 ` 2p2πq2dγ2κ|σ1,m|2 ˙ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This is a second order polynomial equation for λ and the roots are given by λm,˘ “ ´ik ¨ m ˘ |m| 2 b ´m2 ` 4γ2κp2πq2d|σ1,m|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If the smallness condition (9) holds, the argument of the square root is negative for any m P Zd∖t0u, and thus the roots λ are all purely imaginary (and we note that λ´m,˘ “ λm,¯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' More precisely, we have the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2 (Spectral stability for the Hartree equation) Let k, m P Zd and Lk,m defined as in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' λ0 “ 0 is the unique eigenvalue of Lk,0 and KerpLk,0q “ span "ˆ 0 1 ˙* ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' for any m P Zd ∖ t0u, the eigenvalue of Lk,m are λm,˘ “ ´ik ¨ m ˘ |m| 2 b ´m2 ` 4γ2κp2πq2d|σ1,m|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (a) if 4γ2κp2πq2d |σ1,m|2 m2 ď 1, then λm,˘ P iR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (b) if 4γ2κp2πq2d |σ1,m|2 m2 ą 1, then λm,˘ P C ∖ iR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, Repλm,`q ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Now, (9) implies 4γ2κp2πq2d |σ1,m|2 m2 ă 1 for all m P Zd ∖ t0u, so that σpLkq Ă iR and uωpt, xq “ eiωt1pxq is spectrally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Conversely, if σ1, σ2 and γ are such that there exists m˚ P Zd ∖ t0u 10 verifying 4γ2κp2πq2d |σ1,m˚|2 m2 ˚ ą 1, then the plane wave uω is spectrally unstable for any k P Zd and ω ą 0 that satisfy the dispersion relation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This proves Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We observe that this result is consistent with the linear stability analysis of (7), see [40, The- orem 1], when replacing formally Σ by the delta-Dirac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The analogy should be considered with caution, though, since the functional difficulties are substantially different: here u ÞÑ ´ 1 2∆Tdu ´ 2γ2κΣ‹Repuq is a compact perturbation of ´ 1 2∆Td, which has a compact resolvent hence a spectral decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It is important to remark that the analysis of eigenproblems for Lk has consequences on the behavior of solutions to (17) of the particular form Qpt, xq “ eλtqpxq, Ppt, xq “ eλtppxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We warn the reader that spectral stability excludes the exponential growth of the solutions of the linearized problem when the smallness condition (9) holds, but a slower growth is still possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This can be seen by direct inspection for the mode m “ 0: we have BtQ0 “ 0, so that Q0ptq “ Q0p0q and BtP0 “ 2p2πq2dκ @ σ1 D2 TdQ0p0q which shows that the solution can grow linearly in time P0ptq “ P0p0q ` 2p2πq2dγ2κ @ σ1 D2 TdQ0p0qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In fact, excluding the mode m “ 0 suffices to guaranty the linearized stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3 (Linearized stability for the Hartree equation) Suppose (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let w be the solution of (16) associated to an initial data wInit P H1pTdq such that ´ Td wInit dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, there exists a constant C ą 0 such that suptě0 }wpt, ¨q}H1 ď C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that if ´ Td wInit dx “ 0 then the corresponding Fourier coefficients Q0p0q and P0p0q are equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, Q0ptq “ P0ptq “ 0 for all t ě 0, so that ´ Td wpt, xq dx “ 0 for all t ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The proof follows from energetic consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, we observe that, on the one hand, 1 2 d dt ˆ Td |∇w|2 dx “ ´γ2κ 2i ˆ Td Σ ‹ pw ` wq∆pw ´ wq dx, and, on the other hand, 1 2 d dt ˆ Td Σ ‹ pw ` wqpw ` wq dx “ ´ 1 2i ˆ Td Σ ‹ pw ` wq∆pw ´ wq dx ´ k ¨ ˆ Td ∇pw ` wqΣ ‹ pw ` wq dx, where we get rid of the last term in the right hand side by assuming k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This leads to the following energy conservation property d dt "1 2 ˆ Td |∇w|2 dx ´ γ2κ 2 ˆ Td Σ ‹ pw ` wqpw ` wq dx “ 0 which holds for k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We denote by E0 the energy of the initial data wInit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, we can simply estimate ˇˇˇˇ ˆ Td Σ ‹ pw ` wqpw ` wq dx ˇˇˇˇ ď }Σ ‹ pw ` wq}L2}w ` w}L2 ď }Σ}L1}w ` w}2 L2 ď 4}Σ}L1}w}2 L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 11 To conclude, we use the Poincaré-Wirtinger estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, since we have already remarked that the condition ´ Td wInit dx “ 0 implies ´ Td wpt, xq dx “ 0 for any t ě 0, we can write }wpt, ¨q}2 L2 “ ›››wpt, ¨q ´ 1 p2πqd ˆ Td wpt, yq dy ››› 2 L2 “ p2πqd ÿ mPZd∖t0u |cmpwpt, ¨qq|2 ď p2πqd ÿ mPZd∖t0u m2|cmpwpt, ¨qq|2 “ }∇wpt, ¨q}2 L2 for any t ě 0, where the cmpwpt, ¨qq’s are the Fourier coefficients of the function x P Td ÞÑ wpt, xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, for any solution with zero mean, we infer, for all t ě 0, 2E0 “ ˆ Td |∇w|2pt, xq dx´γ2κ ˆ Td Σ‹pw `wqpw `wqpt, xq dx ě p1´4γ2κ}Σ}L1q ˆ Td |∇wpt, xq|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, if (9) is satisfied, we obtain sup tě0 }wpt, ¨q}H1 ď 2 d E0 1 ´ 4γ2κ}Σ}L1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The stability estimate extends to the situation where k � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, from the solution w of (16), we set vpt, xq “ wpt, x ` tkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It satisfies iBtv ` 1 2∆xv “ ´2γ2κΣ ‹ Repvq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, repeating the previous argument, }vpt, ¨q}H1 “ }wpt, ¨q}H1 remains uniformly bounded on p0, 8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This step of the proof relies on the Galilean invari- ance of (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' it could have been used from the beginning, but it does not apply for the Schrödinger- Wave system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 The analysis applies mutadis mutandis to any equation of the form (1a), with the potential defined by a kernel Σ and a strength encoded by the constant γ2κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, the stability criterion is set on the quantity 4γ2κp2πqd |pΣm| m2 For instance, the elementary solution of pa2´∆xqΣ “ δx“0 with periodic boundary condition has its Fourier coefficients given by pΣm “ 1 p2πqdpa2`m2q ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Coming back to the physical variable, in the one-dimension case, the function Σ reads Σpxq “ e´a|x| 2a ` coshpaxq ape2aπ ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The linearized stability thus holds provided 4γ2κp2πq2d 1 a2`1 ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2 Orbital stability In this subsection, we wish to establish the orbital stability of the plane wave uωpt, xq “ eiωt1pxq as a solution to (13) for k P Zd and ω ą 0 that satisfy the dispersion relation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As pointed out before, (13) is invariant under multiplications by a phase factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This leads to define the corresponding orbit through upxq “ 1pxq by O1 “ teiθ, θ P Ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 12 Intuitively, orbital stability means that the solutions of (13) associated to initial data close enough to the constant function x P Td ÞÑ 1 “ 1pxq remain at a close distance to the set O1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Stability analysis then amounts to the construction of a suitable Lyapounov functional satisfying a coercivity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This functional should be a constant of the motion and be invariant under the action of the group that generates the orbit O1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, the construction of such a functional relies on the invariants of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, the plane wave has to be a critical point on the Lyapounov functional so that the coercivity can be deduced from the properties of its second variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The difficulty here is that, in general, the bilinear symmetric form defining the second variation of the Lyapounov function is not positive on the whole space: according to the strategy designed in [22], see also the review [47], it will be enough to prove the coercivity on an appropiate subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Here and below, we adopt the framework presented in [4] (see also [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Inspired by the strategy designed in [4, Section 8 & 9], we introduce, for any k P Zd and ω ą 0 satisfying the dispersion relation (12), the set Sω “ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' u P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq, Fpuq “ Fp1q “ p2πqd 2 “ p2πqd k2{2 ` ω 2γ2κ @ σ1 D2 Td ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Sω is therefore the level set of the solutions of (13), associated to the plane wave pt, xq ÞÑ uωpt, xq “ eiωt1pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we introduce the functional Lωpuq “ H puq ` ωFpuq ´ dÿ j“1 kjGjpuq, (23) which is conserved by the solutions of (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We have BuLωpuqpvq “ Re ˆ1 2 ˆ Tdp´∆uqv dx ` k2 2 ˆ Td uv dx ´γ2κ ¨ TdˆTd Σpx ´ yq|upyq|2upxqvpxq dy dx`ω ˆ Td uv dx ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a matter of fact, we observe that BuLωp1q “ 0 owing to the dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we get B2 uLωpuqpv, wq “ Re ˆ1 2 ˆ Tdp´∆ ` k2qwv dx ´2γ2κ ¨ TdˆTd Σpx ´ yqRe ` upyqwpyq ˘ upxqvpxq dy dx ´γ2κ ¨ TdˆTd Σpx ´ yq|upyq|2wpxqvpxq dy dx ` ω ˆ Td wv dx ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Still by using the dispersion relation, we obtain B2 uLωp1qpv, wq “ Re ¨ ˚ ˚ ˚ ˝ ˆ Td ˆ ´∆w 2 ´ 2γ2κΣ ‹ Repwq ˙ looooooooooooooooomooooooooooooooooon :“Sw vpxq dx ˛ ‹‹‹‚“ xSw|vy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 13 S : H2pTdq Ă L2pTdq Ñ L2pTdq is an unbounded linear operator and its spectral properties will play an important role for the orbital stability of uω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that the operator S is the linearized operator (18), up to the advection term k ¨ ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The main result of this subsection is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5 (Orbital stability for the Hartree equation) Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose (9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then the plane wave uωpt, xq “ eiωt1pxq is orbitally stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' @ε ą 0, Dδ ą 0, @vInit P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq, }vInit ´ 1}H1 ă δ ñ sup tě0 distpvptq, O1q ă ε (24) where distpv, O1q “ infθPr0,2πr }v ´ eiθ1} and pt, xq ÞÑ vpt, xq P C0pr0, 8q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1pTdqq stands for the solution of (13) with Cauchy data vInit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The key ingredient to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5 is the following coercivity estimate on the Lyapounov functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='6 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose that there exist η ą 0 and c ą 0 such that @w P Sω, dpw, O1q ă η ñ Lωpwq ´ Lωp1q ě c distpw, O1q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (25) Then the the plane wave uωpt, xq “ eiωt1pxq is orbitally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Assume that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='6 holds and suppose, by contradiction, that uω is not orbitally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, there exists 0 ă ε0 ă 2 3η such that @n P N ∖ t0u, DuInit n P H1pTdq, }uInit n ´ 1}H1 ă 1 n and Dtn P r0, `8r, distpunptnq, O1q “ ε0, pt, xq ÞÑ unpt, xq P C0pr0, 8q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1pTdqq being the solution of (13) with Cauchy data uInit n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To apply the coercivity estimate of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='6, we define zn “ ´ F p1q F punptnqq ¯1{2 unptnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It is clear that zn P Sω since Fpznq “ Fp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, ` unptnq ˘ nPN∖t0u is a bounded sequence in H1pTdq and limnÑ`8 Fpunptnqq “ Fp1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, on the one hand, there exists γ P r0, 2πr such that }unptnq}H1 ď }unptnq ´ eiθ1}H1 ` }eiθ1}H1 ď 2dpunptnq, O1q ` }eiθ1}H1 “ 2ε0 ` }1}H1 and, on the other hand, |Fpunptnqq ´ Fp1q| “ 1 2|}unptnq}2 L2 ´ }1}2 L2| ď }unptnq ´ 1}L2pε0 ` }1}H1q ă 1 npε0 ` }1}H1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, limnÑ`8 }zn ´ unptnq}H1 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This implies for n P N large enough, ε0 2 ď dpzn, O1q ď 3ε0 2 ă η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, thanks to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='6, we obtain LωpuInit n q ´ Lωp1q “ Lωpunptnqq ´ Lωp1q “ Lωpunptnqq ´ Lωpznq ` Lωpznq ´ Lωp1q ě Lωpunptnqq ´ Lωpznq ` cdpzn, O1q2 ě Lωpunptnqq ´ Lωpznq ` c 4ε2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 14 Finally, using the fact that BuLωp1q “ 0 and B2 uLωp1qpw, wq ď C}w}2 H1, we deduce that lim nÑ`8pLωpuInit n q ´ Lωp1qq “ 0, lim nÑ`8pLωpunptnqq ´ Lωpznqq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We are thus led to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since BuLωp1q “ 0, the coercivity estimate (25) can be obtained from a similar estimate on the bilinear form B2 uLωp1qpw, wq for any w P H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As pointed out before, the difficulty lies in the fact that, in general, this bilinear form is not positive on the whole space H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The following lemma states that it is enough to have a coercivity estimate on B2 uLωp1qpw, wq for any w P T1SωXpT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Recall that the tangent set to Sω is given by T1Sω “ tu P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq, BuFp1q “ 0u “ " pq, pq P H1pTd, Rq ˆ H1pTd, Rq, A ˆ q p ˙ ˇˇˇ ˆ 1 0 ˙ E “ 0 This set is the orthogonal to 1 with respect to the inner product defined in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The tangent set to O1 (which is the orbit generated by the phase multiplication) is T1O1 “ spanRti1u so that pT1O1qK “ tu P H1pTd, Cq, xu, i1y “ 0u “ " pq, pq : Td Ñ R, A ˆ q p ˙ ˇˇˇ ˆ 0 1 ˙ E “ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='7 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose that there exists ˜c ą 0 B2 uLωp1qpu, uq ě ˜c}u}2 H1 (26) for any u P T1S1 X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then there exist η ą 0 and c ą 0 such that (25) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let w P Sω such that distpw, O1q ă η with η ą 0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' By means of an implicit function theorem argument (see [4, Section 9, Lemma 8]), we obtain that there exists θ P r0, 2πr and v P pT1O1qK such that eiθw “ 1 ` v, distpw, O1q ď }v}H1 ď Cdistpw, O1q for some positive constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we use the fact that H1pTdq “ T1Sω ‘ spanRt1u to write v “ v1 ` v2 with v1 P T1Sω X pT1O1qK and v2 P spanRt1u X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since v “ eiθw ´ 1 and Fpwq “ Fp1q, we obtain 0 “ Fpeiθwq ´ Fp1q “ 1 2 ˆ Td |v|2 dx ` Re ˆ Tdpv1 ` v2q1 dx “ 1 2 ˆ Td |v|2 dx ` Re ˆ Td v21 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since v2 P spanRt1u, it follows that }v2}H1 ď }v}2 H1 }1}L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This implies }v1}H1 “ }v ´ v2}H1 ě }v}H1 ´ 1 }1}L2 }v}2 H1 ě 1 2}v}H1 15 provided }v}H1 ď }1}L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, if }v}H1 is small enough, using that B2 uLωp1qpw, zq ď C}w}H1}z}H1, we obtain B2 uLωp1qpv1, v2q ď C}v}3 H1, B2 uLωp1qpv2, v2q ď C}v}4 H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This leads to B2 uLωp1qpv, vq “ B2 uLωp1qpv1, v1q ` op}v}2 H1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, let w P Sω be such that dpw, O1q ă η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We have Lωpwq ´ Lωp1q “ Lωpeiθwq ´ Lωp1q “ 1 2B2 uLωp1qpv, vq ` op}v}2 H1q “ 1 2B2 uLωp1qpv1, v1q ` op}v}2 H1q ě ˜c}v1}2 H1 ` op}v}2 H1q ě ˜c 2}v}2 H1 ` op}v}2 H1q ě ˜c 4distpw, O1q2 where we use BuLωp1q “ 0 and v1 P T1Sω X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' At the end of the day, to prove the orbital stability of the plane wave uωpt, xq “ eiωt1pxq it is enough to prove (26) for any u P T1S1 X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This can be done by studying the spectral properties of the operator S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However, in the simpler case of the Hartree equation, the coercivity of B2 uLωp1q on T1S1 X pT1O1qK can be also obtained directly from the expression B2 uLωp1qpu, uq “ Re ˆˆ Td ˆ ´∆u 2 ´ 2γ2κΣ ‹ Repuq ˙ upxq dx ˙ “ xSu|uy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (27) Let u P T1S1 X pT1O1qK and write u “ q ` ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This leads to B2 uLωp1qpu, uq “ 1 2 ˆ Td |∇q|2 dx ´ 2γ2κ ˆ TdpΣ ‹ qqq dx ` 1 2 ˆ Td |∇p|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, since u P T1S1 X pT1O1qK, we have ˆ Td q dx “ 0 and ˆ Td p dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, thanks to the Poincaré-Wirtinger inequality, we deduce B2 uLωp1qpu, uq ě 1 2 ˆ Td |∇q|2 dx ´ 2γ2κ ˆ TdpΣ ‹ qqq dx ` 1 4}p}2 H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (28) Next, we expand q and Σ in Fourier series, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' qpxq “ ÿ mPZd qmeim¨x and Σpxq “ ÿ mPZd Σmeim¨x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that, if Σ “ σ1 ‹ σ1, then Σm “ p2πqdσ2 1,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, ´ Td q dx “ 0 implies q0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, 1 2 ˆ Td |∇q|2 dx ´ 2γ2κ ˆ TdpΣ ‹ qqq dx “ p2πqd ÿ mPZd∖t0u ˆm2 2 ´ 2γ2κp2πqdΣm ˙ q2 m “ p2πqd ÿ mPZd∖t0u ˆ 1 ´ 4γ2κp2πqd Σm m2 ˙ m2 2 q2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (29) As a consequence, we obtain the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 16 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='8 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose that there exists δ P p0, 1q such that 4γ2κp2πq2d σ2 1,m m2 ď δ (30) for all m P Zd ∖ t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, there exists ˜c ą 0 such that B2 uLωp1qpu, uq ě ˜c}u}2 H1 (31) for any u P T1S1 X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If (30) holds, then (28)-(29) lead to B2 uLωp1qpu, uq ě 1 ´ δ 2 p2πqd ÿ mPZd∖t0u m2q2 m ` 1 4}p}H1 “ 1 ´ δ 2 }∇q}2 L2 ` 1 4}p}2 H1 ě 1 ´ δ 4 }u}2 H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' where in the last inequality we used the Poincaré-Wirtinger inequality together with the fact that ´ Td q dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='9 By decomposing the linear operator S into real and imaginary part and by using Fourier series, one can study its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, S has exactly one negative eigenvalue λ´ “ ´2γ2κ @ Σ D Td with eigenspace spanRt1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, KerpSq “ spanRti1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, if (30) is satisifed, then infpσpSq X p0, 8qq ě 1´δ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, by applying the same arguments as in [5, Section 6], we can recover the coercivity of B2 uLωp1q on T1S1 X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='8 together with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='7 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='6, gives Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5 and the orbital stability of the plane wave uω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 4 Stability analysis of the Schrödinger-Wave system: the case k “ 0 Like in the case of the Hartree system, to study the stability of the plane wave solutions of the Schrödinger-Wave system (3a)-(3c), it is useful to write its solutions in the form Upt, xq “ eik¨xupt, xq with pt, x, zq ÞÑ pupt, xq, Ψpt, x, zqq solution to iBtu ` 1 2∆xu ´ k2 2 u ` ik ¨ ∇xu “ ´ ˆ γσ1 ‹ ˆ Rn σ2Ψ dz ˙ u, 1 c2 B2 ttΨ ´ ∆zΨ “ ´γσ2σ1 ‹ |u|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (32) If k P Zd and ω ą 0 satisfy the dispersion relation (12), uωpt, xq “ eiωt1pxq, Ψ˚pt, x, zq “ ´γΓpzq @ σ1 D Td, Π˚pt, x, zq “ BtΨ˚pt, x, zq “ 0 17 with Γ the solution of ´∆zΓ “ σ2 (see (11)), is a solution to (32) with initial condition uωp0, tq “ 1pxq, Ψ˚p0, x, zq “ ´γΓpzq @ σ1 D Td, Π˚p0, x, zq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For the time being, we stick to the framework identified for the study of the asymptotic Hartree equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Problem (32) has a natural Hamiltonian symplectic structure when considered on the real Banach space H1pTdqˆH1pTdqˆL2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqqˆL2pTd ˆRnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, if we write u “ q `ip, with p, q real-valued, we obtain Bt ¨ ˚ ˚ ˝ q p Ψ Π ˛ ‹‹‚“ ˆJ 0 0 ´J ˙ ∇pq,p,Ψ,ΠqHSW pq, p, Ψ, Πq with J “ ˆ 0 1 ´1 0 ˙ and HSWpq, p, Ψ, Πq “ 1 2 ˆ1 2 ˆ Td |∇q|2 ` |∇p|2 dx ` k2 2 ˆ Tdpp2 ` q2q dx ´ ˆ Td pk ¨ ∇q dx ` ˆ Td qk ¨ ∇p dx ˙ ` 1 4 ˆ TdˆRn ˆΠ2 c2 ` |∇zΨ|2 ˙ dx dz ` γ 2 ˆ Td ˆˆ TdˆRnpσ1px ´ yqσ2pzqΨpt, y, zq dy dz ˙ pp2 ` q2qpxq dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Coming back to u “ q ` ip, we can write HSWpu, Ψ, Πq “ 1 2 ˆ1 2 ˆ Td |∇u|2 dx ` k2 2 ˆ Td |upxq|2 dx ` ˆ Td k ¨ p´i∇uqu dx ˙ ` 1 4 ˆ TdˆRn ˆΠ2 c2 ` |∇zΨ|2 ˙ dx dz ` γ 2 ˆ Td ˆˆ TdˆRnpσ1px ´ yqσ2pzqΨpt, y, zq dy dz ˙ |upxq|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (33) As a consequence, HSW is a constant of the motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, it is clear that (32) is invariant under multiplications by a phase factor of u so that Fpuq “ 1 2}u}2 L2 is conserved by the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However, now, the quantities Gjpuq “ 1 2 ˆ Td ˆ1 i Bxju ˙ u dx (34) are not constants of the motion: d dtGjpuqptq “ γ 2 ˆ Td ˆ Td Bxjσ1px ´ yq ˆˆ Rn σ2pzqΨpt, y, zq dz ˙ |u|2pt, xq dy dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, they cannot be used in the construction of the Lyapounov functional as we did for the Hartree system (see (23)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 18 Finally, we consider the Banach space H1pTdqˆH1pTdqˆL2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqqˆL2pTdˆRnq endowed with the inner product C ¨ ˚ ˚ ˝ q p Ψ Π ˛ ‹‹‚ ˇˇˇ ¨ ˚ ˚ ˝ q1 p1 Ψ1 Π1 ˛ ‹‹‚ G “ ˆ Td ` pp1 ` qq1q dx ` ˆ TdˆRnp∇zΨ∇zΨ1 ` ΠΠ1q dx dz that can be also interpreted as an inner product for complex valued functions: xpu, Ψ, Πq|pu1, Ψ1, Π1qy “ Re ˆ Td uu1 dx ` ˆ TdˆRnp∇zΨ ¨ ∇zΨ1 ` ΠΠ1q dx dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (35) We denote by } ¨ } the norm on H1pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq ˆ L2pTd ˆ Rnq induced by this inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 Preliminary results for the linearized problem: spectral sta- bility when k “ 0 As before, we linearize the system (3a)-(3c) around the plane wave solution obtained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Namely, we expand Upt, xq “ Ukpt, xqp1 ` upt, xqq, Ψpt, x, zq “ ´γ @ σ1 D TdΓpzq ` ψpt, x, zq and, assuming that u, ψ and their derivatives are small, we are led to the following equations for the fluctuation pt, xq ÞÑ upt, xq P C, pt, x, zq ÞÑ ψpt, x, zq P R iBtu ` 1 2∆xu ` ik ¨ ∇xu “ γΦ, ´ 1 c2 B2 ttψ ´ ∆zψ ¯ pt, x, zq “ ´γσ2pzqσ1 ‹ ρpt, xq, ρpt, xq “ 2Re ` upt, xq ˘ , Φpt, xq “ ¨ TdˆRn σ1px ´ yqσ2pzqψpt, y, zq dz dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (36) We split the solution into real and imaginary parts upt, xq “ qpt, xq ` ippt, xq, qpt, xq “ Repupt, xqq, ppt, xq “ Impupt, xqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We obtain pBtq ` 1 2∆xp ` k ¨ ∇xqqpt, xq “ 0, pBtp ´ 1 2∆xq ` k ¨ ∇xpqpt, xq “ ´γ ˆ σ1 ‹ ˆ Rn σ2pzqψpt, ¨, zq dz ˙ pxq, ´ 1 c2 B2 ttψ ´ ∆zψ ¯ pt, x, zq “ ´2γσ2pzqσ1 ‹ qpt, xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (37) 19 It is convenient to set π “ ´ 1 2c2 Btψ, in order to rewrite the wave equation as a first order system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We obtain Bt ¨ ˚ ˚ ˝ q p ψ π ˛ ‹‹‚“ Lk ¨ ˚ ˚ ˝ q p ψ π ˛ ‹‹‚ (38) where Lk is the operator defined by Lk : ¨ ˚ ˚ ˝ q p ψ π ˛ ‹‹‚ÞÝÑ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˝ ´1 2∆xp ´ k ¨ ∇xq 1 2∆xq ´ k ¨ ∇xp ´ γσ1 ‹ ˆˆ Rn σ2ψ dz ˙ ´2c2π ´1 2∆zψ ` γσ2σ1 ‹ q ˛ ‹‹‹‹‹‹‚ For the next step, we proceed via Fourier analysis as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We expand q, p, ψ, π and σ1 by means of their Fourier series: ψpt, x, zq “ ÿ mPZd ψmpt, zqeim¨x, ψmpt, zq “ 1 p2πqd ˆ Td ψpt, x, zqe´im¨x dx, πpt, x, zq “ ÿ mPZd πmpt, zqeim¨x, πmpt, zq “ 1 p2πqd ˆ Td πpt, x, zqe´im¨x dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, recall that σ1 being real and radially symmetric, (19) holds and, by definition, @ σ1 D Td “ p2πqdσ1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, since the Fourier modes are uncoupled, the Fourier coefficients pQmptq, Pmptq, ψmpt, zq, πmpt, zqq satisfy Bt ¨ ˚ ˚ ˝ Qm Pm ψm πm ˛ ‹‹‚“ Lk,m ¨ ˚ ˚ ˝ Qm Pm ψm πm ˛ ‹‹‚ (39) where Lk,m stands for the operator defined by Lk,m ¨ ˚ ˚ ˝ Qm Pm ψm πm ˛ ‹‹‚“ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˝ ´ik ¨ mQm ` m2 2 Pm ´m2 2 Qm ´ ik ¨ mPm ´ γp2πqdσ1,m ˆ Rn σ2pzqψm dz ´2c2πm γp2πqdσ2pzqσ1,mQm ´ 1 2∆zψm ˛ ‹‹‹‹‹‹‹‚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Like for the Hartree equation, the behavior of the mode m “ 0 can be analysed explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 20 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 (The mode m “ 0) For any k P Zd, the kernel of Lk,0 is spanned by p0, 1, 0, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, equation (39) for m “ 0 admits solutions which grow linearly with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let pQ0, P0, ψ0, π0q P KerpLk,0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It means that $ ’ ’ ’ & ’ ’ ’ % γp2πqdσ1,0 ˆ Rn σ2pzqψ0pzq dz “ 0, π0 “ 0, ∆zψ0 “ 2γp2πqdσ2pzqσ1,0Q0, which yields ψ0pzq “ ´2γ @ σ1 D TdQ0Γpzq with Γpzq “ p´∆q´1σ2pzq so that ´2γ2@ σ1 D2 TdκQ0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It implies that Q0 “ 0, ψ0 “ 0 while P0 is left undetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For m “ 0, the first equation in (39) tells us that Q0ptq “ Q0p0q P C is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we get Btψ0 “ ´2c2π0 which leads to B2 ttψ0 ´ c2∆zψ0 “ ´σ2pzq 2γc2@ σ1 D TdQ0p0q looooooooomooooooooon :“C1 (40) The solution of (40) with initial condition pψ0pzq, π0pzq “ ´ 1 2c2Btψp0, zqq P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqˆL2pRnq satisfies pψ0pt, ξq “ pψ0p0, ξq cospc|ξ|tq ´ 2c2pπ0pξqsinpc|ξ|tq c|ξ| ´ ˆ t 0 sinpc|ξ|sq c|ξ| pσ2pξqC1 ds where pψ0pt, ξq and pπ0pt, ξq are the Fourier transforms of z ÞÑ ψpt, zq and z ÞÑ πpt, zq respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, integrating BtP0 “ ´γ @ σ1 D Td loooomoooon :“C2 ˆ Rn σ2pzqψ0pzq dz we obtain P0ptq “ P0p0q ` C2 ˆ Rn pσ2pξq pψ0p0, ξqsinpc|ξ|tq c|ξ| dξ p2πqn ´ 2c2C2 ˆ Rn pσ2pξqpπ0p0, ξq1 ´ cospc|ξ|tq c2|ξ|2 dξ p2πqn ´ C1C2 ˆ t 0 ˆ s 0 pcpτq dτ ds where pcpτq “ ˆ Rd |pσ2pξq|2 sinpc|ξ|τq c|ξ| dξ p2πqn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This kernel already appears in the analysis performed in [10, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The contribution involving the initial data of the vibrational field can be uniformly bounded by 1 p2πqn ˆˆ Rd |pσ2pξq|2 c2|ξ|2 dξ ˙1{2 #ˆˆ Rd | pψ0p0, ξq|2 dξ ˙1{2 ` 4c2 ˆˆ Rd |pπ0p0, ξq|2 c2|ξ|2 dξ ˙1{2+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, as a consequence of (H2), it turns out that pc is compactly supported, with ´ 8 0 pcpτq dτ “ κ c2, see [10, Lemma 14] and [19, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It follows that ˆ t 0 ˆ s 0 pcpτq dτ ds “ ˆ t 0 pcpτq ˆˆ t τ ds ˙ dτ “ ˆ t 0 pt ´ τqpcpτq dτ „ tÑ8 t κ c2 ´ ˆ 8 0 τpcpτq dτ, 21 which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' When k “ 0, basic estimates based on the energy conservation allow us to justify the stability of the solutions with zero mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However, in contrast to what has been established for the Hartree system, this analysis does not extend to any mode k � 0, since the system is not Galilean invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2 (Linearized stability for the Schrödinger-Wave system when k “ 0) Let k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose (9) and let pu, ψ, πq be the solution of (36) associated to an initial data uInit P H1pTdq, ψInit P L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq, πInit P L2pTd ˆ Rnq such that ´ Td uInit dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, there exists a constant C ą 0 such that suptě0 }upt, ¨q}H1 ď C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Again, we use the energetic properties of the linearized equation (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We have already remarked that ´ Td upt, xq dx “ 0 for any t ě 0 when ´ Td uInit dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We start by computing d dt "1 2 ˆ Td |∇xu|2 dx ` 1 2 ˆ TdˆRn ´|Btψ|2 c2 ` |∇zψ|2¯ dz dx “ ´iγ 2 ˆ Td Φ∆xpu ´ uq dx ´ γ ˆ TdˆRn Btψσ2σ1 ‹ pu ` uq dz dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we get d dt ˆ Td Φpu ` uq dx “ ˆ TdˆRn Btψσ2σ1 ‹ pu ` uq dz dx ` i 2 ˆ Td Φ∆xpu ´ uq dx ´ ˆ Td Φk ¨ ∇xpu ` uq dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We get rid of the last term by assuming k “ 0 and we arrive in this case at d dt "1 2 ˆ Td |∇xu|2 dx ` 1 2 ˆ TdˆRn ´|Btψ|2 c2 ` |∇zψ|2¯ dz dx ` γ ˆ Td Φpu ` uq dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We estimate the coupling term as follows ˇˇˇˇ ˆ Td Φpu ` uq dx ˇˇˇˇ “ ˇˇˇˇ ˆ TdˆRn σ2pzqψpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' zqσ1 ‹ pu ` uqpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' xq dz dx ˇˇˇˇ ď }σ1 ‹ pu ` uq}L2 ˆ ˆˆ Td ˇˇˇ ˆ Rn σ2pzqψpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' zq dz ˇˇˇ 2 dx ˙1{2 ď }σ1}L1}u ` u}L2 ˆ ˆˆ Td ˇˇˇ ˆ Rn pσ2pξq pψpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ξq dξ p2πqn ˇˇˇ 2 dx ˙1{2 ď 2}σ1}L1}u}L2 ˆ ˆˆ Td ˇˇˇ ˆ Rn pσ2pξq |ξ| |ξ|| pψpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ξq| dξ p2πqn ˇˇˇ 2 dx ˙1{2 ď 2}σ1}L1}u}L2 ˆ ˆˆ Rn |pσ2pξq|2 |ξ|2 dξ ˙1{2 ˆ ˆˆ TdˆRn |ξ|2| pψpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ξq|2 dξ p2πqn dx ˙1{2 ď 2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='κ}σ1}L1}u}L2 ˆ ˆˆ TdˆRn |∇zψpt, x, ξq|2 dz dx ˙1{2 “ 2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='κ}σ1}L1}u}L2}∇zψ}L2 ď 1 2γ }∇zψ}2 L2 ` 2κγ}σ1}2 L1}u}2 L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 22 By using the Poincaré-Wirtinger inequality }u}L2 ď }∇xu}L2, we deduce that 1 2 ˆ Td |∇xupt, xq|2 dx ď E0 1 ´ 4γ2κ}σ1}2 L1 , where E0 depends on the energy of the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' While it is natural to start with the linearized operator Lk in (38), it turns out that this formulation is not well-adapted to study the spectral stability issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The difficulties relies on the fact that the wave part of the system induces an essential spectrum, reminiscent to the fact that σessp´∆zq “ r0, 8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For instance, this is even an obstacle to set up a perturbation argument from the Hartree equation, in the spirit of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We shall introduce later on a more adapted formulation of the linearized equation, which will allow us to overcome these difficulties (and also to go beyond a mere perturbation analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2 Orbital stability for the Schrödinger-Wave system when k “ 0 In this subsection, we wish to establish the orbital stability of the plane wave solution to (32) obtained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1, namely uωpt, xq “ eiωt1pxq, Ψ˚pt, x, zq “ ´γΓpzq @ σ1 D Td, Π˚pt, x, zq “ 0 with k P Zd and ω ą 0 that satisfy the dispersion relation (12) and Γpzq “ p´∆q´1σ2pzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The system (32) being invariant under multiplications of u by a phase factor, we define the corresponding orbit through p1pxq, ´γΓpzq @ σ1 D Td, 0q by O1 “ tpeiθ, ´γΓpzq @ σ1 D Td, 0q, θ P Ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As before, orbital stability intuitively means that the solutions of (32) associated to initial data close enough to p1pxq, ´γΓpzq @ σ1 D Td, 0q remain at a close distance to the set O1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let us introduce, for any k P Zd and ω ą 0 satisfying the dispersion relation (12), the set Sω “ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' pu, Ψ, Πq P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq ˆ L2pTd, L2pRnqq, Fpuq “ Fp1q “ p2πqd 2 ) , and the functional Lω,kpu, Ψ, Πq “ HSWpu, Ψ, Πq ` ωFpuq, (41) intended to serve as a Lyapounov functional, where HSW is the constant of motion defined in (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For further purposes, we simply denote Lω “ Lω,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that Lω,kpu, Ψ, Πq “ HSWpu, Ψ, Πq ` 1 2i ˆ Td k ¨ ∇u ¯u dx loooooooooomoooooooooon “ dÿ j“1 kjGjpuq ` ´ ω ` k2 2 ¯ Fpuq with HSW defined in (8) and Gjpuq defined in (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Thanks to the dispersion relation (12), only the second term of this expression depends on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Unfortunately, as pointed out before, the quantities 23 Gjpuq are not constants of the motion so that the dependence on k of the Lyapounov functional (41) cannot be disregarded, in contrast to what we did for the Hartree system in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, as in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2, we need to evaluate the first and second order variations of Lω,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We compute Bpu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='Ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='ΠqHSWpu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Πqpv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' τq “ Re ˆ1 2 ˆ Tdp´∆uqv dx ` γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqΨpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' zq dz dy ˙ upxqvpxq dx ˙ ` γ 2 ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqφpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' zq dz dy ˙ |upxq|2 dy dx ` 1 2 ¨ TdˆRn ´ 1 c2 Π τ ` p´∆zΨq φ dz ¯ dx and B2 pu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='Ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='ΠqHSWpu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Πq ` pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' τq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' pv1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' τ 1q ˘ “ Re "1 2 ˆ Tdp´∆vqv1 dx `γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqpφpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' zqv1pxq ` φ1pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' zqvpxqq dz dy ˙ upxq dx ˙ `γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqΨpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' zq dz dy ˙ vpxqv1pxq dx ˙* ` 1 2 ¨ TdˆRn ´ 1 c2 τ τ 1 ` p´∆zφq φ1 dz ¯ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Besides, we have BuFpuqpvq “ Re ˆˆ Td uv dx ˙ , B2 uFpuqpv, v1q “ Re ˆˆ Td vv1 dx ˙ , BuGjpuqpvq “ Im `´ Td Bxjuv dx ˘ , B2 uGpuqpv, v1q “ Im `´ Td Bxjv1v dx ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Accordingly, we are led to Bpu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0qpv, φ, τq “ Re ˆ ´γ2@ σ1 D2 Tdκ ˆ Td v dx ` ´ ω ` k2 2 ¯ ˆ Td v dx ` γ 2 @ σ1 D Td ¨ TdˆRnpσ2 ` ∆zΓq φ dz dx ˙ “ 0 thanks to the dispersion relation (12) and the definition of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Similarly, the second order derivative 24 casts as B2 pu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0q ` pv, φ, τq, pv, φ, τq ˘ “ Re ˆ1 2 ˆ Tdp´∆vqv dx ` 1 2 ¨ TdˆRn ´τ 2 c2 ` p´∆zφq φ dz ¯ dx ` 2γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy ˙ vpxq dx ´ γ2@ σ1 D Td ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqΓpzq dz dy ˙ vpxqvpxq dx ` ´ ω ` k2 2 ¯ ˆ Td vpxqvpxq dx ˙ ` Im ˜ dÿ j“1 kj ˆ Td Bxjvv dx ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The forth and fifth integrals combine as ˆ Td ´ ω ` k2 2 ´ γ2κ @ σ1 D2 Td ¯ vpxqvpxq dx “ 0 which cancels out by virtue of the dispersion relation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence we get B2 pu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0q ` pv, φ, τq, pv, φ, τq ˘ “ Re ˆ1 2 ˆ Tdp´∆vqv dx ` 1 2 ¨ TdˆRn ´τ 2 c2 ` p´∆zφq φ dz ¯ dx ` 2γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy ˙ vpxq dx ´ i ˆ Td k ¨ ∇v v dx ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3 Note that the following continuity estimate holds: for any pv, φ, τq P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq ˆ L2pTd ˆ Rnq, B2 pu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0q ` pv, φ, τq, pv, φ, τq ˘ ď 1 2}∇v}2 L2 ` 1 2c2 }τ}2 L2 ` 1 2}φ}2 L2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1z ` 2γκ1{2}σ1}L1}v}L2}φ}L2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1z ` |k|}∇v}L2}v}L2 ď 1 2 ˆ p1 ` |k|q}v}2 H1 ` 1 c2 }τ}2 L2 ` C}φ}2 L2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1z ˙ ď maxp1{c2, 1 ` |k|, Cq 2 }pv, φ, τq}2 with C “ 1 ` 4γ2κ}σ1}2 L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The functional Lω,k is conserved by the solutions of (32);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' however the difficulty relies on justifying its coercivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We are only able to answer positively in the specific case k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, the main result of this subsection restricts to this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 (Orbital stability for the Schrödinger-Wave system) Let k “ 0 and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose (9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then the plane wave solution peiωt1pxq, ´γΓpzq @ σ D Td, 0q is orbitally stable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' @ε ą 0, Dδ ą 0, @pvInit, φInit, τ Initq P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq ˆ L2pTd ˆ Rnq, }vInit ´ 1}H1 ` }φInit ` γΓ @ σ D Td}L2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1z ` }τ Init}L2 ă δ ñ sup tě0 distppvptq, φptq, τptqq, O1q ă ε (42) 25 where distppv, φ, τq, O1q “ infθPr0,2πr }v ´ eiθ1}H1 ` }φ ` γΓ @ σ D Td}L2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1z ` }τ}L2 and pt, x, zq ÞÑ pvpt, xq, φpt, x, zq, τpt, x, zqq stands for the solution of (32) with Cauchy data pvInit, φInit, τ Initq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Using the same argument as in the case of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5, we can reduce the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 to the following coercivity estimate on the Lyapounov functional (and this is where we use that Lω,k is a conserved quantity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose that there exist η ą 0 and c ą 0 such that @pw, ψ, χq P Sω, distppw, ψ, χq, O1q ă η ñ Lω,kppw, ψ, χqq ´ Lω,kpp1pxq, ´γΓpzq @ σ D Td, 0qq ě cdistppw, ψ, χq, O1q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (43) Then the the plane wave solution peiωt1pxq, ´γΓpzq @ σ D Td, 0q is orbitally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As we have seen before, since Bpu,ψ,ΠqLω,kpp1, ´γΓpzq @ σ D Td, 0qq “ 0, the coercivity estimate (43) can be obtained from an estimate on the bilinear form B2 pu,ψ,ΠqLω,kpp1, ´γ @ σ1 D TdΓ, 0qqppu, φ, τq, pu, φ, τqq for any pu, φ, τq P T1Sω X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Here the tangent set to Sω is given by T1Sω “ " u P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq, Re ˆˆ Td upxq1pxq dx ˙ “ 0 ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq ˆ L2pTd ˆ Rnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This set is the orthogonal to p1, 0, 0q with respect to the inner product defined in (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The tangent set to O1 (which is the orbit generated by the phase multiplications of 1) is T1O1 “ spanRtpi1, 0, 0qu so that pT1O1qK “ " u P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq, Re ˆ i ˆ Td upxq1pxq dx ˙ “ 0 ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq ˆ L2pTd ˆ Rnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='6 Let k P Zd and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose that there exists ˜c ą 0 B2 pu,ψ,ΠqLω,kpp1, ´γΓpzq @ σ D Td, 0qqppu, φ, τq, pu, φ, τqq ě ˜cp}u}2 H1 ` }φ}2 L2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1z ` }τ}2 L2q “ ˜c}pu, φ, τq}2 (44) for any pu, φ, τq P T1S1 X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then there exist η ą 0 and c ą 0 such that (43) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let pw, ψ, χq P Sω such that distppw, ψ, χq, O1q ă η with η ą 0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, infθPr0,2πq }w´eiθ1} ă η and, by means of an implicit function theorem argument (see [4, Section 9, Lemma 8]), we obtain that there exists θ P r0, 2πq and v P ␣ u P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq, Re ` i ´ Td upxq dx ˘ “ 0 ( such that eiθw “ 1 ` v, inf θPr0,2πq }w ´ eiθ1} ď }v}H1 ď C inf θPr0,2πq }w ´ eiθ1} 26 for some positive constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Denote by φpx, zq “ ψpx, zq`γΓpzq @ σ1 D Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then pv, φ, χq P pT1O1qK and }pv, φ, χq} ď Cη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we use the fact that H1pTdq “ ␣ u P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Cq, Re `´ Td upxq dx ˘ “ 0 ( ‘ spanRt1u to write pv, φ, χq “ pv1, φ, χq ` pv2, 0, 0q with pv1, φ, χq P T1Sω X pT1O1qK and v2 P spanRt1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, }v2}H1 ď }v}2 H1 }1}L2 and }v1}H1 ě 1 2}v}H1 provided }v}H1 ď }1}L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, if }v}H1 is small enough, using that B2 pu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0q ` pv, φ, τq, pv1, φ1, τ 1q ˘ ď C}pv, φ, τq}}pv1, φ1, τ 1q}, we obtain B2 pu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0q ` pv1, φ, χq, pv2, 0, 0q ˘ ď C}pv, φ, χq} }v}2 H1 ď C}pv, φ, χq}3, B2 pu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0q ` pv2, 0, 0q, pv2, 0, 0q ˘ ď C}v}4 H1 ď C}pv, φ, χq}4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This leads to B2 pu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0q ` pv, φ, χq, pv, φ, χq ˘ “ B2 pu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0q ` pv1, φ, χq, pv1, φ, χq ˘ ` op}pv, φ, χq}2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' let pw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq P Sω such that dppw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' O1q ă η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' we have Lω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='kppw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χqq ´ Lω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='kpp1pxq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ´γΓpzq @ σ D Td,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 0qq “ Lω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='kppeiθw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χqq ´ Lω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='kpp1pxq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ´γΓpzq @ σ D Td,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 0qq “ B2 pu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='Ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='ΠqLω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='kp1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ´γ @ σ1 D TdΓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 0q ` pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq ˘ ` op}pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq}2q “ B2 pu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='Ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='ΠqLω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='kp1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ´γ @ σ1 D TdΓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 0q ` pv1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' pv1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq ˘ ` op}pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq}2q ě ˜c}pv1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' τq}2 ` op}pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq}2q ě ˜c 2}pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' τq}2 ` op}pv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq}2q ě ˜c 6dppw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' O1q2 where we use Bpu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='Ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='ΠqLω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='kp1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ´γ @ σ1 D TdΓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 0q “ 0 and pv1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' χq P T1Sω X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As before, to prove the orbital stability of the plane solution peiωt1pxq, ´γΓpzq @ σ D Td, 0q it is enough to prove (44) for any pu, φ, τq P T1S1 XpT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let pu, φ, τq P T1S1 XpT1O1qK and write u “ q ` ip with q, p P H1pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then B2 pu,Ψ,ΠqLω,kp1, ´γ @ σ1 D TdΓ, 0q ` pu, φ, τq, pu, φ, τq ˘ “ Re ˆ1 2 ˆ Tdp´∆uqu dx ` 1 2 ¨ TdˆRn ´τ 2 c2 ` p´∆zφq φ dz ¯ dx ` 2γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy ˙ upxq dx ´ i ˆ Td k ¨ ∇u u dx ˙ (45) can be reinterpreted as a quadratic form acting on the 4-uplet W “ pq, p, φ, τq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To be specific, it 27 expresses as the following quadratic form on W, QpW, Wq “1 2 ˆ Td |∇p|2 dx ` 1 2c2 ¨ TdˆRn |τ|2 dz dx ` 1 2 ˆ Td |∇q|2 dx ` 1 2 ¨ TdˆRnp´∆zφq φ dx dz ` 2γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dyqpxq dx ˙ ` 2 ˆ Td qk ¨ ∇p dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The crossed term ´ Td qk ¨ ∇p dx is an obstacle for proving a coercivity on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For this reason, let us focus on the case k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since pu, φ, τq P T1S1 X pT1O1qK, we have ˆ Td q dx “ 0 and ˆ Td p dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, thanks to the Poincaré-Wirtinger inequality, we deduce, when k “ 0 QpW, Wq ě1 4}p}2 H1 ` 1 2c2 }τ}2 L2 ` 1 2 ˆ Td |∇q|2 dx ` 1 2 ¨ TdˆRnp´∆zφq φ dx dz ` 2γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy ˙ qpxq dx (46) Next, we expand q, σ1 and φp¨, zq in Fourier series, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' qpxq “ ÿ mPZd qmeim¨x, φpx, zq “ ÿ mPZd φmpzqeim¨x and σ1pxq “ ÿ mPZd σ1,meim¨x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that σ1,m “ σ1,m “ σ1,´m since σ1 is real and radially symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, ´ Td q dx “ 0 implies q0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, ˆ Td ˆˆ TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy ˙ qpxq dx “ p2πq2dRe ¨ ˝ ÿ mPZd∖t0u σ1,mqm ˆ Rn σ2pzqφmpzq dz ˛ ‚ which implies 1 2 ˆ Td |∇q|2 dx ` 1 2 ¨ TdˆRnp´∆zφq φ dx dz ` 2γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy ˙ qpxq dx “ p2πqd ÿ mPZd∖t0u Re ˆm2 2 q2 m ` 1 2 ˆ Rn |∇zφm|2 dz ` 2p2πqdγσ1,mqm ˆ Rn σ2pzqφmpzq dz ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we remark that for any m P Zd, ˇˇˇˇRe ˆ 2p2πqdγσ1,mqm ˆ Rn σ2pzqφmpzq dz ˙ˇˇˇˇ ď 2p2πqdγσ1,m|qm| ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='κ}∇φm}L2 ď 1 2˜δp4γ2κp2πq2dσ2 1,mqq2 m ` ˜δ 2}∇φm}2 L2 28 for any ˜δ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, for any ˜δ P p0, 1q, we get 1 2 ˆ Td |∇q|2 dx ` 1 2 ¨ TdˆRnp´∆zφq φ dx dz ` 2γ ˆ Td ˆ¨ TdˆRn σ1px ´ yqσ2pzqφpt, y, zq dz dy ˙ qpxq dx ě p2πqd ÿ mPZd ˆˆm2 2 ´ 1 2˜δ p4γ2κp2πq2dσ2 1,mq ˙ q2 m ` 1 ´ ˜δ 2 }∇φm}2 L2 ˙ (47) As a consequence, we obtain the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='7 Let k “ 0 and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose that there exists δ P p0, 1q such that 4γ2κp2πq2d σ2 1,m m2 ď δ (48) for all m P Zd ∖ t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, there exists ˜c ą 0 such that B2 pu,Ψ,ΠqLωp1, ´γ @ σ1 D TdΓ, 0q ` pu, φ, τq, pu, φ, τq ˘ ě ˜c}pu, φ, τq}2 (49) for any pu, φ, τq P T1S1 X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If (48) holds, then, for any ˜δ P pδ, 1q, (45)-(46)-(47) lead to B2 pu,Ψ,ΠqLωp1, ´γ @ σ1 D TdΓ, 0q ` pu, φ, τq, pu, φ, τq ˘ ě 1 4}p}2 H1 ` 1 2c2 }τ}2 L2 ` ˜δ ´ δ 2˜δ p2πqd ÿ mPZd∖t0u m2q2 m ` 1 ´ ˜δ 2 p2πqd ÿ mPZd }∇φm}2 L2 “ 1 4}p}2 H1 ` 1 2c2 }τ}2 L2 ` ˜δ ´ δ 2˜δ }∇q}L2 ` 1 ´ ˜δ 2 }φ}L2xH1z ě ˜c}pu, φ, τq}2 where in the last inequality we used the Poincaré-Wirtinger inequality together with the fact that ´ Td q dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='7 together with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='6 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5 gives Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 and the orbital stability of the plane wave solution peiωt1pxq, ´γΓpzq @ σ D Td, 0q in the case k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='8 The coercivity of B2 pu,Ψ,ΠqLωp1, ´γ @ σ1 D TdΓ, 0q ` pu, φ, τq, pu, φ, τq ˘ on T1S1XpT1O1qK can be recovered from the spectral properties of a convenient unbounded linear operator S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, as we have seen before, by decomposing u into real and imaginary part, the quadratic form defined by (45) (with k “ 0) can be written as QpW, Wq “ 1 2 ˆ Td |∇p|2 dx ` 1 2c2 ¨ TdˆRn |τ|2 dz dx ` B S ˆq φ ˙ ˇˇˇ ˆq φ ˙F 29 with S : H2pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq Ă L2pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq Ñ L2pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq the unbounded linear operator given by S ˆ q φ ˙ “ ¨ ˚ ˝ ´1 2∆xq ` γσ1 ‹ ˆ Rn σ2φ dz 1 2φ ` γΓσ1 ‹ q ˛ ‹‚ (where we remind the reader that Γ “ p´∆q´1σ2q) and the inner product Bˆ q φ ˙ ˇˇˇ ˆ q1 φ1 ˙F “ ˆ Td qq1 dx` ˆ TdˆRn ∇zφ¨∇zφ1 dz dx “ ˆ Td qq1 dx` ˆ TdˆRn ˆφpx, ξqˆφ1px, ξq|ξ|2 dξ p2πqn dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that L2pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq is an Hilbert space with this inner product since n ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since ˆ T d ˆ σ1 ‹ ˆ Rn σ2φ dz ˙ pxqq1pxq dx “ ˆ Td ˆˆ TdˆRn σ1px ´ yqσ2pzqφpy, zq dz dy ˙ q1pxq dx “ ˆ TdˆRn φpx, zqσ2pzqpσ1 ‹ q1qpxq dx dz “ ˆ TdˆRn ˆφpx, ξq ˆσ2pξq |ξ|2 pσ1 ‹ q1qpxq dx|ξ|2 dξ p2πqn we can check that S is a self-adjoint operator on L2pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, σpSq Ă R and one can easily study the spectrum of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' More precisely, using Fourier series, we find that if λ is an eigenvalue of S then there exists at least one m P Zd such that for some pqm, φmq � p0, 0q there holds $ ’ ’ & ’ ’ % ˆm2 2 ´ λ ˙ qm ` γp2πqdσ1,m ˆ Rn σ2pzqφmpzq dz “ 0, ˆ1 2 ´ λ ˙ φmpzq ` γp2πqdΓpzqσ1,mqm “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let λ � 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, for any m P Zd, qm “ 0 implies φmpzq “ 0 for any z P Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, we may assume qm � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This leads to φmpzq “ ´ γp2πqdσ1,mqm 1{2´λ Γpzq and ˆm2 2 ´ λ ˙ ˆ1 2 ´ λ ˙ ´ γ2p2πq2dσ2 1,mκ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' By solving this equation, we obtain λ˘,m “ ´ m2`1 2 ¯ ˘ c´ m2´1 2 ¯2 ` 4γ2p2πq2dσ2 1,mκ 2 so that λ`,m ě 1 4 for any m P Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we remark that λ´,0 “ 1 2 ´ b 1 4 ` 4γ2p2πq2dσ2 1,0κ 2 ă 0 30 since 4γ2κp2πq2dσ2 1,0 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This eigenvalue corresponds to an eigenfunction p˜q, ˜φq with ˜q P spanRt1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, ´ Td ˜qpxq dx � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, if (30) holds, λ´,m ě ´ m2`1 2 ¯ ´ c´ m2´1 2 ¯2 ` δm2 2 ě 1 ´ δ 5 for any m P Zd ∖ t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We conclude that B S ˆq φ ˙ ˇˇˇ ˆq φ ˙F “ C¨ ˚ ˝ ´1 2∆xq ` γσ1 ‹ ˆ Rn σ2φ dz 1 2φ ` γΓσ1 ‹ q ˛ ‹‚ ˇˇˇ ˆq φ ˙G ě min ˆ1 2, 1 ´ δ 5 ˙ p}q}2 L2 ` }φ}L2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1z q for all pq, φq P tq P L2pTdq, ´ T d q dx “ 0u ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This, together with the Poincaré- Wirtinger inequality, proves the coercivity of B2 pu,Ψ,ΠqLωp1, ´γ @ σ1 D TdΓ, 0q ` pu, φ, τq, pu, φ, τq ˘ on T1S1 X pT1O1qK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 5 Discussion about the case k � 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 A new symplectic form of the linearized Schrödinger-Wave system We go back to the linearized problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The viewpoint presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 looks quite natural;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' however, it misses some structural properties of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In order to work in a unified functional framework, we find convenient to change the wave unknown ψ, which is naturally valued in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnq, into p´∆q´1{2φ, where the new unknown φ now lies in L2pRnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, the linearized problem is rephrased as BtX “ LX, where X stands for the 4-uplet pq, p, φ, πq and LX “ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˝ ´1 2∆xp ´ k ¨ ∇xq 1 2∆xq ´ k ¨ ∇xp ´ γσ1 ‹ ˆˆ Rnp´∆q´1{2σ2φ dz ˙ ´2c2p´∆q1{2π 1 2p´∆q1{2φ ` γσ2σ1 ‹ q ˛ ‹‹‹‹‹‹‹‚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The operator L is seen as an operator on the Hilbert space V “ L2pTdq ˆ L2pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pRnqq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pRnqq, with domain DpLq “ H2pTdq ˆ H2pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1pRnqq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H1pRnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We can start with the following basic information, which has the consequence that the spectral stability amounts to justify that σpLq Ă iR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 31 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 Let pλ, Xq be an eigenpair of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let Y : px, zq ÞÑ pqp´xq, ´pp´xq, φp´x, zq, ´πp´x, zqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, pλ, Xq, p´λ, Y q and p´λ, Y q are equally eigenpairs of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since L has real coefficients, LX “ λX implies LX “ λX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we check that LY px, zq “ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˝ 1 2∆xp ` k ¨ ∇xq 1 2∆xq ´ k ¨ ∇xp ´ γσ1 ‹ ˆˆ Rnp´∆q´1{2σ2φ dz1 ˙ 2c2p´∆q1{2π 1 2p´∆q1{2φ ` γσ2σ1 ‹ q ˛ ‹‹‹‹‹‹‹‚ p´x, zq “ λ ¨ ˚ ˚ ˝ ´qp´x, zq pp´x, zq ´φp´x, zq πp´x, zq ˛ ‹‹‚“ ´λY px, zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we make a new symplectic structure appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To this end, let us introduce the blockwise operator J “ ˆJ1 0 0 J2 ˙ , J1 “ ˆ 0 1 ´1 0 ˙ , J2 “ ˆ 0 ´p´∆q1{2 p´∆q1{2 0 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We are thus led to L “ J L with L X “ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˝ ´1 2∆xq ` k ¨ ∇xp ` γσ1 ‹ ˆˆ Rnp´∆q´1{2σ2φ dz ˙ ´1 2∆xp ´ k ¨ ∇xq 1 2φ ` γp´∆q´1{2σ2σ1 ‹ q 2c2π ˛ ‹‹‹‹‹‹‚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (50) For further purposes, we also set Ă J “ ˆ ˜ J1 0 0 ˜ J2 ˙ , ˜ J1 “ ˆ0 ´1 1 0 ˙ , ˜ J2 “ ˆ 0 p´∆q´1{2 ´p´∆q´1{2 0 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (51) The operator J has 0 in its essential spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' nevertheless Ă J plays the role of its inverse since J Ă J “ I “ Ă J J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2 The operator L is an unbounded self adjoint operator on V with domain DpL q “ H2pTdq ˆ H2pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pRnqq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pRnqq, and the operator J is skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The space V is endowed with the standard L2 inner product ` X|X1q “ ˆ Tdpqq1 ` pp1q dx ` ¨ TdˆRnpφφ1 ` ππ1q dx dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 32 We get ` L X|X1˘ “ ˆ Td !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´ ´ 1 2∆xq ` k ¨ ∇xp ¯ q1 ` ´ ´ 1 2∆xp ´ k ¨ ∇xq ¯ p1 ) dx `γ ˆ Td σ1 ‹ ˆˆ Rnp´∆q´1{2σ2φ dz ˙ q1 dx ` ¨ TdˆRn ´1 2φφ1 ` 2c2ππ1 ¯ dx dz `γ ¨ TdˆRn ´ p´∆q´1{2σ2σ1 ‹ q ¯ φ1 dx dz “ ˆ Td !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' q ´ ´ 1 2∆xq1 ` k ¨ ∇xp1 ¯ ` p ´ ´ 1 2∆xp1 ´ k ¨ ∇xq1 ¯) dx `γ ¨ TdˆRn φp´∆q´1{2σ2σ1 ‹ q1 dz dx ` ¨ TdˆRn ´1 2φφ1 ` 2c2ππ1 ¯ dx dz `γ ˆ Td qσ1 ‹ ˆˆ Rnp´∆q´1{2σ2φ1 dz ˙ dx “ ` X|L X1˘ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' and ` J X|X1˘ “ ¨ Td ´ pq1 ´ qp1 ¯ dx ` ¨ TdˆRn ´ ´ p´∆q1{2πφ1 ` p´∆q1{2φπ1 ¯ dx dz “ ´ ¨ Td ´ qp1 ´ pq1 ¯ dx ´ ¨ TdˆRn ´ ´ φp´∆q1{2π1 ` πp´∆q1{2φ1 ¯ dx dz “ ´ ` X|J X1˘ As said above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' justifying the spectral stability for the Schrödinger-Wave equation reduces to verify that the spectrum σpLq is purely imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However, the coupling with the wave equation induces delicate subtleties and a direct approach is not obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Instead, based on the expression L “ J L , we can take advantage of stronger structural properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, the functional framework adopted here allows us to overcome the difficulties related to the essential spectrum induced by the wave equation, which ranges over all the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This approach is strongly inspired by the methods introduced by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Pelinovsky and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Chugunova [9, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The workplan can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It can be shown that the eigenproblem LX “ λX for L is equivalent to a generalized eigenvalue problem AW “ αKW, with α “ ´λ2, see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5 below, where the auxiliary operators A and K depend on J , L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, we need to identify negative eigenvalues and complex but non real eigenvalues of the generalized eigenproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To this end, we appeal to a counting statement due to [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2 Spectral properties of the operator L The stability analysis relies on the spectral properties of L , collected in the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3 Let L the linear operator defined by (50) on DpL q Ă V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, the following assertions hold: 33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' σesspL q “ ␣ 1 2, 2c2( , 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L has a finite number of negative eigenvalues, with eigenfunctions in DpL q, given by npL q “ 1 ` #tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ă 0 and σ1,m “ 0u `#tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ď 0 and σ1,m � 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, npL q “ 1 when k “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The eigenspaces associated to the negative eigenvalues are all finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' With X0 “ p0, 1, 0, 0q, we have spanRtX0u Ă KerpL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, given k P Zd∖t0u, let K˚ “ tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 “ 0 and σ1,m “ 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, we get dimpKerpL qq “ 1 ` #K˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We remind the reader that σ1 is assumed radially symmetric, see (H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Consequently σ1,m “ σ1,´m “ σ1,˘m and both #K˚ and #tm P Zd ∖t0u, m4 ´4pk¨mq2 ď 0 and σ1,m � 0u are necessarily even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since L is self-adjoint, σpL q Ă R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let us study the eigenproblem for L : λX “ L X means $ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ % λq “ ´1 2∆xq ` k ¨ ∇xp ` γσ1 ‹ ˆˆ Rnp´∆q´1{2σ2φ dz ˙ , λp “ ´1 2∆xp ´ k ¨ ∇xq, λφ “ 1 2φ ` γp´∆q´1{2σ2σ1 ‹ q, λπ “ 2c2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (52) Clearly λ “ 2c2 is an eigenvalue with eigenfunctions of the form p0, 0, 0, πq, π P L2pTd ˆ Rnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, dimpKerpL ´ 2c2Iqq is not finite and 2c2 P σesspL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We turn to the case λ � 2c2, where the last equation imposes π “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Using Fourier series, we obtain λqm “ m2 2 qm ` ik ¨ mpm ` γp2πqdσ1,m ˆˆ Rnp´∆q´1{2σ2φm dz ˙ , λpm “ m2 2 pm ´ ik ¨ mqm, λφm “ 1 2φm ` γp2πqdp´∆q´1{2σ2σ1,mqm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (53) where qm, pm P C are the Fourier coefficients of q, p P L2pTdq while φmpzq “ 1 p2πqd ´ Td φpx, zqe´im¨x dx for all z P Rn and φ P L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pRnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We split the discussion into several cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Case m “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For m “ 0, the equations (53) degenerate to λq0 “ γp2πqdσ1,0 ˆˆ Rnp´∆q´1{2σ2φ0 dz ˙ , λp0 “ 0, ´ λ ´ 1 2 ¯ φ0 “ γp2πqdp´∆q´1{2σ2σ1,0q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 34 Combining the first and the third equation yields λ ´ λ ´ 1 2 ¯ q0 “ γ2p2πq2dσ2 1,0κq0, still with κ “ ´ p´∆q´1σ2σ2 dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It permits us to identify the following eigenvalues: λ “ 0 is an eigenvalue associated to the eigenfunction p0, 1, 0, 0q, since σ1,0 “ 1 p2πqd ´ Td σ1 dx � 0, and p´∆q´1{2σ2 � 0, λ “ 1{2 is an eigenvalue associated to eigenfunctions p0, 0, φ, 0q, for any function z ÞÑ φpzq orthogonal to p´∆q´1{2σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As before, since dimpKerpL ´ 1 2Iqq is not finite, 1 2 P σesspL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' the roots of λ ´ λ ´ 1 2 ¯ ´ γ2p2πq2dσ2 1,0κ “ λ2 ´ λ 2 ´ γ2p2πq2dσ2 1,0κ “ 0, provide two additional eigenvalues λ˘ “ 1{2 ˘ b 1{4 ` 4γ2p2πq2dσ2 1,0κ 2 , associated to the eigenfunctions p1, 0, γp2πqdσ1,0p´∆q´1{2σ2 λ˘´1{2 , 0q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To sum up, the Fourier mode m “ 0 gives rise to two positive eigenvalues (1/2 and λ`), one negative eigenvalue (λ´) and the eigenvalue 0, the last two being both one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It tells us that dimpKerpL qq ě 1 and npL q ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Case m � 0 with σ1,m “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In this case, the m-mode equations (53) for the particle and the wave are uncoupled pλ ´ 1{2qφm “ 0, pMm ´ λq ˆ qm pm ˙ “ 0 where we have introduced the 2 ˆ 2 matrix Mm “ ˆ m2{2 ik ¨ m ´ik ¨ m m2{2 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (54) We identify the following eigenvalues: λ “ 1{2 is an eigenvalue associated to the eigenfunction p0, 0, eim¨xφpzq, 0q, for any φ P L2pRnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Once again, this tells us that 1 2 P σesspL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' the eigenvalues λ˘ “ m2˘2k¨m 2 P R of the 2 ˆ 2 matrix Mm, associated to the eigenfunctions peim¨x, ¯ieim¨x, 0, 0q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since trpMmq ą 0, at most only one of these eigenvalues can be negative, which occurs when detpMmq “ m4 4 ´ pk ¨ mq2 ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Given k P Zd, we conclude this case by asserting npL q ě 1 ` #tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ă 0, σ1,m “ 0u, 35 and dimpKerpL qq ě 1 ` #tm P Zd ∖ t0u, m2 “ ˘2k ¨ m, σ1,m “ 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Case m � 0 with σ1,m � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Again, we distinguish several subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' if λ “ 1{2, the third equation on (53) imposes qm “ 0, and we are led to 1 ´ m2 2 pm “ 0, ik ¨ mpm ` γp2πqdσ1,m ˆˆ Rnp´∆q´1{2σ2φm dz ˙ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Thus, λ “ 1{2 is an eigenvalue associated to the eigenfunctions: p0, 0, eim¨xφpzq, 0q, for any function z ÞÑ φpzq orthogonal to p´∆q´1{2σ2, (we recover the same eigenfunctions as for the case m “ 0), p0, eim¨x, 0, 0q if k ¨ m “ 0, m2 “ 1, and ´ 0, ´γp2πqdκσ1,m ik ¨ m eim¨x, p´∆q´1{2σ2pzqeim¨x, 0 ¯ if k ¨ m � 0, m2 “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' if λ “ m2 2 � 1 2, (53) becomes 0 “ ik ¨ mpm ` γp2πqdσ1,m ˆˆ Rnp´∆q´1{2σ2φm dz ˙ , 0 “ ´ik ¨ mqm, m2 ´ 1 2 φm “ γp2πqdp´∆q´1{2σ2σ1,mqm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' There is no non-trivial solution when k ¨ m � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Otherwise, we see that λ “ m2{2 is an eigenvalue associated to the eigenfunctions: p0, eim¨x, 0, 0q if λ � t1 2, m2 2 u, we set µ “ λ ´ m2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We see that a non trivial solution of (53) exists if its component qm does not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We combine the equations in (53) to obtain Ppµqqm “ 0 where P is the third order polynomial Ppµq “ µ3 ` bµ2 ` cµ ` d, b “ m2 ´ 1 2 ě 0, c “ ´ppk ¨ mq2 ` γ2κp2πq2dσ2 1,mq ă 0, d “ ´pk ¨ mq2 m2 ´ 1 2 ď 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Observe that d “ ´pk ¨ mq2b and pk ¨ mq2 ă |c| ă pk ¨ mq2 ` 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We thus need to examine the roots of this polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To this end, we compute the discriminant D “ 18bcd ´ 4b3d ` b2c2 ´ 4c3 ´ 27d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' A tedious, but elementary, computation allows us to reorganize terms as follows D “ 4pk ¨ mq2` pk ¨ mq2 ´ b2˘2 ` b2σ2 1,mγp20pk ¨ mq2 ` γσ2 1,mq `4pk ¨ mq2σ2 1,mγp2pk ¨ mq2 ` γσ2 1,mq ` 4σ2 1,mγ ` pk ¨ mq4 ` 2pk ¨ mq2σ2 1,mγ ` σ4 1,mγ2˘ , 36 where we have set γ “ γ2κp2πq2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since σ1,m � 0, we thus have D ą 0 and P has 3 distinct real roots, µ1 ă µ2 ă µ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In order to bring further information about the location of the roots, we observe that limµÑ˘8 Ppµq “ ˘8 while Pp0q “ d ď 0 and P 1p0q “ c ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, studying the zeroes of P 1pµq “ 3µ2 ` 2bµ ` c, we see that µmax “ ´b´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' b2´3c 3 ă 0 is a local maximum and µmin “ ´b` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' b2´3c 3 ą 0 is a local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, P 2pµq “ 6µ ` 2b, showing that P is convex on the domain p´pm2 ´ 1q{6, `8q, concave on p´8, ´pm2 ´ 1q{6q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' A typical shape of the polynomial P is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' From this discussion, we infer µ1 ă µmax ă µ2 ď 0 ă µmin ă µ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 6 5 4 3 2 1 0 1 2 3 40 30 20 10 0 10 20 30 40 50 P( ) Figure 1: Typical graph for µ ÞÑ Ppµq, with its roots µ1 ă µ2 ă µ3 and local extrema µmax, µmin Coming back to the issue of counting the negative eigenvalues of L , we are thus wondering whether or not λj “ µj ` m2{2 is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We already know that µ3 ą µmin ą 0, hence µ3 ą ´m2{2 and we have at most 2 negative eigenvalues for each Fourier mode m � 0 such that σ1,m � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To decide how many negative eigenvalues should be counted, we look at the sign of Pp´m2{2q (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 1): i) if Pp´m2{2q ą 0 then µ1 ă ´m2{2 ă µ2, ii) if Pp´m2{2q “ 0 then either ´m2{2 ă µmax, in which case µ1 “ ´m2{2 ă µ2, or ´m2{2 ą µmax, in which case µ2 “ ´m2{2 ą µ1, iii) if Pp´m2{2q ă 0 then either ´m2{2 ă µmax, in which case ´m2{2 ă µ1 ă µ2, or ´m2{2 ą µmax, in which case µ1 ă µ2 ă ´m2{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However, we remark that Pp´m2{2q “ ´m6 8 ` m4pm2 ´ 1q 8 ` m2 2 ppk ¨ mq2 ` γσ2 1,mq ´ m2 ´ 1 2 pk ¨ mq2 “ ´m4 8 ´ 1 ´ 4γσ2 1,m m2 ¯ ` pk ¨ mq2 2 “ ´1 8pm4 ´ 4pk ¨ mq2 ´ 4m2γσ2 1,mq, (55) 37 where, by virtue of (9), m � 0 and σ1m � 0, 1 ą 4 γσ2 1,m m2 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This can be combined together with P 1p´m2{2q “3m4 4 ´ m2pm2 ´ 1q 2 ´ pk ¨ mq2 ´ γσ2 1,m “ m4 4 ` m2 2 ´ pk ¨ mq2 ´ γσ2 1,m “1 4 ` m4 ´ 4pk ¨ mq2 ´ 4m2γσ2 1,m ˘ ` m2γσ2 1,m ` m2 2 ´ γσ2 1,m “ ´ 2Pp´m2{2q ` m2 2 ` pm2 ´ 1qγσ2 1,m ą ´2Pp´m2{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, P 2p´m2{2q “ ´2m2 ´ 1 ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, Pp´m2{2q ă 0 implies P 1p´m2{2q ą 0, while P 2p´m2{2q ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This shows that ´m2{2 ă µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore, in case iii), the only remaining possibility is the situation where Pp´m2{2q ă 0 with ´m2{2 ă µ1 ă µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a conclusion, if Pp´m2{2q ă 0, all eigenvalues λj are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we claim that case ii) cannot occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, Pp´m2{2q “ 0 if and only if pm2 ´ 2k ¨ mqpm2 ` 2k ¨ mq “ 4m2γσ2 1,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, the term on the left hand side of this equality has to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This is possible if and only if both factors, which belong to Z, are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In this case, according to the sign of k ¨ m, one of them is ě m2 so that m2 ď 4m2γσ2 1,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This contradicts the smallness condition (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that Pp´m2{2q � 0 implies λj � 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' m-modes with m � 0 and σ1,m � 0 cannot generate elements of KerpL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a conclusion, negative eigenvalues only come from case i) and for each m-mode such that Pp´m2{2q ą 0 we have exactly one negative eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Going back to (55), in this case, we have pm4 ´ 4pk ¨ mq2q “ pm2 ´ 2k ¨ mqpm2 ` 2k ¨ mq ă m24γσ2 1,m ă m2 owing to (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This excludes the possibility that m4´4pk¨mq2 ą 0, since we noticed above that whenever this term is positive, it is ě m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, case i) holds if and only if m4´4pk¨mq2 ď 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This ends the counting of the negative eigenvalues of L in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that the associated eigenspaces are spanned by ´ eim¨x, ´ ik ¨ m λ ´ m2{2eim¨x, eim¨x σ1,mγp2πqdp´∆zq´1{2σ2 λ ´ 1{2 , 0 ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The discussion has permitted us to find the elements of KerpL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To be specific, the equation 38 L X “ 0 yields π “ 0 and the following relations for the Fourier coefficients m2 2 pm ´ ik ¨ mqm “ 0, φm 2 ` p2πqdγp´∆q´1{2σ2σ1,mqm “ 0, m2 2 qm ` ik ¨ mpm ` p2πqdγσ1,m ˆ p´∆q´1{2σ2φm dz “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We have seen that the mode m “ 0 gives rise the eigenspace spanned by p0, 1, 0, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For m � 0, ele- ments of KerpL q can be obtained only in the case σ1,m “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, the condition m2 “ ˘2k ¨m has to be fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In such a case, peim¨x, ¯ieim¨x, 0, 0q P KerpL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, it remains to prove that σesspL q “ ␣1 2, 2c2( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We have already noticed that ␣ 1 2, 2c2( Ă σesspL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose, by contradiction, that there exists λ P σesspL q ∖ ␣ 1 2, 2c2( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, by Weyl’s criterion [42, Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='14], there exists a sequence pXνqνPN with Xν “ pqν, pν, φν, πνq P DpL q such that, as ν goes to 8, }pL ´ λIqXν} Ñ 0, }Xν} “ 1 and Xν á 0 weakly in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (56) Since λ � 1 2 and λ � 2c2, from (52) and (56) we have }πν}L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='L2pRnqq Ñ 0 and φν “ ´ ˆ1 2 ´ λ ˙´1 γp´∆q´1{2σ2σ1 ‹ qν ` εν with εν P L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pRnqq such that limνÑ8 }εν}L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='L2pRnqq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This leads to ››››´1 2∆xqν ´ λqν ` k ¨ ∇xpν ´ γ2κ 1{2 ´ λΣ ‹ qν ` γσ1 ‹ ˆˆ Rnp´∆q´1{2σ2εν dz ˙›››› L2pTdq ÝÝÝÑ νÑ8 0, ››››´1 2∆xpν ´ λpν ´ k ¨ ∇xqν ›››› L2pTdq ÝÝÝÑ νÑ8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Using the fact that the sequence ppqν, pν, ενqqνPN is bounded in L2pTdq ˆ L2pTdq ˆ L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pRnqq, we deduce that pqν, pνqνPN is bounded in H2pTdq ˆ H2pTdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, reasoning on Fourier series, this amounts to estimate ÿ mPZd |m|4p|qν,m|2 ` |pν,m|2q ď 2 ÿ mPZd ` |m2qν,m ` 2ik ¨ mpν,m|2 ` |m2pν,m ´ 2ik ¨ mqν,m|2q `8 ÿ mPZd p|k ¨ mpν,m|2 ` |k ¨ mqν,m|2q ď 2 ›› ´ ∆xqν ` 2k ¨ ∇xpν ›› L2pTdq ` 2 ›› ´ ∆xpν ´ 2k ¨ ∇xqν ›› L2pTdq `4 δ|k|4 ÿ mPZd ` |qν,m|2 ` |pν,m|2˘ ` 4δ ÿ mPZd |m|4p|qν,m|2 ` |pν,m|2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Choosing 0 ă δ ă 1{4 and using the already known estimates, we conclude that }∆xqν}2 L2 ` }∆xpν}2 L2 “ ř mPZd |m|4` |qν,m|2 ` |pν,m|2˘ is bounded, uniformly with respect to ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, because of the compact Sobolev embedding of H2pTdq into L2pTdq, we have that pqν, pνqνPN has a (strongly) convergent subsequence in L2pTdq ˆ L2pTdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, the sequence pXνqνPN has a conver- 39 gent subsequence in V , which contradicts (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' A consequence of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3 is that 0 is an isolated eigenvalue of L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since the restriction of L to the subspace pKerpL qqK is, by definition, injective, it makes sense to define on it its inverse L ´1, with domain RanpL q Ă pKerpL qqK Ă V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In fact, 0 being an isolated eigenvalue, RanpL q is closed and coincides with pKerpL qqK, [42, Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This can be shown by means of spectral measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Given X P pKerpL qqK, the support of the associated spectral measure dµX does not meet the interval p´ǫ, `ǫq for ǫ ą 0 small enough, independent of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Accordingly, we get }L X}2 “ ˆ `8 ´8 λ2 dµXpλq “ ˆ |λ|ěǫ λ2 dµXpλq ě ǫ2}X}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, the Fredholm alternative applies: for any Y P pKerpL qqK, there exists a unique X P pKerpL qqK such that L X “ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We will denote X “ L ´1Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For further purposes, let us set X0 “ p0, 1, 0, 0q P KerpL q and Y0 “ ´J X0 “ p1, 0, 0, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that Y0 P pKerpL qqK, so that it makes sense to consider the equation L U0 “ Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We find πm “ 0, φm “ ´2γp2πqdp´∆q´1{2σ2σ1,mqm, m2pm “ 2ik ¨ mqm, and m2qm ` 2ik ¨ mpm ` 2γp2πqdσ1,m ˆ p´∆q´1{2σ2φm dz “ δ0,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It yields, for m � 0, pm4 4 ´ pk ¨ mq2 ´ γ|σ1,m|2m2¯ qm “ 0 and q0 “ ´ 1 2γ2p2πq2d|σ1,0|2κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore, we can set U0 “ L ´1Y0 “ ´ 1 2γ2p2πq2d|σ1,0|2κ ` 1, 0, ´2γp2πqdp´∆q´1{2σ2σ1,0, 0 ˘ , solution of L U0 “ Y0 such that U0 P pKerpL qqK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We note that pU0, Y0q “ ´ 1 2γ2p2πqd|σ1,0|2κ ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (57) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3 Reformulation of the eigenvalue problem, and counting theo- rem The aim of the section is to introduce several reformulations of the eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This will allow us to make use of general counting arguments, set up by [9, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 Let us set M “ ´J L J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The coupled system M Y “ ´λX, L X “ λY, (58) admits a solution with λ � 0, X P DpL q ∖ t0u, Y P DpJ L J q ∖ t0u iff there exists two vectors X˘ P DpLq ∖ t0u that satisfy LX˘ “ ˘λX˘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 40 Let P stand for the orthogonal projection from V to pKerpL qqK Ă V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5 Let us set A “ PM P and K “ PL ´1P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let us define the following Hilbert space H “ DpM q X pKerpL qqK Ă V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The coupled system (58) has a pair of non trivial solutions p˘λ, X, ˘Y q, with λ � 0 iff the gener- alized eigenproblem AW “ αKW, W P H , (59) admits the eigenvalue α “ ´λ2 � 0, with at least two linearly independent eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Recall that the plane wave solution obtained Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 is spectrally stable, if the spectrum of L is contained in iR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In view of Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5, this happens if and only if all the eigenvalues of the generalized eigenproblem (59) are real and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In other words, the presence of spectrally unstable directions corresponds to the existence of negative eigenvalues or complex but non real eigenvalues of the generalized eigenproblem (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Our goal is then to count the eigenvalues α of the generalized eigenvalue problem (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular we define the following quantities: N ´ n , the number of negative eigenvalues N 0 n, the number of eigenvalues zero N p n, the number of positive eigenvalues of (59), counted with their algebraic multiplicity, the eigenvectors of which are associated to non- positive values of the the quadratic form W ÞÑ pKW|Wq “ pL ´1PW|PWq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, let NC` be the number of eigenvalues α P C with Impαq ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As pointed out above, the eigenvalues counted by N ´ n and NC` correspond to cases of instabil- ities for the linearized problem (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that to prove the spectral stability, it is enough to show that the generalized eigenproblem (59) does not have negative eigenvalues and NC` “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Indeed, as a consequence of Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1, if α P C ∖ R is an eigenvalue of (59), then ¯α is an eigenvalue too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, if NC` “ 0, then the generalized eigenproblem (59) does not have solutions in C ∖ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, for using the counting argument introduce by Chugunova and Pelinovsky in [9], we need the following information on the essential spectrum of A, see [43, Lemma 2-(H1’) and Lemma 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='6 Let M “ ´J L J be defined on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then σesspM q “ r0, `8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let A “ PM P and K “ PL ´1P be defined on H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then σesspAq “ r0, `8q and we can find δ˚, d˚ ą 0 such that for any real number 0 ă δ ă δ˚, A`δK admits a bounded inverse and we have σesspA`δKq Ă rd˚δ, `8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We check that J L J X “ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˝ ∆xq 2 ´ k ¨ ∇xp ∆xp 2 ` k ¨ ∇xq ` γσ1 ‹ ˆ p´∆zq´1{2σ2p´∆zq1{2π dz 2c2∆zφ ∆zπ 2 ` γσ2σ1 ‹ p ˛ ‹‹‹‹‹‹‚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 41 As a matter of fact, for any φ P H2pRnq, the vector Xe “ p0, 0, φ, 0q lies in pKerpL qqK and satisfies J L J Xe “ ¨ ˚ ˚ ˝ 0 0 2c2∆zφ 0 ˛ ‹‹‚P pKerpL qqK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Consequently M Xe “ AXe “ ´J L J Xe “ p0, 0, ´2c2∆zφ, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It indicates that a Weyl sequence for A ´ λI, λ ą 0, can be obtained by adapting a Weyl sequence for p´∆z ´ µIq, µ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let us consider a sequence of smooth functions ζν P C8 c pRnq such that supppζνq Ă Bp0, ν ` 1q, ζνpzq “ 1 for x P Bp0, νq and }∇zζν}L8pRnq ď C0 ă 8, }D2 zζν}L8pRnq ď C0 ă 8, uniformly with respect to ν P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We set φνpzq “ ζνpzqeiξ¨z{p ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 2cq for some ξ P Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We get p´|ξ|2 ´ 2c2∆zqφνpzq “ eiξ¨z{p ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 2cq´ 2i ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 2cξ ¨ ∇zζν ` 2c2∆zζν ¯ pzq, which is thus bounded in L8pRnq and supported in Bp0, ν ` 1q ∖ Bp0, νq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It follows that }p´|ξ|2 ´ 2c2∆zqφν}2 L2pRnq ≲ νn´1, while }φν}2 L2pRnq ≳ νn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Accordingly, we obtain }φν}2 L2pRnq }p´|ξ|2´2c2∆zqφν}2 L2pRnq ≳ ν Ñ 8 as ν Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore, φν equally provides a Weyl sequence for M ´ |ξ|2I and A ´ |ξ|2I, showing the inclusions r0, 8q Ă σesspM q and r0, 8q Ă σesspAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, let λ � r0, 8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We suppose that we can find a Weyl sequence pXνqνPN for M , such that M Xν ´ λXν “ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˝ ´λqν ´ ∆xqν 2 ` k ¨ ∇xpν ´λpν ´ ∆xpν 2 ´ k ¨ ∇xqν ´ γσ1 ‹ ˆ p´∆zq´1{2σ2p´∆zq1{2πν dz ´λφν ´ 2c2∆zφν ´λπν ´ ∆zπν 2 ´ γσ2σ1 ‹ pν ˛ ‹‹‹‹‹‹‚ “ ¨ ˚ ˚ ˝ q1 ν p1 ν φ1 ν π1 ν ˛ ‹‹‚ÝÝÝÑ νÑ8 0, with, moreover, }Xν} “ 1 and Xν á 0 weakly in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, we can set x φνpx, ξq “ x φ1νpx, ξq 2c2|ξ|2 ´ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (60) It defines a sequence which tends to 0 strongly L2pTd ˆ Rnq since, writing λ “ a ` ib P C ∖ r0, 8q, we get |2c2|ξ|2 ´λ|2 “ |2c2|ξ|2 ´a|2 `b2 which is ě b2 ą 0 when λ � R, and, in case b “ 0, ě a2 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Similarly, we can write x πνpx, ξq “ 2x π1νpx, ξq |ξ|2 ´ 2λ loooomoooon “hνpx,ξqPL2pTdˆRnq `γ 2x σ2pξq |ξ|2 ´ 2λ loooomoooon PL2pRnq σ1 ‹ pν, (61) 42 where hν tends to 0 strongly L2pTd ˆ Rnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We are led to the system ¨ ˚ ˝ ´ ´ λ ` ∆x 2 ¯ qν ` k ¨ ∇xpν ´k ¨ ∇xqν ´ ´ λ ` ∆x 2 ¯ pν ´ 2γ2 ˆ |x σ2|2 p2πqnp|ξ|2 ´ 2λq dξ ˆ Σ ‹ pν ˛ ‹‚ “ ¨ ˝ q1 ν p1 ν ´ γσ1 ‹ ˆ x σ2pξq |ξ| hνpx, ξq dξ p2πqn ˛ ‚ÝÝÝÑ νÑ8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (62) Reasoning as in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3-1), we conclude that Xν converges strongly to 0 in V , a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, λ P C ∖ r0, 8q cannot belong to σesspM q and the identification σesspM q “ r0, 8q holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3-3) identifies KerpL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let us introduce the mapping Ă P : ˆq p ˙ P L2pTdqˆL2pTdq ÞÝÑ ¨ ˚ ˚ ˝ ÿ mPK˚, k¨mą0 pqm ´ ipmqeim¨x ` ÿ mPK˚, k¨mă0 pqm ` ipmqeim¨x p0 ` i ÿ mPK˚, k¨mą0 pqm ´ ipmqeim¨x ´ i ÿ mPK˚, k¨mă0 pqm ` ipmqeim¨x ˛ ‹‹‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, X “ ¨ ˚ ˚ ˝ q p φ π ˛ ‹‹‚ÞÝÑ ¨ ˚ ˚ ˝ Ă P ˆ q p ˙ 0 0 ˛ ‹‹‚ is the projection of V on KerpL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Accordingly, we realize that P does not modify the last two components of a vector X “ pq, p, φ, πq P V , and for X P pKerpL qqK, we have p0 “ 0, and qm “ ˘ipm for any m P K˚, depending on the sign of k ¨ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Now, let λ P C ∖ r0, 8q and suppose that we can exhibit a Weyl sequence pXνqνPN for A ´ λI: Xν P H Ă pKerpL qqK, PXν “ Xν, }Xν} “ 1, Xν á 0 in V and limνÑ8 }pA ´ λIqXν} “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We can apply the same reasoning as before for the last two components of pA ´ λIqXν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' it leads to (60) and (61), where, using λ � r0, 8q, φν and hν converge strongly to 0 in L2pTd ˆ Rnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We arrive at the following analog to (62) pI ´ Ă Pq ¨ ˚ ˝ ´ ´ λ ` ∆x 2 ¯ qν ` k ¨ ∇xpν ´k ¨ ∇xqν ´ ´ λ ` ∆x 2 ¯ pν ´ 2γ2 ˆ |x σ2|2 p2πqnp|ξ|2 ´ 2λq dξ ˆ Σ ‹ pν ˛ ‹‚ “ ˆq1 ν p1 ν ˙ ´ pI ´ Ă Pq ¨ ˝ 0 γσ1 ‹ ˆ x σ2pξq |ξ| hνpx, ξq dξ p2πqn ˛ ‚ÝÝÝÑ νÑ8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (63) In order to derive from (63) an estimate in a positive Sobolev space as we did in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3-1), we should consider the Fourier coefficients arising from ´ 1 2∆xqν ` k ¨ ∇xpν and ´ 1 2∆xpν ´k¨∇xqν, namely Qm “ m2 2 qν,m`ik¨mpν,m and Pm “ m2 2 pν,m´ik¨mqν,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Only the coef- ficients belonging to K˚ are affected by the action of Ă P, which leads to Qm ´ pQm ¯ iPmq “ ˘iPm and Pm ¯ ipQm ¯ iPmq “ ¯iQm, according to the sign of k ¨ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However, we bear in mind that qm “ ˘ipm when m P K˚ with ˘k ¨ m ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, for coefficients in K˚, the contributions of the differential operators reduces to ˘im2pm “ ˘m2qm and ¯im2qm “ ˘m2pm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note 43 also that for these coefficients there is no contributions coming from the convolution with σ1 in (63) since σ1,m “ 0 for m P K˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore, reasoning as in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3-1) for coefficients m P Zd ∖ K˚, we can obtain a uniform bound on ř mPZd |m|4p|qν,m|2 ` |pν,m|2q, which provides a uniform H2 bound on qν and pν, leading eventually to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We conclude that σesspAq “ r0, 8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let δ ą 0 and consider the shifted operator A ` δK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='10, we will see that KerpA ` δKq “ t0u for any δ ą 0: 0 is not an eigenvalue for A ` δK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' let us justify it does not belong to the essential spectrum neither.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To this end, we need to detail the expression of the operator K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Given X P H , we wish to find X1 P H satisfying L X1 “ ¨ ˚ ˚ ˚ ˚ ˚ ˚ ˝ ´1 2∆xq1 ` k ¨ ∇xp1 ` γσ1 ‹ ˆˆ Rnp´∆q´1{2σ2φ1 dz ˙ ´1 2∆xp1 ´ k ¨ ∇xq1 1 2φ1 ` γp´∆q´1{2σ2σ1 ‹ q1 2c2π1 ˛ ‹‹‹‹‹‹‚ “ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We infer π1 “ π 2c2 and the relation φ1 “ 2φ ´ 2γp´∆zq´1{2σ2σ1 ‹ q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In turn, the Fourier coefficients of q1, p1 are required to satisfy ˆ m2{2 ´ 2γ2κp2πq2d|σ1,m|2 ik ¨ m ´ik ¨ m m2{2 ˙ ˆ q1 m p1 m ˙ “ ¨ ˝qm ´ 2γp2πqdσ1,m ˆ p´∆q´1{2σ2φm dz pm ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' When m � 0, m � K˚, the matrix of this system has its determinant equal to det “ m4 4 ` 1 ´ 4γ2κp2πq2d |σ1,m|2 m2 ˘ ´ pk ¨ mq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Owing to (9), since pk ¨ mq2 takes values in N, it does not vanish and we obtain q1 m, p1 m by solving the system q1 m “ 1 det ˆm2 2 ´ qm ´ 2γp2πqdσ1,m ˆ p´∆q´1{2σ2φm dz ¯ ´ ik ¨ mpm ˙ , p1 m “ 1 det ˆ `ik ¨ m ´ qm ´ 2γp2πqdσ1,m ˆ p´∆q´1{2σ2φm dz ¯ ` ´m2 2 ´ 2γ2κp2πq2d|σ1,m|2¯ pm ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If m P K˚ we find a solution in pKerpL qqK by setting p1 m “ pm m2 , q1 m “ ˘ip1 m, according to the sign of k ¨m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' if m “ 0, we set p1 0 “ 0 and q1 0 “ 1 2γ2κp2πq2d|σ1,0|2 ` q0 ´2γp2πqdσ1,0 ´ p´∆q´1{2σ2φ0 dz ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This defines X1 “ KX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore, the last two components of pA ` δK ´ λIqX read p2δ ´ λqφ ´ 2c2∆zφ ´ 2δγp´∆q´1{2σ2σ1 ‹ q1, ´ δ 2c2 ´ λ ¯ π ´ 1 2∆zπ ´ γσ2σ1 ‹ p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, when λ does not belong to rδd˚, 8q, with d˚ “ minp2, 1 2c2q, we can repeat the analysis performed above to establish that λ � σesspA ` δKq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular the essential spectrum of A has been shifted away from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We are now able to apply the results of Chugunova and Pelinovsky [9] (see also [43]), to obtain 44 the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='7 [9, Theorem 1] Let L be defined by (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' With the operators M , A, K defined as in Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5, the following identity holds N ´ n ` N 0 n ` N ` n ` NC` “ npL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let us now detail the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5, adapted from [43, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 1 & Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof of Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The goal is to establish connections between the following three problems: (Ev) the eigenvalue problem LX “ λX, with L “ J L , (Co) the coupled problem L X “ λY , M Y “ ´λX, with M “ ´J L J , (GEv) the generalized eigenvalue problem AW “ αKW, with A “ PM P, K “ PL ´1P, the projection P on pKerpL qqK, and W P H “ DpM q X pKerpL qqK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The proof of Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5 follows from the following sequence of arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (i) By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1, we already know that if there exists a solution pλ, X`q of (Ev), with λ � 0 and X` � 0, then, there exists X´ � 0, such that p´λ, X´q satisfies (Ev).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Being eigenvectors associated to distinct eigenvalues, X` and X´ are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that only this part of the proof uses the specific structure of the operator L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (ii) From these eigenpairs for L, we set X “ X` ` X´ 2 , Y “ Ă J ˆX` ´ X´ 2 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since X` and X´ are linearly independent, we have X � 0, Y � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, X “ X``X´ 2 and J Y “ X`´X´ 2 are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We get L X “ Ă J LX “ Ă J ˆλ 2pX` ´ X´q ˙ “ λY, M Y “ ´J L ˆX` ´ X´ 2 ˙ “ ´L ˆX` ´ X´ 2 ˙ “ ´λ 2pX` ` X´q “ ´λX, so that pλ, X, Y q satisfies (Co).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (iii) If pλ, X, Y q is a solution (Co), then p´λ, X, ´Y q satisfies (Co) too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (iv) Let pλ, X, Y q be a solution (Co).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Set X1 “ J Y, Y 1 “ Ă J X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We observe that M Y 1 “ ´J L J Ă J X “ ´J L X “ ´J pλY q “ ´λX1, L X1 “ L J Y “ Ă J J L J Y “ ´ Ă J M Y “ λ Ă J X “ λY 1, which means that pλ, J Y, Ă J Xq is a solution of (Co).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, if X and J Y are linearly independent, Y and Ă J X are linearly independent too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 45 (v) Let pλ, X, Y q be a solution (Co), with X � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We get LpX ˘ J Y q “ J L X ˘ J L J Y “ J L X ¯ M Y “ J pλY q ˘ λX “ ˘λpX ˘ J Y q, so that p˘λ, X ˘ J Y q satisfy (Ev).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In the situation where X and J Y are linearly inde- pendent, we have X ˘ J Y � 0 and p˘λ, X ˘ J Y q are eigenpairs for L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Otherwise, one of the vectors X ˘ J Y might vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Nevertheless, since only one of these two vectors can be 0, we still obtain an eigenvector X˘ � 0 of L, associated to either ˘λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Coming back to i), we conclude that ¯λ is an eigenvalue too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Items i) to v) justify the equivalence stated in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (vi) Let pλ, X, Y q be a solution (Co).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' From L X “ λY , we infer Y P RanpL q Ă pKerpL qqK so that PY “ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The relation thus recasts as X “ λPL ´1PY ` ˜Y, ˜Y P KerpL q, P ˜Y “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (Here, PL ´1PY stands for the unique solution of L Z “ Y which lies in pKerpL qqK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=') We obtain PM Y “ Pp ´ λXq “ ´λPpλPL ´1PY ` ˜Y q “ ´λ2PL ´1PY “ ´λ2KY “ PM PY “ AY, so that p´λ2, Y q satisfies (GEv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Going back to iv), we know that p´λ2, Ă J Xq is equally a solution to (GEv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If X and J Y are linearly independent, we obtain this way two linearly independent vectors, Y and Ă J X, solutions of (GEv) with α “ ´λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (vii) Let pα, Wq satisfy (GEv), with α � 0, W � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We set X “ ´M W ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We have Ă J X “ ´ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α Ă J M W “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α Ă J J L J W “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αL J W which lies in RanpL q Ă pKerpL qqK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Thus, using P Ă J X “ Ă J X, we compute K Ă J X “ PL ´1P Ă J X “ PL ´1 Ă J X “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αPL ´1L J W “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αPJ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we observe that A Ă J X “ PM P Ă J X “ ´PJ L J Ă J X “ ´PJ L X “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αPJ L M W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However, we can use PW “ W (since W P H Ă pKerpL qqK) and the fact that, for any vector Z, L Z “ L pI ´ PqZ ` L PZ “ 0 ` L PZ, which yields A Ă J X “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αPJ L PM PW “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αPJ L AW “ ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ´αPJ L KW “ ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αPJ L PL ´1PW “ ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αPJ L L ´1W “ ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αPJ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We conclude that A Ă J X “ αK Ă J X: pα, Ă J Xq satisfies (GEv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (viii) Let pα, Wq satisfy (GEv), with α � 0, W � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We have PpM PW ´ αL ´1PWq “ 0 and thus M PW ´ αL ´1PW “ ˜Y P KerpL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 46 Let us set Y “ PW P pKerpL qqK, X “ ´M PW ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α “ ´1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αp˜Y ` αL ´1PWq, so that L X “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ´αPW “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ´αY, M Y “ M PW “ ´ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' ´αX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore p ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α, X, Y q satisfies (Co).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' By v), p˘ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α, X ˘ J Y q satisfy (Ev), and at least one of the vectors X ˘J Y does not vanish;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' using i), we thus obtain eigenpairs p˘ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α, X˘q of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' With ii), we construct solutions of (Co) under the form ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α, X``X´ 2 , Ă J ` X`´X´ 2 ˘˘ , which, owing to iv) and vi), provide the linearly independent solutions ` α, Ă J `X`˘X´ 2 ˘˘ of (GEv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The dimension of the linear space of solutions of (GEv) is at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' At least one of these vectors X˘ is given by the formula ˜X˘ “ ´ M W ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α ˘ J W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' By the way, we indeed note that AW “ αKW, with W P H , can be cast as L J L J W “ ´αW (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='8 below) so that L ´ ´ M W ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α ˘ J W ¯ “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αJ pL J L J Wq ˘ J L J W “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´αJ W ¯ M W “ ˘ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α ´ ´ M W ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α ˘ J W ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' With these manipulations we have checked that p˘ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='´α, ˜X˘q satisfy (Ev).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If both vectors ˜X˘ are non zero, we get X˘ “ ˜X˘ and we recover W “ Ă J ` X`´X´ 2 ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If ˜X˘ “ 0, then, we get ˜X¯ “ ¯J W � 0, and we directly obtain X¯ “ ˜X¯, W “ ¯ Ă J X¯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In any cases, W lies in the space spanned by X` and X´ and the dimension of the space of solutions of (GEv) is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This ends the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='4 Spectral instability We are going to compute the terms arising in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Eventually, it will allow us to identify the possible unstable modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In what follows, we find convenient to work with the operator M ´αL ´1 instead of PpM ´ αL ´1qP “ A ´ αK, owing to to the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='8 Let α � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In the space H “ DpM q X pKerpL qqK, the two subspaces KerpA ´ αKq and KerpM ´ αL ´1q coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let X P H satisfy M X “ αL ´1X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, we have X “ PX and, thus, pA ´ αKqX “ PpM ´ αL ´1qPX “ PpM X ´ αL ´1Xq “ 0, showing the inclusion KerpM ´ αL ´1q X H Ă KerpA ´ αKq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Conversely, the equation pA ´ αKqX “ 0, with X “ PX P pKerpL qqK means that pM ´ αL ´1qX “ Y P KerpL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Applying L then yields L M X “ αX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since both terms of this relation lie in pKerpL qqK, it is legitimate to apply L ´1, showing that M X “ αL ´1X: we have shown KerpA ´ αKq X H Ă KerpM ´ αL ´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 47 Therefore, we shall consider the solutions of the generalized eigenvalue problem M X “ αL ´1X, with X P H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We rewrite the equation by introducing an auxiliary unknown: M X “ α ˜X, L ˜X “ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='9 Suppose (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' N 0 n “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We are interested in the solutions of ´1 2∆xq ` k ¨ ∇xp “ 0, ´1 2∆xp ´ k ¨ ∇xq ´ γσ1 ‹ ˆ σ2π dz “ 0, ´2c2∆zφ “ 0, ´1 2∆zπ ´ γσ2σ1 ‹ p “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We infer φpx, zq “ 0 and pπpx, ξq “ 2γ x σ2pξq |ξ|2 σ1 ‹ ppxq, and, next, ´1 2∆xq ` k ¨ ∇xp “ 0, ´1 2∆xp ´ k ¨ ∇xq ´ 2γ2κΣ ‹ p “ 0 with Σ “ σ1 ‹ σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In terms of Fourier coefficients, it becomes m2 2 qm ` ik ¨ mpm “ 0, m2 2 pm ´ ik ¨ mqm ´ 2p2πq2dγ2κ|σ1,m|2pm “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For m “ 0, we get p0 “ 0 and we find the eigenfunction p1, 0, 0, 0q “ Y0 “ ´J X0 with X0 “ p0, 1, 0, 0q P KerpL q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For m � 0 with σ1,m � 0, we get m4 ´ 4pk ¨ mq2 “ 2p2πq2dγ2κ|σ1,m|2 loooooooooomoooooooooon Pp0,1q m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' which cannot hold (see the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For m � 0 with σ1,m “ 0, we get Mm ˆqm pm ˙ “ 0 with Mm defined in (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As far as m4 ´4pk ¨mq2 � 0, Mm is invertible and the only solution is pm “ 0 “ qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If m4 ´4pk ¨mq2 “ 0, we find the eigenfunctions peik¨m, ˘ieik¨m, 0, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' These functions belong to KerpL q, and thus do not lie in the working space H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We conclude that KerpM q “ spanRtY0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, this vector Y0 does not belong to RanpM q so that the algebraic multiplicity of the eigenvalue 0 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Finally, bearing in mind (57), which can be recast as pKY0|Y0q ă 0, we arrive at N 0 n “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='10 Suppose (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The generalized eigenproblem (59) does not admit negative eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, N ´ n “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 48 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let α ă 0, X “ pq, p, φ, πq and ˜X “ p˜q, ˜p, ˜φ, ˜πq satisfy ´1 2∆xq ` k ¨ ∇xp “ α˜q, ´1 2∆xp ´ k ¨ ∇xq ´ γσ1 ‹ ˆ σ2π dz “ α˜p, ´2c2∆zφ “ α˜φ, ´1 2∆zπ ´ γσ2σ1 ‹ p “ α˜π, (64) where q “ ´1 2∆x˜q ` k ¨ ∇x˜p ` γσ1 ‹ ˆ p´∆zq´1{2σ2 ˜φ dz, p “ ´1 2∆x˜p ´ k ¨ ∇x˜q, φ “ 1 2 ˜φ ` γp´∆zq´1{2σ2σ1 ‹ ˜q, π “ 2c2˜π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (65) This leads to solve an elliptic equation for π ´|α| c2 ´ ∆z ¯ π “ 2γσ2σ1 ‹ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In other words, we get, by means of Fourier transform pπpx, ξq “ 2γσ1 ‹ ppxq ˆ x σ2pξq |ξ|2 ` |α|{c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' On the same token, we obtain ´|α| c2 ´ ∆z ¯ ˜φ “ ´2γp´∆zq1{2σ2σ1 ‹ ˜q, which yields p˜φpx, ξq “ ´2γσ1 ‹ ˜qpxq ˆ |ξ|x σ2pξq |ξ|2 ` |α|{c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For λ ą 0, we introduce the symbol 0 ď κλ “ ˆ |x σ2pξq|2 |ξ|2 ` λ ď κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It turns out that ´1 2∆xq ` k ¨ ∇xp “ α˜q, ´1 2∆xp ´ k ¨ ∇xq ´ 2γ2κ|α|{c2Σ ‹ p “ α˜p, with q “ ´1 2∆x˜q ` k ¨ ∇x˜p ´ 2γ2κ|α|{c2Σ ‹ ˜q, p “ ´1 2∆x˜p ´ k ¨ ∇x˜q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 49 For the Fourier coefficients, it casts as m2 2 qm ` ik ¨ mpm “ α˜qm, m2 2 pm ´ ik ¨ mqm ´ 2γ2κ|α|{c2p2πq2d|σ1,m|2pm “ α˜pm, with qm “ m2 2 ˜qm ` ik ¨ m˜pm ´ 2γ2κ|α|{c2p2πq2d|σ1,m|2˜qm, pm “ m2 2 ˜pm ´ ik ¨ m˜qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We are going to see that these equations do not have non trivial solutions with α ă 0: If m “ 0, we get p0 “ 0, ˜q0 “ 0, and, consequently, ˜p0 “ 0, q0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, for α ă 0, we cannot find an eigenvector with a non trivial 0-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If m � 0 and σ1,m “ 0, we see that pqm, pmq and p˜qm, ˜pmq are related by Mm ˆqm pm ˙ “ α ˆ˜qm ˜pm ˙ , ˆqm pm ˙ “ Mm ˆ˜qm ˜pm ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (66) It means that α is an eigenvalue of M2 m “ ˜ m4 4 ` pk ¨ mq2 im2k ¨ m ´im2k ¨ m m4 4 ` pk ¨ mq2 ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The roots of the characteristic polynomial of M2 m are pm2 2 ˘ k ¨ mq2 ě 0, which contradicts the assumption α ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For the case where m � 0 and σ1,m � 0, we introduce the shorthand notation am “ 2γ2p2πq2d|σ1,m|2κ|α|{c2, bearing in mind that 0 ă am ă m2 2 by virtue of the smallness condi- tion (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We are led to the systems ˆ Mm ´ ˆ 0 0 0 am ˙˙ ˆ qm pm ˙ “ α ˆ ˜qm ˜pm ˙ , ˆ qm pm ˙ “ ˆ Mm ´ ˆ am 0 0 0 ˙˙ ˆ ˜qm ˜pm ˙ , which imply that α is an eigenvalue of the matrix ˆ Mm ´ ˆ 0 0 0 am ˙˙ ˆ Mm ´ ˆ am 0 0 0 ˙˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' However the eigenvalues of this matrix read ` b m2 2 pm2 2 ´ amq ˘ pk ¨ mq2˘2 ě 0, contradicting that α is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='11 Suppose (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' N ` n “ #tm P Zd ∖ t0u, σ1,m “ 0, and m4 ´ 4pk ¨ mq2 ă 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 50 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We should consider the system of equations (64)-(65), now with α ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For Fourier coefficients, it casts as m2 2 qm ` ik ¨ mpm “ α˜qm, m2 2 pm ´ ik ¨ mqm ´ γp2πqdσ1,m ˆ σ2πm dz “ α˜pm, ´2c2∆zφm “ α˜φm, ´1 2∆zπm ´ γp2πqdσ1,mσ2pm “ α˜πm, where qm “ m2 2 ˜qm ` ik ¨ m˜pm ` γp2πqdσ1,m ˆ p´∆zq´1{2σ2 ˜φm dz, pm “ m2 2 ˜pm ´ ik ¨ m˜qm, φm “ 1 2 ˜φm ` γp2πqdp´∆zq´1{2σ2σ1,m˜qm, πm “ 2c2˜πm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' For m “ 0, we obtain p0 “ 0, ˜q0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence π0 satisfies p´α{c2 ´ ∆zqπ0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Here, `α{c2 lies in the essential spectrum of ´∆z and the only solution in L2 of this equation is π0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In turn, this implies ˜p0 “ 0, p´α{c2 ´ ∆zqφ0 “ 0, and thus φ0 “ 0, q0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, for α ą 0, we cannot find an eigenvector with a non trivial 0-mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' When m � 0 and σ1,m “ 0, we are led to p´α{c2 ´ ∆qφm “ 0, p´α{c2 ´ ∆qπm “ 0 that imply φm “ 0, πm “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In turn, we get (66) for qm, pm, ˜qm, ˜pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This holds iff α is an eigenvalue of M2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' If m4 � 4pk ¨ mq2, we find two eigenvalues αm,˘ “ pm2 2 ˘ k ¨ mq2 ą 0, with associated eigenvectors Xm,˘ “ peim¨x, ¯ieim¨x, 0, 0q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' To decide whether these modes should be counted, we need to evaluate the sign of pL ´1Xm,˘|Xm,˘q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We start by solving L X1 m,˘ “ Xm,˘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It yields φ1 m,˘ 2 “ 0, 2c2π1 m,˘ “ 0 and Mm ˆq1 m,˘ p1 m,˘ ˙ “ ˆ 1 ¯i ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We obtain q1 m,˘ “ 2 m2 ˘ 2k ¨ m, π1 m,˘ “ ¯2i m2 ˘ 2k ¨ m, so that pL ´1Xm,˘|Xm,˘q “ 2 m2 ˘ 2k ¨ m ˆˆ Td eim¨xe´im¨x dx ` ˆ Tdp¯iqeim¨x˘ie´im¨x dx ˙ “ 4p2πqd m2 ˘ 2k ¨ m, the sign of which is determined by the sign of m2 ˘2k¨m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We count only the situation where these quantities are negative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' reproducing a discussion made in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3, we conclude that N ` n ě #tm P Zd ∖ t0u, σ1,m “ 0 and m4 ´ 4pk ¨ mq2 ă 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 51 When m4 ´ 4pk ¨ mq2 “ 0, the eigenvalues of M2 n are 0 and m4, and we just have to consider the positive eigenvalue α “ m4, associated to the eigenvector Xm “ peim¨x, ˘ieim¨x, 0, 0q (depending whether m2 2 “ ¯k ¨ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The equation L Ym “ Xm has infinitely many solu- tions of the form p2{m2eim¨x, 0, 0, 0q ` zp˘ieim¨x, eim¨x, 0, 0q, with z P C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We deduce that pL ´1Xm|Xmq “ 2p2πqd m2 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Thus these modes do not affect the counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' When m � 0 and σ1,m � 0, we are led to the relations p´α{c2 ´ ∆zqπm “ 2σ2γp2πqdσ1,mpm, p´α{c2 ´∆zq˜φm “ ´2p´∆zq1{2σ2γp2πqdσ1,m˜qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The only solutions with square integrability on Rn are πm “ 0, ˜φm “ 0, pm “ 0, ˜qm “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This can be seen by means of Fourier transform: p´α{c2 ´ ∆zqφ “ σ amounts to pφpξq “ pσpξq |ξ|2´α{c2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' due to (H4) this function has a singularity which cannot be square-integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In turn, this equally implies φm “ 0 and ˜πm “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, we arrive at m2 2 qm “ 0 and ´ik ¨ mqm “ α˜pm, together with qm “ ik ¨ m˜pm and m2 2 ˜pm “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We conclude that α ą 0 cannot be an eigenvalue associated to a m-mode such that m � 0 and σ1,m � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We can now make use of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='7, together with Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This leads to 0 ` 1 ` #tm P Zd ∖ t0u, σ1,m “ 0, and m4 ´ 4pk ¨ mq2 ă 0u ` NC` “ N ´ n ` N 0 n ` N ` n ` NC` “ npL q “ 1 ` #tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ă 0 and σ1,m “ 0u `#tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ď 0 and σ1,m � 0u so that NC` “ #tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ď 0 and σ1,m � 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since the eigenvalue problem (59) does not have negative (real) eigenvalues, this is the only source of instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a matter of fact, when k “ 0, we obtain NC` “ 0, which yields the following statement, (hopefully!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=') consistent with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='12 Let k “ 0 and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose (9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then the plane wave solution peiωt1pxq, ´γΓpzq @ σ D Td, 0q is spectrally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In contrast to what happens for the Hartree equation, for which the eigenvalues are purely imaginary, see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2, we can find unstable modes, despite the smallness condition (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let us consider the following two examples in dimension d “ 1, with k P Z ∖ t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='13 Suppose σ1,0 � 0, and σ1,1 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, the set tm P Z ∖ t0u, m4 ´ 4k2m2 ď 0 and σ1,m � 0u contains t´1, `1u (since 4k2 ě 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let k P Z ∖ t0u and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then the plane wave solution peiωteikx, ´γΓpzq @ σ1 D Td, 0q is spectrally unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='14 Let m˚ P Z ∖ t0u be the first Fourier mode such that σ1,m˚ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let k P Z and ω ą 0 such that the dispersion relation (12) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then, for all k P Z such that 4k2 ă m2 ˚, the plane wave solution peiωteikx, ´γΓpzq @ σ D Td, 0q is spectrally stable, while for all k P Z such that 4k2 ě m2 ˚, the plane wave solution peiωteikx, ´γΓpzq @ σ1 D Td, 0q is spectrally unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 52 In general, if k P Zd ∖ t0u, the set tm P Zd ∖ t0u, m4 ´ 4pk ¨ mq2 ď 0 and σ1,m � 0u contains ´k and k provided σ1,k � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='15 Let k P Zd ∖ t0u and ω ą 0 such that the dispersion relation (12) is satis- fied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Suppose (9) holds and σ1,m � 0 for all m P Zd ∖ t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Then the plane wave solution peipωt`k¨xq, ´γΓpzq @ σ1 D Td, 0q is spectrally unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='16 (Orbital instability) Given Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='15, it is natural to ask whether in this case the plane wave solution peipωt`k¨xq, ´γΓpzq @ σ1 D Td, 0q is orbitally unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Note that, if σ1,m � 0 for all m P Zd ∖ t0u, we deduce from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3 that npLq ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' As a consequence, the arguments used in [22] to prove the orbital instability (see also [38, 41]) do not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It seems then necessary to work directly with the propagator generated by the linearized operator as in [23, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, one has to establish Strichartz type estimates for the propagator of L (a task we leave for future work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' A Scaling of the model and physical interpretation It is worthwhile to discuss the meaning of the parameters that govern the equations and the asymptotic issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Going back to physical units, the system reads ˆ iℏBtU ` ℏ2 2m∆xU ˙ pt, xq “ ˆˆ TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dy dz ˙ upt, xq, (67a) pB2 ttΨ ´ κ2∆zΨqpt, x, zq “ ´σ2pzq ˆˆ Td σ1px ´ yq|Upt, yq|2 dy ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (67b) The quantum particle is described by the wave function pt, xq ÞÑ Upt, xq: given Ω Ă Td, the integral ´ Ω |Upt, xq|2 dx gives the probability of presence of the quantum particle at time t in the domain Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' this is a dimensionless quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In (67a), ℏ stands for the Planck constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' its homogeneity is MassˆLength2 Time (and its value is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='055 ˆ 10´34 Js) and m is the inertial mass of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let us introduce mass, length and time units of observations: M, L and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It helps the intuition to think of the z directions as homogeneous to a length, but in fact this is not necessarily the case: we denote by Ψ and Z the (unspecified) units for Ψ and the zj’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, κ is homogeneous to the ratio Z T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The coupling between the vibrational field and the particle is driven by the product of the form functions σ1σ2, which has the same homogeneity as ℏ TΨLdZn from (67a) and as Ψ LdT2 from (67b), both are thus measured with the same units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' From now on, we denote by ς this coupling unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore, we are led to the following dimensionless quantities U1pt1, x1q “ Upt1T, x1Lq c Ld m M, Ψ1pt1, x1, z1q “ 1 ΨΨpt1T, x1L, z1Zq, σ1 1px1qσ2pz1q “ 1 ς σ1px1Lqσ2pz1Zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Bearing in mind that u is a probability density, we note that ˆ Td |U1pt1, x1q|2 dx1 “ m M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 53 Dropping the primes, (67a)-(67b) becomes, in dimensionless form, ˆ iBtU ` ℏT mL2 1 2∆xU ˙ pt, xq “ ςΨLdZnT ℏ ˆˆ TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dy dz ˙ Upt, xq, (68a) ´ B2 ttΨ ´ κ2T2 Z2 ∆zΨ ¯ pt, x, zq “ ´ςT2 Ψ M mσ2pzq ˆˆ Td σ1px ´ yq|Upt, yq|2 dy ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (68b) Energy conservation plays a central role in the analysis of the system: the total energy is defined by using the reference units and we obtain E0 “ ´ ℏT mL2 ¯2 1 2 ˆ Td |∇xU|2 dx ` Ψ2LdZn ML2 1 2 ¨ TdˆRn ´ |BtΨ|2 ` κ2T2 Z2 |∇zΨ|2¯ dz dx `ς ΨLdZnT2 mL2 ¨ TdˆRn |U|2σ2σ1 ‹ Ψ dz dx, with E0 dimensionless (hence the total energy of the original system is E0 ML2 T2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore, we see that the dynamics is encoded by four independent parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In what follows, we get rid of a parameter by assuming ℏT mL2 “ 1, and we work with the following three independent parameters α “ ςΨLdZnT2 mL2 mL2 ℏT , β “ ςZ2 κ2Ψ M m, c “ κT Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It leads to ˆ iBtU ` 1 2∆xU ˙ pt, xq “ α ˆˆ TdˆRn σ1px ´ yqσ2pzqΨpt, y, zq dy dz ˙ Upt, xq, (69a) ´ 1 c2 B2 ttΨ ´ ∆zΨ ¯ pt, x, zq “ ´βσ2pzq ˆˆ Td σ1px ´ yq|Upt, yq|2 dy ˙ (69b) together with E0 “ 1 2 ˆ Td |∇xU|2 dx ` 1 2 α β ¨ TdˆRn ´ 1 c2 |BtΨ|2 ` |∇zΨ|2¯ dz dx `α ¨ TdˆRn |U|2σ2σ1 ‹ Ψ dz dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This relation allows us to interpret the scaling parameters as weights in the energy balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Now, for notational convenience, we decide to work with a m M b α β Ψ instead of Ψ and b M mU instead of U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' it leads to (3a)-(3c) and (8) with γ “ b M m ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Accordingly, we shall implicitly work with solutions with amplitude of magnitude unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The regime where c Ñ 8, with α, β fixed leads, at least formally, to the Hartree system (1a)-(1b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' arguments are sketched in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The smallness condition (9) makes a threshold appear on the coefficients in order to guaranty the stability: since it involves the product M mαβ, it can be interpreted as a condition on the strength of the coupling between the particle and the environment, and on the amplitude of the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We shall see in the proof that a sharper condition can be derived, expressed by means of the Fourier coefficients of the form function σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 54 B From Schödinger-Wave to Hartree In this Section we wish to justify that solutions – hereafter denoted Uc – of (3a)-(3c) converge to the solution of (1a)-(1b) as c Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We adapt the ideas in [10] where this question is investigated for Vlasov equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Throughout this section we consider a sequence of initial data UInit c , ΨInit c , ΠInit c such that sup cą0 ˆ Td |UInit c |2 dx “ M0 ă 8, (70a) sup cą0 ˆ Td |∇xUInit c |2 dx “ M1 ă 8, (70b) sup cą0 " 1 2c2 ¨ TdˆRn |ΠInit c |2 dz dx ` 1 2 ¨ TdˆRn |∇zΨInit c |2 dz dx “ M2 ă 8, (70c) sup cą0 ¨ |UInit c |2σ1 ‹ σ2|ΨInit c | dz dx “ M3 ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (70d) There are several direct consequences of these assumptions: The total energy is initially bounded uniformly with respect to c ą 0, In fact, we shall see that the last assumption can be deduced from the previous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since the L2 norm of Uc is conserved by the equation, we already know that Uc is bounded in L8p0, 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pTdqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we reformulate the expression of the potential, separating the contribution due to the initial data of the wave equation and the self-consistent part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' By using the linearity of the wave equation, we can split Φc “ ΦInit,c ` ΦCou,c where ΦInit,c is defined from the free-wave equation on Rn and initial data ΨInit c , ΠInit c : 1 c2 B2 ttΥc ´ ∆zΨ “ 0, pΥc, BtΥcq ˇˇ t“0 “ pΨInit c , ΠInit c q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (71) Namely, we set ΦInit,cpt, xq “ ˆ Rn σ2pzqσ1 ‹ Υcpt, x, zq dz “ ˆ Rn ´ cospc|ξ|tqσ1 ‹ pΨInit c px, |ξq ` sinpc|ξ|t c|ξ| σ1 ‹ pΨInit c px, |ξq ¯pσ2pξq dξ p2πqn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Accordingly rΨc “ Ψc ´ Υc satisfies 1 c2 B2 tt rΨc ´ ∆z rΨc´ “ ´γσ2σ1 ‹ |Uc|2, prΨc, Bt rΨcq ˇˇ t“0 “ p0, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' (72) 55 and we get ΦCou,cpt, xq “ γ ˆ Rn σ2pzqσ1 ‹ rΨcpt, x, zq dz “ γ2c2 ˆ t 0 ˆ Rn sinpc|ξ|sq c|ξ| Σ ‹ |Uc|2pt ´ s, xq|pσ2pξq|2 dξ p2πqn ds “ γ2 ˆ ct 0 ˆˆ Rn sinpτ|ξ|q |ξ| |pσ2pξq|2 dξ p2πqn ˙ looooooooooooooooooomooooooooooooooooooon “ppτq Σ ‹ |Uc|2pt ´ τ{c, xq dτ, where it is known that the kernel p is integrable on r0, 8q [10, Lemma 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='1 There exists a constant Mw ą 0 such that sup c,t,x |ΦInit,cpt, xq| ď Mw, sup c,t,x |ΦCou,cpt, xq| ď Mw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Combining the Sobolev embedding theorem (mind the condition n ě 3) and the standard energy conservation for the free linear wave equation, we obtain }Υc}L8p0,8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='L2n{pn´2qpRnqqq ď C}∇zΥc}L8p0,8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='L2pTdˆRnqq ď C a 2M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Applying Hölder’s inequality, we are thus led to: |ΦInit,cpt, xq| ď C}σ2}L2n{pn`2qpRnq}σ1}L2pRdq a 2M2, (73) which proves the first part of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Incidentally, it also shows that (70d) is a consequence of (70a) and (70c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we get |ΦCou,cpt, xq| ď γ}Σ}L8pTdq}Uc}L8pr0,8q,L2pTdqq ˆ 8 0 |ppτq| dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2 There exists a constant MS ą 0 such that sup c,t }∇Ucpt, ¨q}L2pTdq ď MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This is a consequence of the energy conservation (the total energy being bounded by virtue of (70b)-(70d)) where the coupling term ˆ TdpΦInit,c ` ΦCou,cq|Uc|2 dx can be dominated by 2MwM0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Coming back to BtUc “ ´ 1 2i∆xUc ` γ i pΦInit,c ` ΦCou,cqUc (74) 56 we see that BtUc is bounded in L2p0, 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' H´1pTdqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Combining the obtained estimates with Aubin- Simon’s lemma [44, Corollary 4], we deduce that Uc is relatively compact in in C0pr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' LppTdqq, 1 ď p ă 2d d ´ 2, for any 0 ă T ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Therefore, possibly at the price of extracting a subsequence, we can suppose that Uc converges strongly to U in C0pr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pTdqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It remains to pass to the limit in (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The difficulty consists in letting c go to 8 in the potential term and to justify the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3 For any ζ P C8 c pp0, 8q ˆ Tdq, we have lim cÑ8 ˆ 8 0 ˆ TdpΦInit,c ` ΦCou,cqUcζ dx dt “ γκ ˆ 8 0 ˆ Td Σ ‹ |Uc|2 Ucζ dx dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We expect that ΦCou,c converges to γκΣ ‹ |U|2: ˇˇΦCou,cpt, xq ´ γκΣ ‹ |U|2pt, xq ˇˇ “ γ ˇˇˇˇ ˆ ct 0 Σ ‹ |Uc|2pt ´ τ{c, xqppτq dτ ´ κΣ ‹ |U|2pt, xq ˇˇˇˇ ď γ ˆ ct 0 ˇˇˇΣ ‹ |Uc|2pt ´ τ{c, xq ´ Σ ‹ |U|2pt, xq ˇˇˇ |ppτq| dτ ` γ ˆ 8 ct |ppτq| dτ ˆ }Σ ‹ |U|2}L8pp0,8qˆTdq ď γ ˆ ct 0 Σ ‹ ˇˇ|Uc|2 ´ |U|2ˇˇpt ´ τ{c, xq |ppτq| dτ `γ ˆ ct 0 Σ ‹ ˇˇ|U|2pt ´ τ{c, xq ´ |U|2pt, xq ˇˇ |ppτq| dτ `γ ˆ 8 ct |ppτq| dτ }Σ}L8pTdq}U}L8pp0,8q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='L2pTdqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Let us denote by Icpt, xq, IIcpt, xq, IIIcptq, the three terms of the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since p P L1pr0, 8qq, for any t ą 0, IIIcptq tends to 0 as c Ñ 8, and it is dominated by }p}L1pr0,8q}Σ}L8pTdqM0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we have |Icpt, xq| ď }p}L1pr0,8q}Σ}L8pTdq sup sě0 ˆ Td ˇˇ|Uc|2 ´ |U|2ˇˇps, yq dy ď }p}L1pr0,8q}Σ}L8pTdq sup sě0 ˆˆ Td |Uc ´ U|2ps, yq dy ` 2Re ˆ TdpUc ´ UqUps, yq dy ˙ which also goes to 0 as c Ñ 8 and is dominated by 2M0}p}L1pr0,8qq}Σ}L8pTdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Eventually, we get |IIcpt, xq| ď }Σ}L8pTdq ˆ ct 0 ˆˆ Td ˇˇ|U|2pt ´ τ{c, yq ´ |U|2pt, yq ˇˇ dy ˙ |ppτq| dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since U P C0pr0, 8q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pTdqq, with }Upt, ¨q}L2pTdq ď M0, we can apply the Lebesgue theorem to show that IIcpt, xq tends to 0 for any pt, xq fixed, and it is dominated by 2M0}p}L1pr0,8qq}Σ}L8pTdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This allows us to pass to the limit in ˆ 8 0 ˆ Td ΦCou,cUcζ dx dt ´ κ ˆ 8 0 ˆ Td Σ ‹ |U|2Uζ dx dt “ ˆ 8 0 ˆ Td ΦCou,cpUc ´ Uqζ dx dt ` ˆ 8 0 ˆ Td ´ ΦCou,c ´ γκΣ ‹ |U|2¯ Uζ dx dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 57 It remains to justify that lim cÑ8 ˆ 8 0 ˆ Td Φinit,cUcζ dx dt “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The space variable x is just a parameter for the free wave equation (71), which is equally satisfied by σ1 ‹ Υc, with initial data σ1 ‹ pΨInit c , ΠInit c q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We appeal to the Strichartz estimate for the wave equation, see [26, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='3] or [45, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='2, for the case n “ 3],which yields c1{p ˜ˆ 8 0 ˆˆ Rn |σ1 ‹ Υcpt, x, yq|q dy ˙p{q dt ¸1{p ď C ˆ 1 c2 ˆ Rn |σ1 ‹ ΠInit c px, zq|2 dz ` ˆ Rn |σ1 ‹ ∇yΨInit c px, zq|2 dz ˙1{2 , for any admissible pair: 2 ď p ď q ď 8, 1 p ` n q “ n 2 ´ 1, 2 p ` n ´ 1 q ď n ´ 1 2 , pp, q, nq � p2, 8, 3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The L2 norm with respect to the space variable of the right hand side is dominated by b }σ1}L1pTdq M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It follows that ˆ Td ˜ˆ 8 0 ˆˆ Rn |σ1 ‹ Υcpt, x, zq|q dz ˙p{q dt ¸2{p dx ď C2}σ1}L1pRdq M2 1 c2{p ÝÝÝÑ cÑ8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Repeated use of the Hölder inequality (with 1{p ` 1{p1 “ 1) leads to ˇˇˇˇ ˆ 8 0 ˆ Td UcζΦInit,c dx dt ˇˇˇˇ ď ˜ˆ Td ˆˆ 8 0 |Ucζpt, xq|p1 dt ˙2{p1 dx ¸1{2 ˜ˆ Td ˆˆ 8 0 |ΦInit,cpt, xq|p dt ˙2{p dx ¸1{2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' On the one hand, assuming that ζ is supported in r0, Rs ˆ Td and p ą 2, we have ˆ Td ˆˆ 8 0 |Ucζ|p1 dt ˙2{p1 dx ď ˆ Td ˆˆ R 0 |Uc|2 dt ˙ ˆˆ R 0 |ζ|2p1{p2´p1q dt ˙p2´p1q{p1 dx ď R1`p2´p1q{p1}ζ}L8pp0,8qˆTdq}Uc}L8pp0,8q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='L2pTdqq which is thus bounded uniformly with respect to c ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' On the other hand, we get ˆ Td ˆˆ 8 0 |ΦInit,cpt, xq|p dt ˙2{p dx “ ˆ Td ˆˆ 8 0 ˇˇˇ ˆ Rn σ2pzqσ1 ‹ Υcpt, x, zq dz ˇˇˇ p dt ˙2{p dx ď }σ2}Lq1pRnq ˆ Td ˆˆ 8 0 ˇˇˇ ˆ Rn |σ1 ‹ Υcpt, x, zq|q dz ˇˇˇ p{q dt ˙2{p dx which is of the order Opc´2{pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 58 C Well-posedness of the Schrödinger-Wave system The well-posedness of the Schrödinger-Wave system is justified by means of a fixed point argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The method described here works as well for the problem set on Rd, and it is simpler than the approach in [21] since it avoids the use of “dual” Strichartz estimates for the Schrödinger and the wave equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We define a mapping that associates to a function pt, xq P r0, Ts ˆ Td ÞÑ V pt, xq P C: first, the solution Ψ of the linear wave equation 1 c2 B2 ttΨ ´ ∆zΨ “ ´σ2σ1 ‹ |V |2, pΨ, BtΨq ˇˇ t“0 “ pΨ0, Ψ1q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' next, the potential Φ “ σ1 ‹ ´ Rn σ2Ψ dz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' and finally the solution of the linear Schrödinger equation iBtU ` 1 2∆xU “ γΦU, U ˇˇ t“0 “ UInit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' These successive steps define a mapping S : V ÞÝÑ U and we wish to show that this mapping admits a fixed point in C0pr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pTdqq, which, in turn, provides a solution to the non linear system (3a)-(3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In this discussion, the initial data UInit, Ψ0, Ψ1 are fixed once for all in the space of finite energy: UInit P H1pTdq, Ψ0 P L2pTd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='H1pRnqq, Ψ1 P L2pTd ˆ Rnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We observe that d dt ˆ Td |U|2 dx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence, the mapping S applies the ball Bp0, }UInit}L2pTdqq of C0pr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pTdqq in itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' we thus consider U “ SpV q for V P C0pr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pTdqq such that }V pt, ¨q}L2pTdq ď }UInit}L2pTdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Moreover, we can split Ψ “ Υ ` rΨ with Υ solution of the free wave equation 1 c2 B2 ttΥ ´ ∆zΥ “ 0, pΥ, BtΥq ˇˇ t“0 “ pΨ0, Ψ1q, and 1 c2 B2 tt rΨ ´ ∆z rΨ “ 0, pΥ, Bt rΨq ˇˇ t“0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We write Φ “ ΦI ` rΦ for the associated splitting of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular, the standard energy conservation for the wave equation tells us that 1 2c2 ¨ TdˆRn |BtΥ|2 dz dx ` 1 2 ¨ TdˆRn |∇zΥ|2 dz dx “ 1 2c2 ¨ TdˆRn |Ψ1|2 dz dx ` 1 2 ¨ TdˆRn |∇zΨ0|2 dz dx “ M2 59 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It follows that |ΦIpt, xq| ď C}σ2}L2n{pn`2pRnq}σ1}L2pTdq a 2M2 by using Sobolev’s embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Next, we obtain rΦpt, xq “ ˆ Rn σ2pzqσ1 ‹ rΨpt, x, zq dz “ γ ˆ ct 0 ˆˆ Rn sinpτ|ξ|q |ξ| |pσ2pξq|2 dξ p2πqn ˙ looooooooooooooooooomooooooooooooooooooon “ppτq Σ ‹ |V |2pt ´ τ{c, xq dτ, which thus satisfies sup xPTd |rΦpt, xq| ď γ}Σ}L8pTdq ˆ ct 0 |ppτq| ˆˆ Td |V |2pt ´ τ{c, yq dy ˙ dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' In particular |rΦpt, xq| ď γ}Σ}L8pTdq}p}L1pp0,8qq}V }C0pr0,Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='L2pTdqq ď γ}Σ}L8pTdq}p}L1pp0,8qq}UInit}L2pTdq lies in L8pp0, Tq ˆ Tdq, and thus Φ P L8pp0, Tq ˆ Rdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' This observation guarantees that U “ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='SpV q is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Thus, let us pick V1, V2 in this ball of C0pr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pTdqq and consider Uj “ SpVjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We have iBtpU2 ´ U1q ` 1 2∆xpU2 ´ U1q “ γΦ2pU2 ´ U1q ` γpΦ2 ´ Φ1qU1, pU2 ´ U1q ˇˇ t“0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' It follows that d dt ˆ Td |U2 ´ U1|2 dx “ 2γIm ˆˆ TdpΦ2 ´ Φ1qU 1pU2 ´ U1q dx ˙ ď 2γ}U1}L2pTdq }U2 ´ U1}L2pTdq }Φ2 ´ Φ1}L8pTdq “ 2γ}U1}L2pTdq }U2 ´ U1}L2pTdq }rΦ2 ´ rΦ1}L8pTdq ď 2γ2}Σ}L8pTdq}UInit}L2pTdq }U2 ´ U1}L2pTdq ˆ ct 0 |ppτq| ˆˆ Td ˇˇ|V2|2 ´ |V1|2ˇˇpt ´ τ{c, yq dy ˙ dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We use the elementary estimate ˆ Td ˇˇ|V2|2´|V1|2ˇˇ dy “ ˆ Td ˇˇ|V2´V1|2`2RepV2´V1qV1 ˇˇ dy ď }V2´V1}2 L2pTdq`2}V2´V1}L2pTdq }V1}L2pTdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Combining this with Cauchy-Schwarz and Young inequalities, we arrive at d dt ˆ Td |U2 ´ U1|2 dx ď 2γ2}Σ}L8pTdq}UInit}L2pTdq ˆ 2}UInit}L2pTdq ˆ ct 0 |ppτq|}V2 ´ V1}2pt ´ τ{cqL2pTdq dτ `}U2 ´ U1}L2pTdq2}UInit}L2pTdq ˆ ct 0 |ppτq|}V2 ´ V1}pt ´ τ{cqL2pTdq dτ ˙ ď 2γ2}Σ}L8pTdq}UInit}2 L2pTdq ´ }U2 ´ U1}2 L2pTdq `p2 ` }p}L1pp0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='8qq ˆ ct 0 |ppτq|}V2 ´ V1}2pt ´ τ{cqL2pTdq dτ ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 60 Set L “ 2γ2}Σ}L8pTdq}UInit}2 L2pTdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We deduce that }U2 ´ U1}ptq2 L2pTdq ď p2 ` }p}L1pp0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='8qqL ˆ t 0 eLpt´sq ˆ cs 0 |ppτq|}V2 ´ V1}2ps ´ τ{cqL2pTdq dτ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' We use this estimate for 0 ď t ď T ă 8 and we obtain }U2 ´ U1}ptq2 L2pTdq ď p4 ` }p}L1pp0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='8qqLTeLT }p}L1pp0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content='8q sup 0ďsďT }V2 ´ V1}2psqL2pTdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Hence for T small enough, S is a contraction in C0pr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' L2pTdqq, and consequently it admits a unique fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Since the fixed point still has its L2 norm equal to }UInit}L2pTdq, the solution can be extended on the whole interval r0, 8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' The argument can be adapted to handle the Hartree system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' References [1] B.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' [51] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Zhang and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Travelling solitary waves for boson stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' El.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=', 2019:73: 1–12, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} +page_content=' 64' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FAT4oBgHgl3EQflB1V/content/2301.08614v1.pdf'} diff --git a/39AyT4oBgHgl3EQfP_aH/content/2301.00036v1.pdf b/39AyT4oBgHgl3EQfP_aH/content/2301.00036v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4567a0ed7a488adf2171b9b9bba9a5edf32b86c2 --- /dev/null +++ b/39AyT4oBgHgl3EQfP_aH/content/2301.00036v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e02d526b52fba9c7e1542cc47d63361547c7bc9faeb85e9c97721f2a1a58cf90 +size 791302 diff --git a/3dFQT4oBgHgl3EQfGzVw/content/tmp_files/2301.13246v1.pdf.txt b/3dFQT4oBgHgl3EQfGzVw/content/tmp_files/2301.13246v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e8fd52604b933585e8844bb3a7fcf1a0b1a1dd1f --- /dev/null +++ b/3dFQT4oBgHgl3EQfGzVw/content/tmp_files/2301.13246v1.pdf.txt @@ -0,0 +1,867 @@ +CONVERSATIONAL AUTOMATED PROGRAM REPAIR +Chunqiu Steven Xia, Lingming Zhang +University of Illinois at Urbana-Champaign +{chunqiu2, lingming}@illinois.edu +ABSTRACT +Automated Program Repair (APR) can help developers automatically generate +patches for bugs. Due to the impressive performance obtained using Large Pre- +Trained Language Models (LLMs) on many code related tasks, researchers have +started to directly use LLMs for APR. However, prior approaches simply repeat- +edly sample the LLM given the same constructed input/prompt created from the +original buggy code, which not only leads to generating the same incorrect patches +repeatedly but also miss the critical information in testcases. To address these lim- +itations, we propose conversational APR, a new paradigm for program repair that +alternates between patch generation and validation in a conversational manner. +In conversational APR, we iteratively build the input to the model by combining +previously generated patches with validation feedback. As such, we leverage the +long-term context window of LLMs to not only avoid generating previously incor- +rect patches but also incorporate validation feedback to help the model understand +the semantic meaning of the program under test. We evaluate 10 different LLM +including the newly developed ChatGPT model to demonstrate the improvement +of conversational APR over the prior LLM for APR approach. +1 +INTRODUCTION +Bugs in software can cause significant financial losses Matteson (2018) and create dangerous health +and safety problems Hanbury (2019). Due to the high manual cost of fixing bugs O’Dell (2017), +Automated Program Repair (APR) Gazzola et al. (2019) is a promising solution to reduce developer +work by automatically generating patches given the buggy code and failing testcases. +Traditionally, APR approaches commonly use the paradigm of Generate and Validate (G&V), where +APR tools will first generate a list of candidate patches given the original buggy code and then +validate each one sequentially until a plausible patch that passes all the testcases is found. Plausible +patch is then passed on to a human developer where they have to determine if this is a correct +patch that correctly fixes the underlying bug. Traditional APR approaches such as template-based +tools Ghanbari et al. (2019); Liu et al. (2019); Lou et al. (2020) have been proven useful in fixing +bugs with pre-defined templates to match buggy and corresponding fix code patterns. Recently, +researchers have designed learning-based APR tools Ye et al. (2022); Zhu et al. (2021); Jiang et al. +(2021) which build a Neural Machine Translation (NMT) model by training on pairs of buggy and +patch code. However, these learning-based APR tools suffer from lack of patch variety as it can +only repair the types of bugs that are a part of the buggy/patch training data. Furthermore, these bug +fixing datasets can be difficult to construct as it require scraping open-source bug fix commits which +may contain many false positives, adding noise to the dataset. +Recognizing the limitation of prior learning-based APR tools, researchers have started to look +into directly leveraging Large Pre-Trained Language Models (LLMs) for APR without fine-tuning. +LLMs have proven their ability in various code generation tasks Austin et al. (2021). Xia & Zhang +(2022) first introduced cloze-style APR where a LLM directly fill-in the correct code given its sur- +rounding context. Other studies Prenner et al. (2022); Kolak et al. (2022); Xia et al. (2022) have also +investigated directly applying different types of LLMs for APR by smartly applying prompts or giv- +ing original buggy code as context. Typically, directly applying LLMs for APR involves creating a +common prompt/prefix which can be just the buggy context (zero-shot) or combining buggy context +with a few examples of bug fixes (few-shot) as input to the model. Following the G&V paradigm, +1 +arXiv:2301.13246v1 [cs.SE] 30 Jan 2023 + +prior approach will sample the LLMs multiple times to obtain candidate patches. However, this +pipeline has the following limitations: +First, sampling from the same prefix/prompt multiple times can lead to many repeated patches due +to the probabilistic nature of sampling. This means the LLMs may waste a lot of compute and +time generating the same patches which have already been validated as incorrect by the testsuite. +Second, prompts provided to the LLMs for APR are created only from the original buggy code and +does not include any of the testcase information. Such information like the expected input and output +examples that can help LLMs understand the functionality of the buggy program are not provided. +Third, prior approaches also fail to consider the outputs produced by the generated incorrect patches. +Previously incorrect patches may fail on a particular corner case, which can be exposed by looking +at the test output and providing it to the LLM to address it in future patches. +Our Work. We propose conversational APR – a new paradigm of using LLMs for APR that di- +rectly leverages the testcase validation information to provide feedback to LLMs in a conversational +manner. In conversational APR, we interleave patch generation with validation where LLM first +generates a patch, we then validate it against testsuite to provide feedback and prompt LLM with +the new feedback information to generate a new patch. While in this paper we consider simple test- +case input/output/error validation feedback, one can apply conversational APR with a wild range of +possible feedback information such as human evaluation of the patch. We refer to the process of +generating a patch followed by validation as a turn where a conversation chain is made up of mul- +tiple turns in sequence. In the start of the conversation chain, we begin with an initial prompt and +sample the LLM to obtain a candidate patch. As we continue the conversation, the input given to the +LLM in each turn is a concatenation of all previously incorrect patches along with their associated +testcase feedback within the same conversation chain. A conversational chain is terminated once a +patch that passes all the testcases are found or the maximum chain length is reached (i.e., maximum +number of turns). In the latter case, we start a new conversation chain with the initial prompt again. +Compared with prior LLM for APR tools which only use the buggy code snippet as inputs, conver- +sational APR incorporates patch validation in the form of validation feedback to help the model un- +derstand the reason why previously generated patches are incorrect. Such feedback can contain the +incorrect and expected test outputs or indicate if the generated patch contains compilation/runtime +errors. Furthermore, while prior LLM for APR tools continuously sample from the same input, our +approach iteratively builds the input by including previously incorrect patches. As such, the LLM, +through its long context window, can recognize previous generations and avoid repeatedly generat- +ing an already validated incorrect patch. We evaluated our conversational APR by using 10 popular +LLMs, where we found that our approach not only improves the number of bugs fixed but also +can arrive at the correct patch faster compared with sampling-based baseline. Furthermore, we also +evaluate the recently developed ChatGPT Schulman et al. (2022)1, a dialogue focused LLM trained +using reinforcement learning and highlight the performance of conversational APR when using a +LLM designed for conversation/dialogue. +2 +BACKGROUND & RELATED WORK +2.1 +LLMS FOR APR +To combat the reliance on training using bug-fixing datasets to build learning-based APR tools based +on NMT models, researchers directly applied LLMs for APR without any fine-tuning. Xia & Zhang +(2022) proposed AlphaRepair, the first cloze-style APR to directly leverage LLMs for APR in a +zero-shot setting by removing the buggy line and replacing it with masked tokens. AlphaRepair +then queries the CodeBERT Feng et al. (2020) model to fill-in the masked tokens with the correct +tokens to generate patches. Prenner et al. (2022) investigated the ability for Codex Chen et al. (2021) +to repair bugs using a simple prompting method to generate a complete patched function given the +original buggy function. Kolak et al. (2022) evaluated the scaling effect of LLMs for APR by using +4 LLMs of different model sizes to generate a single line fix given only the original buggy prefix +(i.e., removing all lines after and including the buggy line of the buggy function). Recently, Xia et al. +(2022) conducted an extensive study on directly applying LLMs for APR. In the study, they adopt +1While we perform repair using ChatGPT, no part of this paper is written by ChatGPT. :) +2 + +several repair settings, including few-shot generation using a few examples of bug fixes, cloze-style +APR and also single line generation. +The findings across these prior work is consistent in showing that directly using LLMs for APR +achieves comparable if not better performance compared to prior APR tools. However, these pro- +posed LLMs for APR techniques almost exclusively use sampling where patches are generated by +sampling from the same input over and over again, leading to many repeated patches. Furthermore, +the inputs to the LLMs are only constructed from the original buggy function, missing the rich infor- +mation in the form of testcases. In this work, our conversational APR approach aims to bridge these +limitations in LLMs for APR by constructing new inputs based on prior incorrect patches to avoid +sampling repeated patches and providing the validation feedback to add another dimension of input +apart from original buggy code to help the model understand the semantic meaning of the program. +2.2 +MULTI-STEP PROGRAM REASONING AND SYNTHESIS USING LLMS +A related research direction is in applying multi-step reasoning for code understanding and synthe- +sis. Nye et al. (2021) trains a LLM designed for program understanding by introducing the idea +of a “scratchpad” in which the LLM predicts the intermediate states of a program along with the +final execution results. Chen et al. (2022) extends the chain-of-thoughts Wei et al. (2022) prompting +style in NLP to propose program-of-thoughts where the prompt contains an explicit command to +construct the program step-by-step. However, these work still generates a complete result (i.e., final +program execution or code), albeit with intermediate results, in one shot, whereas our conversational +APR samples multiple times LLMs with different inputs to obtain one output plausible patch. +Different from one-shot methods, Austin et al. (2021) investigated the ability for LLMs to use hu- +man feedback in a conversational manner for program synthesis. The approach works by keeping a +conversation of previously generated code and correcting any mistake using natural language feed- +back provided by human developers. Nijkamp et al. (2022) manually created a multi-step synthesis +dataset where each target program is broken down into multiple smaller steps where only a few lines +of code needs to be generated. They then sample the model multiple times to iteratively complete +each smaller step and concatenate them together to form the final program. While these described +techniques involve iteratively sampling from the model with new feedback similar to a conversa- +tional manner, our work can automatically create this feedback through testcase execution without +any human-in-the-loop. +3 +CONVERSATIONAL APR +We propose a conversational APR approach to prompt LLM patch generation by combining previ- +ously generated patches and validation feedback in a conversational manner. Contrasting with the +classic Generate and Validate (G&V) APR approach that first generates a large number of candidate +patches and then validate each one to find a list of plausible patches, conversational APR interleaves +generation and validation to provide immediate feedback for the new candidate patch. Different +from previous APR tools which make use of LLMs through sampling given the same prefix/context +for each bug, conversational APR approach aims to incorporate feedback information after each +generation (if the candidate patch failed to pass all tests) as new context for subsequent generations. +Specifically, the feedback information includes both the incorrect generated patch and its associated +failed testcase information. +Conversational APR involves iteratively obtaining new candidate patches from the LLM by using +previously generated patches/validation results as feedback. We refer to this process as a turn, where +each turn includes three different steps: 1) construct new a prompt based on prior feedback 2) sam- +ple the model to produce a sample output function 3) validate the sample output function against +testcases to obtain validation feedback. Multiple turns in sequence is defined as a chain. The ter- +minating conditions are that the sample output patch is able to pass all testcases (i.e., a plausible +patch is obtained) or the maximum number of turns (length of the chain) is reached. Note that each +turn (all three steps) are done automatically without needing any human-in-the-loop, this allows +conversational APR to be an automatic approach for program repair. +3 + +Turn 1: +Turn 2: +The following code is buggy. +def sieve(max): + primes = [] + for n in range(2, max): + if any(n%p for p in primes): + primes.append(n) + return primes +Please provide a fixed version. +def sieve(max): + primes = [] + for n in range(2, max+1): + if not any(n%p for p in primes): + primes.append(n) + return primes +The fixed version is still not correct. +def sieve(max): + primes = [] + for n in range(2, max): + if all(n%p for p in primes): + primes.append(n) + return primes +def sieve(max): + primes = [] + for n in range(2, max+1): + if all(n%p for p in primes): + primes.append(n) + return primes +sieve(4) returns [2, 4] but it should return [2, 3] +Please provide a fixed version. +The fixed version is still not correct. +sieve(2) returns [] but it should return [2] +Please provide a fixed version. +S +I +F1 +I +S1 +F1 +concatenate +S2 +F2 +I +S1 +F1 +S2 +F2 +concatenate +S3 +Turn 3: +F3 +Initial +Prompt +sample +output +validation +feedback +sample model +sample model +sample model +run +testcase +run +testcase +run +testcase +sample +output +validation +feedback +sample +output +The fixed version is correct! +validation +feedback +def sieve(max): + primes = [] + for n in range(2, max): + if any(n%p for p in primes): + primes.append(n) + return primes +def sieve(max): + primes = [] + for n in range(2, max): + if all(n % p for p in primes): + primes.append(n) + return primes +original buggy function +plausible patch +S1 +I +S2 +S3 +F3 +F2 +F1 +I +S1 +I +S1 +F1 +F1 +S2 +F2 +F3 +S3 +I +S1 +F1 +S2 +F2 +Figure 1: Overview of conversational APR with an illustrative example in fixing the buggy +sieve function +3.1 +PIPELINE & EXAMPLE +Figure 1 shows an illustrative example of a conversation chain (multiple turns) and an overview +of the pipeline of the conversational APR approach. We first take in as input the original buggy +function and a set of testcases which contains some failing tests that expose the underlying bug. +In the example, the buggy function (sieve) attempts to use to sieve algorithm to calculate the list +of prime numbers below the integer input (max). The location of the bug occurs on line 4 where +the buggy function incorrectly uses any instead of all. This bug is exposed by the testcase of +sieve(2) = [2] where the buggy function incorrectly returns an empty array []. +• Turn 1: We first create an initial prompt +I using the original buggy function which contains +natural language description to indicate that the function is buggy (The following code is +buggy) and the task we want the LLM to solve (Please provide a fixed version). We +then sample the model using the initial prompt +I to obtain the first sample output function S1 . +The change is made to line 4 where the function in S1 negated the original if condition. We then +validate S1 against the list of tests and found that while the new patch is able to successfully pass +the previous failing test of sieve(2) = [2], it returns [2, 4] for sieve(4) when the correct +output should be [2, 3]. This validation information F1 is collected as feedback to use during +the next conversation turn. +• Turn 2: Different from turn 1, where the input to the LLM is just the initial prompt +I , now we +provide the model also with the previously generated patch and its failing testcase. In short, we +construct the validation feedback F1 by using the failing testcase and indicate to the model that the +previous sample S1 is still not correct (The fixed version is still not correct) and +the new task (Please provide another fixed version). We then concatenate the initial +prompt, first sample output function and the validation feedback { I , S1 , F1 } together as the input +to the LLM. As such, the model is able to not only use the original buggy function but also use the +previously generated sample and its testcase feedback to generate a new patched function. Similar +to turn 1, we obtain S2 and F2 where the correct line 4 is obtained (switching any to all) but the +candidate patch function incorrectly reduced the upper range of the for loop by 1. +4 + +• Turn 3: Similar to turn 2, we first construct the new validation feedback F2 from the previous +failing test case. We then concatenate all previously sampled output along with its validation +feedback in sequence to produce { I , S1 , F1 , S2 , F2 }. Using this input, we then sample the LLM +again to produce the next candidate patch S3 . We observe that this candidate patch correctly fixes +the underlying bug and this is indicated by its validation F3 where it is able to pass all the testcases. +The program repair process is then terminated as we have obtained our plausible patch S3 . +Compared to prior approach in APR based on LLMs which simply samples from a pre-defined +prompt/context, conversational APR leverages the previously missing key feedback information in +the form of testcase results to prompt future patch generations. The testcase feedback not only tells +the LLM that the previous patches are incorrect (i.e. leading to more unique patches) but also pro- +vides input and output examples which helps the model to understand the underlying functionality +of the function (i.e. leading to more correct patches). +3.2 +DESIGN DECISIONS +In the above example illustrated in Figure 1, we show the overall pipeline of conversational APR. +However, there are different design decisions which can impact the performance of the approach: +Prompt engineering. Prompting has been shown to be an effective way of leveraging LLMs on +various downstream tasks without needing any explicit fine-tuning. In conversational APR approach, +we follow the style of prior work Xia et al. (2022) in providing a short and concise prompt with +respect to the description of the input and the task we want to model to solve. Additionally, we +follow prior guidelines and kept the prompt to be open-ended rather than to restrict the generation +with a close-ended prompt. One particular important prompt constructing is validation feedback +in providing the failing testcase to the LLM. In the Figure 1 example, we provide a functional +prompt that directly invokes the function and highlight the discrepancy between output and expected +testcase output. We refer to this as functional prompt since it directly calls the function with input +parameters similar to what one would do in code. In Section 6.2, we compare this style of validation +prompting with other methods including without any testcase information to demonstrate the benefit +of including validation feedback to the model. +Maximum chain length. Recall that a conversation chain refers to the continuous sequence of turns +to fix a bug. A chain is demonstrated in Figure 1 with a chain length of 3. Along with finding a +plausible patch, a preset value for the maximum chain length is also a terminating condition since +the LLM used will have a maximum context window and cannot take in arbitrary length inputs. +Once this maximum chain length is reached, conversational APR will restart from the beginning +(i.e., by crafting initial prompt again) with a new chain conversation. The maximum chain length +is a parameter which controls how much history the LLM may receive. A maximum chain length +of 1 refers to the base case of sampling from the initial prompt over and over again, meaning the +model does not know any of the previously generated incorrect patches. A higher maximum chain +length means the model can see multiple previously failed patches, however this also may not be +beneficial as it can cause the LLM to repeat some of the earlier patches or get stuck on a particular +implementation of the function. In Section 6.2, we evaluate the effect of the chain length has on +repair performance. +4 +DATASETS +In this section, we describe the LLMs used in our evaluation and also the repair benchmark used to +evaluate our proposed technique. +4.1 +LLMS +In our work, we evaluate 10 different LLMs to not only demonstrate the effect of scaling behavior +on our proposed conversational APR approach but also to evaluate how different pre-training and +model design contribute to the overall effectiveness. Table 1 presents an overview of the studied +LLMs. Column Model is the model name, #Parameters indicates the number of model parameters, +Context Window represents the size of the context window, and Training Strategy refers to the +training strategy used. +5 + +Table 1: Evaluation LLM overview +Model +#Parameters +Context Window +Training Strategy +CODEGEN-MONO +350M/2B/6B/16B +2048 +Unsupervised CLM +CODEGEN-MULTI +350M/2B/6B/16B +2048 +Unsupervised CLM +Codex +12B +4096 +Unsupervised CLM +ChatGPT +∼175B +∼4000 +Reinforcement Learning +from Human Feedback + CLM +bitcount.py +bitcount.java +fixed line +fixed line +testcase +Figure 2: Example bug in both Python and Java in QuixBugs along with the testcases +• CODEGEN Nijkamp et al. (2022). A family of autoregressive LLMs trained using Causal Lan- +guage Modeling (CLM) objective (next-token-prediction) ranging from 350M to 16B in parameter +size. CODEGEN is first trained on the open-source ThePile Gao et al. (2020), containing 22 diverse +text-based datasets. The models are then trained on BigQuery BigQuery, a dataset of open-source +code from 6 programming languages. We refer to these models (trained on ThePile then Big- +Query) as CODEGEN-MULTI. CODEGEN-MULTI is then further trained on a dataset containing +large amounts of Python GitHub code to produce CODEGEN-MONO. In our experiments, we +use CODEGEN-MONO for repair benchmarks in Python and CODEGEN-MULTI for repair bench- +marks in other programming languages by refer to them both as CODEGEN for simplicity. +• Codex Chen et al. (2021). A programming language focused autoregressive model based on the +GPT-3 architecture Brown et al. (2020). Codex is first initialized with GPT-3 weights from training +on natural language corpus and then fine-tuned using next-token-prediction on a large dataset of +code files. While Codex also contains a version which can take in suffix tokens (i.e., fill-in code +in the middle), for our experiments, we only use Codex by providing the prefix context. +• ChatGPT Schulman et al. (2022). A conversational-based LLM first initialized from GPT-3.5 +model and then fine-tuned using Reinforcement Learning from Human Feedback (RLHF) Ziegler +et al. (2019). ChatGPT is first fine-tuned based on supervised learning where human provides +example responses to prompts in the dataset. Using this fine-tuned model, a reward model is +then trained by sampling multiple outputs of the model from a given prompt and again using a +human to rank the outputs. The reward model is used in the reinforcement learning step where +Proximal Policy Optimization Schulman et al. (2017) is used to fine-tune ChatGPT. Different from +the Codex and CODEGEN, ChatGPT through the usage of RLHF and fine-tuning data is designed +for conversation where the usage encourages a dialogue format. Note that much of the ChatGPT +model detail is unknown to the public, therefore, we can only provide an approximate value for +the number of parameters2 and context window size OpenAI (2022) according to verified sources. +4.2 +BENCHMARKS +We use the QuixBugs Lin et al. (2017) repair benchmark to evaluate our proposed conversational +APR approach. +QuixBugs has been widely used to evaluate many repair tools including both +learning-based Ye et al. (2022); Zhu et al. (2021); Jiang et al. (2021); Drain et al. (2021) and LLM for +APR Xia & Zhang (2022); Xia et al. (2022); Kolak et al. (2022); Prenner et al. (2022) approaches. +QuixBugs dataset contains the same 40 bugs and it associated correct patch in both Python and +Java. These bugs are self contained functions based on classic algorithms and it usually only takes +a single line change to fix the underlying bug. Each bug comes with a set of testcases which the +buggy function failed to pass and can be used to evaluate any candidate patch generated. Figure 2 +shows an example bug for the bitcount function in QuixBugs for both Java and Python. The bug +occurs inside the while loop where the code incorrectly uses the ˆ operator instead of & operator. We +also show the example testcases for bitcount where it contains example inputs and the expected +outputs when evaluated using the function. +2As ChatGPT is fine-tuned on GPT-3.5, we assume a similar number of parameters as GPT-3.5 +6 + +Out of the 40 bugs in QuixBugs, we further filter out 10 bugs which includes testcases that are +difficult to represent with our validation feedback prompt. For example, testcases for detect cycle +involves a graph as an input to the function. In total, we use 60 bugs (30 and 30 respectively for Java +and Python) for our evaluation. +5 +EXPERIMENTAL SETUP +In this section, we describe the key research questions that our evaluation seek to answer, the evalu- +ation metrics used and also describe the implementation details. +5.1 +RESEARCH QUESTIONS +We aim to investigate the following research questions: +• RQ1: What is the effectiveness of applying conversational APR? +• RQ2: How do different components of conversational APR effect performance? +In RQ1, we first compare the performance of conversational APR with a baseline approach used +in prior LLM for APR work where the patches are generated by continuously sampling from the +same initial prompt. We further evaluate both the scaling effective of LLM as we increase the size +of the model and also investigate the difference in performance of different pre-training strategies +(e.g., ChatGPT vs. Codex). In RQ2, we dive deeper into the different parameters of conversational +APR. Specifically, we evaluate how the length of the conversational chain and different validation +feedback prompts affect the performance. +5.2 +EVALUATION METRICS +Our evaluation metric consist of the standard metric used to evaluate APR tools: number of plausible +patches: patches which passes all the testcases and correct patches: patches which are semantically +equivalent to the reference developer patch. Additionally, since we are using sampling LLMs, we +also define tries as the number of samples needed to obtain a plausible/correct patch. This metric is +useful when comparing two approaches/models that achieve similar number of bugs fixed, the one +with fewer number of tries is preferred as we want to limit the number of times we have to sample +the LLM. +5.3 +IMPLEMENTATION +We implemented the LLM generation pipeline in Python using Hugging Face HuggingFace imple- +mentation of the CODEGEN models. We access Codex through the OpenAI API by querying the +code-davinci-002 engine. Since ChatGPT is not open-sourced and does not provide an official API +endpoint (like Codex), we manually input the prompt and extract the outputs. For all models apart +from ChatGPT, we use a default generation setting of nucleus sampling with top p = 0.95, tempera- +ture = 1, 50 samples per bug with a maximum chain length of 3. We generate and evaluate patches +on a 32-Core workstation with AMD Ryzen Threadripper PRO 5975WX CPU, 256 GB RAM and 3 +NVIDIA GeForce RTX 3090 GPUs, running Ubuntu 22.04.1 LTS. +6 +RESULTS +6.1 +RQ1: CONVERSATIONAL APR EFFECTIVENESS +We first evaluate the effectiveness of applying conversational APR using validation feedback com- +pared to prior method of sampling given the same prompt without any feedback. Table 2 shows the +results on QuixBugs-Python and QuixBugs-Java. We observe that by applying our feedback driven +conversational APR, we are able to improve the # of correct and plausible patches for all unsupervis- +edly trained LLM across all model sizes. Additionally, conversational APR is also able to decrease +the # of tries (# of samples) needed before obtaining the first plausible/correct patch. Compared +to traditional sampling method of producing patches, conversational APR is able to leverage the +7 + +Table 2: Conversational APR performance on both QuixBugs-Python and QuixBugs-Java +compared with baseline sampling method. #c/#p refers to the number of correct / plausible +patches. +Models +QuixBugs-Python +QuixBugs-Java +Sampling +Conversational +Sampling +Conversational +#c/#p +#tries +#c/#p +#tries +#c/#p +#tries +#c/#p +#tries +CODEGEN-350M +7 / 10 +20.5 +8 / 11 +18.4 +4 / 4 +24.2 +5 / 5 +23.5 +CODEGEN-2B +22 / 23 +16.6 +25 / 26 +14.3 +12 / 14 +18.8 +15 / 16 +16.4 +CODEGEN-6B +22 / 24 +14.0 +27 / 28 +12.1 +18 / 20 +19.8 +22 / 22 +13.5 +CODEGEN-16B +29 / 29 +5.6 +30 / 30 +4.8 +24 / 25 +14.5 +28 / 29 +13.2 +Codex +29 / 30 +4.6 +30 / 30 +3.8 +28 / 30 +7.2 +29 / 30 +5.7 +Table 3: ChatGPT and Codex comparison on QuixBugs-Python and QuixBugs-Java where +each cell indicates the number of correct / plausible patches +Models +QuixBugs-Python +QuixBugs-Java +one try +two tries +three tries +one try +two tries +three tries +Codex +16 / 16 +21 / 21 +24 / 24 +11 / 12 +18 / 19 +21 / 22 +ChatGPT +24 / 24 +27 / 28 +28 / 29 +24 / 24 +26 / 26 +26 / 26 +model’s understanding of natural language feedback to indicate why the patch is incorrect. LLMs +can use this validation feedback information to generate new patches that try to pass the previ- +ously failed testcase. Furthermore, conversational APR also helps to reduce the number of repeated +patches from sampling using the same prompt over and over again. By using the large context size +of many state-of-the-art LLMs, conversational APR can use recently generated incorrect patches as +previous context to prompt the model to generate a new patch that is different. +ChatGPT evaluation. We now evaluate the performance of ChatGPT when using conversational +APR. Due to the requirement of manually inputting and extracting outputs from ChatGPT, we only +use a single conversation chain with at most 3 tries (i.e. maximum chain length of 3). We compare +with the best performing LLM of Codex from previous results under the same setting in Table 3. +We observe that compared to Codex, which is trained in an unsupervised manner, ChatGPT which +is fine-tuned using Reinforcement Learning from Human Feedback (RLHF) performed much better +across the two repair datasets. This improvement in result can be partially attributed to increase +in model parameter size, but we believe this is also due to the dialogue-based fine-tuning dataset +used in ChatGPT. Conversational APR relies on the model understanding the validation feedback +to condition the future generation in trying to generate a patch that passes the testcase. A more +dialogue-oriented model such as ChatGPT is well suited for this task as both the training data and +algorithm contain feedback driven loops. As ChatGPT and other dialogue-based LLMs become +more popular, we believe conversational APR can also be further improved through more usage of +these LLMs. +6.2 +RQ2: COMPONENT ANALYSIS +Maximum chain length. We first investigate the effect of different maximum chain length has on +the repair performance. Figure 3 shows the number of plausible patches when we vary the maximum +chain length from 1 to 6 for the 4 CODEGEN models. Recall from Section 3 that chain length refers +Figure 3: Number of plausible patches for the 4 different CODEGEN models as we vary the +maximum chain length on QuixBugs-Python +8 + +CodeGen-350M +CodeGen-2B +CodeGen-6B +CodeGen-16B +12 +28 +Patches +30 +30 +10 +26 +28 +28 +8 +24 +Plausible +26 +26 +22 +24 +24 +20 +# +6 +2 +3 +4 +6 +1 +2 +3 +Maximum ChainLengthTable 4: Prompting Style Evaluation on QuixBugs-Python with each cell showing the number +of plausible patches +Models +no testcase +natural language +functional +CODEGEN-350M +9 +11 +11 +CODEGEN-2B +20 +25 +26 +CODEGEN-6B +24 +27 +28 +CODEGEN-16B +27 +30 +30 +Codex +29 +30 +30 +to the number of turns (each turn consist of generating and validating a new patch) in a conversation +chain. A maximum chain length of 1 is the simple sampling from the same initial prompt baseline +(used in prior LLM for APR tools). As we increase chain length, the model has to take in more +and more previous context in the form of prior generations and feedbacks. We observe that the +performance increase as we start from a small chain length and reaches the maximum around 3 or 4 +and then decrease as chain length continue to increase. The decrease in number of plausible patches +once we reach a high chain length is because the context may be too much for the model to handle +since it can include multiple previously failed patches. We also observe that this decrease is more +significant in smaller models, where larger models are less affected by longer chain length, showing +the ability for larger models to better capture the long term context dependencies. This shows that +the optimal chain length to use for conversational APR can be dependent on the individual LLM +used. +Feedback prompting style. +We now evaluate the effect of the feedback prompting style +used in our conversational APR. Table 4 shows the number of plausible patches using differ- +ent validation prompts in QuixBugs-Python. +Column no testcase does not include any test- +case feedback (only states that the patch is not correct), natural language describes the failing +testcase (e.g., when input is 2, the patch incorrectly returns [] but it should +return [2]) and functional which is the default prompting style discussed in Section 3. We ob- +serve that different prompting style does have an effect on the final performance of conversational +APR. Starting from no testcase prompt, we can improve performance by adding specific testcase +feedback information on top of telling the LLM that the patch is not correct. We also observe that +the functional prompting style, using the buggy/patch function name and passing parameters (see +Figure 1), performs the best. Functional prompting style conveys the failing testcase information in +a more concise and natural way by phrasing the testcase input and expected output relationship as a +function call. +7 +CONCLUSION +We propose conversational APR, a new paradigm for program repair that interleaves patch gener- +ation with validation to provide immediate feedback for LLMs to better prompt future generated +patches. Compared to previous LLM for APR approaches that only sample from the same input, +conversational APR iteratively builds the input by concatenating previously incorrect patches and +validation feedback. This allows for the model to avoid generating previously incorrect patches and +also understand the semantic meaning of the function through validation feedback. Our evaluation +on 10 different LLMs shows the improvement of conversational APR over the baseline sampling +method used in prior LLM for APR tools. Furthermore, we demonstrate the promising future of ap- +plying ChatGPT, a conversational/dialogue driven LLM, for conversational APR, or APR in general +for the first time. +REFERENCES +Jacob Austin, Augustus Odena, Maxwell Nye, Maarten Bosma, Henryk Michalewski, David Dohan, +Ellen Jiang, Carrie Cai, Michael Terry, Quoc Le, and Charles Sutton. Program synthesis with large +language models, 2021. arXiv:2108.07732. +BigQuery. +Bigquery github repos, 2022. +https://console.cloud.google.com/ +marketplace/details/github/github-repos. +Tom B. 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Fine-tuning language models from human preferences, 2019. +arXiv:1909.08593. +11 + diff --git a/3dFQT4oBgHgl3EQfGzVw/content/tmp_files/load_file.txt b/3dFQT4oBgHgl3EQfGzVw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..316b56fdd6bdd8bc578b850707ce9443a5c312d5 --- /dev/null +++ b/3dFQT4oBgHgl3EQfGzVw/content/tmp_files/load_file.txt @@ -0,0 +1,645 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf,len=644 +page_content='CONVERSATIONAL AUTOMATED PROGRAM REPAIR Chunqiu Steven Xia, Lingming Zhang University of Illinois at Urbana-Champaign {chunqiu2, lingming}@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='edu ABSTRACT Automated Program Repair (APR) can help developers automatically generate patches for bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Due to the impressive performance obtained using Large Pre- Trained Language Models (LLMs) on many code related tasks, researchers have started to directly use LLMs for APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' However, prior approaches simply repeat- edly sample the LLM given the same constructed input/prompt created from the original buggy code, which not only leads to generating the same incorrect patches repeatedly but also miss the critical information in testcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' To address these lim- itations, we propose conversational APR, a new paradigm for program repair that alternates between patch generation and validation in a conversational manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' In conversational APR, we iteratively build the input to the model by combining previously generated patches with validation feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' As such, we leverage the long-term context window of LLMs to not only avoid generating previously incor- rect patches but also incorporate validation feedback to help the model understand the semantic meaning of the program under test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We evaluate 10 different LLM including the newly developed ChatGPT model to demonstrate the improvement of conversational APR over the prior LLM for APR approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 1 INTRODUCTION Bugs in software can cause significant financial losses Matteson (2018) and create dangerous health and safety problems Hanbury (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Due to the high manual cost of fixing bugs O’Dell (2017), Automated Program Repair (APR) Gazzola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2019) is a promising solution to reduce developer work by automatically generating patches given the buggy code and failing testcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Traditionally, APR approaches commonly use the paradigm of Generate and Validate (G&V), where APR tools will first generate a list of candidate patches given the original buggy code and then validate each one sequentially until a plausible patch that passes all the testcases is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Plausible patch is then passed on to a human developer where they have to determine if this is a correct patch that correctly fixes the underlying bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Traditional APR approaches such as template-based tools Ghanbari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Lou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2020) have been proven useful in fixing bugs with pre-defined templates to match buggy and corresponding fix code patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Recently, researchers have designed learning-based APR tools Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2021) which build a Neural Machine Translation (NMT) model by training on pairs of buggy and patch code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' However, these learning-based APR tools suffer from lack of patch variety as it can only repair the types of bugs that are a part of the buggy/patch training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Furthermore, these bug fixing datasets can be difficult to construct as it require scraping open-source bug fix commits which may contain many false positives, adding noise to the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Recognizing the limitation of prior learning-based APR tools, researchers have started to look into directly leveraging Large Pre-Trained Language Models (LLMs) for APR without fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' LLMs have proven their ability in various code generation tasks Austin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Xia & Zhang (2022) first introduced cloze-style APR where a LLM directly fill-in the correct code given its sur- rounding context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Other studies Prenner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Kolak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2022) have also investigated directly applying different types of LLMs for APR by smartly applying prompts or giv- ing original buggy code as context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Typically, directly applying LLMs for APR involves creating a common prompt/prefix which can be just the buggy context (zero-shot) or combining buggy context with a few examples of bug fixes (few-shot) as input to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Following the G&V paradigm, 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='13246v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='SE] 30 Jan 2023 prior approach will sample the LLMs multiple times to obtain candidate patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' However, this pipeline has the following limitations: First, sampling from the same prefix/prompt multiple times can lead to many repeated patches due to the probabilistic nature of sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' This means the LLMs may waste a lot of compute and time generating the same patches which have already been validated as incorrect by the testsuite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Second, prompts provided to the LLMs for APR are created only from the original buggy code and does not include any of the testcase information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Such information like the expected input and output examples that can help LLMs understand the functionality of the buggy program are not provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Third, prior approaches also fail to consider the outputs produced by the generated incorrect patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Previously incorrect patches may fail on a particular corner case, which can be exposed by looking at the test output and providing it to the LLM to address it in future patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Our Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We propose conversational APR – a new paradigm of using LLMs for APR that di- rectly leverages the testcase validation information to provide feedback to LLMs in a conversational manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' In conversational APR, we interleave patch generation with validation where LLM first generates a patch, we then validate it against testsuite to provide feedback and prompt LLM with the new feedback information to generate a new patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' While in this paper we consider simple test- case input/output/error validation feedback, one can apply conversational APR with a wild range of possible feedback information such as human evaluation of the patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We refer to the process of generating a patch followed by validation as a turn where a conversation chain is made up of mul- tiple turns in sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' In the start of the conversation chain, we begin with an initial prompt and sample the LLM to obtain a candidate patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' As we continue the conversation, the input given to the LLM in each turn is a concatenation of all previously incorrect patches along with their associated testcase feedback within the same conversation chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' A conversational chain is terminated once a patch that passes all the testcases are found or the maximum chain length is reached (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=', maximum number of turns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' In the latter case, we start a new conversation chain with the initial prompt again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Compared with prior LLM for APR tools which only use the buggy code snippet as inputs, conver- sational APR incorporates patch validation in the form of validation feedback to help the model un- derstand the reason why previously generated patches are incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Such feedback can contain the incorrect and expected test outputs or indicate if the generated patch contains compilation/runtime errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Furthermore, while prior LLM for APR tools continuously sample from the same input, our approach iteratively builds the input by including previously incorrect patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' As such, the LLM, through its long context window, can recognize previous generations and avoid repeatedly generat- ing an already validated incorrect patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We evaluated our conversational APR by using 10 popular LLMs, where we found that our approach not only improves the number of bugs fixed but also can arrive at the correct patch faster compared with sampling-based baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Furthermore, we also evaluate the recently developed ChatGPT Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2022)1, a dialogue focused LLM trained using reinforcement learning and highlight the performance of conversational APR when using a LLM designed for conversation/dialogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 2 BACKGROUND & RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='1 LLMS FOR APR To combat the reliance on training using bug-fixing datasets to build learning-based APR tools based on NMT models, researchers directly applied LLMs for APR without any fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Xia & Zhang (2022) proposed AlphaRepair, the first cloze-style APR to directly leverage LLMs for APR in a zero-shot setting by removing the buggy line and replacing it with masked tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' AlphaRepair then queries the CodeBERT Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2020) model to fill-in the masked tokens with the correct tokens to generate patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Prenner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2022) investigated the ability for Codex Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2021) to repair bugs using a simple prompting method to generate a complete patched function given the original buggy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Kolak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2022) evaluated the scaling effect of LLMs for APR by using 4 LLMs of different model sizes to generate a single line fix given only the original buggy prefix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=', removing all lines after and including the buggy line of the buggy function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Recently, Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2022) conducted an extensive study on directly applying LLMs for APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' In the study, they adopt 1While we perform repair using ChatGPT, no part of this paper is written by ChatGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' :) 2 several repair settings, including few-shot generation using a few examples of bug fixes, cloze-style APR and also single line generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' The findings across these prior work is consistent in showing that directly using LLMs for APR achieves comparable if not better performance compared to prior APR tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' However, these pro- posed LLMs for APR techniques almost exclusively use sampling where patches are generated by sampling from the same input over and over again, leading to many repeated patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Furthermore, the inputs to the LLMs are only constructed from the original buggy function, missing the rich infor- mation in the form of testcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' In this work, our conversational APR approach aims to bridge these limitations in LLMs for APR by constructing new inputs based on prior incorrect patches to avoid sampling repeated patches and providing the validation feedback to add another dimension of input apart from original buggy code to help the model understand the semantic meaning of the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='2 MULTI-STEP PROGRAM REASONING AND SYNTHESIS USING LLMS A related research direction is in applying multi-step reasoning for code understanding and synthe- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Nye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2021) trains a LLM designed for program understanding by introducing the idea of a “scratchpad” in which the LLM predicts the intermediate states of a program along with the final execution results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2022) extends the chain-of-thoughts Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2022) prompting style in NLP to propose program-of-thoughts where the prompt contains an explicit command to construct the program step-by-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' However, these work still generates a complete result (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=', final program execution or code), albeit with intermediate results, in one shot, whereas our conversational APR samples multiple times LLMs with different inputs to obtain one output plausible patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Different from one-shot methods, Austin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2021) investigated the ability for LLMs to use hu- man feedback in a conversational manner for program synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' The approach works by keeping a conversation of previously generated code and correcting any mistake using natural language feed- back provided by human developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Nijkamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2022) manually created a multi-step synthesis dataset where each target program is broken down into multiple smaller steps where only a few lines of code needs to be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' They then sample the model multiple times to iteratively complete each smaller step and concatenate them together to form the final program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' While these described techniques involve iteratively sampling from the model with new feedback similar to a conversa- tional manner, our work can automatically create this feedback through testcase execution without any human-in-the-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 3 CONVERSATIONAL APR We propose a conversational APR approach to prompt LLM patch generation by combining previ- ously generated patches and validation feedback in a conversational manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Contrasting with the classic Generate and Validate (G&V) APR approach that first generates a large number of candidate patches and then validate each one to find a list of plausible patches, conversational APR interleaves generation and validation to provide immediate feedback for the new candidate patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Different from previous APR tools which make use of LLMs through sampling given the same prefix/context for each bug, conversational APR approach aims to incorporate feedback information after each generation (if the candidate patch failed to pass all tests) as new context for subsequent generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Specifically, the feedback information includes both the incorrect generated patch and its associated failed testcase information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Conversational APR involves iteratively obtaining new candidate patches from the LLM by using previously generated patches/validation results as feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We refer to this process as a turn, where each turn includes three different steps: 1) construct new a prompt based on prior feedback 2) sam- ple the model to produce a sample output function 3) validate the sample output function against testcases to obtain validation feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Multiple turns in sequence is defined as a chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' The ter- minating conditions are that the sample output patch is able to pass all testcases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=', a plausible patch is obtained) or the maximum number of turns (length of the chain) is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Note that each turn (all three steps) are done automatically without needing any human-in-the-loop, this allows conversational APR to be an automatic approach for program repair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 3 Turn 1: Turn 2: The following code is buggy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' def sieve(max): primes = [] for n in range(2, max): if any(n%p for p in primes): primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='append(n) return primes Please provide a fixed version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' def sieve(max): primes = [] for n in range(2, max+1): if not any(n%p for p in primes): primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='append(n) return primes The fixed version is still not correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' def sieve(max): primes = [] for n in range(2, max): if all(n%p for p in primes): primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='append(n) return primes def sieve(max): primes = [] for n in range(2, max+1): if all(n%p for p in primes): primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='append(n) return primes sieve(4) returns [2, 4] but it should return [2, 3] Please provide a fixed version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' The fixed version is still not correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' sieve(2) returns [] but it should return [2] Please provide a fixed version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' S I F1 I S1 F1 concatenate S2 F2 I S1 F1 S2 F2 concatenate S3 Turn 3: F3 Initial Prompt sample output validation feedback sample model sample model sample model run testcase run testcase run testcase sample output validation feedback sample output The fixed version is correct!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' validation feedback def sieve(max): primes = [] for n in range(2, max): if any(n%p for p in primes): primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='append(n) return primes def sieve(max): primes = [] for n in range(2, max): if all(n % p for p in primes): primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='append(n) return primes original buggy function plausible patch S1 I S2 S3 F3 F2 F1 I S1 I S1 F1 F1 S2 F2 F3 S3 I S1 F1 S2 F2 Figure 1: Overview of conversational APR with an illustrative example in fixing the buggy sieve function 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='1 PIPELINE & EXAMPLE Figure 1 shows an illustrative example of a conversation chain (multiple turns) and an overview of the pipeline of the conversational APR approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We first take in as input the original buggy function and a set of testcases which contains some failing tests that expose the underlying bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' In the example, the buggy function (sieve) attempts to use to sieve algorithm to calculate the list of prime numbers below the integer input (max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' The location of the bug occurs on line 4 where the buggy function incorrectly uses any instead of all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' This bug is exposed by the testcase of sieve(2) = [2] where the buggy function incorrectly returns an empty array [].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Turn 1: We first create an initial prompt I using the original buggy function which contains natural language description to indicate that the function is buggy (The following code is buggy) and the task we want the LLM to solve (Please provide a fixed version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We then sample the model using the initial prompt I to obtain the first sample output function S1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' The change is made to line 4 where the function in S1 negated the original if condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We then validate S1 against the list of tests and found that while the new patch is able to successfully pass the previous failing test of sieve(2) = [2], it returns [2, 4] for sieve(4) when the correct output should be [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' This validation information F1 is collected as feedback to use during the next conversation turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Turn 2: Different from turn 1, where the input to the LLM is just the initial prompt I , now we provide the model also with the previously generated patch and its failing testcase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' In short, we construct the validation feedback F1 by using the failing testcase and indicate to the model that the previous sample S1 is still not correct (The fixed version is still not correct) and the new task (Please provide another fixed version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We then concatenate the initial prompt, first sample output function and the validation feedback { I , S1 , F1 } together as the input to the LLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' As such, the model is able to not only use the original buggy function but also use the previously generated sample and its testcase feedback to generate a new patched function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Similar to turn 1, we obtain S2 and F2 where the correct line 4 is obtained (switching any to all) but the candidate patch function incorrectly reduced the upper range of the for loop by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 4 Turn 3: Similar to turn 2, we first construct the new validation feedback F2 from the previous failing test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We then concatenate all previously sampled output along with its validation feedback in sequence to produce { I , S1 , F1 , S2 , F2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Using this input, we then sample the LLM again to produce the next candidate patch S3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We observe that this candidate patch correctly fixes the underlying bug and this is indicated by its validation F3 where it is able to pass all the testcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' The program repair process is then terminated as we have obtained our plausible patch S3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Compared to prior approach in APR based on LLMs which simply samples from a pre-defined prompt/context, conversational APR leverages the previously missing key feedback information in the form of testcase results to prompt future patch generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' The testcase feedback not only tells the LLM that the previous patches are incorrect (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' leading to more unique patches) but also pro- vides input and output examples which helps the model to understand the underlying functionality of the function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' leading to more correct patches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='2 DESIGN DECISIONS In the above example illustrated in Figure 1, we show the overall pipeline of conversational APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' However, there are different design decisions which can impact the performance of the approach: Prompt engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Prompting has been shown to be an effective way of leveraging LLMs on various downstream tasks without needing any explicit fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' In conversational APR approach, we follow the style of prior work Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2022) in providing a short and concise prompt with respect to the description of the input and the task we want to model to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Additionally, we follow prior guidelines and kept the prompt to be open-ended rather than to restrict the generation with a close-ended prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' One particular important prompt constructing is validation feedback in providing the failing testcase to the LLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' In the Figure 1 example, we provide a functional prompt that directly invokes the function and highlight the discrepancy between output and expected testcase output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We refer to this as functional prompt since it directly calls the function with input parameters similar to what one would do in code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='2, we compare this style of validation prompting with other methods including without any testcase information to demonstrate the benefit of including validation feedback to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Maximum chain length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Recall that a conversation chain refers to the continuous sequence of turns to fix a bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' A chain is demonstrated in Figure 1 with a chain length of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Along with finding a plausible patch, a preset value for the maximum chain length is also a terminating condition since the LLM used will have a maximum context window and cannot take in arbitrary length inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Once this maximum chain length is reached, conversational APR will restart from the beginning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=', by crafting initial prompt again) with a new chain conversation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' The maximum chain length is a parameter which controls how much history the LLM may receive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' A maximum chain length of 1 refers to the base case of sampling from the initial prompt over and over again, meaning the model does not know any of the previously generated incorrect patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' A higher maximum chain length means the model can see multiple previously failed patches, however this also may not be beneficial as it can cause the LLM to repeat some of the earlier patches or get stuck on a particular implementation of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='2, we evaluate the effect of the chain length has on repair performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 4 DATASETS In this section, we describe the LLMs used in our evaluation and also the repair benchmark used to evaluate our proposed technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='1 LLMS In our work, we evaluate 10 different LLMs to not only demonstrate the effect of scaling behavior on our proposed conversational APR approach but also to evaluate how different pre-training and model design contribute to the overall effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Table 1 presents an overview of the studied LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Column Model is the model name, #Parameters indicates the number of model parameters, Context Window represents the size of the context window, and Training Strategy refers to the training strategy used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 5 Table 1: Evaluation LLM overview Model #Parameters Context Window Training Strategy CODEGEN-MONO 350M/2B/6B/16B 2048 Unsupervised CLM CODEGEN-MULTI 350M/2B/6B/16B 2048 Unsupervised CLM Codex 12B 4096 Unsupervised CLM ChatGPT ∼175B ∼4000 Reinforcement Learning from Human Feedback + CLM bitcount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='py bitcount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='java fixed line fixed line testcase Figure 2: Example bug in both Python and Java in QuixBugs along with the testcases CODEGEN Nijkamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' A family of autoregressive LLMs trained using Causal Lan- guage Modeling (CLM) objective (next-token-prediction) ranging from 350M to 16B in parameter size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' CODEGEN is first trained on the open-source ThePile Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2020), containing 22 diverse text-based datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' The models are then trained on BigQuery BigQuery, a dataset of open-source code from 6 programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We refer to these models (trained on ThePile then Big- Query) as CODEGEN-MULTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' CODEGEN-MULTI is then further trained on a dataset containing large amounts of Python GitHub code to produce CODEGEN-MONO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' In our experiments, we use CODEGEN-MONO for repair benchmarks in Python and CODEGEN-MULTI for repair bench- marks in other programming languages by refer to them both as CODEGEN for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Codex Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' A programming language focused autoregressive model based on the GPT-3 architecture Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Codex is first initialized with GPT-3 weights from training on natural language corpus and then fine-tuned using next-token-prediction on a large dataset of code files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' While Codex also contains a version which can take in suffix tokens (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=', fill-in code in the middle), for our experiments, we only use Codex by providing the prefix context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' ChatGPT Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' A conversational-based LLM first initialized from GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='5 model and then fine-tuned using Reinforcement Learning from Human Feedback (RLHF) Ziegler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' ChatGPT is first fine-tuned based on supervised learning where human provides example responses to prompts in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Using this fine-tuned model, a reward model is then trained by sampling multiple outputs of the model from a given prompt and again using a human to rank the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' The reward model is used in the reinforcement learning step where Proximal Policy Optimization Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2017) is used to fine-tune ChatGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Different from the Codex and CODEGEN, ChatGPT through the usage of RLHF and fine-tuning data is designed for conversation where the usage encourages a dialogue format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Note that much of the ChatGPT model detail is unknown to the public, therefore, we can only provide an approximate value for the number of parameters2 and context window size OpenAI (2022) according to verified sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='2 BENCHMARKS We use the QuixBugs Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2017) repair benchmark to evaluate our proposed conversational APR approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' QuixBugs has been widely used to evaluate many repair tools including both learning-based Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Drain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2021) and LLM for APR Xia & Zhang (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Kolak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Prenner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' (2022) approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' QuixBugs dataset contains the same 40 bugs and it associated correct patch in both Python and Java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' These bugs are self contained functions based on classic algorithms and it usually only takes a single line change to fix the underlying bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Each bug comes with a set of testcases which the buggy function failed to pass and can be used to evaluate any candidate patch generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Figure 2 shows an example bug for the bitcount function in QuixBugs for both Java and Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' The bug occurs inside the while loop where the code incorrectly uses the ˆ operator instead of & operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We also show the example testcases for bitcount where it contains example inputs and the expected outputs when evaluated using the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 2As ChatGPT is fine-tuned on GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='5, we assume a similar number of parameters as GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='5 6 Out of the 40 bugs in QuixBugs, we further filter out 10 bugs which includes testcases that are difficult to represent with our validation feedback prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' For example, testcases for detect cycle involves a graph as an input to the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' In total, we use 60 bugs (30 and 30 respectively for Java and Python) for our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 5 EXPERIMENTAL SETUP In this section, we describe the key research questions that our evaluation seek to answer, the evalu- ation metrics used and also describe the implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='1 RESEARCH QUESTIONS We aim to investigate the following research questions: RQ1: What is the effectiveness of applying conversational APR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' RQ2: How do different components of conversational APR effect performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' In RQ1, we first compare the performance of conversational APR with a baseline approach used in prior LLM for APR work where the patches are generated by continuously sampling from the same initial prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We further evaluate both the scaling effective of LLM as we increase the size of the model and also investigate the difference in performance of different pre-training strategies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=', ChatGPT vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Codex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' In RQ2, we dive deeper into the different parameters of conversational APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Specifically, we evaluate how the length of the conversational chain and different validation feedback prompts affect the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='2 EVALUATION METRICS Our evaluation metric consist of the standard metric used to evaluate APR tools: number of plausible patches: patches which passes all the testcases and correct patches: patches which are semantically equivalent to the reference developer patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Additionally, since we are using sampling LLMs, we also define tries as the number of samples needed to obtain a plausible/correct patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' This metric is useful when comparing two approaches/models that achieve similar number of bugs fixed, the one with fewer number of tries is preferred as we want to limit the number of times we have to sample the LLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='3 IMPLEMENTATION We implemented the LLM generation pipeline in Python using Hugging Face HuggingFace imple- mentation of the CODEGEN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We access Codex through the OpenAI API by querying the code-davinci-002 engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Since ChatGPT is not open-sourced and does not provide an official API endpoint (like Codex), we manually input the prompt and extract the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' For all models apart from ChatGPT, we use a default generation setting of nucleus sampling with top p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='95, tempera- ture = 1, 50 samples per bug with a maximum chain length of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We generate and evaluate patches on a 32-Core workstation with AMD Ryzen Threadripper PRO 5975WX CPU, 256 GB RAM and 3 NVIDIA GeForce RTX 3090 GPUs, running Ubuntu 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='1 LTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 6 RESULTS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='1 RQ1: CONVERSATIONAL APR EFFECTIVENESS We first evaluate the effectiveness of applying conversational APR using validation feedback com- pared to prior method of sampling given the same prompt without any feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Table 2 shows the results on QuixBugs-Python and QuixBugs-Java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We observe that by applying our feedback driven conversational APR, we are able to improve the # of correct and plausible patches for all unsupervis- edly trained LLM across all model sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Additionally, conversational APR is also able to decrease the # of tries (# of samples) needed before obtaining the first plausible/correct patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Compared to traditional sampling method of producing patches, conversational APR is able to leverage the 7 Table 2: Conversational APR performance on both QuixBugs-Python and QuixBugs-Java compared with baseline sampling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' #c/#p refers to the number of correct / plausible patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Models QuixBugs-Python QuixBugs-Java Sampling Conversational Sampling Conversational #c/#p #tries #c/#p #tries #c/#p #tries #c/#p #tries CODEGEN-350M 7 / 10 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='5 8 / 11 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='4 4 / 4 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='2 5 / 5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='5 CODEGEN-2B 22 / 23 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='6 25 / 26 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='3 12 / 14 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='8 15 / 16 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='4 CODEGEN-6B 22 / 24 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='0 27 / 28 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='1 18 / 20 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='8 22 / 22 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='5 CODEGEN-16B 29 / 29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='6 30 / 30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='8 24 / 25 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='5 28 / 29 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='2 Codex 29 / 30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='6 30 / 30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='8 28 / 30 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='2 29 / 30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='7 Table 3: ChatGPT and Codex comparison on QuixBugs-Python and QuixBugs-Java where each cell indicates the number of correct / plausible patches Models QuixBugs-Python QuixBugs-Java one try two tries three tries one try two tries three tries Codex 16 / 16 21 / 21 24 / 24 11 / 12 18 / 19 21 / 22 ChatGPT 24 / 24 27 / 28 28 / 29 24 / 24 26 / 26 26 / 26 model’s understanding of natural language feedback to indicate why the patch is incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' LLMs can use this validation feedback information to generate new patches that try to pass the previ- ously failed testcase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Furthermore, conversational APR also helps to reduce the number of repeated patches from sampling using the same prompt over and over again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' By using the large context size of many state-of-the-art LLMs, conversational APR can use recently generated incorrect patches as previous context to prompt the model to generate a new patch that is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' ChatGPT evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We now evaluate the performance of ChatGPT when using conversational APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Due to the requirement of manually inputting and extracting outputs from ChatGPT, we only use a single conversation chain with at most 3 tries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' maximum chain length of 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We compare with the best performing LLM of Codex from previous results under the same setting in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We observe that compared to Codex, which is trained in an unsupervised manner, ChatGPT which is fine-tuned using Reinforcement Learning from Human Feedback (RLHF) performed much better across the two repair datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' This improvement in result can be partially attributed to increase in model parameter size, but we believe this is also due to the dialogue-based fine-tuning dataset used in ChatGPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Conversational APR relies on the model understanding the validation feedback to condition the future generation in trying to generate a patch that passes the testcase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' A more dialogue-oriented model such as ChatGPT is well suited for this task as both the training data and algorithm contain feedback driven loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' As ChatGPT and other dialogue-based LLMs become more popular, we believe conversational APR can also be further improved through more usage of these LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='2 RQ2: COMPONENT ANALYSIS Maximum chain length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We first investigate the effect of different maximum chain length has on the repair performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Figure 3 shows the number of plausible patches when we vary the maximum chain length from 1 to 6 for the 4 CODEGEN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Recall from Section 3 that chain length refers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='Figure 3: Number of plausible patches for the 4 different CODEGEN models as we vary the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='maximum chain length on QuixBugs-Python ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='CodeGen-350M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='CodeGen-2B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='CodeGen-6B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='CodeGen-16B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='Patches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='Plausible ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='Maximum ChainLengthTable 4: Prompting Style Evaluation on QuixBugs-Python with each cell showing the number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='of plausible patches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='Models ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='no testcase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='natural language ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='functional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='CODEGEN-350M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='CODEGEN-2B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='CODEGEN-6B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='CODEGEN-16B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='Codex ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='to the number of turns (each turn consist of generating and validating a new patch) in a conversation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' A maximum chain length of 1 is the simple sampling from the same initial prompt baseline (used in prior LLM for APR tools).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' As we increase chain length, the model has to take in more and more previous context in the form of prior generations and feedbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We observe that the performance increase as we start from a small chain length and reaches the maximum around 3 or 4 and then decrease as chain length continue to increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' The decrease in number of plausible patches once we reach a high chain length is because the context may be too much for the model to handle since it can include multiple previously failed patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We also observe that this decrease is more significant in smaller models, where larger models are less affected by longer chain length, showing the ability for larger models to better capture the long term context dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' This shows that the optimal chain length to use for conversational APR can be dependent on the individual LLM used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Feedback prompting style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We now evaluate the effect of the feedback prompting style used in our conversational APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Table 4 shows the number of plausible patches using differ- ent validation prompts in QuixBugs-Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Column no testcase does not include any test- case feedback (only states that the patch is not correct), natural language describes the failing testcase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=', when input is 2, the patch incorrectly returns [] but it should return [2]) and functional which is the default prompting style discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We ob- serve that different prompting style does have an effect on the final performance of conversational APR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Starting from no testcase prompt, we can improve performance by adding specific testcase feedback information on top of telling the LLM that the patch is not correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' We also observe that the functional prompting style, using the buggy/patch function name and passing parameters (see Figure 1), performs the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Functional prompting style conveys the failing testcase information in a more concise and natural way by phrasing the testcase input and expected output relationship as a function call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' 7 CONCLUSION We propose conversational APR, a new paradigm for program repair that interleaves patch gener- ation with validation to provide immediate feedback for LLMs to better prompt future generated patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Compared to previous LLM for APR approaches that only sample from the same input, conversational APR iteratively builds the input by concatenating previously incorrect patches and validation feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' This allows for the model to avoid generating previously incorrect patches and also understand the semantic meaning of the function through validation feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Our evaluation on 10 different LLMs shows the improvement of conversational APR over the baseline sampling method used in prior LLM for APR tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Furthermore, we demonstrate the promising future of ap- plying ChatGPT, a conversational/dialogue driven LLM, for conversational APR, or APR in general for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' REFERENCES Jacob Austin, Augustus Odena, Maxwell Nye, Maarten Bosma, Henryk Michalewski, David Dohan, Ellen Jiang, Carrie Cai, Michael Terry, Quoc Le, and Charles Sutton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Program synthesis with large language models, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='07732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' BigQuery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Bigquery github repos, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' https://console.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content='com/ marketplace/details/github/github-repos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Tom B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhari- wal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, 9 Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFQT4oBgHgl3EQfGzVw/content/2301.13246v1.pdf'} +page_content=' Ziegler, Jeffrey Wu, Clemens Winter, Christopher Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei.' metadata={'source': 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As a +consequence, transport companies are responsible not only of scheduling driving plans aligned with +laws that define the legal behaviour of a driver, but also of monitoring and identifying as soon as +possible problematic patterns that can incur in costs due to sanctions. Transport experts are fre- +quently in charge of many drivers and lack time to analyse the vast amount of data recorded by +the onboard sensors, and companies have grown accustomed to pay sanctions rather than predict +and forestall wrongdoings. This paper exposes an application for summarising raw driver activity +logs according to these regulations and for explaining driver behaviour in a human readable format. +The system employs planning, constraint, and clustering techniques to extract and describe what +the driver has been doing while identifying infractions and the activities that originate them. Fur- +thermore, it groups drivers based on similar driving patterns. An experimentation in real world data +indicates that recurring driving patterns can be clustered from short basic driving sequences to whole +drivers working days. +1 +Introduction +World wide transport authorities are imposing complex Hours of Service (from now on, HoS) regulations to drivers +(Meyer 2011, Goel and Vidal 2013), which constraint the amount of working, driving and resting time when delivering +a service. As a consequence, transport companies are responsible not only of scheduling driving plans aligned with +laws that define the legal behaviour of a driver, but also of monitoring and identifying as soon as possible problematic +patterns that can incur in costs due to sanctions. +Fortunately, the widespread adoption of onboard IoT devices in vehicle fleets enables recording of the driver activities +in event logs, but the large amount of data ingested makes difficult for transport experts to understand what happened +and to make actions that forestall illegal behaviour. For this reason, an important technical challenge is to come up +with easily interpretable descriptive models that help understand the huge amount of information stored in such event +logs. The main objective not only consists of finding out if drivers workplan complies with the HoS regulation, but +also summarising their activities in a concise but representative way. Additionally, these underlying patterns in the +event log could be analysed in order to discover driving styles which could make possible the suggestion of routes or +tasks more aligned to the driver preferences. +The creation of driver profiles based on driving styles with HoS can be extremely useful for managers, as they could +assign transport routes to the most appropriate drivers, given the length of the route and the proximity of the deadline. +For example, drivers who maximise their driving hours could be preferred for long distance routes and drivers who +tend to take split rest to on-city deliveries. +arXiv:2301.05082v1 [cs.AI] 12 Jan 2023 + +Discovering and Explaining Driver Behaviour under HoS Regulations +A PREPRINT +Weekly Driving +Normal +Daily +Driving +Extended +Daily +Driving +Driving +Sequence +Driving +Sequence +Driving +Sequence +Driving +Sequence +Uninterrupted +Split 1 +Split 2 +Activities +Break +Type 1 +Rest +Day +Activities +Activities +Break +Type 2 +... +Driving +Sequence +Weekly +Rest +... +Figure 1: Partial example of the HoS tree. At the upmost level, a weekly driving period is formed by several daily +periods, which must end with a weekly rest. Similarly, daily driving periods are separated by daily rests, and according +to the accumulative hours of driving in them can be classified as Normal Daily Driving periods (up to 9 hours) or +Extended Daily Driving periods (more than 9 hours). Because each driving sequence should not surpass 4.5 hours of +driving time, they can be distinguished by the number of driving sequences in them. +Therefore, in this paper we present a method that, starting from real event logs extracted from a tachograph device, 1) +labels driver activities according to the HoS regulation, 2) identifies infractions and their cause, 3) extract summarised +information about the log while clustering driving sequences based on similar behaviour patterns, and 4) group drivers +by similarity of those clustered patterns. As a results, experts are provided with an understandable analysis of what +the driver has been doing in multiple levels of granularity, from a detailed description of the activities and infractions +under the HoS regulation to a categorisation with similar tendencies. +The remainder of this paper shows, firstly, a description of the problem addressed and some background concepts to +it. Then, we present the methodology of the approach, followed by details of experimentation conducted over a proof +of concept of the application. Finally, we conclude discussing related and future work. +2 +Problem Description +We are collaborating with a company which provides decision support based on prediction services to its customers. +Ultimately they want to help them govern the behaviour of their drivers by predicting whether a driver is close to +committing an infraction, as well as characterising drivers according to their driving style with respect to the HoS +regulation. +They handed us tachograph logs of multiple drivers with thousands of activities and asked us to develop a system to +analyse driver behaviour. Due to the regulation imposing additional difficulties at interpreting the data, and the high +volume that is constantly being generated, experts cannot interpret directly the original tachograph logs and require +summarisation of what a driver has been doing during that period of time to make business decisions. A tachograph +(Baldini et al. 2018) is an automated recording device fitted into a vehicle that extracts information from the driving +activities such as speed, duration and distance. +Our dataset represented an event log where every activity is a tuple (id, start, end, dur, a), each component +referring to: driver identifier id; start and end timestamps; activity duration dur; and activity identifier a, respectively. +A value for a is any of the labels [Driving, Other, Break, Idle] meaning that the driver is either Driving, performing +2 + +Discovering and Explaining Driver Behaviour under HoS Regulations +A PREPRINT +Tachograph +Log +Activity Recognition +and +HOS anaylsis +Labelled +Log +Infringement +Analysis +Extended +Labelled +Log +Driver Behaviour +Analysis +Day +summary +Driver +Patterns +Clustering +Day +summary +... +Clustering +... +... +... +Figure 2: General overview of our approach. +Another Work, at Break or Idle during dur minutes, between start and end. The semantics of each event is completed +with the definitions provided by the HoS regulation, which are detailed in the following paragraphs. +Although the HoS standard is applied in several countries, in this work we focus on the European Union regulation +(EC) No 561/2006, which has been extensively analysed in (Goel and Vidal 2013, Meyer 2011). The basic terms refer +to four types of driver activities as break (short period for recuperation), rest (free disposal period with enough time to +sleep), driving (time during which the driver is operating a vehicle) and other work (time devoted to any work except +driving, like loading). +These activities are hierarchically grouped up to weekly intervals, based on the duration of the events contained in +them. To ease the explanation of this article we are referring at the whole structures as HoS trees. In Figure 1 we +exemplify a portion of a HoS tree displaying a Normal Daily Driving (NDD) period on the first day and a Extended +Daily Driving (EDD) period on the last. +At the lower levels activities are joined in different types of driving sequences. A basic driving sequence is composed +of a totally ordered set of the elements of [Driving, Other, Break, Idle] constrained so that the duration of any +Break is less than 15 minutes. More constraints are defined over the duration of the rests and breaks, and over the +accumulated duration of driving sequences. +The regulation provides a set of basic and optional rules, should the former not be satisfied, thus allowing more +flexibility to generate and interpret driving schedules under such constraints. For example, either a break of 45 min +has to be taken after 4.5 hours of accumulated driving or it can be taken split in two parts of at least 15 min and +30 min respectively. This feature is good for drivers since it provides flexibility to their work, but complicates the +interpretability of what they are doing. The regulation also defines additional constraints (for example, the maximum +number of occurrences of a reduced rest in a weekly driving period), and relationships between the different types of +sub-sequences, as well as their internal structure. +3 +Background +Automated planning (Ghallab et al. 2016) is a branch of A.I. concerned with the study of agent acting techniques. +However, its uses can be broaden and as we show in this paper planning can also be applied to recognition tasks. +Two elements are required in a planning environment: (i) the action models existing in the world, referred as domain; +and (ii) a description of the initial state of the world, the objects involved in it and the desired goals, called problem. +These two inputs are provided to a planner, a search-based algorithm that determines the plan (sequence of actions) +that achieve the goals from the starting state. +Our proposed methodology employs hierarchical planning, more commonly referred as Hierarchical Task Networks +(HTN). HTNs forms a branch of classical planning where the domain can be decomposed in hierarchical structures +of tasks/subtasks, with low level tasks representing temporally annotated actions, and compound tasks representing +temporal ordering strategies between those actions. +4 +Application Overall Description +To solve the problem of explaining and summarising a driver’s tachograph log and its compliance with the HoS +regulation we propose a modular architecture divided in three main components, as seen in Figure 2: +• First, an initial planning process to label the input tachograph log according to the HoS regulation. +• Then, a system to identify and explain the causes of driver infractions extending the previous labelled log. +3 + +Discovering and Explaining Driver Behaviour under HoS Regulations +A PREPRINT +Tachograph +Log +Transformation +to HTN +problem +Labelled +Log +HPDL +problem +HPDL +domain +Planner +HoS +Rules +Transformation +to HTN +domain +Temporal +Observations +Atrribute +Grammar +1 +2 +3 +5 +4 +Figure 3: Labelling process for a tachograph log. +Table 1: Labelling output for legal activities. This example shows the second (and last) driving sequence in a normal +daily driving period, where the required break has been taken in two parts, a small break in the first split and a second +one extended as a daily rest. +Original Log +Annotated Labels +Driver +Start +End +Duration +Activity +Week +Day +DayType +Sequence +BreakType +Token +Legal +driver1 +11/01/2017 17:33 +11/01/2017 17:37 +4 +Driving +1 +4 +ndd +second +split 1 +A +yes +driver1 +11/01/2017 17:37 +11/01/2017 18:16 +39 +Break +B T2 +yes +driver1 +11/01/2017 18:16 +11/01/2017 18:17 +1 +Driving +split 2 +A +yes +driver1 +11/01/2017 18:17 +11/01/2017 18:25 +8 +Other +A +yes +driver1 +11/01/2017 18:25 +11/01/2017 19:54 +89 +Driving +A +yes +driver1 +11/01/2017 19:54 +11/01/2017 19:57 +3 +Break +B T0 +yes +driver1 +11/01/2017 19:57 +11/01/2017 19:58 +1 +Driving +A +yes +driver1 +11/01/2017 19:58 +11/01/2017 20:01 +3 +Other +A +yes +driver1 +11/01/2017 20:01 +12/01/2017 07:06 +665 +Break +DR T1 +yes +• Thirdly, a module to analyse driver behaviour via summarisation of driving sequences. +• Lastly, summarised driving days are used as training data to clusterize drivers by similar driving patterns. +The following subsections provide a detailed explanation of each component. +4.1 +Labelling Driver Activities +To label our logs with HoS terms we employ our previously developed methodology proposed in (Vellido-Exp´osito. +et al. 2022), where a HTN domain serves to both recognise and tag activities from a tachograph log. We provide a +brief summary below, but we refer the reader to the original paper for an in depth explanation of the methodology. The +overall steps of this system, represented in Figure 3, are: +1. Generate a set of ordered temporal observations from the tachograph activity log, which are part of the initial +state of a HTN problem. +2. Represent the recognition of a driver activity as a temporal HTN problem, where an activity is added to +the plan if (i) the temporal information of the activity is consistent with the domain, and (ii) the temporal +constraints of the activity are consistent with the rest of temporal constraints of the actions already added to +the plan. +3. Codify a HoS tree in an attribute grammar (Knuth 1968) as an intermediate representation, with HoS rules as +productions. +4. Translate the grammar into a temporal HTN domain, aimed at representing the parsing of the activity log as +a HTN problem where (i) terminal symbols are recognised as temporal events and (ii) nonterminal symbols +are recognised according to grammar rules. +5. Extend the domain to both recognise and label activities from the log to be easily interpretable. The resulting +log contains five new labels according to the contexts DayType (Normal or Extended Daily Driving period), +Sequence (if the activity belongs to the first, second or third sequence in the day), BreakType (if breaks are +taken in one or two parts), Token (the type of activity at the lowest level of in the HoS tree1) and Legal +1Many types of categories exist at the lowest level, based on the duration of the action. For example, A indicates a working +activity, B T0 a break of less than 15 minutes, DR T1 a daily rest of more than 11 hours, and WR T1 a weekly rest with more than +45 hours. +4 + +Discovering and Explaining Driver Behaviour under HoS Regulations +A PREPRINT +Labelled +Log +Labelled Log +with +Infringements +Test +Evaluation +Labels +Comparison +Test +List +Original +Tachograph +Log +Relaxed +HPDL +domain +Relaxed +Labelled +Log +Labelling +Process +Figure 4: Infringement analysis process for a labelled log. +(whether the activity complies or not with the regulation), as well as two counter columns for the day and the +week processed. An output example can be seen in Table 1. +In summary, the recognition problem is solved with a planning process where the domain walks through an activity log +and its internal HTN structure simultaneously, the latter codifying the HoS tree. If activities comply with the temporal +and formal restrictions they are labelled with the appropriate terms, in other case contexts are tagged as unrecognised. +Nevertheless, the domain is designed to label as many contexts as possible. If a higher (more general) context cannot +be identified, the domain still attempts to identify lower contexts before ignoring the action. That means that when +a bigger sequence cannot be grouped and labelled together (e.g. when the driver exceeds the maximum number +of driving hours and the DayType column cannot be tagged), the domain tries to tag smaller sequences with their +corresponding label. An example is shown below in Table 3, where although DayType and Sequence tags could not be +recognised the system still identifies both BreakType splits and includes the appropriate labels, as well as the correct +Token contexts. +4.2 +Explaining Infringements +The previous recognition process labels the tachograph log considering the terms defined by the HoS regulation, its +compliance with it and details of their position in the HoS tree. However, when drivers commit infractions this system +by itself cannot provide an explanation of the cause and the exact root activity, due to the fact that planning techniques +rely on backtracking (that is, the ability to retract while exploring the planning graph) and there is not a simple way to +distinguish between a genuine backtracking step while walking through the HTN domain or a forced one by an illegal +activity in the log. +Therefore we found a need to further analyse the labelled log and explain these information to users without requiring +them to inspect all activities not recognised in the log. We solved this problem from two perspectives, each one +concerned with different kinds of violations, which are explained in the following subsections. Figure 4 shows an +overview of the approaches. +4.2.1 +Test evaluation +On one hand we represent rules from the HoS regulation as tests and applied them to the sequences the labelling +process found unrecognisable events (i.e., those missing at least a label). These tests, as exemplified in the left part +of Table 2, codify limits and restrictions in the duration of driving sequences and breaks. Whenever a test flags a +sequence the system marks it and provides an explanation of the infringement, as seen in Table 3. +5 + +Discovering and Explaining Driver Behaviour under HoS Regulations +A PREPRINT +Table 2: Tests applied to driving sequences in the log in order to identify infringement causes. +Test +Infraction type +dt seq > 4.5h +Excessive Driving without breaks +dt day > 9h +∧ +EDDs this week > 2 +Excessive Driving in day (NDD) +dt day > 10h +Excessive Driving in day (EDD) +Token day before = DR T3 +∧ +Token = ¬ (DR T4 or WR) +Missing other half of split daily rest +Token = DR or WR +∧ +Legal = No +∧ +Remaining +contexts = ¬ Unknown +Rest past the daily/weekly deadline +Table 3: Labelling output example for illegal activities and the infraction detected by the tests list. +Original Log +Annotated Labels +Driver +Start +End +Duration +Activity +Week +Day +DayType +Sequence +BreakType +Token +Legal +Infraction +driver39 +10/01/2017 12:12 +10/01/2017 14:17 +125 +Driving +1 +5 +unkown +unkown +split 1 +A +no +Surpassed NDD driving time +driver39 +10/01/2017 14:17 +10/01/2017 14:40 +23 +Break +B T2 +no +driver39 +10/01/2017 14:40 +10/01/2017 16:52 +132 +Driving +split 2 +A +no +driver39 +10/01/2017 16:52 +10/01/2017 17:25 +33 +Break +B T3 +no +driver39 +10/01/2017 17:25 +10/01/2017 20:27 +182 +Driving +ndd +first +split 1 +A +yes +driver39 +10/01/2017 20:27 +10/01/2017 20:42 +15 +Break +B T2 +yes +driver39 +10/01/2017 20:42 +10/01/2017 21:54 +72 +Driving +split 2 +A +yes +driver39 +10/01/2017 21:54 +10/01/2017 21:59 +5 +Break +B T0 +yes +driver39 +10/01/2017 21:59 +10/01/2017 22:00 +1 +Driving +A +yes +driver39 +10/01/2017 22:00 +10/01/2017 22:37 +37 +Break +B T3 +yes +driver39 +10/01/2017 22:37 +10/01/2017 23:21 +44 +Driving +second +uninterrupted +A +yes +driver39 +10/01/2017 23:21 +11/01/2017 08:53 +572 +Break +DR T2 +yes +Tests takes the form of logic constraints +f(astart, aend) o V +(1) +being: +• f a function applied over the sequence defined between activities astart and aend (e.g. sum, context value). +• o a logic operator. +• V either the value of a context (e.g. Token, DayType), a scalar or a duration. +As an example, the first constraint in Table 2 could be rewritten as duration(seqstart, seqend) > 4.5h. +It is important to note that to correctly identify the infraction some tests may consider not only the illegal activities +but also prior activities of other days, a situation frequently present in reduced breaks and rests, where sometimes +compensation breaks are not fulfilled. Therefore the interval of activities checked by the tests depends on the test +itself. +Because tests are encoded as logic constraints, it is easy to extend the system with additional expert provided rules or +modify them if the regulation changes. +4.2.2 +Re-labelling +A second approach consists of re-labelling the log using a domain with relaxed duration intervals, that is, the limits +imposed by the regulation are softened (e.g. maximum driving time or minimum break time are enlarged up and down) +and the system looks for changes between the new log and the original tagged log. +This process helps to discover infringements caused by a slightly borderline duration, like the driver surpassing (prob- +ably unconsciously) the restriction by a small amount. These kind of situations are not easily identified by the tests, +due to the fact that the activity by itself could still be legal but labelled differently, becoming an infraction later on. +6 + +Discovering and Explaining Driver Behaviour under HoS Regulations +A PREPRINT +Table 4: Identifying infringements with a relaxed domain. In this example the fourth activity surpasses by one minute +the duration limit to be considered B T0, making the whole sequence illegal. +Original Labelled Log +Duration +Activity +DayType +Sequence +BreakType +Token +Legal +57 +Driving +unknown +unknown +split 1 +A +no +3 +Break +B T0 +no +2 +Driving +A +no +16 +Break +B T2 +no +Relaxed Labelled Log +Duration +Activity +DayType +Sequence +BreakType +Token +Legal +57 +Driving +ndd +first +split 1 +A +yes +3 +Break +B T0 +yes +2 +Driving +A +yes +16 +Break +B T0 +yes + +Extended +Labelled +Log +Clustered +Log +Paragraph +Vector +Model +HDBSCAN +Model +Vectors +Legal +Days +Illegal +Days +Paragraph +Vector +Model +Vectors +HDSBSCAN +Model +Centroids +Encoding +Encoding +Figure 5: Clustering process for a labelled log. +For example, a driver could surpass the maximum limit for a pause before being considered a break by a few minutes +without noticing, and proceeding like a break has not been consumed. As a consequence, that action will be valid, but +after the next breaks infractions may arise because the driver is not following its plan as expected, and such actions +may not fit correctly under the HoS tree. +If the violation is related with this type of mistake, the new relabelled log will contain less illegal sequences than the +original and we can compare the Token contexts (concerning the type of activity at the lowest level in the HoS tree) to +understand which changes make the sequence legal. Table 4 shows an example with a driver exceeding the break time +by two minutes. +Therefore, this method allows us to (a) discover new infringements not considered by the test list, and (b) analyse how +the activity should have been to avoid infractions. +4.3 +Analysing Driver Behaviour +The two previous steps provide a way to understand a driver log and its compliance with the HoS regulation. However, +experts are usually responsible of dozens of drivers and its not feasible to analyse the substantial logs of each one of +them in order to detect problematic tendencies. +Therefore, we developed a module that clusters behaviour patterns in driver activities and summarises each cluster +with expert knowledge. This method helps to separate standard driving days from unusual ones without inspecting the +driver log, and let users concentrate their efforts in analysing only the problematic sequences. +In order to do that, we considered our problem as an NLP (Natural Language Processing) task, where activities from +the log are treated as words and daily sequences as documents. That way we can employ NLP oriented techniques to +transform sequences of varying length into fixed dimensions and measure similarity between them. +Figure 5 shows an overview of the process, consisting of the following steps: +7 + +Discovering and Explaining Driver Behaviour under HoS Regulations +A PREPRINT +Table 5: Partial output of the clustering process. The system identifies the most similar centroid to the input sequence +and the description associated with it. +Labelled Log +Activity +DayType +Sequence +BreakType +Token +Legal +Cluster +Driving +ndd +unique +uninterrupted +A +yes +2 +Other +A +yes +Break +DR T1 +yes +Most similar centroid +Activity +DayType +Sequence +BreakType +Token +Legal +Cluster +Driving +ndd +unique +uninterrupted +A +yes +2 +Other +A +yes +Break +DR T3 +yes +Description +Legal and standard daily driving formed by a unique and uninterrupted driving sequence +1. First, a preprocessing step is applied in which the dataset is split in two parts depending on whether the days +contains or not illegal activities. The reason behind this process is that an infraction recognition process +is already provided by the previous module, and thus there is no need for our clustering model to learn to +distinguish between legal and illegal sequences. On the contrary, we are providing a prior separation to help +the model extract more interesting patterns that are not related with the legality of the sequence. +2. The subset of labelled columns (i.e. contexts) that describe the action from an overall point of view are +selected, these are (Activity, DayType, BreakType, Token). For the illegal subset, the Infraction column is also +included to generate clusters and centroids associated with already identified infringement. Specific details +about duration and timestamps are not relevant to summarise the days. Nevertheless, some of the information +they provided is encoded in the labels, as it is used by the labelling process. This step could be consider as +cleaning a document prior an NLP topic categorisation task. +3. Because columns contains categorical features not suitable for computation they are transformed into numer- +ical, and then joined together using a special character as a separator. After this step we can consider each +entry in our log as a word. +4. Both previous steps are repeated for each activity in our dataset, and activities of the same day are grouped +into documents. As a result, we have a collection of documents each one encoding the activities in a driving +day sequence as words. +5. We then use Paragraph Vector (Le and Mikolov 2014) (also known as Doc2Vec) models to obtain dense rep- +resentations of fixed dimensions2. Although one model could be trained for both data splits (and reasonable +so, as both encodings are subset of the same language), we obtained better results finetuning one model for +each split, but ultimately both transforming a document into a 200 sized output vector. +6. The resulting representations are now suitable for clustering techniques. We obtained our best results using +HDBSCAN (Campello et al. 2013) thanks to its robustness to noise, and choosing the number of clusters +based on both expert knowledge and metrics results (Silhouette Coefficient, Calinski-Harabasz and Davies- +Bouldin indexs), setting on a final value of 8 clusters for legal data and 7 for days with infractions. In the next +section we display a comparative analysis of other techniques under this data. +7. Lastly, days are clustered and presented with the decoded centroids, which are described by an expert with a +meaningful description, as shown in Table 5. +4.4 +Generating Driver Profiles +Similarly to the working days clustering previously explained, we performed categorisation of drivers based on similar +behaviour with the idea of extracting driver profiles. With enough data, we saw that the large amount of activities +contained in event logs can be summarised in different types of driving days as described in the previous sections, +and such types encode enough information to extract a characterisation of the driver that can be informative for the +transport company. +2Other techniques like Word2Vec or Bag of Words could be used, but we considered the paragraph weight extracted by Paragraph +Vector a useful source of information in our task. +8 + +Discovering and Explaining Driver Behaviour under HoS Regulations +A PREPRINT +Table 6: Example input data for extracting driver profiles. Each row encodes how frequently a driver perform one of +four types of driving days. Given the uneven number of data of each driver values are expressed as percentages and +all rows sums to one. +Driver +Driving day type +Split Sequences Normal Rest +Uninterrupted Sequences Normal Rest +Split Sequences Reduced Rest +Uninterrupted Sequences Reduced Rest +. . . +1 +0.5 +0.1 +0.3 +0.1 +2 +0.2 +0.8 +0.0 +0.0 +3 +0.15 +0.5 +0.3 +0.15 +We performed the following steps to categorise drivers: +1. Drop days with infractions: Tests introducing violations gave us results who grouped drivers by similar ratio +of infractions and by day types (e.g., those who tended to excess their break time ended up in the same cluster). +However, we opted to not include such information as they did not align with the purposes of the application. +Because the context around the infraction cannot be extracted exclusively from the tachograph data (e.g., was +the infringement voluntary, due to lack of correct planning or caused by unexpected circumstances on the +road?) we believe managers should analyse case by case rather than making decisions without understanding +the real motive behind the infraction. +2. Create frequency table: For each driver its daily logs are processed following the methodology explained +in subsections 4.1 and 4.3, keeping the day type predicted by the clustering model. The training dataset is +created counting the frequencies the driver has performed each type of driving day. Due to the fact that in +our data the amount of information varies for each driver, these frequencies are transformed into percentages. +Ultimately, we obtain a table D × C as shown in Table 6, being D the number of drivers and C the number +of different days categories (i.e., the number of clusters discovered in the previous section). +3. Training: We perform clustering with the resulting table. From a multiple of techniques our best results were +obtained with a Gaussian mixture model trained with the Expectation-Maximization algorithm (Fraley and +Raftery 2002). +The deciding factor at choosing the best partition was based exclusively on expert knowledge. Because we +were informed that our data was for drivers who performed similar routes on the same country, we looked for +a few number of clusters that separated nicely the data. +As a closing point, we would like to note that driver profiles based only on tachograph data could be misleading, as +they do not account for the specific routes they perform. We believe a better approach would be to combine the cluster +information with route details like distance, type of vehicle or type of cargo, and as a result get a categorisation of +the driver given the type of route. That way, decisions based on these profiles will not be biased, and traffic managers +could assess their drivers for each particular service. We intend to explore those options in future work. +5 +Experimentation +We have validated our methodology with an experimentation using real tachograph logs provided by an industrial +collaborator. We were provided with a dataset formed by two-weeks-long sequences of activities from 290 different +drivers. +Because the architecture is composed of three different components, each one was validated individually. The labelling +process was verified against multiple driving sequences selected at random, both legal and illegal, manually verifying +that the output was the appropriate under the HoS regulation. For the infringement analysis system multiple tests for +each kind of infraction were carried out, confirming that not only the cause, but also the subsequence containing the +infraction was detected. +Lastly, due to the fact that the clustering in our problem is an unsupervised task, we experimented with different +techniques and hyperparametrization to discover the best possible clusters. The quality of each partition was measured +with the Silhouette Coefficient and both Calinski-Harabasz and Davies-Bouldin indexs. The final clustering result was +selected between the best performing tests, and after expert inspection of the resulting clusters and centroids. +Figure 6 shows the performance of multiple algorithms in data with and without infractions, respectively. The algo- +rithms are: Gaussian Mixture models, having each component its own covariance matrix, and controlling the number +of mixture components as the number of clusters; HDBSCAN (Campello et al. 2013), a hierarchical clustering model +employing density based measures; classical agglomerative clustering using average, complete and ward criteria; and +K-Means with cosine similarity as distance metric. +9 + +Discovering and Explaining Driver Behaviour under HoS Regulations +A PREPRINT +Data with infractions +Data without infractions +2 +4 +6 +8 +10 +2 +4 +6 +8 +10 +12 +14 +0.2 +0.3 +0.4 +0.5 +0.6 +0.1 +0.2 +0.3 +0.4 +0.5 +Silhouette coefficent +2 +4 +6 +8 +10 +2 +4 +6 +8 +10 +12 +14 +0 +500 +1000 +1500 +0 +100 +200 +300 +Calinski−Harabarsz score +2 +4 +6 +8 +10 +2 +4 +6 +8 +10 +12 +14 +1 +2 +3 +4 +5 +0.9 +1.2 +1.5 +Davies−Bouldin score +Number of Clusters +Algorithm +Gaussian Mixture +HDBSCAN +Hierarchical (avg) +Hierarchical (complete) +Hierarchical (ward) +KMeans +Figure 6: Silhouette Coefficient, Calinski-Harabasz and Davies-Bouldin indexs as a function of the number of clusters +for multiple clustering algorithms. Notice that the y-axis scalings differ among the different panels of this figure. +Some insights can be extracted from the graphs. HDBSCAN is without doubt the best algorithm under both subsets of +this data, but we believe that the reason relies mostly due to its robustness to noise points. Furthermore, results on data +with infringements are, as expected, more variable, as this subset combines multiple types of infractions with driving +sequences that can be perfectly legal. The runner-up model is not clear, as results vary greatly with the number of +clusters. +For fully legal data we see hierarchical clustering with average and complete linkage method vastly underperforming. +The graphs for the rest of techniques take similar shape, mostly agreeing in that 8 clusters seems an appropriate +partition for this data. Nevertheless, as the results are intended for human interpretation, it is important to remind that +the clusters should be reviewed by an expert whenever possible before setting on a final value. +Finally, we believe is worth mentioning our experimentation with the LDA (Latent Dirichlet Allocation) (Blei et al. +2003). This technique is frequently used in NLP tasks to summarise a document with a set of topics. Due to a small +vocabulary size in our data as opposed to an NLP task, most words (i.e. driver activities) are present in many different +clusters (with the exception of illegal activities), and although the most relevant topics could be ranked and considered +as centroids there is no assurance that these topics are understandable (e.g. a B T2 break only makes sense if followed +10 + +Discovering and Explaining Driver Behaviour under HoS Regulations +A PREPRINT +Table 7: Clustering results for driver profiles and they interpretation. +Cluster +Interpretation +Proportion +1 +No extended days and mostly +takes rests uninterrupted +8.6% +2 +Usually splits rests as much as possible +and rarely takes extended days +51.8% +3 +Neither takes many extended days +or splits rests +20.2% +4 +No clear tendency, +driver seems to be flexible +14.4% +5 +Tends to split rests as much as possible +and frequently takes extended days +5.0% +by a B T3 break. The presence of only one of them as a topic does not clarify if the driver completed the sequence or +committed an infraction). +For our driver clustering experimentation Table 7 shows 5 resulting clusters and their interpretation after training. +Given that our training data is compromised of mostly event logs of national deliveries in Spain, we can see that more +than half of our drives prefer to spent their rests split in two. Nevetheless, as mentioned above, the lack of data about +the routes performed in the tachograph hinders the expressivenes, but experts welcome any information that could help +them assign the best driver to a service as easily as possible. +The methodology and experimental results are encapsulated in an web application publicly available at https:// +github.com/IgnacioVellido/Driver-Assistance-System. +6 +Related Work +This project is an extension of authors prior work (Fernandez-Olivares and Perez 2020) focused on the recognition +and labelling of driver activities under the HoS regulation. The novel contributions provided in this paper go a step +forward in our goal of developing an intelligent assistant to drivers and traffic managers, proposing a planning and +constraint based analysis of infractions causes and summarisation of driver behaviour with NLP techniques. +Regarding applications concerned with the HoS regulation, many approaches have been developed aimed to solve +route planning problems under these rules while minimising transportation costs (Mbiydzenyuy 2015, Omelianenko +et al. 2019, Goel 2018, Goel and Irnich 2017). Nonetheless, the authors have not found works that extract insights that +can be useful for experts in analysing and understanding driver activities from a legal perspective. +As for driver behaviour modelling from tachograph data, proposals like (Zhou and Zhang 2019) employs data mining +techniques to categorise truck drivers and analyse dangerous tendencies. Their approach is similar to ours in that +clusters are manually studied and labelled. However, PCA for dimensionally reduction and DBSCAN for clustering +are directly used instead due to the fact that their data does not contain categorical variables. +Lastly, word embedding techniques like Paragraph Vector models has been previously applied in non textual data like +web user activities (Tagami et al. 2015) and server logs (Mimura and Tanaka 2018) as a way to transform sequential +data of variable length into dimensionally fixed data. Similarly, although oriented to process mining applications, +the trace2vec model proposed in (De Koninck et al. 2018) uses embedding techniques for discovery, monitoring and +clustering of sequences of activities. Nevertheless, up to the author’s knowledge there have not been prior research +with tachograph logs. +7 +Conclusion +We have presented a novel planning application that brings the worlds of Data Analytics, IoT and Automated Planning +and Scheduling together. The approach provides support to experts on the task of interpreting what drivers are or have +been doing by recognising and summarising their activity recorded in an event log. +11 + +Discovering and Explaining Driver Behaviour under HoS Regulations +A PREPRINT +Using as a basis our prior work in driver activity recognition, the main contributions exposed in this paper are an +infringement analysis process with a planning and constraint based approach, the summarisation of temporal activity +logs using word embeddings and clustering, and the creation of driver profiles based on such summaries. The over- +all system provides a human readable summary of the driver behaviour under the HoS regulation while explaining +infractions and the root cause. +Regarding future work, it is worth noting that the main interest and the ultimate goal of the company is to build an +intelligent assistant to provide decision support services to both drivers and companies decision makers. This is a +research direction aligned with the concept of assistive interaction (Freedman and Zilberstein 2017), that advocates +for the integration of plan recognition and planning. In this way, the recognition of driver’s intent is a previous stage +needed to respond with a generated plan adapted to the currently recognised task. +For our next steps we intent to enrich the driver profiling model adding non-tachograph data about the transport service, +like type of vehicle and cargo. Additionally, we are focused on integrating descriptive support to the assistant, being +able to suggest drivers plans of actions in compliance with the HoS regulation and considering preference patterns +extracted from previous personal behaviour. +References +Baldini, G., Sportiello, L., Chiaramello, M. and Mahieu, V.: 2018, Regulated applications for the road transportation +infrastructure: The case study of the smart tachograph in the european union, International Journal of Critical +Infrastructure Protection 21, 3–21. +Blei, D. M., Ng, A. Y. and Jordan, M. I.: 2003, Latent dirichlet allocation, Journal of machine Learning research +3(Jan), 993–1022. +Campello, R. J. G. B., Moulavi, D. and Sander, J.: 2013, Density-based clustering based on hierarchical density +estimates, in J. Pei, V. S. Tseng, L. Cao, H. Motoda and G. 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2022, Analyzing driver behavior compli- +ance under hos regulations, Proceedings of the 8th International Conference on Vehicle Technology and Intelligent +Transport Systems - VEHITS,, INSTICC, SciTePress, pp. 463–470. +Zhou, T. and Zhang, J.: 2019, Analysis of commercial truck drivers’ potentially dangerous driving behaviors based on +11-month digital tachograph data and multilevel modeling approach, Accident Analysis & Prevention 132, 105256. +13 + diff --git a/6tE4T4oBgHgl3EQfcQy_/content/tmp_files/load_file.txt b/6tE4T4oBgHgl3EQfcQy_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7167c26306a2be67aee3508ad153452817bc39a4 --- /dev/null +++ b/6tE4T4oBgHgl3EQfcQy_/content/tmp_files/load_file.txt @@ -0,0 +1,717 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf,len=716 +page_content='DISCOVERING AND EXPLAINING DRIVER BEHAVIOUR UNDER HOS REGULATIONS A PREPRINT Ignacio Vellido1, Juan Fdez-Olivares1, and Ra´ul P´erez1 1Department of Computer Science and Artificial Intelligence, University of Granada, Spain ignaciovellido@ugr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='es, {faro, fgr}@decsai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='ugr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='es January 13, 2023 ABSTRACT World wide transport authorities are imposing complex Hours of Service regulations to drivers, which constraint the amount of working, driving and resting time when delivering a service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' As a consequence, transport companies are responsible not only of scheduling driving plans aligned with laws that define the legal behaviour of a driver, but also of monitoring and identifying as soon as possible problematic patterns that can incur in costs due to sanctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Transport experts are fre- quently in charge of many drivers and lack time to analyse the vast amount of data recorded by the onboard sensors, and companies have grown accustomed to pay sanctions rather than predict and forestall wrongdoings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' This paper exposes an application for summarising raw driver activity logs according to these regulations and for explaining driver behaviour in a human readable format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The system employs planning, constraint, and clustering techniques to extract and describe what the driver has been doing while identifying infractions and the activities that originate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Fur- thermore, it groups drivers based on similar driving patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' An experimentation in real world data indicates that recurring driving patterns can be clustered from short basic driving sequences to whole drivers working days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 1 Introduction World wide transport authorities are imposing complex Hours of Service (from now on, HoS) regulations to drivers (Meyer 2011, Goel and Vidal 2013), which constraint the amount of working, driving and resting time when delivering a service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' As a consequence, transport companies are responsible not only of scheduling driving plans aligned with laws that define the legal behaviour of a driver, but also of monitoring and identifying as soon as possible problematic patterns that can incur in costs due to sanctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Fortunately, the widespread adoption of onboard IoT devices in vehicle fleets enables recording of the driver activities in event logs, but the large amount of data ingested makes difficult for transport experts to understand what happened and to make actions that forestall illegal behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' For this reason, an important technical challenge is to come up with easily interpretable descriptive models that help understand the huge amount of information stored in such event logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The main objective not only consists of finding out if drivers workplan complies with the HoS regulation, but also summarising their activities in a concise but representative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Additionally, these underlying patterns in the event log could be analysed in order to discover driving styles which could make possible the suggestion of routes or tasks more aligned to the driver preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The creation of driver profiles based on driving styles with HoS can be extremely useful for managers, as they could assign transport routes to the most appropriate drivers, given the length of the route and the proximity of the deadline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' For example, drivers who maximise their driving hours could be preferred for long distance routes and drivers who tend to take split rest to on-city deliveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='05082v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='AI] 12 Jan 2023 Discovering and Explaining Driver Behaviour under HoS Regulations A PREPRINT Weekly Driving Normal Daily Driving Extended Daily Driving Driving Sequence Driving Sequence Driving Sequence Driving Sequence Uninterrupted Split 1 Split 2 Activities Break Type 1 Rest Day Activities Activities Break Type 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Driving Sequence Weekly Rest .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Figure 1: Partial example of the HoS tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' At the upmost level, a weekly driving period is formed by several daily periods, which must end with a weekly rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Similarly, daily driving periods are separated by daily rests, and according to the accumulative hours of driving in them can be classified as Normal Daily Driving periods (up to 9 hours) or Extended Daily Driving periods (more than 9 hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Because each driving sequence should not surpass 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='5 hours of driving time, they can be distinguished by the number of driving sequences in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Therefore, in this paper we present a method that, starting from real event logs extracted from a tachograph device, 1) labels driver activities according to the HoS regulation, 2) identifies infractions and their cause, 3) extract summarised information about the log while clustering driving sequences based on similar behaviour patterns, and 4) group drivers by similarity of those clustered patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' As a results, experts are provided with an understandable analysis of what the driver has been doing in multiple levels of granularity, from a detailed description of the activities and infractions under the HoS regulation to a categorisation with similar tendencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The remainder of this paper shows, firstly, a description of the problem addressed and some background concepts to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Then, we present the methodology of the approach, followed by details of experimentation conducted over a proof of concept of the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Finally, we conclude discussing related and future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 2 Problem Description We are collaborating with a company which provides decision support based on prediction services to its customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Ultimately they want to help them govern the behaviour of their drivers by predicting whether a driver is close to committing an infraction, as well as characterising drivers according to their driving style with respect to the HoS regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' They handed us tachograph logs of multiple drivers with thousands of activities and asked us to develop a system to analyse driver behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Due to the regulation imposing additional difficulties at interpreting the data, and the high volume that is constantly being generated, experts cannot interpret directly the original tachograph logs and require summarisation of what a driver has been doing during that period of time to make business decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' A tachograph (Baldini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 2018) is an automated recording device fitted into a vehicle that extracts information from the driving activities such as speed, duration and distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Our dataset represented an event log where every activity is a tuple (id, start, end, dur, a), each component referring to: driver identifier id;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' start and end timestamps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' activity duration dur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' and activity identifier a, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' A value for a is any of the labels [Driving, Other, Break, Idle] meaning that the driver is either Driving, performing 2 Discovering and Explaining Driver Behaviour under HoS Regulations A PREPRINT Tachograph Log Activity Recognition and HOS anaylsis Labelled Log Infringement Analysis Extended Labelled Log Driver Behaviour Analysis Day summary Driver Patterns Clustering Day summary .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Clustering .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Figure 2: General overview of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Another Work, at Break or Idle during dur minutes, between start and end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The semantics of each event is completed with the definitions provided by the HoS regulation, which are detailed in the following paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Although the HoS standard is applied in several countries, in this work we focus on the European Union regulation (EC) No 561/2006, which has been extensively analysed in (Goel and Vidal 2013, Meyer 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The basic terms refer to four types of driver activities as break (short period for recuperation), rest (free disposal period with enough time to sleep), driving (time during which the driver is operating a vehicle) and other work (time devoted to any work except driving, like loading).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' These activities are hierarchically grouped up to weekly intervals, based on the duration of the events contained in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' To ease the explanation of this article we are referring at the whole structures as HoS trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' In Figure 1 we exemplify a portion of a HoS tree displaying a Normal Daily Driving (NDD) period on the first day and a Extended Daily Driving (EDD) period on the last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' At the lower levels activities are joined in different types of driving sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' A basic driving sequence is composed of a totally ordered set of the elements of [Driving, Other, Break, Idle] constrained so that the duration of any Break is less than 15 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' More constraints are defined over the duration of the rests and breaks, and over the accumulated duration of driving sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The regulation provides a set of basic and optional rules, should the former not be satisfied, thus allowing more flexibility to generate and interpret driving schedules under such constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' For example, either a break of 45 min has to be taken after 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='5 hours of accumulated driving or it can be taken split in two parts of at least 15 min and 30 min respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' This feature is good for drivers since it provides flexibility to their work, but complicates the interpretability of what they are doing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The regulation also defines additional constraints (for example, the maximum number of occurrences of a reduced rest in a weekly driving period), and relationships between the different types of sub-sequences, as well as their internal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 3 Background Automated planning (Ghallab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 2016) is a branch of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' concerned with the study of agent acting techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' However, its uses can be broaden and as we show in this paper planning can also be applied to recognition tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Two elements are required in a planning environment: (i) the action models existing in the world, referred as domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' and (ii) a description of the initial state of the world, the objects involved in it and the desired goals, called problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' These two inputs are provided to a planner, a search-based algorithm that determines the plan (sequence of actions) that achieve the goals from the starting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Our proposed methodology employs hierarchical planning, more commonly referred as Hierarchical Task Networks (HTN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' HTNs forms a branch of classical planning where the domain can be decomposed in hierarchical structures of tasks/subtasks, with low level tasks representing temporally annotated actions, and compound tasks representing temporal ordering strategies between those actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 4 Application Overall Description To solve the problem of explaining and summarising a driver’s tachograph log and its compliance with the HoS regulation we propose a modular architecture divided in three main components, as seen in Figure 2: First, an initial planning process to label the input tachograph log according to the HoS regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Then, a system to identify and explain the causes of driver infractions extending the previous labelled log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 3 Discovering and Explaining Driver Behaviour under HoS Regulations A PREPRINT Tachograph Log Transformation to HTN problem Labelled Log HPDL problem HPDL domain Planner HoS Rules Transformation to HTN domain Temporal Observations Atrribute Grammar 1 2 3 5 4 Figure 3: Labelling process for a tachograph log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Table 1: Labelling output for legal activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' This example shows the second (and last) driving sequence in a normal daily driving period, where the required break has been taken in two parts, a small break in the first split and a second one extended as a daily rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Original Log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Annotated Labels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Driver ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Start ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='End ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Duration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Activity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Week ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Day ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='DayType ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Sequence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='BreakType ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Token ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Legal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='driver1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='12/01/2017 07:06 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='665 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Break ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='DR T1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Thirdly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' a module to analyse driver behaviour via summarisation of driving sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Lastly, summarised driving days are used as training data to clusterize drivers by similar driving patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The following subsections provide a detailed explanation of each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='1 Labelling Driver Activities To label our logs with HoS terms we employ our previously developed methodology proposed in (Vellido-Exp´osito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 2022), where a HTN domain serves to both recognise and tag activities from a tachograph log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' We provide a brief summary below, but we refer the reader to the original paper for an in depth explanation of the methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The overall steps of this system, represented in Figure 3, are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Generate a set of ordered temporal observations from the tachograph activity log, which are part of the initial state of a HTN problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Represent the recognition of a driver activity as a temporal HTN problem, where an activity is added to the plan if (i) the temporal information of the activity is consistent with the domain, and (ii) the temporal constraints of the activity are consistent with the rest of temporal constraints of the actions already added to the plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Codify a HoS tree in an attribute grammar (Knuth 1968) as an intermediate representation, with HoS rules as productions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Translate the grammar into a temporal HTN domain, aimed at representing the parsing of the activity log as a HTN problem where (i) terminal symbols are recognised as temporal events and (ii) nonterminal symbols are recognised according to grammar rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Extend the domain to both recognise and label activities from the log to be easily interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The resulting log contains five new labels according to the contexts DayType (Normal or Extended Daily Driving period), Sequence (if the activity belongs to the first, second or third sequence in the day), BreakType (if breaks are taken in one or two parts), Token (the type of activity at the lowest level of in the HoS tree1) and Legal 1Many types of categories exist at the lowest level, based on the duration of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' For example, A indicates a working activity, B T0 a break of less than 15 minutes, DR T1 a daily rest of more than 11 hours, and WR T1 a weekly rest with more than 45 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 4 Discovering and Explaining Driver Behaviour under HoS Regulations A PREPRINT Labelled Log Labelled Log with Infringements Test Evaluation Labels Comparison Test List Original Tachograph Log Relaxed HPDL domain Relaxed Labelled Log Labelling Process Figure 4: Infringement analysis process for a labelled log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' (whether the activity complies or not with the regulation), as well as two counter columns for the day and the week processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' An output example can be seen in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' In summary, the recognition problem is solved with a planning process where the domain walks through an activity log and its internal HTN structure simultaneously, the latter codifying the HoS tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' If activities comply with the temporal and formal restrictions they are labelled with the appropriate terms, in other case contexts are tagged as unrecognised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Nevertheless, the domain is designed to label as many contexts as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' If a higher (more general) context cannot be identified, the domain still attempts to identify lower contexts before ignoring the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' That means that when a bigger sequence cannot be grouped and labelled together (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' when the driver exceeds the maximum number of driving hours and the DayType column cannot be tagged), the domain tries to tag smaller sequences with their corresponding label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' An example is shown below in Table 3, where although DayType and Sequence tags could not be recognised the system still identifies both BreakType splits and includes the appropriate labels, as well as the correct Token contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='2 Explaining Infringements The previous recognition process labels the tachograph log considering the terms defined by the HoS regulation, its compliance with it and details of their position in the HoS tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' However, when drivers commit infractions this system by itself cannot provide an explanation of the cause and the exact root activity, due to the fact that planning techniques rely on backtracking (that is, the ability to retract while exploring the planning graph) and there is not a simple way to distinguish between a genuine backtracking step while walking through the HTN domain or a forced one by an illegal activity in the log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Therefore we found a need to further analyse the labelled log and explain these information to users without requiring them to inspect all activities not recognised in the log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' We solved this problem from two perspectives, each one concerned with different kinds of violations, which are explained in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Figure 4 shows an overview of the approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='1 Test evaluation On one hand we represent rules from the HoS regulation as tests and applied them to the sequences the labelling process found unrecognisable events (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=', those missing at least a label).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' These tests, as exemplified in the left part of Table 2, codify limits and restrictions in the duration of driving sequences and breaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Whenever a test flags a sequence the system marks it and provides an explanation of the infringement, as seen in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 5 Discovering and Explaining Driver Behaviour under HoS Regulations A PREPRINT Table 2: Tests applied to driving sequences in the log in order to identify infringement causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Test Infraction type dt seq > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='5h Excessive Driving without breaks dt day > 9h ∧ EDDs this week > 2 Excessive Driving in day (NDD) dt day > 10h Excessive Driving in day (EDD) Token day before = DR T3 ∧ Token = ¬ (DR T4 or WR) Missing other half of split daily rest Token = DR or WR ∧ Legal = No ∧ Remaining contexts = ¬ Unknown Rest past the daily/weekly deadline Table 3: Labelling output example for illegal activities and the infraction detected by the tests list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Original Log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Annotated Labels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Driver ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='second ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='uninterrupted ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='driver39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='10/01/2017 23:21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='11/01/2017 08:53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='572 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Break ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='DR T2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Tests takes the form of logic constraints ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='f(astart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' aend) o V (1) being: f a function applied over the sequence defined between activities astart and aend (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' sum, context value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' o a logic operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' V either the value of a context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Token, DayType), a scalar or a duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' As an example, the first constraint in Table 2 could be rewritten as duration(seqstart, seqend) > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='5h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' It is important to note that to correctly identify the infraction some tests may consider not only the illegal activities but also prior activities of other days, a situation frequently present in reduced breaks and rests, where sometimes compensation breaks are not fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Therefore the interval of activities checked by the tests depends on the test itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Because tests are encoded as logic constraints, it is easy to extend the system with additional expert provided rules or modify them if the regulation changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='2 Re-labelling A second approach consists of re-labelling the log using a domain with relaxed duration intervals, that is, the limits imposed by the regulation are softened (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' maximum driving time or minimum break time are enlarged up and down) and the system looks for changes between the new log and the original tagged log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' This process helps to discover infringements caused by a slightly borderline duration, like the driver surpassing (prob- ably unconsciously) the restriction by a small amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' These kind of situations are not easily identified by the tests, due to the fact that the activity by itself could still be legal but labelled differently, becoming an infraction later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 6 Discovering and Explaining Driver Behaviour under HoS Regulations A PREPRINT Table 4: Identifying infringements with a relaxed domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' In this example the fourth activity surpasses by one minute the duration limit to be considered B T0, making the whole sequence illegal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Original Labelled Log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Paragraph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Vector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='HDBSCAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Vectors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Legal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Days ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Illegal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Days ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Paragraph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Vector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Vectors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='HDSBSCAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Centroids ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Encoding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Encoding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='Figure 5: Clustering process for a labelled log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' For example, a driver could surpass the maximum limit for a pause before being considered a break by a few minutes without noticing, and proceeding like a break has not been consumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' As a consequence, that action will be valid, but after the next breaks infractions may arise because the driver is not following its plan as expected, and such actions may not fit correctly under the HoS tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' If the violation is related with this type of mistake, the new relabelled log will contain less illegal sequences than the original and we can compare the Token contexts (concerning the type of activity at the lowest level in the HoS tree) to understand which changes make the sequence legal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Table 4 shows an example with a driver exceeding the break time by two minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Therefore, this method allows us to (a) discover new infringements not considered by the test list, and (b) analyse how the activity should have been to avoid infractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='3 Analysing Driver Behaviour The two previous steps provide a way to understand a driver log and its compliance with the HoS regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' However, experts are usually responsible of dozens of drivers and its not feasible to analyse the substantial logs of each one of them in order to detect problematic tendencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Therefore, we developed a module that clusters behaviour patterns in driver activities and summarises each cluster with expert knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' This method helps to separate standard driving days from unusual ones without inspecting the driver log, and let users concentrate their efforts in analysing only the problematic sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' In order to do that, we considered our problem as an NLP (Natural Language Processing) task, where activities from the log are treated as words and daily sequences as documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' That way we can employ NLP oriented techniques to transform sequences of varying length into fixed dimensions and measure similarity between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Figure 5 shows an overview of the process, consisting of the following steps: 7 Discovering and Explaining Driver Behaviour under HoS Regulations A PREPRINT Table 5: Partial output of the clustering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The system identifies the most similar centroid to the input sequence and the description associated with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Labelled Log Activity DayType Sequence BreakType Token Legal Cluster Driving ndd unique uninterrupted A yes 2 Other A yes Break DR T1 yes Most similar centroid Activity DayType Sequence BreakType Token Legal Cluster Driving ndd unique uninterrupted A yes 2 Other A yes Break DR T3 yes Description Legal and standard daily driving formed by a unique and uninterrupted driving sequence 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' First, a preprocessing step is applied in which the dataset is split in two parts depending on whether the days contains or not illegal activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The reason behind this process is that an infraction recognition process is already provided by the previous module, and thus there is no need for our clustering model to learn to distinguish between legal and illegal sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' On the contrary, we are providing a prior separation to help the model extract more interesting patterns that are not related with the legality of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The subset of labelled columns (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' contexts) that describe the action from an overall point of view are selected, these are (Activity, DayType, BreakType, Token).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' For the illegal subset, the Infraction column is also included to generate clusters and centroids associated with already identified infringement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Specific details about duration and timestamps are not relevant to summarise the days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Nevertheless, some of the information they provided is encoded in the labels, as it is used by the labelling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' This step could be consider as cleaning a document prior an NLP topic categorisation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Because columns contains categorical features not suitable for computation they are transformed into numer- ical, and then joined together using a special character as a separator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' After this step we can consider each entry in our log as a word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Both previous steps are repeated for each activity in our dataset, and activities of the same day are grouped into documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' As a result, we have a collection of documents each one encoding the activities in a driving day sequence as words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' We then use Paragraph Vector (Le and Mikolov 2014) (also known as Doc2Vec) models to obtain dense rep- resentations of fixed dimensions2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Although one model could be trained for both data splits (and reasonable so, as both encodings are subset of the same language), we obtained better results finetuning one model for each split, but ultimately both transforming a document into a 200 sized output vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The resulting representations are now suitable for clustering techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' We obtained our best results using HDBSCAN (Campello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 2013) thanks to its robustness to noise, and choosing the number of clusters based on both expert knowledge and metrics results (Silhouette Coefficient, Calinski-Harabasz and Davies- Bouldin indexs), setting on a final value of 8 clusters for legal data and 7 for days with infractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' In the next section we display a comparative analysis of other techniques under this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Lastly, days are clustered and presented with the decoded centroids, which are described by an expert with a meaningful description, as shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='4 Generating Driver Profiles Similarly to the working days clustering previously explained, we performed categorisation of drivers based on similar behaviour with the idea of extracting driver profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' With enough data, we saw that the large amount of activities contained in event logs can be summarised in different types of driving days as described in the previous sections, and such types encode enough information to extract a characterisation of the driver that can be informative for the transport company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 2Other techniques like Word2Vec or Bag of Words could be used, but we considered the paragraph weight extracted by Paragraph Vector a useful source of information in our task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 8 Discovering and Explaining Driver Behaviour under HoS Regulations A PREPRINT Table 6: Example input data for extracting driver profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Each row encodes how frequently a driver perform one of four types of driving days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Given the uneven number of data of each driver values are expressed as percentages and all rows sums to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Driver Driving day type Split Sequences Normal Rest Uninterrupted Sequences Normal Rest Split Sequences Reduced Rest Uninterrupted Sequences Reduced Rest .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='0 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='15 We performed the following steps to categorise drivers: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Drop days with infractions: Tests introducing violations gave us results who grouped drivers by similar ratio of infractions and by day types (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=', those who tended to excess their break time ended up in the same cluster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' However, we opted to not include such information as they did not align with the purposes of the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Because the context around the infraction cannot be extracted exclusively from the tachograph data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=', was the infringement voluntary, due to lack of correct planning or caused by unexpected circumstances on the road?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=') we believe managers should analyse case by case rather than making decisions without understanding the real motive behind the infraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Create frequency table: For each driver its daily logs are processed following the methodology explained in subsections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='3, keeping the day type predicted by the clustering model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The training dataset is created counting the frequencies the driver has performed each type of driving day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Due to the fact that in our data the amount of information varies for each driver, these frequencies are transformed into percentages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Ultimately, we obtain a table D × C as shown in Table 6, being D the number of drivers and C the number of different days categories (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=', the number of clusters discovered in the previous section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Training: We perform clustering with the resulting table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' From a multiple of techniques our best results were obtained with a Gaussian mixture model trained with the Expectation-Maximization algorithm (Fraley and Raftery 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The deciding factor at choosing the best partition was based exclusively on expert knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Because we were informed that our data was for drivers who performed similar routes on the same country, we looked for a few number of clusters that separated nicely the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' As a closing point, we would like to note that driver profiles based only on tachograph data could be misleading, as they do not account for the specific routes they perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' We believe a better approach would be to combine the cluster information with route details like distance, type of vehicle or type of cargo, and as a result get a categorisation of the driver given the type of route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' That way, decisions based on these profiles will not be biased, and traffic managers could assess their drivers for each particular service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' We intend to explore those options in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 5 Experimentation We have validated our methodology with an experimentation using real tachograph logs provided by an industrial collaborator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' We were provided with a dataset formed by two-weeks-long sequences of activities from 290 different drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Because the architecture is composed of three different components, each one was validated individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The labelling process was verified against multiple driving sequences selected at random, both legal and illegal, manually verifying that the output was the appropriate under the HoS regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' For the infringement analysis system multiple tests for each kind of infraction were carried out, confirming that not only the cause, but also the subsequence containing the infraction was detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Lastly, due to the fact that the clustering in our problem is an unsupervised task, we experimented with different techniques and hyperparametrization to discover the best possible clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The quality of each partition was measured with the Silhouette Coefficient and both Calinski-Harabasz and Davies-Bouldin indexs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The final clustering result was selected between the best performing tests, and after expert inspection of the resulting clusters and centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Figure 6 shows the performance of multiple algorithms in data with and without infractions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The algo- rithms are: Gaussian Mixture models, having each component its own covariance matrix, and controlling the number of mixture components as the number of clusters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' HDBSCAN (Campello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 2013), a hierarchical clustering model employing density based measures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' classical agglomerative clustering using average, complete and ward criteria;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' and K-Means with cosine similarity as distance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 9 Discovering and Explaining Driver Behaviour under HoS Regulations A PREPRINT Data with infractions Data without infractions 2 4 6 8 10 2 4 6 8 10 12 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='5 Silhouette coefficent 2 4 6 8 10 2 4 6 8 10 12 14 0 500 1000 1500 0 100 200 300 Calinski−Harabarsz score 2 4 6 8 10 2 4 6 8 10 12 14 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='5 Davies−Bouldin score Number of Clusters Algorithm Gaussian Mixture HDBSCAN Hierarchical (avg) Hierarchical (complete) Hierarchical (ward) KMeans Figure 6: Silhouette Coefficient, Calinski-Harabasz and Davies-Bouldin indexs as a function of the number of clusters for multiple clustering algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Notice that the y-axis scalings differ among the different panels of this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Some insights can be extracted from the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' HDBSCAN is without doubt the best algorithm under both subsets of this data, but we believe that the reason relies mostly due to its robustness to noise points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Furthermore, results on data with infringements are, as expected, more variable, as this subset combines multiple types of infractions with driving sequences that can be perfectly legal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The runner-up model is not clear, as results vary greatly with the number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' For fully legal data we see hierarchical clustering with average and complete linkage method vastly underperforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The graphs for the rest of techniques take similar shape, mostly agreeing in that 8 clusters seems an appropriate partition for this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Nevertheless, as the results are intended for human interpretation, it is important to remind that the clusters should be reviewed by an expert whenever possible before setting on a final value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Finally, we believe is worth mentioning our experimentation with the LDA (Latent Dirichlet Allocation) (Blei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' This technique is frequently used in NLP tasks to summarise a document with a set of topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Due to a small vocabulary size in our data as opposed to an NLP task, most words (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' driver activities) are present in many different clusters (with the exception of illegal activities), and although the most relevant topics could be ranked and considered as centroids there is no assurance that these topics are understandable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' a B T2 break only makes sense if followed 10 Discovering and Explaining Driver Behaviour under HoS Regulations A PREPRINT Table 7: Clustering results for driver profiles and they interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Cluster Interpretation Proportion 1 No extended days and mostly takes rests uninterrupted 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='6% 2 Usually splits rests as much as possible and rarely takes extended days 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='8% 3 Neither takes many extended days or splits rests 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='2% 4 No clear tendency, driver seems to be flexible 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='4% 5 Tends to split rests as much as possible and frequently takes extended days 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='0% by a B T3 break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The presence of only one of them as a topic does not clarify if the driver completed the sequence or committed an infraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' For our driver clustering experimentation Table 7 shows 5 resulting clusters and their interpretation after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Given that our training data is compromised of mostly event logs of national deliveries in Spain, we can see that more than half of our drives prefer to spent their rests split in two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Nevetheless, as mentioned above, the lack of data about the routes performed in the tachograph hinders the expressivenes, but experts welcome any information that could help them assign the best driver to a service as easily as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The methodology and experimental results are encapsulated in an web application publicly available at https:// github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content='com/IgnacioVellido/Driver-Assistance-System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 6 Related Work This project is an extension of authors prior work (Fernandez-Olivares and Perez 2020) focused on the recognition and labelling of driver activities under the HoS regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The novel contributions provided in this paper go a step forward in our goal of developing an intelligent assistant to drivers and traffic managers, proposing a planning and constraint based analysis of infractions causes and summarisation of driver behaviour with NLP techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Regarding applications concerned with the HoS regulation, many approaches have been developed aimed to solve route planning problems under these rules while minimising transportation costs (Mbiydzenyuy 2015, Omelianenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 2019, Goel 2018, Goel and Irnich 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Nonetheless, the authors have not found works that extract insights that can be useful for experts in analysing and understanding driver activities from a legal perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' As for driver behaviour modelling from tachograph data, proposals like (Zhou and Zhang 2019) employs data mining techniques to categorise truck drivers and analyse dangerous tendencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Their approach is similar to ours in that clusters are manually studied and labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' However, PCA for dimensionally reduction and DBSCAN for clustering are directly used instead due to the fact that their data does not contain categorical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Lastly, word embedding techniques like Paragraph Vector models has been previously applied in non textual data like web user activities (Tagami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 2015) and server logs (Mimura and Tanaka 2018) as a way to transform sequential data of variable length into dimensionally fixed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Similarly, although oriented to process mining applications, the trace2vec model proposed in (De Koninck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 2018) uses embedding techniques for discovery, monitoring and clustering of sequences of activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Nevertheless, up to the author’s knowledge there have not been prior research with tachograph logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 7 Conclusion We have presented a novel planning application that brings the worlds of Data Analytics, IoT and Automated Planning and Scheduling together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The approach provides support to experts on the task of interpreting what drivers are or have been doing by recognising and summarising their activity recorded in an event log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 11 Discovering and Explaining Driver Behaviour under HoS Regulations A PREPRINT Using as a basis our prior work in driver activity recognition, the main contributions exposed in this paper are an infringement analysis process with a planning and constraint based approach, the summarisation of temporal activity logs using word embeddings and clustering, and the creation of driver profiles based on such summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' The over- all system provides a human readable summary of the driver behaviour under the HoS regulation while explaining infractions and the root cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Regarding future work, it is worth noting that the main interest and the ultimate goal of the company is to build an intelligent assistant to provide decision support services to both drivers and companies decision makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' This is a research direction aligned with the concept of assistive interaction (Freedman and Zilberstein 2017), that advocates for the integration of plan recognition and planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' In this way, the recognition of driver’s intent is a previous stage needed to respond with a generated plan adapted to the currently recognised task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' For our next steps we intent to enrich the driver profiling model adding non-tachograph data about the transport service, like type of vehicle and cargo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' Additionally, we are focused on integrating descriptive support to the assistant, being able to suggest drivers plans of actions in compliance with the HoS regulation and considering preference patterns extracted from previous personal behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' References Baldini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=', Sportiello, L.' metadata={'source': 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on 11-month digital tachograph data and multilevel modeling approach, Accident Analysis & Prevention 132, 105256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} +page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE4T4oBgHgl3EQfcQy_/content/2301.05082v1.pdf'} diff --git a/79E4T4oBgHgl3EQf2g20/content/2301.05299v1.pdf b/79E4T4oBgHgl3EQf2g20/content/2301.05299v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..eecc34d3d746832eb6f1564d098dece2c65a2cb7 --- /dev/null +++ b/79E4T4oBgHgl3EQf2g20/content/2301.05299v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c96752c05f34fd83ed72891e4fd3b21d456a98e64058dc8ad7bf96145b8e702e +size 16460806 diff --git a/79E4T4oBgHgl3EQf2g20/vector_store/index.pkl b/79E4T4oBgHgl3EQf2g20/vector_store/index.pkl new file mode 100644 index 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a/B9FKT4oBgHgl3EQfXi52/content/tmp_files/2301.11795v1.pdf.txt b/B9FKT4oBgHgl3EQfXi52/content/tmp_files/2301.11795v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..18ee0d42d966b6a96f32a64d22ee7cb2bde32cf6 --- /dev/null +++ b/B9FKT4oBgHgl3EQfXi52/content/tmp_files/2301.11795v1.pdf.txt @@ -0,0 +1,4110 @@ +arXiv:2301.11795v1 [math.AP] 27 Jan 2023 +Higher regularity for weak solutions +to degenerate parabolic problems +Andrea Gentile - Antonia Passarelli di Napoli∗ +Dipartimento di Matematica e Applicazioni “R. Caccioppoli” +Universitá di Napoli “Federico II”, via Cintia - 80126 Napoli +e-mail: andrea.gentile@unina.it,antpassa@unina.it +January 30, 2023 +Abstract +In this paper, we study the regularity of weak solutions to the following strongly degen- +erate parabolic equation +ut − div +� +(|Du| − 1)p−1 ++ +Du +|Du| +� += f +in ΩT , +where Ω is a bounded domain in Rn for n ≥ 2, p ≥ 2 and ( · )+ stands for the positive +part. We prove the higher differentiability of a nonlinear function of the spatial gradient +of the weak solutions, assuming only that f ∈ L2 +loc (ΩT ). This allows us to establish the +higher integrability of the spatial gradient under the same minimal requirement on the +datum f. +Key words. Widely degenerate problems. Second order regularity. Higher integrability. +AMS Classification. 35B45, 35B65, 35D30, 35K10, 35K65 +1 +Introduction +In this paper, we study the regularity properties of weak solutions u : ΩT → R to the following +parabolic equation +ut − div +� +(|Du| − 1)p−1 ++ +Du +|Du| +� += f +in ΩT = Ω × (0, T ), +(1.1) +which appears in gas filtration problems taking into account the initial pressure gradient. For a precise +description of this motivation we refer to [1] and [3, Section 1.1]. +The main feature of this equation is that it possesses a wide degeneracy, coming from the fact that +its modulus of ellipticity vanishes at all points where |Du| ≤ 1 and hence its principal part of behaves +like a p-Laplacian operator only at infinity. +In this paper we address two interrelated aspects of the regularity theory for solutions to parabolic +problems, namely the higher differentiability and the higher integrability of the weak solutions to +(1.1), with the main aim of weakening the assumption on the datum f with respect to the available +literature. +∗Aknowledgments. The work of the authors is supported by GNAMPA (Gruppo Nazionale per l’Analisi +Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica). The +authors have been also supported by the Universitá degli Studi di Napoli “Federico II” through the project +FRA-000022-ALTRI-CDA-752021-FRA-PASSARELLI. +1 + +2 +These questions have been exploited in case of non degenerate parabolic problems with quadratic +growth by Campanato in [9], by Duzaar et al. in [13] in case of superquadratic growth, while Scheven in +[17] faced the subquadratic growth case. In the above mentioned papers, the problem have been faced +or in case of homogeneous equations or considering sufficiently regular datum. It is worth mentioning +that the higher integrability of the gradient of the solution is achieved through an interpolation +argument, once its higher differentiability is established. +This strategy has revealed to be successful also for degenerate equations as in (1.1). Indeed the higher +integrability of the spatial gradient of weak solutions to equation (1.1), has been proven in [3] , under +suitable assumptions on the datum f in the scale of Sobolev spaces. +We’d like to recall that a common feature for nonlinear problems with growth rate p > 2 is that the +higher differentiability is proven for a nonlinear expression of the gradient which takes into account +the growth of the principal part of the equation. +Indeed, already for the non degenerate p-Laplace equation, the higher differentiability refers to the +function Vp (Du) = +� +1 + |Du|2� p−2 +4 Du. In case of widely degenerate problems, this phenomenon +persists, and higher differentiability results, both for the elliptic and the parabolic problems, hold true +for the function H p +2 (Du) = (|Du| − 1) +p +2 ++ +Du +|Du|. It is worth noticing that, as it can be expected, this +function of the gradient doesn’t give information on the second regularity of the solutions in the set +where the equation degenerates. Actually, since every 1-Lipschitz continuous function is a solution to +the elliptic equation +div (Hp−1 (Du)) = 0, +where Hp−1 (Du) = (|Du| − 1)p−1 ++ +Du +|Du|, no more than Lipschitz regularity can be expected. +Moreover, it is well known that in case of degenerate problems (already for the degenerate p-Laplace +equation, with p > 2) a Sobolev regularity is required for the datum f in order to get the higher +differentiability of the solutions (see, for example [8] for elliptic and [3] for parabolic equations). +Actually, the sharp assumption for the datum in the elliptic setting has been determined in [8] as a +fractional Sobolev regularity suitably related to the growth exponent p and the dimension n. +The main aim of this paper is to show that without assuming any kind of Sobolev regularity for the +datum, but assuming only f ∈ L2, we are still able to obtain higher differentiability for the weak +solutions but outside a set larger than the degeneracy set of the problem. It is worth mentioning that, +while for the p-Laplace equation the degeneracy appears for p > 2, here, even in case p = 2, under +a L2 integrability assumption on the datum f, the local W 2,2 regularity of the solutions cannot be +obtained. +Actually, we shall prove the following +Theorem 1.1. Let n ≥ 2, p ≥ 2 and f ∈ L2 +loc (ΩT ). Moreover, let us assume that +u ∈ C0 � +0, T ; L2 (Ω) +� +∩ Lp +loc +� +0, T ; W 1,p +loc (Ω) +� +is a weak solution to (1.1). Then, for any δ ∈ (0, 1), we have +Gδ +� +(|Du| − 1 − δ)+ +� +∈ L2 +loc +� +0, T ; W 1,2 +loc (Ω) +� +, +where +Gδ(t) := +ˆ t +0 +s(s + δ) +p−2 +2 +√ +1 + δ + s2 ds, +for every t ≥ 0. +Moreover the following estimate +ˆ +Q R +16 +��D +� +Gδ +� +(|Du| − δ − 1)+ +����2 dz +≤ +c (n, p) +R2δ2 +�ˆ +QR +(|Du|p + 1) dz + 1 +δp +ˆ +QR +|f|2 dz +� +, +(1.2) +holds for any R > 0 such that QR = QR (z0) ⋐ ΩT . + +3 +As already mentioned, the weak solutions of (1.1) are not twice differentiable, and hence it is not +possible in general to differentiate the equation to estimate the second derivative of the solutions. We +overcome this difficulty by introducing a suitable family of approximating problems whose solutions +are regular enough by the standard theory ([11]). The major effort in the proof of previous Theorem +is to establish suitable estimates for the solutions of the regularized problems that are uniform with +respect to the approximation’s parameter. Next, we take advantage from these uniform estimates +in the use of a comparison argument aimed to bound the difference quotient of a suitable nonlinear +function of the gradient of the solution that vanishes in the set { |Du| ≤ 1 + δ }, with δ > 0. +Roughly speaking, due to the weakness of our assumption on the datum, we only get the higher +differentiability of a nonlinear function of the gradient of the solutions that vanishes in a set which is +larger with respect to that of the degeneracy of the problem. This is quite predictable, since the same +kind of phenomenon occurs in the setting of widely degenerate elliptic problems (see, for example +[10]). +Anyway, as a consequence of the higher differentiability result in Theorem 1.1, we establish a higher +integrability result for the spatial gradient of the solution to equation (1.1), which is the following +Theorem 1.2. Under the assumptions of Theorem 1.1, we have +Du ∈ L +p+ 4 +n +loc +(ΩT ) +with the following estimate +ˆ +Q ρ +2 +|Du|p + 4 +n dz ≤ c (n, p) +ρ +2(n+2) +n +�ˆ +Q2ρ +� +1 + |Du|p + |f|2� +dz +� 2 +n +1 +, +(1.3) +for every parabolic cylinder Q2ρ (z0) ⋐ ΩT , with a constant c = c(n, p). +The proof of previous Theorem consists in using an interpolation argument with the aim of establishing +an estimate for the Lp+ 4 +n norm of the gradient of the solutions to the approximating problems that +is preserved in the passage to the limit. +We conclude mentioning that the elliptic version of our equation naturally arises in optimal transport +problems with congestion effects, and the regularity properties of its weak solutions have been widely +investigated (see e.g. [2, 4, 6, 8]). Moreover, we’d like to stress that, for sake of clarity, we confine +ourselves to equation (1.1), but we believe that our techniques apply as well to a general class of +equations with a widely degenerate structure. +2 +Notations and preliminaries +In this paper we shall denote by C or c a general positive constant that may vary on different +occasions. Relevant dependencies on parameters will be properly stressed using parentheses or sub- +scripts. The norm we use on Rn will be the standard Euclidean one and it will be denoted by | · |. In +particular, for the vectors ξ, η ∈ Rn, we write ⟨ξ, η⟩ for the usual inner product and |ξ| := ⟨ξ, ξ⟩ +1 +2 for +the corresponding Euclidean norm. +For points in space-time, we will use abbreviations like z = (x, t) or z0 = (x0, t0), for spatial variables +x, x0 ∈ Rn and times t, t0 ∈ R. We also denote by B (x0, ρ) = Bρ (x0) = { x ∈ Rn : |x − x0| < ρ } the +open ball with radius ρ > 0 and center x0 ∈ Rn; when not important, or clear from the context, we +shall omit to indicate the center, denoting: Bρ ≡ B (x0, ρ). Unless otherwise stated, different balls in +the same context will have the same center. Moreover, we use the notation +Qρ (z0) := Bρ (x0) × +� +t0 − ρ2, t0 +� +, +z0 = (x0, t0) ∈ Rn × R, +ρ > 0, +for the backward parabolic cylinder with vertex (x0, t0) and width ρ. We shall sometimes omit the +dependence on the vertex when the cylinders occurring share the same vertex. Finally, for a cylinder +Q = A × (t1, t2), where A ⊂ Rn and t1 < t2, we denote by +∂parQ := (A × { t1 }) ∪ (∂A × [t1, t2]) +the usual parabolic boundary of Q, which is nothing but the standard topological boundary without +the upper cap A × { t2 }. + +4 +We now recall some tools that will be useful to prove our results. +For the auxiliary function Hλ : Rn → Rn defined as +Hλ(ξ) := + + + + + + + +(|ξ| − 1)λ ++ +ξ +|ξ| +if +ξ ∈ Rn \ {0} , +0 +if +ξ = 0, +(2.1) +where λ > 0 is a parameter, we record the following estimates (see [7, Lemma 4.1]): +Lemma 2.1. If 2 ≤ p < ∞, then for every ξ, η ∈ Rn it holds +⟨Hp−1(ξ) − Hp−1(η), ξ − η⟩ ≥ 4 +p2 +���H p +2 (ξ) − H p +2 (η) +��� +2 +, +|Hp−1(ξ) − Hp−1(η)| ≤ (p − 1) +����H p +2 (ξ) +��� +p−2 +p ++ +���H p +2 (η) +��� +p−2 +p � ���H p +2 (ξ) − H p +2 (η) +��� . +we record the following estimates (see [4, Lemma 2.8]) +Lemma 2.2. Let ξ, η ∈ Rk with |ξ| > 1. Then, we have +|Hp−1(ξ) − Hp−1(η)| ≤ c(p) +� +(|ξ| − 1) + (|η| − 1)+ +�p−1 +|ξ| − 1 +|ξ − η| +and +⟨Hp−1(η) − Hp−1(ξ), ·η − ξ⟩ ≥ min { 1, p − 1 } +2p+1 +(|ξ| − 1)p +|ξ| (|ξ| + |η|) |η − ξ|2 . +Definition 2.3. With the use of (2.1), a function u ∈ C0 � +0, T ; L2 (Ω) +� +∩Lp � +0, T ; W 1,p (Ω) +� +is a weak +solution of equation (1.1) if +ˆ +ΩT +(u · ∂tϕ − ⟨Hp−1 (Du) , Dϕ⟩) dz = − +ˆ +ΩT +fϕ dz +(2.2) +for every ϕ ∈ C∞ +0 (ΩT ). +In the following, we shall also use the well known auxiliary function Vp : Rn → Rn defined as +Vp(ξ) := +� +1 + |ξ|2� p−2 +4 ξ, +where p ≥ 2. We have the following result. +Lemma 2.4. +For every ξ, η ∈ Rn there hold +1 +c1 (p) |Vp(ξ) − Vp(η)|2 +≤ +� +1 + |ξ|2 + |η|2� p−2 +2 |ξ − η|2 +≤ +c1(p) +�� +1 + |ξ|2� p−2 +2 ξ − +� +1 + |η|2� p−2 +2 η, ξ − η +� +, +We refer to [16, Chapter 12] or to [15, Lemma 9.2] for a proof of these fundamental inequalities. +For further needs, we also record the following interpolation inequality whose proof can be found +in [12, Proposition 3.1] +Lemma 2.5. +Assume that the function v : Qr(z0) ∪ ∂parQr(z0) → R satisfies +v ∈ L∞ � +t0 − r2, t0; Lq (Br (x0)) +� +∩ Lp � +t0 − r2, t0; W 1,p +0 +(Br (x0)) +� +for some exponents 1 ≤ p, q < ∞ . Then the following estimate +ˆ +Qr(z0) +|v|p+ pq +n dz ≤ c +� +sup +s∈(t0−r2,t0) +ˆ +Br(x0) +|v(x, s)|q dx +� p +n ˆ +Qr(z0) +|Dv|p dz +holds true for a positive constant c depending at most on n, p and q. + +5 +2.1 +Difference quotients +We recall here the definition and some elementary properties of the difference quotients (see, for ex- +ample, [15, Chapter 8]). +Definition 2.6. For every function F : Rn → RN the finite difference operator in the direction xs is +defined by +τs,hF(x) = F (x + hes) − F(x), +where h ∈ R, es is the unit vector in the direction xs and s ∈ {1, . . . , n}. +The difference quotient of F with respect to xs is defined for h ∈ R \ {0} as +∆s,hF(x) = τs,hF(x) +h +. +We shall omit the index s when it is not necessary, and simply write τhF(x) = F(x + h) − F(x) and +|∆hF(x)| = |τhF(x)| +|h| +for h ∈ Rn. +Proposition 2.7. Let F ∈ W 1,p (Ω), with p ≥ 1, and let us set +Ω|h| := { x ∈ Ω : dist (x, ∂Ω) > |h| } . +Then: +(a) ∆hF ∈ W 1,p � +Ω|h| +� +and +Di(∆hF) = ∆h(DiF), +for every i ∈ {1, . . . , n} . +(b) If at least one of the functions F or G has support contained in Ω|h|, then +ˆ +Ω +F ∆hG dx = − +ˆ +Ω +G ∆−hF dx. +(c) We have +∆h (FG) (x) = F (x + hes) ∆hG(x) + G(x)∆hF(x). +The next result about the finite difference operator is a kind of integral version of Lagrange Theorem +(see [15, Lemma 8.1]). +Lemma 2.8. If 0 < ρ < R, |h| < R − ρ +2 +, 1 < p < +∞, and F ∈ W 1,p � +BR, RN� +, then +ˆ +Bρ +|τhF(x)|p dx ≤ cp(n) |h|p +ˆ +BR +|DF(x)|p dx. +Moreover +ˆ +Bρ +|F(x + hes)|p dx ≤ +ˆ +BR +|F(x)|p dx. +We conclude this section with the following fundamental result, whose proof can be found in [15, +Lemma 8.2]: +Lemma 2.9. Let F : Rn → RN, F ∈ Lp � +BR, RN� +with 1 < p < +∞. Suppose that there exist +ρ ∈ (0, R) and a constant M > 0 such that +n +� +s=1 +ˆ +Bρ +|τs,hF(x)|p dx ≤ M p |h|p +for every h, with |h| < R − ρ +2 +. Then F ∈ W 1,p � +Bρ, RN� +and +∥DF∥Lp(Bρ) ≤ M. +Moreover +∆s,hF → DsF +strongly in Lp +loc (BR) , as h → 0, +for each s ∈ {1, . . . , n}. + +6 +2.2 +Some auxiliary functions and related algebraic inequalities +In this section we introduce some auxiliary functions and we list some of their properties, that will be +used in what follows. +For any k > 1 and for s ∈ [0, +∞), let us consider the function +gk(s) = +s2 +k + s2 , +(2.3) +for which we record the following +Lemma 2.10. Let k > 1, and let gk be the function defined by (2.3). Then for every A, B ≥ 0 the +following Young’s type inequality +A · B[s · g′ +k ((s − k)+)] ≤ 2 +√ +2k +� +αA2gk ((s − k)+) + ασA2 + cαB2� +, +(2.4) +holds for every parameters α, σ > 0 with a constant cα independent of σ. Moreover, there exists a +constant ck > 0, depending on k, such that +sg′ +k +�� +s2 − k +� ++ +� +≤ ck, +∀s ≥ 0. +(2.5) +Proof. Since +g′ +k(s) = +2ks +(k + s2)2 , +(2.6) +both the conclusions trivially hold for s ≤ +√ +k. Now assume that s > +√ +k and note that Young’s +inequality implies +A · B [s · g′ +k ((s − k)+)] += +A · B · s · g′ +k ((s − k)+) [σ + (s − k)+] +1 +2 +[σ + (s − k)+] +1 +2 +≤ +αA2s · g′ +k ((s − k)+) [σ + (s − k)+] + cα +B2s · g′ +k ((s − k)+) +[σ + (s − k)+] += +αA2s · g′ +k ((s − k)+) (s − k)+ + ασA2s · g′ +k ((s − k)+) ++cα +B2s · g′ +k ((s − k)+) +[σ + (s − k)+] +≤ +αA2 +2ks(s − k)2 ++ +� +k + (s − k)2 ++ +�2 + ασA2 +2ks(s − k)+ +� +k + (s − k)2 ++ +�2 ++cαB2 +2ks +� +k + (s − k)2 ++ +�2 +(s − k)+ +[σ + (s − k)+], +(2.7) +where we used the explicit expression of g′ +k(s) at (2.6). Recalling (2.3) and since +t +k + t2 ≤ 1, from +(2.7) we deduce +A · B [s · g′ +k ((s − k)+)] +≤ +αA2 +2ks +k + (s − k)2 ++ +gk ((s − k)+) ++ασA2 +2ks +k + (s − k)2 ++ ++ cαB2 +2ks +k + (s − k)2 ++ +. +(2.8) +Setting h(s) = +s +k + (s − k)2 ++ +, we can easily check that +h(k) = 1, +lim +s→+∞ h(s) = 0, +max +s∈[k,+∞) h(s) = h +�� +k2 + k +� += 1 +2 +� +1 + +� +1 + 1 +k +� +< +√ +2 + +7 +and so +2ks +k + (s − k)2 ++ +≤ 2 +√ +2k +∀s > k. +Inserting this in (2.8), we get (2.4). +In order to prove (2.5), let us notice that, recalling (2.6), we have +sg′ �� +s2 − k +� ++ +� += +2ks +� +s2 − k +� ++ +� +k + (s2 − k)2 ++ +�2 . +So, since the function sg′ �� +s2 − k +� ++ +� +is continuous in the interval +� +s ≥ 0 +�� s2 > k +� += +�√ +k, +∞ +� +and +lim +s→+∞ +2ks +� +s2 − k +� ++ +� +k + (s2 − k)2 ++ +�2 = 0, +then there exists a constant ck > 0 such that +sg′ �� +s2 − k +� ++ +� +≤ ck +for every s ≥ 0, +which is the conclusion. +For any δ > 0, let us define +Gδ(t) := +ˆ t +0 +s(s + δ) +p−2 +2 +√ +1 + δ + s2 ds, +for t ≥ 0, +(2.9) +and observe that +G′ +δ(t) = t(t + δ) +p−2 +2 +√ +1 + δ + t2 . +(2.10) +Next Lemma relates the function Gδ (|ξ|) with H p +2 (ξ). +Lemma 2.11. Let Gδ be the function defined by (2.9) and H p +2 be the one defined in (2.1) with λ = p +2. +Then we have +��Gδ +� +(|ξ| − δ − 1)+ +� +− Gδ +� +(|η| − δ − 1)+ +���2 ≤ cp +���H p +2 (ξ) − H p +2 (η) +��� +2 +(2.11) +for any ξ, η ∈ Rn. +Proof. If |ξ| < 1 + δ and |η| < 1 + δ there is nothing to prove. So will assume that |ξ| > 1 + δ, and +without loss of generality we may suppose that |η| ≤ |ξ|. Since Gδ(t) is increasing, we have +��Gδ (|ξ| − 1 − δ) − Gδ +� +(|η| − 1 − δ)+ +��� += +Gδ (|ξ| − 1 − δ) − Gδ +� +(|η| − 1 − δ)+ +� += +ˆ |ξ|−1−δ +(|η|−1−δ)+ +s(s + δ) +p−2 +2 +√ +1 + δ + s2 ds +≤ +ˆ |ξ|−1−δ +(|η|−1−δ)+ +(s + δ) +p−2 +2 ds += +2 +p +� +(|ξ| − 1) +p +2 − +� +(|η| − δ − 1)+ + δ +� p +2 � +. +Now, it can be easily checked that +(|ξ| − 1) +p +2 − +� +(|η| − δ − 1)+ + δ +� p +2 + +8 += + + + + + +(|ξ| − 1) +p +2 − δ +p +2 +if +|ξ| > δ + 1 and |η| ≤ δ + 1 +(|ξ| − 1) +p +2 − (|η| − 1) +p +2 +if +|ξ| > δ + 1 and |η| > δ + 1. +In the first case, we have +���(|ξ| − 1) +p +2 − δ +p +2 +��� += +(|ξ| − 1) +p +2 − δ +p +2 ≤ (|ξ| − 1) +p +2 − (|η| − 1) +p +2 ++ += +���H p +2 (ξ) +��� − +���H p +2 (η) +��� ≤ +���H p +2 (η) − H p +2 (ξ) +��� , +while, in the second, +(|ξ| − 1) +p +2 − +� +(|η| − δ − 1)+ + δ +� p +2 = +���H p +2 (ξ) +��� − +���H p +2 (η) +��� ≤ +���H p +2 (η) − H p +2 (ξ) +��� . +Therefore, +��Gδ +� +(|ξ| − δ − 1)+ +� +− Gδ +� +(|η| − δ − 1)+ +���2 ≤ cp +���H p +2 (ξ) − H p +2 (η) +��� +2 +for every ξ, η ∈ Rn, which is (2.11). +Arguing as in [14, Lemma 2.1], we prove the following. +Lemma 2.12. Let 0 < δ ≤ 1 and p ≥ 2. Then the following inequalities hold +cp,δ(t + δ) +p +2 − ˜cp,δ ≤ Gδ(t) ≤ 2 +p(t + δ) +p +2 +with constants ˜cp,δ and cp,δ < 2 +p depending on p and δ. +Proof. If p = 2, one can easily calculate +Gδ(t) = +ˆ t +0 +s +√ +1 + δ + s2 ds = +�� +1 + δ + s2 +�t +0 = +� +1 + δ + t2 − +√ +1 + δ, +from which immediately follows +1 +2 (t + δ) − 1 +2 +�√ +1 + δ + δ +� +≤ Gδ(t) ≤ t + δ. +Let p > 2. The right inequality is a simple consequence of the trivial bound +s +√ +1+δ+s2 < 1. For the +left inequality we start observing that +� +1 + δ + s2 ≤ +√ +1 + δ + s +=⇒ +Gδ(t) ≥ +ˆ t +0 +s (s + δ) +p−2 +2 +√ +1 + δ + s ds. +Now, we calculate the integral in previous formula. By the change of variable r = +√ +1 + δ + s, we get +ˆ t +0 +s (s + δ) +p−2 +2 +√ +1 + δ + s ds = +ˆ t+ +√ +1+δ +√ +1+δ +� +r − +√ +1 + δ +� � +r − +√ +1 + δ + δ +� p−2 +2 +r +ds += +ˆ t+ +√ +1+δ +√ +1+δ +� +r − +√ +1 + δ + δ +� p−2 +2 +ds − +√ +1 + δ +ˆ t+ +√ +1+δ +√ +1+δ +� +r − +√ +1 + δ + δ +� p−2 +2 +r +ds +≥ +2 +p +�� +r − +√ +1 + δ + δ +� p +2 �t+√1+δ +√1+δ +− +√ +1 + δ +ˆ t+ +√ +1+δ +√ +1+δ +� +r − +√ +1 + δ + δ +� p +2 −2 +ds, +since 0 < δ ≤ 1, we have δ ≤ +√ +1 + δ and therefore r − +√ +1 + δ + δ ≤ r. Calculating the last integral +in previous formula, we get +ˆ t +0 +s(s + δ) +p−2 +2 +√ +1 + δ + s ds + +9 +≥ +2 +p +�� +r − +√ +1 + δ + δ +� p +2 �t+√1+δ +√1+δ +− 2 +√ +1 + δ +p − 2 +�� +r − +√ +1 + δ + δ +� p +2 −1�t+√1+δ +√1+δ += +2 +p +� +(t + δ) +p +2 − δ +p +2 +� +− 2 +√ +1 + δ +p − 2 +� +(t + δ) +p +2 −1 − δ +p +2 −1� += +2 +p(t + δ) +p +2 − 2 +√ +1 + δ +p − 2 (t + δ) +p +2 −1 + 2 +√ +1 + δ +p − 2 δ +p +2 −1 − 2 +pδ +p +2 . +Therefore the lemma will be proven if there exists a constant cp,δ < 2 +p such that +cp,δ(t + δ) +p +2 ≤ 2 +p(t + δ) +p +2 − 2 +√ +1 + δ +p − 2 (t + δ) +p +2 −1 + 2 +√ +1 + δ +p − 2 δ +p +2 −1 − 2 +pδ +p +2 +which, setting +h(t) = 2 +√ +1 + δ +p − 2 (t + δ) +p +2 −1 + +� +cp,δ − 2 +p +� +(t + δ) +p +2 , +is equivalent to prove that there exists cp,δ such that +h(t) ≤ 2 +√ +1 + δ +p − 2 δ +p +2 −1 − 2 +pδ +p +2 . +It is easy to check that h(t) attains his maximum for t + δ = 2 +√ +1 + δ +2 − pcp,δ +and so +h(t) ≤ h +� 2 +√ +1 + δ +2 − pcp,δ +− δ +� += +� +2 +√ +1 + δ +� p +2 � +1 +2 − pcp,δ +� p−2 +2 +2 +p (p − 2) +Therefore, to complete the proof it’s enough to solve the following equation +� +2 +√ +1 + δ +� p +2 � +1 +2 − pcp,δ +� p−2 +2 +2 +p (p − 2) = 2 +√ +1 + δ +p − 2 δ +p +2 −1 − 2 +pδ +p +2 +which is equivalent to +1 +2 − pcp,δ += +� +δ +2 +√ +1 + δ +� +p +p−2 � +p +�√ +1 + δ − δ +� +δ ++ 2 +� +2 +p−2 +that, for 0 < δ < 1, admits a unique solution cp,δ < 2 +p. +3 +The regularization +For ε > 0, we introduce the sequence of operators +Aε(ξ) := (|ξ| − 1)p−1 ++ +ξ +|ξ| + ε +� +1 + |ξ|2� p−2 +2 ξ +and by +uε ∈ C0 � +t0 − R2, t0; L2 (BR) +� +∩ Lp � +t0 − R2, t0; u + W 1,p +0 +(BR) +� +we denote the unique solution to the corresponding problems + + + +uε +t − div (Aε (Duε)) = f ε +in QR (z0) +uε = u +in ∂parQR (z0) +(3.1) +where QR (z0) ⋐ ΩT with R < 1, f ε = f ∗ ρε with ρε the usual sequence of mollifiers. One can easily +check that the operator Aε satisfies p-growth and p-ellipticity assumptions with constants depending + +10 +on ε. +Therefore, by the results in [13], we have +Vp (Duε) ∈ L2 +loc +� +0, T ; W 1,2 +loc (BR (x0) , Rn) +� +and +|Duε| ∈ L +p+ 4 +n +loc +(QR) +and, by the definition of Vp(ξ), this yields +DVp (Duε) ≈ +� +1 + |Duε|2� p−2 +4 D2uε ∈ L2 +loc +� +QR; Rn×n� +=⇒ +��D2uε�� ∈ L2 +loc (QR) +(3.2) +By virtue of [3, Theorem 1.1], we also have H p +2 (Duε) ∈ L2 +loc +� +0, T ; W 1,2 +loc (Ω, Rn) +� +and, by the definition +of H p +2 (ξ), it follows +���DH p +2 (Du) +��� ≤ cp (|Duε| − 1) +p−2 +2 ++ +|D2uε| ∈ L2 +loc +� +QR; Rn×n� +. +(3.3) +3.1 +Uniform a priori estimates +The first step in the proof of Theorem 1.1 is the following estimate for solutions to the regularized +problem (3.1). +Lemma 3.1. Let uε ∈ C0 � +t0 − R2, t0; L2 (BR) +� +∩ Lp � +t0 − R2, t0; u + W 1,p +0 +(BR) +� +be the unique solu- +tion to (3.1). Then the following estimate +sup +τ∈(t0−4ρ2,t0) +ˆ +Bρ +� +|Duε(x, τ)|2 − 1 − δ +� ++ dx ++ +ˆ +Qρ +��D +� +Gδ +� +(|Duε| − δ − 1)+ +����2 dz +≤ +c +ρ2 +�ˆ +Q2ρ +(1 + |Duε|p) dz + δ2−p +ˆ +Q2ρ +|f ε|2 dz +� +(3.4) +holds for any ε ∈ (0, 1] and for every Qρ ⋐ Q2ρ ⋐ QR, with a constant c = c(n, p) independent of ε. +Proof. The weak formulation of (3.1) reads as +ˆ +QR +(uε · ∂tϕ − ⟨Aε (Duε) , Dϕ⟩) dz = − +ˆ +QR +f ε · ϕ dz +for any test function ϕ ∈ C∞ +0 (QR). +Recalling the notation used in (2.2), and replacing ϕ with +∆−hϕ = τ−hϕ +h +for a sufficiently small h ∈ R \ { 0 }, by virtue of the properties of difference quotients, +we have +ˆ +QR +� +∆huε · ∂tϕ − ⟨∆hHp−1 (Duε) , Dϕ⟩ − ε +� +∆h +�� +1 + |Duε|2� p−2 +2 Duε +� +, Dϕ +�� +dz += +− +ˆ +QR +f ε · ∆−hϕ dz. +(3.5) +Arguing as in [13, Lemma 5.1], from (3.5) we get +ˆ +QR +∂t∆huε · ϕ dz + +ˆ +QR +⟨∆hHp−1 (Duε) , Dϕ⟩ dz ++ε +ˆ +QR +� +∆h +�� +1 + |Duε|2� p−2 +2 Duε +� +, Dϕ +� +dz = +ˆ +QR +f ε · ∆−hϕ dz. +For Φ ∈ W 1,∞ +0 +(QR) non negative and g ∈ W 1,∞ (R) non negative and non decreasing, we choose +ϕ = Φ · ∆huε · g +� +|∆huε|2� +in previous identity, thus getting +ˆ +QR +∂t (∆huε) ∆huε · g +� +|∆huε|2� +Φ dz + +11 ++ +ˆ +QR +� +∆hHp−1 (Duε) , D +� +Φ∆huεg +� +|∆huε|2��� +dz ++ε +ˆ +QR +� +∆h +�� +1 + |Duε|2� p−2 +2 Duε +� +, D +� +Φ∆hug +� +|∆huε|2��� +dz += +ˆ +QR +f ε · ∆−h +� +Φ∆huε · g +� +|∆huε|2�� +dz, +i.e. +ˆ +QR +∂t (∆huε) ∆huε · g +� +|∆huε|2� +Φ dz ++ +ˆ +QR +Φ +� +∆hHp−1 (Duε) , ∆hDuε · g +� +|∆huε|2�� +dz ++ε +ˆ +QR +Φ +� +∆h +�� +1 + |Duε|2� p−2 +2 Duε +� +, ∆hDuε · g +� +|∆huε|2�� +dz ++2 +ˆ +QR +Φ +� +∆hHp−1 (Duε) , |∆huε|2 ∆hDuε · g′ � +|∆huε|2�� +dz ++2ε +ˆ +QR +Φ +� +∆h +�� +1 + |Duε|2� p−2 +2 Duε +� +, |∆huε|2 ∆hDuε · g′ � +|∆huε|2�� +dz += +− +ˆ +QR +� +∆hHp−1 (Duε) , DΦ · ∆huε · g +� +|∆huε|2�� +dz +−ε +ˆ +QR +� +∆h +�� +1 + |Duε|2� p−2 +2 Duε +� +, DΦ · ∆huε · g +� +|∆huε|2�� +dz ++ +ˆ +QR +f ε · ∆−h +� +Φ∆huε · g +� +|∆huε|2�� +dz, +(3.6) +that we rewrite as follows +Jh,1 + Jh,2 + Jh,3 + Jh,4 + Jh,5 = −Jh,6 − Jh,7 + Jh,8. +Arguing as in [5],the first integral in equation (3.6) can be expressed as follows +Jh,1 += +ˆ +QR +∂t (∆huε) ∆huε · g +� +|∆huε|2� +Φ dz = 1 +2 +ˆ +QR +∂t +� +|∆huε|2� +· g +� +|∆huε|2� +Φ dz += +1 +2 +ˆ +QR +∂t +�ˆ |∆huε|2 +0 +g(s) ds +� +Φ dz = −1 +2 +ˆ +QR +�ˆ |∆huε|2 +0 +g(s) ds +� +∂tΦ dz. +Using Lemma 2.2, since Φ, g are non negative, we have +Jh,2 ≥ +ˆ +QR +Φ · g +� +|∆huε|2� +|∆hDuε|2 +(|Duε| − 1)p +|Duε| (|Duε| + |Duε(x + h)|) dz. +The right inequality in the assertion of Lemma 2.4 yields +Jh,3 ≥ εcp +ˆ +QR +Φ · g +� +|∆huε|2� +|∆hVp (Duε)|2 dz +Moreover, again by Lemmas 2.2 and 2.4 and the fact that g′(s) ≥ 0, we infer +Jh,4 + Jh,5 ≥ 0. +Therefore (3.6) implies +−1 +2 +ˆ +QR +�ˆ |∆huε|2 +0 +g(s) ds +� +∂tΦ dz ++ +ˆ +QR +Φ · g +� +|∆huε|2� +|∆hDuε|2 +(|Duε| − 1)p +|Duε| (|Duε| + |Duε(x + h)|) dz + +12 ++cpε +ˆ +QR +Φ · g +� +|∆huε|2� +|∆hVp (Duε)|2 dz +≤ +ˆ +QR +|DΦ| |∆hHp−1 (Duε)| |∆huε| · g +� +|∆huε|2� +dz ++ε +ˆ +QR +|DΦ| +����∆h +�� +1 + |Duε|2� p−2 +2 Duε +����� |∆huε| · g +� +|∆huε|2� +dz ++ +ˆ +QR +|f ε| +���∆−h +� +Φ∆huε · g +� +|∆huε|2����� dz. +(3.7) +Now let us consider a parabolic cylinder Qρ (z0) ⋐ Q2ρ (z0) ⋐ QR (z0) with ρ < 2ρ < R and t0 > 0. +For a fixed time τ ∈ +� +t0 − 4ρ2, t0 +� +and θ ∈ (0, t0 − τ), we choose Φ(x, t) = η2(x)χ(t)˜χ(t) with η ∈ +C∞ +0 (B2ρ (x0)), 0 ≤ η ≤ 1, χ ∈ W 1,∞ ([0, T ]) with ∂tχ ≥ 0 and ˜χ a Lipschitz continuous function +defined, for 0 < τ < τ + θ < T , as follows +˜χ(t) = + + + + + + + + + + + + + +1 +if +t ≤ τ +1 − t − τ +θ +if +τ < t ≤ τ + θ +0 +if +τ + θ < t ≤ T +so that (3.7) yields +Ih,1 + Ih,2 + Ih,3 +:= +1 +2 +ˆ +B2ρ +η2χ(τ) +�ˆ |∆huε(x,τ)|2 +0 +g(s) ds +� +dx ++cp +ˆ +Qτ η2χ(t) · g +� +|∆huε|2� +|∆hDuε|2 +(|Duε| − 1)p +|Duε| (|Duε| + |Duε(x + h)|) dz ++cpε +ˆ +Qτ η2χ(t)g +� +|∆huε|2� +|∆hVp (Duε)|2 dz +≤ +2 +ˆ +Qτ ηχ(t) |Dη| |∆hHp−1 (Duε)| |∆huε| · g +� +|∆huε|2� +dz ++2ε +ˆ +Qτ ηχ(t) |Dη| +����∆h +�� +1 + |Duε|2� p−2 +2 Duε +����� |∆huε| · g +� +|∆huε|2� +dz ++ +ˆ +Qτ χ(t) |f ε| +���∆−h +� +η2∆huε · g +� +|∆huε|2����� dz ++1 +2 +ˆ +Qτ η2∂tχ(t) +�ˆ |∆huε|2 +0 +g(s) ds +� +dz +=: +Ih,4 + Ih,5 + Ih,6 + Ih,7, +(3.8) +where we used the notation Qτ = B2ρ (x0) × +� +t0 − 4ρ2, τ +� +. +Since g ∈ W 1,∞ ([0, ∞)), by (3.2), by the last assertion of Lemma 2.9 and by Fatou’s Lemma, we have +lim inf +h→0 (Ih,1 + Ih,2 + Ih,3) +≤ +1 +2 +ˆ +B2ρ +η2χ(τ) +�ˆ |Duε(x,τ)|2 +0 +g(s) ds +� +dx ++cp +ˆ +Qτ η2χ(t) · g +� +|Duε|2� ��D2uε��2 (|Duε| − 1)p +|Duε|2 +dz ++cpε +ˆ +Qτ η2χ(t)g +� +|Duε|2� +|DVp (Duε)|2 dz. +(3.9) +and +lim +h→0 Ih,7 = 1 +2 +ˆ +Qτ η2∂tχ(t) +�ˆ |Duε|2 +0 +g(s) ds +� +dz. +(3.10) +Now let us observe that +|DHp−1 (Duε)| ≤ cp (|Duε| − 1)p−2 ++ +��D2uε�� +(3.11) + +13 +and, using Hölder’s inequality with exponents +� +2(p−1) +p−2 , 2(p−1) +p +� +, we have +ˆ +BR +|DHp−1 (Duε)| +p +p−1 dx +≤ +cp +ˆ +BR +� +(|Duε| − 1)p−2 ++ +��D2uε�� +� +p +p−1 dx +≤ +cp +�ˆ +BR +(|Duε| − 1)p ++ dx +� +p−2 +2(p−1) +· +�ˆ +BR +� +(|Duε| − 1) +p−2 +2 ++ +��D2uε�� +�2 +dx +� +p +2(p−1) +, +and since, by (3.3), the right hand side of previous inequality is finite again by Lemma 2.9, we have +∆hHp−1 (Duε) → DHp−1 (Duε) +strongly in +L2 � +0, T ; L +p +p−1 (BR) +� +as h → 0, +which, since ∆huε → Duε strongly in L2 (0, T ; Lp (BR)) as h → 0, implies +lim +h→0 Ih,4 = 2 +ˆ +Qτ ηχ(t) |Dη| |DHp−1 (Duε)| |Duε| g +� +|Duε|2� +dz. +(3.12) +Using similar arguments, we can check that +lim +h→0 Ih,5 = 2ε +ˆ +Qτ ηχ(t) |Dη| +����D +�� +1 + |Duε|2� p−2 +2 Duε +����� |Duε| · g +� +|Duε|2� +dz. +(3.13) +Now, by Proposition 2.7(c), it holds +���∆−h +� +η2∆huε · g +� +|∆huε|2����� +≤ +c∥Dη∥∞ |∆huε| +���g +� +|∆huε|2���� ++c |∆−h (∆huε)| +���g +� +|∆huε|2���� ++c |∆huε|2 ���g′ � +|∆huε|2���� |∆hDuε| . +and choosing g such that +sg′ � +s2� +≤ M, +(3.14) +for a positive constant M, we have +���∆−h +� +η2∆huε · g +� +|∆huε|2����� +≤ +c∥Dη∥∞ |∆huε| +���g +� +|∆huε|2���� ++c |∆−h (∆huε)| +���g +� +|∆huε|2���� ++cM |∆huε| |∆−hDuε| +(3.15) +Since ∆huε → Duε, ∆−h (∆huε) → D2uε, ∆−hDuε → D2uε strongly in L2 � +0, T ; L2 +loc (Ω) +� +as h → 0, +and f ε ∈ C∞ (ΩT ), thanks to (3.15), we have +lim +h→0 Ih,6 = +ˆ +Qτ χ(t) |f ε| +���D +� +η2Duε · g +� +|Duε|2����� dz. +(3.16) +So, collecting (3.9), (3.10), (3.12), (3.13) and (3.16), we can pass to the limit as h → 0 in (3.8), thus +getting +1 +2 +ˆ +B2ρ +η2χ(τ) +�ˆ |Duε(x,τ)|2 +0 +g(s) ds +� +dx ++cp +ˆ +Qτ η2χ(t) · g +� +|Duε|2� ��D2uε��2 (|Duε| − 1)p +|Duε|2 +dz ++cpε +ˆ +Qτ η2χ(t)g +� +|Duε|2� +|DVp (Duε)|2 dz + +14 +≤ +2 +ˆ +Qτ ηχ(t) |Dη| |DHp−1 (Duε)| |Duε| · g +� +|Duε|2� +dz ++2ε +ˆ +Qτ ηχ(t) |Dη| +����D +�� +1 + |Duε|2� p−2 +2 Duε +����� |Duε| · g +� +|Duε|2� +dz ++ +ˆ +Qτ χ(t) |f ε| +���D +� +η2Duε · g +� +|Duε|2����� dz ++1 +2 +ˆ +Qτ η2∂tχ(t) +�ˆ |Duε|2 +0 +g(s) ds +� +dz +=: +˜I1 + ˜I2 + ˜I3 + ˜I4, +(3.17) +for every g ∈ W 1,∞(0, +∞) such that (3.14) holds true. Now, by (3.11) and by Young’s inequality, +we have +˜I1 + ˜I2 +≤ +cp +ˆ +Qτ ηχ(t) |Dη| (|Duε| − 1)p−2 ++ +��D2uε�� |Duε| · g +� +|Duε|2� +dz ++cp · ε +ˆ +Qτ ηχ(t) |Dη| +� +1 + |Duε|2� p−1 +2 ��D2uε�� · g +� +|Duε|2� +dz +≤ +σ +ˆ +Qτ η2χ(t)(|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 · g +� +|Duε|2� +dz ++σε +ˆ +Qτ η2χ(t) +� +1 + |Duε|2� p−2 +2 ��D2uε��2 · g +� +|Duε|2� +dz ++cσ +ˆ +Qτ χ(t) |Dη|2 (|Duε| − 1)p−4 ++ +|Duε|4 · g +� +|Duε|2� +dz ++cp,σ · ε +ˆ +Qτ χ(t) |Dη|2 � +1 + |Duε|2� p +2 · g +� +|Duε|2� +dz +≤ +σ +ˆ +Qτ η2χ(t)(|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 · g +� +|Duε|2� +dz ++σε +ˆ +Qτ η2χ(t) |DVp (Duε)|2 · g +� +|Duε|2� +dz ++cσ,p ∥Dη∥2 +L∞ ∥g∥L∞ +ˆ +Qτ χ(t) (1 + |Duε|)p dz, +(3.18) +where we used (3.2), and where σ > 0 is a parameter that will be chosen later. +Now, using Young’s Inequality, we estimate the term ˜I3, as follows +˜I3 +≤ +c +ˆ +Qτ χ(t) |f ε| η |Dη| |Duε| · g +� +|Duε|2� +dz ++c +ˆ +Qτ χ(t) |f ε| η2 ��D2uε�� · g +� +|Duε|2� +dz ++c +ˆ +Qτ χ(t) |f ε| η2 |Duε|2 ��D2uε�� · g′ � +|Duε|2� +dz +≤ +c ∥Dη∥∞ ∥g∥L∞ +ˆ +Qτ ηχ(t) |f ε|2 dz ++c ∥Dη∥∞ ∥g∥L∞ +ˆ +Qτ ηχ(t) |Duε|2 dz ++c +ˆ +Qτ η2χ(t) |f ε| +��D2uε�� · g +� +|Duε|2� +dz ++c +ˆ +Qτ η2χ(t) |f ε| |Duε|2 ��D2uε�� · g′ � +|Duε|2� +dz. +(3.19) +Plugging (3.18) and (3.19) into (3.17), we get +1 +2 +ˆ +B2ρ +η2χ(τ) +�ˆ |Duε(x,τ)|2 +0 +g(s) ds +� +dx + +15 ++cp +ˆ +Qτ η2χ(t) · g +� +|Duε|2� (|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 dz ++cpε +ˆ +Qτ η2χ(t)g +� +|Duε|2� +|DVp (Duε)|2 dz +≤ +σ +ˆ +Qτ η2χ(t)(|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 · g +� +|Duε|2� +dz ++σε +ˆ +Qτ η2χ(t) |DVp (Duε)|2 · g +� +|Duε|2� +dz ++cp,σ ∥Dη∥∞ ∥g∥L∞ +ˆ +Qτ ηχ(t) |f ε|2 dz ++cp,σ∥Dη∥∞ ∥g∥L∞ +ˆ +Qτ ηχ(t) (1 + |Duε|)p dz ++c +ˆ +Qτ η2χ(t) |f ε| +��D2uε�� · g +� +|Duε|2� +dz ++c +ˆ +Qτ η2χ(t) |f ε| |Duε|2 ��D2uε�� · g′ � +|Duε|2� +dz ++1 +2 +ˆ +Qτ η2∂tχ(t) +�ˆ |Duε|2 +0 +g(s) ds +� +dz, +which, for a sufficiently small σ, gives +1 +2 +ˆ +B2ρ +η2χ(τ) +�ˆ |Duε(x,τ)|2 +0 +g(s) ds +� +dx ++cp +ˆ +Qτ η2χ(t) · g +� +|Duε|2� (|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 dz ++cpε +ˆ +Qτ η2χ(t)g +� +|Duε|2� +|DVp (Duε)|2 dz +≤ +cp∥Dη∥∞ ∥g∥L∞ +ˆ +Qτ ηχ(t) |f ε|2 dz ++cp∥Dη∥∞ ∥g∥L∞ +ˆ +Qτ ηχ(t) (1 + |Duε|)p dz ++c +ˆ +Qτ η2χ(t) |f ε| +��D2uε�� · g +� +|Duε|2� +dz ++c +ˆ +Qτ η2χ(t) |f ε| |Duε|2 ��D2uε�� · g′ � +|Duε|2� +dz ++1 +2 +ˆ +Qτ η2∂tχ(t) +�ˆ |Duε|2 +0 +g(s) ds +� +dz, +that, neglecting the third integral in the left hand side, implies +1 +2 +ˆ +B2ρ +η2χ(τ) +�ˆ |Duε(x,τ)|2 +0 +g(s) ds +� +dx ++cp +ˆ +Qτ η2χ(t) · g +� +|Duε|2� (|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 dz +≤ +cp∥Dη∥∞ ∥g∥L∞ +ˆ +Qτ ηχ(t) |f ε|2 dz ++cp∥Dη∥∞ ∥g∥L∞ +ˆ +Qτ ηχ(t) (1 + |Duε|)p dz ++c +ˆ +Qτ η2χ(t) |f ε| +��D2uε�� · g +� +|Duε|2� +dz ++c +ˆ +Qτ η2χ(t) |f ε| |Duε|2 ��D2uε�� · g′ � +|Duε|2� +dz ++1 +2 +ˆ +Qτ η2∂tχ(t) +�ˆ |Duε|2 +0 +g(s) ds +� +dz, +(3.20) + +16 +Now, for δ ∈ (0, 1), recalling the notation in (2.3), we choose +g(s) = g1+δ +� +(s − 1 − δ)+ +� +that is +g(s) = +(s − 1 − δ)2 ++ +1 + δ + (s − 1 − δ)2 ++ +, +that is legitimate since g ∈ W 1,∞([0, +∞)). +Moreover, with this choice, we have g(s) ∈ [0, 1], for every s ≥ 0, and thanks to (2.5), there exists a +constant cδ > 0 such that +sg′ � +s2� +≤ cδ +for every s ≥ 0, +so that (3.14) holds. Therefore, since g(s) vanishes on the set where s ≤ 1 + δ and g(s) ≤ 1 for every +s, (3.20) becomes +1 +2 +ˆ +B2ρ +η2χ(τ) +�ˆ |Duε(x,τ)|2 +0 +g(s) ds +� +dx ++cp +ˆ +Qτ η2χ(t) · g +� +|Duε|2� (|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 dz +≤ +c +ˆ +Qτ∩{|Duε|2>1+δ} +η2χ(t) |f ε| +��D2uε�� (|Duε| − 1) +p +2 ++ +|Duε| +|Duε| +(|Duε| − 1) +p +2 ++ +· g +� +|Duε|2� +dz ++c +ˆ +Qτ ∩{|Duε|2>1+δ} +η2χ(t) |f ε| |Duε|2 (|Duε| − 1) +p +2 ++ +|Duε| +|Duε| +(|Duε| − 1) +p +2 ++ +��D2uε�� g′ � +|Duε|2� +dz ++cp∥Dη∥∞ ∥χ∥L∞ +ˆ +Qτ +� +1 + |Duε|p + |f ε|2� +dz + +ˆ +Qτ η2∂tχ(t) +�ˆ |Duε|2 +0 +g(s) ds +� +dz +≤ +cp +δ +p +2 +ˆ +Qτ η2χ(t) |f ε| +��D2uε�� (|Duε| − 1) +p +2 ++ +|Duε| +· g +� +|Duε|2� +dz ++ cp +δ +p +2 +ˆ +Qτ η2χ(t) |f ε| |Duε|2 (|Duε| − 1) +p +2 ++ +|Duε| +��D2uε�� g′ � +|Duε|2� +dz ++cp∥Dη∥∞ ∥χ∥L∞ +ˆ +Qτ +� +1 + |Duε|p + |f ε|2� +dz + +ˆ +Qτ η2∂tχ(t) +�ˆ |Duε|2 +0 +g(s) ds +� +dz, +where we used that +sup +x∈( +√ +1+δ,+∞) +x +(x − 1) +p +2 = +√ +1 + δ +�√ +1 + δ − 1 +� p +2 = +√ +1 + δ +�√ +1 + δ + 1 +� p +2 +δ +p +2 +≤ cp +δ +p +2 , +since δ < 1. Using Young’s inequality in the first integral in the right hand, previous estimate yields +1 +2 +ˆ +B2ρ +η2χ(τ) +�ˆ |Duε(x,τ)|2 +0 +g(s) ds +� +dx ++cp +ˆ +Qτ η2χ(t) · g +� +|Duε|2� (|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 dz +≤ +cp(β) +δp +ˆ +Qτ η2χ(t) |f ε|2 · g +� +|Duε|2� +dz ++β +ˆ +Qτ η2χ(t)(|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 · g +� +|Duε|2� +dz ++ cp +δ +p +2 +ˆ +Qτ η2χ(t) |f ε| |Duε|2 (|Duε| − 1) +p +2 ++ +|Duε| +��D2uε�� g′ � +|Duε|2� +dz ++cp∥Dη∥∞ ∥χ∥L∞ +ˆ +Qτ +� +1 + |Duε|p + |f ε|2� +dz + +17 ++ +ˆ +Qτ η2∂tχ(t) +�ˆ |Duε|2 +0 +g(s) ds +� +dz. +Choosing β sufficiently small, reabsorbing the second integral in the right hand side by the left hand +side and using that g(s) ≤ 1, we get +ˆ +B2ρ +η2χ(τ) +�ˆ |Duε(x,τ)|2 +0 +g(s) ds +� +dx ++ +ˆ +Qτ η2χ(t) · g +� +|Duε|2� (|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 dz +≤ +c cp +δ +p +2 +ˆ +Qτ η2χ(t) |f ε| |Duε|2 (|Duε| − 1) +p +2 ++ +|Duε| +��D2uε�� g′ � +|Duε|2� +dz ++ +ˆ +Qτ η2∂tχ(t) +�ˆ |Duε|2 +0 +g(s) ds +� +dz +c ∥Dη∥2 +∞ ∥χ∥∞ +ˆ +Qτ (1 + |Duε|)p dz ++c ∥χ∥L∞ +�cp +δp + ∥Dη∥L∞ +� ˆ +Qτ |f ε|2 dz. +(3.21) +We now estimate the first integral in the right side of previous inequality with the use of (2.4) with +s = |Duε|2, A = (|Duε| − 1) +p +2 ++ +|Duε| +��D2uε��, B = cp +δ +p +2 |f ε| and k = 1 + δ, thus getting +cp +δ +p +2 +ˆ +Qτ η2χ(t) |f ε| |Duε|2 (|Duε| − 1) +p +2 ++ +|Duε| +��D2uε�� g′ � +|Duε|2� +dz +≤ +2α +ˆ +Qτ η2χ(t)(|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 g +� +|Duε|2� +dz ++2ασ +ˆ +Qτ η2χ(t)(|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 dz ++cα,p +δp +ˆ +Qτ η2χ(t) |f ε|2 dz, +with constants c, cα both independent of σ and where we used that δ < 1. By virtue of (3.3), taking +the limit as σ → 0 in previous inequality, we have +cp +δ +p +2 +ˆ +Qτ η2χ(t) |f ε| |Duε|2 (|Duε| − 1) +p +2 ++ +|Duε| +��D2uε�� g′ � +|Duε|2� +dz +≤ +2α +ˆ +Qτ η2χ(t)(|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 g +� +|Duε|2� +dz ++cα,p +δp +ˆ +Qτ η2χ(t) |f ε|2 dz, +(3.22) +Inserting (3.22) in (3.21), we find +ˆ +B2ρ +η2χ(τ) +�ˆ |Duε(x,τ)|2 +0 +g(s) ds +� +dx ++ +ˆ +Qτ η2χ(t) · g +� +|Duε|2� (|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 dz +≤ +2α +ˆ +Qτ η2χ(t)(|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 g +� +|Duε|2� +dz ++cα,p +δp +ˆ +Qτ η2χ(t)|f ε|2 dz + +18 ++ +ˆ +Qτ η2∂tχ(t) +�ˆ |Duε|2 +0 +g(s) ds +� +dz +c ∥Dη∥2 +∞ ∥χ∥∞ +ˆ +Qτ (1 + |Duε|)p dz ++c ∥χ∥L∞ +�cp +δp + ∥Dη∥L∞ +� ˆ +Qτ |f ε|2 dz. +Choosing α = 1 +4 , we can reabsorb the first integral in the right hand side by the left hand side, thus +obtaining +ˆ +B2ρ +η2χ(τ) +�ˆ |Duε(x,τ)|2 +0 +g(s) ds +� +dx ++ +ˆ +Qτ η2χ(t) · g +� +|Duε|2� (|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 dz +≤ +c ∥Dη∥2 +∞ ∥χ∥∞ +ˆ +Qτ (1 + |Duε|)p dz ++ c +δp ∥χ∥L∞ (1 + ∥Dη∥L∞) +ˆ +Qτ |f ε|2 dz ++c +ˆ +Qτ η2∂tχ(t) +�ˆ |Duε|2 +0 +g(s) ds +� +dz. +(3.23) +By the definition of g, we have +ˆ ζ +0 +g(s) ds = + + + + + + + +0 +if +0 < ζ ≤ 1 + δ +ˆ ζ +1+δ +(s − 1 − δ)2 +1 + δ + (s − 1 − δ)2 ds +if +ζ > 1 + δ, +and so it is easy to check that +ˆ ζ +0 +g(s) ds = + + + + + + + +0 +if +0 < ζ ≤ 1 + δ +ζ − 1 − δ − +√ +1 + δ arctan +�ζ − 1 − δ +√ +1 + δ +� +if +ζ > 1 + δ, +that is +ˆ ζ +0 +g(s) ds = (ζ − 1 − δ)+ − +√ +1 + δ arctan +�(ζ − 1 − δ)+ +√ +1 + δ +� +. +Therefore, by previous equality and the properties of χ and η, (3.23) implies +ˆ +B2ρ +η2χ(τ) +� +|Duε(x, τ)|2 − 1 − δ +� ++ dx ++ +ˆ +Qτ η2χ(t) · g +� +|Duε|2� (|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 dz +≤ +c ∥Dη∥2 +∞ ∥χ∥∞ +ˆ +Qτ (1 + |Duε|)p dz ++ c +δp ∥χ∥L∞ (1 + ∥Dη∥L∞) +ˆ +Qτ |f ε|2 dz ++c +ˆ +Qτ η2∂tχ(t) +� +|Duε|2 − 1 − δ +� ++ dz ++c ∥∂tχ∥∞ |Qτ| + c ∥χ∥∞ |BR| , +(3.24) +which holds for almost every τ ∈ +� +t0 − 4ρ2, t0 +� +. +We now choose a cut-off function η ∈ C∞ (B2ρ (x0)) with η ≡ 1 on Bρ (x0) such that 0 ≤ η ≤ 1 and + +19 +|Dη| ≤ c +ρ. For the cut-off function in time, we choose χ ∈ W 1,∞ � +t0 − R2, t0, [0, 1] +� +such that χ ≡ 0 +on +� +t0 − R2, t0 − 4ρ2� +, χ ≡ 1 on +� +t0 − ρ2, t0 +� +and ∂tχ ≤ c +ρ2 on +� +t0 − 4ρ2, t0 − ρ2� +. With these choices, +(3.24) gives +sup +τ∈(t0−4ρ2,t0) +ˆ +Bρ +χ(τ) +� +|Duε(x, τ)|2 − 1 − δ +� ++ dx ++ +ˆ +Qρ +g +� +|Duε|2� (|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 dz +≤ +c +ρ2 +ˆ +Q2ρ +(1 + |Duε|p) dz + +c +ρ2δp +ˆ +Q2ρ +|f ε|2 dz ++c |Q2ρ| +ρ2 ++ c |B2ρ| , +and since ρ < 2ρ < R < 1, and Q2ρ = Bρ × +� +t0 − 4ρ2, t0 +� +, we have +sup +τ∈(t0−4ρ2,t0) +ˆ +Bρ +� +|Duε(x, τ)|2 − 1 − δ +� ++ dx ++ +ˆ +Qρ +g +� +|Duε|2� (|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 dz +≤ +c +ρ2 +ˆ +Q2ρ +(1 + |Duε|p) dz + +c +ρ2δp +ˆ +Q2ρ +|f ε|2 dz. +(3.25) +Now, with Gδ(t) defined at (2.9), recalling (2.10), we have +��D +� +Gδ +� +(|Duε| − δ − 1)+ +����2 +≤ +(|Duε| − δ − 1)2 ++ +1 + δ + (|Duε| − δ − 1)2 ++ +� +(|Duε| − δ − 1)+ + δ +�p−2 ��D2uε��2 += +g (|Duε|) +� +(|Duε| − δ − 1)+ + δ +�p−2 ��D2uε��2 . +Since g(s) is nondecreasing, we have g(s) ≤ g +� +s2� +, and therefore +��D +� +Gδ +� +(|Duε| − δ − 1)+ +����2 ≤ g +� +|Duε|2� +(|Duε| − 1)p−2 ++ +��D2uε��2 +≤ +cp +δ2 g +� +|Duε|2� (|Duε| − 1)p ++ +|Duε|2 +��D2uε��2 , +(3.26) +where we also used that g(s) = 0, for 0 < s ≤ 1 + δ. Using (3.26) in the left hand side of (3.25), we +obtain +sup +τ∈(t0−4ρ2,t0) +ˆ +Bρ +� +|Duε(x, τ)|2 − 1 − δ +� ++ dx ++ +ˆ +Qρ +��D +� +Gδ +� +(|Duε| − δ − 1)+ +����2 dz +≤ +c +ρ2δ2 +�ˆ +Q2ρ +(1 + |Duε|p) dz + 1 +δp +ˆ +Q2ρ +|f ε|2 dz +� +, +which is (3.4). +Combining Lemma 3.1 and Lemma 2.8, we have the following. +Corollary 3.2. Let uε ∈ C0 � +t0 − R2, t0; L2 (BR) +� +∩ Lp � +t0 − R2, t0; u + W 1,p +0 +(BR) +� +be the unique +solution to (3.1). Then the following estimate +ˆ +Q ρ +2 +��τh +� +Gδ +� +(|Duε| − δ − 1)+ +����2 dz + +20 +≤ +c|h|2 +ρ2δ2 +�ˆ +Q2ρ +(1 + |Duε|p) dz + 1 +δp +ˆ +Q2ρ +|f ε|2 dz +� +(3.27) +holds for |h| < ρ +4, for any parabolic cylinder Q2ρ ⋐ QR (z0). +4 +Proof of Theorem 1.1 +This section is devoted to the proof of Theorem 1.1, that will be divided in two steps. +In the first one we shall establish an estimate that will allow us to measure the L2-distance between +H p +2 (Du) and H p +2 (Duε) in terms of the L2-distance between f and f ε. +In the second one, we conclude combining this comparison estimate with the one obtained for the +difference quotient of the solution to the regularized problem at (3.27). +Proof of Theorem 1.1. Step 1: the comparison estimate. +We formally proceed by testing equations (1.1) and (3.1) with the map ϕ = k(t) (uε − u), where +k ∈ W 1,∞ (R) is chosen such that +k(t) = + + + + + + + + + + + + + +1 +if +t ≤ t2, +− 1 +ω (t − t2 − ω) +if +t2 < t < t2 + ω, +0 +if +t ≥ t2 + ω, +with t0 − R2 < t2 < t2 + ω < t0, and then letting ω → 0. We observe that, at this stage, it is +important that uε and u agree on the parabolic boundary ∂parQR (z0). +Proceeding in a standard way (see for example [13]), for almost every t2 ∈ +� +t0 − R2, t0 +� +, we find +1 +2 +ˆ +BR(x0) +|uε (x, t2) − u (x, t2)|2 dx ++ +ˆ +QR,t2 +⟨Hp−1 (Duε) − Hp−1 (Du) , Duε − Du⟩ dz ++ε +ˆ +QR,t2 +�� +1 + |Duε|2� p−2 +2 Duε, Duε − Du +� +dz += +ˆ +QR,t2 +(f − f ε) (uε − u) dz, +(4.1) +where we used the abbreviation QR,t2 = BR (x0) × +� +t0 − R2, t2 +� +. +Using Lemma 2.1, the Cauchy- +Schwarz inequality as well as Young’s inequality, from (4.1) we infer +λp +sup +t∈(t0−R2,t0) +∥uε(·, t) − u(·, t)∥2 +L2(BR(x0)) ++λp +ˆ +QR +���H p +2 (Duε) − H p +2 (Du) +��� +2 +dz + ε +ˆ +QR(z0) +|Duε|p dz +≤ +ˆ +QR +|f − f ε| |uε − u| dz + ε +ˆ +QR +|Duε|p−1 |Du| dz +≤ +ˆ +QR +|f − f ε| |uε − u| dz + ε · cp +ˆ +QR +|Du|p dz ++1 +2 · ε +ˆ +QR +|Duε|p dz, +(4.2) +where we set λp = min +� 1 +2, 4 +p2 +� +. Reabsorbing the last integral in the right-hand side of (4.2) by the +left-hand side, we arrive at +sup +t∈(t0−R2,t0) +∥uε(·, t) − u(·, t)∥2 +L2(BR(x0)) + +21 ++ +ˆ +QR +���H p +2 (Duε) − H p +2 (Du) +��� +2 +dz + +ε +2λp +ˆ +QR +|Duε|p dz +≤ +ε cp +ˆ +QR +|Du|p dz + cp +ˆ +QR +|f − f ε| |uε − u| dz. +(4.3) +Using in turn Hölder’s inequality and Lemma 2.5, we get +˜I +:= +ˆ +QR +|f − f ε| |uε − u| dz +≤ +C (R, n, p) ∥f − f ε∥L2(QR) · +�ˆ +QR +|uε − u|p+ 2p +n dz +� +n +p(n+2) +≤ +c (n, p, R) ∥f − f ε∥L2(QR) · +�ˆ +QR +|Duε − Du|p dz +� +n +p(n+2) +· +� +sup +t∈(t0−R2,t0) +∥uε(·, t) − u(·, t)∥2 +L2(BR(x0)) +� +1 +n+2 +(4.4) +Now, let us notice that +ˆ +QR +|Duε − Du|p dz += +ˆ +QR∩{|Duε|≥1} +(|Duε| − 1 + 1)p dz + +ˆ +QR∩{|Duε|<1} +|Duε|p dz + +ˆ +QR +|Du|p dz +≤ +cp +ˆ +QR +� +(|Duε| − 1)p ++ +� +dz + +ˆ +QR +(|Du|p + 1) dz +≤ +cp +ˆ +QR +����H p +2 (Duε) − H p +2 (Du) + H p +2 (Du) +��� +2� +dz + cp +ˆ +QR +(|Du|p + 1) dz +≤ +cp +ˆ +QR +���H p +2 (Duε) − H p +2 (Du) +��� +2 +dz + cp +ˆ +QR +(|Du|p + 1) dz. +(4.5) +Inserting (4.5) in (4.4), we get +˜I +≤ +c (n, p, R) ∥f − f ε∥L2(QR(z0)) +· +�ˆ +QR +���H p +2 (Duε) − H p +2 (Du) +��� +2 +dz + +ˆ +QR +(|Du|p + 1) dz +� +n +p(n+2) +· +� +sup +t∈(t0−R2,t0) +∥uε(·, t) − u(·, t)∥2 +L2(BR(x0)) +� +1 +n+2 +≤ +c (n, p, R) ∥f − f ε∥L2(QR) · +�ˆ +QR +���H p +2 (Duε) − H p +2 (Du) +��� +2 +dz +� +n +p(n+2) +· +� +sup +t∈(t0−R2,t0) +∥uε(·, t) − u(·, t)∥2 +L2(BR(x0)) +� +1 +n+2 ++c (n, p, R) ∥f − f ε∥L2(QR) · +�ˆ +QR +(|Du|p + 1) dz +� +n +p(n+2) +· +� +sup +t∈(t0−R2,t0) +∥uε(·, t) − u(·, t)∥2 +L2(BR(x0)) +� +1 +n+2 +and, by Young’s inequality, we get +˜I +≤ +β +ˆ +QR +���H p +2 (Duε) − H p +2 (Du) +��� +2 +dz + β +sup +t∈(t0−R2,t0) +∥uε(·, t) − u(·, t)∥2 +L2(BR(x0)) ++c (n, p, R, β) ∥f − f ε∥ +n+2 +n+1 +L2(QR) · +�ˆ +QR +(|Du|p + 1) dz +� +n +p(n+1) + +22 ++c (n, p, R, β) ∥f − f ε∥ +p(n+2) +n(p−1)+p +L2(QR) +. +(4.6) +Inserting (4.6) in (4.3), we obtain +sup +t∈(t0−R2,t0) +∥uε(·, t) − u(·, t)∥2 +L2(BR(x0)) ++ +ˆ +QR +���H p +2 (Duε) − H p +2 (Du) +��� +2 +dz + +ε +2λp +ˆ +QR +|Duε|p dz +≤ +β +ˆ +QR +���H p +2 (Duε) − H p +2 (Du) +��� +2 +dz + β +sup +t∈(t0−R2,t0) +∥uε(·, t) − u(·, t)∥2 +L2(BR(x0)) ++c (n, p, R, β) ∥f − f ε∥ +n+2 +n+1 +L2(QR) · +�ˆ +QR +(|Du|p + 1) dz +� +n +p(n+1) ++c (n, p, R, β) ∥f − f ε∥ +p(n+2) +n(p−1)+p +L2(QR) ++ ε cp +ˆ +QR +|Du|p dz. +(4.7) +Choosing β = 1 +2 and neglecting the third non negative term in the left hand side of (4.7), we get +sup +t∈(t0−R2,t0) +∥uε(·, t) − u(·, t)∥2 +L2(BR(x0)) + +ˆ +QR +���H p +2 (Duε) − H p +2 (Du) +��� +2 +dz +≤ +c (n, p, R) ∥f − f ε∥ +n+2 +n+1 +L2(QR) · +�ˆ +QR +(|Du|p + 1) dz +� +n +p(n+1) ++c (n, p, R) ∥f − f ε∥ +p(n+2) +n(p−1)+p +L2(QR) ++ ε cp +ˆ +QR +|Du|p dz. +(4.8) +For further needs, we also record that, combining (4.5) and (4.8), we have +ˆ +QR +|Duε|p dz +≤ +c (n, p, R) ∥f − f ε∥ +n+2 +n+1 +L2(QR) · +�ˆ +QR +(|Du|p + 1) dz +� +n +p(n+1) ++c (n, p, R) ∥f − f ε∥ +p(n+2) +n(p−1)+p +L2(QR) ++ ε cp +ˆ +QR +|Du|p dz ++cp +ˆ +QR +(|Du|p + 1) dz. +(4.9) +Step 2: The conclusion. +Let us fix ρ > 0 such that Q2ρ ⊂ QR. We start observing that +ˆ +Q ρ +2 +��τh +� +Gδ +� +(|Du| − δ − 1)+ +����2 dz +≤ +c +ˆ +Q ρ +2 +��τh +� +Gδ +� +(|Duε| − δ − 1)+ +����2 dz ++c +ˆ +Qρ +��Gδ +� +(|Duε| − δ − 1)+ +� +− Gδ +� +(|Du| − δ − 1)+ +���2 dz. +We estimate the right hand side of previous inequality using (3.27) and (2.11), as follows +ˆ +Q ρ +2 +��τh +� +Gδ +� +(|Du| − δ − 1)+ +����2 dz +≤ +c|h|2 +ρ2 +�ˆ +Q2ρ +(1 + |Duε|p) dz + δ2−p +ˆ +Q2ρ +|f ε|2 dz +� ++cp +ˆ +Q2ρ +���H p +2 (Duε) − H p +2 (Du) +��� +2 +dz +that, thanks to (4.8), implies +ˆ +Q ρ +2 +��τh +� +Gδ +� +(|Du| − δ − 1)+ +����2 dz + +23 +≤ +c|h|2 +ρ2 +�ˆ +Q2ρ +(1 + |Duε|p) dz + δ2−p +ˆ +Q2ρ +|f ε|2 dz +� ++c (n, p, R) ∥f − f ε∥ +n+2 +n+1 +L2(QR) · +�ˆ +QR +(|Du|p + 1) dz +� +n +p(n+1) ++c (n, p, R) ∥f − f ε∥ +p(n+2) +n(p−1)+p +L2(QR) ++ ε cp +ˆ +QR +|Du|p dz. +(4.10) +Now, using (4.9), we get +ˆ +Q2ρ +(1 + |Duε|p) dz +≤ +c (n, p, R) ∥f − f ε∥ +n+2 +n+1 +L2(QR) · +�ˆ +QR +(|Du|p + 1) dz +� +n +p(n+1) ++c (n, p, R) ∥f − f ε∥ +p(n+2) +n(p−1)+p +L2(QR) ++ ε cp +ˆ +QR +|Du|p dz ++cp +ˆ +QR +(|Du|p + 1) dz +which, combined with (4.10), implies +ˆ +Q ρ +2 +��τh +� +Gδ +� +(|Du| − δ − 1)+ +����2 dz +≤ +c (n, p) |h|2 +ρ2 +� +c(R) ∥f − f ε∥ +n+2 +n+1 +L2(QR) · +�ˆ +QR +(|Du|p + 1) dz +� +n +p(n+1) ++c(R) ∥f − f ε∥ +p(n+2) +n(p−1)+p +L2(QR) ++ ε +ˆ +QR +|Du|p dz ++ +ˆ +QR +(|Du|p + 1) dz + δ2−p +ˆ +QR +|f ε|2 dz +� +. +Taking the limit as ε → 0, and since f ε → f strongly in L2 (BR), we obtain +ˆ +Q ρ +2 +��τh +� +Gδ +� +(|Du| − δ − 1)+ +����2 dz +≤ +c (n, p) |h|2 +ρ2 +�ˆ +QR +(|Du|p + 1) dz + δ2−p +ˆ +QR +|f|2 dz +� +, +and thanks to Lemma 2.9, we have Gδ +� +(|Du| − δ − 1)+ +� +∈ L2 � +t0 − ρ2, t0; W 1,2 (Bρ) +� +with the follow- +ing estimate +ˆ +Q ρ +2 +��D +� +Gδ +� +(|Du| − δ − 1)+ +����2 dz +≤ +c (n, p) +ρ2 +�ˆ +QR +(|Du|p + 1) dz + δ2−p +ˆ +QR +|f|2 dz +� +. +Since previous estimate holds true for any ρ > 0 such that 4ρ < R, we may choose ρ = R +8 thus getting +(1.2). +5 +Proof of Theorem 1.2 +The higher differentiability result of Theorem 1.1 allows us to argue as in [13, Lemma 5.3] and [17, +Lemma 3.2] to obtain the proof of Theorem 1.2. +Proof of Theorem 1.2. We start observing that +���D +�� +Gδ +� +(|Duε| − 1 − δ)+ +�� 4 +np + 1���� + +24 +≤ +c +��Gδ +� +(|Duε| − 1 − δ)+ +��� +4 +np ��D +� +Gδ +� +(|Duε| − 1 − δ)+ +���� , +(5.1) +where c ≡ c(n, p) > 0 and Gδ(t) is the function defined at (2.9). +With the notation we used in the previous sections, for B2ρ (x0) ⋐ BR (x0), let ϕ ∈ C∞ +0 (Bρ (x0)) and +χ ∈ W 1,∞ ((0, T )) be two non-negative cut-off functions with χ(0) = 0 and ∂tχ ≥ 0. Now, we fix a +time t0 ∈ (0, T ) and apply the Sobolev embedding theorem on the time slices Σt := Bρ(x0) × {t} for +almost every t ∈ (0, t0), to infer that +ˆ +Σt +ϕ2 �� +Gδ +� +(|Duε| − 1 − δ)+ +�� 4 +np + 1�2 +dx +≤ +c +�ˆ +Σt +���D +� +ϕ +� +Gδ +� +(|Duε| − 1 − δ)+ +�� 4 +np + 1���� +2n +n+2 dx +� n+2 +n +≤ +c +�ˆ +Σt +���ϕ D +�� +Gδ +� +(|Duε| − 1 − δ)+ +�� 4 +np + 1���� +2n +n+2 dx +� n+2 +n ++c +�ˆ +Σt +��� +��Gδ +� +(|Duε| − 1 − δ)+ +��� +4 +np + 1 Dϕ +��� +2n +n+2 dx +� n+2 +n +=: +c I1(t) + c I2(t), +where, in the second to last line, we have applied Minkowski’s and Young’s inequalities one after the +other. We estimate I1(t) and I2(t) separately. Let us first consider I1(t). Using (5.1), Lemma 2.12 +and Hölder’s inequality with exponents +�n + 2 +n +, n + 2 +2 +� +, we deduce +I1(t) +≤ +c +�ˆ +Σt +ϕ +2n +n+2 +� +(|Duε| − 1) +2 +n ++ +��DGδ +� +(|Duε| − 1 − δ)+ +��� +� 2n +n+2 dx +� n+2 +n +≤ +c +ˆ +Σt +ϕ2 ��DGδ +� +(|Duε| − 1 − δ)+ +���2 dx +�ˆ +supp(ϕ) +(|Duε| − 1)2 ++ dx +� 2 +n +≤ +c +ˆ +Σt +ϕ2 ��DGδ +� +(|Duε| − 1 − δ)+ +���2 dx +�ˆ +supp(ϕ) +|Duε|2 dx +� 2 +n +. +We now turn our attention to I2(t). Lemma 2.12 and Hölder’s inequality yield +I2(t) +≤ +c +�ˆ +Σt +(|Duε| − 1) +np + 4 +n+2 ++ +|Dϕ| +2n +n+2 dx +� n+2 +n +≤ +c +�ˆ +Σt +� +|Dϕ|2 |Duε|p� +n +n+2 |Du| +4 +n+2 dx +� n+2 +n +≤ +c +ˆ +Σt +|Dϕ|2 |Duε|p dx +�ˆ +supp(ϕ) +|Duε|2 dx +� 2 +n +. +Putting together the last three estimates, using Lemma 2.12 in the left hand side, and integrating +with respect to time, we obtain +ˆ +Qt0 +χϕ2 (|Duε| − 1) +p + 4 +n ++ +dz +≤ +c +ˆ t0 +0 +χ +�ˆ +supp(ϕ) +|Duε(x, t)|2 dx +� 2 +n +· +· +�ˆ +Σt +� +ϕ2 ��DGδ +� +(|Duε| − 1 − δ)+ +���2 + |Dϕ|2 |Du|p� +dx +� +dt +≤ +c +ˆ +Qt0 +χ +� +ϕ2 ��DGδ +� +(|Duε| − 1 − δ)+ +���2 + |Dϕ|2 |Du|p� +dz + +25 +· +� +sup +0 0, there exists a polynomial p so that +||f − p||K := sup +z∈K +|f(z) − p(z)| < ε. +This famous result does not say much about what the polynomial approximant p looks +like off the compact set. For various applications, it would be useful to understand the global +behavior of p and, in particular, the location of the critical points and values of p. To this +end, we state our first result (Theorem A below) after introducing the following notation. +Notation 1.2. For any compact set K ⊂ C we denote by fill(K) the union of K with all +bounded components of C \ K. We say K is full if C \ K is connected. We let CP(f) denote +the set of critical points of an analytic function f, and let CV(f) := f(CP(f)) denote its +critical values. A domain in �C is an open, connected subset of �C. +Theorem A. (Polynomial Runge+) +Let K ⊂ C be compact and full, D a domain +containing K, and suppose f is a function analytic in a neighborhood of K. Then for all +ε > 0, there exists a polynomial p so that ||p − f||K < ε and: +(1) CP(p) ⊂ D, +(2) CV(f|K) ⊂ CV(p) ⊂ fill{z : d(z, f(K)) ≤ ε}. +2020 Mathematics Subject Classification. Primary: 30C10, 30C62, 30E10, Secondary: 41A20. +Key words and phrases. uniform approximation, polynomials, rational functions, Blaschke products, +Runge’s Theorem, Weierstrass’s Theorem, Mergelyan’s Theorem. +The first author is partially supported by NSF Grant DMS 1906259. +1 +arXiv:2301.02888v1 [math.CV] 7 Jan 2023 + +2 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Analogous improvements of the polynomial approximation theorems of Mergelyan and +Weierstrass will be stated and proved in Section 9 (see Theorem 9.8 and Corollary 9.9). +When K is not full, uniform approximation by polynomials is not always possible, and so +we turn to rational approximation. We denote the Hausdorff distance between two sets X, +Y , by dH(X, Y ). +Theorem B. (Rational Runge+) Let K ⊂ C be compact, D a domain containing K, f +a function analytic in a neighborhood of K, and suppose P ⊂ �C \ K contains exactly one +point from each component of �C \ K. Then there exists P ′ ⊂ P so that for all ε > 0, there +is a rational function r so that ||r − f||K < ε and: +(1) dH(r−1(∞), P ′) < ε and |r−1(∞)| = |P ′|, +(2) CP(r) ⊂ D, +(3) CV(f|K) ⊂ CV(r) ⊂ fill{z : d(z, f(K)) ≤ ε}. +The behavior of p off K is of particular interest in applications, such as in complex dy- +namics where approximation results have been used to prove the existence of various dy- +namical behaviors for entire functions (see, for example, [EL87], [BT21], [MRW21], [ERS22], +[MRW22], [BEF+22]). However, not understanding the critical points and values of p means +it has not been known whether these behaviors can occur within restricted classes of en- +tire functions, such as the well studied Speiser or Eremenko-Lyubich classes (see the survey +[Six18]). +Our approach is based on two ideas. The first is to show that on a compact subset K ⊂ C +of a finitely connected domain Ω ⊂ C, any bounded analytic function can be approximated +uniformly by an analytic function B : Ω → C having the property that |B| is constant on +each component of ∂Ω. This extends a classical theorem of Carath´eodory [Car54] concerning +finite Blaschke products on the unit disk to more general regions, and may be of independent +interest. +The second idea is an extension of quasiconformal folding, a type of quasiconformal surgery +introduced in [Bis15], to extend the (generalized) Blaschke product B from Ω to a quasireg- +ular mapping g : �C → �C with specified poles. The map g may be taken close to holomorphic +in a suitable sense, and so the Measurable Riemann Mapping Theorem (MRMT for brevity) +will imply there is a quasiconformal mapping φ so that g ◦ φ−1 is the desired polynomial or +rational approximant. +This approach yields not only information on the critical points and values of the approx- +imants as in Theorems A and B, but more broadly a detailed description of the geometric +structure of these approximants. +We end the introduction by describing this geometric +structure in a few cases. First we introduce some more notation. +Notation 1.3. Let D ⊂ �C be a simply connected domain so that ∞ ̸∈ ∂D. +We let +ψD : E → D denote a Riemann mapping, where E = D is D is bounded and E = D∗ := �C\D +if D is unbounded, in which case we specify ψD(∞) = ∞. + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +3 +First consider the case when K is full and connected, and f is holomorphic in a neighbor- +hood of K satisfying ||f||K < 1. Let Ω, Ω′ be analytic Jordan domains containing K, f(K), +respectively, so that f is holomorphic in Ω. Then the mapping +F := ψ−1 +Ω′ ◦ f ◦ ψΩ : D → D +is holomorphic, and by Carath´eodory’s theorem for the disk (see Theorem 2.1), there is +a finite Blaschke product b : D → D that approximates F on the compact set ψ−1 +Ω (K). +Therefore +B := ψΩ′ ◦ b ◦ ψ−1 +Ω : Ω → Ω′ +is a holomorphic function that approximates f on K, and moreover B restricts to an analytic, +finite-to-1 map of Γ := ∂Ω onto Γ′ := ∂Ω′. +In this paper, we will show that B can be approximated on Ω by a polynomial p so that +p−1(Γ′) is an approximation of Γ. More precisely, p−1(Γ′) is connected, and consists of a +finite union of Jordan curves {γj}n +0 bounding pairwise disjoint Jordan domains {Ωj}n +0 (see +Figure 1): the {Ωj}n +0 are precisely the connected components of p−1(Ω′). There is one “large” +component Ω0 that approximates Ω in the Hausdorff metric. The other components {Ωj}n +1 +can be made as small as we wish and to lie in any given neighborhood of ∂K. Moreover, the +collection {Ωj}n +0 forms a tree structure with any two boundaries ∂Ωj, ∂Ωk either disjoint or +intersecting at a single point, and with Ω0 as the “root” of the tree as in Figure 1. Let Ω∞ +denote the unbounded component of C \ p−1(Γ′), so that +(1.1) +C \ p−1(Γ′) = Ω0 ⊔ +� +⊔n +j=1Ωj +� +⊔ Ω∞. +Recalling Notation 1.3, the polynomial p has the following simple structure with respect to +the domains in (1.1). +(1) p(Ω0) = Ω′ and ψ−1 +Ω′ ◦ p ◦ ψΩ0 is a finite Blaschke product. +(2) p(Ωj) = Ω′ and p is conformal on Ωj for 1 ≤ j ≤ n. +(3) p(Ω∞) = C \ Ω′ and p = ψC\Ω′ ◦ (z �→ zm) ◦ ψ−1 +Ω∞ on Ω∞ for m = deg(p|Ω0) + n. +In other words, up to conformal changes of coordinates, p is simply a Blaschke product in +Ω0, a conformal map in each Ωj, 1 ≤ j ≤ n, and a power map z �→ zm in Ω∞. The only finite +critical points of p are either in Ω0, or at a point where two of the curves (γj)n +j=1 intersect, +in which case the corresponding critical value lies on ∂Ω′. +Next suppose K is connected, but C \ K has more than one component. In this case, in +order to prove Theorem B, we will need to let Ω be a multiply connected analytic domain +containing K, and Ω′ an analytic Jordan domain containing f(K). We cannot proceed as in +the case C\K is connected, however, without a multiply connected version of Carath´eodory’s +Theorem for the disk (Theorem 2.1). The usual proof of Carath´eodory’s Theorem is based +on power series, and it does not extend to the multiply connected setting. However, there +is an alternate proof based on potential theory that does extend. We now briefly sketch the +idea. + +4 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +K +p +Ω∞ +f(K) +Γ′ +Ω0 +{Ωj}n +1 +Figure 1. This figure illustrates the geometry of the approximant p in Theorem A +when K is connected. The notation is explained in the text. Both domain and co- +domain are colored so that regions with the same color correspond to one another +under p. +Suppose f is holomorphic in a simply connected domain D. To simplify matters, suppose +f(D) is compactly contained in D \ {0}. Then u := − log |f| is a positive harmonic function +on D, and u is also bounded and bounded away from zero. Thus u is the Poisson integral +of its boundary values. +The Poisson kernel Px(z) associated to any point x ∈ ∂D is a +limit of normalized Green’s functions on D: Px(z) ≈ G(z, wn)/G(0, wn) with wn → x. +Approximating the Poisson integral by a Riemann sum gives an approximation of u on +any compact set K ⊂ D by a sum of Green’s functions H(z) := � +j G(z, wj) with poles +distributed on {z : d(z, ∂D) = δ}, and with δ as small as we wish. Figure 2 shows the +function u(x, y) = 1 +2 + xy being approximated by a sum of 25 Green’s functions on D = D. +Since D is simply connected, H has a well defined harmonic conjugate �H, and after adding +a constant to �H if necessary, +B := exp(−H − i �H) +approximates f on K. Moreover, B satisfies: +(1) B is holomorphic on D. +(2) |B| extends continuously to a constant function on ∂D, with ||B||∂D = 1. +We call such a function B on a (not necessarily simply connected) domain D a (generalized) +finite Blaschke product on D (see Definition 2.2). When D = D this definition coincides with + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +5 +Figure 2. +On the left is the positive harmonic function u(x, y) = 1 +2 + xy +on the unit disk. On the right is a sum of 25 Green’s functions with poles +on the circle of radius .98. On D(0, 1 +2) the two functions agree to within .03. +As expected, the poles are closer together where u is large and farther apart +where u is small. +the usual definition of Blaschke product, and the above argument yields Carath´eodory’s The- +orem. In fact, the above argument yields several technical improvements of Carath´eodory’s +Theorem (see Theorem 2.6) which we will need in order to prove Theorem A. +This argument generalizes to finitely connected domains D, except that the sum of Green’s +functions H may not have a well defined harmonic conjugate (even modulo 2π). We will fix +this by adding a small harmonic function h that is constant on each boundary component +of D and whose periods match the periods of −H around each boundary component of +D. Exponentiating the sum of the modified function H + h with its harmonic conjugate +(now well-defined modulo 2π) then gives a (generalized) finite Blaschke product B which +approximates the given function f on the desired compact set K ⊂ D. This gives a version +of Carath´eodory’s theorem on finitely connected domains. By choosing H correctly, we can +take ∥h∥D as small as we wish, and hence |B| is constant on each connected component of +∂D, with values that can all be taken as close to 1 as desired. +Now we return to the description of our rational approximant in the case that K is +connected, but C\K has more than one component, recalling that Ω is a multiply connected +analytic domain containing K, and Ω′ is an analytic Jordan domain containing f(K). By +the multiply connected version of Caratheodory’s Theorem, there exists a (generalized) finite +Blaschke product b approximating ψ−1 +Ω′ ◦ f on K, so that B := ψΩ′ ◦ b is a holomorphic map +approximating f on K, and B restricts to an analytic, finite-to-1 mapping of each component +of Γ := ∂Ω onto ψΩ′(|z| = t) for t = 1 or t ≈ 1, where t may depend on the component of Γ. + +1.5 +0.5 ~ +0 +0.5 +-0.5 +-0.5 +0.51.6 +1.4 +1.2 ~ +0.8 +0.6 - +0.2 - +0.5 +0.5 +0.56 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +We will show B can be approximated on Ω by a rational map r so that each component of ∂Ω +can be approximated by a component of r−1 ◦ ψΩ′(|z| = t) for t as above. These components +of r−1 ◦ ψΩ′(|z| = t) bound Jordan domains which form a decomposition of the plane as in +the previously described polynomial setting, and in the interior of each such domain again r +behaves either as a (generalized) finite Blaschke product, a conformal mapping, or a power +mapping (up to conformal changes of coordinates). +Lastly, the case when K has more than one connected component is more intricate, and +we will leave the precise description to later in the paper. (Briefly, quasiconformal folding is +applied not just along the boundary of a neighborhood of K, but also along specially chosen +curves that connect different connected components of this neighborhood.) +We remark that while Theorem A strictly improves on Runge’s Theorem on polynomial +approximation, the relationship between Theorem B and Runge’s Theorem on rational ap- +proximation is more subtle. Both show existence of rational approximants, and only The- +orem B describes the critical point structure of the approximant, however the poles of the +approximant in Theorem B are specified only up to a small perturbation, whereas in Runge’s +Theorem they are specified exactly. We do not know whether it is necessary to consider per- +turbations of P ′ in Theorem B, or if the improvement r−1(∞) = P ′ is possible (a related +problem appears in [BL19], [DKM20], [BLU], where it is known no such improvement is +possible). +Acknowledgements. The authors would like to thank Malik Younsi and Oleg Ivrii for useful +discussions related to this manuscript, and Xavier Jarque for his comments on a preliminary +version of this manuscript. +2. Interior Approximation +The following classical result of Carath´eodory (referenced in the introduction) allows for +approximation by Blaschke products in simply-connected domains. +Theorem 2.1. ([Car54]) Let f : D → C be holomorphic and suppose ||f||D ≤ 1. Then there +exists a sequence of finite Blaschke products on D converging to f uniformly on compact +subsets of D. +The proof of Theorem 2.1 is elementary and may be found, for example, in Theorem I.2.1 of +[Gar81] or Theorem 5.1 of the survey [GMR17]. In order to prove Theorem A, we will need +to prove a version of Theorem 2.1 in which the Blaschke products satisfy certain boundary +regularity conditions, and in order to prove Theorem B, we will need to prove a multiply- +connected version of Theorem 2.1. These improvements are stated below in Theorem 2.6. +First we need several definitions: +Definition 2.2. Let D ⊂ C be a finitely connected domain. We say a non-constant holo- +morphic function B : D → C is a finite Blaschke product on D if |B| extends continuously +to a non-zero, constant function on each component of ∂D and ||B||D = 1. + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +7 +Remark 2.3. When D = D, the definition above corresponds with the usual definition of +finite Blaschke product. +Notation 2.4. For a finite Blaschke product B on a finitely connected domain D, we let +IB denote the connected components of ∂D \ {z : B(z) ∈ R}. In other words, IB are the +preimages (under B) of the open upper and lower half-circle components of B(∂D) ∩ H, +B(∂D) ∩ (−H). We will frequently be dealing with sequences of finite Blaschke products +(Bn)∞ +n=1 on D, in which case we abbreviate IBn by In. +Definition 2.5. We call a domain D ⊂ C an analytic domain if D is finitely connected, and +each component of ∂D is an analytic Jordan curve. +We remark that a boundary component of an analytic domain D cannot be a single point. +Theorem 2.6. Let D ⊂ C be an analytic domain, suppose K ⊂ D is compact, and let f be +a function analytic in a neighborhood of D satisfying ||f||D < 1 and {z : f(z) = 0}∩∂D = ∅. +Then there exists M < ∞ and a sequence (Bn)∞ +n=1 of finite Blaschke products on D satisfying: +(2.1) +inf +z∈∂D |Bn(z)| +n→∞ +−−−→ 1, +(2.2) +||Bn − f||K +n→∞ +−−−→ 0, +(2.3) +sup +I∈In +diam(I) +n→∞ +−−−→ 0, and +(2.4) +sup +I,J∈In +diam(I)/diam(J) < M and +(2.5) +sup +I∈In +supz∈I |B′ +n(z)| +infz∈I |B′ +n(z)| < M for all n. +Theorem 2.6 will suffice for the proofs of Theorems A and B, although we prove a slightly +stronger result in Section 12 (see Theorem 12.2). The proof of Theorem 2.6 will be delayed +until Sections 10-12, which are independent of Sections 2-9. +We now turn to applying +Theorem 2.6 to produce Blaschke approximants as described in the introduction. Given a +Jordan curve γ ⊂ C, we denote the bounded component of �C \ γ by int(γ). +Notation 2.7. We refer to Figure 3 for a summary of the following. For the remainder of +this section, we will fix a compact set K, an analytic domain D containing K, and a function +f holomorphic in a neighborhood of D satisfying ||f||D < 1. Fix ε > 0. We assume that +(2.6) +d(z, K) < ε/2 and d(f(z), f(K)) < ε/2 for every z ∈ ∂D. +Definition 2.8. We let γ be an analytic Jordan curve surrounding f(D) such that +(2.7) +dist(w, f(K)) < ε for every w ∈ γ, +and let Ψ : D → int(γ) denote a Riemann mapping. + +8 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Figure 3. This figure illustrates Notations 2.4, 2.7 and Theorem 2.6. The +vertices pictured on ∂D are B−1(±1), and the components IB of Notation 2.4 +are the edges along ∂D connecting these vertices. +Remark 2.9. By enlarging D slightly we may ensure {z : Ψ−1 ◦ f(z) = 0} ∩ ∂D = ∅, so +that Theorem 2.6 applies to the triple D, K, Ψ−1 ◦ f to produce M < ∞ and a sequence of +finite Blaschke products (Bn)∞ +n=1 on D satisfying the conclusions of Theorem 2.6. +Note that in Remark 2.9 we are applying Theorem 2.6 to Ψ−1 ◦ f (and not f). This will +ensure that the critical values of the approximant Ψ ◦ Bn ≈ f are close to fill(f(K)) as +needed for Theorem B. We will now quasiconformally perturb the sequence (Bn) so as to +ensure that we can later prove the conclusion CV(f|K) ⊂ CV(r) of Theorem B: +Theorem 2.10. We may take the sequence (Bn)∞ +n=1 of Remark 2.9 so that it also satisfies: +(2.8) +CV(Ψ−1 ◦ f|K) ⊂ CV(Bn) for all large n. +Proof. Let K′ ⊂ D be a compact set satisfying K ⊂ int(K′) ⊂ D. Apply Theorem 2.6 to +the triple D, K′, Ψ−1 ◦ f to obtain a sequence of Blaschke products (Bn)∞ +n=1 on D. Let +z ∈ CP(f|K). Since +(2.9) +||Bn − Ψ−1 ◦ f||K′ +n→∞ +−−−→ 0, +by Hurwitz’s Theorem there exists a sequence (wn +z )∞ +n=1 such that +B′ +n(wn +z ) = 0 and wn +z +n→∞ +−−−→ z. +Let r < 1 be so that ψ−1 ◦ f(K) ⊂ rD. Define a homeomorphism hn : D → D to be an +interpolation of +(2.10) +hn(Bn(wn +z )) := Ψ−1 ◦ f(z) for all z ∈ CP(f|K), and + +int() +f(K) +Y +Izl<1 +D +K +B ~ +ofA GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +9 +(2.11) +hn(z) = z for r ≤ |z| ≤ 1. +By (2.9), we have that +|Bn(wn +z ) − Ψ−1 ◦ f(z)| +n→∞ +−−−→ 0 for all z ∈ CP(f|K), +and hence hn may be taken to satisfy: +(2.12) +||(hn)z/(hn)z||D +n→∞ +−−−→ 0. +By the Measurable Riemann Mapping Theorem (see [Ahl06]), there is a quasiconformal +φn : D → D so that: +Bn := hn ◦ Bn ◦ φ−1 +n +is holomorphic. We normalize each φn by specifying φn(p) = p and φ′ +n(p) > 0 for some +r < |p| < 1. Note that +(2.13) +Bn(∂D) ⊂ {z : r ≤ |z| ≤ 1} for large n +by (2.1), so it follows from (2.11) that Bn is a Blaschke product on D for large n. We claim +the sequence (Bn) satisfies the conclusions of Theorem 2.6, as well as (2.8). Indeed, we have +φn(wn +z ) ∈ CP(Bn) and +Bn(φn(wn +z )) = Ψ−1 ◦ f(z). +Thus (2.8) is satisfied. By (2.12), we have that +(2.14) +||φn(z) − z||D +n→∞ +−−−→ 0, +and hence (2.2) follows. The relation (2.1) follows from (2.11), and (2.3) follows from (2.14). +To prove (2.4) and (2.5), we first note that since (hn)z = 0 in r < |z| < 1, it follows from +(2.13) that φn extends to a holomorphic function in a neighborhood U of ∂D, where U does +not depend on n. By (2.14) and the Cauchy integral formula, it follows that φ′ +n(z) +n→∞ +−−−→ 1 +uniformly for z ∈ ∂D and hence we deduce (2.4). Similarly, φ′′ +n(z) +n→∞ +−−−→ 0 uniformly for +z ∈ ∂D and so (2.5) follows. +□ +Recall from the introduction that we plan to extend the definition of the approximant +Ψ ◦ Bn ≈ f from D to all of C, where we recall Ψ was defined in Definition 2.8. To this end, +it will be useful to define the following graph structure on ∂D. +Definition 2.11. For any n ∈ N, we define a set of vertices on ∂D by Vn := (Bn|∂D)−1(R), +where each vertex v is labeled black or white according to whether Bn(v) > 0 or Bn(v) < 0, +respectively. The curve ∂D will be considered as a graph with edges defined by In (recall +from Notation 2.4 that In is precisely the collection of components of ∂D \ Vn). We will +sometimes write Dn in place of D when we wish to emphasize the dependence of the graph +∂D on n. +Definition 2.12. We define a holomorphic mapping gn in D by the formula +(2.15) +gn(z) := Ψ ◦ Bn(z). + +10 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +In Sections 3-7 we will quasiregularly extend the definition of gn to C, and then in Section +9 we apply the MRMT to produce the rational approximant of Theorem B as described in +the introduction. +Remark 2.13. Recall that in Notation 2.7, we fixed ε > 0, a compact set K contained in +an analytic domain D, and a function f holomorphic in D (we note ε, K, D, f also satisfied +extra conditions specified in Notation 2.7). The objects γ, Ψ, Bn, Vn, gn we then defined in +this section were determined by our initial choice of ε, K, D, f. In future sections, it will be +useful to think of γ, Ψ, Bn, Vn, gn as defining functions which take as input some quadruple +(ε, K, D, f) (for any ε, K, D, f as in Notation 2.7), and output whatever object we defined +in this section. For instance, Vn defines a function which takes as input any (ε, K, D, f) +as in Notation 2.7 and outputs (via Definition 2.11) a set of vertices Vn(ε, K, D, f) on ∂D. +Similarly, Bn takes as input any (ε, K, D, f) as in Notation 2.7 and outputs (via Theorem +2.10) a Blaschke product Bn(ε, K, D, f) on D. Likewise for γ, Ψ, gn. +3. Quasiconformal Folding +Given a compact set K ⊂ C and a function f holomorphic in a domain D containing +K, we showed in Section 2 how to approximate f by a holomorphic function gn defined in +D (see Definition 2.12). Moreover gn is just a Blaschke product in D. If f is a function +holomorphic in an arbitrary analytic neighborhood U (where U need not be connected) of a +compact set K, then one can apply the results of Section 2 to each component of U which +intersects K (this is done precisely in Definition 4.1): this yields a holomorphic approximant +of f defined in a finite union of domains, so that the approximant is just a finite Blaschke +product on each domain (recall Definition 2.2). In Sections 3-7, we will build the apparatus +necessary to systematically extend this holomorphic approximant to a quasiregular function +of C which is holomorphic outside a small set. +It was convenient to assume in Notation 2.7 that the compact set K was covered by a +single domain D, however we now begin to work more generally: +Remark 3.1. We refer to Figure 6 for a summary of the following. Throughout Sections +3-7, we will fix ε > 0, a compact set K ⊂ C, a domain D containing K, a disjoint collection +of analytic domains (Di)k +i=1 such that K ⊂ U := ∪iDi ⊂ D, and a function f holomorphic in +a neighborhood of U satisfying ||f||U < 1. We assume that the following analog of Equation +(2.6) holds +(3.1) +d(z, K ∩ Di) < ε/2 and d(f(z), f(K ∩ Di)) < ε/2 for all z ∈ ∂Di and 1 ≤ i ≤ k. +Applying the methods of the previous section to each component Di of U, we can define a +sequence of finite Blaschke products (Bn)∞ +n=1 on each Di (see Remark 2.13). We will let Bn +denote the corresponding function defined on U. In particular, (Bn)∞ +n=1 gives the following +definition of vertices on the boundary of U := ∪iDi (see Definition 2.11 and Remark 2.13). + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +11 +Figure 4. Illustrated is the Definition of the domain V in the proof of Propo- +sition 3.3 +Definition 3.2. For every n ∈ N, we define a set of vertices Vn on ∂U by +Vn := +k� +i=1 +Vn(ε, K ∩ Di, Di, f|Di) = +k� +i=1 +(Bn|∂Di)−1(R). +We now extend the graph structure on ∂U by connecting the different components of +U by curves {Γi}k−1 +i=1 in Proposition 3.3 below, and defining vertices along these curves in +Definition 3.4. We will need to prove a certain level of regularity for these curves and vertices +in order to ensure that the dilatations of quasiconformal adjustments we will make later do +not degenerate as n → ∞. We will denote the curves by {Γi}k−1 +i=1 , and we remark that the +curves depend on n, although we suppress this from the notation. +Proposition 3.3. For each n ∈ N, there exists a collection of disjoint, closed, analytic +Jordan arcs {Γi}k−1 +i=1 in (�C \ U) ∩ D satisfying the following properties: +(1) Each endpoint of Γi is a vertex in Vn, +(2) Each Γi meets ∂U at right angles, +(3) U ∪ (∪iΓi) is connected, and +(4) For each 1 ≤ i ≤ k − 1, the sequence (in n) of curves Γi has an analytic limit. +Proof. The set (�C \ U) ∩ D must contain at least one simply-connected region V with the +property that there are distinct i, j with both ∂V ∩∂Di and ∂V ∩∂Dj containing non-trivial +arcs (see Figure 4). By (2.3), for all sufficiently large n both ∂V ∩ ∂Di, ∂V ∩ ∂Dj contain +vertices of Vn which we denote by vi ∈ ∂Di, vj ∈ ∂Dj, respectively. Consider a conformal +map φ : D → V , and define Γ1 to be the image under φ of the hyperbolic geodesic connecting +φ−1(vi), φ−1 +i (vj) in D. +We now proceed recursively, making sure at step l we pick a V which connects two com- +ponents of U not already connected by a Γ1, ..., Γl−1, and so that V is disjoint from Γ1, ..., +Γl−1. The curves Γi satisfy conclusions (1)-(3) of the proposition. We may ensure that for +each 1 ≤ i ≤ k − 1, the sequence (in n) of curves Γi has an analytic limit by choosing vi, vj +above to converge as n → ∞. +□ + +D +D2 +V +IJ +D112 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Figure 5. Illustrated is Definition 3.4. +Definition 3.4. Consider the vertices Vn ⊂ ∂U of Definition 3.2. We will augment Vn to +include vertices on the curves (Γi)k−1 +i=1 as follows (see Figure 5). Let Γ ∈ (Γi)k−1 +i=1 denote both +the curve as a subset of C and the arclength parameterization of the curve, and suppose Γ +connects vertices +Γ(0) = vi ∈ ∂Di, Γ(length(Γ)) = vj ∈ ∂Dj. +For k = i, j, let εk denote the minimum length of the two edges with endpoint vk in ∂Dk, +and suppose without loss of generality εj < εi. Let l be so that +εj/2 ≤ εi/2l ≤ 2εj. +We place vertices at Γ(εi/2), ..., Γ(εi/2l), and we place vertices along Γ([εi/2l, length(Γ)]) +at equidistributed points. We can label the vertices black/white along Γ so that vertices +connect only to vertices of the opposite color by adding one extra vertex at the midpoint of +the segment having vj as an endpoint, if need be. +We introduce the following notation. +Notation 3.5. Throughout Sections 3-7, we will let Ω denote a fixed (arbitrary) component +of +(3.2) +�C \ +� +U ∪ +k−1 +� +i=1 +Γi +� +, +and p ∈ Ω. Note that Ω is simply connected by Proposition 3.3(3). Denote D∗ := �C\D, and +let σ denote any conformal mapping +(3.3) +σ : D∗ → Ω +satisfying σ(∞) = p. For z ∈ Ω, we define τ(z) := σ−1(z). The map τ induces a partition of +T which we denote by Vn := τ(Vn). + +D1 +E1 +82A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +13 +|z|>1 +Ω +τ +3 +D +D2 +Γ2 +1Γ +Γ3 +D4 +D1 +Figure 6. This figure illustrates Remark 3.1 and Notation 3.5. As pictured, +U has four components (Di)4 +i=1 which are connected by curves (Γi)3 +i=1. Recall +K ⊂ U (the compact set K is not shown in the figure). +The unbounded +component Ω of (3.2) is pictured in dark grey. +The map τ : Ω → D∗ is +a conformal mapping, and sends the vertices on ∂Ω to (possibly unevenly +spaced) vertices on the unit circle. +Remark 3.6. We will sometimes write Ωn, D∗ +n in place of Ω, D∗, respectively, when we wish +to emphasize the dependence of the vertices Vn ⊂ ∂Ω, Vn ⊂ ∂D∗ on the parameter n. +Proposition 3.7. For the graph ∂Ωn, we have: +max{diam(e) : e is an edge of ∂Ωn} +n→∞ +−−−→ 0. +Proof. This follows from (2.3) and Definition 3.4. +□ +As explained in the introduction, in order to prove uniform approximation in Theorem +B, we will need to prove that our quasiregular extension is holomorphic outside a region of +small area. This will usually mean proving the following condition holds. +Definition 3.8. Suppose V ⊂ C is an analytic domain, and ∂V is a graph. Let C > 0. We +say a quasiregular mapping φ : V → φ(V ) is C-vertex-supported if +(3.4) +supp(φz) ⊂ +� +e∈∂V +{z : dist(z, e) < C · diam(e)} +(see Figure 7), where the union in (3.4) is taken over all edges e on ∂V . +It will also be useful to have the following definition. +Definition 3.9. Suppose e, f are rectifiable Jordan arcs, and h : e → f is a homeomorphism. +We say that h is length-multiplying on e if the push-forward (under h) of arc-length measure +on e coincides with the arc-length measure on f multiplied by length(f)/length(e). + +14 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Figure 7. Shown as a black curve is part of a graph G, and in light gray the +neighborhood ∪e∈G{z : dist(z, e) < C · diam(e)} of G. +First we will adjust the conformal map τ so as to be length-multiplying along edges of +∂Ω. Recall the vertices Vn ⊂ T defined in Notation 3.5. +Proposition 3.10. For every n, there is a K-quasiconformal mapping λ : D∗ +n → D∗ +n so that: +(1) λ is C-vertex-supported for some C > 0, +(2) λ(z) = z on Vn and off of supp(λz), +(3) λ ◦ τ is length-multiplying on every component of ∂Ω \ Vn, +(4) C, K do not depend on n. +Proof. This is a consequence of Theorem 4.3 of [Bis15]. Indeed, recall τ := σ−1 and consider +the 2πi-periodic covering map +(3.5) +φ := σ ◦ exp : Hr �→ Ω. +The map φ induces a periodic partition φ−1(Vn) of ∂Hr which has bounded geometry (see +the introduction of [Bis15], or Section 2 of [BL19]) with constants independent of n by +Proposition 3.3(2) and Definition 3.4. Thus Theorem 4.3 of [Bis15] applies to produce a +2πi-periodic, C vertex-supported, and K-quasiconformal map β : Hr → Hr so that φ ◦ β is +length-multiplying on edges of Hr, and C, K are independent of n. Thus, the inverse +β−1 ◦ log ◦τ +is length-multiplying, and since exp is length-multiplying on vertical edges, the well-defined +map +λ := exp ◦β−1 ◦ log : D∗ → D∗ +satisfies the conclusions of the Proposition. +□ +The main idea in defining the quasiregular extension in Ω is to send each edge of ∂Di +to the upper or lower half of the unit circle by following λ ◦ τ with a power map z �→ zn + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +15 +Figure 8. This figure illustrates the Folding Theorem 3.13 and Notation 3.14. +The simply connected domain Ω′ +n is obtained by removing from Ω certain trees +based at the vertices along ∂Ω. +of appropriate degree. The main difficulty in this approach, however, is that the images of +different edges of ∂Di under λ ◦ τ may differ significantly in size, so that there is no single +n with z �→ zn achieving the desired behavior. The solution is to modify the domain Ω by +removing certain “decorations” from the domain Ω, so that each edge of ∂Di is sent to an +arc of roughly the same size under λ ◦ τ. This is formalized below in Theorem 3.13 (see also +Figures 8, 9), and is an application of the main technical result of [Bis15] (see Lemma 5.1). +The “decorations” are the trees in the following definition. +Definition 3.11. Let V ⊂ T be a discrete set. We call a domain W ⊂ D∗ a tree domain +rooted at V if W consists of the complement in D∗ of a collection of disjoint trees, one rooted +at each vertex of V (see the center of Figure 8). +Notation 3.12. For m ∈ N, we let +Z± +m := {z ∈ T : zm = ±1}, +Zm := Z+ +m ∪ Z− +m. +In other words, Z+ +m denotes the mth roots of unity, and Z− +m the mth roots of −1. +Theorem 3.13. For every n, there exists a tree domain Wn rooted at Vn, an integer m = +m(n), and a K-quasiconformal mapping ψ : Wn → D∗ so that: +(1) ψ is C-vertex-supported for some C > 0, and ψ(z) = z off of supp(ψz), +(2) on any edge e of ∂Wn ∩ T, ψ is length-multiplying and ψ(e) is an edge in T \ Zm, +(3) for any edge e of ∂Wn ∩D∗, ψ(e) consists of two edges in T\Zm. Moreover, if x ∈ e, +the two limits limWn∋z→x ψ(z) ∈ T are equidistant from Z+ +m, and from Z− +m, and +(4) C, K do not depend on n. +Proof. We consider the 2π-periodic covering map +(3.6) +φ := σ ◦ λ ◦ exp ◦(z �→ −iz) : H �→ Ω. + +Izl>1 +ov16 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Figure 9. For any x ∈ ∂Wn ∩ D∗, there are two limits limWn∋z→x ψ(z) ∈ T +as illustrated in this figure. Theorem 3.13(3) says that these two limits are +equidistant from the nearest black vertex, and are equidistant from the nearest +white vertex. +inducing a periodic partition φ−1(Vn) of ∂H. +By (2.4), Definition 3.4, and Proposition +3.10(2), any two edges of H have comparable lengths with constant independent of n. There- +fore, Lemma 5.1 of [Bis15] applies to yield a 2π-periodic K-quasiconformal map Ψn of H +onto a subdomain Ψn(H) ⊊ H, with K independent of n. We let +Wn := exp(−iΨn(H)) +and +(3.7) +ψ := exp ◦ − iΨ−1 +n ◦ i log : Wn → D∗. +The map (3.7) is well-defined, and the conclusions of the theorem follow from Lemma 5.1 of +[Bis15]. +□ +Notation 3.14. We will use the notation Ω′ +n := (λ ◦ τ)−1(Wn). +4. Annular Interpolation Between the Identity and a Conformal Mapping +Recall from Notation 3.1 that we have fixed ε > 0, a compact set K, disjoint analytic +domains (Di)k +i=1 so that U := ∪iDi contains K, and f holomorphic in a neighborhood of U +with ||f||U < 1. In this section, we briefly define two useful interpolations in Lemmas 4.3 +and 4.4. +Since the domain Di contains the compact set K∩Di, the definitions and results of Section +2 apply to (ε, K ∩ Di, Di, f|Di) for each 1 ≤ i ≤ k (see Notation 2.7). Thus Remark 2.13 +applies to define (4.1), (4.2) and (4.3) in the following. +Definition 4.1. Let 1 ≤ i ≤ k. We define the Jordan curve +(4.1) +γi := γ(ε, K ∩ Di, Di, f|Di). +Recalling that int(γi) denotes the bounded component of �C \ γi, we define +(4.2) +Ψi := Ψ(ε, K ∩ Di, Di, f|Di) +to be a Riemann mapping Ψi : D → int(γi). Lastly, we define the finite Blaschke products +(4.3) +Bn := Bn(ε, K ∩ Di, Di, f|Di) on Di, + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +17 +Figure 10. Illustrated is the map ηΨ +i : D∗ → �C \ Ψi(riD) of Lemma 4.3 in +the case ri = 1. The dotted circle on the right depicts the unit circle. +where we suppress the dependence of (Bn)∞ +n=1 on i from the notation. +Recall that in Section 3, we defined curves {Γi}k−1 +i=1 connecting the domains Di, and in +Notation 3.5 we fixed a component Ω of the complement of U ∪ ∪k−1 +i=1 Γi. +Notation 4.2. After relabeling the (Di)k +i=1 if necessary, there exists 1 ≤ ℓ ≤ k so that +∂Di ∩ ∂Ω ̸= ∅ if and only if i ≤ ℓ (see Figure 6 for example). For each 1 ≤ i ≤ ℓ, note that +the intersection ∂Di ∩ ∂Ω consists of a single Jordan curve. We let ri := |Bn(∂Di ∩ ∂Ω)|, so +that 1 − ε ≤ ri ≤ 1 by (2.1). +The two interpolations we will need are given in Lemmas 4.3 and 4.4 below. In Lemma +4.3, we define an interpolation ηΨ +i between z �→ z on |z| = 2 with z �→ Ψi(riz) on |z| = 1 +(see Figure 10), and in Lemma 4.4 we modify ηΨ +i to define a map ηi so that ηi(z) = ηi(z) for +|z| = 1. +Lemma 4.3. For each 1 ≤ i ≤ ℓ, there is a quasiconformal mapping ηΨ +i : D∗ → �C \ Ψi(riD) +satisfying the relations: +(4.4) +ηΨ +i (z) = z for |z| ≥ 2 and +(4.5) +ηΨ +i (z) = Ψi(riz) for all |z| = 1. +Moreover, if Di, Dj for 1 ≤ i, j ≤ ℓ are connected by one of the curves (Γi)k−1 +i=1 , then +(4.6) +ηΨ +i ([−2, −1]) ∩ ηΨ +j ([1, 2]) = ηΨ +i ([1, 2]) ∩ ηΨ +j ([−2, −1]) = ∅. +Proof. The existence of ηΨ +i +satisfying (4.4) and (4.5) follows from a standard lemma on +the extension of quasisymmetric maps between boundaries of quasiannuli (see, for instance, +Proposition 2.30(b) of [BF14]). If (4.6) fails for the collection (ηΨ +i )i∈I thus defined, we can +renormalize the conformal mappings (Ψi)ℓ +i=1 appropriately (to rotate the points Ψi(±ri) +along the curve Ψi(riT)), and post-compose a subcollection of the ηΨ +i by diffeomorphisms of + +DV +n!18 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +2D \ Ψi(riD) so that (4.6) is satisfied for the composition when i, j ∈ I, and (4.4) and (4.5) +still hold. +□ +Lemma 4.4. For each 1 ≤ i ≤ ℓ, there is a quasiconformal mapping +ηi : D∗ → C \ Ψi([−ri, ri]) +satisfying the relations +(4.7) +ηi(z) = z for |z| ≥ 2, +(4.8) +ηi(z) = ηi(z) for |z| = 1, and +(4.9) +ηi(z) = ηΨ +i (z) for z ∈ R ∩ D∗. +Proof. Define +γ+ +i := ηΨ +i (∂(A(1, 2) ∩ H)). +Let η be a quasisymmetric mapping of T∩H onto [−1, 1] fixing ±1 (one can take η := M|T∩H +where M is a Mobius transformation mapping −1, 1, i to −1, 1, 0, respectively). Define a +mapping g on γ+ +i by: +(4.10) +g(z) := +� +Ψi ◦ η ◦ Ψ−1 +i (z) +z ∈ Ψi(T ∩ H) +z +otherwise +Since g is a quasisymmetric mapping, a standard lemma on extension of quasisymmetric +maps between boundaries of quasidisks (see, for instance, Proposition 2.30(a) of [BF14]) +implies that g may be extended to a quasiconformal mapping of ηΨ +i (A(1, 2) ∩ H). Define g +similarly in ηΨ +i (A(1, 2) ∩ (−H)). We let ηi := g ◦ ηΨ +i . It is then straightforward to check that +ηi satisfies (4.7)-(4.9). +□ +Remark 4.5. Lemmas 4.3 and 4.4 define 2ℓ many quasiconformal mappings: {ηΨ +i }ℓ +i=1 and +{ηi}ℓ +i=1. The definition of the mappings ηΨ +i , ηi depend on the objects ε, K, (Di)k +i=1, f as +fixed in Notation 3.1, but not on the parameter n in (4.3). Thus we record the trivial but +important observation that the mappings {ηΨ +i }ℓ +i=1 and {ηi}ℓ +i=1 are quasiconformal with a +constant independent of n. +5. Annular Interpolation Between a Blaschke Product and a Power Map +Recall that we have fixed ε > 0, a compact set K, disjoint analytic domains (Di)k +i=1 so +that U := ∪iDi contains K, and f holomorphic in a neighborhood of U with ||f||U < 1. The +curves {Γi}k−1 +i=1 connect the domains (Di)k +i=1, and Ω is a component of the complement of +U ∪ ∪k−1 +i=1 Γi with τ : Ω → D∗ conformal. Recall that the domain Ω′ +n was defined in Theorem +3.13 and Notation 3.14 by removing from Ω a collection of trees rooted at the vertices along +∂Ω, and the map ψ ◦ λ ◦ τ maps Ω′ +n onto D∗ (see Proposition 3.10 and Theorem 3.13). + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +19 +Notation 5.1. Recall from Notation 4.2 that ∂Di ∩ ∂Ω ̸= ∅ if and only if 1 ≤ i ≤ ℓ. Hence +exactly ℓ − 1 of the curves (Γi)k−1 +i=1 intersect ∂Ω. By relabelling the (Γi)k−1 +i=1 if necessary, we +may assume Γj intersects ∂Ω if and only if 1 ≤ j ≤ ℓ − 1. +Let m = m(n) be as in Theorem 3.13. To prove our main results, we will need to modify +z �→ zm in D∗ so that, roughly speaking, (z �→ zm) ◦ ψ ◦ λ ◦ τ(z) agrees with the Blaschke +products Bn (see Definition 4.1) along ∂Di. This is done in Theorem 5.3 below. Its proof +uses the following. +Proposition 5.2. Suppose φ1, φ2 are C1 homeomorphisms of a C1 Jordan arc e such that: +(1) φ1(e) = φ2(e), +(2) φ1, φ2 agree on the two endpoints of e, and +(3) |φ′ +1(z)| = |φ′ +2(z)| for all z ∈ e. +Then φ1 = φ2 on e. +The proof of Proposition 5.2 is a consequence of the Fundamental Theorem of Calculus and +is left to the reader. Recall the constant ri := |Bn(∂Di ∩ ∂Ω)| of Notation 4.2. +Theorem 5.3. For every n, there exists a locally univalent K-quasiregular mapping hn : +D∗ → D∗ so that: +(1) hn(z) = zm for |z| ≥ +m√ +2 where m := m(n) is as in Theorem 3.13, +(2) hn ◦ ψ ◦ λ ◦ τ(z) = Bn(z)/ri for every z ∈ ∂Di and 1 ≤ i ≤ l, and +(3) K is independent of n. +Proof. Fix the standard branch of log. Given an edge e ∈ ∂Di, we have by Theorem 3.13 +that +(5.1) +log ◦ψ ◦ λ ◦ τ(e) = {0} × +�jπ +m , (j + 1)π +m +� +for some 0 ≤ j ≤ 2m − 1. +Denote the vertical line segment in (5.1) by ve. Let f : ve �→ e be a length-multiplying, C1 +homeomorphism so that f −1 agrees with log ◦ψ ◦ λ ◦ τ on the two endpoints of e. Consider +the maps: +(5.2) +z �→ mz for z ∈ +�log 2 +m +� +× +�jπ +m , (j + 1)π +m +� +, +(5.3) +z �→ log ◦r−1 +i Bn ◦ f for z ∈ {0} × +�jπ +m , (j + 1)π +m +� +. +For each 1 ≤ i ≤ l, the Blaschke products Bn are orientation-preserving on the unique outer +boundary component of Di, and orientation-reserving on all other boundary components of +Di. This implies that we may choose the branch of log in (5.3) so that the images of (5.2) +and (5.3) are horizontal translates of one another (recall Bn(e) is a circular arc of angle +π), and the derivative of (5.3) is strictly positive for all z ∈ ve. Since the derivative of + +20 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Figure 11. Illustrated is the proof of Theorem 5.3. In logarithmic coordi- +nates the desired interpolation is denoted φ, and hn is then defined by (5.5) +and (5.6). +(5.2) is also strictly positive, this means the linear interpolation between (5.2) and (5.3) is a +homeomorphism. +By (2.5), we have that |B′ +n| is comparable at all points of e with constant independent of e +and n. Thus, since f is length-multiplying and log is length-multiplying on Euclidean circles +centered at 0, we conclude that the derivative of (5.3) is comparable to m at all points of +ve with constant independent of e and n. Thus, we conclude that the linear interpolation +between (5.2) and (5.3) in the rectangle +(5.4) +� +0, log 2 +m +� +× +�jπ +m , (j + 1)π +m +� +is K-quasiconformal with K independent of e and n (see, for instance, Theorem A.1 of +[MPS20]). Denote the linear interpolation by φ (see Figure 11). +We define +(5.5) +hn := exp ◦φ ◦ log in {z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(e)} ∩ {|z| ≤ +m√ +2}. +The equation (5.5) defines hn(z) for z in {z : 1 ≤ |z| ≤ +m√ +2} and sharing a common angle +with the image under ψ ◦ λ ◦ τ of an edge on some ∂Di. We finish the definition of hn by +simply setting: +(5.6) +hn(z) := zm in {z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(∂Ω′ +n \ (∪i∂Di))}. +The conclusion (1) now follows by definition of hn, and (3) follows since hn is a composition +of holomorphic mappings and a K-quasiconformal interpolation where we have already noted +that K is independent of n. + +h +nA GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +21 +We now show that conclusion (2) follows from Proposition 5.2. Fix an edge e on ∂Di. +Recall ve := log ◦ψ ◦ λ ◦ τ(e). Thus, by (5.3) and (5.5) we have that: +(5.7) +hn ◦ ψ ◦ λ ◦ τ = r−1 +i Bn ◦ f ◦ log ◦ψ ◦ λ ◦ τ on e. +First note that (5.7) agrees set-wise with r−1 +i Bn on e and at the endpoints of e. The map +ψ ◦ λ ◦ τ is length-multiplying (by Proposition 3.10(3) and Theorem 3.13(2)), log is length- +multiplying on the circular segment ψ ◦ λ ◦ τ(e), and f is length-multiplying by definition. +Thus the modulus of the derivative of f ◦log ◦ψ◦λ◦τ is constant on e, and so the derivatives +of (5.7) and r−1 +i Bn have the same modulus at each point of e. Conclusion (2) now follows +from Proposition 5.2. +□ +6. Joining Different Types of Boundary Arcs: the Map En +Recall that in Section 4 we defined the maps ηΨ +i , ηi where 1 ≤ i ≤ ℓ, and in Section 5 +we defined the map hn for all n ∈ N. In this section we define a map En in D∗ which is +roughly given by either z �→ ηΨ +i ◦ hn(z) or z �→ ηi(z) ◦ hn(z), where i is allowed to depend +on arg(z) and which of ηΨ +i , ηi we post-compose hn with is also allowed to depend on arg(z). +Thus, we will need a way to interpolate between the definitions of ηΨ +i , ηi, for different i. The +interpolation regions are defined in Definition 6.1 below, and the map En in Proposition 6.2. +It will be useful to keep Figure 12 in mind for the remainder of this section. +Definition 6.1. Mark one edge ei on Γi for each 1 ≤ i ≤ ℓ − 1. Label the ℓ components of +∂Ω′ +n \ ∪iei as (Gi)ℓ +i=1, where ∂Di ⊂ Gi. Let +(1) J D +i +denote those edges in ∂Di, +(2) J G +i +denote those edges in Gi \ J D +i , +(3) J e denote the edges (ei)ℓ−1 +i=1. +In other words, J D +i +are the edges shared by ∂Ω′ +n and ∂Di, J e consists of ℓ − 1 edges: one +on each of the curves (Γi)ℓ +i=1, and J G +i +are the remaining edges on Gi. Thus we have: +∂Ω′ +n = J e ∪ +� +i +� +J D +i +∪ J G +i +� +. +For z ∈ D∗, we define: +(6.1) +En(z) := +� +ηψ +i ◦ hn(z) +if +z/|z| ∈ ψ ◦ λ ◦ τ(J D +i ) +ηi ◦ hn(z) +if +z/|z| ∈ ψ ◦ λ ◦ τ(J G +i ) +It remains to define En(z) for z ∈ D∗ satisfying z/|z| ∈ ψ ◦ λ ◦ τ(J e). We do so in the +following Proposition. +Proposition 6.2. The map En extends to a locally univalent K-quasiregular mapping En : +D∗ → C satisfying En(z) = zm for |z| ≥ +m√ +2, where m = m(n) is as in Theorem 3.13. +Moreover, K does not depend on n. + +22 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Figure 12. Illustrated is Definition 6.1. The curves Γ1, Γ2 are depicted as +black dotted lines, except for the edges e1 ⊂ Γ1, e2 ⊂ Γ2 which are in thick +black. +Proof. Consider (6.1). Note that if En is defined at z and |z| ≥ +m√ +2, then En(z) = zm +by Theorem 5.3(1) and (4.4), (4.7). Thus, setting En(z) := zm for |z| ≥ +m√ +2 extends the +definition of En. +It remains to extend the definition of En to: +(6.2) +{z : 1 ≤ |z| ≤ +m√ +2 and z/|z| ∈ ψ ◦ λ ◦ τ(ei)}, for 1 ≤ i ≤ ℓ − 1. +Each of the ℓ − 1 sets in (6.2) consists of 2 quadrilaterals which we denote by Q± +i . The +curve Γi connects two distinct elements of (Di)ℓ−1 +i=1. In order to avoid complicating notation +significantly, we will assume without loss of generality that Γi connects Di to Di+1. Let +γi ⊂ 2D be a smooth Jordan arc connecting ηΨ +i (1) to ηΨ +i+1(−1). Moreover, by (4.6), we can +choose γi so that the union of the arcs +(6.3) +ηΨ +i ([1, 2]), 2T ∩ H, ηΨ +i+1([−2, −1]), γi +forms a topological quadrilateral we denote by Q+ +i (in particular none of the arcs in (6.3) +intersect except at common endpoints). +Define a quasisymmetric homeomorphism g+ +i : ∂Q+ +i → ∂(A(1, 2) ∩ H) (see Figure 13) by +g+ +i (z) = z for z ∈ 2T +g+ +i (z) = (ηΨ +i )−1(z) for z ∈ ηΨ +i ([1, 2]) +g+ +i (z) = (ηΨ +i+1)−1(z) for z ∈ ηΨ +i+1([−2, −1]), +and extending g+ +i to a quasisymmetric homeomorphism of γi to T ∩ H. The mapping g+ +i +extends to a quasiconformal homeomorphism g+ +i : Q+ +i → A(1, 2) ∩ H (see Lemma 2.24 of + +D1 +" +G1A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +23 +Figure 13. Illustrated is the quadrilateral Q+ +i and the map g+ +i in the proof +of Proposition 6.2. +[BF14]). We define En(z) := (g+ +i )−1(zm) for z ∈ Q+ +i . A similar definition of g− +i : Q− +i → +A(1, 2) ∩ −H is given (using the same curve γi) so that +g+ +i (z) = g− +i (z) for z ∈ T ∩ H. +We let En(z) := (g− +i )−1(zm) for z ∈ Q− +i . +To summarize, we have defined En in each of the three regions +{z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(J D +i )}, +(6.4) +{z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(J G +i )}, +(6.5) +{z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(J e +i )}, +(6.6) +Indeed, the definition of En in (6.4) and (6.5) was given already in (6.1), and in this proof we +have defined En in (6.6). The definitions of En in each of (6.4), (6.5), (6.6) agree along any +common boundary, and thus by removability of analytic arcs for quasiregular mappings, it +follows that En is quasiregular on D∗. Moreover, En has no branched points in D∗, and hence +En is locally quasiconformal. The dilatation of the map En depends only on the dilatation +of hn (which is independent of n by Theorem 5.3(3)) and the dilatations of the the finite +collection of quasiconformal maps used in its definition: ηΨ +i , ηi, g+ +i , g− +i , and hence we may +take K independent of n. +□ +7. Defining gn in Ω′ +n +First we recall our setup. +We have fixed ε > 0, a compact set K, disjoint, analytic +domains (Di)k +i=1 so that K ⊂ U := ∪iDi, and f holomorphic in a neighborhood of U with +||f||U < 1. We defined curves {Γi}k−1 +i=1 connecting the domains (Di)k +i=1, and we denoted by +Ω a component of the complement of U ∪ ∪k−1 +i=1 Γi with τ : Ω → D∗ conformal. The domain +Ω′ +n is contained in Ω, and ψ ◦ λ ◦ τ maps Ω′ +n onto D∗. In Section 6 we defined the map En. +Definition 7.1. We define the mapping gn : Ω′ +n → �C by +(7.1) +gn := En ◦ ψ ◦ λ ◦ τ. + +Izl=2 +Izl=2 +Izl=1 +Di +D +i+124 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +We will now record at which points the function gn|Ω′n is locally n : 1 for n > 1. +Definition 7.2. Let g be a quasiregular function, defined in a neighborhood of a point +z ∈ C. We say that z is a branched point of g if for any sufficiently small neighborhood U +of z, the map g|U is n : 1 onto its image for n > 1. We say w ∈ C is a branched value of g +if w = g(z) for a branched point z of g. We denote the branched points of a quasiregular +mapping g by BP(g), and the branched values by BV(g). +Remark 7.3. Recall that in Notation 3.5 we fixed a point p ∈ Ω satisfying τ(p) = ∞. +Proposition 7.4. The mapping gn : Ω′ +n → C of Definition 7.1 is K-quasiregular and C- +vertex supported for K, C independent of n. Moreover, g−1 +n (∞) = {p}, +(7.2) +BP(gn) ⊂ +� +e∈∂Ωn +{z : dist(z, e) < C · diam(e)}, and +(7.3) +BV(gn) ⊂ +k� +i=1 +Ψi(riT). +Proof. Since each of the mappings in the composition (7.1) are K-quasiregular and C-vertex +supported for K, C independent of n, the same is true of gn. The only points where the +mapping gn is locally l : 1 for l > 1 are a subset of the vertices of the graph ∂Ω′ +n. By +Theorem 3.13, the vertices of ∂Ω′ +n all lie in +� +e∈∂Ωn +{z : dist(z, e) < C · diam(e)}. +Thus, (7.2) is proven. Moreover, any vertex of ∂Ω′ +n is mapped to a point on one of the curves +Ψi(riT) by gn. Hence, (7.3) follows since BV(gn) = gn(BP(gn)). It remains to show: +(7.4) +g−1 +n (∞) = {p}. +Indeed, note that En◦ψ◦λ fixes ∞ and has no finite poles. The map τ : Ω → D∗ is conformal +and hence only one point p is mapped to ∞. The relation (7.4) now follows. +□ +It will be useful to record the following result. +Proposition 7.5. Let r > 1. Then for all sufficiently large n, we have: +(7.5) +gn(z) = τ(z)m for any z ∈ τ −1({z : |z| > r}). +Proof. Consider the functional equation (7.1) defining gn. +The maps λ, ψ are vertex- +supported, and moreover λ (respectively, ψ) is the identity outside of the support of λz, +(respectively, ψz). By Proposition 3.7, we therefore have that ψ ◦ λ(z) = z if z ∈ τ −1({z : +|z| > r}) and n is sufficiently large. The relation (7.5) now follows from (7.1) and Theorem +5.3(1) since m → ∞ as n → ∞. +□ + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +25 +Remark 7.6. As in Remark 2.13, we note that our Definition 7.1 of gn is determined by a +choice of the objects K, U, D, f, ε, Ω, p we fixed in Notations 3.1 and 3.5. When we wish +to emphasize this dependence, we will write gn(K, U, D, f, ε, Ω, p). In particular, it will be +useful in the next section to think of gn as a function taking as input any choice of K, U, +D, f, ε, Ω, p satisfying the conditions in Notations 3.1, 3.5, and outputting (via Definition +7.1) a quasiregular function gn(K, U, D, f, ε, Ω, p) defined on Ω′ +n. +8. Verifying gn is Quasiregular on �C +In this section we combine our efforts in Sections 2-7 to define an approximant gn : �C → �C +of a given f. The approximant gn will not be holomorphic as required in Theorems A and +B, but we will solve this problem in the next section by applying the Measurable Riemann +Mapping Theorem. We fix the following for Sections 8-9. +Notation 8.1. Fix K, f, D, ε, P as in the statement of Theorem B. Denote by U the +neighborhood of K in which f is holomorphic. Define +(8.1) +P ′ := {p ∈ P : p is contained in a component V of �C \ K such that V ̸⊆ U}. +Compactness of K implies that U contains all but finitely many components of �C \ K, and +so the set P ′ is finite. Moreover, P ′ does not depend on ε. By shrinking U if necessary, we +may assume that: +(1) U ∩ P ′ = ∅, +(2) P ′ contains exactly one point in each component of �C \ U, +(3) f is holomorphic in a neighborhood of U ⊂ D, and +(4) the components of U are a finite collection of analytic Jordan domains (Di)k +i=1 so +that (2.6) holds for each Di. +Let K′ be a compact set such that K ⊂ int(K′) ⊂ K′ ⊂ U. We will assume for now that +||f||U < 1. +We now define a quasiregular approximation gn of f by applying the Blaschke product +construction of Section 2 in each Di, and by applying the folding construction of Sections +3-7 in each complementary component of ∪i(Di ∪ Γi): +Definition 8.2. For every n, we define a quasiregular mapping gn as follows. Recalling +Remark 2.13, we first set +(8.2) +gn := gn(ε, K′ ∩ Di, Di, f|Di) in Di for 1 ≤ i ≤ k +The equation (8.2) defines the curves (Γi)k +i=1 by way of Proposition 3.3, and we enumerate +the components of +�C \ +� +U ∪ +k−1 +� +i=1 +Γi +� + +26 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +by (Ω(i))ℓ +i=1. Recalling Remark 7.6 and Notation 3.14, we extend the definition of gn to the +open set +(8.3) +Ω := �C \ +� ℓ� +i=1 +∂Ω′ +n(i) +� +by the formula +(8.4) +gn := gn(K′, U, D, f, ε, Ω(i), P ′ ∩ Ω(i)) in Ω′ +n(i) for 1 ≤ i ≤ ℓ. +Proposition 8.3. The quasiregular function gn is C-vertex supported and K-quasiregular +for C, K independent of n. +Proof. For gn|Ω′n(i) this is exactly Proposition 7.4, and so the conclusion follows since gn is +holomorphic in U. +□ +The function gn is now defined on all of �C except for the edges of each ∂Ω′ +n(i). We show in +Propositions 8.4, 8.5 below that gn in fact extends continuously across each edge of ∂Ω′ +n(i), +and deduce in Corollary 8.6 that gn extends quasiregularly across ∂Ω. +Proposition 8.4. The K-quasiregular function gn : Ω → �C extends to a continuous function +gn : Ω ∪ e → �C for any edge e ⊂ ∂Ω ∩ ∂U. +Proof. Let i be so that e ⊂ ∂Di and denote the unique element of (Ω(i))ℓ +i=1 that contains e +on its boundary by Ω(j). Recall by Definitions 2.12 and 7.1 that +(8.5) +gn|Di = Ψi ◦ Bn, +(8.6) +gn|Ω′n(j) = En ◦ ψ ◦ λ ◦ τ. +Assume that i ∈ I (the reasoning in the case i ̸∈ I will be the same), so that by Theorem +5.3(2) we have +hn ◦ ψ ◦ λ ◦ τ = r−1 +i Bn on e. +By (4.5) and the definition (6.1) of En, it follows from (8.6) that +gn|Ω′n(j)(z) = Ψi ◦ Bn(z) for z ∈ e, +in other words gn|Ω′n(j) and gn|Di agree pointwise on e. +□ +Proposition 8.5. The K-quasiregular function gn : Ω → �C extends to a continuous function +gn : Ω ∪ e → �C for any edge e ⊂ ∂Ω ∩ (∪ℓ +i=1Ω(i)). +Proof. Let j be so that e ⊂ ∂Ω′ +n(j), and as in the proof of Proposition 8.4, recall that +(8.7) +gn|Ω′n(j) = En ◦ ψ ◦ λ ◦ τ. +Let x ∈ e. There are two limits +lim +Ω′n(j)∋z→x ψ ◦ λ ◦ τ(z), + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +27 +each lying on the unit circle. Denote them by ζ±. By Theorem 3.13(3), +ζm ++ = ζm +− . +Thus, by (4.8) and (6.1), we conclude that there is a unique limit +lim +Ω′n(j)∋z→x En ◦ ψ ◦ λ ◦ τ(z). +Hence, setting +gn(x) := +lim +Ω′n(j)∋z→x En ◦ ψ ◦ λ ◦ τ(z) +defines a continuous extension of gn across the edge e. +□ +Corollary 8.6. The K-quasiregular function gn : Ω → �C extends to a K-quasiregular func- +tion gn : �C → �C. +Proof. The set �C \ Ω = ∂Ω consists of a finite collection of analytic arcs: the edges of +the graphs ∂Ω′ +n(i) over 1 ≤ i ≤ ℓ. Thus, by removability of analytic arcs for quasiregular +mappings, it suffices to show that gn : Ω → �C extends continuously across each such edge. +There are two types of edges to check: those that lie on the boundary of a domain Di, and +those that lie in the interior of a domain Ω(i). We have already checked continuity across +both types of edges in Propositions 8.4, 8.5, and so the proof is complete. +□ +9. Proof of the Main Theorems +In Section 9 we prove Theorems A and B, modulo the proof of Theorem 2.6 which is left +to Sections 10-12. +Recall that in Section 8 we fixed the objects K, f, D, ε, P as in Theorem B (see Notation +8.1), and we defined a quasiregular approximation gn to f in Definition 8.2. We also showed +in Section 8 that gn in fact extends to a quasiregular function gn : �C → �C. Now we apply +the MRMT below in Definition 9.1 to obtain the rational maps rn : �C → �C which we will +prove satisfy the conclusions of Theorems A and B for large n. +Definition 9.1. The mapping gn induces a Beltrami coefficient µn := (gn)z/(gn)z, which, by +way of the MRMT, defines a quasiconformal mapping φn : �C → �C such that rn := gn ◦ φ−1 +n +is holomorphic. We normalize φn so that φn(∞) = ∞ and φn(z) = z + O(1/|z|) as z → ∞. +We now begin deducing that for large n, the maps rn satisfy the various conclusions in +Theorems A and B. +Proposition 9.2. The function rn of Definition 9.1 is rational, and r−1 +n (∞) = φn(P ′). In +particular, if K is full and P = {∞}, then rn is a polynomial. +Proof. The function rn is holomorphic on �C and takes values in �C: the only such functions are +rational. Note that g−1 +n (∞)∩U = ∅ since gn is bounded on U. Thus, by Proposition 7.4 and +(8.4), we have that g−1 +n (∞) = P ′. Since rn := gn ◦ φ−1 +n , we conclude that r−1 +n (∞) = φn(P ′). + +28 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +The last statement of the proposition follows since we normalized φn(∞) = ∞, and the only +rational functions with a unique pole at ∞ are polynomials. +□ +Proposition 9.3. For all R < ∞, the mapping φn satisfies: +(9.1) +||φn(z) − z||R·D +n→∞ +−−−→ 0. +Proof. Since gn is C-vertex supported by Proposition 8.3, we conclude from Proposition 3.7 +that +(9.2) +Area(supp(µn)) +n→∞ +−−−→ 0. +The relation (9.1) now follows from (9.2) since ||µn||L∞ ≤ K for all n by Proposition 8.3. +□ +Theorem 9.4. For all sufficiently large n, the mapping rn satisfies CP(rn) ⊂ D. +Proof. By Proposition 3.7, we have +max +� +diam(e) : e is an edge of +ℓ� +i=1 +∂Ω(i) +� +n→∞ +−−−→ 0. +Thus, since D is a domain containing ∪ℓ +i=1∂Ω(i), we have by (7.2) that BP(gn) \ U ⊂ D for +large n. Since U ⊂ D we conclude that BP(gn) ⊂ D for large n. The result now follows from +Proposition 9.3 since φ(BP(gn)) = CP(rn). +□ +Theorem 9.5. For all sufficiently large n, we have +||rn − f||K < 3ε. +Proof. First we note that since f is uniformly continuous on K′, there exists δ > 0 so that +if z, w ∈ K′ and |z − w| < δ, then |f(z) − f(w)| < ε. By Proposition 9.3, we can conclude +that +(9.3) +||φn(z) − z||K < min(δ, dist(K, ∂K′)) +for all sufficiently large n. +Let z ∈ K, w := φ−1 +n (z) and suppose j is such that z ∈ Dj. Then +|rn(z) − gn(z)| = |gn(w) − gn(z)| ≤ |gn(w) − f(w)| + |f(w) − f(z)| + |f(z) − gn(z)|. +(9.4) +Let +C := sup +z∈D +1≤i≤k +|Ψ′ +i(z)|. +Then +||gn − f||K′ ≤ C · ||Ψ−1 +j +◦ gn − Ψ−1 +j +◦ f||K′, +and we deduce by (2.15) that: +||gn − f||K′ ≤ C · ||Bn − Ψ−1 +j +◦ f||K′. + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +29 +Applying Theorem 2.10 (see also Remark 2.9), we conclude that +(9.5) +||gn − f||K′ +n→∞ +−−−→ 0. +Next, we deduce from (9.3), (9.4) and (9.5) that +|rn(z) − gn(z)| < 2ε +for sufficiently large n. It follows that for sufficiently large n: +||rn − f||K ≤ ||rn − gn||K + ||gn − f||K < 3ε. +□ +Theorem 9.6. For all sufficiently large n we have CV(f|K) ⊂ CV(rn). +Proof. Let z ∈ CP(f|K), and let i be so that z ∈ Di. Then Ψ−1 +i ◦f(z) ∈ CV(Bn) by Theorem +2.10. Thus f(z) ∈ CV(Ψi ◦ Bn). Thus, by the Definition 2.12 of gn, we have for large n that +f(z) ∈ BV(gn) = CV(rn). +□ +Theorem 9.7. For all sufficiently large n, we have +CV(rn) ⊂ fill{z : d(z, f(K)) < ε}. +Proof. Since +CV(rn) = BV(gn), +it suffices to show that for every z ∈ BP(gn) and sufficiently large n, we have +(9.6) +gn(z) ∈ fill{z : d(z, f(K)) < ε}. +For z ∈ BP(gn) \ U, (9.6) follows from (7.3). +For z ∈ BP(gn) ∩ Di, (9.6) follows from +Definition 2.12 of gn|Di. +□ +Proof of Theorem B: In the special case that ||f||K < 1, we have already proven that the +mappings rn satisfy the conclusions of Theorem B for all sufficiently large n. Indeed, Theorem +9.5 says that ||rn − f||K < ε, conclusion (2) in Theorem B is Theorem 9.4, and conclusion +(3) is Theorems 9.6, 9.7. Conclusion (1) follows from Propositions 9.2, 9.3. The general case +follows by applying the above special case to an appropriately rescaled f. +□ +Proof of Theorem A: When K is full, we may take P = {∞} and apply Theorem B, in which +case Proposition 9.2 guarantees that the maps rn are polynomials. +□ +Theorem 9.8. (Mergelyan+) Let K ⊂ C be full, suppose f ∈ C(K) is holomorphic in +int(K), and let D be a domain containing K. For every ε > 0, there exists a polynomial p +so that ||p − f||K < ε and: +(1) CP(p) ⊂ D, +(2) CV(p) ⊂ fill{z : d(z, f(K)) ≤ ε}. + +30 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Proof : By the usual version of Mergelyan’s Theorem, there exists a polynomial q so that +||q − f||K < ε/2. Apply Theorem A to K, D, q, ε/2 to obtain an approximant of q which +we denote by p. The polynomial p satisfies the conclusions of Theorem 9.8. +□ +Corollary 9.9. (Weierstrass+) Suppose that I ⊂ R is a closed interval, f : I → R is +continuous, and U, V ⊂ C are planar domains containing I, f(I), respectively. Then, for +every ε > 0, there exists a polynomial p with real coefficients so that ∥f − p∥I ≤ ε, and +(1) CP(p) ⊂ U, +(2) CV(p) ⊂ V . +Proof. Let I = [a, b], and f, U, V as in the statement of the corollary. By Theorem 9.8, +there exists a complex polynomial q so that ||q − f||[a,b] < ε/2. The real polynomial +Q(z) := q(z) + q(z) +2 +satisfies Q(x) = Re(q(x)) for x ∈ R and hence ||Q − f||[a,b] < ε/2. We will use the symbol +⋐ to mean compactly contained. Let V1 ⋐ V be a sufficiently small, R-symmetric domain +containing f(I) so that there is a component of Q−1(V1) (which we denote by U1) satisfying +U1 ⋐ U. Let U2 be a R-symmetric, analytic domain satisfying I ⋐ U1 ⋐ U2 ⋐ U. Recall +Notation 8.1 and consider: +(1) the compact set U1, +(2) the analytic function Q, +(3) the analytic domain U2 containing U1, +(4) min{ε/2, dist(∂V1, ∂V )}, +(5) P = {∞}. +Applying Definition 8.2 to (1)-(5) yields quasiregular mappings gn with R-symmetric Bel- +trami coefficient, so that +(9.7) +pn := gn ◦ φ−1 +n +is a real polynomial approximant of Q for large n satisfying: +(1) ||pn − f||[a,b] < ε, +(2) CP(pn) ⊂ U2, +(3) CV(pn) ⊂ V . +Thus pn satisfies the conclusion of Corollary 9.9 for large n. +□ +Remark 9.10. If we make further assumptions on the compact set K, the conclusion +CV(r) ⊂ fill{z : d(z, f(K)) < ε} +of Theorems A and B can be improved to +(9.8) +CV(r) ⊂ fill(f(K)), + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +31 +which is equivalent to +CV(r) ⊂ f(K) +if f(K) is full. Indeed, if for instance the interiors of K, f(K) are analytic domains and +f : int(K) → int(f(K)) is proper, then a similar strategy as in the proofs of Theorems A +and B but replacing Ψ in (2.15) with a conformal map D �→ int(K) can be used to prove +(9.8). +Recall the notation Ω(i) from Definition 8.2, and let τi : Ω(i) → D∗ be the conformal +mapping satisfying τ −1 +i +(∞) = P ′ ∩ Ω(i) as in Notation 3.5. The following fact justifies part +of our description in the introduction of the behavior of the rational approximants off K. +Proposition 9.11. Let 1 < r < R < ∞. Then, for all sufficiently large n, we have +(9.9) +rn ◦ φn(z) = τi(z)m and +(9.10) +|rn(z)| > R +for all z ∈ τ −1 +i +({z : |z| > r}). +Proof. Fix R < ∞ and r > 1. From (7.5) and the functional equation (8.4) defining gn in +Ω(i), it follows that: +(9.11) +gn(z) = τi(z)m for all z ∈ τ −1 +i +({z : |z| > (r + 1)/2}) +for all large n. Since +(9.12) +rn ◦ φn = gn, +The relation (9.9) follows. Moreover, we have by Proposition 9.3 that: +(9.13) +φn ◦ τ −1 +i +({z : |z| > r}) ⊂ τ −1 +i +({z : |z| > (r + 1)/2}) +for all sufficiently large n. +Since ((r + 1)/2)m > R for large n, the relation (9.10) also +follows. +□ +10. Some Estimates Involving Harmonic Measure and Green’s Functions +In Sections 10-12 we turn to the proof of Theorem 2.6. In fact, we will prove a slightly +stronger result (Theorem 12.2 in Section 12) from which Theorem 2.6 follows. We begin by +recalling a few standard facts, and sketch the proofs for the convenience of the reader. Let +D(z, r) denote the disk of radius r centered at z. Lemma 10.1 is illustrated in Figure 14. +Lemma 10.1. If K ⊂ D is continuum connecting 0 to T, then ω(z, K, D \ K) ≥ c > 0 for +all |z| < 1/4 and some c > 0 independent of K. + +32 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Figure 14. Illustrated is the compact set K of Lemma 10.1. +Proof. Exercise III.10 of [GM08] says that if E is a continuum connecting {|z| = 1 +2} to T +in D, then ω(0, E, D \ E) ≥ c = +2 +π tan−1(1/ +√ +8). +Apply the exercise and the maximum +principle to the disk D = D(z, 1 +2) to deduce the lemma. [GM08] gives a simple direct proof +of the exercise, but it also follows from Beurling’s projection theorem, e.g., Theorem II.9.2 +of [GM08]. +□ +Lemma 10.2. Suppose u, v are harmonic functions on D such that supD |u|, supD |v| ≤ M +and that |u − v| < ε on some continuum K connecting 0 to T. Then |u − v| ≤ εcM 1−c on +D(0, 1/4), where c > 0 is the constant from Lemma 10.1. +Proof. Consider the subharmonic function log |u−v|. It is less than log ε on K and less than +log M on ∂D. Thus for |z| ≤ 1/4, Lemma 10.1 implies +log |u(z) − v(z)| +≤ +ω(z, K, D \ K) log ε + ω(z, T, D \ K) log M +≤ +c log ε + (1 − c) log M, +so |u(z) − v(z)| ≤ εcM 1−c, as desired. +□ +Lemma 10.3. Suppose Ω is a planar domain, K ⊂ Ω is compact and connected, and 0 < +ε, M < ∞. Then there is a δ > 0 so that if h = u + i�u is holomorphic on Ω, �u vanishes at +some point of K, supΩ |h| ≤ M and supK |u| < δ, then supK |�u| < ε. +Proof. If K is single point, this is trivial since �u = 0 there by assumption, so assume K is +non-trivial. Choose η > 0 so that η < dist(K, ∂Ω) and η < diameter(K). Then for any +radius η disk D centered at a point of K, u is less than δ on a continuum connecting the +center of D to its boundary. This implies |u| ≤ δcM 1−c (for c as in Lemma 10.1) on a +η/4-neighborhood of K. Thus |∇u| = O(δcM 1−c/η) on the (η/8)-neighborhood U of K (this +uses the Cauchy estimate for |∇u|, e.g., Theorem 2.4 of [ABR01]). Since |∇�u| = |∇u| and + +Iwl=1 +K +.0 +Z +wl=1/4A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +33 +Figure 15. Illustrated in gray is the set Uδ of Lemma 10.4. +U is path connected, this implies �u is within ε of zero on K if δ is small enough (depending +on ε, η, M and the diameter of U in the path metric). +□ +We will use this later in the situation that if u and v are harmonic functions on Ω that +are close enough on K ⊂ Ω, then f = exp(u + i�u) and g = exp(v + i�v) are holomorphic +functions on Ω that are close on K (if �v is chosen to agree with �u at some point of K). +Next, we recall the well known boundary Harnack inequality (e.g., see Theorem 7.18 of +[Mar19] or Exercise I.6 of [GM08]). +Lemma 10.4. Suppose u and v are positive harmonic functions on D which extend contin- +uously to the boundary T, and suppose furthermore that u and v are both equal to zero on an +arc I ⊂ T. For δ > 0 let Uδ = {z ∈ D : dist(z, T \ I) > δ} (see Figure 15). Then for z ∈ Uδ, +δ2 +4 · u(0) +v(0) ≤ u(z) +v(z) ≤ 4 +δ2 · u(0) +v(0). +Note that, under these conditions, u and v have well defined inward normal derivatives on +I. By letting z → T radially, the inequalities above imply that ∂u +∂n and ∂v +∂n are comparable +(with the same constants as above) at any point of Uδ ∩ T. +Corollary 10.5. Suppose u is a positive harmonic function on D which extends continuously +to the boundary T and that equals zero on an arc I ⊂ T. Suppose a, b ∈ I both have distance +> δ from T \ I. Then +δ2 +4 · ∂u +∂n(a) ≤ ∂u +∂n(b) ≤ 4 +δ2 · ∂u +∂n(a). +Proof. We simply compare u to a rotation of itself. Let v(z) = u( b +az). Then v and u both +vanish on J = I ∩ a +b · I, and a ∈ J is distance > δ from either endpoint of J. Hence the +normal derivatives of u and v at a are comparable by the boundary Harnack principle, and +hence so are the normal derivatives of u at a and b. +□ + +Us +134 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Suppose Ω is an analytic domain (see Definition 2.5). For w ∈ Ω, let G(z, w) be the +Green’s function on Ω with pole at w, i.e., G is harmonic on Ω \ {w}, vanishes identically +on ∂Ω and G(z, w) + log |z − w| is bounded in a neighborhood of w. Our assumptions on Ω +imply that Ω is regular for the Dirichlet problem, and hence that the Green’s function exists +and is unique for any w ∈ Ω (e.g., see Sections II.1 and II.2 of [GM08]). +Lemma 10.6. Suppose Ω is an analytic domain. If r > 0 is small enough (depending only +on Ω), x ∈ ∂Ω, and y ∈ Ω \ D(x, r), then the normal derivative of the Green’s function with +pole at y has comparable size at all points of ∂Ω ∩ D(x, r/2) with a constant independent of +y and Ω. +Proof. Choose r small enough that W = D(x, r) ∩ Ω is a Jordan domain whose boundary +consists of a sub-arc γ of ∂Ω and an arc of the circle ∂D(x, r). Let �γ = ∂Ω ∩ D(x, r/2). +Choose a point z ∈ W that is about distance r from ∂W and choose a conformal map +ϕ : W → D taking z to 0. If r is small enough, ϕ extends analytically across γ to all of +D(x, r) and by the Koebe distortion theorem it has comparable derivative at all points of +�γ. Also, since γ and each component of γ \ �γ has harmonic measure with respect to z that +is bounded away from zero, the image arcs on T all have lengths bounded uniformly from +below. Thus u(z) = G(ϕ−1(z), y) is a positive harmonic function on D vanishing on ϕ(γ). +By Corollary 10.5 the normal derivatives of u are comparable at all points of ϕ(�γ). Since +the values of |ϕ′| are comparable at all points of �γ, we can deduce the lemma from the chain +rule. +□ +Corollary 10.7. Suppose Ω is an analytic domain. If r > 0 is small enough (depending only +on Ω), x ∈ ∂Ω, and y ∈ Ω, then ∂G(x, y)/∂n is comparable at all points of γ = ∂Ω ∩ D(x, r) +with a constant depending only on an upper bound for +M = max +� +1, +ℓ(γ) +dist(y, ∂Ω) +� +, +where ℓ(γ) denotes the length of γ). Moreover, ∂G(x, y)/∂n = O(1/r) on γ. +Proof. We can cover γ by at most O(M) disks Dj = D(xj, rj) whose doubles are all disjoint +from y. +Lemma 10.6 implies the normal derivatives are comparable with some uniform +constant C on each corresponding arc, so they are comparable with constant CO(M) on γ. +(In fact, if r is so small that ∂Ω looks “straight” on scale r, then only O(log M) disks are +needed, since they become geometrically larger as we move away from y.) +The final claim follows because the integral of the normal derivative over the whole bound- +ary is 2π, and hence the integral over γ is ≤ 2π. Since the size of the normal derivative is +comparable at all points of γ, this implies it is bounded above by O(1/ℓ(γ)) = O(1/r) (recall +ℓ denotes length). +□ +The following two lemmas relate harmonic measure to Green’s function: we refer to Figure +16 for an illustration. + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +35 +(a) +(b) +Figure 16. In (A) the setup for Lemma 10.8 is shown, and in (B) the setup +for Lemma 10.9. +Lemma 10.8. There is a constant C1 < ∞ so the following holds. Suppose Ω is an analytic +domain, that Γ is a connected component of ∂Ω and that w ∈ Ω satisfies +dist(w, Γ) = dist(w, ∂Ω) ≤ diameter(Γ). +Then G(z, w) ≤ C1 on σ = {z : |z − w| = 1 +2 dist(w, Γ)}. +Proof. Let Ω′ be the component of �C \ Γ containing Ω. Then Ω ⊂ Ω′, so by the maximum +principle the Green’s function for Ω is less than the Green’s function for Ω′. Thus it suffices +to prove the lemma for the simply connected domain Ω′. But by Koebe’s distortion theorem, +σ contains a ball of fixed hyperbolic radius around w and hence its image contains a ball of +fixed radius if we conformally map Ω′ to the disk with w going to zero. On the disk, the +Green’s function is log 1 +|z| which is clearly bounded by some C outside a fixed ball around +the origin. +□ +Lemma 10.9. There is a constant C2 < ∞ so that the following holds. Suppose that Ω is +an analytic domain, z ∈ Ω, that γ ⊂ ∂Ω is a subarc, and that w ∈ Ω satisfies +(1) dist(w, Γ) = dist(w, ∂Ω) ≤ diameter(Γ), where Γ is the component of ∂Ω containing +γ, +(2) |z − w| ≥ dist(w, ∂Ω). +Then G(z, w) ≤ C2ω(z, γ, Ω)/ω(w, γ, Ω). +Proof. Let C1 and σ = {z : |z − w| = 1 +2 dist(w, Γ)} be as in Lemma 10.8. By assumption z +is in the component Ω′ of Ω \ σ not containing w and by the maximum principle applied to +Ω′, ω(z, σ, Ω′) ≥ G(z, w)/C1. By the maximum principle (again applied to Ω′), +ω(z, γ, Ω) +≥ +ω(z, σ, Ω′) · min +x∈σ ω(x, γ, Ω). + +r +5 +Wr +Z +W +5 +Y36 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +By Harnack’s inequality (see Theorem 7.17 of [Mar19]), all the values of ω(x, γ, Ω) are +comparable on σ and hence there is an ε > 0 so that +ω(z, γ, Ω) +≥ +ω(z, σ, Ω′) · ε · ω(w, γ, Ω). +Finally, by Lemma 10.8 we have +ω(z, γ, Ω) +≥ +(ε/C1) · G(z, w) · ω(w, γ, Ω), +which is the desired inequality with C2 = C1/ε. +□ +Corollary 10.10. Suppose Ω is an analytic domain, z ∈ Ω and ε > 0. Suppose also that +{γk}n +1 ⊂ ∂Ω is a collection of disjoint arcs and {wk}n +1 ⊂ Ω is a collection of points, so that +for all k we have: +(1) dist(wk, Γk) = dist(wk, ∂Ω) ≤ diameter(Γk), where Γk is the component of ∂Ω con- +taining γk, +(2) |z − wk| ≥ dist(wk, ∂Ω), +(3) ω(wk, γk, Ω) ≥ ε > 0. +Then �n +k=1 G(z, wk) ≤ C2/ε where C2 is the constant from Lemma 10.9. +Proof. By Lemma 10.9, � +k G(z, wk) ≤ (C2/ε) � +k ω(z, γk, Ω) and since the arcs {γk} are +disjoint, we have � +k ω(z, γk, Ω) ≤ 1. +□ +11. Periods of Harmonic Functions +Suppose Ω is an analytic domain with N +1 boundary components Γ0, Γ1, . . . , ΓN, so that +Ω is regular for the Dirichlet problem. Suppose h is harmonic in Ω. In any sub-disk D ⊂ Ω, +h has a harmonic conjugate �h that is well defined up to an additive constant. If γ is a closed +curve in Ω, then we can analytically continue �h along γ until we return to the starting point. +The period of h along γ is the difference between the starting and ending values of �h. If γ is +homologous to a point, the period is zero, but if γ is homologous to a boundary component +of Ω, then the period may be non-zero. The sum of the periods corresponding to all N + 1 +boundary components is always zero (the union of all boundary curves is homologous to zero +in Ω). +Next we consider the periods of certain special functions. For j = 0, . . . , N let ωj be the +harmonic function on Ω that has boundary value 1 on Γj and is 0 on the other boundary +components. Since the boundary components are analytic, each ωj extends to be analytic +across ∂Ω, so the normal and tangential derivatives are well defined, and themselves analytic, +at every boundary point. The period of ωj along Γk is the integral of the tangential derivative +of �ωj around Γk, and this equals the integral of the normal derivative of ωj, i.e., the period +is +λjk = +� +Γk +∂ωj +∂n ds. + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +37 +Note that since 0 < ωj < 1 in Ω and ωj = 1 on Γj, the inward normal derivative of ωj on Γj +is non-positive (and strictly negative by analyticity: in that case, G can’t have critical point +on the boundary). Similarly, the inward normal derivative of ωj is strictly positive on Γk for +k ̸= j. Thus λjj < 0 and λjk > 0 for k ̸= j. Since the periods of ωj sum to zero we have +N +� +k=0 +λjk = 0 for every j, +and since � +j ωj is the constant function 1 on Ω, we have +N +� +j=0 +λjk = 0 for every k. +Thus for each j we have: +λjj = − +� +k≥0:k̸=j +λjk. +In other words, the row and column sums are all zero for the (N +1)×(N +1) matrix (λjk), +0 ≤ j, k ≤ N. +If we drop the first row and column of this matrix, we are removing all positive terms +(except for λ00), so we have j = 1, . . . , N, +λjj < − +� +k≥1;k̸=j +λjk. +Thus the N × N matrix Λ = (λjk), 1 ≤ j, k ≤ N is diagonally dominant, and this implies it +is invertible by the Levy-Desplanques theorem: if the kernel of Λ contains a non-zero vector +v = (v1, . . . , vN), then � +k≥1 vkλjk = 0, and if |vk| is the largest component of v, then +|vkλkk| = | +� +j≥1:j̸=k +λjkvj| ≤ |vk| · +� +j≥1:j̸=k +|λjk| < |vk| · |λkk|, +which is a contradiction. See Olga Taussky-Todd’s paper [Tau49] for some history of this +oft-rediscovered fact. Also note that ∥Λ∥ and ∥Λ−1∥ only depend on Ω. +We have now proven the following result (due in a slightly different form to Heins [Hei50] +and in greater generality to Khavinson [Kha84]). +Lemma 11.1. Suppose Ω, Λ and {ωj}N +1 are as above and suppose we assign real numbers v = +(v1, . . . , vN) to the N boundary components Γ1, . . . , ΓN. Then there is a linear combination +h = �n +j=1 ajωj so that the period of h around Γj is exactly vj for j = 1, . . . , N. +The +coefficients a = (a1, . . . , aN) are solutions of the linear equation Λa = v and hence ∥a∥ ≤ +∥v∥ · ∥Λ−1∥. Thus if ∥v∥ ≤ ε then ∥a∥ = O(ε) with a constant that depends only on Ω. +We say that the periods of a harmonic function h on Ω are well defined modulo 2π if +every period is some integer multiple of 2π. In this case, f = exp(h + i�h) is a well defined + +38 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +analytic function on Ω. For example, if Ω is the complement of a finite set of points {zk}n +1, +then �n +k=1 log |z − zn| has periods that are well defined modulo 2π. +The corresponding +holomorphic function is a polynomial with zeros at {zk} (and hence extends holomorphically +from Ω to the whole plane). Note that ∇�u is always well defined even if �u is not, since any +two different branches of �u differ by a constant. +Corollary 11.2. Suppose Ω is an analytic domain and K ⊂ Ω is a compact set that contains +curves {σj}N +0 homologous to each of the boundary components {Γj}N +0 . Suppose K ⊂ W ⊂ Ω +is open, let η = dist(K, ∂W) and set U = {z ∈ W : dist(z, ∂W) > η/2}. Suppose P ⊂ Ω is a +finite set and suppose u and H are harmonic functions on Ω \ P, and each is either bounded +or has a logarithmic pole at each point of P. Suppose |u − H| is bounded by M on U, and +that |u − H| < ε on K. If u has a well defined harmonic conjugate modulo 2π on Ω, then +there is an harmonic h on Ω so that +(1) h + H also has a well defined harmonic conjugate modulo 2π on Ω \ Z, +(2) |h| ≤ Cεc on all of Ω (not just K), +(3) h is constant on each component of ∂Ω. +The constant c is the same as in Lemma 10.1 and C depends only on Ω and K. +Proof. Since v = u − H is bounded on U, it extends to be harmonic at each point of P ∩ U. +Since |v| < ε on K, it is bounded by O(εcM 1−c) on U and hence the gradient of v is bounded +by O(εcM 1−c/η) on U. Thus the gradient of the (possibly multi-valued) harmonic conjugate +of v is bounded by the same quantity. We deduce that the harmonic conjugates of H and u +differ by δ = O(LεcM 1−c/η) on U, where L is the diameter of U in the path metric. Thus +the periods of H on the {σj} differ from multiples of 2π by at most δ. Now apply Lemma +11.1 to define a harmonic function h on Ω that is bounded by O(δ) and has exactly the +periods of −v. +□ +12. The Generalized Caratheodory Theorem on Blaschke Approximation +If B is a finite Blaschke product (see Definition 2.2) on an analytic domain Ω, then it has +non-zero, continuous boundary values, and hence can have only finitely many zeros inside +Ω. Moreover, the Schwarz reflection principle implies B extends holomorphically across ∂Ω. +The following result extends Carath´eodory’s Theorem (Theorem 2.1), and its statement uses +the notation IB (see Notation 2.4), and the following notation: +Notation 12.1. If Ω ⊂ C is an analytic domain and Γ := ∂Ω, we denote Γδ := {z ∈ Ω : +dist(z, Γ) = δ}. +Theorem 12.2. Suppose that Ω ⊂ C is an analytic domain, K ⊂ Ω is compact, and f is +holomorphic on a neighborhood of Ω with supΩ |f| ≤ 1. Let Γ := ∂Ω. Then for any ε > 0 +and sufficiently small δ > 0, there is a finite Blaschke product B on Ω so that +(1) supK |f − B| ≤ ε, + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +39 +(2) 1 − ε < |B| ≤ 1 on ∂Ω, +(3) Every component of Γδ \ {B = 0} has length comparable to δ and adjacent connected +components have length comparable to within a factor of 1 + ε. +(4) All components of IB have length which is comparable to δ with constants that depend +only on ε and Ω. +(5) on each component γ of IB, the ratio maxγ |B′|/ minγ |B′| is bounded depending only +on ε and Ω. +Proof. Without loss of generality, we may assume K is connected. Otherwise, replace K by +a compact, connected superset, for instance, its closed convex hull in the hyperbolic metric. +As before, let G(z, w) denote the Green’s function on Ω with pole at w. +Since f is holomorphic on a neighborhood of Ω, it only has finitely many zeros in Ω. We +consider g := (1 − a) · f + b for some constants 0 < a < ε and |b| < ε such that |g| < 1 − ε +on Ω and g has no zeros on ∂Ω. If we construct a finite Blaschke product B approximating +g to within ε on K, then it approximates f to within 3ε. Fix a finite number of smooth +curves {σk} that are homologous to each boundary curve of Ω. By enlarging K, if necessary, +we may assume it contains all these curves. +Let {zk}N +1 be the zeros of g, counted with +multiplicity. By enlarging K again, if necessary, we may assume all the zeros of g are in K. +Let W be an open domain with K ⊂ W ⊂ W ⊂ Ω. By compactness we have min∂W |g| ≥ η +for some η > 0. In what follows, δ > 0 will always be chosen so small that W is disjoint +from {z ∈ Ω : dist(z, ∂Ω) < 2δ}. In particular, g has no zeros in this neighborhood of the +boundary. +Let u(z) = − log |g(z)|. Then u ≥ − log(1 − ε) ≥ ε is positive and harmonic on Ω except +for finitely many logarithmic poles at the {zk}N +1 , the zeros of g listed with multiplicity. +Let p(z) = �N +k=1 G(z, zk) be the sum of the Green’s functions with these poles. +Then +v(z) = u(z)−p(z) is harmonic on Ω, and equals u on ∂Ω. Thus v is continuous and non-zero +on ∂Ω, and hence it is bounded and bounded away from zero there, say m ≤ v ≤ M on ∂Ω. +Hence m ≤ v ≤ M on all of Ω by the maximum principle. +By Theorem II.2.5 of [GM08] v is the Poisson integral of its boundary values, i.e., +v(z) = 1 +2π +� +∂Ω +v(w)∂G(w, z) +∂n +ds(w) +where +∂ +∂n is the inward normal and ds denotes length measure on the boundary. For w ∈ Γδ, +denote by w∗ ∈ ∂Ω the closest point to w on ∂Ω. For z ∈ K and w ∈ Γδ we have +G(w, z) = δ · ∂G(w∗, z) +∂n ++ O(δ2), +where the constant depends on z, but is uniformly bounded as long as z is in the compact +set K (the constant in the “big-Oh” depends on a bound for |∇2G| between Γδ and ∂Ω and +since G is harmonic and extends analytically across ∂Ω, this is bounded as long as the pole + +40 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +of Green’s function is not too close to ∂Ω). It follows that +v(z) = +1 +2πδ +� +Γδ +v(w)G(w, z)ds(w) + O(δ). +(12.1) +Next use the identity G(z, w) = G(w, z) (e.g., Theorem II.2.8 of [GM08]), to deduce +v(z) = +1 +2πδ +� +Γδ +v(w)G(z, w)ds(w) + O(δ). +We now discretize the integral by cutting Γδ into disjoint subarcs {γk} chosen so that +1 ≤ +� +γk +v(w) +2πδ ds(w) ≤ 1 + O(δ). +(12.2) +This is possible since the integral over a component Γk +δ of Γδ is at least A = m · ℓ(Γk +δ)/2πδ +and this tends to infinity as δ ↘ 0. We can therefore cut each boundary curve into sub-arcs +where the integral is between 1 and 1+O(1/A) = 1+O(δ), i.e., we can make the sub-integrals +all as close to 1 as we wish, by taking δ small enough. +The left side of Equation (12.2) implies that for each k, we have ℓ(γk)M/2πδ ≥ +� +γk v/2π ≥ +1 and hence ℓ(γk) ≥ 2πδ/M. Similarly, the other side implies ℓ(γk) ≤ (1 + O(δ))2πδ/m. +Thus every such arc has length comparable to δ. If δ is small enough, then the continuity +implies that on the union of two adjacent arcs with common endpoint x, v is close to v(x), +and hence by (12.2), the lengths of these intervals are both close to 2πδ/v(x), and hence are +close to each other. This fact, together with each γk have length comparable to δ, will imply +part (3) of the Theorem once we have defined the generalized Blaschke product B. +Adding and subtracting a term from (12.1), have +v(z) += +1 +2πδ +� +j +G(z, wj) +� +γj +v(w)ds(w) +(12.3) ++ 1 +2πδ +� +j +� +γj +v(w)[G(z, w) − G(z, wj)]ds(w) + O(δ). +The curve Γδ is parallel to the boundary, which is the level line G(z, w) = 0 of the Green’s +function with pole at w. Thus, the gradient of G along Γδ is nearly perpendicular to Γδ. +Denoting by wj the center of γj, we conclude: +|G(z, w) − G(z, wj)| = O(δ2). +(12.4) +Using (12.4) in the last term of Equation 12.3 gives +v(z) += +1 +2πδ +� +j +G(z, wj) +� +γj +v(w)ds(w) + +� +j +1 +2πδ +� +γj +v(w)O(δ2)ds(w) + O(δ). + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +41 +Simplifying, we get +v(z) += +1 +2πδ +� +j +G(z, wj) +� +γj +v(w)ds(w) + O(δ) +� +j +� +γj +v(w)ds(w) + O(δ) += +� +j +G(z, wj) +� +γj +v(w) +2πδ ds(w) + O(δ) += +� +j +G(z, wj)(1 + O(δ)) + O(δ), += +� +j +G(z, wj) + O(δ), +where in the last line we have used Corollary 10.10 to bound the sum of Green’s functions by +O(1). Therefore, v is approximated on K by a finite sum of Green’s functions on Ω (indeed, +we can even take the approximation to hold on the larger compact set W). +If Ω is simply connected, then we are essentially done. In this case, +H(z) += +� +k +G(z, zk) + +� +j +G(z, wj) = p(z) + +� +j +G(z, wj) +is harmonic except for a finite number of logarithmic poles at P = {∪kzk}∪{∪jwj}. Therefore +H has a harmonic conjugate �H that is well defined modulo 2π on Ω \ P. Then +B(z) = exp(−H − i �H) +is holomorphic on Ω \ P, but tends to zero at each point of P, so B is holomorphic on all of +Ω with zeros exactly at the points of P. Moreover, on ∂Ω we have |B| = exp(0) = 1, so B +is a finite Blaschke product on Ω. Moreover, +H(z) = p(z) + +� +j +G(z, wj) = u(z) − v(z) + +� +j +G(z, wj) ≈ u(z) = − log |g(z)|, +so log |B| = −H approximates log |g| on W as closely as wish. In particular, we may assume +|B| ≥ η/2 on ∂W. In this case, g/B is holomorphic on W (since every zero of B inside W is +also a zero of g of the same multiplicity), and so by the maximum principle |g/B| is bounded +on K by max∂K |g/B| ≤ maxK 2|g|/η. Now apply Lemma 10.3 to h = log g/B = u+ �u on W +to deduce that arg(B) approximates arg(g) on K, at least if we add an appropriate constant +to arg(B). Therefore B (times an appropriate unit scalar) uniformly approximates g on K. +This extends Carath´eodory’s theorem to simply connected domains. +However, if Ω is multiply connected, then H need not have a well defined harmonic con- +jugate modulo 2π on Ω \ P. Therefore B = exp(−H − i �H) need not be well defined: if +we analytically continue B along one of the closed loops σk, we return to the same absolute +value, but possibly a different value of the argument. The change in the argument is as small +as we wish, tending to zero as the difference between log |B| = H and log |g| tends to zero + +42 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +on K. This is because g has a well defined harmonic conjugate modulo 2π on Ω \ P, and +so the discussion following Lemma 10.3 applies. In order to get a well defined (generalized) +finite Blaschke product B on Ω, we apply Corollary 11.2 to u and H to construct h so that +(1) h is harmonic on all of Ω, and +(2) h + H has a well defined harmonic conjugate modulo 2π on Ω \ P. +(Note the H was constructed exactly so that the corollary can be applied: u−H is harmonic +except for poles on Γδ which are outside W, and we may make u − H is as close to zero on +K as we wish, while keeping it uniformly bounded on W.) +Now set F = −H − h. This function has all periods equal to zero modulo 2π, so B = +exp(F + i �F) is a well defined holomorphic function on Ω. Since |h| = O(δ), we can deduce +that B still approximates g on the compact set K. Moreover, since H = 0 on ∂Ω and h = aj +on Γj, we see that |B| = exp(aj) = 1 + O(δ) on Γj, so |B| is constant on each boundary +component. Dividing by the largest such value, we get another finite Blaschke product that +satisfies (2) and still approximates g to within O(δ) on K. +To prove (4), it suffices to show that the modulus of the tangential derivative of B along +∂Ω (which is equal to |B′| since B is holomorphic) is comparable to 1/δ everywhere: then +the preimage of a half-circle will have length comparable to δ. Note that on ∂Ω, |B′| is also +equal to the normal derivative of h + H. The function h is a linear combination of fixed +functions ωj that depend only on Ω, and although the coefficients of the combination may +depend on δ, they remain small as δ ↘ 0. Thus the normal derivative of h remains bounded +on ∂Ω as δ ↘ 0, and this is negligible compared to 1/δ. +The function H is a sum of two sets of Green’s functions, one with poles {zj} corresponding +to the zeros of g and the other with poles {wk} along the curve Γδ. The first set of poles is +fixed independent of δ and their contribution to the normal derivative of H is also bounded +independent of δ. Again, these terms are negligible. +The main contribution to the normal derivative of H comes from the poles {wk} lying on +Γδ. For each such point wk consider the arc σk = D(wk, 2δ) ∩ ∂Ω; the harmonic measure of +this arc with respect to wk is bounded uniformly away from zero, i.e., ω(wk, σk, Ω) > c > 0. +This harmonic measure is the integral over σk of the normal derivative of the Green’s function +with pole at wk, and thus it is less than the integral of the normal derivative of H, since this +Green’s function is one term of the sum defining H. Thus the integral of |B′| over σk is > c +and so any single arc lj in IB can contain at most a bounded number of arcs of the form σk. +By Part (3) of the theorem, adjacent points wk are at most distance O(δ) apart and hence +lj can have length at most O(δ) (otherwise it would cover too many of the arcs σk). +Next, we need to prove ℓ(lj) is bounded below by a multiple of δ, and this is equivalent to +proving an upper bound |B′(x)| = O(1/δ) for any x ∈ ∂Ω. As noted above, the contributions +to |B′| from h and from the poles of H coming from the zeros of g are both bounded +independent of δ. To deal with the poles {wk} of H on Γk, we choose a point z ∈ Γδ that is + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +43 +close to x ∈ ∂Ω, say |x − z| ≤ 2δ and note that +∂ +∂nG(x, wk) ≃ 1 +δG(z, wk) +Thus by Corollary 10.10, the total contribution of all the poles {wk} to |B′| is at most +� +k +1 +δG(wk, z) = O(1 +δ), +and hence (4) is proven. +Let γ denote a component of IB. By Corollary 10.7 and conclusion (4) just proven, the +normal derivative of Green’s function with a pole at least distance δ from ∂Ω has comparable +values at all points of γ, and so the same holds for any finite sum of such functions (with +the same constant). Since on ∂Ω we have |B′| = | � ∂G +∂n |, we deduce (5). +□ +Remark 12.3. If Ω = D, then it suffices to assume f is holomorphic on Ω instead of a +neighborhood of Ω. In that case if r < 1 then g(z) = f(rz) is holomorphic on a neighborhood +of D and approximates f on a compact set K ⊂ D if r is close enough to 1. So if B is a finite +Blaschke product on Ω approximating g to within ε/2, and r is chosen so that g approximates +f to within ε/2, then B approximates f to within ε, as desired. +Remark 12.4. The analyticity of f on a neighborhood of Ω is only used to deduce that f +has a finite number of zeros inside Ω. The same proof would work if we assumed that f is +holomorphic on Ω, extends continuously to ∂Ω, and is non-zero on ∂Ω. +Remark 12.5. If we make the previous assumption on f, then it suffices to assume Ω is +bounded and finitely connected. If so, and any component of ∂Ω is a single point p, then +by the Riemann removable singularity theorem, f extends to be holomorphic at p, and +it suffices the prove the theorem for the extended function on the domain Ω′ = Ω ∪ {p}. +By removing all the point components of ∂Ω, we may assume every component of Ω is +non-trivial. By repeated applications of the Riemann mapping theorem to the complement +of each complementary component of Ω, the domain Ω can be mapped to a domain Ω′ +bounded by a finite number of analytic curves indeed, with more work, the Koebe circle +domain theorem says it can be mapped to a domain bounded by circles. Transferring f and +K to Ω we can use Theorem 12.2 to construct an approximating finite Blaschke product on +Ω′, and then transfer this to a finite Blaschke on Ω that approximates f on K. +Remark 12.6. We placed the poles of our Green’s function all at the same distance δ from +∂Ω, but this was not necessary. If we fix z0 ∈ Ω and let w ∈ Ω approach x ∈ ∂Ω, then +G(z, w)/G(z0) approaches a positive harmonic function on Ω with zero boundary values on +∂Ω, except at x, where it blows up. Thus the limiting function must be a multiple of the +Poisson kernel on Ω with respect to x ∈ ∂Ω. Thus it is easy to find finite weighted sums +of Green’s functions (with positive real weights) that approximate the Poisson integral of v. + +44 +CHRISTOPHER J. BISHOP AND KIRILL LAZEBNIK +Then one must cluster the poles to approximate this sum by a sum of Green’s functions with +integral weights; exponentiating such a sum gives a finite Blaschke product on Ω. Possibly +this extra flexibility would be useful in other problems, such trying to minimize the number +of poles needed. +Proof of Theorem 2.6: The hypotheses of Theorem 2.6 are stronger than those of Theorem +12.2: namely we assume in Theorem 2.6 that ||f||Ω < 1 (rather than ≤ 1) and that the +zeros of f are disjoint from ∂Ω. Under these additional assumptions, there is no need in the +second paragraph of the proof of Theorem 12.2 to replace f by g := (1 − a) · f + b since +we already have |f| < 1 − ε and f has no zeros on ∂Ω. Since the B produced in Theorem +12.2 approximates g to within O(δ) and we may take δ → 0, the conclusions of Theorem 2.6 +follow from the conclusions of Theorem 12.2. +□ +References +[ABR01] +Sheldon Axler, Paul Bourdon, and Wade Ramey. Harmonic function theory, volume 137 of Grad- +uate Texts in Mathematics. Springer-Verlag, New York, second edition, 2001. +[Ahl06] +Lars V. Ahlfors. Lectures on quasiconformal mappings, volume 38 of University Lecture Series. +American Mathematical Society, Providence, RI, second edition, 2006. With supplemental chap- +ters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. +[BEF+22] Anna Miriam Benini, Vasiliki Evdoridou, N´uria Fagella, Philip J. Rippon, and Gwyneth M. +Stallard. Classifying simply connected wandering domains. Math. Ann., 383(3-4):1127–1178, 2022. +[BF14] +Bodil Branner and N´uria Fagella. Quasiconformal surgery in holomorphic dynamics, volume 141 +of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2014. +With contributions by Xavier Buff, Shaun Bullett, Adam L. Epstein, Peter Ha¨ıssinsky, Christian +Henriksen, Carsten L. Petersen, Kevin M. Pilgrim, Tan Lei and Michael Yampolsky. +[Bis15] +Christopher J. Bishop. Constructing entire functions by quasiconformal folding. Acta Math., +214(1):1–60, 2015. +[BL19] +Christopher J. Bishop and Kirill Lazebnik. Prescribing the postsingular dynamics of meromorphic +functions. Math. Ann., 375(3-4):1761–1782, 2019. +[BLU] +Christopher J. Bishop, Kirill Lazebnik, and Mariusz Urba´nski. Equilateral Triangulations and +The Postcritical Dynamics of Meromorphic Functions. to appear in Math. Ann. +[BT21] +Luka Boc Thaler. On the geometry of simply connected wandering domains. Bull. Lond. Math. +Soc., 53(6):1663–1673, 2021. +[Car54] +C. Caratheodory. Theory of functions of a complex variable. Vol. 2. Chelsea Publishing Co., New +York, 1954. Translated by F. Steinhardt. +[DKM20] Laura G. DeMarco, Sarah C. Koch, and Curtis T. McMullen. On the postcritical set of a rational +map. Math. Ann., 377(1-2):1–18, 2020. +[EL87] +Alexandre `E. Er¨emenko and Misha Yu. Ljubich. Examples of entire functions with pathological +dynamics. J. London Math. Soc. (2), 36(3):458–468, 1987. +[ERS22] +Vasiliki Evdoridou, Philip J. Rippon, and Gwyneth M. Stallard. Oscillating simply connected +wandering domains. Ergodic Theory and Dynamical Systems, page 1–30, 2022. +[Gar81] +John B. Garnett. Bounded analytic functions, volume 96 of Pure and Applied Mathematics. Aca- +demic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. +[GM08] +John B. Garnett and Donald E. Marshall. Harmonic measure, volume 2 of New Mathematical +Monographs. Cambridge University Press, Cambridge, 2008. Reprint of the 2005 original. + +A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION +45 +[GMR17] Stephan Ramon Garcia, Javad Mashreghi, and William T. Ross. Finite Blaschke products: a +survey. In Harmonic analysis, function theory, operator theory, and their applications, volume 19 +of Theta Ser. Adv. Math., pages 133–158. Theta, Bucharest, 2017. +[Hei50] +Maurice Heins. A lemma on positive harmonic functions. Ann. of Math. (2), 52:568–573, 1950. +[Kha84] +D. Khavinson. On removal of periods of conjugate functions in multiply connected domains. +Michigan Math. J., 31(3):371–379, 1984. +[Mar19] +Donald E. Marshall. Complex analysis. Cambridge Mathematical Textbooks. Cambridge Univer- +sity Press, Cambridge, 2019. +[MPS20] +David Mart´ı-Pete and Mitsuhiro Shishikura. Wandering domains for entire functions of finite +order in the Eremenko-Lyubich class. Proc. Lond. Math. Soc. (3), 120(2):155–191, 2020. +[MRW21] David Mart´ı-Pete, Lasse Rempe, and James Waterman. Eremenko’s conjecture, wandering Lakes +of Wada, and maverick points. arXiv e-prints, page arXiv:2108.10256, August 2021. +[MRW22] David Mart´ı-Pete, Lasse Rempe, and James Waterman. Bounded Fatou and Julia components of +meromorphic functions. arXiv e-prints, page arXiv:2204.11781, April 2022. +[Run85] +C. Runge. Zur Theorie der Eindeutigen Analytischen Functionen. Acta Math., 6(1):229–244, 1885. +[Six18] +David J. Sixsmith. Dynamics in the Eremenko-Lyubich class. Conform. Geom. Dyn., 22:185–224, +2018. +[Tau49] +Olga Taussky. A recurring theorem on determinants. Amer. Math. Monthly, 56:672–676, 1949. +C.J. Bishop, Mathematics Department, Stony Brook University, Stony Brook, NY 11794- +3651 +Email address: bishop@math.stonybrook.edu +Kirill Lazebnik, Mathematics Department, University of North Texas, Denton, TX, 76205 +Email address: Kirill.Lazebnik@unt.edu + diff --git a/D9E1T4oBgHgl3EQfEQOb/content/tmp_files/load_file.txt b/D9E1T4oBgHgl3EQfEQOb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..defc98da8607b08c886ef00715dbb3f51ad7a0ea --- /dev/null +++ b/D9E1T4oBgHgl3EQfEQOb/content/tmp_files/load_file.txt @@ -0,0 +1,1600 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf,len=1599 +page_content='A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We strengthen the classical approximation theorems of Weierstrass, Runge and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In particular, when approximating a function f on a compact set K, all the critical points of our approximants lie close to K, and all the critical values lie close to f(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Our proofs rely on extensions of (1) the quasiconformal folding method of the first author, and (2) a theorem of Carath´eodory on approximation of bounded analytic functions by finite Blaschke products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Introduction The following is Runge’s classical theorem on polynomial approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' [Run85] Let f be a function analytic on a neighborhood of a compact set K ⊂ C, and suppose C \\ K is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For all ε > 0, there exists a polynomial p so that ||f − p||K := sup z∈K |f(z) − p(z)| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This famous result does not say much about what the polynomial approximant p looks like off the compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For various applications, it would be useful to understand the global behavior of p and, in particular, the location of the critical points and values of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' To this end, we state our first result (Theorem A below) after introducing the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Notation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For any compact set K ⊂ C we denote by fill(K) the union of K with all bounded components of C \\ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We say K is full if C \\ K is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We let CP(f) denote the set of critical points of an analytic function f, and let CV(f) := f(CP(f)) denote its critical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A domain in �C is an open, connected subset of �C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (Polynomial Runge+) Let K ⊂ C be compact and full, D a domain containing K, and suppose f is a function analytic in a neighborhood of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then for all ε > 0, there exists a polynomial p so that ||p − f||K < ε and: (1) CP(p) ⊂ D, (2) CV(f|K) ⊂ CV(p) ⊂ fill{z : d(z, f(K)) ≤ ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Primary: 30C10, 30C62, 30E10, Secondary: 41A20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' uniform approximation, polynomials, rational functions, Blaschke products, Runge’s Theorem, Weierstrass’s Theorem, Mergelyan’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The first author is partially supported by NSF Grant DMS 1906259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='02888v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='CV] 7 Jan 2023 2 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Analogous improvements of the polynomial approximation theorems of Mergelyan and Weierstrass will be stated and proved in Section 9 (see Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8 and Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' When K is not full, uniform approximation by polynomials is not always possible, and so we turn to rational approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We denote the Hausdorff distance between two sets X, Y , by dH(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (Rational Runge+) Let K ⊂ C be compact, D a domain containing K, f a function analytic in a neighborhood of K, and suppose P ⊂ �C \\ K contains exactly one point from each component of �C \\ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then there exists P ′ ⊂ P so that for all ε > 0, there is a rational function r so that ||r − f||K < ε and: (1) dH(r−1(∞), P ′) < ε and |r−1(∞)| = |P ′|, (2) CP(r) ⊂ D, (3) CV(f|K) ⊂ CV(r) ⊂ fill{z : d(z, f(K)) ≤ ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The behavior of p off K is of particular interest in applications, such as in complex dy- namics where approximation results have been used to prove the existence of various dy- namical behaviors for entire functions (see, for example, [EL87], [BT21], [MRW21], [ERS22], [MRW22], [BEF+22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' However, not understanding the critical points and values of p means it has not been known whether these behaviors can occur within restricted classes of en- tire functions, such as the well studied Speiser or Eremenko-Lyubich classes (see the survey [Six18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Our approach is based on two ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The first is to show that on a compact subset K ⊂ C of a finitely connected domain Ω ⊂ C, any bounded analytic function can be approximated uniformly by an analytic function B : Ω → C having the property that |B| is constant on each component of ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This extends a classical theorem of Carath´eodory [Car54] concerning finite Blaschke products on the unit disk to more general regions, and may be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The second idea is an extension of quasiconformal folding, a type of quasiconformal surgery introduced in [Bis15], to extend the (generalized) Blaschke product B from Ω to a quasireg- ular mapping g : �C → �C with specified poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The map g may be taken close to holomorphic in a suitable sense, and so the Measurable Riemann Mapping Theorem (MRMT for brevity) will imply there is a quasiconformal mapping φ so that g ◦ φ−1 is the desired polynomial or rational approximant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This approach yields not only information on the critical points and values of the approx- imants as in Theorems A and B, but more broadly a detailed description of the geometric structure of these approximants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We end the introduction by describing this geometric structure in a few cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' First we introduce some more notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Notation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let D ⊂ �C be a simply connected domain so that ∞ ̸∈ ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We let ψD : E → D denote a Riemann mapping, where E = D is D is bounded and E = D∗ := �C\\D if D is unbounded, in which case we specify ψD(∞) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 3 First consider the case when K is full and connected, and f is holomorphic in a neighbor- hood of K satisfying ||f||K < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let Ω, Ω′ be analytic Jordan domains containing K, f(K), respectively, so that f is holomorphic in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then the mapping F := ψ−1 Ω′ ◦ f ◦ ψΩ : D → D is holomorphic, and by Carath´eodory’s theorem for the disk (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1), there is a finite Blaschke product b : D → D that approximates F on the compact set ψ−1 Ω (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Therefore B := ψΩ′ ◦ b ◦ ψ−1 Ω : Ω → Ω′ is a holomorphic function that approximates f on K, and moreover B restricts to an analytic, finite-to-1 map of Γ := ∂Ω onto Γ′ := ∂Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In this paper, we will show that B can be approximated on Ω by a polynomial p so that p−1(Γ′) is an approximation of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' More precisely, p−1(Γ′) is connected, and consists of a finite union of Jordan curves {γj}n 0 bounding pairwise disjoint Jordan domains {Ωj}n 0 (see Figure 1): the {Ωj}n 0 are precisely the connected components of p−1(Ω′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' There is one “large” component Ω0 that approximates Ω in the Hausdorff metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The other components {Ωj}n 1 can be made as small as we wish and to lie in any given neighborhood of ∂K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, the collection {Ωj}n 0 forms a tree structure with any two boundaries ∂Ωj, ∂Ωk either disjoint or intersecting at a single point, and with Ω0 as the “root” of the tree as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let Ω∞ denote the unbounded component of C \\ p−1(Γ′), so that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) C \\ p−1(Γ′) = Ω0 ⊔ � ⊔n j=1Ωj � ⊔ Ω∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recalling Notation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3, the polynomial p has the following simple structure with respect to the domains in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (1) p(Ω0) = Ω′ and ψ−1 Ω′ ◦ p ◦ ψΩ0 is a finite Blaschke product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (2) p(Ωj) = Ω′ and p is conformal on Ωj for 1 ≤ j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (3) p(Ω∞) = C \\ Ω′ and p = ψC\\Ω′ ◦ (z �→ zm) ◦ ψ−1 Ω∞ on Ω∞ for m = deg(p|Ω0) + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In other words, up to conformal changes of coordinates, p is simply a Blaschke product in Ω0, a conformal map in each Ωj, 1 ≤ j ≤ n, and a power map z �→ zm in Ω∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The only finite critical points of p are either in Ω0, or at a point where two of the curves (γj)n j=1 intersect, in which case the corresponding critical value lies on ∂Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Next suppose K is connected, but C \\ K has more than one component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In this case, in order to prove Theorem B, we will need to let Ω be a multiply connected analytic domain containing K, and Ω′ an analytic Jordan domain containing f(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We cannot proceed as in the case C\\K is connected, however, without a multiply connected version of Carath´eodory’s Theorem for the disk (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The usual proof of Carath´eodory’s Theorem is based on power series, and it does not extend to the multiply connected setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' However, there is an alternate proof based on potential theory that does extend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We now briefly sketch the idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 4 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK K p Ω∞ f(K) Γ′ Ω0 {Ωj}n 1 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This figure illustrates the geometry of the approximant p in Theorem A when K is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The notation is explained in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Both domain and co- domain are colored so that regions with the same color correspond to one another under p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose f is holomorphic in a simply connected domain D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' To simplify matters, suppose f(D) is compactly contained in D \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then u := − log |f| is a positive harmonic function on D, and u is also bounded and bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus u is the Poisson integral of its boundary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The Poisson kernel Px(z) associated to any point x ∈ ∂D is a limit of normalized Green’s functions on D: Px(z) ≈ G(z, wn)/G(0, wn) with wn → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Approximating the Poisson integral by a Riemann sum gives an approximation of u on any compact set K ⊂ D by a sum of Green’s functions H(z) := � j G(z, wj) with poles distributed on {z : d(z, ∂D) = δ}, and with δ as small as we wish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Figure 2 shows the function u(x, y) = 1 2 + xy being approximated by a sum of 25 Green’s functions on D = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since D is simply connected, H has a well defined harmonic conjugate �H, and after adding a constant to �H if necessary, B := exp(−H − i �H) approximates f on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, B satisfies: (1) B is holomorphic on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (2) |B| extends continuously to a constant function on ∂D, with ||B||∂D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We call such a function B on a (not necessarily simply connected) domain D a (generalized) finite Blaschke product on D (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' When D = D this definition coincides with A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' On the left is the positive harmonic function u(x, y) = 1 2 + xy on the unit disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' On the right is a sum of 25 Green’s functions with poles on the circle of radius .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' On D(0, 1 2) the two functions agree to within .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' As expected, the poles are closer together where u is large and farther apart where u is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' the usual definition of Blaschke product, and the above argument yields Carath´eodory’s The- orem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In fact, the above argument yields several technical improvements of Carath´eodory’s Theorem (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) which we will need in order to prove Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This argument generalizes to finitely connected domains D, except that the sum of Green’s functions H may not have a well defined harmonic conjugate (even modulo 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will fix this by adding a small harmonic function h that is constant on each boundary component of D and whose periods match the periods of −H around each boundary component of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Exponentiating the sum of the modified function H + h with its harmonic conjugate (now well-defined modulo 2π) then gives a (generalized) finite Blaschke product B which approximates the given function f on the desired compact set K ⊂ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This gives a version of Carath´eodory’s theorem on finitely connected domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By choosing H correctly, we can take ∥h∥D as small as we wish, and hence |B| is constant on each connected component of ∂D, with values that can all be taken as close to 1 as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Now we return to the description of our rational approximant in the case that K is connected, but C\\K has more than one component, recalling that Ω is a multiply connected analytic domain containing K, and Ω′ is an analytic Jordan domain containing f(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By the multiply connected version of Caratheodory’s Theorem, there exists a (generalized) finite Blaschke product b approximating ψ−1 Ω′ ◦ f on K, so that B := ψΩ′ ◦ b is a holomorphic map approximating f on K, and B restricts to an analytic, finite-to-1 mapping of each component of Γ := ∂Ω onto ψΩ′(|z| = t) for t = 1 or t ≈ 1, where t may depend on the component of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 ~ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='56 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK We will show B can be approximated on Ω by a rational map r so that each component of ∂Ω can be approximated by a component of r−1 ◦ ψΩ′(|z| = t) for t as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' These components of r−1 ◦ ψΩ′(|z| = t) bound Jordan domains which form a decomposition of the plane as in the previously described polynomial setting, and in the interior of each such domain again r behaves either as a (generalized) finite Blaschke product, a conformal mapping, or a power mapping (up to conformal changes of coordinates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lastly, the case when K has more than one connected component is more intricate, and we will leave the precise description to later in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (Briefly, quasiconformal folding is applied not just along the boundary of a neighborhood of K, but also along specially chosen curves that connect different connected components of this neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=') We remark that while Theorem A strictly improves on Runge’s Theorem on polynomial approximation, the relationship between Theorem B and Runge’s Theorem on rational ap- proximation is more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Both show existence of rational approximants, and only The- orem B describes the critical point structure of the approximant, however the poles of the approximant in Theorem B are specified only up to a small perturbation, whereas in Runge’s Theorem they are specified exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We do not know whether it is necessary to consider per- turbations of P ′ in Theorem B, or if the improvement r−1(∞) = P ′ is possible (a related problem appears in [BL19], [DKM20], [BLU], where it is known no such improvement is possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The authors would like to thank Malik Younsi and Oleg Ivrii for useful discussions related to this manuscript, and Xavier Jarque for his comments on a preliminary version of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Interior Approximation The following classical result of Carath´eodory (referenced in the introduction) allows for approximation by Blaschke products in simply-connected domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' ([Car54]) Let f : D → C be holomorphic and suppose ||f||D ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then there exists a sequence of finite Blaschke products on D converging to f uniformly on compact subsets of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 is elementary and may be found, for example, in Theorem I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 of [Gar81] or Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 of the survey [GMR17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In order to prove Theorem A, we will need to prove a version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 in which the Blaschke products satisfy certain boundary regularity conditions, and in order to prove Theorem B, we will need to prove a multiply- connected version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' These improvements are stated below in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' First we need several definitions: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let D ⊂ C be a finitely connected domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We say a non-constant holo- morphic function B : D → C is a finite Blaschke product on D if |B| extends continuously to a non-zero, constant function on each component of ∂D and ||B||D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 7 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' When D = D, the definition above corresponds with the usual definition of finite Blaschke product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For a finite Blaschke product B on a finitely connected domain D, we let IB denote the connected components of ∂D \\ {z : B(z) ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In other words, IB are the preimages (under B) of the open upper and lower half-circle components of B(∂D) ∩ H, B(∂D) ∩ (−H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will frequently be dealing with sequences of finite Blaschke products (Bn)∞ n=1 on D, in which case we abbreviate IBn by In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We call a domain D ⊂ C an analytic domain if D is finitely connected, and each component of ∂D is an analytic Jordan curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We remark that a boundary component of an analytic domain D cannot be a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let D ⊂ C be an analytic domain, suppose K ⊂ D is compact, and let f be a function analytic in a neighborhood of D satisfying ||f||D < 1 and {z : f(z) = 0}∩∂D = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then there exists M < ∞ and a sequence (Bn)∞ n=1 of finite Blaschke products on D satisfying: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) inf z∈∂D |Bn(z)| n→∞ −−−→ 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) ||Bn − f||K n→∞ −−−→ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) sup I∈In diam(I) n→∞ −−−→ 0, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) sup I,J∈In diam(I)/diam(J) < M and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) sup I∈In supz∈I |B′ n(z)| infz∈I |B′ n(z)| < M for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 will suffice for the proofs of Theorems A and B, although we prove a slightly stronger result in Section 12 (see Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 will be delayed until Sections 10-12, which are independent of Sections 2-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We now turn to applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 to produce Blaschke approximants as described in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Given a Jordan curve γ ⊂ C, we denote the bounded component of �C \\ γ by int(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We refer to Figure 3 for a summary of the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For the remainder of this section, we will fix a compact set K, an analytic domain D containing K, and a function f holomorphic in a neighborhood of D satisfying ||f||D < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Fix ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We assume that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) d(z, K) < ε/2 and d(f(z), f(K)) < ε/2 for every z ∈ ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We let γ be an analytic Jordan curve surrounding f(D) such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7) dist(w, f(K)) < ε for every w ∈ γ, and let Ψ : D → int(γ) denote a Riemann mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 8 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This figure illustrates Notations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The vertices pictured on ∂D are B−1(±1), and the components IB of Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4 are the edges along ∂D connecting these vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By enlarging D slightly we may ensure {z : Ψ−1 ◦ f(z) = 0} ∩ ∂D = ∅, so that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 applies to the triple D, K, Ψ−1 ◦ f to produce M < ∞ and a sequence of finite Blaschke products (Bn)∞ n=1 on D satisfying the conclusions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Note that in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9 we are applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 to Ψ−1 ◦ f (and not f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This will ensure that the critical values of the approximant Ψ ◦ Bn ≈ f are close to fill(f(K)) as needed for Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will now quasiconformally perturb the sequence (Bn) so as to ensure that we can later prove the conclusion CV(f|K) ⊂ CV(r) of Theorem B: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We may take the sequence (Bn)∞ n=1 of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9 so that it also satisfies: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8) CV(Ψ−1 ◦ f|K) ⊂ CV(Bn) for all large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let K′ ⊂ D be a compact set satisfying K ⊂ int(K′) ⊂ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 to the triple D, K′, Ψ−1 ◦ f to obtain a sequence of Blaschke products (Bn)∞ n=1 on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let z ∈ CP(f|K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9) ||Bn − Ψ−1 ◦ f||K′ n→∞ −−−→ 0, by Hurwitz’s Theorem there exists a sequence (wn z )∞ n=1 such that B′ n(wn z ) = 0 and wn z n→∞ −−−→ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let r < 1 be so that ψ−1 ◦ f(K) ⊂ rD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Define a homeomorphism hn : D → D to be an interpolation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10) hn(Bn(wn z )) := Ψ−1 ◦ f(z) for all z ∈ CP(f|K), and int() f(K) Y Izl<1 D K B ~ ofA GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 9 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='11) hn(z) = z for r ≤ |z| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9), we have that |Bn(wn z ) − Ψ−1 ◦ f(z)| n→∞ −−−→ 0 for all z ∈ CP(f|K), and hence hn may be taken to satisfy: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='12) ||(hn)z/(hn)z||D n→∞ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By the Measurable Riemann Mapping Theorem (see [Ahl06]), there is a quasiconformal φn : D → D so that: Bn := hn ◦ Bn ◦ φ−1 n is holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We normalize each φn by specifying φn(p) = p and φ′ n(p) > 0 for some r < |p| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Note that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13) Bn(∂D) ⊂ {z : r ≤ |z| ≤ 1} for large n by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1), so it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='11) that Bn is a Blaschke product on D for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We claim the sequence (Bn) satisfies the conclusions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6, as well as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Indeed, we have φn(wn z ) ∈ CP(Bn) and Bn(φn(wn z )) = Ψ−1 ◦ f(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='12), we have that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='14) ||φn(z) − z||D n→∞ −−−→ 0, and hence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='11), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' To prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5), we first note that since (hn)z = 0 in r < |z| < 1, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13) that φn extends to a holomorphic function in a neighborhood U of ∂D, where U does not depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='14) and the Cauchy integral formula, it follows that φ′ n(z) n→∞ −−−→ 1 uniformly for z ∈ ∂D and hence we deduce (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Similarly, φ′′ n(z) n→∞ −−−→ 0 uniformly for z ∈ ∂D and so (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Recall from the introduction that we plan to extend the definition of the approximant Ψ ◦ Bn ≈ f from D to all of C, where we recall Ψ was defined in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' To this end, it will be useful to define the following graph structure on ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For any n ∈ N, we define a set of vertices on ∂D by Vn := (Bn|∂D)−1(R), where each vertex v is labeled black or white according to whether Bn(v) > 0 or Bn(v) < 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The curve ∂D will be considered as a graph with edges defined by In (recall from Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4 that In is precisely the collection of components of ∂D \\ Vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will sometimes write Dn in place of D when we wish to emphasize the dependence of the graph ∂D on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We define a holomorphic mapping gn in D by the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='15) gn(z) := Ψ ◦ Bn(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 10 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK In Sections 3-7 we will quasiregularly extend the definition of gn to C, and then in Section 9 we apply the MRMT to produce the rational approximant of Theorem B as described in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall that in Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7, we fixed ε > 0, a compact set K contained in an analytic domain D, and a function f holomorphic in D (we note ε, K, D, f also satisfied extra conditions specified in Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The objects γ, Ψ, Bn, Vn, gn we then defined in this section were determined by our initial choice of ε, K, D, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In future sections, it will be useful to think of γ, Ψ, Bn, Vn, gn as defining functions which take as input some quadruple (ε, K, D, f) (for any ε, K, D, f as in Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7), and output whatever object we defined in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For instance, Vn defines a function which takes as input any (ε, K, D, f) as in Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7 and outputs (via Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='11) a set of vertices Vn(ε, K, D, f) on ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Similarly, Bn takes as input any (ε, K, D, f) as in Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7 and outputs (via Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10) a Blaschke product Bn(ε, K, D, f) on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Likewise for γ, Ψ, gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Quasiconformal Folding Given a compact set K ⊂ C and a function f holomorphic in a domain D containing K, we showed in Section 2 how to approximate f by a holomorphic function gn defined in D (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover gn is just a Blaschke product in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If f is a function holomorphic in an arbitrary analytic neighborhood U (where U need not be connected) of a compact set K, then one can apply the results of Section 2 to each component of U which intersects K (this is done precisely in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1): this yields a holomorphic approximant of f defined in a finite union of domains, so that the approximant is just a finite Blaschke product on each domain (recall Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In Sections 3-7, we will build the apparatus necessary to systematically extend this holomorphic approximant to a quasiregular function of C which is holomorphic outside a small set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' It was convenient to assume in Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7 that the compact set K was covered by a single domain D, however we now begin to work more generally: Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We refer to Figure 6 for a summary of the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Throughout Sections 3-7, we will fix ε > 0, a compact set K ⊂ C, a domain D containing K, a disjoint collection of analytic domains (Di)k i=1 such that K ⊂ U := ∪iDi ⊂ D, and a function f holomorphic in a neighborhood of U satisfying ||f||U < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We assume that the following analog of Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) holds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) d(z, K ∩ Di) < ε/2 and d(f(z), f(K ∩ Di)) < ε/2 for all z ∈ ∂Di and 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Applying the methods of the previous section to each component Di of U, we can define a sequence of finite Blaschke products (Bn)∞ n=1 on each Di (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will let Bn denote the corresponding function defined on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In particular, (Bn)∞ n=1 gives the following definition of vertices on the boundary of U := ∪iDi (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='11 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 11 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Illustrated is the Definition of the domain V in the proof of Propo- sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For every n ∈ N, we define a set of vertices Vn on ∂U by Vn := k� i=1 Vn(ε, K ∩ Di, Di, f|Di) = k� i=1 (Bn|∂Di)−1(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We now extend the graph structure on ∂U by connecting the different components of U by curves {Γi}k−1 i=1 in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 below, and defining vertices along these curves in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will need to prove a certain level of regularity for these curves and vertices in order to ensure that the dilatations of quasiconformal adjustments we will make later do not degenerate as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will denote the curves by {Γi}k−1 i=1 , and we remark that the curves depend on n, although we suppress this from the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For each n ∈ N, there exists a collection of disjoint, closed, analytic Jordan arcs {Γi}k−1 i=1 in (�C \\ U) ∩ D satisfying the following properties: (1) Each endpoint of Γi is a vertex in Vn, (2) Each Γi meets ∂U at right angles, (3) U ∪ (∪iΓi) is connected, and (4) For each 1 ≤ i ≤ k − 1, the sequence (in n) of curves Γi has an analytic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The set (�C \\ U) ∩ D must contain at least one simply-connected region V with the property that there are distinct i, j with both ∂V ∩∂Di and ∂V ∩∂Dj containing non-trivial arcs (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3), for all sufficiently large n both ∂V ∩ ∂Di, ∂V ∩ ∂Dj contain vertices of Vn which we denote by vi ∈ ∂Di, vj ∈ ∂Dj, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Consider a conformal map φ : D → V , and define Γ1 to be the image under φ of the hyperbolic geodesic connecting φ−1(vi), φ−1 i (vj) in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We now proceed recursively, making sure at step l we pick a V which connects two com- ponents of U not already connected by a Γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', Γl−1, and so that V is disjoint from Γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', Γl−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The curves Γi satisfy conclusions (1)-(3) of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We may ensure that for each 1 ≤ i ≤ k − 1, the sequence (in n) of curves Γi has an analytic limit by choosing vi, vj above to converge as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ D D2 V IJ D112 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Illustrated is Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Consider the vertices Vn ⊂ ∂U of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will augment Vn to include vertices on the curves (Γi)k−1 i=1 as follows (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let Γ ∈ (Γi)k−1 i=1 denote both the curve as a subset of C and the arclength parameterization of the curve, and suppose Γ connects vertices Γ(0) = vi ∈ ∂Di, Γ(length(Γ)) = vj ∈ ∂Dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For k = i, j, let εk denote the minimum length of the two edges with endpoint vk in ∂Dk, and suppose without loss of generality εj < εi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let l be so that εj/2 ≤ εi/2l ≤ 2εj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We place vertices at Γ(εi/2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', Γ(εi/2l), and we place vertices along Γ([εi/2l, length(Γ)]) at equidistributed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We can label the vertices black/white along Γ so that vertices connect only to vertices of the opposite color by adding one extra vertex at the midpoint of the segment having vj as an endpoint, if need be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We introduce the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Throughout Sections 3-7, we will let Ω denote a fixed (arbitrary) component of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) �C \\ � U ∪ k−1 � i=1 Γi � , and p ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Note that Ω is simply connected by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Denote D∗ := �C\\D, and let σ denote any conformal mapping (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) σ : D∗ → Ω satisfying σ(∞) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For z ∈ Ω, we define τ(z) := σ−1(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The map τ induces a partition of T which we denote by Vn := τ(Vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' D1 E1 82A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 13 |z|>1 Ω τ 3 D D2 Γ2 1Γ Γ3 D4 D1 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This figure illustrates Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 and Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' As pictured, U has four components (Di)4 i=1 which are connected by curves (Γi)3 i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall K ⊂ U (the compact set K is not shown in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The unbounded component Ω of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) is pictured in dark grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The map τ : Ω → D∗ is a conformal mapping, and sends the vertices on ∂Ω to (possibly unevenly spaced) vertices on the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will sometimes write Ωn, D∗ n in place of Ω, D∗, respectively, when we wish to emphasize the dependence of the vertices Vn ⊂ ∂Ω, Vn ⊂ ∂D∗ on the parameter n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For the graph ∂Ωn, we have: max{diam(e) : e is an edge of ∂Ωn} n→∞ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) and Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ As explained in the introduction, in order to prove uniform approximation in Theorem B, we will need to prove that our quasiregular extension is holomorphic outside a region of small area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This will usually mean proving the following condition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose V ⊂ C is an analytic domain, and ∂V is a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We say a quasiregular mapping φ : V → φ(V ) is C-vertex-supported if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) supp(φz) ⊂ � e∈∂V {z : dist(z, e) < C · diam(e)} (see Figure 7), where the union in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) is taken over all edges e on ∂V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' It will also be useful to have the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose e, f are rectifiable Jordan arcs, and h : e → f is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We say that h is length-multiplying on e if the push-forward (under h) of arc-length measure on e coincides with the arc-length measure on f multiplied by length(f)/length(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 14 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Shown as a black curve is part of a graph G, and in light gray the neighborhood ∪e∈G{z : dist(z, e) < C · diam(e)} of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' First we will adjust the conformal map τ so as to be length-multiplying along edges of ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall the vertices Vn ⊂ T defined in Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For every n, there is a K-quasiconformal mapping λ : D∗ n → D∗ n so that: (1) λ is C-vertex-supported for some C > 0, (2) λ(z) = z on Vn and off of supp(λz), (3) λ ◦ τ is length-multiplying on every component of ∂Ω \\ Vn, (4) C, K do not depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This is a consequence of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 of [Bis15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Indeed, recall τ := σ−1 and consider the 2πi-periodic covering map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) φ := σ ◦ exp : Hr �→ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The map φ induces a periodic partition φ−1(Vn) of ∂Hr which has bounded geometry (see the introduction of [Bis15], or Section 2 of [BL19]) with constants independent of n by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3(2) and Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 of [Bis15] applies to produce a 2πi-periodic, C vertex-supported, and K-quasiconformal map β : Hr → Hr so that φ ◦ β is length-multiplying on edges of Hr, and C, K are independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, the inverse β−1 ◦ log ◦τ is length-multiplying, and since exp is length-multiplying on vertical edges, the well-defined map λ := exp ◦β−1 ◦ log : D∗ → D∗ satisfies the conclusions of the Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ The main idea in defining the quasiregular extension in Ω is to send each edge of ∂Di to the upper or lower half of the unit circle by following λ ◦ τ with a power map z �→ zn A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 15 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This figure illustrates the Folding Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13 and Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The simply connected domain Ω′ n is obtained by removing from Ω certain trees based at the vertices along ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' of appropriate degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The main difficulty in this approach, however, is that the images of different edges of ∂Di under λ ◦ τ may differ significantly in size, so that there is no single n with z �→ zn achieving the desired behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The solution is to modify the domain Ω by removing certain “decorations” from the domain Ω, so that each edge of ∂Di is sent to an arc of roughly the same size under λ ◦ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This is formalized below in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13 (see also Figures 8, 9), and is an application of the main technical result of [Bis15] (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The “decorations” are the trees in the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let V ⊂ T be a discrete set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We call a domain W ⊂ D∗ a tree domain rooted at V if W consists of the complement in D∗ of a collection of disjoint trees, one rooted at each vertex of V (see the center of Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For m ∈ N, we let Z± m := {z ∈ T : zm = ±1}, Zm := Z+ m ∪ Z− m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In other words, Z+ m denotes the mth roots of unity, and Z− m the mth roots of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For every n, there exists a tree domain Wn rooted at Vn, an integer m = m(n), and a K-quasiconformal mapping ψ : Wn → D∗ so that: (1) ψ is C-vertex-supported for some C > 0, and ψ(z) = z off of supp(ψz), (2) on any edge e of ∂Wn ∩ T, ψ is length-multiplying and ψ(e) is an edge in T \\ Zm, (3) for any edge e of ∂Wn ∩D∗, ψ(e) consists of two edges in T\\Zm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, if x ∈ e, the two limits limWn∋z→x ψ(z) ∈ T are equidistant from Z+ m, and from Z− m, and (4) C, K do not depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We consider the 2π-periodic covering map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) φ := σ ◦ λ ◦ exp ◦(z �→ −iz) : H �→ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Izl>1 ov16 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For any x ∈ ∂Wn ∩ D∗, there are two limits limWn∋z→x ψ(z) ∈ T as illustrated in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13(3) says that these two limits are equidistant from the nearest black vertex, and are equidistant from the nearest white vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' inducing a periodic partition φ−1(Vn) of ∂H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4), Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4, and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10(2), any two edges of H have comparable lengths with constant independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' There- fore, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 of [Bis15] applies to yield a 2π-periodic K-quasiconformal map Ψn of H onto a subdomain Ψn(H) ⊊ H, with K independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We let Wn := exp(−iΨn(H)) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7) ψ := exp ◦ − iΨ−1 n ◦ i log : Wn → D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7) is well-defined, and the conclusions of the theorem follow from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 of [Bis15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will use the notation Ω′ n := (λ ◦ τ)−1(Wn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Annular Interpolation Between the Identity and a Conformal Mapping Recall from Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 that we have fixed ε > 0, a compact set K, disjoint analytic domains (Di)k i=1 so that U := ∪iDi contains K, and f holomorphic in a neighborhood of U with ||f||U < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In this section, we briefly define two useful interpolations in Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since the domain Di contains the compact set K∩Di, the definitions and results of Section 2 apply to (ε, K ∩ Di, Di, f|Di) for each 1 ≤ i ≤ k (see Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13 applies to define (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We define the Jordan curve (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) γi := γ(ε, K ∩ Di, Di, f|Di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recalling that int(γi) denotes the bounded component of �C \\ γi, we define (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) Ψi := Ψ(ε, K ∩ Di, Di, f|Di) to be a Riemann mapping Ψi : D → int(γi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lastly, we define the finite Blaschke products (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) Bn := Bn(ε, K ∩ Di, Di, f|Di) on Di, A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 17 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Illustrated is the map ηΨ i : D∗ → �C \\ Ψi(riD) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 in the case ri = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The dotted circle on the right depicts the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' where we suppress the dependence of (Bn)∞ n=1 on i from the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall that in Section 3, we defined curves {Γi}k−1 i=1 connecting the domains Di, and in Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 we fixed a component Ω of the complement of U ∪ ∪k−1 i=1 Γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' After relabeling the (Di)k i=1 if necessary, there exists 1 ≤ ℓ ≤ k so that ∂Di ∩ ∂Ω ̸= ∅ if and only if i ≤ ℓ (see Figure 6 for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For each 1 ≤ i ≤ ℓ, note that the intersection ∂Di ∩ ∂Ω consists of a single Jordan curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We let ri := |Bn(∂Di ∩ ∂Ω)|, so that 1 − ε ≤ ri ≤ 1 by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The two interpolations we will need are given in Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3, we define an interpolation ηΨ i between z �→ z on |z| = 2 with z �→ Ψi(riz) on |z| = 1 (see Figure 10), and in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4 we modify ηΨ i to define a map ηi so that ηi(z) = ηi(z) for |z| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For each 1 ≤ i ≤ ℓ, there is a quasiconformal mapping ηΨ i : D∗ → �C \\ Ψi(riD) satisfying the relations: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) ηΨ i (z) = z for |z| ≥ 2 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) ηΨ i (z) = Ψi(riz) for all |z| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, if Di, Dj for 1 ≤ i, j ≤ ℓ are connected by one of the curves (Γi)k−1 i=1 , then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) ηΨ i ([−2, −1]) ∩ ηΨ j ([1, 2]) = ηΨ i ([1, 2]) ∩ ηΨ j ([−2, −1]) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The existence of ηΨ i satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) follows from a standard lemma on the extension of quasisymmetric maps between boundaries of quasiannuli (see, for instance, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='30(b) of [BF14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) fails for the collection (ηΨ i )i∈I thus defined, we can renormalize the conformal mappings (Ψi)ℓ i=1 appropriately (to rotate the points Ψi(±ri) along the curve Ψi(riT)), and post-compose a subcollection of the ηΨ i by diffeomorphisms of DV n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='18 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK 2D \\ Ψi(riD) so that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) is satisfied for the composition when i, j ∈ I, and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) still hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For each 1 ≤ i ≤ ℓ, there is a quasiconformal mapping ηi : D∗ → C \\ Ψi([−ri, ri]) satisfying the relations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7) ηi(z) = z for |z| ≥ 2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8) ηi(z) = ηi(z) for |z| = 1, and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9) ηi(z) = ηΨ i (z) for z ∈ R ∩ D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Define γ+ i := ηΨ i (∂(A(1, 2) ∩ H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let η be a quasisymmetric mapping of T∩H onto [−1, 1] fixing ±1 (one can take η := M|T∩H where M is a Mobius transformation mapping −1, 1, i to −1, 1, 0, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Define a mapping g on γ+ i by: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10) g(z) := � Ψi ◦ η ◦ Ψ−1 i (z) z ∈ Ψi(T ∩ H) z otherwise Since g is a quasisymmetric mapping, a standard lemma on extension of quasisymmetric maps between boundaries of quasidisks (see, for instance, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='30(a) of [BF14]) implies that g may be extended to a quasiconformal mapping of ηΨ i (A(1, 2) ∩ H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Define g similarly in ηΨ i (A(1, 2) ∩ (−H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We let ηi := g ◦ ηΨ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' It is then straightforward to check that ηi satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4 define 2ℓ many quasiconformal mappings: {ηΨ i }ℓ i=1 and {ηi}ℓ i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The definition of the mappings ηΨ i , ηi depend on the objects ε, K, (Di)k i=1, f as fixed in Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1, but not on the parameter n in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus we record the trivial but important observation that the mappings {ηΨ i }ℓ i=1 and {ηi}ℓ i=1 are quasiconformal with a constant independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Annular Interpolation Between a Blaschke Product and a Power Map Recall that we have fixed ε > 0, a compact set K, disjoint analytic domains (Di)k i=1 so that U := ∪iDi contains K, and f holomorphic in a neighborhood of U with ||f||U < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The curves {Γi}k−1 i=1 connect the domains (Di)k i=1, and Ω is a component of the complement of U ∪ ∪k−1 i=1 Γi with τ : Ω → D∗ conformal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall that the domain Ω′ n was defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13 and Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='14 by removing from Ω a collection of trees rooted at the vertices along ∂Ω, and the map ψ ◦ λ ◦ τ maps Ω′ n onto D∗ (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 19 Notation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall from Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 that ∂Di ∩ ∂Ω ̸= ∅ if and only if 1 ≤ i ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Hence exactly ℓ − 1 of the curves (Γi)k−1 i=1 intersect ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By relabelling the (Γi)k−1 i=1 if necessary, we may assume Γj intersects ∂Ω if and only if 1 ≤ j ≤ ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let m = m(n) be as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' To prove our main results, we will need to modify z �→ zm in D∗ so that, roughly speaking, (z �→ zm) ◦ ψ ◦ λ ◦ τ(z) agrees with the Blaschke products Bn (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) along ∂Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This is done in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Its proof uses the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose φ1, φ2 are C1 homeomorphisms of a C1 Jordan arc e such that: (1) φ1(e) = φ2(e), (2) φ1, φ2 agree on the two endpoints of e, and (3) |φ′ 1(z)| = |φ′ 2(z)| for all z ∈ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then φ1 = φ2 on e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 is a consequence of the Fundamental Theorem of Calculus and is left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall the constant ri := |Bn(∂Di ∩ ∂Ω)| of Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For every n, there exists a locally univalent K-quasiregular mapping hn : D∗ → D∗ so that: (1) hn(z) = zm for |z| ≥ m√ 2 where m := m(n) is as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13, (2) hn ◦ ψ ◦ λ ◦ τ(z) = Bn(z)/ri for every z ∈ ∂Di and 1 ≤ i ≤ l, and (3) K is independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Fix the standard branch of log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Given an edge e ∈ ∂Di, we have by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13 that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) log ◦ψ ◦ λ ◦ τ(e) = {0} × �jπ m , (j + 1)π m � for some 0 ≤ j ≤ 2m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Denote the vertical line segment in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) by ve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let f : ve �→ e be a length-multiplying, C1 homeomorphism so that f −1 agrees with log ◦ψ ◦ λ ◦ τ on the two endpoints of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Consider the maps: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) z �→ mz for z ∈ �log 2 m � × �jπ m , (j + 1)π m � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) z �→ log ◦r−1 i Bn ◦ f for z ∈ {0} × �jπ m , (j + 1)π m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For each 1 ≤ i ≤ l, the Blaschke products Bn are orientation-preserving on the unique outer boundary component of Di, and orientation-reserving on all other boundary components of Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This implies that we may choose the branch of log in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) so that the images of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) are horizontal translates of one another (recall Bn(e) is a circular arc of angle π), and the derivative of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) is strictly positive for all z ∈ ve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since the derivative of 20 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Illustrated is the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In logarithmic coordi- nates the desired interpolation is denoted φ, and hn is then defined by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) is also strictly positive, this means the linear interpolation between (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5), we have that |B′ n| is comparable at all points of e with constant independent of e and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, since f is length-multiplying and log is length-multiplying on Euclidean circles centered at 0, we conclude that the derivative of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) is comparable to m at all points of ve with constant independent of e and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, we conclude that the linear interpolation between (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) in the rectangle (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) � 0, log 2 m � × �jπ m , (j + 1)π m � is K-quasiconformal with K independent of e and n (see, for instance, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 of [MPS20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Denote the linear interpolation by φ (see Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We define (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) hn := exp ◦φ ◦ log in {z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(e)} ∩ {|z| ≤ m√ 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) defines hn(z) for z in {z : 1 ≤ |z| ≤ m√ 2} and sharing a common angle with the image under ψ ◦ λ ◦ τ of an edge on some ∂Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We finish the definition of hn by simply setting: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) hn(z) := zm in {z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(∂Ω′ n \\ (∪i∂Di))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The conclusion (1) now follows by definition of hn, and (3) follows since hn is a composition of holomorphic mappings and a K-quasiconformal interpolation where we have already noted that K is independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' h nA GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 21 We now show that conclusion (2) follows from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Fix an edge e on ∂Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall ve := log ◦ψ ◦ λ ◦ τ(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) we have that: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7) hn ◦ ψ ◦ λ ◦ τ = r−1 i Bn ◦ f ◦ log ◦ψ ◦ λ ◦ τ on e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' First note that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7) agrees set-wise with r−1 i Bn on e and at the endpoints of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The map ψ ◦ λ ◦ τ is length-multiplying (by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10(3) and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13(2)), log is length- multiplying on the circular segment ψ ◦ λ ◦ τ(e), and f is length-multiplying by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus the modulus of the derivative of f ◦log ◦ψ◦λ◦τ is constant on e, and so the derivatives of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7) and r−1 i Bn have the same modulus at each point of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Conclusion (2) now follows from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Joining Different Types of Boundary Arcs: the Map En Recall that in Section 4 we defined the maps ηΨ i , ηi where 1 ≤ i ≤ ℓ, and in Section 5 we defined the map hn for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In this section we define a map En in D∗ which is roughly given by either z �→ ηΨ i ◦ hn(z) or z �→ ηi(z) ◦ hn(z), where i is allowed to depend on arg(z) and which of ηΨ i , ηi we post-compose hn with is also allowed to depend on arg(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, we will need a way to interpolate between the definitions of ηΨ i , ηi, for different i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The interpolation regions are defined in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 below, and the map En in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' It will be useful to keep Figure 12 in mind for the remainder of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Mark one edge ei on Γi for each 1 ≤ i ≤ ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Label the ℓ components of ∂Ω′ n \\ ∪iei as (Gi)ℓ i=1, where ∂Di ⊂ Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let (1) J D i denote those edges in ∂Di, (2) J G i denote those edges in Gi \\ J D i , (3) J e denote the edges (ei)ℓ−1 i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In other words, J D i are the edges shared by ∂Ω′ n and ∂Di, J e consists of ℓ − 1 edges: one on each of the curves (Γi)ℓ i=1, and J G i are the remaining edges on Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus we have: ∂Ω′ n = J e ∪ � i � J D i ∪ J G i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For z ∈ D∗, we define: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) En(z) := � ηψ i ◦ hn(z) if z/|z| ∈ ψ ◦ λ ◦ τ(J D i ) ηi ◦ hn(z) if z/|z| ∈ ψ ◦ λ ◦ τ(J G i ) It remains to define En(z) for z ∈ D∗ satisfying z/|z| ∈ ψ ◦ λ ◦ τ(J e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We do so in the following Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The map En extends to a locally univalent K-quasiregular mapping En : D∗ → C satisfying En(z) = zm for |z| ≥ m√ 2, where m = m(n) is as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, K does not depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 22 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Illustrated is Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The curves Γ1, Γ2 are depicted as black dotted lines, except for the edges e1 ⊂ Γ1, e2 ⊂ Γ2 which are in thick black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Consider (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Note that if En is defined at z and |z| ≥ m√ 2, then En(z) = zm by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3(1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, setting En(z) := zm for |z| ≥ m√ 2 extends the definition of En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' It remains to extend the definition of En to: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) {z : 1 ≤ |z| ≤ m√ 2 and z/|z| ∈ ψ ◦ λ ◦ τ(ei)}, for 1 ≤ i ≤ ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Each of the ℓ − 1 sets in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) consists of 2 quadrilaterals which we denote by Q± i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The curve Γi connects two distinct elements of (Di)ℓ−1 i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In order to avoid complicating notation significantly, we will assume without loss of generality that Γi connects Di to Di+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let γi ⊂ 2D be a smooth Jordan arc connecting ηΨ i (1) to ηΨ i+1(−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6), we can choose γi so that the union of the arcs (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) ηΨ i ([1, 2]), 2T ∩ H, ηΨ i+1([−2, −1]), γi forms a topological quadrilateral we denote by Q+ i (in particular none of the arcs in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) intersect except at common endpoints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Define a quasisymmetric homeomorphism g+ i : ∂Q+ i → ∂(A(1, 2) ∩ H) (see Figure 13) by g+ i (z) = z for z ∈ 2T g+ i (z) = (ηΨ i )−1(z) for z ∈ ηΨ i ([1, 2]) g+ i (z) = (ηΨ i+1)−1(z) for z ∈ ηΨ i+1([−2, −1]), and extending g+ i to a quasisymmetric homeomorphism of γi to T ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The mapping g+ i extends to a quasiconformal homeomorphism g+ i : Q+ i → A(1, 2) ∩ H (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='24 of D1 " G1A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 23 Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Illustrated is the quadrilateral Q+ i and the map g+ i in the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' [BF14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We define En(z) := (g+ i )−1(zm) for z ∈ Q+ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A similar definition of g− i : Q− i → A(1, 2) ∩ −H is given (using the same curve γi) so that g+ i (z) = g− i (z) for z ∈ T ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We let En(z) := (g− i )−1(zm) for z ∈ Q− i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' To summarize, we have defined En in each of the three regions {z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(J D i )}, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) {z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(J G i )}, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) {z ∈ D∗ : z/|z| ∈ ψ ◦ λ ◦ τ(J e i )}, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) Indeed, the definition of En in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) was given already in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1), and in this proof we have defined En in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The definitions of En in each of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) agree along any common boundary, and thus by removability of analytic arcs for quasiregular mappings, it follows that En is quasiregular on D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, En has no branched points in D∗, and hence En is locally quasiconformal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The dilatation of the map En depends only on the dilatation of hn (which is independent of n by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3(3)) and the dilatations of the the finite collection of quasiconformal maps used in its definition: ηΨ i , ηi, g+ i , g− i , and hence we may take K independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Defining gn in Ω′ n First we recall our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We have fixed ε > 0, a compact set K, disjoint, analytic domains (Di)k i=1 so that K ⊂ U := ∪iDi, and f holomorphic in a neighborhood of U with ||f||U < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We defined curves {Γi}k−1 i=1 connecting the domains (Di)k i=1, and we denoted by Ω a component of the complement of U ∪ ∪k−1 i=1 Γi with τ : Ω → D∗ conformal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The domain Ω′ n is contained in Ω, and ψ ◦ λ ◦ τ maps Ω′ n onto D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In Section 6 we defined the map En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We define the mapping gn : Ω′ n → �C by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) gn := En ◦ ψ ◦ λ ◦ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Izl=2 Izl=2 Izl=1 Di D i+124 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK We will now record at which points the function gn|Ω′n is locally n : 1 for n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let g be a quasiregular function, defined in a neighborhood of a point z ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We say that z is a branched point of g if for any sufficiently small neighborhood U of z, the map g|U is n : 1 onto its image for n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We say w ∈ C is a branched value of g if w = g(z) for a branched point z of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We denote the branched points of a quasiregular mapping g by BP(g), and the branched values by BV(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall that in Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 we fixed a point p ∈ Ω satisfying τ(p) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The mapping gn : Ω′ n → C of Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 is K-quasiregular and C- vertex supported for K, C independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, g−1 n (∞) = {p}, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) BP(gn) ⊂ � e∈∂Ωn {z : dist(z, e) < C · diam(e)}, and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) BV(gn) ⊂ k� i=1 Ψi(riT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since each of the mappings in the composition (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) are K-quasiregular and C-vertex supported for K, C independent of n, the same is true of gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The only points where the mapping gn is locally l : 1 for l > 1 are a subset of the vertices of the graph ∂Ω′ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13, the vertices of ∂Ω′ n all lie in � e∈∂Ωn {z : dist(z, e) < C · diam(e)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, any vertex of ∂Ω′ n is mapped to a point on one of the curves Ψi(riT) by gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Hence, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) follows since BV(gn) = gn(BP(gn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' It remains to show: (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) g−1 n (∞) = {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Indeed, note that En◦ψ◦λ fixes ∞ and has no finite poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The map τ : Ω → D∗ is conformal and hence only one point p is mapped to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The relation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) now follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ It will be useful to record the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let r > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then for all sufficiently large n, we have: (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) gn(z) = τ(z)m for any z ∈ τ −1({z : |z| > r}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Consider the functional equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) defining gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The maps λ, ψ are vertex- supported, and moreover λ (respectively, ψ) is the identity outside of the support of λz, (respectively, ψz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7, we therefore have that ψ ◦ λ(z) = z if z ∈ τ −1({z : |z| > r}) and n is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The relation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) now follows from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3(1) since m → ∞ as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 25 Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' As in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13, we note that our Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 of gn is determined by a choice of the objects K, U, D, f, ε, Ω, p we fixed in Notations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' When we wish to emphasize this dependence, we will write gn(K, U, D, f, ε, Ω, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In particular, it will be useful in the next section to think of gn as a function taking as input any choice of K, U, D, f, ε, Ω, p satisfying the conditions in Notations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5, and outputting (via Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) a quasiregular function gn(K, U, D, f, ε, Ω, p) defined on Ω′ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Verifying gn is Quasiregular on �C In this section we combine our efforts in Sections 2-7 to define an approximant gn : �C → �C of a given f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The approximant gn will not be holomorphic as required in Theorems A and B, but we will solve this problem in the next section by applying the Measurable Riemann Mapping Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We fix the following for Sections 8-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Notation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Fix K, f, D, ε, P as in the statement of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Denote by U the neighborhood of K in which f is holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Define (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) P ′ := {p ∈ P : p is contained in a component V of �C \\ K such that V ̸⊆ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Compactness of K implies that U contains all but finitely many components of �C \\ K, and so the set P ′ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, P ′ does not depend on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By shrinking U if necessary, we may assume that: (1) U ∩ P ′ = ∅, (2) P ′ contains exactly one point in each component of �C \\ U, (3) f is holomorphic in a neighborhood of U ⊂ D, and (4) the components of U are a finite collection of analytic Jordan domains (Di)k i=1 so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) holds for each Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let K′ be a compact set such that K ⊂ int(K′) ⊂ K′ ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will assume for now that ||f||U < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We now define a quasiregular approximation gn of f by applying the Blaschke product construction of Section 2 in each Di, and by applying the folding construction of Sections 3-7 in each complementary component of ∪i(Di ∪ Γi): Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For every n, we define a quasiregular mapping gn as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recalling Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13, we first set (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) gn := gn(ε, K′ ∩ Di, Di, f|Di) in Di for 1 ≤ i ≤ k The equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) defines the curves (Γi)k i=1 by way of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3, and we enumerate the components of �C \\ � U ∪ k−1 � i=1 Γi � 26 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK by (Ω(i))ℓ i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recalling Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 and Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='14, we extend the definition of gn to the open set (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) Ω := �C \\ � ℓ� i=1 ∂Ω′ n(i) � by the formula (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) gn := gn(K′, U, D, f, ε, Ω(i), P ′ ∩ Ω(i)) in Ω′ n(i) for 1 ≤ i ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The quasiregular function gn is C-vertex supported and K-quasiregular for C, K independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For gn|Ω′n(i) this is exactly Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4, and so the conclusion follows since gn is holomorphic in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ The function gn is now defined on all of �C except for the edges of each ∂Ω′ n(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We show in Propositions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 below that gn in fact extends continuously across each edge of ∂Ω′ n(i), and deduce in Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 that gn extends quasiregularly across ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The K-quasiregular function gn : Ω → �C extends to a continuous function gn : Ω ∪ e → �C for any edge e ⊂ ∂Ω ∩ ∂U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let i be so that e ⊂ ∂Di and denote the unique element of (Ω(i))ℓ i=1 that contains e on its boundary by Ω(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall by Definitions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='12 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 that (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) gn|Di = Ψi ◦ Bn, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) gn|Ω′n(j) = En ◦ ψ ◦ λ ◦ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Assume that i ∈ I (the reasoning in the case i ̸∈ I will be the same), so that by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3(2) we have hn ◦ ψ ◦ λ ◦ τ = r−1 i Bn on e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) and the definition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) of En, it follows from (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) that gn|Ω′n(j)(z) = Ψi ◦ Bn(z) for z ∈ e, in other words gn|Ω′n(j) and gn|Di agree pointwise on e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The K-quasiregular function gn : Ω → �C extends to a continuous function gn : Ω ∪ e → �C for any edge e ⊂ ∂Ω ∩ (∪ℓ i=1Ω(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let j be so that e ⊂ ∂Ω′ n(j), and as in the proof of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4, recall that (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7) gn|Ω′n(j) = En ◦ ψ ◦ λ ◦ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let x ∈ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' There are two limits lim Ω′n(j)∋z→x ψ ◦ λ ◦ τ(z), A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 27 each lying on the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Denote them by ζ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13(3), ζm + = ζm − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1), we conclude that there is a unique limit lim Ω′n(j)∋z→x En ◦ ψ ◦ λ ◦ τ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Hence, setting gn(x) := lim Ω′n(j)∋z→x En ◦ ψ ◦ λ ◦ τ(z) defines a continuous extension of gn across the edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The K-quasiregular function gn : Ω → �C extends to a K-quasiregular func- tion gn : �C → �C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The set �C \\ Ω = ∂Ω consists of a finite collection of analytic arcs: the edges of the graphs ∂Ω′ n(i) over 1 ≤ i ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, by removability of analytic arcs for quasiregular mappings, it suffices to show that gn : Ω → �C extends continuously across each such edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' There are two types of edges to check: those that lie on the boundary of a domain Di, and those that lie in the interior of a domain Ω(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We have already checked continuity across both types of edges in Propositions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5, and so the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof of the Main Theorems In Section 9 we prove Theorems A and B, modulo the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 which is left to Sections 10-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall that in Section 8 we fixed the objects K, f, D, ε, P as in Theorem B (see Notation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1), and we defined a quasiregular approximation gn to f in Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We also showed in Section 8 that gn in fact extends to a quasiregular function gn : �C → �C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Now we apply the MRMT below in Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 to obtain the rational maps rn : �C → �C which we will prove satisfy the conclusions of Theorems A and B for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The mapping gn induces a Beltrami coefficient µn := (gn)z/(gn)z, which, by way of the MRMT, defines a quasiconformal mapping φn : �C → �C such that rn := gn ◦ φ−1 n is holomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We normalize φn so that φn(∞) = ∞ and φn(z) = z + O(1/|z|) as z → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We now begin deducing that for large n, the maps rn satisfy the various conclusions in Theorems A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The function rn of Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 is rational, and r−1 n (∞) = φn(P ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In particular, if K is full and P = {∞}, then rn is a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The function rn is holomorphic on �C and takes values in �C: the only such functions are rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Note that g−1 n (∞)∩U = ∅ since gn is bounded on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4 and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4), we have that g−1 n (∞) = P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since rn := gn ◦ φ−1 n , we conclude that r−1 n (∞) = φn(P ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 28 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK The last statement of the proposition follows since we normalized φn(∞) = ∞, and the only rational functions with a unique pole at ∞ are polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For all R < ∞, the mapping φn satisfies: (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) ||φn(z) − z||R·D n→∞ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since gn is C-vertex supported by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3, we conclude from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7 that (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) Area(supp(µn)) n→∞ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The relation (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) now follows from (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) since ||µn||L∞ ≤ K for all n by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For all sufficiently large n, the mapping rn satisfies CP(rn) ⊂ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7, we have max � diam(e) : e is an edge of ℓ� i=1 ∂Ω(i) � n→∞ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, since D is a domain containing ∪ℓ i=1∂Ω(i), we have by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) that BP(gn) \\ U ⊂ D for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since U ⊂ D we conclude that BP(gn) ⊂ D for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The result now follows from Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 since φ(BP(gn)) = CP(rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For all sufficiently large n, we have ||rn − f||K < 3ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' First we note that since f is uniformly continuous on K′, there exists δ > 0 so that if z, w ∈ K′ and |z − w| < δ, then |f(z) − f(w)| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3, we can conclude that (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) ||φn(z) − z||K < min(δ, dist(K, ∂K′)) for all sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let z ∈ K, w := φ−1 n (z) and suppose j is such that z ∈ Dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then |rn(z) − gn(z)| = |gn(w) − gn(z)| ≤ |gn(w) − f(w)| + |f(w) − f(z)| + |f(z) − gn(z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) Let C := sup z∈D 1≤i≤k |Ψ′ i(z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then ||gn − f||K′ ≤ C · ||Ψ−1 j gn − Ψ−1 j f||K′, and we deduce by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='15) that: ||gn − f||K′ ≤ C · ||Bn − Ψ−1 j f||K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 29 Applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10 (see also Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9), we conclude that (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) ||gn − f||K′ n→∞ −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Next, we deduce from (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) that |rn(z) − gn(z)| < 2ε for sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' It follows that for sufficiently large n: ||rn − f||K ≤ ||rn − gn||K + ||gn − f||K < 3ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For all sufficiently large n we have CV(f|K) ⊂ CV(rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let z ∈ CP(f|K), and let i be so that z ∈ Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then Ψ−1 i ◦f(z) ∈ CV(Bn) by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus f(z) ∈ CV(Ψi ◦ Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, by the Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='12 of gn, we have for large n that f(z) ∈ BV(gn) = CV(rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For all sufficiently large n, we have CV(rn) ⊂ fill{z : d(z, f(K)) < ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since CV(rn) = BV(gn), it suffices to show that for every z ∈ BP(gn) and sufficiently large n, we have (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) gn(z) ∈ fill{z : d(z, f(K)) < ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For z ∈ BP(gn) \\ U, (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) follows from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For z ∈ BP(gn) ∩ Di, (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6) follows from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='12 of gn|Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Proof of Theorem B: In the special case that ||f||K < 1, we have already proven that the mappings rn satisfy the conclusions of Theorem B for all sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Indeed, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 says that ||rn − f||K < ε, conclusion (2) in Theorem B is Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4, and conclusion (3) is Theorems 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Conclusion (1) follows from Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The general case follows by applying the above special case to an appropriately rescaled f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Proof of Theorem A: When K is full, we may take P = {∞} and apply Theorem B, in which case Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 guarantees that the maps rn are polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (Mergelyan+) Let K ⊂ C be full, suppose f ∈ C(K) is holomorphic in int(K), and let D be a domain containing K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For every ε > 0, there exists a polynomial p so that ||p − f||K < ε and: (1) CP(p) ⊂ D, (2) CV(p) ⊂ fill{z : d(z, f(K)) ≤ ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 30 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Proof : By the usual version of Mergelyan’s Theorem, there exists a polynomial q so that ||q − f||K < ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Apply Theorem A to K, D, q, ε/2 to obtain an approximant of q which we denote by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The polynomial p satisfies the conclusions of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (Weierstrass+) Suppose that I ⊂ R is a closed interval, f : I → R is continuous, and U, V ⊂ C are planar domains containing I, f(I), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then, for every ε > 0, there exists a polynomial p with real coefficients so that ∥f − p∥I ≤ ε, and (1) CP(p) ⊂ U, (2) CV(p) ⊂ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let I = [a, b], and f, U, V as in the statement of the corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8, there exists a complex polynomial q so that ||q − f||[a,b] < ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The real polynomial Q(z) := q(z) + q(z) 2 satisfies Q(x) = Re(q(x)) for x ∈ R and hence ||Q − f||[a,b] < ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We will use the symbol ⋐ to mean compactly contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let V1 ⋐ V be a sufficiently small, R-symmetric domain containing f(I) so that there is a component of Q−1(V1) (which we denote by U1) satisfying U1 ⋐ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let U2 be a R-symmetric, analytic domain satisfying I ⋐ U1 ⋐ U2 ⋐ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall Notation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 and consider: (1) the compact set U1, (2) the analytic function Q, (3) the analytic domain U2 containing U1, (4) min{ε/2, dist(∂V1, ∂V )}, (5) P = {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Applying Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 to (1)-(5) yields quasiregular mappings gn with R-symmetric Bel- trami coefficient, so that (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7) pn := gn ◦ φ−1 n is a real polynomial approximant of Q for large n satisfying: (1) ||pn − f||[a,b] < ε, (2) CP(pn) ⊂ U2, (3) CV(pn) ⊂ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus pn satisfies the conclusion of Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9 for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If we make further assumptions on the compact set K, the conclusion CV(r) ⊂ fill{z : d(z, f(K)) < ε} of Theorems A and B can be improved to (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8) CV(r) ⊂ fill(f(K)), A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 31 which is equivalent to CV(r) ⊂ f(K) if f(K) is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Indeed, if for instance the interiors of K, f(K) are analytic domains and f : int(K) → int(f(K)) is proper, then a similar strategy as in the proofs of Theorems A and B but replacing Ψ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='15) with a conformal map D �→ int(K) can be used to prove (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Recall the notation Ω(i) from Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2, and let τi : Ω(i) → D∗ be the conformal mapping satisfying τ −1 i (∞) = P ′ ∩ Ω(i) as in Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The following fact justifies part of our description in the introduction of the behavior of the rational approximants off K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let 1 < r < R < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then, for all sufficiently large n, we have (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9) rn ◦ φn(z) = τi(z)m and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10) |rn(z)| > R for all z ∈ τ −1 i ({z : |z| > r}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Fix R < ∞ and r > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' From (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5) and the functional equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) defining gn in Ω(i), it follows that: (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='11) gn(z) = τi(z)m for all z ∈ τ −1 i ({z : |z| > (r + 1)/2}) for all large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='12) rn ◦ φn = gn, The relation (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, we have by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 that: (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='13) φn ◦ τ −1 i ({z : |z| > r}) ⊂ τ −1 i ({z : |z| > (r + 1)/2}) for all sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since ((r + 1)/2)m > R for large n, the relation (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10) also follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Some Estimates Involving Harmonic Measure and Green’s Functions In Sections 10-12 we turn to the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In fact, we will prove a slightly stronger result (Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 in Section 12) from which Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We begin by recalling a few standard facts, and sketch the proofs for the convenience of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let D(z, r) denote the disk of radius r centered at z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 is illustrated in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If K ⊂ D is continuum connecting 0 to T, then ω(z, K, D \\ K) ≥ c > 0 for all |z| < 1/4 and some c > 0 independent of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 32 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Illustrated is the compact set K of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Exercise III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10 of [GM08] says that if E is a continuum connecting {|z| = 1 2} to T in D, then ω(0, E, D \\ E) ≥ c = 2 π tan−1(1/ √ 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Apply the exercise and the maximum principle to the disk D = D(z, 1 2) to deduce the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' [GM08] gives a simple direct proof of the exercise, but it also follows from Beurling’s projection theorem, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 of [GM08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose u, v are harmonic functions on D such that supD |u|, supD |v| ≤ M and that |u − v| < ε on some continuum K connecting 0 to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then |u − v| ≤ εcM 1−c on D(0, 1/4), where c > 0 is the constant from Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Consider the subharmonic function log |u−v|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' It is less than log ε on K and less than log M on ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus for |z| ≤ 1/4, Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 implies log |u(z) − v(z)| ≤ ω(z, K, D \\ K) log ε + ω(z, T, D \\ K) log M ≤ c log ε + (1 − c) log M, so |u(z) − v(z)| ≤ εcM 1−c, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose Ω is a planar domain, K ⊂ Ω is compact and connected, and 0 < ε, M < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then there is a δ > 0 so that if h = u + i�u is holomorphic on Ω, �u vanishes at some point of K, supΩ |h| ≤ M and supK |u| < δ, then supK |�u| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If K is single point, this is trivial since �u = 0 there by assumption, so assume K is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Choose η > 0 so that η < dist(K, ∂Ω) and η < diameter(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then for any radius η disk D centered at a point of K, u is less than δ on a continuum connecting the center of D to its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This implies |u| ≤ δcM 1−c (for c as in Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) on a η/4-neighborhood of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus |∇u| = O(δcM 1−c/η) on the (η/8)-neighborhood U of K (this uses the Cauchy estimate for |∇u|, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4 of [ABR01]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since |∇�u| = |∇u| and Iwl=1 K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='0 Z wl=1/4A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 33 Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Illustrated in gray is the set Uδ of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' U is path connected, this implies �u is within ε of zero on K if δ is small enough (depending on ε, η, M and the diameter of U in the path metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ We will use this later in the situation that if u and v are harmonic functions on Ω that are close enough on K ⊂ Ω, then f = exp(u + i�u) and g = exp(v + i�v) are holomorphic functions on Ω that are close on K (if �v is chosen to agree with �u at some point of K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Next, we recall the well known boundary Harnack inequality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', see Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='18 of [Mar19] or Exercise I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 of [GM08]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose u and v are positive harmonic functions on D which extend contin- uously to the boundary T, and suppose furthermore that u and v are both equal to zero on an arc I ⊂ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For δ > 0 let Uδ = {z ∈ D : dist(z, T \\ I) > δ} (see Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then for z ∈ Uδ, δ2 4 · u(0) v(0) ≤ u(z) v(z) ≤ 4 δ2 · u(0) v(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Note that, under these conditions, u and v have well defined inward normal derivatives on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By letting z → T radially, the inequalities above imply that ∂u ∂n and ∂v ∂n are comparable (with the same constants as above) at any point of Uδ ∩ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose u is a positive harmonic function on D which extends continuously to the boundary T and that equals zero on an arc I ⊂ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose a, b ∈ I both have distance > δ from T \\ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then δ2 4 · ∂u ∂n(a) ≤ ∂u ∂n(b) ≤ 4 δ2 · ∂u ∂n(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We simply compare u to a rotation of itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let v(z) = u( b az).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then v and u both vanish on J = I ∩ a b · I, and a ∈ J is distance > δ from either endpoint of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Hence the normal derivatives of u and v at a are comparable by the boundary Harnack principle, and hence so are the normal derivatives of u at a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Us 134 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Suppose Ω is an analytic domain (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For w ∈ Ω, let G(z, w) be the Green’s function on Ω with pole at w, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', G is harmonic on Ω \\ {w}, vanishes identically on ∂Ω and G(z, w) + log |z − w| is bounded in a neighborhood of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Our assumptions on Ω imply that Ω is regular for the Dirichlet problem, and hence that the Green’s function exists and is unique for any w ∈ Ω (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', see Sections II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 of [GM08]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose Ω is an analytic domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If r > 0 is small enough (depending only on Ω), x ∈ ∂Ω, and y ∈ Ω \\ D(x, r), then the normal derivative of the Green’s function with pole at y has comparable size at all points of ∂Ω ∩ D(x, r/2) with a constant independent of y and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Choose r small enough that W = D(x, r) ∩ Ω is a Jordan domain whose boundary consists of a sub-arc γ of ∂Ω and an arc of the circle ∂D(x, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let �γ = ∂Ω ∩ D(x, r/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Choose a point z ∈ W that is about distance r from ∂W and choose a conformal map ϕ : W → D taking z to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If r is small enough, ϕ extends analytically across γ to all of D(x, r) and by the Koebe distortion theorem it has comparable derivative at all points of �γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Also, since γ and each component of γ \\ �γ has harmonic measure with respect to z that is bounded away from zero, the image arcs on T all have lengths bounded uniformly from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus u(z) = G(ϕ−1(z), y) is a positive harmonic function on D vanishing on ϕ(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 the normal derivatives of u are comparable at all points of ϕ(�γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since the values of |ϕ′| are comparable at all points of �γ, we can deduce the lemma from the chain rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose Ω is an analytic domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If r > 0 is small enough (depending only on Ω), x ∈ ∂Ω, and y ∈ Ω, then ∂G(x, y)/∂n is comparable at all points of γ = ∂Ω ∩ D(x, r) with a constant depending only on an upper bound for M = max � 1, ℓ(γ) dist(y, ∂Ω) � , where ℓ(γ) denotes the length of γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, ∂G(x, y)/∂n = O(1/r) on γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We can cover γ by at most O(M) disks Dj = D(xj, rj) whose doubles are all disjoint from y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 implies the normal derivatives are comparable with some uniform constant C on each corresponding arc, so they are comparable with constant CO(M) on γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (In fact, if r is so small that ∂Ω looks “straight” on scale r, then only O(log M) disks are needed, since they become geometrically larger as we move away from y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=') The final claim follows because the integral of the normal derivative over the whole bound- ary is 2π, and hence the integral over γ is ≤ 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since the size of the normal derivative is comparable at all points of γ, this implies it is bounded above by O(1/ℓ(γ)) = O(1/r) (recall ℓ denotes length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ The following two lemmas relate harmonic measure to Green’s function: we refer to Figure 16 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 35 (a) (b) Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In (A) the setup for Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8 is shown, and in (B) the setup for Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' There is a constant C1 < ∞ so the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose Ω is an analytic domain, that Γ is a connected component of ∂Ω and that w ∈ Ω satisfies dist(w, Γ) = dist(w, ∂Ω) ≤ diameter(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then G(z, w) ≤ C1 on σ = {z : |z − w| = 1 2 dist(w, Γ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let Ω′ be the component of �C \\ Γ containing Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then Ω ⊂ Ω′, so by the maximum principle the Green’s function for Ω is less than the Green’s function for Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus it suffices to prove the lemma for the simply connected domain Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' But by Koebe’s distortion theorem, σ contains a ball of fixed hyperbolic radius around w and hence its image contains a ball of fixed radius if we conformally map Ω′ to the disk with w going to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' On the disk, the Green’s function is log 1 |z| which is clearly bounded by some C outside a fixed ball around the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' There is a constant C2 < ∞ so that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose that Ω is an analytic domain, z ∈ Ω, that γ ⊂ ∂Ω is a subarc, and that w ∈ Ω satisfies (1) dist(w, Γ) = dist(w, ∂Ω) ≤ diameter(Γ), where Γ is the component of ∂Ω containing γ, (2) |z − w| ≥ dist(w, ∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then G(z, w) ≤ C2ω(z, γ, Ω)/ω(w, γ, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let C1 and σ = {z : |z − w| = 1 2 dist(w, Γ)} be as in Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By assumption z is in the component Ω′ of Ω \\ σ not containing w and by the maximum principle applied to Ω′, ω(z, σ, Ω′) ≥ G(z, w)/C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By the maximum principle (again applied to Ω′), ω(z, γ, Ω) ≥ ω(z, σ, Ω′) · min x∈σ ω(x, γ, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' r 5 Wr Z W 5 Y36 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK By Harnack’s inequality (see Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='17 of [Mar19]), all the values of ω(x, γ, Ω) are comparable on σ and hence there is an ε > 0 so that ω(z, γ, Ω) ≥ ω(z, σ, Ω′) · ε · ω(w, γ, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Finally, by Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8 we have ω(z, γ, Ω) ≥ (ε/C1) · G(z, w) · ω(w, γ, Ω), which is the desired inequality with C2 = C1/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose Ω is an analytic domain, z ∈ Ω and ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose also that {γk}n 1 ⊂ ∂Ω is a collection of disjoint arcs and {wk}n 1 ⊂ Ω is a collection of points, so that for all k we have: (1) dist(wk, Γk) = dist(wk, ∂Ω) ≤ diameter(Γk), where Γk is the component of ∂Ω con- taining γk, (2) |z − wk| ≥ dist(wk, ∂Ω), (3) ω(wk, γk, Ω) ≥ ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then �n k=1 G(z, wk) ≤ C2/ε where C2 is the constant from Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='9, � k G(z, wk) ≤ (C2/ε) � k ω(z, γk, Ω) and since the arcs {γk} are disjoint, we have � k ω(z, γk, Ω) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Periods of Harmonic Functions Suppose Ω is an analytic domain with N +1 boundary components Γ0, Γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' , ΓN, so that Ω is regular for the Dirichlet problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose h is harmonic in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In any sub-disk D ⊂ Ω, h has a harmonic conjugate �h that is well defined up to an additive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If γ is a closed curve in Ω, then we can analytically continue �h along γ until we return to the starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The period of h along γ is the difference between the starting and ending values of �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If γ is homologous to a point, the period is zero, but if γ is homologous to a boundary component of Ω, then the period may be non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The sum of the periods corresponding to all N + 1 boundary components is always zero (the union of all boundary curves is homologous to zero in Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Next we consider the periods of certain special functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' , N let ωj be the harmonic function on Ω that has boundary value 1 on Γj and is 0 on the other boundary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since the boundary components are analytic, each ωj extends to be analytic across ∂Ω, so the normal and tangential derivatives are well defined, and themselves analytic, at every boundary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The period of ωj along Γk is the integral of the tangential derivative of �ωj around Γk, and this equals the integral of the normal derivative of ωj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', the period is λjk = � Γk ∂ωj ∂n ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 37 Note that since 0 < ωj < 1 in Ω and ωj = 1 on Γj, the inward normal derivative of ωj on Γj is non-positive (and strictly negative by analyticity: in that case, G can’t have critical point on the boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Similarly, the inward normal derivative of ωj is strictly positive on Γk for k ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus λjj < 0 and λjk > 0 for k ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since the periods of ωj sum to zero we have N � k=0 λjk = 0 for every j, and since � j ωj is the constant function 1 on Ω, we have N � j=0 λjk = 0 for every k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus for each j we have: λjj = − � k≥0:k̸=j λjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In other words, the row and column sums are all zero for the (N +1)×(N +1) matrix (λjk), 0 ≤ j, k ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If we drop the first row and column of this matrix, we are removing all positive terms (except for λ00), so we have j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' , N, λjj < − � k≥1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='k̸=j λjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus the N × N matrix Λ = (λjk), 1 ≤ j, k ≤ N is diagonally dominant, and this implies it is invertible by the Levy-Desplanques theorem: if the kernel of Λ contains a non-zero vector v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' , vN), then � k≥1 vkλjk = 0, and if |vk| is the largest component of v, then |vkλkk| = | � j≥1:j̸=k λjkvj| ≤ |vk| · � j≥1:j̸=k |λjk| < |vk| · |λkk|, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' See Olga Taussky-Todd’s paper [Tau49] for some history of this oft-rediscovered fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Also note that ∥Λ∥ and ∥Λ−1∥ only depend on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We have now proven the following result (due in a slightly different form to Heins [Hei50] and in greater generality to Khavinson [Kha84]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose Ω, Λ and {ωj}N 1 are as above and suppose we assign real numbers v = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' , vN) to the N boundary components Γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' , ΓN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then there is a linear combination h = �n j=1 ajωj so that the period of h around Γj is exactly vj for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The coefficients a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' , aN) are solutions of the linear equation Λa = v and hence ∥a∥ ≤ ∥v∥ · ∥Λ−1∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus if ∥v∥ ≤ ε then ∥a∥ = O(ε) with a constant that depends only on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We say that the periods of a harmonic function h on Ω are well defined modulo 2π if every period is some integer multiple of 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In this case, f = exp(h + i�h) is a well defined 38 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK analytic function on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For example, if Ω is the complement of a finite set of points {zk}n 1, then �n k=1 log |z − zn| has periods that are well defined modulo 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The corresponding holomorphic function is a polynomial with zeros at {zk} (and hence extends holomorphically from Ω to the whole plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Note that ∇�u is always well defined even if �u is not, since any two different branches of �u differ by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose Ω is an analytic domain and K ⊂ Ω is a compact set that contains curves {σj}N 0 homologous to each of the boundary components {Γj}N 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose K ⊂ W ⊂ Ω is open, let η = dist(K, ∂W) and set U = {z ∈ W : dist(z, ∂W) > η/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose P ⊂ Ω is a finite set and suppose u and H are harmonic functions on Ω \\ P, and each is either bounded or has a logarithmic pole at each point of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose |u − H| is bounded by M on U, and that |u − H| < ε on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If u has a well defined harmonic conjugate modulo 2π on Ω, then there is an harmonic h on Ω so that (1) h + H also has a well defined harmonic conjugate modulo 2π on Ω \\ Z, (2) |h| ≤ Cεc on all of Ω (not just K), (3) h is constant on each component of ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The constant c is the same as in Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 and C depends only on Ω and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since v = u − H is bounded on U, it extends to be harmonic at each point of P ∩ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since |v| < ε on K, it is bounded by O(εcM 1−c) on U and hence the gradient of v is bounded by O(εcM 1−c/η) on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus the gradient of the (possibly multi-valued) harmonic conjugate of v is bounded by the same quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We deduce that the harmonic conjugates of H and u differ by δ = O(LεcM 1−c/η) on U, where L is the diameter of U in the path metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus the periods of H on the {σj} differ from multiples of 2π by at most δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Now apply Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1 to define a harmonic function h on Ω that is bounded by O(δ) and has exactly the periods of −v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The Generalized Caratheodory Theorem on Blaschke Approximation If B is a finite Blaschke product (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) on an analytic domain Ω, then it has non-zero, continuous boundary values, and hence can have only finitely many zeros inside Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, the Schwarz reflection principle implies B extends holomorphically across ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The following result extends Carath´eodory’s Theorem (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1), and its statement uses the notation IB (see Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4), and the following notation: Notation 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If Ω ⊂ C is an analytic domain and Γ := ∂Ω, we denote Γδ := {z ∈ Ω : dist(z, Γ) = δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Suppose that Ω ⊂ C is an analytic domain, K ⊂ Ω is compact, and f is holomorphic on a neighborhood of Ω with supΩ |f| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let Γ := ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then for any ε > 0 and sufficiently small δ > 0, there is a finite Blaschke product B on Ω so that (1) supK |f − B| ≤ ε, A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 39 (2) 1 − ε < |B| ≤ 1 on ∂Ω, (3) Every component of Γδ \\ {B = 0} has length comparable to δ and adjacent connected components have length comparable to within a factor of 1 + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (4) All components of IB have length which is comparable to δ with constants that depend only on ε and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (5) on each component γ of IB, the ratio maxγ |B′|/ minγ |B′| is bounded depending only on ε and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Without loss of generality, we may assume K is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Otherwise, replace K by a compact, connected superset, for instance, its closed convex hull in the hyperbolic metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' As before, let G(z, w) denote the Green’s function on Ω with pole at w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since f is holomorphic on a neighborhood of Ω, it only has finitely many zeros in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We consider g := (1 − a) · f + b for some constants 0 < a < ε and |b| < ε such that |g| < 1 − ε on Ω and g has no zeros on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If we construct a finite Blaschke product B approximating g to within ε on K, then it approximates f to within 3ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Fix a finite number of smooth curves {σk} that are homologous to each boundary curve of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By enlarging K, if necessary, we may assume it contains all these curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let {zk}N 1 be the zeros of g, counted with multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By enlarging K again, if necessary, we may assume all the zeros of g are in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let W be an open domain with K ⊂ W ⊂ W ⊂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By compactness we have min∂W |g| ≥ η for some η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In what follows, δ > 0 will always be chosen so small that W is disjoint from {z ∈ Ω : dist(z, ∂Ω) < 2δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In particular, g has no zeros in this neighborhood of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let u(z) = − log |g(z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then u ≥ − log(1 − ε) ≥ ε is positive and harmonic on Ω except for finitely many logarithmic poles at the {zk}N 1 , the zeros of g listed with multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let p(z) = �N k=1 G(z, zk) be the sum of the Green’s functions with these poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then v(z) = u(z)−p(z) is harmonic on Ω, and equals u on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus v is continuous and non-zero on ∂Ω, and hence it is bounded and bounded away from zero there, say m ≤ v ≤ M on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Hence m ≤ v ≤ M on all of Ω by the maximum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5 of [GM08] v is the Poisson integral of its boundary values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', v(z) = 1 2π � ∂Ω v(w)∂G(w, z) ∂n ds(w) where ∂ ∂n is the inward normal and ds denotes length measure on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For w ∈ Γδ, denote by w∗ ∈ ∂Ω the closest point to w on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For z ∈ K and w ∈ Γδ we have G(w, z) = δ · ∂G(w∗, z) ∂n + O(δ2), where the constant depends on z, but is uniformly bounded as long as z is in the compact set K (the constant in the “big-Oh” depends on a bound for |∇2G| between Γδ and ∂Ω and since G is harmonic and extends analytically across ∂Ω, this is bounded as long as the pole 40 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK of Green’s function is not too close to ∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' It follows that v(z) = 1 2πδ � Γδ v(w)G(w, z)ds(w) + O(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1) Next use the identity G(z, w) = G(w, z) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='8 of [GM08]), to deduce v(z) = 1 2πδ � Γδ v(w)G(z, w)ds(w) + O(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We now discretize the integral by cutting Γδ into disjoint subarcs {γk} chosen so that 1 ≤ � γk v(w) 2πδ ds(w) ≤ 1 + O(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) This is possible since the integral over a component Γk δ of Γδ is at least A = m · ℓ(Γk δ)/2πδ and this tends to infinity as δ ↘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We can therefore cut each boundary curve into sub-arcs where the integral is between 1 and 1+O(1/A) = 1+O(δ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', we can make the sub-integrals all as close to 1 as we wish, by taking δ small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The left side of Equation (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2) implies that for each k, we have ℓ(γk)M/2πδ ≥ � γk v/2π ≥ 1 and hence ℓ(γk) ≥ 2πδ/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Similarly, the other side implies ℓ(γk) ≤ (1 + O(δ))2πδ/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus every such arc has length comparable to δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If δ is small enough, then the continuity implies that on the union of two adjacent arcs with common endpoint x, v is close to v(x), and hence by (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2), the lengths of these intervals are both close to 2πδ/v(x), and hence are close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This fact, together with each γk have length comparable to δ, will imply part (3) of the Theorem once we have defined the generalized Blaschke product B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Adding and subtracting a term from (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='1), have v(z) = 1 2πδ � j G(z, wj) � γj v(w)ds(w) (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3) + 1 2πδ � j � γj v(w)[G(z, w) − G(z, wj)]ds(w) + O(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The curve Γδ is parallel to the boundary, which is the level line G(z, w) = 0 of the Green’s function with pole at w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus, the gradient of G along Γδ is nearly perpendicular to Γδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Denoting by wj the center of γj, we conclude: |G(z, w) − G(z, wj)| = O(δ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) Using (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4) in the last term of Equation 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 gives v(z) = 1 2πδ � j G(z, wj) � γj v(w)ds(w) + � j 1 2πδ � γj v(w)O(δ2)ds(w) + O(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 41 Simplifying, we get v(z) = 1 2πδ � j G(z, wj) � γj v(w)ds(w) + O(δ) � j � γj v(w)ds(w) + O(δ) = � j G(z, wj) � γj v(w) 2πδ ds(w) + O(δ) = � j G(z, wj)(1 + O(δ)) + O(δ), = � j G(z, wj) + O(δ), where in the last line we have used Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10 to bound the sum of Green’s functions by O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Therefore, v is approximated on K by a finite sum of Green’s functions on Ω (indeed, we can even take the approximation to hold on the larger compact set W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If Ω is simply connected, then we are essentially done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In this case, H(z) = � k G(z, zk) + � j G(z, wj) = p(z) + � j G(z, wj) is harmonic except for a finite number of logarithmic poles at P = {∪kzk}∪{∪jwj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Therefore H has a harmonic conjugate �H that is well defined modulo 2π on Ω \\ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Then B(z) = exp(−H − i �H) is holomorphic on Ω \\ P, but tends to zero at each point of P, so B is holomorphic on all of Ω with zeros exactly at the points of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, on ∂Ω we have |B| = exp(0) = 1, so B is a finite Blaschke product on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, H(z) = p(z) + � j G(z, wj) = u(z) − v(z) + � j G(z, wj) ≈ u(z) = − log |g(z)|, so log |B| = −H approximates log |g| on W as closely as wish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In particular, we may assume |B| ≥ η/2 on ∂W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In this case, g/B is holomorphic on W (since every zero of B inside W is also a zero of g of the same multiplicity), and so by the maximum principle |g/B| is bounded on K by max∂K |g/B| ≤ maxK 2|g|/η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Now apply Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 to h = log g/B = u+ �u on W to deduce that arg(B) approximates arg(g) on K, at least if we add an appropriate constant to arg(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Therefore B (times an appropriate unit scalar) uniformly approximates g on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This extends Carath´eodory’s theorem to simply connected domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' However, if Ω is multiply connected, then H need not have a well defined harmonic con- jugate modulo 2π on Ω \\ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Therefore B = exp(−H − i �H) need not be well defined: if we analytically continue B along one of the closed loops σk, we return to the same absolute value, but possibly a different value of the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The change in the argument is as small as we wish, tending to zero as the difference between log |B| = H and log |g| tends to zero 42 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This is because g has a well defined harmonic conjugate modulo 2π on Ω \\ P, and so the discussion following Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In order to get a well defined (generalized) finite Blaschke product B on Ω, we apply Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 to u and H to construct h so that (1) h is harmonic on all of Ω, and (2) h + H has a well defined harmonic conjugate modulo 2π on Ω \\ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' (Note the H was constructed exactly so that the corollary can be applied: u−H is harmonic except for poles on Γδ which are outside W, and we may make u − H is as close to zero on K as we wish, while keeping it uniformly bounded on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=') Now set F = −H − h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This function has all periods equal to zero modulo 2π, so B = exp(F + i �F) is a well defined holomorphic function on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since |h| = O(δ), we can deduce that B still approximates g on the compact set K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Moreover, since H = 0 on ∂Ω and h = aj on Γj, we see that |B| = exp(aj) = 1 + O(δ) on Γj, so |B| is constant on each boundary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Dividing by the largest such value, we get another finite Blaschke product that satisfies (2) and still approximates g to within O(δ) on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' To prove (4), it suffices to show that the modulus of the tangential derivative of B along ∂Ω (which is equal to |B′| since B is holomorphic) is comparable to 1/δ everywhere: then the preimage of a half-circle will have length comparable to δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Note that on ∂Ω, |B′| is also equal to the normal derivative of h + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The function h is a linear combination of fixed functions ωj that depend only on Ω, and although the coefficients of the combination may depend on δ, they remain small as δ ↘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus the normal derivative of h remains bounded on ∂Ω as δ ↘ 0, and this is negligible compared to 1/δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The function H is a sum of two sets of Green’s functions, one with poles {zj} corresponding to the zeros of g and the other with poles {wk} along the curve Γδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The first set of poles is fixed independent of δ and their contribution to the normal derivative of H is also bounded independent of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Again, these terms are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The main contribution to the normal derivative of H comes from the poles {wk} lying on Γδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' For each such point wk consider the arc σk = D(wk, 2δ) ∩ ∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' the harmonic measure of this arc with respect to wk is bounded uniformly away from zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=', ω(wk, σk, Ω) > c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' This harmonic measure is the integral over σk of the normal derivative of the Green’s function with pole at wk, and thus it is less than the integral of the normal derivative of H, since this Green’s function is one term of the sum defining H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus the integral of |B′| over σk is > c and so any single arc lj in IB can contain at most a bounded number of arcs of the form σk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Part (3) of the theorem, adjacent points wk are at most distance O(δ) apart and hence lj can have length at most O(δ) (otherwise it would cover too many of the arcs σk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Next, we need to prove ℓ(lj) is bounded below by a multiple of δ, and this is equivalent to proving an upper bound |B′(x)| = O(1/δ) for any x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' As noted above, the contributions to |B′| from h and from the poles of H coming from the zeros of g are both bounded independent of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' To deal with the poles {wk} of H on Γk, we choose a point z ∈ Γδ that is A GEOMETRIC APPROACH TO POLYNOMIAL AND RATIONAL APPROXIMATION 43 close to x ∈ ∂Ω, say |x − z| ≤ 2δ and note that ∂ ∂nG(x, wk) ≃ 1 δG(z, wk) Thus by Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='10, the total contribution of all the poles {wk} to |B′| is at most � k 1 δG(wk, z) = O(1 δ), and hence (4) is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Let γ denote a component of IB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='7 and conclusion (4) just proven, the normal derivative of Green’s function with a pole at least distance δ from ∂Ω has comparable values at all points of γ, and so the same holds for any finite sum of such functions (with the same constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since on ∂Ω we have |B′| = | � ∂G ∂n |, we deduce (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' □ Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If Ω = D, then it suffices to assume f is holomorphic on Ω instead of a neighborhood of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' In that case if r < 1 then g(z) = f(rz) is holomorphic on a neighborhood of D and approximates f on a compact set K ⊂ D if r is close enough to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' So if B is a finite Blaschke product on Ω approximating g to within ε/2, and r is chosen so that g approximates f to within ε/2, then B approximates f to within ε, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The analyticity of f on a neighborhood of Ω is only used to deduce that f has a finite number of zeros inside Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' The same proof would work if we assumed that f is holomorphic on Ω, extends continuously to ∂Ω, and is non-zero on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If we make the previous assumption on f, then it suffices to assume Ω is bounded and finitely connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If so, and any component of ∂Ω is a single point p, then by the Riemann removable singularity theorem, f extends to be holomorphic at p, and it suffices the prove the theorem for the extended function on the domain Ω′ = Ω ∪ {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By removing all the point components of ∂Ω, we may assume every component of Ω is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' By repeated applications of the Riemann mapping theorem to the complement of each complementary component of Ω, the domain Ω can be mapped to a domain Ω′ bounded by a finite number of analytic curves indeed, with more work, the Koebe circle domain theorem says it can be mapped to a domain bounded by circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Transferring f and K to Ω we can use Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 to construct an approximating finite Blaschke product on Ω′, and then transfer this to a finite Blaschke on Ω that approximates f on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' We placed the poles of our Green’s function all at the same distance δ from ∂Ω, but this was not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' If we fix z0 ∈ Ω and let w ∈ Ω approach x ∈ ∂Ω, then G(z, w)/G(z0) approaches a positive harmonic function on Ω with zero boundary values on ∂Ω, except at x, where it blows up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus the limiting function must be a multiple of the Poisson kernel on Ω with respect to x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Thus it is easy to find finite weighted sums of Green’s functions (with positive real weights) that approximate the Poisson integral of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' 44 CHRISTOPHER J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' BISHOP AND KIRILL LAZEBNIK Then one must cluster the poles to approximate this sum by a sum of Green’s functions with integral weights;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' exponentiating such a sum gives a finite Blaschke product on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Possibly this extra flexibility would be useful in other problems, such trying to minimize the number of poles needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6: The hypotheses of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 are stronger than those of Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2: namely we assume in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='6 that ||f||Ω < 1 (rather than ≤ 1) and that the zeros of f are disjoint from ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Under these additional assumptions, there is no need in the second paragraph of the proof of Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='2 to replace f by g := (1 − a) · f + b since we already have |f| < 1 − ε and f has no zeros on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content=' Since the B produced in Theorem 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Bishop, Mathematics Department, Stony Brook University, Stony Brook, NY 11794- 3651 Email address: bishop@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='edu Kirill Lazebnik, Mathematics Department, University of North Texas, Denton, TX, 76205 Email address: Kirill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='Lazebnik@unt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E1T4oBgHgl3EQfEQOb/content/2301.02888v1.pdf'} diff --git a/D9E2T4oBgHgl3EQfSgcI/content/2301.03792v1.pdf 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+interactions. One of the main ingredients in the proof is a rigorous implementation of the +fermionic cluster expansion of Gaudin, Gillespie and Ripka (Nucl. Phys. A, 176.2 (1971), +pp. 237–260). +Contents +1 +Introduction and main results +2 +1.1 +Precise statement of results +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2 +Preliminary computations +6 +2.1 +The scattering function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +2.2 +The “Fermi polyhedron” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +2.3 +Reduced densities of the Slater determinant +. . . . . . . . . . . . . . . . . . . . +15 +3 +Gaudin-Gillespie-Ripka-expansion +17 +3.0.1 +Calculation of the normalization constant . . . . . . . . . . . . . . . . . . +17 +3.0.2 +Calculation of the 1-particle reduced density . . . . . . . . . . . . . . . . +19 +3.0.3 +Calculation of the 2-particle reduced density . . . . . . . . . . . . . . . . +20 +3.0.4 +Calculation of the 3-particle reduced density . . . . . . . . . . . . . . . . +21 +3.0.5 +Summarising the results +. . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +3.1 +Absolute convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +3.1.1 +Absolute convergence of the Γ-sum . . . . . . . . . . . . . . . . . . . . . +23 +3.1.2 +Absolute convergence of the Γ1-sum . . . . . . . . . . . . . . . . . . . . . +27 +3.1.3 +Absolute convergence of the Γ2-sum . . . . . . . . . . . . . . . . . . . . . +28 +3.1.4 +Absolute convergence of the Γ3-sum . . . . . . . . . . . . . . . . . . . . . +30 +∗alaurits@ist.ac.at +†robert.seiringer@ist.ac.at +1 + +4 +Energy of the trial state +31 +4.1 +Thermodynamic limit via a box method +. . . . . . . . . . . . . . . . . . . . . . +34 +4.2 +Subleading 2-particle diagrams (proof of Lemma 4.1) +. . . . . . . . . . . . . . . +35 +4.3 +Subleading 3-particle diagrams (proof of Lemma 4.2) +. . . . . . . . . . . . . . . +42 +5 +One and two dimensions +43 +5.1 +Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +43 +5.2 +One dimension +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +47 +A Small diagrams +54 +A.1 Small 2-particle diagrams (proof of Lemmas 4.6 and 4.8) . . . . . . . . . . . . . +54 +A.2 Small 3-particle diagrams (proof of Lemma 4.11) . . . . . . . . . . . . . . . . . . +57 +A.3 Small diagrams in 1 dimension (proof of Lemma 5.21) . . . . . . . . . . . . . . . +58 +B Derivative Lebesgue constants (proof of Lemma 4.9) +61 +B.1 Reduction to simpler tetrahedron . . . . . . . . . . . . . . . . . . . . . . . . . . +62 +B.2 Reduction from d = 3 to d = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . +63 +B.3 Reduction from d = 2 to d = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . +69 +B.4 Bounding the one-dimensional integrals . . . . . . . . . . . . . . . . . . . . . . . +70 +B.5 Bounding the j = 3 two-dimensional integral . . . . . . . . . . . . . . . . . . . . +72 +1 +Introduction and main results +We consider a Fermi gas of N particles in a box Λ = ΛL = [−L/2, L/2]d in d dimensions, +d = 1, 2, 3. We will mostly focus on the case d = 3. The particles interact via a two-body +interaction v, which we assume to be positive, radial and of compact support. In particular we +allow for v to have a hard core, i.e. v(x) = ∞ for |x| ≤ r for some r > 0. In natural units +where ℏ = 1 and the mass of the particles is m = 1/2 the Hamiltonian of the system takes the +form +HN = +N +� +i=1 +−∆xi + +� +j 0 (i.e., the error-term in the +theorem, O((a3ρ)2/3+1/21| log(a3ρ)|6) could be replaced by Oε((a3ρ)1−ε)). This is similar to the +recent work on the Bose gas [BCGOPS22]. We shall discuss this further in Remark 4.10. +We consider the lower-dimensional problems next. +We start with 2 dimensions, where the +scattering length is defined as follows. +Definition 1.6. The (2-dimensional) p-wave scattering length a of the interaction v is defined +by the minimization problem +4πa2 = inf +�ˆ +R2 |x|2 +� +|∇f0(x)|2 + 1 +2v(x)|f0(x)|2 +� +dx : f0(x) → 1 as |x| → ∞ +� +The minimizer f0 is the (2-dimensional) (p-wave) scattering function. +With this, we may state the 2-dimensional analogue of Theorem 1.3. +Theorem 1.7 (Two dimensions). Suppose that v ≥ 0 is radial and compactly supported. Then, +for sufficiently small a2ρ, the ground-state energy density satisfies +ed=2(ρ) ≤ π +8 ρ2 + π2 +4 a2ρ3 +� +1 + O +� +a2ρ| log(a2ρ)|2�� +. +We sketch in Section 5.1 how to adapt the proof in the 3-dimensional setting to 2 dimensions. +Finally, we consider the 1-dimensional problem. The scattering length is defined as follows. +Definition 1.8. The (1-dimensional) p-wave scattering length a of the interaction v is defined +by the minimization problem +2a = inf +�ˆ +R +|x|2 +� +|∂f0(x)|2 + 1 +2v(x)|f0(x)|2 +� +dx : f0(x) → 1 as |x| → ∞ +� +The minimizer f0 is the (1-dimensional) (p-wave) scattering function. +5 + +We show in Proposition 5.12 that Definition 1.8 agrees with the (seemingly different) definition +of the scattering length in [ARS22]. With this, we may state the 1-dimensional analogue of +Theorem 1.3. +Theorem 1.9 (One dimension). Suppose that v ≥ 0 is even and compactly supported. Suppose +moreover that +´ � 1 +2vf 2 +0 + |∂f0|2� +dx < ∞, where f0 denotes the (p-wave) scattering function. +Then, for sufficiently small aρ, the ground-state energy density satisfies +ed=1(ρ) ≤ π2 +3 ρ3 + 2π2 +3 aρ4 +� +1 + O +� +(aρ)13/17�� +. +We remark that Agerskov, Reuvers and Solovej [ARS22] recently showed (almost) the same +result with a matching lower bound ed=1(ρ) ≥ +π2 +3 ρ3 + 2π2 +3 aρ4(1 + o(1)). Compared to their +result we treat a slightly different class of potentials and obtain an improved error bound. The +conjectured next contribution is of order a2ρ5, see [ARS22]. +Remark 1.10 (On the assumptions on v). Any smooth interaction or an interaction with a hard +core (meaning that v(x) = +∞ for |x| ≤ a0 for some a0 > 0) satisfies +´ �1 +2vf 2 +0 + |∂f0|2� +dx < ∞, +see Propositions 5.13 and 5.14. +We sketch in Section 5.2 how to adapt the proof in the 3-dimensional setting to 1 dimension. +This turns out to be more involved than adapting the argument to 2 dimensions. +The paper is structured as follows. In Section 2 we give some preliminary computations +and in particular we introduce the “Fermi polyhedron”, a polyhedral approximation to the +Fermi ball. In Section 3 we introduce the fermionic cluster expansion of Gaudin, Gillespie +and Ripka [GGR71] and we find conditions on absolute convergence of the resulting formulas. +In the subsequent Section 4 we compute the energy of a Jastrow-type trial state and glue +many of them together using a box method to form trial states of arbitrary many particles. +Finally, in Section 5 we sketch how to adapt the argument to the lower-dimensional settings. In +Appendix A we give computations of “small diagrams” needed for some bounds in Sections 4 +and 5.2 and in Appendix B we give the proof of Lemma 4.9, an important lemma used in +Section 4. +2 +Preliminary computations +We will construct a trial state using a box method, and bound the energy of such trial state. +To use such a box method we need to use Dirichlet boundary conditions in each smaller box. In +Lemma 4.3 we show that we may construct trial states with Dirichlet boundary condition out of +trial states with periodic boundary conditions. We will thus use periodic boundary conditions +in the box Λ = [−L/2, L/2]3. For periodic boundary conditions, the Hamiltonian is given by +HN = Hper +N,L = +N +� +j=1 +−∆j + +� +i R0, the range of v, is some cut-off to be chosen later, PF is a polyhedral +approximation to the Fermi ball BF of radius kF described in Section 2.2, and the number of +particles is N = #PF, the number of points in PF. We choose b to be larger than the range of +v; in particular, then f is continuous. (Note that the metric on the torus is d(x, y) = |x − y|. +We will abuse notation slightly and denote by |·| also the absolute value of some number or the +norm on R3.) +Before going further with the proof we first fix some notation. +Notation 2.1. We introduce the following. +• For any function h and edge (of some graph) e = (i, j) we will write he = hij = h(xi − xj). +• We denote by C a generic positive constant whose value may change line by line. +• For expressions A, B we write A ≲ B if there exists some constant C > 0 such that A ≤ CB. +If both A ≲ B and B ≲ A we write A ∼ B. +• For a vector x = (x1, . . . , xd) ∈ Rd we write x1, . . . , xd for its components. +We will fix the Fermi momentum kF and then choose L, N large but finite depending on kF. +The density of particles in the trial state ψN is ρ := N/L3. The limit of small density a3ρ → 0 +will be realized as kFa → 0. +To compute the energy of the trial state ψN note that for (real-valued) functions F, G we have +ˆ +|∇(FG)|2 = +ˆ +|∇F|2|G|2 − +ˆ +|F|2G∆G. +Using this on F = � +i a. +We give a short proof here for completeness. +Proof. From the radial Euler-Lagrange equation (2.4) we have ∂r(r4∂rf0) = vr4f0/2 ≥ 0. +Denote by fhc = +� +1 − a3 +|x|3 +� ++ the solution for a hard core potential of range a. Then +r4∂rfhc = +� +3a3 +r > a +0 +r < a +In particular ∂r(r4∂rfhc) = 0 for r > a. We thus see that ∂rf0 ≤ ∂rfhc = 3a3r−4 and f0 ≥ fhc +for r > a by integrating. Trivially f0 ≥ 0 = fhc for r ≤ a. +Remark 2.3. A hard core interaction of range R0 > 0, +vhc(x) = +� ++∞ +|x| ≤ R0, +0 +|x| > R0, +has f0(x) = fhc(x) = +� +1 − a3 +|x|3 +� ++ and thus a0 = a = R0. +2.2 +The “Fermi polyhedron” +We now introduce a polyhedral approximation to the Fermi ball BF = {k ∈ 2π +L Z3 : |k| ≤ kF}. +We discuss why we need this in Remark 3.5. The problem is that +ˆ +[0,L]3 +1 +L3 +������ +� +k∈B(kF )∩ 2π +L Z3 +eikx +������ +dx = +1 +(2π)3 +ˆ +[0,2π]3 +������ +� +q∈B(cN1/3)∩Z3 +eiqu +������ +du ∼ N1/3 +for large N (see [GL19; Lif06] and references therein) is too big for our purposes. +Note +that this behaviour is a consequence of taking the absolute value. +In fact we have that +1 +L3 +´ � +k∈B(kF )∩ 2π +L Z3 eikx dx = 1. +8 + +This type of quantity is referred to as the Lebesgue constant [GL19; Lif06] of some domain +Ω, +L(Ω) := +1 +(2π)3 +ˆ +[0,2π]3 +������ +� +q∈Ω∩Z3 +eiqu +������ +du +These kinds of integrals appear in estimates in Sections 3.1 and 4. For an overview of such +Lebesgue constants, see [GL19; Lif06]. +Of particular relevance for us is the fact that the +Lebesgue constants are much smaller for polyhedral domains than for balls. Hence we introduce +the polyhedron P = P(N) as an approximation of the unit ball. +Then the scaled version +PF = kFP ∩ 2π +L Z3 approximates the Fermi ball. We will refer to PF as the Fermi polyhedron. +In [KL18, Theorem 4.1] it is shown that for any fixed convex polyhedron P ′ of s vertices +L(RP ′) = +1 +(2π)3 +ˆ +[0,2π]3 +������ +� +q∈RP ′∩Z3 +eiqu +������ +du ≤ Cs(log R)3 + C(s)(log R)2 +(2.5) +for any R > 2, in particular for R ∼ N1/3, where C(s) is some unknown function of s. We +will improve on this bound for the specific polyhedron P = P(N) to control the s-dependence +of the subleading (in R) terms, i.e. of C(s). We first give an almost correct definition of the +polyhedron P. +“Definition” 2.4 (Simple definition). The polyhedron P is chosen to be the convex hull of +s = s(N) points κ1, . . . , κs on a sphere of radius 1+δ, where δ is chosen such that Vol(P) = 4π/3. +We moreover choose the set of points to have the following properties. +• The points are evenly distributed, meaning that the distance d between any pair of points +satisfies d ≳ s−1/2, and that for any k on the sphere of radius 1 + δ the distance from k to +the closest point is ≲ s−1/2. That is, for some constants c, C > 0 we have d ≥ cs−1/2 and +infj |k − κj| ≤ Cs−1/2. +• P is invariant under any map (k1, k2, k3) �→ (±ka, ±kb, ±kc) for {a, b, c} = {1, 2, 3}, i.e. +reflection in or permutation of any of the axes. +The Fermi polyhedron is the rescaled version defined as PF := kFP ∩ 2π +L Z3, where L is chosen +large (depending on kF) such that kFL is large. +Remark 2.5. Note that the symmetry constraint adds a restriction on s. For instance, a +generic point away from any plane of symmetry (i.e. +k1, k2, k3 all different and non-zero) +has 48 images (including itself) when reflected by the maps (k1, k2, k3) �→ (±ka, ±kb, ±kc) for +{a, b, c} = {1, 2, 3}. +For s points on a sphere of radius 1 + δ, the natural lengthscale is (1 + δ)s−1/2 ∼ s−1/2. The +requirement that the points are evenly distributed then ensures that all pairs of close points +(for any reasonable definition of “close points”) have a pairwise distance of this order. +Remark 2.6. For all purposes apart from the technical argument in Appendix B one may +take this as the definition. In particular, the convergence criterion of the cluster expansion +formulas of Gaudin, Gillespie and Ripka [GGR71], given in Theorem 3.4, holds also for this +simpler definition of P. We provide this simpler definition to better give an intuition of the +construction. +We now give the actual definition of P. We first give the construction. Then in Remark 2.9 we +give a few comments and in Remark 2.10 we give a short motivation. +9 + +Definition 2.7 (Actual definition). The polyhedron P with s corners and the “centre” z is +constructed as follows. +• First, choose a big number Q, the “size of the primes” satisfying +Q−1/4 ≤ Cs−1, +N4/3 ≪ Q ≤ CNC +in the limit N → ∞. +• Pick three large distinct primes Q1, Q2, Q3 with Qj ∼ Q. +• Place s evenly distributed points κR +1 , . . . , κR +s on the sphere of radius Q−3/4 and such that the +set of points {κR +1 , . . . , κR +s } is invariant under the symmetries (k1, k2, k3) �→ (±ka, ±kb, ±kc) +for {a, b, c} = {1, 2, 3}. +Here, evenly distributed means that the distance between any pair of points is d ≳ s−1/2Q−3/4 +and that for any k on the sphere of radius Q−3/4 the distance from k to the nearest point is +≲ s−1/2Q−3/4. That is, d ≥ cs−1/2Q−3/4 and infj +��k − κR +j +�� ≤ Cs−1/2Q−3/4 for some constants +c, C > 0. +• Find points κ1, . . . , κs of the form +κj = +� p1 +j +Q1 +, p2 +j +Q2 +, p3 +j +Q3 +� +, +pµ +j ∈ Z, +µ = 1, 2, 3, +j = 1, . . . , s, +(2.6) +such that +��κj − κR +j +�� ≲ Q−1 for all j = 1, . . . , s and such that the set of points {κ1, . . . , κs} is +invariant under the symmetries (k1, k2, k3) �→ (±k1, ±k2, ±k3). +• Define ˜P as the convex hull of all the points κ1, . . . , κs. That is, ˜P = conv{κ1, . . . , κs}. +• Define P as σ ˜P, where σ is chosen such that Vol(P) = 4π/3. We will refer to the scaled +points σκj = σ(p1 +j/Q1, p2 +j/Q2, p3 +j/Q3) for j = 1, . . . , s as corners of P. +• Define P R = σ conv{κR +1 , . . . , κR +s } as the scaled convex hull of all the initial points κR +1 , . . . , κR +s . +• Define the centre as z = σ(1/Q1, 1/Q2, 1/Q3). +The Fermi polyhedron is the rescaled version defined as PF := kFP ∩ 2π +L Z3, where L is chosen +large (depending on kF) such that kF L +2π is rational and large. +We additionally define P R +F := kFP R ∩ 2π +L Z3. +Remark 2.8. We choose N := #PF, so that the Fermi polyhedron is filled. The dependence +in N of, for instance, Q should therefore more precisely be given in terms of a dependence on +kFL. Note that N = ρL3 ∼ (kFL)3 and kF = (6π2ρ)1/3(1 + O(N−1/3)). +We will choose also s depending on N (i.e. on kFL) satisfying s → ∞ as N → ∞. +Remark 2.9 (Comments on and properties of the construction). We collect here some prop- +erties of the Fermi polyhedron, some of which will only be needed in Appendix B. +• The points κ1, . . . , κs are evenly distributed on a thickened sphere of radius Q−3/4 – their +radial coordinates are |κj| = Q−3/4 + O(Q−1). +Indeed, the points κR +1 , . . . , κR +s are evenly +distributed and Q−1 ≪ s−1/2Q−3/4. For s points on a thickened sphere of radius Q−3/4, the +natural lengthscale between points is s−1/2Q−3/4. +10 + +• There is some constraint on the number of points s. A generic point κ ∈ S (with κ1, κ2, κ3 +all different and non-zero) has 48 images, including itself. The constraint on s is more or less +the same as for the simpler ‘‘Definition” 2.4. +• By choosing the points κ1, . . . , κs as in Equation (2.6) we break the symmetries of permuting +the coordinates, i.e. (k1, k2, k3) �→ (ka, kb, kc) if (a, b, c) ̸= (1, 2, 3). These symmetries are +however still almost satisfied, see Lemma 2.11. +• We choose s, Q such that Q−1/4 ≪ s−1/2 in the limit of large N. Hence, for N sufficiently +large, all the chosen points {κ1, . . . , κs} are extreme points of ˜P, i.e. all corners are extreme +points of the polyhedron P. That is, the name “corner” is well-chosen, and we do not have +any superfluous points in the construction. +• For any three points (xi, yi, zi) ∈ R3, i = 1, 2, 3 the plane through them is given by the +equation + + +(y2 − y1)(z3 − z1) − (y3 − y1)(z2 − z1) +(z2 − z1)(x3 − x1) − (z3 − z1)(x2 − x1) +(x2 − x1)(y3 − y1) − (x3 − x1)(y2 − y1) + + · + + +x +y +z + + = const. +Hence, for three points K1, K2, K3 of the form Ki = (p1 +i /Q1, p2 +i /Q2, p3 +i /Q3), pµ +i ∈ Z, i, µ = +1, 2, 3 the plane through them is given by +α1 +Q2Q3 +k1 + +α2 +Q1Q3 +k2 + +α3 +Q1Q2 +k3 = γ ∈ Q, +(2.7) +where +α1 = (p2 +2 − p2 +1)(p3 +3 − p3 +1) − (p2 +3 − p2 +1)(p3 +2 − p3 +1) ∈ Z +and similarly for α2, α3. From these formulas it is immediate that |αj| ≤ C√Q for j = 1, 2, 3. +For some planes we may have αj = 0 for some j. +• We claim that σ = Q3/4(1 + O(s−1)). In particular, that any point on the boundary ∂P has +radial coordinate 1 + O(s−1). To see this, note that Q3/4 ˜P is a polyhedron whose corners are +evenly spaced and have radial coordinates r with r = 1 + O(Q−1/4). Thus, by scaling Q3/4 ˜P +by 1 − CQ−1/4 we get that (1 − CQ−1/4)Q3/4 ˜P ⊂ B1(0) so that this has volume ≤ 4π +3 . It +follows that σ ≥ Q3/4(1 − CQ−1/4). On the other hand, scaling Q3/4 ˜P by 1 + Cs−1 we have +that (1 + Cs−1)Q3/4 ˜P ⊃ B1(0). Indeed, since the distance from any point k on the sphere of +radius 1 to any corner of Q3/4 ˜P is ≲ s−1/2, and the sphere is locally quadratic, the smallest +radial coordinate r of a point on the boundary ∂(Q3/4 ˜P) is r ≥ 1 − Cs−1. It follows that +σ ≤ (1 + Cs−1)Q3/4. Since Q−1/4 ≤ Cs−1 this shows the desired. +• Note moreover that σ is irrational. Indeed, the volume of a polyhedron with rational corners +is rational. (This is easily seen for tetrahedra, of which any polyhedron is an essentially +disjoint union.) Thus σ3 = πr for a rational r. Hence the equations of the planes defined by +corners of P (i.e. scaled points) are of the form Equation (2.7) with an irrational constant σγ +on the right-hand side. Indeed, the corners of P (and the central point z) are all scaled by +σ compared to points of the form (p1/Q1, p2/Q2, p3/Q3). The equation of the plane through +three scaled points only differ by scaling the constant term. Since σ is irrational, and the +constant term was rational for the unscaled points, this shows the desired. +• We now construct a triangulation of ∂P. For all (2-dimensional) triangular faces of P simply +consider these as part of the triangulation. That is, we construct edges between any pair of +11 + +the three corners of such a triangle. Some of the (2-dimensional) faces of P may be polygons +of more than 3 sides (1-dimensional faces). Construct edges between all pairs of corners +sharing a side (i.e. a 1-dimensional face) and choose one corner and construct edges from +this corner to all other corners of the polygon. +Doing this constructs a triangulation of ∂P and we will refer to all pairs of corners with +an edge between them as close or neighbours. Since the points {κ1, . . . , κs} are evenly dis- +tributed, that the distance between any pair of close corners is d ∼ s−1/2. +• Additionally, one may note that the corners of P have ≤ C many neighbours since the points +are evenly distributed. +• The reason we need +LkF +2π +rational will only become apparent in Appendix B and will be +explained there. +Remark 2.10 (Motivation of construction). The purpose of the construction is twofold. Firstly +we avoid a casework argument as in the proof of [KL18, Lemma 3.5] of whether the coefficients +of the planes are rational or not. The argument in Lemmas B.6 and B.7 is heavily inspired by +[KL18, Lemmas 3.6, 3.9], where such casework is required. Secondly we have good control over +how many (and which) lattice points (i.e. points in 2π +L Z3) can lie on each plane (or, rather, a +closely related plane, see Appendix B for the details). +These are technical details only needed in Appendix B. We reiterate, that apart from the +arguments in Appendix B, the reader may have the simpler ‘‘Definition” 2.4 in mind instead. +As mentioned in Remark 2.9 the Fermi polyhedron is almost symmetric under permutation of +the axes. This is formalized as follows. +Lemma 2.11. For µ ̸= ν let Fµν be the map that permutes kµ and kν (i.e. F12(k1, k2, k3) = +(k2, k1, k3), etc.). Then for any function t ≥ 0 we have +� +k∈ 2π +L Z3 +��χ(k∈PF ) − χ(k∈Fµν(PF )) +�� t(k) ≲ Q−1/4N sup +|k|∼kF +t(k) ≲ N2/3 sup +|k|∼kF +t(k), +where Q is as in Definition 2.7 and χ denotes the indicator function. +Proof. Note that +� +k∈ 2π +L Z3 +��χ(k∈PF ) − χ(k∈Fµν(PF )) +�� t(k) +≤ +� +k∈ 2π +L Z3 +���χ(k∈PF ) − χ(k∈P R +F ) +��� t(k) + +� +k∈ 2π +L Z3 +���χ(k∈Fµν(PF )) − χ(k∈Fµν(P R +F )) +��� t(k) +(2.8) +since P R +F is invariant under permutation of the axes, i.e. Fµν(P R +F ) = P R +F . The points {κj}j=1,...,s +only differ from {κR +j }j=1,...,s by at most ∼ Q−1 thus the points {σκj}j=1,...,s (the corners of P) +only differ from the points {σκR +j }j=1,...,s by ∼ Q−1/4. Hence, the support of χ(k∈PF ) − χ(k∈P R +F ) is +contained in a shell of width ∼ kFQ−1/4 around the surface ∂(kFP). That is, +supp +� +χ(k∈PF ) − χ(k∈P R +F ) +� +⊂ +� +k ∈ 2π +L Z3 : dist(k, ∂(kFP)) ≲ kFQ−1/4 +� +. +The surface ∂(kF P) has area ∼ k2 +F so +Vol +�� +k ∈ R3 : dist(k, ∂(kFP)) ≲ kFQ−1/4�� +≲ k3 +FQ−1/4. +12 + +The spacing between the k’s in 2π +L Z3 is ∼ L−1 and any k with dist(k, ∂(kFP)) ≲ kFQ−1/4 has +|k| ∼ kF. Thus +� +k∈PF +���χ(k∈PF ) − χ(k∈P R +F ) +��� t(k) ≲ L3k3 +FQ−1/4 sup +|k|∼kF +t(k) ∼ Q−1/4N sup +|k|∼kF +t(k). +The same argument applies to the second summand in Equation (2.8). We conclude the desired. +We now improve on Equation (2.5) for our polyhedron. +Lemma 2.12. The Lebesgue constant of the Fermi polyhedron satisfies +ˆ +Λ +1 +L3 +����� +� +k∈PF +eikx +����� dx = +1 +(2π)3 +ˆ +[0,2π]3 +������� +� +q∈ +� LkF +2π P +� +∩Z3 +eiqu +������� +du ≤ Cs(log N)3. +The proof is (almost) the same as given in [KL18, Theorem 4.1]. We need to be a bit more +careful in the decomposition into tetrahedra. +Proof. Define R = LkF +2π . We decompose RP into tetrahedra using the “central” point z from +the construction of P. We triangulate the surface of RP as in Remark 2.9. For each triangle +in the triangulation add the point Rz to form a tetrahedron. Note that Rz /∈ Z3 since |Rz| ≤ +CRQ−1/4 ≪ 1 and z ̸= 0. This gives m = O(s) many (closed) tetrahedra {Tj} such that +RP = � Tj and that Tj ∩ Tj′ is a tetrahedron of lower dimension (i.e. the central point Rz, +a line segment or a triangle). Then, as in [KL18, Theorem 4.1] by the inclusion–exclusion +principle we have +L(RP) = +1 +(2π)3 +ˆ ������ +� +j +� +q∈Tj∩Z3 +eiqu − +� +j N we have ∆p = 0 for p > N. +17 + +Thus we may extend the summation to ∞. We now expand out the determinant and the Wp. +That is +CN +N! = 1 + +∞ +� +p=2 +1 +p! +� +G∈Gp +π∈Sp +(−1)π +˙ � +e∈G +ge +p +� +j=1 +γ(1) +N (xj, xπ(j)) dx1 . . . dxp, +where Sp denotes the symmetric group on p elements. We will consider π and G together as +a diagram (π, G). We give a slightly more general definition for what a diagram is, as we will +need such for the calculation of the reduced densities. +Definition 3.1. Define the set Gq +p as the set of all graphs on q “external” vertices {1, . . . , q} +and p “internal” vertices {q + 1, . . . , q + p} such that all internal vertices have degree at least +1, i.e. each internal vertex has at least one incident edge. The external edges are allowed to +have degree zero, i.e. have no incident edges. For q = 0 we recover G0 +p = Gp. +A diagram (π, G) on q “external” and p “internal” vertices is a pair of a permutation +π ∈ Sq+p (viewed as a directed graph on {1, . . . , q + p}) and a graph G ∈ Gq +p. We denote the +set of all diagrams on q “external” vertices and p “internal” vertices by Dq +p. +We will sometimes refer to edges in G as g-edges, directed edges in π as γ(1) +N -edges and the +graph G as a g-graph. The value of a diagram (π, G) ∈ Dq +p is the function +Γq +π,G(x1, . . . , xq) := (−1)π +˙ � +e∈G +ge +q+p +� +j=1 +γ(1) +N (xj, xπ(j)) dxq+1 . . . dxq+p. +For q = 0 we write Γπ,G = Γ0 +π,G and Dp = D0 +p. +A diagram (π, G) is said to be linked if the graph ˜G with edges the union of edges in G +and directed edges in π is connected. The set of all linked diagrams on q “external” and p +“internal” vertices is denoted Lq +p. For q = 0 we write Lp = L0 +p. +By the translation invariance we have that Γ1 +π,G is a constant for any diagram (π, G). +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +(π1, G1) +(π2, G2) +(π3, G3) +Figure 3.1: A diagram (π, G) decomposed into linked components. The dashed lines +denote g-edges and the arrows (i, j) denote that π(i) = j. +In terms of diagrams we thus have +CN +N! = 1 + +∞ +� +p=2 +1 +p! +� +(π,G)∈Dp +Γπ,G. +If (π, G) is not linked we may decompose it into its linked components. Here the integration +factorizes. +We split the sum according to the number of linked components. +Each linked +18 + +component has at least 2 vertices, since each vertex must be connected to another vertex with +an edge in the corresponding graph. We get +CN +N! = 1 + +∞ +� +p=2 +∞ +� +k=1 +���� +# lnk. cps. +1 +k! +� +p1≥2 +· · · +� +pk≥2 +� +�� +� +sizes linked cps. +χ(� pℓ=p) +� +(π1,G1)∈Lp1 +· · · +� +(πk,Gk)∈Lpk +� +�� +� +linked components +Γπ1,G1 +p1! +· · · Γπk,Gk +pk! +(3.1) +Here χ is the indicator function. The factor 1/(k!) comes from counting the possible labellings of +the k linked components. The factors 1/(p1!), . . . , 1/(pk!) come from counting how to distribute +the p vertices {1, . . . , p} between the linked components of prescribed sizes p1, . . . , pk. This +gives the factor +� +p +p1,...,pk +� += +p! +p1!...pk!, which together with the factor 1/p! already present gives the +claimed formula. +We want to pull the p-summation inside the p1, . . . , pk-summation. This is allowed once we +check that � +p +1 +p! +� +(π,G)∈Lp Γπ,G is absolutely convergent. More precisely we need that the p-sum +is absolutely convergent, i.e. � +p +1 +p! +��� +� +(π,G)∈Lp Γπ,G +��� < ∞. This is the content of Lemma 3.2 +below. We conclude that if the assumptions of Lemma 3.2 are satisfied then +CN +N! = 1 + +∞ +� +k=1 +1 +k! +� +p1≥2 +· · · +� +pk≥2 +� +(π1,G1)∈Lp1 +· · · +� +(πk,Gk)∈Lpk +Γπ1,G1 +p1! +· · · Γπk,Gk +pk! += 1 + +∞ +� +k=1 +1 +k! + + +∞ +� +p=2 +1 +p! +� +(π,G)∈Lp +Γπ,G + + +k += exp + + +∞ +� +p=2 +1 +p! +� +(π,G)∈Lp +Γπ,G + + . +(3.2) +3.0.2 +Calculation of the 1-particle reduced density +We consider now the 1-particle reduced density of the Jastrow trial state. We have by the +translation invariance that ρ(1) +Jas = ρ(1) = ρ. We nonetheless compute it here, as we need the +formula in terms of (linked) diagrams. We have similarly as before +ρ(1) +Jas(x1) = N +CN +˙ +� +2≤i≤N +f 2 +1i +� +2≤i N. +We again expand out the determinant and the X1 +p’s. For each summand π ∈ Sp+1 and +G ∈ G1 +p we again think of them together as a diagram (π, G) ∈ D1 +p. The formula for ρ(1) +Jas in +terms of diagrams is +ρ(1) +Jas = N! +CN +∞ +� +p=0 +1 +p! +� +(π,G)∈D1p +Γ1 +π,G = N! +CN + +ρ(1) + +∞ +� +p=1 +1 +p! +� +(π,G)∈D1p +Γ1 +π,G + + . +19 + +As for the normalization we write out the diagrams in terms of their linked components. There +is a distinguished linked component, namely the one containing the vertex {1}. We will write +its size as p∗. It is convenient to take “size” to mean number of internal vertices, i.e. p∗ = 0 +if {1} is not connected to any other vertex by either an edge in G or an edge in π. Similarly +“number of linked components” means disregarding the distinguished one. +Analogously to the computation in Equation (3.1) we thus get for any p ≥ 0 +1 +p! +� +(π,G)∈D1p +Γ1 +π,G = +� +(π∗,G∗)∈L1p +Γ1 +π∗,G∗ +p! ++ + + +∞ +� +k=1 +1 +k! +� +p∗≥0 +� +p1≥2 +· · · +� +pk≥2 +χ(� +ℓ∈{∗,1,...,k} pℓ=p) +� +(π∗,G∗)∈L1p∗ +Γ1 +π∗,G∗ +p∗! +× +� +(π1,G1)∈Lp1 +· · · +� +(πk,Gk)∈Lpk +Γπ1,G1 +p1! +· · · Γπk,Gk +pk! + + , +where the superscript 1 refers to the slightly modified structure as described in Definition 3.1, +where there may be no g-edges connecting to {1}, and there is no integration over x1. Note +that (π1, G1) ∈ Lp1, . . . , (πk, Gk) ∈ Lpk only deal with internal vertices. +Again here we take the sum over p’s. We are allowed to permute the p-sum inside of the +p∗- and p1, . . . , pk-sums if the sums over linked diagrams are absolutely summable. That is, if +� +p≥0 +1 +p! +������ +� +(π,G)∈L1p +Γ1 +π,G +������ +< ∞, +� +p≥2 +1 +p! +������ +� +(π,G)∈Lp +Γπ,G +������ +< ∞, +then we have, as for the normalization in Equation (3.2), that +ρ(1) +Jas = N! +CN +∞ +� +p=0 +1 +p! +� +(π,G)∈D1p +Γ1 +π,G += N! +CN + +� +p∗≥0 +� +(π∗,G∗)∈Lp∗ +Γ1 +π∗,G∗ +p∗! + + × + + +∞ +� +k=0 +1 +k! + +� +p≥2 +� +(π,G)∈Lp +Γπ,G +p! + + +k + += +� +p≥0 +� +(π,G)∈L1p +Γ1 +π,G +p! += ρ(1) + +� +p≥1 +1 +p! +� +(π,G)∈L1p +Γ1 +π,G, +where we used Equation (3.2) and that the p = 0 term just gives the 1-particle density of the +Slater determinant. Thus, by translation invariance, we have +ρ = ρ(1) = ρ(1) +Jas = +� +p≥0 +1 +p! +� +(π,G)∈L1p +Γ1 +π,G. +3.0.3 +Calculation of the 2-particle reduced density +Let us now compute the 2-particle reduced density. As before by expanding all the f 2 = 1 + g +factors apart from the factor f12 we get +ρ(2) +Jas = N! +CN +f 2 +12 +∞ +� +p=0 +ˆ +X2 +p∆p+2 dx3 . . . dxp+2, +X2 +p = +� +G∈G2p +� +e∈G +ge, +20 + +where G2 +p is as in Definition 3.1. +We again decompose the diagrams into linked components. However, we need to distinguish +between the cases where {1} and {2} are both in the same component or in two different +components. The computation is analogous to the computation above. We get +ρ(2) +Jas(x1, x2) = f 2 +12 + + + + +� +p1,p2≥0 +� +(π1,G1)∈L1 +p1 +(π2,G2)∈L1 +p2 +Γ1 +π1,G1(x1)Γ1 +π2,G2(x2) +p1!p2! +� +�� +� +{1} and {2} in different linked components ++ +� +p12≥0 +� +(π12,G12)∈L2p12 +Γ2 +π12,G12(x1, x2) +p12! +� +�� +� +{1} and {2} in same linked component + + + + += f 2 +12 + +ρ(1) +Jas(x1)ρ(1) +Jas(x2) + +� +p12≥0 +1 +p12! +� +(π12,G12)∈L2p12 +Γ2 +π12,G12(x1, x2) + + +after pulling in the sum over p ≥ 0. The p12 = 0 term together with the term ρ(1) +Jas(x1)ρ(1) +Jas(x2) = +ρ(1)(x1)ρ(1)(x2) = ρ2 gives ρ(2) by Wick’s rule. The condition of absolute convergence is +� +p≥2 +1 +p! +������ +� +(π,G)∈Lp +Γπ,G +������ +< ∞, +� +p≥0 +1 +p! +������ +� +(π,G)∈L1p +Γ1 +π,G +������ +< ∞, +� +p≥0 +1 +p! +������ +� +(π,G)∈L2p +Γ2 +π,G +������ +< ∞. +3.0.4 +Calculation of the 3-particle reduced density +The calculation of the 3-particle reduced density follows along the same arguments as for the +2-particle reduced density. We introduce the relevant diagrams and decompose these according +to their linked components. As for the 2-particle reduced density we distinguish between the +cases according to whether the external vertices {1, 2, 3} are in the same or different linked +components. They are either in 1, 2 or 3 different components. Thus, schematically +ρ(3) +Jas = f 2 +12f 2 +13f 2 +23 +� +� +all in different +Γ1Γ1Γ1 + +� � +2 in one +Γ1(x1)Γ2(x2, x3) + permutations +� ++ +� +all in same +Γ3 +� +. +Any case where one external vertex is in its own linked component, the contribution for such a +linked component is ρ(1) +Jas = ρ(1) = ρ (assuming absolute convergence). Thus, +ρ(3) +Jas(x1, x2, x3) = f 2 +12f 2 +13f 2 +23 +� +ρ3 + ρ +� +p≥0 +1 +p! +� +(π,G)∈L2p +� +Γ2 +π,G(x1, x2) + Γ2 +π,G(x1, x3) + Γ2 +π,G(x2, x3) +� ++ +� +p≥0 +1 +p! +� +(π,G)∈L3p +Γ3 +π,G(x1, x2, x3) +� +. +All the p = 0-terms together give ρ(3) by Wick’s rule. The condition for absolute convergence +is that for any q ≤ 3 we have � +p≥0 +1 +p! +��� +� +(π,G)∈Lq +p Γq +π,G +��� < ∞. +3.0.5 +Summarising the results +For the absolute convergence we have +21 + +Lemma 3.2. There exists a constant c > 0 such that if sa3ρ log(b/a)(log N)3 < c, then +� +p≥0 +1 +p! +������ +� +(π,G)∈Lq +p +Γq +π,G +������ +< ∞ +for any 0 ≤ q ≤ 3. +Remark 3.3. As mentioned in the beginning of the section, the calculation just given is still +valid if we replace f by some general function h ≥ 0 and replace |DN|2 by some more general +determinant det[γ(xi − xj)]1≤i,j≤N, where γ(x − y) is the kernel of some rank N projection (for +instance the one-particle density matrix of a Slater determinant of N particles). The criterion +for absolute convergence reads +sup +x1,...,xn +� +1≤i 0, where the first condition is the “stability condition” of the tree- +graph bound [PU09, Proposition 6.1; Uel18] and ˆγ(k) := +´ +Λ γ(x)e−ikx dx. +We give the proof of Lemma 3.2 in Section 3.1. Thus, we have the following. +Theorem 3.4. There exists a constant c > 0 such that if sa3ρ log(b/a)(log N)3 < c, then +CN +N! = exp + + +∞ +� +p=2 +1 +p! +� +(π,G)∈Lp +Γπ,G + + , +ρ(1) +Jas = ρ(1) + +∞ +� +p=1 +1 +p! +� +(π,G)∈L1p +Γ1 +π,G, +ρ(2) +Jas = f 2 +12 + +ρ(2) + +∞ +� +p=1 +1 +p! +� +(π,G)∈L2p +Γ2 +π,G + + . +ρ(3) +Jas = f 2 +12f 2 +13f 2 +23 +� +ρ(3) + ρ +� +p≥1 +1 +p! +� +(π,G)∈L2p +� +Γ2 +π,G(x1, x2) + Γ2 +π,G(x1, x3) + Γ2 +π,G(x2, x3) +� ++ +� +p≥1 +1 +p! +� +(π,G)∈L3p +Γ3 +π,G +� +. +(3.3) +The first three formulas are the same as those of [GGR71, Equations (3.19), (4.9) and (8.4)]. +Our main contribution is to give a criterion for convergence, and hence for validity of the +formulas. +Remark 3.5. The factor s(log N)3 results from the bound in Lemma 2.12. If we had not +introduced the Fermi polyhedron, and instead used the Fermi ball, we would instead have a +factor N1/3 as mentioned in Section 2.2. That is, the condition for absolute convergence would +be N1/3a3ρ log(b/a) < c for some constant c > 0. +In either case, the N-dependence prevents us from taking a thermodynamic limit directly, +and we instead use a box method of gluing together multiple smaller boxes, where we may put +some finite number of particles in each box, see Section 4.1. For the case of using a Slater +22 + +determinant with momenta in the Fermi ball, there is no way to choose the number of particles +in each smaller box so that both the absolute convergence holds (N1/3a3ρ log(b/a) < c), and +the finite-size error made in the kinetic energy (∼ N2/3ρ2/3, see Lemma 2.13) is smaller than +the claimed energy contribution from the interaction (∼ Na3ρ5/3, see Theorem 1.3). For this +reason we need the polyhedron of Section 2.2. +Remark 3.6. The formulas for ρ(2) +Jas and ρ(3) +Jas only hold for periodic boundary conditions, since +in this case ρ(1) +Jas = ρ(1) = ρ. For different boundary conditions, one has to take into account +that this equality is not valid. In general one has for ρ(2) +Jas that +ρ(2) +Jas = f 2 +12 + +ρ(2) + +∞ +� +p=1 +1 +p! +� +(π,G)∈L2p +Γ2 +π,G + +� +ρ(1) +Jas(x1)ρ(1) +Jas(x2) − ρ(1)(x1)ρ(1)(x2) +� + + += f 2 +12 + +ρ(2) + +∞ +� +p=1 +1 +p! +� +(π,G)∈L2p +Γ2 +π,G + +� +p1,p2≥0 +p1+p2≥1 +1 +p1!p2! +� +(π1,G1)∈L1 +p1 +(π2,G2)∈L1 +p2 +Γ1 +π1,G1(x1)Γ1 +π2,G2(x2) + + . +One of the reasons we work with periodic boundary conditions is that by doing so, we don’t +have the complication of dealing with the additional term. +Remark 3.7. By following the same procedure as in the previous sections, one can equally +well get formulas for the higher order reduced particle densities. Similarly one can extend the +absolute convergence, Lemma 3.2, to any q, only one may have to change the constant c > 0 +to depend on q. +3.1 +Absolute convergence +We now prove Lemma 3.2, i.e. that the appropriate sums are absolutely convergent. +Proof of Lemma 3.2. We consider the four sums � +p≥0 +1 +p! +��� +� +(π,G)∈Lq +p Γq +π,G +���, q = 0, 1, 2, 3 one by +one. +3.1.1 +Absolute convergence of the Γ-sum +Consider first +1 +p! +� +(π,G)∈Lp Γπ,G. Split the sum according to the number of connected compo- +nents of G, labelled as (G1, . . . , Gk) of sizes n1, . . . , nk. We call these clusters. (Note that +“connected” only refers to the graph G, and is independent of the permutation π.) Name the +vertices in G1 as {1, . . . , n1}, in G2 as {n1 +1, . . . , n1 +n2} and so on. Then we have (for p ≥ 2) +1 +p! +� +(π,G)∈Lp +Γπ,G = +∞ +� +k=1 +1 +k! +� +n1,...,nk≥2 +1 +n1! · · ·nk!χ(� nℓ=p) +� +G1,...,Gk +Gℓ∈Cnℓ +� +π∈Sp +(−1)πχ((π,∪Gℓ)∈Lp) +× +˙ +k +� +ℓ=1 +� +e∈Gℓ +ge +p +� +j=1 +γ(1) +N (xj; xπ(j)) dx1 . . . dxp, +where Cn denotes the set of connected graphs on n (labelled) vertices. The factorial factors are +similar to those of Equation (3.1). Indeed, the factor 1/(k!) comes from counting the possible +23 + +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +G1 +G2 +G3 +Figure 3.2: A linked diagram (π, G) decomposed into clusters G1, G2, G3. Dashed +lines denote g-edges, and arrows (i, j) denote that π(i) = j. +labelling of the clusters, and the factors 1/(n1!), . . . , 1/(nk!) come from counting the number of +ways to distribute the p = � nℓ vertices into the clusters and using the factor 1/(p!) already +present. +For the analysis we will need the following. +Definition 3.8. Let A1, . . . , Ak denote disjoint non-empty sets. +The truncated correlation +function is +ρ(A1,...,Ak) +t +:= +� +π∈S∪ℓAℓ +(−1)πχ((π,∪Gℓ) linked) +� +j∈∪ℓAℓ +γ(1) +N (xj; xπ(j)). +(3.4) +for some choice of connected graphs Gℓ ∈ CAℓ. The definition does not depend on the choice of +graphs Gℓ. +If the underlying sets A1, . . . , Ak are clear we will simply denote the truncated correlation +by their sizes, +ρ(|A1|,...,|Ak|) +t += ρ(A1,...,Ak) +t +. +The truncated correlation functions are also sometimes referred to as connected correlation +functions [GMR21, Appendix D]. +Remark 3.9. We write the characteristic function in Equation (3.4) as χ((π,∪Gℓ) linked) for ease +of generalizability to the cases in Sections 3.1.2 and 3.1.3 where we will need the notion of trun- +cated correlations also for some of the vertices being external. For the truncated correlations +it doesn’t matter which (if any) vertices are external, only which vertices are in which clusters. +Since 0 ≤ f ≤ 1 we have −1 ≤ g ≤ 0. Thus, by the tree-graph bound [PU09, Proposition 6.1; +Uel18] we have +����� +� +G∈Cn +� +e∈G +ge +����� ≤ +� +T∈Tn +� +e∈T +|ge|, +where Tn is the set of all trees on n (labelled) vertices. Thus we get +∞ +� +p=2 +1 +p! +������ +� +(π,G)∈Lp +Γπ,G +������ +≤ +∞ +� +k=1 +1 +k! +� +n1,...,nk≥2 +1 +n1! · · ·nk! +˙ +� +T1,...,Tk +Tℓ∈Tnℓ +k +� +ℓ=1 +� +e∈Tℓ +|ge| +���ρ(n1,...,nk) +t +��� dx1 . . . dx� nℓ. +(3.5) +24 + +Here the vertices in Tℓ are the same as in Gℓ, i.e. T1 has vertices {1, . . . , n1}, T2 has vertices +{n1 + 1, . . . , n1 + n2} and so on. +In [GMR21, Equation (D.53)] the following formula, known as the Brydges-Battle-Federbush +(BBF) formula, is shown for the truncated correlation functions +ρ(A1,...,Ak) +t += +� +τ∈A(A1,...,Ak) +� +(i,j)∈τ +γ(1) +N (xi; xj) +ˆ +dµτ(r) det N (r), +(3.6) +where A(A1,...,Ak) is the set of all anchored trees on k clusters with vertices A1, . . . , Ak. (If the sets +A1, . . . , Ak are clear we will write A(|A1|,...,|Ak|) = A(A1,...,Ak) as for the truncated correlations.) +An anchored tree is a directed graph on all the ∪ℓAℓ vertices, such that each vertex has at +most one incoming and at most one outgoing edge (note that these are all γ(1) +N -edges, and that +the g-edges don’t matter for this construction) and such that upon identifying all vertices in +each cluster, the resulting graph is a (directed) tree. The measure µτ is a probability measure +on {(rℓℓ′)1≤ℓ≤ℓ′≤k : 0 ≤ rℓℓ′ ≤ 1} = [0, 1]k(k−1)/2 and depends only on τ but not on the factors +γ(1) +N (xi; xj). Finally, N is an I × J (square) matrix with entries Nij = rc(i)c(j)γ(1) +N (xi; xj), where +c(i) is the (label of the) cluster containing the vertex {i} and rℓℓ′ := rℓ′ℓ if ℓ > ℓ′. Here +I = +� +i ∈ �k +ℓ=1 Aℓ : ∄j : (i, j) ∈ τ +� +, +J = +� +j ∈ �k +ℓ=1 Aℓ : ∄i : (i, j) ∈ τ +� +, +are the set of i’s (respectively j’s) not appearing as i’s (respectively j’s) in the anchored tree +τ. +T1 +T2 +T3 +T4 +T5 +T6 +Figure 3.3: An anchored tree τ (arrows) and trees T1, . . . , T6 (dashed lines). +From [GMR21, Equation (D.57)] it follows that |det N | ≤ ρ +� nℓ−(k−1). To see this, one has +to adapt the argument in [GMR21, Lemma D.2] slightly. We sketch the argument here. +Lemma 3.10 ([GMR21, Lemmas D.2 and D.6]). The matrix N (r) satisfies |det N (r)| ≤ +ρ +� nℓ−(k−1) for all r ∈ [0, 1]k(k−1)/2. +Proof. First we bound ρ(p) = det[γ(1) +N (xi; xj)]1≤i,j≤p following the strategy of [GMR21, Lemma +D.2]. This is done by writing (as in [GMR21, Equations (D.8), (D.9)]) +γ(1) +N (xi; xj) = ⟨αi|βj⟩ℓ2((2π/L)Z3) , +where for k ∈ 2π +L Z3 +αi(k) = L−3/2e−ikxiχ(k∈PF ) +βj(k) = L−3/2e−ikxjχ(k∈PF ) = αj(k). +25 + +By the the Gram-Hadamard inequality [GMR21, Lemma D.1] we have +��ρ(p)�� = +���det[γ(1) +N (xi; xj)]1≤i,j≤p +��� ≤ +p +� +i=1 +∥αi∥ℓ2((2π/L)Z3) ∥βi∥ℓ2((2π/L)Z3) = ρp. +By modifying this argument exactly as described in the proof of [GMR21, Lemma D.6] and +noting that rℓℓ′ ≤ 1 one concludes the desired. +Remark 3.11. We denote the functions as αj and βj (even though they denote the same +function) for ease of modifying the argument later in order to prove Equation (4.16). +In particular one concludes the bound +���ρ(n1,...,nk) +t +��� ≤ ρ +� nℓ−(k−1) +� +τ∈A(n1,...,nk) +� +(i,j)∈τ +���γ(1) +N (xi; xj) +��� . +(3.7) +Plugging this into Equation (3.5) above we get +∞ +� +p=2 +1 +p! +������ +� +(π,G)∈Lp +Γπ,G +������ +≤ +∞ +� +k=1 +1 +k! +� +n1,...,nk≥2 +1 +n1! · · · nk!ρ +� nℓ−(k−1) � +T1,...,Tk +Tℓ∈Tnℓ +� +τ∈A(n1,...,nk) +× +˙ +dx1 . . . dx� nℓ +k +� +ℓ=1 +� +e∈Tℓ +|ge| +� +(i,j)∈τ +���γ(1) +N (xi; xj) +��� . +To compute these integrals we note that by Lemma 2.2 +ˆ +|g(x)| dx = +ˆ � +1 − f(x)2� +dx +≤ +4π +(1 − a3/b3)2 +ˆ b +a +�� +1 − a3 +b3 +�2 +− +� +1 − a3 +r3 +�2� +r2 dr ≤ Ca3 log b +a. +That is, each factor of ge gives a contribution Ca3 log(b/a) after integration. The γ(1) +N -factors +we can bound by Lemma 2.12 as +ˆ ���γ(1) +N (x; y) +��� dy ≤ Cs(log N)3. +This takes care of all but one integration, which gives the volume factor L3. We shall compute +the integrations in the following order: +(1.) Pick any leaf {j0} of the anchored tree τ lying in some cluster ℓ, meaning that there is +exactly one edge of τ incident in ℓ. +(2.) Consider {j0} as the root of Tℓ and pick any leaf {j} of Tℓ and integrate over xj. Since +{j} is a leaf of Tℓ and {j0} is a leaf of τ we have that the only place xj appears in the +integrand is in some factor gij for {i} the unique vertex connected to {j} by a g-edge. +Hence the xj-integral contributes +´ +|g| by the translation invariance. +Remove {j} and its incident edge from Tℓ. +Repeat for all vertices in the cluster until only {j0} remain. (At this point the entirety +of Tℓ has been removed.) +26 + +(3.) Integrate over xj0. Since {j0} is a leaf of τ the only place xj0 appears (in the remaining +integrand) is in the γ(1) +N -factor from τ. Thus, the xj0-integral gives a contribution +´ +|γ(1) +N | +by the translation invariance. +Remove {j0} and its incident edge from τ. +(4.) Repeat steps (1.)-(3.) until all integrals have been computed. The final integral gives the +volume factor L3. +Steps (1.)-(3.) compute all integrations in one cluster. Repeating this process we integrate +over the clusters one by one and thus compute all the integrals. Note that each integration is +always over a coordinate associated to a leaf of the relevant graphs. This is a key point, since +then by translation invariance each integration contributes exactly +´ +|g| or +´ +|γ(1) +N | whichever is +appropriate. In total we thus have the bound +˙ +dx1 . . . dx� nℓ +k +� +ℓ=1 +� +e∈Tℓ +|ge| +� +(i,j)∈τ +���γ(1) +N (xi; xj) +��� ≤ +� +Ca3 log(b/a) +�� nℓ−k � +Cs(log N)3�k−1 L3. +This bound is for each summand τ, T1, . . . , Tk. By Cayley’s formula #Tn = nn−2 ≤ Cnn!, and +by [GMR21, Appendix D.5] #A(n1,...,nk) ≤ k!4 +� nℓ. Thus, we get +∞ +� +p=2 +1 +p! +������ +� +(π,G)∈Lp +Γπ,G +������ +≤ CN +∞ +� +k=1 +� +Cs(log N)3�k−1 +� ∞ +� +n=2 +(Ca3ρ log(b/a))n−1 +�k +≤ CNa3ρ log(b/a) +∞ +� +k=1 +� +Csa3ρ log(b/a)(log N)3�k−1 +≤ CNa3ρ log(b/a) < ∞, +for sa3ρ log(b/a)(log N)3 sufficiently small. This shows that � +p +1 +p! +� +(π,G)∈Lp Γπ,G is absolutely +convergent under this condition. +3.1.2 +Absolute convergence of the Γ1-sum +Consider now +1 +p! +� +(π,G)∈L1p Γ1 +π,G. The argument is almost identical to the argument above. We +again split the sum according to the connected components of G. Call these G∗, G1, . . . , Gk, +where G∗ is the distinguished connected component (cluster) containing the distinguished vertex +{1}. Exactly as for +1 +p! +� +(π,G)∈Lp Γπ,G we have that (for p = 0 one has to interpret the empty +product of integrals as 1, so � +(π,G)∈L1 +0 Γ1 +π,G = ρ) +1 +p! +� +(π,G)∈L1p +Γ1 +π,G(x1) += +∞ +� +k=0 +1 +k! +� +n∗≥0 +� +n1,...,nk≥2 +1 +n∗!n1! · · · nk!χ( +� +ℓ{∗,1,...,k} nℓ=p) +� +G1∈Cn1,...,Gk∈Cnk +G∗∈Cn∗+1 +� +π∈Sp+1 +(−1)π +× χ((π,∪ℓ∈{∗,1,...,k}Gℓ)∈L1p) +˙ +� +ℓ∈{∗,1,...,k} +� +e∈Gℓ +ge +p+1 +� +j=1 +γ(1) +N (xj; xπ(j)) dx2 . . . dxp+1. +27 + +Here for the k = 0 term one should think of the n1, . . . , nk-sums as being an empty product +before it is an empty sum, i.e. it should give a factor 1. That is, the k = 0 term reads +� +G∗∈Cp+1 +� +π∈Sp+1 +(−1)π +˙ +� +e∈G∗ +ge +p+1 +� +j=1 +γ(1) +N (xj; xπ(j)) dx2 . . . dxp+1, +since (π, G∗) is trivially linked, since G∗ is connected. From here on, we won’t write out the +k = 0 term separately to make the formulas more concise. As before we use the tree-graph +inequality and the truncated correlation function (see Remark 3.9) to get +� +p≥0 +1 +p! +������ +� +(π,G)∈L1p +Γ1 +π,G +������ +≤ +∞ +� +k=0 +1 +k! +� +n∗≥0 +� +n1,...,nk≥2 +1 +n∗!n1! · · ·nk! +� +T1∈Tn1,...,Tk∈Tnk +T∗∈Tn∗+1 +× +˙ +� +ℓ∈{∗,1,...,k} +� +e∈Tℓ +|ge| +���ρ(n∗+1,n1,...,nk) +t +��� dx2 . . . dx� +ℓ∈{∗,1,...,k} nℓ+1. +To bound this we use the same bound, Equation (3.7), on the truncated correlations as before. +It reads +���ρ(n∗+1,n1,...,nk) +t +��� ≤ ρ +� +ℓ∈{∗,1,...,k} nℓ+1−(k+1−1) +� +τ∈A(n∗+1,n1,...,nk) +� +(i,j)∈τ +���γ(1) +N (xi; xj) +��� . +Computing the integrals is as before, with a few differences. During each repeat (apart from +the last one) of step (1.) we pick not just any leaf j0 but a leaf j0 not in the cluster containing +{1}. (Since any tree has at least 2 leaves, this is always possible.) For each of these repeats, +the argument is the same as before. For the last repeat of step (1.) where only the cluster +containing {1} remains we follow step (2.) with the slight change, that the root is chosen to be +{1}. (There are no γ(1) +N -factors left, so we are free to choose any vertex as the root.) There is +then no step (3.) since we do not integrate over x1. +This has the following effect. First, the last variable x1 is not integrated over, so there is +no volume factor L3. And second, there are k integrals +´ +|γ(1) +N | instead of k − 1 (since there are +k + 1 many clusters including the distinguished one). For the bounds of the sum of all terms +we use that #A(n∗+1,n1,...,nk) ≤ (k + 1)!4 +� +ℓ∈{∗,1,...,k} nℓ+1. Thus, uniformly in x1 +� +p≥0 +1 +p! +������ +� +(π,G)∈L1p +Γ1 +π,G +������ +≤ Cρ +� ∞ +� +n∗=0 +� +Ca3ρ log(b/a) +�n∗ +�  + +∞ +� +k=0 +(k + 1) +� +Cs(log N)3�k +� ∞ +� +n=2 +(Ca3ρ log(b/a))n−1 +�k + +≤ Cρ < ∞, +for sa3ρ log(b/a)(log N)3 sufficiently small. This shows that � +p +1 +p! +� +(π,G)∈L1p Γ1 +π,G is absolutely +convergent under this condition. +3.1.3 +Absolute convergence of the Γ2-sum +For the third sum the argument is mostly analogous. +There are a few changes needed for +the argument. First, one has to distinguish between the two cases of whether or not the two +28 + +distinguished vertices {1, 2} are in the same connected component (cluster) of the graph or not. +One computes +1 +p! +� +(π,G)∈L2p +Γ2 +π,G = Σdifferent + Σsame, +(3.8) +where +Σdifferent = +∞ +� +k=0 +1 +k! +� +n∗,n∗∗≥0 +� +n1,...,nk≥2 +1 +� +ℓ nℓ!χ(� nℓ=p) +� +G1∈Cn1,...,Gk∈Cnk +G∗∈Cn∗+{1} +G∗∗∈Cn∗∗+{2} +× +˙ � +ℓ +� +e∈Gℓ +geρ(n∗+{1},n∗∗+{2},n1,...,nk) +t +dx3 . . . dxp+2, +Σsame = +∞ +� +k=0 +1 +k! +� +n∗≥1 +� +n1,...,nk≥2 +1 +� +ℓ nℓ!χ(� nℓ=p) +� +G1∈Cn1,...,Gk∈Cnk +G∗∈Cn∗+{1,2} +(1,2)/∈G∗ +× +˙ � +ℓ +� +e∈Gℓ +geρ(n∗+{1,2},n1,...,nk) +t +dx3 . . . dxp+2. +(3.9) +Here � +ℓ and � +ℓ are over ℓ ∈ {∗, ∗∗, 1, . . ., k} or ℓ ∈ {∗, 1, . . . , k}, whichever is appropriate. +With a slight abuse of notation we write n∗ +{1} for the set of vertices in the cluster containing +the external vertex {1} (similarly for n∗∗ + {2}, n∗ + {1, 2}). This set has exactly n∗ internal +vertices. For p = 0 one has to interpret the empty product of integrals as a factor 1. +The first part is the contribution where {1} and {2} are in distinct clusters (labelled ∗ and +∗∗), the second part is the contribution from where they are in the same (labelled ∗). Note +that in the second contribution we have n∗ ≥ 1. Indeed, {1} and {2} are connected, but not +by an edge. Hence they must be connected by a path of length ≥ 2, which necessarily goes +through at least one vertex {j}, j ̸= 1, 2. +We treat the two cases separately. In the case where the two distinguished vertices are +in different clusters we may readily apply both the tree-graph bound and the bound on the +truncated correlation Equation (3.7). The latter reads +���ρ(n∗+{1},n∗∗+{2},n1,...,nk) +t +��� ≤ ρ(� +ℓ∈{∗,∗∗,1,...,k} +2)−(k+2−1) +� +τ∈A(n∗+{1},n∗∗+{2},n1,...,nk) +� +(i,j)∈τ +���γ(1) +N (xi; xj) +��� . +The integration procedure is slightly modified compared to that of Section 3.1.2. In the anchored +tree there is a path between (the cluster containing) {1} and (the cluster containing) {2}. For +the edge incident to (the cluster containing) {1} on this path, we bound |γ(1) +N | ≤ ρ. This cuts +the anchored tree into two anchored trees τ1, τ2 such that (with a slight abuse of notation) +1 ∈ τ1 and 2 ∈ τ2. We may follow the integration procedure exactly as for the Γ1-sum for each +of the anchored trees τ1 and τ2. Recall the bound +#A(n∗+{1},n∗∗+{2},n1,...,nk) ≤ (k + 2)!4 +� +ℓ∈{∗,∗∗,1,...,k} nℓ+2 ≤ C(k2 + 1)k!4 +� +ℓ∈{∗,∗∗,1,...,k} nℓ. +We thus get for the contribution of all terms where the two distinguished vertices are in different +29 + +clusters (assuming that sa3ρ log(b/a)(log N)3 is sufficiently small) +|Σdifferent| +≤ Cρ2 +� ∞ +� +n∗=0 +� +Ca3ρ log(b/a) +�n∗ +�2  + +∞ +� +k=0 +(k2 + 1) +� +Cs(log N)3�k +� ∞ +� +n=2 +(Ca3ρ log(b/a))n−1 +�k + +≤ Cρ2 < ∞. +(3.10) +Now we consider the case where {1} and {2} are in the same distinguished cluster. Here we +may readily apply the bound in Equation (3.7) on the truncated correlation but we need to +be a bit careful in applying the tree-graph bound. Indeed, then the sum over graphs in the +cluster containing the two vertices is not � +G∗∈Cn∗+2, but instead � +G∗∈Cn∗+2,(1,2)/∈G∗, since in the +construction, no g-edges are allowed between {1} and {2}. To still apply the tree-graph bound, +we define +˜ge := +� +ge +e ̸= (1, 2) +0 +e = (1, 2). +Then −1 ≤ ˜ge ≤ 0 so we can apply the tree-graph bound with these edge-weights to get +������ +� +G∗∈Cn∗+2,(1,2)/∈G∗ +� +e∈G∗ +ge +������ += +������ +� +G∗∈Cn∗+2 +� +e∈G∗ +˜ge +������ +≤ +� +T∗∈Tn∗+2 +� +e∈T∗ +|˜ge| = +� +T∗∈Tn∗+2,(1,2)/∈T∗ +� +e∈T∗ +|ge|. +We again have to modify the integrations slightly. +The integrations over all clusters apart +from the distinguished one may be computed as for the Γ- and Γ1-sums. For the distinguished +cluster with {1} and {2} there is some path of g-edges connecting them. Pick the unique edge +on this path incident with {1} and bound |g| ≤ 1 for this factor. This splits the tree T∗ into +two trees T 1 +∗ and T 2 +∗ with 1 ∈ T 1 +∗ and 2 ∈ T 2 +∗ . We may compute the integrations over all the +variables with index in the distinguished cluster exactly as for the Γ1-sum for each tree T 1 +∗ and +T 2 +∗ separately. One gets for the contribution (assuming that sa3ρ log(b/a)(log N)3 is sufficiently +small) +|Σsame| +≤ Cρ2 +� ∞ +� +n∗=1 +� +Ca3ρ log(b/a) +�n∗ +�  + +∞ +� +k=0 +(k + 1) +� +Cs(log N)3�k +� ∞ +� +n=2 +(Ca3ρ log(b/a))n−1 +�k + +≤ Ca3ρ3 log(b/a) < ∞. +(3.11) +We conclude that +� +p≥0 +1 +p! +������ +� +(π,G)∈L2p +Γ2 +π,G +������ +≤ Cρ2 < ∞, +uniformly in x1, x2 for sufficiently small sa3ρ log(b/a)(log N)3. +3.1.4 +Absolute convergence of the Γ3-sum +The argument for the last sum is completely analogous to the argument for the Γ2-sum. We have +to distinguish between different cases of the clusters containing the external vertices {1, 2, 3}. +Either there is one cluster containing all of them, one cluster containing two of them and one +30 + +cluster containing the last vertex, or they are all in distinct clusters. One then deals with the +different cases exactly as we did for the Γ2-sum. We skip the details. This concludes the proof +of Lemma 3.2. +4 +Energy of the trial state +In this section we bound the energy of the trial state ψN defined in Equation (2.1). Recall +Equation (2.2). By Theorem 3.4 we have (for sa3ρ log(b/a)(log N)3 sufficiently small) +ρ(2) +Jas(x1, x2) = f(x1 − x2)2 + +ρ(2)(x1, x2) + +∞ +� +p=1 +1 +p! +� +(π,G)∈L2p +Γ2 +π,G + + . +We can expand ρ(2) in x1 − x2 using Lemma 2.14. The second term is an error term we have to +control. Additionally, also the three-body term is an error we have to control. We claim that +Lemma 4.1. There exist constants c, C > 0 such that if sa3ρ log(b/a)(log N)3 < c and N = +#PF > C, then +������ +∞ +� +p=1 +1 +p! +� +(π,G)∈L2p +Γ2 +π,G +������ +≤ Ca6ρ4(log(b/a))2� +s3a6ρ2(log b/a)2(log N)9 + 1 +� ++ Ca3ρ3+2/3|x1 − x2|2� +s5a12ρ4(log(b/a))5(log N)16 + b4ρ4/3 + log(b/a) +� +. +Lemma 4.2. There exists a constant c > 0 such that if sa3ρ log(b/a)(log N)3 < c, then +ρ(3) +Jas = f 2 +12f 2 +13f 2 +23 +� +ρ(3) + O +� +a3ρ4 log(b/a) +� +s3a6ρ2(log(b/a))2(log N)9 + 1 +��� +where the error is uniform in x1, x2, x3. +We give the proof of these lemmas in Sections 4.2 and 4.3 below. For the three-body term, we +additionally have the bound ρ(3) ≤ Cρ5|x1 − x2|2|x1 − x3|2|x2 − x3|2 by Lemma 2.15. Com- +bining now Lemmas 2.13, 2.14, 4.1 and 4.2, Theorem 3.4 and Equation (2.2) we thus get (for +sa3ρ log(b/a)(log N)3 sufficiently small and N sufficiently large) +⟨ψN|HN|ψN⟩ = 3 +5(6π)2/3ρ2/3N +� +1 + O(N−1/3) + O(s−2) +� ++ L3 +ˆ +dx +� +|∇f(x)|2 + 1 +2v(x)f(x)2 +� +× +� +(6π2)2/3 +5 +ρ8/3|x|2 +� +1 − 3(6π2)2/3 +35 +ρ2/3|x|2 ++ O(N−1/3) + O(s−2) + O(N−1/3ρ2/3|x|2) + O(ρ4/3|x|4) +� ++ O +� +a6ρ4(log(b/a))2� +s3a6ρ2(log b/a)2(log N)9 + 1 +�� ++ O +� +a3ρ3+2/3|x|2� +s5a12ρ4(log(b/a))5(log N)16 + b4ρ4/3 + log(b/a) +�� � ++ +˚ +dx1 dx2 dx3f12∇f12f23∇f23f 2 +13 +� +O(ρ5|x1 − x2|2|x1 − x3|2|x2 − x3|2) ++ O +� +a3ρ4 log(b/a) +� +s3a6ρ2(log(b/a))2(log N)9 + 1 +��� +. +(4.1) +31 + +We will choose N (really L, see Remark 2.8) some large negative power of a3ρ, so errors with +N−1/3 are subleading. We may compute +ˆ +dx +� +|∇f(x)|2 + 1 +2v(x)f(x)2 +� +|x|2 += +1 +(1 − a3/b3)2 +ˆ +|x|≤b +dx +� +|∇f0(x)|2 + 1 +2v(x)f0(x)2 +� +|x|2 +≤ 12πa3 � +1 + O(a3/b3) +� +, +(4.2) +by Definition 1.1 since f = +1 +1−a3/b3f0 for |x| ≤ b and b > R0, the range of v. For the higher +moments we recall that |∇f0| ≤ |∇fhc| = 3a3 +|x|4 for |x| ≥ a by Lemma 2.2. Then we have (for +n = 4, 6) +ˆ +dx +� +|∇f(x)|2 + 1 +2v(x)f(x)2 +� +|x|n += +1 +(1 − a3/b3)2 +ˆ +|x|≤b +dx +� +|∇f0(x)|2 + 1 +2v(x)f0(x)2 +� +|x|n +≤ 1 +2Rn−2 +0 +ˆ +v|f0|2|x|2 dx + +ˆ +|x|≥a +� 3a3 +|x|4 +�2 +|x|n dx + an−2 +ˆ +|x|≤a +|∇f0|2|x|2 dx +≲ CRn−2 +0 +a3. +(4.3) +For n = 4 we have more precisely +ˆ +dx +� +|∇f(x)|2 + 1 +2v(x)f(x)2 +� +|x|4 ≤ +ˆ +dx +� +|∇f0(x)|2 + 1 +2v(x)f0(x)2 +� +|x|4 � +1 + O(a3b−3) +� += 36πa3a2 +0 + O(a6a2 +0b−3). +For the lower moment, we have by Equation (2.4) +ˆ +dx +� +|∇f(x)|2 + 1 +2v(x)f(x)2 +� += 4π +ˆ b +0 +� +|∂rf|2r2 + r2f∂2 +rf + 4rf∂rf +� +dr += 12πa3/b2 +1 − a3/b3 + 8π +ˆ b +0 +rf∂rf dr +(4.4) +where ∂r denotes the radial derivative, and we integrated by parts using that f(r) = 1−a3/r3 +1−a3/b3 +outside the support of v. By Lemma 2.2 we have +2 +ˆ b +0 +rf∂rf dr = b − +ˆ b +0 +f 2 dr ≤ b − +1 +(1 − a3/b3)2 +ˆ b +a +� +1 − a3 +r3 +�2 +dr ≤ Ca. +(4.5) +Hence +ˆ +dx +� +|∇f(x)|2 + 1 +2v(x)f(x)2 +� +≤ Ca. +This concludes the bounds on all the terms in Equation (4.1) arising from the 2-body term. To +bound those arising from the 3-body term we bound |x1 − x3| ≤ 2b in the support of ∇f12∇f23 +and f13 ≤ 1. By the translation invariance one integration gives a volume factor L3. The +32 + +remaining two integrals then both give the same contribution. That is, +˚ +dx1 dx2 dx3f12∇f12f23∇f23f 2 +13 +� +O(ρ5|x1 − x2|2|x1 − x3|2|x2 − x3|2) ++ O +� +a3ρ4 log(b/a) +� +s3a6ρ2(log(b/a))2(log N)9 + 1 +��� +≤ CNρ4b2 +�ˆ +|x|2f∂rf dx +�2 ++ CNa3ρ3 log(b/a) +� +s3a6ρ2(log(b/a))2(log N)9 + 1 +� �ˆ +f∂rf dx +�2 +. +Using integration by parts and Lemma 2.2, we have that +1 +4π +ˆ +|x|nf∂rf dx = +ˆ b +0 +rn+2f∂rf dr = bn+2 +2 +− n + 2 +2 +ˆ b +0 +rn+1f 2 dr +≤ bn+2 +2 +− n + 2 +2 +ˆ b +a +rn+1 +�1 − a3/r3 +1 − a3/b3 +�2 +dr ≤ +� +Ca2 +n = 0, +Ca3b +n = 2. +(4.6) +Plugging all this into Equation (4.1) we thus get for the energy density +⟨ψN|HN|ψN⟩ +L3 += 3 +5(6π)2/3ρ5/3 + 12π +5 (6π2)2/3a3ρ8/3 − 108π(6π2)4/3 +175 +a3a2 +0ρ10/3 ++ O +� +s−2ρ5/3� ++ O +� +N−1/3ρ5/3� ++ O +� +a6b−3ρ8/3� ++ O +� +a6a2 +0b−3ρ10/3� ++ O +� +R4 +0a3ρ4� ++ O +� +a7ρ4(log(b/a))2� +s3a6ρ2(log b/a)2(log N)9 + 1 +�� ++ O +� +a6ρ3+2/3� +s5a12ρ4(log(b/a))5(log N)16 + b4ρ4/3 + log(b/a) +�� ++ O +� +a6b4ρ5� ++ O +� +a7ρ4 log(b/a) +� +s3a6ρ2(log(b/a))3(log N)9 + 1 +�� +. +(4.7) +We choose L ∼ a(a3ρ)−10 still ensuring that +LkF +2π +is rational. +(More precisely one chooses +L ∼ a(kFa)−30, since ρ is defined in terms of L.) Then N ∼ (a3ρ)−29. Choose moreover +b = a(a3ρ)−β, +s ∼ (a3ρ)−α| log(a3ρ)|−δ. +Note that we need +2 +9 < β < 5 +12, +5 +6 < α < 13 +15 +for the error terms to be smaller than the desired accuracy of order a3a2 +0ρ10/3. We get +⟨ψN|HN|ψN⟩ +L3 += 3 +5(6π)2/3ρ5/3 + 12π +5 (6π2)2/3a3ρ8/3 − 108π(6π2)4/3 +175 +a3a2 +0ρ10/3 ++ O(ρ5/3(a3ρ)γ1| log(a3ρ)|γ2). +where +γ1 = min +� +2α, 1 + 3β, 13 +3 − 3α, 6 − 5α, 10 +3 − 4β +� +, +and γ2 is given by the power of the logarithmic factors of the largest error term. Optimising in +α, β, δ we see that for the choice +β = 1 +3, +α = 6 +7, +δ = 3 +33 + +we have γ1 = 12 +7 and γ2 = 6, i.e. +⟨ψN|HN|ψN⟩ +L3 += 3 +5(6π)2/3ρ5/3 ++ 12π +5 (6π2)2/3a3ρ8/3 +� +1 − 9 +35(6π2)2/3a2 +0ρ2/3 + O((a3ρ)2/3+1/21| log(a3ρ)|6) +� +(4.8) +for a3ρ small enough. Note that for this choice of s, N we have s ∼ N6/203(log N)3. Thus any +Q with N4/3 ≪ Q ≤ CNC satisfies the condition Q−1/4 ≤ Cs−1 of Definition 2.7. +4.1 +Thermodynamic limit via a box method +In this section we construct a trial state in the thermodynamic limit using a box method of +gluing trial states for finite n together. First we show that we may choose periodic boundary +conditions in the small boxes instead of using Dirichlet boundary conditions. The setting and +argument is due to Robinson [Rob71, Lemmas 2.1.12 and 2.1.13]. We present a slightly modified +version in [MS20, Section C]. +Lemma 4.3 ([MS20; Rob71]). Let 0 < d < L/2 be a cut-off, let HD +N,L+2d = �N +j=1 −∆D +j,L+2d + +� +i 0 with ˜f (n) ց f. +37 + +Then one readily applies the Lebesgue dominated convergence theorem to exchange the limit +˜f (n) → f with the relevant sums and integrals.) Taking x1 = x2 in this we have DN = 0 and +ρ(2) = 0. This shows Equation (4.13). We may thus bound the zeroth order term of ξsmall,0 and +ξ0 by +|ξsmall,0(x2, x2) + ξ0(x2, x2)| +≤ |ξsmall,≥1(x2, x2)| + |ξ≥1(x2, x2)| +≤ Ca6ρ4(log(b/a))2 � +s3(log N)9a6ρ(log(b/a))2 + s(log N)a3ρ log(b/a) + 1 +� +≤ Ca6ρ4(log(b/a))2 � +s3(log N)9a6ρ(log(b/a))2 + 1 +� +. +(4.14) +Since both ξsmall,0 and ξ0 are symmetric in x1 and x2 all first order terms vanish. We are left +with bounding the second derivatives. For ξsmall,0 we have +Lemma 4.8. For any µ, ν = 1, 2, 3 we have +��∂µ +x1∂ν +x1ξsmall,0 +�� ≤ Ca3ρ3+2/3 log(b/a) +uniformly in x1, x2. Here ∂µ +x1 denotes the derivative in the xµ +1-direction. +The proof of this lemma is a (not very insightful) computation. We give it in Appendix A.1. +Next we consider ∂µ +x1∂ν +x1ξ0. We write ξ0 in terms of truncated densities as in Equation (3.9), +i.e. +ξ0 = +∞ +� +k=0 +1 +k! +� +n1,...,nk≥2 +χ(k≥5,nℓ=2 or k≥1,� nℓ=2k+1) +1 +� +ℓ nℓ! +� +G1,...,Gk +Gℓ∈Cnℓ +× +˙ � +ℓ +� +e∈Gℓ +geρ({1},{2},n1,...,nk) +t +dx3 . . . dxp+2. +Since we consider terms with n∗ = n∗∗ = 0, there are no g-factors that depend on x1 and +thus all derivatives are of ρ({1},{2},n1,...,nk) +t +. We thus need to calculate ∂µ +x1∂ν +x1ρ({1},{2},n1,...,nk) +t +. For +this we use the definition in Equation (3.4) rather than the formula in Equation (3.6). In +Equation (3.4) the variable x1 appears exactly twice: Once in an outgoing γ(1) +N -edge from {1} +and once in an incoming γ(1) +N -edge to {1}. Taking the derivatives then amounts to replacing +either one of these edges by its second derivative or both of them by their first derivatives. +Thus, using that γ(1) +N (xi; xj) = γ(1) +N (xj; xi) since γ(1) +N is real, and that for (π, ∪Gℓ) to be linked +necessarily π(1) ̸= 1, we have (for p = 2 + � +ℓ nℓ) +∂µ +x1∂ν +x1ρ({1},{2},n1,...,nk) +t += ∂µ +x1∂ν +x1 +� +π∈Sp +(−1)πχ((π,∪Gℓ)∈Lp) +p +� +j=1 +γ(1) +N (xj; xπ(j)) += +� +π∈Sp +(−1)πχ((π,∪Gℓ)∈Lp) +� +j̸=1,j̸=π−1(1) +γ(1) +N (xj; xπ(j)) +� +2∂µ +x1∂ν +x1γ(1) +N (x1; xπ(1))γ(1) +N (xπ−1(1); x1) ++2∂µ +x1γ(1) +N (x1; xπ(1))∂ν +x1γ(1) +N (xπ−1(1); x1) +� +. +(4.15) +With this formula we may then redo the computation of [GMR21, Equation (D.53)] only now +some of the γ(1) +N -factors (precisely 1 or 2 of them) carry derivatives. +The γ(1) +N -factors with +derivatives may end up in the anchored tree, or they may end up in the matrix N (r). If they +end up in N (r) it is explained around [GMR21, Equation (D.9)] how to modify Lemma 3.10. +38 + +One simply includes factors ikµ in the definition of (some of) the functions αi (and not of βj) +in the proof of Lemma 3.10. Since we may bound |k| ≤ Cρ1/3 for k ∈ PF we get +���det ˜ +N(r) +��� ≤ + + + + + +ρ +� nℓ+2−(k+2−1) +if no derivatives end up in N , +Cρ +� nℓ+2−(k+2−1)+1/3 +if one derivative ends up in N, +Cρ +� nℓ+2−(k+2−1)+2/3 +if two derivatives end up in N , +(4.16) +where ˜ +N(r) is the appropriate modification of N (r). To get the formula for ρt we need also to +consider two cases for how the anchored tree looks. There could be both an incoming and an +outgoing edge to/from the vertex {1}. And if there is just one edge to/from {1} it could be +either an incoming or an outgoing edge. Since γ(1) +N is real, incoming and outgoing edges gives +the same factor γ(1) +N (x1; xj). A simple calculation (essentially just undoing the product rule) +then shows that +∂µ +x1∂ν +x1ρ({1},{2},n1,...,nk) +t += +� +∂∈{1,∂µ +x1,∂νx1,∂µ +x1∂νx1} + + +� +τ∈A({1},{2},n1,...,nk) +two edges to/from {1} +∂ +� +γ(1) +N (xj2, x1)γ(1) +N (x1; xj1) +� ++ +� +τ∈A({1},{2},n1,...,nk) +one edge to/from {1} +∂γ(1) +N (x1; xj1) + + +� +(i,j)∈τ +i,j̸=1 +γ(1) +N (xi; xj) +ˆ +dµτ(r) det ˜ +N∂(r), +(4.17) +where j1 and j2 denote the vertices connected to {1} by the relevant edges in τ and ˜ +N∂ is +the appropriately modified version of N , where the derivatives not in ∂ end up in N , i.e. +Equation (4.16) reads +���det ˜ +N∂(r) +��� ≤ Cρ +� nℓ+2−(k+2−1)+(2−#∂)/3, +where #∂ denotes the number of derivatives in ∂, i.e. #1 = 0, #∂µ +x1 = 1 and #∂µ +x1∂ν +x1 = 2. +We denote the contribution of the two terms in Equation (4.17) to ∂µ +x1∂ν +x1ξ0 by (∂µ +x1∂ν +x1ξ0)→•→ +and (∂µ +x1∂ν +x1ξ0)•→ respectively. +We first deal with the second term of Equation (4.17) where there is just one edge to/from +{1} in the anchored tree. We may bound the contribution of this term almost exactly as in the +proof of Lemma 3.2. We give a sketch here. Using Equation (4.16) we get the bound +≤ C +� +∂∈{1,∂µ +x1,∂νx1,∂µ +x1∂νx1} +ρ(2−#∂)/3 +� +τ∈A({1},{2},n1,...,nk) +���∂γ(1) +N (x1; xj1) +��� +× +� +(i,j)∈τ +i,j̸=1 +���γ(1) +N (xi; xj) +��� ρ(� +ℓ nℓ+2)−(k+2−1), +(4.18) +where again #∂ denotes the number of derivatives in ∂. +To bound the integrations we again follow the strategy of the proof of the Γ2-sum of +Lemma 3.2, Section 3.1.3. The only difference is that the γ(1) +N -edge on the path in the an- +chored tree between {1} and {2} incident to {1} is the edge with derivatives, ∂γ(1) +N (x1; xj1). +This we bound by |∂γ(1) +N | ≤ Cρ1+#∂/3. The integrations can then be performed exactly as in +39 + +Section 3.1.3. We conclude the bound +˙ +k +� +ℓ=1 +� +e∈Tℓ +|ge| +���∂γ(1) +N (x1; xj1) +��� +� +(i,j)∈τ +i,j̸=1 +���γ(1) +N (xi; xj) +��� dx3 . . . dx� nℓ+2 +≤ ρ1+#∂/3 � +Ca3 log(b/a) +�� nℓ−k � +Cs(log N)3�k . +Again, as in the proof of Lemma 3.2 we have by Cayley’s formula that #Tn = nn−2 ≤ Cnn! +and by [GMR21, Appendix D.5] that #A({1},{2},n1,...,nk) ≤ (k + 2)!4 +� nℓ+2 ≤ C(k2 + 1)k!4 +� nℓ. +Following the same arguments as for Equation (4.10) and recalling that the diagrams in ξ0 have +either k ≥ 5 or ng ≥ 1 we get the contribution to ∂µ +x1∂ν +x1ξ0 of +��(∂µ +x1∂ν +x1ξ0)•→�� ≤ Cρ2+2/3� +(s(log N)3)5(a3ρ log(b/a))5 +� +�� +� +type (A) diagrams ++ s(log N)3(a3ρ log(b/a))2 +� +�� +� +type (B) diagrams +� +≤ Ca6ρ4+2/3(log(b/a))2s(log N)3 � +s4(log N)12a9ρ3(log(b/a))3 + 1 +� +(4.19) +uniformly in x1, x2. +Next consider the first term of Equation (4.17). The argument is almost the same, only +we have to distinguish between which γ(1) +N -factor(s) the derivatives in ∂ hits. We consider the +case ∂ = ∂µ +x1∂ν +x1. The other cases are similar. Suppose that the γ(1) +N -edge on the path (in the +anchored tree) from {1} to {2} is γ(1) +N (x1; xj1) and the γ(1) +N -factor not on the path is γ(1) +N (xj2; x1). +We distinguish between three cases: +1. If both derivatives are on γ(1) +N (x1; xj1) we may bound this exactly as above. +2. If one derivative is on γ(1) +N (x1; xj1) (say ∂ν +x1) and one derivative (say ∂µ +x1) is on γ(1) +N (xj2; x1) +we bound |∂ν +x1γ(1) +N (x1; xj1)| ≤ Cρ4/3. +Then the argument is similar, only now one of +the γ(1) +N -integrations is with ∂µγ(1) +N +instead. Thus, in the computation leading to Equa- +tion (4.19) we should replace one factor Csρ1/3(log N)3 with +´ +|∂µγ(1) +N | dx. +3. If both derivatives are on γ(1) +N (xj2; x1) then analogously we bound |γ(1) +N (x1; xj1)| ≤ Cρ and +in the computation leading to Equation (4.19) we should replace one factor Csρ2/3(log N)3 +with +´ +|∂µ∂νγ(1) +N | dx. +In total we have the contribution to ∂µ +x1∂ν +x1ξ0 of +≤ Cρ2 +� +ρ2/3s(log N)3 + ρ1/3 +ˆ +Λ +���∂µγ(1) +N +��� dx + ρ1/3 +ˆ +Λ +���∂νγ(1) +N +��� dx + +ˆ +Λ +���∂µ∂νγ(1) +N +��� dx +� +× +� +(s(log N)3)4(a3ρ log(b/a))5 + (a3ρ log(b/a))2� +uniformly in x1, x2. One may do a similar computation for the other cases of ∂ and conclude +that +��(∂µ +x1∂ν +x1ξ0)→•→�� +≤ Cρ2 +� +ρ2/3s(log N)3 + ρ1/3 +ˆ +Λ +���∂µγ(1) +N +��� dx + ρ1/3 +ˆ +Λ +���∂νγ(1) +N +��� dx + +ˆ +Λ +���∂µ∂νγ(1) +N +��� dx +� +× +� +(s(log N)3)4(a3ρ log(b/a))5 + (a3ρ log(b/a))2� +(4.20) +40 + +uniformly in x1, x2. Thus, we need to bound the integrals +ˆ +Λ +���∂µγ(1) +N +��� dx = +ˆ +Λ +1 +L3 +����� +� +k∈PF +kµeikx +����� dx = +1 +(2π)2L +ˆ +[0,2π]3 +������� +� +q∈ +� LkF +2π P +� +∩Z3 +qµeiqu +������� +du, +and +ˆ +Λ +���∂µ∂νγ(1) +N +��� dx = +ˆ +Λ +1 +L3 +����� +� +k∈PF +kµkνeikx +����� dx = +1 +2πL2 +ˆ +[0,2π]3 +������� +� +q∈ +� LkF +2π P +� +∩Z3 +qµqνeiqu +������� +du. +Here we have +Lemma 4.9. The polyhedron P from Definition 2.7 satisfies for any µ, ν = 1, 2, 3 that +ˆ +[0,2π]3 +������� +� +q∈ +� LkF +2π P +� +∩Z3 +qµeiqu +������� +du ≤ CsN1/3(log N)3, +ˆ +[0,2π]3 +������� +� +q∈ +� LkF +2π P +� +∩Z3 +qµqνeiqu +������� +du ≤ CsN2/3(log N)4 +for sufficiently large N. +The proof of Lemma 4.9 is a long and technical computation, which we give in Appendix B. +Applying the lemma we conclude that +ˆ +Λ +���∂µγ(1) +N +��� dx ≤ Csρ1/3(log N)3, +ˆ +Λ +���∂µ∂νγ(1) +N +��� dx ≤ Csρ2/3(log N)4. +By combining this with Equations (4.19) and (4.20) we get +��∂µ +x1∂ν +x1ξ0 +�� ≤ Ca6ρ4+2/3(log(b/a))2s(log N)4 � +s4(log N)12a9ρ3(log(b/a))3 + 1 +� +(4.21) +uniformly in x1, x2. Combining Lemmas 4.6 and 4.8 and Equations (4.12), (4.14) and (4.21) +and using that for any real number t > 0 and integer n ≥ 1 we may bound t ≲ tn + 1 this +shows the desired. +Remark 4.10 (Treating more diagrams as small). One can improve the error bound in The- +orem 1.3 slightly by treating more diagrams as small. +This is similar to what is done in +[BCGOPS22] for the dilute Bose gas. We sketch the overall idea. +If we choose s ∼ (a3ρ)−1+ε/2 then the error from the s-dependent term in the kinetic energy +to the energy density is ρ5/3(a3ρ)2−ε. Then choose as “large” the diagrams for which the bound +in Equation (4.10) gives contributions to the energy density much smaller than ρ5/3(a3ρ)2−ε. +This happens for k0 > K for some large K ∼ ε−1. We can then evaluate all small diagrams +as in Appendix A and conclude that their contributions are as given in Appendix A only with +some K-dependent constants, since there is some K-dependent number of small diagrams. In +total we would then get an error of size Oε(ρ5/3(a3ρ)2−ε) in Theorem 1.3. +41 + +4.3 +Subleading 3-particle diagrams (proof of Lemma 4.2) +Proof of Lemma 4.2. Recall the formula for ρ(3) +Jas of Theorem 3.4. In this formula there are terms +like ρ � +p≥1 +1 +p! +� +(π,G)∈L2p Γ2 +π,G(x2, x3). We have ρ = ρ(1) +Jas(x1) = � +p′≥1 +1 +p′! +� +(π′,G′)∈L1 +p′ Γ1 +π′,G′(x1) by +translation invariance. Joining the two diagrams (π, G) ∈ L2 +p and (π′, G′) ∈ L1 +p′ we get a new +(no longer linked) diagram (π′′, G′′) ∈ D3 +p+p′ with two linked components, one of which contains +the vertices {2} and {3} and one of which contains the vertex {1}. Doing this for all three +terms of this type, we are led to define the set +˜L3 +p := L3 +p ∪ +� +q+q′=p +(L2 +q ⊕ L1 +q′), +where ⊕ refers to the operation of joining two diagrams as above. The set ˜L3 +p is then the set of +diagrams on 3 external and p internal vertices such that there is at most two linked components, +and that each linked component contains at least one external vertex. With this, the formula +for ρ(3) +Jas of Theorem 3.4 reads (assuming that sa3ρ log(b/a)(log N)3 is sufficiently small) +ρ(3) +Jas = f 2 +12f 2 +13f 2 +23 +� +ρ(3) + +� +p≥1 +1 +p! +� +(π,G)∈ ˜L3p +Γ3 +π,G +� +. +We split the diagrams in ˜L3 +p into two groups, large and small similarly to the proof of Lemma 4.1. +To do this, we similarly define for a diagram (π, G) ∈ ˜L3 +p the number k = k(π, G) as the number +of clusters entirely containing internal vertices. (This k is exactly the same k as in the proof of +Lemma 3.2.) Then we define +ng = ng(π, G) = �k +ℓ=1 nℓ − 2k, +n∗ +g = n∗ +g(π, G) = n∗ + n∗∗ + n∗∗∗, +where we understand n∗∗ = 0 and/or n∗∗∗ = 0 if {1, 2, 3} are not all in different clusters. (One +defines n∗∗∗ as the number of internal vertices in the cluster containing {3} if all {1, 2, 3} are in +different clusters, exactly as for the n∗∗ and n∗ of Sections 3.1.2 and 3.1.3.) We may still think +of ng +n∗ +g as the “number of added vertices”. As for Equation (4.10) we have (for p = 2k0+ng0) +������������ +1 +p! +� +(π,G)∈ ˜L3 +p +ng(π,G)+n∗ +g(π,G)=ng0 +k(π,G)=k0 +Γ3 +π,G +������������ +≤ Cρ3(Cs(log N)3)k0(Ca3ρ log(b/a))k0+ng0. +(4.22) +The main difference compared to Equation (4.10) is that we here allow diagrams that are not +linked. +This doesn’t matter, since when we compute the integrals (as in Section 3.1.3) we +anyway have to cut the diagram up into 3 parts (either by bounding g-edges or γ(1) +N -edges) as +described in Section 3.1.3. We split the diagrams into two (exhaustive) groups: +1. Small diagrams with 1 ≤ k ≤ 2, ng = 0, n∗ +g = 0, and +2. Large diagrams as the rest, i.e. with +(A) k ≥ 3, or +42 + +(B) ng + n∗ +g ≥ 1. +As in Section 4.2, the splitting is motivated by counting powers in Equation (4.22). Note that +for p ≥ 1 the diagrams with k = 0, ng = 0, n∗ +g = 0 are not present. We write +� +p≥1 +1 +p! +� +(π,G)∈ ˜L3p +Γ3 +π,G = ξ3 +small + ξ3 +large, +where ξ3 +small and ξ3 +large are the contributions of small and large diagrams respectively. Exactly +as in Equation (4.12) we may bound, using Equation (4.22) +|ξ3 +large| ≤ Cρ3(s(log N)3)3(a3ρ log(b/a))3 +� +�� +� +type (A) diagrams ++ Cρ3a3ρ log(b/a) +� +�� +� +type (B) diagrams +≤ Ca3ρ4 log(b/a) +� +s3a6ρ2(log(b/a))2(log N)9 + 1 +� +. +For the small diagrams we have +Lemma 4.11. We have +|ξ3 +small| ≤ Ca3ρ4 log(b/a) +uniformly in x1, x2, x3. +As with Lemma 4.8, the proof is simply a computation, which we give in Appendix A.2. We +conclude the desired. +5 +One and two dimensions +In this section we sketch the necessary changes one needs to make for the argument to apply in +dimensions d = 1 and d = 2. We will abuse notation slightly and denote by the same symbols +as in Sections 2, 3 and 4 the relevant 1- and 2-dimensional analogues. +5.1 +Two dimensions +Similarly to the 3-dimensional setting, the p-wave scattering function f0 in 2 dimensions is +radial and solves the equation +− ∂2 +rf0 − 3 +r∂rf0 + 1 +2vf0 = 0, +(5.1) +see Section 2.1 and recall Definition 1.6. Thus, it is the same as the s-wave scattering function +in 4 dimensions. In particular it satisfies the bound +Lemma 5.1 ([LY01, Lemma A.1], Lemma 2.2). The scattering function satisfies +� +1 − a2 +|x|2 +� ++ ≤ +f0(x) ≤ 1 for all x and |∇f0(x)| ≤ 2a2 +|x|3 for |x| > a. +As for the 3-dimensional setting we consider the trial state +ψN(x1, . . . , xN) = +1 +√CN +� +i 0 such that if sa2ρ log(b/a)(log N)2 < c, then +the formulas in Equation (3.3) hold (with ρ(n) +Jas and Γn +π,G interpreted as appropriate in the two- +dimensional setting). +The analogues of Lemmas 4.1 and 4.2 read +Lemma 5.8. There exist constants c, C > 0 such that if sa2ρ log(b/a)(log N)2 < c and N = +#PF > C, then +������ +∞ +� +p=1 +1 +p! +� +(π,G)∈L2p +Γ2 +π,G +������ +≤ Ca4ρ4(log(b/a))2� +s3a4ρ2(log b/a)2(log N)6 + 1 +� ++ Ca2ρ4|x1 − x2|2� +s5a8ρ4(log(b/a))5(log N)11 + b4ρ2 + log(b/a) +� +, +and +ρ(3) +Jas ≤ Cf 2 +12f 2 +13f 2 +23 +� +ρ6|x1 − x2|2|x1 − x3|2|x2 − x3|2 ++ a2ρ4 log(b/a) +� +s3a4ρ2(log(b/a))2(log N)6 + 1 +�� +. +45 + +The proof is again similar to the 3-dimensional case replacing the bounds on +´ +|g| and +´ +|γ(1) +N | +as in Equation (5.2) above. Apart from this, there are two main changes. The first is in the +proof of the analogue of Lemma 4.6, namely Equation (A.1), where one bounds +´ +|x|2|1 − f 2|. +In two dimensions this bound is, using Lemma 5.1, +ˆ +R2 +� +1 − f(x)2� +|x|2 dx ≤ Ca4 + +C +(1 − a2/b2)2 +ˆ b +a +�� +1 − a2 +b2 +�2 +− +� +1 − a2 +r2 +�2� +r3 dr ≤ Ca2b2. +The other main difference is for the analogue of Lemma 4.9. Here the 2-dimensional analogue +reads +Lemma 5.9. The polygon P from Definition 5.2 satisfies for any µ, ν = 1, 2 that +ˆ +[0,2π]2 +������� +� +q∈ +� LkF +2π P +� +∩Z2 +qµeiqu +������� +du ≤ CsN1/2(log N)2, +ˆ +[0,2π]2 +������� +� +q∈ +� LkF +2π P +� +∩Z2 +qµqνeiqu +������� +du ≤ CsN(log N)3 +for sufficiently large N. +The proof is similar to that of Lemma 4.9 given in Appendix B only one skips Appendix B.2 +and notes that R = LkF +2π ∼ N1/2. +Putting together the formulas in Theorem 5.7 with the bounds in Lemmas 5.5, 5.6 and 5.8 +we easily find the analogue of Equation (4.1). We then need to bound a few terms. Following +the type of arguments of Section 4, namely Equations (4.2), (4.3), (4.4), (4.5) and (4.6) and +using Lemma 5.1 we get the bounds +ˆ � +|∇f(x)|2 + 1 +2v(x)f(x)2 +� +|x|n dx ≤ + + + + + +C, +n = 0, +4πa2 + O(a4b−2), +n = 2, +Ca4 log(b/a) + CR2 +0a2, +n = 4, +ˆ +|x|nf∂rf dx ≤ +� +Ca, +n = 0, +Ca2b, +n = 2. +Plugging this into the analogue of Equation (4.1) we get the analogue of Equation (4.7), +⟨ψN|HN|ψN⟩ +L2 += π +8 ρ2 + π2 +4 a2ρ3 ++ O +� +s−4ρ2� ++ O +� +N−1/2ρ2� ++ O +� +a4b−2ρ3� ++ O +� +a4ρ4 log(b/a) +� ++ O +� +R2 +0a2ρ4� ++ O +� +a4ρ4(log(b/a))2� +s3a4ρ2(log b/a)2(log N)6 + 1 +�� ++ O +� +a4ρ4� +s5a8ρ4(log(b/a))5(log N)11 + b4ρ2 + log(b/a) +�� ++ O +� +a4b4ρ6� ++ O +� +a4ρ4 log(b/a) +� +s3a4ρ2(log(b/a))3(log N)6 + 1 +�� +. +(5.3) +As above, we can choose L ∼ a(a2ρ)−10 still ensuring that LkF +2π is rational. (More precisely one +chooses L ∼ a(kFa)−20, since ρ is defined in terms of L.) Then N ∼ (a2ρ)−19. Choose moreover +b = a(a2ρ)−β, +s ∼ (a2ρ)−α| log(a2ρ)|−γ. +46 + +Optimising in α, β, γ we see that for the choice +β = 1 +2, +α = 4 +7, +γ = 10 +7 +we have +⟨ψN|HN|ψN⟩ +L3 += π +8ρ2 + π2 +4 a2ρ3 +� +1 + O +� +a2ρ| log(a2ρ)|2�� +(5.4) +for a2ρ small enough. Note that for this choice of s, N we have s ∼ N4/133(log N)10/7. Thus +any Q with N3/2 ≪ Q ≤ CNC satisfies the condition Q−1/2 ≤ Cs−2 of Definition 5.2. +The extension to the thermodynamic limit of Section 4.1 is readily generalized. We thus +conclude the proof of Theorem 1.7. +5.2 +One dimension +Similarly to the 2- and 3-dimensional settings, the p-wave scattering function f0 in 1 dimension +is even and solves the equation (here ∂2 denotes the second derivative) +− ∂2f0 − 2 +r∂f0 + 1 +2vf0 = 0, +(5.5) +see Section 2.1 and recall Definition 1.8. Thus, it is the same as the s-wave scattering function +in 3 dimensions. In particular it satisfies the bound +Lemma 5.10 ([LY01, Lemma A.1], Lemma 2.2). The scattering function satisfies +� +1 − +a +|x| +� ++ ≤ +f0(x) ≤ 1 for all x and |∂f0(x)| ≤ +a +|x|2 for |x| > a. +Before giving the proof of Theorem 1.9 we first compare our definition of the scattering length +to that of [ARS22]. In [ARS22] the following definition is given. +Definition 5.11 ([ARS22, Section 1.3]). The odd-wave scattering length aodd is given by +4 +R − aodd += inf +�ˆ R +−R +� +2|∂h|2 + v|h|2� +dx : h(R) = −h(−R) = 1 +� +for any R > R0, the range of v. +The value of aodd is independent of R > R0 so aodd is well-defined. We claim that +Proposition 5.12. The p-wave scattering length a defined in Definition 1.8 and the odd-wave +scattering length aodd defined in Definition 5.11 agree, i.e. a = aodd. +Proof. Note first that h �→ E(h) = +´ R +−R (2|∂h|2 + v|h|2) dx is convex, so by replacing h by +(h(x) − h(−x))/2 we can only lower its value. Thus, we have +4 +R − aodd += inf +�ˆ R +−R +� +2|∂h|2 + v|h|2� +dx : h(x) = −h(−x), h(R) = 1 +� +. +Any h we write as h(x) = xf(x) +R . Using this and integration by parts we get +4 +R − aodd += 1 +R2 inf +�ˆ R +−R +� +2|f|2 + 4xf∂f + 2|x|2|∂f|2 + v|f|2|x|2� +dx : f(x) = f(−x), f(R) = 1 +� += 4 +R + 2 +R2 inf +�ˆ R +−R +� +|∂f|2 + 1 +2v|f|2 +� +|x|2 dx : f(x) = f(−x), f(R) = 1 +� +. +47 + +That is, +2R +� +1 +1 − aodd/R − 1 +� += inf +�ˆ R +−R +� +|∂f|2 + 1 +2v|f|2 +� +|x|2 dx : f(x) = f(−x), f(R) = 1 +� +. +Taking R → ∞ in this we recover the definition of a. We conclude that a = aodd. +Concerning the assumption on v that +´ �1 +2vf 2 +0 + |∂f0|2� +dx < ∞ we have the following two +propositions. +Proposition 5.13. Suppose that v ≥ 0 is even and compactly supported and that for some +interval [x1, x2], 0 ≤ x1 < x2 we have v(x) = ∞ for x1 ≤ x ≤ x2. Then +´ � 1 +2vf 2 +0 + |∂f0|2� +dx < +∞, where f0 denotes the p-wave scattering function. +Proof. Let [x1, x2] be an interval where v(x) = ∞ for x1 ≤ x ≤ x2 and note that f0(x) = 0 for +all |x| ≤ x2. Then we have +ˆ �1 +2vf 2 +0 + |∂f0|2 +� +dx ≤ 1 +x2 +2 +ˆ +|x|≥x2 +�1 +2vf 2 +0 + |∂f0|2 +� +|x|2 dx = 2ax−2 +2 +< ∞. +Proposition 5.14. Suppose that v ≥ 0 is even, compactly supported and smooth. +Then +´ � 1 +2vf 2 +0 + |∂f0|2� +dx < ∞, where f0 denotes the p-wave scattering function. +Proof. For smooth v also the scattering function f0 is smooth. Recall the scattering equation +(5.5). Then a simple calculation using integration by parts shows that +ˆ �1 +2vf 2 +0 + |∂f0|2 +� +dx = 2 +ˆ ∞ +0 +� +f0∂2f0 + 2f0∂f0 +x ++ (∂f0)2 +� +dx = 2 +ˆ ∞ +0 +f0(x)2 − f0(0)2 +x2 +dx. +The function f0 is smooth and even. Thus for small x we have f(x) = f(0) + O(|x|2), hence +the integral converges around 0. By the decay of +1 +x2 the integral converges at ∞. We conclude +the desired. +We now give the proof of Theorem 1.9. We consider the trial state given in Equation (2.1) +where f is a rescaled scattering function +f(x) = +� +1 +1−a/bf0(|x|) +|x| ≤ b, +1 +|x| ≥ b +and +DN(x1, . . . , xN) = det[uk(xi)]1≤i≤N +k∈BF +, +uk(x) = +1 +L1/2eikx, +N = #BF. +In 1 dimension, there is no difference between a ball and a polyhedron, so we may use the Fermi +ball BF = {k ∈ 2π +L Z : |k| ≤ kF} for the momenta in the Slater determinant. In this case we +have (see [KL18, Lemma 3.2] or Lemma B.11) +Lemma 5.15. The Lebesgue constant of the Fermi ball satisfies +ˆ L/2 +−L/2 +1 +L +����� +� +k∈BF +eikx +����� dx = 1 +2π +ˆ 2π +0 +������� +� +q∈ +� +B +� LkF +2π +�� +∩Z2 +eiqu +������� +du ≤ C log N. +48 + +As for the 2-dimensional setting one easily generalizes the computation of the kinetic energy in +Lemma 2.13 and the calculation of the 2-particle reduced density for a Slater determinant in +Lemma 2.14. That is, +Lemma 5.16. The kinetic energy of the (Slater determinant with momenta in the) Fermi ball +satisfies +� +k∈BF +|k|2 = π2 +3 ρ2N +� +1 + O(N−1) +� +. +Lemma 5.17. The 2-particle reduced density of the (normalized) Slater determinant satisfies +ρ(2)(x1, x2) = π2 +3 ρ4|x1 − x2|2 � +1 + O(N−1) + O(ρ2|x1 − x2|2) +� +. +For the Gaudin-Gillespie-Ripka-expansion we replace occurrences of g and γ(1) +N +with their 1- +dimensional analogues as for the 2-dimensional setting. Here we have the bounds (using Lem- +mas 5.10 and 5.15) +ˆ +Λ +|g| ≲ a log(b/a), +ˆ +Λ +|γ(1) +N | ≲ log N. +(5.6) +Then, the 1-dimensional analogue of Theorem 3.4 reads +Theorem 5.18. There exists a constant c > 0 such that if aρ log(b/a) log N < c, then the +formulas in Equation (3.3) hold (with ρ(n) +Jas and Γn +π,G interpreted as appropriate for the 1- +dimensional setting.) +For the analogues of Lemmas 4.1 and 4.2 we have to a bit more careful. In order to get errors +smaller than the desired accuracy of the leading interaction term (of order aρ4 for the energy +density) we need to also do a Taylor expansion of (some of) the 3-particle diagrams. (Pointwise +we only have the bound +��Γ3 +π,G +�� ≤ Caρ4 log(b/a) log N (see Section 4.3 and Appendix A.2) for +any subleading diagram (π, G), i.e. for (π, G) ∈ ˜L3 +p with p ≥ 1.) +Remark 5.19 (Why this was not a problem for dimensions d = 2, 3). In dimensions d = 1, 2, 3 +the analoguous bound reads +��Γ3 +π,G +�� ≤ Csadρ4 log(b/a)(log N)d (if d = 1 then there is no s) for +any subleading diagram, see Equation (4.22). This bound should be compared to the energy +density of the leading interaction term of order adρ2+2/d. Considering just the power of ρ, we +see that such terms are subleading compared to the interaction term for d ̸= 1. +Similarly the argument for Γ2 is also slightly different compared to that of Lemma 4.1. We +have the bounds +Lemma 5.20. There exists a constant c > 0 such that if aρ log(b/a) log N < c, then +������ +∞ +� +p=1 +1 +p! +� +(π,G)∈L2p +Γ2 +π,G +������ +≤ Ca2ρ4 � +aρ(log(b/a))3(log N)3 + b4ρ4 � +1 + b2ρ2�� ++ Caρ5|x1 − x2|2 � +b4ρ4 + Nab6ρ7 + log(b/a) +� +and +ρ(3) +Jas ≤ Cf 2 +12f 2 +13f 2 +23 +� +ρ9|x1 − x2|2|x1 − x3|2|x2 − x3|2 + a2ρ5(log(b/a))2(log N)2 ++ aρ6 � +(b4ρ4 + log(b/a)) +� � +|x1 − x2|2 + |x1 − x3|2 + |x2 − x3|2�� +. +49 + +The proof is similar to that of Lemmas 4.1 and 4.2. We postpone it to the end of this section. +Note here that the N-dependence is not just via logarithmic factors. Thus, we need to be more +careful in choosing the size of the smaller boxes when applying the box method arguments of +Section 4.1. With this we get the analogue of Equation (4.1) in 1 dimension, +⟨ψN|HN|ψN⟩ += π2 +3 ρ2N +� +1 + O(N−1) +� ++ L +ˆ +dx +� +|∂f(x)|2 + 1 +2v(x)f(x)2 +� +× +� +π2 +3 ρ4|x|2 � +1 + O(N−1) + O(ρ2|x|2) +� ++ O +� +aρ5|x|2� +b4ρ4 + Nab6ρ7 + log(b/a) +�� ++ O +� +a2ρ4� +aρ(log b/a)3(log N)3 + b4ρ4 � +1 + b2ρ2��� � ++ +˚ +dx1 dx2 dx3 f12∂f12f23∂f23f 2 +13 +× +� +O(ρ9|x1 − x2|2|x1 − x3|2|x2 − x3|2) + O +� +a2ρ5(log(b/a))2(log N)2� ++ O +� +aρ6 � +b4ρ4 + log(b/a) +� � +|x1 − x2|2 + |x1 − x3|2 + |x2 − x3|2�� +� +. +(5.7) +For the 2-body error terms we may follow the type of arguments of Section 4, namely Equa- +tions (4.2), (4.3), (4.4), (4.5) and (4.6) exactly as for the 2-dimensional case. +By using +Lemma 5.10 we get the bounds +ˆ � +|∂f(x)|2 + 1 +2v(x)f(x)2 +� +|x|n dx ≤ +� +2a, +n = 2, +Ca2b, +n = 4, +ˆ +|x|nf∂f dx ≤ + + + + + +C, +n = 0, +Ca log(b/a), +n = 1, +Cab, +n = 2. +Define a0 by +1 +2a0 += +ˆ � +|∂f(x)|2 + 1 +2v(x)f(x)2 +� +dx +and recall by assumption on v that a0 > 0, i.e. that 1/a0 < ∞. For the 3-body terms we may +do as for the 3-dimensional case, Section 4. For the first term we bound |x1 − x3| ≤ 2b in the +support of ∂f12∂f23 and f13 ≤ 1. For the other terms we bound f13 ≤ 1. By the translation +50 + +invariance one integration gives a volume (i.e. length) factor L. That is, +˚ +dx1 dx2 dx3 +��f12∂f12f23∂f23f 2 +13 +�� +× +� +O(ρ9|x1 − x2|2|x1 − x3|2|x2 − x3|2) + O +� +a2ρ5(log(b/a))2(log N)2� ++ O +� +aρ6 � +b4ρ4 + log(b/a) +� � +|x1 − x2|2 + |x1 − x3|2 + |x2 − x3|2�� +� +≤ CNb2ρ8 +�ˆ b +0 +|x|2f∂f dx +�2 ++ CNa2ρ4(log(b/a))2(log N)2 +�ˆ b +0 +f∂f +�2 ++ CNaρ5 � +b4ρ4 + log(b/a) +� �ˆ b +0 +|x|2f∂f dx +� �ˆ b +0 +f∂f dx +� ++ CNaρ5 � +b4ρ4 + log(b/a) +� +��ˆ b +0 +|x|2f∂f dx +� �ˆ b +0 +f∂f dx +� ++ +�ˆ b +0 +|x|f∂f dx +�2� +. +≤ CNa2ρ4 � +b4ρ4 + (log(b/a))2(log N)2 + bρ +� +b4ρ4 + log(b/a) +�� +. +We conclude the analogue of Equation (4.7) in dimension 1 +⟨ψN|HN|ψN⟩ +L += π2 +3 ρ3 + 2π2 +3 aρ4 + O +� +N−1ρ3� ++ O +� +a2b−1ρ4� ++ O +� +a2bρ6� ++ O +� +a2ρ4a−1 +0 +� +aρ(log(b/a))3(log N)3 + b4ρ4 � +1 + b2ρ2��� ++ O +� +a2ρ5 � +b4ρ4 + Nab6ρ7 + log(b/a) +�� ++ O +� +a2ρ5 � +b4ρ4 + (log(b/a))2(log N)2 + bρ +� +b4ρ4 + log(b/a) +��� +. +(5.8) +We need to be careful how we choose N (i.e. how we choose L), since the error depends on N +not just via logarithmic terms. We choose +N = (aρ)−α, +α > 1 +b = a(aρ)−β, +0 < β < 1 +where the bounds on α, β are immediate for all the error-terms to be smaller than the desired +accuracy (there is similarly also an upper limit for α, which we do not write). Keeping then +only the leading error terms we get +⟨ψN|HN|ψN⟩ +L += π2 +3 ρ3 + 2π2 +3 aρ4 + O +� +N−1ρ3� ++ O +� +a2b−1ρ4� ++ O +� +a2a−1 +0 b4ρ8� ++ O +� +Na3b6ρ12� +. +(5.9) +Using the box method similarly as in Section 4.1 we also have to be careful with how we choose +the parameter d. As in Equation (4.9) we get +e(˜ρ) ≤ ⟨ψn|Hn|ψn⟩ +ℓ +[1 + O(d/ℓ) + O(b/ℓ)] + O +� +ρd−2� +≤ π2 +3 ρ3 + 2π2 +3 aρ4 + O +� +n−1ρ3� ++ O +� +a2b−1ρ4� ++ O +� +a2a−1 +0 b4ρ8� ++ O +� +na3b6ρ12� ++ O +� +dℓ−1ρ3� ++ O +� +bℓ−1ρ3� ++ O +� +ρd−2� +. +Here we change notation from N to n and choose d = a(aρ)−δ. To get the error smaller than +desired, we see that we need to choose δ > 3/2. In particular then the error is O(ρ3(aρ)γ), +where +γ = min{1 + β, 5 − 4β, 9 − α − 6β, α + 1 − δ, 2δ − 2}. +51 + +Then, also ˜ρ = ρ (1 + O((aρ)γ)) so ρ = ˜ρ(1 + O((a˜ρ)γ)). Optimising in α, β, δ we see that for +α = 12 +5 , +β = 13 +17, +δ = 32 +17 +(5.10) +we get γ = 30/17, i.e. +e(˜ρ) ≤ π2 +3 ˜ρ3 + 2π2 +3 a˜ρ4 � +1 + O +� +(a˜ρ)13/17�� +. +This concludes the proof of Theorem 1.9. +It remains to give the +Proof of Lemma 5.20. Note first that, completely analogously to Equations (4.10) and (4.22), +we have +1 +p! +����������� +� +(π,G)∈L2 +p +ng(π,G)+n∗ +g(π,G)=ng0 +k(π,G)=k0 +Γ2 +π,G +����������� +≤ Cρ2(C log N)k0(Caρ log(b/a))k0+ng0, +p = 2k0 + ng0 +1 +p! +������������ +� +(π,G)∈ ˜L3 +p +ng(π,G)+n∗ +g(π,G)=ng0 +k(π,G)=k0 +Γ3 +π,G +������������ +≤ Cρ3(C log N)k0(Caρ log(b/a))k0+ng0, +p = 2k0 + ng0. +(5.11) +We will use this to split the diagrams of L2 +p and ˜L3 +p into groups. We split diagrams in L2 +p into +three (exhaustive) groups: +1. Small diagrams with 1 ≤ k + ng + n∗ +g ≤ 2, {1} and {2} in different clusters +(A) and k ≥ 1, +(B) and k = 0, n∗ +g = 1. +2. Small diagrams with 1 ≤ k + ng + n∗ +g ≤ 2 and +(A) {1} and {2} in different clusters and k = 0, n∗ +g = 2, +(B) {1} and {2} in the same cluster, +3. Large diagrams with k + ng + n∗ +g ≥ 3. +We then split +∞ +� +p=1 +1 +p! +� +(π,G)∈L2p +Γ2 +π,G = ξsmall,0 + ξsmall,≥1 + ξ≥1, +where ξsmall,0 is the contribution of all small diagram in the first group, ξsmall,≥1 is the contribu- +tion of all small diagrams in the second group and ξ≥1 is the contribution of all large diagrams. +We will then do a Taylor expansion of ξsmall,0 but not of the other terms. +We split diagrams in ˜L3 +p into three (exhaustive) groups: +52 + +1. Small diagrams with k + ng + n∗ +g = 1 and {1}, {2} and {3} in 3 different clusters. (Then +ng = 0.) +2. Small diagrams with k + ng + n∗ +g = 1 and {1}, {2} and {3} in < 3 different clusters. +(Then k = ng = 0.) +3. Large diagrams with k + ng + n∗ +g ≥ 2. +We then split +∞ +� +p=1 +1 +p! +� +(π,G)∈ ˜L3p +Γ3 +π,G = ξ3 +small,0 + ξ3 +small,≥1 + ξ3 +≥1, +where ξ3 +small,0 is the contribution of all small diagram in the first group, ξ3 +small,≥1 is the contribu- +tion of all small diagrams in the second group and ξ3 +≥1 is the contribution of all large diagrams. +Again, we do a Taylor expansion of ξ3 +small,0 but not of the other terms. For simplicity we will +only compute the derivatives ∂2 +x1. With this bound the error term for the energy density is +O(a2bρ6 log(b/a)) and so it is even smaller than the accuracy a2ρ5 with b chosen as in Equa- +tion (5.10). (By the symmetry, we could bound ξsmall,0 by bounding its 6th derivative ∂2 +x1∂2 +x2∂2 +x3 +instead.) To keep the result symmetric in x1, x2, x3 we will symmetrize the result afterwards. +We have immediately by Equation (5.11) that +|ξ≥1| ≤ Ca3ρ5(log(b/a))3(log N)3, +��ξ3 +≥1 +�� ≤ Ca2ρ5(log(b/a))2(log N)2. +(5.12) +Similarly as in the proof of Lemma 4.1 we have for x1 = x2 +ξsmall,0(x2, x2) + ξsmall,≥1(x2, x2) + ξ≥1(x2, x2) = 0, +ξ3 +small,0(x2, x2, x3) + ξ3 +small,≥1(x2, x2, x3) + ξ3 +≥1(x2, x2, x3) = 0. +Hence we may bound the zeroth order by +|ξsmall,0(x2, x2)| ≤ |ξsmall,≥1(x2, x2)| + |ξ≥1(x2, x2)| , +��ξ3 +small,0(x2, x2, x3) +�� ≤ +��ξ3 +small,≥1(x2, x2, x3) +�� + +��ξ3 +≥1(x2, x2, x3) +�� . +For the diagrams in ξsmall,0 and ξ3 +small,0 we have similarly to Lemma 4.8 that +��∂2 +x1ξsmall,0 +�� ≤ Caρ5 log(b/a), +��∂2 +x1ξ3 +small,0 +�� ≤ Caρ6 log(b/a) +(5.13) +uniformly in x1, x2, x3. +For the diagrams in ξsmall,≥1 and ξ3 +small,≥1 the analysis is somewhat +similar to the proof of Lemma 4.6. We have +Lemma 5.21. For the small diagrams in ξsmall,≥1 and ξ3 +small,≥1 we have the bounds +|ξsmall,≥1| ≤ Ca2b4ρ8(1 + b2ρ2) + Cab4ρ9|x1 − x2|2 � +1 + Nab2ρ3� +, +(5.14) +��ξ3 +small,≥1 +�� ≤ Cab4ρ10 � +|x1 − x2|2 + |x1 − x3|2 + |x2 − x3|2� +(5.15) +uniformly in x1, x2, x3. +We give the proof of Lemma 5.21 in Appendix A.3. +Combining Lemma 5.21 and Equa- +tions (5.12) and (5.13) concludes the proof of Lemma 5.20. +Acknowledgements +A.B.L. would like to thank Johannes Agerskov and Jan Philip Solovej for valuable discus- +sions. We thank Alessandro Giuliani for helpful discussions and for pointing out the reference +[GMR21]. +Funding from the European Union’s Horizon 2020 research and innovation pro- +gramme under the ERC grant agreement No 694227 is acknowledged. +53 + +A +Small diagrams +In this appendix we compute the contributions of all the small diagrams of Lemmas 4.6, 4.8, +4.11 and 5.21. We first consider those of Lemmas 4.6 and 4.8. +A.1 +Small 2-particle diagrams (proof of Lemmas 4.6 and 4.8) +Recall from the proof of Lemma 4.1, Section 4.2 that +ξsmall,0 + ξsmall,≥1 = +∞ +� +p=1 +1 +p! +� +(π,G)∈L2 +p +(π,G) small +Γ2 +π,G. +The criterion for being small is defined in the proof of Lemma 4.1 around Equation (4.11), +and will be recalled below. The diagrams are split into types (A), (B) and (C) according their +underlying graphs G as in the proof of Lemma 4.1. We further split the type (B) into two types +(B1) and (B2). The diagrams of type (B1) are those diagrams for which the extra vertex {3} in +the distinguished clusters is in the cluster containing {1}, i.e. connected to {1}. The diagrams +of type (B2) are those diagrams for which the extra vertex {3} is in the cluster containing {2}, +i.e. connected to {2}. That is, the different types are as follows. See also Figure A.1. +(A) {1} and {2} in different clusters and 1 ≤ k ≤ 4, ng = 0, n∗ +g = 0, +(B) {1} and {2} in different clusters and 0 ≤ k ≤ 2, ng = 0, n∗ +g = 1, +(B1) and n∗ = 1, n∗∗ = 0, +(B2) and n∗ = 0, n∗∗ = 1, +(C) {1} and {2} in the same cluster and 0 ≤ k ≤ 2, ng = 0, n∗ +g = 1. +k +1 +2 +(a) Type (A), 1 ≤ k ≤ 4 +k +1 +2 +3 +(b) Type (B1), 0 ≤ k ≤ 2 +k +1 +2 +3 +(c) Type (B2), 0 ≤ k ≤ 2 +k +1 +2 +3 +(d) Type (C), 0 ≤ k ≤ 2 +Figure A.1: g-graphs of small diagrams of different types. For each diagram only +the graph G is drawn. The relevant diagrams come with permutations π such that +the diagrams are linked. +We first give the +Proof of Lemma 4.6. Consider first all diagrams of type (C) of smallest size, i.e. with g-graph +G0 = +1 +2 +54 + +Since this graph is connected, all π ∈ S3 give rise to a linked diagram (π, G0). By Wick’s rule, +the π-sum then gives the factor ρ(3). That is, +� +π∈S3:(π,G0)∈L2 +1 +Γ2 +π,G0 = +ˆ +g13g23 +� +π∈S3 +(−1)π +3 +� +j=1 +γ(1) +N (xj, xπ(j)) dx3 = +ˆ +g13g23ρ(3) dx3. +Recall the bound ρ(3) ≤ Cρ5|x1 − x2|2|x1 − x3|2|x2 − x3|2 from Lemma 2.15. Now we bound +|g23| ≤ 1 and |x2 − x3| ≤ b in the support of g23. Thus +������ +� +π∈S3:(π,G0)∈L2 +1 +Γ2 +π,G0 +������ +≤ Cb2ρ5|x1 − x2|2 +ˆ � +1 − f(x)2� +|x|2 dx. +Recalling Lemma 2.2 we may bound +ˆ � +1 − f(x)2� +|x|2 dx ≤ Ca5 + +C +(1 − a3/b3)2 +ˆ b +a +�� +1 − a3 +b3 +�2 +− +� +1 − a3 +r3 +�2� +r4 dr ≤ Ca3b2. +(A.1) +We conclude that all diagrams of smallest size contribute ≤ Ca3b4ρ5|x1 − x2|2. +For the larger diagrams, we consider an example diagram +(π, G) = +1 +2 +3 +For this diagram we have +Γ2 +π,G = (−1)π +˚ +γ(1) +N (x1; x4)γ(1) +N (x4; x3)γ(1) +N (x3; x1)γ(1) +N (x2; x5)γ(1) +N (x5; x2)g13g23g45 dx3 dx4 dx5 += −1 +L15 +� +k1,...,k5∈PF +˚ +eik1(x1−x4)eik2(x4−x3)eik3(x3−x1)eik4(x2−x5)eik5(x5−x2)g13g23g45 dx3 dx4 dx5 += −1 +L15 +� +k1,...,k5∈PF +ei(k1−k3)x1ei(k4−k5)x2 +ˆ +dx3 ei(k3−k2)x3g(x1 − x3)g(x2 − x3) +× +ˆ +dx4 +� +ei(k2−k1+k5−k4)x4 +ˆ +dx5 e−i(k5−k4)(x4−x5)g(x4 − x5) +� += −1 +L12 +� +k1,...,k5∈PF +ei(k1−k3)x1ei(k4−k5)x2 +ˆ +dx3 ei(k3−k2)x3g(x1 − x3)g(x2 − x3) +× χ(k2−k1=k4−k5)ˆg(k5 − k4), +where ˆg(k) := +´ +Λ g(x)e−ikx dx. Bounding |g23| ≤ 1 and |ˆg(k)| ≤ +´ +|g| ≤ a3 log(b/a) we get that +��Γ2 +π,G +�� ≤ Ca6ρ4(log(b/a))2. +One may do a similar computation for all the remaining diagrams. By computing the integra- +tions of the vertices in the internal clusters first, these give some factor ˆg(ki − kj) and a factor +L3χ(ki−kj=ki′−kj′). By bounding as above we conclude that the contribution of small diagrams +of type (C) is bounded as desired. +55 + +1 +2 +(a) Example of a type (A) di- +agram of smallest size +1 +2 +3 +(b) Example of a type (B1) di- +agram of smallest size +1 +2 +3 +(c) Example of a type (B2) di- +agram of smallest size +Figure A.2: Exemplary diagrams of types (A), (B1) and (B2). The dashed lines +denote g-edges, and the arrows denote (directed) edges of the permutation. +Proof of Lemma 4.8. As with the larger diagrams of type (C) we only give calculations for a +few example diagrams and explain how the calculation for the remaining diagrams are similar. +We consider the examples in Figure A.2. +The contribution of the diagram in Figure A.2a to ∂µ +x1∂ν +x1ξsmall is +1 +2∂µ +x1∂ν +x1Γ2 +π,G += −1 +2L12 +� +k1,...,k4∈PF +(kµ +1 − kµ +2)(kν +1 − kν +2) +¨ +eik1(x1−x3)eik2(x3−x1)eik3(x2−x4)eik4(x4−x2)g34 dx3 dx4 += −1 +2L12 +� +k1,...,k4∈PF +(kµ +1 − kµ +2)(kν +1 − kν +2)ei(k1−k2)x1ei(k3−k4)x2 +× +¨ +e−i(k4−k3)(x3−x4)ei(k2−k1+k4−k3)x3g(x3 − x4) dx3 dx4 += −1 +2L9 +� +k1,...,k4∈PF +(kµ +1 − kµ +2)(kν +1 − kν +2)ei(k1−k2)x1ei(k3−k4)x2χ(k2−k1=k3−k4)ˆg(k4 − k3) += O(ρ3+2/3a3 log(b/a)) +using that ˆg(k) = +´ +Λ g(x)e−ikx dx satisfies |ˆg(k)| ≤ +´ +|g| ≤ Ca3 log(b/a). The same type of +computation is valid for all other diagrams of type (A). +Consider now the diagram in Figure A.2c of type (B2). This contributes +∂µ +x1∂ν +x1 +−1 +L9 +� +k1,k2,k3∈PF +ˆ +eik1(x1−x2)eik2(x2−x1)g(x2 − x3) dx3 += 1 +L9 +� +k1,k2,k3∈PF +(kµ +1 − kµ +2)(kν +1 − kν +2)ei(k1−k2)x1ei(k2−k1)x2 +ˆ +g(x2 − x3) dx3 += O(ρ3+2/3a3 log(b/a)) +exactly as for type (A). Similarly, all other diagrams of type (B2) may be bounded using the +same method as for types (A). +56 + +Finally, we consider the diagram in Figure A.2b of type (B1). Here we have +∂µ +x1∂ν +x1Γ2 +π,G = ∂µ +x1∂ν +x1 +1 +L9 +� +k1,k2,k3∈PF +ˆ +eik1(x1−x3)eik2(x3−x2)eik3(x2−x1)g(x1 − x3) dx3 += ∂µ +x1∂ν +x1 +1 +L9 +� +k1,k2,k3∈PF +ei(k2−k3)x1ei(k3−k2)x2 +ˆ +e−i(k2−k1)(x1−x3)g(x1 − x3) dx3 += −1 +L9 +� +k1,k2,k3∈PF +(kµ +2 − kµ +3)(kν +2 − kν +3)ei(k2−k3)x1ei(k3−k2)x2ˆg(k2 − k1) += O(ρ3+2/3a3 log(b/a)). +All larger diagrams of type (B1) may be bounded similarly. We conclude the desired. +A.2 +Small 3-particle diagrams (proof of Lemma 4.11) +We now give the +Proof of Lemma 4.11. Recall that +ξ3 +small = +∞ +� +p=1 +1 +p! +� +(π,G)∈ ˜L3 +p +(π,G) small +Γ3 +π,G, +where “small” refers to diagrams with G-graph +G = +k = 1, 2 +k +1 +2 +3 +and permutation π such that (π, G) has at most two linked components, both of which contain +at least one external vertex. +As in the proof of Lemmas 4.6 and 4.8 in Appendix A.1 we +compute the value of a few examples and explain how to compute the value of the remaining +diagrams. We consider the examples of Figure A.3 +1 +2 +3 +(a) Example of a diagram of smallest size +with one linked component +1 +2 +3 +(b) Example of a diagram of smallest size +with two linked components +Figure A.3: Exemplary small diagrams in ˜L3 +2 +The contribution of the diagram in Figure A.3a is +Γ3 +π,G = −1 +L15 +� +k1,...,k5∈PF +¨ +eik1(x1−x4)eik2(x4−x2)eik3(x2−x1)eik4(x3−x5)eik5(x5−x3)g45 dx4 dx5 += −1 +L12 +� +k1,...,k5∈PF +ei(k1−k3)x1ei(k3−k2)x2ei(k4−k5)x3χ(k2−k1=k4−k5)ˆg(k5 − k4) += O(a3ρ4 log(b/a)). +57 + +Similarly, the contribution of the diagram in Figure A.3b is +Γ3 +π,G = −1 +L15 +� +k1,...,k5∈PF +¨ +eik1(x1−x4)eik2(x4−x5)eik3(x5−x1)eik4(x2−x3)eik5(x3−x2)g45 dx4 dx5 += −1 +L12 +� +k1,...,k5∈PF +ei(k1−k3)x1ei(k4−k5)x2ei(k5−k4)x3χ(k2−k1=k2−k3)ˆg(k3 − k2) += O(a3ρ4 log(b/a)). +One may follow this kind of computation for any diagram. The central property we used is +that the internal vertices are all in the same linked component as some external vertex. This +means that the integrals over internal vertices either gives a factor of ˆg(ki − kj) or a factor of +L3χ(ki−kj=ki′−kj′). We conclude the desired. +A.3 +Small diagrams in 1 dimension (proof of Lemma 5.21) +We now give the +Proof of Lemma 5.21. We first give the proof of Equation (5.14). We split the two cases (A) +and (B) of small diagrams further. They are given as follows. +(A) {1} and {2} in different clusters and k = 0, n∗ +g = 2, +(A1) n∗ = 2, n∗∗ = 0 (or n∗ = 0, n∗∗ = 2), +(A2) n∗ = 1, n∗∗ = 1. +(B) {1} and {2} in the same cluster and 1 ≤ k + ng + n∗ +g ≤ 2, +(B1) k = 0, +(B2) k = 1. +See also Figure A.4. +1 +2 +(a) Type (A1) +1 +2 +(b) Type (A2) +1 +2 +(c) Type (B1) +1 +2 +(d) Type (B2) +Figure A.4: g-graphs of small diagrams of different types. For each diagram only +the graph G is drawn. The relevant diagrams come with permutations π such the +the diagrams are linked. The diagrams of type (A1) and (B1) may have some of +the drawn g-edges not present, but the same connected components. +Moreover, +the diagrams of type (B1) may have one of the internal vertices drawn not present +(indicated by a ◦). With the modification of the drawings described here these are +all small diagrams. +We will consider some examples of diagrams. Namely those drawn in Figure A.4 (but not +modified as described in the caption), except for the diagram of type (B1), where we will +consider diagrams of smallest size, with g-graph +1 +2 +G0 = +(A.2) +58 + +All other diagrams can be treated in a similar fashion. For the argument we will need a different +formula for ρt. Recall the definition in Equation (3.4). We may write the characteristic function +as +χ((π, ∪Gℓ) linked) = 1 − χ((π, ∪Gℓ) not linked). +That is, +ρ(A1,...,Ak) +t += +� +π∈S∪Aℓ +(−1)π � +j∈∪Aℓ +γ(1) +N (xj; xπ(j)) − +� +π∈S∪Aℓ +(−1)πχ((π,∪Gℓ) not linked) +� +j∈∪Aℓ +γ(1) +N (xj; xπ(j)). +For our case we only need to consider cases where there are at most two clusters. If there is +just one cluster then ρ(A) +t += ρ(|A|)((xj)j∈A). So suppose we have two clusters A1, A2. Here, all +the π’s for which (π, ∪Gℓ) is not linked are exactly those arising as products π = π1π2, where +π1 ∈ SA1 and π2 ∈ SA1 are permutations of the vertices in the 2 clusters. Thus, +ρ(A1,A2) +t +((xj)j∈A1∪A2) = +� +π∈SA1∪A2 +(−1)π +� +j∈A1∪A2 +γ(1) +N (xj; xπ(j)) +− +� +π1∈SA1 +(−1)π1 � +j∈A1 +γ(1) +N (xj; xπ1(j)) +� +π2∈SA2 +(−1)π2 � +j∈A2 +γ(1) +N (xj; xπ2(j)) += ρ(|A1|+|A2|)((xj)j∈A1∪A2) − ρ(|A1|)((xj)j∈A1)ρ(|A2|)((xj)j∈A2). +(A.3) +We now consider the diagrams in Figures A.4a, A.4b and A.4d and (A.2). We get +Type (A1) : +� +π∈S4:(π,G0)∈L2 +2 +Γ2 +π,G0 = +¨ +g13g14g34ρ({1,3,4},{2}) +t +dx3 dx4, +Type (A2) : +� +π∈S4:(π,G0)∈L2 +2 +Γ2 +π,G0 = +¨ +g13g24ρ({1,3},{2,4}) +t +dx3 dx4, +Type (B1) : +� +π∈S3:(π,G0)∈L2 +1 +Γ2 +π,G0 = +ˆ +g13g23ρ(3) dx3, +Type (B2) : +� +π∈S5:(π,G0)∈L2 +3 +Γ2 +π,G0 = +˚ +g13g23g45ρ({1,2,3},{4,5}) +t +dx3 dx4 dx5. +(A.4) +Using Equation (A.3) and (the 1-dimensional versions of) Lemmas 2.14 and 2.15 and simi- +lar bounds for the 4- and 5-particle reduced densities we get the bounds on the truncated +correlations +(A1) +���ρ({1,3,4},{2}) +t +��� ≤ ρ(4)(x1, . . . , x4) + ρ(3)(x1, x3, x4)ρ(1)(x2) +≤ Cρ10|x1 − x3|2|x1 − x4|2|x3 − x4|2, +(A2) +���ρ({1,3},{2,4}) +t +��� ≤ ρ(4)(x1, . . . , x4) + ρ(2)(x1, x3)ρ(2)(x2, x4) +≤ Cρ8|x1 − x3|2|x2 − x4|2, +(B1) +ρ(3) ≤ Cρ9|x1 − x2|2|x1 − x3|2|x2 − x3|2, +(B2) +���ρ({1,2,3},{4,5}) +t +��� ≤ ρ(5)(x1, . . . , x5) + ρ(3)(x1, x2, x3)ρ(2)(x4, x5) +≤ Cρ13|x1 − x2|2|x1 − x3|2|x2 − x3|2|x4 − x5|2. +59 + +Bounding moreover, g34|x3−x4|2 ≤ b2 for the diagram of type (A1) we thus get by the translation +invariance +�������� +� +π∈S4:(π,G0)∈L2 +2 +type (A1) +Γ2 +π,G0 +�������� +≤ Cb2ρ10 +�ˆ +|g(x)||x|2 dx +�2 +. +For the diagram of type (A2) we get +�������� +� +π∈S4:(π,G0)∈L2 +2 +type (A2) +Γ2 +π,G0 +�������� +≤ Cρ8 +�ˆ +|g(x)||x|2 dx +�2 +. +For the diagram of type (B1) we get by bounding g23|x2−x3|2 ≤ b2 (as in the proof of Lemma 4.6) +�������� +� +π∈S3:(π,G0)∈L2 +1 +type (B1) +Γ2 +π,G0 +�������� +≤ Cb2ρ9|x1 − x2|2 +ˆ +|g(x)||x|2 dx. +Finally, for the diagram of type (B2) we get in the same way +�������� +� +π∈S5:(π,G0)∈L2 +3 +type (B2) +Γ2 +π,G0 +�������� +≤ CNb2ρ12|x1 − x2|2 +�ˆ +|g(x)||x|2 dx +�2 +. +We may bound +´ +|x|2|g| dx similarly as in 3 and 2 dimensions, +ˆ +R +� +1 − f(x)2� +|x|2 dx ≤ Ca3 + +C +(1 − a/b)2 +ˆ b +a +�� +1 − a +b +�2 +− +� +1 − a +r +�2� +r2 dr ≤ Cab2. +The other diagrams of types (A1) and (B1) (there are no other diagrams of type (A2) or (B2)) +we may treat similarly by bounding some of the g-edges by |g| ≤ 1. Combining these bounds +we conclude the proof of Equation (5.14). +To prove Equation (5.15) we recall that we consider all diagrams with g-graph +G0 = +1 +2 +3 +or +G1 = +1 +2 +3 +(and graphs that look like G0 where {1, 2, 3} are permuted). One may treat this similarly as +the diagrams above, with the result that +������ +� +π∈S4:(π,G0)∈L3 +1 +Γ3 +π,G0 +������ +≤ +ˆ +|g14||g24| +���ρ({1,2,4},{3}) +t +��� dx4 ≤ Cab4ρ10|x1 − x2|2 +������ +� +π∈S4:(π,G1)∈L3 +1 +Γ3 +π,G1 +������ +≤ +ˆ +|g14||g24||g34|ρ(4) dx4 ≤ Cab4ρ10|x1 − x2|2. +Summing this over all the permutations of {1, 2, 3} we conclude the proof of Equation (5.15). +60 + +B +Derivative Lebesgue constants (proof of Lemma 4.9) +In this appendix we give the proof of Lemma 4.9. We recall the statement in slightly different +notation for convenience. +Lemma 4.9. The polyhedron P from Definition 2.7 satisfies for any µ, ν = 1, 2, 3 that +ˆ +[0,2π]3 +����� +� +k∈RP ∩Z3 +kµeikx +����� dx ≤ CsR(log R)3, +ˆ +[0,2π]3 +����� +� +k∈RP ∩Z3 +kµkνeikx +����� dx ≤ CsR2(log R)4 +for sufficiently large R = LkF +2π . +Recall that by construction R ∼ N1/3 is rational. +The proof follows quite closely the argument in [KL18]. In particular the structure is that +of induction. The 3-dimensional integral is bounded one dimension at a time. We start by +introducing some notation from [KL18]. +Notation B.1. For any real number x we will write [x] for either ⌊x⌋ or ⌈x⌉. Similarly we will +write ⟨x⟩ = x − [x], i.e. ⟨x⟩ is either the fractional part {x} = x − ⌊x⌋ or x − ⌈x⌉. For any +computation we do below, the definition of [x] is fixed, but the computations hold with either +choice. +Additionally for a d-dimensional vector x = (x1, . . . , xd) we write x( ˜d) = (x1, . . . , x +˜d) for the +first ˜d ≤ d components. +We emphasize that expressions like k2, x3, . . . do not denote squares or cubes of numbers +k, x, but instead refer to coordinates of vectors k, x. The instances where we do want to denote +a square, cube or higher power should be clear. +By potentially relabelling the coordinates it suffices to consider the cases µ = 1, µ = ν = 1 +and µ = 1, ν = 2. +(Alternatively, by appealing to Lemma 2.11 and choosing Q ≳ N4 in +Definition 2.7 we have a symmetry of coordinates up to error-terms which are subleading +compared to Lemma 4.9.) Hence define +t1(k) = k1, +t2(k) = k1k1 = (k1)2, +t3(k) = k1k2. +We want to show that +ˆ +[0,2π]3 +����� +� +k∈RP ∩Z3 +tj(k)eikx +����� dx ≤ +� +CsR(log R)3 +j = 1, +CsR2(log R)4 +j = 2, 3. +As in the proof of Lemma 2.12 we write RP as a union of O(s) closed tetrahedra. We also recall +that Rz /∈ Z3. As in the proof of Lemma 2.12 we get by the inclusion exclusion principle O(s) +terms with tetrahedra of lower dimension (triangles or line segments). All the 3-dimensional +(closed) tetrahedra are convex and hence of the form +T = +� +k ∈ Z3 : λ1 ≤ k1 ≤ Λ1, λ2(k1) ≤ k2 ≤ Λ2(k1), λ3(k1, k2) ≤ k3 ≤ Λ3(k1, k2) +� +, +for some piecewise affine functions λi, Λi, i = 1, 2, 3. +They are the equations of the planes +bounding the tetrahedron T. Since any k ∈ T has integer coordinates we can replace Λj by +⌊Λj⌋ and λj by ⌈λj⌉. It will be convenient to not distinguish between ⌊·⌋ and ⌈·⌉ and use +instead the notation [·] introduced in Notation B.1. Then the tetrahedra are of the form +T = +� +k ∈ Z3 : [λ1] ≤ k1 ≤ [Λ1], [λ2(k1)] ≤ k2 ≤ [Λ2(k1)], [λ3(k1, k2)] ≤ k3 ≤ [Λ3(k1, k2)] +� +, +(B.1) +61 + +where we allow [·] to be different in any of the 6 instances it appears. +Sums over lower-dimensional tetrahedra can be written as differences of sums over 3- +dimensional tetrahedra (with potentially different meanings of [·]). We will thus only consider +3-dimensional tetrahedra. That is, for a tetrahedron T of the form Equation (B.1), we need to +bound +ˆ +[0,2π]3 +����� +� +k∈T∩Z3 +tj(k)eikx +����� dx ≤ +� +CR(log R)3 +j = 1, +CR2(log R)4 +j = 2, 3. +(B.2) +Gluing together tetrahedra as in Lemma 2.12 we conclude the desired bound, Lemma 4.9. The +remainder of this section gives the proof of Equation (B.2). +B.1 +Reduction to simpler tetrahedron +We first reduce to the case of a simpler tetrahedron T. Consider what happens by shifting all +k’s by some fixed lattice vector κ ∈ Z3 with |κ| ≤ CR. For t2 we have +� +k∈T∩Z3 +(k1)2eikx = +� +k∈(T−κ)∩Z3 +(k1 + κ1)2eikx += +� +k∈(T−κ)∩Z3 +(k1)2eikx + 2κ1 +� +k∈(T−κ)∩Z3 +k1eikx + (κ1)2 +� +k∈(T−κ)∩Z3 +eikx. +A similar computation holds for t1, t3. We may bound |κ| ≤ CR and thus we may assume that +T ⊂ [0, CR]3. (Recall that +´ +[0,2π]3 +��� +k∈T∩Z3 eikx�� dx ≤ C(log R)3 by [KL18, Theorem 4.1], see +the proof of Lemma 2.12.) +For any tetrahedron of the form (B.1) we may write the k-sum as three 1-dimensional sums +� +k∈T∩Z3 += +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +[Λ3(k1,k2)] +� +k3=[λ3(k1,k2)] += +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] + + +[Λ3(k1,k2)] +� +k3=0 +− +[λ3(k1,k2)−1] +� +k3=0 + + , +where the λj’s and Λj’s are the equations of the planes bounding the tetrahedron T, i.e. piece- +wise affine functions. As in Equation (B.1) each instance of [·] may be either of the definitions +of Notation B.1. By splitting the k1, k2 sums into at most 4 parts, we may ensure that both Λ3 +and λ3 − 1 are only from one bounding plane, i.e. they are affine functions. When we do this +splitting, we have to choose (in each new tetrahedron) which definition of [·] to use for the new +bounding plane. This may give rise to some “boundary term”, if we choose definitions of [·] in +the new tetrahedra such that the k’s on the splitting face are either in both or in neither of +the two tetrahedra sharing this face. These boundary terms are sums over lower-dimensional +tetrahedra, and may thus be bounded by sums over 3-dimensional ones as above. +Remark B.2. One may similarly let the k1- and k2-sums go from 0 by writing e.g. +[Λ2(k1)] +� +k2=[λ2(k1)] += +[Λ2(k1)] +� +k2=0 +− +[λ2(k1)−1] +� +k2=0 +. +However, the upper limits Λ3(k1, k2) and λ3(k1, k2)−1 for the k3-sum may become much larger +than R for k2 ≤ λ2(k1). This is why we don’t do this. +The terms with Λ3 and λ3 − 1 may be treated the same way, so we just look at the one with +Λ3. We thus want to bound +ˆ +[0,2π]3 +������ +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +tj(k1, k2) +[Λ3(k1,k2)] +� +k3=0 +eikx +������ +dx ≲ +� +R(log R)3 +j = 1, +R2(log R)4 +j = 2, 3. +62 + +B.2 +Reduction from d = 3 to d = 2 +We show that we may bound the three-dimensional integrals by analogous two-dimensional +integrals up to a factor of (log R + log Q) ∼ log N. +First, before shifting by a constant κ ∈ Z3, Λ3 is given by either the plane through 3 close +corners of RP (points Rσ(p1/Q1, p2/Q2, p3/Q3)) or of two close corners and the centre Rz. This +follows from the construction of P in Definition 2.7, since forming the edges between pairs of +close points constructs a triangulation of P. +The equation for a plane through the three points Rσ(p1 +j/Q1, p2 +j/Q2, p3 +j/Q3), j = 1, 2, 3 is +given by +α1 +Q2Q3 +k1 + +α2 +Q1Q3 +k2 + +α3 +Q1Q2 +k3 = Rσγ +where by construction of P, see Definition 2.7, we have +σ /∈ Q, +γ ∈ Q, +αj ∈ Z, +|αj| ≤ C +� +Q, +j = 1, 2, 3. +We might have that αj = 0. If α3 = 0 then this plane is parallel to the k3-axis and so does not +give rise to a bound on the k3-sum. Hence α3 ̸= 0. By choice of L, we have that R is rational, +and so Rσγ /∈ Q. (The choice of L such that R is rational, is exactly so that Rσγ /∈ Q.) The +equation for Λ3 is an integer shift of this plane, hence it is of the form +Λ3(k1, k2) = n3 − m1k1 − m2k2 = n3 − Q1α1 +Q3α3 +k1 − Q2α2 +Q3α3 +k2, +n3 /∈ Q, +|αj| ≤ C +� +Q, j = 1, 2, 3. +(B.3) +Define for j = 1, 2, 3 the quantities +Dj +3(x) := +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +tj(k1, k2) +[Λ3(k1,k2)] +� +k3=0 +eik(3)x(3), +˜Dj +2(x) := +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +tj(k1, k2)eik(2)x(2), +Gj +3(x) := +1 +eix3 − 1 +� +ei(n3+1)x3 ˜Dj +2(x(2) − m(2)x3) − ˜Dj +2(x(2)) +� +, +F j +3 (x) := ei(n3+1)x3 +eix3 − 1 +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +tj(k1, k2)eik(2)(x(2)−m(2)x3) � +e−i⟨Λ3(k(2))⟩x3 − 1 +� +, +(B.4) +where m(2) = (m1, m2) is defined in Equation (B.3). We shall prove the following bound. +Lemma B.3. We have for some k(2) +0 +∈ Z2, some (non-zero) κ = κ(2) ∈ Z2 and a h ∈ Z, h ≥ 0 +with |k(2) +0 | ≤ CR and h|κ(2)| ≤ CR that for any j = 1, 2, 3 +ˆ +[0,2π]3 +��Dj +3(x(3)) +�� dx(3) +≲ (log R + log Q) +ˆ +[0,2π]2 +��� ˜Dj +2(x(2)) +��� dx(2) + 1 + +ˆ 2π +0 +����� +h +� +τ=0 +tj +� +k(2) +0 ++ τκ(2)� +eiτ|κ(2)|x1 +����� dx1. +63 + +As a first step, consider the case where both α1 = α2 = 0 in Equation (B.3). Then the k3-sum +and x3-integral in Lemma B.3 factors out. Using [KL18, Lemma 3.2] to evaluate the k3-sum +and x3-integral we conclude the desired. Hence we can assume that at most one of α1, α2 is 0. +(This will be relevant for Lemma B.8, but only then.) +A simple calculation shows that [KL18, Lemma 3.1] +Dj +3(x) = Gj +3(x) + F j +3(x), +j = 1, 2, 3. +(B.5) +By a straightforward modification of the argument in [KL18, Lemma 3.3] (including the factor +tj) we have +Lemma B.4 ([KL18, Lemma 3.3]). For any j = 1, 2, 3 we have +ˆ +[0,2π]3 +��Gj +3(x) +�� dx ≲ log R +ˆ +[0,2π]2 +��� ˜Dj +2(x(2)) +��� dx(2). +We thus want to bound the integral of F j +3. Again, by a straightforward modification of the +argument in [KL18, Lemma 3.7] (including the factor tj) we have +Lemma B.5 ([KL18, Lemma 3.7]). For any j = 1, 2, 3 we have +ˆ +[0,2π]3 +��F j +3 (x) +�� dx ≲ +∞ +� +r=1 +(2π)r +r! +ˆ +[0,2π]2 +������ +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +tj(k1, k2)eik(2)x(2) � +Λ3(k(2)) +�r +������ +dx(2). +To bound the right hand side of Lemma B.5 we bound either definition of ⟨·⟩ by the fractional +part {·}. This follows the strategy in [KL18]. In analogy with [KL18, Lemma 3.6] we have +Lemma B.6 ([KL18, Lemma 3.6]). For either definition of ⟨·⟩ we have the bound +ˆ +[0,2π]2 +������ +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +tj(k1, k2)eik(2)x(2) � +Λ3(k(2)) +�r +������ +dx(2) +≤ +ˆ +[0,2π]2 +��� ˜Dj +2(x(2)) +��� dx(2) + +r +� +ν=1 +�r +ν +� ˆ +[0,2π]2 +������ +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +tj(k1, k2)eik(2)x(2){Λ3(k(2))}ν +������ +dx(2) +uniformly in (integer) r ≥ 1. +Proof. If ⟨·⟩ = {·} this is clear. Hence suppose that ⟨x⟩ = x − ⌈x⌉. Then +⟨x⟩ = {x} − 1 + +� +1 +if x ∈ Z +0 +otherwise. +By construction Λ3(k1, k2) /∈ Z for k1, k2 ∈ Z. Thus, ⟨Λ3(k1, k2)⟩ = {Λ3(k1, k2)} − 1. Then +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +tj(k1, k2)eik(2)x(2) � +Λ3(k(2)) +�r += +r +� +ν=0 +(−1)r−ν +�r +ν +� +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +tj(k1, k2)eik(2)x(2){Λ3(k(2))}ν += (−1)r ˜Dj +2(x(2)) + +r +� +ν=1 +(−1)r−ν +�r +ν +� +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +tj(k1, k2)eik(2)x(2){Λ3(k(2))}ν. +64 + +We now bound the second summand of Lemma B.6 similarly to [KL18, Lemmas 3.8 and 3.9]. +We first define ˜Λ3, a rational approximation of Λ3. Recall the definition of Λ3 in Equation (B.3). +By Dirichlet’s approximation theorem we may for any Q∞ find integers p, q with 1 ≤ q ≤ Q∞ +such that +γ3 := n3 − p +q +satisfies +|γ3| < +1 +qQ∞ +. +We will choose Q∞ = Q3α3. Define then +˜Λ3(k(2)) = Λ3(k(2)) − γ3 = p +q − Q1α1 +Q3α3 +k1 − Q2α2 +Q3α3 +k2. +(B.6) +Note that this takes values in +1 +qQ∞Z for integers k1, k2. +In particular (for integers k1, k2) +{˜Λ3(k(2))} ∈ {0, +1 +qQ∞, . . . , qQ∞−1 +qQ∞ }. Thus, since |γ3| < +1 +qQ∞ we have +{Λ3(k(2))} = γ3 + {˜Λ3(k(2))} + +� +1 +if γ3 < 0 and ˜Λ3(k(2)) ∈ Z, +0 +otherwise. +(B.7) +We claim that +Lemma B.7. For N sufficiently large, we have uniformly in (integer) r ≥ 1 that +ˆ +[0,2π]2 +������ +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +tj(k1, k2)eik(2)x(2){Λ3(k(2))}r +������ +dx(2) +≲ log(rQ) +ˆ +[0,2π]2 +��� ˜Dj +2(x(2)) +��� dx(2) + +ˆ +[0,2π]2 +��������� +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +˜Λ3(k1,k2)∈Z +tj(k1, k2)eik(2)x(2) +��������� +dx(2) + 2r +The proof differs from that of [KL18, Lemmas 3.8 and 3.9] in a few key location, so we give it +here. +Proof. Using Equation (B.7) we have +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +tj(k1, k2)eik(2)x(2){Λ3(k(2))}r += +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +tj(k1, k2)eik(2)x(2) � +γ3 + {˜Λ3(k(2))} +�r ++ χ(γ3<0) +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +˜Λ3(k1,k2)∈Z +tj(k1, k2)eik(2)x(2) + mixed terms. +All the mixed terms have at least one power of γ3 + {˜Λ3(k(2))} = γ3. (Indeed, in the mixed +terms we have ˜Λ3(k(2)) ∈ Z so {˜Λ3(k(2))} = 0.) Since |γ3| < 1/(qQ∞) ≤ 1/Q the sum of all +mixed terms may be bounded by 2rR4Q−1 ≲ 2r for N sufficiently large (independent of r) by +65 + +our choice of Q, see Definition 2.7. Similarly expanding the first summand, all the terms with +at least one power of γ3 may be bounded the same way. We thus have +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +tj(k1, k2)eik(2)x(2){Λ3(k(2))}r += +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +tj(k1, k2)eik(2)x(2){˜Λ3(k(2))}r ++ χ(γ3<0) +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +˜Λ3(k1,k2)∈Z +tj(k1, k2)eik(2)x(2) + O(2r), +(B.8) +where the error is O(2r) uniform in x(2). For the first summand we have by a simple modification +of [KL18, Lemma 3.8] (including the factor tj) that +ˆ +[0,2π]2 +������ +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +tj(k1, k2)eik(2)x(2){˜Λ3(k(2))}r +������ +dx(2) ≲ log(rqQ∞) +ˆ +[0,2π]2 +��Dj +2(x(2)) +�� dx(2). +This importantly uses that {˜Λ3(k(2))} ∈ {0, +1 +qQ∞, . . . , qQ∞−1 +qQ∞ } for integers k1, k2, so that on +can find some smartly chosen function h(u) ≈ ur on [0, 1] but with a smooth cut-off at 1 and +h({˜Λ3(k(2))}) = {˜Λ3(k(2))}r for which one can bound Fourier coefficients, see [KL18, Lemma +3.8]. +We have q ≤ Q∞ = Q3α3 ≤ CQ3/2. We conclude the desired. +Next we bound the second term in Lemma B.7, where ˜Λ3 is integer. If there are no valid choices +of k1, k2 for which ˜Λ3(k1, k2) is an integer, then this term is clearly zero. Otherwise we have +the following. +Lemma B.8. Let N be sufficiently large and suppose that the set +I0 = +� +(k1, k2) ∈ Z2 : [λ1] ≤ k1 ≤ [Λ1], [λ2(k1)] ≤ k2 ≤ [Λ2(k1)], ˜Λ3(k1, k2) ∈ Z +� +is non-empty. Then we may find a point k(2) +0 +∈ I0, a (non-zero) lattice vector κ = κ(2) ∈ Z2 +and an integer h ≥ 0 with k(2) +0 ++ hκ ∈ I0 (in particular h|κ| ≲ R) such that I0 = {k(2) +0 ++ τκ(2) : +τ ∈ {0, . . . , h}}. In particular +ˆ +[0,2π]2 +��������� +[Λ1] +� +k1=[λ1] +[Λ2(k1)] +� +k2=[λ2(k1)] +˜Λ3(k1,k2)∈Z +tj(k1, k2)eik(2)x(2) +��������� +dx(2) ≲ +ˆ 2π +0 +����� +h +� +τ=0 +tj +� +k(2) +0 ++ τκ(2)� +eiτ|κ(2)|x +����� dx. +(B.9) +The proof is an exercise in elementary number theory analysing the set I0. +Proof. Define k(2) +0 +to be any point in the (non-empty) set I0. Recall Equation (B.6), and that +���˜Λ3(k1, k2) +��� ≤ CR for any k(2) ∈ I0. (This follows since the relevant tetrahedron is contained +in [0, CR]3.) By redefining αj as αj/ gcd(α1, α2, α3) we may assume that α1, α2, α3 have no +shared prime factors. (This only decreases their values, so that still |αj| ≤ C√Q.) In case one +of the αj’s is zero we will use the convention that gcd(α, β, 0) = gcd(α, β) and gcd(α, 0) = α +for α, β > 0. +66 + +Solving the general problem. +We first consider the general problem of finding all k1, k2 ∈ Z +for which ˜Λ3(k1, k2) is an integer. This set has the form k(2) +0 +Γ for some two-dimensional lattice +Γ. We now find spanning lattice vectors of Γ. +Define αij = gcd(αi, αj) for i ̸= j. (Note that the αj’s are not necessarily pairwise coprime, +only all 3 αj’s have no shared factor by the reduction above. Also, since α1 and α2 are not +both 0, we have α12 ̸= 0 is well-defined.) Shifting k(2) +0 +by κ0 := (Q2 +α2 +α12 , −Q1 +α1 +α12) we have +˜Λ3(k(2) +0 ++ bκ0) = ˜Λ3(k(2) +0 ) ∈ Z, +b ∈ Z +and κ0 is the shortest lattice vector with this property. One should note here that κ0 is not +“short”. Indeed |κ0| ≳ Q since both Q1, Q2 ≳ Q, see Definition 2.7, and α1, α2 are not both +0. We now look for the lattice vector in Γ giving the smallest possible (integer) increase of ˜Λ3. +This lattice vector together with κ0 spans Γ. Note that +δ˜Λ3(κ) := ˜Λ3(k(2) +0 ++ κ) − ˜Λ3(k(2) +0 ) = −Q1α1κ1 − Q2α2κ2 +Q3α3 +. +(B.10) +Suppose first that either α1 = 0 or α2 = 0, say α2 = 0. Now, Q1 ̸= Q3 and |αj| ≤ C√Q so Qj +is not a factor of αi for any i = 1, 3, j = 1, 2, 3. Thus, gcd(Q3α3, Q1α1) = gcd(α1, α3) = 1 since +α2 = 0. For the ratio δ˜Λ3(κ) to be an integer we need that the numerator is some multiple of +Q3α3, and thus that |κ| ≳ Q3 ≫ R. Thus there is at most one k(2) +0 +∈ I0 and the lemma is clear. +Suppose then that α1 ̸= 0, α2 ̸= 0. Varying κ ∈ Z2 we have by B´ezout’s lemma that the +numerator in Equation (B.10) assumes as values all multiples of gcd(Q1α1, Q2α2). We have +gcd(Q1α1, Q2α2) = gcd(α1, α2) = α12. For the ratio δ˜Λ3(κ) to be an integer we need that the +numerator is some multiple of Q3α3. Since by assumption there are no prime factors shared by +all αj’s and Q3 is not a factor of α12 we have gcd(α12, Q3α3) = 1. Thus, the smallest integer +increase of ˜Λ3 is α12 ≥ 1 and this happens along some lattice vector κ1. Immediately then +Γ ⊃ {aκ1 + bκ0 : a, b ∈ Z}. To see that Γ ⊂ {aκ1 + bκ0 : a, b ∈ Z} note that by B´ezout’s lemma +the (integer) solutions to the equation +−Q1α1κ1 − Q2α2κ2 = Q3α3A, +for some integer A ∈ Z, is exactly (κ1, κ2) ∈ +� +A +α12 κ1 + bκ0 : b ∈ Z +� +if α12 divides A and there +are no solutions otherwise. In summary then +Γ = {aκ1 + bκ0 : a, b ∈ Z}, +˜Λ3(k(2) +0 ++ aκ1 + bκ0) = ˜Λ3(k(2) +0 ) + aα12, +a, b ∈ Z. +(B.11) +Moreover +I0 = +� +k(2) +0 ++ Γ +� +∩ +� +(k1, k2) ∈ Z2 : [λ1] ≤ k1 ≤ [Λ1], [λ2(k1)] ≤ k2 ≤ [Λ2(k1)] +� +. +Finding the candidate for κ. +We now find the candidate for the κ in the lemma. Either +I0 = {k(2) +0 }, in which case the lemma is clear (take h = 0), or there exists some (non-zero) +κ = aκ1 + bκ0 ∈ Γ such that k(2) +0 ++ κ ∈ I0. For such κ we have (for sufficiently large N) that +a ̸= 0 as |κ0| ≳ Q ≫ R and any such κ has |κ| ≤ CR. Let κ2 = a2κ1 + b2κ0 be the κ such +that k(2) +0 ++ κ ∈ I0 with minimal value of |a2|. (κ2 is unique up to potentially a sign if both +k(2) +0 +− κ2 ∈ I0 and k(2) +0 ++ κ2 ∈ I0.) It follows from Equation (B.11) that |a2| ≤ CR/α12 ≤ CR +since |δ˜Λ3(κ2)| ≤ CR as the tetrahedron is contained in [0, CR]3. +If b2 = 0 then a2 = ±1, else if b2 ̸= 0 then gcd(a2, b2) = 1. Indeed, if a2 and b2 shared some +common factor, we could factor this out to find a κ with smaller value |a| contradicting the +minimality of |a2|. +67 + +Characterizing all allowed κ’s. +We claim that by potentially redefining k(2) +0 +to k(2) +0 +− aκ2 +with a ∈ Z largest such that still k(2) +0 +− aκ2 ∈ I0 we have that +I0 = {k(2) +0 ++ τκ2 : τ ∈ {0, . . . , h}}, +for some h ∈ Z, h ≥ 0. +(B.12) +(The intuition for the remainder of the argument is as follows. +Essentially, if some κ had +k(2) +0 ++ κ ∈ I0 but was not a multiple of κ2, it would have to differ from some multiple of κ2 by +at least κ0 or κ1. Since |κ0| ≫ R and either κ1 = κ2 or |κ1| ≫ R, this is impossible.) +To prove Equation (B.12) we first introduce the following notation. We view a lattice vector +κ ∈ Z2 as a vector κ ∈ R2 and write κ∥ for its component parallel to κ0. Note that κ∥ need not +have integer coordinates. Define the constant A such that κ∥ +1 = Aκ0. (Note that A need not +be an integer.) Let 0 ̸= κ = aκ1 + bκ0 ∈ Γ with k(2) +0 ++ κ ∈ I0. We have +κ∥ = aκ∥ +1 + bκ0 = (aA + b)κ0. +Thus, since |κ0| ≳ Q, |κ| ≲ CR and |a| ≥ 1 (since κ ̸= 0) we have +�� b +a + A +�� ≤ CR +Q . +Using this also for κ2 = a2κ1 + b2κ0 we get +|ba2 − b2a| = +���� +b +a − b2 +a2 +���� |aa2| ≤ |aa2| +����� +b +a + A +���� + +����−b2 +a2 +− A +���� +� +≤ CR2R +Q ≪ 1. +But ba2 − b2a is an integer. Hence (for N sufficiently large) we have ba2 = b2a. Now, if b2 = 0 +then b = 0 and so a2 = ±1 is a divisor of a so κ = ±aκ2. If b2 ̸= 0 then gcd(a2, b2) = 1 and +thus a2 is again a divisor of a and a/a2 = b/b2. Then κ = +a +a2κ2 is a multiple of κ2. This shows +the desired. +Integral form. +To prove Equation (B.9) we do the following. Define e2 = κ2/|κ2| as the unit +vector parallel to κ2 and e⊥ +2 as the unit vector perpendicular to κ2. Then define the domain +S0 := +� +x(2) ∈ R2 : +��x(2) · e2 +�� ≤ 4π, +��x(2) · e⊥ +2 +�� ≤ 4π +� +and note that [0, 2π]2 ⊂ S0. Thus, using Equation (B.12) +ˆ +[0,2π]2 +����� +� +k∈I0 +tj(k1, k2)eik(2)x(2) +����� dx(2) ≤ +ˆ +S0 +����� +h +� +τ=0 +tj +� +k(2) +0 ++ τκ(2)� +eiτκ(2)x(2) +����� dx(2). +The integrand is constant in the e⊥ +2 -direction, and 2π-periodic in the e2-direction. Thus, com- +puting the integral in these coordinates we have +ˆ +S0 +����� +h +� +τ=0 +tj +� +k(2) +0 ++ τκ(2)� +eiτκ(2)x(2) +����� dx(2) = 32π +ˆ 2π +0 +����� +h +� +τ=0 +tj +� +k(2) +0 ++ τκ(2)� +eiτ|κ(2)|x +����� dx. +This concludes the proof. +Combining Lemmas B.5, B.6, B.7 and B.8 the r- and ν-sums in Lemmas B.5 and B.6 are readily +bounded because of the factor 1/r! from Lemma B.5. We conclude that +ˆ +[0,2π]3 +��F j +3(x) +�� dx ≲ log Q +ˆ +[0,2π]2 +��� ˜Dj +2(x) +��� dx + 1 + +ˆ 2π +0 +����� +h +� +τ=0 +tj +� +k(2) +0 ++ τκ(2)� +eiτ|κ(2)|x1 +����� dx1, +where k(2) +0 +and κ(2) are as in Lemma B.8. If the set I0 from Lemma B.8 is empty, then the +bound is valid without the last term. In particular it is valid with any k(2) +0 +∈ [0, CR]2, (non- +zero) κ = κ(2) ∈ Z2 and h = 0. Thus, by Lemma B.4 and Equation (B.5) we prove the desired +bound, Lemma B.3. +68 + +B.3 +Reduction from d = 2 to d = 1 +For j = 1, 2 we will do one more step reducing the dimension. The argument is basically the +same as for going from dimension d = 3 to d = 2 in Appendix B.2. We sketch the main +differences. +As we did in Appendix B.1 for d = 3 by adding and subtracting the lower tail of the sum, +we may assume that the k2-sum is �[Λ2(k1)] +k2=0 +. +Remark B.9. It is valid here to make the k2-sum go from 0, since now the k2-sum is the inner- +most sum and we do not risk values of k3 much larger that R by doing so (as in Remark B.2). +Indeed, we already computed the sum over the relevant k3. We could at this point also do +the same splitting of the k1-sum, but we would have the same problems that Λ2(k1) or λ2(k1) +might be much larger than R for k1 ≤ λ1 as in Remark B.2. +Additionally, by splitting the k1-sum into at most 2 parts, we may assume that Λ2 is just the +equation for a line. Here again one needs to be careful with what to do with the boundary +terms. This gives some sums over 1-dimensional tetrahedra (i.e. line segments), which we can +write as differences of sums over 2-dimensional tetrahedra exactly as for the 3-dimensional case. +We are led to define the quantities +Dj +2(x) := +[Λ1] +� +k1=[λ1] +tj(k1) +[Λ2(k1)] +� +k2=0 +eik(2)x(2), +˜Dj +1(x) := +[Λ1] +� +k1=[λ1] +tj(k1)eik1x1, +Gj +2(x) := +1 +eix2 − 1 +� +ei(n2+1)x2 ˜Dj +1(x1 − m1x2) − ˜Dj +1(x1) +� +, +F j +2(x) := ei(n2+1)x2 +eix2 − 1 +[Λ1] +� +k1=[λ1] +tj(k1)eik1(x1−m1x2) � +e−i⟨Λ2(k1)⟩x2 − 1 +� +. +We claim the following inductive bound. +Lemma B.10. For j = 1, 2 we have for N sufficiently large that +ˆ +[0,2π]2 +��Dj +2(x(2)) +�� dx(2) ≲ (log R + log Q) +ˆ +[0,2π] +��� ˜Dj +1(x1) +��� dx1 + +� +R +j = 1, +R2 +j = 2. +Proof. As for Λ3, we have that the equation of a line between any two points (p1 +i /Q1, p2 +i /Q2), +i = 1, 2 is given by +p1 +1 − p1 +2 +Q2 +k1 + p2 +2 − p2 +1 +Q1 +k2 = const . +If we choose the points to be either corners of RP or the central point Rz we get the equation +α1 +Q2 +k1 + α2 +Q1 +k2 = Rσγ /∈ Q. +Here we might have that α1 = 0 or α2 = 0. +If α2 = 0 this line is parallel to the k2-axis and so does not give rise to a bound for the +k2-sum. Thus α2 ̸= 0. If α1 = 0 the sum in Dj +2(x) and integral thereof factorizes, and hence by +69 + +[KL18, Lemma 3.2] we have that +ˆ +[0,2π]2 +��Dj +2(x(2)) +�� dx(2) ≤ C log R +ˆ 2π +0 +��� ˜Dj +1(x1) +��� dx1. +Hence, this case yields the desired inductive bound, Lemma B.10. Suppose then α1, α2 ̸= 0. +Then +Λ2(k1) = n2 − m1k1 = n2 − Q1α1 +Q2α2 +k1, +n2 /∈ Q, +|αj| ≤ CQ1/4, +j = 1, 2. +Lemmas B.4, B.5 and B.6 are readily adapted and proven as before. +The adaptation of +Lemma B.7 is then mostly analogous. One chooses Q∞ = Q2α2 and finds the rational ap- +proximation of Λ2 as +˜Λ2(k1) = Λ2(k1) − γ2 = p +q − Q1α1 +Q2α2 +k1, +|γ2| < +1 +qQ∞ +≤ Q−1. +The rest of the argument follows exactly as for d = 3 only that the extra term of the sum where +˜Λ2(k1) ∈ Z may be bounded as follows. +��������� +[Λ1] +� +k1=[λ1] +˜Λ2(k1)∈Z +tj(k1)eik1x1 +��������� +≤ +� +R +j = 1 +R2 +j = 2 +since there is at most one k1 such that ˜Λ2(k1) is an integer. To see this note that gcd(α1, Q2) = 1 +since |α1| ≤ CQ1/4 ≪ Q2, hence the change in k1 to change ˜Λ2(k1) by an integer is at least +Q3 ≫ R. We thus conclude the desired bound. +B.4 +Bounding the one-dimensional integrals +Now we bound +´ +| ˜Dj +1| and +´ ��� +�h +τ=0 tj +� +k(2) +0 ++ τκ +� +eiτ|κ|x��� dx from the right-hand-sides of Lem- +mas B.3 and B.10. For ˜Dj +1 we may assume that the lower bound of the summations are at 0 +by the same procedure as in Appendix B.1. Expanding tj +� +k(2) +0 ++ τκ +� +we see that j = 1 gives +an affine expression in τ and j = 2, 3 give quadratic expressions in τ. For instance, +t2 +� +k(2) +0 ++ τκ +� += (k1 +0)2 + 2k1 +0κ1τ + (κ1)2τ 2. +Thus, bounding both the integrals amounts to bounding the following: +Lemma B.11. Let M ≥ 2 be an integer. Then +(1) +ˆ 2π +0 +����� +M +� +k=0 +eikx +����� dx ≤ C log M, +(2) +ˆ 2π +0 +����� +M +� +k=0 +keikx +����� dx ≤ CM log M, +(3) +ˆ 2π +0 +����� +M +� +k=0 +k2eikx +����� dx ≤ CM2 log M. +70 + +Proof. The bound (1) is elementary, see also [KL18, Lemma 3.2]. For any M ∈ N and q ∈ C\{1} +we have +M +� +k=0 +qk = qM+1 − 1 +q − 1 +M +� +k=0 +kqk = +q +(q − 1)2 +� +qM(Mq − M − 1) + 1 +� +M +� +k=0 +k2qk = +q +(q − 1)3 +� +qM � +M2(q − 1)2 − 2M(q − 1) + q + 1 +� +− q − 1 +� +. +(B.13) +Consider now the integrals (2) and (3). By symmetry of complex conjugation +´ 2π +0 += 2 +´ π +0 . We +split the integrals according according to whether x ≤ 1/M or x ≥ 1/M. For x ≤ 1/M we have +ˆ 1/M +0 +����� +M +� +k=0 +keikx +����� dx ≲ +ˆ 1/M +0 +M2 dx ≲ M, +ˆ 1/M +0 +����� +M +� +k=0 +k2eikx +����� dx ≲ +ˆ 1/M +0 +M3 dx ≲ M2. +For x ≥ 1/M we use Equation (B.13) and note that |eix − 1| ≥ cx for x ≤ π. Expanding the +exponentials eix = 1 + O(x) we thus have +ˆ π +1/N +����� +M +� +k=0 +keikx +����� dx ≲ +ˆ π +1/M +1 +x2 +� +MeiMx(eix − 1) + 1 − eiMx� +dx +≲ +ˆ π +1/M +�M +x + 1 +x2 +� +dx ≲ M log M +and +ˆ π +1/N +����� +M +� +k=0 +k2eikx +����� dx ≲ +ˆ π +1/M +1 +x3 +� +eiMx � +M2(eix − 1)2 − 2M(eix − 1) + eix + 1 +� +− eix − 1 +� +dx +≲ +ˆ π +1/M +�M2 +x + M +x2 + 1 +x3 +� +dx ≲ M2 log M. +This concludes the proof. +With this we may thus bound for (j = 2, say) +ˆ 2π +0 +����� +h +� +τ=0 +t2 +� +k(2) +0 ++ τκ +� +eiτ|κ|x +����� dx += +ˆ 2π +0 +����� +h +� +τ=0 +� +(k1 +0)2 + 2k1 +0κ1τ + (κ1)2τ 2� +eiτ|κ|x +����� dx +≤ CR2 +ˆ 2π +0 +����� +h +� +τ=0 +eiτ|κ|x +����� dx + CR|κ| +ˆ 2π +0 +����� +h +� +τ=0 +τeiτ|κ|x +����� dx + C|κ|2 +ˆ 2π +0 +����� +h +� +τ=0 +τ 2eiτ|κ|x +����� dx . +Substituting y = |κ|x, using Lemma B.11 and recalling that h|κ| ≲ R and |κ| ≥ 1 by Lemma B.8 +we may bound this by R2 log R. An analoguous bound holds for j = 1. This takes care of all +the one-dimensional integrals. In combination with Lemmas B.3 and B.10 we get the bounds +for j = 1, 2 of Equation (B.2). It remains to consider the two-dimensional integral for j = 3. +71 + +B.5 +Bounding the j = 3 two-dimensional integral +We are left with bounding the integral +´ +| ˜D3 +2| on the right-hand-side of Lemma B.3. +We +first reduce to the case of a simpler tetrahedron (triangle). By shifting the sums by a fixed +κ = (κ1, 0) ∈ Z2 and using the bounds in Lemma B.11 to evaluate the extra contributions of the +shift, we may assume that the k1-sum starts at 0. By splitting the k2-sum as in Appendix B.1 +we may assume that that k2-sum also starts at 0. That is, we need to evaluate the integral +¨ +[0,2π]2 +������ +[Λ1] +� +k=0 +[Λ2(k)] +� +ℓ=0 +kℓeikxeiℓy +������ +dx dy, +where Λ2(k) = n2 − Q1α1 +Q2α2k for an irrational n2. Recall that |Λ1| ≤ CR and for any 0 ≤ k ≤ [Λ1] +we have |Λ2(k)| ≤ CR. +The analysis given here is in spirit the same as given in Appendices B.2, B.3 and B.4. It is +sufficiently different that we find it easier to do the arguments separately. We shall show the +following. +Lemma B.12. We have the following bound +¨ +[0,2π]2 +������ +[Λ1] +� +k=0 +[Λ2(k)] +� +ℓ=0 +kℓeikxeiℓy +������ +dx dy ≤ CR2(log R)2 log Q. +Combining then Lemmas B.3, B.10, B.11 and B.12 and choosing Q some sufficiently large +power of N as required in Definition 2.7 we conclude the proof of Equation (B.2) and thus of +Lemma 4.9. It remains to give the proof of Lemma B.12. +Proof. Denote M = [Λ1] and recall Λ2(k) = n2−m1k = n2− Q1α1 +Q2α2k. First note that by mapping +ℓ �→ [Λ2(k)] − ℓ we may assume that m1 ≥ 0 . If m1 = 0 the sum factors, and so does the +integral into two one-dimensional sums/integrals. These may be bounded using Lemma B.11. +In this case we get the bound ≤ CR2(log R)2 as desired. Hence assume that m1 > 0. Moreover, +if n2 > m1M we may split the (k, ℓ)-sum into two parts, +M +� +k=0 +[Λ2(k)] +� +ℓ=0 += +M +� +k=0 +[n2−m1M] +� +ℓ=0 ++ +M +� +k=0 +[Λ2(k)] +� +ℓ=[n2−m1M]+1 +. +The first sum factors into one-dimensional integrals which we may bound using Lemma B.11 +again. The second we may shift by a constant ℓ (again then using Lemma B.11 to evaluate the +contribution of the shift) and assume that the lower limit of the ℓ-sum is 0. The upper limit +then becomes [Λ(k)], where +Λ(k) = n2 − ([n2 − m1M] + 1) − m1k := n − mk . +Geometrically, this means that the domain of the (k, ℓ)-sum is a triangle with two sides along +the axes. We thus need to bound +¨ +[0,2π]2 +������ +M +� +k=0 +[Λ(k)] +� +ℓ=0 +kℓeikxeiℓy +������ +dx dy, +where +Λ(k) = n − mk, +M ≤ R, +mM = n + O(1), +n ≤ R. +72 + +By the symmetries of translation invariance and complex conjugation we may integrate over +the domain [−π, π] × [0, π] instead. We evaluate the ℓ-sum using Equation (B.13). Recall that +[Λ(k)] = Λ(k) − ⟨Λ(k)⟩. We thus have +[Λ(k)] +� +ℓ=0 +ℓeiℓy = +eiy +(eiy − 1)2 +� +ei[Λ(k)]y([Λ(k)]eiy − [Λ(k)] − 1) + 1 +� += +eiy +(eiy − 1)2 +�� +eiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1 +� ++ +� +e−i⟨Λ(k)⟩y − 1 +� � +eiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1 +� +− +� +⟨Λ(k)⟩ ei(Λ(k)−⟨Λ(k)⟩)y(eiy − 1) + (e−i⟨Λ(k)⟩y − 1) +�� +=: (I) + (II) + (III). +The third summand (III) may be calculated as +−eiy +(eiy − 1)2 +� +⟨Λ(k)⟩ +� +ei[Λ(k)]y − 1 +� +iy + O(y2) +� +. +The factor +−eiy +(eiy−1)2 may be bounded by 1/y2. For this term we split the y-integral according to +whether y ≤ 1/n or y ≥ 1/n. For y ≤ 1/n we expand additionally ei[Λ(k)]y − 1 = O(ny). We +get the contribution +ˆ π +−π +dx +ˆ 1/n +0 +dy 1 +y2 +����� +M +� +k=0 +keikx � +⟨Λ(k)⟩ +� +ei[Λ(k)]y − 1 +� +y + O(y2) +� +����� ≲ 1 +nM2n + 1 +nM2 ≲ R2. +For y ≥ 1/n we bound ei[Λ(k)]y − 1 = O(1). We get +ˆ π +−π +dx +ˆ π +1/n +dy 1 +y2 +����� +M +� +k=0 +keikx � +⟨Λ(k)⟩ +� +ei[Λ(k)]y − 1 +� +y + O(y2) +� +����� ≲ (log n)M2 + M2 ≲ R2 log R. +For the second summand (II) we again split the integral according to whether y ≤ 1/n or +y ≥ 1/n. If y ≤ 1/n we have +eiy +(eiy − 1)2 +� +e−i⟨Λ(k)⟩y − 1 +� � +eiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1 +� += O(Λ(k)2y) = O(n). +Hence this contributes the term +ˆ π +−π +dx +ˆ 1/n +0 +dy +����� +M +� +k=0 +keikxO(Λ(k)2y) +����� ≲ 1 +nM2n ≲ R2. +For y ≥ 1/n we write +eiy +(eiy − 1)2 +� +e−i⟨Λ(k)⟩y − 1 +� � +eiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1 +� += +eiy +(eiy − 1)2 +∞ +� +ν=1 +(−iy)ν +ν! +⟨Λ(k)⟩ν � +eiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1 +� +. +73 + +Again we bound the factor +eiy +(eiy−1)2 as 1/y2. We treat each summand similarly as in Lemmas B.6 +and B.7 (or rather, the 2-dimensional version of these as used in Appendix B.3.) Completely +analogously to Lemma B.6 we see that for any integer r ≥ 1 we have +¨ ����� +M +� +k=0 +keikxeiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1 +y2 +⟨Λ(k)⟩r +����� dx dy +≤ +¨ ����� +M +� +k=0 +keikxeiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1 +y2 +����� dx dy ++ +r +� +ν=1 +�r +ν +� ¨ ����� +M +� +k=0 +keikxeiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1 +y2 +{Λ(k)}ν +����� dx dy, +for either definition of ⟨·⟩ (i.e. +either ⟨·⟩ = {·} or ⟨·⟩ = · − ⌈·⌉). +Also the application of +Lemma B.7 is analogous to its use in Appendix B.3. There is at most one k such that ˜Λ(k) ∈ Z +for the appropriate rational approximation ˜Λ of Λ. Using that eiy = 1 + O(y) we obtain the +bound +����keikxeiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1 +y2 +���� ≲ M ny + 1 +y2 +≲ R2 +y + R +y2, +valid for any k. Hence this error term contributes at most +ˆ π +−π +dx +ˆ π +1/n +dy +�R2 +y + R +y2 +� +≲ R2 log n + Rn ≲ R2 log R. +The rest of the argument in Lemma B.7 is the same. We conclude that we may bound the +contribution of the term (II) by that of (I) up to a factor of log Q and an error R2 log R, i.e. +¨ ����� +� +k +keikx(II) +����� dx dy ≲ log Q +¨ ����� +� +k +keikx(I) +����� dx dy + R2 log R. +In particular +¨ +[0,2π]2 +������ +M +� +k=0 +[Λ(k)] +� +ℓ=0 +kℓeikxeiℓy +������ +dx dy +≲ log Q +ˆ π +−π +dx +ˆ π +0 +dy +����� +M +� +k=0 +keikxeiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1 +y2 +����� + R2 log R. +(B.14) +In order to evaluate the integral on the right-hand side, we split the integration domain into 5 +regions, see Figure B.1. +I1 = {|x| ≤ 2/M, y ≤ 2/n}, +I2 = {|x| ≤ 1/M, y ≥ 2/n}, +I3 = {y ≤ 1/n, |x| ≥ 2/M}, +I4 = {y ≥ 1/n, |x| ≥ 1/M, |x − my| ≥ 1/M} +I5 = {|x − my| ≤ 1/M, (x, y) /∈ I1}. +We will be a bit sloppy with notation and refer to both the domain of integration and the value +of the integration over that domain by Ij. +(I1). We expand +(∗) := +M +� +k=0 +keikxeiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1 +y2 +74 + +x +y +2/M +−2/M +2/n +x = my +I1 +I2 +I3 +I3 +I4 +I4 +I4 +I5 +Figure B.1: Decomposition of the domain [−π, π] × [0, π] into different regions. +(or rather the numerator) to second order in y. Using that Λ(k) = O(n) we get that (∗) ≲ M2n2. +Thus the integral gives +I1 ≲ +ˆ 2/M +−2/M +dx +ˆ 2/n +0 +dyM2n2 ≲ Mn ≲ R2. +(I2). +We expand eiy = 1 + O(y) in (∗). +Then (∗) ≲ +M2n +y ++ M2 +y2 . +The integral is then +I2 ≲ R2 log R. +(I3, I4, I5). For the remaining integrals we use the explicit formula for Λ(k) = n−mk. Then +(∗) = 1 +y2 +M +� +k=0 +keikx(eiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1) += 1 +y2 +M +� +k=0 +� +keikx + keik(x−my)einy(neiy − n − 1) + k2eik(x−my)einy(m − meiy) +� += 1 +y2 +� +−i∂D(x) − ieiny(neiy − n − 1)∂D(x − my) + meiny(eiy − 1)∂2D(x − my) +� +, +(B.15) +where we introduced D(z) = �M +k=0 eikz = ei(M+1)z−1 +eiz−1 +. From Equation (B.13) we conclude that +we may bound derivatives of D as +|∂D(z)| ≲ M +z + 1 +z2, +|∂2D(z)| ≲ M2 +z ++ M +z2 + 1 +z3, +|∂3D(z)| ≲ M3 +z ++ . . . + 1 +z4. +(B.16) +(I3). We have y ≤ 1/n and |x| ≥ 2/M. We expand Equation (B.15) to second order in y. +Expanding first the exponentials and then derivatives of D where needed we get +(B.15) = 1 +y2 +� +− i∂D(x) + i∂D(x − my) + imy∂2D(x − my) ++ O(n2y2∂D(x − my)) + O(nmy2∂2D(x − my)) +� +≲ n2 sup +y |∂D(x − my)| + nm sup +y |∂2D(x − my)| + m2 sup +y |∂3D(x − my)|. +75 + +Now we use the bounds Equation (B.16) and use that z := x − my has |z| ≥ |x| − m/n = +|x| − 1/M + O(1/(Mn)) (recall that mM = n + O(1)) and |x| ≥ 2/M. Thus +I3 ≲ 1 +n +ˆ π +1/M +dz +� +n2|∂D(z)| + nm|∂2D(z)| + m2|∂3D(z)| +� +≲ R2 log R. +(I4). We expand the exponentials eiy = 1 + O(y). Then +|(B.15)| ≤ |∂D(x)| +y2 ++ n|∂D(x − my)| +y ++ |∂D(x − my)| +y2 ++ m|∂2D(x − my)| +y +. +Using the bounds Equation (B.16) as before and noting that |x| ≥ 2/M and z = x − my has +|z| ≥ 1/M one easily sees that I4 ≲ R2(log R)2. +(I5). Again, expanding the exponentials eiy = 1 + O(y) we have as for I4 that +|(B.15)| ≤ |∂D(x)| +y2 ++ n|∂D(x − my)| +y ++ |∂D(x − my)| +y2 ++ m|∂2D(x − my)| +y +. +We use the bounds +|∂D(z)| = +����� +M +� +k=0 +keikz +����� ≤ M2, +|∂2D(z)| = +����� +M +� +k=0 +k2eikz +����� ≤ M3. +Thus +I5 ≲ 1 +M +ˆ π +1/n +M2 +y2 + nM2 + mM3 +y +dy ≲ R2 log R. +We conclude that +ˆ π +−π +dx +ˆ π +0 +dy +����� +M +� +k=0 +keikxeiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1 +y2 +����� ≲ R2(log R)2. +Together with Equation (B.14) this concludes the proof. +References +[ARS22] +J. Agerskov, R. Reuvers, and J. P. Solovej. “Ground state energy of dilute Bose gases in 1D”. +2022. doi: 10.48550/arXiv.2203.17183. +[BCGOPS22] +G. Basti, S. Cenatiempo, A. Giuliani, A. Olgiati, G. 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Phys. 136.3 (2009), pp. 453–503. doi: 10.1007/s10955-009-9792-3. +78 + diff --git a/ENE4T4oBgHgl3EQfGgwU/content/tmp_files/load_file.txt b/ENE4T4oBgHgl3EQfGgwU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..95c014cb38b098c95332b2842eadf4057654b72f --- /dev/null +++ b/ENE4T4oBgHgl3EQfGgwU/content/tmp_files/load_file.txt @@ -0,0 +1,4098 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf,len=4097 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='04894v1 [math-ph] 12 Jan 2023 Ground state energy of the dilute spin-polarized Fermi gas: Upper bound via cluster expansion Asbjørn Bækgaard Lauritsen∗ and Robert Seiringer† IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria 13 January 2023 Abstract We prove an upper bound on the ground state energy of the dilute spin-polarized Fermi gas capturing the leading correction to the kinetic energy resulting from repulsive interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' One of the main ingredients in the proof is a rigorous implementation of the fermionic cluster expansion of Gaudin, Gillespie and Ripka (Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' A, 176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='2 (1971), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' 237–260).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' Contents 1 Introduction and main results 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='1 Precise statement of results .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' 15 3 Gaudin-Gillespie-Ripka-expansion 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='1 Calculation of the normalization constant .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='2 Calculation of the 1-particle reduced density .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='3 Calculation of the 2-particle reduced density .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='4 Calculation of the 3-particle reduced density .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='3 Absolute convergence of the Γ2-sum .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' 30 ∗alaurits@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='at †robert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='seiringer@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='at 1 4 Energy of the trial state 31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='1 Thermodynamic limit via a box method .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' 34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='2 Subleading 2-particle diagrams (proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' 42 5 One and two dimensions 43 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='1 Two dimensions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' 43 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='2 One dimension .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' 47 A Small diagrams 54 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='1 Small 2-particle diagrams (proof of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='8) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' 54 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='2 Small 3-particle diagrams (proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='11) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' 57 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='3 Small diagrams in 1 dimension (proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='21) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' 58 B Derivative Lebesgue constants (proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='9) 61 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='1 Reduction to simpler tetrahedron .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' 62 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='2 Reduction from d = 3 to d = 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' 70 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='5 Bounding the j = 3 two-dimensional integral .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' 72 1 Introduction and main results We consider a Fermi gas of N particles in a box Λ = ΛL = [−L/2, L/2]d in d dimensions, d = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' We will mostly focus on the case d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' The particles interact via a two-body interaction v, which we assume to be positive, radial and of compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' In particular we allow for v to have a hard core, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' v(x) = ∞ for |x| ≤ r for some r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'} +page_content=' In natural units where ℏ = 1 and the mass of the particles is m = 1/2 the Hamiltonian of the system takes the form HN = N � i=1 −∆xi + � j 𝜖 +17: x𝑛+1 ← x, v𝑛+1 ← (x − x𝑛)/ℎ +18: return x𝑛+1, v𝑛+1 +Similarly, one can also separate elasticity from contact energy +using the contact proxy. In this fashion, we would have a three-phase +(fluid, solid, and contact) time splitting scheme +x𝑛+1/3 = arg min +x +1 +2 ∥x − ˆx𝑛∥2 +M + ℎ2(𝑃(x) + ˆ𝐶sf(x)), +x𝑛+2/3 = arg min +x +1 +2 ∥x − x𝑛+1/3∥2 +M + ℎ2(Ψ(x) + ˆ𝐶sf(x) + ˆ𝐶ss(x)), +x𝑛+1 = arg min +x +1 +2 ∥x − x𝑛+2/3∥2 +M + ℎ2( 1 +3𝐶sf(x) + 1 +2𝐶ss(x)), +(26) +where ˆ𝐶sf(x) and ˆ𝐶ss(x) are the 2nd-order Taylor expansion of +1 +3𝐶sf(x) and 1 +2𝐶ss(x) respectively. However, this aggressive split- +ting scheme only applies to inversion-robust constitutive models, +e.g. the fixed corotated model [Stomakhin et al. 2012]. While inver- +sion can be prevented with guarantee at the solid phase where the +elasticity energy is considered, it may not hold at the contact phase. +Despite this limitation, the three-phase splitting scheme can still +work properly for inversion-robust constitutive models in practice +to further accelerate the simulation. + +HoudiniaiupnohiupnohA Contact Proxy Splitting Method for Lagrangian Solid-Fluid Coupling +• +7 +4.2 +Solving Linear Systems +In our time splitting scheme, solving large sparse linear systems +dominates both the computational and memory costs of each phase. +We thus devise matrix-free and Schur-complement based strategies +to solve them efficiently. +4.2.1 +Fluid Phase. Since 2-ring neighbors of SPH particles need to +be considered in our formulation, both constructing and directly +factorizing the Hessian matrix can cost a significant amount of time +and memory. Therefore, we devise a matrix-free conjugate gradient +(CG) solver to efficiently solve for the intermediate state of fluids. +As all energy potentials are quadratic in this phase, the energy +gradient g(x) is merely a linear function of x with constant coeffi- +cient matrix H(x). Thus, the product between H(x) and an arbitrary +vector p can be expressed as +H(x)p = g(p) − g(0). +(27) +This allows us to compute gradients to evaluate the matrix-vector +product, and we only need to acquire the 3 × 3 diagonal blocks of +the Hessian for block-Jacobi preconditioning in our CG solver. +4.2.2 +Solid-Coupling Phase. As the fluid energy potential is not +included in this phase, the components of the Hessian matrix become +H𝑓 = 𝜕2𝐶(x) +𝜕x2 +𝑓 +, +G = 𝜕2𝐶(x) +𝜕x𝑠𝜕x𝑓 +, +H𝑠 = 𝜕2𝐶(x) +𝜕x2𝑠 ++ 𝜕2Ψ(x) +𝜕x2𝑠 +. +(28) +Although this linear system is no longer that intractable, it is not +optimal to directly factorize the whole system given the considerable +amount of nonzeros in H𝑓 and G when fluid resolution is high. +We thus design a domain decomposed linear solver that treats +H𝑓 and H𝑠 separately. Based on Schur complement [Zhang 2006], +the inverse of our Hessian matrix can be expressed as +H−1 = +� +H−1 +𝑓 ++ H−1 +𝑓 G(H/H𝑓 )−1G𝑇 H𝑓 +−H−1 +𝑓 G(H/H𝑓 )−1 +−(H/H𝑓 )−1G𝑇 H𝑓 +(H/H𝑓 )−1 +� +, (29) +where H/H𝑓 = H𝑠 − G𝑇 H−1 +𝑓 G is the Schur complement of block +H𝑓 . Since the nonzeros of H𝑓 only exist in the diagonal blocks, it +is trivial to obtain its inverse matrix H−1 +𝑓 . We can then apply the +CHOLMOD [Chen et al. 2008] LLT solver to factorize H/H𝑓 , which +is only in the size of solid DOFs, and then the search direction can +be computed via matrix-vector products and back-solves. When +there is no solid-fluid interaction, H/H𝑓 ’s sparsity pattern remains +identical with H𝑠. Only when two solid nodes 𝑖 and 𝑗 are interacting +with the same fluid particle, the 3 × 3 block (H/H𝑓 )𝑖,𝑗 (in 3D) will +become non-zero. Typically, this only happens for neighboring mesh +primitives and thus the sparsity pattern of H/H𝑓 is mostly nice. +Note that when the three-phase time splitting scheme (Eq. 26) is +used, our domain decomposed solver can also be applied to the solid +and contact phases since their systems share a similar structure +with the solid-coupling phase here. +5 +EXPERIMENTS AND EVALUATION +Our code is implemented in C++ with Eigen for basic linear algebra +operations and Intel TBB for multi-threading. The time step size +of all our simulations is adaptively chosen by the SPH CFL con- +dition and a user-defined upper bound. We set the support radius +Initial +(A) +(B) +(C) +(a) A viscous armadillo dropped onto the ground. +(A) +(B) +(C) +(b) Cube on cloth. An elastic cube is dropped onto a square cloth with +four corners fixed. +Fig. 7. Simulation results of (A) Joint Optimization, (B) Time Splitting with +Contact Proxy, and (C) Baseline Time Splitting. While directly applying time +splitting results in instability at the boundaries, our results with contact +proxy are consistent with joint optimization. +of our SPH kernel function to 2𝑑, where 𝑑 is the particle diameter. +In our implementation, we use the cubic Spline kernel for density +estimation and the Spiky kernel for gradient calculation. For Fig. +5, 4, 2 and 6, we employ our three-phase time splitting scheme, +showing its efficacy when the constitutive models are compatible +with mesh inversion. For the rest of the simulations, we stick with +our two-phase time splitting scheme. Most experiments are per- +formed on a 24-core 3.50GHz Intel i9-10920X machine, except for +the comparative study with ElastoMonolith [Takahashi and Batty +2022]. We demonstrate that our method achieves efficient and robust +solid-fluid coupling. The parameters and timing breakdown of all +simulations are provided in Table 1 and Fig. 12 respectively. +5.1 +Ablation Study +5.1.1 +Time Splitting Evaluation. Three simulations (Fig. 6 and Fig. +7) are performed to demonstrate the efficiency of time splitting and +the efficacy of our proposed contact proxy on maintaining stability. +To begin with, we need to take care of choosing a proper time step +ℎ. First of all, it has to be restricted by the CFL condition. Otherwise, +severe volume loss may be observed due to SPH approximation +error. Additionally, in contrast to the joint optimization (Eq. 19), the +time splitting scheme usually requires smaller time steps to stay +stable, which imposes a second time step constraint. However, we +observed that in practice, even using the largest CFL time step, our +proxy-assisted time splitting can still work properly and produce +stable simulation results. Hence, for comparison, we use the largest +CFL time step for both schemes to maximize their performance +as smaller ℎ typically takes more Newton’s iterations in total to +simulate a frame. For joint optimization, since direct factorization is +intractable, we solve Eq. 20 using the block-Jacobi preconditioned +conjugate gradient solver with the fluid part matrix free. + +HoudiniaHoudiniaHoudiniaHoudinia8 +• +Xie et al. +Scene +Δ𝑡frame +Δ𝑡 +𝑁fluid +𝑁solid +𝑘𝐼 +𝜈𝑓 +𝑑 +𝜌𝑓 +E +𝜈𝑠 +𝜌𝑠 +𝑇 +Fig. 6 Bob +1/24 +4 × 10−3 +97K +2.3K +2 × 105 +0 +15 +1000 +1 × 105 +0.3 +500 +0.3 +Fig. 7a Viscous Armadillo +1/48 +4 × 10−3 +238K +0 +1 × 105 +100 +10 +1200 +- +- +- +0.4 +Fig. 2 Shot Armadillo +1/24 +4 × 10−3 +103K +16K +1 × 105 +0 +10 +1000 +1 × 105 +0.3 +200 +1.3 +Fig. 9 Dam Break +1/24 +5 × 10−3 +280K +0 +2 × 105 +0.005 +25 +1000 +- +- +- +0.4 +Fig. 10a Liquid Bunnys +1/50 +4 × 10−3 +52K +3.7K +1 × 105 +0 +10 +1000 +4 × 103 +0.49 +200 +0.4 +Fig. 10b Liquid Bunnys +1/50 +4 × 10−3 +101K +4.5K +6 × 104 +0 +6.4 +1000 +1 × 103 +0.49 +200 +1.0 +Fig. 4 Buoyancy +1/24 +5 × 10−3 +787K +66K +2 × 105 +1 +10 +1000 +1 × 105 +0.4 +200/700/1200 +5.9 +Fig. 5 Twist Cylinder* +1/24 +5 × 10−3 +486K +12K +4 × 104 +0 +5 +1000 +- +- +500 +7.9 +Fig. 3 Cream +1/24 +4 × 10−3 +159K +9K +3 × 104 +25 +3 +1000 +5 × 108 +0.49 +1000 +1.8 +Fig. 13 Angry Cow* +1/24 +5 × 10−3 +789K +13K +1 × 105 +0.2 +10 +1000 +1 × 105 +0.45 +100/700 +4.9 +Fig. 1 Kick Water* +1/24 +6 × 10−3 +1M +43K +2.5 × 105 +0.1 +25 +1000 +- +- +500 +37.9 +Table 1. Simulation statistics including duration of each frame (Δ𝑡frame, [𝑠]), time step size upperbound (Δ𝑡, [𝑠]), number of fluid particles (𝑁fluid), number +of solid vertices (𝑁solid), incompressibility coefficient (𝑘𝐼, [𝑃𝑎]), dynamic viscosity (𝜈𝑓 , [𝑃𝑎 ·𝑠]), fluid particle diameter (𝑑, [𝑚𝑚]), fluid density (𝜌𝑓 , [𝑘𝑔/𝑚3]), +Young’s modulus (𝐸, [𝑃𝑎]), Possion’s ratio (𝜈𝑠), solid density (𝜌𝑠, [𝑘𝑔/𝑚3]) and the average simulation time for each frame (𝑇, [𝑚𝑖𝑛]). Timing statistics are +measured on a 24-core 3.50GHz Intel i9-10920X machine except for Fig. 10, which is tested on the “e2-standard-8” (8 cores with 32GB RAM) Google Compute +Engine. Note that examples marked with * contain codimensional materials, whose parameter settings are not covered here. +Scene +Scheme +Sec/Frame +# Newton Iter./Frame +Fig. 6 +Joint/TS/TSCP +66.1 / 38.0 / 22.5 +63.5 / 117.3 / 37.1 +Fig. 7a +Joint/TS/TSCP +41.3 / 32.3 / 25.5 +16.5 / 29.0 / 10.5 +Table 2. Statistics of different time stepping schemes: Joint Optimiza- +tion (Joint), Baseline Time Splitting (TS) and Time Splitting with Contact +Proxy (TSCP). Our proposed TSCP is much faster than both the Joint and +TS. +As shown in Table 2, even in these simple examples, our time split- +ting scheme is significantly (up to 3×) faster than joint optimization, +especially for cases (e.g. Fig. 6) involving contacts between fluids +and deformable solids. This improvement stems from no longer +having to solve for incompressibility of fluids repeatedly within a +time step. Moreover, one can also find out that Newton’s iterations +are much less with our proxy-assisted time splitting scheme. As dis- +cussed in § 4.1.2, this is because the challenging high-speed impacts +are already partially resolved in the fluid phase. Another benefit +of time splitting is the support of different error tolerances for the +two phases. Errors in the fluid phase are sourced from the solution +deviation of the CG solver, while in the solid phase they are directly +controlled by the tolerance of Newton’s method. Typically, setting a +slightly higher tolerance for fluids yields better performance while +still producing visually plausible results. +Aside from efficiency, our proposed contact proxy also improves +the stability of time splitting scheme. Though simulation results of +the baseline time splitting scheme look fine in the case of inviscid +fluids, situations get worse when it is applied to viscous fluids. In Fig. +7a, a viscous armadillo is dropped to the ground. In this example, +the baseline time splitting scheme produces severe sticky artifacts at +the boundary, and the fluid surface could not finally calm down. By +consistently applying our contact proxy to exert boundary pressure +in the fluid phase, the artifacts can be well resolved as demonstrated +in Fig. 7a. Similarly, our idea of contact proxy is also applicable +𝑘! = 1e4 +𝑘! = 1e5 +𝑘! = 1e6 +Fig. 8. Statistics of simulations with different stiffness parameter 𝑘𝐼 . +A larger 𝑘𝐼 preserves volume better but results in more CG iterations. In +this case, a proper 𝑘𝐼 = 105 Pa can be set to balance the computational cost +and visual artifacts. +to further separate elasticity from IPC contact while maintaining +stability, leading to our three-phase scheme (Fig. 7b). +5.1.2 +Linear Solver Evaluation. For the fluid phase, we designed +a matrix-free conjugate gradient (CG) solver that calculates the +matrix-vector product via gradient computation to avoid the expen- +sive computational and memory costs of direct factorization (§ 4.2.1). +However, the performance improvement from this approach will be +less significant if the number of CG iterations required for conver- +gence is too large, making the cost of computing gradients higher +than constructing the Hessian once. In our fluid phase, the number +of CG iterations is proportional to the stiffness 𝑘𝐼 of the incompress- +ibility energy. A larger 𝑘𝐼 can better preserve the volume of the +fluids but also results in a worse-conditioned system, demanding +more iterations to converge (Fig. 8). In practice, by setting 𝑘𝐼 to a + +80 +70 +60 +CG +40 +# +30 +20 +10 +4.00 +4.25 +4.50 +4.75 +5.00 +5.25 +5.50 +5.75 +6.00 +log10(ki)1200 +k, = 1e4 +1175 +k, = 1e5 +1150 +k, = 1e6 +1125 +1100 +1075 +1050 +1025 +1000 +0 +i +2 +3 +4 +5 +Time[s]A Contact Proxy Splitting Method for Lagrangian Solid-Fluid Coupling +• +9 +Solver +Fluid Phase +Solid Phase +Contact Phase +Mem. +hess +solve +solve +solve +CG + LLT +14.9 +0.49 +1.45 +0.43 +12375 +Ours +0.15 +0.59 +1.11 +0.25 +1469 +Table 3. Time and memory cost of different solvers in example 2. The +costs are measured per time step in units [s] and [MB] respectively. The +baseline uses the Conjugate Gradient (CG) method for the fluid phase and +CHOLMOD LLT for the solid and contact phases. Our method instead em- +ploys a matrix-free CG solver for the fluid phase and a domain-decomposed +solver for the solid and contact phases, thereby improving efficiency and +saving memory. +Initial +Ours +IISPH +DFSPH +Fig. 9. Dam break with 280K SPH particles. Our weakly compressible +formulation produces stable fluid dynamics without visually evident volume +loss. Compared to incompressible SPH solvers IISPH [Ihmsen et al. 2013] +and DFSPH [Bender and Koschier 2015], our simulation results demonstrate +more smooth particle distribution. +proper value, we can efficiently solve the systems within 50 CG +iterations without obvious fluid volume loss. +We test the performance of our matrix-free CG solver together +with the domain-decomposed solver we designed for the solid- +coupling phase on the Shot Armadillo example (Fig. 2), and present +the results in Table 3. Our matrix-free CG solver significantly boosts +efficiency (20× faster) and reduces memory costs by avoiding the +construction of the Hessian matrix. On the other hand, our domain +decomposed solver is 40% faster than directly factorizing the solid +and contact systems. +5.2 +Comparisons +In this section, we compare our method with several popular SPH +fluid solvers and a state-of-the-art solid-fluid coupling method Elas- +toMonolith [Takahashi and Batty 2022]. We leveraged the open- +source library SPlisHSPlasH1 to implement the SPH fluid simula- +tors. To compare our method with ElastoMonolith, we set up two +scenes from their paper with identical parameters and run all the +simulations using “e2-standard-8” (8 cores with 32GB RAM) Google +Compute Engine for fairness. +1https://github.com/InteractiveComputerGraphics/SPlisHSPlasH +(a) A liquid bunny dropped into a bowl. +(b) A liquid bunny and an elastic bowl dropped onto a static torus. +Fig. 10. Liquid Bunnys. Compared to ElastoMonolith [Takahashi and +Batty 2022], our method achives an over 5× speedup for both of these two +examples with exactly the same scene setups. +5.2.1 +Fluid Dynamics. While most existing SPH fluid solvers focus +on incompressible fluids, our formulation treats fluids as weakly +compressible, allowing us to couple fluids with deformable solids in +a unified framework. We run a dam break simulation to compare +our method with two SPH fluid solvers IISPH [Ihmsen et al. 2013] +and DFSPH [Bender and Koschier 2015]. These methods typically +use particle resampling [Akinci et al. 2013, 2012] or implicit repre- +sentation [Bender et al. 2019; Koschier and Bender 2017] to exert +boundary counter-forces. Our method instead employs IPC [Li et al. +2020] for more robust solid-fluid coupling, with penetration-free +guarantee. We uniformly enforce the same CFL condition for all +methods along with an upperbound at 5𝑚𝑠, and use the volume map +[Bender et al. 2019] for their boundary handling. As shown in Fig. +9, though our formulation does not strictly enforce incompressibil- +ity, it produces natural fluid dynamics without visually observable +volume loss. On the other hand, our method (0.45 min/frame) is +slower than IISPH (0.31 min/frame) and DFSPH (0.15 min/frame) +due to the more sophisticated boundary handling strategy. How- +ever, our proposed approach can couple SPH fluids and elastic solids +with arbitrary constitutive models, while most existing SPH meth- +ods [Kugelstadt et al. 2021; Peer et al. 2018] treat elastic solids as +incompressible, which is not generally applicable. +5.2.2 +Solid-Fluid Coupling. We then compare our method with +ElastoMonolith [Takahashi and Batty 2022], which couples Eulerian +fluids with Lagrangian solids in a monolithic manner. Following +their experiment setting, we run two solid-fluid coupling simula- +tions with identical parameters using our method (Fig. 10). The +timing of our method for these two scenes are 24.1 sec/frame and +62.8 sec/frame respectively, both of which are over 5× faster than +ElastoMonolith according to their reported timings (253.2 sec/frame +and 352.0 sec/frame). Coupling Eulerian fluids with Lagrangian +solids requires dealing with geometric differences and it is often + +HoudiniaHoudiniaHoudiniaHoudiniaHoudiniaHoudinia10 +• +Xie et al. +Fig. 11. Varying friction. Three viscous bunnies are dropped onto the slope with different coefficients of friction 𝜇 (from left to right: 0.5, 0.03, 0). Our method +supports adjustable solid-fluid boundary friction. +Fig. 12. Timing breakdown. We show the timing profile of different simulation phases and plot the proportions of the major routines. Examples marked with +* are simulated using our three-phase time splitting scheme. Other examples are generated with the two-phase scheme. In particular, SPH update (including +neighborhood search and density update) only occurs in the fluid phase, line search happens in the solid and contact phases for non-linear optimization, and +continuous collision detection (CCD) is counted when IPC contact energy is considered. +needed to perform SPD reformulation to make the linear system +tractable. As stated in ElastoMonolith, this SPD reformulation can +introduce many additional non-zeros to the system, especially when +contacts are rich and solids are intricately shaped. Conversely, our +method treats solids and fluids from a unified Lagrangian viewpoint, +where solid-solid and solid-fluid contacts are resolved in a unified +manner. +5.3 +Complex Scenarios +We then evaluate the efficiency and robustness of our method in +more complicated scenarios. We demonstrate our method correctly +captures the buoyancy behavior in Fig. 4. Viscous fluids can also be +naturally simulated (Fig. 3), even with adjustable boundary friction +(Fig. 11). In addition, two-way coupling with thin shells (Fig. 5 and +Fig. 1) is also well supported with penetration-free guarantee. In +Fig. 13, we show that our framework can even simulate a tightly +coupled system of geometries in arbitrary codimensions (0, 1, 2, and +3). Detailed parameter settings can be found in the appendix. +6 +CONCLUSION +We presented a unified two-way strong coupling framework for +weakly-compressible SPH fluids and nonlinear elastic FEM solids. +To achieve this, we modeled solid-fluid interactions as contact forces +between SPH particles and FEM boundary elements, applying IPC +for guaranteed non-penetration and stability. As we track the vol- +ume change of SPH particles in an updated Lagrangian fashion, +the incompressibility energy stays quadratic and nice particle dis- +tributions are maintained. Utilizing a symmetric approximation of +discrete viscosity forces, we proposed a viscosity potential that fit- +ted into optimization time integration. We then proposed a time +splitting scheme with a contact proxy to efficiently solve the time +integration optimization while maintaining robustness. The per- +formance is further boosted by our matrix-free conjugate gradient +method and a domain-decomposed solver based on Schur comple- +ment. +Compared to existing works [Takahashi and Batty 2022; Zarifi and +Batty 2017] coupling Eulerian fluids with Lagrangian elastic solids, + +SPH Update +Gradient +Hessian +Linear System +Line Search +CCD +Z Fluid Phase + SolidPhase +ContactPhase +Twist Cylinder* +Buoyancy +Shot Armadillo* +Bob* +KickWater +Angry Cow +Inviscid Bunny +ViscousBunny +DamBreak +Viscous Armadillo +0% +20% +40% +60% +80% +100%HoudiniaHoudiniaHoudiniaA Contact Proxy Splitting Method for Lagrangian Solid-Fluid Coupling +• +11 +Fig. 13. Angry cow. We show our method can simulate the coupling of materials in arbitrary codimensions, including fluid particles, rods (the rubber bands), +thin shells (the leather pad), deformable solids (the cow), and rigid bodies (the cubes). We launch an angry cow with a slingshot, and the cow hits through the +wall and then falls into the water. Interactions between various materials are all accurately captured. +our method treats both fluids and solids in a Langrangian manner, +avoiding the need to handle different spatial discretizations. Under +such a unified view, our method achieves more convenient and +robust two-way coupling, even between fluids and codimensional +solids. Likewise, different from existing SPH methods [Kugelstadt +et al. 2021; Peer et al. 2018] that treat all materials as SPH particles, +our formulation enjoys both the efficiency of SPH fluids and the +accuracy of FEM solids. +There are many meaningful future research directions. 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Springer Science +& Business Media. +A +DERIVATIVES OF FLUID POTENTIALS +The gradient and Hessian of the incompressibility potential w.r.t +the fluid particle position are +𝜕𝑃𝐼 (x) +𝜕x += +∑︁ +𝑖 +𝑘𝐼𝑉0(𝐽𝑖 − 1)𝐽𝑛 +𝑖 ℎ +𝜕(∇ · v𝑛+1 +𝑖 +) +𝜕x +, +𝜕2𝑃𝐼 (x) +𝜕x2 += +∑︁ +𝑖 +𝑘𝐼𝑉0(𝐽𝑛 +𝑖 )2ℎ2 𝜕(∇ · v𝑛+1 +𝑖 +) +𝜕x +� +𝜕(∇ · v𝑛+1 +𝑖 +) +𝜕x +�𝑇 +. +(30) +This constant Hessian matrix is obviously positive semi-definite +(PSD) since it is simply the sum of the outer product of some vector +with positive coefficients. +Similarly, the gradient and Hessian of the viscosity potential w.r.t +the fluid particle position are +𝜕𝑃𝑉 (x) +𝜕x𝑖 += 𝜈 +∑︁ +𝑗 +V𝑖𝑗v𝑛+1 +𝑖𝑗 , +𝜕2𝑃𝑉 (x) +𝜕x𝑖𝜕x𝑗 += +� +𝜈 +ˆℎ +� +𝑘 V𝑖𝑘, 𝑗 = 𝑖 +− 𝜈 +ˆℎ V𝑖𝑗, 𝑗 ≠ 𝑖 +. +(31) + +A Contact Proxy Splitting Method for Lagrangian Solid-Fluid Coupling +• +13 +Since V𝑖𝑗 = −4(𝑑 + 2) 𝑚𝑖𝑚𝑗 +𝜌𝑖+𝜌𝑗 +∇𝑖𝑊𝑖𝑗 (x𝑛 +𝑖𝑗)𝑇 +∥x𝑛 +𝑖𝑗 ∥2+0.01ℏ2 ∝ x𝑛 +𝑖𝑗 (x𝑛 +𝑖𝑗)𝑇 is a 3 × 3 +constant PSD matrix for any particle pair within a time step, the +Hessian of the viscosity potential is a constant PSD matrix as well. +B +TIME SPLITTING +B.1 +Baseline Time Splitting +Applying time splitting, we can split the original time integration +into a fluid phase + + +v𝑛+1/2 +𝑓 += v𝑛 +𝑓 + ℎM−1 +𝑓 (−∇𝑓 𝑃([(x𝑛+1/2 +𝑓 +)𝑇, (x𝑛𝑠 )𝑇 ]𝑇 ) + f𝑓 ) +x𝑛+1/2 +𝑓 += x𝑛 +𝑓 + ℎv𝑛+1/2 +𝑓 +(32) +and a solid-coupling phase + + +v𝑛+1 = +� +v𝑛+1/2 +𝑓 +v𝑛𝑠 +� ++ ℎM−1(−∇Ψ(x𝑛+1) − ∇𝐶(x𝑛+1) + +� +0 +f𝑠 +� +) +x𝑛+1 = x𝑛 + ℎv𝑛+1 +, +(33) +where f𝑓 and f𝑠 are the external forces on the fluids and the solids +respectively. The two phases of this baseline time splitting scheme +have equivalent optimization forms +x𝑛+1/2 +𝑓 += 𝑎𝑟𝑔 min +x𝑓 +1 +2 ∥x𝑓 − ˆx𝑛 +𝑓 ∥2 +M𝑓 + ℎ2𝑃([x𝑇 +𝑓 , (x𝑛 +𝑠 )𝑇 ]𝑇 ) +x𝑛+1 = 𝑎𝑟𝑔 min +x +1 +2 ∥x − ˆx𝑛+1/2∥2 +M + ℎ2(Ψ(x) + 𝐶(x)) +(34) +where ˆx𝑛 +𝑓 = x𝑛 +𝑓 +ℎv𝑛 +𝑓 +ℎ2M−1 +𝑓 f𝑓 and ˆx𝑛+1/2 = x𝑛+ℎ[(v𝑛+1/2 +𝑓 +)𝑇, (v𝑛𝑠 )𝑇 ]𝑇 + +ℎ2M−1[0𝑇, f𝑇𝑠 ]𝑇 with v𝑛+1/2 +𝑓 += (x𝑛+1/2 +𝑓 +− x𝑛 +𝑓 )/ℎ. The details of the +optimization algorithm can be found in Alg. 2. +Algorithm 2 Baseline Time Splitting +1: x ← x𝑛, ˆx𝑛 +𝑓 ← x𝑛 +𝑓 + ℎv𝑛 +𝑓 + ℎ2M−1 +𝑓 f𝑓 +2: SPH Neighbor Search & Density Update +3: // Fluid Phase +4: H𝑓 ← ℎ2∇2 +𝑓 𝑃([x𝑇 +𝑓 , (x𝑛𝑠 )𝑇 ]) + M𝑓 +5: p𝑓 ← −H−1 +𝑓 +� +ℎ2∇𝑓 𝑃([x𝑇 +𝑓 , (x𝑛𝑠 )𝑇 ]𝑇 ) + M𝑓 (x𝑓 − ˆx𝑛 +𝑓 ) +� +6: x𝑓 ← x𝑓 + p𝑓 +7: x𝑛+1/2 +𝑓 +← x𝑓 , v𝑛+1/2 +𝑓 += (x𝑓 − x𝑛 +𝑓 )/ℎ +8: // Solid Coupling Phase +9: ˆx𝑛+1/2 ← x𝑛 + ℎ[(v𝑛+1/2 +𝑓 +)𝑇, (v𝑛𝑠 )𝑇 ]𝑇 + ℎ2M−1[0𝑇, f𝑇𝑠 ]𝑇 +10: do +11: +H ← ℎ2 �∇2Ψ(x) + ∇2𝐶(x)� + M +12: +p ← −H−1 � +ℎ2(∇Ψ(x) + ∇𝐶(x)) + M(x − ˆx𝑛+1/2) +� +13: +𝛼 ← Backtracking Line Search with CCD +14: +x ← x + 𝛼p +15: while 1 +ℎ ∥p∥ > 𝜖 +16: x𝑛+1 ← x, v𝑛+1 ← (x − x𝑛)/ℎ +17: return x𝑛+1, v𝑛+1 +B.2 +Error Analysis +The position update of implicit Euler and our proxy-enhanced time +splitting scheme can be respectively expressed as +x𝑛+1 +IE += x𝑛 + ℎv𝑛 + ℎ2M−1� +−∇𝑃(x𝑛+1 +IE ) +− ∇Ψ(x𝑛+1 +IE ) − ∇𝐶sf(x𝑛+1 +IE ) − ∇𝐶ss(x𝑛+1 +IE ) + fext +� +x𝑛+1 = x𝑛 + ℎv𝑛 + ℎ2M−1� +−∇𝑃(x𝑛+1/2) − ∇Ψ(x𝑛+1) +− ∇ ˆ𝐶sf(x𝑛+1/2) − 1 +2∇𝐶sf(x𝑛+1) − ∇𝐶ss(x𝑛+1) + fext +� +. +(35) +If we define +𝑒(x) = +���x𝑛 + ℎv𝑛 + ℎ2M−1� +−∇𝑃(x) − ∇Ψ(x) +− ∇𝐶sf(x) − ∇𝐶ss(x) + fext +� +− x +���, +(36) +x𝑛+1 +IE +given by implicit Euler satisfies 𝑒(x𝑛+1 +IE ) = 0, while for x𝑛+1 +from our scheme, we have 𝑒(x𝑛+1) = O(ℎ4). Specifically, +𝑒(x𝑛+1) = ℎ2���M−1(∇𝑃(x𝑛+1/2) − ∇𝑃(x𝑛+1) ++ ∇ ˆ𝐶sf(x𝑛+1/2) − 1 +2∇𝐶sf(x𝑛+1)) +��� += O +� +ℎ3���M−1� +∇2𝑃(x𝑛)(v𝑛+1/2 − v𝑛+1)+ +1 +2∇2𝐶sf(x𝑛)(v𝑛+1/2 − v𝑛+1) +���� +� += O +� +ℎ4���M−1� +∇2𝑃(x𝑛) + 1 +2∇2𝐶sf(x𝑛) +� +M−1� +∇Ψ(x𝑛) ++ 1 +2∇𝐶sf(x𝑛) + ∇𝐶ss(x𝑛) +���� +� += O(ℎ4). +(37) +Here we assume that, in our discretized domain, the distance be- +tween any pair of primitives (particle-particle pair, particle-triangle +pair and triangle-triangle pair) has a lower bound 𝜖. Thus ∇2𝑃(x𝑛), +∇2𝐶sf(x𝑛), ∇Ψ(x𝑛), ∇𝐶sf(x𝑛) and ∇𝐶ss(x𝑛) are all bounded, and +this indicates our method has an O(ℎ4) difference compared to +implicit Euler solution. Since implicit Euler has an O(ℎ2) error com- +pared to the PDE solution, our proposed time splitting scheme shares +the same order of accuracy with implicit Euler when it is stable. +C +EXPERIMENT OF COMPLEX SCENARIOS +In this section, we describe the experiment settings of our simula- +tions in various complex scenarios and briefly discuss the results. +Buoyancy. We drop three elastic elephants with varying densities +into the water (1000 𝑘𝑔/𝑚3). The light grey elephant (200 𝑘𝑔/𝑚3) +floats on the surface; the blue elephant (700 𝑘𝑔/𝑚3) is around half +immersed in the water; and the red elephant (1200 𝑘𝑔/𝑚3) sinks into +the bottom. This demonstrates that our method correctly captures +the buoyancy behavior. + +14 +• +Xie et al. +Varying Friction. We drop three viscous bunnies onto the slope +with different coefficients of friction (orange bunny: 0.5, green +bunny: 0.03, blue bunny: 0). All three bunnies share the same dy- +namic viscosity coefficients 100 𝑘𝑔/𝑚3 and the angle of slope is +30◦. +Twist Cylinder. Coupling fluids with thin shells is challenging +since penetration can easily happen without careful treatments. As +stated in [Zarifi and Batty 2017], Eulerian fluids may flow through +solids if their thickness is less than a grid cell size. Conversely, our +approach adopts a unified Lagrangian view and penetration-free is +guaranteed by IPC. In this example, we simulate twisting a cylinder +full of water. The cylinder is modeled as a thin shell with a 2𝑚𝑚 +thickness, and there are two holes in the front and back sides of this +cylinder respectively. The left side and right side are rotated at 72◦/𝑠 +and are slowly moved towards each other at 2𝑐𝑚/𝑠. As we twist +the cylinder, the water gets squeezed out through the holes. This +simulation demonstrates our method produces stable simulation +results with penetration-free guarantee. +Cream. This example exhibits the coupling behaviors of viscous +fluids and elastic solids. We use an elastic spoon to stir the cream +in a porcelain bowl. The spoon handle rotates around y-axis at +360◦/𝑠 (0.2𝑚/𝑠) while the bowl is fixed at the table. As shown in our +simulation results, the spoon gets deformed due to the resistance +forces it receives from the viscous cream while stirring. +Angry Cow. We then show our framework can simulate natural +physical behaviors of geometries in arbitrary codimensions (0, 1, 2, +and 3) as well as their interactions. In this scene, the codimensional- +0,1,2 objects respectively refer to fluid particles, rubber bands and the +leather pad. A deformable cow is launched by the slingshot, hitting +the wall consisting of rigid cubes, and then falling into the water +pool, producing interesting physical behaviors. The density of the +rigid cubes and the cow are 100𝑘𝑔/𝑚3 and 700𝑘𝑔/𝑚3 respectively. +Kick Water. In this example, we show a scene where a mannequin +dressed in a multilayer skirt kicks in a large water pool, involv- +ing complex interactions between fluid particles and garments. As +the mannequin moves in the water, our method produces natural +deformation of the skirt caused by the contact with water; as it +kicks out of the water at a high speed, the resulting water splash +is also correctly captured. Our method well resolves the contacts +among fluids particles, thin garments and rapidly moving complex +boundaries with penetration-free guarantee. + diff --git a/F9A0T4oBgHgl3EQfBf9X/content/tmp_files/load_file.txt b/F9A0T4oBgHgl3EQfBf9X/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1363ccec1d0ff5bd619ff6cf817b114c731164de --- /dev/null +++ b/F9A0T4oBgHgl3EQfBf9X/content/tmp_files/load_file.txt @@ -0,0 +1,1017 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf,len=1016 +page_content='A Contact Proxy Splitting Method for Lagrangian Solid-Fluid Coupling TIANYI XIE, MINCHEN LI, YIN YANG, and CHENFANFU JIANG Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Kick water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Our method accurately captures the complex interactions between the water, the multi-layer skirt, and the mannequin body without any interpenetration as the mannequin wearing the skirt kicks in a swimming pool and sends water flying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We present a robust and efficient method for simulating Lagrangian solid- fluid coupling based on a new operator splitting strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We use variational formulations to approximate fluid properties and solid-fluid interactions, and introduce a unified two-way coupling formulation for SPH fluids and FEM solids using interior point barrier-based frictional contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We split the resulting optimization problem into a fluid phase and a solid-coupling phase using a novel time-splitting approach with augmented contact proxies, and propose efficient custom linear solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Our technique accounts for fluids interaction with nonlinear hyperelastic objects of different geome- tries and codimensions, while maintaining an algorithmically guaranteed non-penetrating criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Comprehensive benchmarks and experiments demonstrate the efficacy of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Additional Key Words and Phrases: Lagrangian Solid-Fluid Coupling, Time Splitting, Contact Proxy 1 INTRODUCTION The coupling of solids and fluids is common in nature but chal- lenging to simulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' While solids are typically simulated using La- grangian meshes, fluids are often discretized using Eulerian grids to accommodate topology changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' To accurately couple these dis- tinct discretizations, sophisticated algorithms such as the method by Zarifi and Batty [2017] are often necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Unfortunately, they can be expensive and do not handle thin shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Purely Eulerian [Teng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Valkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2015] or SPH [Akinci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Gissler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2019] schemes have demonstrated successful two-way coupling by same-view discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' They do not easily extend to nonlinear elastodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Hybrid methods like MPM [Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2016] can simulate mixed materials, but can experience artificial stickiness unless resolved with more expensive schemes [Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Furthermore, these methods do not ensure non-intersecting trajectories and often require additional correction procedures to handle accidentally penetrated fluids during advection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We take the Lagrangian path and present a new method for cou- pling FEM solids and SPH fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' By approximating solid, fluid, Authors’ address: Tianyi Xie, tianyixie77@g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Minchen Li, minchernl@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Yin Yang, yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='yang@utah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Chenfanfu Jiang, chenfanfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='jiang@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' and interaction terms with potentials, we formulate two-way cou- pling as an optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Specifically, we draw inspiration from position-based fluids [Macklin and Müller 2013] and model weak incompressibility using a quadratic energy and a new updated Lagrangian update rule to track volume changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We further sym- metrize the discrete Laplacian-based viscosity and propose a discrete quadratic potential for better accuracy and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We follow the Incremental Potential Contact [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2020] model to enforce guaranteed separable boundary conditions and resolve frictional contacts at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The proposed formulation achieves strong coupling, but can be exceedingly inefficient when solved with Newton’s method due to the huge and dense Hessian of the fluid part, which is a result of the need for many particle neighbors for an accurate SPH discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' This causes a significant computing bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' To tackle this issue, we propose a robust proxy contact energy formulation, spitting the time integration into a fluid phase and a solid-coupling phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The fluid phase requires only one Newton iter- ation per time step, resulting in increased efficiency with nonlinear optimization occurring only during the solid-coupling phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' One of the key advantages of our quadratic proxy is its ability to effectively resolve instability caused by time splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' This is achieved through its asymptotic approximation to the solid-fluid contact force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Ad- ditionally, time integration is maintained consistent through the cancellation of the proxy’s contribution in the solid-coupling phase, resulting in only a small splitting error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Finally, we design a matrix- free conjugate gradient solver and a domain-decomposed solver to further enhance the computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2 RELATED WORK Traditional fluid solvers typically use an Eulerian grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Many existing works focus on coupling Eulerian fluids with Lagrangian solids by resolving interactions between the grids and irregular mesh bound- aries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The ghost fluid method [Fedkiw 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Fedkiw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 1999] was proposed to additionally discretize the Eulerian/Lagrangian arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='01976v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='GR] 5 Jan 2023 HoudiniHoudinia2 Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Early works considered weak coupling [Guendelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2005], which advances the solids and fluids alternatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Strong coupling [Klingner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2006] on the other hand solves a monolithic system and is often more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The cut-cell method [Roble et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2005] is another widely used solution, often through the usage of virtual nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Batty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' [2007] proposed a variational framework to strongly couple fluids and rigid bodies by casting the pressure solve into minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Subsequent extensions support deformable objects and thin shells [Robinson-Mosher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2011, 2008], where elastic forces are explicitly applied and the coupling step is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Assuming corotated linear elasticity, Zarifi and Batty [2017] incorpo- rated the implicit solid dynamics into pressure projection, obtaining a symmetric positive-definite system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Later works also explored rigid-rigid [Takahashi and Batty 2020] and rigid-fluid [Takahashi and Batty 2021] frictional contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Eulerian solids [Levin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2011] were also explored, where coupling can be conveniently achieved in a purely Eulerian fashion [Teng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Valkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' However they face challenges in numerical dissipation, volume con- servation, and handling structures thinner than a grid cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' More recently, Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' [2019] built upon the immersed boundary method [Peskin 2002] and proposed a reduced solver to simulate real-time coupling, focusing on incompressible elastic materials and no-slip boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Fluids can also be directly modeled with Lagrangian meshes [Batty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Clausen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Klingner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2020], enabling explicit coupling with solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' However, remeshing tends to become a bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Using particles is another popular strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' SPH [Koschier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2022] uses spatial sampling to approxi- mate continuous functions and has been shown compelling for fluid dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Pioneering works [Becker and Teschner 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Monaghan 1994] used the Equation of State (EOS) for weakly compressible fluids, where the pressure is proportional to the density deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' An explicit formulation may strictly restrict time step sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Incom- pressibility has also been enforced by solving a Pressure Poisson Equation (PPE) [Bender and Koschier 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Ihmsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Solen- thaler and Pajarola 2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' This approach seeks to cancel out density or velocity divergence deviations caused by non-pressure forces through the use of pressure accelerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' SPH boundary handling techniques have been developed to prevent penetrations of fluid par- ticles near solid boundaries [Becker and Teschner 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2009b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Ihmsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' One such method, proposed by Akinci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' [2012], uses a single layer of boundary samples and has been applied to the coupling of fluids with both rigid bodies and elastic solids [Akinci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Gissler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' [2019] proposed a global formulation that unifies rigid body and fluid dynamics, in which the fluid pressure solver is linked to a second artificial pressure solver for rigid body particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Koschier and Bender [2017] introduced an alternative method using density maps to represent dynamic rigid boundaries, eliminating the need for boundary particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Bender et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' [2019] proposed using the volume contribution of boundary geometry to compute boundary forces, which reduces the cost of precomputation but cannot be applied to deformable bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Solenthaler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' [2007] used SPH to approximate the deforma- tion gradient of linear elastic materials, but the resulting gradient is not rotation invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' [2009a] addressed this issue by using shape matching to determine orientation and calculating Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Shooting armadillo with a high-speed water jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' forces in a rotated configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Peer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' [2018] proposed an implicit scheme and applied kernel gradient correction [Bonet and Lok 1999] to obtain a first-order consistent SPH formulation for the deformation gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Incorporating solid particles into the preexist- ing fluid pressure solver can resolve contact handling, but SPH still faces numerical issues such as the zero-mode [Ganzenmüller 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Kugelstadt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2021] when simulating elastic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Additionally, the pressure solver will treat solid objects as incompressible under compression, which may not be applicable in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The Material-Point Method (MPM) [Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Sulsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 1995] combines Lagrangian and Eulerian representations to capture solid-fluid coupling [Fei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Stomakhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2018] and mixture [Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Tampubolon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' [2020] proposed a free-slip treatment, but did not consider separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Recently, a FEM-MPM coupling method based on a variational barrier formulation [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2020] has been proposed for coupling frictional and separable elastic materials [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Our approach for solid-fluid coupling is inspired by this method and uses a similar purely Lagrangian framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 3 FORMULATION Here we derive a time integrator for a coupled system of solids and fluids by starting with the governing equations and then performing discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Subscripts 𝑠 and 𝑓 represent solid and fluid quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1 Governing Equations The governing equations for the coupled system are 𝜌𝑠 𝐷v𝑠 𝐷𝑡 = ∇ · 𝜎 + 𝜌𝑠g + f𝑠�𝑠 + f𝑓 �𝑠, (1) 𝜌𝑓 𝐷v𝑓 𝐷𝑡 = −∇𝑝 + 𝜇∇2v𝑓 + 𝜌𝑓 g − f𝑓 �𝑠, (2) ∇ · v𝑓 = 0, (3) where 𝜌 is density, g is gravity, f𝑠�𝑠 is the self-contact force of solids, f𝑓 �𝑠 is the contact force exerted by fluids, 𝜎 is Cauchy stress, 𝑝 is pressure and 𝜇 is the dynamic viscosity [Bridson 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' At the interface between solids and fluids, we enforce the separa- ble boundary condition 0 ≤ (v𝑠 − v𝑓 ) · n𝑓 ⊥ (f𝑓 �𝑠 · n𝑓 ) ≥ 0 (4) to prevent penetration while allowing separation [Batty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2007].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' This condition helps determine the normal component of f𝑓 �𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' For the tangential component (friction), let u = (I − n𝑓 ⊗ n𝑓 )(v𝑠 − v𝑓 ) HoudiniaHoudiniaA Contact Proxy Splitting Method for Lagrangian Solid-Fluid Coupling 3 be the tangential relative velocity, we have (I − n𝑓 ⊗ n𝑓 )f𝑓 �𝑠 = arg min 𝜷 𝜷 · u s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' ∥𝜷∥ ≤ 𝜇𝑡f𝑓 �𝑠 · n𝑓 and 𝜷 · n𝑓 = 0 (5) following the Maximum Dissipation Principle [Moreau 2011], where 𝜇𝑡 is the friction coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We enforce exact mass conservation by adopting Lagrangian methods to discretize both domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2 Solid Domain We focus on nonlinear hyperelastic solids, where the elastic force is the negative gradient of an elastic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' After discretizing the solid domain Ω𝑠 as Lagrangian linear finite elements (triangles in 2D and tetrahedra in 3D), the total elastic potential is a piecewise constant summation of an elastic energy density function 𝜓𝑠 (F) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' neo-Hookean) over the mesh domain: Ψ𝑠 (x) = � 𝑒 𝑉𝑒𝜓𝑠 (F𝑒), where 𝑉𝑒 is the rest volume of tetrahedron 𝑒, and F = 𝜕x(X,𝑡) 𝜕X is the deformation gradient with X and x the material and world space coordinates respectively [Sifakis and Barbic 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' For f𝑠�𝑠, we follow Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' [2020]’s smooth barrier approach that guarantees non-penetration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We leave the discussion of f𝑓 �𝑠 to § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='3 Fluid Domain Following SPH literature [Becker and Teschner 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Bender and Koschier 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Ihmsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Macklin and Müller 2013], we dis- cretize the fluid domain Ω𝑓 with Lagrangian particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' To integrate fluids with optimization-based time integration, we approximate both the pressure and viscosity forces as conservative forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We verify in the appendix that these proposed potential energies are both convex and quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1 Incompressibility Potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Pressure forces help preserve the volume of incompressible fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We thus model the incompressibil- ity via a quadratic energy density function𝜓𝑓 ,𝐼 (𝐽) = 𝑘𝐼 2 (𝐽 −1)2 that penalizes the deviation of volume ratio 𝐽 = 𝜌0/𝜌 from 1, where 𝜌0 is the initial density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The use of a large stiffness value (𝑘𝐼 ) in a con- vergent solve results in negligible visual compression, eliminating the need for higher degree polynomials in nearly incompressible fluids [Hyde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The incompressibility potential is obtained by integrating 𝜓𝑓 ,𝐼 (𝐽) over the fluid domain Ω0 𝑓 in material space: 𝑃𝐼 (x) = ∑︁ 𝑖 𝑘𝐼 2 𝑉0(𝐽𝑖 (x) − 1)2, (6) where we assumed all fluid particles have equal rest volume 𝑉0, and 𝐽𝑖 denotes the volume ratio of the 𝑖-th particle as a function of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Updated Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' SPH literature often relate 𝜌 to x through density summation in the world space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' To obtain a linear relation between 𝐽 and x so that the incompressibility potential stays qua- dratic in terms of x, we track 𝐽 in an updated Lagrangian fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Treating Ω𝑛 as an intermediate reference space and differentiating the deformation map between Ω𝑛 and Ω𝑛+1 results in an update rule 𝐽𝑛+1 𝑖 = 𝐽𝑛 𝑖 (1 + ℎ∇ · v𝑛+1 𝑖 ), (7) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Cream is stirred, causing spoon deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' where 𝐽𝑛 𝑖 and ∇ · v𝑛+1 𝑖 can be approximated as 𝐽𝑛 𝑖 = 𝜌0 � 𝑗 𝑚𝑗𝑊𝑖𝑗 , ∇ · v𝑛+1 𝑖 = ∑︁ 𝑗 𝑚𝑗 𝜌𝑛 𝑗 (v𝑛+1 𝑗 − v𝑛+1 𝑖 ) · ∇𝑖𝑊𝑖𝑗 (8) via SPH, and 𝑊𝑖𝑗 = 𝑊 (x𝑖 − x𝑗) is a kernel function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Cubic Spline kernel [Monaghan 1992, 2005] or Spiky kernel [Müller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2003]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Here 𝐽𝑛 𝑖 denote the reinitialized volume ratio of the 𝑖-th fluid particle at the beginning of time step 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Such reinitialization avoids accumulated density and particle distribution errors commonly seen in other updated Lagrangian solvers like MPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2 Viscosity Potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Modeling viscosity via strain rate tensors [Bender and Koschier 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Peer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Peer and Teschner 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2015] is possible, but may suffer from artifacts at the surface due to particle deficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We follow Monaghan [2005] to use the more robust velocity Laplacian [Weiler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2018] and derive its energy form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Combining SPH 1st-order derivatives and finite differences, the viscosity force can be computed as f𝑖 (x) = 𝜈𝑚𝑖∇2v𝑛+1 𝑖 = 2𝜈(𝑑 + 2) ∑︁ 𝑗 𝑚𝑖𝑚𝑗 𝜌𝑗 ∇𝑖𝑊𝑖𝑗 (x𝑛 𝑖𝑗)𝑇 ∥x𝑛 𝑖𝑗 ∥2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='01ℏ2 v𝑛+1 𝑖𝑗 , where x𝑛 𝑖𝑗 = x𝑛 𝑖 − x𝑛 𝑗 , v𝑛+1 𝑖𝑗 = v𝑛+1 𝑖 − v𝑛+1 𝑗 , ℏ is the support radius of the kernel, 𝜈 and 𝑑 ∈ {2, 3} denote the kinematic viscosity and spatial dimension respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Directly applying this force violates momentum conservation as the mutual interaction forces are not equal and opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Thus, we perform a further approximation f𝑖 (x) ≈ 4𝜈(𝑑 + 2) ∑︁ 𝑗 𝑚𝑖𝑚𝑗 𝜌𝑖 + 𝜌𝑗 ∇𝑖𝑊𝑖𝑗 (x𝑛 𝑖𝑗)𝑇 ∥x𝑛 𝑖𝑗 ∥2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='01ℏ2 v𝑛+1 𝑖𝑗 (9) to solve this issue and also make the force integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Let V𝑖𝑗 = 4(𝑑 + 2) 𝑚𝑖𝑚𝑗 𝜌𝑖+𝜌𝑗 (−∇𝑖𝑊𝑖𝑗) (x𝑛 𝑖𝑗)𝑇 ∥x𝑛 𝑖𝑗 ∥2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='01ℏ2 , we can now gather and integrate all viscosity forces and obtain a quadratic viscosity potential 𝑃𝑉 (x) = 1 4𝜈 ˆℎ ∑︁ 𝑖 ∑︁ 𝑗 ∥v𝑛+1 𝑖𝑗 ∥2 V𝑖𝑗, (10) 4 Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Buoyancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Three elastic elephants with different densities (from left to right: 200, 700, and 1200 𝑘𝑔/𝑚3) fall into the water, demonstrating buoyancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' where ˆℎ is a constant scalar related to the time integration scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' For example, ˆℎ = ℎ for implicit Euler as v𝑛+1 = (x𝑛+1 − x𝑛)/ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='4 Coupling 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1 Barrier Potential for Non-penetration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' To couple the solid domain Ω𝑠 with the fluid domain Ω𝑓 , we use the separable boundary condition (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 4), which enforces non-interpenetration constraints between these two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' To model these constraints, we first define a distance function 𝑑(𝜕Ω𝑡 𝑠, x𝑓 ) = min x𝑠 ∥x𝑠 − x𝑓 ∥, x𝑠 ∈ 𝜕Ω𝑡 𝑠, x𝑓 ∈ Ω𝑡 𝑓 , (11) which measures the distance between x𝑓 , a point in the fluid domain, and the surface of the solid domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Then the primal component of the constraints can be expressed as 𝑑(𝜕Ω𝑡 𝑠, x𝑓 ) ≥ 0, ∀𝑡 ≥ 0, ∀x𝑓 ∈ Ω𝑡 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' (12) We then adopt the barrier formulation from Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' [2020] to model all the constraints in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 4 between solids and fluids, and obtain a barrier potential ∫ 𝜕Ω0 𝑓 𝑏(𝑑(𝜕Ω𝑡 𝑠, x𝑓 ), ˆ𝑑)𝑑X𝑓 , (13) where the barrier energy density 𝑏(𝑑, ˆ𝑑) is piecewise smooth and only activated when 𝑑 < ˆ𝑑, improving efficiency and approximately satisfying the complimentarity slackness condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' As𝑑 approaches 0, the value of 𝑏(𝑑, ˆ𝑑) monotonically increases to infinity, providing arbitrarily large repulsion to avoid interpenetration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Since our solids and fluids domains are respectively discretized as meshes and particles, the barrier potential (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 13) in 3D can be numerically integrated as 𝐵sf(x𝑠, x𝑓 ) = ∑︁ 𝑞∈Q𝑓 𝑠𝑞𝑏( min 𝑒 ∈B𝑠 𝑑𝑃𝑇 (x𝑞,𝑒), ˆ𝑑) = ∑︁ 𝑞∈Q𝑓 𝑠𝑞 max 𝑒 ∈B𝑠 𝑏(𝑑𝑃𝑇 (x𝑞,𝑒), ˆ𝑑), (14) where Q𝑓 is the set of all SPH fluid particles, B𝑠 is the set of all boundary triangles of the solids, 𝑠𝑞 = 𝜋( 3𝑉0 4𝜋 ) 2 3 is the integration weight (boundary area) of each fluid particle, and 𝑑𝑃𝑇 (x𝑞,𝑒) mea- sures the distance between particle x𝑞 and triangle 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Here the min-max transformation is based on the non-ascending property of the barrier function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' However, the max operator here makes the barrier potential challenging to be efficiently optimized by gradient- based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Fortunately, due to the local support of the barrier function 𝑏(𝑑, ˆ𝑑) as ˆ𝑑 is small, we can simply approximate the barrier potential as 𝐵sf(x𝑠, x𝑓 ) = ∑︁ 𝑞∈Q𝑓 ∑︁ 𝑒 ∈B𝑠 𝑠𝑞𝑏(𝑑𝑃𝑇 (x𝑞,𝑒), ˆ𝑑), (15) which may result in overestimated contact forces near the edges and nodes on the mesh boundary, but we have not observed any artifacts in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2 Friction Potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Following Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' [2020], we model the local friction forces f𝑘 for every active solid-fluid contact pair k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Formally, the friction force is defined as f𝑘 (x𝑠, x𝑓 ) = −𝜇𝑡𝜆𝑘𝑇𝑘 (x𝑠, x𝑓 )𝑓1(∥u𝑘 ∥) u𝑘 ∥u𝑘 ∥ , (16) where 𝜆𝑘 is the contact force magnitude, 𝑇𝑘 (x𝑠, x𝑓 ) ∈ R3𝑛×2 is the consistently oriented sliding basis, and u𝑘 is the relative sliding dis- placement, which can be computed as u𝑘 = 𝑇𝑘 (x𝑠, x𝑓 )𝑇 ([x𝑇𝑠 , x𝑇 𝑓 ]𝑇 − [x𝑛𝑠 , x𝑛 𝑓 ]𝑇 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Here 𝑓1 is a smoothly approximated function designed for the smooth transition between sticking and sliding modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' To make this friction formulation fit into optimization time integra- tion, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' [2020] further approximated the sliding basis𝑇 (x𝑠, x𝑓 ) and contact force 𝜆𝑘 (x𝑠, x𝑓 ) explicitly as 𝑇 (x𝑛𝑠 , x𝑛 𝑓 ) and 𝜆𝑘 (x𝑛𝑠 , x𝑛 𝑓 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Then the semi-implicit friction force is integrable with the friction potential computed as 𝐷sf(x𝑠, x𝑓 ) = ∑︁ 𝑘 ∈A𝑛 𝜇𝑡𝜆𝑛 𝑘 𝑓0(∥u𝑘 ∥), (17) where 𝑓0 is defined by the relation 𝑓 ′ 0 = 𝑓1 and A𝑛 is the set con- taining all activate particle-triangle contact pairs at the previous time step 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='5 Optimization Time Integrator With the above potential energies modeling all the solid and fluid forces, now we can build a unified two-way solid-fluid coupling framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' By stacking all nodal positions and velocities of SPH particles and FEM nodes as x = [x𝑇 𝑓 , x𝑇𝑠 ]𝑇 and v = [v𝑇 𝑓 , v𝑇𝑠 ]𝑇 , we define Ψ(x) = Ψ𝑠 (x𝑠), 𝑃(x) = 𝑃𝐼 (x𝑓 ) + 𝑃𝑉 (x𝑓 ) and 𝐶sf(x) = 𝐵sf(x𝑠, x𝑓 ) + 𝐷sf(x𝑠, x𝑓 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Combined with the solid-solid contact po- tential 𝐶ss(x) from IPC, our solid-fluid coupling problem can be HoudiniaHoudiniaHoudiniaA Contact Proxy Splitting Method for Lagrangian Solid-Fluid Coupling 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Twist cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' A cylindrical cloth with four holes is twisted, squeezing out water from the inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' solved in a monolithic manner applying implicit Euler time integra- tion � v𝑛+1 = v𝑛 + ℎM−1(fext − ∇𝑃(x𝑛+1) − ∇Ψ(x𝑛+1) − ∇𝐶(x𝑛+1)) x𝑛+1 = x𝑛 + ℎv𝑛+1 , (18) which is equivalent to x𝑛+1 = arg min x 1 2 ∥x − ˆx𝑛∥2 M + ℎ2(𝑃(x) + Ψ(x) + 𝐶(x)) (19) with the mass matrix M, time step size ℎ, the predictive position ˆx𝑛 = x𝑛 + ℎv𝑛 + ℎ2M−1fext and the total contact potential 𝐶(x) = 𝐶sf(x) + 𝐶ss(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 4 EFFICIENT SOLVER A straightforward way to robustly solve the time-stepping optimiza- tion problem (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 19) is to apply the projected Newton’s method with line search [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' At every iteration, the search direction p can be computed by solving the linear system �H𝑓 G G𝑇 H𝑠 � p = �g𝑓 g𝑠 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' (20) Here H𝑓 and H𝑠 are the (projected) Hessian matrices w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' the posi- tion of fluids and solids respectively, and G = 𝜕2𝐸 𝜕x𝑓 𝜕x𝑠 denotes the coupling submatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Nevertheless, solving this linear system can be a severe bottleneck in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' One reason is that SPH techniques need sufficient neighbors to accurately approximate physical quanti- ties, which results in a much larger and denser fluid Hessian matrix H𝑓 compared to the solid one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In addition, the optimization may require many iterations to converge due to the sharpness of barrier energy, especially in contact-rich cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Since our fluid energies are all quadratic, we separate them from the highly nonlinear solids and contact energies via a robust time splitting scheme (§ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1) so that the fluid part can be solved within a single Newton iteration per time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We then propose efficient methods to solve the domain-decomposed linear systems (§ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1 Time Splitting 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1 Baseline Time Splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Intuitively, we can split the original time integration into a fluid phase \uf8f1\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f3 v𝑛+1/2 𝑓 = v𝑛 𝑓 + ℎM−1 𝑓 (−∇𝑓 𝑃([(x𝑛+1/2 𝑓 )𝑇, (x𝑛𝑠 )𝑇 ]𝑇 ) + f𝑓 ) x𝑛+1/2 𝑓 = x𝑛 𝑓 + ℎv𝑛+1/2 𝑓 (21) and a solid-coupling phase \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 v𝑛+1 = � v𝑛+1/2 𝑓 v𝑛𝑠 � + ℎM−1(−∇Ψ(x𝑛+1) − ∇𝐶(x𝑛+1) + � 0 f𝑠 � ) x𝑛+1 = x𝑛 + ℎv𝑛+1 , (22) where f𝑓 and f𝑠 are the external forces on the fluids and the solids respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In the fluid phase, we solve for an intermediate state for the fluid particles in a single Newton’s iteration, ignoring contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Then the highly nonlinear barrier force is resolved in the solid- coupling phase along with elasticity, where the fluid Hessian H𝑓 reduces to a block-diagonal matrix 𝜕2𝐶 (x) 𝜕x2 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In this setting, nonlinear optimization only happens for fluid boundaries and solid DOFs in the solid-coupling phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The details of this Baseline Time Splitting Scheme can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Although this baseline splitting strategy indeed brings a signifi- cant performance gain, severe instabilities can happen at the solid- fluid interface if the time step size is not sufficiently small, especially when simulating viscous fluids (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' For example, fluid particles may stick to the solid boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' This is an artifact also seen in existing SPH fluid solvers, and it is typically addressed by sampling particles at solid boundaries to exert boundary pressures [Akinci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2009b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Ihmsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In light of this, we consistently augment the fluid phase with proxy forces for solid-fluid contact to improve stability while avoiding any particle sampling overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2 Time Splitting with Contact Proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We introduce a solid-fluid contact proxy energy ˆ𝐶sf(x) into the fluid phase to efficiently exert approximated interaction forces between the boundaries of solids and fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In the following discussions, we will also write contact energy 𝐶(x) as the sum of the solid-fluid part (𝐶sf(x)) and the solid- solid part (𝐶ss(x)) for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' To ensure consistency with the original HoudiniaHoudiniaHoudiniaHoudinia6 Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' (A) (B) (C) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Bob simulated with (A) Joint Optimization, (B) Time Splitting with Contact Proxy, and (C) Baseline Time Splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' For this example, baseline time splitting can also produce visually plausible results, and our proxy-assisted scheme is 3× faster than joint optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' PDE, we cancel the contribution of this contact proxy in the solid- coupling phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The resulting time integration becomes � v𝑛+1/2 = v𝑛 + ℎM−1(−∇𝑃(x𝑛+1/2) + fext − ∇ ˆ𝐶sf(x𝑛+1/2)) x𝑛+1/2 = x𝑛 + ℎv𝑛+1/2 � v𝑛+1 = v𝑛+1/2 + ℎM−1(−∇Ψ(x𝑛+1) − ∇𝐶(x𝑛+1) + ∇ ˆ𝐶sf(x𝑛+1)) x𝑛+1 = x𝑛 + ℎv𝑛+1 (23) where the fluid phase now also implicitly updates the solid boundary nodes near the fluids to an intermediate state and explicitly update all other solid nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' For ˆ𝐶sf(x), a straightforward choice is ˆ𝐶sf(x) = 1 2𝐶sf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' But to ensure our fluid phase still only contains linear forces, we apply the 2nd-order Taylor expansion of 1 2𝐶sf(x) at x𝑛 for the approximation in the fluid phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' ˆ𝐶sf(x) = 1 2 � 𝐶sf(x𝑛) + ∇𝐶sf(x𝑛)(x − x𝑛) + 1 2 ∥x − x𝑛∥2 ∇2𝐶sf(x𝑛) � , (24) while in the solid-coupling phase, we simply use ˆ𝐶sf(x) = 1 2𝐶sf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In the appendix, we prove that our time splitting scheme with con- tact proxy only has an O(ℎ4) mismatch compared to implicit Euler solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Reformulating both phases (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 23) as optimization prob- lems, we obtain x𝑛+1/2 = arg min x 1 2 ∥x − ˆx𝑛∥2 M + ℎ2(𝑃(x) + ˆ𝐶sf(x)), x𝑛+1 = arg min x 1 2 ∥x − x𝑛+1/2∥2 M + ℎ2(Ψ(x) + 1 2𝐶sf(x) + 𝐶ss(x)), (25) where ˆx𝑛 = x𝑛 + ℎv𝑛 + ℎ2M−1fext with fext = [f𝑇 𝑓 , f𝑇𝑠 ]𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In addition to avoiding fluid particle sticking issues without ex- tra expensive costs, another benefit of our method is that it helps reduce the number of Newton’s iterations for solving the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Typically, the barrier method takes many Newton iterations when resolving high-speed impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' With our scheme, when high-speed fluid particles are colliding with a deformable object, their speed will be significantly reduced after the fluid phase due to the contact proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The reduced speed will then be taken into the solid-coupling phase, which makes the nonlinear optimization easier to solve (by having less contact constraint set changes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The details of our proxy- based time splitting scheme can be found in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Algorithm 1 Time Splitting with Contact Proxy 1: x ← x𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' ˆx𝑛 ← x𝑛 + ℎv𝑛 + ℎ2M−1fext ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2: SPH Neighbor Search & Density Update ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='3: ˆ𝐶sf(x) ← 2nd Taylor Expansion of 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2𝐶sf(x) at x = x𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='4: // Fluid Phase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='5: H ← ℎ2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='∇2𝑃(x) + ∇2 ˆ𝐶sf(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='+ M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='6: p ← −H−1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='ℎ2(∇𝑃(x) + ∇ ˆ𝐶sf(x)) + M(x − ˆx𝑛) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='7: x ← x + p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='8: x𝑛+1/2 ← x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='9: // Solid-Coupling Phase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='10: do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='11: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='H ← ℎ2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='∇2Ψ(x) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2∇2𝐶sf(x) + ∇2𝐶ss(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='+ M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='12: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='g ← ℎ2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='∇Ψ(x) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2∇𝐶sf(x) + ∇𝐶ss(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='+ M(x − x𝑛+1/2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='13: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='p ← −H−1g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='14: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='𝛼 ← Backtracking Line Search with CCD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='15: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='x ← x + 𝛼p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='16: while 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='ℎ ∥p∥ > 𝜖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='17: x𝑛+1 ← x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' v𝑛+1 ← (x − x𝑛)/ℎ 18: return x𝑛+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' v𝑛+1 Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' one can also separate elasticity from contact energy using the contact proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In this fashion, we would have a three-phase (fluid, solid, and contact) time splitting scheme x𝑛+1/3 = arg min x 1 2 ∥x − ˆx𝑛∥2 M + ℎ2(𝑃(x) + ˆ𝐶sf(x)), x𝑛+2/3 = arg min x 1 2 ∥x − x𝑛+1/3∥2 M + ℎ2(Ψ(x) + ˆ𝐶sf(x) + ˆ𝐶ss(x)), x𝑛+1 = arg min x 1 2 ∥x − x𝑛+2/3∥2 M + ℎ2( 1 3𝐶sf(x) + 1 2𝐶ss(x)), (26) where ˆ𝐶sf(x) and ˆ𝐶ss(x) are the 2nd-order Taylor expansion of 1 3𝐶sf(x) and 1 2𝐶ss(x) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' However, this aggressive split- ting scheme only applies to inversion-robust constitutive models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' the fixed corotated model [Stomakhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' While inver- sion can be prevented with guarantee at the solid phase where the elasticity energy is considered, it may not hold at the contact phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Despite this limitation, the three-phase splitting scheme can still work properly for inversion-robust constitutive models in practice to further accelerate the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' HoudiniaiupnohiupnohA Contact Proxy Splitting Method for Lagrangian Solid-Fluid Coupling 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2 Solving Linear Systems In our time splitting scheme, solving large sparse linear systems dominates both the computational and memory costs of each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We thus devise matrix-free and Schur-complement based strategies to solve them efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1 Fluid Phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Since 2-ring neighbors of SPH particles need to be considered in our formulation, both constructing and directly factorizing the Hessian matrix can cost a significant amount of time and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Therefore, we devise a matrix-free conjugate gradient (CG) solver to efficiently solve for the intermediate state of fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' As all energy potentials are quadratic in this phase, the energy gradient g(x) is merely a linear function of x with constant coeffi- cient matrix H(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Thus, the product between H(x) and an arbitrary vector p can be expressed as H(x)p = g(p) − g(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' (27) This allows us to compute gradients to evaluate the matrix-vector product, and we only need to acquire the 3 × 3 diagonal blocks of the Hessian for block-Jacobi preconditioning in our CG solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2 Solid-Coupling Phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' As the fluid energy potential is not included in this phase, the components of the Hessian matrix become H𝑓 = 𝜕2𝐶(x) 𝜕x2 𝑓 , G = 𝜕2𝐶(x) 𝜕x𝑠𝜕x𝑓 , H𝑠 = 𝜕2𝐶(x) 𝜕x2𝑠 + 𝜕2Ψ(x) 𝜕x2𝑠 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' (28) Although this linear system is no longer that intractable, it is not optimal to directly factorize the whole system given the considerable amount of nonzeros in H𝑓 and G when fluid resolution is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We thus design a domain decomposed linear solver that treats H𝑓 and H𝑠 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Based on Schur complement [Zhang 2006], the inverse of our Hessian matrix can be expressed as H−1 = � H−1 𝑓 + H−1 𝑓 G(H/H𝑓 )−1G𝑇 H𝑓 −H−1 𝑓 G(H/H𝑓 )−1 −(H/H𝑓 )−1G𝑇 H𝑓 (H/H𝑓 )−1 � , (29) where H/H𝑓 = H𝑠 − G𝑇 H−1 𝑓 G is the Schur complement of block H𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Since the nonzeros of H𝑓 only exist in the diagonal blocks, it is trivial to obtain its inverse matrix H−1 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We can then apply the CHOLMOD [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2008] LLT solver to factorize H/H𝑓 , which is only in the size of solid DOFs, and then the search direction can be computed via matrix-vector products and back-solves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' When there is no solid-fluid interaction, H/H𝑓 ’s sparsity pattern remains identical with H𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Only when two solid nodes 𝑖 and 𝑗 are interacting with the same fluid particle, the 3 × 3 block (H/H𝑓 )𝑖,𝑗 (in 3D) will become non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Typically, this only happens for neighboring mesh primitives and thus the sparsity pattern of H/H𝑓 is mostly nice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Note that when the three-phase time splitting scheme (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 26) is used, our domain decomposed solver can also be applied to the solid and contact phases since their systems share a similar structure with the solid-coupling phase here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 5 EXPERIMENTS AND EVALUATION Our code is implemented in C++ with Eigen for basic linear algebra operations and Intel TBB for multi-threading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The time step size of all our simulations is adaptively chosen by the SPH CFL con- dition and a user-defined upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We set the support radius Initial (A) (B) (C) (a) A viscous armadillo dropped onto the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' (A) (B) (C) (b) Cube on cloth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' An elastic cube is dropped onto a square cloth with four corners fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Simulation results of (A) Joint Optimization, (B) Time Splitting with Contact Proxy, and (C) Baseline Time Splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' While directly applying time splitting results in instability at the boundaries, our results with contact proxy are consistent with joint optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' of our SPH kernel function to 2𝑑, where 𝑑 is the particle diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In our implementation, we use the cubic Spline kernel for density estimation and the Spiky kernel for gradient calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' For Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 5, 4, 2 and 6, we employ our three-phase time splitting scheme, showing its efficacy when the constitutive models are compatible with mesh inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' For the rest of the simulations, we stick with our two-phase time splitting scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Most experiments are per- formed on a 24-core 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='50GHz Intel i9-10920X machine, except for the comparative study with ElastoMonolith [Takahashi and Batty 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We demonstrate that our method achieves efficient and robust solid-fluid coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The parameters and timing breakdown of all simulations are provided in Table 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 12 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1 Ablation Study 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1 Time Splitting Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Three simulations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 7) are performed to demonstrate the efficiency of time splitting and the efficacy of our proposed contact proxy on maintaining stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' To begin with, we need to take care of choosing a proper time step ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' First of all, it has to be restricted by the CFL condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Otherwise, severe volume loss may be observed due to SPH approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Additionally, in contrast to the joint optimization (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 19), the time splitting scheme usually requires smaller time steps to stay stable, which imposes a second time step constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' However, we observed that in practice, even using the largest CFL time step, our proxy-assisted time splitting can still work properly and produce stable simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Hence, for comparison, we use the largest CFL time step for both schemes to maximize their performance as smaller ℎ typically takes more Newton’s iterations in total to simulate a frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' For joint optimization, since direct factorization is intractable, we solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 20 using the block-Jacobi preconditioned conjugate gradient solver with the fluid part matrix free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' HoudiniaHoudiniaHoudiniaHoudinia8 Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Scene Δ𝑡frame Δ𝑡 𝑁fluid 𝑁solid 𝑘𝐼 𝜈𝑓 𝑑 𝜌𝑓 E 𝜈𝑠 𝜌𝑠 𝑇 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 6 Bob 1/24 4 × 10−3 97K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='3K 2 × 105 0 15 1000 1 × 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='3 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 7a Viscous Armadillo 1/48 4 × 10−3 238K 0 1 × 105 100 10 1200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2 Shot Armadillo 1/24 4 × 10−3 103K 16K 1 × 105 0 10 1000 1 × 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='3 200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 9 Dam Break 1/24 5 × 10−3 280K 0 2 × 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='005 25 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 10a Liquid Bunnys 1/50 4 × 10−3 52K 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='7K 1 × 105 0 10 1000 4 × 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='49 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 10b Liquid Bunnys 1/50 4 × 10−3 101K 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='5K 6 × 104 0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='4 1000 1 × 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='49 200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 4 Buoyancy 1/24 5 × 10−3 787K 66K 2 × 105 1 10 1000 1 × 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='4 200/700/1200 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 5 Twist Cylinder* 1/24 5 × 10−3 486K 12K 4 × 104 0 5 1000 500 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 3 Cream 1/24 4 × 10−3 159K 9K 3 × 104 25 3 1000 5 × 108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='49 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 13 Angry Cow* 1/24 5 × 10−3 789K 13K 1 × 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2 10 1000 1 × 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='45 100/700 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 1 Kick Water* 1/24 6 × 10−3 1M 43K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='5 × 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1 25 1000 500 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='9 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Simulation statistics including duration of each frame (Δ𝑡frame, [𝑠]), time step size upperbound (Δ𝑡, [𝑠]), number of fluid particles (𝑁fluid), number of solid vertices (𝑁solid), incompressibility coefficient (𝑘𝐼, [𝑃𝑎]), dynamic viscosity (𝜈𝑓 , [𝑃𝑎 ·𝑠]), fluid particle diameter (𝑑, [𝑚𝑚]), fluid density (𝜌𝑓 , [𝑘𝑔/𝑚3]), Young’s modulus (𝐸, [𝑃𝑎]), Possion’s ratio (𝜈𝑠), solid density (𝜌𝑠, [𝑘𝑔/𝑚3]) and the average simulation time for each frame (𝑇, [𝑚𝑖𝑛]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Timing statistics are measured on a 24-core 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='50GHz Intel i9-10920X machine except for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 10, which is tested on the “e2-standard-8” (8 cores with 32GB RAM) Google Compute Engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Note that examples marked with * contain codimensional materials, whose parameter settings are not covered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Scene Scheme Sec/Frame # Newton Iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='/Frame Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 6 Joint/TS/TSCP 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1 / 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='0 / 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='5 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='5 / 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='3 / 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 7a Joint/TS/TSCP 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='3 / 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='3 / 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='5 / 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='0 / 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='5 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Statistics of different time stepping schemes: Joint Optimiza- tion (Joint), Baseline Time Splitting (TS) and Time Splitting with Contact Proxy (TSCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Our proposed TSCP is much faster than both the Joint and TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' As shown in Table 2, even in these simple examples, our time split- ting scheme is significantly (up to 3×) faster than joint optimization, especially for cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 6) involving contacts between fluids and deformable solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' This improvement stems from no longer having to solve for incompressibility of fluids repeatedly within a time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Moreover, one can also find out that Newton’s iterations are much less with our proxy-assisted time splitting scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' As dis- cussed in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2, this is because the challenging high-speed impacts are already partially resolved in the fluid phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Another benefit of time splitting is the support of different error tolerances for the two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Errors in the fluid phase are sourced from the solution deviation of the CG solver, while in the solid phase they are directly controlled by the tolerance of Newton’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Typically, setting a slightly higher tolerance for fluids yields better performance while still producing visually plausible results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Aside from efficiency, our proposed contact proxy also improves the stability of time splitting scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Though simulation results of the baseline time splitting scheme look fine in the case of inviscid fluids, situations get worse when it is applied to viscous fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 7a, a viscous armadillo is dropped to the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In this example, the baseline time splitting scheme produces severe sticky artifacts at the boundary, and the fluid surface could not finally calm down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' By consistently applying our contact proxy to exert boundary pressure in the fluid phase, the artifacts can be well resolved as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Similarly, our idea of contact proxy is also applicable 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' = 1e4 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' = 1e5 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' = 1e6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Statistics of simulations with different stiffness parameter 𝑘𝐼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' A larger 𝑘𝐼 preserves volume better but results in more CG iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In this case, a proper 𝑘𝐼 = 105 Pa can be set to balance the computational cost and visual artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' to further separate elasticity from IPC contact while maintaining stability, leading to our three-phase scheme (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 7b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2 Linear Solver Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' For the fluid phase, we designed a matrix-free conjugate gradient (CG) solver that calculates the matrix-vector product via gradient computation to avoid the expen- sive computational and memory costs of direct factorization (§ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' However, the performance improvement from this approach will be less significant if the number of CG iterations required for conver- gence is too large, making the cost of computing gradients higher than constructing the Hessian once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In our fluid phase, the number of CG iterations is proportional to the stiffness 𝑘𝐼 of the incompress- ibility energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' A larger 𝑘𝐼 can better preserve the volume of the fluids but also results in a worse-conditioned system, demanding more iterations to converge (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In practice, by setting 𝑘𝐼 to a 80 70 60 CG 40 # 30 20 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='75 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='00 log10(ki)1200 k, = 1e4 1175 k, = 1e5 1150 k, = 1e6 1125 1100 1075 1050 1025 1000 0 i 2 3 4 5 Time[s]A Contact Proxy Splitting Method for Lagrangian Solid-Fluid Coupling 9 Solver Fluid Phase Solid Phase Contact Phase Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' hess solve solve solve CG + LLT 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='43 12375 Ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='25 1469 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Time and memory cost of different solvers in example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The costs are measured per time step in units [s] and [MB] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The baseline uses the Conjugate Gradient (CG) method for the fluid phase and CHOLMOD LLT for the solid and contact phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Our method instead em- ploys a matrix-free CG solver for the fluid phase and a domain-decomposed solver for the solid and contact phases, thereby improving efficiency and saving memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Initial Ours IISPH DFSPH Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Dam break with 280K SPH particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Our weakly compressible formulation produces stable fluid dynamics without visually evident volume loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Compared to incompressible SPH solvers IISPH [Ihmsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2013] and DFSPH [Bender and Koschier 2015], our simulation results demonstrate more smooth particle distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' proper value, we can efficiently solve the systems within 50 CG iterations without obvious fluid volume loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We test the performance of our matrix-free CG solver together with the domain-decomposed solver we designed for the solid- coupling phase on the Shot Armadillo example (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2), and present the results in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Our matrix-free CG solver significantly boosts efficiency (20× faster) and reduces memory costs by avoiding the construction of the Hessian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' On the other hand, our domain decomposed solver is 40% faster than directly factorizing the solid and contact systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2 Comparisons In this section, we compare our method with several popular SPH fluid solvers and a state-of-the-art solid-fluid coupling method Elas- toMonolith [Takahashi and Batty 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We leveraged the open- source library SPlisHSPlasH1 to implement the SPH fluid simula- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' To compare our method with ElastoMonolith, we set up two scenes from their paper with identical parameters and run all the simulations using “e2-standard-8” (8 cores with 32GB RAM) Google Compute Engine for fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='com/InteractiveComputerGraphics/SPlisHSPlasH (a) A liquid bunny dropped into a bowl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' (b) A liquid bunny and an elastic bowl dropped onto a static torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Liquid Bunnys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Compared to ElastoMonolith [Takahashi and Batty 2022], our method achives an over 5× speedup for both of these two examples with exactly the same scene setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1 Fluid Dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' While most existing SPH fluid solvers focus on incompressible fluids, our formulation treats fluids as weakly compressible, allowing us to couple fluids with deformable solids in a unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We run a dam break simulation to compare our method with two SPH fluid solvers IISPH [Ihmsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2013] and DFSPH [Bender and Koschier 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' These methods typically use particle resampling [Akinci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2013, 2012] or implicit repre- sentation [Bender et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Koschier and Bender 2017] to exert boundary counter-forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Our method instead employs IPC [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2020] for more robust solid-fluid coupling, with penetration-free guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We uniformly enforce the same CFL condition for all methods along with an upperbound at 5𝑚𝑠, and use the volume map [Bender et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2019] for their boundary handling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 9, though our formulation does not strictly enforce incompressibil- ity, it produces natural fluid dynamics without visually observable volume loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' On the other hand, our method (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='45 min/frame) is slower than IISPH (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='31 min/frame) and DFSPH (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='15 min/frame) due to the more sophisticated boundary handling strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' How- ever, our proposed approach can couple SPH fluids and elastic solids with arbitrary constitutive models, while most existing SPH meth- ods [Kugelstadt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Peer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2018] treat elastic solids as incompressible, which is not generally applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2 Solid-Fluid Coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We then compare our method with ElastoMonolith [Takahashi and Batty 2022], which couples Eulerian fluids with Lagrangian solids in a monolithic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Following their experiment setting, we run two solid-fluid coupling simula- tions with identical parameters using our method (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The timing of our method for these two scenes are 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1 sec/frame and 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='8 sec/frame respectively, both of which are over 5× faster than ElastoMonolith according to their reported timings (253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2 sec/frame and 352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='0 sec/frame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Coupling Eulerian fluids with Lagrangian solids requires dealing with geometric differences and it is often HoudiniaHoudiniaHoudiniaHoudiniaHoudiniaHoudinia10 Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Varying friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Three viscous bunnies are dropped onto the slope with different coefficients of friction 𝜇 (from left to right: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='03, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Our method supports adjustable solid-fluid boundary friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Timing breakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We show the timing profile of different simulation phases and plot the proportions of the major routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Examples marked with are simulated using our three-phase time splitting scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Other examples are generated with the two-phase scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In particular, SPH update (including neighborhood search and density update) only occurs in the fluid phase, line search happens in the solid and contact phases for non-linear optimization, and continuous collision detection (CCD) is counted when IPC contact energy is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' needed to perform SPD reformulation to make the linear system tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' As stated in ElastoMonolith, this SPD reformulation can introduce many additional non-zeros to the system, especially when contacts are rich and solids are intricately shaped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Conversely, our method treats solids and fluids from a unified Lagrangian viewpoint, where solid-solid and solid-fluid contacts are resolved in a unified manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='3 Complex Scenarios We then evaluate the efficiency and robustness of our method in more complicated scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We demonstrate our method correctly captures the buoyancy behavior in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Viscous fluids can also be naturally simulated (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 3), even with adjustable boundary friction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In addition, two-way coupling with thin shells (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 1) is also well supported with penetration-free guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 13, we show that our framework can even simulate a tightly coupled system of geometries in arbitrary codimensions (0, 1, 2, and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Detailed parameter settings can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 6 CONCLUSION We presented a unified two-way strong coupling framework for weakly-compressible SPH fluids and nonlinear elastic FEM solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' To achieve this, we modeled solid-fluid interactions as contact forces between SPH particles and FEM boundary elements, applying IPC for guaranteed non-penetration and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' As we track the vol- ume change of SPH particles in an updated Lagrangian fashion, the incompressibility energy stays quadratic and nice particle dis- tributions are maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Utilizing a symmetric approximation of discrete viscosity forces, we proposed a viscosity potential that fit- ted into optimization time integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We then proposed a time splitting scheme with a contact proxy to efficiently solve the time integration optimization while maintaining robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The per- formance is further boosted by our matrix-free conjugate gradient method and a domain-decomposed solver based on Schur comple- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Compared to existing works [Takahashi and Batty 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Zarifi and Batty 2017] coupling Eulerian fluids with Lagrangian elastic solids, SPH Update Gradient Hessian Linear System Line Search CCD Z Fluid Phase SolidPhase ContactPhase Twist Cylinder* Buoyancy Shot Armadillo* Bob* KickWater Angry Cow Inviscid Bunny ViscousBunny DamBreak Viscous Armadillo 0% 20% 40% 60% 80% 100%HoudiniaHoudiniaHoudiniaA Contact Proxy Splitting Method for Lagrangian Solid-Fluid Coupling 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Angry cow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We show our method can simulate the coupling of materials in arbitrary codimensions, including fluid particles, rods (the rubber bands), thin shells (the leather pad), deformable solids (the cow), and rigid bodies (the cubes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We launch an angry cow with a slingshot, and the cow hits through the wall and then falls into the water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Interactions between various materials are all accurately captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' our method treats both fluids and solids in a Langrangian manner, avoiding the need to handle different spatial discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Under such a unified view, our method achieves more convenient and robust two-way coupling, even between fluids and codimensional solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Likewise, different from existing SPH methods [Kugelstadt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Peer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2018] that treat all materials as SPH particles, our formulation enjoys both the efficiency of SPH fluids and the accuracy of FEM solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' There are many meaningful future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' First, when fluid DOFs dominate, building and querying the spatial hash for each fluid particle can become a considerable cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In fact, since there is no solid-fluid contact for interior particles, we can construct the spatial data structure only in the intersection between the extended bounding boxes of the fluids and each solid for better efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In addition, the adhesion between solids and fluids is also an interesting behavior to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Similar to the barrier energy, adhesion forces can be exerted on close solid-fluid primitive pairs but in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Modeling adhesion via resolving the surface tension of fluids is also an interesting future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' REFERENCES Nadir Akinci, Jens Cornelis, Gizem Akinci, and Matthias Teschner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2013.' metadata={'source': 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+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Monolith: a monolithic pressure- viscosity-contact solver for strong two-way rigid-rigid rigid-fluid coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Tetsuya Takahashi and Christopher Batty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' FrictionalMonolith: a monolithic optimization-based approach for granular flow with contact-aware rigid-body cou- pling.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Tetsuya Takahashi, Yoshinori Dobashi, Issei Fujishiro, Tomoyuki Nishita, and Ming C Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Implicit formulation for SPH-based viscous fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In Computer Graphics Forum, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Wiley Online Library, 493–502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Andre Pradhana Tampubolon, Theodore Gast, Gergely Klár, Chuyuan Fu, Joseph Teran, Chenfanfu Jiang, and Ken Museth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Multi-species simulation of porous sand and water mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' ACM Transactions on Graphics (TOG) 36, 4 (2017), 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Yun Teng, David IW Levin, and Theodore Kim.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Xiao Yan, C-F Li, X-S Chen, and S-M Hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' MPM simulation of interacting fluids and solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In Computer Graphics Forum, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Wiley Online Library, 183–193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Omar Zarifi and Christopher Batty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' A positive-definite cut-cell method for strong two-way coupling between fluids and deformable bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Fuzhen Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The Schur complement and its applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Springer Science & Business Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' A DERIVATIVES OF FLUID POTENTIALS The gradient and Hessian of the incompressibility potential w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='t the fluid particle position are 𝜕𝑃𝐼 (x) 𝜕x = ∑︁ 𝑖 𝑘𝐼𝑉0(𝐽𝑖 − 1)𝐽𝑛 𝑖 ℎ 𝜕(∇ · v𝑛+1 𝑖 ) 𝜕x , 𝜕2𝑃𝐼 (x) 𝜕x2 = ∑︁ 𝑖 𝑘𝐼𝑉0(𝐽𝑛 𝑖 )2ℎ2 𝜕(∇ · v𝑛+1 𝑖 ) 𝜕x � 𝜕(∇ · v𝑛+1 𝑖 ) 𝜕x �𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' (30) This constant Hessian matrix is obviously positive semi-definite (PSD) since it is simply the sum of the outer product of some vector with positive coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Similarly, the gradient and Hessian of the viscosity potential w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='t the fluid particle position are 𝜕𝑃𝑉 (x) 𝜕x𝑖 = 𝜈 ∑︁ 𝑗 V𝑖𝑗v𝑛+1 𝑖𝑗 , 𝜕2𝑃𝑉 (x) 𝜕x𝑖𝜕x𝑗 = � 𝜈 ˆℎ � 𝑘 V𝑖𝑘, 𝑗 = 𝑖 − 𝜈 ˆℎ V𝑖𝑗, 𝑗 ≠ 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' (31) A Contact Proxy Splitting Method for Lagrangian Solid-Fluid Coupling 13 Since V𝑖𝑗 = −4(𝑑 + 2) 𝑚𝑖𝑚𝑗 𝜌𝑖+𝜌𝑗 ∇𝑖𝑊𝑖𝑗 (x𝑛 𝑖𝑗)𝑇 ∥x𝑛 𝑖𝑗 ∥2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='01ℏ2 ∝ x𝑛 𝑖𝑗 (x𝑛 𝑖𝑗)𝑇 is a 3 × 3 constant PSD matrix for any particle pair within a time step, the Hessian of the viscosity potential is a constant PSD matrix as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' B TIME SPLITTING B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='1 Baseline Time Splitting Applying time splitting, we can split the original time integration into a fluid phase \uf8f1\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f3 v𝑛+1/2 𝑓 = v𝑛 𝑓 + ℎM−1 𝑓 (−∇𝑓 𝑃([(x𝑛+1/2 𝑓 )𝑇, (x𝑛𝑠 )𝑇 ]𝑇 ) + f𝑓 ) x𝑛+1/2 𝑓 = x𝑛 𝑓 + ℎv𝑛+1/2 𝑓 (32) and a solid-coupling phase \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 v𝑛+1 = � v𝑛+1/2 𝑓 v𝑛𝑠 � + ℎM−1(−∇Ψ(x𝑛+1) − ∇𝐶(x𝑛+1) + � 0 f𝑠 � ) x𝑛+1 = x𝑛 + ℎv𝑛+1 , (33) where f𝑓 and f𝑠 are the external forces on the fluids and the solids respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The two phases of this baseline time splitting scheme have equivalent optimization forms x𝑛+1/2 𝑓 = 𝑎𝑟𝑔 min x𝑓 1 2 ∥x𝑓 − ˆx𝑛 𝑓 ∥2 M𝑓 + ℎ2𝑃([x𝑇 𝑓 , (x𝑛 𝑠 )𝑇 ]𝑇 ) x𝑛+1 = 𝑎𝑟𝑔 min x 1 2 ∥x − ˆx𝑛+1/2∥2 M + ℎ2(Ψ(x) + 𝐶(x)) (34) where ˆx𝑛 𝑓 = x𝑛 𝑓 +ℎv𝑛 𝑓 +ℎ2M−1 𝑓 f𝑓 and ˆx𝑛+1/2 = x𝑛+ℎ[(v𝑛+1/2 𝑓 )𝑇, (v𝑛𝑠 )𝑇 ]𝑇 + ℎ2M−1[0𝑇, f𝑇𝑠 ]𝑇 with v𝑛+1/2 𝑓 = (x𝑛+1/2 𝑓 − x𝑛 𝑓 )/ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The details of the optimization algorithm can be found in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Algorithm 2 Baseline Time Splitting 1: x ← x𝑛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' ˆx𝑛 𝑓 ← x𝑛 𝑓 + ℎv𝑛 𝑓 + ℎ2M−1 𝑓 f𝑓 2: SPH Neighbor Search & Density Update 3: // Fluid Phase 4: H𝑓 ← ℎ2∇2 𝑓 𝑃([x𝑇 𝑓 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' (x𝑛𝑠 )𝑇 ]) + M𝑓 5: p𝑓 ← −H−1 𝑓 � ℎ2∇𝑓 𝑃([x𝑇 𝑓 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' (x𝑛𝑠 )𝑇 ]𝑇 ) + M𝑓 (x𝑓 − ˆx𝑛 𝑓 ) � 6: x𝑓 ← x𝑓 + p𝑓 7: x𝑛+1/2 𝑓 ← x𝑓 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' v𝑛+1/2 𝑓 = (x𝑓 − x𝑛 𝑓 )/ℎ 8: // Solid Coupling Phase 9: ˆx𝑛+1/2 ← x𝑛 + ℎ[(v𝑛+1/2 𝑓 )𝑇,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' (v𝑛𝑠 )𝑇 ]𝑇 + ℎ2M−1[0𝑇,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' f𝑇𝑠 ]𝑇 10: do 11: H ← ℎ2 �∇2Ψ(x) + ∇2𝐶(x)� + M 12: p ← −H−1 � ℎ2(∇Ψ(x) + ∇𝐶(x)) + M(x − ˆx𝑛+1/2) � 13: 𝛼 ← Backtracking Line Search with CCD 14: x ← x + 𝛼p 15: while 1 ℎ ∥p∥ > 𝜖 16: x𝑛+1 ← x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' v𝑛+1 ← (x − x𝑛)/ℎ 17: return x𝑛+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' v𝑛+1 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2 Error Analysis The position update of implicit Euler and our proxy-enhanced time splitting scheme can be respectively expressed as x𝑛+1 IE = x𝑛 + ℎv𝑛 + ℎ2M−1� −∇𝑃(x𝑛+1 IE ) − ∇Ψ(x𝑛+1 IE ) − ∇𝐶sf(x𝑛+1 IE ) − ∇𝐶ss(x𝑛+1 IE ) + fext � x𝑛+1 = x𝑛 + ℎv𝑛 + ℎ2M−1� −∇𝑃(x𝑛+1/2) − ∇Ψ(x𝑛+1) − ∇ ˆ𝐶sf(x𝑛+1/2) − 1 2∇𝐶sf(x𝑛+1) − ∇𝐶ss(x𝑛+1) + fext � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' (35) If we define 𝑒(x) = ���x𝑛 + ℎv𝑛 + ℎ2M−1� −∇𝑃(x) − ∇Ψ(x) − ∇𝐶sf(x) − ∇𝐶ss(x) + fext � − x ���, (36) x𝑛+1 IE given by implicit Euler satisfies 𝑒(x𝑛+1 IE ) = 0, while for x𝑛+1 from our scheme, we have 𝑒(x𝑛+1) = O(ℎ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Specifically, 𝑒(x𝑛+1) = ℎ2���M−1(∇𝑃(x𝑛+1/2) − ∇𝑃(x𝑛+1) + ∇ ˆ𝐶sf(x𝑛+1/2) − 1 2∇𝐶sf(x𝑛+1)) ��� = O � ℎ3���M−1� ∇2𝑃(x𝑛)(v𝑛+1/2 − v𝑛+1)+ 1 2∇2𝐶sf(x𝑛)(v𝑛+1/2 − v𝑛+1) ���� � = O � ℎ4���M−1� ∇2𝑃(x𝑛) + 1 2∇2𝐶sf(x𝑛) � M−1� ∇Ψ(x𝑛) + 1 2∇𝐶sf(x𝑛) + ∇𝐶ss(x𝑛) ���� � = O(ℎ4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' (37) Here we assume that, in our discretized domain, the distance be- tween any pair of primitives (particle-particle pair, particle-triangle pair and triangle-triangle pair) has a lower bound 𝜖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Thus ∇2𝑃(x𝑛), ∇2𝐶sf(x𝑛), ∇Ψ(x𝑛), ∇𝐶sf(x𝑛) and ∇𝐶ss(x𝑛) are all bounded, and this indicates our method has an O(ℎ4) difference compared to implicit Euler solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Since implicit Euler has an O(ℎ2) error com- pared to the PDE solution, our proposed time splitting scheme shares the same order of accuracy with implicit Euler when it is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' C EXPERIMENT OF COMPLEX SCENARIOS In this section, we describe the experiment settings of our simula- tions in various complex scenarios and briefly discuss the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Buoyancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We drop three elastic elephants with varying densities into the water (1000 𝑘𝑔/𝑚3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The light grey elephant (200 𝑘𝑔/𝑚3) floats on the surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' the blue elephant (700 𝑘𝑔/𝑚3) is around half immersed in the water;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' and the red elephant (1200 𝑘𝑔/𝑚3) sinks into the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' This demonstrates that our method correctly captures the buoyancy behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' 14 Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Varying Friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We drop three viscous bunnies onto the slope with different coefficients of friction (orange bunny: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='5, green bunny: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='03, blue bunny: 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' All three bunnies share the same dy- namic viscosity coefficients 100 𝑘𝑔/𝑚3 and the angle of slope is 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Twist Cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Coupling fluids with thin shells is challenging since penetration can easily happen without careful treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' As stated in [Zarifi and Batty 2017], Eulerian fluids may flow through solids if their thickness is less than a grid cell size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Conversely, our approach adopts a unified Lagrangian view and penetration-free is guaranteed by IPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In this example, we simulate twisting a cylinder full of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The cylinder is modeled as a thin shell with a 2𝑚𝑚 thickness, and there are two holes in the front and back sides of this cylinder respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The left side and right side are rotated at 72◦/𝑠 and are slowly moved towards each other at 2𝑐𝑚/𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' As we twist the cylinder, the water gets squeezed out through the holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' This simulation demonstrates our method produces stable simulation results with penetration-free guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Cream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' This example exhibits the coupling behaviors of viscous fluids and elastic solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We use an elastic spoon to stir the cream in a porcelain bowl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The spoon handle rotates around y-axis at 360◦/𝑠 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content='2𝑚/𝑠) while the bowl is fixed at the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' As shown in our simulation results, the spoon gets deformed due to the resistance forces it receives from the viscous cream while stirring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Angry Cow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' We then show our framework can simulate natural physical behaviors of geometries in arbitrary codimensions (0, 1, 2, and 3) as well as their interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In this scene, the codimensional- 0,1,2 objects respectively refer to fluid particles, rubber bands and the leather pad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' A deformable cow is launched by the slingshot, hitting the wall consisting of rigid cubes, and then falling into the water pool, producing interesting physical behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' The density of the rigid cubes and the cow are 100𝑘𝑔/𝑚3 and 700𝑘𝑔/𝑚3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Kick Water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' In this example, we show a scene where a mannequin dressed in a multilayer skirt kicks in a large water pool, involv- ing complex interactions between fluid particles and garments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' As the mannequin moves in the water, our method produces natural deformation of the skirt caused by the contact with water;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' as it kicks out of the water at a high speed, the resulting water splash is also correctly captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} +page_content=' Our method well resolves the contacts among fluids particles, thin garments and rapidly moving complex boundaries with penetration-free guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9A0T4oBgHgl3EQfBf9X/content/2301.01976v1.pdf'} diff --git a/FNFLT4oBgHgl3EQfFy-9/content/tmp_files/2301.11989v1.pdf.txt b/FNFLT4oBgHgl3EQfFy-9/content/tmp_files/2301.11989v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..28c68c156c780fbfb67cee1f1c7bc4ab39fbe2c6 --- /dev/null +++ b/FNFLT4oBgHgl3EQfFy-9/content/tmp_files/2301.11989v1.pdf.txt @@ -0,0 +1,2288 @@ +Practical Differentially Private +Hyperparameter Tuning with Subsampling +Antti Koskela and Tejas Kulkarni +Nokia Bell Labs +Espoo, Finland +Abstract +Tuning all the hyperparameters of differentially private (DP) machine learning (ML) algorithms +often requires use of sensitive data and this may leak private information via hyperparameter values. +Recently, Papernot and Steinke (2022) proposed a certain class of DP hyperparameter tuning algo- +rithms, where the number of random search samples is randomized itself. Commonly, these algorithms +still considerably increase the DP privacy parameter ε over non-tuned DP ML model training and +can be computationally heavy as evaluating each hyperparameter candidate requires a new training +run. We focus on lowering both the DP bounds and the computational complexity of these methods +by using only a random subset of the sensitive data for the hyperparameter tuning and by extrapo- +lating the optimal values from the small dataset to a larger dataset. We provide a Rényi differential +privacy analysis for the proposed method and experimentally show that it consistently leads to better +privacy-utility trade-off than the baseline method by Papernot and Steinke (2022). +1 +Introduction +Our aim is two-fold: to decrease the computational cost as well as the privacy cost of hyperparameter +tuning of DP ML models. The reasons for this are clear. As the dataset sizes grow and models get +more complex, blackbox optimization of hyperparameters becomes more expensive since evaluation of +a single set of hyperparameters often requires retraining a new model. On the other hand, tuning the +hyperparameters often depends on the use of sensitive data, so it requires privacy protection as well, +as illustrated by the counterexample by Papernot and Steinke (2022). +Intuitively, the leakage from +hyperparameters is much smaller than from the model parameters, however providing tuning algorithms +with low additional DP cost has turned out challenging. Current best algorithms (Papernot and Steinke, +2022) still come with a considerable DP cost overhead. +Although our methods and results are applicable to general DP mechanisms, we will in particular focus +on tuning of the DP stochastic gradient descent (DP-SGD) (Song et al., 2013; Bassily et al., 2014; Abadi +et al., 2016) which has become the most widely used method to train ML models with DP guarantees. +Compared to plain SGD, DP brings additional hyperparameters to tune: the noise level σ and the clipping +constant C. Additionally, also the subsampling ratio γ affects the DP guarantees, as well as length of the +training. Tuning all the hyperparameters of DP-SGD commonly requires use of sensitive data. +We use the results by Papernot and Steinke (2022) as building blocks of our methods. Their work +was based on the analysis of Liu and Talwar (2019) who provided the first results for DP black-box +1 +arXiv:2301.11989v1 [cs.LG] 27 Jan 2023 + +optimization of hyperparameters, where, if the base training algorithm is (ε, 0)-DP, then the tuned model +is approximately (3ε, 0)-DP. Papernot and Steinke (2022) gave Rényi differential privacy (RDP) analysis +for a class of black-box tuning algorithms, where the number of random search samples is randomized +itself. As the privacy bounds are in terms of RDP and assume only RDP bounds about the candidate +model training algorithms, they are particularly suitable to tuning DP-SGD. However, still, running these +algorithms increase the ε-values two or three-fold or more, and they can be computationally heavy as +evaluating each candidate model requires training a new model. Our novelty is to consider using only +a random subset of the sensitive data for the tuning part and use the output hyperparameter values +(and potentially the model) for training subsequent models. Using a random subset for the privacy and +compute costly part automatically leads to both lower DP privacy leakage as well as computational cost. +We also consider ways to appropriately extrapolate the optimal value from the small subset of data to a +larger dataset. +The RDP bounds for the DP tuning methods by Papernot and Steinke (2022) assume that the RDP- +values of the candidate model training algorithms are fixed. We also consider ways to use these bounds +for tuning hyperparameters that affect the RDP-values of the base algorithm, being the noise level σ, the +subsampling ratio γ and the length of training in case of DP-SGD. +1.1 +Related Work on Hyperparameter Tuning +Chaudhuri and Vinterbo (2013) were the first ones focus on DP bounds for hyperparameter tuning. +An improvement was made by Liu and Talwar (2019) who considered black-box tuning of (ε, δ)-DP +mechanisms. +Mohapatra et al. (2022) showed that for reasonable numbers of adaptively chosen private +candidates a naive RDP accounting (i.e., RDP parameters grow linearly w.r.t. number of model eval- +uations) often leads to lower DP bounds then the methods by Liu and Talwar (2019). Papernot and +Steinke (2022) gave RDP bounds for black-box tuning algorithms that grow only logarithmically w.r.t. +number of model evaluations. In non-DP setting, hyperparameter tuning with random subsamples has +been considered for SVMs (Horváth et al., 2017), for large datasets in healthcare (Waring et al., 2020).. +Small random subsets of data have been used in Bayesian optimization of hyperparameters (Swersky +et al., 2013; Klein et al., 2017). +1.2 +Our Contributions +• We propose a novel subsampling strategy to lower the privacy and compute cost of DP hyperparam- +eter tuning. Our privacy analysis is in terms of RDP and we use existing results for tuning Papernot +and Steinke (2022) and DP-SGD (Zhu and Wang, 2019) as building blocks, however our methods +and results are independent of those. We provide a tailored RDP analysis for the proposed strategy. +• We propose algorithms to tune hyperparemeters that affect the RDP guarantees of the base model +training algorithms. We provide a rigorous RDP analysis for these algorithms. +• We carry out experiments on a variety of datasets, where we are able to consistently improve upon +the baseline tuning method by a clear margin. +2 + +2 +Background +2.1 +DP and DP-SGD +We first give the basic definitions. +An input dataset containing n data points is denoted as X = +(x1, . . . , xn) ∈ X n, where xi ∈ X, 1 ≤ i ≤ n. +We say X and Y are neighbours if we get one by +adding or removing one data element to or from the other (denoted X ∼ Y ). Consider a randomized +mechanism M : X n → O, where O denotes the output space. The (ε, δ)-definition of DP can be given +as follows (Dwork, 2006). +Definition 1. Let ε > 0 and δ ∈ [0, 1]. We say that a mechanism M is (ε, δ)-DP, if for all neighbouring +datasets X and Y and for every measurable set E ⊂ O we have: +Pr(M(X) ∈ E) ≤ eεPr(M(Y ) ∈ E) + δ. +We will also use the Rényi differential privacy (RDP) (Mironov, 2017) which is defined as follows. +Rényi divergence of order λ > 1 between two distributions P and Q is defined as +Dλ(P||Q) = +1 +λ − 1 log +� �P(t) +Q(t) +�λ +Q(t) dt. +(2.1) +Definition 2. We say that a mechanism M is (λ, ε)-RDP, if for all neighbouring datasets X and Y , the +output distributions M(X) and M(Y ) have Rényi divergence of order λ at most ε, i.e., +max +X∼Y Dλ +� +M(X)||M(Y ) +� +≤ ε. +We can convert from Rényi DP to approximate DP using, for example, the following formula: +Lemma 3 (Canonne et al. 2020). Suppose the mechanism M is +� +λ, ε′� +-RDP. Then M is also (ε, δ(ε))-DP +for arbitrary ε ≥ 0 with +δ(ε) = exp +� +(λ − 1)(ε′ − ε) +� +λ +� +1 − 1 +λ +�λ−1 +. +(2.2) +As is common, in practice we carry out the RDP accounting such that we do bookkeeping of total ε(λ)- +values for a list of RDP-orders (e.g. integer λ’s) and in the end convert to (ε, δ)-guarantees by minimizing +over the values given by the formula (2.2). RDP accounting for compositions of DP mechanisms is carried +using standard RDP composition results (Mironov, 2017). +DP-SGD differs from SGD such that sample-wise gradients of a random mini-batch are clipped to +have L2-norm at most C and normally distributed noise with variance σ2 is added to the sum of the +gradients of the mini-batch (Abadi et al., 2016). One iteration is given by +θj+1 = θj − ηj +� 1 +|B| +� +i∈Bj +∇f(xi, θj) + Zj +� +, +(2.3) +where Zj ∼ N(0, C2σ2 +|B|2 Id), and ηj denotes the learning rate hyperparameter and |B| is the expected batch +size (if we carry out Poisson subsampling of mini-batches, |Bj| varies). There are several RDP results +that enable the RDP analysis of DP-SGD iterations (Abadi et al., 2016; Balle et al., 2018; Zhu and Wang, +2019), and nowadays DP-SGD is an indispensable part of frameworks such as Opacus (Yousefpour et al., +3 + +2021) and Tensorflow Privacy (Papernot et al., 2020). The following result by Zhu and Wang (2019) is +directly applicable to analyzing DP-SGD, however we also use it for analyzing one of our hyperparameter +tuning strategies. +Theorem 4 (Zhu and Wang 2019). Suppose M is a +� +λ, ε(λ) +� +-RDP mechanism, w.r.t. to the add/remove +neighbourhood relation. +Consider the subsampled mechanism (M ◦ subsamplePoisson(γ))(X). +If M is +� +λ, ε(λ) +� +-RDP then M ◦ subsamplePoisson(q) is +� +λ, ε′(λ) +� +-RDP (λ ≥ 2 is an integer), where +ε′(λ) = +1 +λ − 1 log +� +(1 − γ)λ−1(λγ − γ + 1) + +� +λ +2 +� +γ2(1 − γ)λ−2e ε(2) + 3 +λ +� +j=3 +� +λ +j +� +γj(1 − γ)λ−je (j−1)ε(j) +� +. +We remark that the recent works (Koskela et al., 2020; Gopi et al., 2021; Zhu et al., 2022) give methods +to carry out (ε, δ)-analysis of DP-SGD tightly. +2.2 +DP Hyperparameter Tuning +When tuning the hyperparameters using sensitive data, as Papernot and Steinke (2022) show, private +information may leak, especially in case the hyperparameters are chosen based on non-private ML model +training. Intuitively, the leakage from hyperparameters is much smaller than from the model parameters, +however considering it in the final accounting is essential to ensure rigorous DP guarantees. As described +in Section 1.1, currently the most practical (ε, δ)-guarantees for DP hyperparameter tuning algorithms +are those of Papernot and Steinke (2022). +In the results of Papernot and Steinke (2022) important is that the number of candidate models K is +randomized. They analyze various distributions for drawing K, however we focus on using the Poisson +distribution as it is the most concentrated around the mean among all the alternatives. The corresponding +hyperparameter tuning algorithm and its privacy guarantees are given as follows. +First recall: K is distributed according to a Poisson distribution with mean µ > 0, if for all non- +negative integer values k: P(K = k) = e−µ · µk +k! . +Theorem 5 (Papernot and Steinke 2022). Let Q : X N → Y be a randomized algorithm satisfying +� +λ, ε(λ) +� +-RDP and (�ε, �δ)-DP for some λ ∈ (1, ∞) and ε, �ε, �δ ≥ 0. Assume Y is totally ordered. Let the +Poisson distribution parameter µ > 0. Define the hyperparameter tuning algorithm A : X N → Y as +follows. Draw K from a Poisson distribution with mean µ. Run Q(X) for K times. Then A(X) returns +the best value of those K runs (both the hyperparameters and the model parameters). If K = 0, A(X) +returns some arbitrary output. If e �ε ≤ 1 + +1 +λ−1, then A satisfies +� +λ, ε′(λ) +� +-RDP, where +ε′(λ) = ε(λ) + µ · �δ + log µ +λ − 1. +Remark 6. Notice that in a common analysis of DP-SGD, the guarantees hold not only for the best +hyperparameters and the corresponding models, but for the whole histories of models: when the output +A(X) of the base mechanism is a sequence of DP gradients, post-processing of that sequence gives the +sequence of models +� +θ1, . . . , θT +� +. Thus, in the tuning algorithm, we are free to do model check-pointing +and choose the best model along the histories. +4 + +3 +DP Hyperparameter Tuning with a Random Subset +We next consider our main tools: we carry out the private hyperparameter tuning on a random subset, +and if needed, extrapolate the found hyperparameter values to larger datasets that we use for training +subsequent models. We consider two strategies: in the first one the subset of data used for tuning is +possibly smaller than the data used for training for subsequent models and thus we extrapolate the +hyperparameter values. In the other approach, the subsequent models are trained with datasets of the +same size as the tuning set. +3.1 +Strategy 1: Small Random Subset for Tuning +In the first strategy, we carry out the following steps: +1. Use Poisson subsampling to draw X1 ⊂ X: draw a random subset X1 such that each x ∈ X is +included in X1 with probability q. +2. Compute (θ1, t1) = M1(X1), where M1 is a hyperparameter tuning algorithm (e.g., method by +Papernot and Steinke, 2022) that outputs the vector of optimal hyperparameters t1 and the corre- +sponding model θ1. +3. If needed, extrapolate the hyperparameters t1 to the dataset X \ X1: t1 → t2. +4. Compute θ2 = M2(t2, X \ X1), where M2 is the base mechanism (e.g., DP-SGD training run). +Denote the whole mechanism by M. Then, we may write +M(X) = +� +M1(X1), M2 +� +M1(X), X \ X1 +� +. +(3.1) +Notice that since M2 and M1 use the same randomness about X1, (3.1) cannot be analyzed using stan- +dard RDP composition results. We provide a tailored analysis for this method in Section 3.3. However, +we can consider alternative third and fourth steps which use the whole dataset X: +3. If needed, extrapolate the hyperparameters t1 to the dataset X: t1 → t2. +4. Compute θ2 = M2(t2, X). +Then, the procedure can be written as a composition +M(X) = +� +� +M1(X), M2 +� � +M1(X), X +� +, +(3.2) +where � +M1(X) = (M1◦subsample)(X), and the RDP guarantees are obtained using standard subsampling +and composition results. +3.1.1 +Extrapolating the Hyperparameters +In this work, we use simple heuristics to transfer the optimal hyperparameter values found for the small +subset of data to a larger dataset. When using DP-SGD, we use the heuristics used by (van der Veen +et al., 2018): we scale the learning rate η with the dataset size. I.e., if we carry out the hyperparameter +tuning using a subset of size m and find an optimal value η∗, we multiply η∗ by n/m when transferring +5 + +to the dataset of size n. Clipping constant C, the noise level σ and the subsampling ratio γ are kept +constant in this transfer. +This can be also heuristically motivated as follows. Consider T steps of the DP-SGD (2.3). With the +above rules, the distribution of the noise that gets injected into the model is +T +� +j=1 +Zj ∼ N +� +0, +T ·( n +m η∗) +2σ2C2 +(γ·n)2 +� +∼ N +� +0, T ·η∗2σ2C2 +(γ·m)2 +� +which is exactly the distribution of the noise added to the model trained with the subsample of size m. +Note that such linear scaling of learning rate has also been considered in non-DP SGD (Goyal et al., +2017) to obtain similar validation errors for both small and large minibatch training. +Training of certain models benefits from use of adaptive optimizers such as Adam (Kingma and Ba, +2014) or RMSProp, e.g., due to sparse gradients. Then the above extrapolation rules for DP-SGD are +not meaningful anymore. In our experiments, when training a neural network classifier for the IMDB +dataset and using Adam with DP-SGD gradients, we found that keeping the value of learning rate fixed +in the transfer to the larger dataset lead to better results than increasing it as in case of DP-SGD. We +mention that there are principled ways of extrapolating the hyperparameters in non-DP setting such as +those of (Klein et al., 2017). +3.2 +Strategy 2: Model Ensemble with Disjoint Shards +Strategy 1 can be directly extended to the case, where we divide the dataset into multiple disjoint sets of +the same size and train a model for each of the datasets separately. Similar sharding has been considered +also in the machine unlearning framework SISA by Bourtoule et al. (2021) and also in the framework +PATE Papernot et al. (2016) where student models are trained on disjoint datasets. We randomly pick +only one of the shards for hyperparameter tuning. +Our strategy 2 is described as follows: +1. Divide X into k shards such that for each data element x of X, the probability of ending up to +shard i, i = 1, . . . , k, is 1/k. +2. First compute (θi, t) = M1(Xi) using some shard Xi, i ∈ [k] and then θj = M2(t, Xj) for all +j ∈ [k]\{i} (can be computed in parallel). +3. Perform predictions using an ensemble of models, i.e., if f(x, θ) denotes the model output for a +data element x with model parameters θ, we carry out predictions using the weighted average +1 +k +�k +j=1 f(x, θj). +It is reasonable to assume that the optimal hyperparameters for the rest of the shards are not far +from those of the shard Xi, i.e., no extrapolation is needed. +3.3 +RDP-Analysis of Strategy 1 and Strategy 2 +We first consider our strategy 1, i.e., the mechanisms (3.1) and (3.2). The mechanism (3.2) can be analyzed +using subsampling and composition results for RDP. Using the RDP values given by Thm. 5 for M1 and +a subsampling amplification result such as Thm. 4, we obtain RDP bounds for (M1◦subsamplePoisson(γ)). +Using RDP bounds for M2 (e.g., DP-SGD) and composition results, we further get RDP bounds for M +6 + +of (3.2). However, if we only use the rest of the data X\X1 for M2, i.e. the mechanism (3.1), we can get +even tighter RDP bounds. +Let X ∈ X n and Y = X ∪ {x′} for some x′ ∈ X. To bound the RDP-parameters for the mechanism +M of (3.1), it is sufficient that we find X− and x′−indepedent bounds for +ε(λ) = max{Dλ +� +M(Y )||M(X) +� +, Dλ +� +M(X)||M(Y ) +� +} +for different orders of λ.These bounds should be obtained in terms of RDP values for M1 (the hyper- +parameter tuning algorithm, RDP bounds obtained, e.g., using Thm. 5) and for M2 (DP-SGD, RDP +bounds obtained using, e.g., Thm. 4). +The following theorem gives tailored RDP bounds for the mechanism (3.1). Similarly to the analy- +sis of Zhu and Wang (2019) for the Poisson subsampled Gaussian mechanism, we obtain RDP bounds +for (3.1) using the RDP bounds of the mechanisms M1 and M2 and by using binomial expansions (proof +given in the Appendix C.2). We can also use this theorem to analyse the ensemble method described +in Section 3.2, since the remaining submodels can be thought of as a parallel composition for which we +obtain RDP bounds using Lemma D.1 and its Corollary D.2 of Appendix D. +Theorem 7. Let X ∈ X n and Y = X ∪ {x′} for some x′ ∈ X. Let M(X) be the mechanism described +in Section 3.1, such that X1 is sampled with sampling ratio q, 0 ≤ q ≤ 1. Let λ > 1. Denote by εP (λ) +and εQ(λ) the RDP-values of mechanisms M1 and M2, respectively. We have that +Dλ +� +M(Y )||M(X) +� +≤ +1 +λ − 1 log +� +qλ exp +� +(λ − 1)εP (λ) +� ++ (1 − q)λ exp +� +(λ − 1)εQ(λ) +� ++ λ · qλ−1 · (1 − q) · exp +� +(λ − 2)εP (λ − 1) +� ++ λ · q · (1 − q)λ−1 · exp +� +(λ − 2)εQ(λ − 1) +� ++ +λ−2 +� +j=2 +�λ +j +� +· qλ−j · (1 − q)j · exp +� +(λ − j − 1)εP (λ − j) +� +exp +� +(j − 1)εQ(j) +�� +(3.3) +and +Dλ +� +M(X)||M(Y ) +� +≤ +1 +λ − 1 log +� +(1 − q)λ−1 exp +� +(λ − 1)εQ(λ) ++ qλ−1 exp +� +(λ − 1)εP (λ) ++ +λ−2 +� +j=1 +�λ − 1 +j +� +· qj · (1 − q)λ−1−j · exp +� +j · εP (j + 1) +� +· exp +� +(λ − j − 1)εQ(λ − j) +�� +. +(3.4) +Remark 8. We can initialize the subsequent model training M2 using the model θ1. This adaptivity is +included in the RDP analyses of both strategies (3.1) and (3.2). +3.4 +Computational Savings +The expected number of required gradient evaluation for our strategy 1 is bounded by µ · q · n· epochs ++n· epochs, whereas the baseline requires in expectation µ · n· epochs evaluations. For example, in our +experiments with µ = 45 and q = 0.1, the baseline method requires +µ +µ·q+1 ≈ 8 times more gradient +evaluations than our strategy 1. +7 + +4 +Dealing with Hyperparameters that Affect the DP Guarantees +Theorem 5 gives RDP-parameters of order λ for the tuning algorithm, assuming the underlying candidate +picking algorithm is +� +λ, ε(λ) +� +-RDP. If we are tuning, for example, the DP-SGD hyperparameters learning +rate η or clipping constant C, and fix rest of the hyperparameters, these +� +λ, ε(λ) +� +-RDP bounds are +fixed for all hyperparameter candidates. However, if we tune the hyperparameters that affect the DP +guarantees, i.e., the subsampling ratio γ, the noise level σ or the length of the training T, it is less +straightforward to determine what are the uniform ε(λ)-upper bounds. +As is common practice, we fix a grid Λ of λ-orders for RDP bookkeeping (e.g. integer values of λs). +If for each random choice of hyperparameters, and for each λ ∈ Λ, the RDP value �ε(λ) of each candidate +model is bounded by a value ε(λ), then we can use ε(λ) to obtain RDP-bounds of the hyperparameter +tuning algorithm with Theorem 5. This is formally shown below in Lemma 9. For example, when we +tune DP-SGD and vary the values of γ and σ, we need to somehow ensure that the RDP values of the +candidates stay below the upper bound. We below consider two algorithms for finding random candidates +and appropriate upper RDP bounds ε(λ) for the candidate picking algorithm Q. +4.1 +Algorithm 1: Grid Search with Randomization +In the first approach we first set an approximative DP target value (ε, δ) which we use to adjust the +hyperparameters. For example, if we are tuning the DP-SGD parameters subsampling ratio γ and noise +level σ, we can, for each choice of (γ, σ), adjust the length of the training T so that the resulting training +iteration is at most (ε, δ)-DP. Vice versa, we may tune the γ and T, and take minimal value of σ such +that the resulting training iteration is at most (ε, δ)-DP. +Consider first tuning of DP-SGD and how to set T for each choice of hyperparameters γ and σ. We +first fix ε, δ > 0 which represent the target approximative DP bound for each candidate model. Denote +by ε(T, δ, γ, σ) the approximate DP ε of the subsampled Gaussian mechanism with parameter values γ, σ +and T. For each value of (γ, σ), we attach a number of iterations Tγ,σ such that it is the largest number +with which the resulting composition is (ε, δ)-DP: +Tγ,σ = max{T ∈ N : ε(T, δ, γ, σ) ≤ ε}. +As the RDP values increase monotonously w.r.t. number of compositions, it is straightforward to find +Tγ,σ, e.g., using the bisection method Wang et al. (2019). +Alternatively, we could fix a few values of T, and to each combination of (γ, T), attach the smallest +σ (denoted σγ,T ) such that the target (ε, δ)-bound holds. We prefer this option in our experiments to +suit our computational constraints. By the data-processing inequality, the privacy parameters decrease +monotonously w.r.t. σ, so that again, e.g., the bisection method can be used to find σ. +We consider a finite grid Γ of possible hyperparameter values t (e.g., in case of DP-SGD, t = (γ, σ, T), +where T is adjusted to γ and σ). Then, for all t ∈ Γ, we compute the corresponding RDP value εt(λ) for +each λ ∈ Λ. Finally, for each λ ∈ Λ, we set +ε(λ) = max +t∈Γ εt(λ). +Then, for each random draw of t, the DP-SGD trained candidate model is is ε(λ)-RDP, and by Lemma 9 +below, the candidate picking algorithm Q is also ε(λ)-RDP. +8 + +4.2 +Algorithm 2: Random Search +Here, we assume we are given some distributions of the hyperparameter candidates and the algorithm +Q draws hyperparameters using them. In order to adjust the number of iterations for each candidate, +we take a λ-line as an RDP upper bound. More specifically, we require that the candidate models are +� +λ, c · λ +� +-RDP for some c > 0 and for all λ ∈ Λ. Then the number of iterations Tγ,σ for each draw of +(γ, σ) is the maximum number of iterations for which the +� +λ, c · λ +� +-RDP bound holds, i.e., +Tγ,σ = max{T ∈ N : T · εγ,σ(λ) ≤ c · λ for all λ ∈ Λ}. +Similarly, we can find the minimal σ based on T and γ such that the mechanism is +� +λ, c · λ +� +-RDP for all +λ ∈ Λ. +Again, by Lemma 9 below, the candidate picking algorithm Q is then c · λ-RDP and we may use +Thm. 5 to obtain RDP bounds for the tuning algorithm +4.3 +RDP Analysis of Algorithms 1 and 2 +Lemma 9. Denote by C the random variable of which outcomes are the hyperparameter candidates +(drawing either randomly from a grid or from given distributions). Consider an algorithm Q, that first +randomly picks hyperparameters t ∼ C, then runs a randomised mechanism M(t, X). Suppose M(t, X) +is +� +λ, ε(λ) +� +-RDP for all t. Then, Q is +� +λ, ε(λ) +� +-RDP. +Proof. Suppose the hyperparameters t are outcomes of a random variable C. Let X and Y be neighbouring +datasets. Then, if p(t, s) and q(t, s) (as functions of s) give the density functions of M(t, X) and M(t, Y ), +respectively, we have that +Q(X) ∼ Et∼C p(t, s) +and +Q(Y ) ∼ Et∼C q(t, s). +Since for any distributions p and q, and for any λ > 1, +exp +� +(λ − 1)Dλ(p||q) +� += +� �p(t) +q(t) +�λ +q(t) dt +gives an f-divergence (for f(x) = xλ), it is also jointly convex w.r.t. p and q (Liese and Vajda, 2006). +Thus, using Jensen’s inequality, we have +exp +� +(λ − 1)Dλ +� +Q(X)||Q(Y ) +�� += +� �Et∼C p(t, s) +Et∼C q(t, s) +�λ +· Et∼C q(t, s) ds +≤ Et∼C +� �p(t, s) +q(t, s) +�λ +· q(t, s) ds += Et∼C exp +� +(λ − 1)Dλ +� +M(t, X)||M(t, Y ) +�� +≤ Et∼C exp ((λ − 1)ε(λ)) += exp ((λ − 1)ε(λ)) +from which the claim follows. +9 + +4.4 +Accuracy of Algorithms 1 and 2 +Success of both of these algorithms can be explained by the observation that Poisson subsampled Gaussian +mechanism has approximately RDP guarantees of the Gaussian mechanism, i.e., they are actually almost +lines w.r.t. the RDP order λ. This is discussed in more detail in Appendix A. For example, as shown +in (Thm. 38, Steinke, 2022), if the underlying mechanism is ρ-zCDP, then the Poisson subsampled version +with subsampling ratio γ is +� +λ, 10γ2ρλ +� +-RDP for λ sufficiently small. The larger the value of σ and the +smaller the value of γ, the closer the RDPs are to a line, as illustrated in Fig. 3. +5 +Experiments +Quality Metric and Evaluation. In all of our experiments, we choose the best model based on the test +accuracy. Training and test sets are disjoint, and the quality metric is usually a low sensitivity function. +Therefore, even for a private test set, parallel composition can accommodate DP evaluation of a quality +metric in the training budget itself. However, we assume that only the training dataset is private and +the test data is public for simplicity. This relaxation also allows us to take the best checkpoint over all +epochs for each model. +Our plots show the test accuracy of the final model against the final approximate DP ε of the tuning +process for several target ε’s, varying from {0.4, 0.6, .., 2.0}. We fix q = 0.1 for strategy 1 and k = 2 +for strategy 2. We mention that in several cases smaller value of q would have lead to better privacy- +accuracy trade-off (see Appendix E for additional experiments), however we use the same value q = 0.1 +in all experiments. The carry out RDP accounting using RDP orders {2, . . . , 64} and use δ = 10−5 in all +experiments. +Methods. We consider the two variants of strategy 1 in our experiments, one uses the entire dataset +X for the final model (green curves, the mechanism (3.2)) and the other uses X\X1 (blue curves, the +mechanism (3.1)). The final ε(λ)’s for the first and the second version are obtained by combining Thm. 5 +with Thm. 4 and Thm. 7, respectively. We expect the first variant to be more accurate for a slightly +larger final ε compared to second, since the final models obtained with the mechanism (3.2) use slightly +more data than the one obtained with the mechanism (3.1). As discussed in Section 3.1.1, we scale the +best learning rate obtained from the first model by the dataset size for training the final model. We do +not perform any learning rate scaling in strategy 2 (black curves) because all shards have the same size. +In strategy 2, we take the weighted average of the outcomes from both models as the prediction, with +weights as the reciprocals of the shard sizes. The method by Papernot and Steinke (2022) described in +Thm. 5 is the baseline (red curves). +Datasets and Models. We carry out our experiments on the following standard benchmark datasets +for classification: CIFAR-10 (Krizhevsky and Hinton, 2009), MNIST (LeCun et al., 1998), FashionM- +NIST (Xiao et al., 2017) and IMDB (Maas et al., 2011). For MNIST and IMDB, we use the convolutional +neural networks from the examples provided in the Opacus library Yousefpour et al. (2021). For Fash- +ionMNIST, we consider a simple feedforward 3-layer ReLU network with hidden layers of width 120. For +CIFAR-10, we use a Resnet20 pretrained on CIFAR-100 (Krizhevsky and Hinton, 2009) dataset so that +only the last fully connected layer is trained. We minimize the cross-entropy loss in all models. Following +the Opacus example, we optimize with DP-Adam for IMDB dataset, but use DP-SGD for others. +Hyperparameters. For these datasets, in one of the experiments we tune only the learning rate, +and in the other one the learning rate, batch size, and the number of epochs, while fixing the clipping +constant C. The number of trainable parameters and the hyperparameter ranges used are provided in +10 + +Table 2 (Appendix B). The numbers of epochs are chosen to suit our computational constraints. Following +the procedure from Section 4.1, we compute the smallest σ satisfying a target (ε, δ) bound for each (γ, +epoch) pair. The only purpose of target ε’s is to facilitate the mapping from epochs to σ, and retain +comparability across methods. +Initializing with the First Model. We are free to use the first model beyond hyperparameter +extraction, since its privacy cost is already accounted for in the privacy analysis. Therefore, we use it to +initialize the final model in strategy 1, and the subsequent models in the ensemble in strategy 2. +Implementation. +For the implementation of DP-SGD, we use the Opacus library (Yousefpour +et al., 2021). For scalability, we explore the hyperparameter spaces with Ray Tune (Liaw et al., 2018) on +a multi-GPU cluster. We spell out additional details in the corresponding sections. +5.1 +Tuning Learning Rate +The learning rate is among the most critical hyperaparameters, and thus we start with a learning rate +optimization experiment. We fix the subsampling ratio γ and the number of epochs to the values given +in Table 1 (Appendix B) for all models. For example, for q = 0.1, γ = 0.0213 on MNIST dataset, the +Poisson subsampling of DP-SGD gives in expectation batch sizes of 128 and 1150 in strategy 1, 640 for +strategy 2 with k = 2, and 1280 for the baseline method. The learning rate grid size is either 5 or 6, and +we use µ = 10, which is sufficiently large to include a good candidate with a large probability. +Plot Description. Figure 1 plots the test accuracy against the final ε for all 4 datasets. The labels +with ’strategy 1’ refer to the mechanism given in (3.1) (’X1 + X \ X1’) and (3.2) (’X1 + X’). The label +’baseline’ refers to the baseline method by (Papernot and Steinke, 2022) described in Thm. 5 and the +label ’strategy 2’ refers to the ensemble method described in Section 3.2. The left panel for each dataset +considers the case when the first model output by the tuning algorithm is used to initialize the subsequent +model and the right panel shows the results when the first model is not used for initialization. Each plot +contains 9 points for each method, one for each target ε (ε ∈ {0.4, 0.6, .., 2.0}) for each model training +run. +Findings. Our main observation across the board is that the baseline method is generally more +accurate for target ε ≤ 0.5 (at a much higher final ε), but gets easily matched or outperformed by both +variants of strategy 1 for higher target ε’s. Moreover, as expected, the variant of strategy 1 which uses +the entire data in the final model (green plot) performs marginally better than the alternative (blue plot). +Strategy 2 is the least accurate method due to lower privacy amplification with subsampling (q = 0.5) +compared to strategy 1 (q = 0.1). +The impact of initializing the final model with the first one on accuracy is not conclusive. For MNIST, +FashionMNIST, and CIFAR-10, strategy 2 is slightly more accurate for the same final ε when the model +trained on shard 2 is initialized with the first model. For IMDB, however, the opposite is true. Strategy +1 does not seem to then benefit from the initialization either. However, we experimentally observed that +the final model achieves its peak accuracy much earlier when initialized with the first model. +5.2 +Tuning All Hyperaparameters +Next, we also tune the batch size and the number of epochs in addition to the learning rate. The remaining +setup is the same as the previous experiment. The choices are listed in Table 2 (Appendix B). Figure 2 +shows the same quantities as Figure 1, however higher values of µ are used to accommodate to increased +hyperaparameter spaces. Strategy 1 is more accurate compared to the baseline for low target ε-values +and there is slight degradation in accuracy for higher target ε’s. For a fixed target ε-value of candidate +11 + +models, the final ε-values for strategy 1 are roughly 25 − 30% lower than those of the baseline. We also +notice smaller standard errors for CIFAR-10 and FashionMNIST, specially for strategy 2 which might +be due to increased chances of finding favorable combinations of hyperaparameters. We also note that +for final ε-value 2.0, our test accuracies are at least as good as those of (Abadi et al., 2016) in case of +MNIST (95% for Abadi et al., 2016) for and approximately the same in case of CIFAR-10 (67% for Abadi +et al., 2016), although the results of (Abadi et al., 2016) do not include the DP cost of hyperparameter +tuning. +6 +Conclusions +We have considered a simple strategy for lowering the privacy and compute cost of DP hyperparameter +tuning: we carry out tuning using a random subset of data and extrapolate the optimal values to the +larger training set used to train the final model. We have also provided methods to tune the hyper- +parameters that affect the DP guarantees of the model training themselves, those being the noise level +and subsampling ratio in case of DP-SGD. Our experiments show a clear improvement over the baseline +method by Papernot and Steinke (2022), when tuning DP-SGD for neural networks and using simple +heuristics for the extrapolation. An interesting avenue of future work is to find more sophisticated ways +to extrapolate the hyperparameters, something that has been considered in non-DP case (see e.g. Klein +et al., 2017). +We have also considered using a weighted ensemble of submodels trained on random disjoint equally- +sized subsets of data, in which case extrapolation is not needed. We found it leading to a slightly worse +privacy-utility trade-off, albeit with computational speed-ups. Exploring further ways to improve this +strategy is also left for future work. +7 +Acknowledgments +We would like thank our colleague Laith Zumot for discussions about practical hyperparameter tuning +methods at the initial stages of the project. +Bibliography +Abadi, M., Chu, A., Goodfellow, I., McMahan, H. B., Mironov, I., Talwar, K., and Zhang, L. (2016). Deep +learning with differential privacy. 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Here ’strategy 1’ is ∼ 5 times faster than the baseline method by Papernot and Steinke +(2022). +16 + +0.97 +0.96 +.- +0.95 +t6'0 +0.93 +baseline +Z6'0 +strategy 1 ---Xi+X\Xi +strategy 1 --- X1+X +0.91 +strategy 2 --- k =2 +1 +2 +E +4 +5 +6 +1 +2 +E +4 +5 +60.84 +0.82 +- +0.80 +0.78 +baseline +0.76 +strategy1---Xi+XXi +strategy 1 --- X1+X +0.74 +strategy 2 --- k=2 +2 +E +4 +5 +6 +1 +2 +E +4 +5 +60.680 +0.675 +0.670 +0.665 +0.660 +0.655 +baseline +strategy1 ---Xi+X\Xi +0.650 +strategy 1 --- Xi+X +strategy 2 ---k=2 +0.645 +3 +4 +5 +6 +1 +2 +E +4 +5 +6 +na0.74 +0.72 +0.70 +0.68 +0.66 +0.64 +baseline +0.62 +strategy1---Xi+XXi +0.60 +strategy 1 --- Xi+X +strategy 2 --- k=2 +2 +3 +4 +5 +6 +1 +2 +3 +4 +5 +na(a) MNIST with µ = 45 +(b) FashionMNIST with µ = 50 +(c) CIFAR-10 with µ = 40 +(d) IMDB with µ = 45 +Figure 2: Tuning of subsampling ratio, training length, and learning rate. Test accuracies averaged across +5 independent runs. The error bars denote the std. error of the mean. Each curve contains 9 points, one +for each target ε ∈ {0.4, 0.6, .., 2.0}. Here ’strategy 1’ is ∼ 8 times faster than the method by Papernot +and Steinke (2022). +17 + +0.97 +0.96 +0.95 ++6'0 +baseline +strategy 1 ---Xi+X\Xi +strategy 1 --- Xi+X +E6'0 +strategy 2 --- k = 2 +4 +6 +8 +10 +12 +2 +4 +6 +8 +10 +12 +fina0.84 +0.82 +0.80 +baseline +0.78 +strategy1---Xi+XXi +strategy1 --- Xi+X +strategy2 --- k =2 +0.76 +2 +4 +5 +8 +10 +12 +2 +4 +6 +8 +10 +12 +6na0.680 +0.675 +0.670 +- +0.665 +0.660 +0.655 +baseline +0.650 +strategy 1 ---Xi+XXi +strategy 1 --- X1+X +0.645 +strategy 2 --- k=2 +2 +4 +6 +8 +10 +7. +4 +6 +8 +10 +naLE +naLE0.72 +0.70 +0.68 +0.66 +0.64 +baseline +strategy1---Xi+X\Xi +0.62 +strategy 1---Xi+X +0.60 +strategy 2--- k = 2 +2 +4 +8 +10 +12 +2 +4 +6 +8 +10 +12 +6naAppendix +A +Adjusting the Parameters T and σ for DP-SGD +It is often a good approximation to say that the RDP-guarantees of the Poisson subsampled Gaussian +mechanism are lines as functions of the RDP order λ, i.e., that the guarantees are those a Gaussian +mechanism with some sensitivity and noise level values. For example, Mironov et al. (Thm. 11, 2019) +show that the Poisson subsampled Gaussian mechanism is +� +λ, 2γ2λ/σ2� +-RDP when λ is sufficiently small. +Also, (Thm. 38, +Steinke, 2022) show that if the underlying mechanism is ρ-zCDP, then the Poisson +subsampled version with subsampling ratio γ is +� +λ, 10γ2ρλ +� +-RDP when λ is sufficiently small. Notice +that the Gaussian mechanism with L2-sensitivity ∆ and noise level σ is (∆2/2σ2)-zCDP (Bun and Steinke, +2016). +We numerically observe, that the larger the noise level σ and the smaller the subsampling ratio γ, the +better the line approximation of the RDP-guarantees (see Figure 3). +In case the privacy guarantees (either (ε, δ)-DP or RDP) are approximately those of a Gaussian mech- +anisms with some sensitivity and noise level values, both of the methods for tuning the hyperparameters +γ, σ and T described in Section 4 would lead to very little slack. This is because for the Gaussian mech- +anism, both the RDP guarantees (Mironov, 2017) and (ε, δ)-DP guarantees (Dong et al., 2022) depend +monotonously on the scaled parameter +�σ = +σ +∆ · +√ +T +. +This means that if we adjust the training length T based on values of σ by having some target (δ, ε)- +bound for the candidate model (Algorithm 1 of Section 4), the resulting RDP upper bounds of different +candidates will not be far from each other (and similarly for adjusting σ based on value of T). Similarly, +for Algorithm 2 of Section 4, when adjusting T based on values of σ, the RDP guarantees of all the +candidate models would be close to the upper bound (c · λ, c > 0), i.e., they would not be far from each +other. +5 +10 +15 +20 +25 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +( ) +=2, T = 1000 +Gauss RDP-fit ( =2) +=4, T = 4000 +Gauss RDP-fit ( =4) +=6, T = 8000 +Gauss RDP-fit ( =6) +5 +10 +15 +20 +25 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +( ) +=2, T = 1000 +Gauss RDP-fit ( =2) +=4, T = 4000 +Gauss RDP-fit ( =4) +=6, T = 8000 +Gauss RDP-fit ( =6) +Figure 3: DP-SGD RDP curves for different values of noise level σ and number of compostions T. Left: +γ = 1/100, right: γ = 1/50 and the corresponding lines with the smallest slope that give upper bounds +for the RDP orders up to λ = 24. +18 + +B +Hyperparameter Tables for Experiments of Figures 1 and 2 +MNIST +FashionMNIST +CIFAR-10 +IMDB +γ = B +N +0.0213 +0.0213 +0.0256 +0.0256 +epochs +40 +40 +40 +110 +Table 1: Tuning η: rest of the hyperparameters fixed to these values. +train/test set +parameters +C +B +learning rate +epochs +MNIST +60k/10k +∼26k +1 +{128, 256} +{10−i}i∈{2,1.5,1,0.5,0} +{10,20,30,40} +FashionMNIST +60k/10k +∼109k +3 +{128, 256} +{10−i}i∈{2,1.5,1,0.5,0,−0.5} +{10,20,30,40} +CIFAR-10 +50k/10k +0.65k +3 +{64, 128} +{10−i}i∈{2,1.5,1,0.5,0,−0.5} +{20,30,40} +IMDB +25k/25k +∼464k +1 +{64, 128} +{0.02,0.1,0.2,1,1.1} +{50,70,90,110} +Table 2: Tuning σ, η and T: datasets used and the corresponding hyperparameter grids. +C +Proof of Theorem 7 +C.1 +Auxiliary Lemma +We will need the following inequality for the proof of Theorem 7. +Lemma C.1 (Lemma 35, Steinke 2022). For all p ∈ [0, 1] and x ∈ (0, ∞), +1 +1 − p + p +x +≤ 1 − p + p · x. +Remark C.2. An alternative proof for this result can be obtained using so called Bergström’s inequality, +which states that for all xk ∈ R, ak > 0, k ∈ [n], +(x1 + . . . + xn)2 +a1 + . . . + an +≤ x2 +1 +a1 ++ . . . + x2 +n +an +. +(C.1) +In particular, for n = 2 and a1 = q +x, a2 = 1 − q, x1 = q, x2 = 1 − q, this gives +1 +1 − q + q +x +≤ 1 − q + q · x. +As the proof of Theorem 7 shows, it could be generalized to the case of random k-way split (k > 2) using +the inequality (C.1). +19 + +C.2 +Proof of Thm. 7 +Proof. We first consider the case Dλ +� +M(Y )||M(X) +� +. Let X ∈ X n and x′ ∈ X. Denote +ε1(λ) = Dλ +� +M(X ∪ {x′})||M(X) +� +. +Looking at Strategy 1 which uses Poisson subsampling with subsampling ratio q to obtain the dataset +X1, and conditioning the output on the randomness in choosing X1, we can write the mechanism as a +mixture over all possible choices of X1 as +M(X) = +� +X1 +pX1 · +� +M1(X1), M2(M(X1), X\X1) +� +, +(C.2) +where pX1 is the probability of sampling X1. Since each data element is in X1 with probability q, we can +furthermore write M(X ∪ {x′}) as a mixture +M(X ∪ {x′}) = +� +X1 +pX1 · +� +q · +� +M1(X1 ∪ {x′}), M2(M1(X1 ∪ {x′}), X\X1) +� ++ (1 − q) · +� +M1(X1), M2(M(X1), X\X1 ∪ {x′}) +�� +. +(C.3) +From the quasi-convexity of the Rényi divergence (Van Erven and Harremos, 2014) and the expressions +(C.2) and (C.3), it follows that +Dλ +� +M(X ∪ {x′})||M(X) +� +≤ sup +X1 +Dλ +� +q · +� +M1(X1 ∪ {x′}), M2(M1(X1 ∪ {x′}), X\X1) +� ++ (1 − q) · +� +M1(X1), M2(M(X1), X\X1 ∪ {x′}) +� +|| +� +M1(X1), M2(M(X1), X\X1) +�� +. +(C.4) +Our aim is to express the right-hand side of (C.4) in terms of RDP parameters of M1 and M2. To +this end, take an arbitrary X1 ⊂ X, and denote by +• �P(t) the density function of M1(X1 ∪ {x′}), +• P(t) the density function of M1(X1), +• �Q(t, s) the density function of M2(t, X\X1 ∪ {x′}) for auxiliary variable t (the output of M1), +• Q(t, s) the density function of M2(t, X\X1) for auxiliary variable t. +Then, we see that +P +�� +M1(X1), M2(M(X1), X\X1) +� += (t, s) +� += P(t) · Q(t, s) +and similarly that +P +� +q · +� +M1(X1 ∪ {x′}), M2(M1(X1 ∪ {x′}), X\X1) +� ++ (1 − q) · +� +M1(X1), M2(M(X1), X\X1 ∪ {x′}) +� += (t, s) +� += q · P +�� +M1(X1 ∪ {x′}), M2(M1(X1 ∪ {x′}), X\X1) +� += (t, s) +� ++ (1 − q) · P +�� +M1(X1), M2(M(X1), X\X1 ∪ {x′}) +� += (t, s) +� += q · �P(t) · Q(t, s) + (1 − q) · P(t) · �Q(t, s). +20 + +By the definition of the Rényi divergence, we have that +exp +� +(λ − 1)Dλ +� +q · +� +M1(X1 ∪ {x′}), M2(M1(X1 ∪ {x′}), X\X1) +� ++ (1 − q) · +� +M1(X1), M2(M(X1), X\X1 ∪ {x′}) +� +|| +� +M1(X1), M2(M(X1), X\X1) +��� += +� � � +q · �P(t) · Q(t, s) + (1 − q) · P(t) · �Q(t, s) +P(t) · Q(t, s) +�λ +· P(t) · Q(t, s) dt ds. +(C.5) +which can be expanded as +� � � +q · �P(t) · Q(t, s) + (1 − q) · P(t) · �Q +P(t) · Q(t, s) +�λ +P(t) · Q(t, s) dt ds += +� � � +q · +�P(t) +P(t) + (1 − q) · +�Q(t, s) +Q(t, s) +�λ +P(t) · Q(t, s) dt ds += +� � +qλ +� �P(t) +P(t) +�λ +P(t) · Q(t, s) dt ds ++ +� � +(1 − q)λ +� �Q(t, s) +Q(t, s) +�λ +P(t) · Q(t, s) dt ds ++ +� � +λ · qλ−1 · (1 − q) · +� �P(t) +P(t) +�λ−1 +P(t) · �Q(t, s) dt ds ++ +� � +λ · q · (1 − q)λ−1 · +� �Q(t, s) +Q(t, s) +�λ−1 +Q(t, s) · �P(t) dt ds ++ +� � +λ−2 +� +j=2 +�λ +j +� +· qλ−j · (1 − q)j · +� +� +� �P(t) +P(t) +�λ−j +P(t) +� +� +� +� +� �Q(t, s) +Q(t, s) +�j +Q(t, s) +� +� dt ds. +(C.6) +We next bound the five integrals on the right hand side of (C.6). For the first two integrals, we use +the RDP-bounds for M1 and M2 to obtain +� � � �P(t) +P(t) +�λ +P(t)Q(t, s) dt ds = +� � � �P(t) +P(t) +�λ +P(t) dt +≤ exp +� +(λ − 1)εP (λ) +� +. +(C.7) +and +� � � �Q(t, s) +Q(t, s) +�λ +Q(t, s)P(t) ds dt ≤ +� � +exp +� +(λ − 1)εQ(λ) +� +P(t) dt += exp +� +(λ − 1)εQ(λ), +(C.8) +21 + +where εP and εQ give the RDP-parameters of order λ for M1 and M2, respectively. The third and +fourth integral can be bounded analogously. In the second inequality we have also used the fact that the +RDP-parameters of M2 are independent of the auxiliary variable t. Similarly, for the third integral, we +have +� � � +� +� �P(t) +P(t) +�λ−j +P(t) +� +� +� +� +� �Q(t, s) +Q(t, s) +�j +Q(t, s) +� +� ds dt ≤ +� � � +� +� �P(t) +P(t) +�λ−j +P(t) +� +� exp +� +(j − 1)εQ(j) +� +dt +≤ exp +� +(λ − j − 1)εP (λ − j) +� +· exp +� +(j − 1)εQ(j) +� +. +(C.9) +Substituting (C.7), (C.8) (and similar expressions for the third and fourth integral) and (C.9) to (C.6), +we get a bound for (C.5). Since X1 ⊂ X was arbitrary, we arrive at the claim via (C.4). +Next, we consider bounding Dλ +� +M(X)||M(Y ) +� +. The proof goes similarly as the one for Dλ +� +M(Y )||M(X) +� +. +Denote +ε2(λ) = Dλ +� +M(X)||M(X ∪ {x′}) +� +. +With the notation of proof of Thm. 7, we see that, instead of (C.5), we need to bound +exp +� +(λ − 1)ε2(λ) +� += +� � � +P(t) · Q(t, s) +q · �P(t) · Q(t, s) + (1 − q) · P(t) · �Q(t, s) +�λ +(q · �P(t) · Q(t, s) + (1 − q) · P(t) · �Q(t, s)) dt ds. +In order to use here the series approach, we need to use Lemma C.1: +� +P · Q +q · �P · Q + (1 − q) · P · �Q +�λ +(q · �P · Q + (1 − q) · P · �Q) += +� +P · Q +q · �P · Q + (1 − q) · P · �Q +�λ−1 +· P · Q += +� +q · +�P +P + (1 − q) · +�Q +Q +�1−λ +· P · Q += +� +q · +�P +P +Q +�Q ++ (1 − q) +�1−λ +· +� �Q +Q +�1−λ +· P · Q += +� +q · +�P +P +Q +�Q ++ (1 − q) +�1−λ +· +�Q +�Q +�λ−1 +· P · Q +≤ +� +q · P +�P +�Q +Q + (1 − q) +�λ−1 +· +�Q +�Q +�λ−1 +· P · Q += +� +q · P +�P +�Q +Q + (1 − q) +�λ−1 +· +�Q +�Q +�λ +· P · �Q, +(C.10) +22 + +where in the inequality we have used Lemma C.1. Now we can expand +� +q · P +� +P +� +Q +Q + 1 − q +�λ−1 +: +� +1 − q + q · P +�P +�Q +Q +�λ−1 +· +�Q +�Q +�λ +· P · �Q = +� +� +λ−1 +� +j=0 +�λ − 1 +j +� +qj · (1 − q)λ−1−j · +�P +�P +�j � �Q +Q +�j� +� · +�Q +�Q +�λ +· P · �Q += +� +� +λ−1 +� +j=0 +�λ − 1 +j +� +qj · (1 − q)λ−1−j · +�P +�P +�j �Q +�Q +�λ−j +� +� · P · �Q += +λ−1 +� +j=0 +�λ − 1 +j +� +qj · (1 − q)λ−1−j · +�P +�P +�j+1 +�P · +�Q +�Q +�λ−j +�Q. +Then, we use the known εP (λ) and εQ(λ)-values as in (C.9) to arrive at the claim. +23 + +D +f-Divergence of Parallel Compositions +We first formulate the parallel composition result for general f-divergences (Lemma D.1). We then obtain +the RDP bound for parallel compositions as a corollary (Cor. D.2). +Our Lemma D.1 below can be seen as an f-divergence version of the (ε, 0)-DP result given in (Thm. 4 +McSherry, 2009). Corollary 2 by Smith et al. (2022) gives the corresponding result in terms of µ-Gaussian +differential privacy (GDP), and it is a special case of our Lemma D.1 as µ-GDP equals the (ε, δ)-DP (i.e., +the hockey-stick divergence) of the Gaussian mechanism with a certain noise scale (Cor. 1, Dong et al., +2022). +We define f-divergence for distributions on Rd as follows. Consider two probability densities P and +Q defined on Rd, such that if Q(x) = 0 then also P(x) = 0, and a convex function f : [0, ∞) → R. Then, +an f-divergence (Liese and Vajda, 2006) is defined as +Df(P||Q) = +� +f +�P(t) +Q(t) +� +Q(t) dt. +In case the data is divided into disjoint shards and separate mechanisms are applied to each shard, the f- +divergence upper bound for two neighbouring datasets can be obtained from the individual f-divergence +upper bounds: +Lemma D.1. Suppose a dataset X ∈ X N is divided into k disjoint shards Xi, i ∈ [k], and mechanisms +Mi, i ∈ [k], are applied to the shards, respectively. Consider the mechanism +M(X) = +� +M1(X1), . . . , Mk(Xk) +� +. +Then, we have that +max +X∼Y Df +� +M(X)||M(Y ) +� +≤ max +i∈[k] max +X∼Y Df +� +Mi(X)||Mi(Y ) +� +. +Proof. Let X be divided into k shards as described above and suppose Y is a neighbouring dataset such +that X1 ∼ Y1 and Y = {Y1, X2, . . . , Xk}. +Then, we see that +P +� +M(X) = (a1, . . . , ak) +� +P +� +M(Y ) = (a1, . . . , ak) +� = P +� +M1(X1) = a1 +� +· P +� +M1(X2, a1) = a2 +� +· · · P +� +Mk(Xk, a1, . . . , ak−1) = ak +� +P +� +M1(Y1) = a1 +� +· P +� +M1(X2, a1) = a2 +� +· · · P +� +Mk(Xk, a1, . . . , ak−1) = ak +� += P +� +M1(X1) = a1 +� +P +� +M1(Y1) = a1 +� . +and furthermore, denoting a = (a1, . . . , ak), +Df +� +M(X)||M(Y )) +� += +� +f +� +P +� +M(X) = a +� +P +� +M(Y ) = a +� +� +P +� +M(Y ) = a +� +da += +� +f +� +P +� +M1(X1) = (a1) +� +P +� +M1(Y1) = (a1) +� +� +P +� +M(Y ) = a +� +da += +� +f +� +P +� +M1(X1) = (a1) +� +P +� +M1(Y1) = (a1) +� +� +P +� +M(Y1) = (a1) +� +da1 += Df(M1(X1)||M1(Y1)) +� +. +24 + +Thus, +Df(M(X)||M(Y )) = Df(M1(X1)||M1(Y1)) ≤ max +X∼Y Df(M1(X)||M1(Y )). +Similarly, if Xi ∼ Yi, i = 2, . . . , k and +Y = +� +X1, . . . Xi−1, Yi, Xi+1, . . . , Xk +� +, +we see that +Df(M(X)||M(Y )) = Df(Mi(Xi)||Mi(Yi)) ≤ max +X∼Y Df(Mi(X)||Mi(Y )). +Thus, we have that +max +X∼Y Df(M(X)||M(Y )) = max +i∈[k] max +X∼Y Df(Mi(X)||Mi(Y )). +Corollary D.2. Suppose a dataset X ∈ X N is divided into k disjoint shards Xi, i ∈ [k], and mechanisms +Mi, i ∈ [k], are applied to the shards, respectively. Consider the mechanism +M(X) = +� +M1(X1), . . . , Mk(Xk) +� +. +Suppose each Mi is +� +λ, εi(λ) +� +-RDP, respectively. Then, M is +� +λ, maxi∈[k] εi(λ) +� +-RDP. +Proof. This follows from Lemma D.1 since +exp +� +(λ − 1)Dλ(M(X)||M(Y )) +� += +� � +P +� +M(X) = a +� +P +� +M(Y ) = a +� +�λ +P +� +M(Y ) = a +� +da +is an f-divergence for f(x) = xλ. Thus, by Lemma D.1 we have that +max +X∼Y exp +� +(λ − 1)Dλ(M(X)||M(Y )) +� +≤ max +i∈[k] max +X∼Y exp +� +(λ − 1)Dλ(Mi(X)||Mi(Y )) +� +from which it follows that +max +X∼Y Dλ(M(X)||M(Y )) ≤ max +i∈[k] max +X∼Y Dλ(Mi(X)||Mi(Y )) = max +i∈[k] εi(λ). +25 + +E +Additional Experiments +E.1 +Comparison of (ε, δ)-Bounds +Figure 4: Final ε as a function of q for µ = 50, when the base mechanism is the subsampled Gaussian +mechanism with γ = 64/6000, number of epochs = 50 and σ = 2.0. The value of q varies from 0.05 to +0.9. Strategy 1 has considerably smaller ε’s for q ≤ 0.1 compared to the baseline. However, with q ≤ 0.1, +strategy 1 may not provide good utility on small data sets (e.g., IMDB). Our proposed Thm. 7 used for +the mechanism (3.1) (’X1 + X \ X1’) is suboptimal since for small values of q (e.g. q = 0.05) the bound +of the mechanism (3.2) obtained with Thm. 4 is tighter. +E.2 +Vary k in Strategy 2 +For strategy 2, higher k’s provide more estimators in the ensemble. On the other hand, each submodel +is trained on a shard of smaller size. To experimentally see the effect of k, we carry out an experiment +similar to the one described in Section 5.1 for a different values of k. Figure 5 shows test accuracy as +a function of target ε, k, and final ε for strategy 2. The value of k in each plot varies from 2 to 10. +The first observation is that higher values of k reduce the final ε (at the cost of lower accuracy). This +is due to smaller values of q (q = 1 +k) in Theorem 7. The k = 2 case generally yields the peak accuracy +on all datasets, although in some cases higher k would have lead to better privacy-utility trade-off. We +fix k = 2 in the experiments of the main text. In strategy 2, the compute cost of the tuning part also +lowers the overall compute cost. Moreover, larger k’s reduce the training time for each submodel and the +submodels can be trained in parallel. +26 + +N=6000. ep0chs=50 . B=64 . μ=50, 0=2.0 +10.5 +10.0 +9.5 +9.0 +8.5 +8.0 +baseline (Papernotand Steinke) +7.5 +strategy1---Xi+X(Zhu&Wang) +7.0 +strategy 1 and 2 ---Xi +XXi (our bound) +0.2 +0.4 +0.6 +0.8E.3 +Vary q in Strategy 1 +In strategy 1, when q increases, the hyperparameter tuning mechanism trains with a larger dataset which +means also weaker final privacy guarantees. Additionally, the best learning rate from the first model is +also scaled with a smaller factor in the final model. We perform for strategy 1 an experiment similar as +described in E.2 to compute the final test accuracies and ε-values as a function of q. +Figure 6 shows the test accuracy as a function of target ε for the base mechanism (DP-SGD), q, final +ε. The value of q in each plot varies from 0.1 to 0.8. As expected, the accuracy for the variant that use +X\X1 in the final model drops when q increases. The performance of the other variant that trains the +final model with the full dataset remains relatively steady for all models. In IMDB, we use DP-Adam to +train the IMDB model and do not scale the best initial learning rate. +27 + +(a) MNIST with µ = 10 +(b) FashionMNIST with µ = 10 +(c) CIFAR-10 with µ = 10 +(d) IMDB with µ = 10 +Figure 5: Tuning learning rate for strategy 2: We keep the batch size and the number of epochs fixed and +only tune the learning rate, for various values of k. The test accuracies are averaged across 5 independent +runs. The error bars denotes the std. error of the mean. Each plot contains 9 points for strategy 2, one +for each k ∈ {2, 3, .., 10} (right to left). We also add the baseline method for comparison. The k = 2 case +generally yields the peak accuracy on all datasets. +28 + +MNIsT,P=1o,rgtE=1.6 +↓ baseline +0.96 +■ +strategy2 +.-- various k's +- +0.94 +10%表 +0.92 +5 0.90 +10 +0.88 +10 +0.86 +0.6 +0.7 +0.8 +0.9 +10 +11 +12 +1.4 +16 +18 +2.0 +2.2 +2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 +2.8 3.0 3.2 3.5 3.8 4.0 4.2 4.5 +3.8 4.0 4.2 4.5 4.8 5.0 5.2 5.5 5.8 +EFhionMNisT,p=1o,rgatE=d8 +FhionMNIST,μ=10, barget E=1.2 +T=FLSINHUOIy +0.84 + baseline +strategy 2 --- various k's +0.82 +0.80 +0.78 ++ +HI +L +0 0.76 +0.74 +0.6 +0.7 +0.8 +0.9 +10 +11 +12 +1.4 +1.6 +1a +2.0 +2.2 +2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 +2.8 3.0 3.2 3.5 3.8 4.0 4.2 4.5 +3.8 4.0 4.2 4.5 4.8 5.0 5.2 5.5 5.8*FAA-10,H=10,trgatE=0.4 +FAA-10,H=1o,trgatE=0.8 +FAA-10,μ=10,trgatE=1.2 +FAA-10,H=10,trgatE=1.6 +*FAA-10,=10,trgatE=2.0 +0.67 ++baseline +strategy 2 --- various k's +HII +I12 +x10.11 +0.66 +0.65 +0.64 +0.63 +0.62 +0.61 +10 +410 +0.60 +0.6 +0.7 +0.8 +0.9 +10 +11 +12 +14 +16 +2.0 +2.2 +2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 +2.8 3.0 3.2 3.5 3.8 4.0 4.2 4.5 +3.8 4.0 4.2 4.5 4.8 5.0 5.2 5.5 5.8 +E +nalEMDBH=1o,rgatE=0.8 +MDB,=10,trgatE=1.2 +MDB,P=10,trgotE=1.6 +IMDB,=10,trgat E=2.0 +0.750 +baseline +■ +0.725 +strategy 2 -* +various k's +0.675 +0.650 +0.625 +. +0.600 +161 +0.575 +0.6 +0.7 +0.8 +0.9 +10 +11 +1.2 +14 +1.6 +18 +2.0 +2.2 +2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 +2.8 3.0 3.2 3.5 3.8 4.0 4.2 4.5 +4.0 4.2 4.5 4.8 5.0 5.2 5.5 5.8(a) MNIST with µ = 10 +(b) FashionMNIST with µ = 10 +(c) CIFAR-10 with µ = 10 +(d) IMDB with µ = 10 +Figure 6: Tuning learning rate for strategy 1: We keep the batch size, the number of epochs fixed and +only tune the learning rate for strategy 1 for various values of q’. Test accuracies are averaged across +5 independent runs. The error bars denotes the std. error of the mean. Each plot contains 8 points +for strategy 1, one for each q ∈ {0.1, 0.2, .., 0.8} (left to right). We also add the baseline method for +comparison. The q = 0.1 case generally yields the peak accuracy on all datasets. +29 + +0.98 +MNIST, μ=10, rgat E=0.4 +MMIsT, μ=10, rgat E=0.8 +ANST,1O,gatE=1.6 +091 +96°0 +0.8 +18 ++6'0 2 +0. +0L8 +0.8 +0.92 +0.90 +0.8 +L +0 0.88 +baseline +strategy 1 --- Xi +XXi +0.86 +0..6 +07 +0.9 +10 +11 +12 +1.4 +1.6 +18 +2.02.2 +2.0 2.2 2.52.83.03.23.5 +3.0 +4.0 +4.5 +4.0 +4.5 +5.0 +5.5 +6.0 +bnaLE +naLE +naE +fnal +FFahionMNIST,p=1o,target =0.4 +FhionMNisT, μ=10, brgeat E=1.6 +FhionMNIST, μ=10, brgat E=2.0 ++8'0 +cPil +10011 +8 +HH +0.82 +0.8 +0.80 +0.78 +10.8 +No.8 +No.8 +T +0.76 +￥0.8 +0.74 +baseline +0.72 +strategy 1 -.- Xi +XXi +0.8 +0.70 +f strategy 1 -.- Xi +X +0.68 +0.6 +0.7 +0.9 +10 +11 +1214 +18 +2.02.2 +2.0 2.2 2.5 2.8 3.0 3.2 3.5 +3.0 +4.0 +4.5 +4.0 +4.5 +5.0 +5.5 +6.0 +hnalE +naE +fnal +F+FAR-10, μ=10, trget E=0.8 ++FAA-10, μ=10, trgat E=1.2 +*FAR-10,P=10,rgatE=1.6 +*FAR-10,P=10,rgatE=2.0 +0.68 +0.8 +0071 +0.8 +OD +0.8 +0.66 +20.8 ++ 0.8 +0.8 +0.64 +0.8 +0.8 +.CO +0.62 +baseline +0.60 +strategy 1 --- Xi +XXi +↑ strategy1 --- Xi+×0.8 +0.6 +0.7 +0.8 +0.9 +10 +11 +12 +14 +16 +2.02.2 +2.0 2.2 2.5 2.8 3.0 3.2 3.5 +3.0 +4.0 +4.5 +4.0 +4.5 +5.0 +5.5 +6.0 +bna +hnaLEMDB,=10,rgtE=0.4 +IMbB, μ=10, t=rgat E=0.8 +IMDB, p=1o, trgat E=1.2 +IMbB, μ=10, t=rgat E=1.6 +MDB,P=10,rgat E=2.0 +0.75 +baseline +.0.1 +0.D.1 +strategy 1 --- Xi +XX1 +00u1 +0.8 +strategy 1 -.- Xi +X +0. +.8 +二 +0.65 +0 +5 0.60 +0.8 +0.8 +0.8 +8'0. +0.8 +0.55 +8'0 +0.60.7 +0.9 +10 +11 +12 +1.4 +1.6 +2.0 2.2 2.5 2.8 3.0 3.2 3.5 3.8 +3.0 +3.5 +4.0 +4.5 +5.0 +4.0 +4.5 +5.0 +5.5 +6.0 +hnaLE +hnalE +naE +6naLE(a) MNIST with µ = 10 +(b) FashionMNIST with µ = 10 +(c) CIFAR-10 with µ = 10 +(d) IMDB with µ = 10 +Figure 7: Tuning learning rate for strategy 1 for q ≤ 0.1: We keep the batch size and the number of +epochs fixed and only tune the learning rate for strategy 1 for various values of q ≤ 0.1. Test accuracies +are averaged across 5 independent runs. The error bars denotes the std. error of the mean. Each plot +contains 4 points for strategy 1, i.e. when q ∈ {0.05, 0.067, 0.083, 0.1}. As shown in Fig. 4, the RDP +bound of Thm. 7 is suboptimal and therefore the final ε-value for the variant of strategy 1 that uses the +entire data is is lower in some cases. +30 + +strategy 1 --- Xi + XX1 +0.05 +0.05 +0.05 +0.05 +0.1 +strategy 1 --- Xi +X +0.1 +5050.0 +0.1 +0.05 +10.05strategy 1 --- Xi +XX1 +strategy 1 --- Xi +X +5.05 +10.05strategy 1 --- Xi +XXi +0.1 +strategy 1 -.- Xi +X +0.05 +0.1 +0.0505 +0.05 +0.05 +0.05 +0.1 +5.05 +To.05 +0.05 +0.05 +D.1strategy 1 -.- Xi +XX, +0.05 +0.1 +0.05 +strategy 1 --- Xi +X +0.05050.1 +0.05 +005 +0.05 +J0.05 \ No newline at end of file diff --git a/FNFLT4oBgHgl3EQfFy-9/content/tmp_files/load_file.txt b/FNFLT4oBgHgl3EQfFy-9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..79c63bed950cfb222102c9c89146028bf5955e5d --- /dev/null +++ b/FNFLT4oBgHgl3EQfFy-9/content/tmp_files/load_file.txt @@ -0,0 +1,1521 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf,len=1520 +page_content='Practical Differentially Private Hyperparameter Tuning with Subsampling Antti Koskela and Tejas Kulkarni Nokia Bell Labs Espoo, Finland Abstract Tuning all the hyperparameters of differentially private (DP) machine learning (ML) algorithms often requires use of sensitive data and this may leak private information via hyperparameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Recently, Papernot and Steinke (2022) proposed a certain class of DP hyperparameter tuning algo- rithms, where the number of random search samples is randomized itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Commonly, these algorithms still considerably increase the DP privacy parameter ε over non-tuned DP ML model training and can be computationally heavy as evaluating each hyperparameter candidate requires a new training run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We focus on lowering both the DP bounds and the computational complexity of these methods by using only a random subset of the sensitive data for the hyperparameter tuning and by extrapo- lating the optimal values from the small dataset to a larger dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We provide a Rényi differential privacy analysis for the proposed method and experimentally show that it consistently leads to better privacy-utility trade-off than the baseline method by Papernot and Steinke (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 1 Introduction Our aim is two-fold: to decrease the computational cost as well as the privacy cost of hyperparameter tuning of DP ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The reasons for this are clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' As the dataset sizes grow and models get more complex, blackbox optimization of hyperparameters becomes more expensive since evaluation of a single set of hyperparameters often requires retraining a new model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' On the other hand, tuning the hyperparameters often depends on the use of sensitive data, so it requires privacy protection as well, as illustrated by the counterexample by Papernot and Steinke (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Intuitively, the leakage from hyperparameters is much smaller than from the model parameters, however providing tuning algorithms with low additional DP cost has turned out challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Current best algorithms (Papernot and Steinke, 2022) still come with a considerable DP cost overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Although our methods and results are applicable to general DP mechanisms, we will in particular focus on tuning of the DP stochastic gradient descent (DP-SGD) (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Bassily et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Abadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2016) which has become the most widely used method to train ML models with DP guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Compared to plain SGD, DP brings additional hyperparameters to tune: the noise level σ and the clipping constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Additionally, also the subsampling ratio γ affects the DP guarantees, as well as length of the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Tuning all the hyperparameters of DP-SGD commonly requires use of sensitive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We use the results by Papernot and Steinke (2022) as building blocks of our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Their work was based on the analysis of Liu and Talwar (2019) who provided the first results for DP black-box 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='11989v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='LG] 27 Jan 2023 optimization of hyperparameters, where, if the base training algorithm is (ε, 0)-DP, then the tuned model is approximately (3ε, 0)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Papernot and Steinke (2022) gave Rényi differential privacy (RDP) analysis for a class of black-box tuning algorithms, where the number of random search samples is randomized itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' As the privacy bounds are in terms of RDP and assume only RDP bounds about the candidate model training algorithms, they are particularly suitable to tuning DP-SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' However, still, running these algorithms increase the ε-values two or three-fold or more, and they can be computationally heavy as evaluating each candidate model requires training a new model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Our novelty is to consider using only a random subset of the sensitive data for the tuning part and use the output hyperparameter values (and potentially the model) for training subsequent models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Using a random subset for the privacy and compute costly part automatically leads to both lower DP privacy leakage as well as computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We also consider ways to appropriately extrapolate the optimal value from the small subset of data to a larger dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The RDP bounds for the DP tuning methods by Papernot and Steinke (2022) assume that the RDP- values of the candidate model training algorithms are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We also consider ways to use these bounds for tuning hyperparameters that affect the RDP-values of the base algorithm, being the noise level σ, the subsampling ratio γ and the length of training in case of DP-SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 Related Work on Hyperparameter Tuning Chaudhuri and Vinterbo (2013) were the first ones focus on DP bounds for hyperparameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' An improvement was made by Liu and Talwar (2019) who considered black-box tuning of (ε, δ)-DP mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Mohapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (2022) showed that for reasonable numbers of adaptively chosen private candidates a naive RDP accounting (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', RDP parameters grow linearly w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' number of model eval- uations) often leads to lower DP bounds then the methods by Liu and Talwar (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Papernot and Steinke (2022) gave RDP bounds for black-box tuning algorithms that grow only logarithmically w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' number of model evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' In non-DP setting, hyperparameter tuning with random subsamples has been considered for SVMs (Horváth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2017), for large datasets in healthcare (Waring et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='. Small random subsets of data have been used in Bayesian optimization of hyperparameters (Swersky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Klein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2 Our Contributions We propose a novel subsampling strategy to lower the privacy and compute cost of DP hyperparam- eter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Our privacy analysis is in terms of RDP and we use existing results for tuning Papernot and Steinke (2022) and DP-SGD (Zhu and Wang, 2019) as building blocks, however our methods and results are independent of those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We provide a tailored RDP analysis for the proposed strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We propose algorithms to tune hyperparemeters that affect the RDP guarantees of the base model training algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We provide a rigorous RDP analysis for these algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We carry out experiments on a variety of datasets, where we are able to consistently improve upon the baseline tuning method by a clear margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 2 2 Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 DP and DP-SGD We first give the basic definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' An input dataset containing n data points is denoted as X = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' , xn) ∈ X n, where xi ∈ X, 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We say X and Y are neighbours if we get one by adding or removing one data element to or from the other (denoted X ∼ Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Consider a randomized mechanism M : X n → O, where O denotes the output space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The (ε, δ)-definition of DP can be given as follows (Dwork, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Let ε > 0 and δ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We say that a mechanism M is (ε, δ)-DP, if for all neighbouring datasets X and Y and for every measurable set E ⊂ O we have: Pr(M(X) ∈ E) ≤ eεPr(M(Y ) ∈ E) + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We will also use the Rényi differential privacy (RDP) (Mironov, 2017) which is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Rényi divergence of order λ > 1 between two distributions P and Q is defined as Dλ(P||Q) = 1 λ − 1 log � �P(t) Q(t) �λ Q(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1) Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We say that a mechanism M is (λ, ε)-RDP, if for all neighbouring datasets X and Y , the output distributions M(X) and M(Y ) have Rényi divergence of order λ at most ε, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', max X∼Y Dλ � M(X)||M(Y ) � ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We can convert from Rényi DP to approximate DP using, for example, the following formula: Lemma 3 (Canonne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Suppose the mechanism M is � λ, ε′� RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Then M is also (ε, δ(ε))-DP for arbitrary ε ≥ 0 with δ(ε) = exp � (λ − 1)(ε′ − ε) � λ � 1 − 1 λ �λ−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2) As is common, in practice we carry out the RDP accounting such that we do bookkeeping of total ε(λ)- values for a list of RDP-orders (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' integer λ’s) and in the end convert to (ε, δ)-guarantees by minimizing over the values given by the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' RDP accounting for compositions of DP mechanisms is carried using standard RDP composition results (Mironov, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' DP-SGD differs from SGD such that sample-wise gradients of a random mini-batch are clipped to have L2-norm at most C and normally distributed noise with variance σ2 is added to the sum of the gradients of the mini-batch (Abadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' One iteration is given by θj+1 = θj − ηj � 1 |B| � i∈Bj ∇f(xi, θj) + Zj � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='3) where Zj ∼ N(0, C2σ2 |B|2 Id), and ηj denotes the learning rate hyperparameter and |B| is the expected batch size (if we carry out Poisson subsampling of mini-batches, |Bj| varies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' There are several RDP results that enable the RDP analysis of DP-SGD iterations (Abadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Balle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Zhu and Wang, 2019), and nowadays DP-SGD is an indispensable part of frameworks such as Opacus (Yousefpour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 3 2021) and Tensorflow Privacy (Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The following result by Zhu and Wang (2019) is directly applicable to analyzing DP-SGD, however we also use it for analyzing one of our hyperparameter tuning strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Theorem 4 (Zhu and Wang 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Suppose M is a � λ, ε(λ) � RDP mechanism, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' to the add/remove neighbourhood relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Consider the subsampled mechanism (M ◦ subsamplePoisson(γ))(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' If M is � λ, ε(λ) � RDP then M ◦ subsamplePoisson(q) is � λ, ε′(λ) � RDP (λ ≥ 2 is an integer), where ε′(λ) = 1 λ − 1 log � (1 − γ)λ−1(λγ − γ + 1) + � λ 2 � γ2(1 − γ)λ−2e ε(2) + 3 λ � j=3 � λ j � γj(1 − γ)λ−je (j−1)ε(j) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We remark that the recent works (Koskela et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Gopi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2022) give methods to carry out (ε, δ)-analysis of DP-SGD tightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2 DP Hyperparameter Tuning When tuning the hyperparameters using sensitive data, as Papernot and Steinke (2022) show, private information may leak, especially in case the hyperparameters are chosen based on non-private ML model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Intuitively, the leakage from hyperparameters is much smaller than from the model parameters, however considering it in the final accounting is essential to ensure rigorous DP guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' As described in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1, currently the most practical (ε, δ)-guarantees for DP hyperparameter tuning algorithms are those of Papernot and Steinke (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' In the results of Papernot and Steinke (2022) important is that the number of candidate models K is randomized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' They analyze various distributions for drawing K, however we focus on using the Poisson distribution as it is the most concentrated around the mean among all the alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The corresponding hyperparameter tuning algorithm and its privacy guarantees are given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' First recall: K is distributed according to a Poisson distribution with mean µ > 0, if for all non- negative integer values k: P(K = k) = e−µ · µk k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Theorem 5 (Papernot and Steinke 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Let Q : X N → Y be a randomized algorithm satisfying � λ, ε(λ) � RDP and (�ε, �δ)-DP for some λ ∈ (1, ∞) and ε, �ε, �δ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Assume Y is totally ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Let the Poisson distribution parameter µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Define the hyperparameter tuning algorithm A : X N → Y as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Draw K from a Poisson distribution with mean µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Run Q(X) for K times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Then A(X) returns the best value of those K runs (both the hyperparameters and the model parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' If K = 0, A(X) returns some arbitrary output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' If e �ε ≤ 1 + 1 λ−1, then A satisfies � λ, ε′(λ) � RDP, where ε′(λ) = ε(λ) + µ · �δ + log µ λ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Notice that in a common analysis of DP-SGD, the guarantees hold not only for the best hyperparameters and the corresponding models, but for the whole histories of models: when the output A(X) of the base mechanism is a sequence of DP gradients, post-processing of that sequence gives the sequence of models � θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' , θT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Thus, in the tuning algorithm, we are free to do model check-pointing and choose the best model along the histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 4 3 DP Hyperparameter Tuning with a Random Subset We next consider our main tools: we carry out the private hyperparameter tuning on a random subset, and if needed, extrapolate the found hyperparameter values to larger datasets that we use for training subsequent models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We consider two strategies: in the first one the subset of data used for tuning is possibly smaller than the data used for training for subsequent models and thus we extrapolate the hyperparameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' In the other approach, the subsequent models are trained with datasets of the same size as the tuning set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 Strategy 1: Small Random Subset for Tuning In the first strategy, we carry out the following steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Use Poisson subsampling to draw X1 ⊂ X: draw a random subset X1 such that each x ∈ X is included in X1 with probability q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Compute (θ1, t1) = M1(X1), where M1 is a hyperparameter tuning algorithm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', method by Papernot and Steinke, 2022) that outputs the vector of optimal hyperparameters t1 and the corre- sponding model θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' If needed, extrapolate the hyperparameters t1 to the dataset X \\ X1: t1 → t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Compute θ2 = M2(t2, X \\ X1), where M2 is the base mechanism (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', DP-SGD training run).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Denote the whole mechanism by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Then, we may write M(X) = � M1(X1), M2 � M1(X), X \\ X1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1) Notice that since M2 and M1 use the same randomness about X1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1) cannot be analyzed using stan- dard RDP composition results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We provide a tailored analysis for this method in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' However, we can consider alternative third and fourth steps which use the whole dataset X: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' If needed, extrapolate the hyperparameters t1 to the dataset X: t1 → t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Compute θ2 = M2(t2, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Then, the procedure can be written as a composition M(X) = � � M1(X), M2 � � M1(X), X � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2) where � M1(X) = (M1◦subsample)(X), and the RDP guarantees are obtained using standard subsampling and composition results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 Extrapolating the Hyperparameters In this work, we use simple heuristics to transfer the optimal hyperparameter values found for the small subset of data to a larger dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' When using DP-SGD, we use the heuristics used by (van der Veen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2018): we scale the learning rate η with the dataset size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', if we carry out the hyperparameter tuning using a subset of size m and find an optimal value η∗, we multiply η∗ by n/m when transferring 5 to the dataset of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Clipping constant C, the noise level σ and the subsampling ratio γ are kept constant in this transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' This can be also heuristically motivated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Consider T steps of the DP-SGD (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' With the above rules, the distribution of the noise that gets injected into the model is T � j=1 Zj ∼ N � 0, T ·( n m η∗) 2σ2C2 (γ·n)2 � ∼ N � 0, T ·η∗2σ2C2 (γ·m)2 � which is exactly the distribution of the noise added to the model trained with the subsample of size m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Note that such linear scaling of learning rate has also been considered in non-DP SGD (Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2017) to obtain similar validation errors for both small and large minibatch training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Training of certain models benefits from use of adaptive optimizers such as Adam (Kingma and Ba, 2014) or RMSProp, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', due to sparse gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Then the above extrapolation rules for DP-SGD are not meaningful anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' In our experiments, when training a neural network classifier for the IMDB dataset and using Adam with DP-SGD gradients, we found that keeping the value of learning rate fixed in the transfer to the larger dataset lead to better results than increasing it as in case of DP-SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We mention that there are principled ways of extrapolating the hyperparameters in non-DP setting such as those of (Klein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2 Strategy 2: Model Ensemble with Disjoint Shards Strategy 1 can be directly extended to the case, where we divide the dataset into multiple disjoint sets of the same size and train a model for each of the datasets separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Similar sharding has been considered also in the machine unlearning framework SISA by Bourtoule et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (2021) and also in the framework PATE Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (2016) where student models are trained on disjoint datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We randomly pick only one of the shards for hyperparameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Our strategy 2 is described as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Divide X into k shards such that for each data element x of X, the probability of ending up to shard i, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' , k, is 1/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' First compute (θi, t) = M1(Xi) using some shard Xi, i ∈ [k] and then θj = M2(t, Xj) for all j ∈ [k]\\{i} (can be computed in parallel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Perform predictions using an ensemble of models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', if f(x, θ) denotes the model output for a data element x with model parameters θ, we carry out predictions using the weighted average 1 k �k j=1 f(x, θj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' It is reasonable to assume that the optimal hyperparameters for the rest of the shards are not far from those of the shard Xi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', no extrapolation is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='3 RDP-Analysis of Strategy 1 and Strategy 2 We first consider our strategy 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', the mechanisms (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The mechanism (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2) can be analyzed using subsampling and composition results for RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Using the RDP values given by Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 5 for M1 and a subsampling amplification result such as Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 4, we obtain RDP bounds for (M1◦subsamplePoisson(γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Using RDP bounds for M2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', DP-SGD) and composition results, we further get RDP bounds for M 6 of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' However, if we only use the rest of the data X\\X1 for M2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' the mechanism (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1), we can get even tighter RDP bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Let X ∈ X n and Y = X ∪ {x′} for some x′ ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' To bound the RDP-parameters for the mechanism M of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1), it is sufficient that we find X− and x′−indepedent bounds for ε(λ) = max{Dλ � M(Y )||M(X) � , Dλ � M(X)||M(Y ) � } for different orders of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='These bounds should be obtained in terms of RDP values for M1 (the hyper- parameter tuning algorithm, RDP bounds obtained, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', using Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 5) and for M2 (DP-SGD, RDP bounds obtained using, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The following theorem gives tailored RDP bounds for the mechanism (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Similarly to the analy- sis of Zhu and Wang (2019) for the Poisson subsampled Gaussian mechanism, we obtain RDP bounds for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1) using the RDP bounds of the mechanisms M1 and M2 and by using binomial expansions (proof given in the Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We can also use this theorem to analyse the ensemble method described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2, since the remaining submodels can be thought of as a parallel composition for which we obtain RDP bounds using Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 and its Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2 of Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Let X ∈ X n and Y = X ∪ {x′} for some x′ ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Let M(X) be the mechanism described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1, such that X1 is sampled with sampling ratio q, 0 ≤ q ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Let λ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Denote by εP (λ) and εQ(λ) the RDP-values of mechanisms M1 and M2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We have that Dλ � M(Y )||M(X) � ≤ 1 λ − 1 log � qλ exp � (λ − 1)εP (λ) � + (1 − q)λ exp � (λ − 1)εQ(λ) � + λ · qλ−1 · (1 − q) · exp � (λ − 2)εP (λ − 1) � + λ · q · (1 − q)λ−1 · exp � (λ − 2)εQ(λ − 1) � + λ−2 � j=2 �λ j � qλ−j · (1 − q)j · exp � (λ − j − 1)εP (λ − j) � exp � (j − 1)εQ(j) �� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='3) and Dλ � M(X)||M(Y ) � ≤ 1 λ − 1 log � (1 − q)λ−1 exp � (λ − 1)εQ(λ) + qλ−1 exp � (λ − 1)εP (λ) + λ−2 � j=1 �λ − 1 j � qj · (1 − q)λ−1−j · exp � j · εP (j + 1) � exp � (λ − j − 1)εQ(λ − j) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='4) Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We can initialize the subsequent model training M2 using the model θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' This adaptivity is included in the RDP analyses of both strategies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='4 Computational Savings The expected number of required gradient evaluation for our strategy 1 is bounded by µ · q · n· epochs +n· epochs, whereas the baseline requires in expectation µ · n· epochs evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' For example, in our experiments with µ = 45 and q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1, the baseline method requires µ µ·q+1 ≈ 8 times more gradient evaluations than our strategy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 7 4 Dealing with Hyperparameters that Affect the DP Guarantees Theorem 5 gives RDP-parameters of order λ for the tuning algorithm, assuming the underlying candidate picking algorithm is � λ, ε(λ) � RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' If we are tuning, for example, the DP-SGD hyperparameters learning rate η or clipping constant C, and fix rest of the hyperparameters, these � λ, ε(λ) � RDP bounds are fixed for all hyperparameter candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' However, if we tune the hyperparameters that affect the DP guarantees, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', the subsampling ratio γ, the noise level σ or the length of the training T, it is less straightforward to determine what are the uniform ε(λ)-upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' As is common practice, we fix a grid Λ of λ-orders for RDP bookkeeping (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' integer values of λs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' If for each random choice of hyperparameters, and for each λ ∈ Λ, the RDP value �ε(λ) of each candidate model is bounded by a value ε(λ), then we can use ε(λ) to obtain RDP-bounds of the hyperparameter tuning algorithm with Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' This is formally shown below in Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' For example, when we tune DP-SGD and vary the values of γ and σ, we need to somehow ensure that the RDP values of the candidates stay below the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We below consider two algorithms for finding random candidates and appropriate upper RDP bounds ε(λ) for the candidate picking algorithm Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 Algorithm 1: Grid Search with Randomization In the first approach we first set an approximative DP target value (ε, δ) which we use to adjust the hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' For example, if we are tuning the DP-SGD parameters subsampling ratio γ and noise level σ, we can, for each choice of (γ, σ), adjust the length of the training T so that the resulting training iteration is at most (ε, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Vice versa, we may tune the γ and T, and take minimal value of σ such that the resulting training iteration is at most (ε, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Consider first tuning of DP-SGD and how to set T for each choice of hyperparameters γ and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We first fix ε, δ > 0 which represent the target approximative DP bound for each candidate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Denote by ε(T, δ, γ, σ) the approximate DP ε of the subsampled Gaussian mechanism with parameter values γ, σ and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' For each value of (γ, σ), we attach a number of iterations Tγ,σ such that it is the largest number with which the resulting composition is (ε, δ)-DP: Tγ,σ = max{T ∈ N : ε(T, δ, γ, σ) ≤ ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' As the RDP values increase monotonously w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' number of compositions, it is straightforward to find Tγ,σ, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', using the bisection method Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Alternatively, we could fix a few values of T, and to each combination of (γ, T), attach the smallest σ (denoted σγ,T ) such that the target (ε, δ)-bound holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We prefer this option in our experiments to suit our computational constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' By the data-processing inequality, the privacy parameters decrease monotonously w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' σ, so that again, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', the bisection method can be used to find σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We consider a finite grid Γ of possible hyperparameter values t (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', in case of DP-SGD, t = (γ, σ, T), where T is adjusted to γ and σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Then, for all t ∈ Γ, we compute the corresponding RDP value εt(λ) for each λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Finally, for each λ ∈ Λ, we set ε(λ) = max t∈Γ εt(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Then, for each random draw of t, the DP-SGD trained candidate model is is ε(λ)-RDP, and by Lemma 9 below, the candidate picking algorithm Q is also ε(λ)-RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2 Algorithm 2: Random Search Here, we assume we are given some distributions of the hyperparameter candidates and the algorithm Q draws hyperparameters using them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' In order to adjust the number of iterations for each candidate, we take a λ-line as an RDP upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' More specifically, we require that the candidate models are � λ, c · λ � RDP for some c > 0 and for all λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Then the number of iterations Tγ,σ for each draw of (γ, σ) is the maximum number of iterations for which the � λ, c · λ � RDP bound holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', Tγ,σ = max{T ∈ N : T · εγ,σ(λ) ≤ c · λ for all λ ∈ Λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Similarly, we can find the minimal σ based on T and γ such that the mechanism is � λ, c · λ � RDP for all λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Again, by Lemma 9 below, the candidate picking algorithm Q is then c · λ-RDP and we may use Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 5 to obtain RDP bounds for the tuning algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='3 RDP Analysis of Algorithms 1 and 2 Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Denote by C the random variable of which outcomes are the hyperparameter candidates (drawing either randomly from a grid or from given distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Consider an algorithm Q, that first randomly picks hyperparameters t ∼ C, then runs a randomised mechanism M(t, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Suppose M(t, X) is � λ, ε(λ) � RDP for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Then, Q is � λ, ε(λ) � RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Suppose the hyperparameters t are outcomes of a random variable C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Let X and Y be neighbouring datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Then, if p(t, s) and q(t, s) (as functions of s) give the density functions of M(t, X) and M(t, Y ), respectively, we have that Q(X) ∼ Et∼C p(t, s) and Q(Y ) ∼ Et∼C q(t, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Since for any distributions p and q, and for any λ > 1, exp � (λ − 1)Dλ(p||q) � = � �p(t) q(t) �λ q(t) dt gives an f-divergence (for f(x) = xλ), it is also jointly convex w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' p and q (Liese and Vajda, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Thus, using Jensen’s inequality, we have exp � (λ − 1)Dλ � Q(X)||Q(Y ) �� = � �Et∼C p(t, s) Et∼C q(t, s) �λ Et∼C q(t, s) ds ≤ Et∼C � �p(t, s) q(t, s) �λ q(t, s) ds = Et∼C exp � (λ − 1)Dλ � M(t, X)||M(t, Y ) �� ≤ Et∼C exp ((λ − 1)ε(λ)) = exp ((λ − 1)ε(λ)) from which the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='4 Accuracy of Algorithms 1 and 2 Success of both of these algorithms can be explained by the observation that Poisson subsampled Gaussian mechanism has approximately RDP guarantees of the Gaussian mechanism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', they are actually almost lines w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' the RDP order λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' This is discussed in more detail in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' For example, as shown in (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 38, Steinke, 2022), if the underlying mechanism is ρ-zCDP, then the Poisson subsampled version with subsampling ratio γ is � λ, 10γ2ρλ � RDP for λ sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The larger the value of σ and the smaller the value of γ, the closer the RDPs are to a line, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 5 Experiments Quality Metric and Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' In all of our experiments, we choose the best model based on the test accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Training and test sets are disjoint, and the quality metric is usually a low sensitivity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Therefore, even for a private test set, parallel composition can accommodate DP evaluation of a quality metric in the training budget itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' However, we assume that only the training dataset is private and the test data is public for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' This relaxation also allows us to take the best checkpoint over all epochs for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Our plots show the test accuracy of the final model against the final approximate DP ε of the tuning process for several target ε’s, varying from {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='., 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We fix q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 for strategy 1 and k = 2 for strategy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We mention that in several cases smaller value of q would have lead to better privacy- accuracy trade-off (see Appendix E for additional experiments), however we use the same value q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The carry out RDP accounting using RDP orders {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' , 64} and use δ = 10−5 in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We consider the two variants of strategy 1 in our experiments, one uses the entire dataset X for the final model (green curves, the mechanism (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2)) and the other uses X\\X1 (blue curves, the mechanism (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The final ε(λ)’s for the first and the second version are obtained by combining Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 5 with Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 4 and Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We expect the first variant to be more accurate for a slightly larger final ε compared to second, since the final models obtained with the mechanism (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2) use slightly more data than the one obtained with the mechanism (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1, we scale the best learning rate obtained from the first model by the dataset size for training the final model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We do not perform any learning rate scaling in strategy 2 (black curves) because all shards have the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' In strategy 2, we take the weighted average of the outcomes from both models as the prediction, with weights as the reciprocals of the shard sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The method by Papernot and Steinke (2022) described in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 5 is the baseline (red curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Datasets and Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We carry out our experiments on the following standard benchmark datasets for classification: CIFAR-10 (Krizhevsky and Hinton, 2009), MNIST (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 1998), FashionM- NIST (Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2017) and IMDB (Maas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' For MNIST and IMDB, we use the convolutional neural networks from the examples provided in the Opacus library Yousefpour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' For Fash- ionMNIST, we consider a simple feedforward 3-layer ReLU network with hidden layers of width 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' For CIFAR-10, we use a Resnet20 pretrained on CIFAR-100 (Krizhevsky and Hinton, 2009) dataset so that only the last fully connected layer is trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We minimize the cross-entropy loss in all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Following the Opacus example, we optimize with DP-Adam for IMDB dataset, but use DP-SGD for others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' For these datasets, in one of the experiments we tune only the learning rate, and in the other one the learning rate, batch size, and the number of epochs, while fixing the clipping constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The number of trainable parameters and the hyperparameter ranges used are provided in 10 Table 2 (Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The numbers of epochs are chosen to suit our computational constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Following the procedure from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1, we compute the smallest σ satisfying a target (ε, δ) bound for each (γ, epoch) pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The only purpose of target ε’s is to facilitate the mapping from epochs to σ, and retain comparability across methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Initializing with the First Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We are free to use the first model beyond hyperparameter extraction, since its privacy cost is already accounted for in the privacy analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Therefore, we use it to initialize the final model in strategy 1, and the subsequent models in the ensemble in strategy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' For the implementation of DP-SGD, we use the Opacus library (Yousefpour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' For scalability, we explore the hyperparameter spaces with Ray Tune (Liaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2018) on a multi-GPU cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We spell out additional details in the corresponding sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 Tuning Learning Rate The learning rate is among the most critical hyperaparameters, and thus we start with a learning rate optimization experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We fix the subsampling ratio γ and the number of epochs to the values given in Table 1 (Appendix B) for all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' For example, for q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0213 on MNIST dataset, the Poisson subsampling of DP-SGD gives in expectation batch sizes of 128 and 1150 in strategy 1, 640 for strategy 2 with k = 2, and 1280 for the baseline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The learning rate grid size is either 5 or 6, and we use µ = 10, which is sufficiently large to include a good candidate with a large probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Plot Description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Figure 1 plots the test accuracy against the final ε for all 4 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The labels with ’strategy 1’ refer to the mechanism given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1) (’X1 + X \\ X1’) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2) (’X1 + X’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The label ’baseline’ refers to the baseline method by (Papernot and Steinke, 2022) described in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 5 and the label ’strategy 2’ refers to the ensemble method described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The left panel for each dataset considers the case when the first model output by the tuning algorithm is used to initialize the subsequent model and the right panel shows the results when the first model is not used for initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Each plot contains 9 points for each method, one for each target ε (ε ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='., 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0}) for each model training run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Our main observation across the board is that the baseline method is generally more accurate for target ε ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5 (at a much higher final ε), but gets easily matched or outperformed by both variants of strategy 1 for higher target ε’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Moreover, as expected, the variant of strategy 1 which uses the entire data in the final model (green plot) performs marginally better than the alternative (blue plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Strategy 2 is the least accurate method due to lower privacy amplification with subsampling (q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5) compared to strategy 1 (q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The impact of initializing the final model with the first one on accuracy is not conclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' For MNIST, FashionMNIST, and CIFAR-10, strategy 2 is slightly more accurate for the same final ε when the model trained on shard 2 is initialized with the first model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' For IMDB, however, the opposite is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Strategy 1 does not seem to then benefit from the initialization either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' However, we experimentally observed that the final model achieves its peak accuracy much earlier when initialized with the first model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2 Tuning All Hyperaparameters Next, we also tune the batch size and the number of epochs in addition to the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The remaining setup is the same as the previous experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The choices are listed in Table 2 (Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Figure 2 shows the same quantities as Figure 1, however higher values of µ are used to accommodate to increased hyperaparameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Strategy 1 is more accurate compared to the baseline for low target ε-values and there is slight degradation in accuracy for higher target ε’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' For a fixed target ε-value of candidate 11 models, the final ε-values for strategy 1 are roughly 25 − 30% lower than those of the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We also notice smaller standard errors for CIFAR-10 and FashionMNIST, specially for strategy 2 which might be due to increased chances of finding favorable combinations of hyperaparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We also note that for final ε-value 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0, our test accuracies are at least as good as those of (Abadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2016) in case of MNIST (95% for Abadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2016) for and approximately the same in case of CIFAR-10 (67% for Abadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2016), although the results of (Abadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2016) do not include the DP cost of hyperparameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 6 Conclusions We have considered a simple strategy for lowering the privacy and compute cost of DP hyperparameter tuning: we carry out tuning using a random subset of data and extrapolate the optimal values to the larger training set used to train the final model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We have also provided methods to tune the hyper- parameters that affect the DP guarantees of the model training themselves, those being the noise level and subsampling ratio in case of DP-SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Our experiments show a clear improvement over the baseline method by Papernot and Steinke (2022), when tuning DP-SGD for neural networks and using simple heuristics for the extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' An interesting avenue of future work is to find more sophisticated ways to extrapolate the hyperparameters, something that has been considered in non-DP case (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Klein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We have also considered using a weighted ensemble of submodels trained on random disjoint equally- sized subsets of data, in which case extrapolation is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We found it leading to a slightly worse privacy-utility trade-off, albeit with computational speed-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Exploring further ways to improve this strategy is also left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 7 Acknowledgments We would like thank our colleague Laith Zumot for discussions about practical hyperparameter tuning methods at the initial stages of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Bibliography Abadi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', Chu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', Goodfellow, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', McMahan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Optimal accounting of differential privacy via characteristic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' In International Conference on Artificial Intelligence and Statistics, pages 4782–4817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' and Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Poisson subsampled Rényi differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' In International Con- ference on Machine Learning, pages 7634–7642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 15 (a) MNIST with µ = 10 (b) FashionMNIST with µ = 10 (c) CIFAR-10 with µ = 10 (d) IMDB with µ = 10 Figure 1: Tuning only the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Test accuracies averaged across 5 independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The error bars denote the std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' error of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Each curve contains 9 points, one for each target ε ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='., 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Here ’strategy 1’ is ∼ 5 times faster than the baseline method by Papernot and Steinke (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='97 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='78 baseline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='76 strategy1---Xi+XXi strategy 1 --- X1+X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='74 strategy 2 --- k=2 2 E 4 5 6 1 2 E 4 5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='680 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='670 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='660 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='655 baseline strategy1 ---Xi+X\\Xi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='650 strategy 1 --- Xi+X strategy 2 ---k=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='645 3 4 5 6 1 2 E 4 5 6 na0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='64 baseline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='62 strategy1---Xi+XXi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='60 strategy 1 --- Xi+X strategy 2 --- k=2 2 3 4 5 6 1 2 3 4 5 na(a) MNIST with µ = 45 (b) FashionMNIST with µ = 50 (c) CIFAR-10 with µ = 40 (d) IMDB with µ = 45 Figure 2: Tuning of subsampling ratio, training length, and learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Test accuracies averaged across 5 independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The error bars denote the std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' error of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Each curve contains 9 points, one for each target ε ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='., 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Here ’strategy 1’ is ∼ 8 times faster than the method by Papernot and Steinke (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content="95 +6'0 baseline strategy 1 ---Xi+X\\Xi strategy 1 --- Xi+X E6'0 strategy 2 --- k = 2 4 6 8 10 12 2 4 6 8 10 12 fina0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='80 baseline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='78 strategy1---Xi+XXi strategy1 --- Xi+X strategy2 --- k =2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='76 2 4 5 8 10 12 2 4 6 8 10 12 6na0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='680 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='670 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='660 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='655 baseline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='650 strategy 1 ---Xi+XXi strategy 1 --- X1+X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='645 strategy 2 --- k=2 2 4 6 8 10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 4 6 8 10 naLE naLE0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='64 baseline strategy1---Xi+X\\Xi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='62 strategy 1---Xi+X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='60 strategy 2--- k = 2 2 4 8 10 12 2 4 6 8 10 12 6naAppendix A Adjusting the Parameters T and σ for DP-SGD It is often a good approximation to say that the RDP-guarantees of the Poisson subsampled Gaussian mechanism are lines as functions of the RDP order λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', that the guarantees are those a Gaussian mechanism with some sensitivity and noise level values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' For example, Mironov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 11, 2019) show that the Poisson subsampled Gaussian mechanism is � λ, 2γ2λ/σ2� RDP when λ is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Also, (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 38, Steinke, 2022) show that if the underlying mechanism is ρ-zCDP, then the Poisson subsampled version with subsampling ratio γ is � λ, 10γ2ρλ � RDP when λ is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Notice that the Gaussian mechanism with L2-sensitivity ∆ and noise level σ is (∆2/2σ2)-zCDP (Bun and Steinke, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We numerically observe, that the larger the noise level σ and the smaller the subsampling ratio γ, the better the line approximation of the RDP-guarantees (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' In case the privacy guarantees (either (ε, δ)-DP or RDP) are approximately those of a Gaussian mech- anisms with some sensitivity and noise level values, both of the methods for tuning the hyperparameters γ, σ and T described in Section 4 would lead to very little slack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' This is because for the Gaussian mech- anism, both the RDP guarantees (Mironov, 2017) and (ε, δ)-DP guarantees (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2022) depend monotonously on the scaled parameter �σ = σ ∆ · √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' This means that if we adjust the training length T based on values of σ by having some target (δ, ε)- bound for the candidate model (Algorithm 1 of Section 4), the resulting RDP upper bounds of different candidates will not be far from each other (and similarly for adjusting σ based on value of T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Similarly, for Algorithm 2 of Section 4, when adjusting T based on values of σ, the RDP guarantees of all the candidate models would be close to the upper bound (c · λ, c > 0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', they would not be far from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='35 ( ) =2, T = 1000 Gauss RDP-fit ( =2) =4, T = 4000 Gauss RDP-fit ( =4) =6, T = 8000 Gauss RDP-fit ( =6) 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='6 ( ) =2, T = 1000 Gauss RDP-fit ( =2) =4, T = 4000 Gauss RDP-fit ( =4) =6, T = 8000 Gauss RDP-fit ( =6) Figure 3: DP-SGD RDP curves for different values of noise level σ and number of compostions T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Left: γ = 1/100, right: γ = 1/50 and the corresponding lines with the smallest slope that give upper bounds for the RDP orders up to λ = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 18 B Hyperparameter Tables for Experiments of Figures 1 and 2 MNIST FashionMNIST CIFAR-10 IMDB γ = B N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0213 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0213 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0256 epochs 40 40 40 110 Table 1: Tuning η: rest of the hyperparameters fixed to these values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' train/test set parameters C B learning rate epochs MNIST 60k/10k ∼26k 1 {128, 256} {10−i}i∈{2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5,1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5,0} {10,20,30,40} FashionMNIST 60k/10k ∼109k 3 {128, 256} {10−i}i∈{2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5,1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5,0,−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5} {10,20,30,40} CIFAR-10 50k/10k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='65k 3 {64, 128} {10−i}i∈{2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5,1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5,0,−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5} {20,30,40} IMDB 25k/25k ∼464k 1 {64, 128} {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='02,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2,1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1} {50,70,90,110} Table 2: Tuning σ, η and T: datasets used and the corresponding hyperparameter grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' C Proof of Theorem 7 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 Auxiliary Lemma We will need the following inequality for the proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 (Lemma 35, Steinke 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' For all p ∈ [0, 1] and x ∈ (0, ∞), 1 1 − p + p x ≤ 1 − p + p · x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Remark C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' An alternative proof for this result can be obtained using so called Bergström’s inequality, which states that for all xk ∈ R, ak > 0, k ∈ [n], (x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' + xn)2 a1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' + an ≤ x2 1 a1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' + x2 n an .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1) In particular, for n = 2 and a1 = q x, a2 = 1 − q, x1 = q, x2 = 1 − q, this gives 1 1 − q + q x ≤ 1 − q + q · x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' As the proof of Theorem 7 shows, it could be generalized to the case of random k-way split (k > 2) using the inequality (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 19 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2 Proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We first consider the case Dλ � M(Y )||M(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Let X ∈ X n and x′ ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Denote ε1(λ) = Dλ � M(X ∪ {x′})||M(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Looking at Strategy 1 which uses Poisson subsampling with subsampling ratio q to obtain the dataset X1, and conditioning the output on the randomness in choosing X1, we can write the mechanism as a mixture over all possible choices of X1 as M(X) = � X1 pX1 · � M1(X1), M2(M(X1), X\\X1) � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2) where pX1 is the probability of sampling X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Since each data element is in X1 with probability q, we can furthermore write M(X ∪ {x′}) as a mixture M(X ∪ {x′}) = � X1 pX1 · � q · � M1(X1 ∪ {x′}), M2(M1(X1 ∪ {x′}), X\\X1) � + (1 − q) · � M1(X1), M2(M(X1), X\\X1 ∪ {x′}) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='3) From the quasi-convexity of the Rényi divergence (Van Erven and Harremos, 2014) and the expressions (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='3), it follows that Dλ � M(X ∪ {x′})||M(X) � ≤ sup X1 Dλ � q · � M1(X1 ∪ {x′}), M2(M1(X1 ∪ {x′}), X\\X1) � + (1 − q) · � M1(X1), M2(M(X1), X\\X1 ∪ {x′}) � || � M1(X1), M2(M(X1), X\\X1) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='4) Our aim is to express the right-hand side of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='4) in terms of RDP parameters of M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' To this end, take an arbitrary X1 ⊂ X, and denote by �P(t) the density function of M1(X1 ∪ {x′}), P(t) the density function of M1(X1), �Q(t, s) the density function of M2(t, X\\X1 ∪ {x′}) for auxiliary variable t (the output of M1), Q(t, s) the density function of M2(t, X\\X1) for auxiliary variable t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Then, we see that P �� M1(X1), M2(M(X1), X\\X1) � = (t, s) � = P(t) · Q(t, s) and similarly that P � q · � M1(X1 ∪ {x′}), M2(M1(X1 ∪ {x′}), X\\X1) � + (1 − q) · � M1(X1), M2(M(X1), X\\X1 ∪ {x′}) � = (t, s) � = q · P �� M1(X1 ∪ {x′}), M2(M1(X1 ∪ {x′}), X\\X1) � = (t, s) � + (1 − q) · P �� M1(X1), M2(M(X1), X\\X1 ∪ {x′}) � = (t, s) � = q · �P(t) · Q(t, s) + (1 − q) · P(t) · �Q(t, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 20 By the definition of the Rényi divergence, we have that exp � (λ − 1)Dλ � q · � M1(X1 ∪ {x′}), M2(M1(X1 ∪ {x′}), X\\X1) � + (1 − q) · � M1(X1), M2(M(X1), X\\X1 ∪ {x′}) � || � M1(X1), M2(M(X1), X\\X1) ��� = � � � q · �P(t) · Q(t, s) + (1 − q) · P(t) · �Q(t, s) P(t) · Q(t, s) �λ P(t) · Q(t, s) dt ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5) which can be expanded as � � � q · �P(t) · Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' s) + (1 − q) · P(t) · �Q P(t) · Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' s) �λ P(t) · Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' s) dt ds = � � � q · �P(t) P(t) + (1 − q) · �Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' s) Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' s) �λ P(t) · Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' s) dt ds = � � qλ � �P(t) P(t) �λ P(t) · Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' s) dt ds + � � (1 − q)λ � �Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' s) Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' s) �λ P(t) · Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' s) dt ds + � � λ · qλ−1 · (1 − q) · � �P(t) P(t) �λ−1 P(t) · �Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' s) dt ds + � � λ · q · (1 − q)λ−1 · � �Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' s) Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' s) �λ−1 Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' s) · �P(t) dt ds + � � λ−2 � j=2 �λ j � qλ−j · (1 − q)j · � � � �P(t) P(t) �λ−j P(t) � � � � � �Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' s) Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' s) �j Q(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' s) � � dt ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='6) We next bound the five integrals on the right hand side of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' For the first two integrals, we use the RDP-bounds for M1 and M2 to obtain � � � �P(t) P(t) �λ P(t)Q(t, s) dt ds = � � � �P(t) P(t) �λ P(t) dt ≤ exp � (λ − 1)εP (λ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='7) and � � � �Q(t, s) Q(t, s) �λ Q(t, s)P(t) ds dt ≤ � � exp � (λ − 1)εQ(λ) � P(t) dt = exp � (λ − 1)εQ(λ), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='8) 21 where εP and εQ give the RDP-parameters of order λ for M1 and M2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The third and fourth integral can be bounded analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' In the second inequality we have also used the fact that the RDP-parameters of M2 are independent of the auxiliary variable t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Similarly, for the third integral, we have � � � � � �P(t) P(t) �λ−j P(t) � � � � � �Q(t, s) Q(t, s) �j Q(t, s) � � ds dt ≤ � � � � � �P(t) P(t) �λ−j P(t) � � exp � (j − 1)εQ(j) � dt ≤ exp � (λ − j − 1)εP (λ − j) � exp � (j − 1)εQ(j) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='9) Substituting (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='7), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='8) (and similar expressions for the third and fourth integral) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='9) to (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='6), we get a bound for (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Since X1 ⊂ X was arbitrary, we arrive at the claim via (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Next, we consider bounding Dλ � M(X)||M(Y ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The proof goes similarly as the one for Dλ � M(Y )||M(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Denote ε2(λ) = Dλ � M(X)||M(X ∪ {x′}) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' With the notation of proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 7, we see that, instead of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5), we need to bound exp � (λ − 1)ε2(λ) � = � � � P(t) · Q(t, s) q · �P(t) · Q(t, s) + (1 − q) · P(t) · �Q(t, s) �λ (q · �P(t) · Q(t, s) + (1 − q) · P(t) · �Q(t, s)) dt ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' In order to use here the series approach, we need to use Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1: � P · Q q · �P · Q + (1 − q) · P · �Q �λ (q · �P · Q + (1 − q) · P · �Q) = � P · Q q · �P · Q + (1 − q) · P · �Q �λ−1 P · Q = � q · �P P + (1 − q) · �Q Q �1−λ P · Q = � q · �P P Q �Q + (1 − q) �1−λ � �Q Q �1−λ P · Q = � q · �P P Q �Q + (1 − q) �1−λ �Q �Q �λ−1 P · Q ≤ � q · P �P �Q Q + (1 − q) �λ−1 �Q �Q �λ−1 P · Q = � q · P �P �Q Q + (1 − q) �λ−1 �Q �Q �λ P · �Q, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='10) 22 where in the inequality we have used Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Now we can expand � q · P � P � Q Q + 1 − q �λ−1 : � 1 − q + q · P �P �Q Q �λ−1 �Q �Q �λ P · �Q = � � λ−1 � j=0 �λ − 1 j � qj · (1 − q)λ−1−j · �P �P �j � �Q Q �j� � · �Q �Q �λ P · �Q = � � λ−1 � j=0 �λ − 1 j � qj · (1 − q)λ−1−j · �P �P �j �Q �Q �λ−j � � · P · �Q = λ−1 � j=0 �λ − 1 j � qj · (1 − q)λ−1−j · �P �P �j+1 �P · �Q �Q �λ−j �Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Then, we use the known εP (λ) and εQ(λ)-values as in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='9) to arrive at the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 23 D f-Divergence of Parallel Compositions We first formulate the parallel composition result for general f-divergences (Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We then obtain the RDP bound for parallel compositions as a corollary (Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Our Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 below can be seen as an f-divergence version of the (ε, 0)-DP result given in (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 4 McSherry, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Corollary 2 by Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' (2022) gives the corresponding result in terms of µ-Gaussian differential privacy (GDP), and it is a special case of our Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 as µ-GDP equals the (ε, δ)-DP (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', the hockey-stick divergence) of the Gaussian mechanism with a certain noise scale (Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 1, Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We define f-divergence for distributions on Rd as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Consider two probability densities P and Q defined on Rd, such that if Q(x) = 0 then also P(x) = 0, and a convex function f : [0, ∞) → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Then, an f-divergence (Liese and Vajda, 2006) is defined as Df(P||Q) = � f �P(t) Q(t) � Q(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' In case the data is divided into disjoint shards and separate mechanisms are applied to each shard, the f- divergence upper bound for two neighbouring datasets can be obtained from the individual f-divergence upper bounds: Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Suppose a dataset X ∈ X N is divided into k disjoint shards Xi, i ∈ [k], and mechanisms Mi, i ∈ [k], are applied to the shards, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Consider the mechanism M(X) = � M1(X1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' , Mk(Xk) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Then, we have that max X∼Y Df � M(X)||M(Y ) � ≤ max i∈[k] max X∼Y Df � Mi(X)||Mi(Y ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Let X be divided into k shards as described above and suppose Y is a neighbouring dataset such that X1 ∼ Y1 and Y = {Y1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' , Xk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Then, we see that P � M(X) = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' , ak) � P � M(Y ) = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' , ak) � = P � M1(X1) = a1 � P � M1(X2, a1) = a2 � · · P � Mk(Xk, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' , ak−1) = ak � P � M1(Y1) = a1 � P � M1(X2, a1) = a2 � · · P � Mk(Xk, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' , ak−1) = ak � = P � M1(X1) = a1 � P � M1(Y1) = a1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' and furthermore, denoting a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' , ak), Df � M(X)||M(Y )) � = � f � P � M(X) = a � P � M(Y ) = a � � P � M(Y ) = a � da = � f � P � M1(X1) = (a1) � P � M1(Y1) = (a1) � � P � M(Y ) = a � da = � f � P � M1(X1) = (a1) � P � M1(Y1) = (a1) � � P � M(Y1) = (a1) � da1 = Df(M1(X1)||M1(Y1)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 24 Thus, Df(M(X)||M(Y )) = Df(M1(X1)||M1(Y1)) ≤ max X∼Y Df(M1(X)||M1(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Similarly, if Xi ∼ Yi, i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' , k and Y = � X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Xi−1, Yi, Xi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' , Xk � , we see that Df(M(X)||M(Y )) = Df(Mi(Xi)||Mi(Yi)) ≤ max X∼Y Df(Mi(X)||Mi(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Thus, we have that max X∼Y Df(M(X)||M(Y )) = max i∈[k] max X∼Y Df(Mi(X)||Mi(Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Suppose a dataset X ∈ X N is divided into k disjoint shards Xi, i ∈ [k], and mechanisms Mi, i ∈ [k], are applied to the shards, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Consider the mechanism M(X) = � M1(X1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' , Mk(Xk) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Suppose each Mi is � λ, εi(λ) � RDP, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Then, M is � λ, maxi∈[k] εi(λ) � RDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' This follows from Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 since exp � (λ − 1)Dλ(M(X)||M(Y )) � = � � P � M(X) = a � P � M(Y ) = a � �λ P � M(Y ) = a � da is an f-divergence for f(x) = xλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Thus, by Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 we have that max X∼Y exp � (λ − 1)Dλ(M(X)||M(Y )) � ≤ max i∈[k] max X∼Y exp � (λ − 1)Dλ(Mi(X)||Mi(Y )) � from which it follows that max X∼Y Dλ(M(X)||M(Y )) ≤ max i∈[k] max X∼Y Dλ(Mi(X)||Mi(Y )) = max i∈[k] εi(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 25 E Additional Experiments E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 Comparison of (ε, δ)-Bounds Figure 4: Final ε as a function of q for µ = 50, when the base mechanism is the subsampled Gaussian mechanism with γ = 64/6000, number of epochs = 50 and σ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The value of q varies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='05 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Strategy 1 has considerably smaller ε’s for q ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 compared to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' However, with q ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1, strategy 1 may not provide good utility on small data sets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=', IMDB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Our proposed Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 7 used for the mechanism (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1) (’X1 + X \\ X1’) is suboptimal since for small values of q (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='05) the bound of the mechanism (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2) obtained with Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 4 is tighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2 Vary k in Strategy 2 For strategy 2, higher k’s provide more estimators in the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' On the other hand, each submodel is trained on a shard of smaller size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' To experimentally see the effect of k, we carry out an experiment similar to the one described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 for a different values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Figure 5 shows test accuracy as a function of target ε, k, and final ε for strategy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The value of k in each plot varies from 2 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The first observation is that higher values of k reduce the final ε (at the cost of lower accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' This is due to smaller values of q (q = 1 k) in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The k = 2 case generally yields the peak accuracy on all datasets, although in some cases higher k would have lead to better privacy-utility trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We fix k = 2 in the experiments of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' In strategy 2, the compute cost of the tuning part also lowers the overall compute cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Moreover, larger k’s reduce the training time for each submodel and the submodels can be trained in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 26 N=6000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' ep0chs=50 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' B=64 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' μ=50, 0=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0 baseline (Papernotand Steinke) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5 strategy1---Xi+X(Zhu&Wang) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0 strategy 1 and 2 ---Xi +XXi (our bound) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='8E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='3 Vary q in Strategy 1 In strategy 1, when q increases, the hyperparameter tuning mechanism trains with a larger dataset which means also weaker final privacy guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Additionally, the best learning rate from the first model is also scaled with a smaller factor in the final model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We perform for strategy 1 an experiment similar as described in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2 to compute the final test accuracies and ε-values as a function of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Figure 6 shows the test accuracy as a function of target ε for the base mechanism (DP-SGD), q, final ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The value of q in each plot varies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' As expected, the accuracy for the variant that use X\\X1 in the final model drops when q increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The performance of the other variant that trains the final model with the full dataset remains relatively steady for all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' In IMDB, we use DP-Adam to train the IMDB model and do not scale the best initial learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 27 (a) MNIST with µ = 10 (b) FashionMNIST with µ = 10 (c) CIFAR-10 with µ = 10 (d) IMDB with µ = 10 Figure 5: Tuning learning rate for strategy 2: We keep the batch size and the number of epochs fixed and only tune the learning rate, for various values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The test accuracies are averaged across 5 independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The error bars denotes the std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' error of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Each plot contains 9 points for strategy 2, one for each k ∈ {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='., 10} (right to left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We also add the baseline method for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The k = 2 case generally yields the peak accuracy on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 28 MNIsT,P=1o,rgtE=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='6 ↓ baseline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='96 ■ strategy2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content="-- various k's 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='94 10%表 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='92 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='90 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='88 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='9 10 11 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='4 16 18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='8 EFhionMNisT,p=1o,rgatE=d8 FhionMNIST,μ=10, barget E=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2 T=FLSINHUOIy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content="84 baseline strategy 2 --- various k's 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='82 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='8(a) MNIST with µ = 10 (b) FashionMNIST with µ = 10 (c) CIFAR-10 with µ = 10 (d) IMDB with µ = 10 Figure 6: Tuning learning rate for strategy 1: We keep the batch size, the number of epochs fixed and only tune the learning rate for strategy 1 for various values of q’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Test accuracies are averaged across 5 independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The error bars denotes the std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' error of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Each plot contains 8 points for strategy 1, one for each q ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='., 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='8} (left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' We also add the baseline method for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 case generally yields the peak accuracy on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='98 MNIST, μ=10, rgat E=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='4 MMIsT, μ=10, rgat E=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='8 ANST,1O,gatE=1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0 hnaLE hnalE naE 6naLE(a) MNIST with µ = 10 (b) FashionMNIST with µ = 10 (c) CIFAR-10 with µ = 10 (d) IMDB with µ = 10 Figure 7: Tuning learning rate for strategy 1 for q ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1: We keep the batch size and the number of epochs fixed and only tune the learning rate for strategy 1 for various values of q ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Test accuracies are averaged across 5 independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' The error bars denotes the std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' error of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' Each plot contains 4 points for strategy 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' when q ∈ {0.' metadata={'source': 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and therefore the final ε-value for the variant of strategy 1 that uses the entire data is is lower in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content=' 30 strategy 1 --- Xi + XX1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 strategy 1 --- Xi +X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 5050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='05 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='05strategy 1 --- Xi +XX1 strategy 1 --- Xi +X 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='05 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='05strategy 1 --- Xi +XXi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 strategy 1 -.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='- Xi +X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='0505 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNFLT4oBgHgl3EQfFy-9/content/2301.11989v1.pdf'} +page_content='05 0.' metadata={'source': 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b/GdAyT4oBgHgl3EQf5Prw/content/tmp_files/2301.00803v1.pdf.txt @@ -0,0 +1,2322 @@ +ASYMPTOTICALLY COMPATIBILITY OF A CLASS OF NUMERICAL +SCHEMES FOR A NONLOCAL TRAFFIC FLOW MODEL +KUANG HUANG∗ AND QIANG DU† +Abstract. +This paper considers numerical discretization of a nonlocal conservation law modeling vehicular traffic flows +involving nonlocal inter-vehicle interactions. The nonlocal model involves an integral over the range measured by +a horizon parameter and it recovers the local Lighthill-Richards-Whitham model as the nonlocal horizon parameter +goes to zero. Good numerical schemes for simulating these parameterized nonlocal traffic flow models should be +robust with respect to the change of the model parameters but this has not been systematically investigated in +the literature. We fill this gap through a careful study of a class of finite volume numerical schemes with suitable +discretizations of the nonlocal integral, which include several schemes proposed in the literature and their variants. +Our main contributions are to demonstrate the asymptotically compatibility of the schemes, which includes both +the uniform convergence of the numerical solutions to the unique solution of nonlocal continuum model for a given +positive horizon parameter and the convergence to the unique entropy solution of the local model as the mesh size +and the nonlocal horizon parameter go to zero simultaneously. It is shown that with the asymptotically compatibility, +the schemes can provide robust numerical computation under the changes of the nonlocal horizon parameter. +1. Introduction. In this work, we study the numerical discretization of a nonlocal analog of +the classical Lighthill-Richards-Whitham (LWR) model [36,37]. The latter, given by +∂tρ(t, x) + ∂x(ρ(t, x)v(ρ(t, x))) = 0, +(1.1) +for a density ρ = ρ(t, x) and a velocity v = v(ρ(t, x)), has been widely used in the study of traffic +flows. To study the dynamics of traffic flows in the presence of nonlocal inter-vehicle interactions +[5,26], the following nonlocal LWR model has been developed in recent years +∂tρ(t, x) + ∂x(ρ(t, x)vδ(ρ(t, ·), t, x)) = 0, +x ∈ R, t > 0. +(1.2) +In contrast to (1.1), the nonlocal LWR model (1.2) adopts a modeling assumption that in a fleet +of vehicles driving on a highway, each vehicle decides its driving speed not by the local information +but rather through a nonlocal weighted average of traffic information within a road segment of +length δ > 0 ahead of the vehicle’s current location. More specifically, the velocity vδ = vδ(ρ, t, x) +takes on the form +vδ(ρ(t, ·), t, x) = v(qδ(ρ(t, ·), t, x)), +with +qδ(ρ(t, ·), t, x) = +� δ +0 +ρ(t, x + s)wδ(s) ds, +(1.3) +where the integral kernel w = wδ(s) is assumed to be a probability density function defined on the +interval [0, δ]. Alternatively, one may also consider the nonlocal velocity given by [22] +vδ(ρ(t, ·), t, x) = +� δ +0 +v(ρ(t, x + s))wδ(s) ds. +(1.4) +The equation (1.2) is solved with the initial condition: +ρ(0, x) = ρ0(x), +x ∈ R, +(1.5) +∗Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027; +kh2862@columbia.edu +†Department of Applied Physics and Applied Mathematics, and Data Science Institute, Columbia University, +New York, NY 10027; qd2125@columbia.edu +1 +arXiv:2301.00803v1 [math.NA] 2 Jan 2023 + +where ρ0 : R → [0, 1] represents the initial traffic density. The case ρ0 ≡ 0 indicates that the road +is empty and the case ρ0 ≡ 1 corresponds to fully congested traffic. +The equation (1.2) leads to a nonlocal conservation law due to the nonlocal dependence of the +velocity on the density. Consider the rescaled kernel wδ(s) = w(s/δ)/δ such that wδ converges to +a Dirac point mass as δ → 0, it is clear that the nonlocal LWR model (1.2), with either choices +of the velocity given by (1.3) or (1.4), formally recovers the local model (1.1) by taking the limit +δ → 0. For more rigorous analysis of the nonlocal LWR model (1.2), we refer to a number of +existing studies in the literature, including the model well-posedness [5, 8, 14, 22, 26], traveling +wave solutions [38,40], the asymptotic stability of uniform flows [28], and nonlocal-to-local limit as +δ → 0 [7,8,12,14–16,22,32,33]. +The numerical discretization of the nonlocal LWR model (1.2) has also been studied in [5,10, +13,23,24,26]. However, there was no systematic study on the dependence of numerical solutions on +the parameter δ and their behavior under the limit δ → 0. In the present work, we aim to fill this +gap by designing and analysing finite volume numerical schemes for the nonlocal LWR model (1.2) +such that they are able to correctly resolve both the nonlocal model for a given δ > 0 and also the +local model (1.1) when δ → 0. Such schemes are in the spirit of asymptotically compatible schemes, +which can offer robust numerical computation under the changes of δ; see [42, 43] for discussions +on asymptotically compatibility of numerical discretizations of more general nonlocal models. The +main contributions of our work here are the rigorous proofs of the asymptotically compatibility of +the schemes, which include both the uniform convergence of the numerical solutions to the unique +solution of nonlocal continuum model for a given positive horizon parameter and the convergence to +the unique entropy solution of the local model as the mesh size and the nonlocal horizon parameter +go to zero simultaneously. These results are established for the first time in the literature. The main +ingredients of the proofs are the compactness in the BVloc space and the entropy admissibility of +numerical solutions. The analysis provided in [5,26] was based on a priori L∞ and total variation +estimates for a fixed δ > 0, but the resulting total variation bound blows up to infinity as δ → 0. In +this work, a novelty is our use of a different approach to prove that numerical solutions produced by +the proposed schemes satisfy an one-sided Lipschitz condition when δ is close to zero, which enforces +both the boundedness of total variation and the entropy admissibility. Such an approach has been +used to study numerical schemes for the local model (1.1), see [6, 41], but to our best knowledge, +has not been used for nonlocal models. Numerical experiments are also reported to complement the +theoretical investigation. Note that while the current work is motivated by modeling traffic flows +with nonlocal vehicle interactions, let us mention that conservation laws with nonlocal fluxes were +also studied in the modeling of pedestrian traffic [9, 17], sedimentation [4], and material flow on +conveyor belts [27,39]; see [1–3,11,18,21,25,31] for more relevant studies. Thus, our study here can +be useful in the numerical simulations of a broad range of problems in various application domains. +To summarize the paper, in the remainder of this Section 1, after briefly describing the assump- +tions on the nonlocal model and some basic mathematical properties, we introduce the numerical +discretization schemes and summarize the main theorems on their convergence behavior and the +asymptotic compatibility. The detailed proofs of the main theorems are given in Section 2. We +present results of some numerical experiments in Section 3 and offer some conclusions in Section 4. +1.1. A review of well-posedness and nonlocal-to-local limit. Let us first state some +assumptions on the model. +Assumption 1. (i) The nonlocal kernel is given by wδ(s) = w(s/δ)/δ for s ∈ [0, δ], where +w = w(s) is a C1 smooth, strictly decreasing, and nonnegative probability density function defined +2 + +on [0, 1], and it satisfies the normalization condition +� 1 +0 w(s) ds = 1. +(ii) The velocity function is v(ρ) = 1 − ρ. Consequently, (1.3) and (1.4) produce the same outcome. +(iii) The initial data ρ0 ∈ L∞(R) and it satisfies 0 ≤ ρ0(x) ≤ 1 for all x ∈ R. In addition, ρ0 has +bounded total variation. +Concerning the mathematical analysis of the nonlocal LWR model (1.2), we recall that the +existence and uniqueness of weak solutions have been shown with general choices of the nonlocal +kernel, the velocity function, and the initial data, see for example, [5, 8, 14, 26]. For our case, the +following proposition summarizes the known results in the above works. +Proposition 1. Under Assumption 1, the nonlocal LWR model (1.2) admits a unique weak +solution ρ ∈ L∞([0, +∞) × R) such that +� ∞ +0 +� +R +ρ(t, x)∂tφ(t, x) + ρ(t, x)vδ(ρ(t, ·), t, x))∂xφ(t, x) dxdt + +� +R +ρ0(x)φ(0, x) dx = 0, +(1.6) +for all φ ∈ C1 +c([0, +∞) × R), where vδ(ρ(t, ·), t, x) is given by (1.3). Moreover, the solution satisfies +the maximum principle +inf +x∈R ρ0(x) ≤ ρ(t, x) ≤ sup +x∈R +ρ0(x), +(t, x) ∈ [0, +∞) × R. +(1.7) +The convergence of solutions of the nonlocal LWR model (1.2) as δ → 0 has also been extensively +studied. In the literature, it was usually assumed that the nonlocal kernel w = wδ(s) is defined for +s ∈ [0, +∞) and the nonlocal density is defined by +qδ(ρ(t, ·), t, x) = +� ∞ +0 +ρ(t, x + s)wδ(s) ds. +[7,8] considered the exponential kernels wδ(s) = δ−1e− s +δ and showed convergence from the solutions +of the nonlocal model (1.2) to the unique weak entropy solution of the local model (1.1), assuming +that the initial data ρ0 is uniformly positive. [14] generalized the convergence result for a class +of nonlocal kernels with exponential decay rate but under one additional assumption that ρ0 is +one-sided Lipschitz continuous. In [14], the authors also provided counterexamples to show that +the uniform positivity of the initial data is essential to the convergence result. In the subsequent +works [12,15,22,32,33], convergence results concerning the nonlocal quantity qδ(ρ(t, ·), t, x) as δ → 0 +were given without assuming the initial data being uniformly positive. +In the present work, we adopt an approach similar as that in [14] and make the following +additional assumption on the initial data, which basically requires the initial data to be uniformly +positive and to have no negative jumps. +Assumption 2. The initial data ρ0 satisfies +ρ0(x) ≥ ρmin > 0 +∀x ∈ R, +−ρ0(y) − ρ0(x) +y − x +≤ L +∀x ̸= y ∈ R, +(1.8) +for some constants ρmin > 0 and L > 0. +In our case, the same arguments as in [14] can be applied to give the nonlocal-to-local limit re- +sult, as stated in the following Proposition 2, with very little modifications for compactly supported +nonlocal kernels. +3 + +Proposition 2. Suppose Assumptions 1 and 2 are satisfied. As δ → 0, the solution of the +nonlocal LWR model (1.2) converges in L1 +loc([0, +∞) × R) to the weak entropy solution of the local +model (1.1) that satisfies +� ∞ +0 +� +R +ρ(t, x)∂tφ(t, x) + ρ(t, x)v(ρ(t, x))∂xφ(t, x) dxdt + +� +R +ρ0(x)φ(0, x) dx = 0, +(1.9) +for all φ ∈ C1 +c([0, +∞) × R), and +−ρ(t, y) − ρ(t, x) +y − x +≤ 1 +2t +∀x ̸= y ∈ R, t > 0. +(1.10) +In Proposition 2, the inequality (1.10), which is known as the Oleinik’s entropy condition, is +used to select the unique entropy admissible solution of the scalar conservation law (1.1), see [34]. +As a constraint on the one-sided Lipschitz constant of the solution, the entropy condition (1.10) +yields that the solution can only have positive jumps. +1.2. Finite volume approximations. Now let us consider the numerical discretization of the +nonlocal LWR model (1.2). With finite volume approximations, the numerical solution is defined +as a piecewise constant function: +ρ(t, x) = +� +j∈Z +∞ +� +n=0 +ρn +j 1Cj×T n(t, x), +(1.11) +where Cj = (xj−1/2, xj+1/2), T n = (tn, tn+1) are spatial and temporal cells. The grid points are +xj = jh and tn = nτ, where h and τ are spatial and temporal mesh sizes. At the initial time t0 = 0, +the initial data is discretized as: +ρ0 +j = 1 +h +� +Cj +ρ0(x) dx, +j ∈ Z. +(1.12) +Denote F n +j−1/2 and F n +j+1/2 the numerical fluxes across cell boundaries xj−1/2 and xj+1/2 during +time tn to tn+1. Specifying appropriate boundary fluxes, the finite volume scheme is: +ρn+1 +j += ρn +j + λ(F n +j−1/2 − F n +j+1/2), +(1.13) +where the CFL ratio λ = τ/h is taken to be a fixed constant. To specify the numerical fluxes, we +need to evaluate the nonlocal density qδ(ρ(t, ·), t, x) given in (1.3). Let us take +qn +j = +m−1 +� +k=0 +wkρn +j+k, +(1.14) +where m = ⌈ δ +h⌉ is the number of cells involved in the nonlocal integral, and {wk}m−1 +k=0 is a set of +numerical quadrature weights, such that: +wδ,h(s) = +m−1 +� +k=0 +wk1[kh,(k+1)h](s), +s ∈ [0, δ], +(1.15) +is a piecewise constant approximation of the nonlocal kernel wδ = wδ(s). +Given the discretized nonlocal densities {qn +j }n≥0 +j∈Z , the nonlocal fluxes in (1.13) can be con- +structed in a number of different ways. Let us mention the following examples. +4 + +• In [5,26], a Lax-Friedrichs type scheme was developed with the numerical fluxes: +F n +j−1/2 = 1 +2 +� +ρn +j−1v +�m−1 +� +k=0 +wkρn +j+k−1 +� ++ ρn +j v +�m−1 +� +k=0 +wkρn +j+k +�� ++ α +2 (ρn +j−1 − ρn +j ), +(1.16) +where α > 0 is a numerical viscosity constant and the numerical quadrature weights are +given by the left endpoint values: +[Left endpoint] +wk = wδ(kh)h, +k = 0, · · · , m − 1. +(1.17) +• In [24], a Godunov type scheme was proposed with the numerical fluxes defined by: +F n +j−1/2 = ρn +j−1v +�m−1 +� +k=0 +wkρn +j+k +� +, +(1.18) +where the numerical quadrature weights are given by the exact quadrature: +[Exact quadrature] +wk = +� min{(k+1)h,δ} +kh +wδ(s) ds, +k = 0, · · · , m − 1. +(1.19) +• Inspired by both (1.16) and (1.18), we also consider the following Lax-Friedrichs type fluxes: +F n +j−1/2 = 1 +2 +� +ρn +j−1 + ρn +j +� +v +�m−1 +� +k=0 +wkρn +j+k +� ++ α +2 (ρn +j−1 − ρn +j ), +(1.20) +where the numerical quadrature weights are given by either the left endpoint values or the +exact quadrature. +In the present work, we consider a family of finite volume schemes: +ρn+1 +j += H +� +ρn +j−1, ρn +j , ρn +j+1, · · · , ρn +j+m +� += ρn +j + λ(F n +j−1/2 − F n +j+1/2) +(1.21) += ρn +j + λ +� +g(ρn +j−1, ρn +j , qn +j−1, qn +j ) − g(ρn +j , ρn +j+1, qn +j , qn +j+1) +� +, +(1.22) +where qn +j is given by (1.14), λ = τ/h is the CFL ratio, and g = g(ρL, ρR, qL, qR) is a numerical +flux function that depends on both local densities ρL, ρR and nonlocal densities qL, qR. We remark +that, by taking qL = ρL and qR = ρR, g = g(ρL, ρR, ρL, ρR) becomes a numerical flux function for +the local model (1.1), and the respective numerical scheme: +ρn+1 +j += ρn +j + λ +� +g(ρn +j−1, ρn +j , ρn +j−1, ρn +j ) − g(ρn +j , ρn +j+1, ρn +j , ρn +j+1) +� +, +(1.23) +can be viewed as the local counterpart of (1.21)-(1.22). +It is worthwhile to mention that the aforementioned schemes, with numerical fluxes in (1.16), +(1.18) and (1.20) respectively, all belong to the above family (1.21)-(1.22), with the numerical flux +functions given by: +[Lax-Friedrichs] +g(ρL, ρR, qL, qR) = 1 +2(ρLv(qL) + ρRv(qR)) + α +2 (ρL − ρR), +(1.24a) +[Godunov] +g(ρL, ρR, qL, qR) = ρLv(qR), +(1.24b) +[modified Lax-Friedrichs] +g(ρL, ρR, qL, qR) = 1 +2(ρL + ρR)v(qR) + α +2 (ρL − ρR), +(1.24c) +respectively. Now we make the following assumptions on the numerical quadrature weights and the +numerical flux function. +5 + +Assumption 3. The numerical quadrature weights {wk}0≤k≤m−1 satisfy +wδ(kh)h ≥ wk ≥ wδ((k + 1)h)h +and +wk − wk+1 ≥ cm−2, +(1.25) +for some constant c > 0 only depending on the kernel function w = w(s). Moreover, {wk}0≤k≤m−1 +satisfy the normalization condition: +m−1 +� +k=0 +wk = 1. +(1.26) +Assumption 4. (i) The numerical flux function g is a quadratic function. (ii) When ρL = ρR +and qL = qR, g(ρL, ρL, qL, qL) = ρL(1 − qL). (iii) Denote γij, 1 ≤ i, j ≤ 4 the second order partial +derivatives of g, they satisfy +γ11 = γ12 = γ22 = 0, +γ33 = γ34 = γ44 = 0, +(1.27) +γ13, γ23, γ14, γ24 ≤ 0, +γ13 + γ23 + γ14 + γ24 = −1. +(1.28) +(iv) Denote θ(i), 1 ≤ i ≤ 4 the first order partial derivatives of g with respect to its four arguments +ρL, ρR, qL, qR. For any 0 ≤ ρL, ρR, qL, qR ≤ 1: +θ(1)(qL, qR) ≥ 0, θ(2)(qL, qR) ≤ 0, θ(3)(ρL, ρR) ≤ 0, θ(4)(ρL, ρR) ≤ 0, +(1.29) +θ(1)(qL, qR) + θ(3)(ρL, ρR) + 2(γ13 + γ23) ≥ 0, θ(2)(qL, qR) − 2(γ23 + γ24) ≤ 0, +(1.30) +θ(3)(ρL, ρR) + θ(4)(ρL, ρR) ≤ − min{ρL, ρR}. +(1.31) +1.3. Main results. This section summarizes the main results. We note that all the theorems +are subject to Assumptions 1-4. To clarify the notation, we denote: +• ρδ: the continuum solution of the nonlocal LWR model (1.2); +• ρ0: the continuum solution of the local LWR model (1.1); +• ρδ,h: the numerical solution of the nonlocal LWR model; and +• ρ0,h: the numerical solution of the local LWR model. +There are two sets of parameters: the nonlocal horizon parameter δ and the mesh size parameter h. +In the present work, we are interested in establishing relations between those solutions when δ → 0 +and h → 0 along various limiting paths, as shown in Fig. 1.1. +1. The numerical convergence for the nonlocal model: ρδ,h → ρδ when h → 0 with fixed δ > 0 +can be proved following the approach in [26]. The proof is based on a priori L∞ and total +variation estimates of the numerical solution. In Theorem 3, we provide a stronger result +stating uniform numerical convergence with respect to δ. +2. The numerical convergence for the local model: ρ0,h → ρ0 when h → 0 is a classical result, +see for example [35]. +3. The nonlocal-to-local limit: ρδ → ρ0 when δ → 0 is given in Proposition 2. +4. The nonlocal-to-local limit of numerical discretizations: ρδ,h → ρ0,h as δ → 0 with fixed h +follows from Proposition 4. We also provide a uniform convergence result in Theorem 4. +To complete the convergence diagram in Fig. 1.1, one would ask whether ρδ,h → ρ0 when both +δ → 0 and h → 0 simultaneously. If that is the case, we say that the numerical scheme (1.21)-(1.22) +is asymptotically compatible [42,43] with its local limit. +Our key contribution is to prove the asymptotically compatibility of the proposed scheme +(1.21)-(1.22), which is given in Theorem 2. The proof is base on the a priori L∞ and total variation +estimates given in Theorem 1, which are uniform to the nonlocal horizon parameter δ. +6 + +ρδ,h +δ → 0 +Proposition 4, Theorem 4 +� +δ → 0, h → 0 +Theorem 2 +� +h → 0 +Theorem 3 +� +ρ0,h +h → 0 +� +ρδ +Proposition 2 +δ → 0 +� ρ0 +Fig. 1.1: Diagram of various limiting paths +Theorem 1. Suppose Assumptions 1-4 are satisfied, and that the spatial mesh size h > 0, the +CFL ratio λ satisfies λ �4 +i=1 +��θ(i)�� +∞ < 1 where θ(i), 1 ≤ i ≤ 4 are as in Assumption 4, and +0 < δ ≤ δ0 .= +cρmin +2Lw(0), +(1.32) +where the constant c is as in (1.25) and the constants ρmin, L are as in (1.8). Then the numerical +solution ρδ,h produced by the scheme (1.21)-(1.22) satisfies the maximum principle +inf +x∈R ρ0(x) ≤ ρδ,h(t, x) ≤ sup +x∈R +ρ0(x), +(t, x) ∈ [0, +∞) × R. +(1.33) +Moreover, the total variation of the numerical solution in space TV(ρδ,h(t, ·)) is a non-increasing +function of t ∈ [0, +∞), and +TV(ρδ,h; [0, T] × R) ≤ T · TV(ρ0) +∀T > 0. +(1.34) +Theorem 2. Suppose Assumptions 1-4 are satisfied, and that the CFL ratio λ satisfies +λ �4 +i=1 +��θ(i)�� +∞ < 1 where θ(i), 1 ≤ i ≤ 4 are as in Assumption 4. When δ → 0 and h → 0 +simultaneously, the numerical solution ρδ,h produced by the scheme (1.21)-(1.22) converges in +L1 +loc([0, +∞) × R) to the weak entropy solution ρ0 of the local model (1.1) as defined in Propo- +sition 2. +Based on the asymptotically compatibility of the scheme, we can show numerical convergence +from ρδ,h to ρδ uniformly in δ, which guarantees robustness of numerical computation when using +the scheme (1.21)-(1.22) under changes to δ. Moreover, we can also give uniform convergence from +ρδ,h to ρ0,h with respect to the mesh size h. Such a property is referred to as asymptotic preserving +in the literature [20,29,30]. +Theorem 3. Suppose Assumptions 1-4 are satisfied, and that the CFL ratio λ satisfies +λ �4 +i=1 +��θ(i)�� +∞ < 1 where θ(i), 1 ≤ i ≤ 4 are as in Assumption 4, and δ satisfies the condition +(1.32). When h → 0, the numerical solution ρδ,h produced by the scheme (1.21)-(1.22) converges in +L1 +loc([0, +∞) × R) to the weak solution ρδ of the nonlocal model (1.2) as defined in Proposition 1. +Moreover, the convergence is uniform with respect to δ ∈ (0, δ0] where δ0 is as in (1.32): +lim +h→0 sup +δ∈(0,δ0] +��ρδ,h − ρδ�� +L1(U) = 0 +for any bounded U ⊂ [0, +∞) × R. +(1.35) +7 + +Let us make some remarks on the convergence rates in the above Theorem 2 and Theorem 3. +On one hand, the scheme (1.21)-(1.22) is expected to be at most first order accurate because it is +based on a piecewise constant approximation. On the other hand, for scalar conservation laws, it is +known that a first order monotone scheme may have a O(h1/2) convergence rate for discontinuous +solutions [35]. In the numerical experiments in Section 3, we test the scheme with both smooth +initial data and discontinuous ones, the results validate the O(h) convergence rate to the local +solution (as in Theorem 2) when δ = mh for a fixed integer m > 0, and the O(h) convergence +rate to the nonlocal solution uniformly in δ (as in Theorem 3). We leave the rigorous analysis of +convergence rates along various limiting paths in the future works. +Finally, we can also obtain the nonlocal-to-local limit of numerical discretizations, in particular, +the following uniform convergence result. +Theorem 4. Suppose Assumptions 1-4 are satisfied, and that the CFL ratio λ satisfies +λ �4 +i=1 +��θ(i)�� +∞ < 1 where θ(i), 1 ≤ i ≤ 4 are as in Assumption 4. +Let ρδ,h be the numerical +solution produced by the scheme (1.21)-(1.22) and ρ0,h be the one produced by (1.23). For any +h0 > 0, we have +lim +δ→0 +sup +h∈(0,h0] +��ρδ,h − ρ0,h�� +L1(U) = 0 +for any bounded U ⊂ [0, +∞) × R. +(1.36) +1.4. Comments on numerical quadrature weights and numerical flux functions. Let +us make some remarks on the choice of the numerical quadrature weights {wk}0≤k≤m−1. Provided +that the nonlocal kernel wδ = wδ(s) is C1 smooth and decreasing, one can write the numerical +quadrature weights as +wk = w(ξk)h +δ , +ξk ∈ +� +k h +δ , (k + 1)h +δ +� +, +k = 0, · · · , m − 1, +where {ξk}0≤k≤m−1 can be viewed as sampling points of a Riemann sum quadrature on [0, 1]. +The condition (1.25) in Assumption 3 basically requires that the sampling points should not +be too close to each other, and the condition is used to derive the necessary a priori estimates on +numerical solutions as in Theorem 1. To demonstrate the meaning of the constant c and the factor +m−2 in (1.25), let us illustrate with the left endpoint quadrature weights in (1.17). In this case, +wk−1 − wk = h +δ +� +w +� +(k − 1)h +δ +� +− w +� +k h +δ +�� +≥ +� +min +s∈[0,1] −w′(s) +��h +δ +�2 +≥ cm−2, +where the constant c = mins∈[0,1] −w′(s) > 0. +The condition (1.26) in Assumption 3 is the normalization condition for the numerical quadra- +ture weights, which is essential to the consistency between the scheme (1.21)-(1.22) and the local +model (1.1). To demonstrate potential risks when the normalization condition (1.26) is violated, +let us consider the case δ = mh where m is a fixed positive integer. Then the scheme (1.21)-(1.22) +can be viewed as a m + 2-point conservative scheme of the local model (1.1) with the numerical +flux function: +glocal(ρj, · · · , ρj+m) = g +� +ρj, ρj+1, +m−1 +� +k=0 +wkρj+k, +m−1 +� +k=0 +wkρj+k+1 +� +, +8 + +where g is as in Assumption 4. Suppose ρj = · · · = ρj+m = ¯ρ, to make glocal consistent to the local +model (1.1), it is necessary to have +glocal(¯ρ, · · · , ¯ρ) = ¯ρ +� +1 − ¯ρ +m−1 +� +k=0 +wk +� += ¯ρ(1 − ¯ρ), +which requires the normalization condition (1.26). In contrast, if the condition (1.26) is violated +and η .= �m−1 +k=0 wk ̸= 1, the numerical solutions will formally converge to a solution of the equation +∂tρ(t, x) + ∂x(ρ(t, x)(1 − ηρ(t, x))) = 0, +other than the desired equation (1.1) with v(ρ) = 1 − ρ. +This means that the absence of the +normalization condition (1.26) for some numerical quadrature weights may lead to incorrect limit +solutions when δ → 0 and h → 0 simultaneously. Hence, we introduce the following normalized left +endpoint quadrature weights: +[Normalized left endpoint] +wk = +wδ(kh)h +�m−1 +k=0 wδ(kh)h +, +k = 0, · · · , m − 1, +(1.37) +and give the following proposition. +Proposition 3. The normalized left endpoint quadrature weights (1.37) and the exact quadra- +ture weights (1.19) both satisfy the Assumption 3, with the constant c in the condition (1.25) given +by c = +1 +1+w(0) mins∈[0,1] −w′(s) and c = mins∈[0,1] −w′(s), respectively. The left endpoint quadrature +weights satisfy the condition (1.25) with the constant c = mins∈[0,1] −w′(s) but they do not satisfy +the normalization condition (1.26). +A comparison between the different choices of numerical quadrature weights is made through nu- +merical experiments in Section 3. +Concerning the Assumption 4 on the numerical flux function g, with the velocity function +v(ρ) = 1 − ρ, the flux function in the continuum model (1.2) is ρ(1 − q), which is a quadratic +polynomial of (ρ, q) with the only quadratic term being −ρq. It is then reasonable to assume that +the numerical flux function g is quadratic with its second order derivatives satisfying the condition +(iii). The condition (ii) guarantees the consistency of the scheme (1.21)-(1.22) to the model (1.2). +The condition (iv) is used to show that the scheme is monotone under all Assumptions 1-4, see +the Theorem 1. It is natural to ask if the results in this work can be extended to more general +numerical flux functions, e.g., g is not quadratic. We leave the study of such an extension to future +works. +Let us mention that the numerical flux functions given in (1.24a)-(1.24c) all satisfy Assump- +tion 4. For the two Lax-Friedrichs type numerical flux functions (1.24a) and (1.24c), the numerical +viscosity constant should satisfy α ≥ 2. +We also remark that, in the case of 0 < δ ≤ h, i.e., the nonlocal horizon is within one spatial +mesh cell, it holds that qL = ρL and qR = ρR by Assumption 3. Suppose g satisfies Assumption 4, +the numerical flux function g = g(ρL, ρR, ρL, ρR) for the local model (1.1) is non-decreasing with +respect to ρL and non-increasing with respect to ρR. Therefore, the scheme (1.23) is a monotone +scheme for the local model (1.1). As a consequence, we give the following result. +Proposition 4. Suppose Assumptions 1, 3 and 4 are satisfied, and that the spatial mesh size +h > 0. Let ρδ,h be the numerical solution produced by the scheme (1.21)-(1.22) and ρ0,h be the one +9 + +produced by (1.23). It holds that: +ρδ,h = ρ0,h +when +0 < δ ≤ h. +Moreover, ρ0,h converges in L1 +loc([0, +∞) × R) to the weak entropy solution ρ0 of the local model +(1.1) as defined in Proposition 2. +2. Proof of theorems. This section aims to give the proofs of our main results. First, in +Section 2.1, we show the maximum principle for numerical solutions. Then we present an one-sided +Lipschitz estimate for numerical solutions in Section 2.2, the monotonicity of the numerical scheme +(1.21)-(1.22) and the total variation estimate for numerical solutions follow as corollaries. These two +subsections constitute the proof of Theorem 1. In Section 2.3, we let h → 0 and show convergence +of numerical solutions to the proper nonlocal or local solution, which gives the proofs of Theorem 2 +and Theorem 3. In Section 2.4, we give the proof of Theorem 4 on the nonlocal-to-local limit of +numerical discretizations. +2.1. Maximum principle. In this subsection, we aim to show the maximum principle (1.33) +in Theorem 1. By Assumption 1 and (1.12), the numerical solution at the initial time {ρ0 +j}j∈Z +satisfies 0 ≤ ρ0 +j ≤ 1 for all j ∈ Z. Then the maximum principle (1.33) can be proved by induction +using the following Lemma 4.1. +Lemma 4.1. Suppose all conditions in Theorem 1 are given, and that 0 ≤ ρmin ≤ ρn +j+k ≤ +ρmax ≤ 1 for k = −1, 0, 1, · · · , m. Then we have +ρmin ≤ H(ρn +j−1, ρn +j , ρn +j+1, · · · , ρn +j+m) ≤ ρmax, +(2.1) +where the operator H is as defined in (1.21)-(1.22). +Let us first check the monotonicity of the scheme defined by (1.21)-(1.22). Denote +θn,(i) +j += θ(i)(qn +j−1, qn +j ), i = 1, 2; +θn,(i) +j += θ(i)(ρn +j−1, ρn +j ), i = 3, 4 +for +j ∈ Z, n ≥ 0. +A direct calculation gives: +∂H +∂ρn +j−1 += λ +� +θn,(1) +j ++ w0θn,(3) +j +� +; +(2.2a) +∂H +∂ρn +j += 1 + λ +� +θn,(2) +j +− θn,(1) +j+1 + w1θn,(3) +j ++ w0θn,(4) +j +− w0θn,(3) +j+1 +� +; +(2.2b) +∂H +∂ρn +j+1 += λ +� +w2θn,(3) +j ++ w1θn,(4) +j +− θn,(2) +j+1 − w1θn,(3) +j+1 − w0θn,(4) +j+1 +� +; +(2.2c) +∂H +∂ρn +j+k += λ +� +wk+1θn,(3) +j +− wkθn,(3) +j+1 + wkθn,(4) +j +− wk−1θn,(4) +j+1 +� +, +k = 2, · · · , m; +(2.2d) +where we make the convention that wm = wm+1 = 0. +In (2.2d) that corresponds to the nonlocal dependence of the flux on the solution, it is possible +that θn,(3) +j +< 0, θn,(4) +j +< 0 while θn,(3) +j+1 += θn,(4) +j+1 += 0 at some point j = j0, e.g., if we consider the +Riemann type solution: +ρn +j = 1, j ≤ j0; +ρn +j = 0, j > j0. +10 + +In this case, +∂H +∂ρn +j+k < 0 for k = 2, · · · , m − 1. Therefore, one can not deduce (2.1) by showing +(1.21)-(1.22) is a monotone scheme. Here we prove (2.1) in an alternative way, which was also used +in [24,26]. +Proof of Lemma 4.1. We observe the identity H(ρmin, ρmin, ρmin, · · · , ρmin) = ρmin thus we can +write the term H(ρn +j−1, ρn +j , ρn +j+1, · · · , ρn +j+m) − ρmin as the summation of two parts: +∆H1 =H(ρn +j−1, ρn +j , ρn +j+1, ρn +j+2 · · · , ρn +j+m) − H(ρmin, ρmin, ρn +j+1, ρn +j+2, · · · , ρn +j+m), +∆H2 =H(ρmin, ρmin, ρn +j+1, ρn +j+2, · · · , ρn +j+m) − H(ρmin, ρmin, ρmin, ρmin · · · , ρmin). +By the mean value theorem, +∆H1 = +� +k=−1,0 +∂H +∂ρn +j+k +(˜ρn +j−1, ˜ρn +j , ρn +j+1, ρn +j+2 · · · , ρn +j+m)(ρn +j+k − ρmin), +∆H2 = +� +1≤k≤m +∂H +∂ρn +j+k +(ρmin, ρmin, ˜ρn +j+1, ˜ρn +j+2 · · · , ˜ρn +j+m)(ρn +j+k − ρmin), +where 0 ≤ ρmin ≤ ˜ρn +j+k ≤ ρmax ≤ 1 ∀k = −1, 0, 1, · · · , m. +Let us use (2.2a)-(2.2d) with θn,(i) +j +replaced by ˜θn,(i) +j +that is with respect to ˜ρn +j+k. +By Assumption 4, we have ˜θn,(1) +j ++ ˜θn,(3) +j +≥ 0 giving that the term with respect to k = −1 in +∆H1 is nonnegative. Moreover, the CFL condition λ �4 +i=1 +��θ(i)�� +∞ < 1 gives that the term with +respect to k = 0 in ∆H1 is nonnegative. For ∆H2, we note that +˜θn,(3) +j+1 = ˜θn,(3) +j ++ γ23(˜ρn +j+1 − ρmin) ≤ ˜θn,(3) +j +, +˜θn,(4) +j+1 = ˜θn,(4) +j ++ γ24(˜ρn +j+1 − ρmin) ≤ ˜θn,(4) +j +, +which yields that +wk+1˜θn,(3) +j +− wk˜θn,(3) +j+1 + wk˜θn,(4) +j +− wk−1˜θn,(4) +j+1 ≥ (wk+1 − wk)˜θn,(3) +j ++ (wk − wk−1)˜θn,(4) +j +≥ 0, +for k = 1, · · · , m. Hence +∂H +∂ ˜ρn +j+k ≥ 0 for k = 1, · · · , m. Then we deduce that +H(ρn +j−1, ρn +j , ρn +j+1, · · · , ρn +j+m) − ρmin = ∆H1 + ∆H2 ≥ 0. +Similarly one can show the upper bound estimate H(ρn +j−1, ρn +j , ρn +j+1, · · · , ρn +j+m) − ρmax ≤ 0. +2.2. One-sided Lipschitz estimate. We now derive an one-sided Lipschitz estimate for +numerical solutions as given in Lemma 4.2. Then we can deduce that the scheme (1.21)-(1.22) is +monotone and obtain total variation estimates for numerical solutions as given in Lemma 4.4. The +total variation diminishing property and the estimate (1.34) in Theorem 1 are direct corollaries of +Lemma 4.4. +Lemma 4.2. Suppose all conditions in Theorem 1 are given, and that {ρn +j }n≥0 +j∈Z is the numerical +solution produced by the scheme (1.21)-(1.22). The numerical differences +rn +j = ρn +j+1 − ρn +j , +j ∈ Z, n ≥ 0, +(2.3) +satisfy +rn +j ≥ −Lh, +j ∈ Z, n ≥ 0. +(2.4) +11 + +Proof. It follows from the definition of the scheme (1.21)-(1.22) that +rn+1 +j += rn +j + λ +� +2g(ρn +j , ρn +j+1, qn +j , qn +j+1) − g(ρn +j−1, ρn +j , qn +j−1, qn +j ) − g(ρn +j+1, ρn +j+2, qn +j+1, qn +j+2) +� +. +(2.5) +Noting that g is a quadratic function, we can do Taylor’s expansions to get +g(ρn +j , ρn +j+1, qn +j , qn +j+1) − g(ρn +j−1, ρn +j , qn +j−1, qn +j ) += θn,(1) +j +rn +j−1 + θn,(2) +j +rn +j + θn,(3) +j +(qn +j − qn +j−1) + θn,(4) +j +(qn +j+1 − qn +j ) ++ γ13rn +j−1(qn +j − qn +j−1) + γ14rn +j−1(qn +j+1 − qn +j ) + γ23rn +j (qn +j − qn +j−1) + γ24rn +j (qn +j+1 − qn +j ), +g(ρn +j+1, ρn +j+2, qn +j+1, qn +j+2) − g(ρn +j−1, ρn +j , qn +j−1, qn +j ) += θn,(1) +j +(rn +j−1 + rn +j ) + θn,(2) +j +(rn +j + rn +j+1) + θn,(3) +j +(qn +j+1 − qn +j + qn +j − qn +j−1) ++ θn,(4) +j +(qn +j+2 − qn +j+1 + qn +j+1 − qn +j ) + γ13(rn +j−1 + rn +j )(qn +j+1 − qn +j + qn +j − qn +j−1) ++ γ14(rn +j−1 + rn +j )(qn +j+2 − qn +j+1 + qn +j+1 − qn +j ) + γ23(rn +j + rn +j+1)(qn +j+1 − qn +j + qn +j − qn +j−1) ++ γ24(rn +j + rn +j+1)(qn +j+2 − qn +j+1 + qn +j+1 − qn +j ), +where {θn,(i) +j += θ(i)(qn +j−1, qn +j )}2 +i=1 and {θn,(i) +j += θ(i)(ρn +j−1, ρn +j )}4 +i=3 for j ∈ Z and n ≥ 0. Moreover, +from the definition of qn +j give in (1.14) we obtain: +qn +j+1 − qn +j = +m−1 +� +k=0 +wkrn +j+k. +Therefore (2.5) can be rewritten as +rn+1 +j += λθn,(1) +j +rn +j−1 + +� +1 + λθn,(2) +j +− λθn,(1) +j +� +rn +j − λθn,(2) +j +rn +j+1 ++ λ +� +θn,(3) +j ++ γ13rn +j−1 + (γ23 − γ13)rn +j − γ23rn +j+1 +� m−1 +� +k=0 +wkrn +j+k−1 ++ λ +� +θn,(4) +j +− θn,(3) +j ++ (γ14 − γ13)rn +j−1 + (γ24 − γ13 − γ14 − γ23)rn +j − (γ23 + γ24)rn +j+1 +� m−1 +� +k=0 +wkrn +j+k +− λ +� +θn,(4) +j ++ γ14rn +j−1 + (γ14 + γ24)rn +j + γ24rn +j+1 +� m−1 +� +k=0 +wkrn +j+k+1. +In the above expression, rn+1 +j +is represented as a linear combination of rn +j−1, · · · , rn +j+m. By a direct +calculation, the summation of the coefficients before the terms rn +j−1, · · · , rn +j+m is +S = 1 − 2λ +� +(γ13 + γ14)rn +j + (γ23 + γ24)rn +j+1 +� +, +where the fact γ13 + γ14 + γ23 + γ24 = −1 is used. +Since the summation does not equal one, we split two quadratic terms with respect to rn +j and +rn +j+1, which gives the form +rn+1 +j += +� +−1≤k≤m +cn +j,krn +j+k − 2λ(γ13 + γ14)(rn +j )2 − 2λ(γ23 + γ24)(rn +j+1)2, +(2.6) +12 + +such that � +−1≤k≤m cn +j,k = 1. The coefficients {cn +j,k}−1≤k≤m are given by: +cn +j,−1 = λθn,(1) +j ++ λw0 +� +θn,(3) +j ++ γ13rn +j−1 + (γ23 − γ13)rn +j − γ23rn +j+1 +� +; +(2.7a) +cn +j,0 = 1 + λ +� +θn,(2) +j +− θn,(1) +j +� ++ λpn +j,0 + 2λ(γ13 + γ14)rn +j ; +(2.7b) +cn +j,1 = −λθn,(2) +j ++ λpn +j,1 + 2λ(γ23 + γ24)rn +j+1; +(2.7c) +cn +j,k = λpn +j,k, +k = 2, · · · , m; +(2.7d) +where +pn +j,k =wk+1 +� +θn,(3) +j ++ γ13rn +j−1 + (γ23 − γ13)rn +j − γ23rn +j+1 +� +(2.8) ++ wk +� +θn,(4) +j +− θn,(3) +j ++ (γ14 − γ13)rn +j−1 + (γ24 − γ13 − γ14 − γ23)rn +j − (γ23 + γ24)rn +j+1 +� +− wk−1 +� +θn,(4) +j ++ γ14rn +j−1 + (γ14 + γ24)rn +j + γ24rn +j+1 +� +, +and we make the convention that w−1 = wm = wm+1 = 0. +The initial one-sided Lipschitz condition (1.8) gives r0 +j ≥ −Lh for all j ∈ Z. We next show that +if (2.4) holds for any n ≥ 0, then it is also true for n + 1. Then (2.4) follows by induction. +Let us use (2.6)-(2.8). By Assumptions 3-4 and the CFL condition λ �4 +i=1 +��θ(i)�� +∞ < 1, we have +cn +j,k ≥ 0 for k = −1, 0 and −λθn,(2) +j ++ 2λ(γ23 + γ24)rn +j+1 ≥ 0. To show cn +j,k ≥ 0 for all −1 ≤ k ≤ m, +it suffices to show pn +j,k ≥ 0 for all k = 1, · · · , m. By Assumptions 3-4, we have that +wk+1 ≤ wk ≤ wk−1 ≤ w(0)m−1, +wk−1 − wk ≥ cm−2, +wk − wk+1 ≥ cm−2, +and θn,(3) +j ++ θn,(4) +j +≤ −ρmin, where the constant c is as in (1.25) and the constant ρmin is as in (1.8). +Then we deduce that +pn +j,k ≥ − 2w(0)m−1[(γ13 + γ14)rn +j + (γ23 + γ24)rn +j+1] + cm−2(θn,(3) +j ++ θn,(4) +j +) +≥ − 2w(0)m−1Lh + cm−2ρmin ≥ m−2(cρmin − 2w(0)δL) ≥ 0, +provided 0 < δ ≤ +cρmin +2Lw(0). +Now we have that the coefficients {cn +j,k}−1≤k≤m are all nonnegative and the sum of the coeffi- +cients � +−1≤k≤m cn +j,k = 1. Therefore rn+1 +j +is a convex combination of rn +j−1, rn +j , rn +j+1, · · · , rn +j+m plus +the nonnegative quadratic terms −2λ(γ13 + γ14)(rn +j )2 − 2λ(γ23 + γ24)(rn +j+1)2. Hence we have: +inf +j∈Z rn+1 +j +≥ inf +j∈Z rn +j ≥ −Lh, +which completes the proof. +Based on Lemma 4.2, a more careful analysis gives the following sharper estimate corresponding +to the entropy condition (1.10). +Lemma 4.3. Suppose all conditions in Theorem 1 are given, and that 0 < h < h0 with h0 > 0 +only depending on 1 − λ �4 +i=1 +��θ(i)�� +∞ and +cρmin +2Lw(0) − δ. We have: +Ln ≤ +1 +1 +L0 + 2nτ ≤ +1 +2nτ , +n ≥ 1, +(2.9) +13 + +where +Ln ≜ sup +j∈Z +max +� +−rn +j +h , 0 +� +, +n ≥ 0. +(2.10) +Proof. We still start with (2.6). For k ̸= 0, 1, we use the estimate +cn +j,krn +j+k ≥ −cn +j,kLnh. +(2.11) +For k = 0 and k = 1, we consider the following quadratic functions: +b0(rn +j ) .= cn +j,0rn +j − 2λ(γ13 + γ14)(rn +j )2, +b1(rn +j+1) .= cn +j,1rn +j+1 − 2λ(γ23 + γ24)(rn +j+1)2, +respectively. One can verify that +b′ +0(rn +j ) = cn +j,0 − 4λ(γ13 + γ14)rn +j ≥ cn +j,0 + 4λ(γ13 + γ14)Lnh ≥ C0 − 4λLh, +b′ +1(rn +j+1) = cn +j,1 − 4λ(γ23 + γ24)rn +j+1 ≥ cn +j,1 − 4λ(γ23 + γ24)Lnh ≥ C1 − 4λLh, +when rn +j , rn +j+1 ≥ −Lnh, where the constant C0 > 0 only depends on 1 − λ �4 +i=1 +��θ(i)�� +∞ and the +constant C1 > 0 only depends on +cρmin +2Lw(0) − δ. Therefore there exists h0 > 0 only depending on +1 − λ �4 +i=1 +��θ(i)�� +∞ and +cρmin +2Lw(0) − δ such that b′ +0(rn +j ) ≥ 0, b′ +1(rn +j+1) ≥ 0 whenever h < h0. In this +case, we have +b0(rn +j ) ≥ −cn +j,0Lnh − 2λ(γ13 + γ14)(Lnh)2, +b1(rn +j+1) ≥ −cn +j,1Lnh − 2λ(γ23 + γ24)(Lnh)2. +(2.12) +Summing up (2.11) for k ̸= 0, 1 and (2.12) for k = 0, 1, and noting that γ13 + γ14 + γ23 + γ24 = −1, +we obtain: +rn+1 +j +≥ − +� +� +� +−1≤k≤m +cn +j,k +� +�Lnh + 2λ(Lnh)2 = −Lnh + 2(Ln)2hτ, +which yields Ln+1 ≤ Ln − 2(Ln)2τ. Then (2.9) follows by induction. +Now let us go back to check the monotonicity of the scheme (1.21)-(1.22). With the derived +one-sided Lipschitz estimate (2.4), a calculation similar to that in the proof of Lemma 4.2 gives: +∂H +∂ρn +j+k += λ +� +wk+1θn,(3) +j +− wkθn,(3) +j+1 + wkθn,(4) +j +− wk−1θn,(4) +j+1 +� +≥ λm−2(cρmin − 2w(0)δL) ≥ 0, +for k = 1, · · · , m. In this case, the scheme (1.21)-(1.22) is monotone with respect to each of its +arguments. As a direct corollary, it is total variation diminishing (TVD). So we have the following +lemma. +Lemma 4.4. Under the same conditions as in Lemma 4.2, the numerical solution {ρn +j }n≥0 +j∈Z +produced by the scheme (1.21)-(1.22) satisfies: +� +j∈Z +|rn +j | ≤ +� +j∈Z +|r0 +j| ≤ TV(ρ0), +n ≥ 0; +(2.13) +� +j∈Z +|ρn+1 +j +− ρn +j | ≤λ∥∇g∥∞ +� +j∈Z +|rn +j | ≤ TV(ρ0), +n ≥ 0. +(2.14) +The proof of the Lemma is similar to that given in [35] for monotone schemes of scalar conser- +vation laws. The total variation estimate (1.34) follows immediately from the above lemma. +14 + +2.3. Convergence. In this subsection, we are going to give the proofs of Theorem 2 and +Theorem 3. We recall that the numerical solution is defined as: +ρδ,h(t, x) = +� +j∈Z +∞ +� +n=0 +ρn +j 1Cj×T n(t, x), +(2.15) +where Cj = (xj−1/2, xj+1/2) ∀j ∈ Z and T n = (tn, tn+1) ∀n ≥ 0. +Proof of Theorem 2. Let us consider the family of numerical solutions {ρδ,h}0<δ≤δ0,0 0 is a constant only depending on φ. Then we can pass the limit +hτ +� +n≥0 +� +j∈Z +φn +j+1 − φn +j +h +ρn +j v(ρn +j ) → +� ∞ +0 +� +R +ρ∗(t, x)v(ρ∗(t, x))∂xφ(t, x) dxdt +as h → 0 by using the L1 +loc convergence from ρδ,h to ρ∗, and the a priori L∞ bound as given in +(1.33), and deduce that ρ∗ satisfies (1.9). +For the entropy condition, let us consider numerical solutions ˜ρδ,h that are constructed by +linear interpolation rather than the piecewise constant reconstruction as defined in (2.15). Then by +Lemma 4.3, ˜ρδ,h satisfies the one-sided Lipschitz estimate: +− ˜ρδ,h(t, y) − ˜ρδ,h(t, x) +y − x +≤ 1 +2t +∀x ̸= y ∈ R, t > 0. +(2.20) +Noting that ˜ρδ,h converges to the same limit function ρ∗ pointwise, we can show that ρ∗ satisfies +the Oleinik’s entropy condition (1.10) by passing the limit on (2.20). +To prove Theorem 3, we first prove the following lemma. +Lemma 4.5. Suppose Assumptions 1-4 are satisfied, and that the CFL ratio λ satisfies +λ �4 +i=1 +��θ(i)�� +∞ < 1 where θ(i), 1 ≤ i ≤ 4 are as in Assumption 4 and δ satisfies the condition +(1.32). When δ → δ∗ > 0 and h → 0, the numerical solution ρδ,h produced by the scheme (1.21)- +(1.22) converges in L1 +loc([0, +∞) × R) to the weak solution ρδ∗ of the nonlocal LWR model (1.2) as +defined in Proposition 1. +Proof. Similarly as in the proof of Theorem 2, when taking the limit δ → δ∗ and h → 0, there +exists a sequence {ρδl,hl} converging to a limit function ρ∗∗ in the L1 +loc norm with δl → δ∗, hl → 0. +Noting that Proposition 1 already gives the solution uniqueness, we only need to show that the +limit function ρ∗∗ satisfies the weak form (1.6). +With similar calculations to those in the proof of Theorem 2, we have (2.16)-(2.18) for ρ∗∗. But +here we only use (2.19) and the convergence: +� +j∈Z +∞ +� +n=0 +qn +j 1Cj×T n(t, x) → +� δ +0 +ρδ,h(t, x + s)wδ(s) ds, +(2.21) +in the L1 +loc norm. The proof of (2.21) is similar to that given in [5], we omit the details here. Then +we have +hτ +� +n≥0 +� +j∈Z +φn +j+1 − φn +j +h +g(ρn +j , ρn +j+1, qn +j , qn +j+1) +→ +� ∞ +0 +� +R +ρ∗∗(t, x)v +�� δ +0 +ρ∗∗(t, x + s)wδ(s) ds +� +∂xφ(t, x) dxdt, +16 + +which implies that ρ∗∗ satisfies (1.6). +We now give the proof of Theorem 3. +Proof of Theorem 3. For any bounded set U ⊂ [0, +∞) × R, suppose (1.35) is not true, there +exists a sequence of δl and hl where δl ∈ (0, δ0] and hl → 0 as l → ∞, and ε > 0, such that +��ρδl,hl − ρδl�� +L1(U) ≥ ε. +By possibly selecting a subsequence we suppose δl → δ∗ ∈ [0, δ0]. If δl → 0, both ρδl,hl and ρδl +converge to ρ0; If δl → δ∗ > 0, by Lemma 4.5, ρδl,hl → ρδ∗, and by applying the same arguments +on continuum solutions, it holds that ρδl → ρδ∗. In either case there is a contradiction. +2.4. Local limit of numerical discretizations. We now present the proof of Theorem 4. +Proof of Theorem 4. For any bounded set U ⊂ [0, +∞) × R, suppose (1.36) is not true, there +exists a sequence of δl and hl where hl ∈ (0, h0] and δl → 0 as l → ∞, and ε > 0, such that +��ρδl,hl − ρ0,hl�� +L1(U) ≥ ε. +By possibly selecting a subsequence we suppose hl → h∗ ∈ [0, h0]. If hl → 0, both ρδl,hl and ρ0,hl +converge to ρ0; If hl → h∗ > 0, by Proposition 4 it holds that ρδl,hl = ρ0,hl when l is large enough. +In either case there is a contradiction. +3. Numerical experiments. In this section, we test the presented numerical scheme (1.21)- +(1.22) in several numerical experiments to demonstrate the established results. In the implemen- +tation of the scheme (1.21)-(1.22), the numerical flux function g is chosen from the ones given in +(1.24a)-(1.24c), and the numerical quadrature weights {wk}0≤k≤m−1 are chosen from the ones given +in (1.17)-(1.37)-(1.19). We fix the CFL ratio λ = 0.25. For the Lax-Friedrichs type numerical flux +functions (1.24a) and (1.24c), we fix the numerical viscosity constant α = 2. In all but the final +experiments, we use the linear decreasing kernel wδ(s) = +2 +δ2 (δ −s). Assuming δ = mh where m is a +positive integer, the numerical quadrature weights for the linear decreasing kernel computed from +(1.17)-(1.37)-(1.19) are given respectively by +• (Left endpoint) wk = 2(m−k) +m2 +for 0 ≤ k ≤ m − 1, with �m−1 +k=0 wk = 1 + 1 +m; +• (Normalized left endpoint) wk = 2(m−k) +m(m+1) for 0 ≤ k ≤ m − 1, with �m−1 +k=0 wk = 1; +• (Exact quadrature) wk = 2(m−k)−1 +m2 +for 0 ≤ k ≤ m − 1, with �m−1 +k=0 wk = 1. +The velocity function is chosen to be v(ρ) = 1 − ρ. Two sets of initial data ρ0 are used, one is +a bell-shaped curve: +ρ0(x) = 0.4 + 0.4 exp +� +−100(x − 0.5)2� +, +x ∈ R, +(3.1) +while the other represents the Riemann data: +ρ0(x) = +� +ρL, +x < 0.5 +ρR, +x > 0.5, +x ∈ R, +(3.2) +we take ρL = 0.1 and ρR = 0.6 in all the experiments. The numerical solutions are presented on the +spatial domain x ∈ [0, 1] and in the time horizon t ∈ [0, 1] even though the numerical computations +are done on a larger spatial domain with the constant extension on both sides. In the first three +17 + +experiments, we examine the asymptotically compatibility and uniform numerical convergence of +the scheme (1.21)-(1.22) with different numerical quadrature weights. In the last experiment, we +test the scheme with different choices of the nonlocal kernel. +Experiment 1. We first present numerical solutions ρδ,h computed with the Lax-Friedrichs +numerical flux function (1.24a) and different numerical quadrature weights. For each initial data +and each set of numerical quadrature weights, we compute the numerical solution ρδ,h with δ = +0.005, h = 0.001 and plot its snapshots at selected times t = 0, 0.5, 1. Moreover, the snapshot of the +numerical solution ρδ,h at time t = 1 is compared with that of the solution ρ0 of the local model +(1.1). In this experiment, the local solution ρ0 is also computed numerically because the analytical +solution is not always available. The numerical computation is done on a fine grid with h = 0.0002 +using a Lax-Friedrichs scheme for (1.1) with the numerical flux function +glocal(ρL, ρR) = 1 +2(ρLv(ρL) + ρRv(ρR)) + α +2 (ρL − ρR), +(3.3) +that is the local counterpart of (1.24a). The snapshot of the local solution ρ0 at time t = 1 is +plotted with dashed line. See Figure 3.1. +Fig. 3.1: Experiment 1: Snapshots of computed solutions for the bell-shaped initial data (top) and +the Riemann initial data (bottom) corresponding to the left endpoint quadrature weights (left), the +normalized left endpoint quadrature weights (middle), and the exact quadrature weights (right). +For the bell-shaped initial data, we observe from the top row of Figure 3.1 that the numerical +solutions of the nonlocal model preserve the smoothness of the initial data while the local solution +develops a shock. At time t = 1, the numerical solutions of the nonlocal model computed with +the normalized left endpoint quadrature weights and the exact quadrature weights are close to +the local solution, especially in the region away from the shock of the local solution. This means +that the numerical solution ρδ,h with both δ, h small provides a good approximation to the local +solution ρ0, which validates the conclusion of Theorem 2. We also observe from the top left figure +of Figure 3.1 that the numerical solution of the nonlocal model computed with the left endpoint +18 + +t=0.00 +0.8 +t=0.50 +t=1.00 +t=1.00 local +0.7 +density +0.6 +0.5 +0.4 +0.2 +0.4 +0.6 +0.8 +0.0 +1.0 +Xt=0.00 +0.8 +t=0.50 +t=1.00 +t=1.00 local +0.7 +density +0.6 +0.5 +0.4 +0.2 +0.4 +0.6 +0.8 +0.0 +1.0 +Xt=0.00 +0.8 +t=0.50 +t=1.00 +t=1.00 local +0.7 +density +0.6 +0.5 +0.4 +0.2 +0.4 +0.6 +0.8 +0.0 +1.0 +Xt=0.00 +0.6 +t=0.50 +t=1.00 +0.5 +t=1.00 local +density +0.4 +0.3 +0.2 +0.1 +0.4 +0.8 +0.0 +0.2 +0.6 +1.0 +Xt=0.00 +0.6 +t=0.50 +t=1.00 +0.5 +t=1.00 local +density +0.4 +0.3 +0.2 +0.1 +0.2 +0.4 +0.6 +0.8 +0.0 +1.0 +Xt=0.00 +0.6 +t=0.50 +t=1.00 +0.5 +t=1.00 local +density +0.4 +0.3 +0.2 +0.1 +0.2 +0.4 +0.6 +0.8 +0.0 +1.0 +Xquadrature weights is very different from the local solution at time t = 1. Although the numerical +solution of the nonlocal model still approximates a shock profile at time t = 1, the shock position +is incorrect. The comparison between the three sets of numerical quadrature weights emphasizes +the significance of the normalization condition (1.26) for numerical quadrature weights. +For the Riemann initial data, the local solution ρ0 is a traveling wave moving at the constant +speed v = 1 − (ρL + ρR) = 0.3. We observe from the bottom row of Figure 3.1 that the numerical +solutions of the nonlocal model computed with the normalized left endpoint quadrature weights and +the exact quadrature weights are close to the local solution at time t = 1. Meanwhile, in contrast to +the discontinuity of the local solution, the nonlocal solutions get smoothed because of the nonlocal +effects. We also observe from the bottom left figure of Figure 3.1 that the numerical solution of +the nonlocal model computed with the left endpoint quadrature weights is very different from the +local solution at time t = 1. While the former still approximates a Riemann data at time t = 1, +the position of the jump from ρL = 0.1 to ρR = 0.6 is incorrect. The comparison again emphasizes +the significance of the normalization condition (1.26) for numerical quadrature weights. +Experiment 2. We next check the asymptotically compatibility of the scheme (1.21)-(1.22) +by plotting +��ρδ,h − ρ0�� +L1 with δ ∝ h → 0. We take δ = mh where m = 1, 2, 5 and h = 0.01×2−l for +l = 0, 1, 2, 3, and compute numerical solutions ρδ,h using the Lax-Friedrichs numerical flux function +(1.24a) and different numerical quadrature weights. The local solution ρ0 is numerically solved on +a fine grid with h = 0.01 × 2−5 using a Lax-Friedrichs scheme for (1.1) with the numerical flux +function (3.3). For each initial data and each set of numerical quadrature weights, we compute the +L1 error +��ρδ,h − ρ0�� +L1 with an interpolation of ρδ,h onto the fine grid on which ρ0 is computed, +and plot +��ρδ,h − ρ0�� +L1 against h−1 in the log-log scale for δ = h, δ = 2h, and δ = 5h in different +colors. We also plot a dashed line with the slope −1 to represent the linear convergence rate. See +the results in Figure 3.2. +Fig. 3.2: Experiment 2: Convergence from ρδ,h to ρ0 for the bell-shaped initial data (top) and the +Riemann initial data (bottom) corresponding to the left endpoint quadrature weights (left), the +normalized left endpoint quadrature weights (middle), and the exact quadrature weights (right). +19 + +10-1 +Ilod +S=h +=2h +6= 5h +first order +10-3 +102 +10310-1 +=h +=2h +=5h +first order +10-2 +10-3 +102 +10310-1 +=h +=2h +=5h +first order +10-2 +10-3 +102 +10310-1 +10-2 +=h +=2h +=5h +first order +10-3 +102 +10310-1 +=h +=2h +=5h +first order +10-2 +10-3 +102 +103 +h-110-1 +=h +=2h +=5h +first order +10-2 +10-3 +102 +103 +h-1We observe from Figure 3.2 that: for the normalized left endpoint quadrature weights and the +exact quadrature weights, the error +��ρδ,h − ρ0�� +L1 has a linear decay rate with respect to h for both +initial data and δ = mh for m = 1, 2, 5. This means that ρδ,h converges to ρ0 along the limiting +paths δ = mh → 0 for m = 1, 2, 5, which validates the conclusion of Theorem 2. Moreover, the +numerical results show that the convergence is of first order with the particular choices of the initial +data and the limiting paths. In contrast, for the left endpoint numerical quadrature weights, the +error +��ρδ,h − ρ0�� +L1 stagnates on the scale of 10−1 for both initial data and δ = mh for m = 1, 2, 5. +This is due to the convergence of ρδ,h to an incorrect solution when δ = mh → 0, further highlighting +the importance of asymptotically compatibility via the normalization condition (1.26). +Experiment 3. +We now check the uniform convergence of the scheme (1.21)-(1.22) with +respect to δ by plotting +��ρδ,h − ρδ�� +L1 with h → 0 for different choices of δ. We take δ = 0.01 × 2−l +for l = 0, 1, 2 and h = 0.01 × 2−l for l = 0, 1, 2, 3, and compute numerical solutions ρδ,h using the +Lax-Friedrichs numerical flux function (1.24a) and different numerical quadrature weights. The +reference solution ρδ is numerically solved on a fine grid with h = 0.01 × 2−5 using the same +scheme. For each initial data and each set of numerical quadrature weights, we compute the L1 +error +��ρδ,h − ρδ�� +L1 with an interpolation of ρδ,h onto the fine grid, and plot +��ρδ,h − ρδ�� +L1 with +respect to h−1 in the log-log scale for δ = 0.01, δ = 0.005, and δ = 0.0025 in different colors. A +dashed line with the slope −1 is again provided. See the results in Figure 3.3. +Fig. 3.3: Experiment 3: Convergence from ρδ,h to ρδ for the bell-shaped initial data (top) and the +Riemann initial data (bottom) corresponding to the left endpoint quadrature weights (left), the +normalized left endpoint quadrature weights (middle), and the exact quadrature weights (right). +From Figure 3.3, we see that for the normalized left endpoint quadrature weights and the exact +quadrature weights, the error +��ρδ,h − ρδ�� +L1 has a linear decay rate with respect to h for both +initial data and all choices of δ. Moreover, the plots of +��ρδ,h − ρδ�� +L1 with respect to h−1 have +very little change for δ = 0.01, δ = 0.005, and δ = 0.0025. This means that ρδ,h converges to ρδ +as h → 0 uniformly in δ, which validates the conclusion of Theorem 3. In addition, the numerical +results show that the convergence is of first order with the particular choices of the initial data +20 + +10-1 +10-2 +6=0.01 +§ = 0.005 +6 = 0.0025 +first order +10-3 +102 +1036=0.01 +10-1 +§=0.005 +§ = 0.0025 +first order +10-2 +10-3 +102 +1036=0.01 +10-1 +§=0.005 +§ = 0.0025 +first order +10-2 +10-3 +102 +10310-1 +6=0.01 +§ = 0.005 +6 = 0.0025 +first order +10-3 +102 +1036=0.01 +10-1 +§=0.005 +§ = 0.0025 +first order +10-2 +10-3 +102 +1036=0.01 +10-1 +§=0.005 +§ = 0.0025 +first order +10-2 +10-3 +102 +103and the parameter δ. In contrast, for the left endpoint numerical quadrature weights, the error +��ρδ,h − ρδ�� +L1 stagnates on the scale of 10−1 when h ≥ δ for both initial data and all choices of δ. +This may because ρδ approximates ρ0 well when δ is small while ρδ,h = ρ0,h when h ≥ δ and ρ0,h +is not a consistent numerical approximation to ρ0. We also observe that, in each case, the error +decays when h < δ. However, the error increases when δ decreases from 0.01 to 0.0025 for any fixed +mesh size h. One can infer that the convergence from ρδ,h to ρδ as h → 0 will become slower and +slower as δ → 0, and the uniform convergence cannot hold, which is again showing the importance +of the normalization condition (1.26) for the uniform convergence of the scheme (1.21)-(1.22). +Experiment 4. We finally test the scheme (1.21)-(1.22) with different choices of the nonlocal +kernel. Besides the linear decreasing kernel considered before, we also use the exponential kernel +wδ(s) = +e− s +δ +δ(1−e−1) and the constant kernel wδ(s) = +1 +δ , and adopt the exact quadrature weights +(1.19). We take δ = mh where m = 1, 2, 5 and h = 0.01 × 2−l for l = 0, 1, 2, 3, and compute numer- +ical solutions ρδ,h using the Lax-Friedrichs numerical flux function (1.24a) and different numerical +quadrature weights. The local solution ρ0 is numerically solved on a fine grid with h = 0.01 × 2−5 +using a Lax-Friedrichs scheme for (1.1) with the numerical flux function (3.3). For each initial data +and each nonlocal kernel, we compute the L1 error +��ρδ,h − ρ0�� +L1. A dashed line with the slope −1 +is again provided. See the results in Figure 3.4. +Fig. 3.4: Experiment 4: Convergence from ρδ,h to ρ0 for the bell-shaped initial data (top) and the +Riemann initial data (bottom) corresponding to the linear decreasing kernel (left), the exponential +kernel (middle), and the constant kernel (right). +We observe from Figure 3.4 that: for all the three nonlocal kernels, the error +��ρδ,h − ρ0�� +L1 has +a linear decay rate with respect to h for both initial data and in all cases δ = mh for m = 1, 2, 5. +Moreover, the plots for the three nonlocal kernels have little difference. For the linear decreasing +kernel and the exponential kernel, the convergence result validates the conclusion of Theorem 2. +For the constant kernel, it does not satisfy the condition that w = wδ(s) is strictly decreasing, and +(1.25) does not hold because wk−1 − wk = 0 for all k = 1, · · · , m − 1. In this case, the analysis +used in the proof of Theorem 1 cannot give the necessary estimates on numerical solutions but the +21 + +10-1 +=h +=2h +=5h +first order +10-2 +10-3 +102 +103 +h-110-1 +=h +=2h +=5h +first order +10-2 +10-3 +102 +10310-1 +=h +=2h +θ=5h +first order +10-2 +10-3 +102 +10310-1 +=h +=2h +=5h +first order +10-2 +10-3 +102 +10310-1 +§=h +=2h +=5h +first order +10-2 +10-3 +102 +10310-1 +§=h +=2h +=5h +first order +10-2 +10-3 +102 +103numerical results show that the conclusion of Theorem 2 may still be true. +4. Conclusions and future work. In this work, finite volume numerical schemes (1.21)- +(1.22) are studied for solving the nonlocal LWR model (1.2) with a parameter δ that measures the +range of information exchange. An important observation is that, based on both numerical analysis +and computational experiments, certain numerical quadrature weights that provide consistent ap- +proximations in the case of a given δ > 0 may lead to consistency between the scheme (1.21)-(1.22) +and the local limit (1.1) of the nonlocal model (1.2) as δ → 0 and h → 0. For properly selected +numerical quadrature weights, we are able to prove, under reasonable assumptions that the numer- +ical solutions of the nonlocal model converge to the continuum solution of the nonlocal model with +a fixed δ > 0 as h → 0, while they converge to the entropy solution of the local continuum model +(1.1) as δ → 0 and h → 0 simultaneously. That is, such schemes are asymptotically compatible +with its local limit. We are able to demonstrate that these asymptotically compatible schemes can +offer robust numerical simulations under the changes in δ due to the uniform convergence when the +values of δ are within a proper range. +Our established results are based on the a priori estimates on the numerical solutions as given in +Theorem 1, subject to assumptions alluded to above. As shown in the computational experiments, +the normalization condition for numerical quadrature weights is essential to the asymptotically +compatibility of the scheme (1.21)-(1.22). The experiments also suggest that the results of this work +may be extended to the cases with more general nonlocal kernels and numerical flux functions. It +might also be possible to establish the results with more general velocity functions v = v(ρ) other +than the linear one v(ρ) = 1−ρ used here and also more general initial data that may have negative +jumps. 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LeFloch, Hyperbolic Systems of Conservation Laws: The theory of classical and nonclassical shock +waves, Springer Science & Business Media, 2002. +[35] R. J. LeVeque et al., Finite volume methods for hyperbolic problems, vol. 31, Cambridge university press, +2002. +[36] M. J. Lighthill and G. B. Whitham, On kinematic waves II. A theory of traffic flow on long crowded roads, +Proc. R. Soc. Lond. A, 229 (1955), pp. 317–345. +[37] P. I. Richards, Shock waves on the highway, Operations research, 4 (1956), pp. 42–51. +[38] J. Ridder and W. Shen, Traveling waves for nonlocal models of traffic flow, Discrete & Continuous Dynamical +Systems-A, 39 (2019), p. 4001. +23 + +[39] E. Rossi, J. Weißen, P. Goatin, and S. G¨ottlich, Well-posedness of a non-local model for material flow +on conveyor belts, ESAIM: Mathematical Modelling and Numerical Analysis, 54 (2020), pp. 679–704. +[40] W. Shen, Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads, Networks +and Heterogeneous Media, 14 (2019), pp. 709–732. +[41] E. Tadmor, The large-time behavior of the scalar, genuinely nonlinear lax-friedrichs scheme, Mathematics of +computation, 43 (1984), pp. 353–368. +[42] X. Tian and Q. Du, Asymptotically compatible schemes and applications to robust discretization of nonlocal +models, SIAM Journal on Numerical Analysis, 52 (2014), pp. 1641–1665. +[43] +, Asymptotically compatible schemes for robust discretization of parametrized problems with applications +to nonlocal models, SIAM Review, 62 (2020), pp. 199–227. +24 + diff --git a/GdAyT4oBgHgl3EQf5Prw/content/tmp_files/load_file.txt b/GdAyT4oBgHgl3EQf5Prw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..95c4a82c8ad3ff9f6e0b81ba5700ee91a7a44121 --- /dev/null +++ b/GdAyT4oBgHgl3EQf5Prw/content/tmp_files/load_file.txt @@ -0,0 +1,1200 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf,len=1199 +page_content='ASYMPTOTICALLY COMPATIBILITY OF A CLASS OF NUMERICAL SCHEMES FOR A NONLOCAL TRAFFIC FLOW MODEL KUANG HUANG∗ AND QIANG DU† Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' This paper considers numerical discretization of a nonlocal conservation law modeling vehicular traffic flows involving nonlocal inter-vehicle interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The nonlocal model involves an integral over the range measured by a horizon parameter and it recovers the local Lighthill-Richards-Whitham model as the nonlocal horizon parameter goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Good numerical schemes for simulating these parameterized nonlocal traffic flow models should be robust with respect to the change of the model parameters but this has not been systematically investigated in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' We fill this gap through a careful study of a class of finite volume numerical schemes with suitable discretizations of the nonlocal integral, which include several schemes proposed in the literature and their variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Our main contributions are to demonstrate the asymptotically compatibility of the schemes, which includes both the uniform convergence of the numerical solutions to the unique solution of nonlocal continuum model for a given positive horizon parameter and the convergence to the unique entropy solution of the local model as the mesh size and the nonlocal horizon parameter go to zero simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' It is shown that with the asymptotically compatibility, the schemes can provide robust numerical computation under the changes of the nonlocal horizon parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In this work, we study the numerical discretization of a nonlocal analog of the classical Lighthill-Richards-Whitham (LWR) model [36,37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The latter, given by ∂tρ(t, x) + ∂x(ρ(t, x)v(ρ(t, x))) = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1) for a density ρ = ρ(t, x) and a velocity v = v(ρ(t, x)), has been widely used in the study of traffic flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' To study the dynamics of traffic flows in the presence of nonlocal inter-vehicle interactions [5,26], the following nonlocal LWR model has been developed in recent years ∂tρ(t, x) + ∂x(ρ(t, x)vδ(ρ(t, ·), t, x)) = 0, x ∈ R, t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2) In contrast to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1), the nonlocal LWR model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2) adopts a modeling assumption that in a fleet of vehicles driving on a highway, each vehicle decides its driving speed not by the local information but rather through a nonlocal weighted average of traffic information within a road segment of length δ > 0 ahead of the vehicle’s current location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' More specifically, the velocity vδ = vδ(ρ, t, x) takes on the form vδ(ρ(t, ·), t, x) = v(qδ(ρ(t, ·), t, x)), with qδ(ρ(t, ·), t, x) = � δ 0 ρ(t, x + s)wδ(s) ds, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='3) where the integral kernel w = wδ(s) is assumed to be a probability density function defined on the interval [0, δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Alternatively, one may also consider the nonlocal velocity given by [22] vδ(ρ(t, ·), t, x) = � δ 0 v(ρ(t, x + s))wδ(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='4) The equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2) is solved with the initial condition: ρ(0, x) = ρ0(x), x ∈ R, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='5) ∗Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' kh2862@columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='edu †Department of Applied Physics and Applied Mathematics, and Data Science Institute, Columbia University, New York, NY 10027;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' qd2125@columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='00803v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='NA] 2 Jan 2023 where ρ0 : R → [0, 1] represents the initial traffic density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The case ρ0 ≡ 0 indicates that the road is empty and the case ρ0 ≡ 1 corresponds to fully congested traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2) leads to a nonlocal conservation law due to the nonlocal dependence of the velocity on the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Consider the rescaled kernel wδ(s) = w(s/δ)/δ such that wδ converges to a Dirac point mass as δ → 0, it is clear that the nonlocal LWR model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2), with either choices of the velocity given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='3) or (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='4), formally recovers the local model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1) by taking the limit δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' For more rigorous analysis of the nonlocal LWR model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2), we refer to a number of existing studies in the literature, including the model well-posedness [5, 8, 14, 22, 26], traveling wave solutions [38,40], the asymptotic stability of uniform flows [28], and nonlocal-to-local limit as δ → 0 [7,8,12,14–16,22,32,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The numerical discretization of the nonlocal LWR model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2) has also been studied in [5,10, 13,23,24,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' However, there was no systematic study on the dependence of numerical solutions on the parameter δ and their behavior under the limit δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In the present work, we aim to fill this gap by designing and analysing finite volume numerical schemes for the nonlocal LWR model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2) such that they are able to correctly resolve both the nonlocal model for a given δ > 0 and also the local model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1) when δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Such schemes are in the spirit of asymptotically compatible schemes, which can offer robust numerical computation under the changes of δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' see [42, 43] for discussions on asymptotically compatibility of numerical discretizations of more general nonlocal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The main contributions of our work here are the rigorous proofs of the asymptotically compatibility of the schemes, which include both the uniform convergence of the numerical solutions to the unique solution of nonlocal continuum model for a given positive horizon parameter and the convergence to the unique entropy solution of the local model as the mesh size and the nonlocal horizon parameter go to zero simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' These results are established for the first time in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The main ingredients of the proofs are the compactness in the BVloc space and the entropy admissibility of numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The analysis provided in [5,26] was based on a priori L∞ and total variation estimates for a fixed δ > 0, but the resulting total variation bound blows up to infinity as δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In this work, a novelty is our use of a different approach to prove that numerical solutions produced by the proposed schemes satisfy an one-sided Lipschitz condition when δ is close to zero, which enforces both the boundedness of total variation and the entropy admissibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Such an approach has been used to study numerical schemes for the local model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1), see [6, 41], but to our best knowledge, has not been used for nonlocal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Numerical experiments are also reported to complement the theoretical investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Note that while the current work is motivated by modeling traffic flows with nonlocal vehicle interactions, let us mention that conservation laws with nonlocal fluxes were also studied in the modeling of pedestrian traffic [9, 17], sedimentation [4], and material flow on conveyor belts [27,39];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' see [1–3,11,18,21,25,31] for more relevant studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Thus, our study here can be useful in the numerical simulations of a broad range of problems in various application domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' To summarize the paper, in the remainder of this Section 1, after briefly describing the assump- tions on the nonlocal model and some basic mathematical properties, we introduce the numerical discretization schemes and summarize the main theorems on their convergence behavior and the asymptotic compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The detailed proofs of the main theorems are given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' We present results of some numerical experiments in Section 3 and offer some conclusions in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' A review of well-posedness and nonlocal-to-local limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Let us first state some assumptions on the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (i) The nonlocal kernel is given by wδ(s) = w(s/δ)/δ for s ∈ [0, δ], where w = w(s) is a C1 smooth, strictly decreasing, and nonnegative probability density function defined 2 on [0, 1], and it satisfies the normalization condition � 1 0 w(s) ds = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (ii) The velocity function is v(ρ) = 1 − ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Consequently, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='3) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='4) produce the same outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (iii) The initial data ρ0 ∈ L∞(R) and it satisfies 0 ≤ ρ0(x) ≤ 1 for all x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In addition, ρ0 has bounded total variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Concerning the mathematical analysis of the nonlocal LWR model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2), we recall that the existence and uniqueness of weak solutions have been shown with general choices of the nonlocal kernel, the velocity function, and the initial data, see for example, [5, 8, 14, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' For our case, the following proposition summarizes the known results in the above works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Under Assumption 1, the nonlocal LWR model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2) admits a unique weak solution ρ ∈ L∞([0, +∞) × R) such that � ∞ 0 � R ρ(t, x)∂tφ(t, x) + ρ(t, x)vδ(ρ(t, ·), t, x))∂xφ(t, x) dxdt + � R ρ0(x)φ(0, x) dx = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='6) for all φ ∈ C1 c([0, +∞) × R), where vδ(ρ(t, ·), t, x) is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Moreover, the solution satisfies the maximum principle inf x∈R ρ0(x) ≤ ρ(t, x) ≤ sup x∈R ρ0(x), (t, x) ∈ [0, +∞) × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='7) The convergence of solutions of the nonlocal LWR model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2) as δ → 0 has also been extensively studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In the literature, it was usually assumed that the nonlocal kernel w = wδ(s) is defined for s ∈ [0, +∞) and the nonlocal density is defined by qδ(ρ(t, ·), t, x) = � ∞ 0 ρ(t, x + s)wδ(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' [7,8] considered the exponential kernels wδ(s) = δ−1e− s δ and showed convergence from the solutions of the nonlocal model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2) to the unique weak entropy solution of the local model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1), assuming that the initial data ρ0 is uniformly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' [14] generalized the convergence result for a class of nonlocal kernels with exponential decay rate but under one additional assumption that ρ0 is one-sided Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In [14], the authors also provided counterexamples to show that the uniform positivity of the initial data is essential to the convergence result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In the subsequent works [12,15,22,32,33], convergence results concerning the nonlocal quantity qδ(ρ(t, ·), t, x) as δ → 0 were given without assuming the initial data being uniformly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In the present work, we adopt an approach similar as that in [14] and make the following additional assumption on the initial data, which basically requires the initial data to be uniformly positive and to have no negative jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The initial data ρ0 satisfies ρ0(x) ≥ ρmin > 0 ∀x ∈ R, −ρ0(y) − ρ0(x) y − x ≤ L ∀x ̸= y ∈ R, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='8) for some constants ρmin > 0 and L > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In our case, the same arguments as in [14] can be applied to give the nonlocal-to-local limit re- sult, as stated in the following Proposition 2, with very little modifications for compactly supported nonlocal kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 3 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Suppose Assumptions 1 and 2 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' As δ → 0, the solution of the nonlocal LWR model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2) converges in L1 loc([0, +∞) × R) to the weak entropy solution of the local model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1) that satisfies � ∞ 0 � R ρ(t, x)∂tφ(t, x) + ρ(t, x)v(ρ(t, x))∂xφ(t, x) dxdt + � R ρ0(x)φ(0, x) dx = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='9) for all φ ∈ C1 c([0, +∞) × R), and −ρ(t, y) − ρ(t, x) y − x ≤ 1 2t ∀x ̸= y ∈ R, t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='10) In Proposition 2, the inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='10), which is known as the Oleinik’s entropy condition, is used to select the unique entropy admissible solution of the scalar conservation law (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1), see [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' As a constraint on the one-sided Lipschitz constant of the solution, the entropy condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='10) yields that the solution can only have positive jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Finite volume approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Now let us consider the numerical discretization of the nonlocal LWR model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' With finite volume approximations, the numerical solution is defined as a piecewise constant function: ρ(t, x) = � j∈Z ∞ � n=0 ρn j 1Cj×T n(t, x), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='11) where Cj = (xj−1/2, xj+1/2), T n = (tn, tn+1) are spatial and temporal cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The grid points are xj = jh and tn = nτ, where h and τ are spatial and temporal mesh sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' At the initial time t0 = 0, the initial data is discretized as: ρ0 j = 1 h � Cj ρ0(x) dx, j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='12) Denote F n j−1/2 and F n j+1/2 the numerical fluxes across cell boundaries xj−1/2 and xj+1/2 during time tn to tn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Specifying appropriate boundary fluxes, the finite volume scheme is: ρn+1 j = ρn j + λ(F n j−1/2 − F n j+1/2), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='13) where the CFL ratio λ = τ/h is taken to be a fixed constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' To specify the numerical fluxes, we need to evaluate the nonlocal density qδ(ρ(t, ·), t, x) given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Let us take qn j = m−1 � k=0 wkρn j+k, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='14) where m = ⌈ δ h⌉ is the number of cells involved in the nonlocal integral, and {wk}m−1 k=0 is a set of numerical quadrature weights, such that: wδ,h(s) = m−1 � k=0 wk1[kh,(k+1)h](s), s ∈ [0, δ], (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='15) is a piecewise constant approximation of the nonlocal kernel wδ = wδ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Given the discretized nonlocal densities {qn j }n≥0 j∈Z , the nonlocal fluxes in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='13) can be con- structed in a number of different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Let us mention the following examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 4 In [5,26], a Lax-Friedrichs type scheme was developed with the numerical fluxes: F n j−1/2 = 1 2 � ρn j−1v �m−1 � k=0 wkρn j+k−1 � + ρn j v �m−1 � k=0 wkρn j+k �� + α 2 (ρn j−1 − ρn j ), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='16) where α > 0 is a numerical viscosity constant and the numerical quadrature weights are given by the left endpoint values: [Left endpoint] wk = wδ(kh)h, k = 0, · · · , m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='17) In [24], a Godunov type scheme was proposed with the numerical fluxes defined by: F n j−1/2 = ρn j−1v �m−1 � k=0 wkρn j+k � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='18) where the numerical quadrature weights are given by the exact quadrature: [Exact quadrature] wk = � min{(k+1)h,δ} kh wδ(s) ds, k = 0, · · · , m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='19) Inspired by both (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='16) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='18), we also consider the following Lax-Friedrichs type fluxes: F n j−1/2 = 1 2 � ρn j−1 + ρn j � v �m−1 � k=0 wkρn j+k � + α 2 (ρn j−1 − ρn j ), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='20) where the numerical quadrature weights are given by either the left endpoint values or the exact quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In the present work, we consider a family of finite volume schemes: ρn+1 j = H � ρn j−1, ρn j , ρn j+1, · · · , ρn j+m � = ρn j + λ(F n j−1/2 − F n j+1/2) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21) = ρn j + λ � g(ρn j−1, ρn j , qn j−1, qn j ) − g(ρn j , ρn j+1, qn j , qn j+1) � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22) where qn j is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='14), λ = τ/h is the CFL ratio, and g = g(ρL, ρR, qL, qR) is a numerical flux function that depends on both local densities ρL, ρR and nonlocal densities qL, qR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' We remark that, by taking qL = ρL and qR = ρR, g = g(ρL, ρR, ρL, ρR) becomes a numerical flux function for the local model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1), and the respective numerical scheme: ρn+1 j = ρn j + λ � g(ρn j−1, ρn j , ρn j−1, ρn j ) − g(ρn j , ρn j+1, ρn j , ρn j+1) � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='23) can be viewed as the local counterpart of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' It is worthwhile to mention that the aforementioned schemes, with numerical fluxes in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='16), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='18) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='20) respectively, all belong to the above family (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22), with the numerical flux functions given by: [Lax-Friedrichs] g(ρL, ρR, qL, qR) = 1 2(ρLv(qL) + ρRv(qR)) + α 2 (ρL − ρR), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='24a) [Godunov] g(ρL, ρR, qL, qR) = ρLv(qR), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='24b) [modified Lax-Friedrichs] g(ρL, ρR, qL, qR) = 1 2(ρL + ρR)v(qR) + α 2 (ρL − ρR), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='24c) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Now we make the following assumptions on the numerical quadrature weights and the numerical flux function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 5 Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The numerical quadrature weights {wk}0≤k≤m−1 satisfy wδ(kh)h ≥ wk ≥ wδ((k + 1)h)h and wk − wk+1 ≥ cm−2, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='25) for some constant c > 0 only depending on the kernel function w = w(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Moreover, {wk}0≤k≤m−1 satisfy the normalization condition: m−1 � k=0 wk = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='26) Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (i) The numerical flux function g is a quadratic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (ii) When ρL = ρR and qL = qR, g(ρL, ρL, qL, qL) = ρL(1 − qL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (iii) Denote γij, 1 ≤ i, j ≤ 4 the second order partial derivatives of g, they satisfy γ11 = γ12 = γ22 = 0, γ33 = γ34 = γ44 = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='27) γ13, γ23, γ14, γ24 ≤ 0, γ13 + γ23 + γ14 + γ24 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='28) (iv) Denote θ(i), 1 ≤ i ≤ 4 the first order partial derivatives of g with respect to its four arguments ρL, ρR, qL, qR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' For any 0 ≤ ρL, ρR, qL, qR ≤ 1: θ(1)(qL, qR) ≥ 0, θ(2)(qL, qR) ≤ 0, θ(3)(ρL, ρR) ≤ 0, θ(4)(ρL, ρR) ≤ 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='29) θ(1)(qL, qR) + θ(3)(ρL, ρR) + 2(γ13 + γ23) ≥ 0, θ(2)(qL, qR) − 2(γ23 + γ24) ≤ 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='30) θ(3)(ρL, ρR) + θ(4)(ρL, ρR) ≤ − min{ρL, ρR}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='31) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' This section summarizes the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' We note that all the theorems are subject to Assumptions 1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' To clarify the notation, we denote: ρδ: the continuum solution of the nonlocal LWR model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' ρ0: the continuum solution of the local LWR model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' ρδ,h: the numerical solution of the nonlocal LWR model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' and ρ0,h: the numerical solution of the local LWR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' There are two sets of parameters: the nonlocal horizon parameter δ and the mesh size parameter h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In the present work, we are interested in establishing relations between those solutions when δ → 0 and h → 0 along various limiting paths, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The numerical convergence for the nonlocal model: ρδ,h → ρδ when h → 0 with fixed δ > 0 can be proved following the approach in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The proof is based on a priori L∞ and total variation estimates of the numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In Theorem 3, we provide a stronger result stating uniform numerical convergence with respect to δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The numerical convergence for the local model: ρ0,h → ρ0 when h → 0 is a classical result, see for example [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The nonlocal-to-local limit: ρδ → ρ0 when δ → 0 is given in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The nonlocal-to-local limit of numerical discretizations: ρδ,h → ρ0,h as δ → 0 with fixed h follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' We also provide a uniform convergence result in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' To complete the convergence diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1, one would ask whether ρδ,h → ρ0 when both δ → 0 and h → 0 simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' If that is the case, we say that the numerical scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22) is asymptotically compatible [42,43] with its local limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Our key contribution is to prove the asymptotically compatibility of the proposed scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22), which is given in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The proof is base on the a priori L∞ and total variation estimates given in Theorem 1, which are uniform to the nonlocal horizon parameter δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 6 ρδ,h δ → 0 Proposition 4, Theorem 4 � δ → 0, h → 0 Theorem 2 � h → 0 Theorem 3 � ρ0,h h → 0 � ρδ Proposition 2 δ → 0 � ρ0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1: Diagram of various limiting paths Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Suppose Assumptions 1-4 are satisfied, and that the spatial mesh size h > 0, the CFL ratio λ satisfies λ �4 i=1 ��θ(i)�� ∞ < 1 where θ(i), 1 ≤ i ≤ 4 are as in Assumption 4, and 0 < δ ≤ δ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='= cρmin 2Lw(0), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='32) where the constant c is as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='25) and the constants ρmin, L are as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Then the numerical solution ρδ,h produced by the scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22) satisfies the maximum principle inf x∈R ρ0(x) ≤ ρδ,h(t, x) ≤ sup x∈R ρ0(x), (t, x) ∈ [0, +∞) × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='33) Moreover, the total variation of the numerical solution in space TV(ρδ,h(t, ·)) is a non-increasing function of t ∈ [0, +∞), and TV(ρδ,h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' [0, T] × R) ≤ T · TV(ρ0) ∀T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='34) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Suppose Assumptions 1-4 are satisfied, and that the CFL ratio λ satisfies λ �4 i=1 ��θ(i)�� ∞ < 1 where θ(i), 1 ≤ i ≤ 4 are as in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' When δ → 0 and h → 0 simultaneously, the numerical solution ρδ,h produced by the scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22) converges in L1 loc([0, +∞) × R) to the weak entropy solution ρ0 of the local model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1) as defined in Propo- sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Based on the asymptotically compatibility of the scheme, we can show numerical convergence from ρδ,h to ρδ uniformly in δ, which guarantees robustness of numerical computation when using the scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22) under changes to δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Moreover, we can also give uniform convergence from ρδ,h to ρ0,h with respect to the mesh size h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Such a property is referred to as asymptotic preserving in the literature [20,29,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Suppose Assumptions 1-4 are satisfied, and that the CFL ratio λ satisfies λ �4 i=1 ��θ(i)�� ∞ < 1 where θ(i), 1 ≤ i ≤ 4 are as in Assumption 4, and δ satisfies the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' When h → 0, the numerical solution ρδ,h produced by the scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22) converges in L1 loc([0, +∞) × R) to the weak solution ρδ of the nonlocal model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2) as defined in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Moreover, the convergence is uniform with respect to δ ∈ (0, δ0] where δ0 is as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='32): lim h→0 sup δ∈(0,δ0] ��ρδ,h − ρδ�� L1(U) = 0 for any bounded U ⊂ [0, +∞) × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='35) 7 Let us make some remarks on the convergence rates in the above Theorem 2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' On one hand, the scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22) is expected to be at most first order accurate because it is based on a piecewise constant approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' On the other hand, for scalar conservation laws, it is known that a first order monotone scheme may have a O(h1/2) convergence rate for discontinuous solutions [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In the numerical experiments in Section 3, we test the scheme with both smooth initial data and discontinuous ones, the results validate the O(h) convergence rate to the local solution (as in Theorem 2) when δ = mh for a fixed integer m > 0, and the O(h) convergence rate to the nonlocal solution uniformly in δ (as in Theorem 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' We leave the rigorous analysis of convergence rates along various limiting paths in the future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Finally, we can also obtain the nonlocal-to-local limit of numerical discretizations, in particular, the following uniform convergence result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Suppose Assumptions 1-4 are satisfied, and that the CFL ratio λ satisfies λ �4 i=1 ��θ(i)�� ∞ < 1 where θ(i), 1 ≤ i ≤ 4 are as in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Let ρδ,h be the numerical solution produced by the scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22) and ρ0,h be the one produced by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' For any h0 > 0, we have lim δ→0 sup h∈(0,h0] ��ρδ,h − ρ0,h�� L1(U) = 0 for any bounded U ⊂ [0, +∞) × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='36) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Comments on numerical quadrature weights and numerical flux functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Let us make some remarks on the choice of the numerical quadrature weights {wk}0≤k≤m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Provided that the nonlocal kernel wδ = wδ(s) is C1 smooth and decreasing, one can write the numerical quadrature weights as wk = w(ξk)h δ , ξk ∈ � k h δ , (k + 1)h δ � , k = 0, · · · , m − 1, where {ξk}0≤k≤m−1 can be viewed as sampling points of a Riemann sum quadrature on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='25) in Assumption 3 basically requires that the sampling points should not be too close to each other, and the condition is used to derive the necessary a priori estimates on numerical solutions as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' To demonstrate the meaning of the constant c and the factor m−2 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='25), let us illustrate with the left endpoint quadrature weights in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In this case, wk−1 − wk = h δ � w � (k − 1)h δ � − w � k h δ �� ≥ � min s∈[0,1] −w′(s) ��h δ �2 ≥ cm−2, where the constant c = mins∈[0,1] −w′(s) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='26) in Assumption 3 is the normalization condition for the numerical quadra- ture weights, which is essential to the consistency between the scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22) and the local model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' To demonstrate potential risks when the normalization condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='26) is violated, let us consider the case δ = mh where m is a fixed positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Then the scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22) can be viewed as a m + 2-point conservative scheme of the local model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1) with the numerical flux function: glocal(ρj, · · · , ρj+m) = g � ρj, ρj+1, m−1 � k=0 wkρj+k, m−1 � k=0 wkρj+k+1 � , 8 where g is as in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Suppose ρj = · · · = ρj+m = ¯ρ, to make glocal consistent to the local model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1), it is necessary to have glocal(¯ρ, · · · , ¯ρ) = ¯ρ � 1 − ¯ρ m−1 � k=0 wk � = ¯ρ(1 − ¯ρ), which requires the normalization condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In contrast, if the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='26) is violated and η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='= �m−1 k=0 wk ̸= 1, the numerical solutions will formally converge to a solution of the equation ∂tρ(t, x) + ∂x(ρ(t, x)(1 − ηρ(t, x))) = 0, other than the desired equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1) with v(ρ) = 1 − ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' This means that the absence of the normalization condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='26) for some numerical quadrature weights may lead to incorrect limit solutions when δ → 0 and h → 0 simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Hence, we introduce the following normalized left endpoint quadrature weights: [Normalized left endpoint] wk = wδ(kh)h �m−1 k=0 wδ(kh)h , k = 0, · · · , m − 1, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='37) and give the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The normalized left endpoint quadrature weights (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='37) and the exact quadra- ture weights (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='19) both satisfy the Assumption 3, with the constant c in the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='25) given by c = 1 1+w(0) mins∈[0,1] −w′(s) and c = mins∈[0,1] −w′(s), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The left endpoint quadrature weights satisfy the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='25) with the constant c = mins∈[0,1] −w′(s) but they do not satisfy the normalization condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' A comparison between the different choices of numerical quadrature weights is made through nu- merical experiments in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Concerning the Assumption 4 on the numerical flux function g, with the velocity function v(ρ) = 1 − ρ, the flux function in the continuum model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2) is ρ(1 − q), which is a quadratic polynomial of (ρ, q) with the only quadratic term being −ρq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' It is then reasonable to assume that the numerical flux function g is quadratic with its second order derivatives satisfying the condition (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The condition (ii) guarantees the consistency of the scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22) to the model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The condition (iv) is used to show that the scheme is monotone under all Assumptions 1-4, see the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' It is natural to ask if the results in this work can be extended to more general numerical flux functions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=', g is not quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' We leave the study of such an extension to future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Let us mention that the numerical flux functions given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='24a)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='24c) all satisfy Assump- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' For the two Lax-Friedrichs type numerical flux functions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='24a) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='24c), the numerical viscosity constant should satisfy α ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' We also remark that, in the case of 0 < δ ≤ h, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=', the nonlocal horizon is within one spatial mesh cell, it holds that qL = ρL and qR = ρR by Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Suppose g satisfies Assumption 4, the numerical flux function g = g(ρL, ρR, ρL, ρR) for the local model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1) is non-decreasing with respect to ρL and non-increasing with respect to ρR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Therefore, the scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='23) is a monotone scheme for the local model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' As a consequence, we give the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Suppose Assumptions 1, 3 and 4 are satisfied, and that the spatial mesh size h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Let ρδ,h be the numerical solution produced by the scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22) and ρ0,h be the one 9 produced by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' It holds that: ρδ,h = ρ0,h when 0 < δ ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Moreover, ρ0,h converges in L1 loc([0, +∞) × R) to the weak entropy solution ρ0 of the local model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1) as defined in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Proof of theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' This section aims to give the proofs of our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' First, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1, we show the maximum principle for numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Then we present an one-sided Lipschitz estimate for numerical solutions in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2, the monotonicity of the numerical scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22) and the total variation estimate for numerical solutions follow as corollaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' These two subsections constitute the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='3, we let h → 0 and show convergence of numerical solutions to the proper nonlocal or local solution, which gives the proofs of Theorem 2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='4, we give the proof of Theorem 4 on the nonlocal-to-local limit of numerical discretizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Maximum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In this subsection, we aim to show the maximum principle (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='33) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' By Assumption 1 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='12), the numerical solution at the initial time {ρ0 j}j∈Z satisfies 0 ≤ ρ0 j ≤ 1 for all j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Then the maximum principle (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='33) can be proved by induction using the following Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Suppose all conditions in Theorem 1 are given, and that 0 ≤ ρmin ≤ ρn j+k ≤ ρmax ≤ 1 for k = −1, 0, 1, · · · , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Then we have ρmin ≤ H(ρn j−1, ρn j , ρn j+1, · · · , ρn j+m) ≤ ρmax, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1) where the operator H is as defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Let us first check the monotonicity of the scheme defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Denote θn,(i) j = θ(i)(qn j−1, qn j ), i = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' θn,(i) j = θ(i)(ρn j−1, ρn j ), i = 3, 4 for j ∈ Z, n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' A direct calculation gives: ∂H ∂ρn j−1 = λ � θn,(1) j + w0θn,(3) j � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2a) ∂H ∂ρn j = 1 + λ � θn,(2) j − θn,(1) j+1 + w1θn,(3) j + w0θn,(4) j − w0θn,(3) j+1 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2b) ∂H ∂ρn j+1 = λ � w2θn,(3) j + w1θn,(4) j − θn,(2) j+1 − w1θn,(3) j+1 − w0θn,(4) j+1 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2c) ∂H ∂ρn j+k = λ � wk+1θn,(3) j − wkθn,(3) j+1 + wkθn,(4) j − wk−1θn,(4) j+1 � , k = 2, · · · , m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2d) where we make the convention that wm = wm+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2d) that corresponds to the nonlocal dependence of the flux on the solution, it is possible that θn,(3) j < 0, θn,(4) j < 0 while θn,(3) j+1 = θn,(4) j+1 = 0 at some point j = j0, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=', if we consider the Riemann type solution: ρn j = 1, j ≤ j0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' ρn j = 0, j > j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 10 In this case, ∂H ∂ρn j+k < 0 for k = 2, · · · , m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Therefore, one can not deduce (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1) by showing (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22) is a monotone scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Here we prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1) in an alternative way, which was also used in [24,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' We observe the identity H(ρmin, ρmin, ρmin, · · · , ρmin) = ρmin thus we can write the term H(ρn j−1, ρn j , ρn j+1, · · · , ρn j+m) − ρmin as the summation of two parts: ∆H1 =H(ρn j−1, ρn j , ρn j+1, ρn j+2 · · · , ρn j+m) − H(ρmin, ρmin, ρn j+1, ρn j+2, · · · , ρn j+m), ∆H2 =H(ρmin, ρmin, ρn j+1, ρn j+2, · · · , ρn j+m) − H(ρmin, ρmin, ρmin, ρmin · · · , ρmin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' By the mean value theorem, ∆H1 = � k=−1,0 ∂H ∂ρn j+k (˜ρn j−1, ˜ρn j , ρn j+1, ρn j+2 · · · , ρn j+m)(ρn j+k − ρmin), ∆H2 = � 1≤k≤m ∂H ∂ρn j+k (ρmin, ρmin, ˜ρn j+1, ˜ρn j+2 · · · , ˜ρn j+m)(ρn j+k − ρmin), where 0 ≤ ρmin ≤ ˜ρn j+k ≤ ρmax ≤ 1 ∀k = −1, 0, 1, · · · , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Let us use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2a)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2d) with θn,(i) j replaced by ˜θn,(i) j that is with respect to ˜ρn j+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' By Assumption 4, we have ˜θn,(1) j + ˜θn,(3) j ≥ 0 giving that the term with respect to k = −1 in ∆H1 is nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Moreover, the CFL condition λ �4 i=1 ��θ(i)�� ∞ < 1 gives that the term with respect to k = 0 in ∆H1 is nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' For ∆H2, we note that ˜θn,(3) j+1 = ˜θn,(3) j + γ23(˜ρn j+1 − ρmin) ≤ ˜θn,(3) j , ˜θn,(4) j+1 = ˜θn,(4) j + γ24(˜ρn j+1 − ρmin) ≤ ˜θn,(4) j , which yields that wk+1˜θn,(3) j − wk˜θn,(3) j+1 + wk˜θn,(4) j − wk−1˜θn,(4) j+1 ≥ (wk+1 − wk)˜θn,(3) j + (wk − wk−1)˜θn,(4) j ≥ 0, for k = 1, · · · , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Hence ∂H ∂ ˜ρn j+k ≥ 0 for k = 1, · · · , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Then we deduce that H(ρn j−1, ρn j , ρn j+1, · · · , ρn j+m) − ρmin = ∆H1 + ∆H2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Similarly one can show the upper bound estimate H(ρn j−1, ρn j , ρn j+1, · · · , ρn j+m) − ρmax ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' One-sided Lipschitz estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' We now derive an one-sided Lipschitz estimate for numerical solutions as given in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Then we can deduce that the scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22) is monotone and obtain total variation estimates for numerical solutions as given in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The total variation diminishing property and the estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='34) in Theorem 1 are direct corollaries of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Suppose all conditions in Theorem 1 are given, and that {ρn j }n≥0 j∈Z is the numerical solution produced by the scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The numerical differences rn j = ρn j+1 − ρn j , j ∈ Z, n ≥ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='3) satisfy rn j ≥ −Lh, j ∈ Z, n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='4) 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' It follows from the definition of the scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22) that rn+1 j = rn j + λ � 2g(ρn j , ρn j+1, qn j , qn j+1) − g(ρn j−1, ρn j , qn j−1, qn j ) − g(ρn j+1, ρn j+2, qn j+1, qn j+2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='5) Noting that g is a quadratic function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' we can do Taylor’s expansions to get g(ρn j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' ρn j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' qn j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' qn j+1) − g(ρn j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' ρn j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' qn j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' qn j ) = θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='(1) j rn j−1 + θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='(2) j rn j + θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='(3) j (qn j − qn j−1) + θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='(4) j (qn j+1 − qn j ) + γ13rn j−1(qn j − qn j−1) + γ14rn j−1(qn j+1 − qn j ) + γ23rn j (qn j − qn j−1) + γ24rn j (qn j+1 − qn j ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' g(ρn j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' ρn j+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' qn j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' qn j+2) − g(ρn j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' ρn j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' qn j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' qn j ) = θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='(1) j (rn j−1 + rn j ) + θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='(2) j (rn j + rn j+1) + θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='(3) j (qn j+1 − qn j + qn j − qn j−1) + θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='(4) j (qn j+2 − qn j+1 + qn j+1 − qn j ) + γ13(rn j−1 + rn j )(qn j+1 − qn j + qn j − qn j−1) + γ14(rn j−1 + rn j )(qn j+2 − qn j+1 + qn j+1 − qn j ) + γ23(rn j + rn j+1)(qn j+1 − qn j + qn j − qn j−1) + γ24(rn j + rn j+1)(qn j+2 − qn j+1 + qn j+1 − qn j ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' where {θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='(i) j = θ(i)(qn j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' qn j )}2 i=1 and {θn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='(i) j = θ(i)(ρn j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' ρn j )}4 i=3 for j ∈ Z and n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Moreover, from the definition of qn j give in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='14) we obtain: qn j+1 − qn j = m−1 � k=0 wkrn j+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Therefore (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='5) can be rewritten as rn+1 j = λθn,(1) j rn j−1 + � 1 + λθn,(2) j − λθn,(1) j � rn j − λθn,(2) j rn j+1 + λ � θn,(3) j + γ13rn j−1 + (γ23 − γ13)rn j − γ23rn j+1 � m−1 � k=0 wkrn j+k−1 + λ � θn,(4) j − θn,(3) j + (γ14 − γ13)rn j−1 + (γ24 − γ13 − γ14 − γ23)rn j − (γ23 + γ24)rn j+1 � m−1 � k=0 wkrn j+k − λ � θn,(4) j + γ14rn j−1 + (γ14 + γ24)rn j + γ24rn j+1 � m−1 � k=0 wkrn j+k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In the above expression, rn+1 j is represented as a linear combination of rn j−1, · · · , rn j+m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' By a direct calculation, the summation of the coefficients before the terms rn j−1, · · · , rn j+m is S = 1 − 2λ � (γ13 + γ14)rn j + (γ23 + γ24)rn j+1 � , where the fact γ13 + γ14 + γ23 + γ24 = −1 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Since the summation does not equal one, we split two quadratic terms with respect to rn j and rn j+1, which gives the form rn+1 j = � −1≤k≤m cn j,krn j+k − 2λ(γ13 + γ14)(rn j )2 − 2λ(γ23 + γ24)(rn j+1)2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='6) 12 such that � −1≤k≤m cn j,k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The coefficients {cn j,k}−1≤k≤m are given by: cn j,−1 = λθn,(1) j + λw0 � θn,(3) j + γ13rn j−1 + (γ23 − γ13)rn j − γ23rn j+1 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='7a) cn j,0 = 1 + λ � θn,(2) j − θn,(1) j � + λpn j,0 + 2λ(γ13 + γ14)rn j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='7b) cn j,1 = −λθn,(2) j + λpn j,1 + 2λ(γ23 + γ24)rn j+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='7c) cn j,k = λpn j,k, k = 2, · · · , m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='7d) where pn j,k =wk+1 � θn,(3) j + γ13rn j−1 + (γ23 − γ13)rn j − γ23rn j+1 � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='8) + wk � θn,(4) j − θn,(3) j + (γ14 − γ13)rn j−1 + (γ24 − γ13 − γ14 − γ23)rn j − (γ23 + γ24)rn j+1 � − wk−1 � θn,(4) j + γ14rn j−1 + (γ14 + γ24)rn j + γ24rn j+1 � , and we make the convention that w−1 = wm = wm+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The initial one-sided Lipschitz condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='8) gives r0 j ≥ −Lh for all j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' We next show that if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='4) holds for any n ≥ 0, then it is also true for n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='4) follows by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Let us use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='6)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' By Assumptions 3-4 and the CFL condition λ �4 i=1 ��θ(i)�� ∞ < 1, we have cn j,k ≥ 0 for k = −1, 0 and −λθn,(2) j + 2λ(γ23 + γ24)rn j+1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' To show cn j,k ≥ 0 for all −1 ≤ k ≤ m, it suffices to show pn j,k ≥ 0 for all k = 1, · · · , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' By Assumptions 3-4, we have that wk+1 ≤ wk ≤ wk−1 ≤ w(0)m−1, wk−1 − wk ≥ cm−2, wk − wk+1 ≥ cm−2, and θn,(3) j + θn,(4) j ≤ −ρmin, where the constant c is as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='25) and the constant ρmin is as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Then we deduce that pn j,k ≥ − 2w(0)m−1[(γ13 + γ14)rn j + (γ23 + γ24)rn j+1] + cm−2(θn,(3) j + θn,(4) j ) ≥ − 2w(0)m−1Lh + cm−2ρmin ≥ m−2(cρmin − 2w(0)δL) ≥ 0, provided 0 < δ ≤ cρmin 2Lw(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Now we have that the coefficients {cn j,k}−1≤k≤m are all nonnegative and the sum of the coeffi- cients � −1≤k≤m cn j,k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Therefore rn+1 j is a convex combination of rn j−1, rn j , rn j+1, · · · , rn j+m plus the nonnegative quadratic terms −2λ(γ13 + γ14)(rn j )2 − 2λ(γ23 + γ24)(rn j+1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Hence we have: inf j∈Z rn+1 j ≥ inf j∈Z rn j ≥ −Lh, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Based on Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2, a more careful analysis gives the following sharper estimate corresponding to the entropy condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Suppose all conditions in Theorem 1 are given, and that 0 < h < h0 with h0 > 0 only depending on 1 − λ �4 i=1 ��θ(i)�� ∞ and cρmin 2Lw(0) − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' We have: Ln ≤ 1 1 L0 + 2nτ ≤ 1 2nτ , n ≥ 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='9) 13 where Ln ≜ sup j∈Z max � −rn j h , 0 � , n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' We still start with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' For k ̸= 0, 1, we use the estimate cn j,krn j+k ≥ −cn j,kLnh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='11) For k = 0 and k = 1, we consider the following quadratic functions: b0(rn j ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='= cn j,0rn j − 2λ(γ13 + γ14)(rn j )2, b1(rn j+1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='= cn j,1rn j+1 − 2λ(γ23 + γ24)(rn j+1)2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' One can verify that b′ 0(rn j ) = cn j,0 − 4λ(γ13 + γ14)rn j ≥ cn j,0 + 4λ(γ13 + γ14)Lnh ≥ C0 − 4λLh, b′ 1(rn j+1) = cn j,1 − 4λ(γ23 + γ24)rn j+1 ≥ cn j,1 − 4λ(γ23 + γ24)Lnh ≥ C1 − 4λLh, when rn j , rn j+1 ≥ −Lnh, where the constant C0 > 0 only depends on 1 − λ �4 i=1 ��θ(i)�� ∞ and the constant C1 > 0 only depends on cρmin 2Lw(0) − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Therefore there exists h0 > 0 only depending on 1 − λ �4 i=1 ��θ(i)�� ∞ and cρmin 2Lw(0) − δ such that b′ 0(rn j ) ≥ 0, b′ 1(rn j+1) ≥ 0 whenever h < h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In this case, we have b0(rn j ) ≥ −cn j,0Lnh − 2λ(γ13 + γ14)(Lnh)2, b1(rn j+1) ≥ −cn j,1Lnh − 2λ(γ23 + γ24)(Lnh)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='12) Summing up (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='11) for k ̸= 0, 1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='12) for k = 0, 1, and noting that γ13 + γ14 + γ23 + γ24 = −1, we obtain: rn+1 j ≥ − � � � −1≤k≤m cn j,k � �Lnh + 2λ(Lnh)2 = −Lnh + 2(Ln)2hτ, which yields Ln+1 ≤ Ln − 2(Ln)2τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='9) follows by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Now let us go back to check the monotonicity of the scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' With the derived one-sided Lipschitz estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='4), a calculation similar to that in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2 gives: ∂H ∂ρn j+k = λ � wk+1θn,(3) j − wkθn,(3) j+1 + wkθn,(4) j − wk−1θn,(4) j+1 � ≥ λm−2(cρmin − 2w(0)δL) ≥ 0, for k = 1, · · · , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In this case, the scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22) is monotone with respect to each of its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' As a direct corollary, it is total variation diminishing (TVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' So we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Under the same conditions as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='2, the numerical solution {ρn j }n≥0 j∈Z produced by the scheme (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='21)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='22) satisfies: � j∈Z |rn j | ≤ � j∈Z |r0 j| ≤ TV(ρ0), n ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='13) � j∈Z |ρn+1 j − ρn j | ≤λ∥∇g∥∞ � j∈Z |rn j | ≤ TV(ρ0), n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='14) The proof of the Lemma is similar to that given in [35] for monotone schemes of scalar conser- vation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' The total variation estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='34) follows immediately from the above lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' In this subsection, we are going to give the proofs of Theorem 2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' We recall that the numerical solution is defined as: ρδ,h(t, x) = � j∈Z ∞ � n=0 ρn j 1Cj×T n(t, x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content='15) where Cj = (xj−1/2, xj+1/2) ∀j ∈ Z and T n = (tn, tn+1) ∀n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdAyT4oBgHgl3EQf5Prw/content/2301.00803v1.pdf'} +page_content=' Let us consider the family of numerical solutions {ρδ,h}0<δ≤δ0,0 0 and σa +j with a = x, y, z represent Pauli ma- +trices at site j of the honeycomb lattice, which has two +basis sites A and B per unit cell. Each spin has three +nearest-neighbors, and ⟨i, j⟩a sums over nearest-neighbor +pairs connected by an a-bond (see Fig. 1). +The Ki- +taev model is exactly solvable because every honeycomb +plaquette p hosts a flux operator ˆWp = σx +1σy +2σz +3σx +4σy +5σz +6 +(j = 1, . . . , 6 label the sites around the plaquette) that +commutes with the Hamiltonian and with all other Wp′. +The flux operator has eigenvalues wp = ±1 and a pla- +quette is flux-free if wp = +1 and has a flux otherwise. +Kitaev’s solution involves writing the spin operators +σa +j = iba +j cj using four Majorana fermions bx +j , by +j , bz +j, cj, +which satisfy {ba +i , bd +j} = 2δijδad, {ci, cj} = 2δij, and +{ba +i , cj} = 0. +One refers to ba +j as bond fermions and +to cj as matter fermions. The introduction of four Ma- +joranas per site doubles the Hilbert space and leads to +a local Z2 gauge field, which poses the main challenge +when computing correlation functions [26, 45, 46]. The +constraint Dj = bx +j by +j bz +jcj = 1 ∀j restores the physical +Hilbert space. In terms of Majorana fermions, the spin +Hamiltonian takes the form Hˆu = i +2 +� +j,k ˆAjkcjck. Here, +ˆAjk = Jaˆujk if j and k sites are connected by an a-bond +and zero otherwise. The bond operators ˆujk = iba +j ba +k (j +is always an A site) commute with the Hamiltonian and +among themselves. They can thus be replaced by their +eigenvalues ujk = ±1 and a particular bond configura- +tion u ≡ {uij} determines the gauge-independent fluxes +via ˆWp = � +⟨j,k⟩∈∂p ˆujk. +Since the fluxes are static, we can work in a particu- +lar gauge configuration u, where the Hamiltonian Hu is +quadratic in matter fermions cj. Even though the matter +spectrum and eigenstates depend on u, it is convenient to +write the eigenstates using a tensor product notation as +|ψ⟩ = |F⟩⊗|M u⟩ with the flux state |F⟩ ≡ |{Wp}⟩ set by +u and the matter state |M u⟩ consisting of matter excita- +tions on top of the vacuum |M u +0 ⟩. It is useful to introduce +complex bond fermions as χa +jk = 1 +2(ba +j + iba +k) , where j +is an A site and k is connected to j by an a-bond. We +choose the convention ˆujk = 2(χa +jk)†χa +jk−1 where the flux +free state |F0⟩ corresponds to all bond fermions occupied +and a general flux state reads |F⟩ = χan +jnkn · · · χa1 +j1k1 |F0⟩. +Once the gauge field state is determined, one can diago- +nalize the matter part Hu = � +λ ϵλ[2(au +λ)†au +λ−1] in terms +of complex fermion eigenmodes au +λ and write its state as +|M u⟩ = (au +λs)† · · · (au +λ1)† |M u +0 ⟩ with vacuum |M u +0 ⟩ (see +Supplementary Material [47] for details). +Second-order 2DCS response.– The ground state of +Eq. (1) is a spin liquid, which is gapless for Jx + Jy > Jz +(and permutations) and gapped otherwise. In the follow- +ing we focus on the isotropic point Jx = Jy = Jz ≡ J +(see [47] for anisotropic couplings) and compute the +second-order 2DCS response +χabc(τ1, τ2) += i2 +N θ(τ1)θ(τ2)⟨[[M a(τ1 + τ2), M b(τ1)], M c(0)]⟩ . +(2) +Here, N is the total number of unit cells and M a(t) = +� +j σa +j (t) is the a-th component of the total magnetiza- +tion in the Heisenberg picture, and the Heaviside θ func- +tions guarantee the causality of the response. The ex- +pectation value is taken in the many-body ground state. +This response corresponds to the nonlinear part of the +magnetization M a induced by a sequence of two mag- +netic field pulses B(t) = Bc +1δ(t) + Bb +2δ(t − τ1) shown in +Fig. 1 [16, 23]. The second-order susceptibility χabc is +finite only when a, b, c are all different and we consider +χyzx in the following. At the isotropic point, all other +nonzero components can be related by symmetry [47]. +Expansion of the commutator in Eq. (2) shows that χabc +consists of two contributions, +χyzx(τ1, τ2) = − 1 +N θ(τ1)θ(τ2)[R1(τ1, τ2)−R2(τ1, τ2)+c.c.], +where +R1(τ1, τ2) = ⟨M y(τ1 + τ2)M z(τ1)M x(0)⟩, +R2(τ1, τ2) = ⟨M z(τ1)M y(τ1 + τ2)M x(0)⟩. +(3) +We represent the processes in R1 and R2 with the Li- +ouville pathways shown in Fig. 2 [5]. The system starts +in the ground state density matrix |0⟩ ⟨0| with energy +E0, where |0⟩ is constructed with zero flux and mat- +ter fermions. +We note that while this is not a physi- +cal state for our choice of periodic boundary conditions +and geometry, which is required to contain one matter +fermion |0⟩phys = |F0⟩⊗a† +1 |M0⟩ [48], it is well known that +physical and unphysical states yield identical results for + +3 +FIG. 2. Liouville pathways for (a) R1 and (b) R2 processes. +Time evolves from bottom to top, and dots represent bra +or ket operations on the density matrix by the Pauli oper- +ators (summation over sites m, l, k is done in the end). |0⟩ +is the ground state; |P⟩ and |Q⟩ denote excited states of the +Hamiltonian, and the exponentials describe the phases ac- +quired during time evolution over intervals τ1 and τ2. +large enough system size [26, 48]. Using the zero matter +ground state reduces the complexity of the calculations +and facilitates the interpretation of the results. Since the +entire spectrum of the Kitaev Hamiltonian (1) is known, +we can use the Lehmann representation and insert two +resolutions of identity � +P |P⟩ ⟨P| = � +Q |Q⟩ ⟨Q| = 1: +χyzx +R1 = −2 +N Re +� +P Q +� +klm +⟨0|σy +k|Q⟩ ⟨Q|σz +l |P⟩ ⟨P|σx +m|0⟩ +× θ(τ1)θ(τ2)e−iτ1(EP −E0)e−iτ2(EQ−E0) , +(4) +χyzx +R2 = 2 +N Re +� +P Q +� +klm +⟨0|σz +l |Q⟩ ⟨Q|σy +k|P⟩ ⟨P|σx +m|0⟩ +× θ(τ1)θ(τ2)e−iτ1(EP −E0)e−iτ2(EP −EQ) . +(5) +Here, +each +pathway +is +combined +with +its +time- +reversed partner as χyzx +Rn (τ1, τ2) = θ(τ1)θ(τ2)[Rn(τ1, τ2)+ +R∗ +n(τ1, τ2)]/N, and the states |P⟩ and |Q⟩ are eigenstates +of the Hamiltonian (1) with energy EP and EQ. +The states |P⟩ are connected to the flux-free ground +state |0⟩ and the first pulse at t = 0 is polarized in +the x-direction. +Therefore, nonzero matrix elements +only occur if |P⟩ contains a pair of x-fluxes |P⟩ = +χx +mn |F0⟩ ⊗ (au +λs)† · · · (au +λ1)† |M u +0 ⟩. Here, |M u +0 ⟩ is the vac- +uum for u with one x bond flipped, umn = −1. A phase +e−iτ1(EP −E0) is acquired during the time evolution by +τ1. Note that we truncate the matter fermion number +in the intermediate state |P⟩ to one, which is known to +be an excellent approximation [26]. +Next, a pulse po- +larized in the z-direction arrives at time τ1 and creates +a pair of z-fluxes at site l via application of σz +l . Since +after the measurement of σy +k at time τ1 + τ2 the system +must return to either the initial state (for pathway R1) +or a diagonal state |Q⟩ ⟨Q| (for R2), the sites l, k must be +in proximity to site m such that the fluxes overlap and +partially annihilate each other [47]. As a result, the state +|Q⟩ contains a pair of y-fluxes when computing R1 and +a pair of z-fluxes that can be obtained by application of +σy +k to |P⟩ for pathway R2. We also truncate the number +of matter fermions in state |Q⟩ to be maximally one. It +is worth highlighting another difference between the R1 +and R2 processes. For R1 the z-polarized pulse induces +a ket operation, leading to a transition from |P⟩ to state +|Q⟩ and the phase acquired during τ2 is e−iτ2(EQ−E0). In +contrast, for R2 the z-polarized pulse induces a bra op- +eration onto the density matrix and creates a coherence +|P⟩ ⟨Q|. The phase accumulated during time evolution +τ2 is thus e−iτ2(EP −EQ). +Results and Discussion.– We analyze χyzx in frequency +space and label by ω1 and ω2 the frequencies conjugate +to the time intervals τ1 and τ2, respectively. +The re- +sponses are written in terms of a product of matrix el- +ements and the function g(x) = i/(x + iΓ), with the +broadening Γ coming from the scattering of quasiparti- +cles. The second-order response involves the product of +two g-functions g(x1)g(x2) [47]. In the small Γ limit, it +leads to the terms δ(x1)δ(x2)−P 1 +x1 P 1 +x2 in the imaginary +part of the response, because the product of matrix ele- +ments is purely imaginary. The real part of the response +contains terms mixing principle values and delta-like con- +tributions δ(x1)P 1 +x2 +δ(x2)P 1 +x1 . Such mixing is a general +feature in nonlinear response functions [24, 26]. Taking +into account the time-reversal partners, we notice that +the real parts of the 2D spectra are symmetric about the +origin, while the imaginary part is antisymmetric. +We plot the 2D spectrum of Imχyzx(ω1, ω2) in Fig. 3 for +a lattice with 100 × 100 unit cells, N = L2 = 104. Panel +(a) shows the contribution of pathway R1 and panel (b) +the one from pathway R2 (see [47] for the real parts and +the sum of both pathways). Given the symmetry prop- +erties of the response functions, we only show results for +ω1 > 0. Panel (c) shows χyzx +R2 in the frequency window +0.3J < ω1, ω2 < 0.7J on a logarithmic scale. We start +by analyzing the results for pathway R1. The peaks of +the R1 process appear near the diagonal ω1 = ω2. Inves- +tigating the spectrum over a wider frequency range [47] +shows that the largest response occurs in the shown fre- +quency range. +The dashed box indicates the flux gap +Eg = min(EP (Q)) − E0 = 0.263J in the thermodynamic +limit, which is the minimal energy cost of excitations. +Due to the product of g-functions, the peaks occur at +ω1 = EP − E0 and ω2 = EQ − E0. As discussed above, +for a given site m in M x (or M y), the states P (or Q) have +fluxes at honeycomb plaquettes neighboring site m con- +nected by x (or y) bonds. Since we consider the isotropic +case, x and y fluxes cost the same energy and the sig- +nal vanishes inside the region |ω1,2| < Eg. Interestingly, +we find the strongest signal along the diagonal, centered +around energies ω1 = ω2 ≈ 0.5J, even though the joint +density of states in this region is not large [see Fig. 4(b)]. +This implies that the response is due to the matrix ele- +ments being large for these processes. Below, we show + +4 +FIG. 3. 2D spectrum of second-order response χyzx(ω1, ω2) +for 100×100 lattice, N = 104, Ja = 1, Γ = 0.01. (a) Imaginary +part of χyzx +R1 shows peaks along the diagonal ω1 = ω2. Vertical +and horizontal streaks are due to the principal value parts +from g functions. The dashed box indicates the flux gap Eg +below which the response vanishes. (b) Imaginary part of χyzx +R2 +exhibits vertical streaks at energies ω1 = EP − E0. (c) Inset +zooms into low-energy region of panel (b). Below the flux gap +(dashed line), no vertical streaks appear. (d) Absolute value +of χyzx +R1 +on a logarithmic scale in region 0.25J ≤ ω1,2 ≤ J. +Grey dots denote excitation energies of localized states with +high IPR. +that it indeed derives from localized matter Majorana +states that are trapped around plaquettes with nonzero +flux. This is in sharp contrast to results of the third-order +response functions, where a strong diagonal peak arises +from a constructive interference effect [26]. +In Fig. 3(c), we plot the absolute value of χyzx +R1 +on a +logarithmic scale to highlight the presence of off-diagonal +FIG. 4. (a) A representative localized state |P⟩ state trapped +near an x-flux pair (green). The color and size of each site +denote the amplitudes u and v of the Majorana matter wave- +function. (b) Density of states and maximal IPR for states +within a given energy bin. +A few high IPR states (order +∼ 1/N of the total states) at low energies make up for most +of the linear and nonlinear response. +peaks, which are due to transitions between |P⟩ and |Q⟩ +states with different energies. These peaks are only about +a factor of ten smaller than the diagonal ones, which in- +dicates the locality of those states. Otherwise the matrix +element of a local operator ⟨P|σz +l |Q⟩ could not be large +between the orthogonal states |P⟩ and |Q⟩. +We now analyze R2 shown in Fig 3(b), which exhibits +vertical stripes that are centered at energies ω1 = EP − +E0. +The reason is that peaks along the ω2-axis occur +at energy differences EQ − EP and thus densely overlap. +As detailed in the inset panel (c), the signal below the +flux gap Eg stems purely from the principal values as +we observe no vertical tails for ω1 < Eg. The strongest +vertical streaks occur at the same energies ω1 as in R1 +and arise from the large overlap of localized Majorana +states as we show next. +To quantitatively characterize the localization of the +matter Majorana wavefunctions we present their inverse +participation ratio (IPR) in Fig. 4(b). The IPR is of or- +der 1 for localized states and of order 1/L2 for extended +states [47]. The IPR distribution separates into two re- +gions with a few states at low energy having a much +higher IPR. We find that these states are indeed localized + +yzr +Im +3~t3) +(3~-3) +1 +1.0 +(a) +(6) +400 +0.5 +200 +- 0 +-200 +(C) +-0.5 +-400 +0.0 +0.25 +0.5 +-1.0 - +0.0 +0.5 +0.5 +1.0 0.0 +1.0 +w1 +w1 +(d) +log10 +(1,2) +0.7 +3.0 +0.6 + 2.5 +30.5 +2.0 +0.4 +-1.5 +0.3 +1.0 +0.3 +0.4 +0.5 +0.6 +0.7 +w1a +- 0.025 +2 +- 0.020 +0.015 +0.010 +0.005 +(6) +600 +0.002 +IPR +Bin Max IPR +Dos +400 +Dos +0.001 +200 +0.000 +2 +6 +4 +Ep-EGS5 +around non-zero fluxes. Fig. 4(a) shows a representative +example at energy E ≈ 0.5J. The dominance of the low- +energy peaks in Imχyzx for ω1,2 <∼ J and the presence of +the off-diagonal peaks in this region can thus be under- +stood in terms of the large wavefunction overlap matrix +elements ⟨P|σz +k|Q⟩ of high-IPR states. We note that this +also accounts for most of the peak intensity in linear re- +sponse χaa [47], which exclusively probes the diagonal +elements ⟨P|σa +k|P⟩. The second-order response addition- +ally contains information about the off-diagonal matrix +elements ⟨Q|σz|P⟩ with P ̸= Q. By taking the ratio of +second- to first-order response, χ(2)/χ(1), we can extract +the size of this element from experiment. Being only one +order of magnitude smaller in the low-energy region is a +clear indication of the localized nature of the Majorana +wavefunctios at these energies as discussed above [47]. +Finally, we briefly comment on results away from the +isotropic case. +If Jx ̸= Jy the center of the peaks in +Imχyzx +R1 shift away from the diagonal, reflecting the dif- +ferent energy costs of creating x and y fluxes. This can +be used as a sensitive probe of exchange anisotropies. +The sharp features originating from the localized matter +fermions are still present [47]. +Conclusions.– A primary challenge in the experimen- +tal search for spin liquids is to find their unique and +observable signatures, and one promising path is to di- +rectly probe properties of their fractionalized excitations. +The Kitaev spin liquids host two different types of frac- +tionalized excitations, Z2 fluxes and matter Majorana +fermions, which are not clearly separable in linear re- +sponse, where a broad continuum of excitations occurs +above the flux gap. +In contrast, we demonstrate that +they can be disentangled in the second-order nonlinear +susceptibility χabc with non-repeating indices. +In ad- +dition, off-diagonal peaks in the 2D spectrum directly +indicate the presence of localized Majorana matter ex- +citations trapped by fluxes, and the 2DCS peak sizes +are quantitatively related to the IPRs of their wavefunc- +tions. Involving the lowest nonlinear response, our pro- +posal of using crossed-polarization pulses to probe the +off-diagonal second-order susceptibility χyzx is the ex- +perimentally most straightforward way of using 2DCS to +probe fractionalized excitations in Kitaev spin liquids. +We acknowledge valuable discussions with N. Pe- +ter Armitage, Yueqing Chang, Elio Koenig, Milan Ko- +rnjaˇca, Ana-Marija Nedi´c, Natalia Perkins, Nicholas Sir- +ica, Yuriy Sizyuk, and Yuan Wan. +V.L.Q., T.V.T., +and P.P.O. acknowledge support from the Research Cor- +poration for Science Advancement via P.P.O.’s Cottrell +Scholar Award. +Y.Q. was supported by the U.S. De- +partment of Energy, Office of Science, National Quan- +tum Information Science Research Centers, Supercon- +ducting Quantum Materials and Systems Center (SQMS) +under the contract No. DE-AC02-07CH11359. 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Knolle, Dynamics of a Quantum Spin Liquid, Springer +Theses (Springer International Publishing, Cham, 2016). + +7 +SUPPLEMENTAL MATERIAL +This Supplemental Material includes symmetry analysis of the second-order response tensor and the constraints +among different components (Section S1), a derivation of the higher-order response functions (Section S2), and a +detailed computation of the matrix elements (Section S3). By comparing the peak positions and their intensities, we +demonstrate the role played by the novel matrix element (Section S4). We also present results for the anisotropic +case (Section S5). +The definition and analysis of the inverse participation ratio of our problem are explained in +(Section S6). +SYMMETRY ANALYSIS OF THE SECOND-ORDER CORRELATION FUNCTIONS +The crystal symmetries, combined with time reversal, lead to the following relations that hold, in general, for any +choice of the couplings Jx, Jy and Jz, +χx,y,z (ω1, ω2) = −χy,x,z (ω2, ω1) , +χx,z,y (ω1, ω2) = −χz,x,y (ω2, ω1) , +χy,x,z (ω1, ω2) = −χx,y,z (ω2, ω1) , +χyzx (ω1, ω2) = −χz,y,x (ω2, ω1) , +χz,x,y (ω1, ω2) = −χx,z,y (ω2, ω1) , +χz,y,x (ω1, ω2) = −χyzx (ω2, ω1) . +At the isotropic point, the symmetry constraints impose that there is only one independent component of the tensor, +and all other components can be related to that one in the following way: +χx,y,z (ω1, ω2) = −χx,z,y (ω1, ω2) , += −χy,x,z (ω1, ω2) , += χyzx (ω1, ω2) , += χz,x,y (ω1, ω2) , += −χz,y,x (ω1, ω2) . +FOURIER TRANSFORMATION OF THE RESPONSE FUNCTIONS +Here, we perform the necessary Fourier transformation from the time to the frequency domain. +Having these +expressions analytically in frequency avoids the need for Fourier transformation numerically. +In general, the re- +sponse function χabc(τ1, τ2) in Lehmann representation can be separated into contributions based on pathways. The +contribution from pathway 1 is +χabc +R1 (τ1, τ2) = − 1 +N θ(τ1)θ(τ2)[R(1)(τ1, τ2) + R(1)(τ1, τ2)∗] += − 1 +N θ(τ1)θ(τ2)2ℜ +� +P Q +� +klm +⟨0| σa +k |Q⟩ ⟨Q| σb +l |P⟩ ⟨P| σc +m |0⟩ e−iτ1(EP −E0)e−iτ2(EQ−E0). +(6) +while the contribution from pathway 2 is +χabc +R2 (τ1, τ2) = + 1 +N θ(τ1)θ(τ2)[R(2)(τ1, τ2) + R(2)(τ1, τ2)∗] += + 1 +N θ(τ1)θ(τ2)2ℜ +� +P Q +� +klm +⟨0| σb +l |Q⟩ ⟨Q| σa +k |P⟩ ⟨P| σc +m |0⟩ e−iτ1(EP −E0)e−iτ2(EP −EQ), +(7) +where |P⟩, |Q⟩ are eigenstates of the Hamiltonian. + +8 +In frequency space, introducing g(x) = +i +x+iΓ, where Γ takes into account the level broadening, we rewrite +χabc(ω1, ω2) = +� ∞ +−∞ +dτ1 +� ∞ +−∞ +dτ2 eiω1τ1eiω2τ2χabc(τ1, τ2) += − 1 +N +� ∞ +0 +dτ1 +� ∞ +0 +dτ2[R(1)(τ1, τ2) + R(1)(τ1, τ2)∗ − R(2)(τ1, τ2) − R(2)(τ1, τ2)∗]eiω1τ1eiω2τ2 +(8) +Performing the Fourier transformations and calling E0 the ground state energy, we find, for the R1 = R(1) + R(1)∗ +contributions, +χabc +R1 (ω1, ω2) = − 1 +N +� +P Q +� +klm +⟨0| σa +k |Q⟩ ⟨Q| σb +l |P⟩ ⟨P| σc +m |0⟩ +i +ω1 − (EP − E0) + iΓ +i +ω2 − (EQ − E0) + iΓ +− 1 +N +� +P Q +� +klm +(⟨0| σa +k |Q⟩ ⟨Q| σb +l |P⟩ ⟨P| σc +m |0⟩)∗ +i +ω1 + (EP − E0) + iΓ +i +ω2 + (EQ − E0) + iΓ += − 1 +N +� +P Q +� +klm +⟨0| σa +k |Q⟩ ⟨Q| σb +l |P⟩ ⟨P| σc +m |0⟩ g(ω1 − (EP − E0))g(ω2 − (EQ − E0)) +− 1 +N +� +P Q +� +klm +(⟨0| σa +k |Q⟩ ⟨Q| σb +l |P⟩ ⟨P| σc +m |0⟩)∗g(ω1 + (EP − E0))g(ω2 + (EQ − E0)), +(9) +Proceeding along the same lines for R2, +χabc +R2 (ω1, ω2) = + 1 +N +� +P Q +� +klm +⟨0| σb +l |Q⟩ ⟨Q| σa +k |P⟩ ⟨P| σc +m |0⟩ +i +ω1 − (EP − E0) + iΓ +i +ω2 − (EP − EQ) + iΓ ++ 1 +N +� +P Q +� +klm +(⟨0| σb +l |Q⟩ ⟨Q| σa +k |P⟩ ⟨P| σc +m |0⟩)∗ +i +ω1 + (EP − E0) + iΓ +i +ω2 + (EP − EQ) + iΓ += + 1 +N +� +P Q +� +klm +⟨0| σb +l |Q⟩ ⟨Q| σa +k |P⟩ ⟨P| σc +m |0⟩ g(ω1 − (EP − E0))g(ω2 − (EP − EQ)) ++ 1 +N +� +P Q +� +klm +(⟨0| σb +l |Q⟩ ⟨Q| σa +k |P⟩ ⟨P| σc +m |0⟩)∗g(ω1 + (EP − E0))g(ω2 + (EP − EQ)). +(10) +Putting them together, +χabc(ω1, ω2) = χabc +R1 (ω1, ω2) + χabc +R2 (ω1, ω2). +(11) +This is the expression analyzed in the main text. +DETAILS OF THE SOLUTION OF THE HAMILTONIAN AND THE MATRIX ELEMENTS OF THE +SECOND-ORDER CORRELATION FUNCTIONS +In this Section, we give further details on how to solve the Hamiltonian and how to calculate the matrix elements +entering the response functions. +Diagonalization of the Hamiltonian and gauge structure +Following Kitaev’s solution [44], we introduce the complex bond fermions +χa +⟨jk⟩ = 1 +2 +� +ba +j + iba +k +� +, +(χa +⟨jk⟩)† = 1 +2 +� +ba +j − iba +k +� +. +(12) +For simplicity we denote χa +⟨jk⟩ as χjk from now on. Since ujk = 2χ† +jkχjk − 1, the χ fermion describes the occupation +of a bond ⟨j, k⟩ along the direction a. A bond ⟨j, k⟩ is occupied by χ fermion if ujk = +1 or empty if ujk = −1. +Together with the fermionic mapping, we express the spin operators as + +9 +FIG. 5. Total contribution of R1 and R2. L = 100, Ji = 1.0. +FIG. 6. Real and imaginary part of χyzx +R1 and χyzx +R1 with large frequency range. L = 80, Ji = 1.0. We see the signals mostly +appear in the low-frequency region. +σa +i = i +� +χij + χ† +ij +� +cAi +σa +j = +� +χij − χ† +ij +� +cBj +(13) +The action of a spin operator can be viewed as applying a matter Majorana fermion and flipping the value of the +bond ujk variable. The latter action corresponds to introducing two fluxes in plaquettes adjacent to the a-type bond. +The problem becomes to write the eigenstates of the matter fermions moving in a background of fluxes characterized +by the set {u}. +The ground state lies in the flux-free sector for large enough systems with spatial translational +invariance [48]. The conventional choice for the ground state gauge is u = +1. Obviously, all gauge configurations +leading to the same flux sector will be equivalent and gives the same fermion energies. +The singular-value-decomposition (SVD) of the matter Hamiltonian leads to a natural definition of complex matter + +Re xyza(w1, w2) +Im xyza(w1, w2) +1.0 +0.50.751.0 +600 +0.75 +400 +0.5 +- 200 +0 0'0-0-90- 20- 0' +0 +-200 +-400 +-600 +-1.0 -0.75 -0.5 -0.25 0.0 0.25 0.5 0.75 1.0 +-1.0 -0.75 -0.5 -0.25 0.0 0.25 0.5 0.75 1.0 +w1 +w1Im XR2 +(w1,2 +5.0 +300 +200 +2.5- +- 100 +-0 +30.0 +-100 +200 +-2.5 +一 +-300 +-5.0 - +0.0 +2.5 +5.0 0.0 +2.5 +5.0 0.0 +2.5 +5.0 0.0 +2.5 +5.0 +w1 +w1 +w1 +w110 +fermions [48], +H{u} = i +2 +� +cT +A cT +B +� � +0 +M +−M T +0 +� � cA +cB +� += i +2 +� +cT +A cT +B +� � +0 +USV T +−V SU T +0 +� � cA +cB +� += i +2 +� +cT +A cT +B +� � U +0 +0 V +� � 0 +S +−S 0 +� � U 0 +0 +V +�T � cA +cB +� += i +2 +� +(e′)T (e′′)T � � 0 +S +−S 0 +� � e′ +e′′ +� += i +N−1 +� +m=0 +εme′ +me′′ +m = +N−1 +� +m=0 +2εm +� +a† +mam − 1 +2 +� +, +(14) +where em are Majorana modes and am are matter fermion excitations with am = 1 +2(e′ +m + ie′′ +m). The vector cA(B) is +of length N (N ≡ L2 is the number of unit cells). We call the ground state complex matter excitation a† +m, related to +the matter Majoranas c by +cAi = +� +m +(U0)im(a† +m + am), +cBj = +� +m +(iV0)jm(a† +m − am). +(15) +As a consequence of enlarging the Hilbert space, not all possible occupations of a are physically acceptable [44]. As +mentioned in the main text, the Majorana operators act on the extended 4-dimensional Fock space +˜ +M, whereas the +physical Hilbert space M of a spin is a subspace of +˜ +M defined by [44, 46, 48] +|ξ⟩ ∈ M ⇐⇒ Dj|ξ⟩ = |ξ⟩ +∀j, +Dj = bx +j by +j bz +jcj. +(16) +This constraint also ensures the Majorana representation of the spins satisfies the SU(2) algebra. A state |Φ⟩ is +physical if P |Φ⟩ = |Φ⟩, where the projection operator |P⟩ is [48] +P = +2N +� +i=1 +�1 + Di +2 +� += +1 +22N +� +{j} +� +i∈{j} +Di += +� +� +1 +22N−1 +� +{j}′ +� +i∈{j}′ +Di +� +� · +� +1 + �2N +i=1 Di +2 +� += S · P0 +(17) +where {j} runs over all possible subsets of site index set λ, while {j}′ is restricted to half of it (meaning {j}′ +will not be {i} and the complementary set λ − {i} at the same time; these 22N−1 terms give all the inequivalent +transformations).Here S symmetrically sums over physically equivalent eigenstates and P0 projects out the unphysical +states. Di, the gauge transformation operator acting on-site i, can be rewritten in terms of complec fermions as +DiA = +� +χx +ij + +� +χx +ij +�†� � +χy +ij + +� +χy +ij +�†� � +χz +ij + +� +χz +ij +�†� +ci, +DjB = i +� +χx +ij − +� +χx +ij +�†� � +χy +ij − +� +χy +ij +�†� � +χz +ij − +� +χz +ij +�†� +cj, +(18) +The matrix elements +As discussed in the main text, purely from the flux constraints, the only possible non-vanishing polarization com- +binations are a, b, c = x, y, z and their permutations. Below we show an example where a = y, b = z, c = x. The local +structure of the flux operations also simplifies the summation over sites � +k,l,m, as demonstrated below. +As an example, we show how to simplify the first line in χyzx +R1 , Eq. (9) and compute its matrix elements. The other +matrix elements in Eq. (9) and Eq. (10) can be computed similarly. The first line of χyzx +R1 is called χyzx +R1,1(ω1, ω2) and +given by +χyzx +R1,1(ω1, ω2) = − 1 +N +� +P Q +� +klm +⟨0| σy +k |Q⟩ ⟨Q| σz +l |P⟩ ⟨P| σx +m |0⟩ g(ω1 − (EP − E0))g(ω2 − (EQ − E0)). +(19) + +11 +The strategy is to fix the site m and look at all possible neighboring sites contributing to the sum. We label the +unit cell associated with site m as ν and call the two sublattices A and B. To understand which are the unit cells +neighboring ν that contribute to the sum, the easiest is to draw the lattice and its connection, as shown in Fig. 7. +The expression becomes +χyzx +R1,1(ω1, ω2) = − 1 +N +� +P Q +� +ν +[⟨0| σy +νA + σy +ν−a2B |Q⟩ ⟨Q| σz +νA + σz +νB |P⟩ ⟨P| σx +νA + σx +ν−a1B |0⟩ ++ ⟨0| σy +νB + σy +ν+a2A |Q⟩ ⟨Q| σz +νA + σz +νB |P⟩ ⟨P| σx +νB + σx +ν+a1A |0⟩] +× g(ω1 − (EP − E0))g(ω2 − (EQ − E0)), +(20) +where ν ± ai labels the neighboring unit cells in ±ai directions with ai the basis vectors for honeycomb lattice. This +important simplification reduced the sum over three indices to a sum over a single index and is a clear consequence of +the fluxes getting created locally. By translation symmetry, the last sum can also be reduced to the structure shown +in Fig. 7, which means that all we have to calculate involves a fixed value of ν and multiplying by the number of unit +cells. In fact, we can compute the lower four sites (Benz star) and multiply the result by two, given the symmetry +around the z vertical bond. The factor of N unit cells is canceled by the 1/N factor in the definition of the response +function. +FIG. 7. Summation over unit cell index ν involved in Eq. (20). a1 and a2 are base vectors. Solid (hollow) dots represent sites +belonging to A(B) sublattice. The sites (ν, A) and (ν, B) are connected by a z bond. +We consider the intermediate states |P⟩ and |Q⟩ to have two fluxes, as the ground state is flux-free and each +spin operator adds two fluxes to the system. As for the matter sector, we consider up to one matter fermion in +these intermediate states. It has been argued that considering one particle in the matter sector provides a good +approximation to capture the main physics [26]. +The complex matter excitation in the 2-flux sector b† +λ(≡ au2flux) is related to the complex matter excitation a† +m(≡ +auflux-free) in the flux-free sector [49] by +bλ = +� +m +(X2 +0)∗ +λmam + (Y 2 +0 )∗ +λma† +m, +b† +λ = +� +m +(X2 +0)λma† +m + (Y 2 +0 )λmam, +(21) +where +X2∗ +0 += 1 +2(U † +2U0 + V † +2 V0), +Y 2∗ +0 += 1 +2(U † +2U0 − V † +2 V0), +(22) +with U0(2), V0(2) the orthogonal matrices given by the SVD transformation, Eq. (14). + +12 +We also relate the matter vacuum of the 2-flux state |M 2 +0 ⟩ to that of ground state |M0⟩ [50], +|M 2 +0 ⟩ = | det X2 +0| +1 +2 e− 1 +2 +� a† +mFmna† +n |M0⟩ , +(23) +with +F = (X2∗ +0 )−1Y 2∗ +0 , +(F T = −F). +(24) +To compute the matrix elements, we use the following relations derived using Wick’s theorem, +⟨M0| cAib† +λ |M 2 +0 ⟩ = | det X2 +0| +1 +2 [U0(X2 +0)−1]iλ +⟨M 2 +0 | bλcBj |M0⟩ = | det X2 +0| +1 +2 [iV0(X2 +0)−1]jλ +⟨M0| cBjb† +λ |M 2 +0 ⟩ = | det X2 +0| +1 +2 [−iV0(X2 +0)−1]jλ +⟨M 2 +0 | bλcAi |M0⟩ = | det X2 +0| +1 +2 [U0(X2 +0)−1]iλ. +(25) +We devote particular attention to how to compute the middle matrix element ⟨P| PσzP |Q⟩. The states |P⟩ and |Q⟩ +have the same number of bond fermions and of matter excitations. The operator σz in the middle changes the bond +and matter fermion number by one and, therefore, without the projector operators, this matrix element would vanish. +The projection operator plays, therefore, an important role in making this element finite. Computing the matrix +element explicitly for unit cell ν = 0, and sublattice site A, we find +⟨Q| Pσz +0AP |P⟩ = ⟨Q| [i(χz +0A0B + χz† +0A0B)c0A]P |P⟩ += ⟨M 2′ +0 | dµ ⟨FGS| χy† +0A,L(L−1)B[i(χz +0A0B + χz† +0A0B)c0A](1 + +� +j +Dj + +� +j Jx, and that in R1 process EQ − E0 is probed by ω2, so the peaks now appear above ω2 = ω1 +diagonal. +THE INVERSE PARTICIPATION RATIO OF THE MATTER STATES +Here, we give further details on differentiating the localized and extended matter states according to their IPR. +From Eq. (15), we express zero flux complex matter excitation in terms of real Majorana matter fermions as a† +m = +1 +2 +�(U T +micAi − iV T +mjcBj). Thus for 0-flux matter state wave-function ψ(0) +m (r) = ⟨r| a† +m |0⟩, we associate (U T +0 )mν (or +(V T +0 )mν) as the amplitude of applying Majorana c fermion on A (or B) site in unit cell ν. Generally, for the l-flux +sector, we define the inverse participation ratio (IPR) of a real space wavefunction the according to (U T +l ) and (V T +l ) +as +IPR(ψ(l) +m ) = +� +k(U T +l )4 +m,k + (V T +l )4 +m,k +� +k(U T +l )2 +m,k + (V T +l )2 +m,k +, +(33) +where l labels what type of flux we have for state ψ(l) +m . For example, for R(1) process the |P⟩ states have (l = 2, x) +while |Q⟩ states have (l = 2, y). + +15 +FIG. 9. Contributions for the first nonlinear susceptibility χyzx, R1 (a and b) and R2 (c and d), for the choice of parameters +L = 100, Ji = 1 and Γ = 0.01. (a and b) Real and imaginary parts of R1(ω1, ω2). The black dashed box indicates the flux gap +in the thermodynamic limit. The proximity to the diagonal is determined by how anisotropic the coupling constants are. The +strong peak along the diagonal comes from the overlap of states with two neighboring fluxes and trapped Majorana matter +fermions. (c and d) Real and imaginary part of R(2)(ω1, ω2)). In this case, ω2 measures the energy difference between fluxes +neighboring x and y bonds. In the isotropic case that we are considering, the peaks are along the ω2 = 0 line. The position of +the peak at ω1 is the same as for R1. + +(α) +(6) +Im X (w1, w2) +Re XR (w1, w2) +1.0 +1.0 +600 +0.75 +0.75 + 400 +0.5 +0.5 +0.25 +0.25 +200 +0.0 +3 +0.0 +W2 +- 0 +-1.0 -0.75 -0.5 -0.25 +-1.0 -0.75 -0.5 -0.25 +-200 +-400 +-600 + -0.75 -0.5 -0.25 0.0 0.25 0.5 0.75 1.0 +-1.0 -0.75 -0.5 -0.25 0.0 0.25 0.5 0.75 1.0 +1.0 +Im +wi +(c) +(d) +D. +1.0 +1.0 +0.50.75 +0.75 +600 +0.5 + 400 +0.25 +0.25 +200 +-1.0 -0.75 -0.5 -0.25 0.0 +0.0 +2 +3 +- 0 +-1.0 -0.75 -0.5 -0.25 +-200 +400 +600 +-1.0 -0.75 -0.5 -0.25 0.0 0.25 0.5 0.75 1.0 +-1.0 +) -0.75 -0.5 -0.250.0 +0.25 +0.5 +0.751.0 +1 +Im16 +FIG. 10. Anisotropic zoomed-in plot for 0.25 < ω1,2 < 0.75; L = 100, Jx = 0.86, Jy = 0.95, Jz = 1.19. The black dashed line +indicates the diagonal ω1 = ω2. + +yza +Re +400 +0 +- 200 +15 +0 +3 +:0 +0.45 +--200 +3 + -400 +5 +2 +0.25 +0.35 +0.45 +0.55 +0.65 +0.75 +w117 +FIG. 11. Cut plots for the 2D spectra. + +diagonal cut +vertical cut; wi = 0.5 +3.5 +500 +1.0 +3.0 - +0.5 +0.0 +2.5 +1.0 +-500 +-0.5 +三 2.0 +0.5 +-1000 +0.0 +0.0 0.5 1.0 +-1500 +-0.5 +1.0 +0 +2000 +0.5 +0.0 0.5 1.0 +0.0 +0.2 +0.4 +0.6 +1.0 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.8 +W2 +horizontal cut; w2 = 0.5 +horizontal cut; w2 = 0.5 +3.5 +0 +3.0 +-500 +0.0 +1000 +0 +RJ2.5 +-0.5 +5. +1.0 +0 +-1500 += 2.0 - +10g1 +0.0 +0.0 0.5 1.0 +-2000 +-0.5 +1.5 +-2500 +1.0 +0.0 0.5 1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +w1 +w1 \ No newline at end of file diff --git a/HNFIT4oBgHgl3EQfXStx/content/tmp_files/load_file.txt b/HNFIT4oBgHgl3EQfXStx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..600b72dfb8602ccd47ffa06991d954a00a2731fd --- /dev/null +++ b/HNFIT4oBgHgl3EQfXStx/content/tmp_files/load_file.txt @@ -0,0 +1,1165 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf,len=1164 +page_content='Probing Majorana wavefunctions in Kitaev honeycomb spin liquids with second-order two-dimensional spectroscopy Yihua Qiang,1, 2 Victor L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Quito,1, 2 Tha´ıs V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Trevisan,1, 2, ∗ and Peter P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Orth1, 2, 3 1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA 2Ames National Laboratory, Ames, Iowa 50011, USA 3Department of Physics, Saarland University, 66123 Saarbr¨ucken, Germany (Dated: January 27, 2023) Two-dimensional coherent terahertz spectroscopy (2DCS) emerges as a valuable tool to probe the nature, couplings, and lifetimes of excitations in quantum materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' It thus promises to identify unique signatures of spin liquid states in quantum magnets by directly probing properties of their exotic fractionalized excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Here, we calculate the second-order 2DCS of the Kitaev honeycomb model and demonstrate that distinct spin liquid fingerprints appear already in this lowest-order nonlinear response χ(2) yzx(ω1, ω2) when using crossed light polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' We further relate the off- diagonal 2DCS peaks to the localized nature of the matter Majorana excitations trapped by Z2 flux excitations and show that 2DCS thus directly probes the inverse participation ratio of Majorana wavefunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' By providing experimentally observable features of spin liquid states in the 2D spectrum, our work can guide future 2DCS experiments on Kitaev magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='– Spectroscopic techniques are among the most powerful interrogation methods of quantum materi- als by directly measuring electronic Green’s functions [1– 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' While much insight can be gained in linear response, nonlinear response functions often provide a wealth of additional information that is inaccessible in the lin- ear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Examples include nonlinear conductivities that probe the Berry phase and quantum geometry of the electronic wavefunction in solids [6–9] and second- harmonic generation that is extremely sensitive to a sys- tem’s symmetry [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Another striking example is two-dimensional coherent spectroscopy (2DCS), which exposes the system to a sequence of coherent light pulses in order to measure a higher-order retarded Green’s func- tion [5, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' It provides a detailed two-dimensional excitation map of two frequencies that can be used to extract the nature, couplings and lifetimes of elementary excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' This technique has long been used in the radio and optical frequency range and has only recently been extended to terahertz (THz) frequencies, which are ideal for the study of excitations and collective modes in quantum materials [15–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Being able to disentangle different types of excitations and to discriminate between intrinsic and inhomogeneous broadening, THz 2DCS has been proposed to provide unique fingerprints of fractionalized excitations in exotic quantum magnets [23–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' A previous theoretical study of 2DCS in the Kitaev honeycomb spin liquid [26], for example, has shown that the third-order diagonal sus- ceptibility χ(3) zzzz contains signatures of the two types of fractionalized excitations in the Kitaev model: static Z2 gauge fluxes and itinerant Majorana fermion excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Here, we demonstrate that marks of fractionalization are already present in the lower second-order off-diagonal re- sponse tensor element χ(2) yzx, which is much larger in in- tensity and thus experimentally easier accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' We find clear evidence of the presence of a nonzero flux gap and a broad continuum of Majorana fermion excitations, whose intrinsic lifetimes can be extracted from the 2D spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' In addition, we show that χ(2) provides di- rect evidence of the trapping of Majorana wavefunctions around static Z2 flux excitations and that the ratio of second and first-order response, χ(2)/χ(1), is a quantita- tive measure of the overlap of such localized Majorana wavefunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Our work thus directly links localized Majorana states trapped around Z2 gauge fluxes to ob- servable peaks in the 2D spectrum, and we relate the inverse participation ratios of the wavefunctions to the peak sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Finally, we show how exchange anisotropies modify the 2D spectrum, which can be used as a sensitive experimental probe of anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Magnetic field pulse sequence B(t) used to measure χyzx(τ1, τ2) in the Kitaev honeycomb model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Different bond colors denote the a-bonds (a = x, y, z) and p labels plaquettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The figure shows effect of the pulses on the flux configuration in the R1 process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Initially, the system is in the flux-free ground state |0⟩, when at t = 0 an x-polarized pulse creates a pair of x-fluxes (green) next to a spin at site j (black dot), resulting in state |P⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' At τ1, a z-polarized pulse creates a pair of z-fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Since the system needs to return to the flux- free state in the end, the z-bond must be connected to the same spin j, resulting in a y-flux pair (blue) in state |Q⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Measurement of the magnetization M y(τ1 + τ2) removes the y-flux pair and system returns to a flux-free state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='11243v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='str-el] 26 Jan 2023 p2 Identifying unique fingerprints of spin liquid states with 2DCS promises to become a fruitful direction in the experimental study of Kitaev magnets [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Anisotropic compass-like Kitaev spin interactions are found in d-electron materials with strong crystal field and spin-orbit interactions [30–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Proposals for possible re- alizations of a Kitaev spin liquid on the honeycomb lat- tice include α-RuCl3 [34–36], iridates [37–39] and cobal- tates [40–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The main challenge is to differentiate the phenomena associated with the Kitaev exchange from those due to Heisenberg and other exchange interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' While the latter often drive the system into a magnet- ically ordered ground state, unusual spin-liquid-like be- havior has been observed in the presence of a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' To this end, we here calculate the second-order 2DCS response of the pure Kitaev honeycomb model in order to provide clear signatures of the spin liquid state that can guide experimental studies of Kitaev magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Kitaev model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='– The ferromagnetic Kitaev spin model on the honeycomb lattice is defined as [44], H = −Jx � ⟨i,j⟩x σx i σx j − Jy � ⟨i,j⟩y σy i σy j − Jz � ⟨i,j⟩z σz i σz j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (1) Here, Ja > 0 and σa j with a = x, y, z represent Pauli ma- trices at site j of the honeycomb lattice, which has two basis sites A and B per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Each spin has three nearest-neighbors, and ⟨i, j⟩a sums over nearest-neighbor pairs connected by an a-bond (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The Ki- taev model is exactly solvable because every honeycomb plaquette p hosts a flux operator ˆWp = σx 1σy 2σz 3σx 4σy 5σz 6 (j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' , 6 label the sites around the plaquette) that commutes with the Hamiltonian and with all other Wp′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The flux operator has eigenvalues wp = ±1 and a pla- quette is flux-free if wp = +1 and has a flux otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Kitaev’s solution involves writing the spin operators σa j = iba j cj using four Majorana fermions bx j , by j , bz j, cj, which satisfy {ba i , bd j} = 2δijδad, {ci, cj} = 2δij, and {ba i , cj} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' One refers to ba j as bond fermions and to cj as matter fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The introduction of four Ma- joranas per site doubles the Hilbert space and leads to a local Z2 gauge field, which poses the main challenge when computing correlation functions [26, 45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The constraint Dj = bx j by j bz jcj = 1 ∀j restores the physical Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' In terms of Majorana fermions, the spin Hamiltonian takes the form Hˆu = i 2 � j,k ˆAjkcjck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Here, ˆAjk = Jaˆujk if j and k sites are connected by an a-bond and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The bond operators ˆujk = iba j ba k (j is always an A site) commute with the Hamiltonian and among themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' They can thus be replaced by their eigenvalues ujk = ±1 and a particular bond configura- tion u ≡ {uij} determines the gauge-independent fluxes via ˆWp = � ⟨j,k⟩∈∂p ˆujk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Since the fluxes are static, we can work in a particu- lar gauge configuration u, where the Hamiltonian Hu is quadratic in matter fermions cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Even though the matter spectrum and eigenstates depend on u, it is convenient to write the eigenstates using a tensor product notation as |ψ⟩ = |F⟩⊗|M u⟩ with the flux state |F⟩ ≡ |{Wp}⟩ set by u and the matter state |M u⟩ consisting of matter excita- tions on top of the vacuum |M u 0 ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' It is useful to introduce complex bond fermions as χa jk = 1 2(ba j + iba k) , where j is an A site and k is connected to j by an a-bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' We choose the convention ˆujk = 2(χa jk)†χa jk−1 where the flux free state |F0⟩ corresponds to all bond fermions occupied and a general flux state reads |F⟩ = χan jnkn · · · χa1 j1k1 |F0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Once the gauge field state is determined, one can diago- nalize the matter part Hu = � λ ϵλ[2(au λ)†au λ−1] in terms of complex fermion eigenmodes au λ and write its state as |M u⟩ = (au λs)† · · · (au λ1)† |M u 0 ⟩ with vacuum |M u 0 ⟩ (see Supplementary Material [47] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Second-order 2DCS response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='– The ground state of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (1) is a spin liquid, which is gapless for Jx + Jy > Jz (and permutations) and gapped otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' In the follow- ing we focus on the isotropic point Jx = Jy = Jz ≡ J (see [47] for anisotropic couplings) and compute the second-order 2DCS response χabc(τ1, τ2) = i2 N θ(τ1)θ(τ2)⟨[[M a(τ1 + τ2), M b(τ1)], M c(0)]⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (2) Here, N is the total number of unit cells and M a(t) = � j σa j (t) is the a-th component of the total magnetiza- tion in the Heisenberg picture, and the Heaviside θ func- tions guarantee the causality of the response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The ex- pectation value is taken in the many-body ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' This response corresponds to the nonlinear part of the magnetization M a induced by a sequence of two mag- netic field pulses B(t) = Bc 1δ(t) + Bb 2δ(t − τ1) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 1 [16, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The second-order susceptibility χabc is finite only when a, b, c are all different and we consider χyzx in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' At the isotropic point, all other nonzero components can be related by symmetry [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Expansion of the commutator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (2) shows that χabc consists of two contributions, χyzx(τ1, τ2) = − 1 N θ(τ1)θ(τ2)[R1(τ1, τ2)−R2(τ1, τ2)+c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' ], where R1(τ1, τ2) = ⟨M y(τ1 + τ2)M z(τ1)M x(0)⟩, R2(τ1, τ2) = ⟨M z(τ1)M y(τ1 + τ2)M x(0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (3) We represent the processes in R1 and R2 with the Li- ouville pathways shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 2 [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The system starts in the ground state density matrix |0⟩ ⟨0| with energy E0, where |0⟩ is constructed with zero flux and mat- ter fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' We note that while this is not a physi- cal state for our choice of periodic boundary conditions and geometry, which is required to contain one matter fermion |0⟩phys = |F0⟩⊗a† 1 |M0⟩ [48], it is well known that physical and unphysical states yield identical results for 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Liouville pathways for (a) R1 and (b) R2 processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Time evolves from bottom to top, and dots represent bra or ket operations on the density matrix by the Pauli oper- ators (summation over sites m, l, k is done in the end).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' |0⟩ is the ground state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' |P⟩ and |Q⟩ denote excited states of the Hamiltonian, and the exponentials describe the phases ac- quired during time evolution over intervals τ1 and τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' large enough system size [26, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Using the zero matter ground state reduces the complexity of the calculations and facilitates the interpretation of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Since the entire spectrum of the Kitaev Hamiltonian (1) is known, we can use the Lehmann representation and insert two resolutions of identity � P |P⟩ ⟨P| = � Q |Q⟩ ⟨Q| = 1: χyzx R1 = −2 N Re � P Q � klm ⟨0|σy k|Q⟩ ⟨Q|σz l |P⟩ ⟨P|σx m|0⟩ × θ(τ1)θ(τ2)e−iτ1(EP −E0)e−iτ2(EQ−E0) , (4) χyzx R2 = 2 N Re � P Q � klm ⟨0|σz l |Q⟩ ⟨Q|σy k|P⟩ ⟨P|σx m|0⟩ × θ(τ1)θ(τ2)e−iτ1(EP −E0)e−iτ2(EP −EQ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (5) Here, each pathway is combined with its time- reversed partner as χyzx Rn (τ1, τ2) = θ(τ1)θ(τ2)[Rn(τ1, τ2)+ R∗ n(τ1, τ2)]/N, and the states |P⟩ and |Q⟩ are eigenstates of the Hamiltonian (1) with energy EP and EQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The states |P⟩ are connected to the flux-free ground state |0⟩ and the first pulse at t = 0 is polarized in the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Therefore, nonzero matrix elements only occur if |P⟩ contains a pair of x-fluxes |P⟩ = χx mn |F0⟩ ⊗ (au λs)† · · · (au λ1)† |M u 0 ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Here, |M u 0 ⟩ is the vac- uum for u with one x bond flipped, umn = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' A phase e−iτ1(EP −E0) is acquired during the time evolution by τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Note that we truncate the matter fermion number in the intermediate state |P⟩ to one, which is known to be an excellent approximation [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Next, a pulse po- larized in the z-direction arrives at time τ1 and creates a pair of z-fluxes at site l via application of σz l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Since after the measurement of σy k at time τ1 + τ2 the system must return to either the initial state (for pathway R1) or a diagonal state |Q⟩ ⟨Q| (for R2), the sites l, k must be in proximity to site m such that the fluxes overlap and partially annihilate each other [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' As a result, the state |Q⟩ contains a pair of y-fluxes when computing R1 and a pair of z-fluxes that can be obtained by application of σy k to |P⟩ for pathway R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' We also truncate the number of matter fermions in state |Q⟩ to be maximally one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' It is worth highlighting another difference between the R1 and R2 processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' For R1 the z-polarized pulse induces a ket operation, leading to a transition from |P⟩ to state |Q⟩ and the phase acquired during τ2 is e−iτ2(EQ−E0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' In contrast, for R2 the z-polarized pulse induces a bra op- eration onto the density matrix and creates a coherence |P⟩ ⟨Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The phase accumulated during time evolution τ2 is thus e−iτ2(EP −EQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Results and Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='– We analyze χyzx in frequency space and label by ω1 and ω2 the frequencies conjugate to the time intervals τ1 and τ2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The re- sponses are written in terms of a product of matrix el- ements and the function g(x) = i/(x + iΓ), with the broadening Γ coming from the scattering of quasiparti- cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The second-order response involves the product of two g-functions g(x1)g(x2) [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' In the small Γ limit, it leads to the terms δ(x1)δ(x2)−P 1 x1 P 1 x2 in the imaginary part of the response, because the product of matrix ele- ments is purely imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The real part of the response contains terms mixing principle values and delta-like con- tributions δ(x1)P 1 x2 +δ(x2)P 1 x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Such mixing is a general feature in nonlinear response functions [24, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Taking into account the time-reversal partners, we notice that the real parts of the 2D spectra are symmetric about the origin, while the imaginary part is antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' We plot the 2D spectrum of Imχyzx(ω1, ω2) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 3 for a lattice with 100 × 100 unit cells, N = L2 = 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Panel (a) shows the contribution of pathway R1 and panel (b) the one from pathway R2 (see [47] for the real parts and the sum of both pathways).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Given the symmetry prop- erties of the response functions, we only show results for ω1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Panel (c) shows χyzx R2 in the frequency window 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='3J < ω1, ω2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='7J on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' We start by analyzing the results for pathway R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The peaks of the R1 process appear near the diagonal ω1 = ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Inves- tigating the spectrum over a wider frequency range [47] shows that the largest response occurs in the shown fre- quency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The dashed box indicates the flux gap Eg = min(EP (Q)) − E0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='263J in the thermodynamic limit, which is the minimal energy cost of excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Due to the product of g-functions, the peaks occur at ω1 = EP − E0 and ω2 = EQ − E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' As discussed above, for a given site m in M x (or M y), the states P (or Q) have fluxes at honeycomb plaquettes neighboring site m con- nected by x (or y) bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Since we consider the isotropic case, x and y fluxes cost the same energy and the sig- nal vanishes inside the region |ω1,2| < Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Interestingly, we find the strongest signal along the diagonal, centered around energies ω1 = ω2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='5J, even though the joint density of states in this region is not large [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 4(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' This implies that the response is due to the matrix ele- ments being large for these processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Below, we show 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 2D spectrum of second-order response χyzx(ω1, ω2) for 100×100 lattice, N = 104, Ja = 1, Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (a) Imaginary part of χyzx R1 shows peaks along the diagonal ω1 = ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Vertical and horizontal streaks are due to the principal value parts from g functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The dashed box indicates the flux gap Eg below which the response vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (b) Imaginary part of χyzx R2 exhibits vertical streaks at energies ω1 = EP − E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (c) Inset zooms into low-energy region of panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Below the flux gap (dashed line), no vertical streaks appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (d) Absolute value of χyzx R1 on a logarithmic scale in region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='25J ≤ ω1,2 ≤ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Grey dots denote excitation energies of localized states with high IPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' that it indeed derives from localized matter Majorana states that are trapped around plaquettes with nonzero flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' This is in sharp contrast to results of the third-order response functions, where a strong diagonal peak arises from a constructive interference effect [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 3(c), we plot the absolute value of χyzx R1 on a logarithmic scale to highlight the presence of off-diagonal FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (a) A representative localized state |P⟩ state trapped near an x-flux pair (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The color and size of each site denote the amplitudes u and v of the Majorana matter wave- function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (b) Density of states and maximal IPR for states within a given energy bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' A few high IPR states (order ∼ 1/N of the total states) at low energies make up for most of the linear and nonlinear response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' peaks, which are due to transitions between |P⟩ and |Q⟩ states with different energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' These peaks are only about a factor of ten smaller than the diagonal ones, which in- dicates the locality of those states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Otherwise the matrix element of a local operator ⟨P|σz l |Q⟩ could not be large between the orthogonal states |P⟩ and |Q⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' We now analyze R2 shown in Fig 3(b), which exhibits vertical stripes that are centered at energies ω1 = EP − E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The reason is that peaks along the ω2-axis occur at energy differences EQ − EP and thus densely overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' As detailed in the inset panel (c), the signal below the flux gap Eg stems purely from the principal values as we observe no vertical tails for ω1 < Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The strongest vertical streaks occur at the same energies ω1 as in R1 and arise from the large overlap of localized Majorana states as we show next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' To quantitatively characterize the localization of the matter Majorana wavefunctions we present their inverse participation ratio (IPR) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The IPR is of or- der 1 for localized states and of order 1/L2 for extended states [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The IPR distribution separates into two re- gions with a few states at low energy having a much higher IPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' We find that these states are indeed localized yzr Im 3~t3) (3~-3) 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0 (a) (6) 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='5 200 0 200 (C) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='5 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0 w1 w1 (d) log10 (1,2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='7 w1a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='025 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='005 (6) 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='002 IPR Bin Max IPR Dos 400 Dos 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='001 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='000 2 6 4 Ep-EGS5 around non-zero fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 4(a) shows a representative example at energy E ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='5J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The dominance of the low- energy peaks in Imχyzx for ω1,2 <∼ J and the presence of the off-diagonal peaks in this region can thus be under- stood in terms of the large wavefunction overlap matrix elements ⟨P|σz k|Q⟩ of high-IPR states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' We note that this also accounts for most of the peak intensity in linear re- sponse χaa [47], which exclusively probes the diagonal elements ⟨P|σa k|P⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The second-order response addition- ally contains information about the off-diagonal matrix elements ⟨Q|σz|P⟩ with P ̸= Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' By taking the ratio of second- to first-order response, χ(2)/χ(1), we can extract the size of this element from experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Being only one order of magnitude smaller in the low-energy region is a clear indication of the localized nature of the Majorana wavefunctios at these energies as discussed above [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Finally, we briefly comment on results away from the isotropic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' If Jx ̸= Jy the center of the peaks in Imχyzx R1 shift away from the diagonal, reflecting the dif- ferent energy costs of creating x and y fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' This can be used as a sensitive probe of exchange anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The sharp features originating from the localized matter fermions are still present [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='– A primary challenge in the experimen- tal search for spin liquids is to find their unique and observable signatures, and one promising path is to di- rectly probe properties of their fractionalized excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The Kitaev spin liquids host two different types of frac- tionalized excitations, Z2 fluxes and matter Majorana fermions, which are not clearly separable in linear re- sponse, where a broad continuum of excitations occurs above the flux gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' In contrast, we demonstrate that they can be disentangled in the second-order nonlinear susceptibility χabc with non-repeating indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' In ad- dition, off-diagonal peaks in the 2D spectrum directly indicate the presence of localized Majorana matter ex- citations trapped by fluxes, and the 2DCS peak sizes are quantitatively related to the IPRs of their wavefunc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Involving the lowest nonlinear response, our pro- posal of using crossed-polarization pulses to probe the off-diagonal second-order susceptibility χyzx is the ex- perimentally most straightforward way of using 2DCS to probe fractionalized excitations in Kitaev spin liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' We acknowledge valuable discussions with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Pe- ter Armitage, Yueqing Chang, Elio Koenig, Milan Ko- rnjaˇca, Ana-Marija Nedi´c, Natalia Perkins, Nicholas Sir- ica, Yuriy Sizyuk, and Yuan Wan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=', and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' acknowledge support from the Research Cor- poration for Science Advancement via P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='O.’s Cottrell Scholar Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='S.' metadata={'source': 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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Orth, Supplementary Material (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' [48] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Zschocke and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Vojta, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' B 92, 014403 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' [49] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Blaizot and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Ripka, Quantum Theory of Finite Sys- tems (The MIT Press, Cambridge, Massachusetts, 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' [50] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Knolle, Dynamics of a Quantum Spin Liquid, Springer Theses (Springer International Publishing, Cham, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 7 SUPPLEMENTAL MATERIAL This Supplemental Material includes symmetry analysis of the second-order response tensor and the constraints among different components (Section S1), a derivation of the higher-order response functions (Section S2), and a detailed computation of the matrix elements (Section S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' By comparing the peak positions and their intensities, we demonstrate the role played by the novel matrix element (Section S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' We also present results for the anisotropic case (Section S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The definition and analysis of the inverse participation ratio of our problem are explained in (Section S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' SYMMETRY ANALYSIS OF THE SECOND-ORDER CORRELATION FUNCTIONS The crystal symmetries, combined with time reversal, lead to the following relations that hold, in general, for any choice of the couplings Jx, Jy and Jz, χx,y,z (ω1, ω2) = −χy,x,z (ω2, ω1) , χx,z,y (ω1, ω2) = −χz,x,y (ω2, ω1) , χy,x,z (ω1, ω2) = −χx,y,z (ω2, ω1) , χyzx (ω1, ω2) = −χz,y,x (ω2, ω1) , χz,x,y (ω1, ω2) = −χx,z,y (ω2, ω1) , χz,y,x (ω1, ω2) = −χyzx (ω2, ω1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' At the isotropic point, the symmetry constraints impose that there is only one independent component of the tensor, and all other components can be related to that one in the following way: χx,y,z (ω1, ω2) = −χx,z,y (ω1, ω2) , = −χy,x,z (ω1, ω2) , = χyzx (ω1, ω2) , = χz,x,y (ω1, ω2) , = −χz,y,x (ω1, ω2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' FOURIER TRANSFORMATION OF THE RESPONSE FUNCTIONS Here, we perform the necessary Fourier transformation from the time to the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Having these expressions analytically in frequency avoids the need for Fourier transformation numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' In general, the re- sponse function χabc(τ1, τ2) in Lehmann representation can be separated into contributions based on pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The contribution from pathway 1 is χabc R1 (τ1, τ2) = − 1 N θ(τ1)θ(τ2)[R(1)(τ1, τ2) + R(1)(τ1, τ2)∗] = − 1 N θ(τ1)θ(τ2)2ℜ � P Q � klm ⟨0| σa k |Q⟩ ⟨Q| σb l |P⟩ ⟨P| σc m |0⟩ e−iτ1(EP −E0)e−iτ2(EQ−E0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (6) while the contribution from pathway 2 is χabc R2 (τ1, τ2) = + 1 N θ(τ1)θ(τ2)[R(2)(τ1, τ2) + R(2)(τ1, τ2)∗] = + 1 N θ(τ1)θ(τ2)2ℜ � P Q � klm ⟨0| σb l |Q⟩ ⟨Q| σa k |P⟩ ⟨P| σc m |0⟩ e−iτ1(EP −E0)e−iτ2(EP −EQ), (7) where |P⟩, |Q⟩ are eigenstates of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 8 In frequency space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' introducing g(x) = i x+iΓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' where Γ takes into account the level broadening,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' we rewrite χabc(ω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' ω2) = � ∞ −∞ dτ1 � ∞ −∞ dτ2 eiω1τ1eiω2τ2χabc(τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' τ2) = − 1 N � ∞ 0 dτ1 � ∞ 0 dτ2[R(1)(τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' τ2) + R(1)(τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' τ2)∗ − R(2)(τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' τ2) − R(2)(τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' τ2)∗]eiω1τ1eiω2τ2 (8) Performing the Fourier transformations and calling E0 the ground state energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' we find,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' for the R1 = R(1) + R(1)∗ contributions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' χabc R1 (ω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' ω2) = − 1 N � P Q � klm ⟨0| σa k |Q⟩ ⟨Q| σb l |P⟩ ⟨P| σc m |0⟩ i ω1 − (EP − E0) + iΓ i ω2 − (EQ − E0) + iΓ − 1 N � P Q � klm (⟨0| σa k |Q⟩ ⟨Q| σb l |P⟩ ⟨P| σc m |0⟩)∗ i ω1 + (EP − E0) + iΓ i ω2 + (EQ − E0) + iΓ = − 1 N � P Q � klm ⟨0| σa k |Q⟩ ⟨Q| σb l |P⟩ ⟨P| σc m |0⟩ g(ω1 − (EP − E0))g(ω2 − (EQ − E0)) − 1 N � P Q � klm (⟨0| σa k |Q⟩ ⟨Q| σb l |P⟩ ⟨P| σc m |0⟩)∗g(ω1 + (EP − E0))g(ω2 + (EQ − E0)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (9) Proceeding along the same lines for R2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' χabc R2 (ω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' ω2) = + 1 N � P Q � klm ⟨0| σb l |Q⟩ ⟨Q| σa k |P⟩ ⟨P| σc m |0⟩ i ω1 − (EP − E0) + iΓ i ω2 − (EP − EQ) + iΓ + 1 N � P Q � klm (⟨0| σb l |Q⟩ ⟨Q| σa k |P⟩ ⟨P| σc m |0⟩)∗ i ω1 + (EP − E0) + iΓ i ω2 + (EP − EQ) + iΓ = + 1 N � P Q � klm ⟨0| σb l |Q⟩ ⟨Q| σa k |P⟩ ⟨P| σc m |0⟩ g(ω1 − (EP − E0))g(ω2 − (EP − EQ)) + 1 N � P Q � klm (⟨0| σb l |Q⟩ ⟨Q| σa k |P⟩ ⟨P| σc m |0⟩)∗g(ω1 + (EP − E0))g(ω2 + (EP − EQ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (10) Putting them together, χabc(ω1, ω2) = χabc R1 (ω1, ω2) + χabc R2 (ω1, ω2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (11) This is the expression analyzed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' DETAILS OF THE SOLUTION OF THE HAMILTONIAN AND THE MATRIX ELEMENTS OF THE SECOND-ORDER CORRELATION FUNCTIONS In this Section, we give further details on how to solve the Hamiltonian and how to calculate the matrix elements entering the response functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Diagonalization of the Hamiltonian and gauge structure Following Kitaev’s solution [44], we introduce the complex bond fermions χa ⟨jk⟩ = 1 2 � ba j + iba k � , (χa ⟨jk⟩)† = 1 2 � ba j − iba k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (12) For simplicity we denote χa ⟨jk⟩ as χjk from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Since ujk = 2χ† jkχjk − 1, the χ fermion describes the occupation of a bond ⟨j, k⟩ along the direction a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' A bond ⟨j, k⟩ is occupied by χ fermion if ujk = +1 or empty if ujk = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Together with the fermionic mapping, we express the spin operators as 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Total contribution of R1 and R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' L = 100, Ji = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Real and imaginary part of χyzx R1 and χyzx R1 with large frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' L = 80, Ji = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' We see the signals mostly appear in the low-frequency region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' σa i = i � χij + χ† ij � cAi σa j = � χij − χ† ij � cBj (13) The action of a spin operator can be viewed as applying a matter Majorana fermion and flipping the value of the bond ujk variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The latter action corresponds to introducing two fluxes in plaquettes adjacent to the a-type bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The problem becomes to write the eigenstates of the matter fermions moving in a background of fluxes characterized by the set {u}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The ground state lies in the flux-free sector for large enough systems with spatial translational invariance [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The conventional choice for the ground state gauge is u = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Obviously, all gauge configurations leading to the same flux sector will be equivalent and gives the same fermion energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The singular-value-decomposition (SVD) of the matter Hamiltonian leads to a natural definition of complex matter Re xyza(w1, w2) Im xyza(w1, w2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='50.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='0 w1 w1 w1 w110 fermions [48], H{u} = i 2 � cT A cT B � � 0 M −M T 0 � � cA cB � = i 2 � cT A cT B � � 0 USV T −V SU T 0 � � cA cB � = i 2 � cT A cT B � � U 0 0 V � � 0 S −S 0 � � U 0 0 V �T � cA cB � = i 2 � (e′)T (e′′)T � � 0 S −S 0 � � e′ e′′ � = i N−1 � m=0 εme′ me′′ m = N−1 � m=0 2εm � a† mam − 1 2 � , (14) where em are Majorana modes and am are matter fermion excitations with am = 1 2(e′ m + ie′′ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The vector cA(B) is of length N (N ≡ L2 is the number of unit cells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' We call the ground state complex matter excitation a† m, related to the matter Majoranas c by cAi = � m (U0)im(a† m + am), cBj = � m (iV0)jm(a† m − am).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (15) As a consequence of enlarging the Hilbert space, not all possible occupations of a are physically acceptable [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' As mentioned in the main text, the Majorana operators act on the extended 4-dimensional Fock space ˜ M, whereas the physical Hilbert space M of a spin is a subspace of ˜ M defined by [44, 46, 48] |ξ⟩ ∈ M ⇐⇒ Dj|ξ⟩ = |ξ⟩ ∀j, Dj = bx j by j bz jcj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (16) This constraint also ensures the Majorana representation of the spins satisfies the SU(2) algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' A state |Φ⟩ is physical if P |Φ⟩ = |Φ⟩, where the projection operator |P⟩ is [48] P = 2N � i=1 �1 + Di 2 � = 1 22N � {j} � i∈{j} Di = � � 1 22N−1 � {j}′ � i∈{j}′ Di � � · � 1 + �2N i=1 Di 2 � = S · P0 (17) where {j} runs over all possible subsets of site index set λ, while {j}′ is restricted to half of it (meaning {j}′ will not be {i} and the complementary set λ − {i} at the same time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' these 22N−1 terms give all the inequivalent transformations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='Here S symmetrically sums over physically equivalent eigenstates and P0 projects out the unphysical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Di, the gauge transformation operator acting on-site i, can be rewritten in terms of complec fermions as DiA = � χx ij + � χx ij �†� � χy ij + � χy ij �†� � χz ij + � χz ij �†� ci, DjB = i � χx ij − � χx ij �†� � χy ij − � χy ij �†� � χz ij − � χz ij �†� cj, (18) The matrix elements As discussed in the main text, purely from the flux constraints, the only possible non-vanishing polarization com- binations are a, b, c = x, y, z and their permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Below we show an example where a = y, b = z, c = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The local structure of the flux operations also simplifies the summation over sites � k,l,m, as demonstrated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' As an example, we show how to simplify the first line in χyzx R1 , Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (9) and compute its matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The other matrix elements in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (9) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (10) can be computed similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The first line of χyzx R1 is called χyzx R1,1(ω1, ω2) and given by χyzx R1,1(ω1, ω2) = − 1 N � P Q � klm ⟨0| σy k |Q⟩ ⟨Q| σz l |P⟩ ⟨P| σx m |0⟩ g(ω1 − (EP − E0))g(ω2 − (EQ − E0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (19) 11 The strategy is to fix the site m and look at all possible neighboring sites contributing to the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' We label the unit cell associated with site m as ν and call the two sublattices A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' To understand which are the unit cells neighboring ν that contribute to the sum, the easiest is to draw the lattice and its connection, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The expression becomes χyzx R1,1(ω1, ω2) = − 1 N � P Q � ν [⟨0| σy νA + σy ν−a2B |Q⟩ ⟨Q| σz νA + σz νB |P⟩ ⟨P| σx νA + σx ν−a1B |0⟩ + ⟨0| σy νB + σy ν+a2A |Q⟩ ⟨Q| σz νA + σz νB |P⟩ ⟨P| σx νB + σx ν+a1A |0⟩] × g(ω1 − (EP − E0))g(ω2 − (EQ − E0)), (20) where ν ± ai labels the neighboring unit cells in ±ai directions with ai the basis vectors for honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' This important simplification reduced the sum over three indices to a sum over a single index and is a clear consequence of the fluxes getting created locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' By translation symmetry, the last sum can also be reduced to the structure shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 7, which means that all we have to calculate involves a fixed value of ν and multiplying by the number of unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' In fact, we can compute the lower four sites (Benz star) and multiply the result by two, given the symmetry around the z vertical bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The factor of N unit cells is canceled by the 1/N factor in the definition of the response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Summation over unit cell index ν involved in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' a1 and a2 are base vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Solid (hollow) dots represent sites belonging to A(B) sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The sites (ν, A) and (ν, B) are connected by a z bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' We consider the intermediate states |P⟩ and |Q⟩ to have two fluxes, as the ground state is flux-free and each spin operator adds two fluxes to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' As for the matter sector, we consider up to one matter fermion in these intermediate states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' It has been argued that considering one particle in the matter sector provides a good approximation to capture the main physics [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The complex matter excitation in the 2-flux sector b† λ(≡ au2flux) is related to the complex matter excitation a† m(≡ auflux-free) in the flux-free sector [49] by bλ = � m (X2 0)∗ λmam + (Y 2 0 )∗ λma† m, b† λ = � m (X2 0)λma† m + (Y 2 0 )λmam, (21) where X2∗ 0 = 1 2(U † 2U0 + V † 2 V0), Y 2∗ 0 = 1 2(U † 2U0 − V † 2 V0), (22) with U0(2), V0(2) the orthogonal matrices given by the SVD transformation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' 12 We also relate the matter vacuum of the 2-flux state |M 2 0 ⟩ to that of ground state |M0⟩ [50], |M 2 0 ⟩ = | det X2 0| 1 2 e− 1 2 � a† mFmna† n |M0⟩ , (23) with F = (X2∗ 0 )−1Y 2∗ 0 , (F T = −F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (24) To compute the matrix elements, we use the following relations derived using Wick’s theorem, ⟨M0| cAib† λ |M 2 0 ⟩ = | det X2 0| 1 2 [U0(X2 0)−1]iλ ⟨M 2 0 | bλcBj |M0⟩ = | det X2 0| 1 2 [iV0(X2 0)−1]jλ ⟨M0| cBjb† λ |M 2 0 ⟩ = | det X2 0| 1 2 [−iV0(X2 0)−1]jλ ⟨M 2 0 | bλcAi |M0⟩ = | det X2 0| 1 2 [U0(X2 0)−1]iλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' (25) We devote particular attention to how to compute the middle matrix element ⟨P| PσzP |Q⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The states |P⟩ and |Q⟩ have the same number of bond fermions and of matter excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The operator σz in the middle changes the bond and matter fermion number by one and, therefore, without the projector operators, this matrix element would vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' The projection operator plays, therefore, an important role in making this element finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' Computing the matrix element explicitly for unit cell ν = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' and sublattice site A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content=' we find ⟨Q| Pσz 0AP |P⟩ = ⟨Q| [i(χz 0A0B + χz† 0A0B)c0A]P |P⟩ = ⟨M 2′ 0 | dµ ⟨FGS| χy† 0A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFIT4oBgHgl3EQfXStx/content/2301.11243v1.pdf'} +page_content='L(L−1)B[i(χz 0A0B + χz† 0A0B)c0A](1 + � j Dj + � j. +tical scenarios where centralized coordination is not possi- +ble. Communication constraints may appear due to large +amounts of data or due to privacy constraints (Koneˇcn´y +et al., 2016) and are determined by the structure of the net- +work. Decentralized optimization is widely used in dis- +tributed machine learning (Rabbat & Nowak, 2004; Forero +et al., 2010; Nedi´c, 2020; Gorbunov et al., 2022), dis- +tributed control (Ram et al., 2009; Gan et al., 2012) and +distributed sensing (Bazerque & Giannakis, 2009). +1.1. Time-Varying Networks +We study the setting when the network is time-varying. +That means that the links between the nodes may appear +and disappear from time to time. In practice, the changes +in the links may occur due to loss of wireless connection +between the agents or other technical malfunctions. Note +that while the set of edges may change, the set of vertices +stays the same. +1.2. Related work +In this paper, we assume the objective f(x) in (1) to be +L-smooth and µ-strongly convex. Complexity bounds for +decentralized optimization include two quantities: objec- +tive condition number κg = L/µ and network condition +number χ. In case of the time-varying network, χ denotes +the worst-case condition number over time steps. +Lower complexity bounds for optimization over static +graphs +were +proposed +in +(Scaman +et +al., +2017). +The +lower +communication +complexity +bound +is +Ω(κ1/2 +g +χ1/2 log(1/ε)). +The +corresponding +optimal +algorithms are MSDA (Scaman et al., 2017) (using dual +oracle) and OPAPC (Kovalev et al., 2020) (using primal +oracle). +For time-varying networks, the lower communication +complexity bound is Ω(κ1/2 +g +χ log(1/ε)) (Kovalev et al., +2021a). The corresponding optimal algorithms with primal +oracle are ADOM+ (Kovalev et al., 2021a) and Acc-GT (Li +& Lin, 2021) with multi-step communication. An optimal +dual algorithm is ADOM (Kovalev et al., 2021b). Prior to +optimal algorithms, several non-accelerated schemes like +DIGing (Nedic et al., 2017) and sub-optimal methods with +arXiv:2301.11817v1 [math.OC] 27 Jan 2023 + +Slowly Time-Varying Networks +additional logarithmic factor, i.e. APM-C (Li et al., 2020; +Dvinskikh & Gasnikov, 2021; Rogozin et al., 2021) and +Mudag (Ye et al., 2020) were proposed. +Optimal algorithms both for static and time-varying scenar- +ios use a multi-step consensus scheme. In the time-static +case, the communication matrix is replaced by a Cheby- +shev polynomial of it (Scaman et al., 2017). The degree of +polynomial is ⌈χ1/2⌉ and its condition number is O(1). In +the time-varying case, after each oracle call multiplication +is performed by χ matrices in a row instead of only one +matrix (Kovalev et al., 2021a). +1.3. Contributions +The case when arbitrarily many edges can change at each +time step is well-studied. +However, we think that such +a setting is not realistic and in practice the magnitude of +graph changes may be limited. We investigate several types +of such restrictions and derive lower complexity bounds for +each case. This constitutes the first part of our work (see +Table 1). We show that it is sufficient to change a poly- +nomial number of vertices (i.e. O(nα) for some α > 0) +at each iteration in order to slow consensus speed down to +factor χ. Moreover, if a logarithmic number of edges is +changed (i.e. O(log n)), the consensus is slowed down to +χ/ log χ. Finally, our results suggest that a partial consen- +sus acceleration (i.e. dependency on χ in power between +1/2 and 1) is possible if the number of changes is bounded +by a constant. +Table 1. Known lower communication complexity bounds for de- +centralized optimization and our results. Here α > 0 is a scalar +and d ∈ N is a constant. The complexity depends on the maxi- +mum number of changes in links allowed at each iteration. +Number of +Lower bound +Reference +changes +no +changes +Ω +� +χ1/2κ1/2 +g +log 1 +ε +� +(Scaman et al., 2017) +O(n) +Ω +� +χκ1/2 +g +log 1 +ε +� +(Kovalev et al., 2021a) +O(nα) +Ω +� +χκ1/2 +g +log 1 +ε +� +This paper, Th. 3.1 +O(log n) +Ω +� +χ +log χκ1/2 +g +log 1 +ε +� +This paper, Th. 3.3 +12(d − 1) +Ω +� +χd/(d+1)κ1/2 +g +log 1 +ε +� +This paper, Th. 3.5 +In the setting where a constant number of edges changes +at each iteration, our results allow to establish the known +lower bounds for static graphs and time-varying graphs +with arbitrary changes. The corresponding results are pre- +sented in the last line of Table 1. Putting d = 1 leads to the +static case and the lower bound coincides with the one in +(Scaman et al., 2017). In the opposite case, taking d → ∞ +leads to the scenario with arbitrary changes, and the corre- +sponding lower bound approaches the results in (Kovalev +et al., 2021a). In other words, our results suggest an inter- +polation between two edge cases: static graphs and time- +varying graphs with arbitrary changes. +In the second part of our paper, we address a multi-step +consensus technique for time-varying graphs. More pre- +cisely, we apply Nesterov acceleration technique to time- +varying consensus. The acceleration is attained under an +additional assumption: we assume that all graphs have a +common connected subgraph that we call a skeleton. On +the one hand, this assumption is more strict then requiring +the network to stay connected all the time. On the other +hand, we think that such an assumption may be realistic in +practical scenarios. +The consensus procedure for graphs with connected skele- +ton is slightly modified: the two nodes stop communicating +to each other if the connection between them has been lost +at least once. In other words, the active links in the com- +munication graph are not restored after they have failed at +least once, and therefore the network is ”monotonically de- +creasing”. +Summing up, this paper makes a step in the direction of op- +timization over special classes of time-varying networks. +Our lower bounds show that acceleration communication +protocol is hard to be designed even over slowly time- +varying graphs. On the other hand, we show a specific class +of networks over which accelerated consensus is reachable. +The paper is organized as follows. In Section 2 we intro- +duce notation, definitions and assumptions. In Section 3, +we present our main results on lower bounds. After that, in +Section 4 we describe the accelerated gossip protocol over +time-varying networks with connected skeleton. +2. Definitions and Assumptions +We denote Kronecker product by ⊗. +The nullspace of +matrix A is denoted ker A and the range of A is de- +noted range A. Moreover, if A is symmetric and positive +semi-definite, we denote its largest eigenvalue λmax(A), +its minimal nonzero eigenvalue λ+ +min(A) and its condi- +tion number χ(A) = λmax(A)/λ+ +min(A). +For vectors +x1, . . . , xn ∈ Rd, we introduce a column stacked vector +x = col[x1, . . . , xn] = (x⊤ +1 . . . x⊤ +n )⊤ ∈ Rnd. We also +denote N = {1, 2, . . .} to be the set of positive integers. +2.1. Objective Functions +Let H be an arbitrary Hilbert space, let ∥·∥ be the norm on +H and let ∥·∥∗ denote the conjugate norm. +Definition 2.1. Function h(x) : +H → R is called L- +smooth if for any x, y ∈ H it holds +∥∇h(y) − ∇h(x)∥∗ ≤ L ∥y − x∥ . + +Slowly Time-Varying Networks +Definition 2.2. Function h(x) : +H → R is called µ- +strongly convex if for any x, y ∈ H it holds +h(y) ≥ h(x) + ⟨∇h(x), y − x⟩ + µ +2 ∥y − x∥2 . +Throughout the paper we only work with H = l2. Al- +though the lower bounds are meat for optimization in a fi- +nite dimension, they are derived for l2, as it is typically +done in optimization of strongly convex smooth functions +(Nesterov, 2004). +2.2. Decentralized Communication +We assume that distributed communication is performed +via a series of communication rounds. +In each of the +rounds, the nodes interact through a network represented +by an undirected communication graph Gk = (V, Ek), +where k ∈ N is the current iteration number. The nodes +can only communicate to their immediate neighbors in the +corresponding network. +Note that the set of nodes V does not change over time. +Throughout the paper we only consider the case when all +graphs Gk are connected. +In the literature, analysis of optimization algorithms in the +decentralized setting as well as the lower bounds are usu- +ally based on the condition number of gossip matrices. +Assumption 2.3. Matrix Wk ∈ Rn×n is called a gossip +matrix of undirected graph Gk = (V, Ek) if the following +properties are satisfied: +1. Wk is symmetric positive semi-definite, +2. [Wk]i,j = 0 if i ̸= j and (i, j) ̸∈ Ek, +3. ker Wk = {(x1, . . . , xn) ∈ Rn : x1 = · · · = xn}. +Given a gossip matrix Wk, we define Wk = Wk⊗Id. Then +Wk is also symmetric and positive semi-definite, multi- +plication by Wk represents one communication round and +ker Wk = {x = col[x1, . . . , xn] ∈ Rnd : x1 +. . .+xn = +0}. +For a given gossip matrix W, introduce its condition num- +ber χ(W) = λmax(W)/λ+ +min(W). +A common example of a communication matrix is the +graph Laplacian L(Gk) += +D(Gk) − A(Gk), where +A(Gk) denotes the adjacency matrix of Gk and D(Gk) = +diag(� +i Aij) is a diagonal matrix with degrees of the +nodes at diagonal. Laplacian matrix L(Gk) satisfies As- +sumption 2.3. +Further in the paper we will use only Laplacian matrices, +therefore by slight abuse of notation we denote χ(G) = +χ(L(G)). +2.3. Decentralized Problem and Slowly-Changing +Setup +The main results of our work are lower bounds for decen- +tralized optimization problems. We formalize the definition +of decentralized problem and its characteristics. +Definition 2.4. Let us define DP (decentralized time- +varying problem) as a pair ({Gk}∞ +k=1, {fi}n +i=1). +Firstly, +DP includes a sequence of undirected connected graphs +{Gk}∞ +k=1 with a common set of vertices V = {1, 2, . . . , n} +and edge sets {Ek}∞ +k=1. +Let n(DP) = n, χ(DP) = +supk∈N χ(Gk). Secondly, DP includes a set of objective +functions {fi}n +i=1. We refer to f(x) = 1 +n +�n +i=1 fi(x) as a +global function. +Definition 2.5. Decentralized time-varying problem DP is +called L-smooth if global function f is L-smooth. +Definition 2.6. Decentralized time-varying problem DP +is called convex (µ-strongly convex) if global function f is +convex (µ-strongly convex). +Definition +2.7. +Let +∆(DP) +denote +the +maximum +amount +of +edges +that +change +between +consequent +communication +rounds. +Particularly, +∆(DP) += +maxk∈N +�� +i 0, L > 24, α > 0, c > +0, M > 0 there exists a constant K(α, c) > 0 and L- +smooth µ-strongly convex decentralized problem DP with +n(DP) = n > M, χ(DP) = χ > M, ∆(DP) ≤ cnα, +such that for any first-order decentralized algorithm for all +p ∈ N we have +∥xp − x∗∥2 ≥ +� +1 − 2 +√ +6 +� µ +L +� K(α,c)p +χ ++2 +∥x0 − x∗∥2 . +Corollary 3.2. For any L ≥ µ > 0, L > 24, α > +0, c > 0 there exists L-smooth µ-strongly convex decen- +tralized problem DP with sufficiently large χ(DP) = +χ, n(DP) = n, such that ∆(DP) ≤ cnα, and for any +first-order decentralized algorithm the number of commu- +nication rounds to find an ε-accurate solution of the prob- +lem 1 is lower bounded by +Ω +� +χ +� +L/µ log 1 +ε +� +. +The following theorem corresponds to the case where the +number of edges that can change per iteration is at most +logarithmic in the number of nodes. +Theorem 3.3. For any L ≥ µ > 0, L > 10, M > 0 there +exists L-smooth µ-strongly convex decentralized problem +DP with n(DP) = n > M, χ(DP) = χ > M, such that +∆(DP) ≤ 12 log2(n) and for any first-order decentralized +algorithm for all p ∈ N we have +∥xp − x∗∥2 ≥ +� +1 − +√ +10 +� +µ +L +� 12 log2(χ/2)p +χ ++2 +∥x0 − x∗∥2 . +Corollary 3.4. For any L ≥ µ > 0, L > 10 there exists L +smooth and µ-strongly convex decentralized problem DP +with sufficiently large χ(DP) = χ, n(DP) = n, such that +∆(DP) ≤ 12 log2 n, and for any first-order decentralized +algorithm the number of communication rounds to find an +ε-accurate solution of the problem 1 is lower bounded by +Ω +� +χ +log χ +� +L/µ log 1 +ε +� +. +As we can see, although the logarithmic constraints are +tighter than the polynomial ones, the problem cannot be +solved much faster. The benefit we get from logarithmic +constraints is only the logarithmic factor log χ, which is +typically small compared to the main term χ. +The following theorem describes the case when the con- +straints on changes for sequential iteration are constant. +Theorem 3.5. For any L ≥ µ > 0, L > 24, M > 0, +d ∈ N there exists a constant K(d) > 0 and L-smooth µ- +strongly convex decentralized problem DP with n(DP) = +n > M, χ(DP) = χ > M, ∆(DP) ≤ 12(d − 1), such +that for any first-order decentralized algorithm for all p ∈ +N we have +∥xp − x∗∥2 ≥ +� +1 − 2 +√ +6 +� µ +L +�K(d)pχ +− +d +d+1 +2 +∥x0 − x∗∥2 . +Corollary 3.6. For any L ≥ µ > 0, L > 24, d ∈ N there +exists L-smooth µ-strongly convex decentralized problem +DP with sufficiently large χ(DP) = χ and n(DP) = n, +such that ∆(DP) ≤ 12(d − 1), and for any first-order de- +centralized algorithm the number of communication rounds +to find an ε-accurate solution of the problem 1 is lower +bounded by +Ω +� +χ +d +d+1 � +L/µ log 1 +ε +� +. + +Slowly Time-Varying Networks +This result shows that even with constant constraints, the +lower estimates approach the estimates without constraints. +This indicates that the criterion based on the Laplacian ma- +trix condition number is very sensitive to the degree of +graph variability. +3.3. Discussion +In summary, lower bounds on the number of communica- +tions have been obtained for the decentralized optimization +of smooth strongly convex functions with restrictions on +network change rate. In particular, three modes were con- +sidered, which differ in the speed of changing of the com- +munication graph. +Polynomial change. The first mode assumes a polynomial +maximum change in the number of edges in the graph; as +it turned out, such a setting gives the lower bounds in this +case coincide with lower bounds when no additional con- +ditions are imposed on the change of the graph. +The second mode considers a maximum logarithmic +change in the number of edges in the graph. In this case, +the lower bounds turned out to be close to the estimates in +the case of time-varying networks without restrictions. +The third mode considers a constant change in the num- +ber of edges; here, for each value of the constant d, an +individual estimate is obtained. When d = 1, we restore +the lower bound for static graphs (Scaman et al., 2017) and +when d → ∞, we approach a lower bound for time-varying +networks with no restrictions on the speed of changes (Ko- +valev et al., 2021a). In other words, the lower bounds for +the last mode can be seen as an interpolation between static +and time-varying networks without restrictions. +Meaning of lower bounds. It is worth noting that the inter- +pretation of our lower complexity bounds is different from +previous results in (Kovalev et al., 2021a) and (Scaman +et al., 2017). The mentioned papers build an example of a +optimization problem for any χ > 0, while our results sug- +gest that a counterexample exists only if χ is sufficiently +large. Let us illustrate how to interpret the lower bounds in +the regime of polynomial change. Theorem 3.1 means that +A decentralized optimization method that solves any de- +centralized problem with polynomial bound on changes in +O(χp� +L/µ log(1/ε)), p < 1 communication rounds does +not exist. +In other words, our results are asymptotic, but they are suf- +ficient to restrict the area for future research. +4. Accelerated Gossip for Time-Varying +Graphs with Connected Skeleton +4.1. Time-Varying Graphs with Connected Skeleton +It is known that the number of communications cannot be +enhanced on the class of time-varying graphs that are al- +lowed to change arbitrarily but stay connected at each it- +eration. The corresponding lower complexity bounds have +been proposed in (Kovalev et al., 2021a). However, the +lower bounds in (Kovalev et al., 2021b) are built using a +graph where O(n) edges change at each time step. Namely, +a ”bad” graph is a star graph where the center of a star +changes at each iteration (see Figure 1). +In practice, a situation where O(n) edges change at every +iteration may not always occur. The amplitude of network +malfunctions may be not so large. We let all of the graphs +in the sequence have a common subgraph (a skeleton) that +remains connected through time. +Assumption 4.1. Graph sequence {Gk = (V, Ek)}∞ +k=0 has +a connected skeleton: there exists a connected graph ˆG = +(V, ˆE) such that for all k = 0, 1, . . . we have ˆE ⊆ Ek. +Assumption 4.2. For each k += +0, 1, . . . we have +λmax(L(Gk)) ≤ λmax. +Moreover, we have λ+ +min +≤ +λ+ +min( ˆG). +Assumption 4.1 is more strict then the assumption on the +graph staying connected at each iteration. However, under +Assumption 4.1 we propose an accelerated consensus pro- +cedure over time-varying graphs. +4.2. Accelerated Gossip with Non-Recoverable Links +A common approach to accelerated consensus over static +graphs is Chebyshev acceleration proposed in (Scaman +et al., 2017). A gossip matrix W with condition number +χ(W) can be replaced by a matrix polynomial PK(W) +of degree K += ⌈(χ(W))1/2⌉ with condition number +χ(PK(W)) = O(1). The construction of PK(W) is based +on Chebyshev polynomials of first type. Then, the con- +dition number of communication matrix is reduced from +χ(W) to O(1) at the cost of performing ⌈(χ(W))1/2⌉ +communication rounds instead of one. +However, Chebyshev method is only known to be applied +to consensus over static networks. In our work, we propose +an accelerated gossip scheme over time varying graphs +based on Nesterov acceleration. The acceleration is possi- +ble because of assumption on connected skeleton (Assump- +tion 4.1) and due to a specific consensus strategy. +We use the following approach to tackle with time-varying +graphs that have a connected skeleton. Let agent i in the +network stop exchanging information to agent j once the +connection between i and j has been lost at any commu- + +Slowly Time-Varying Networks +nication round. In other words, if a link fails once, the +communication through it is not recovered afterwards. This +procedure is referred to as accelerated gossip with non- +recoverable links. +Algorithm 1 Accelerated Gossip with Non-Recoverable +Links +Require: Initial guess x ∈ Rnd, stepsizes η, β > 0. Set +y0 = x0 = x. +1: Every node i = 1, . . . , n initializes set of neighbors +Ni = N 0 +i . +2: for t = 0, 1, . . . , T − 1 do +3: +Every node i does +4: +Update the set of nodes to which the node commu- +nicates: Ni = Ni ∩ N k +i +5: +yk+1 +i += xk +i − η(|Ni|xk +i − � +j∈Ni xk +j ) +6: +xk+1 +i += (1 + β) yk+1 +i +− βyk +i +7: end for +8: return CT (x) = x − xT +Note that the output of Algorithm 1 is CT (x). We claim +that CT (x) is a linear operator that is a time-varying ana- +logue of PK(W)x. +Theorem 4.3. Let Assumptions 4.1 and 4.2 hold. Denote +χ = λmax/λ+ +min and set the parameters of Algorithm 1 to +η = 1/λmax, β = (√χ − 1)/(√χ + 1). Then operator +CT (x) defined in Algorithm 1 has the following properties. +1. CT (x) is linear. +2. range CT (x) = L⊤ = {x ∈ Rmd : x1+. . .+xm = 0}. +3. For T = √χ log(4χ) we have that for any x ∈ L⊤ it +holds (1 − 1/ +√ +2) ∥x∥2 ≤ ∥CT (x)∥2 ≤ (1 + 1/ +√ +2) ∥x∥2. +The meaning of Theorem 4.3 is the following: for ”mono- +tone” graphs we can replace Wk with condition number χ +by CT (·) with condition number O(1). The payment for +reduction of condition number is √χ log(4χ) communica- +tion rounds. +4.3. Accelerated Gossip as Accelerated Method over +Time-Varying Function +Algorithm 1 can be viewed as a gossip algorithm over +a ”monotonic” network where the edges only vanish +and do not appear. +Namely, introduce a sequence of +graphs { ˆGk = (V, ∩k +j=0Ej)}∞ +k=0, corresponding Laplacians +{ ˆW k = L( ˆGk)}∞ +k=0 and denote ˆ +Wk = ˆW k ⊗Id. Then Al- +gorithm 1 writes as +� +yt+1 = xk − η ˆ +Wkxk, +xk+1 = (1 + β)yk+1 − βyk. +(2) +As can be seen from (2), on each time step a multiplication +by ˆ +Wk is performed, which corresponds to one commu- +nication over ˆGk. The edges in sequence { ˆGk}∞ +k=0 only +vanish and do not appear. +Algorithm 1 can also be interpreted as minimization of +a time-varying functional with an accelerated gradient +method. Consider problem +min +x∈Rnm hk(x) = 1 +2x⊤ ˆ +Wkx. +(3) +Algorithm 1 is accelerated Nesterov method with step-size +η and momentum term β applied a time-varying prob- +lem (3). +Remark 4.4. Returning to the case of static networks, it is +worth mentioning that Chebyshev polynomials and accel- +erated gradient methods for quadratic minimization have +a strong connection, see i.e. Chapter 2 of (d’Aspremont +et al., 2021). +In fact, Polyak’s momentum can be de- +rived through application of Chebyshev polynomials to a +quadratic minimization problem. +The analysis of accelerated method over a uniformly non- +increasing time-varying function is based on the Lyapunov +function technique. +Lemma 4.5. Let Assumptions 4.1 and 4.2 hold. Denote +τ = 1/(√χ + 1), zk = 1/τ xk − (1 − τ)/τ yk, γ = +1/(√χ − 1) and introduce potential +Ψk = (1 + γ)k +� +hk(yk) + λ+ +min +2 +��zk − x∗��2 +2 +� +, +where x∗ is a solution of (3). Then Ψk+1 − Ψk ≤ 0. +The proof of Lemma 4.5 is based on standard analysis +proposed in (Bansal & Gupta, 2019) and on the observa- +tion that the objective function in (3) is uniformly non- +increasing, i.e. for any x ∈ Rnd and for any k = 0, 1, . . . +we have hk+1(x) ≤ hk(x). +Indeed, we have ˆW k = +� +(i,j)∈ ˆEk(ei − ej)(ei − ej)⊤, where ei denotes the i-th +coordinate vector of Rm. Therefore, it holds +W k − W k+1 = +� +(i,j)∈ ˆEk\ ˆEk+1 +(ei − ej)(ei − ej)⊤ ⪰ 0. +In other words, for any x ∈ Rnm it holds +hk+1(x) − hk(x) = 1 +2x⊤ ˆ +Wk+1x − 1 +2x⊤ ˆ +Wkx ≤ 0. +The analysis of accelerated gradient method over time- +varying uniformly non-increasing functions is presented in +Appendix D. +5. Conclusion +In this paper, we study new classes of time-varying net- +works that may be more practical then the scenarios previ- +ously studied in the literature. We propose to look into a + +Slowly Time-Varying Networks +new direction of research – slowly time-varying graphs. 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IEEE Transactions on +Control of Network Systems, 7(2):829–841, 2019. + +Slowly Time-Varying Networks +Rogozin, A., Lukoshkin, V., Gasnikov, A., Kovalev, D., and +Shulgin, E. Towards accelerated rates for distributed op- +timization over time-varying networks. In International +Conference on Optimization and Applications, pp. 258– +272. Springer, 2021. +Rojo, O. and Medina, L. +Tight bounds on the al- +gebraic connectivity of bethe trees. +Linear Algebra +and its Applications, 418(2):840–853, 2006. +ISSN +0024-3795. +doi: https://doi.org/10.1016/j.laa.2006.03. +016. URL https://www.sciencedirect.com/ +science/article/pii/S0024379506001649. +Scaman, K., Bach, F., Bubeck, S., Lee, Y. T., and Mas- +souli´e, L. Optimal algorithms for smooth and strongly +convex distributed optimization in networks. In Proceed- +ings of the 34th International Conference on Machine +Learning-Volume 70, pp. 3027–3036. JMLR. org, 2017. +Stevanovi´c, D. Bounding the largest eigenvalue of trees in +terms of the largest vertex degree. Linear Algebra and +its Applications, 360:35–42, 2003. +ISSN 0024-3795. +URL +https://www.sciencedirect.com/ +science/article/pii/S0024379502004421. +Ye, H., Luo, L., Zhou, Z., and Zhang, T. +Multi- +consensus decentralized accelerated gradient descent. +arXiv preprint arXiv:2005.00797, 2020. + +Slowly Time-Varying Networks +Supplementary material +A. Proof of the Theorem 3.1 +Proof. Denote as Bd,k a Bethe tree of degree d and depth k, where the root has a degree of d, vertices at levels from 2 to +k − 1 have a degree of d + 1, and vertices at the k’th level have a degree of 1. Let n = n(d, k) be a number of vertices of +Bd,k. Consider a Bethe tree Bd,k. By simple calculations we get n = dk−1 +d−1 . Suppose k ≥ 2, d ≥ 3, thus using Theorem 2 +and Theorem 3 from (Rojo & Medina, 2006) and considering the asymptotic behavior, we obtain that ∃d0 : ∀d ≥ d0 +(d − 1)2 +dk − 1 ≤ λn−1(L(Bd,k)) ≤ 2(d − 1)2 +dk − 1 . +Figure 2. Example of B3,4 +Using results from (Stevanovi´c, 2003) we get d + 2 ≤ λ1(L(Bd,k)) ≤ ( +√ +d + 1)2, thus we conclude that ∃d1 : ∀d ≥ +d1, k ≥ 2 +n(Bd,k) +2 +≤ χ(Bd,k) ≤ 2n(Bd,k). +(4) +Denote as V1 the set of vertices of type 1, V2 the set of vertices of type 2 (V1 ∩ V2 = ∅), and W the set of remaining +vertices. Let d ≥ t > 2. V1 consists of [ d +t ] subtrees with roots adjacent to the root of Bd,k, and V2 is defined in the same +way. Therefore |V1| = |V2| = [ d +t ] dk−1−1 +d−1 . +Figure 3. Example of splitting a graph into three sets. The vertices of V1 (”Bad” vertices) are indicated in red. The vertices of V2 +(”Good” vertices) are indicated in green. +Denote the vertex functions fv : ℓ2 → R depending on vertex type: +fv(x) = +� +� +� +� +� +µ +2n ∥x∥2 + L−µ +4|V1| +� +(x1 − 1)2 + �∞ +k=1(x2k − x2k+1)2� +, +v ∈ V1 +µ +2n ∥x∥2 + L−µ +4|V2| +�∞ +k=1(x2k−1 − x2k)2, +v ∈ V2 +µ +2n ∥x∥2 , +v ∈ W +. +(5) +Estimate V1 and V2 through n, using that d ≥ t, k ≥ 2, d ≥ 3 we get +|V1| = |V2| ≥ d +2t +dk−1 − 1 +d − 1 +≥ n +4t. +(6) +Estimate the network’s global characteristic number using the local one +κl = +L−µ +2|V1| + µ +n +µ +n +≤ +2t(L−µ) +n ++ µ +n +µ +n += 2t(L − µ) + µ +µ += 2(κg − 1)t + 1, + +Slowly Time-Varying Networks +thus we have +κg ≥ κl − 1 +2t ++ 1. +(7) +Let us now start describing the sequence of edges {Ei}∞ +i=1. Firstly, we introduce the swap(v1, v2) operation. When +applied to the graph G = (V, E), it changes the edges between v1 and v2. Note that this operation changes no more than +2(deg(v1) + deg(v2)) edges (∆ ≤ 2(deg(v1) + deg(v2))). +Next, introduce the following scheme: A graph consists of ”good” and ”bad” vertices. At each iteration, each good vertex +adjacent to at least one bad vertex becomes a bad vertex. After that, we somehow change the edges in the graph, and the +scheme continues. In our case, we call V1 vertices ”bad” and G \ V1 vertices ”good”. Our goal is to make the V2 vertices +remain ”good” as long as possible. The algorithm returns a graph sequence of decentralized problems {DPi}∞ +i=1. The +individual steps of the algorithm work as follows: +Algorithm 2 Inner loop of graph changing scheme with polynomial restrictions +Input: G, V1, V2, i +B = V1 +while NoBadVertices(V2, B) do +U = PotentialBadVertices(B) +for j = 0, 1, . . . , |U| − 1 do +u = U[j] +v = FindCandidate(u) +B = B ∪ {v} +G = swap(G, u, v) +end for +Gi = G +i = i + 1 +end while +We make a few assumptions: +• Each level of the Bethe tree has a natural order of vertices, particularly each vertex has an ordered set of its children. +The root children in V1 are the most minimal, while the root children in V2 are the most maximal. +• B is the set of ”bad” vertices. At moment i, Gi is a graph in the sequence {DP}∞ +i=1. +• i is the state number of the {DP}∞ +i=1 sequence on which we run Algorithm 2. +Each iteration works as follows: +• NoBadVertices checks if the set consists only of ”good” vertices. +• PotentialBadVertices computes the set of ”good” vertices which would become ”bad” ones after an iteration. +• FindCandidate finds a vertex, which would be swapped with the input vertex. The step works as follows: we start +with root vertex and move to the leftmost (lowest numbered) ”good” vertex (not in B), if there is no such vertex, we +stop. Return the vertex where we stopped. +• swap operation swaps the edges of two input vertices, but not the vertices themselves. +Note that V1 is the set of ”bad” vertices, which can be obtained through several While iterations of Algorithm 2, starting +from a graph with no ”bad” vertices. +Lemma A.1. Consider an Algorithm 2, applied to Bd,k. At any moment of this algorithm, the number of new bad vertices +is bounded as follows: +|U| ≤ k − 1. +(8) + +Slowly Time-Varying Networks +Figure 4. Example of a graph change scheme. Potentially ”bad” vertices, i.e. those that will become ”bad” in one move, are indicated in +yellow. +Proof. Note that such iterations support the following invariant: if vertex p is ”bad”, then all its ancestors are ”bad” too +(those that can be reached from p by going from the root). Consider the set of new ”bad” vertices. On the last level, there +are no such vertices, as it would be a contradiction with the first sentence. At other levels, there are at most two. Suppose +the contradiction, let them be vertices v and u. These vertices are adjacent to ”bad” vertices v′ and u′ at the level below. +Consider their common ancestor p. It is ”good” and has two ”good” children v and u, which have ”bad” vertices v′ and u′ +in their subtrees. But then the algorithm would have to ”fill” one of the subtrees on v or u (the one with the lowest number) +with ”bad” vertices, a contradiction. +As a result, at each iteration, we get the upper bound on the change of edges: +∆i ≤ 4|U|dmax = 4(k − 1)(d + 1). +(9) +Using that n = dk−1 +d−1 and assuming d ≥ k we get +∆i ≤ 4(k − 1)(d + 1) ≤ 4kd ≤ 4kn1/(k−1). +(10) +Let T be the number of iterations needed for one of the vertices in V2 to become ”bad” (it equals the number of iterations +of Algorithm 2), we will refer to this value as the ”information flow”. At each iteration, there are at most k − 1 new ”bad” +vertices. Therefore, we get: +T ≥ +� |W| +k − 1 +� ++ 1 ≥ |W| +k − 1. +(11) +Estimate the size of the neutral vertex set W using n = dk−1 +d−1 and the definitions of V1, V2 +|W| = n − |V1| − |V2| ≥ +1 +d − 1 +� +dk − 1 − 2dk − d +t +� +≥ +� +1 − 2 +t +� +n. +Therefore using 4 we get lower bound on the information flow: +T ≥ 1 − 2/t +2(k − 1)χ. +(12) +Now, we are going to define the whole sequence of graphs in DPj for every j ∈ N. Algorithm 2 only defines the graphs +for i = i0, i0+1, . . . , i0+T −1, where i0 is the input state number for the Inner algorithm. We run Algorithm 2 iteratively. + +Slowly Time-Varying Networks +After each iteration, at least one vertex of V2 becomes ”bad”, then we rearrange the sets V1 and V2, reverse the order of +children for each vertex in G, and run the algorithm again. +Algorithm 3 Outer loop of graph changing scheme with polynomial restrictions +Input: G, V1, V2 +i = 1 +while True do +InnerLoop(G, V1, V2, i) +Rearrange(V1, V2) +ReverseOrders(G) +end while +InnerLoop is Algorithm 2. Rearrange changes the pointers for variables V1 and V2. ReverseOrders reverses the order of +children for each vertex. +We have found a sequence of graphs in which information flows slowly. Specifically, to get from V1 to V2, it takes T +iterations. To get back (from V2 to V1) it takes T iterations as well and so on. +Let x0 = 0 be the initial point for the first-order decentralized algorithm. +For every m ≥ 1, we define lm = +min p ≥ 1|∃v : ∃x ∈ Hv(p) : xm ̸= 0 as the first moment when we can get a non-zero element at the m-th place at any +node. +Considering the types of functions on vertices of the graph 26, we can conclude that functions on vertices from V1 can +”transfer” (by calculating the gradient) information (non-zero element) from the even positions (2, 4, 6, . . .) to the next +ones, and functions on vertices from V2 can transfer information from the odd positions (1, 3, 5, . . .) to the next ones. +Therefore, for the network to get a new non-zero element at the next position, a complete iteration of Algorithm 2 is +required, that is T communication iterations. +One of the main ideas is that this ”information” cannot spread faster than ”bad” vertices. +To reach the m-th non-zero element, we need to make at least m local steps and (m−1)T communication steps to transfer +information from gradients between V1 and V2 sets. Therefore, we can estimate lm: +lm ≥ (m − 1)T + m. +(13) +The solution of the global optimization problem is x∗ +p = +� √κg−1 +√κg+1 +�p +. +For any m, p such that lm > p +∥xp − x∗∥2 ≥ (x∗)2 +m + (x∗)2 +m+1 + . . . = +�√κg − 1 +√κg + 1 +�m +∥x0 − x∗∥2 . +Using 13 we can take m = ⌈ +p +T +1⌉ + 1. From 7 we conclude that +√κg−1 +√κg+1 ≥ 1 − 2 +√ +6 +√κl +Therefore using 7, 12 and assign t = 3 we get +∥xp − x∗∥2 ≥ +�√κg − 1 +√κg + 1 +�⌈p/( +1 +6(k−1) χ+1)⌉+1 +∥x0 − x∗∥2 . +Rearranging it, we get +∥xp − x∗∥2 ≥ +� +max +� +1 − 2 +√ +6 +� µ +L +�� 6(k−1)p +χ ++2 +∥x0 − x∗∥2 . +(14) + +Slowly Time-Varying Networks +B. Proof of the Theorem 3.3 +Proof. The proof is very similar to the proof of the Theorem 3.1, but here we fix d = 2 and k → ∞. +Let Bk be a binary tree B2,k. Using the lower and upper bounds on algebraic connectivity (λn−1) of such trees from +(Molitierno et al., 2000) and following the same logic as in 4, we can conclude that there exists a k0 such that for all +k > k0 the following is true +2n(Bk) ≤ χ(Bk) ≤ 6n(Bk). +(15) +Denote the set of vertices of type 1 as V1, the set of vertices of type 2 as V2 (V1 ∩ V2 = ∅), and the set of remaining +vertices as W. Suppose that every non-leaf vertex has a ”left” and ”right” child. Let V1 be a subtree with a ”left-left” root +vertex (it can be reached from the graph’s root by going to the left child and then back to the left child). V2 is defined in +the same way. Therefore |V1| = |V2| = 2k−2 − 1. +Denote the vertex functions fv : ℓ2 → R similarly as in 26. +Estimate V1 and V2 through n, using that k ≥ 4 we get +n +5 ≤ |V1| = |V2| ≤ n +4 . +(16) +Estimate global characteristic number of the network through local one +κl = +L−µ +2|V1| + µ +n +µ +n +≤ +5(L−µ) +2n ++ µ +n +µ +n += 5(L − µ) + µ +2µ += 5 +2(κg − 1) + 1, +thus we have +κg ≥ 2 +5(κl − 1) + 1. +(17) +Next, we will use exactly the same technique to construct a sequence of communication graphs as in the proof of the +Theorem 3.1 (Algorithm 2 and Algorithm 3). As a result, we get something resembling 11 inequality on ”information +flow” defined in previous proof +T ≥ |W| +k − 1. +(18) +Using n = 2k − 1 and the definitions of V1, V2, we can estimate the size of the neutral vertex set W. +|W| = n − |V1| − |V2| = 2k − 1 − 2(2k−2 − 1) ≥ n +2 . +By using 15 and inequality k − 1 ≤ log2 n we derive a lower bound on T: +T ≥ +χ +12(k − 1) ≥ +χ +12log2(χ/2). +(19) +Also we similarly take an upper bound on edge change ∆i in graph sequence +∆i ≤ 4|U|dmax = 12(k − 1) ≤ 12 log2 n. +(20) +Do the same reasoning with lm (defined in the last section). +In order to reach the m-th non-zero element, at least m local steps and (m − 1)T communication steps are required to +transfer information from gradients between V1 and V2 sets. Based on this, we can estimate lm: +lm ≥ (m − 1)T + m. +(21) +The solution of the global optimization problem is x∗ +p = +� √κg−1 +√κg+1 +�p +. +For any m, p such that lm > p +∥xp − x∗∥2 ≥ (x∗)2 +m + (x∗)2 +m+1 + . . . = +�√κg − 1 +√κg + 1 +�m +∥x0 − x∗∥2 . + +Slowly Time-Varying Networks +Using 21 we can take m = ⌈ +p +T +1⌉ + 1. From 17 we conclude that +√κg−1 +√κg+1 ≥ 1 − +√ +10 +√κl +Therefore using 17, 19 we get +∥xp − x∗∥2 ≥ +� +max +� +0, 1 − +√ +10 +� µ +L +��⌈p/ +� +χ +12 log2(χ/2) +1 +� +⌉+1 +∥x0 − x∗∥2 . +Rearranging it, we get +∥xp − x∗∥2 ≥ +� +max +� +0, 1 − +√ +10 +� µ +L +�� 12 log2(χ/2)p +χ ++2 +∥x0 − x∗∥2 . +(22) +C. Proof of the Theorem 3.5 +Firstly, let’s define the structure of the graph that will serve as a counter-example in the case under consideration and study +its properties. +We will define the graph Hd,k through induction. Let H1,k be a path of length k, and call any of its leaf vertices the root. +Then, assuming we have defined Hd,k for all k, we define Hd+1,k as follows: take a path of length k (and call the leaf +vertex of the path the root of the graph), and attach a copy of Hd,k to the root of each vertex in the path. +It can be seen that each tree Hd,k consists of paths of length k with fixed vertices, which we will refer to as roots. Consider +one such path, with a start vertex and an end vertex, where the end vertex is a root and the start vertex is the other leaf +vertex. We will also assign a number from 1 to k to each vertex in the path, corresponding to the distance from the start +vertex, increased by 1. That is, the start and root vertices have numbers 1 and k respectively. +We will divide the tree Hd,k into levels. As the graph was constructed through induction, the first level will consist of +the vertices added in the first iteration of the induction, the second level will consist of the vertices added in the second +iteration of induction, and so on, up to level d. +To move forward, let’s assign coordinates to these vertices as follows: consider the vertex v at level g, then its coordinates +will be the tuple x = (x1, . . . , xg), where xi is the number corresponding to the vertex closest to v at level i. +To proceed, let’s introduce a linear order relation on these vertices as follows: if they have different lengths and the +coordinates of the first vertex is a prefix of the second one, then the second one is considered smaller. Otherwise, the +one with the smallest element that has the first difference from left to right is considered smaller. This way, the vertex +(x1, . . . , xg) will be adjacent to the vertices (x1, . . . , xg − 1), (x1, . . . , xg + 1), (x1, . . . , xg, d) if notations is correct and +to vertex (x1, . . . , xg−1) if xg = d.” +Let us introduce a linear order relation on these vertices: if the coordinate of the first vertex is the prefix of the second one, +then the second one is smaller, otherwise the one with the smallest element, which has the first difference from left to right, +is smaller. Then the vertex (x1, . . . , xg) will be adjacent to the vertices (x1, . . . , xg − 1), (x1, . . . , xg + 1), (x1, . . . , xg, d) +if notations is correct and to vertex (x1, . . . , xg−1) if xg = d. +Consider a graph Hd,k. +For this graph n = k + k2 + . . . + kd. Let D be the diameter of this graph, it can be easily seen that D = (2d − 1)k. +According to Theorem 4.1.1 in (Das, 2004), we can obtain an estimation of λn−1(L(Hd,k)) +λn−1(L(Hd,k)) ≥ +1 +nD ≥ +1 +d(2d − 1)kd+1 . +(23) +Using results from (Stevanovi´c, 2003) we get +4 ≤ λ1(L(Hd,k)) ≤ 3 + 2 +√ +2 ≤ 6. +(24) +As a result, by using 23 and 24 we can obtain an upper bound for χ(Hd,k). +χ(Hd,k) ≤ 6d(2d − 1)kd+1 ≤ 6d(2d − 1)n +d+1 +d . +(25) + +Slowly Time-Varying Networks +Let V1 and V2 be disjoint sets of vertices of type 1 and type 2, respectively, and let W be the set of remaining vertices. V1 +consists of vertices that have a coordinate of at most ( +� k +3 +� +), and V2 consists of vertices that have a coordinate of at least +(k − +� k +3 +� +). Therefore, |V1| = |V2| = +� k +3 +� +(k + k2 + . . . + kd−1). +Denote the vertex functions fv : ℓ2 → R depending on vertex type: +fv(x) = +� +� +� +� +� +µ +2n ∥x∥2 + L−µ +4|V1| +� +(x1 − 1)2 + �∞ +k=1(x2k − x2k+1)2� +, +v ∈ V1 +µ +2n ∥x∥2 + L−µ +4|V2| +�∞ +k=1(x2k−1 − x2k)2, +v ∈ V2 +µ +2n ∥x∥2 , +v ∈ W +. +(26) +Estimate V1 and V2 through n, let k ≥ 3 +|V1| = |V2| ≥ k +6(k + k2 + . . . + kd−1) = n +12. +(27) +Estimate the network’s global characteristic number using the local one +κl = +L−µ +2|V1| + µ +n +µ +n +≤ +6(L−µ) +n ++ µ +n +µ +n += 6(L − µ) + µ +µ += 6(κg − 1) + 1, +thus we have +κg ≥ κl − 1 +6 ++ 1. +(28) +Let us now describe the sequence of edges {Ei}∞ +i=1. Similar to the proof of Theorem 3.1, we will construct an algorithm +that generates a sequence of graphs that works under the same conditions as Algorithm 2 and Algorithm 3. In this scheme, +we will refer to vertices in V1 as ”bad” and the remaining vertices as ”good”. After each iteration, a ”good” vertex that is +adjacent to a ”bad” vertex becomes ”bad”, and the graph is modified in some way. The goal is to keep the vertices in V2 +”good” for as long as possible. Additionally, we will maintain the invariant that after each graph change, a ”good” vertex +cannot be less than a ”bad” vertex. +Algorithm 4 Inner loop of graph changing scheme with constant restrictions +Input: G, V1, V2, i +B = V1 +while NoBadVertices(V2, B) do +U = PotentialBadVertices(B) +AtLastLevel(U, B) +for j = 0, 1, . . . , |U| − 1 do +u = U[j] +v = FindCandidate(u, B) +B = B ∪ {v} +G = swap(G, u, v) +end for +Gi = G +i = i + 1 +end while +Every function works in the same way as in the Algorithm 2, except for FindCandidate and AtLastLevel. +FindCandidate finds a vertex that would be swapped with the input vertex. It finds the smallest ”good” vertex in the +graph. It is simple to check that the invariant is preserved. AtLastLevel checks if there is a vertex at the last level, makes +it ”bad”, and removes it from U. +Then we apply Algorithm 3, but using Algorithm 4 as the inner algorithm, thus obtaining a sequence of graphs. +Lemma C.1. Consider an Algorithm 4, applied to Hd,k. At any moment of this algorithm, the number of new bad vertices +is bounded as follows: +|U| ≤ d. +(29) + +Slowly Time-Varying Networks +Proof. We consider the set U and show that it can contain at most one element from each level. Suppose the converse, +let the vertices v, u ∈ U and belong to level g. Let vertex v have coordinates (x1, . . . , xg) and vertex u have coordinates +(y1, . . . , yg) and v < u. The vertex u is adjacent to the ”bad” vertex b, and b < u, so it can be (y1, . . . , yg − 1) +or (y1, . . . , yg, d). The second case is impossible because the ”bad” vertex (y1, . . . , yg, d) is greater than v, and this +contradicts the invariant. Consider the first case when b has coordinates (y1, . . . , yg − 1). v < u, so v ≤ b, but they cannot +be equal, since at the given iteration, v is only potentially ”bad” (i.e. ”good”) so far, so we are led to the same contradiction +when the ”bad” vertex is greater than the ”good” one. +Note that either U contains no vertices on the last level, in which case |U| ≤ d − 1 (as can be proved similarly to +Lemma C.1), or there is a vertex on the last level, but it need not be swapped. Therefore, we can obtain an upper bound on +the number of edges changed at each iteration. +∆i ≤ 12(d − 1). +(30) +Let T be the number of iterations, needed for one of the vertex in V2 to become bad (it equals to the number of iterations +of Algorithm 4), we wil refer to this value as ”information flow”. According to Lemma C.1 at each While iteration there +are not more than d new bad vertices, therefore we get +T ≥ +�|W| +d +� ++ 1 ≥ |W| +d . +(31) +Using |V1| = |V2| ≤ n +3 to estimate the size of the neutral vertex set cW, we get +|W| = n − |V1| − |V2| ≥ n +3 . +Therefore using 25 we get lower bound on the information flow: +T ≥ n +3d ≥ +χ +d +d+1 +3d(6d(2d − 1)) +d +d+1 . +(32) +To proceed further, we apply a similar approach to determine lm (defined in the the proof of the Theorem 3.1). +To reach the m-th non-zero element, we need to make at least m local steps and (m−1)T communication steps to transfer +information from gradients between V1 and V2 sets. Therefore, we can estimate lm: +lm ≥ (m − 1)T + m. +(33) +The solution of the global optimization problem is x∗ +p = +� √κg−1 +√κg+1 +�p +. +For any m, p such that lm > p +∥xp − x∗∥2 ≥ (x∗)2 +m + (x∗)2 +m+1 + . . . = +�√κg − 1 +√κg + 1 +�m +∥x0 − x∗∥2 . +Using 33 we can take m = ⌈ +p +T +1⌉ + 1. From 28 we conclude that +√κg−1 +√κg+1 ≥ 1 − 2 +√ +6 +√κl +Let C(d) = 3d(6d(2d − 1)) +d +d+1 . Therefore using 28, 32 we get +∥xp − x∗∥2 ≥ +� +max +� +0, 1 − 2 +√ +6 +� µ +L +��⌈p/ +� +C(d)−1χ +d +d+1 +1 +� +⌉+1 +∥x0 − x∗∥2 . +Rearranging it, we get +∥xp − x∗∥2 ≥ +� +max +� +0, 1 − 2 +√ +6 +� µ +L +��C(d)pχ +− +d +d+1 +2 +∥x0 − x∗∥2 . +(34) + +Slowly Time-Varying Networks +D. Accelerated Method over Time-Varying Function +In this section, we show the convergence of accelerated Nesterov method over a uniformly non-increasing time-varying +function. The proof is based on potential analysis in (Bansal & Gupta, 2019), and a similar proof technique was used in +(Rogozin et al., 2019). +Consider a sequence of functions {fk(x)}∞ +k=0 such that the following assumptions hold. +Assumption D.1. For every k = 0, 1, 2, . . ., function fk(x) is L-smooth and µ-strongly convex, that is, for any x, y ∈ Rd +we have +µ +2 ∥y − x∥2 +2 ≤ f(y) − f(x) − ⟨∇f(x), y − x⟩ ≤ L +2 ∥y − x∥2 +2 . +Assumption D.2. Sequence {fk(x)}∞ +k=0 is uniformly non-increasing: for each x ∈ Rd and for any k = 0, 1, 2, . . . we +have +fk+1(x) ≤ fk(x). +Assumption D.3. Functions of sequence {fk(x)}∞ +k=0 have a common minimizer x∗. +Let an accelerated method be run over {fk(x)}∞ +k=0. +yk+1 = xk − 1 +L∇fk(xk), +(35a) +xk+1 = +� +1 + +√κ − 1 +√κ + 1 +� +yk+1 − +√κ − 1 +√κ + 1yk, +(35b) +where κ = L/µ. Then we have the following convergence result. +Theorem D.4. Let accelerated Nesterov method be run over sequence {fk}∞ +k=0 and let Assumptions D.1 and D.2 hold. We +have +fN(yN) − f ∗ ≤ (L + µ)R2 +2 +(1 − 1/√κ)N, +∥yN − y∗∥2 +2 ≤ (L + µ)R2 +µ +(1 − 1/√κ)N. +Before passing to proof of Theorem D.4, we need the following auxiliary lemma. +Lemma D.5. Consider updates in (35) and define +τ = +1 +√κ + 1, and zk+1 = 1 +τ xk+1 − 1 − τ +τ +yk+1. +Then, zk+1 = +1 +1+γ zk + +γ +1+γ xk − +γ +µ(1+γ)∇fk(xk), where γ = +1 +√κ−1. +Proof. By the update rule for xk+1 given in (35) and the definition of τ, we have that +xk+1 = +� +1 + +√κ − 1 +√κ + 1 +� +yk+1 − +√κ − 1 +√κ + 1yk += (2 − 2τ)yk+1 − (1 − 2τ)yk. +Moreover, by the definition of zk+1, it follows that +zk+1 = 1 +τ xk+1 − 1 − τ +τ +yk+1 += 1 +τ ((2 − 2τ)yk+1 − (1 − 2τ)yk) − 1 − τ +τ +yk+1 += 1 +τ ((1 − τ)yk+1 − (1 − 2τ)yk) . + +Slowly Time-Varying Networks +Now we use the update rule for yk+1 given in (35) and also note that xk = (1 − τ)yk + τzk: +zk+1 = 1 +τ +� +(1 − τ)(xk − 1 +L∇fk(xk)) − 1 − 2τ +1 − τ (xk − τzk) +� += 1 − 2τ +1 − τ zk + +τ +1 − τ xk − 1 − τ +Lτ ∇fk(xk) +x= +√κ − 1 +√κ +zk + 1 +√κxk − +1 +µ√κ +y= +1 +1 + γ zk + +γ +1 + γ xk − +γ +µ(1 + γ)∇fk(xk), +where x is obtained by using the definitions of τ and κ, and y is obtained by using the definition of γ. +Lemma D.6. Let {fk(x)}∞ +k=0 be a sequence of functions for which Assumptions D.1 and D.2 hold. Introduce potential +function +Ψk = (1 + γ)k · +� +fk(yk) − f ∗ + µ +2 ∥zk − x∗∥2 +2 +� +, +(36) +Then, it holds that +∆Ψk = Ψk+1 − Ψk ≤ 0. +(37) +Proof. The proof is analogous to the proof in Section 5.4 in (?). We use the definitions of τ, zk given in Lemma D.5. We +have +∆Ψk · (1 + γ)−k = (1 + γ) +� +f(yk+1) − f ∗ + µ +2 ∥zk+1 − x∗∥2 +2 +� +− +� +f(yk) − f ∗ + µ +2 ∥zk − x∗∥2 +2 +� += (1 + γ) +� +fk+1(yk+1) − fk+1(x∗) +� +− +� +fk(yk) − fk(x∗) +� ++ µ +2 +� +(1 + γ)∥zk+1 − x∗∥2 +2 − ∥zk − x∗∥2 +2 +� +. +(38) +Note that from Assumption D.2 and from basic gradient step inequality we have +fk(yk) ≤ fk(yk+1) ≤ fk(xk) − 1 +2L∥∇fk(xk)∥2 +2, +We bound th first term in (38) as follows: +(1 + γ) +� +fk+1(yk+1) − fk+1(x∗) +� +− +� +fk(yk) − fk(x∗) +� +≤ (1 + γ) +� +fk(xk) − 1 +2L∥∇fk(xk)∥2 +2 − f ∗� +− +� +fk(yk) − f ∗� += fk(xk) − fk(yk) + γ(fk(xk) − f ∗) − (1 + γ)∥∇fk(xk)∥2 +2 +2L +≤ ⟨∇fk(xk), xk − yk⟩ + γ +� +⟨∇fk(xk), xk − x∗⟩ − µ +2 ∥xk − x∗∥2 +2 +� +− 1 + γ +2L ∥∇fk(xk)∥2 +2. +(39) +Let us employ Lemma D.5 to get rid of references to yk. We have +zk = +�1 +τ − 1 +� +(xk − yk) + xk = √κ(xk − yk) + xk +γ(zk − x∗) = √κγ(xk − yk) + γ(xk − x∗). + +Slowly Time-Varying Networks +Note that √κγ = 1 + γ. We have +(xk − yk) + γ(xk − x∗) = +1 +1 + γ · +� +γ(zk − x∗) + γ2(xk − x∗) +� +. +After that, we rewrite the expression on the right hand side of (39) as follows: +1 +1 + γ ⟨∇fk(xk), γ(zk − x∗) + γ2(xk − x∗)⟩− +µγ +2 ∥xk − x∗∥2 +2 − 1 + γ +2L ∥∇fk(xk)∥2 +2. +(40) +We bound the second term in (38) similarly to (Bansal & Gupta, 2019). By Lemma D.5: +µ +2 +� +(1 + γ)∥zk+1 − x∗∥2 +2 − ∥zk − x∗∥2 +2 +� += µ +2 (1 + γ) +��� +1 +1 + γ (zk − x∗) + +γ +1 + γ (xk − x∗) − +γ +µ(1 + γ)∇fk(xk) +��� +2 +2 − µ +2 ∥zk − x∗∥2 +2 += µ +2 +1 +1 + γ +� +∥zk − x∗∥2 +2 + γ2∥xk − x∗∥2 +2 + γ2 +µ2 ∥∇fk(xk)∥2 +2 ++ 2γ⟨zk − x∗, xk − x∗⟩ − 2γ +µ ⟨zk − x∗, ∇fk(xk)⟩ +− 2γ2 +µ ⟨xk − x∗, ∇fk(xk)⟩ +� +− µ +2 ∥zk − x∗∥2 +2. +(41) +Adding (40) yields a bound on ∆Ψk. Moreover, note that terms involving ⟨∇fk(xk), xk − x∗⟩ and ⟨∇fk(xk), zk − x∗⟩ +cancel out. +∆Ψk(1 + γ)−k +≤ +� +−1 + γ +2L ++ +γ2 +2µ(1 + γ) +� +∥∇fk(xk)∥2 +2 ++ µγ +2 +� +γ +1 + γ − 1 +� +∥xk − x∗∥2 +2 + µ +2 +� +1 +1 + γ − 1 +� +∥zk − x∗∥2 +2 ++ +µγ +1 + γ ⟨zk − x∗, xk − x∗⟩ +≤ − +µγ +2(1 + γ) +� +∥xk − x∗∥2 +2 + ∥zk − x∗∥2 +2 − 2⟨zk − x∗, xk − x∗⟩ +� += − +µγ +2(1 + γ)∥(xk − x∗) − (zk − x∗)∥2 +2 ≤ 0, +and the proof is complete. +Now we can prove Theorem D.4 using the potentials technique. +Proof of Theorem D.4. Following the definition of Ψk and using the Lemma D.6, we obtain +(1 + γ)N(fN(yN) − f ∗) ≤ ΨN ≤ Ψ0 ≤ (L + µ)R2 +2 +, +fN(yN) − f ∗ ≤ (L + µ)R2 +2(1 + γ)N = (L + µ)R2 +2 +(1 − 1/√κ)N. + +Slowly Time-Varying Networks +E. Missing Proofs from Section 4 +E.1. Proof of Theorem 4.3 +Statement 1 follows directly from Algorithm 1. To see why statement 2 holds it is sufficient to note that range Wk = L⊤. +For statement 3, denote x = 1 +m11⊤ ⊗ I let us apply Theorem D.4 to see that +��xT − x +��2 +2 ≤ 2χ +��x0 − x +��2 +2 (1 − 1/√χ)T . +Taking T = √χ log(4χ) we obtain +��xT − x +��2 +2 ≤ 1/2 +��x0 − x +��2 +2. If x ∈ L⊤, then x = 0 and following the definition +CT (x) = x − xT , we write +∥CT (x)∥2 = +��x − xT �� +2 ≤ ∥x − x∥2 + +��xT − x +�� +2 +≤ (1 + 1/ +√ +2) ∥x − x∥2 , +∥CT (x)∥2 = +��x − xT �� +2 ≥ ∥x − x∥2 − +��xT − x +�� +2 +≥ (1 − 1/ +√ +2) ∥x − x∥2 +E.2. Proof of Lemma 4.5 +The proof directly follows from Lemma D.6 applied to problem (3). + diff --git a/HdFKT4oBgHgl3EQfcy4l/content/tmp_files/load_file.txt b/HdFKT4oBgHgl3EQfcy4l/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0b9d31a9e2af8cc4cc56aa648dc82e961ebecd59 --- /dev/null +++ b/HdFKT4oBgHgl3EQfcy4l/content/tmp_files/load_file.txt @@ -0,0 +1,967 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFKT4oBgHgl3EQfcy4l/content/2301.11817v1.pdf,len=966 +page_content='Is Consensus Acceleration Possible in Decentralized Optimization over Slowly Time-Varying Networks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFKT4oBgHgl3EQfcy4l/content/2301.11817v1.pdf'} +page_content=' Dmitriy Metelev 1 Alexander Rogozin 1 2 Dmitry Kovalev 3 Alexander Gasnikov 1 4 5 Abstract We consider decentralized optimization prob- lems where one aims to minimize a sum of con- vex smooth objective functions distributed be- tween nodes in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFKT4oBgHgl3EQfcy4l/content/2301.11817v1.pdf'} +page_content=' The links in the network can change from time to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFKT4oBgHgl3EQfcy4l/content/2301.11817v1.pdf'} +page_content=' For the setting when the amount of changes is arbitrary, lower complexity bounds and corresponding op- timal algorithms are known, and the consensus acceleration is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFKT4oBgHgl3EQfcy4l/content/2301.11817v1.pdf'} +page_content=' However, in prac- tice the magnitude of network changes may be limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFKT4oBgHgl3EQfcy4l/content/2301.11817v1.pdf'} +page_content=' We derive lower communication com- plexity bounds for several regimes of velocity of networks changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFKT4oBgHgl3EQfcy4l/content/2301.11817v1.pdf'} +page_content=' Moreover, we show how to obtain accelerated communication rates for a cer- tain class of time-varying graphs using a specific consensus algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFKT4oBgHgl3EQfcy4l/content/2301.11817v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFKT4oBgHgl3EQfcy4l/content/2301.11817v1.pdf'} +page_content=' Introduction In this paper we consider a decentralized optimization problem min x∈Rm f(x) = 1 n n � i=1 fi(x), (1) where each function fi is convex, has a Lipschitz gradient and is stored at a separate computational node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFKT4oBgHgl3EQfcy4l/content/2301.11817v1.pdf'} +page_content=' Nodes are connected by a communication network (that may change over time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFKT4oBgHgl3EQfcy4l/content/2301.11817v1.pdf'} +page_content=' Each node is an independent computational agent that can perform local computations based only on the information in its local memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFKT4oBgHgl3EQfcy4l/content/2301.11817v1.pdf'} +page_content=' At each communica- tion step, nodes can only exchange information with their neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFKT4oBgHgl3EQfcy4l/content/2301.11817v1.pdf'} +page_content=' Sum-type problems of type (1) have applications in prac- 1Moscow Institute of Physics and Technology, Moscow, Rus- sia 2HSE University, Moscow, Russia 3Universit´e Catholique de Louvain, Ottignies-Louvain-la-Neuve, Belgium 4ISP RAS Re- search Center for Trusted Artificial Intelligence, Moscow, Rus- sia 5Institute of Information Transmission Problems, Moscow, Russia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFKT4oBgHgl3EQfcy4l/content/2301.11817v1.pdf'} +page_content=' Correspondence to: Alexander Rogozin 1014M⊙, therefore we can refer to these systems +as clusters. The properties of the 319 clusters are estimated +via Virial Analysis (Lopes et al. 2009) applied to the final +member list defined by the Shiftgapper technique (De Car- +vaho et al. 2017). Note that only members within R200 are +considered in the estimate of the velocity dispersion. In this +paper, the radius that characterises each group is given by +MNRAS 000, 1–10 (2023) + +The stellar populations of backsplash galaxies +3 +1 +2 +3 +4 +R/R200 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +|v |/ +25% +50% +75% +95% +Figure 1. Projected phase space diagram showing contours that +engulf a given fraction of the total sample, as labelled. The dashed +lines mark the region defined for this paper to detect backsplash +galaxies, located outside of the virial radius of the structures, split +into high (blue) and low (red) line of sight velocity (compared with +respect to the velocity dispersion of the structure). The sample in +the blue region is expected to have a higher fraction of backsplash +galaxies, with respect to the general population of field/infalling +galaxies. Hereafter, all figures will use this colour coding to sepa- +rate the two subsamples. +R200 defined as the position where the density is 200 times +the critical value, as laid out in Lopes et al. (2009). +Fig. 1 shows the PPS diagram of the sample, as a density +plot. The original set consists of 570,643 galaxies, from which +we have 23,631 satellites with detailed phase space informa- +tion. Modulo projection effects, backsplash galaxies should +be located outside of the group radius. According to Hag- +gar et al. (2020), over half of all galaxies located between +one and two group radii (measured as R200), are backsplash +galaxies, as measured at z=0 in a set of detailed numerical +simulations. This fraction decreases to negligible values at +R≳3 R200. Moreover, this fraction increases in dynamically +relaxed clusters (i.e. Gaussian clusters). Our criterion to split +the sample on the PPS diagram is aimed at testing whether +the spectroscopic signature of the backsplash galaxies can +be determined by focusing on the not-too-distant outskirts +of clusters, split with respect to relative velocity. The ex- +pected quenching effects will be noticeable in the signature of +the stellar populations, although we should also expect addi- +tional factors affecting the stellar populations of the infalling +galaxies. The sample is also split with respect to the dynam- +ical state of the group, into Gaussian (G) and non Gaussian +(nG) clusters, a criterion based on the distribution of veloc- +ities. The latter represent structures with a more recent dy- +namical formation history and a higher diversity regarding +galaxy properties, suggesting higher levels of pre-processing +(De Carvaho et al. 2017; Roberts & Parker 2017; Sampaio +et al. 2021). We note that Gaussian clusters are already viri- +alised systems (at least within R200), where the infall of new +members is, in principle, smaller than in non-Gaussian clus- +ters. Therefore, in the vicinity of nG grous we expect to find +a higher fraction of infalling galaxies, thus with overall higher +levels of pre-processing, as these infallers belong to smaller +groups that are in the process of merging. We also note that +0 +200 +Vel. disp. (km/s) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +× 100 +All +|v| > +|v| < +Field Hom +Field +0 +200 +Vel. disp. (km/s) +Gaussian +0 +200 +Vel. disp. (km/s) +Non-Gaussian +Figure 2. Distribution of stellar velocity dispersion in the subsam- +ples explored in this paper. The “field” reference is also shown in +brown, including the distribution before (dotted) and after (solid) +homogeneisation. +the fraction of interlopers – defined as galaxies that are within +1-3 R200 in projection, but not in 3D radial distance, r200, can +be rather high (see figure 2 of Oman & Hudson 2016). How- +ever, we will see below that the PPS-selected sample shows +strong differences with respect to the field. +We select galaxies in the region of PPS: R/R200 ∈ [1, 3] and +|v∥|/σ < 3 (a total of 12,738 SDSS spectra), split into high- +and low-velocity galaxies at |v∥| = σ, as highlighted in blue +and red, respectively, in Fig. 1. We follow this colour cod- +ing throughout the paper. From this set, we retrieve the line +strength measurements from the SDSS galSpecIndx table, +as defined in Brinchmann et al. (2004). The cross-correlation +produces 11,252 galaxies, with 5,282 (1,951) classified as be- +longing to Gaussian (non Gaussian) clusters. The remaining +4,019 cannot be accurately classified regarding the Gaussian +nature of the velocity dispersion of the cluster, but are also +included in a general sample that comprise all cluster galax- +ies with well-defined PPS information. As comparison with a +general (“field”) sample, we also extract from SDSS (Legacy) +a complete set of spectra that are unconstrained in PPS, +imposing a high enough S/N (>51) in the same redshift win- +dow, 0.03R200, +and thus fully eliminating galaxies inside the cluster. +Since this analysis focuses on the comparison of line +strengths that constrain the stellar population properties, +1 Defined by the snMedian_r parameter in the SpecObjAll table +as the median signal-to-noise over all good pixels in the SDSS r +band, given per pixel, ∼1Å. +MNRAS 000, 1–10 (2023) + +4 +Ferreras et al. +1 +2 +1.4 +1.6 +Dn(4000) +All +1 +2 +0 +2 +H +A +1 +2 +R/R200 +2.0 2.5 3.0 +[MgFe]′ +|v| > +|v| < +Field +1 +2 +1.4 +1.6 +Gaussian +1 +2 +0 +2 +1 +2 +R/R200 +2.0 2.5 3.0 +1 +2 +1.4 +1.6 +Non Gaussian +1 +2 +0 +2 +1 +2 +R/R200 +2.0 2.5 3.0 +1 +2 +10.2 +10.4 +log(Ms/M +) +All +|v| > +|v| < +Field +1 +2 +1.0 +0.5 +log SFR (M +/yr) +1 +2 +R/R200 +0.2 +0.4 +0.6 +fQ +1 +2 +10.2 +10.4 +Gaussian +1 +2 +1.0 +0.5 +1 +2 +R/R200 +0.2 +0.4 +0.6 +1 +2 +10.2 +10.4 +Non Gaussian +1 +2 +1.0 +0.5 +1 +2 +R/R200 +0.2 +0.4 +0.6 +Figure 3. Trends with respect to cluster-centric distance, measured as a fraction of the group radius (R200). In each panel, the samples +with high and low |v∥/σ| are shown in blue and red, respectively. The brown lines correspond to the field sample of SDSS galaxies selected +with the same stellar velocity dispersion as the group-related subsamples in each radial bin. We emphasize that the homogenisation +process is done in individual bins, to make sure we remove any trends in the stellar populations caused by differences in the distribution +of (internal) velocity dispersion. The datapoints correspond to the median within each bin, and the error bars represent the 1 σ error in +the median. The leftmost (rightmost) figures show spectral line strengths, and a number of general galaxy observables, as labelled. In each +figure, the sample is shown separately (in columns) for the whole sample, and for galaxies in Gaussian and non-Gaussian groups (see text +for details). +it is important to ensure that all samples feature the same +distribution of stellar velocity dispersion, which consistently +appears as the dominant driver of population properties in +galaxies (see, e.g. Bernardi et al. 2003; Ferreras et al. 2019b). +We extract subsamples from the original ones, so that the +cross-comparison sets involve galaxies with identical distri- +butions of velocity dispersion. This homogeneisation process +is illustrated in Fig. 2, where we compare the distributions +(from left to right), of the complete sample, galaxies in Gaus- +sian clusters and in non-Gaussian clusters, respectively. In +each panel, galaxies with high- (low-) |v∥|/σ are shown in blue +(red). The original general sample from SDSS (i.e. “field”) is +shown as a brown dashed line, with a clear excess at high +velocity dispersion with respect to the PPS-constrained sam- +ples. The distribution of the homogenised field sample ap- +pears as a solid brown line. We note that hereafter, in each +comparison, all samples are homogenised with respect to the +smaller subset, by random selection of galaxies until the tar- +get distribution is reached. +3 PROBING THE STELLAR POPULATION +PROPERTIES +Fig. 3 (left) shows three of the line strengths – from top +to bottom: Dn(4000), HδA and [MgFe]′. The 4000Å break +strength follows the definition of the index as a ratio of flux +redward and blueward of the break, as defined in Bruzual +(1983), but over a narrower spectral window, as proposed +by Balogh et al. (1999). The Balmer line uses the defi- +nition of Worthey & Ottaviani (1997), and has been cor- +rected for emission contamination: we use the lick_hd_sub +parameter in the SDSS galSpecIndx table, which removes +all emission lines detected in the spectra at the 3σ level +(Brinchmann et al. 2004). The [MgFe]′ index is defined +as +� +Mgb(0.72Fe5270 + 0.28 Fe5335) (Thomas et al. 2003), +where those indices are the standard Lick definitions shown +in Trager et al. (1998). +The results are shown in bins regarding group-centric dis- +tance. The samples within each radial bin are homogeneised +in stellar velocity dispersion as described above. The red and +blue lines correspond to subsamples with a high and low pro- +jected velocity, consistently with the shaded zones of Fig. 1. +For reference, we include as brown lines the results for a gen- +eral (i.e. field) sample of SDSS galaxies with the same distri- +bution in velocity dispersion and stellar mass as the targeted +samples. Within each bin in R/R200, we extract a sample of +SDSS galaxies from the general catalogue, with the same ve- +locity dispersion as the reference, which is always the smaller +sample, i.e. the high |v∥|/σ subset in that radial bin. There- +fore, while the R/R200 has no actual meaning for the SDSS +general sample it simply represents a subset with the same +stellar velocity dispersion as the group-related samples in the +same bin. Variations in the field sample regarding group- +centric distance only reflect differences in the stellar velocity +dispersion of the cluster galaxies. Note the substantial differ- +ence towards higher Dn(4000) and [MgFe]′, and lower HδA, in +the low |v∥|/σ sample, corresponding to older and more metal +rich populations, suggesting these as the typical populations +in backsplash galaxies. +As reference, Fig. 3 (right) shows the trends of stellar mass, +star formation rate and fraction of quiescent galaxies (fQ, de- +fined as the fraction of galaxies with a BPT classification flag +−1 according to the galspecExtra catalogue, along with +weak specific star formation, sSFR<0.03 Gyr−1). The sam- +ples are indistinguishable regarding stellar mass. Note that +MNRAS 000, 1–10 (2023) + +The stellar populations of backsplash galaxies +5 +1.50 1.75 +0.1(u-g) +All +0.8 +0.9 +1.0 +0.1(g-r) +0.400.42 +0.1(r-i) +|v| > +|v| < +Field +1 +2 +R/R200 +0.30 +0.35 +0.1(i-z) +1.50 1.75 +Gaussian +0.8 +0.9 +1.0 +0.400.42 +1 +2 +R/R200 +0.30 +0.35 +1.50 1.75 +Non Gaussian +0.8 +0.9 +1.0 +0.400.42 +1 +2 +R/R200 +0.30 +0.35 +Figure 4. Equivalent of Fig. 3, showing the radial trends of SDSS +colours with respect to cluster-centric distance. The colours are +k-corrected to z=0.1, and are measured within the 3 arcsec fibre +diameter of the SDSS classic spectrograph. +the homogenisation process (Fig. 2) somehow forces this re- +sult in general, to ensure that the differences in stellar popu- +lation properties are not caused by a systematic difference in +stellar velocity dispersion. The star formation rate does show +a substantial difference with respect to |v∥|/σ, with the low +velocity sample having consistently weaker star formation ac- +tivity, a result also confirmed by the increased fraction of qui- +escent galaxies (fQ). In all the plots of Fig. 3 it is also evident +the remarkable difference between galaxies in the “catchment +area” of groups (i.e. both blue and red lines) and the gen- +eral population extracted from SDSS (in brown). The latter +shows substantially weaker Dn(4000), [MgFe]′, and stronger +HδA, reflecting younger and metal-poorer populations, and a +correspondingly higher star formation rate and lower quies- +cent fraction. We emphasize that, by construction, the gen- +eral SDSS sample is defined to have the same distribution of +velocity dispersion as those selected in PPS. Hence, this dif- +ference should be interpreted as pre-processing in the higher +density regions surrounding galaxy groups. Therefore, we see +two levels of variation regarding these samples: the strong dif- +ference from pre-processing (brown vs blue/red), along with +the subtle difference introduced by backsplash galaxies (blue +vs red). +In addition, Fig. 3 shows the results for the full set of +galaxies selected in PPS, and for galaxies in Gaussian and +non-Gaussian clusters. Note that the number of galaxies in +non-Gaussian clusters is smaller in our sample, so that the +Poisson noise is higher. The trends are rather similar be- +tween G and nG clusters. While simulations suggest a higher +fractional contribution from backsplash galaxies in Gaussian +clusters – i.e. already virialised systems that had more time to +produce overshooting orbits – the effect on the line strengths +is not so evident, perhaps reflecting a weaker environment +impact on the stellar populations by the cluster passage, or +a more substantial amount of pre-processing in nG clusters, +thus producing a higher quenched fraction that counterbal- +ances the additional contribution of backsplash galaxies in +1.0 +1.5 +2.0 +0 +1 +2 +All +1.0 +1.5 +2.0 +0 +1 +2 +Gaussian +1.0 +1.5 +2.0 +Dn(4000) +0 +1 +2 +Non-Gaussian +0 +5 +0 +1 +2 +0 +5 +0 +1 +2 +0 +5 +H +A +0 +1 +2 +|v| > +|v| < +Field +2 +4 +0 +1 +2 +3 +2 +4 +0 +1 +2 +3 +2 +4 +[MgFe]′ +0 +1 +2 +3 +× 100 +Figure 5. Histogram of line strengths in galaxies chosen with +projected radial distance R∈[1,3] R200, with high- (low-) velocity +galaxies in blue (red). The field sample from SDSS, homogenised +in stellar velocity dispersion, is shown in brown. +G clusters (Sampaio et al. 2021). It is worth noting that the +comparison includes the Balmer absorption index HδA, which +is especially sensitive to changes in the stellar population con- +tent within the recent ∼ 1 Gyr, i.e. comparable with typical +dynamical timescales in these systems. +Fig. 4 shows the trends of the standard colours from the +SDSS photometric measurements – after correcting for fore- +ground, Milky Way, dust extinction, taking the fluxes within +the 3 arcsec diameter fibers, and k-correcting the colours to a +common redshift z=0.1, following Blanton & Roweis (2007). +The group vs field difference is once more evident, with bluer +colours in the field sample. However, the more subtle changes +regarding backsplash are undistinguishable with broadband +photometry. +Complementary to Fig. 3 – where the median of the dis- +tributions were shown as a function of projected group- +centric distance – we show in Fig. 5 the distribution of tar- +geted line strengths, taking all galaxies within the interval +R/R200 ∈ [1, 3], following the same split regarding |v∥|/σ. +The well-known bimodality (e.g. Strateva et al. 2001; Baldry +et al. 2004; Angthopo et al. 2019) is evident in the 4000Å +break strength, with a more prominent blue cloud (i.e. the +peak at low Dn(4000) in the field sample. Within groups, the +low velocity subset has a slightly stronger red sequence, i.e. +once more reflecting an additional component of quenched +galaxies. This behaviour is also prominent in HδA. We apply +the Anderson-Darling test for k-samples (Scholz & Stephens +1987) to statistically confirm the difference between the sub- +sets regarding velocity, finding low significance (p) values for +the general cluster sample in Dn(4000) (p=0.065) and HδA +(p <0.001) thus rejecting the null hypothesis that both sets +originate from the same parent distribution. Differences re- +garding the metallicity-sensitive index [MgFe]′ are only sub- +stantial between field and groups (being higher in the latter). +However, they are not conclusive when splitting group galax- +ies into high- and low-velocity. We will see below that a bivari- +ate plot involving [MgFe]′ and Dn(4000) produces a cleaner +separation between these two sets (i.e. extending the analysis +beyond a simple, one-dimensional interpretation, akin to that +of a Simple Stellar Population). +MNRAS 000, 1–10 (2023) + +6 +Ferreras et al. +4 THE CONTRIBUTION OF BACKSPLASH +GALAXIES +In the outskirts of groups and clusters, we expect to find +galaxies in different “dynamical phases” concerning their mo- +tion with respect to those. We simplify this scenario by as- +suming that the mixture reduces to two components in our +working sample of galaxies at R≳R200, namely: 1) infalling +galaxies with a substantial amount of pre-processing – rela- +tive to a field sample of galaxies with similar stellar velocity +dispersion, and 2) backsplash galaxies that have undergone +strong environment-related effects during (at least one) pas- +sage through the cluster core. When split with respect to +projected velocity, we expect a higher fraction of backsplash +galaxies at low |v∥|/σ. Unfortunately, the effect of the clus- +ter passage on the absorption spectra may be subtle, so that +we can not pick individual galaxies as representative of this +sample. We follow instead a statistical estimate, imposing +the ansatz that backsplash galaxies can only be found in +the |v∥| < σ subsample (see Fig. 6). We emphasize this is +just an approximation, but it will allow us to produce ro- +bust lower bounds on the contribution of backsplash galaxies +in clusters, as long as the effect of the group/cluster pas- +sage leaves discernible imprints on the spectra. Previous es- +timates impose specific models about how this population +should look like. For instance, Pimbblet (2011) adopt a mix- +ture model, imposing the backsplash population to have the +same quenched features as those found in galaxies at the +cores of the clusters, leading to a fraction around 56% at +the virial radius, slowly decreasing outwards. However, such +an approach would imply that the infalling population should +have no quenched star formation, to be properly discerned in +the mixture, whereas pre-processing of the infallers will also +produce quenched galaxies. In fact, Fig. 3 confirms a large +difference between the field sample and those in the vicinity +of clusters – regardless of their velocity – hence showing that +the presence of quenched features is prevalent even before the +galaxy enters the cluster. Therefore, we emphasize that if we +choose the field sample as reference, any method based on a +prior on quenching signatures is expected to overestimate the +fraction of backsplash galaxies, as they will be contaminated +by (quenched) infallers. +Our method to quantify the presence of backsplash galax- +ies only relies on phase space, with the main assumption +that high velocity galaxies outside of the virial radius are +infallers. In this case, the difference between high- and low- +velocity galaxies outside of the group radius will determine +the fraction of backsplash galaxies. The real scenario will of +course include a number of backsplash galaxies at high veloc- +ity, therefore our estimates will provide a robust lower bound +on this fraction. We use the distributions presented in Fig. 5 +to compute the cumulative fractions fv(> π), i.e. the frac- +tion of galaxies within a given sample that have a spectral +index (π) higher than a chosen value. This function trivially +decreases from 1, at the lowest value of π in the sample, to 0, +at the highest value. Fig. 7 illustrates this definition for the +targeted line strengths adopted in this paper to characterize +the stellar populations of the full sample. To avoid crowd- +ing, we do not show the Gaussian and non Gaussian samples +here. If we define the cumulative function as fv+ for the high +velocity sample, and fv− for the low velocity sample, then we +expect the difference ∆f ≡ fv− − fv+ to represent the excess +High |v|/𝞂 +Low |v|/𝞂 +3R200 +R200 +Quenching +Figure 6. Sketch of the ansatz adopted in our analysis of back- +splash galaxies. We assume that only galaxies with low projected +velocities (|v| < σ) include the backsplash population, in addition +to infalling galaxies. Within the cluster (R ) +|v | < +|v | > +2.5 +0.0 +2.5 +5.0 +H +A +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +2 +3 +4 +5 +[MgFe]′ +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Figure 7. Illustration of the cumulative functions used to define +∆f that allows us to quantify the fraction of backsplash galaxies. +Each panel shows the cumulative function f(> π), where π rep- +resents one of the three spectral indices adopted in this paper, as +labelled. The blue (red) lines correspond to the subsets with high- +(low-) line-of-sight velocity. The small shaded regions are the ex- +pected Poisson uncertainties. Fig. 8 is produced by differences of +these cumulative functions with respect to velocity, reflecting the +contribution of backsplash galaxies (see text for details). +low-redshift clusters, with a strong dependence on cluster- +centric distance and dynamical state. Their results are based +on simulations and allow them to precisely track the orbits of +the cluster members. Therefore, the backsplash “dynamical” +signature is determined accurately in the simulations. Our +observations would suggest that a dominant portion of back- +splash galaxies feature similar stellar population properties +as the infallers, i.e. only a small fraction shows signatures of +stronger quenching caused by having passed already through +the cluster core. The substantial difference between our re- +sults and those from Haggar et al. (2020) – that can cleanly +define the set of backsplash galaxies based on their orbits +– imply that the passage through the cluster core only pro- +duces noticeable differences on the stellar populations in a +relatively small fraction of galaxies. A previous set of simula- +tions (Oman et al. 2013) suggest potentially lower fractions +of backsplash galaxies. Their figure 5 shows the distribution +of infall times in different locations of PPS. If we focus on +group-centric distances larger than the virial radius, and in- +fall times ≳4 Gyr to represent the backsplash population, we +only find fractions comparable with Haggar et al. (2020) at +R∼R200 but not at 1.5 R200. Therefore, this type of observable +constraint can be used to calibrate galaxy formation models, +both on the dynamical properties as well as the hydrodynam- +ics that controls the flow of the gas components that eventu- +ally feed star formation. Environment-related effects on the +star formation histories of galaxies leave a signal on the stel- +lar populations that can be tested in samples defined in pro- +jected phase space in the same way as presented in this paper. +Phenomenological models have suggested extended quench- +ing timescales in agreement with our results (e.g., Reeves +et al. 2022). In other words, the observational data suggests +weak (or delayed) quenching in backsplash galaxies, in line +with mechanisms based on the ’delayed-then-rapid’ environ- +ment quenching (Wetzel et al. 2013), where the slow quench- +ing process can extend over ∼2-4 Gyr, and thus over longer +than the typical dynamical timescales. Therefore, our ansatz +is only sensitive to a fraction of the total sample of backsplash +galaxies. +A more detailed view of the differences found in the +line strengths can be produced with bivariate plots of line +strengths. We show in Fig. 10 the distribution of cluster +1.0 +1.5 +2.0 +Dn(4000) +6 +4 +2 +0 +2 +4 +6 +(fv +-fv + )×100 +2.5 +0.0 +2.5 +5.0 +H +A +6 +4 +2 +0 +2 +4 +6 +nG +G +ALL +2 +3 +4 +[MgFe]′ +6 +4 +2 +0 +2 +4 +6 +Figure 8. Difference between cumulative fractions of low- and +high-|v∥|/σ subsets as a function of the three targeted spectral +indices. The results are shown for the full cluster sample (black) +as well as for Gaussian (green) and non Gaussian clusters (orange). +See text for details. +galaxies on the plane spanned by [MgFe]′ (top) or HδA (bot- +tom) vs 4000Å break strength. These diagrams are powerful +indicators of the population properties, as Dn(4000) preferen- +tially traces the average stellar age, whereas Balmer absorp- +tion is sensitive to recent episodes of star formation (see, e.g. +Kauffmann et al. 2003; Gallazzi et al. 2005). While [MgFe]′ +is usually adopted as a metallicity indicator, it features a sig- +nificant age dependence (see, e.g. fig. 4 of La Barbera et al. +2013). The sample is once more separated into low (|v∥| < σ, +red) and high projected velocity (1 σ < |v∥| < 3 σ, blue). The +contours represent fractions of the total number of galaxies +in each sample, as labelled. The field sample is also shown, +only at the 75% level, to avoid crowding, with a back dashed +line. As reference, we include as a grey line the estimates +for a set of synthethic simple stellar populations (SSP) from +the models of Vazdekis et al. (2010) at solar metallicity. The +star symbols mark the values for ages of 1, 2, 3, 4, 8 and +10 Gyr (with the youngest population represented by a white +star). The high velocity sample – expected to be dominated +by pre-processed, infalling galaxies – has a clear tail towards +younger ages (weaker Dn(4000), and higher HδA absorption) +and lower metallicity (although [MgFe]′ also decreases to- +wards young age), whereas the low velocity sample – which +is expected to include backsplash galaxies – is more concen- +trated towards weaker Balmer absorption and [MgFe]′, as well +as stronger 4000Å break strength, suggesting that this set +has more complex star formation histories, and likely having +an increased fraction of galaxies with quenched star forma- +tion, lacking systems with SSP-equivalent ages younger than +tSSP ≲2 Gyr. Our results align with the work of Mahajan +et al. (2011) where a special set of galaxies, termed GORES +(Galaxies with Ongoing or Recent Efficient Star formation), +was defined based on their 4000Å break strength (>1.5) and +Hδ equivalent width (>2Å), indicative of recent quenching. +Their analysis, combining SDSS data with simulations, found +that ∼19% of backsplash galaxies have a GORES signature, +in contrast with ∼34% for the infallers, which implies quench- +ing operates efficiently after a core passage, but leaves open +the issue that many of the infallers may also have signatures +of quenching due to pre-processing. Our work qualitatively +goes in the same direction, but the effect is more nuanced +as seen in the bivariate distribution on the bottom-left panel +of Fig. 10. Rather than simple thresholds on 4000Å break +strength and Balmer absorption, the differences, rather affect +the offset with respect to the correlation between HδA and +MNRAS 000, 1–10 (2023) + +8 +Ferreras et al. +1.5 +2.0 +2.5 +RMAX/R200 +0 +2 +4 +6 +8 +10 +| f|MAX × 100 +Dn(4000) +nG +G +ALL +1.5 +2.0 +2.5 +RMAX/R200 +0 +2 +4 +6 +8 +10 +H +A +1.5 +2.0 +2.5 +RMAX/R200 +0 +2 +4 +6 +8 +10 +[MgFe]′ +Figure 9. Maximum fractional difference in line strength between +the low- and high-velocity subsamples, a proxy for the fraction +of backsplash galaxies, shown as a function of projected cluster- +centric distance. The results are shown for the full sample (black), +and for Gaussian (green) and non Gaussian clusters (orange). +Dn(4000) typically found in galaxies. The low-velocity sub- +set features a higher scatter. Note that simple star formation +histories, defined by a single burst of star formation, will pop- +ulate the lines traced by the stars in the figure. In contrast, +the presence of recently quenched episodes of star formation +will result in departures away from the locus described in this +diagram by the SSP prediction (see, e.g. figure 7 of Wild et al. +2007). Therefore, a marked difference between the low- and +high-velocity subsets concerns galaxies with a more complex, +recent history of star formation (and quenching), as expected +in a backsplash scenario. +If we take the 75% contour level of the high velocity sub- +sample of the HδA vs Dn(4000) plot as reference (thick red +line) and integrate the number density outwards, we find a +fraction of the total of 25% for the high velocity sample (triv- +ially, by construction), and 58% for the low velocity set, i.e. +an excess of 33%. The equivalent comparison in the [MgFe]′ +vs Dn(4000) plot yields an excess of 19% at low |v∥|/σ, which +could be interpreted as the fractional contribution from back- +splash galaxies. We also note that this result is not depen- +dent on the signal-to-noise threshold imposed: samples cut at +higher values produce a consistent difference in the distribu- +tions of low- and high-velocity: if we restrict the spectra to +S/N>20 (measured as the median value in the SDSS-r band), +the excess fractions are 27% (for the HδA vs Dn(4000) dia- +gram) and 15% (for the [MgFe]′ vs Dn(4000) distribution). +Note that a bivariate analysis is needed to produce this +result, whereas the one dimensional work presented above +gives a lower 5% fractional difference between the high- and +low-velocity subsamples. This is corroborated by the one di- +mensional histograms shown in Fig. 10, where the differences +between the two subsets are much smaller than those in the +bivariate plot. Somehow, the 2D analysis allows us to go be- +yond a simple interpretation of the results as a single param- +eter (say the age of an SSP). A more detailed, but model- +prone, analysis is beyond the scope of this paper, but will be +explored in a future work. +5 CONCLUSIONS +In this paper, we explore a method to assess the role of back- +splash galaxies in the outskirts of galaxy groups, by studying +1.2 +1.4 +1.6 +1.8 +2.0 +1 +2 +3 +4 +[MgFe]′ +25% +50% +75% +25% +50% +75% +1.2 +1.4 +1.6 +1.8 +2.0 +Dn(4000) +4 +2 +0 +2 +4 +6 +H +A +25% +50% +75% +25% +50% +75% +0.00 +0.02 +0.01 0.02 +Figure 10. Bivariate plots with respect to the three line strengths +explored in this paper. The high (blue) and low (red) |v∥|/σ sub- +samples are shown as a density plot encompassing a fraction of the +total, as labelled. Model results for the simple stellar population +models of Vazdekis et al. (2010), at solar metallicity, are shown, +marking the ages with star symbols at 1, 2, 3, 4, 8, and 10 Gyr +– with the youngest age represented by a white star. The 1D his- +tograms of the three line strengths are also shown by the side. For +reference, the black dashed lines correspond to the field sample +– homogenised to the velocity dispersion distribution of the high +|v∥|/σ sample (the contour is only shown at the 75% level to avoid +crowding). +the imprint of environment-related effects on the stellar pop- +ulations. We define a sample of 11,252 galaxy spectra from +the legacy Sloan Digital Sky Survey, cross-matched with the +groups catalogue of Yang et al. (2007), only selecting galax- +ies with cluster-centric distance 1≲R/R200 ≲3, and separate +the sample with respect to line of sight velocity, normalized +by the velocity dispersion of the individual clusters. We split +the sample into low (|v∥|/σ < 1) and high (1< |v∥|/σ <3) +velocity, and adopt the ansatz that the former includes a sub- +stantially higher fraction of backsplash galaxies, as suggested +by simulations (Haggar et al. 2020). A comparison is also +made with a general sample of SDSS galaxy spectra selected +in the same redshift window, but irrespective of environment, +always making sure the samples have the same distribution +in stellar velocity dispersion, which is a well-known driver of +population properties in galaxies (Ferreras et al. 2019b). +Our analysis confirms the trend towards older stellar ages +in the low velocity sample that we interpret as the additional +contribution from the backsplash population. This difference +is consistent in three key spectral indices: Dn(4000), HδA, +and [MgFe]′, but also manifests itself in the gas phase via +lower star formation rates and a higher fraction of quies- +MNRAS 000, 1–10 (2023) + +The stellar populations of backsplash galaxies +9 +cent galaxies in the low velocity subsample. The difference is +small but statistically significant, which implies backsplash +galaxies cannot be the dominant population, at least regard- +ing the observable signatures of interaction. The comparison +with the general sample confirms the very large difference +between field galaxies and those in the catchment area of +clusters, confirming the strong effect of pre-processing on the +stellar content of galaxies (e.g. Roberts & Parker 2017). +By defining the difference between the cumulative fractions +of galaxies with a given spectral indicator, we quantify the +role of backsplash galaxies and conclude that the fraction of +this type of galaxies should be at least 5%. By comparing this +fraction with respect to cluster-centric distance, we confirm +that the difference is highest around R≲2 R200. The extension +of the analysis to bivariate line strength diagrams (Fig. 10), +shows a more marked difference between the two subsamples, +with the low-velocity set featuring a wider scatter in HδA and +higher values of [MgFe]′, interpreted as the contribution of +backsplash galaxies, with a more complex formation history, +consistent with the quenching of a recent episode of star for- +mation. Integrating the 2D number density on these plots, we +semi-quantitatively estimate the contribution of backsplash +galaxies in the range 15-30%, more in line with numerical +simulations. +Our work confirms the presence of the backsplash popula- +tion, and suggests that detailed analysis, beyond simple 1D +distributions, should be adopted to find the subtle features +left by the cluster core passage on the absorption spectra that +reflects the variations in stellar populations with respect to +the infalling galaxies. Pre-processing is found to produce as +strong a signature as in backsplash galaxies, thus compli- +cating observational constraints. Our estimate of the back- +splash fraction is, at most, around ∼25%, whereas simulations +suggest a higher fraction, confirming that various quenching +processes operate in the vicinity of clusters, and backsplash +quenching may be affected by delays longer than the dynam- +ical timescale, hindering accurate estimates of the backsplash +galaxies from spectroscopic data alone. +ACKNOWLEDGEMENTS +IF acknowledges support from the Spanish Research Agency +of the Ministry of Science and Innovation (AEI-MICINN) +under grant PID2019-104788GB-I00. KU acknowledges sup- +port from the Ministry of Science and Technology of Tai- +wan (grant MOST 109-2112-M-001-018-MY3) and from the +Academia Sinica (grant AS-IA-107-M01). VMS acknowl- +edges the FAPESP scholarship programme through grants +2020/16243-3 and 2021/13683-5. Funding for SDSS-III has +been provided by the Alfred P. Sloan Foundation, the Par- +ticipating Institutions, the National Science Foundation, and +the U.S. Department of Energy Office of Science. The SDSS- +III web site is http://www.sdss3.org/. +DATA AVAILABILITY +This work has been fully based on publicly available data: +galaxy spectra were retrieved from the SDSS DR16 archive +(https://www.sdss.org/dr16/) and stellar population synthe- +sis models can be obtained from the respective authors. +REFERENCES +Adhikari S., Dalal N., Chamberlain R. T., 2014, J. Cosmology +Astropart. Phys., 2014, 019 +Angthopo J., Ferreras I., Silk J., 2019, MNRAS, 488, L99 +Baldry I. K., Glazebrook K., Brinkmann J., Ivezić Ž., Lupton +R. H., Nichol R. C., Szalay A. S., 2004, ApJ, 600, 681 +Balogh M. L., Morris S. L., Yee H. K. C., Carlberg R. G., Ellingson +E., 1999, ApJ, 527, 54 +Balogh M. L., Navarro J. F., Morris S. 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G., et al., 2000, AJ, 120, 1579 +This paper has been typeset from a TEX/LATEX file prepared by +the author. +MNRAS 000, 1–10 (2023) + diff --git a/QNAzT4oBgHgl3EQfz_4g/content/tmp_files/load_file.txt b/QNAzT4oBgHgl3EQfz_4g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d0da5c027081d63c51e27b7df0e057aa39e15660 --- /dev/null +++ b/QNAzT4oBgHgl3EQfz_4g/content/tmp_files/load_file.txt @@ -0,0 +1,1088 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf,len=1087 +page_content='MNRAS 000, 1–10 (2023) Preprint 6 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='0 Exploring the stellar populations of backsplash galaxies I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Ferreras1,2,3⋆, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Böhm4, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Umetsu5, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Sampaio6,7, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' de Carvalho6 1 Instituto de Astrofísica de Canarias, Calle Vía Láctea s/n, E38205, La Laguna, Tenerife, Spain 2 Department of Physics and Astronomy, University College London, London WC1E 6BT, UK 3 Departamento de Astrofísica, Universidad de La Laguna, E38206 La Laguna, Tenerife, Spain 4 Department of Astrophysics, University of Vienna, 1180, Vienna, Austria 5 Academia Sinica Institute of Astronomy and Astrophysics (ASIAA), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 1, Section 4, Roosevelt Road, 10617, Taipei, Taiwan 6 NAT-Universidade Cidade de São Paulo, Rua Galvão Bueno, 868, 01506-000 São Paulo, SP, Brazil 7 School of Physics and Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD, UK Accepted 2022 December 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Received 2022 December 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' in original form 2022 July 4 ABSTRACT Backsplash galaxies are those that traverse and overshoot cluster cores as they fall into these structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' They are affected by environment, and should stand out in contrast to the infalling population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' We target galaxies in the vicinity of clusters (R≳R200) and select a sample in projected phase space (PPS), from the compilation of Sampaio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' based on SDSS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' We present a statistical analysis, comparing two regions in PPS, with the same projected distance to the cluster but different velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' The analysis relies on the presence of variations in the stellar population content of backsplash galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' We find a lower limit in the fractional contribution of ∼5% with respect to the general sample of infalling galaxies at similar group-centric distance when using single line strength analysis, or ∼15-30% when adopting bivariate distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' The stellar populations show a subtle but significant difference towards older ages, and a higher fraction of quiescent galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' We also compare this set with a general field sample, where a substantially larger difference in galaxy properties is found, with the field sample being consistently younger, metal poorer and with a lower fraction of quiescent galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Noting that our “cluster” sample is located outside of the virial radius, we expect this difference to be caused by pre-processing of the infalling galaxies in the overall higher density regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Key words: galaxies: clusters: general – galaxies: evolution galaxies:interactions – galaxies: stellar content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 1 INTRODUCTION The evolution of cosmic structure is mostly driven by the growth of dark matter fluctuations in an expanding Universe, that takes shape as an entangled web, where fluctuations cover a wide range of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' In this picture, galaxies are lo- cated within dark matter halos, where gas cools down and forms stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' While the overall properties of galaxies strongly correlate with “local” observables – such as morphology, stel- lar mass or central velocity dispersion – the mass distribution over larger scales (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' “the environment”) also play an impor- tant role: In high density regions, such as the cores of clusters, it is more likely to find galaxies with an early-type morphol- ogy, and with overall older stellar ages with respect to the field, suggesting that the process of galaxy formation is “ac- celerated” in denser places (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Dressler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' A num- ber of environment-related mechanisms have been proposed to explain the observations in groups and clusters, includ- ing ram pressure stripping, strangulation, harassment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Those processes affect the evolution of the gas reservoir in galaxies ⋆ E-mail: iferreras@iac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='es and their subsequent star formation (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=', Pasquali 2015, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Disentangling the different effects is one of the main goals in extragalactic astrophysics, as they link the evolution of galaxies with the larger dark matter ha- los that host them (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=', Weinmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Henriques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Stellar populations can be used as a tracer of these pro- cesses by comparing their properties in carefully selected sam- ples that would appear homogeneous if the environment did not affect their star formation histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Many papers present detailed analyses of spectroscopic indicators of stellar age and metallicity (to name a few: Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Pasquali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' La Barbera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Fitzpatrick & Graves 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Goddard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Scholz-Díaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2022), with the major conclusion being that the largest difference in the stellar population content concerns the central vs satel- lite nature of these galaxies, with the former having slightly younger ages, representative of extended episodes of star for- mation, in contrast with satellites, that feature older pop- ulations, indicative of efficient quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Beyond the cen- tral/satellite dichotomy, the effect of other environment prop- erties – such as host halo mass, or cluster-centric distance – © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='01776v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='GA] 4 Jan 2023 2 Ferreras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' is discernible (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='g, Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2006, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Taranu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Pasquali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Sampaio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' However, the overall effect on the underlying stellar populations ap- pears weaker when the samples are segregated by morphology or evolutionary phase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=', La Barbera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Trussler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' In early-type galaxies, environment effects are subdominant to the more important scaling relations with local observables such as stellar mass or velocity dispersion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Ferreras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Some targeted differences can be found, for instance, in close galaxy pairs, where satellites of the same mass, orbiting more massive primaries tend to be slightly older, a potential indicator of assembly bias (Ferreras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2017, 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' A more detailed view in projected phase space finds subtle but consistent differences between the stel- lar populations of galaxies in clusters with a Gaussian or non Gaussian velocity distribution (Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' De Carvaho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2017), reflecting the state of virialisation, and thus the intensity of environment-related effects on the star formation histories (Sampaio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Backsplash galaxies are defined as those that already fell into the cluster, passing through the dense cluster core, and overshot the structure, so that they are found at larger dis- tances, typically outside of the virial radius (Gill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' In this framework, backsplash galaxies constitute very inter- esting laboratories, as they represent systems that have been strongly affected by environment-related effects, but, due to their motion within the structure, are found in the outer, lower density regions, where such effects are not expected (Balogh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Therefore, the population of backsplash galaxies would stand out with respect to the other galaxies in the same region, mainly infallers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Moreover, galaxies falling towards a cluster were likely part of a lower mass group, and therefore were also affected by the environment of the ear- lier structures, defined as pre-processsing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=', Fujita 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Numerical simulations show that a high fraction of galaxies outside of the virial radius have already plunged within the core of the corresponding cluster, thus being affected by their environment (Mamon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2004), losing a substantial frac- tion of their mass (Gill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Observational studies based on cluster samples from SDSS confirm the presence of backsplash galaxies at projected radii ∼1–2 R200 (Pimbblet 2011), with signs of quenched star formation activity (Maha- jan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Muriel & Coenda 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' The presence of backsplash galaxies is related to a more fundamental property concerning the formation of dark mat- ter halos (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Fillmore & Goldreich 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Due to the colli- sionless nature of dark matter, during the collapse and virial- isation of a structure, the particles will overshoot the nascent core and will reach some distance before falling back again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Such behaviour explains the steep density gradients found in the outer profile of dark matter halos (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Diemer & Kravtsov 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Adhikari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Xhakaj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2020), which some- how represents a limit of the bound particles with respect to the infalling matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' This signature can also be found in ve- locity space (Okumura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Similarly to dark matter, galaxies that fall into a cluster but overshoot them, reach an orbital apocenter further out, defining the so-called splash- back radius (Rsp) that could be detected in the radial profile of the galaxy distribution, where a sharp drop is expected outside of this radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' More et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' (2015) estimate Rsp be- tween 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='8 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='5 R200m (with R200m defined at 200 times the mean density of the Universe), depending on the accre- tion history of the cluster, and therefore allowing for the de- termination of assembly bias, since Rsp will depend on halo parameters other than its mass (see also Busch & White 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Sunayama & More 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Recently, a number of attempts have been made to determine the location of Rsp in galaxy clusters from weak lensing and galaxy clustering observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=', More et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Baxter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Umetsu & Diemer 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Contigiani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Murata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2020) Comparisons with numerical simulations allow us to char- acterize in detail the properties of the backsplash population and its connection to assembly bias (Busch & White 2017),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' finding that over half of all galaxies located,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' at present,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' within R200 and 2 R200 of the cluster centre have already passed through the core – where R200 is defined as the radius where the density reaches 200 times the critical value – with this population assembled over relatively recent times,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' therefore being dependent on the recent formation history of the clus- ter (Haggar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' This paper focuses on a methodology to contrast the spec- troscopic signatures of stellar populations in a sample selected in projected phase space, in order to explore the effect of en- vironment in backsplash galaxies, and constrain their frac- tional contribution to the samples located just outside of the virial radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Our methodology relies on variations of the stel- lar population content, therefore assumes that the effect of the passage through the cluster leaves some imprint on the absorption line spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Section 2 defines the two key sub- sets adopted to find backsplash galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Section 3 explores a number of observable properties, focusing on three targeted line strengths that are strongly dependent on the stellar age of the stellar populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Section 4 describes a simple method- ology based on cumulative distributions to assess the frac- tion of backsplash galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Finally, Section 5 summarises our main conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2 DEFINING THE BACKSPLASH REGION IN PROJECTED PHASE SPACE Our sample of backsplash candidates is retrieved from the compilation of Sampaio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' (2021), who made use of the galaxy group catalogue of Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' (2007), based on the Sloan Digital Sky Survey (York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2000, hereafter SDSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' This catalogue classifies galaxies in projected phase space (PPS) relative to their respective groups, where the peculiar, line-of-sight, velocity of a galaxy – measured as a fraction of the velocity dispersion within the group – is plotted against its 2D projected group-centric distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' By making compar- isons with computer simulations, the PPS diagram can be sliced in a number of zones that characterize the dynamical state of the galaxy within the group, and can give the time since the galaxy fell in that structure (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Oman & Hud- son 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Rhee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Pasquali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' The Sampaio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' (2021) sample targets Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' (2007) groups with mass M200 > 1014M⊙, therefore we can refer to these systems as clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' The properties of the 319 clusters are estimated via Virial Analysis (Lopes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2009) applied to the final member list defined by the Shiftgapper technique (De Car- vaho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Note that only members within R200 are considered in the estimate of the velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' In this paper, the radius that characterises each group is given by MNRAS 000, 1–10 (2023) The stellar populations of backsplash galaxies 3 1 2 3 4 R/R200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='0 |v |/ 25% 50% 75% 95% Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Projected phase space diagram showing contours that engulf a given fraction of the total sample, as labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' The dashed lines mark the region defined for this paper to detect backsplash galaxies, located outside of the virial radius of the structures, split into high (blue) and low (red) line of sight velocity (compared with respect to the velocity dispersion of the structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' The sample in the blue region is expected to have a higher fraction of backsplash galaxies, with respect to the general population of field/infalling galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Hereafter, all figures will use this colour coding to sepa- rate the two subsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' R200 defined as the position where the density is 200 times the critical value, as laid out in Lopes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 1 shows the PPS diagram of the sample, as a density plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' The original set consists of 570,643 galaxies, from which we have 23,631 satellites with detailed phase space informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Modulo projection effects, backsplash galaxies should be located outside of the group radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' According to Hag- gar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' (2020), over half of all galaxies located between one and two group radii (measured as R200), are backsplash galaxies, as measured at z=0 in a set of detailed numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' This fraction decreases to negligible values at R≳3 R200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Moreover, this fraction increases in dynamically relaxed clusters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Gaussian clusters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Our criterion to split the sample on the PPS diagram is aimed at testing whether the spectroscopic signature of the backsplash galaxies can be determined by focusing on the not-too-distant outskirts of clusters, split with respect to relative velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' The ex- pected quenching effects will be noticeable in the signature of the stellar populations, although we should also expect addi- tional factors affecting the stellar populations of the infalling galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' The sample is also split with respect to the dynam- ical state of the group, into Gaussian (G) and non Gaussian (nG) clusters, a criterion based on the distribution of veloc- ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' The latter represent structures with a more recent dy- namical formation history and a higher diversity regarding galaxy properties, suggesting higher levels of pre-processing (De Carvaho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Roberts & Parker 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Sampaio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' We note that Gaussian clusters are already viri- alised systems (at least within R200), where the infall of new members is, in principle, smaller than in non-Gaussian clus- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Therefore, in the vicinity of nG grous we expect to find a higher fraction of infalling galaxies, thus with overall higher levels of pre-processing, as these infallers belong to smaller groups that are in the process of merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' We also note that 0 200 Vel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' disp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' (km/s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='0 × 100 All |v| > |v| < Field Hom Field 0 200 Vel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' disp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' (km/s) Gaussian 0 200 Vel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' disp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' (km/s) Non-Gaussian Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' Distribution of stellar velocity dispersion in the subsam- ples explored in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' The “field” reference is also shown in brown, including the distribution before (dotted) and after (solid) homogeneisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' the fraction of interlopers – defined as galaxies that are within 1-3 R200 in projection, but not in 3D radial distance, r200, can be rather high (see figure 2 of Oman & Hudson 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' How- ever, we will see below that the PPS-selected sample shows strong differences with respect to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' We select galaxies in the region of PPS: R/R200 ∈ [1, 3] and |v∥|/σ < 3 (a total of 12,738 SDSS spectra), split into high- and low-velocity galaxies at |v∥| = σ, as highlighted in blue and red, respectively, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' We follow this colour cod- ing throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' From this set, we retrieve the line strength measurements from the SDSS galSpecIndx table, as defined in Brinchmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' The cross-correlation produces 11,252 galaxies, with 5,282 (1,951) classified as be- longing to Gaussian (non Gaussian) clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' The remaining 4,019 cannot be accurately classified regarding the Gaussian nature of the velocity dispersion of the cluster, but are also included in a general sample that comprise all cluster galax- ies with well-defined PPS information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content=' As comparison with a general (“field”) sample, we also extract from SDSS (Legacy) a complete set of spectra that are unconstrained in PPS, imposing a high enough S/N (>51) in the same redshift win- dow, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfz_4g/content/2301.01776v1.pdf'} +page_content='03 1, a 1D +depthwise convolution [6] is added at the beginning of the +block to form the adjacent relationship (local correlation), +as presented in Tab. 1 with ks is kernel size and r is dilation +rate. +Table 1. Details of each Conv1D block ri in TPS module. +Layer +ks +r +Included +Output size +input +− +− +✓ +T × dk +depthwise conv1d +3 +1 +ri > 1 +T × dk +conv1d +3 +ri +✓ +T × dk +layer norm +− +− +✓ +T × dk +activation (gelu) +− +− +✓ +T × dk +In addition, the output from blocks is also concatenated +and projected to the lower dimension as the same as input, +in which the most important properties are compressed to +produce a comprehensive representation. In total, the sys- +tem uses n + 1 feature vectors with each size of T × dk, +k ∈ {1, 2, 3, 4} corresponding to 4 stages in the feature +extractor step, to calculate the similarity between neighbor +frames with Euclidean distance, as done in [25], for each +stage, as shown in Fig. 2. To incorporate the difference in +spatial dimension, we add a residual connection to connect +each stage output with its input before measuring the simi- +larity between frames. Formally, let F i +k is the output of the +feature vector Ik at stage k with dilation rate ri, we obtain +n similarity vectors by the following equation +Ri +k = +F i +k + Ik +∥F i +k + Ik∥, +(2) +D(q) = ∥Ri +k,t − Ri +k,t+q∥2, +(3) +Di +k = [D(−l), . . . , D(−1), D(1), . . . , D(l)], +(4) +with D(q) denotes the Euclidean distance between frame at +time t and t + q. In this work, we consider neighbors in 1 +3 + +second for each side, in other words, l equals the frames per +second of video. And Rn+1 +k +, the comprehensive representa- +tion of F i +k, is calculated as +P = Fp([F 1 +k , F 2 +k , . . . , F n +k ]), +(5) +Rn+1 +k += +P +∥P∥, +(6) +where Fp is a 1D convolution with kernel size of 1, and P +has the same dimension with F i +k. These similarities, Rk, are +concatenated and projected into higher dimension, +Tk = Ft([R1 +k, R2 +k, . . . , Rn +k, Rn+1 +k +]), +(7) +with Ft is 1D convolution with dout filters of kernel size of +1. Consequently, this module formed a feature vector, Tk, +of size T × dout for each stage output. At this point, we +concatenate the feature vectors of 4 stages and project to a +new dimension to obtain a final feature vector, T , as +T = F([T1, T2, T3, T4]), +(8) +in which, the projection F is a 1D convolution block with +dout filters of kernel size 3 with aim to reduce the dimension +of the concatenation vector and incorporate these informa- +tion together. +Conv1D block +Conv1D block +Conv1D block +... +... +Neighbor +similarity +Neighbor +similarity +Neighbor +similarity +... +... +Concatenate and projection +... +Concatenate +and projection +... +Neighbor +similarity +... +Figure 2. Temporal pyramid similarity module. +3.3. Similarity Decoder +To decode the similarity information for further pre- +diction, ns 1D convolution blocks with dilation rates ri, +i ∈ {1, 2, ..., ns}, are stacked together to form a similar- +ity decoder (SD) block. Due to dilation rate of 1 is applied +to the previous projection block, this module employs di- +lation rates greater than 1 and does not include 1D depth- +wise convolutions. As 1D convolutions are stacked instead +of applying separately as in the previous module, this de- +coder can observe more frames to compute the output at +each timestamp. For simplicity, the same dilation rates as in +the temporal pyramid similarity module are used. In detail, +ns = n − 1 for this part and ri = 2i. Formally, the output +S of SD block is defined as +S = (Frns ◦ Frns−1 ◦ · · · ◦ Fr1) (T ), +(9) +with T from Eq. (8), and Fri denotes the 1D convolution +block with dilation ri. +3.4. Prediction Head +Our network uses two layers of 1D convolutions, fol- +lowed by a sigmoid function, to estimate the boundary score +for each frame. These scores tend to be noisy and difficult to +use only a fixed-threshold (e.g., 0.5) to get exact boundaries. +To address these issues, we use a Gaussian kernel with stan- +dard deviation of 1 and window size equal to the number of +frames in 1 second, as shown in Fig. 3. Then each score +is compared to neighbors in 0.5 seconds and a minimum +threshold of 0.1, which works similar to max pooling, to +consider as a maximum (boundary) or not. +4. Experimental Results +4.1. Dataset and Evaluation +Kinetics-GEBD +Our approach is evaluated primarily on +Kinetics-GEBD, a benchmark dataset for locating the +boundaries of generic events in video, created by Shou et +al. [23]. It consists of around 60 000 10-second videos, with +the training, validation, and testing partition ratio approxi- +mately 1:1:1. The training and testing videos are randomly +selected from Kinetics-400 [33] Train split, while the val- +idation set used all videos in Kinetics-400 Val split. Each +video is annotated by 5 annotators with average 4.77 bound- +aries. Due to the annotation of test set not available publicly, +our network is trained on training set and evaluated on vali- +dation set. +TAPOS +In addition, we conduct experiment on TAPOS +dataset [22] containing Olympics sport videos across 21 +actions. Follow [23], we perform boundaries localization +between sub-actions in each action instance, as annotated +in [22]. The average duration of instances is 9.4 seconds, +and maximum around 5 minutes. TAPOS contains 13 094 +action instances for training and 1790 instances for valida- +tion. Due to the variety of duration over instances, we split +instances into 10-second clips with overlap of 5 seconds +to perform training and evaluation. During evaluation, the +boundary scores from clips are merged with summation to +form a unique score for each timestamp in the whole action +4 + +Table 2. Kinetics-GEBD validation score with ResNet50 as the backbone at Rel.Dis. +Method +F1@0.05 +F1@0.1 +F1@0.15 +F1@0.2 +F1@0.25 +F1@0.3 +F1@0.35 +F1@0.4 +F1@0.45 +F1@0.5 +Avg +BMN [19] +0.186 +0.204 +0.213 +0.220 +0.226 +0.230 +0.233 +0.237 +0.239 +0.241 +0.223 +BMN-StartEnd [19] +0.491 +0.589 +0.627 +0.648 +0.660 +0.668 +0.674 +0.678 +0.681 +0.683 +0.640 +TCN [16] +0.588 +0.657 +0.679 +0.691 +0.698 +0.703 +0.706 +0.708 +0.710 +0.712 +0.685 +PC [23] +0.625 +0.758 +0.804 +0.829 +0.844 +0.853 +0.859 +0.864 +0.867 +0.870 +0.817 +SBoCo [15] +0.732 +0.827 +0.853 +0.877 +0.882 +0.891 +0.894 +0.899 +0.899 +0.907 +0.866 +DDM-Net [25] +0.764 +0.843 +0.866 +0.880 +0.887 +0.892 +0.895 +0.898 +0.900 +0.902 +0.873 +CVRL [17] +0.743 +0.830 +0.857 +0.872 +0.880 +0.886 +0.890 +0.893 +0.896 +0.898 +0.865 +Ours +0.770 +0.842 +0.863 +0.874 +0.880 +0.884 +0.887 +0.890 +0.891 +0.893 +0.867 +Table 3. TAPOS validation score compared to previous works. +Method +F1@0.05 +F1@0.1 +F1@0.15 +F1@0.2 +F1@0.25 +F1@0.3 +F1@0.35 +F1@0.4 +F1@0.45 +F1@0.5 +Avg +ISBA [8] +0.106 +0.170 +0.227 +0.265 +0.298 +0.326 +0.348 +0.369 +0.382 +0.396 +0.302 +TCN [16] +0.237 +0.312 +0.331 +0.339 +0.342 +0.344 +0.347 +0.348 +0.348 +0.348 +0.330 +CTM [12] +0.244 +0.312 +0.336 +0.351 +0.361 +0.369 +0.374 +0.381 +0.383 +0.385 +0.350 +TransParser [22] +0.289 +0.381 +0.435 +0.475 +0.500 +0.514 +0.527 +0.534 +0.540 +0.545 +0.474 +PC [23] +0.522 +0.595 +0.628 +0.646 +0.659 +0.665 +0.671 +0.676 +0.679 +0.683 +0.642 +DDM-Net [25] +0.604 +0.681 +0.715 +0.735 +0.747 +0.753 +0.757 +0.760 +0.763 +0.767 +0.728 +Ours +0.616 +0.701 +0.740 +0.760 +0.772 +0.780 +0.786 +0.790 +0.794 +0.796 +0.754 +instance to get the final boundaries. Similar to Kinetics- +GEBD, our network is trained on the training set and evalu- +ated on the validation set. +Evaluation +Following [23], F1 score with Relative Dis- +tance (Rel.Dis.) measurement is used to evaluate the sys- +tem performance. Rel.Dis. determines whether a detection +is correct or incorrect by calculating the error of detection, +Ed, as +Ed = |Detected point − Ground truth point| +Length of video +, +(10) +and comparing with a threshold, τ. If Ed ≤ τ, the detection +is correct, otherwise incorrect. The reported results are with +thresholds τ from 0.05 to 0.5 with a gap of 0.05. +4.2. Implementation +Because of the variety of frame rate of each video, we +sample frames at 25 fps (frame per second), as done in [2]. +For this task, video at 25 fps may contain redundant infor- +mation. We then sample video to 5 fps that led to 50 frames +per video or T = 50, which is suitable to run the infer- +ence on a usual computing resource. The network is built +with TensorFlow 2.9 and trained end-to-end with ResNet- +50 backbone and Adam optimizer at batch size of 8 in 10 +epochs. We linearly increased the learning rate from 0 to +4 × 10−4 in 2 epochs, and then used a cosine decay sched- +ule to reduce the learning rate to 4 × 10−6 in 8 epochs. The +model from the last epoch is used to generate the prediction +for evaluation. All experiments are conducted on a single +NVIDIA RTX 3090 GPU equipped machine. +4.3. Results +Table 2 shows the comparison of our approach with +the previous method on the Kinetics-GEBD validation +set with the same spatial backbone (ResNet-50). +Com- +pared to PC [23], the GEBD benchmark baseline, our sys- +tem achieves a huge improvement of almost 14.5 percent, +demonstrating the effectiveness of the proposed system. +DDM-Net [25] took consecutive frames with a gap between +them that created a longer view in their system. +Con- +sequently, they also surpass the baseline, but still has a +small gap, 0.6 percent, with our method at strict thresh- +old τ = 0.05 as multiple longer views are exploited in our +method. We also achieved a remarkable improvement com- +pared to CVRL [17] and SBoCo [15] with around 3 percent +at τ = 0.05. +The results of the comparison between our method and +previous methods on TAPOS are summarized in Tab. 3. +Since our approach is able to learn different context in +multiple view, we surpass PC [23] and DDM-Net [25] +as TAPOS contains instances with duration up to 5 min- +utes. We obtain an increasing of around 8.2% and 1.2% +on F1@0.05 compared to PC and DDM-Net, respectively. +These results prove the effectiveness of our method on short +videos, around 10 seconds, and long videos with up to 5 +minutes. +4.4. Ablation study +To evaluate the effectiveness of our approach, several ab- +lations on the network are conducted with the same training +mechanism. +5 + +Effects of stages in the feature extractor +We experiment +our system with different number of feature maps to ana- +lyze how it effects the overall performance. As high-level +feature maps (high stages) contain more specific informa- +tion compared to low-level feature maps (low stages), we +add/remove feature maps from high to low in an orderly +way as in Tab. 4. +The system has increased steadily in +terms of F1@0.05 with a difference of up to 1.22 percent +between the single stage and the combination of 4 stages. +This can be explained because each feature map level con- +tains its own knowledge that makes the different neighbor- +ing frames, which makes their combination useful for the +whole system. +Table 4. Effects of the combination of stages in feature extractor +with results on Kinetics-GEBD. +Stage +Prec@0.05 +Rec@0.05 +F1@0.05 +Stage 4 +0.6965 +0.8330 +0.7587 +Stage 3, 4 +0.6967 +0.8462 +0.7642 +Stage 2, 3, 4 +0.7026 +0.8460 +0.7677 +Stage 1, 2, 3, 4 +0.7076 +0.8450 +0.7702 +Stage 1, 2, 3 +0.7015 +0.8491 +0.7683 +Stage 1, 2 +0.6956 +0.8514 +0.7657 +Stage 1 +0.6924 +0.8373 +0.7580 +Effects of dilation rates +Table 5 shows the results when +adding dilation rates of 2i−1 to compute the similarity in +TPS module. In general, F1 scores increase consistently +around 0.1 percent for each new dilation rate and reach a +total gain of 0.35% based on the improvement 0.5% of pre- +cision. Although these increases are not remarkable, they +still show a potential further extension of how to incorpo- +rate them in an efficient manner and achieve significant im- +provements. +Table 5. Effects of dilation rates in the TPS module with results +on Kinetics-GEBD. +Modification +Prec@0.05 +Rec@0.05 +F1@0.05 +n = 1, ri = 2i−1 +0.7013 +0.8455 +0.7667 +n = 2, ri = 2i−1 +0.7045 +0.8438 +0.7679 +n = 3, ri = 2i−1 +0.7037 +0.8472 +0.7688 +n = 4, ri = 2i−1 +0.7076 +0.8450 +0.7702 +Effects of local correlation +In this study, we incorporate +residual connection (Res.Con.) and 1D depthwise convo- +lution to enrich long-term correlation with local contextual +information in the TPS module. Table 6 shows their effects +on overall system performance. According to that, Res.Con. +achieves an increase of 1.2%, while the other does not have +many effects. Consequently, it encourages refinement of +features with local context in a whole view with Res.Con. +to improve the generalization instead of on individual chan- +nels as to what depthwise convolution is doing. +Table 6. Effects of local correlation in TPS module with results on +Kinetics-GEBD, in which Res.Con. indicates residual connection. +Depthwise +conv1d +Res.Con. Prec@0.05 +Rec@0.05 +F1@0.05 +0.6943 +0.8361 +0.7586 +✓ +0.7063 +0.8470 +0.7703 +✓ +0.6933 +0.8373 +0.7585 +✓ +✓ +0.7076 +0.8450 +0.7702 +Effects of similarity decoder block +We conduct exper- +iments with a fixed dilation rate in SD block along with +the removal of SD block to compare their results, as shown +in Tab. 7. Generally, SD block helps improve performance +at least 0.8% with no dilation and at most 1.2% with pyra- +mid dilation rates. The dilation rates can improve the per- +formance, but large dilation can also cause the drop. In +case of the pyramid dilation rates, it creates balances by in- +creasing the receptive field step by step, which consistently +boosts the performance. +Table 7. +Effects of similarity decoder block with results on +Kinetics-GEBD. In these experiments, ns is set to 3. +Modification +Prec@0.05 +Rec@0.05 +F1@0.05 +Excluded SD block +0.6998 +0.8280 +0.7585 +SD block ri = 1 +0.7059 +0.8395 +0.7669 +SD block ri = 2 +0.7036 +0.8462 +0.7684 +SD block ri = 4 +0.7059 +0.8443 +0.7689 +SD block ri = 8 +0.7060 +0.8333 +0.7644 +SD block ri = 2i +0.7076 +0.8450 +0.7702 +Table 8. Effects of Gaussian smoothing in prediction head with +results on Kinetics-GEBD. +Gaussian smoothing +Prec@0.05 +Rec@0.05 +F1@0.05 +Training +Inference +0.6717 +0.8631 +0.7555 +✓ +0.7426 +0.8038 +0.7720 +✓ +0.6206 +0.9103 +0.7381 +✓ +✓ +0.7076 +0.8450 +0.7702 +Effects of Gaussian smoothing in prediction head +In +this work, the Gaussian filter (GF) is attached and runs on +6 + +0 +10 +20 +30 +40 +50 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +GF-applied score +Estimated score +GF-applied predicted boundaries +Predicted boundaries +Ground truth boundaries +Figure 3. A visualization of detected boundaries with or without GF smoothing. +the GPU together with the other parts of the network dur- +ing training and inference. If GF includes in training, the +learning process will optimize the smoothing scores instead +of the raw ones. To see its impacts on the overall perfor- +mance, we do include and exclude it during training and +inference, respectively. As we can see in Tab. 8, the recall +is very high compared to precision when GF is excluded for +inference, which makes more noise in scores with many lo- +cal maximums and produces many boundaries. On the other +hand, the network focus on the smoothing score and cause +a huge loss, nearly 3.21% that twice compared to the other +(1.65%). Figure 3 shows an example result in which GF re- +duced 4 false positive detection with Rel.Dis. threshold of +0.05. +5. Conclusion +In this work, we proposed an approach based on tem- +poral pyramid similarity to detect the boundary of generic +events in video. Our work exploited the correlation between +frames in different temporal and spatial scales in a pyramid +scheme to compute the distance between adjacent frames, +parallelly. Together, we applied this operation on multi- +ple spatial scales to combine the unique characteristics on +each scale to improve the system performance. Thanks to +1D operation for temporal modeling, we explored the tem- +poral dimension to decode the similarity information and +achieve a considerable improvement for the whole system. +Our method outperforms the previous state-of-the-art meth- +ods on the Kinetics-GEBD and TAPOS benchmarks at a +strict relative distance. More research is needed to expand +our method so that it can work with untrimmed videos, as +we only evaluate our approach for trimmed video with ac- +tion. +References +[1] Roger G Barker and Herbert F Wright. 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Neurocomputing, 358:446–455, +sep 2019. 2 +[33] Andrew Zisserman, Joao Carreira, Karen Simonyan, Will +Kay, Brian Zhang, Chloe Hillier, Sudheendra Vijaya- +narasimhan, Fabio Viola, Tim Green, Trevor Back, Paul Nat- +sev, and Mustafa Suleyman. The kinetics human action video +dataset. 2017. 2, 4 +8 + diff --git a/SdE3T4oBgHgl3EQfDgnR/content/tmp_files/load_file.txt b/SdE3T4oBgHgl3EQfDgnR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..286dd9c278d21e41f92abbb7721431587bf8cc8d --- /dev/null +++ b/SdE3T4oBgHgl3EQfDgnR/content/tmp_files/load_file.txt @@ -0,0 +1,712 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf,len=711 +page_content='Generic Event Boundary Detection in Video with Pyramid Features Van Thong Huynh, Hyung-Jeong Yang, Guee-Sang Lee, Soo-Hyung Kim* Department of AI Convergence, Chonnam National University {vthuynh,hjyang,gslee,shkim}@jnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='kr Abstract Generic event boundary detection (GEBD) aims to split video into chunks at a broad and diverse set of actions as humans naturally perceive event boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' In this study, we present an approach that considers the correlation be- tween neighbor frames with pyramid feature maps in both spatial and temporal dimensions to construct a framework for localizing generic events in video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' The features at mul- tiple spatial dimensions of a pre-trained ResNet-50 are ex- ploited with different views in the temporal dimension to form a temporal pyramid feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Based on that, the similarity between neighbor frames is calculated and pro- jected to build a temporal pyramid similarity feature vec- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' A decoder with 1D convolution operations is used to decode these similarities to a new representation that incor- porates their temporal relationship for later boundary score estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Extensive experiments conducted on the GEBD benchmark dataset show the effectiveness of our system and its variations, in which we outperformed the state-of-the- art approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Additional experiments on TAPOS dataset, which contains long-form videos with Olympic sport ac- tions, demonstrated the effectiveness of our study compared to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Introduction Temporal action localization attempts to spot the range of action instances in timestamps from the video stream with well-known including THUMOS [14], Activi- tyNet [11], HACS [31], FineAction [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' However, with the growth of video contents, the number of classes is expand- ing, and the predefined target classes could not cover com- pletely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' That can be resolved with semi-supervised or unsu- pervised learning, but it still focuses on specific actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' To increase generalization, Shou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' [23] constructed a new benchmark, Generic Event Boundary Detection (GEBD), which increases the ability to detect a board and diverse set of boundaries between taxonomy-free events, as hu- Corresponding author mans naturally perceive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Following cognitive studies [1], GEBD [23] considered four high-level causes that involve changes in subject, action, shot, environment, and object of interaction to form event boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' These changes can occur with varying speed, duration and even do not involve any scene changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' The area of GEBD is attracting considerable interest be- cause of its application in the field of video understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' The findings of GEBD can be applied to uniform sampling a video into a fixed number of frames based on generic boundaries to achieve high classification accuracy [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' It can also provide a cue to select a set of frames for video summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Furthermore, knowledge of GEBD is needed for understanding in downstream tasks [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' For ex- ample, boundary captioning means to generate description for the status change at the boundary, or boundary ground- ing that required to locate the boundary based on given de- scription, or boundary caption-video retrieval aims to re- trieve the video containing boundaries with provided de- scription from video corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Due to its taxonomy-free property, GEBD contains a va- riety of spatial and temporal characteristics that lead to diffi- culties in localizing these events as human perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Still in an early stage, only a few studies have been conducted to solve the GEBD problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' In previous studies [23, 25], the authors considered a set of consecutive frames as input, and its label corresponds to the label of the center frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Although it reduced the computational burden for a single forward pass, but needed to repeat that process many times for the whole video and also lacked the relationships be- tween frames in the longer view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Besides that, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' [17] worked on the entire video but in the compressed domain, which caused deficiencies and reduced their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' In this study, we consider the entire video as an input that takes advantage of the complete semantic relationship between frames to localize event boundaries, and divide the video into meaningful units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' First, based on the observation that the output from different layers of CNN-based architec- ture contains its characteristic to distinguish with the oth- ers, we collected features for each frame on multiple scales of spatial dimension, with a CNN backbone on its way to 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='04288v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='CV] 11 Jan 2023 downsampling the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Second, motivated by the fact that two adjacent frames are almost similar in most of the cases in a video, we compute the similarity between adja- cent frames by exploiting the correlation between frames in short and longer views together with different dilation rates in 1D temporal convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' In addition, our work also ex- ploits the pyramid temporal relationship of the similarities to decode them into boundary scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' In summary, the main contributions of this study are as follows: We exploited the characteristic at different spatial di- mensions to distinguish frames at multiple levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Taking advantage of dilation rate in 1D convolution operation, we propose to use temporal pyramid sim- ilarity (TPS) module that builds upon this operation with dilation rates to construct the similarity between frames in multiple views to solve the GEBD problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' To enrich the information in higher views, a residual connection is used to integrate spatial and temporal in- formation, and a 1D depthwise convolution is inserted at the beginning of these views to incorporate local in- formation to them without increasing too much com- putational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Thanks to stacking multiple 1D convolution layers with different dilation rates to decode the similarity representation, our framework obtained improvements in performance compared to the state-of-the-art ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Related Works Temporal modeling for spatial features Although 3D convolution is designed as a native architecture to process a sequence of images [2,13,26], but it takes a lot of comput- ing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Leveraging 2D CNN to extract spatial fea- tures and then modeling temporal relationship with other architectures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=', 1D CNN, LSTM), is one approach to tackle that problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' The authors in [30] and [32] parti- tioned feature map to various levels and applied pooling operation on partitions, then concatenate results to achieve temporal pyramid features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' [27] was inspired by DeepLab [4, 5], they deployed multiple parallel tempo- ral convolutions with different atrous rates to capture mul- tiscale contextual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' MS-TCN++ [18] stacked multi- ple blocks of dual dilated temporal convolutions to capture temporal dependencies in a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Temporal activity detection in video Early methods [3, 7, 19, 24] formulate the task to proposal generation and ac- tion classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' The generation of action proposals aims to find candidate segments, including a pair of starting and ending boundaries, in a long video that may contain ac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Shou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' [24] deployed 3D convolution operations to classify segments generated from sliding windows of var- ied length into two nodes (action or background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' The au- thors in [3,7,19] used 1D convolution operations to capture context information between features sampling on multiple scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' TURN [9] divided a long video into video units and calculated CNN features for each unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' TURN employed an anchor unit to construct clip pyramid features and compute confidence scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' MGG [20] deployed multiple perspec- tives of granularity, depending on visual characteristics with embedded position information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' [29] employed background constraints that take advantage of the rich in- formation about the action and background to suppress low quality proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Generic event boundary detection Shou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' [23] in- troduced the first benchmark dataset to detect generic event boundaries, based on Kinetics-400 videos [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' They extended the boundary matching (BM) mechanism [19], which considered each pair of starting and ending bound- aries as a proposal and computed their confidence scores, to determine these boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' The authors in [15] exploited the pairwise self-similarity between video frames to con- struct a temporal self-similarity matrix (TSM) for the en- tire video to use as an intermediate stage and integrated with contrastive learning to produce boundary indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' [17] encoded GOPs (group of pictures separated from video with modern codecs) based on their I-frames and P- frames to form unified representations and explore temporal dependence with the temporal contrastive module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' DDM- Net [25] leveraged multi-level features of space and scale for calculating the difference at multiple scales and con- struct multi-level dense difference maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' These maps are then aggregated and fused with progressive attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Method To estimate the boundary score for every frame in a video with T frames, the proposed systems consider the similarity between each frame and their neighbors with the encoded features in a multi-view manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' These similarities are then decoded with 1D CNNs and post-processed with a Gaus- sian filter for smoothing and reducing noise on the estimated scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' An illustration of our system is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Spatial Pyramid Features Over the last decade, many deep learning architectures have been developed to tackle the image classification task, which can be used as a base for the other downstream tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=', activity recognition, video classification), with con- tinuous improvement over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Some of them are con- sidered standard architectures when constructing the base- line model to solve the other problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Following prior 2 Similaritty decoder .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Feature extractor Average pooling Input frame Stage 1 Stage 2 Stage 3 Stage 4 Average pooling Average pooling CNN Backbone Average pooling Concatenate and projection Conv1D block Conv1D block Conv1D block .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Prediction head Temporal pyramid similarity Temporal pyramid similarity Temporal pyramid similarity Temporal pyramid similarity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='. 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='5 T Boundary scores T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' An overview of the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' works [15, 17, 23, 25], our network is built upon ResNet- 50 architecture [10], pre-trained on ImageNet, to construct spatial features for each frame in video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Given a frame re- sized to 224×224, our model leverages the output from the last convolution blocks in which the input is downsampled in a pyramid scheme by 4, 8, 16 and 32 times as they contain the most detail and vital information for each size of spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' The average pooling is applied on the spatial dimensions of these outputs to obtain the 1D feature vec- tor for each of them, as illustrated in Feature extractor part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Based on that, this work takes advantage of 4 fea- ture vectors of size T ×di for a video with T frames, where di = 2i+7 with i ∈ {1, 2, 3, 4} as produced by ResNet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Temporal Pyramid Similarity Based on the observation that two adjacent frames are almost similar in most cases when recording video, we per- form 1D convolution operations with different dilation rates separately to achieve multiple views changing from short- to long-term correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Given a sequence S and a filter K of size 2m + 1 with m ∈ N, convolution G of these inputs with dilation rate r is formulated as follows: Gr i = m � j=−m Kj · Si+r·j (1) where i + r · j accounts for the dependence on the past and future of an output at time i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' This work deploys n = 4 blocks with kernel size of 3 and dilation rates ri = 2i−1 with i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=', n}, implies the system can observe up to 8 frames in the temporal dimension to form the representa- tions for obtaining the similarities between frames and their neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' To exploit the richer relationship between short- term and long-term correlation, for blocks with ri > 1, a 1D depthwise convolution [6] is added at the beginning of the block to form the adjacent relationship (local correlation), as presented in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 1 with ks is kernel size and r is dilation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Details of each Conv1D block ri in TPS module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Layer ks r Included Output size input − − ✓ T × dk depthwise conv1d 3 1 ri > 1 T × dk conv1d 3 ri ✓ T × dk layer norm − − ✓ T × dk activation (gelu) − − ✓ T × dk In addition, the output from blocks is also concatenated and projected to the lower dimension as the same as input, in which the most important properties are compressed to produce a comprehensive representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' In total, the sys- tem uses n + 1 feature vectors with each size of T × dk, k ∈ {1, 2, 3, 4} corresponding to 4 stages in the feature extractor step, to calculate the similarity between neighbor frames with Euclidean distance, as done in [25], for each stage, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' To incorporate the difference in spatial dimension, we add a residual connection to connect each stage output with its input before measuring the simi- larity between frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Formally, let F i k is the output of the feature vector Ik at stage k with dilation rate ri, we obtain n similarity vectors by the following equation Ri k = F i k + Ik ∥F i k + Ik∥, (2) D(q) = ∥Ri k,t − Ri k,t+q∥2, (3) Di k = [D(−l), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' , D(−1), D(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' , D(l)], (4) with D(q) denotes the Euclidean distance between frame at time t and t + q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' In this work, we consider neighbors in 1 3 second for each side, in other words, l equals the frames per second of video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' And Rn+1 k , the comprehensive representa- tion of F i k, is calculated as P = Fp([F 1 k , F 2 k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' , F n k ]), (5) Rn+1 k = P ∥P∥, (6) where Fp is a 1D convolution with kernel size of 1, and P has the same dimension with F i k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' These similarities, Rk, are concatenated and projected into higher dimension, Tk = Ft([R1 k, R2 k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' , Rn k, Rn+1 k ]), (7) with Ft is 1D convolution with dout filters of kernel size of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Consequently, this module formed a feature vector, Tk, of size T × dout for each stage output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' At this point, we concatenate the feature vectors of 4 stages and project to a new dimension to obtain a final feature vector, T , as T = F([T1, T2, T3, T4]), (8) in which, the projection F is a 1D convolution block with dout filters of kernel size 3 with aim to reduce the dimension of the concatenation vector and incorporate these informa- tion together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Conv1D block Conv1D block Conv1D block .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Neighbor similarity Neighbor similarity Neighbor similarity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Concatenate and projection .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Concatenate and projection .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Neighbor similarity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Temporal pyramid similarity module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Similarity Decoder To decode the similarity information for further pre- diction, ns 1D convolution blocks with dilation rates ri, i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=', ns}, are stacked together to form a similar- ity decoder (SD) block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Due to dilation rate of 1 is applied to the previous projection block, this module employs di- lation rates greater than 1 and does not include 1D depth- wise convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' As 1D convolutions are stacked instead of applying separately as in the previous module, this de- coder can observe more frames to compute the output at each timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' For simplicity, the same dilation rates as in the temporal pyramid similarity module are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' In detail, ns = n − 1 for this part and ri = 2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Formally, the output S of SD block is defined as S = (Frns ◦ Frns−1 ◦ · · · ◦ Fr1) (T ), (9) with T from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' (8), and Fri denotes the 1D convolution block with dilation ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Prediction Head Our network uses two layers of 1D convolutions, fol- lowed by a sigmoid function, to estimate the boundary score for each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' These scores tend to be noisy and difficult to use only a fixed-threshold (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='5) to get exact boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' To address these issues, we use a Gaussian kernel with stan- dard deviation of 1 and window size equal to the number of frames in 1 second, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Then each score is compared to neighbors in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='5 seconds and a minimum threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='1, which works similar to max pooling, to consider as a maximum (boundary) or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Experimental Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Dataset and Evaluation Kinetics-GEBD Our approach is evaluated primarily on Kinetics-GEBD, a benchmark dataset for locating the boundaries of generic events in video, created by Shou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' It consists of around 60 000 10-second videos, with the training, validation, and testing partition ratio approxi- mately 1:1:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' The training and testing videos are randomly selected from Kinetics-400 [33] Train split, while the val- idation set used all videos in Kinetics-400 Val split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Each video is annotated by 5 annotators with average 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='77 bound- aries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Due to the annotation of test set not available publicly, our network is trained on training set and evaluated on vali- dation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' TAPOS In addition, we conduct experiment on TAPOS dataset [22] containing Olympics sport videos across 21 actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Follow [23], we perform boundaries localization between sub-actions in each action instance, as annotated in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' The average duration of instances is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='4 seconds, and maximum around 5 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' TAPOS contains 13 094 action instances for training and 1790 instances for valida- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Due to the variety of duration over instances, we split instances into 10-second clips with overlap of 5 seconds to perform training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' During evaluation, the boundary scores from clips are merged with summation to form a unique score for each timestamp in the whole action 4 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Kinetics-GEBD validation score with ResNet50 as the backbone at Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='Dis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Method F1@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 F1@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='1 F1@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='15 F1@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='2 F1@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='25 F1@0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Similar to Kinetics- GEBD, our network is trained on the training set and evalu- ated on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Evaluation Following [23], F1 score with Relative Dis- tance (Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='Dis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=') measurement is used to evaluate the sys- tem performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='Dis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' determines whether a detection is correct or incorrect by calculating the error of detection, Ed, as Ed = |Detected point − Ground truth point| Length of video , (10) and comparing with a threshold, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' If Ed ≤ τ, the detection is correct, otherwise incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' The reported results are with thresholds τ from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='5 with a gap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Implementation Because of the variety of frame rate of each video, we sample frames at 25 fps (frame per second), as done in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' For this task, video at 25 fps may contain redundant infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' We then sample video to 5 fps that led to 50 frames per video or T = 50, which is suitable to run the infer- ence on a usual computing resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' The network is built with TensorFlow 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='9 and trained end-to-end with ResNet- 50 backbone and Adam optimizer at batch size of 8 in 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' We linearly increased the learning rate from 0 to 4 × 10−4 in 2 epochs, and then used a cosine decay sched- ule to reduce the learning rate to 4 × 10−6 in 8 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' The model from the last epoch is used to generate the prediction for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' All experiments are conducted on a single NVIDIA RTX 3090 GPU equipped machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Results Table 2 shows the comparison of our approach with the previous method on the Kinetics-GEBD validation set with the same spatial backbone (ResNet-50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Com- pared to PC [23], the GEBD benchmark baseline, our sys- tem achieves a huge improvement of almost 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='5 percent, demonstrating the effectiveness of the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' DDM-Net [25] took consecutive frames with a gap between them that created a longer view in their system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Con- sequently, they also surpass the baseline, but still has a small gap, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='6 percent, with our method at strict thresh- old τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 as multiple longer views are exploited in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' We also achieved a remarkable improvement com- pared to CVRL [17] and SBoCo [15] with around 3 percent at τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' The results of the comparison between our method and previous methods on TAPOS are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Since our approach is able to learn different context in multiple view, we surpass PC [23] and DDM-Net [25] as TAPOS contains instances with duration up to 5 min- utes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' We obtain an increasing of around 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='2% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='2% on F1@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 compared to PC and DDM-Net, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' These results prove the effectiveness of our method on short videos, around 10 seconds, and long videos with up to 5 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Ablation study To evaluate the effectiveness of our approach, several ab- lations on the network are conducted with the same training mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 5 Effects of stages in the feature extractor We experiment our system with different number of feature maps to ana- lyze how it effects the overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' As high-level feature maps (high stages) contain more specific informa- tion compared to low-level feature maps (low stages), we add/remove feature maps from high to low in an orderly way as in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' The system has increased steadily in terms of F1@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 with a difference of up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='22 percent between the single stage and the combination of 4 stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' This can be explained because each feature map level con- tains its own knowledge that makes the different neighbor- ing frames, which makes their combination useful for the whole system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Effects of the combination of stages in feature extractor with results on Kinetics-GEBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Stage Prec@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 Rec@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 F1@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 Stage 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='6965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7587 Stage 3, 4 0.' 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+page_content='7076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8450 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7702 Stage 1, 2, 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8491 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7683 Stage 1, 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='6956 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8514 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7657 Stage 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='6924 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8373 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7580 Effects of dilation rates Table 5 shows the results when adding dilation rates of 2i−1 to compute the similarity in TPS module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' In general, F1 scores increase consistently around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='1 percent for each new dilation rate and reach a total gain of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='35% based on the improvement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='5% of pre- cision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Although these increases are not remarkable, they still show a potential further extension of how to incorpo- rate them in an efficient manner and achieve significant im- provements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Effects of dilation rates in the TPS module with results on Kinetics-GEBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Modification Prec@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 Rec@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 F1@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 n = 1, ri = 2i−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8455 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7667 n = 2, ri = 2i−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8438 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7679 n = 3, ri = 2i−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8472 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7688 n = 4, ri = 2i−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8450 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7702 Effects of local correlation In this study, we incorporate residual connection (Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='Con.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=') and 1D depthwise convo- lution to enrich long-term correlation with local contextual information in the TPS module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Table 6 shows their effects on overall system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' According to that, Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='Con.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' achieves an increase of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='2%, while the other does not have many effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Consequently, it encourages refinement of features with local context in a whole view with Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='Con.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' to improve the generalization instead of on individual chan- nels as to what depthwise convolution is doing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Effects of local correlation in TPS module with results on Kinetics-GEBD, in which Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='Con.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' indicates residual connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Depthwise conv1d Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='Con.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Prec@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 Rec@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 F1@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='6943 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8361 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7586 ✓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8470 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7703 ✓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='6933 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8373 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7585 ✓ ✓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8450 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7702 Effects of similarity decoder block We conduct exper- iments with a fixed dilation rate in SD block along with the removal of SD block to compare their results, as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Generally, SD block helps improve performance at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8% with no dilation and at most 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='2% with pyra- mid dilation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' The dilation rates can improve the per- formance, but large dilation can also cause the drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' In case of the pyramid dilation rates, it creates balances by in- creasing the receptive field step by step, which consistently boosts the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Effects of similarity decoder block with results on Kinetics-GEBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' In these experiments, ns is set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Modification Prec@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 Rec@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 F1@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 Excluded SD block 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='6998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8280 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7585 SD block ri = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8395 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7669 SD block ri = 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8462 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7684 SD block ri = 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8443 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7689 SD block ri = 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8333 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7644 SD block ri = 2i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8450 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7702 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Effects of Gaussian smoothing in prediction head with results on Kinetics-GEBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Gaussian smoothing Prec@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 Rec@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 F1@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05 Training Inference 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='6717 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8631 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7555 ✓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7426 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7720 ✓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='6206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='9103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7381 ✓ ✓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='8450 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='7702 Effects of Gaussian smoothing in prediction head In this work, the Gaussian filter (GF) is attached and runs on 6 0 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='6 GF-applied score Estimated score GF-applied predicted boundaries Predicted boundaries Ground truth boundaries Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' A visualization of detected boundaries with or without GF smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' the GPU together with the other parts of the network dur- ing training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' If GF includes in training, the learning process will optimize the smoothing scores instead of the raw ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' To see its impacts on the overall perfor- mance, we do include and exclude it during training and inference, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' As we can see in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 8, the recall is very high compared to precision when GF is excluded for inference, which makes more noise in scores with many lo- cal maximums and produces many boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' On the other hand, the network focus on the smoothing score and cause a huge loss, nearly 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='21% that twice compared to the other (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='65%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Figure 3 shows an example result in which GF re- duced 4 false positive detection with Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='Dis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Conclusion In this work, we proposed an approach based on tem- poral pyramid similarity to detect the boundary of generic events in video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Our work exploited the correlation between frames in different temporal and spatial scales in a pyramid scheme to compute the distance between adjacent frames, parallelly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Together, we applied this operation on multi- ple spatial scales to combine the unique characteristics on each scale to improve the system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Thanks to 1D operation for temporal modeling, we explored the tem- poral dimension to decode the similarity information and achieve a considerable improvement for the whole system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Our method outperforms the previous state-of-the-art meth- ods on the Kinetics-GEBD and TAPOS benchmarks at a strict relative distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' More research is needed to expand our method so that it can work with untrimmed videos, as we only evaluate our approach for trimmed video with ac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' References [1] Roger G Barker and Herbert F Wright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' Midwest and its chil- dren: The psychological ecology of an american town.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' The kinetics human action video dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} +page_content=' 2, 4 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SdE3T4oBgHgl3EQfDgnR/content/2301.04288v1.pdf'} diff --git a/StA0T4oBgHgl3EQfD_-h/content/2301.02012v1.pdf b/StA0T4oBgHgl3EQfD_-h/content/2301.02012v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0842c7504cc498b1d9da75d92000667018a09180 --- /dev/null +++ b/StA0T4oBgHgl3EQfD_-h/content/2301.02012v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:020b15d1455bdbee70609a1c9d4d10eb9a79f9af170fe0c138dc8e633ad2e9ff +size 5898888 diff --git a/T9AzT4oBgHgl3EQfJvsR/content/tmp_files/2301.01083v1.pdf.txt b/T9AzT4oBgHgl3EQfJvsR/content/tmp_files/2301.01083v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1b0e173f676549ed35bdf4c70700d148a77f3996 --- /dev/null +++ b/T9AzT4oBgHgl3EQfJvsR/content/tmp_files/2301.01083v1.pdf.txt @@ -0,0 +1,796 @@ +arXiv:2301.01083v1 [hep-ph] 3 Jan 2023 +The dipole cross section by the unintegrated gluon distribution at small x +G.R.Boroun∗ +Department of Physics, Razi University, Kermanshah 67149, Iran +(Dated: January 4, 2023) +We apply a previously developed scheme to consistently include the improved saturation model +for the unintegrated gluon distribution (UGD) in order to evaluate, in the framework of kt +factorization, at small x at the next-to-leading order (NLO) in αs. We start the unintegrated gluon +distribution with a parametrization of the deep inelastic structure function for electromagnetic +scattering with protons, and then extract the color dipole cross section, which preserves its behavior +success in a wide range of k2 +t in comparisons with the UGD models (M. Hentschinski, A. Sabio Vera +and C. Salas (HSS), I.P.Ivanov and N.N.Nikolaev (IN) and G. Watt, A.D. Martin and M.G. Ryskin +(WMR)) . These results show that the geometric scaling holds for the improved saturation model +in a wide kinematic region rQs, and are comparable with the Golec-Biernat-W¨usthoff (GBW) model. +I. Introduction +Although our knowledge of the proton structure at +small-x is very limited, novel opportunities will be +opened at new-generation facilities (Electron-Ion Col- +lider(EIC), High-Luminosity Large Hadron Collider (HL- +LHC), Forward Physics Facility (FPF)). Combining the +information coming from dipole cross sections and pT - +unintegrated densities could play an important role. In +particular, polarized amplitudes and cross sections for +the exclusive electroproduction of ρ and φ mesons at the +Hadron-Electron Ring Accelerator (HERA) and the EIC +are very sensitive to the unintegrated gluon distribution +(UGD) model adopted, whereas forward Drell-Yan dilep- +ton distributions at the Large Hadron Collider beauty +(LHCb) are very sensitive to next-to-leading logarithmic +corrections. Indeed, the dipole cross section is directly +connected via a Fourier transform to the small-x UGD, +whose evolution in x is regulated by the Balitsky- Fadin- +Kuraev-Lipatov (BFKL) equation [1]. The BFKL equa- +tion governs the evolution of the UGD, where the kT - +factorization is used in the high energy limit in which +the QCD interaction is described in terms of the quan- +tity which depends on the transverse momentum of the +gluon [2]. +The object of the BFKL evolution equation at very small +x is the differential gluon structure function1 of proton +f(x, k2 +t ) = ∂xg(x, µ2) +∂lnµ2 +|µ2=k2 +t +(1) +which emerges in the color dipole picture (CDP) of in- +clusive deep inelastic scattering (DIS) and diffractive DIS +into dijets [3]. Here x and k2 +t being the fractional mo- +mentum of proton carried by gluon and the transverse +∗Electronic address: boroun@razi.ac.ir +1 Eq.(1) is modified with the Sudakov form factor as x increase. +momentum of gluon respectively. Unintegrated distribu- +tions are required to describe measurements where trans- +verse momenta are exposed explicitly. +The color dipole picture (CDP) [4] has been introduced +to study a wide variety of small x inclusive and diffractive +processes at HERA. The CDP, at small x, gives a clear +interpretation of the high-energy interactions, where is +characterized by high gluon densities because the proton +structure is dominated by dense gluon systems [5-7] and +predicts that the small x gluons in a hadron wavefunction +should form a Color Glass Condensate [8]. Dipole rep- +resentation provides a convenient description of DIS at +small x. There, the scattering between the virtual pho- +ton γ∗ and the proton is seen as the color dipole where +the transverse dipole size r and the longitudinal momen- +tum fraction z with respect to the photon momentum +are defined. The amplitude for the complete process is +simply the production of these subprocess amplitudes, as +the DIS cross section is factorized into a light-cone wave +function and a dipole cross section. +Using the optical +theorem, this leads to the following expression for the +γ∗p cross-sections +σγ∗p +L,T (x, Q2) = +� +dzd2r|ΨL,T(r, z, Q2)|2σdip(x, r), +(2) +where subscripts L and T referring to the transverse and +longitudinal polarization state of the exchanged boson. +Here ΨL,T are the appropriate spin averaged light-cone +wave functions of the photon and σdip(x, r) is the dipole +cross-section which related to the imaginary part of the +(qq)p forward scattering amplitude. The variable z, with +0 ≤ z ≤ 1, characterizes the distribution of the momenta +between quark and antiquark. The square of the photon +wave function describes the probability for the occurrence +of a (qq) fluctuation of transverse size with respect to the +photon polarization. +Another framework which can be used for calculating +the parton distributions is based on the DGLAP evo- +lution [9]. +Deep inelastic electron-proton scattering is +described in terms of scale dependent parton densities + +2 +q(x, µ2) and g(x, µ2) [10], where the integrated gluon dis- +tribution (xg(x, µ2)) is defined through the unintegrated +gluon distribution (f(x, k2 +t )) by +xg(x, µ2) ≡ +� µ2 dk2 +t +k2 +t +f(x, k2 +t ). +(3) +The unintegrated gluon distribution is related to the +dipole cross section [2, 11] +σ(x, r) = 8π2 +Nc +� dkt +k +3 +t +[1 − J0(ktr)]αsf(x, k2 +t ). +(4) +A novel formulation of the UGD for DIS in a way that +accounts for the leading powers in both the Regge and +Bjorken limits is presented in Ref.[12]. In this way, the +UGD is defined by an explicit dependence on the longi- +tudinal momentum fraction x which entirely spans both +the dipole operator and the gluonic Parton Distribution +Function (PDF). +In addition to the gluon momentum derivative model +(i.e., Eq.(1)), several other models [3-4, 13-15] for the +UGD have also been proposed so far. A comparison be- +tween these models can be found in Refs.[16-17]. The +authors in Ref. [13] presented an x-independent model +(ABIPSW) of the UGD where merely coincides with the +proton impact factor by the following form +f(x, k2 +t ) = +A +4π2M 2 +� +k2 +t +M 2 + k2 +t +� +, +(5) +where M is a characteristic soft scale and A is the nor- +malisation factor. +The authors in Ref. [3] presented a UGD soft-hard model +(IN) in the large and small kt regions by the following +form +f(x, k2 +t ) = f (B) +soft(x, k2 +t ) +k2 +s +k2s + k2 +t ++ fhard(x, k2 +t ) +k2 +t +k2 +h + k2 +t +, (6) +where the soft and the hard components are defined in +[3]. +The UGD model was considered in [14] to used in the +study of DIS structure functions and takes the form of +a convolution between the BFKL gluon Green,s function +and a leading-order (LO) proton impact factor, where +has been employed in the description of single-bottom +quark production at LHC and to investigate the photo- +production of J/Ψ and Υ, by the following form (HSS +model) +f(x, k2 +t , Mh) = +� +∞ +−∞ +dν +2π2 C Γ(δ − iν − 1 +2) +Γ(δ) +( 1 +x)χ( 1 +2 +iν) +×( k2 +t +Q2 +0 +) +1 +2 +iν +� +1 + α2 +sβ0χ0( 1 +2 + iν) +8Nc +log( 1 +x) +× +� +− ψ(1 +2 + iν) − log k2 +t +M 2 +h +�� +, +(7) +where χ0( 1 +2 + iν) and χ(γ) are respectively the LO and +the next-to-leading order (NLO) eigenvalues of the BFKL +kernel and β0 = 11 − 2 +3nf with nf the number of active +quarks. Here αs = 3 +παs(µ2) with µ2 = Q0Mh where Mh +plays the role of the hard scale which can be identified +with the photon virtuality, +� +Q2. +The authors in Ref. [15] presented a UGD model (Watt- +Martin-Ryskin (WMR) model) where depends on an +extra-scale µ, fixed at Q, by the following form +f(x, k2 +t , µ2) = Tg(k2 +t , µ2)αs(k2 +t ) +2π +� 1 +x +dz +� � +q +Pgq(z)x +z q(x +z , k2 +t ) ++Pgg(z)x +z g(x +z , k2 +t )Θ( +µ +µ + kt +− z) +� +, +(8) +where Tg(k2 +t , µ2) gives the probability of evolving from +the scale kt to the scale µ without parton emission and +P , +ijs are the splitting functions. +Golec-Biernat-Wusthoff (GBW) [4] presented a UGD +model where derives from the effective dipole cross sec- +tion σ(x, r) for the scattering of a qq pair of a nucleon +as2 +f(x, k2 +t ) = k4 +t σ0 +R2 +0(x) +2π +e−k2 +t R2 +0(x), +(9) +with R2 +0(x) += +1 +GeV2 ( x +x0 )λp and the following values +σ0 = 23.03 mb, λp = 0.288 and x0 = 3.04×10−4. +Although one of them (the HSS one) was fitted to +reproduce DIS structure functions, the study of other +reactions has provided an evidence that the UGD is +not yet well known. In this paper we use the DGLAP- +improved saturation model with respect to the UGD in +the proton to access the dipole cross section at low x. +II. Method +The interaction of the qq pair with the proton is de- +scribed by the dipole cross section. It is related to the +gluon density in the target by the kT -factorization for- +mula [3-4,11] +σ(x, r) = +� d2kt +k +4 +t +∂xg(x, k2 +t ) +∂lnk2 +t +[1 − eikt.r][1 − e−ikt.r],(10) +where the relation between f(x, k2 +t ) and xg(x, k2 +t ) defined +through Eq.(1). The integrated gluon density is related +2 The reader can be referee to Refs.[3-4, 13-15] for a meticulous +treatment of the parameters. + +3 +to the proton structure function [18] +xg(x, k2 +t ) = +� 1 +x +� +DF 2(ln 1 +y , k2 +t )J(as(k2 +t ), ln y +x) +−F2(ln 1 +y, k2 +t )M(as(k2 +t ), ln y +x) +�dy +y , (11) +where as(k2 +t ) = αs(k2 +t )/4π, DF2(x, k2 +t ) = +∂ +∂lnk2 +t F2(x, k2 +t ) +and F2(x, k2 +t ) is the parametrization of the proton struc- +ture function [19]. Eq.(11) is obtained [18] for the gluon +density at next-to-leading order (NLO) approximation +with respect the Laplace transform method. The stan- +dard representation for QCD couplings in NLO (within +the MS-scheme) approximation reads +αNLO +s +(k2 +t ) = +4π +β0 ln k2 +t +Λ2 +� +1 − β1 +β2 +0 +ln ln k2 +t +Λ2 +ln k2 +t +Λ2 +� +, +(12) +where β0 and β1 are the one and two loop correction to +the QCD β-function and Λ is the QCD cut-off parameter. +The functions J and M in Eq.(11) are defined by the +inverse Laplace transform method as +�J(υ, k2 +t )≡L−1[k(s, k2 +t ); υ], +� +M(υ, k2 +t )≡L−1[h(s, k2 +t ); υ], +(13) +where �f(υ)≡f(e−υ) and υ = ln(1/x). The functions k +and h in s-space reads +k(as(k2 +t ), s) = [< e2 > Θf(as(k2 +t ), s)]−1, +h(as(k2 +t , s) = Φf(as(k2 +t ), s)k(as(k2 +t ), s), +(14) +where < e2 > is the average of the charge e2 for the active +quark flavors nf , < e2 >= n−1 +f +�nf +i=1 e2 +i . The Laplace +transform of the splitting functions (Pqq and Pqg) are +given by +Φf(as(k2 +t ), s) ≡ L[e−υ +1 +� +φ=0 +aφ+1 +s +(k2 +t ) �P (φ) +qq (υ); s], +Θf(as(k2 +t ), s) ≡ L[e−υ +1 +� +φ=0 +aφ+1 +s +(k2 +t ) �P (φ) +qg (υ); s], (15) +where at small x, the kernels at NLO approximation are +given by the following forms +Θf(as, s) ≃ 2nfas +� +1 +1 + s − +2 +2 + s + +2 +3 + s +� ++a2 +sCATf +�40 +9s +� +, +Φf(as, s) ≃ as +� +4 − 8 +3( +1 +1 + s + +1 +2 + s + 2S1(s) +� ++a2 +sCF Tf +�40 +9s +� +, +(16) +where S1(s) = ψ(s+1)+γE, where ψ(x) is the digamma +function and γE = 0.5772156... is Euler constant. Here +CA = Nc = 3, CF = N 2 +c −1 +2Nc += 4 +3 and Tf = 1 +2nf. +In next section we consider the UGD due to the +parametrization of the proton structure function as +f(x, k2 +t ) = +∂ +∂lnk2 +t +� � 1 +x +� +DF 2(ln 1 +y, k2 +t )J(as(k2 +t ), ln y +x) +−F2(ln 1 +y, k2 +t )M(as(k2 +t ), ln y +x) +�dy +y +� +, +(17) +then we obtain the color dipole cross section as +σ(x, r) = +� � 2dktdΘ +k +3 +t +[Eq.(17)][1 − cos(ktrcosΘ)]. (18) +10 +-1 +10 +0 +10 +1 +10 +2 +-2 +-1 +0 +1 +2 +3 +4 +5 +6 +7 +10 +0 +10 +1 +10 +2 +f(x,k +t +2 +) +k +t +2 + [GeV +2 +] + x=10 +-3 + Improved Model + GBW Model +k +t +2 + [GeV +2 +] +x=10 +-3 + + Improved Model +FIG. 1: Left: The UGD as a function of k2 +t with respect to the +improved model compared with the GBW model at x = 10−3. +Right: The error bands are due to the statistical uncertainties +in the coefficients of the parametrization F2 in [19]. +3. Numerical Results +The UGD is obtained directly in terms of the param- +eterization of the structure function F2(x, k2 +t ) and its +derivative. The resulting UGD with the k2 +t -dependence +due to the parametrization of the proton structure +function are shown in Fig. 1. In this figure, we plot the +k2 +t dependence of the UGD at x = 10−3 and compared +the unintegrated gluon distribution behavior due to +the improved saturation model with the GBW model. +An enhancement and then depletion is observable in +the improved and GBW models. These picks occur at +k2 +t ≈10 GeV2 and k2 +t ≈1 GeV2 in the improved and GBW +models respectively. As we can seen, the UGD behavior +in the improved saturation model, in a wide range of +k2 +t , is softer than the GBW model. The error bands in +Fig.1 in the improved saturation model are due to the +uncertainties in the coefficients of the parametrization + +4 +of the proton structure function. +In Fig. 2 we show the kt distributions of three different +10 +-1 +10 +0 +10 +1 +10 +2 +10 +-2 +10 +-1 +10 +0 +10 +1 +10 +2 + + +f(x , k +t +2 +) +k +t +2 + [GeV +2 +] + HSS + IN-CT14 LO + WMR-CT14 LO + Improved Model +FIG. 2: The unintegrated gluon distribution f(x, k2 +t ) obtained +from the improved saturation model (dashed-symbol), as a +function k2 +t at x = 10−3, compared with the HSS [14], IN [3] +and WMR [15] models. +unintegrated gluons at x = 10−3. This figure compares +the results of the improved saturation model, based on +the parametrization of the proton structure function, +with the HSS [14], IN [3] and WMR [15] models. We +observe that the improved saturation result is compa- +rable with the HSS and IN models in a wide range of +k2 +t . The differences are not large, however there is some +suppression due to the models at large and small values +of k2 +t . The continuous behavior of the UGD in our model +with increases of k2 +t is due to the gluon and quark terms +included in the improved saturation model. +In Fig.3, we have calculated the improved saturation +model with respect to the unintegrated gluon distribu- +tion to the ratio σdip/σ0 in a wide range of the dipole +size at the NLO approximation. Results of calculations +and comparison with the GBW model [4] for x = 10−3 +are presented in this figure. +The effective parameters +in the GBW model have been extracted from a fit of +the HERA data as λ = 0.288 and x0 = 3.04×10−4. +These corrections to the ratio of color dipole cross +sections at NLO approximation are comparable with +the GBW model. For rQs≥1, the dashed curve (central +values) merge due to geometric scaling in the dipole +cross section in this region. In the right-hand of Fig.3, +a particular interests present the ratio σdip/σ0 defined +by the scaling variable rQs, which means that the +scattering amplitude and corresponding cross sections +can scale on the dimensionless scale rQs. +In conclusion, we study the unintegrated gluon distri- +bution from a parameterization of the proton structure +10 +-2 +10 +-1 +10 +0 +10 +-4 +10 +-3 +10 +-2 +10 +-1 +10 +0 +10 +1 +10 +-2 +10 +-1 +10 +0 +s/s +0 +r [GeV +-1 +] +x=10 +-3 + Improved Model + GBW Model +rQ +s +FIG. 3: The ratio of the dipole cross sections in the improved +saturation model (dashed lines) as a function of r (left plot) +and rQs (right plot) for x = 10−3 at the NLO approxima- +tion. The solid lines are due to the GBW model [4]. +The +error bands are due to the statistical uncertainties from the +coefficients of the parameterization of F2 in [19]. +function. In this analysis we present the k2 +t -dependence +of the improved UGD model at the NLO approximation +and compare with four models for f(x, k2 +t ), which exhibit +rather different shape of kt-dependence in the region, +10−1≤k2 +t ≤102 GeV2, for a value of the longitudinal +momentum fraction, x = 10−3. We show the behaviour +of our predictions for the UGD are comparable with +HSS and IN models in a wide range of k2 +t . These results +are comparable with the WMR and GBW models at +moderate values of k2 +t . +Then we have presented the improved dipole cross +section when the unintegrated gluon distribution is +derived from the parametrization of the proton structure +function as a function of r and rQs, respectively. The +results according to the UGD are consistent with the +GBW saturation model at the NLO approximation. The +error bars are due to the statistical uncertainties of the +effective parameters and preserved that the NLO results +give a reasonable data description in comparison with +the GBW model. In this method, the large dipole size +part of the dipole cross section retains the features of +the GBW model with the saturation scale. The dipole +cross section at the small dipole size by the presence +of the uninetrgated gluon distribution is modified in +comparison with the saturation scale of the GBW +saturation model. + +5 +ACKNOWLEDGMENTS +The author is grateful to Razi University for the finan- +cial support of this project. I am also very grateful to the +respectable reviewer of the Eur.Phys.J.C82, 740(2022) +article for suggesting this topic. +REFERENCES +1. +V.S.Fadin, +E.A.Kuraev +and +L.N.Lipatov, +Phys.Lett.B 60, 50(1975); L.N.Lipatov, Sov.J.Nucl.Phys. +23, +338(1976); +I.I.Balitsky +and +L.N.Lipatov, +Sov.J.Nucl.Phys. 28, 822(1978). +2. +K.Kutak and A.M.Stasto, Eur.Phys.J.C 41, 343 +(2005). +3. I.P.Ivanov and N.N.Nikolaev, Phys.Rev.D 65, 054004 +(2002). +4. K.Golec-Biernat and M.W¨usthoff, Phys. Rev. D 59, +014017 (1998); K. Golec-Biernat and S.Sapeta, JHEP +03, 102 (2018). +5. +J.Bartels, K.Golec-Biernat and H.Kowalski, Phys. +Rev. D66, 014001 (2002). +6. +B.Sambasivam, T.Toll and T.Ullrich, Phys.Lett.B +803, 135277 (2020). +7. J.R.Forshaw and G.Shaw, JHEP 12, 052 (2004). +8. 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Rev.D +89, no. 9, 094027 (2014). + diff --git a/T9AzT4oBgHgl3EQfJvsR/content/tmp_files/load_file.txt b/T9AzT4oBgHgl3EQfJvsR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1fcfc0e68b30678af93b20059e4250a6da16d53c --- /dev/null +++ b/T9AzT4oBgHgl3EQfJvsR/content/tmp_files/load_file.txt @@ -0,0 +1,351 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf,len=350 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='01083v1 [hep-ph] 3 Jan 2023 The dipole cross section by the unintegrated gluon distribution at small x G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='Boroun∗ Department of Physics, Razi University, Kermanshah 67149, Iran (Dated: January 4, 2023) We apply a previously developed scheme to consistently include the improved saturation model for the unintegrated gluon distribution (UGD) in order to evaluate, in the framework of kt factorization, at small x at the next-to-leading order (NLO) in αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' We start the unintegrated gluon distribution with a parametrization of the deep inelastic structure function for electromagnetic scattering with protons, and then extract the color dipole cross section, which preserves its behavior success in a wide range of k2 t in comparisons with the UGD models (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Hentschinski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Sabio Vera and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Salas (HSS), I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='Ivanov and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='Nikolaev (IN) and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Watt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Martin and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Ryskin (WMR)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' These results show that the geometric scaling holds for the improved saturation model in a wide kinematic region rQs, and are comparable with the Golec-Biernat-W¨usthoff (GBW) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Introduction Although our knowledge of the proton structure at small-x is very limited, novel opportunities will be opened at new-generation facilities (Electron-Ion Col- lider(EIC), High-Luminosity Large Hadron Collider (HL- LHC), Forward Physics Facility (FPF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Combining the information coming from dipole cross sections and pT - unintegrated densities could play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' In particular, polarized amplitudes and cross sections for the exclusive electroproduction of ρ and φ mesons at the Hadron-Electron Ring Accelerator (HERA) and the EIC are very sensitive to the unintegrated gluon distribution (UGD) model adopted, whereas forward Drell-Yan dilep- ton distributions at the Large Hadron Collider beauty (LHCb) are very sensitive to next-to-leading logarithmic corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Indeed, the dipole cross section is directly connected via a Fourier transform to the small-x UGD, whose evolution in x is regulated by the Balitsky- Fadin- Kuraev-Lipatov (BFKL) equation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The BFKL equa- tion governs the evolution of the UGD, where the kT - factorization is used in the high energy limit in which the QCD interaction is described in terms of the quan- tity which depends on the transverse momentum of the gluon [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The object of the BFKL evolution equation at very small x is the differential gluon structure function1 of proton f(x, k2 t ) = ∂xg(x, µ2) ∂lnµ2 |µ2=k2 t (1) which emerges in the color dipole picture (CDP) of in- clusive deep inelastic scattering (DIS) and diffractive DIS into dijets [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Here x and k2 t being the fractional mo- mentum of proton carried by gluon and the transverse ∗Electronic address: boroun@razi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='ir 1 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' (1) is modified with the Sudakov form factor as x increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' momentum of gluon respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Unintegrated distribu- tions are required to describe measurements where trans- verse momenta are exposed explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The color dipole picture (CDP) [4] has been introduced to study a wide variety of small x inclusive and diffractive processes at HERA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The CDP, at small x, gives a clear interpretation of the high-energy interactions, where is characterized by high gluon densities because the proton structure is dominated by dense gluon systems [5-7] and predicts that the small x gluons in a hadron wavefunction should form a Color Glass Condensate [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Dipole rep- resentation provides a convenient description of DIS at small x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' There, the scattering between the virtual pho- ton γ∗ and the proton is seen as the color dipole where the transverse dipole size r and the longitudinal momen- tum fraction z with respect to the photon momentum are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The amplitude for the complete process is simply the production of these subprocess amplitudes, as the DIS cross section is factorized into a light-cone wave function and a dipole cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Using the optical theorem, this leads to the following expression for the γ∗p cross-sections σγ∗p L,T (x, Q2) = � dzd2r|ΨL,T(r, z, Q2)|2σdip(x, r), (2) where subscripts L and T referring to the transverse and longitudinal polarization state of the exchanged boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Here ΨL,T are the appropriate spin averaged light-cone wave functions of the photon and σdip(x, r) is the dipole cross-section which related to the imaginary part of the (qq)p forward scattering amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The variable z, with 0 ≤ z ≤ 1, characterizes the distribution of the momenta between quark and antiquark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The square of the photon wave function describes the probability for the occurrence of a (qq) fluctuation of transverse size with respect to the photon polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Another framework which can be used for calculating the parton distributions is based on the DGLAP evo- lution [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Deep inelastic electron-proton scattering is described in terms of scale dependent parton densities 2 q(x, µ2) and g(x, µ2) [10], where the integrated gluon dis- tribution (xg(x, µ2)) is defined through the unintegrated gluon distribution (f(x, k2 t )) by xg(x, µ2) ≡ � µ2 dk2 t k2 t f(x, k2 t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' (3) The unintegrated gluon distribution is related to the dipole cross section [2, 11] σ(x, r) = 8π2 Nc � dkt k 3 t [1 − J0(ktr)]αsf(x, k2 t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' (4) A novel formulation of the UGD for DIS in a way that accounts for the leading powers in both the Regge and Bjorken limits is presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' In this way, the UGD is defined by an explicit dependence on the longi- tudinal momentum fraction x which entirely spans both the dipole operator and the gluonic Parton Distribution Function (PDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' In addition to the gluon momentum derivative model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' (1)), several other models [3-4, 13-15] for the UGD have also been proposed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' A comparison be- tween these models can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='[16-17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The authors in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' [13] presented an x-independent model (ABIPSW) of the UGD where merely coincides with the proton impact factor by the following form f(x, k2 t ) = A 4π2M 2 � k2 t M 2 + k2 t � , (5) where M is a characteristic soft scale and A is the nor- malisation factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The authors in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' [3] presented a UGD soft-hard model (IN) in the large and small kt regions by the following form f(x, k2 t ) = f (B) soft(x, k2 t ) k2 s k2s + k2 t + fhard(x, k2 t ) k2 t k2 h + k2 t , (6) where the soft and the hard components are defined in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The UGD model was considered in [14] to used in the study of DIS structure functions and takes the form of a convolution between the BFKL gluon Green,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='s function and a leading-order (LO) proton impact factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' where has been employed in the description of single-bottom quark production at LHC and to investigate the photo- production of J/Ψ and Υ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' by the following form (HSS model) f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' k2 t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Mh) = � +∞ −∞ dν 2π2 C Γ(δ − iν − 1 2) Γ(δ) ( 1 x)χ( 1 2 +iν) ×( k2 t Q2 0 ) 1 2 +iν � 1 + α2 sβ0χ0( 1 2 + iν) 8Nc log( 1 x) × � − ψ(1 2 + iν) − log k2 t M 2 h �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' (7) where χ0( 1 2 + iν) and χ(γ) are respectively the LO and the next-to-leading order (NLO) eigenvalues of the BFKL kernel and β0 = 11 − 2 3nf with nf the number of active quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Here αs = 3 παs(µ2) with µ2 = Q0Mh where Mh plays the role of the hard scale which can be identified with the photon virtuality, � Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The authors in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' [15] presented a UGD model (Watt- Martin-Ryskin (WMR) model) where depends on an extra-scale µ, fixed at Q, by the following form f(x, k2 t , µ2) = Tg(k2 t , µ2)αs(k2 t ) 2π � 1 x dz � � q Pgq(z)x z q(x z , k2 t ) +Pgg(z)x z g(x z , k2 t )Θ( µ µ + kt − z) � , (8) where Tg(k2 t , µ2) gives the probability of evolving from the scale kt to the scale µ without parton emission and P , ijs are the splitting functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Golec-Biernat-Wusthoff (GBW) [4] presented a UGD model where derives from the effective dipole cross sec- tion σ(x, r) for the scattering of a qq pair of a nucleon as2 f(x, k2 t ) = k4 t σ0 R2 0(x) 2π e−k2 t R2 0(x), (9) with R2 0(x) = 1 GeV2 ( x x0 )λp and the following values σ0 = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='03 mb, λp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='288 and x0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='04×10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Although one of them (the HSS one) was fitted to reproduce DIS structure functions, the study of other reactions has provided an evidence that the UGD is not yet well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' In this paper we use the DGLAP- improved saturation model with respect to the UGD in the proton to access the dipole cross section at low x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Method The interaction of the qq pair with the proton is de- scribed by the dipole cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' It is related to the gluon density in the target by the kT -factorization for- mula [3-4,11] σ(x, r) = � d2kt k 4 t ∂xg(x, k2 t ) ∂lnk2 t [1 − eikt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='r][1 − e−ikt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='r],(10) where the relation between f(x, k2 t ) and xg(x, k2 t ) defined through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The integrated gluon density is related 2 The reader can be referee to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' [3-4, 13-15] for a meticulous treatment of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' 3 to the proton structure function [18] xg(x, k2 t ) = � 1 x � DF 2(ln 1 y , k2 t )J(as(k2 t ), ln y x) −F2(ln 1 y, k2 t )M(as(k2 t ), ln y x) �dy y , (11) where as(k2 t ) = αs(k2 t )/4π, DF2(x, k2 t ) = ∂ ∂lnk2 t F2(x, k2 t ) and F2(x, k2 t ) is the parametrization of the proton struc- ture function [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' (11) is obtained [18] for the gluon density at next-to-leading order (NLO) approximation with respect the Laplace transform method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The stan- dard representation for QCD couplings in NLO (within the MS-scheme) approximation reads αNLO s (k2 t ) = 4π β0 ln k2 t Λ2 � 1 − β1 β2 0 ln ln k2 t Λ2 ln k2 t Λ2 � , (12) where β0 and β1 are the one and two loop correction to the QCD β-function and Λ is the QCD cut-off parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The functions J and M in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' (11) are defined by the inverse Laplace transform method as �J(υ, k2 t )≡L−1[k(s, k2 t );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' υ], � M(υ, k2 t )≡L−1[h(s, k2 t );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' υ], (13) where �f(υ)≡f(e−υ) and υ = ln(1/x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The functions k and h in s-space reads k(as(k2 t ), s) = [< e2 > Θf(as(k2 t ), s)]−1, h(as(k2 t , s) = Φf(as(k2 t ), s)k(as(k2 t ), s), (14) where < e2 > is the average of the charge e2 for the active quark flavors nf , < e2 >= n−1 f �nf i=1 e2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The Laplace transform of the splitting functions (Pqq and Pqg) are given by Φf(as(k2 t ), s) ≡ L[e−υ 1 � φ=0 aφ+1 s (k2 t ) �P (φ) qq (υ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' s], Θf(as(k2 t ), s) ≡ L[e−υ 1 � φ=0 aφ+1 s (k2 t ) �P (φ) qg (υ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' s], (15) where at small x, the kernels at NLO approximation are given by the following forms Θf(as, s) ≃ 2nfas � 1 1 + s − 2 2 + s + 2 3 + s � +a2 sCATf �40 9s � , Φf(as, s) ≃ as � 4 − 8 3( 1 1 + s + 1 2 + s + 2S1(s) � +a2 sCF Tf �40 9s � , (16) where S1(s) = ψ(s+1)+γE, where ψ(x) is the digamma function and γE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='5772156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' is Euler constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Here CA = Nc = 3, CF = N 2 c −1 2Nc = 4 3 and Tf = 1 2nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' In next section we consider the UGD due to the parametrization of the proton structure function as f(x, k2 t ) = ∂ ∂lnk2 t � � 1 x � DF 2(ln 1 y, k2 t )J(as(k2 t ), ln y x) −F2(ln 1 y, k2 t )M(as(k2 t ), ln y x) �dy y � , (17) then we obtain the color dipole cross section as σ(x, r) = � � 2dktdΘ k 3 t [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' (17)][1 − cos(ktrcosΘ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' (18) 10 1 10 0 10 1 10 2 2 1 0 1 2 3 4 5 6 7 10 0 10 1 10 2 f(x,k t 2 ) k t 2 [GeV 2 ] x=10 3 Improved Model GBW Model k t 2 [GeV 2 ] x=10 3 Improved Model FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' 1: Left: The UGD as a function of k2 t with respect to the improved model compared with the GBW model at x = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Right: The error bands are due to the statistical uncertainties in the coefficients of the parametrization F2 in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Numerical Results The UGD is obtained directly in terms of the param- eterization of the structure function F2(x, k2 t ) and its derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The resulting UGD with the k2 t -dependence due to the parametrization of the proton structure function are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' In this figure, we plot the k2 t dependence of the UGD at x = 10−3 and compared the unintegrated gluon distribution behavior due to the improved saturation model with the GBW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' An enhancement and then depletion is observable in the improved and GBW models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' These picks occur at k2 t ≈10 GeV2 and k2 t ≈1 GeV2 in the improved and GBW models respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' As we can seen, the UGD behavior in the improved saturation model, in a wide range of k2 t , is softer than the GBW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The error bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='1 in the improved saturation model are due to the uncertainties in the coefficients of the parametrization 4 of the proton structure function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' 2 we show the kt distributions of three different 10 1 10 0 10 1 10 2 10 2 10 1 10 0 10 1 10 2 f(x , k t 2 ) k t 2 [GeV 2 ] HSS IN-CT14 LO WMR-CT14 LO Improved Model FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' 2: The unintegrated gluon distribution f(x, k2 t ) obtained from the improved saturation model (dashed-symbol), as a function k2 t at x = 10−3, compared with the HSS [14], IN [3] and WMR [15] models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' unintegrated gluons at x = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' This figure compares the results of the improved saturation model, based on the parametrization of the proton structure function, with the HSS [14], IN [3] and WMR [15] models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' We observe that the improved saturation result is compa- rable with the HSS and IN models in a wide range of k2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The differences are not large, however there is some suppression due to the models at large and small values of k2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The continuous behavior of the UGD in our model with increases of k2 t is due to the gluon and quark terms included in the improved saturation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='3, we have calculated the improved saturation model with respect to the unintegrated gluon distribu- tion to the ratio σdip/σ0 in a wide range of the dipole size at the NLO approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Results of calculations and comparison with the GBW model [4] for x = 10−3 are presented in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The effective parameters in the GBW model have been extracted from a fit of the HERA data as λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='288 and x0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='04×10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' These corrections to the ratio of color dipole cross sections at NLO approximation are comparable with the GBW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' For rQs≥1, the dashed curve (central values) merge due to geometric scaling in the dipole cross section in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' In the right-hand of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='3, a particular interests present the ratio σdip/σ0 defined by the scaling variable rQs, which means that the scattering amplitude and corresponding cross sections can scale on the dimensionless scale rQs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' In conclusion, we study the unintegrated gluon distri- bution from a parameterization of the proton structure 10 2 10 1 10 0 10 4 10 3 10 2 10 1 10 0 10 1 10 2 10 1 10 0 s/s 0 r [GeV 1 ] x=10 3 Improved Model GBW Model rQ s FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' 3: The ratio of the dipole cross sections in the improved saturation model (dashed lines) as a function of r (left plot) and rQs (right plot) for x = 10−3 at the NLO approxima- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The solid lines are due to the GBW model [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The error bands are due to the statistical uncertainties from the coefficients of the parameterization of F2 in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' In this analysis we present the k2 t -dependence of the improved UGD model at the NLO approximation and compare with four models for f(x, k2 t ), which exhibit rather different shape of kt-dependence in the region, 10−1≤k2 t ≤102 GeV2, for a value of the longitudinal momentum fraction, x = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' We show the behaviour of our predictions for the UGD are comparable with HSS and IN models in a wide range of k2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' These results are comparable with the WMR and GBW models at moderate values of k2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' Then we have presented the improved dipole cross section when the unintegrated gluon distribution is derived from the parametrization of the proton structure function as a function of r and rQs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The results according to the UGD are consistent with the GBW saturation model at the NLO approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The error bars are due to the statistical uncertainties of the effective parameters and preserved that the NLO results give a reasonable data description in comparison with the GBW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' In this method, the large dipole size part of the dipole cross section retains the features of the GBW model with the saturation scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' The dipole cross section at the small dipole size by the presence of the uninetrgated gluon distribution is modified in comparison with the saturation scale of the GBW saturation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' 5 ACKNOWLEDGMENTS The author is grateful to Razi University for the finan- cial support of this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content=' I am also very grateful to the respectable reviewer of the Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} +page_content='C82, 740(2022) article for suggesting this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9AzT4oBgHgl3EQfJvsR/content/2301.01083v1.pdf'} 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Copyright may be transferred without notice, after which this version may no longer be accessible. +IEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL. XX, NO. XX, AUGUST XXXX +1 +Learning ultrasound plane pose regression: assessing +generalized pose coordinates in the fetal brain +Chiara Di Vece, Student Member, IEEE, Maela Le Lous, Brian Dromey, Francisco Vasconcelos, Anna L David, +Donald Peebles, Danail Stoyanov Senior Member, IEEE +Abstract—In obstetric ultrasound (US) scanning, the learner’s +ability to mentally build a three-dimensional (3D) map of the fetus +from a two-dimensional (2D) US image represents a significant +challenge in skill acquisition. We aim to build a US plane +localization system for 3D visualization, training, and guidance +without integrating additional sensors. This work builds on top +of our previous work, which predicts the six-dimensional (6D) +pose of arbitrarily-oriented US planes slicing the fetal brain with +respect to a normalized reference frame using a convolutional +neural network (CNN) regression network. Here, we analyze in +detail the assumptions of the normalized fetal brain reference +frame and quantify its accuracy with respect to the acquisition +of transventricular (TV) standard plane (SP) for fetal biometry. +We investigate the impact of registration quality in the training +and testing data and its subsequent effect on trained models. +Finally, we introduce data augmentations and larger training +sets that improve the results of our previous work, achieving +median errors of 3.53 mm and 6.42° for translation and rotation, +respectively. +Index Terms—Fetal ultrasounds, convolutional neural network, +plane localization +I. INTRODUCTION +F +ETAL US is a non-invasive, real-time and cost-effective +diagnostic tool for monitoring fetal growth and anatomy +throughout gestation [1]. During routine mid-trimester fetal US +scan, the sonographer acquires the SP, predefined anatomical +planes defined by scientific committees to promote interna- +tional guidelines for fetal US images [2]. Specifically, the TV +SP, in Figure 1, needs to show the skull shape, the cavum +septum pellucidi, the posterior horn of the lower lateral ven- +tricle, and the anterior horns of the lateral ventricles [3]. This +allows for reliable measurements of specific structures and +reduced inter- and intra-sonographer variability. The correct +identification of SPs is essential in the second-trimester fetal +anatomic survey to investigate the morphological characteris- +tics of the fetus and detect abnormalities or deviations from +the expected growth patterns. Sonographers may struggle to +The work was supported by the Wellcome/EPSRC center for Interven- +tional and Surgical Sciences (WEISS) [203145Z/16/Z]; Engineering and +Physical Sciences Research Council (EPSRC) [EP/P027938/1, EP/R004080 +/1, EP/P012841/1]; The Royal Academy of Engineering Chair in Emerging +Technologies Scheme; and Horizon 2020 FET (GA 863146). For the purpose +of open access, the author has applied a CC BY public copyright licence to +any author accepted manuscript version arising from this submission. +C. Di Vece, F. Vasconcelos and D. Stoyanov are with Wellcome/EPSRC +center for Interventional and Surgical Sciences (WEISS) and the Department +of Computer Science, University College London (UCL), UK. +M. Le Lous, B. Dromey, A.L. David and D.Peebles are with WEISS, Elizabeth +Garrett Anderson Institute for Women’s Health and NIHR University College +London Hospitals Biomedical Research center, UCL, UK +chiara.divece.20@ucl.ac.uk +Fig. 1. TV SP to evaluate fetal biometry in the brain. This plane needs to +show the skull shape, the cavum septum pellucidi, the posterior horn of the +lower lateral ventricle, and the anterior horns of the lateral ventricles +obtain good SPs for various reasons, including inexperience, +limited training, time limitations, and fetal movement [4], +[5]. Most trainees learn on actual patients under the direct +supervision of an expert. Although US simulators have been +developed recently, trainee engagement has been limited due to +competing time priorities [6]. The primary training challenge +faced by all novice sonographers is not related to knowledge +of anatomy or familiarity with the US machine interface. +Rather, the manual navigation of the probe towards acquiring +SP requires the sonographer to build a 3D map of the fetus +from dynamic 2D sectional views while handling the probe. +Measurements of biometric parameters and assessments of the +fetal brain’s anatomy may be erroneous due to mistakes in +locating the 2D scan within the 3D volume. +At present, SPs recognition represents the main focus of +fetal US training. Due to the requirement to interpret variable +and complex images and their spatial relationship, autonomous +probe navigation towards SPs continues to be a challenging +task. Our final aim is to develop a US navigation system that +guides the sonographer towards obtaining SPs with reference +to fetal anatomy. +In [7], we proposed a method to localize a US plane +of a fetal brain directly from its 2D US image. While this +is a promising result towards active guidance during fetal +scanning, a few aspects still require further investigation. +First, it assumes that brains from different fetuses can be +mapped to the same normalized coordinate system, where +each SP always has the same pose coordinates. The accuracy +of this assumption has not been quantified, which limits the +analysis of experimental results. Secondly, the achieved pose +accuracy is still far from optimal due to a lack of variation in +training data. +arXiv:2301.08317v1 [cs.CV] 19 Jan 2023 + +Routine +Har-mid +Transventricular +16 +Gn6 +Plane +C6/M4 +FF2/E1 +SRII10/CRI1 +OFD +BPD +70.1mm +GA +28w1d0.1SD +1211g(2lb11oz) +EFW +GA +28w0d0.3SD +OFD +8.78cm +BPD +28w2d0.5SD +GA +HC +24.90cm +GA +27wOd-1.4SD +80% +CI (BPD/OFD)IEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL. XX, NO. XX, AUGUST XXXX +2 +In this work, we expand on this work with the following +contributions: +• We developed a tool for annotating poses of SPs in 3D US +volumes using the Unity engine and a gaming controller. +We used this tool to obtain ground truth poses of TV +SPs in 3D US volumes (annotated by an obstetrician) of +a publicly available fetal brain dataset. We used this data +to validate the US pose regression and its training data +generation. +• We evaluate to which extent the brains of different fetuses +can be mapped to the same generalized coordinate frame. +We quantify this through the variance in the pose of +annotated TV SPs when represented in the generalized +coordinate frame. This puts strict boundaries on how +accurate our pose regression models can be and adds +additional context to the analysis of its experimental +results. +• The training data consists of 3D US volumes that need +to be well registered. We evaluate the effect of different +registration techniques on the performance of our method. +We demonstrate that high-quality registration is funda- +mental to effective network training and to preserve our +models’ assumptions. +• We outperform the pose regression results obtained in [7] +by introducing additional data augmentation and increas- +ing training sets. +II. RELATED WORK +This section presents the related work for three related but +different tasks: extraction of SPs, slice-to-volume registration, +and localization of SPs. +A. Extraction of Standard Planes +Previous work proposed automating the extraction of SPs +from data acquired with a simplified protocol rather than +assisting operators in acquiring typical freehand 2D SPs. In +one of their initial works, Zhang et al. [8] developed a +system based on two AdaBoost classifiers placed in cascade to +automatically detect in a coarse-to-fine way early gestational +sac SP. Further early studies [9]–[12] detect key abdominal +structures and landmarks in a sequence of 2D ultrasound +fetal images to classify the SPs in each frame of US videos +based on the presence and orientation of the landmarks using +various conventional machine learning (ML) algorithms like +AdaBoost, Random Forest as well as support vector machines. +These methods, however, are only applicable to a subset of +fetal SPs (brain and abdomen); besides, the quality of the +obtained SP cannot be compared with the one achieved with +typical freehand scanning. +Classification methods based on CNNs were used to detect +2D SPs because of their powerful ability to learn hierarchical +representations automatically. To detect the fetal SPs, Chen +et al. [13] fine-tuned a pre-trained classification CNN based +on transfer learning. Baumgartner et al. [14] proposed a +classification model to detect thirteen SPs with unsupervised +learning and then used weakly-supervised learning (SL) based +on image-level labels to locate anatomical structures in each +plane. The study employed extensive data, including videos +longer than those usually collected in clinical practice (roughly +30 minutes). The CNNs are fed with surrounding and addi- +tional information from each US video. To capture temporal +information in 2D US, some works added to the detection of +the three fetal SPs a recurrent neural network (RNN) [15]. +All the methods mentioned above are effective in the +detection of SP images, but they can only determine whether +an image was captured at a SP, not where exactly it is in the +corresponding 3D space. Besides, the models require a high +amount of annotated data to be trained. +B. Slice-to-volume Registration +One approach for US plane localization is to find its +alignment with respect to a pre-acquired 3D volume of the +same anatomy. This is an optimization problem, typically +solved with iterative numerical methods that minimize the +distance between specific landmarks or maximize intensity- +based similarity metrics [16]. Unfortunately, the cost func- +tions associated with these metrics are frequently non-convex, +limiting the capture range of these registration methods. Our +task differs from a classic slice-to-volume registration method, +i.e., it does not require a previously acquired 3D volume of +the same subject being scanned. Instead, we predict the pose +relative to a generalized brain center, i.e., a stable anatomical +brain point across the different, pre-aligned volumes, where +training and test data belong to different subjects. +C. Localization of Standard Planes +Predicting the pose of SPs in 3D volumes can be per- +formed without a patient-specific model and without using +pre-operative data. This has been primarily approached as +a classification problem, where the plane pose space is dis- +cretized into bins, and the estimation boils down to selecting +one of the bins [17], [18]. In fetal magnetic resonance imaging +(MRI) [19], [20] and fetal US [21], the prediction of slice +locations has been previously improved with learning-based +methods. General purpose learning-based methods for pose +estimation approach this as a regression of a 3D translation +and a 3D rotation. 3D rotations can be represented in multiple +conventional ways, such as quaternions, axis-angle, or Euler +angles. Zhou et al. pointed out in [22] that if the entire rotation +space is required, these representations are sub-optimal for +specific angle ranges, and proposed a new 6D representation +for rotations that does not suffer from these issues. This rota- +tion representation has been adopted for US plane localization +in [7], where a regression CNN is proposed to predict the 6D +pose of arbitrarily-oriented planes slicing the fetal brain US +volume without the need for real ground truth data in real-time +or 3D volume scans of the patient beforehand. The proposed +network reliably localizes US planes within the fetal brain in +phantom data and successfully generalizes pose regression for +unseen fetal brains from a similar gestational age (GA) as in +training. The network was tested on real fetal brain images +with a GA ranging from 21 to 25 weeks. Similarly, Yeung et +al. [23] proposed a CNN that takes a set of images as input +and learns to compare them in pairs. The model was tested on + +IEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL. XX, NO. XX, AUGUST XXXX +3 +Fig. 2. Pipeline to train and test the network. 1) The 3D fetal brain US volumes are registered in 3D Slicer using similarity registration; 2) The volumes +are reconstructed into Unity, and synthetic sectional (slice) image representations are generated and saved along with their 6D pose (translation and rotation) +relative to the center of the fetal brain US volume; 3) These images are fed into the network to output the estimated slice 6D pose (translation and rotation) +relative to the same point +fetal brain volumes with a GA ranging from 18 to 22 weeks. +Then, in [24], an unsupervised cycle consistency using the fact +that the overall displacement of a sequence of images in the +3D anatomical atlas is equal to the displacement from the first +image to the last in that sequence was added. +III. MATERIALS AND METHODS +The development of our US pose regression system has been +divided into three main blocks, reported in Figure 2. First, we +align 3D US volumes for the training and validation of our +models. Secondly, we developed a Unity-based simulator to +visualize and manually annotate SPs in 3D US volumes. This +also enables the automated generation of supervised training +data for our pose regression models, i.e., 2D synthetic images +and their ground truth 6D pose relative to the volume center. +Finally, we detail our deep learning (DL)-based plane pose +regression system. +A. Data preparation +1) Dataset: We used seven real fetal brain US volumes +with a GA ranging from 21 to 25 [25] (singleton pregnancy +with no abnormal findings)1 obtained from different fetuses +1Refer to www.datavers.nl for details +(fi, with i = 1, ..., 6). All volumes were processed to be +isotropic with voxel size of 0.5×0.5×0.5 mm and average +size of 249×174×155 mm (coronal×axial×sagittal, actual +size of the acquired volumes). +2) Volume registration: The registration procedure was +performed in 3D Slicer2 as detailed below. The whole process +is depicted in Figure 3 for 3D US real volumes of the fetal +brain. Starting from the initial volumes (Figure 3a), we: +• Used fiducial points to achieve an initial alignment of the +volumes using the Fiducial Registration Wizard module +(Figure 3b); +• Defined a contour mask of the brain using the Segment +Editor module to avoid overfitting on the shape of the +ultrasound volume during registration; +• Used the general registration (BRAIN) module available +in 3D Slicer to register the volumes with a similarity +registration phase (rigid registration + scale for a total of +7 degrees of freedom), as shown in Figure 3c. We chose +the previously obtained masks as a Region of Interest +(ROI) so that the registration algorithm only considers a +specific image region for the registration (the fetal brain). +23D Slicer + +Data preparation: volume registration +Similarity +- Annotation of +A +registration +fiducial points +3DSlicer +- Generation of +masks for the brain +0502-POS +Deep learning-based pose regression system +Unity simulator for volume +reconstruction, annotation + Synthetic +and synthetic data acquisition +sectional +images +Volume +Rendering +6-DoF +Phantom 3D +→ 6D Pose +Network +3D model of the +foetal brain Us +pose +unity +C +fetus starting +from 3D US +Regression Heads +Feature Extraction +Input +ResNet-18 backbone +128x128 +r1,2,3,r4,rs,r6 [5] + Synthetic image +Visualization of +B +(3,3) +the planes slicing +c +the volume +3 +6D Pose +GT = (tx,ty,tz,ax,ay,az) +t'x,t'y,t' +Automated +t1,t2,t3 +Qout +(1.6) +Conv2d +Basic Block 1 +Basic Block O +generation of +(1,3) +(1,3) +L 6D Pose +BatchNorm2d +=3 +supervised data +tx,ty,tz +Epred=(t'x,t'y,t'z,a'x,a'y,a'z) +ReLU +(3,3) +MaxPool2d +unity + Linear +Standard planes +LosSTr +annotation +R'-R²- +LosSTo +LTotIEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL. XX, NO. XX, AUGUST XXXX +4 +Fig. 3. +Registration procedure on real fetal brain US volumes. Workflow +from pre-registration to post-registration: a. Starting US fetal brain volumes +before registration, b. Fiducial points used to achieve an initial alignment of +the volumes, c. Contour mask of the brain used to avoid overfitting on the +shape of the US volume +B. Unity Simulator for Standard Plane Annotations and Syn- +thetic Images Generation +Starting from the open-source project UnityVolumeRen- +dering, we developed our simulator using the game engine +Unity3. The first step is rendering the volume starting from +RAW, PARCHG, or Digital Imaging and Communications in +Medicine (DICOM) datasets. The simulator allows the user to +render the volume using three modes: Isosurface Rendering, +Maximum Intensity Projection, and Direct Volume Rendering. +The latter is the standard rendering mode; it uses transfer +functions (1D or 2D) to determine the color and opacity while +projecting rays across the dataset. Transfer functions translate +density (and gradient magnitude in case of 2D) to a color and +opacity. The simulator allows the user to set a custom transfer +function. +• 1D Transfer Function: the density is represented on the +X-axis, whereas the opacity (alpha) is on the Y-axis. The +user can create a curve for opacity by density by shifting +the grey alpha knots. The bottom gradient-colored panel +maps color to density. +• 2D transfer function: the density is represented on the +X-axis, whereas the gradient magnitude is on the Y-axis. +Using the sliders, the user can define a rectangle shape, +modify its size/position, and the minimum and maximum +values for alpha/opacity. +1) User interface: After loading the DICOM dataset ex- +tracted from 3D Slicer and setting the transfer function, +the clinician can add a plane slicing the reconstructed 3D +US volume (left-hand side of Figure 4) with an arbitrary +orientation and visualize the plane in an external window +(right-hand side, bottom). The plane can be controlled using +a joystick (right-hand side, top), simplifying the annotation +of the SPs. Hence, the clinician can modify the position and +rotation of the plane using the joystick while monitoring the +appearance in the external window. Once the clinicians are +satisfied with the pose of the SP, they can save it using the +last button in the external windows to get a picture of the +slicing plane, and its 6D pose relative to the volume center. +2) Training and testing data generation: To generate train- +ing data for our models, we generated synthetic slices by +applying rotation and translation to a plane placed in the center +of the volume generated with a uniform random distribution +within a fixed range to avoid slices with poor overlap with +the volume. The synthetic images obtained by slicing the +3Unity Real-Time Development Platform +Fig. 4. +Unity simulator for volume reconstruction, SP annotations, and +automatic supervised data generation. While controlling the probe with the +joystick using the suggested commands, the clinician can visualize the slicing +plane in an external window. Once the desired plane is reached, it can be +saved using the “Save plane” button along with its 6D pose relative to the +volume center +volume were saved along with their pose with respect to the +volume center (fetal brain). This provides an automated way +of generating a high amount of training data with reliable +ground truth labels. An obstetrician annotated the position +of the TV SP by directly manipulating a slicing plane with +the joystick and choosing the translation and angle sampling +intervals to avoid sampling of planes at the edges of the +volume containing no information. The nearby planes were +generated by applying small random rotations and translations +(uniform distribution). Specifically, the acquisition interval +between two planes was decreased from 0.1 to 0.001 for +translation (Unity environment, with coordinates normalized +between -1 and 1 so that the pose regression works in a fixed, +normalized range, independent of the real brain size in mm) +and from 7.9° to 1.9° for rotation compared to acquisition of +planes at random coordinates. We acquired 20699 planes with +random orientation per volume and 1330 around the TV SP +for a total of 22029 images for each volume. +C. Deep Learning-based Plane Pose Regression System +We base our 6D pose regression system on the network +proposed in [7] (Figure 2c). We used an 18-layer residual +CNN (ResNet-18) [26] as a backbone for feature extraction +with the pre-trained ImageNet weights [27]. We modified the +network by re-initializing the fully connected layer based on +the representation’s dimension (nine parameters) and adding +a regression head to directly output the rotation and trans- +lation representations. The network receives the US image +I (128×128) obtained by slicing the volume and its 6D +pose with respect to the center of the fetal brain US volume +θGT += +(tx, ty, tz, αx, αy, αz). We use this information as +the ground truth label for network training and validation. +The CNN learns to predict the 6D pose with respect to the +same point θP red = (t′ +x, t′ +y, t′ +z, α′ +x, α′ +y, α′ +z). Specifically, the +network first outputs a vector of nine parameters θOut = +(t1, t2, t3, r1, ..., r6); the first three are used for the translation +and the last six for the rotation. Then, r1, ..., r6 are used +internally by our CNN to reconstruct the rotation matrix R′ +in the forward pass. + +Pre-registration +a +b +Post registratior +cUnity scene +Import RAW dataset +Joystick +ImportPARCHGdataset +Import DIcOM dataset +Edit imported dataset +Edit slicing plane +Slicing plane +Right Stick-RotatePlane +LeftStick-MovePlane +Top/down Arrows -Vertical movement Plane +previous +next +add +remove +save +Button A -Change Shell Transparency +plane +plane +plane +plane +plane +Button B -Reset PoseIEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL. XX, NO. XX, AUGUST XXXX +5 +Fig. 5. Registration procedure on real fetal brain US volumes. First column: results obtained using the automatic similarity registration (rigid registration + +scale for a total of 7 degrees of freedom) provided by 3D Slicer. Second column: results obtained by aligning the volumes with the automatic registration +using the mask to contour the brain used to avoid overfitting on the shape of the US volume. Third column: results obtained using the fiducial points annotated +by the obstetrician, followed by similarity registration using the mask a ROI that contours the brain +Differently from [7], we also perform image intensity aug- +mentation. More specifically, we change brightness, contrast, +and saturation. The detailed parameters of this augmentation +are described in the next section. Indeed, recent work has +found that, in the context of fetal US augmentation strategies, +generating less realistic training images leads to improvement +in generalization capability [28]. +IV. EXPERIMENTS AND RESULTS +In this section, we report the results of three main experi- +ments. First, in Section IV-A, we assess three different US vol- +ume registration methods and investigate their qualitative and +quantitative impact in defining a generalized inter-patient co- +ordinate frame for the fetal brain (Section IV-A1 and IV-A2). +Besides, we illustrate the effect of these registration methods +when training US plane pose regression networks using a fetal +US volume of a single 23-week fetus to train the network and +fetuses with a GA ranging from 21 to 25 weeks for testing +(Section IV-A3). In our second experiment, in Section IV-B, +we investigate how consistent the manual annotations of the +TV SPs are both in terms of quality (Section IV-B1) and +variability (Section IV-B2) and assess their role in evaluating +the quality of volume registration and pose regression. Lastly, +in Section IV-C, we report final pose regression results with +Leave One Out Cross-Validation (LOOCV) when training data +has the best registration alignment available, and intensity data +augmentations are performed. +A. Volume Registration +Before training the pose regression models, we require a +set of well aligned 3D US volumes to generate training and +validation data. Figure 5 reports the results when aligning our +US volumes with three different volume registration methods, +all of them available in 3D Slicer. The first column shows the +results obtained using direct similarity registration on raw US +volume data (rigid registration + scale for a total of 7 degrees + +Automatic +Mask +Fiducials + Mask +Training +23 w +21w +22 w +Testing +23w +24 w +25 wIEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL. XX, NO. XX, AUGUST XXXX +6 +of freedom) that uses the Mattes Mutual Information (MMI) +image comparison cost metric during fitting. We provide an +identity transformation for initialization. The second column +shows the results obtained with the same direct registration +method but performed only on a manually annotated mask, +a ROI that contours the fetal brain. The third column shows +the results obtained using a point-based registration, fiducial +points annotated by an obstetrician, followed by similarity +registration with a fetal brain mask. +Table I reports the evaluation of the different registration +approaches and their effects on US plane pose regression. We +performed three different types of evaluation. +1) Quantitative evaluation: Registration accuracy is usually +evaluated by identifying matching pairs of landmarks anno- +tated by the clinician in the ROI. We report the initial error +between the volumes with a GA ranging from 21 to 15 weeks +and the one used for training (23 weeks); besides, we report +the Root Mean Square (RMS) error between the landmarks +for the same volumes for the three registration approaches; +2) Obstetrician’s evaluation: We asked the obstetrician to +evaluate the registration outcome for the three registration +approaches, shown in Figure 5, by assigning a score between 1 +and 5, without taking into account the quality of the volumes; +3) Pose Regression CNN results: These experiments aim +to evaluate the extent to which the volume registration quality +affects the US plane pose regression results. +Implementation Details: Our framework is implemented +in PyTorch and trained using a single Tesla® A100-SXM4- +40GB hosted on the Computer Science network at University +College London. The network was trained for 50 epochs +with a batch size of K = 64 using Adam optimizer, with a +learning rate of 0.0001 and exponential decay rates β1 and +β2 of 0.9 and 0.999, respectively. We choose the best model +weights considering mean square error (MSE) obtained on the +validation set (20% of the training set). +Experiments: +The network was trained on phantom +data and fine-tuned on real ones [7]. Specifically, we fine- +tuned the network on planes extracted from a fetal brain +US volume with a GA of 23 weeks (f1, 22029 images) +and tested it on planes from five volumes obtained with +a single acquisition of different fetuses (f2, ..., f6) ranging +from a GA of 21 to 25 weeks to understand how well the +model generalizes over different shapes and sizes. Images +were resized to 128×128, preserving the same aspect ratio, +and cropped and centered to avoid visible sharp edges that +could cause overfitting. We augmented the training set by +randomly changing the images’ brightness, contrast, and sat- +uration to a value between 0 and 1. To this aim, we used the +torchvision.transforms.ColorJitter class that +is available in Pytorch for transforming and augmenting im- +ages. To evaluate the translation results, we employed the +Euclidean distance between the two planes, reported in mm. +For rotation, we display errors as the geodesic distance to +ground truth in degrees, more suitable for the geometric inter- +pretation of the distance between two 3D rotations and defined +as ErrorRotation = arccos((R′′ +00 + R′′ +11 + R′′ +22 − 1)/2), +where R′′ += +R′−1. Table I reports the median errors for +translation and rotation obtained on the testing volumes for the +TABLE I +EVALUATION OF THE DIFFERENT REGISTRATION APPROACHES AND +THEIR EFFECTS ON POSE REGRESSION RESULTS. FIRST, WE EVALUATED +THE INITIAL ERROR BETWEEN THE VOLUMES WITH RESPECT TO THE +ONE USED FOR TRAINING AND THE REGISTRATION ERROR FOR THE +THREE REGISTRATION APPROACHES USING THE ROOT MEAN SQUARE +(RMS) ERROR BETWEEN LANDMARKS FOR SIMILARITY REGISTRATION; +SECONDLY, THE OBSTETRICIAN EVALUATED THE QUALITY OF THE +REGISTRATION, PROVIDING A SCORE BETWEEN 1 AND 5 FOR THE +DIFFERENT REGISTRATION APPROACHES; LASTLY, WE REPORT THE +MEDIAN ERROR FOR TRANSLATION (EUCLIDEAN DISTANCE) AND +ROTATION (GEODESIC DISTANCE) FOR THE TESTING VOLUMES +Evaluation +Metric +Registration +Automatic +Mask +Fid + Mask +21 w +Quantitative +(RMS) +6.02 +8.21 +5.82 +5.80 +Obstetrician +(1-5) +4 +4 +5 +CNN +Translation +14.88 +9.42 +10.82 +Rotation +44.12 +19.49 +18.69 +22 w +Quantitative +(RMS) +2.53 +2.54 +2.55679 +2.52 +Obstetrician +(1-5) +4 +4 +5 +CNN +Translation +5.17 +4.19 +6.31 +Rotation +20.36 +14.16 +15.06 +23 w +Quantitative +(RMS) +3.34 +3.56 +3.86 +3.32 +Obstetrician +(1-5) +4 +4 +5 +CNN +Translation +7.29 +5.72 +7.38 +Rotation +26.89 +14.69 +15.65 +24 w +Quantitative +(RMS) +3.45 +4.66 +3.74 +3.00 +Obstetrician +(1-5) +3 +3 +3.5 +CNN +Translation +12.72 +6.62 +5.42 +Rotation +30.49 +14.14 +13.96 +25 w +Quantitative +(RMS) +2.19 +4.18 +3.89 +1.68 +Obstetrician +(1-5) +3 +3 +3.5 +CNN +Translation +7.20 +6.92 +6.14 +Rotation +21.14 +14.06 +14.41 +Avg +Quantitative +(RMS) +3.51 +1.92 +3.46 +3.27 +Obstetrician +(1-5) +3.6 +3.6 +4.4 +CNN +Translation +12.24 +6.57 +7.22 +Rotation +28.6 +15.31 +15.61 +three registration approaches. Figure 6 reports the translation +and rotation error distributions for fetal brain US volumes +ranging from a GA of 21 weeks to 25 weeks to analyze the +generalization capability of the network in the three registra- +tion approaches. Besides, we performed a sanity test using +the manually annotated TV SPs for the registration using the +fiducial points and the mask. The sectional images were saved +and fed into the network to estimate their pose. We plotted the +two planes within the volume in Unity to visually evaluate +the distance between the annotated TV SPs and the predicted +ones. The predicted planes were also fed into SonoNet, a + +IEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL. XX, NO. XX, AUGUST XXXX +7 +(a) Error distributions for the three registration approaches +(b) Sanity test on TV SPs - Fiducial Points + Mask +Fig. 6. +Results obtained for the experiments performed with the regression CNN. (a) Translation and rotation error distributions for both planes acquired +at random coordinates and planes acquired around the annotated TV SP to analyze the generalization capabilities of the network with the three registration +approaches. (b) TV SP prediction performed by the regression CNN. The green and the colored boxes indicate the ground truths and the predictions, respectively. +An obstetrician within the Unity environment manually annotated the ground truth poses of the TV SPs +CNN that can automatically detect 13 fetal standard views +in freehand 2D US data [14], in its Pytorch implementation. +Figure 6 reports the annotated TV SPs (green edges) and the +ones having the pose predicted by the regression network in +their sectional view and within the volumes. +B. Standard Planes Annotations +An obstetrician annotated the SPs for all the 3D US fetal +brain volumes previously registered using the Unity simulator +detailed above. +1) Quality of Annotations: To evaluate the quality of the +annotated TV SPs, we use SonoNet in its Pytorch implementa- +tion. We report the annotated TV SPs for the various volumes +and the registration approaches in Figure 7. SonoNet was able +to classify all the annotated SPs as TV SPs, the brain view +at the posterior horn of the ventricle. We then applied the +coordinates of the TV SP annotated on the training volume +(23 w) to the other planes. The synthetic images obtained +for the different volumes were fed again into SonoNet to +understand if the network could still recognize the planes as +standard views. All the planes were recognized as TV SPs, +except for the volume with a GA of 25 weeks obtained using +the automatic registration. The confidence of the classification +(value between 0 and 1) is reported on top of each image +in Figure 7. For each registration approach, the first column +shows the TV SPs obtained from the annotations, whereas +the second column shows the TV SPs obtained by using the +coordinates of the TV SP annotated on the training volume. At + +Automatic +Mask +Fiducials + Mask +Translation errors - Boxplots +Translation errors - Boxplots +Translation errors - Boxplots +70- +60 +Median +Median +Median +60 +Mean +Mean +Mean +60 +50 +Train +Train +[mm] +Train +Test +Test +Test +40 +++ +40 +H0 + +++ +30 +20 +a10 +10 +10. +0 +· +· +一 +0- +0 +0 +24w +23w +25w +23w +21w +22w +23w +25w +21w +22w +24w +23w +21w +22w +23w +24w +23w +25w +5.171 +0.9 +7.3 +14.882 +12.723 +7.205 +0.898 +9.416 +4.186 +5.716 +6.617 +6.923 +0.904 +10.818 +6.316 +7.381 +5.424 +6.146 +Rotation errors - Boxplots +Rotation errors - Boxplots +Rotation errors - Boxplots +175.0° +175.0° +Median +175.0° +Median +Median + Mean +Mean +Mean +150.0° +Train +Train +Train +Test +Test +125.0° +Test +125.0° +100.0° +jeodesic error[ +100.0° +100.0° +error +75.0° +75.0° +75.0° +Geodesic e +eodesic +50.0° +50.0° +50.0° +25.0° +25.0° +25.0° +G +一 +一 +0.0° +0.0° +0.0° +21w +25w +23w +22w +23w +24w +23w +21w +22w +23w +24w +25w +22w +24w +25w +23w +21w +23w +1.289 +44.126 20.363 +26.894 +30.498 +21.139 +1.287 +19.493 +14.457 +14.696 +14.139 +14.056 +1.29 +18.695 +15.067 +15.651 +13.969 +14.414Annotation +21w +22 w +23w +24w +25 wIEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL. XX, NO. XX, AUGUST XXXX +8 +Fig. 7. Outcome of SPs annotation performed by the obstetrician using the +Unity-based simulator and the joystick. First column: SPs obtained on volumes +registered with the automatic similarity registration (rigid registration + scale +for a total of 7 degrees of freedom) provided by 3D Slicer. Second column: SPs +obtained on volumes registered by aligning the volumes with the automatic +registration using the mask to contour the brain used to avoid overfitting +on the shape of the US volume. Third column: SPs obtained on volumes +registered using the fiducial points annotated by the obstetrician, followed +by similarity registration using the mask a ROI that contours the brain. We +report the confidence of the classification from SonoNet is reported on top of +each image. On the bottom, we report the average score that also includes the +training volume (23 w) for both annotations and the use of the coordinates +from the TV SP annotated of the training volume +the bottom, we report the average score, including the training +volume (23 w). +2) Variability in SPs Annotations: We evaluate the variabil- +ity in annotations by reporting the standard deviation of the +poses of the TV SPs annotated by the obstetrician in the var- +ious volumes. The internal variance of the entire set provides +the quality of the data. We report the variance in translation +and rotation of the annotated TV SPs for each registration +approach. For translation, we computed the coordinates of +the centroid ctransl = xc, yc, zc for each group made of the +training volume and the five testing volumes (i = 1, ..., 6) as +the mean on of three coordinates: +xc = +�6 +i=1 xi +6 +, yc = +�6 +i=1 yi +6 +, zc = +�6 +i=1 zi +6 +(1) +Then, we computed the Euclidean distance between each TV +SP and the centroid: +di,transl = +� +(xi − xc)2 + (yi − yc)2 + (zi − zc)2 +(2) +Lastly, we computed the root mean square of these distances: +RMStransl = +� +� +� +�1 +6 +6 +� +i=1 +d2 +i,transl +(3) +We computed the Chordal L2-averaging of the rotation ma- +trices for rotation, following the approach presented in [29]. +This is achieved by finding the rotation Rc that minimizes +the cost � +(i,j)∈N ∥RijRi − Rj∥2 +F . The above model can be +solved without enforcing the orthogonality constraint as a +least squares problem through vectorization and singular value +decomposition. After that, all the orthogonal constraints are +enforced by finding the nearest orthogonal matrices through +polar decomposition. Then, we computed the geodesic dis- +tance between the rotation matrix of each TV SP and the +average rotation matrix (angle of residual rotation): +di,rot(Ri, Rc) = di,rot(RiRT +c , I) = +��log(RiRT +c ) +�� +2 +(4) +where the norm is the Euclidean norm in R3. The angular +distance function di,rot(Ri, Rc) is equal to the rotation angle +∠(RiRT +c ). Starting from the quaternion representations, it is +possible to compute the angular distance between two rotations +easily. If ri and rc are quaternion representations of Ri and Rc +respectively, and θ = di,rot(Ri, Rc), then θ = 2arccos(|s|), +where (s, v) = r−1 +c +· ri. The absolute value sign in s is +required to account for the sign ambiguity in the quaternion +representation of the rotation RT +i Rc. The positive sign is +chosen so that the angle θ lies in the range 0 ≤ θ ≤ π, +as required. Hence, the distance di,rot(Ri, Rc) is equal to +the angle θ belonging to the rotation RiRT +c . As before, we +computed the root mean square of the distances: +RMSrot = +� +� +� +�1 +6 +6 +� +i=1 +di,rot(Ri, Rc)2 +(5) +The results are reported in Figure 8 along with the appearance +of the TV SPs annotated by the obstetrician for the three +registration approaches. +C. Ablation Study on the Pose Regression CNN +We performed an ablation study on the volumes registered +using the combination of fiducial points and masks. First, we +present the results for the US planes pose regression with and +without data augmentation on the training set. Then, we extend +the training set and combine data augmentation with LOOCV. +1) Data +augmentation: +Data +augmentation +artificially +boosts the size and variance of the training dataset by including +transformed copies of the training examples. This is especially +useful in medical imaging, where data augmentation is applied +to expand training data, address the class imbalance and +improve model generalization. To understand to which extent +data augmentation could benefit our training and increase the +generalization over different shapes and sizes, we augmented +the training set, as detailed above. + +Mask +Automatic +Fiducials + Mask +0.89 +Training +23w +Annotations Same coords. Annotations Same coords.lAnnotations Same coords +1.00 +0.98 +0.99 +0.97 +1.00 +0.99 +21w +0.71 +0.89 +0.78 +0.85 +0.95 +0.89 +22 W +Testing +0.56 +0.99 +0.92 +0.90 +0.99 +0.98 +23w +0.77 +0.98 +0.97 +0.97 +0.98 +0.83 +24 w +0.93 +0.37 +0.90 +0.88 +0.99 +0.96 +25w +Avg +0.88 +0.82 +0.91 +0.91 +0.97 +0.95IEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL. XX, NO. XX, AUGUST XXXX +9 +Fig. 8. TV SPs annotated by the obstetrician for the three registration approaches and variance in translation and rotation coordinates +2) Leave-one-out Cross-Validation: The localization errors +increase when the training distribution is quite different from +the testing one. A better training and testing data distribution +design can accurately reflect the model’s performance dur- +ing application. Hence, to estimate the performance of our +algorithm in making predictions on data not used to train the +model, we performed LOOCV experiments using the volumes +with a GA ranging from 21 to 25 weeks, including the one +originally used for training (23 weeks), for a total of six +volumes (N = 6). LOOCV is a special case of k-fold cross- +validation with k = N, the number of volumes. LOOCV +involves one fold per volume i.e., each volume by itself plays +the role of the validation set. The (N−1) volumes play the role +of the training set. With least-squares linear, a single model +performance cost is the same as a single model. The average +error on the test set is calculated by fitting on the volumes +not used in training and gives us an idea of how well the +model will perform on data it has not previously seen. We +then calculate the error on the test set Test Erravg to be the +average of all the errors on the six test sets Test Erri: +Test Erravg = 1 +N +N +� +i=1 +Test Erri +(6) +Figures 9a-f show the translation and rotation error distribu- +tions for the LOOCV experiments for the different trained +models. Figure 9g reports the results for the sanity test +performed using the manually annotated TV SPs, as previously +detailed in Section IV-A3). Table II reports the median, +mean ± standard deviation, maximum, and minimum errors +and the average error with and without data augmentation +(Equation 6). +V. DISCUSSION +We demonstrate that volume registration quality signifi- +cantly impacts the regression CNN results. This affects both +the trained model’s quality and network evaluation metrics’ +reliability. While this effect was mentioned in [7] as a limita- +tion of the study, here we quantify its impact. A registration +in two steps, initialized with annotated fiducials and refined +with direct iterative registration, leads to the best results. +In this setting, a set of SP from different fetuses, manually +annotated by an obstetrician, have a variance of 0.007 mm and +2.357 degrees in translation and rotation, respectively. These +are promising low values since this measurement provides an +estimate of the uncertainty associated with our volume data in +terms of localizing a specific plane of fetal brain anatomy. It +quantifies to which extent we can assume SPs from different +fetuses will have the same plane coordinates in a generalized +reference frame. This also puts a lower bound on the error we +should expect from our pose regression network. +In our second experiment, we evaluated the quality of the +annotations of the TV SPs. All the planes were recognized as +SPs both when testing the images obtained from the annotated +poses and when using the pose annotated on the reference +volume (training) on all the other volumes. This further +demonstrates the validity of our assumption that the brains +of different fetuses can be mapped to the same generalized +coordinate frame if the volumes are well registered. The +consistency of all annotated SPs with respect to different +criteria also validates the reliability of our new Unity-based +annotation platform. +In our LOOCV study, we outperform the pose regression +results obtained in [7] due to a multitude of factors. These +include a more rigorous volume registration process for the +training and testing data, larger training sets that include +various GAs, and additional image intensity data augmenta- + +Automatic +Mask +Fiducials + Mask +Variance in translation and rotation coordinates +Translation: 0.027 mm +Translation: 0.048 mm +Translation: 0.007 mm +Rotation: 1.110° +Rotation: 9.474° +Rotation: 2.3570 +24w +23w-train +21w +22 w +23w +25wIEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL. XX, NO. XX, AUGUST XXXX +10 +(a) Exp 1 - Test: 23T w +(b) Exp 4 - Test: 23 w +(c) Exp 2 - Test: 21 w +(d) Exp 5 - Test: 24 w +(e) Exp 3 - Test: 22 w +(f) Exp 6 - Test: 25 w +(g) Sanity test on TV SPs with LOOCV - Fiducial Points + Mask +Fig. 9. Results obtained for the experiments performed with the LOOCV. (a-f) Translation and rotation error distributions for the LOOCV experiments for +the six cases. (g) TV SP prediction performed by the regression CNNs. The green and the colored boxes indicate the ground truths and the predictions, +respectively. An obstetrician within the Unity environment manually annotated the ground truth poses of the TV SPs. 23T indicates the volume having a GA +of 23 weeks used for reference in the registration and training in the previous experiments + +Translation errors - Boxplots +Rotation errors - Boxplots +120.0° ++ ++ +Median +Median +80 ++ +Mean +Mean +Geodesic error [degrees] +100.0° ++ +Train +Train ++ +Test +Test +H++++ ++ + +60 +80.0° +lerror +60.0° +40 +Euclidean e +++ +40.0° +20 +20.0° +0 +0.0° +Train +Test +Train +Test +0.982 +3.655 +1.018 +5.159Translation errors - Boxplots +Rotation errors - +Boxplots +70- +140.0° ++ t 1 1 0 hhh +Median ++ +60 +Mean +120.0° +Train +[mm] ++ +50 +100.0° +Test ++ +error +40 +80.0° + + + + +30 +60.0° +20 +40.0° +Median +十 +Mean +10 +20.0° +Train +Test +P +0 +0.0° +Train +Test +Train +Test +1.096 +3.659 +1.007 +4.8Translation errors - Boxplots +Rotation errors - +Boxplots +70 +150.0° +Median +Median +Mean +0 +Mean +60 +Geodesic error [degrees] +125.0° +Train +Train +[ww] +50 +Test +Test +100.0° +error[ + +++ + ++ +40 +75.0° +30 +50.0° +20 +25.0° +10 +G +0 +0.0° +Train +Test +Train +Test +0.922 +3.763 +0.925 +7.54Translation errors - Boxplots +Rotation errors - Boxplots ++ +70 +100.0° ++ ++ ++ +Geodesic error [degrees] +60 ++++++ ++++++ +80.0° +E.50 +error + + +++ +++ + +++ +60.0° +40 +H+ +e+ nt+ +30 +40.0° +20 +Median +Median +Eucl +Mean +20.0° +Mean +10 +Train +Train +Test +Test +0 +0.0° +Train +Test +Train +Test +0.656 +3.57 +0.873 +7.229Translation errors - Boxplots +Rotation errors - Boxplots +70 ++ +Median +120.0° +Mean +++ +60 +Train +[ww] +100.0° ++ ++ +Test +50 ++ +Euclidean error [ +80.0° +40 ++ +60.0° +30 ++ ++ +40.0° ++ +20 +Median +Mean +20.0° +10 +Train +Test +0 +0.0° +Train +Test +Train +Test +1.145 +2.402 +1.011 +5.33Translation errors - Boxplots +Rotation errors - [ +Boxplots +175.0° ++ +Median ++ +60 +Mean +0 ++ ++ +[degrees] +150.0° +Train +50 +Test +125.0° ++ +error +40 ++ ++ +100.0° +Geodesic error [ ++ +30 +丰 +75.0° ++ + + + +20 +50.0° +Median +Mean +10 +25.0° +Train +Test +0 +0.0° +Train +Test +Train +Test +0.677 +4.118 +0.889 +8.44923Tw +21w +23w +24w +22 w +25 w +AnnotationIEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL. XX, NO. XX, AUGUST XXXX +11 +TABLE II +TRANSLATION AND ROTATION ERRORS OF OUR METHOD FOR THE LOOCV EXPERIMENTS ON INCLUDING DATA AUGMENTATION. NORM: EUCLIDEAN +DISTANCE, GE: GEODESIC ERROR, DA: DATA AUGMENTATION, SD: STANDARD DEVIATION, 23T INDICATES THE VOLUME HAVING A GA OF 23 WEEKS +USED FOR REFERENCE IN THE REGISTRATION AND TRAINING IN THE PREVIOUS EXPERIMENTS +Test Volume +DA +Translation - Norm [mm] +Rotation - GE [deg] +Median +Mean±SD +Min +Max +Median +Mean±SD +Min +Max +23T w +No +5.32 +5.72±2.97 +0.03 +21.39 +7.37 +20.32±29.61 +0.19 +157.96 +Yes +3.65 +4.77±3.85 +0.07 +34.52 +5.15 +6.49±5.21 +0.15 +71.42 +21 w +No +4.20 +4.58±2.72 +0.09 +33.96 +9.60 +12.95±111.20 +0.25 +169.52 +Yes +3.76 +4.23±2.63 +0.05 +23.56 +7.53 +9.58±9.61 +0.28 +153.25 +22 w +No +3.16 +3.81±2.47 +0.05 +31.65 +5.73 +6.48±3.79 +0.11 +65.18 +Yes +2.40 +3.00±2.36 +0.06 +31.83 +5.33 +5.99±3.40 +0.25 +88.77 +23 w +No +3.44 +3.80±2.26 +0.01 +69.42 +4.07 +4.61±2.80 +0.14 +30.82 +Yes +3.66 +4.02±2.36 +0.06 +66.07 +4.79 +5.62±4.61 +0.09 +138.82 +24 w +No +3.39 +5.33±7.96 +0.07 +71.75 +6.63 +7.95±6.65 +0.11 +146.72 +Yes +3.57 +5.43±8.23 +0.05 +72.21 +7.23 +8.29±6.44 +0.17 +94.58 +25 w +No +5.35 +8.11±6.95 +0.07 +36.48 +8.07 +10.76±11.61 +0.27 +159.74 +Yes +4.12 +5.20±4.01 +0.07 +40.06 +8.45 +10.34±10.57 +0.15 +162.30 +Test Erravg +No +4.15 +5.23±4.22 +0.05 +44.11 +6.91 +5.18±6.87 +0.18 +121.66 +Yes +3.53 +4.45±3.91 +0.07 +45.71 +6.42 +7.72±6.64 +0.18 +118.19 +tion. The proposed regression CNN successfully generalizes +pose regression to an unseen fetal brain. Specifically, data +augmentation decreased median errors by 14.94% and 7.10% +for translation and rotation, respectively. Besides, by extending +the training sets, we could further decrease the median errors +and obtain a better generalization. Our model is designed to +be size invariant by performing pose regression in normalized +coordinates with respect to the brain limits. However, GA +does not only affect size but also shape. Indeed, the inclusion +of different GAs covers a wide range of shapes and sizes, +enabling us to understand our model’s current generalizability +and limitations. +VI. CONCLUSION +In our previous work, we proposed a regression CNN to +predict the 6D pose of arbitrarily-oriented planes slicing the +fetal brain US volume without the need for real ground +truth data in real-time or 3D volume scans of the patient +beforehand. We assumed that brains from different fetuses +could be mapped to the same normalized coordinate system. In +this paper, we presented a detailed analysis of this assumption +by quantifying the variance in the pose of the TV SPs +annotated by an obstetrician when presented in the generalized +coordinate frame. Results show that this assumption is valid +within a small tolerance for different GAs. However, similarly +to the work in [7], our data only includes 6 volumes. Further +analysis with larger annotated datasets can provide further +insight into our generalized coordinate frame assumptions. +Currently, we can estimate plane poses within the brain with +a median translation error of 3.53mm and a median rotation +error of 6.42 degrees. These results are very promising, given +that we are localizing images of a previously unseen fetus. +However, there are still a few outlier plane pose estimations +with large errors (refer to maximum errors in Table II). 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Li, “Rotation Averaging,” +International Journal of Computer Vision, vol. 103, pp. 267–305, 2012. + diff --git a/WNE_T4oBgHgl3EQfyRyl/content/tmp_files/load_file.txt b/WNE_T4oBgHgl3EQfyRyl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..77571da6d5676e7b42def0df2775afb3f57e9fa6 --- /dev/null +++ b/WNE_T4oBgHgl3EQfyRyl/content/tmp_files/load_file.txt @@ -0,0 +1,1143 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf,len=1142 +page_content='This work has been submitted to the IEEE for possible publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Copyright may be transferred without notice, after which this version may no longer be accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' IEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' AUGUST XXXX 1 Learning ultrasound plane pose regression: assessing generalized pose coordinates in the fetal brain Chiara Di Vece,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Student Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Maela Le Lous,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Brian Dromey,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Francisco Vasconcelos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Anna L David,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Donald Peebles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Danail Stoyanov Senior Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' IEEE Abstract—In obstetric ultrasound (US) scanning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' the learner’s ability to mentally build a three-dimensional (3D) map of the fetus from a two-dimensional (2D) US image represents a significant challenge in skill acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We aim to build a US plane localization system for 3D visualization, training, and guidance without integrating additional sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' This work builds on top of our previous work, which predicts the six-dimensional (6D) pose of arbitrarily-oriented US planes slicing the fetal brain with respect to a normalized reference frame using a convolutional neural network (CNN) regression network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Here, we analyze in detail the assumptions of the normalized fetal brain reference frame and quantify its accuracy with respect to the acquisition of transventricular (TV) standard plane (SP) for fetal biometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We investigate the impact of registration quality in the training and testing data and its subsequent effect on trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Finally, we introduce data augmentations and larger training sets that improve the results of our previous work, achieving median errors of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='53 mm and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='42° for translation and rotation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Index Terms—Fetal ultrasounds, convolutional neural network, plane localization I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' INTRODUCTION F ETAL US is a non-invasive, real-time and cost-effective diagnostic tool for monitoring fetal growth and anatomy throughout gestation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' During routine mid-trimester fetal US scan, the sonographer acquires the SP, predefined anatomical planes defined by scientific committees to promote interna- tional guidelines for fetal US images [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Specifically, the TV SP, in Figure 1, needs to show the skull shape, the cavum septum pellucidi, the posterior horn of the lower lateral ven- tricle, and the anterior horns of the lateral ventricles [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' This allows for reliable measurements of specific structures and reduced inter- and intra-sonographer variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The correct identification of SPs is essential in the second-trimester fetal anatomic survey to investigate the morphological characteris- tics of the fetus and detect abnormalities or deviations from the expected growth patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Sonographers may struggle to The work was supported by the Wellcome/EPSRC center for Interven- tional and Surgical Sciences (WEISS) [203145Z/16/Z];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Engineering and Physical Sciences Research Council (EPSRC) [EP/P027938/1, EP/R004080 /1, EP/P012841/1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The Royal Academy of Engineering Chair in Emerging Technologies Scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' and Horizon 2020 FET (GA 863146).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' For the purpose of open access, the author has applied a CC BY public copyright licence to any author accepted manuscript version arising from this submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Di Vece, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Vasconcelos and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Stoyanov are with Wellcome/EPSRC center for Interventional and Surgical Sciences (WEISS) and the Department of Computer Science, University College London (UCL), UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Le Lous, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Dromey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' David and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='Peebles are with WEISS, Elizabeth Garrett Anderson Institute for Women’s Health and NIHR University College London Hospitals Biomedical Research center, UCL, UK chiara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='divece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='20@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='uk Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' TV SP to evaluate fetal biometry in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' This plane needs to show the skull shape, the cavum septum pellucidi, the posterior horn of the lower lateral ventricle, and the anterior horns of the lateral ventricles obtain good SPs for various reasons, including inexperience, limited training, time limitations, and fetal movement [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Most trainees learn on actual patients under the direct supervision of an expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Although US simulators have been developed recently, trainee engagement has been limited due to competing time priorities [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The primary training challenge faced by all novice sonographers is not related to knowledge of anatomy or familiarity with the US machine interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Rather, the manual navigation of the probe towards acquiring SP requires the sonographer to build a 3D map of the fetus from dynamic 2D sectional views while handling the probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Measurements of biometric parameters and assessments of the fetal brain’s anatomy may be erroneous due to mistakes in locating the 2D scan within the 3D volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' At present, SPs recognition represents the main focus of fetal US training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Due to the requirement to interpret variable and complex images and their spatial relationship, autonomous probe navigation towards SPs continues to be a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Our final aim is to develop a US navigation system that guides the sonographer towards obtaining SPs with reference to fetal anatomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' In [7], we proposed a method to localize a US plane of a fetal brain directly from its 2D US image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' While this is a promising result towards active guidance during fetal scanning, a few aspects still require further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' First, it assumes that brains from different fetuses can be mapped to the same normalized coordinate system, where each SP always has the same pose coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The accuracy of this assumption has not been quantified, which limits the analysis of experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Secondly, the achieved pose accuracy is still far from optimal due to a lack of variation in training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='08317v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='CV] 19 Jan 2023 Routine Har-mid Transventricular 16 Gn6 Plane C6/M4 FF2/E1 SRII10/CRI1 OFD BPD 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='1mm GA 28w1d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='1SD 1211g(2lb11oz) EFW GA 28w0d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='3SD OFD 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='78cm BPD 28w2d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='5SD GA HC 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='90cm GA 27wOd-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='4SD 80% CI (BPD/OFD)IEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, AUGUST XXXX 2 In this work, we expand on this work with the following contributions: We developed a tool for annotating poses of SPs in 3D US volumes using the Unity engine and a gaming controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We used this tool to obtain ground truth poses of TV SPs in 3D US volumes (annotated by an obstetrician) of a publicly available fetal brain dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We used this data to validate the US pose regression and its training data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We evaluate to which extent the brains of different fetuses can be mapped to the same generalized coordinate frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We quantify this through the variance in the pose of annotated TV SPs when represented in the generalized coordinate frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' This puts strict boundaries on how accurate our pose regression models can be and adds additional context to the analysis of its experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The training data consists of 3D US volumes that need to be well registered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We evaluate the effect of different registration techniques on the performance of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We demonstrate that high-quality registration is funda- mental to effective network training and to preserve our models’ assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We outperform the pose regression results obtained in [7] by introducing additional data augmentation and increas- ing training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' RELATED WORK This section presents the related work for three related but different tasks: extraction of SPs, slice-to-volume registration, and localization of SPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Extraction of Standard Planes Previous work proposed automating the extraction of SPs from data acquired with a simplified protocol rather than assisting operators in acquiring typical freehand 2D SPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' In one of their initial works, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' [8] developed a system based on two AdaBoost classifiers placed in cascade to automatically detect in a coarse-to-fine way early gestational sac SP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Further early studies [9]–[12] detect key abdominal structures and landmarks in a sequence of 2D ultrasound fetal images to classify the SPs in each frame of US videos based on the presence and orientation of the landmarks using various conventional machine learning (ML) algorithms like AdaBoost, Random Forest as well as support vector machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' These methods, however, are only applicable to a subset of fetal SPs (brain and abdomen);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' besides, the quality of the obtained SP cannot be compared with the one achieved with typical freehand scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Classification methods based on CNNs were used to detect 2D SPs because of their powerful ability to learn hierarchical representations automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' To detect the fetal SPs, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' [13] fine-tuned a pre-trained classification CNN based on transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Baumgartner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' [14] proposed a classification model to detect thirteen SPs with unsupervised learning and then used weakly-supervised learning (SL) based on image-level labels to locate anatomical structures in each plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The study employed extensive data, including videos longer than those usually collected in clinical practice (roughly 30 minutes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The CNNs are fed with surrounding and addi- tional information from each US video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' To capture temporal information in 2D US, some works added to the detection of the three fetal SPs a recurrent neural network (RNN) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' All the methods mentioned above are effective in the detection of SP images, but they can only determine whether an image was captured at a SP, not where exactly it is in the corresponding 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Besides, the models require a high amount of annotated data to be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Slice-to-volume Registration One approach for US plane localization is to find its alignment with respect to a pre-acquired 3D volume of the same anatomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' This is an optimization problem, typically solved with iterative numerical methods that minimize the distance between specific landmarks or maximize intensity- based similarity metrics [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Unfortunately, the cost func- tions associated with these metrics are frequently non-convex, limiting the capture range of these registration methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Our task differs from a classic slice-to-volume registration method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=', it does not require a previously acquired 3D volume of the same subject being scanned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Instead, we predict the pose relative to a generalized brain center, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=', a stable anatomical brain point across the different, pre-aligned volumes, where training and test data belong to different subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Localization of Standard Planes Predicting the pose of SPs in 3D volumes can be per- formed without a patient-specific model and without using pre-operative data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' This has been primarily approached as a classification problem, where the plane pose space is dis- cretized into bins, and the estimation boils down to selecting one of the bins [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' In fetal magnetic resonance imaging (MRI) [19], [20] and fetal US [21], the prediction of slice locations has been previously improved with learning-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' General purpose learning-based methods for pose estimation approach this as a regression of a 3D translation and a 3D rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 3D rotations can be represented in multiple conventional ways, such as quaternions, axis-angle, or Euler angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' pointed out in [22] that if the entire rotation space is required, these representations are sub-optimal for specific angle ranges, and proposed a new 6D representation for rotations that does not suffer from these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' This rota- tion representation has been adopted for US plane localization in [7], where a regression CNN is proposed to predict the 6D pose of arbitrarily-oriented planes slicing the fetal brain US volume without the need for real ground truth data in real-time or 3D volume scans of the patient beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The proposed network reliably localizes US planes within the fetal brain in phantom data and successfully generalizes pose regression for unseen fetal brains from a similar gestational age (GA) as in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The network was tested on real fetal brain images with a GA ranging from 21 to 25 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Similarly, Yeung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' [23] proposed a CNN that takes a set of images as input and learns to compare them in pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The model was tested on IEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, AUGUST XXXX 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Pipeline to train and test the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 1) The 3D fetal brain US volumes are registered in 3D Slicer using similarity registration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 2) The volumes are reconstructed into Unity, and synthetic sectional (slice) image representations are generated and saved along with their 6D pose (translation and rotation) relative to the center of the fetal brain US volume;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 3) These images are fed into the network to output the estimated slice 6D pose (translation and rotation) relative to the same point fetal brain volumes with a GA ranging from 18 to 22 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Then, in [24], an unsupervised cycle consistency using the fact that the overall displacement of a sequence of images in the 3D anatomical atlas is equal to the displacement from the first image to the last in that sequence was added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' MATERIALS AND METHODS The development of our US pose regression system has been divided into three main blocks, reported in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' First, we align 3D US volumes for the training and validation of our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Secondly, we developed a Unity-based simulator to visualize and manually annotate SPs in 3D US volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' This also enables the automated generation of supervised training data for our pose regression models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=', 2D synthetic images and their ground truth 6D pose relative to the volume center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Finally, we detail our deep learning (DL)-based plane pose regression system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Data preparation 1) Dataset: We used seven real fetal brain US volumes with a GA ranging from 21 to 25 [25] (singleton pregnancy with no abnormal findings)1 obtained from different fetuses 1Refer to www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='datavers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='nl for details (fi, with i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=', 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' All volumes were processed to be isotropic with voxel size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='5×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='5×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='5 mm and average size of 249×174×155 mm (coronal×axial×sagittal, actual size of the acquired volumes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 2) Volume registration: The registration procedure was performed in 3D Slicer2 as detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The whole process is depicted in Figure 3 for 3D US real volumes of the fetal brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Starting from the initial volumes (Figure 3a), we: Used fiducial points to achieve an initial alignment of the volumes using the Fiducial Registration Wizard module (Figure 3b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Defined a contour mask of the brain using the Segment Editor module to avoid overfitting on the shape of the ultrasound volume during registration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Used the general registration (BRAIN) module available in 3D Slicer to register the volumes with a similarity registration phase (rigid registration + scale for a total of 7 degrees of freedom), as shown in Figure 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We chose the previously obtained masks as a Region of Interest (ROI) so that the registration algorithm only considers a specific image region for the registration (the fetal brain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 23D Slicer Data preparation: volume registration Similarity Annotation of A registration fiducial points 3DSlicer Generation of masks for the brain 0502-POS Deep learning-based pose regression system Unity simulator for volume reconstruction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' annotation Synthetic and synthetic data acquisition sectional images Volume Rendering 6-DoF Phantom 3D → 6D Pose Network 3D model of the foetal brain Us pose unity C fetus starting from 3D US Regression Heads Feature Extraction Input ResNet-18 backbone 128x128 r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='r4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='rs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='r6 [5] Synthetic image Visualization of B (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='3) the planes slicing c the volume 3 6D Pose GT = (tx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='ty,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='tz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='ax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='ay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content="az) t'x," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content="t'y," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content="t' Automated t1," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='t3 Qout (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content="6) Conv2d Basic Block 1 Basic Block O generation of (1,3) (1,3) L 6D Pose BatchNorm2d =3 supervised data tx,ty,tz Epred=(t'x,t'y,t'z,a'x,a'y,a'z) ReLU (3,3) MaxPool2d unity Linear Standard planes LosSTr annotation R'-R²- LosSTo LTotIEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, AUGUST XXXX 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Registration procedure on real fetal brain US volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Workflow from pre-registration to post-registration: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Starting US fetal brain volumes before registration, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Fiducial points used to achieve an initial alignment of the volumes, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Contour mask of the brain used to avoid overfitting on the shape of the US volume B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Unity Simulator for Standard Plane Annotations and Syn- thetic Images Generation Starting from the open-source project UnityVolumeRen- dering, we developed our simulator using the game engine Unity3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The first step is rendering the volume starting from RAW, PARCHG, or Digital Imaging and Communications in Medicine (DICOM) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The simulator allows the user to render the volume using three modes: Isosurface Rendering, Maximum Intensity Projection, and Direct Volume Rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The latter is the standard rendering mode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' it uses transfer functions (1D or 2D) to determine the color and opacity while projecting rays across the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Transfer functions translate density (and gradient magnitude in case of 2D) to a color and opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The simulator allows the user to set a custom transfer function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 1D Transfer Function: the density is represented on the X-axis, whereas the opacity (alpha) is on the Y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The user can create a curve for opacity by density by shifting the grey alpha knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The bottom gradient-colored panel maps color to density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 2D transfer function: the density is represented on the X-axis, whereas the gradient magnitude is on the Y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Using the sliders, the user can define a rectangle shape, modify its size/position, and the minimum and maximum values for alpha/opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 1) User interface: After loading the DICOM dataset ex- tracted from 3D Slicer and setting the transfer function, the clinician can add a plane slicing the reconstructed 3D US volume (left-hand side of Figure 4) with an arbitrary orientation and visualize the plane in an external window (right-hand side, bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The plane can be controlled using a joystick (right-hand side, top), simplifying the annotation of the SPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Hence, the clinician can modify the position and rotation of the plane using the joystick while monitoring the appearance in the external window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Once the clinicians are satisfied with the pose of the SP, they can save it using the last button in the external windows to get a picture of the slicing plane, and its 6D pose relative to the volume center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 2) Training and testing data generation: To generate train- ing data for our models, we generated synthetic slices by applying rotation and translation to a plane placed in the center of the volume generated with a uniform random distribution within a fixed range to avoid slices with poor overlap with the volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The synthetic images obtained by slicing the 3Unity Real-Time Development Platform Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Unity simulator for volume reconstruction, SP annotations, and automatic supervised data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' While controlling the probe with the joystick using the suggested commands, the clinician can visualize the slicing plane in an external window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Once the desired plane is reached, it can be saved using the “Save plane” button along with its 6D pose relative to the volume center volume were saved along with their pose with respect to the volume center (fetal brain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' This provides an automated way of generating a high amount of training data with reliable ground truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' An obstetrician annotated the position of the TV SP by directly manipulating a slicing plane with the joystick and choosing the translation and angle sampling intervals to avoid sampling of planes at the edges of the volume containing no information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The nearby planes were generated by applying small random rotations and translations (uniform distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Specifically, the acquisition interval between two planes was decreased from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='001 for translation (Unity environment, with coordinates normalized between -1 and 1 so that the pose regression works in a fixed, normalized range, independent of the real brain size in mm) and from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='9° to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='9° for rotation compared to acquisition of planes at random coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We acquired 20699 planes with random orientation per volume and 1330 around the TV SP for a total of 22029 images for each volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Deep Learning-based Plane Pose Regression System We base our 6D pose regression system on the network proposed in [7] (Figure 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We used an 18-layer residual CNN (ResNet-18) [26] as a backbone for feature extraction with the pre-trained ImageNet weights [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We modified the network by re-initializing the fully connected layer based on the representation’s dimension (nine parameters) and adding a regression head to directly output the rotation and trans- lation representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The network receives the US image I (128×128) obtained by slicing the volume and its 6D pose with respect to the center of the fetal brain US volume θGT = (tx, ty, tz, αx, αy, αz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We use this information as the ground truth label for network training and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The CNN learns to predict the 6D pose with respect to the same point θP red = (t′ x, t′ y, t′ z, α′ x, α′ y, α′ z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Specifically, the network first outputs a vector of nine parameters θOut = (t1, t2, t3, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=', r6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' the first three are used for the translation and the last six for the rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Then, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=', r6 are used internally by our CNN to reconstruct the rotation matrix R′ in the forward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Pre-registration a b Post registratior cUnity scene Import RAW dataset Joystick ImportPARCHGdataset Import DIcOM dataset Edit imported dataset Edit slicing plane Slicing plane Right Stick-RotatePlane LeftStick-MovePlane Top/down Arrows -Vertical movement Plane previous next add remove save Button A -Change Shell Transparency plane plane plane plane plane Button B -Reset PoseIEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, AUGUST XXXX 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Registration procedure on real fetal brain US volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' First column: results obtained using the automatic similarity registration (rigid registration + scale for a total of 7 degrees of freedom) provided by 3D Slicer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Second column: results obtained by aligning the volumes with the automatic registration using the mask to contour the brain used to avoid overfitting on the shape of the US volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Third column: results obtained using the fiducial points annotated by the obstetrician, followed by similarity registration using the mask a ROI that contours the brain Differently from [7], we also perform image intensity aug- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' More specifically, we change brightness, contrast, and saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The detailed parameters of this augmentation are described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Indeed, recent work has found that, in the context of fetal US augmentation strategies, generating less realistic training images leads to improvement in generalization capability [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' EXPERIMENTS AND RESULTS In this section, we report the results of three main experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' First, in Section IV-A, we assess three different US vol- ume registration methods and investigate their qualitative and quantitative impact in defining a generalized inter-patient co- ordinate frame for the fetal brain (Section IV-A1 and IV-A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Besides, we illustrate the effect of these registration methods when training US plane pose regression networks using a fetal US volume of a single 23-week fetus to train the network and fetuses with a GA ranging from 21 to 25 weeks for testing (Section IV-A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' In our second experiment, in Section IV-B, we investigate how consistent the manual annotations of the TV SPs are both in terms of quality (Section IV-B1) and variability (Section IV-B2) and assess their role in evaluating the quality of volume registration and pose regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Lastly, in Section IV-C, we report final pose regression results with Leave One Out Cross-Validation (LOOCV) when training data has the best registration alignment available, and intensity data augmentations are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Volume Registration Before training the pose regression models, we require a set of well aligned 3D US volumes to generate training and validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Figure 5 reports the results when aligning our US volumes with three different volume registration methods, all of them available in 3D Slicer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The first column shows the results obtained using direct similarity registration on raw US volume data (rigid registration + scale for a total of 7 degrees Automatic Mask Fiducials + Mask Training 23 w 21w 22 w Testing 23w 24 w 25 wIEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, AUGUST XXXX 6 of freedom) that uses the Mattes Mutual Information (MMI) image comparison cost metric during fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We provide an identity transformation for initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The second column shows the results obtained with the same direct registration method but performed only on a manually annotated mask, a ROI that contours the fetal brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The third column shows the results obtained using a point-based registration, fiducial points annotated by an obstetrician, followed by similarity registration with a fetal brain mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Table I reports the evaluation of the different registration approaches and their effects on US plane pose regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We performed three different types of evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 1) Quantitative evaluation: Registration accuracy is usually evaluated by identifying matching pairs of landmarks anno- tated by the clinician in the ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We report the initial error between the volumes with a GA ranging from 21 to 15 weeks and the one used for training (23 weeks);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' besides, we report the Root Mean Square (RMS) error between the landmarks for the same volumes for the three registration approaches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 2) Obstetrician’s evaluation: We asked the obstetrician to evaluate the registration outcome for the three registration approaches, shown in Figure 5, by assigning a score between 1 and 5, without taking into account the quality of the volumes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 3) Pose Regression CNN results: These experiments aim to evaluate the extent to which the volume registration quality affects the US plane pose regression results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Implementation Details: Our framework is implemented in PyTorch and trained using a single Tesla® A100-SXM4- 40GB hosted on the Computer Science network at University College London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The network was trained for 50 epochs with a batch size of K = 64 using Adam optimizer, with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0001 and exponential decay rates β1 and β2 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='9 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='999, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We choose the best model weights considering mean square error (MSE) obtained on the validation set (20% of the training set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Experiments: The network was trained on phantom data and fine-tuned on real ones [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Specifically, we fine- tuned the network on planes extracted from a fetal brain US volume with a GA of 23 weeks (f1, 22029 images) and tested it on planes from five volumes obtained with a single acquisition of different fetuses (f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=', f6) ranging from a GA of 21 to 25 weeks to understand how well the model generalizes over different shapes and sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Images were resized to 128×128, preserving the same aspect ratio, and cropped and centered to avoid visible sharp edges that could cause overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We augmented the training set by randomly changing the images’ brightness, contrast, and sat- uration to a value between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' To this aim, we used the torchvision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='ColorJitter class that is available in Pytorch for transforming and augmenting im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' To evaluate the translation results, we employed the Euclidean distance between the two planes, reported in mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' For rotation, we display errors as the geodesic distance to ground truth in degrees, more suitable for the geometric inter- pretation of the distance between two 3D rotations and defined as ErrorRotation = arccos((R′′ 00 + R′′ 11 + R′′ 22 − 1)/2), where R′′ = R′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Table I reports the median errors for translation and rotation obtained on the testing volumes for the TABLE I EVALUATION OF THE DIFFERENT REGISTRATION APPROACHES AND THEIR EFFECTS ON POSE REGRESSION RESULTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' FIRST, WE EVALUATED THE INITIAL ERROR BETWEEN THE VOLUMES WITH RESPECT TO THE ONE USED FOR TRAINING AND THE REGISTRATION ERROR FOR THE THREE REGISTRATION APPROACHES USING THE ROOT MEAN SQUARE (RMS) ERROR BETWEEN LANDMARKS FOR SIMILARITY REGISTRATION;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' SECONDLY, THE OBSTETRICIAN EVALUATED THE QUALITY OF THE REGISTRATION, PROVIDING A SCORE BETWEEN 1 AND 5 FOR THE DIFFERENT REGISTRATION APPROACHES;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' LASTLY, WE REPORT THE MEDIAN ERROR FOR TRANSLATION (EUCLIDEAN DISTANCE) AND ROTATION (GEODESIC DISTANCE) FOR THE TESTING VOLUMES Evaluation Metric Registration Automatic Mask Fid + Mask 21 w Quantitative (RMS) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='02 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='82 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='80 Obstetrician (1-5) 4 4 5 CNN Translation 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='88 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='42 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='82 Rotation 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='12 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='49 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='69 22 w Quantitative (RMS) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='54 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='55679 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='52 Obstetrician (1-5) 4 4 5 CNN Translation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='31 Rotation 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='36 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='16 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='06 23 w Quantitative (RMS) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='34 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='56 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='86 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='32 Obstetrician (1-5) 4 4 5 CNN Translation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='72 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='38 Rotation 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='89 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='69 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='65 24 w Quantitative (RMS) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='66 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='74 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='00 Obstetrician (1-5) 3 3 3.' metadata={'source': 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2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='68 Obstetrician (1-5) 3 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='5 CNN Translation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='46 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='27 Obstetrician (1-5) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='4 CNN Translation 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='24 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='57 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='22 Rotation 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='31 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='61 three registration approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Figure 6 reports the translation and rotation error distributions for fetal brain US volumes ranging from a GA of 21 weeks to 25 weeks to analyze the generalization capability of the network in the three registra- tion approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Besides, we performed a sanity test using the manually annotated TV SPs for the registration using the fiducial points and the mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The sectional images were saved and fed into the network to estimate their pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We plotted the two planes within the volume in Unity to visually evaluate the distance between the annotated TV SPs and the predicted ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The predicted planes were also fed into SonoNet, a IEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, AUGUST XXXX 7 (a) Error distributions for the three registration approaches (b) Sanity test on TV SPs - Fiducial Points + Mask Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Results obtained for the experiments performed with the regression CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' (a) Translation and rotation error distributions for both planes acquired at random coordinates and planes acquired around the annotated TV SP to analyze the generalization capabilities of the network with the three registration approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' (b) TV SP prediction performed by the regression CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The green and the colored boxes indicate the ground truths and the predictions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' An obstetrician within the Unity environment manually annotated the ground truth poses of the TV SPs CNN that can automatically detect 13 fetal standard views in freehand 2D US data [14], in its Pytorch implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Figure 6 reports the annotated TV SPs (green edges) and the ones having the pose predicted by the regression network in their sectional view and within the volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Standard Planes Annotations An obstetrician annotated the SPs for all the 3D US fetal brain volumes previously registered using the Unity simulator detailed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 1) Quality of Annotations: To evaluate the quality of the annotated TV SPs, we use SonoNet in its Pytorch implementa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We report the annotated TV SPs for the various volumes and the registration approaches in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' SonoNet was able to classify all the annotated SPs as TV SPs, the brain view at the posterior horn of the ventricle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We then applied the coordinates of the TV SP annotated on the training volume (23 w) to the other planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The synthetic images obtained for the different volumes were fed again into SonoNet to understand if the network could still recognize the planes as standard views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' All the planes were recognized as TV SPs, except for the volume with a GA of 25 weeks obtained using the automatic registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The confidence of the classification (value between 0 and 1) is reported on top of each image in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' For each registration approach, the first column shows the TV SPs obtained from the annotations, whereas the second column shows the TV SPs obtained by using the coordinates of the TV SP annotated on the training volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' At Automatic Mask Fiducials 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='29 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='695 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='067 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='651 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='969 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='414Annotation 21w 22 w 23w 24w 25 wIEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, AUGUST XXXX 8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Outcome of SPs annotation performed by the obstetrician using the Unity-based simulator and the joystick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' First column: SPs obtained on volumes registered with the automatic similarity registration (rigid registration + scale for a total of 7 degrees of freedom) provided by 3D Slicer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Second column: SPs obtained on volumes registered by aligning the volumes with the automatic registration using the mask to contour the brain used to avoid overfitting on the shape of the US volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Third column: SPs obtained on volumes registered using the fiducial points annotated by the obstetrician, followed by similarity registration using the mask a ROI that contours the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We report the confidence of the classification from SonoNet is reported on top of each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' On the bottom, we report the average score that also includes the training volume (23 w) for both annotations and the use of the coordinates from the TV SP annotated of the training volume the bottom, we report the average score, including the training volume (23 w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 2) Variability in SPs Annotations: We evaluate the variabil- ity in annotations by reporting the standard deviation of the poses of the TV SPs annotated by the obstetrician in the var- ious volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The internal variance of the entire set provides the quality of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We report the variance in translation and rotation of the annotated TV SPs for each registration approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' For translation, we computed the coordinates of the centroid ctransl = xc, yc, zc for each group made of the training volume and the five testing volumes (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=', 6) as the mean on of three coordinates: xc = �6 i=1 xi 6 , yc = �6 i=1 yi 6 , zc = �6 i=1 zi 6 (1) Then, we computed the Euclidean distance between each TV SP and the centroid: di,transl = � (xi − xc)2 + (yi − yc)2 + (zi − zc)2 (2) Lastly, we computed the root mean square of these distances: RMStransl = � � � �1 6 6 � i=1 d2 i,transl (3) We computed the Chordal L2-averaging of the rotation ma- trices for rotation, following the approach presented in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' This is achieved by finding the rotation Rc that minimizes the cost � (i,j)∈N ∥RijRi − Rj∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The above model can be solved without enforcing the orthogonality constraint as a least squares problem through vectorization and singular value decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' After that, all the orthogonal constraints are enforced by finding the nearest orthogonal matrices through polar decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Then, we computed the geodesic dis- tance between the rotation matrix of each TV SP and the average rotation matrix (angle of residual rotation): di,rot(Ri, Rc) = di,rot(RiRT c , I) = ��log(RiRT c ) �� 2 (4) where the norm is the Euclidean norm in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The angular distance function di,rot(Ri, Rc) is equal to the rotation angle ∠(RiRT c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Starting from the quaternion representations, it is possible to compute the angular distance between two rotations easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' If ri and rc are quaternion representations of Ri and Rc respectively, and θ = di,rot(Ri, Rc), then θ = 2arccos(|s|), where (s, v) = r−1 c ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The absolute value sign in s is required to account for the sign ambiguity in the quaternion representation of the rotation RT i Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The positive sign is chosen so that the angle θ lies in the range 0 ≤ θ ≤ π, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Hence, the distance di,rot(Ri, Rc) is equal to the angle θ belonging to the rotation RiRT c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' As before, we computed the root mean square of the distances: RMSrot = � � � �1 6 6 � i=1 di,rot(Ri, Rc)2 (5) The results are reported in Figure 8 along with the appearance of the TV SPs annotated by the obstetrician for the three registration approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Ablation Study on the Pose Regression CNN We performed an ablation study on the volumes registered using the combination of fiducial points and masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' First, we present the results for the US planes pose regression with and without data augmentation on the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Then, we extend the training set and combine data augmentation with LOOCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 1) Data augmentation: Data augmentation artificially boosts the size and variance of the training dataset by including transformed copies of the training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' This is especially useful in medical imaging, where data augmentation is applied to expand training data, address the class imbalance and improve model generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' To understand to which extent data augmentation could benefit our training and increase the generalization over different shapes and sizes, we augmented the training set, as detailed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Mask Automatic Fiducials + Mask 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='89 Training 23w Annotations Same coords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Annotations Same coords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='lAnnotations Same coords 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='98 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='83 24 w 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='96 25w Avg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='95IEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, AUGUST XXXX 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' TV SPs annotated by the obstetrician for the three registration approaches and variance in translation and rotation coordinates 2) Leave-one-out Cross-Validation: The localization errors increase when the training distribution is quite different from the testing one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' A better training and testing data distribution design can accurately reflect the model’s performance dur- ing application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Hence, to estimate the performance of our algorithm in making predictions on data not used to train the model, we performed LOOCV experiments using the volumes with a GA ranging from 21 to 25 weeks, including the one originally used for training (23 weeks), for a total of six volumes (N = 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' LOOCV is a special case of k-fold cross- validation with k = N, the number of volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' LOOCV involves one fold per volume i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=', each volume by itself plays the role of the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The (N−1) volumes play the role of the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' With least-squares linear, a single model performance cost is the same as a single model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The average error on the test set is calculated by fitting on the volumes not used in training and gives us an idea of how well the model will perform on data it has not previously seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We then calculate the error on the test set Test Erravg to be the average of all the errors on the six test sets Test Erri: Test Erravg = 1 N N � i=1 Test Erri (6) Figures 9a-f show the translation and rotation error distribu- tions for the LOOCV experiments for the different trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Figure 9g reports the results for the sanity test performed using the manually annotated TV SPs, as previously detailed in Section IV-A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Table II reports the median, mean ± standard deviation, maximum, and minimum errors and the average error with and without data augmentation (Equation 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' DISCUSSION We demonstrate that volume registration quality signifi- cantly impacts the regression CNN results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' This affects both the trained model’s quality and network evaluation metrics’ reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' While this effect was mentioned in [7] as a limita- tion of the study, here we quantify its impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' A registration in two steps, initialized with annotated fiducials and refined with direct iterative registration, leads to the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' In this setting, a set of SP from different fetuses, manually annotated by an obstetrician, have a variance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='007 mm and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='357 degrees in translation and rotation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' These are promising low values since this measurement provides an estimate of the uncertainty associated with our volume data in terms of localizing a specific plane of fetal brain anatomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' It quantifies to which extent we can assume SPs from different fetuses will have the same plane coordinates in a generalized reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' This also puts a lower bound on the error we should expect from our pose regression network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' In our second experiment, we evaluated the quality of the annotations of the TV SPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' All the planes were recognized as SPs both when testing the images obtained from the annotated poses and when using the pose annotated on the reference volume (training) on all the other volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' This further demonstrates the validity of our assumption that the brains of different fetuses can be mapped to the same generalized coordinate frame if the volumes are well registered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The consistency of all annotated SPs with respect to different criteria also validates the reliability of our new Unity-based annotation platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' In our LOOCV study, we outperform the pose regression results obtained in [7] due to a multitude of factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' These include a more rigorous volume registration process for the training and testing data, larger training sets that include various GAs, and additional image intensity data augmenta- Automatic Mask Fiducials + Mask Variance in translation and rotation coordinates Translation: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='027 mm Translation: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='048 mm Translation: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='007 mm Rotation: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='110° Rotation: 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='474° Rotation: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='3570 24w 23w-train 21w 22 w 23w 25wIEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, AUGUST XXXX 10 (a) Exp 1 - Test: 23T w (b) Exp 4 - Test: 23 w (c) Exp 2 - Test: 21 w (d) Exp 5 - Test: 24 w (e) Exp 3 - Test: 22 w (f) Exp 6 - Test: 25 w (g) Sanity test on TV SPs with LOOCV - Fiducial Points + Mask Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Results obtained for the experiments performed with the LOOCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' (a-f) Translation and rotation error distributions for the LOOCV experiments for the six cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' (g) TV SP prediction performed by the regression CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The green and the colored boxes indicate the ground truths and the predictions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' An obstetrician within the Unity environment manually annotated the ground truth poses of the TV SPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' 23T indicates the volume having a GA of 23 weeks used for reference in the registration and training in the previous experiments Translation errors - Boxplots Rotation errors - Boxplots 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° + + Median Median 80 + Mean Mean Geodesic error [degrees] 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° + Train Train + Test Test H++++ ++ + 60 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° lerror 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° 40 Euclidean e ++ 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° 20 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° Train Test Train Test 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='982 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='655 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='018 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='159Translation errors - Boxplots Rotation errors - Boxplots 70- 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° + t 1 1 0 hhh Median + 60 Mean 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° Train [mm] + 50 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° Test + error 40 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° + + + 30 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° 20 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° Median 十 Mean 10 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° Train Test P 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° Train Test Train Test 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='096 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='659 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='007 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='8Translation errors - Boxplots Rotation errors - Boxplots 70 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° Median Median Mean 0 Mean 60 Geodesic error [degrees] 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° Train Train [ww] 50 Test Test 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° error[ +++ + ++ 40 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° 30 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° 20 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° 10 G 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° Train Test Train Test 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='922 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='763 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='925 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='54Translation errors - Boxplots Rotation errors - Boxplots + 70 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° + + + Geodesic error [degrees] 60 +++++ ++++++ 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='50 error + +++ +++ + +++ 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° 40 H+ +e+ nt+ 30 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° 20 Median Median Eucl Mean 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° Mean 10 Train Train Test Test 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° Train Test Train Test 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='656 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='873 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='229Translation errors - Boxplots Rotation errors - Boxplots 70 + Median 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° Mean ++ 60 Train [ww] 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° + ++ Test 50 + Euclidean error [ 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° 40 + 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° 30 + + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° + 20 Median Mean 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° 10 Train Test 0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° Train 50 Test 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° + error 40 + + 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° Geodesic error [ + 30 丰 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° + + + 20 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° Median Mean 10 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° Train Test 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='0° Train Test Train Test 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='677 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='889 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='44923Tw 21w 23w 24w 22 w 25 w AnnotationIEEE TRANSACTIONS ON MEDICAL ROBOTICS AND BIONICS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' XX, AUGUST XXXX 11 TABLE II TRANSLATION AND ROTATION ERRORS OF OUR METHOD FOR THE LOOCV EXPERIMENTS ON INCLUDING DATA AUGMENTATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' NORM: EUCLIDEAN DISTANCE, GE: GEODESIC ERROR, DA: DATA AUGMENTATION, SD: STANDARD DEVIATION, 23T INDICATES THE VOLUME HAVING A GA OF 23 WEEKS USED FOR REFERENCE IN THE REGISTRATION AND TRAINING IN THE PREVIOUS EXPERIMENTS Test Volume DA Translation - Norm [mm] Rotation - GE [deg] Median Mean±SD Min Max Median Mean±SD Min Max 23T w No 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='32 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='72±2.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='42 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='72±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='18 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='19 tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' The proposed regression CNN successfully generalizes pose regression to an unseen fetal brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Specifically, data augmentation decreased median errors by 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='94% and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='10% for translation and rotation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Besides, by extending the training sets, we could further decrease the median errors and obtain a better generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Our model is designed to be size invariant by performing pose regression in normalized coordinates with respect to the brain limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' However, GA does not only affect size but also shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Indeed, the inclusion of different GAs covers a wide range of shapes and sizes, enabling us to understand our model’s current generalizability and limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' CONCLUSION In our previous work, we proposed a regression CNN to predict the 6D pose of arbitrarily-oriented planes slicing the fetal brain US volume without the need for real ground truth data in real-time or 3D volume scans of the patient beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We assumed that brains from different fetuses could be mapped to the same normalized coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' In this paper, we presented a detailed analysis of this assumption by quantifying the variance in the pose of the TV SPs annotated by an obstetrician when presented in the generalized coordinate frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Results show that this assumption is valid within a small tolerance for different GAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' However, similarly to the work in [7], our data only includes 6 volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Further analysis with larger annotated datasets can provide further insight into our generalized coordinate frame assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Currently, we can estimate plane poses within the brain with a median translation error of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='53mm and a median rotation error of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='42 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' These results are very promising, given that we are localizing images of a previously unseen fetus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' However, there are still a few outlier plane pose estimations with large errors (refer to maximum errors in Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' Future work can potentially help to remove outlier estimations by incorporating temporal models that look at the continuous video rather than single frames (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=', filtering/regularization, LSTMs, transformers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' This work could potentially be generalized to other anatom- ical regions of the fetus, such as the abdomen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' however, the definition of a generalized reference frame would still be challenging due to increased deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' We will also assess the potential of the current work towards active guidance of sonographers during SPs acquisition for fetal biometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE_T4oBgHgl3EQfyRyl/content/2301.08317v1.pdf'} +page_content=' This could include automated feedback signals to guide a novice from an arbitrary US plane towards the target SP.' 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sha256:30d39f95633f0d01b54cfca65736da30e45029c94c5a597c50764200423e45d1 +size 60394 diff --git a/X9E2T4oBgHgl3EQfEAaS/content/tmp_files/2301.03631v1.pdf.txt b/X9E2T4oBgHgl3EQfEAaS/content/tmp_files/2301.03631v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a9b425c0de96b67df671a0521424c80bbcfedfe9 --- /dev/null +++ b/X9E2T4oBgHgl3EQfEAaS/content/tmp_files/2301.03631v1.pdf.txt @@ -0,0 +1,2726 @@ +Bridging quantum criticality via many-body scarring +Aiden Daniel,1 Andrew Hallam,1 Jean-Yves Desaules,1 Ana +Hudomal,1, 2 Guo-Xian Su,3, 4, 5 Jad C. Halimeh,6, 7 and Zlatko Papi´c1 +1School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK +2Institute of Physics Belgrade, University of Belgrade, 11080 Belgrade, Serbia +3Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, +University of Science and Technology of China, Hefei, Anhui 230026, China +4Physikalisches Institut, Ruprecht-Karls-Universit¨at Heidelberg, +Im Neuenheimer Feld 226, 69120 Heidelberg, Germany +5CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, +University of Science and Technology of China, Hefei, Anhui 230026, China +6Department of Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC), +Ludwig-Maximilians-Universit¨at M¨unchen, Theresienstraße 37, D-80333 M¨unchen, Germany +7Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, D-80799 M¨unchen, Germany +(Dated: January 11, 2023) +Quantum dynamics in certain kinetically-constrained systems can display a strong sensitivity +to the initial condition, wherein some initial states give rise to persistent quantum revivals – a +type of weak ergodicity breaking known as ‘quantum many-body scarring’ (QMBS). Recent work +[Phys. Rev. B 105, 125123 (2022)] pointed out that QMBS gets destroyed by tuning the system +to a quantum critical point, echoing the disappearance of long-range order in the system’s ground +state at equilibrium. Here we show that this picture can be much richer in systems that display +QMBS dynamics from a continuous family of initial conditions: as the system is tuned across the +critical point while at the same time deforming the initial state, the dynamical signatures of QMBS +at intermediate times can undergo an apparently smooth evolution across the equilibrium phase +transition point. We demonstrate this using the PXP model – a paradigmatic model of QMBS that +has recently been realized in Rydberg atom arrays as well as ultracold bosonic atoms in a tilted +optical lattice. Using exact diagonalization and matrix product state methods, we map out the +dynamical phase diagram of the PXP model with the quenched chemical potential. We demonstrate +the existence of a continuous family of initial states that give rise to QMBS and formulate a ramping +protocol that can be used to prepare such states in experiment. Our results show the ubiquity of +scarring in the PXP model and highlight its intriguing interplay with quantum criticality. +I. +INTRODUCTION +Quantum many-body scarring (QMBS) is a form of +weak ergodicity breaking in which a small number of +states retain memory of their initial wavefunction de- +spite the rest of the system thermalizing (see recent re- +views [1–4]). The set of models hosting QMBS states has +rapidly expanded in recent years [5–20], including exper- +imental realizations in several cold atom platforms [21– +25]. At the same time, the underlying origin of memory- +retaining initial states remains the subject of on-going +work. Some recently identified mechanisms giving rise +to such phenomena include proximity to an integrable +model [19, 26, 27], dynamical symmetry [5, 28–32] and +eigenstate embedding constructions [33]. +Signatures of QMBSs were initially observed in ex- +periments on Rydberg atom arrays [21], where energy +cost due to van der Waals interactions strongly disfavors +two neighboring atoms occupying excited states – a form +of kinetic constraint called the Rydberg blockade [34]. +When the Rydberg blockade is strong, the atoms are de- +scribed by an effective “PXP” model [35, 36]. This is a +one-dimensional (1D) chain of spin-1/2 degrees of free- +dom, where the spin-up state |1⟩ corresponds to a Ryd- +berg atom occupying an excited state (and, similarly, for +the spin-down state, |0⟩, which denotes an atom in the +ground state). Thus, the number of up spins translates +into the number of Rydberg excitations, and we will use +such nomenclature interchangeably. The PXP Hamilto- +nian for N atoms takes the form (in units ℏ = 1) +HPXP(µ) = Ω +N−1 +� +j=0 +Pj−1XjPj+1 + µ +N−1 +� +j=0 +Qj, +(1) +where X = |1⟩ ⟨0|+|0⟩ ⟨1| is the Pauli-X operator describ- +ing the Rabi flipping of each atom. Below we will set the +Rabi frequency to Ω = 1. The projector P = |0⟩ ⟨0| im- +plements the constraint by preventing the Rabi flip from +generating any neighboring excitations. The complemen- +tary projector, Q = 1−P = |1⟩ ⟨1|, counts the number of +excitations in the system and thus defines the chemical +potential term, µ. We will consider two types of bound- +ary conditions for the Hamiltonian in Eq. (1): for analyti- +cal considerations and exact diagonalization simulations, +we will use periodic boundary conditions (PBCs), which +are implicit in Eq. (1) after identifying site j + N ≡ j. +For matrix product state simulations in large systems, +we will instead use open boundary conditions (OBCs), +where the first and the last flip term are taken to be +X0P1 and PN−2XN−1, respectively. +In the absence of chemical potential (µ=0), the PXP +model displays non-thermalizing dynamics when initial- +arXiv:2301.03631v1 [quant-ph] 9 Jan 2023 + +2 +ized in the N´eel state, |ψ(0)⟩ = |Z2⟩ ≡ |1010...10⟩ [21]. +Evolving this state with respect to the Hamiltonian in +Eq. (1), one observes that the return probability peri- +odically reaches values close to unity [6]. By contrast, +other initial states exhibit fast equilibration, as expected +in a chaotic system. Conversely, this atypical dynamics +is also reflected in ergodicity breaking amongst a subset +of eigenstates of the PXP model [27, 37, 38], even in the +presence of perturbations [39, 40] or in energy transport +at infinite temperature [41]. +The chemical potential term plays a central role in this +paper. +Recent study [24] has found that new QMBS +regimes can emerge for µ > 0. +One prominent exam- +ple is the polarized state, |0⟩ = |000....0⟩. While in the +absence of chemical potential the |0⟩ state is believed to +thermalize [21], at non-zero chemical potential, it starts +to revive, much like the N´eel state. Moreover, periodic +modulation of µ was found to enhance the QMBS be- +havior [22, 42, 43]. +Furthermore, as the chemical po- +tential is tuned to µc ≈ −1.31, the ground state of the +PXP model undergoes an Ising phase transition associ- +ated with a spontaneous breaking of Z2 symmetry [44– +47], whose signatures have also been observed in the +programmable Rydberg atom quantum simulators [48]. +This equilibrium phase transition (referred to as ‘EPT’ +throughout this paper) is in the same universality class +as the one induced by varying the quark mass in the +Schwinger model of quantum electrodynamics in (1+1)- +dimension [49]. +The lattice formulation of the latter, +known as the U(1) quantum link model, exactly maps +to the PXP model in Eq. (1) for the case of spin-1/2 +degrees of freedom [50]. +The EPT has a profound effect on the low-energy +physics of the PXP model, but it is not immediately obvi- +ous that it should directly impact QMBS, which manifest +in the quench dynamics at infinite temperature. Never- +theless, Ref. 51 recently argued that there is a link be- +tween this EPT and QMBS. Namely, when tracing the +eigenstates responsible for the quantum revival of the +|Z2⟩ state, Ref. 51 found that these states merge with +the thermal bulk of the energy spectrum as the EPT is +approached. On the contrary, upon moving away from +the EPT towards µ → −∞, the degenerate ground states +acquire high overlap with the |Z2⟩ state and its partner +translated by one site, +��¯Z2 +� +≡ |0101 . . .⟩. Thus, the |Z2⟩ +state can only thermalize as one approaches the EPT, +suggesting a connection between QMBS and criticality. +This was also demonstrated experimentally in the Bose- +Hubbard quantum simulator [52]. +In this work, we map out the dynamical phase dia- +gram of the PXP model corresponding to global quenches +of the chemical potential from some initial value, µi, to +an arbitrary final value, µf. This provides a means of +probing out-of-equilibrium dynamics from more complex +initial states beyond |Z2⟩ or |0⟩, which had been accessed +in previous experiments by taking the limits µi → ±∞. +We identify QMBS regimes in the dynamical phase dia- +gram based on signatures of ergodicity breaking, such as +the deviation of observable expectation values from the +canonical ensemble predictions and the presence of quan- +tum revivals. Our results show that the previously known +scarring regimes, associated with |Z2⟩ and |0⟩ states, in- +deed break down when approaching the EPT, either via +µi → µc or µf → µc, in agreement with Refs. 51 and +52. However, we also find a new QMBS regime corre- +sponding to the initial state being the ground state near +the EPT. Using the time-dependent variational principle +(TDVP) framework for QMBS, developed in Ref. 53, we +identify a semiclassical picture behind QMBS dynamics. +Across much of the phase diagram away from the EPT +point, the QMBS dynamics can be understood in terms +of a periodic trajectory that passes through the |0⟩ state, +with the radius of the trajectory controlled by the chem- +ical potential. Allowing for a continuous family of initial +states – the ground states of HPXP(µi) – we find sur- +prisingly robust QMBS signatures at intermediate times +that smoothly bridge across the EPT. We work out a +ramping protocol for the preparation of such states, pro- +viding a recipe for probing the dynamical phase diagram +in experiment. +The remainder of this paper is organized as follows. +We start by presenting the results of numerical sim- +ulations of the dynamical phase diagram of the PXP +model for global quenches of the chemical potential in +Sec. II. In Secs. III-V we analyze in detail the vari- +ous regimes of this phase diagram. Sec. III contains a +brief introduction of the TDVP formalism that will be +useful for semiclassical interpretation of the results. In +Sec. IV we focus on QMBS regimes of the phase dia- +gram, while Sec. V discusses the special case when the +system is initialized in the ground state near the EPT. +In Sec. VI, we show how the dynamical phase diagram +can be probed in experiment by preparing the desired +ground states using a ramping protocol. Our conclusions +are presented in Sec. VII, while Appendices contain de- +tails of the TDVP formalism, finite-size scaling analysis, +and additional characterizations of the phase diagram. +II. +DYNAMICAL PHASE DIAGRAM OF THE +PXP MODEL +In this paper we are interested in the following out- +of-equilibrium probe of the PXP model in Eq. (1): start +from the ground state of HPXP(µi) and then evolve with +the same Hamiltonian but generally different chemical +potential value, HPXP(µf). We assume a closed system +evolving under unitary Schr¨odinger dynamics. Since the +energy level spacings in the PXP model are expected +to obey the Wigner-Dyson distribution for all values +of µ [6, 35], the nonequilibrium dynamics induced by +quenching µ should be described by random matrix the- +ory [55]. In particular, quenching the chemical potential +by a large amount ∼O(1) should initialize the system in +a generic high-temperature state, which is expected to +lead to rapid thermalization according to the Eigenstate + +3 +x +x +Figure 1. +Dynamical phase diagram for global quenches +starting in the ground state of HPXP(µi) and evolving with +HPXP(µf). (a) The difference between maximal and minimal +revival fidelity δF over time interval 1 ≤ t ≤ 20 following +the quench. Regions with strong fidelity revivals have been +enumerated (see the text for details). (b) Same as (a) but +the color bar showing the deviation of the excitation density +from the thermal value, Eq. (3). Data is obtained using MPS +simulations [54] for a chain of N = 51 atoms with OBCs, +maximum bond dimension χ = 128 and time step δt = 0.025. +Dashed lines mark the EPT at µc ≈ −1.31. In both plots, +the cross marks the point (µi = −0.76, µf = 1.60) that will be +analyzed in Sec. IV. The diamond marks the optimal reviving +point in the µi = µc plane, which will be discussed in Sec. V. +Thermalization Hypothesis (ETH) [56–58]. This means +that the expectation value of any local observable should +converge towards the value predicted by the canonical en- +semble within any symmetry-resolved sector of the many- +body Hilbert space. Deviation from this prediction, i.e., +ergodicity breaking, can be detected through a number +of probes, two of which we utilize. +One probe of ergodicity breaking, convenient in the +context of QMBS, is quantum fidelity or return proba- +bility of the wavefunction to its initial value, +F(t) = | ⟨ψ(0)|ψ(t)⟩ |2. +(2) +For a thermalizing initial state, F(t) rapidly drops to a +value close to zero and remains exponentially small in +system size at late times. Therefore, if the average fi- +delity over a time interval ≫ Ω−1 is much larger than +∼ O(exp(−N)), we expect non-ergodic behavior. How- +ever, one should exclude trivial cases such as µi ≈ µf +when the ground state of HPXP(µi) is approximately an +eigenstate of HPXP(µf), as this would lead to the system +getting “stuck” in an eigenstate, with fidelity F(t) ≈ 1 +and potentially never decaying. To avoid such cases, we +compute the difference δF between minimum fidelity and +maximum fidelity over a time window t ∈ [t0, t1], with +t0=1 and t1=20. This window is large enough to exceed +the initial relaxation on the scale ≳ Ω−1 (thus excluding +the high fidelity near t = 0), yet small enough (t1 ≲ N/Ω) +to be free of the boundary effects. The obtained δF in +the µi − µf plane is shown in Fig. 1(a). The fidelity has +been evaluated in a system of N = 51 atoms using ma- +trix product state (MPS) [59] simulations based on the +algorithm in Ref. 54, and we have checked that the re- +sults agree closely with exact diagonalization for systems +with N < 30 atoms. +Before we comment on the interesting regimes of the +phase diagram, we note that we have also computed the +deviation of an observable expectation value from the +thermal ensemble prediction, shown in Fig. 1(b). This +provides a complementary probe of ergodicity breaking +that is more amenable to experimental measurements. +For the observable, we chose the density of excitations +in the system, n = (1/N) �N +j=1 Qj, which is readily +available in existing experimental setups [21, 24]. After +quenching the system, we compute the integrated mean- +square deviation of the excitation density from the ther- +mal value over the time window between t0 = 10 and +t1 = 20, +MSD(n) = +1 +t1 − t0 +� t1 +t0 +|⟨ψ(t)|n|ψ(t)⟩ − nth|2 dt. +(3) +The thermal value is defined as +nth = Tr(ρthn), +(4) +where the thermal density matrix is given by the usual +Boltzmann-Gibbs expression, ρth = exp(−βH)/Z, with +the partition function Z = Tr exp(−βH) and the inverse +temperature β determined from the condition +⟨ψ(0)|HPXP(µf)|ψ(0)⟩ = Tr(ρthHPXP). +(5) +The plot of MSD(n) is shown in Fig. 1(b), where the +bright non-ergodic regions match those of high fidelity +in Fig. 1(a). +The color contrast is stronger in the fi- +delity plot due to the exponential sensitivity of that quan- +tity. A few distinct regimes where fidelity displays large- +amplitude oscillations have been marked by (1)-(5) in + +6 +10-2 +4 +2 +0 +10-3 +-2 +-4 +-6 +-6 +-4 +-2 +0 +2 +4 +66 +4 +0.8 +2 +0.6 +0 +0.4 +-2 +0.2 +-4 +-6 +0 +-6 +-4 +-2 +0 +2 +4 +64 +Fig. 1(a). These regions will be analyzed in detail in the +subsequent sections. There, we will argue that regions +(1), (2) and (3) can be identified as QMBS regimes. Re- +gions (1) and (3) fall under the “universality class” of +|Z2⟩ and |0⟩ QMBS behavior, as we explain in Sec. III. +On the other hand, while the dynamics in region (2) has +some similarities with regions (1) and (3), in Sec. IV we +will highlight the distinctions of this QMBS regime. As +it turns out, regions (4), (5), (6) and (7) have a simple +origin, which will be explained briefly in Appendix A. +A few comments are in order. The QMBS fidelity ap- +pears to vary smoothly between regions (1) and (2) in +Fig. 1(a), while they are separated by the EPT (indi- +cated by the dashed line). +In fact, we find the most +robust revivals correspond to the ground state precisely +at the EPT point (highlighted by the diamond in Fig. 1). +This intriguing case will be addressed in detail in Sec. V. +Here we note that we have confirmed the existence of +QMBS across the critical point in much larger systems +(N ≤ 400 spins) using MPS numerics. This is in contrast +to the µf = µc case, where we see no ergodicity breaking +in Fig. 1(a), as also expected from Refs. 51 and 52. +III. +TIME-DEPENDENT VARIATIONAL +PRINCIPLE AND PERIODIC ORBITS FOR +MANY-BODY SCARRING +Without chemical potential, quantum dynamics from +the |Z2⟩ state in the PXP model can be visualized as +a classical periodic orbit [53, 60, 61]. +This is accom- +plished in the framework of the Time-Dependent Vari- +ational Principle (TDVP) [62–64], which we briefly re- +view in this section. +TDVP establishes a parallel be- +tween many-body dynamics in the PXP model and the +analogous dynamical phenomena of a single particle in +a stadium billiard, in which the wavepackets are anoma- +lously long-lived when prepared along the periodic orbits +of the corresponding classical billiard [65, 66]. TDVP will +provide a natural semiclassical language for interpreting +the essential features of the dynamical phase diagram in +Fig. 1. +A. +A brief overview of TDVP formalism +The starting point of TDVP is to specify a variational +manifold of states M, parameterized by some continu- +ous variable, and then project the Schr¨odinger dynamics +into that manifold in a way that manifestly conserves the +energy. The nature of states belonging to M determines +to what extent we can interpret the dynamics as “semi- +classical”. For example, it would be simplest to consider +a manifold spanned by tensor products of spin-coherent +states. This would yield a “mean-field” description for +the dynamics, where each atom precesses independently. +However, the Rydberg blockade intrinsically builds in lo- +cal correlations into the system, due to the fact that any +neighboring excitations, |. . . 11 . . .⟩, are projected out of +the Hilbert space. Ordinary spin-coherent states clearly +violate this blockade condition. +Another way of defining a manifold, which naturally +accommodates the Rydberg blockade constraint, is to +take the span over MPS states with bond dimension χ +controlling the amount of correlations necessary to cap- +ture the projected dynamics [64]. To simplify matters as +much as possible, we will consider the dynamics to be +spatially periodic with a (infinitely repeated) unit cell of +size K (below we will be primarily interested in small +unit cells with K = 1, 2). For a 1D chain of size N, the +resulting MPS ansatz is given by +|ψMPS({x})⟩= +� +{σ} +Tr +�N/K−1 +� +m=0 +Aσ1+Km(x1)Aσ2+Km(x2) +AσK+Km(xK) +� +|σ1σ2σ3 · · · σN⟩ . (6) +Here Aσ(xi) are (χ × χ)-dimensional matrices that de- +pend on variational parameters xi = (θi, φi), where the +angles θi, φi are akin to the Bloch sphere angles of each +spin in the unit cell. +The physical degree of freedom +σi = 0, 1 labels the basis states of a single spin. Follow- +ing Refs. 53 and 67, in order to make things analytically +tractable, we will restrict to χ = 2 and chose +A1(θi, φi) = +� +0 e−iφi +0 +0 +� +, A0(θi, φi) = +� +cos θi 0 +sin θi 0 +� +. (7) +Due to A1A1 = 0, this ansatz ensures that configura- +tions with neighboring spin-up are forbidden, thus our +manifold M = span{|ψMPS(x)⟩ |∀x} is consistent with +the Rydberg blockade. +With the choice of ansatz in Eqs. (6)-(7) and setting +K = 1, we are left with only two variational degrees +of freedom, (θ, φ). +Choosing (0, 0) recovers the state +|0⟩ ≡ |000 . . .⟩, while (π/2, π/2) corresponds to the equal- +weight superposition of the two N´eel states, +��Z+� +≡ +1 +√ +2 +� +|Z2⟩ + +��¯Z2 +�� +. +(8) +Note that with K = 1 unit cell periodicity, the states +|Z2⟩, +��¯Z2 +� +do not individually belong to the manifold. In- +stead, if we extend the ansatz to K = 2, then (θ1, θ2) = +(0, π/2) recovers the |Z2⟩ state. Thus, our manifold with +bond dimension χ = 2 captures the initial product states +that we expect to play an important role for QMBS dy- +namics in the PXP model. +After defining the manifold, the next step is to mini- +mize the difference between exact Hamiltonian dynamics +and its projection to the manifold, +min +{x} +����iℏ ∂ +∂t |ψMPS({x})⟩ − H |ψMPS({x})⟩ +���� . +(9) +This results in the Euler-Lagrange equations of motion +for the classical variables x [64]. In the case of the PXP +model, this step can be performed analytically in the + +5 +Figure 2. Sketch of the TDVP manifold M for the PXP model with chemical potential µ. Red regions represent areas of high +leakage where the TDVP approximation breaks down, as quantified by Eq. (10). The N´eel state is denoted by |Z2⟩ ≡ |1010 . . .⟩ +and its translated partner – the anti-N´eel state is +��¯Z2 +� +≡ |0101 . . .⟩, while +��Z+� += (|Z2⟩ + +��¯Z2 +� +)/ +√ +2. The polarized state is +|0⟩ ≡ |0000 . . .⟩. (a) For a two-site unit cell K = 2 and µ = 0, the |Z2⟩ state lies on a periodic trajectory identified in Ref. 53. +We also illustrate the trajectory of |0⟩ state, which is predicted by TDVP to evolve to +��Z+� +; however, this point lies within +a region of high leakage where the TDVP dynamics does not accurately describe the quantum evolution. This is consistent +with the |0⟩ state thermalizing at µ = 0. (b) Taking K = 1 we focus on the evolution of the |0⟩ state trajectory as µ is varied. +For µ = 0, the trajectory is periodic but passes through a region of high leakage. When µ ̸= 0, the trajectory shrinks, whilst +gradually exiting the high leakage area, and QMBS dynamics starts to emerge in the full system. In this regime, the QMBS +dynamics can be seen as an oscillation between |0⟩ and a new state, |¯0(µ)⟩, defined in Eq. (11). Finally, in the extreme µ → ±∞ +limit, the orbit shrinks to a point. +limit of N → ∞ to obtain the equations of motions for the +θ and φ angles, see Appendix B for K = 1 and Refs. 53 +and 67 for some K = 2 and K = 3 examples. Integrating +this system of differential equations yields the trajectory +in M taken by |ψMPS({θ, φ})⟩ during the course of quan- +tum evolution. Fig. 2 shows a pictorial representation of +the manifold and the projection of exact dynamics into +it, for the cases of interest in the PXP model perturbed +by the chemical potential. +Importantly, beyond equations of motion, it is possible +to estimate “quantum leakage”: the difference between +exact quantum evolution and its projection into the man- +ifold [53]. Quantum leakage, γ, is defined as the instan- +taneous rate at which the exact wave function leaves M: +γ2 = lim +N→∞ +1 +N +����iH |ψMPS(x)⟩ + +� +j +˙xj∂xj |ψMPS(x)⟩ +���� +2 +. +(10) +Red regions in Fig. 2 indicate areas of large γ2. In these +high-leakage regions, the instantaneous TDVP dynamics +is expected to poorly capture the exact dynamics. Con- +sequently, trajectories passing through such regions will +generally be of limited accuracy. On the other hand, as +first noted in Ref. 53, the special property of the PXP +phase space is that it has regions of remarkably low leak- +age, such as the region traversed by the semiclassical or- +bit associated with the |Z2⟩ state. This is depicted in +Fig. 2(a) where the orbit is sketched, lying within a re- +gion of low leakage. Note that, in general, there can exist +multiple periodic orbits within the same manifold [60]. +B. +TDVP interpretation of the dynamical phase +diagram +Much of the PXP dynamical phase diagram in Fig. 1 +can be understood by considering the trajectory of the +polarized state in the TDVP manifold introduced above. +Fig. 2(b) sketches this trajectory for three different values +of the chemical potential µ. Within TDVP, a periodic +orbit exists even for µ = 0. However, the orbit passes +through the superposition of the two N´eel states, |Z+⟩, +which is located in the high-leakage region. The TDVP +dynamics is therefore not a good approximation in this +case, which accounts for the absence of revivals observed +in the full quantum dynamics. +The addition of a finite chemical potential µ contracts +the trajectory and pushes it into a low-leakage region, as +shown in the middle panel of Fig. 2(b), effectively allow- +ing the revivals from the polarized state to emerge. As +we will explain in Sec. IV, in this intermediate range of µ, +the ground state of HPXP(µ) occupies an antipodal posi- +tion on the orbit, corresponding to a chemical-potential +dependent state we label |¯0(µ)⟩, given by Eq. (7) for unit +cell size K = 1: +|¯0(µ)⟩ = |ψMPS(θmax, φmax)⟩ , +(11) +with angles (θmax, φmax) denoting the antipodal point +in the TDVP orbit of the initial polarized state, see +Fig. 2(b). As µ has the effect of deforming the trajec- +tory, the antipodal angles also depend on µ, as will be +specified in Eq. (16) below. Note that the sign of µ has +no effect on the deformation of the particular orbit dis- +cussed here, as we explain in Appendix C. Finally, in the +extreme limit µ → ±∞, the trajectory is restricted to the +vicinity of the initial state and the dynamics is effectively +frozen, as shown in the right panel of Fig. 2(b). +IV. +SCARRING IN GAPPED REGIMES OF +THE PHASE DIAGRAM +In this section we focus on regions (1), (2), and (3) of +the phase diagram in Fig. 1, in particular for the values of +the chemical potential away from the EPT. Based on the + +o←n +o←n +μ≠0 +μ +IH8 +K=2 +K=1 +iHt +(a) +[2) +(b) +Z2/ +(()" +0>Z+> +·Z+> +z+> +/z+> +10) +10)00) +10)* +OJ +Z2)6 +discussion of TDVP in Sec. III and Fig. 2, the origin of +regions (1) and (3) can be understood by examining the +form of the PXP ground state in the presence of chemical +potential. When µi → −∞, excitations are favored and +the ground state is (for PBCs) a superposition of the two +N´eel states, |Z+⟩ in Eq. (8). By contrast, µi → ∞ pe- +nalizes excitations, therefore the ground state is the po- +larized state |0⟩. The superposition state |Z+⟩ is known +to display revivals when quenched to µf = 0 [37], while +the polarized state revives when quenched with µf ̸= 0 +as shown more recently in Refs. 24 and 43. By continu- +ity, these limiting cases explain the mechanism behind +revivals in regions (1) and (3) of Fig. 1. In the remainder +of this section, we focus on the more interesting region (2) +where the pre-quench initial state is an entangled state +with low overlap on both |0⟩ and |Z2⟩ states. +A. +Scarring in region (2) of the phase diagram +We focus on region (2) of the phase diagram in Fig. 1 +and pick (µ∗ +i , µ∗ +f ) = (−0.76, 1.60) as an illustrative point +in this region, marked by the cross in Figs. 1(a)-(b). +QMBS dynamics at this point was first noted in Ref. 24 +and here we will characterize it in detail and explain its +origin. The evolution of fidelity and overlap with the po- +larized and N´eel state are shown in Fig. 3(a), where per- +sistent fidelity revivals can be observed while the overlap +with |Z2⟩ remains negligible throughout the evolution. +Curiously, while the initial state at µ∗ +i has low overlap +with |0⟩, the evolved state does develop a relatively high +overlap with |0⟩ state, approximately half way between +the main revival peaks – see the green line in Fig. 3(a). +This is reminiscent of the |Z2⟩ state, which in the pure +PXP model undergoes state transfer to +��¯Z2 +� +at half the +revival period [53], implying that the ground state of +HPXP(µ∗ +i ) is related to the polarized state. +Another tell-tale signature of QMBS is a slower growth +of entanglement entropy, SE(t), for special initial states. +The entanglement entropy is defined as the von Neu- +mann entropy of the reduced density matrix, ρA = +TrB|ψ(t)⟩⟨ψ(t)|, obtained by tracing out degrees of free- +dom belonging to one half of the chain (denoted B). We +plot the dynamics of SE(t) in Fig. 3(b). Compared to +both |Z+⟩ and a random product state, |σRandom⟩, the +entropy growth from the ground state of HPXP(µ∗ +i ) is +strongly suppressed. Moreover, for the latter state, we +observe clear oscillations in the time series of SE(t), rem- +iniscent of entropy dynamics in the PXP model in the +absence of chemical potential [6]. +We emphasize that the special point (µ∗ +i , µ∗ +f ) is repre- +sentative of the entire region (2) in the phase diagram, +where similar QMBS phenomenology is numerically ob- +served. In the remainder of this section, we use TDVP +to garner a further understanding of this QMBS regime +from a semiclassical point of view. +0.0 +0.5 +1.0 +| +(t)| +2 | +| +(t)|0 | +| +(t)| (0) | +0 +2 +4 +6 +8 +10 +12 +14 +t +1 +2 +3 +S(t) +| random(t) +| ++ +| (t) +Figure 3. +Dynamics of quantum fidelity and entanglement +entropy, following a global quench of the chemical potential, +µ∗ +i = −0.76 → µ∗ +f = 1.6, corresponding to the point marked +by the cross in Fig. 1(a). Quantum fidelity for the initial state +|ψ(0)⟩ defined as the ground state of the PXP model with µ∗ +i . +Also shown is the projection of the time-evolved state on the +|Z2⟩ and |0⟩ states. +While the overlap with the |Z2⟩ state +is low throughout the evolution, the overlap with |0⟩ reaches +relatively high values between the main revival peaks. +(b) +Growth of entanglement entropy, SE(t), for the same initial +state |ψ(0)⟩ as in (a), as well as for a random state |σRandom⟩ +and +��Z+� +state. +The initial state |ψ(0)⟩ has strongly sup- +pressed entanglement growth compared to the other cases. +Data is for system size N = 28 obtained using exact diago- +nalization with PBCs. +B. +TDVP analysis of scarring in region (2) +Before we apply TDVP to extract the semiclassical de- +scription of the dynamics in Fig. 3, we need to make sure +that the PXP ground state in the presence of chemical +potential is represented within the manifold spanned by +states in Eq. (6). In a recent work [68], a method of “op- +timal steering” has been devised to smoothly prepare a +class of PXP ground states based on the minimization of +quantum leakage along the trajectory. To show that the +detuned PXP ground states are captured in the TDVP +manifold, here we follow a simpler approach of optimiz- +ing the overlap |⟨ψMPS({x})|ψ(µi)⟩|2, where |ψ(µi)⟩ is +the ground state of the PXP model in Eq. (1). For a +unit cell size K = 1, we performed exhaustive numerical +sampling at system size N = 20 and found that most +states belonging to the TDVP manifold (> 90% of them) +can be approximated with better than 98% accuracy by a +ground state of Eq. (1). As a side note, we mention that +in order to prepare the states in the TDVP manifold with +unit cell K ≥ 2, we need to make two modifications to +the preparation procedure: (i) we need to allow chemical +potential to be different for different atoms within the +unit cell; (ii) we need to include a unit-cell modulated +pulse in the z-direction. As explained in Appendix D, +after these generalizations, one can also successfully pre- +pare TDVP states with K ≥ 2. While we do not have +a general proof, this provides a numerical confirmation +of the representability of the ground states of the PXP + +7 +Figure 4. +(a) Phase space portrait of quantum dynamics within the K = 1 TDVP manifold for the PXP model with µf=1.6. +Grey shading indicates quantum leakage (darker regions represent larger leakage). The trajectory of the |0⟩ state for the given +value of µf is highlighted in red, while colored symbols indicate the location of the PXP ground states corresponding to various +µi indicated on the color bar. The ground states with µi≈−0.76 can be seen to lie close to the point which is antipodal to the +|0⟩ state in its trajectory. With changing µf, this trajectory either expands or compresses, meaning all ground states will lie on +this antipodal point for some µf. (b) In region (2) of the phase diagram, for a given µf, |¯0(µf)⟩ state is well-approximated by +a detuned PXP ground state with some µi. Color bar shows the highest overlap between the |¯0(µf)⟩ state, given by Eq. (16) +for a range of fixed µf ∈ [−5, 5], and the family of ground states of HPXP(µi). Dashed lines denote the EPT. For negative +chemical potentials, especially relevant for region (1) of the phase diagram, the mapping requires an additional phase pulse, as +described in Appendix D. (c) Matching the detuned PXP ground state with a |¯0⟩ state becomes progressively more difficult at +the critical point (dashed line) as system size N is increased. In contrast to panel (b), here we fix the PXP ground state at µi +and vary µf to find the optimal |¯0(µf)⟩ state with the highest overlap. All plots are obtained using exact diagonalization with +PBCs and system size N = 20 in panels (a)-(b). +model with a suitably-defined generalization of the chem- +ical potential within the TDVP manifold. +Having established that our pre-quench ground state +at arbitrary µi can be approximately mapped to an MPS +state in the K = 1 TDVP manifold for some variational +parameters (θ, φ), we now proceed to describe the dy- +namics from this initial state using the classical dynam- +ical system defined by (θ(t), φ(t)). +From Eq. (9), one +can derive the TDVP equations of motion for K = 1 +and arbitrary chemical potential µ (see Appendix B for +details): +˙θ = − cos θ cos φ +� +1 + sin2 θ +� +, +(12) +˙φ = µ + sin φ +sin θ +� +1 − 4 sin2 θ − sin4 θ +� +. +(13) +Unlike the special case µ = 0, where φ variables can be +set to zero in the flow-invariant subspace [53], for general +values of µ one must consider both θ and φ variables +simultaneously [60]. +Integrating Eqs. (12)-(13), we plot the phase space +θ, φ portrait for the chemical potential value µf = 1.6 in +Fig. 4(a). The greyscale background indicates the quan- +tum leakage at any given point in the manifold, +γ2 = +sin6θ +1 + sin2θ, +(14) +which only depends on θ variable (see Appendix B). By +integrating the equations of motion for µf = 1.6, start- +ing from the polarized state |ψMPS(0, 0)⟩, we obtain the +trajectory plotted in red color in Fig. 4(a). Generally, +for any |µf| ̸= 0, the polarized state has a periodic or- +bit within TDVP. When µf is large, the orbit is pinned +around θ = 0. Decreasing |µf| stretches out the orbit un- +til the maximal point in the trajectory eventually tends +towards the |Z+⟩ superposition state, (θ, φ) ≡ (π/2, π/2). +Due to the presence of a quantum leakage gradient, the +|Z+⟩ point is not reached for any finite time, consis- +tent with the lack of revivals from the polarized state +in full quantum dynamics for sufficiently small values of +µf. Thus, we conclude that the orbit corresponding to +the cross in Fig. 1(a) is a compromise between two com- +peting effects: the orbit is sufficiently stretched so that it +has nontrivial dynamics, while at the same time, by be- +ing not stretched too much, it can avoid the large leakage +in the vicinity of |Z+⟩ state. +To verify this picture across the entire region (2), we +study the projection of the PXP ground state at µi, +|GS(µi)⟩, to the TDVP manifold. We numerically max- +imize the overlap |⟨ψMPS(θ, φ)|GS(µi)⟩|2, with the MPS +state given in Eq. (6). We plot the resulting (θ, φ) phase +space coordinates for a variety of µi in Fig. 4(a), where +the colored dots correspond to the ground states from +our phase diagram in Fig. 1(a). As expected, some of +the ground states are “distant” from |Z+⟩ or |0⟩ but +tend towards either in their respective limits. All suc- +cessfully optimized ground states lie on the same φ plane +in Fig. 1(a), such that the deformation of the trajectory +means they will correspond to some maximum point µf +on the polarized state trajectory, denoted by the state +|¯0⟩. By analogy with the N´eel state, whose translation +partner – the anti-N´eel state – displays identical scarring +behavior [19], here we have a similar relation between |0⟩ +and |¯0(µf)⟩ states. The main difference with the anti-N´eel +state is that |¯0⟩ state depends on the value of µf. + +(b) +(a) +(c) +6. +3+ +5.0 +1.00 +》 +(m)0l +→ +N =14 +0.75 +2.5 +N =16 +4 +2 +0.8 +N=18 +0.0似 +0 +0.50 +S +N =20 +2 +1 +N=22 +2.5 +一 +0.25 +0.6 +N =24 +-5.0 +0.00 +0- +-5 +-0.5 +5 +0 +-1.0 +-1.5 +0.0 +0 +-5 +5 +i +ui8 +To substantiate this further, we analytically derive the +phase-space coordinates corresponding to |¯0(µf)⟩. Using +Eq. (12), we see that the turning point in the gradient of +θ along the trajectory is governed by cos φ. A sign flip +therefore must occur when φ = ±π/2. Because energy is +exactly conserved along a TDVP trajectory, |¯0(µf)⟩ must +have the same energy as the polarized state. For states +belonging to K = 1 TDVP manifold, the energy density +is given by +E(θ, φ)/N = +sin θ +1 + sin2 θ +� +µf sin θ + 2 cos2 θ sin φ +� +. +(15) +For the polarized state, E(0, 0) = 0 and, setting φmax = +π/2, allows us to determine the θmax coordinate of the +|¯0(µf)⟩ turning point: +sin θmax = +� +|µf| − +� +µ2 +f + 16 +� +/4. +(16) +In Fig. 4(b) we test the overlap of the state |¯0(µf)⟩, with +the MPS angles given by Eq. (16), against the family of +ground states of HPXP(µi). +We scan through a set of +values µf ∈ [−5, 5] and, for each µf, plot the maximum +overlap obtained by maximizing over µi. Although µf < 0 +is not particularly relevant for region (2), we note that +the optimization fails there. This, however, can be fixed +by including an additional phase pulse, as explained in +Appendix D. Comparing Fig. 4(b) to Fig. 1(a), we see a +striking correspondence between the successful optimiza- +tion and region (2) in the phase diagram, which confirms +that the QMBS phenomena in region (2) are indeed as- +sociated with |¯0(µf)⟩ state. +Finally, in Fig. 4(c) we study the system size scal- +ing of the mapping between the PXP ground state with +chemical potential and states in the TDVP manifold. +We scan for the maximal overlap of the ground state +at some µi with the set of all |¯0(µf)⟩ states in the in- +terval µf ∈ [−20, 20]. Remarkably, for the vast majority +of region (2) when µi > 0, we see a near perfect over- +lap between the ground state and |¯0(µf)⟩, independent of +system size – suggesting that the TDVP state captures +well the PXP ground state in region (2). Nevertheless, in +Fig. 4(c) we also observe a breakdown of the mapping at +the EPT point µi = µc. This is expected since the ground +state at the critical point develops a diverging entangle- +ment entropy and the χ = 2 MPS approximation must +deteriorate as system size is increased, since an area-law +state cannot capture the critical ground state in the ther- +modynamic limit. This naturally leads to the question: +is the observed scarring in the critical ground state an +artefact of finite size and what is its origin? +V. +INTERPLAY BETWEEN SCARRING AND +CRITICALITY +We now focus on the nature of QMBS regime when +quenching from the critical ground state at µi = µc. De- +spite the complexity of this state, we find robust signa- +tures of ergodicity breaking in the area between regions +(1) and (2) in Fig. 1(a). For example, by fixing µi = µc +and scanning µf to determine the largest δF, we find the +most robust revivals occur at µf = 0.633 – a point that +was marked by the diamond in Fig. 1. This turns out +to be one of the best reviving points in all of regions +(1), (2) and (3), including the |Z2⟩ and |0⟩ initial states. +As discussed above, the TDVP semiclassical formalism +is not well-suited for describing this case as it cannot +capture the diverging entanglement entropy of the initial +state. +This immediately raises the question if the ob- +served QMBS behavior is a finite size effect and whether +one should rather expect a sharp boundary between re- +gions (1) and (2) in Fig. 1 in the thermodynamic limit. +To probe the robustness of QMBS revivals in the ther- +modynamic limit we simulated the quench dynamics µi = +−1.31 → µf = 0.6 in large systems up to N = 401 using +the MPS method [54] in Fig. 5. The fidelity, plotted in +Fig. 5(a), demonstrates that revivals exist in all accessi- +ble system sizes. The fidelity is not an intensive quantity, +therefore it is generically expected to decay in the N → +∞ limit, as indeed can be observed in Fig. 5(a). Thus, +to compare different system sizes, we take the fidelity at +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +N =21 +N =31 +N =41 +N =51 +N =101 +N =151 +N =201 +N =251 +N=301 +N=401 +0 +2 +4 +6 +8 +10 +t +0.6 +0.8 +1.0 +1.2 +1.4 +0 +10 +0.1 +0.5 +0 +0.08 +0.01 +0.02 +Figure 5. +Fidelity and entanglement entropy dynamics for +the quench from the critical ground state with µi = −1.31 +to µf = 0.6. +(a) Fidelity revivals persist up to the largest +system size N = 401. While the fidelity decays with N, the +fidelity density of the first revival peak, − log(F1)/N, plotted +against inverse system size, 1/N, extrapolates to a value close +to 0 (inset), indicating non-ergodic behavior in the thermo- +dynamic limit at a finite time. (b) Dynamics of the half-chain +entanglement entropy SE(t) for the same quench. We scale +the entropy by the critical value given by the Cardy-Calabrese +formula with central charge c = 1/2 [69], which collapses the +data to 1 at t = 0 (inset shows the unscaled entropy). The +growth of entropy is seen to be linear, with pronounced os- +cillations. Data is obtained by MPS simulations with OBCs, +bond dimension χ = 300, and time step δt = 0.025. + +9 +the first revival peak F1 and plot its density, − log(F1)/N +against 1/N, in inset of Fig. 5(a). This serves as an in- +dicator of ergodicity breaking at a finite time that can +be properly scaled to the thermodynamic limit. For a +random state in the constrained Hilbert space of the +PXP model, we expect − log(F1)/N to asymptotically +approach log +� +(1 + +√ +5)/2 +� +≈ 0.48. Contrary to this ex- +pectation, the fidelity density in Fig. 5(a) continues to +decrease as N → ∞, signaling non-ergodicity in the ther- +modynamic limit at a finite time t ∼ 5/Ω, well beyond +the initial relaxation. +In Fig. 5(b) we observe a slow growth of entangle- +ment entropy following the same quench. +In contrast +to previous QMBS cases in the literature, where the sys- +tem was initialized in a product state with zero entropy, +such as |Z2⟩, here we start from a critical ground state +whose entropy is expected to diverge logarithmically with +system size according to the Cardy-Calabrese formula, +Scrit = (c/6) log(N/π) [69]. The universal prefactor is +determined by the central charge c of the conformal field +theory, which is c = 1/2 for our critical point in the Ising +universality class. Scaling the data by Scrit indeed yields +a good collapse at time t = 0. At later times, the en- +tropy grows linearly with time. On top of linear growth, +we observe prominent oscillations that are typically found +in QMBS systems, e.g., the |Z2⟩ initial state in the PXP +model [6]. The amplitude of these oscillations is roughly +independent of system size, as can be seen in the inset +of Fig. 5(b). At much later times, which are inaccessible +to MPS methods, we expect the entropy to saturate to a +value proportional to the volume of the subsystem. +Apart from the diverging entropy of the initial state, +the overall picture from Fig. 5 is broadly similar to pre- +vious studies of QMBS dynamics [1]. What remains to +be explained is why the critical ground state is poised to- +wards QMBS-like dynamics. To identify the microscopic +origin of this robust ergodicity breaking in the vicinity +of µf = 0.633, we plot the overlap of the initial criti- +cal ground state with the eigenstates of the post-quench +Hamiltonian in Fig. 6. The overlap exhibits clear tow- +ers of eigenstates which are emblematic of QMBS. While +these features are present throughout the spectrum, the +dominant contributions to the initial state come from +low-energy eigenstates. +In order to approximate their +characteristics, we can treat them as magnons with a +given momentum k on top of the ground state. +For +µf = 0, this has been shown to give a good approximation +of scarred states even at relatively high energies when us- +ing magnons with momentum k = π [70]. Similarly, we +find this to be true in our case near µf = 0.6, where much +of the low-energy spectrum can be approximately recon- +structed from pairs of non-interacting magnons with mo- +menta k and −k, see the dashed lines in Fig. 6 and in- +set. Note that the PXP model is gapped for µf = 0.633, +hence the ground state and the first tower in Fig. 6 are +separated by a finite energy that is independent of N in +sufficiently large systems. +A detailed analysis of the magnon dispersion as a func- +Figure 6. Overlap between the ground state at the critical +point µi = µc = −1.31 and the eigenstates of the PXP model +with µf = 0.633. +The color indicates the density of data- +points. The red dashed lines indicate multiples of the energy +of a k = π excitation on top of the ground state. This matches +well with the scarred towers in the relevant part of the spec- +trum. +The inset shows the first set of excited states, with +the grey dashed lines indicating the expected energy for non- +interacting pairs of excitations with momenta k and −k. Due +to the flatness of the band near k = π and k = 0, the lines are +denser near the scarred states, leading to sharper towers and +better revivals (see further analysis of the magnon dispersion +in Fig. 7 below). Data is obtained by exact diagonalization +for system size N = 28 with PBCs. +tion of chemical potential is presented in Fig. 7. +The +dispersion relation for several values of µf is shown in +Fig. 7(a). For µf < 0.6, the single-magnon band merges +with the two-magnon continuum, causing the downward +slope near k = 0. +Near µf = 0.6, the band becomes +remarkably flat for small k, coinciding with the one- +magnon and two-magnon bands barely touching. At that +point, the energies of the first excited states at k = 0 are +well approximated by twice the energies of the single- +magnon states, indicating that they correspond to a pair +of two non-interacting magnons with momenta k and −k. +This is illustrated in Fig. 7(b) and the inset of panel +(a). This simple picture of non-interacting excitations al- +lows us to predict the energies of the low-energy excited +states based solely on the dispersion relation of the single- +magnon states. In particular, the flatness of the band +near k = 0 and k = π means that the eigenstates near +the scarred ones have approximately the same energy. +This implies that the towers of states will be sharper, +and that the effective energy spacing, which determines +the dynamics at intermediate times, is the spacing be- +tween the towers. In turn, the fact that the magnons are +very weakly interacting means that the spacing between +these towers will be approximately equal. +In summary, we showed that QMBS in the critical ini- +tial state can persist due to (i) the post-quench Hamil- +tonian HPXP(µf) having a gapped spectrum with a suffi- +ciently flat band of the low-lying magnon excitations; (ii) +the magnons are weakly interacting and their multiplets + +(GS-1.31|E0.633)I2 +10 +10-7 +10-10 +10-13 +10-16 +-10 +10 +20 +E10 +(a) +(b) +-0.5 +0.0 +0.5 +1.0 +1.5 +3 +2 +1 +0 +-6 +-8 +-10 +-12 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +-1 +0 +1 +1/2 +1 +1.0 +Figure 7. (a) Dispersion relation of the low-lying excitations +of the PXP model for several values of the chemical potential +µf, shown in different colors. When µf ≈ 0.6, the dispersion +becomes visibly flat near both k = 0 and k = π momenta. +Inset shows the difference between the actual energies of the +first excited states in the spectrum and their approximation +by a pair of two non-interacting excitations. For all momenta +k, the best agreement between the approximation and exact +energy is attained at µf ≈ 0.6. (b) Low energy spectrum of the +PXP model with µf = 0.6 – the value with the best revivals +when quenching from the critical ground state. The ground +state and first excited states are indicated, along with energies +corresponding to a non-interacting pair of excitations with +momenta k and −k. In this instance, we see the approximate +excitations and exact energy levels lie close to each other. +Data is obtained by exact diagonalization for system size N = +24 with PBCs. +give rise to regularly spaced QMBS-like towers in the +spectrum. While this scenario is reminiscent of Ref. 71, +where quantum revivals in some non-integrable models +were related to the low-lying quasiparticle states, in our +case the chemical potential needs to be finely tuned to +a value µf ≈ 0.6 to meet the conditions (i)-(ii). Indeed, +as seen in Fig. 1, varying µf around this value leads to a +sharp decay of QMBS revivals. In contrast to the PXP +model with µi = 0 and the |Z2⟩ initial state, the QMBS +eigenstates in the µi = µc case are clearly skewed towards +the low-energy part of the spectrum, however this allows +the QMBS revivals to persist in large systems, despite +the highly entangled initial state. +VI. +EXPERIMENTAL PROTOCOL +Finally, in this section we address the experimental ob- +servation of the phase diagram in Fig. 1. The key step is +the preparation of the PXP ground state in Eq. (1). The +protocol below is directly applicable to Rydberg atom +arrays [22], however it can also be adapted to ultracold +bosons in a tilted optical lattice, where the chemical po- +tential µ maps to the energy mismatch between the Hub- +bard interaction and electric field which induces a tilt +potential [24]. +Ground state preparation is accomplished via a “ramp- +ing” procedure utilized in related experiments [21, 46, 48, +72, 73]. This assumes fine control of the chemical poten- +tial that is varied in time, µ = µ(t). Taking the chemical +potential very large, µ → ±∞, one can prepare |0⟩ and +|Z2⟩ states. Starting in one of these states, one can then +ramp to a desired ground state in the interior of our phase +diagram in Fig. 1 by evolving with a time-dependent +PXP Hamiltonian, HPXP(µ(t)), where µ(t) is appropri- +ately parameterized for an adiabatic evolution, as spec- +ified below. The adiabaticity implies that the ramping +will not be able to prepare the critical ground state af- +ter a finite time in the thermodynamic limit. Therefore, +with finite resources, we can only hope to approach the +critical point from different gapped regions of the phase +diagram. We start the ramp either in |Z2⟩ or |0⟩, depend- +ing on whether we are in a ordered (µ < µc) or disordered +(µ > µc) phase, respectively. +Specifically, we make use of the following ramp +µ(t) = +A +(t − B)2 − +A +(t − C)2 + µc, +(17) +where A, B, and C are tunable parameters. One par- +ticularly successful choice was found to be A = ∓40, +when ramping from |0⟩ or |Z2⟩, respectively, B = 30, +and C = −0.1. +An example of this ramping curve is +plotted in the inset of Fig. 8(b). We include µc due to +the need for a much slower ramp as the gap between the +ground state and first excited state closes in the vicinity +of the EPT point. +After specifying the ramp and the +initial state, we evolve by the PXP Hamiltonian in the +presence of chemical potential, Eq. (17), until some time +t. The evolution time is determined by numerically min- +imizing 1 − |⟨ψ(t)|GS(µtarget)⟩|2, where |GS(µtarget)⟩ is +the state we are trying to prepare. +Fig. 8(a) illustrates the success of the ramping proce- +dure. For system sizes ranging from N = 6 to N = 14, we +have ramped to prepare the ground states from µ = ±6, +in increments δµ = 0.5, towards the critical point, µc = +−1.31. Fig. 8(b) shows the time that the ramp took for +each ground state. We see the ramp time is insensitive to +system size in gapped regions of the phase diagram, while +it sharply increases near µc and exhibits strong fluctua- +tions with N. For fixed ramp parameters, we expect it +will take an infinite amount of time to prepare the critical +ground state in the N → ∞ limit. + +11 +0.90 +0.93 +0.95 +0.98 +1.00 +=6 +=8 +=10 +=12 +=14 +0 +5 +10 +15 +4 +2 +6 +0 +2 +4 +6 +6 +4 +2 +0 +2 +4 +6 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +30 +60 +0 +60 +=51 +=75 +=101 +Figure 8. (a) The success of preparing the PXP ground state +at chemical potential µ by ramping the chemical potential ac- +cording to Eq. (17). The total ramp time is varied for each +point to maximize the overlap, which is plotted on the y-axis. +For µ > µc, the initial state is |0⟩ (square symbols), while +for µ < µc we start the ramp in the |Z2⟩ state (triangles). +Separate optimizations were performed for different system +sizes N, shown in the same plot. Black dashed line (in all +the panels) denotes the critical point µc. +Inset: using the +optimal parameters and average ramping time determined +in smaller sizes in the main panel, we prepare the ground +states for the same values of µ in much larger system sizes +N = 51, 75, 101. The preparation in this case was done using +the MPS method with time step δt = 0.025 and maximum +bond dimension χ = 128. While in the gapped phases the +preparation remains successful, there is a visible drop near +the critical point. (b) Total ramp time tramp returned by the +optimizations in the main panel (a). Inset shows the ramping +curve µ(t) in Eq. (17). We observe an increase of the ramp +time and strong finite-size fluctuations at the critical point. +The data in the main panels (a) and (b) was computed using +exact diagonalization in k=0 momentum and p=+1 inversion +symmetry sector with PBCs. +Finally, to verify our preparation scheme in large sys- +tems, we repeated the preparation of the detuned PXP +ground states for system sizes of N = 51, 75 and 101 using +MPS simulations with bond dimension χ = 128 and the +the ramping protocol in Eq. (17), with the same A, B, C +parameters. The inset of Fig. 8(a) demonstrates that the +ramping continues to successfully reproduce the desired +ground state with high fidelity, with the exception of the +critical point where we see a clear drop in overlap with +the target state. This suggests the ramping procedure is +a viable method for generating desired ground states even +in large systems. With this in hand, along with the al- +ready existing capabilities to quench with a detuned PXP +Hamiltonian and conduct measurements of local observ- +ables [21, 22], all the tools are, in principle, available to +reconstruct the dynamical phase diagram in Fig. 1. In +particular, local fidelity measurements [24] can be used +to approximate the numerically computed global fidelity +in Fig. 1(a). This would allow to experimentally verify +the persistence of QMBS across the phase diagram and +its robustness near the critical point. +VII. +CONCLUSIONS AND DISCUSSION +We have mapped out the dynamical phase diagram of +the PXP model, based on ergodicity breaking in its dy- +namics following the global quench of the chemical po- +tential. We have demonstrated the existence of extended +regions which harbor QMBS phenomena, either associ- +ated with the previously studied initial conditions, such +as |Z2⟩ and |0⟩, or with new entangled states such as +|¯0(µ)⟩. The mechanisms giving rise to these QMBS phe- +nomena, in particular the underlying periodic trajecto- +ries, were identified within the TDVP framework. We +have analyzed in detail the robustness of QMBS when +the system is tuned to the EPT point, arguing that this +does not provide an obstacle for QMBS, provided that +the post-quench Hamiltonian is tuned in such a way that +the low-lying quasiparticle excitations are weakly inter- +acting and possess a flat energy-momentum dispersion. +This enables different QMBS regions in the dynamical +phase diagram to connect smoothly, bridging across the +EPT. Finally, we have also outlined an adiabatic prepa- +ration scheme that allows to map out the same phase +diagram in experiments on Rydberg atoms and ultracold +bosons in tilted optical lattices, both of which have re- +cently realized the PXP model in the presence of a tun- +able chemical potential. In light of these experiments, +our discussion of the phase diagram above was restricted +to finite times, however in Appendix F we discuss the +corresponding phase diagram for time t → ∞. We note +that the existence of a continuous family of QMBS states, +tunable by the chemical potential, is of independent in- +terest in quantum-enhanced metrology, for which QMBS +states were shown to be advantageous [74–76]. +One motivation behind this work is the open prob- +lem of identifying all initial conditions associated with +QMBS for a given model. For the pure PXP model it had +originally appeared that only the |Z2⟩ state is special in +this regard [21], however, more recent explorations of the +chemical potential [24] have revealed that the latter can +also stabilize QMBS from a different initial state, |0⟩. In +this paper, we have shown that these two product states +share the semiclassical description and belong to a larger +family, which also includes some other weakly-entangled +states such as |¯0(µ)⟩ state. While we have numerically +related these initial states and their quench dynamics, +it is not obvious how to relate them at the level of a +spectrum-generating su(2) algebra, which has provided +an elegant description of revivals from the |Z2⟩ state in +the pure PXP model [29]. Moreover, our present investi- +gation focused on the dynamics with periodicity K = 1 +and it would be interesting to extend it to K ≥ 2. For ex- +ample, it is known that |Z3⟩ = |100100100 . . . 100⟩ state +also exhibits revivals in the pure PXP model model [37]. +However, this state necessitates a TDVP description with +K = 3 unit cell, which already gives rise to an intricate +phase space at the semiclassical level [60]. It would be +interesting to understand the dynamical phase diagram +associated with such states that have larger unit cells, + +12 +either in the PXP model or analogous models for larger +Rydberg blockade radii. +Finally, our results for the initial state at the critical +point suggest that QMBS dynamics is not necessarily as- +sociated with preparing the system in a product state or +even an area-law entangled state, but in principle allows +for highly-entangled initial states. In this case, QMBS +dynamics is more strongly temperature-dependent, as the +initial state has dominant support on the relatively low- +lying energy eigenstates of the post-quench Hamiltonian. +The key ingredient for making this work was to suppress +the interaction between quasiparticles and flatten their +energy dispersion. It would be interesting to understand +how to engineer such conditions in other models and +thereby realize similar dynamics from highly-entangled +initial states. +ACKNOWLEDGMENTS +This work was supported by the Leverhulme Trust Re- +search Leadership Award RL-2019-015. J.-Y.D. acknowl- +edges support by EPSRC grant EP/R513258/1. A.H. ac- +knowledges funding provided by the Institute of Physics +Belgrade, through the grant by the Ministry of Educa- +tion, Science, and Technological Development of the Re- +public of Serbia. Part of the numerical simulations were +performed at the Scientific Computing Laboratory, Na- +tional Center of Excellence for the Study of Complex +Systems, Institute of Physics Belgrade. +Statement of +compliance with EPSRC policy framework on research +data: This publication is theoretical work that does not +require supporting research data. Z.P. acknowledges sup- +port by the Erwin Schr¨odinger International Institute +for Mathematics and Physics (ESI). J.C.H. acknowledges +funding from the European Research Council (ERC) un- +der the European Union’s Horizon 2020 research and +innovation programm (Grant Agreement no 948141) — +ERC Starting Grant SimUcQuam, and by the Deutsche +Forschungsgemeinschaft (DFG, German Research Foun- +dation) under Germany’s Excellence Strategy – EXC- +2111 – 390814868. +Appendix A: Other regions of the phase diagram +Several regions of the phase diagram in Fig. 1 exhibit +fidelity revivals that have a simple origin that can be +understood without invoking QMBS. Here we explain in +more detail these regions labeled (4), (5), (6) and (7). +It is useful to consider the Inverse Participation Ratio +(IPR), one of the traditional measures of ergodicity of +the eigenfunctions introduced in the context of Anderson +localization [77]. The IPR is defined as +IPR = +1 +� +E +| ⟨E|ψ⟩ |4 , +(A1) +Figure 9. +Logarithm (base 10) of the IPR of the ground +state of HPXP(µi) with respect to the eigenstates of HPXP(µf). +All the labels have the same meaning as in Fig. 1. Data is +obtained using exact diagonalization in the sector with k = 0 +momentum and p = +1 inversion symmetry for size N = 26 +with PBCs. +and it intuitively tells us about how many basis states |E⟩ +the state |ψ⟩ has support on. For example, if |ψ⟩ is a basis +state, its IPR will be 1, while if |ψ⟩ is homogeneously +spread over the entire Hilbert space, the IPR will be equal +to the Hilbert space dimension. Note that IPR is a basis- +dependent quantity and, in our case, we have a natural +choice of eigenstates |E⟩ of HPXP(µf) as the basis states. +The log of IPR for µi ground states with respect to µf +eigenstates is plotted in Fig. 9. This allows us to fur- +ther distinguish between different regions. For conven- +tional |Z2⟩ scarring we expect the IPR to be on the order +of system size N, since the |Z2⟩ state has high overlap +with a band of N + 1 scarred eigenstates of HPXP(0) but +low overlap with the rest. This is evidenced in region +(1) of Fig. 9. On the other hand, the band of scarred +eigenstates associated with |0⟩ state in the detuned PXP +model is “tilted” to one edge of the spectrum, so we ex- +pect the IPR to be smaller. In general, the regions with +high IPR are expected to be ergodic, while the least inter- +esting regimes are characterized by very low IPR, such +as around the µi = µf diagonal and in regions (5) and +(6). The IPR is not as low in parts of regions (4) and (7) +visible in this figure, but it decreases with increasing |µi| +and |µf| as the ground state of HPXP(µi) approaches an +eigenstate of HPXP(µf). +Large |µf| leads to fragmentation of the Hilbert space, +which can effectively trap the initial state in a simple +oscillating superposition. For example, region (4) [i.e., +µi > 0, −µf ≫ 1] roughly corresponds to the polarized +state in the strongly detuned regime, since the initial +ground state has significant overlap with |0⟩ for µi > 0. +In the µi → ∞ limit, it is expected to become the exact +mirror image of region (3), given that the polarized state +has the same dynamics for ±µf (see Appendix C). Simi- + +6 +(7) +2.5 +4 +(2) +2.0 +2 +(3) +X +(1) +1.5 +0 +-2 +1.0 +(6) +-4 +(4) +0.5 +(5) +9- +0.0 +-5.0 +-2.5 +0.0 +2.5 +5.0 +i13 +larly, region (7) [µi < 0, µf ≫ 1] has a simple explanation +in terms of |Z+⟩ state in the strongly detuned regime. +The origin of revivals in region (5) [µf < µi < −1.3] is +perhaps not immediately obvious, since the initial state +in that case does not have high overlap with one of the +previously studied states such as |0⟩ or |Z+⟩. We now +briefly investigate this region. The fidelity and the aver- +age number of excitations after quenching from µi = −2.5 +to µf = −6 can be seen in Figs. 10(a) and (b). +The +quenched state maintains high overlap with the |Z+⟩ +state, with peaks in the middle between the fidelity re- +vivals, see Fig. 10(a). This situation is reminiscent of the +|¯0⟩ state in region (2), which periodically evolves to |0⟩ +and back. Although it oscillates, the overlap with |Z+⟩ +never drops to zero. In contrast, the overlap with |0⟩ is +constantly zero. In Fig. 10(b) we also see that the av- +erage occupation is remarkably stable, fluctuating only +slightly around ≈ 0.47. As explained above for regions +(4) and (7), such behavior arises due to the fact that in +the large-µ limit the Hilbert space becomes fragmented +and the initial state has support on a small number of +eigenstates that are disconnected from the rest. This can +be seen in Fig. 10(c), which shows the overlap of the ini- +tial state and the eigenstates. +The fragmentation and +high overlap with the ground state are apparent. Fur- +ther evidence comes from the inverse participation ratio +(IPR), which we find to be very low in this region, indi- +cating overlap with only a small number of eigenstates, as +will be shown below. Finally, region (6) [µi < µf < −1.3] +has a similar phenomenology to its mirroring region (5). +Figure 10. +Dynamics and eigenstate properties of the PXP +model quenched from µi = −2.5 to µf = −6, corresponding +to region (5) of the phase diagram in Fig. 1. (a) Fidelity of +the initial state |ψ(0)⟩, i.e., the ground state of HPXP(−2.5), +as well as the overlap with both the polarized state |0⟩, and +superposition state +��Z+� +. (b) The average number of excita- +tions remains nearly constant in time. (c) The overlap of the +initial state with eigenstates of HPXP(−6) reveals fragmenta- +tion and large projection on the ground state. Data obtained +by exact diagonalization for N = 28 with PBCs. +In summary, we have argued that regions (4), (7) and +part of (5) correspond to regimes where µf has a large +absolute value, leading to a simple oscillatory dynamics +due to Hilbert space fragmentation, while in regions (5) +and (6), µf ≈ µi causes the initial state to be close to an +eigenstate of the post-quench Hamiltonian. +Appendix B: Derivation of TDVP equations of +motion and quantum leakage +In this section we first derive the TDVP equations of +motion and then compute the instantaneous leakage rate. +These derivations follow Appendices A and C of Ref. 60. +1. +Equations of motion +The TDVP equations of motion can be derived as the +saddle point equations for the following Lagrangian [62, +64]: +L = i +2 +� +⟨ψMPS| ˙ψMPS⟩ − ⟨ ˙ψMPS|ψMPS⟩ +� +− ⟨ψMPS |H| ψMPS⟩, +(B1) +where it will be convenient to split our Hamiltonian into +two terms, H = HPXP+Hµ. Unlike Ref. 60, we restrict to +K = 1 which greatly simplifies the calculation. Through- +out this section we will consider mixed MPS transfer ma- +trices, denoted by +T B +C = +� +σ +¯Bσ ⊗ Cσ, +(B2) +where B and C are arbitrary MPS tensors. The MPS +transfer matrix for the PXP ansatz chosen in the main +text takes the form +T A +A = T = +� +� +� +� +cos2 θ +0 0 1 +cos θ sin θ 0 0 0 +cos θ sin θ 0 0 0 +sin2 θ +0 0 0 +� +� +� +� . +(B3) +The dominant left and right eigenvalues of the transfer +matrix are equal to 1, and the corresponding eigenvectors +are +|R) = +� +� +� +� +1 +cos θ sin θ +cos θ sin θ +sin2 θ +� +� +� +� , +(L| = +� 1 0 0 1 � +, +(B4) +which obey (L|R) = 1 + sin2 θ. We also introduce the +following shorthand for a 3-site local Hamiltonian term +contracted with MPS tensors on every site: +H = HA,A,A +A,A,A = +� +σi +¯Aσ1 ¯Aσ2 ¯Aσ3hσ1,σ2,σ3 +σ4,σ5,σ6Aσ4Aσ5Aσ6. +(B5) + +1.0 +a +<(t) /b(0)> +<2b(t)|Z2) +0.5 + 97% along the +entire trajectory). Inset shows the scaling of the overlap for +the most poorly approximated point on the trajectory as a +function of system size N. The overlap decays slowly and its +extrapolation yields high overlap for this point even in large +systems (e.g., overlap ∼ 90% at size N ∼ 50). +the same for ±µ and that the ceiling state of HPXP(µ) +becomes symmetry-breaking for µ > 1.31. +Similarly, it is important to further note the relation +between µ and −µ with respect to the TDVP equations +of motion, Eq. (12) and Eq. (13). In general, flipping the +sign of µ may not result in identical dynamics, however +this is not the case when considering the dynamics of the +polarized state. As |0⟩ has TDVP angles (0, 0), at this +point ˙φ = µ. On the other hand, ˙θ has no µ dependence +and so is unaffected by the a sign flip and the only de- +pendence on φ comes from the cos(φ) term which has the +property cos(φ) = cos(−φ). Because of this, a sign flip +of µ does not affect the dynamics of θ and simply flips +Eq. (13). This means that the dynamics of |0⟩ are sym- +metric under the sign flip and the shrinking of the orbit +in Fig. 2 occurs for both ±µ. +Appendix D: Preparation of states in the TDVP +manifold +Here we demonstrate that states belonging to the +TDVP manifold with K = 1, 2 unit cell can be rep- +resented as ground states of the PXP model with a +suitably generalized chemical potential term. To show +this correspondence, we numerically optimize the over- +lap |⟨ψMPS({x})|Ψ(w)⟩|2, where |Ψ(w)⟩ is the ground +state of the PXP model with a K-site periodic density + +16 +modulation, +H(w) = +N−1 +� +j=0 +Pj−1XjPj+1 + +N−1 +� +j=0 +wjQj, +(D1) +where w = (w1, w2, . . . , wK, . . . w1, w2, . . . wK) is a gener- +alization of the chemical potential term that is periodic +(with period K) but takes different values for different +atoms within the unit cell. The Hamiltonian H(w) re- +duces to the PXP Hamiltonian with uniform chemical +potential in Eq. (1) for K = 1. +Furthermore, in order to prepare the states in larger +TDVP manifolds with unit cells K ≥ 2, we found it nec- +essary to act on the ground state of Eq. (D1) with a +unit-cell modulated phase pulse: +Θ(γ) = +N/K−1 +� +j=0 +e−iγKZKj+(K−1) · · · e−iγ2ZKj+1e−iγ1ZKj, +(D2) +where Zi denotes the usual Pauli-Z matrix on site i and +γ1, . . . , γK are variational parameters in addition to w. +Our extensive numerical sampling in system sizes N ≤ +18 confirms that the ansatz in Eqs. (D1)-(D2) allows for +an accurate approximation of states in the TDVP mani- +fold after optimizing for (w, γ). As this is performed at +relatively small system sizes, here we verify that these +results can be extended to larger systems. As a test case, +we choose a particularly interesting TDVP trajectory +which starts at (θ1, θ2, φ1, φ2)=(1.25π,2.985,0.166,0.188). +This trajectory was derived in Ref. [60] within a K = 2 +TDVP ansatz and it belongs to a regular region of the +manifold, giving rise to fidelity oscillations in the full +quantum dynamics. We choose this trajectory to show +that the ansatz can capture trajectories of interest in +larger manifolds. We optimize for 30 states evenly spaced +along this TDVP trajectory between time t = 0 and t = 6 +in system sizes ranging from N = 6 to N = 18. The op- +timization yields an overlap close to 1 for all the points +on the trajectory and yields a set of optimal (w1, w2) and +(γ1, γ2) for different N. Over the range of N, we found γ +changes little so we do not reoptimize this in larger N but +simply take the average from smaller sizes. On the other +hand, we find w for different values of N fits well the em- +pirical formula wj = aebN+c + d, where a, b, c and d are +fitting parameters depending on w1 and w2. With this +information, we can calculate (w1, w2), (γ1, γ2) for larger +system sizes via extrapolation. The resulting overlap in +system size N = 22 is shown in Fig. 11. We see that the +ansatz successfully captures the entire trajectory (up to +97% overlap in this system size). In the inset of Fig. 11 +the minimum overlap found along the trajectory is plot- +ted as a function of system size, showing that it decays +very slowly and allows to prepare the TDVP states on +the trajectory with accuracy of 90% or better in large +systems N ∼ 50. +−14 +−12 +−10 +−8 +Ek +µ=0.1 +Ground state +Single mode with k +k and −k pairs +−12 +−10 +−8 +−6 +Ek +µ=0.6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +k/π +−10.0 +−7.5 +−5.0 +−2.5 +Ek +µ=1.2 +Figure 12. Low-energy spectrum of the PXP model for three +values of µ. The red crosses correspond to the energies of a +non-interacting pair of excitations with momenta k and −k. +For µ = 0.1, the first band merges with the two-magnon con- +tinuum. For µ = 1.2, the first excited state with k = 0 has an +energy that differs from that of two non-interacting magnons. +Data is for system size N = 24 with PBCs. +Appendix E: Single mode approximation +In Sec. V we have discussed the revivals from the criti- +cal ground state based on the structure of the low energy +spectrum at µf = 0.633. In this section we provide more +details of this analysis, in particular on the range of µ +that it can be applied to. Ref. 70 showed that for µf = 0, +the scarred states throughout the spectrum could be well +approximated as a collection of magnons with momen- +tum π. Here, we show that this analysis also holds for +µf ≈ 0.6, especially in the low-energy part of the spec- +trum. In turn, the ground state at µi = µc = −1.31 can +be understood as mainly being a superposition of these +multi-magnon states. +In Fig. 12 one can see the low-energy spectrum resolved +by momentum for three different values of µf. The data +for the overlap of the same eigenstates with the ground +state at µf = µc = −1.31 is also plotted in Fig. 13. Note +that, as this ground state has k = 0, only the eigen- +states with the same momentum value will have a non- +zero overlap. For too small values of µ, the one-magnon +states merge into the two-magnon continuum near k = 0, +causing the band to bend downwards. As a consequence, +the non-interacting magnon pairs approximation is less + +17 +0 +5 +E − E0 +10−7 +10−4 +10−1 +|⟨GS−1.31|Eµi⟩|2 +0 +5 +E − E0 +0 +5 +E − E0 +Figure 13. Overlap between the ground sate at µi = µc = +−1.31 and the low-energy eigenstates of the PXP model with +various values of µ for N = 24 and PBCs. The states are +the same as in Fig. 12 with k = 0, and panels correspond to +µf = 0.1, 0.6 and 1.2 respectively (from left to right). The +red lines correspond to the expected energy of two and four +magnons with momentum π on top of the ground state. The +grey line correspond to the expected energy of two magnons +with momentum k and −k on top of the ground state. Due +to the flatness of the band and the weak interactions between +magnons, the towers of states are sharper around µ = 0.633. +accurate for k ̸= π, and the critical ground state has +increased overlap with them. On top of this, the band +being far from flat at the edges means that the towers of +states are not sharp, i.e., states near the top of the towers +have a non-negligible energy difference. As their energy +separation from the ground state is roughly twice that of +a single-magnon with momentum k, the flatter the band +the more similar in energy the states will be. +For µf ≈ 0.6, the single-magnon band barely touches +the two-magnon continuum. The magnon-pair approxi- +mation now holds well for all values of k. Consequently, +one can see that the overlap of the critical ground state +with two-magnon states built out of magnons with mo- +mentum k ̸= π is very low. Among these, the states with +the highest overlap are the ones made from magnons with +momentum close to 0 or π. As the band is flat near these +points, they have approximately the same energy as the +scarred states and so do not lead to dephasing until late +times. +Finally, when µf becomes too large, the nature of the +excitations changes and the π magnons no longer describe +the elementary excitations in the system. +Indeed, for +µf ≫ 1, the ground state is simply the polarized state and +the excitations are just a single flip 1 on top of the back- +ground of 0. So the first excited state with k = 0 is simply +a symmetric superposition of the the state |100 · · · 0⟩ and +its translations. As any kind of excitation with k = π +will need at least one 1 site, adding two of them that +are non-interacting will never lead to the correct excited +state at k = 0. This can already be seen for µf = 1.2 in +the bottom panel of Fig. 12, as the lowest red cross – cor- +responding to the expected energy of two non-interacting +magnons – is far above the actual first excited state with +k = 0. +This again impacts the sharpness of the tow- +ers of states, especially the spacing between the first and +Figure 14. +The norm of the scaled difference of the num- +ber of excitations between the diagonal and canonical ensem- +bles when quenching the initial ground state of HPXP(µi) to +HPXP(µf). All the labels are the same as in Fig. 1. Data is +obtained using exact diagonalization in the momentum k = 0 +and p = +1 inversion symmetry sector for system size N = 28 +with PBCs. +second excited state, which grows with µf. +Appendix F: Dynamical phase diagram in the +infinite-time limit +In the main text, we explored the dynamical phase +diagram using two probes based on the dynamics at in- +termediate time scales: fidelity revivals and the deviation +of average density of excitations from its thermal value. +Here we directly address the long-time behavior of the +system using the latter quantity. We study the average +density of excitations evaluated in the diagonal ensemble: +¯n = lim +T →∞ +1 +T +� T +0 +⟨ψ(t) |n| ψ(t)⟩ dt = +� +j +|cj|2 nj,j, +(F1) +where cj = ⟨Ej|ψ(0)⟩ and nj,k = ⟨Ej| n |Ek⟩. The ini- +tial state |ψ(0)⟩ is the PXP ground state at some µi, +while Ej, |Ej⟩ are the eigenvalues and eigenstates of the +quench Hamiltonian HPXP(µf). In the second equality +of Eq. (F1), we have assumed that the off-diagonal ele- +ments average out to zero in the infinite-T limit. This is +true in the absence of spectral degeneracies, as integrat- +ing off-diagonal contributions over time corresponds to +integrating e−irt with r ̸= 0 being essentially a random +number. Thus, each contribution will give a finite num- +ber that will go to zero as it is multiplied by 1/T and +the limit T → ∞ is taken. The quench Hamiltonian is +generally non-degenerate after resolving the momentum +and inversion symmetries (our calculations are mostly +performed in the sector with k = 0 and p = +1). An +exception to this occurs at µf = 0 where the spectrum +contains an extensive number of “zero modes” [37, 78]. + +6 +0.16 +4. +0.14 +0.12 +2 +0.10 +0.08 +-2 +0.06 +0.04 +-4 +0.02 +-6 +0.00 +-5.0 +-2.5 +0.0 +2.5 +5.0 +i18 +−1.0 +−0.5 +0.0 +0.5 +1.0 +µf +−0.05 +0.00 +0.05 +0.10 +δn/n +(a) +−6 +−4 +−2 +0 +2 +4 +6 +µf +−0.1 +0.0 +0.1 +δn/n +(b) +µi = −∞ +µi = ∞ +µi = −1.31 +14 +18 +22 +26 +30 +N +−5 +0 +5 +µi +Figure 15. Scaled difference of the expectation values between +the diagonal and canonical ensembles. +(a) For µi = µc = +−1.31, there is a large difference around µf = 0.5 that does +not vary much with system size. Notably, we also see that +to the left of that point the difference between the ensembles +increases with system size. (b) Cross cuts through the phase +diagram with a fixed value of µi indicated on the color bar. +The middle peak corresponds to region (1), while the two +negative peaks on the bottom right correspond to regions (2) +and (3), from left to right respectively. Data is obtained by +exact diagonalization for system size N = 26 with PBCs. +In that case, the off-diagonal contributions between all +eigenstates with E = 0 must also be counted. +After evaluating ¯n, we compute the difference between +the diagonal and canonical ensembles, δn = ¯n − nth, +where nth was defined in Eq. (4). This allows to quan- +tify ergodicity breaking via the deviation from the ther- +mal value in the infinite-time limit, as shown in Fig. 14. +Comparing this with the original phase diagram in Fig.1, +we see that the main regions (1),(2),(3) associated with +QMBS still show visible signatures. +In other regions, +such as region (5), the diagonal and canonical ensemble +averages happen to be equal but this does not imply ther- +malization – rather, the difference between ensembles is +small because the dynamics is reduced to a superposition +of only a few eigenstates. Similarly, we notice that region +(2) and region (3) are intersected by a flat line where +|δn| ≪ ¯n, which is completely insensitive to the initial +state (i.e., independent of µi). This line passes through +the vicinity of the diamond point, discussed in Sec. V, +where we emphasized that the relevant dynamics occurs +at lower effective temperatures than the other parts of +region (1) and (2). Consequently, we expect |δn|/¯n to +be suppressed. Indeed, as we discuss in Fig. 15 below, +this apparent discontinuity between regions (1) and (2) +is related to the fact that δn takes opposite signs in the +two regions, thus it crosses zero at their interface. +Fig. 15(a) shows that at the critical point there is still +a sizable difference between the two ensembles in vari- +ous system sizes. The maximum difference is closer to +µf = 0.5 than to the fidelity maximum of 0.633. +The +latter is a compromise between the flatness of the band +and the level of interactions of the magnons. +As the +long time behavior should not depend on the spacings +of the towers, it is not surprising that the optimal µf +is much closer to 0.5, where the level of interactions of +the magnons seems the lowest. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Serbia 3Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Anhui 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' China 4Physikalisches Institut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Ruprecht-Karls-Universit¨at Heidelberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Im Neuenheimer Feld 226,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 69120 Heidelberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Germany 5CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Anhui 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' China 6Department of Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Ludwig-Maximilians-Universit¨at M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Theresienstraße 37,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' D-80333 M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Germany 7Munich Center for Quantum Science and Technology (MCQST),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Schellingstraße 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' D-80799 M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Germany (Dated: January 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 2023) Quantum dynamics in certain kinetically-constrained systems can display a strong sensitivity to the initial condition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' wherein some initial states give rise to persistent quantum revivals – a type of weak ergodicity breaking known as ‘quantum many-body scarring’ (QMBS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Recent work [Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' B 105, 125123 (2022)] pointed out that QMBS gets destroyed by tuning the system to a quantum critical point, echoing the disappearance of long-range order in the system’s ground state at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Here we show that this picture can be much richer in systems that display QMBS dynamics from a continuous family of initial conditions: as the system is tuned across the critical point while at the same time deforming the initial state, the dynamical signatures of QMBS at intermediate times can undergo an apparently smooth evolution across the equilibrium phase transition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We demonstrate this using the PXP model – a paradigmatic model of QMBS that has recently been realized in Rydberg atom arrays as well as ultracold bosonic atoms in a tilted optical lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Using exact diagonalization and matrix product state methods, we map out the dynamical phase diagram of the PXP model with the quenched chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We demonstrate the existence of a continuous family of initial states that give rise to QMBS and formulate a ramping protocol that can be used to prepare such states in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Our results show the ubiquity of scarring in the PXP model and highlight its intriguing interplay with quantum criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' INTRODUCTION Quantum many-body scarring (QMBS) is a form of weak ergodicity breaking in which a small number of states retain memory of their initial wavefunction de- spite the rest of the system thermalizing (see recent re- views [1–4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The set of models hosting QMBS states has rapidly expanded in recent years [5–20], including exper- imental realizations in several cold atom platforms [21– 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' At the same time, the underlying origin of memory- retaining initial states remains the subject of on-going work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Some recently identified mechanisms giving rise to such phenomena include proximity to an integrable model [19, 26, 27], dynamical symmetry [5, 28–32] and eigenstate embedding constructions [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Signatures of QMBSs were initially observed in ex- periments on Rydberg atom arrays [21], where energy cost due to van der Waals interactions strongly disfavors two neighboring atoms occupying excited states – a form of kinetic constraint called the Rydberg blockade [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' When the Rydberg blockade is strong, the atoms are de- scribed by an effective “PXP” model [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This is a one-dimensional (1D) chain of spin-1/2 degrees of free- dom, where the spin-up state |1⟩ corresponds to a Ryd- berg atom occupying an excited state (and, similarly, for the spin-down state, |0⟩, which denotes an atom in the ground state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Thus, the number of up spins translates into the number of Rydberg excitations, and we will use such nomenclature interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The PXP Hamilto- nian for N atoms takes the form (in units ℏ = 1) HPXP(µ) = Ω N−1 � j=0 Pj−1XjPj+1 + µ N−1 � j=0 Qj, (1) where X = |1⟩ ⟨0|+|0⟩ ⟨1| is the Pauli-X operator describ- ing the Rabi flipping of each atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Below we will set the Rabi frequency to Ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The projector P = |0⟩ ⟨0| im- plements the constraint by preventing the Rabi flip from generating any neighboring excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The complemen- tary projector, Q = 1−P = |1⟩ ⟨1|, counts the number of excitations in the system and thus defines the chemical potential term, µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We will consider two types of bound- ary conditions for the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (1): for analyti- cal considerations and exact diagonalization simulations, we will use periodic boundary conditions (PBCs), which are implicit in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (1) after identifying site j + N ≡ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For matrix product state simulations in large systems, we will instead use open boundary conditions (OBCs), where the first and the last flip term are taken to be X0P1 and PN−2XN−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In the absence of chemical potential (µ=0), the PXP model displays non-thermalizing dynamics when initial- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='03631v1 [quant-ph] 9 Jan 2023 2 ized in the N´eel state, |ψ(0)⟩ = |Z2⟩ ≡ |1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='10⟩ [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Evolving this state with respect to the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (1), one observes that the return probability peri- odically reaches values close to unity [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' By contrast, other initial states exhibit fast equilibration, as expected in a chaotic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Conversely, this atypical dynamics is also reflected in ergodicity breaking amongst a subset of eigenstates of the PXP model [27, 37, 38], even in the presence of perturbations [39, 40] or in energy transport at infinite temperature [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The chemical potential term plays a central role in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Recent study [24] has found that new QMBS regimes can emerge for µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' One prominent exam- ple is the polarized state, |0⟩ = |000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='.0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' While in the absence of chemical potential the |0⟩ state is believed to thermalize [21], at non-zero chemical potential, it starts to revive, much like the N´eel state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Moreover, periodic modulation of µ was found to enhance the QMBS be- havior [22, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Furthermore, as the chemical po- tential is tuned to µc ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='31, the ground state of the PXP model undergoes an Ising phase transition associ- ated with a spontaneous breaking of Z2 symmetry [44– 47], whose signatures have also been observed in the programmable Rydberg atom quantum simulators [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This equilibrium phase transition (referred to as ‘EPT’ throughout this paper) is in the same universality class as the one induced by varying the quark mass in the Schwinger model of quantum electrodynamics in (1+1)- dimension [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The lattice formulation of the latter, known as the U(1) quantum link model, exactly maps to the PXP model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (1) for the case of spin-1/2 degrees of freedom [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The EPT has a profound effect on the low-energy physics of the PXP model, but it is not immediately obvi- ous that it should directly impact QMBS, which manifest in the quench dynamics at infinite temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Never- theless, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 51 recently argued that there is a link be- tween this EPT and QMBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Namely, when tracing the eigenstates responsible for the quantum revival of the |Z2⟩ state, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 51 found that these states merge with the thermal bulk of the energy spectrum as the EPT is approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' On the contrary, upon moving away from the EPT towards µ → −∞, the degenerate ground states acquire high overlap with the |Z2⟩ state and its partner translated by one site, ��¯Z2 � ≡ |0101 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Thus, the |Z2⟩ state can only thermalize as one approaches the EPT, suggesting a connection between QMBS and criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This was also demonstrated experimentally in the Bose- Hubbard quantum simulator [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In this work, we map out the dynamical phase dia- gram of the PXP model corresponding to global quenches of the chemical potential from some initial value, µi, to an arbitrary final value, µf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This provides a means of probing out-of-equilibrium dynamics from more complex initial states beyond |Z2⟩ or |0⟩, which had been accessed in previous experiments by taking the limits µi → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We identify QMBS regimes in the dynamical phase dia- gram based on signatures of ergodicity breaking, such as the deviation of observable expectation values from the canonical ensemble predictions and the presence of quan- tum revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Our results show that the previously known scarring regimes, associated with |Z2⟩ and |0⟩ states, in- deed break down when approaching the EPT, either via µi → µc or µf → µc, in agreement with Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 51 and 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' However, we also find a new QMBS regime corre- sponding to the initial state being the ground state near the EPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Using the time-dependent variational principle (TDVP) framework for QMBS, developed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 53, we identify a semiclassical picture behind QMBS dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Across much of the phase diagram away from the EPT point, the QMBS dynamics can be understood in terms of a periodic trajectory that passes through the |0⟩ state, with the radius of the trajectory controlled by the chem- ical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Allowing for a continuous family of initial states – the ground states of HPXP(µi) – we find sur- prisingly robust QMBS signatures at intermediate times that smoothly bridge across the EPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We work out a ramping protocol for the preparation of such states, pro- viding a recipe for probing the dynamical phase diagram in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We start by presenting the results of numerical sim- ulations of the dynamical phase diagram of the PXP model for global quenches of the chemical potential in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' III-V we analyze in detail the vari- ous regimes of this phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' III contains a brief introduction of the TDVP formalism that will be useful for semiclassical interpretation of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' IV we focus on QMBS regimes of the phase dia- gram, while Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' V discusses the special case when the system is initialized in the ground state near the EPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' VI, we show how the dynamical phase diagram can be probed in experiment by preparing the desired ground states using a ramping protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Our conclusions are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' VII, while Appendices contain de- tails of the TDVP formalism, finite-size scaling analysis, and additional characterizations of the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' DYNAMICAL PHASE DIAGRAM OF THE PXP MODEL In this paper we are interested in the following out- of-equilibrium probe of the PXP model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (1): start from the ground state of HPXP(µi) and then evolve with the same Hamiltonian but generally different chemical potential value, HPXP(µf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We assume a closed system evolving under unitary Schr¨odinger dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Since the energy level spacings in the PXP model are expected to obey the Wigner-Dyson distribution for all values of µ [6, 35], the nonequilibrium dynamics induced by quenching µ should be described by random matrix the- ory [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In particular, quenching the chemical potential by a large amount ∼O(1) should initialize the system in a generic high-temperature state, which is expected to lead to rapid thermalization according to the Eigenstate 3 x x Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Dynamical phase diagram for global quenches starting in the ground state of HPXP(µi) and evolving with HPXP(µf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (a) The difference between maximal and minimal revival fidelity δF over time interval 1 ≤ t ≤ 20 following the quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Regions with strong fidelity revivals have been enumerated (see the text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (b) Same as (a) but the color bar showing the deviation of the excitation density from the thermal value, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Data is obtained using MPS simulations [54] for a chain of N = 51 atoms with OBCs, maximum bond dimension χ = 128 and time step δt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Dashed lines mark the EPT at µc ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In both plots, the cross marks the point (µi = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='76, µf = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='60) that will be analyzed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The diamond marks the optimal reviving point in the µi = µc plane, which will be discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Thermalization Hypothesis (ETH) [56–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This means that the expectation value of any local observable should converge towards the value predicted by the canonical en- semble within any symmetry-resolved sector of the many- body Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Deviation from this prediction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=', ergodicity breaking, can be detected through a number of probes, two of which we utilize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' One probe of ergodicity breaking, convenient in the context of QMBS, is quantum fidelity or return proba- bility of the wavefunction to its initial value, F(t) = | ⟨ψ(0)|ψ(t)⟩ |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (2) For a thermalizing initial state, F(t) rapidly drops to a value close to zero and remains exponentially small in system size at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Therefore, if the average fi- delity over a time interval ≫ Ω−1 is much larger than ∼ O(exp(−N)), we expect non-ergodic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' How- ever, one should exclude trivial cases such as µi ≈ µf when the ground state of HPXP(µi) is approximately an eigenstate of HPXP(µf), as this would lead to the system getting “stuck” in an eigenstate, with fidelity F(t) ≈ 1 and potentially never decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' To avoid such cases, we compute the difference δF between minimum fidelity and maximum fidelity over a time window t ∈ [t0, t1], with t0=1 and t1=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This window is large enough to exceed the initial relaxation on the scale ≳ Ω−1 (thus excluding the high fidelity near t = 0), yet small enough (t1 ≲ N/Ω) to be free of the boundary effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The obtained δF in the µi − µf plane is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The fidelity has been evaluated in a system of N = 51 atoms using ma- trix product state (MPS) [59] simulations based on the algorithm in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 54, and we have checked that the re- sults agree closely with exact diagonalization for systems with N < 30 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Before we comment on the interesting regimes of the phase diagram, we note that we have also computed the deviation of an observable expectation value from the thermal ensemble prediction, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This provides a complementary probe of ergodicity breaking that is more amenable to experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For the observable, we chose the density of excitations in the system, n = (1/N) �N j=1 Qj, which is readily available in existing experimental setups [21, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' After quenching the system, we compute the integrated mean- square deviation of the excitation density from the ther- mal value over the time window between t0 = 10 and t1 = 20, MSD(n) = 1 t1 − t0 � t1 t0 |⟨ψ(t)|n|ψ(t)⟩ − nth|2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (3) The thermal value is defined as nth = Tr(ρthn), (4) where the thermal density matrix is given by the usual Boltzmann-Gibbs expression, ρth = exp(−βH)/Z, with the partition function Z = Tr exp(−βH) and the inverse temperature β determined from the condition ⟨ψ(0)|HPXP(µf)|ψ(0)⟩ = Tr(ρthHPXP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (5) The plot of MSD(n) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1(b), where the bright non-ergodic regions match those of high fidelity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The color contrast is stronger in the fi- delity plot due to the exponential sensitivity of that quan- tity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' A few distinct regimes where fidelity displays large- amplitude oscillations have been marked by (1)-(5) in 6 10-2 4 2 0 10-3 2 4 6 6 4 2 0 2 4 66 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='8 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='2 4 6 0 6 4 2 0 2 4 64 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' These regions will be analyzed in detail in the subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' There, we will argue that regions (1), (2) and (3) can be identified as QMBS regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Re- gions (1) and (3) fall under the “universality class” of |Z2⟩ and |0⟩ QMBS behavior, as we explain in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' On the other hand, while the dynamics in region (2) has some similarities with regions (1) and (3), in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' IV we will highlight the distinctions of this QMBS regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' As it turns out, regions (4), (5), (6) and (7) have a simple origin, which will be explained briefly in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' A few comments are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The QMBS fidelity ap- pears to vary smoothly between regions (1) and (2) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1(a), while they are separated by the EPT (indi- cated by the dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In fact, we find the most robust revivals correspond to the ground state precisely at the EPT point (highlighted by the diamond in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This intriguing case will be addressed in detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Here we note that we have confirmed the existence of QMBS across the critical point in much larger systems (N ≤ 400 spins) using MPS numerics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This is in contrast to the µf = µc case, where we see no ergodicity breaking in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1(a), as also expected from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 51 and 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' TIME-DEPENDENT VARIATIONAL PRINCIPLE AND PERIODIC ORBITS FOR MANY-BODY SCARRING Without chemical potential, quantum dynamics from the |Z2⟩ state in the PXP model can be visualized as a classical periodic orbit [53, 60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This is accom- plished in the framework of the Time-Dependent Vari- ational Principle (TDVP) [62–64], which we briefly re- view in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' TDVP establishes a parallel be- tween many-body dynamics in the PXP model and the analogous dynamical phenomena of a single particle in a stadium billiard, in which the wavepackets are anoma- lously long-lived when prepared along the periodic orbits of the corresponding classical billiard [65, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' TDVP will provide a natural semiclassical language for interpreting the essential features of the dynamical phase diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' A brief overview of TDVP formalism The starting point of TDVP is to specify a variational manifold of states M, parameterized by some continu- ous variable, and then project the Schr¨odinger dynamics into that manifold in a way that manifestly conserves the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The nature of states belonging to M determines to what extent we can interpret the dynamics as “semi- classical”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For example, it would be simplest to consider a manifold spanned by tensor products of spin-coherent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This would yield a “mean-field” description for the dynamics, where each atom precesses independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' However, the Rydberg blockade intrinsically builds in lo- cal correlations into the system, due to the fact that any neighboring excitations, |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='⟩, are projected out of the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Ordinary spin-coherent states clearly violate this blockade condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Another way of defining a manifold, which naturally accommodates the Rydberg blockade constraint, is to take the span over MPS states with bond dimension χ controlling the amount of correlations necessary to cap- ture the projected dynamics [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' To simplify matters as much as possible, we will consider the dynamics to be spatially periodic with a (infinitely repeated) unit cell of size K (below we will be primarily interested in small unit cells with K = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For a 1D chain of size N, the resulting MPS ansatz is given by |ψMPS({x})⟩= � {σ} Tr �N/K−1 � m=0 Aσ1+Km(x1)Aσ2+Km(x2) AσK+Km(xK) � |σ1σ2σ3 · · · σN⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (6) Here Aσ(xi) are (χ × χ)-dimensional matrices that de- pend on variational parameters xi = (θi, φi), where the angles θi, φi are akin to the Bloch sphere angles of each spin in the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The physical degree of freedom σi = 0, 1 labels the basis states of a single spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Follow- ing Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 53 and 67, in order to make things analytically tractable, we will restrict to χ = 2 and chose A1(θi, φi) = � 0 e−iφi 0 0 � , A0(θi, φi) = � cos θi 0 sin θi 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (7) Due to A1A1 = 0, this ansatz ensures that configura- tions with neighboring spin-up are forbidden, thus our manifold M = span{|ψMPS(x)⟩ |∀x} is consistent with the Rydberg blockade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' With the choice of ansatz in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (6)-(7) and setting K = 1, we are left with only two variational degrees of freedom, (θ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Choosing (0, 0) recovers the state |0⟩ ≡ |000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='⟩, while (π/2, π/2) corresponds to the equal- weight superposition of the two N´eel states, ��Z+� ≡ 1 √ 2 � |Z2⟩ + ��¯Z2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (8) Note that with K = 1 unit cell periodicity, the states |Z2⟩, ��¯Z2 � do not individually belong to the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In- stead, if we extend the ansatz to K = 2, then (θ1, θ2) = (0, π/2) recovers the |Z2⟩ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Thus, our manifold with bond dimension χ = 2 captures the initial product states that we expect to play an important role for QMBS dy- namics in the PXP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' After defining the manifold, the next step is to mini- mize the difference between exact Hamiltonian dynamics and its projection to the manifold, min {x} ����iℏ ∂ ∂t |ψMPS({x})⟩ − H |ψMPS({x})⟩ ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (9) This results in the Euler-Lagrange equations of motion for the classical variables x [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In the case of the PXP model, this step can be performed analytically in the 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Sketch of the TDVP manifold M for the PXP model with chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Red regions represent areas of high leakage where the TDVP approximation breaks down, as quantified by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The N´eel state is denoted by |Z2⟩ ≡ |1010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='⟩ and its translated partner – the anti-N´eel state is ��¯Z2 � ≡ |0101 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='⟩, while ��Z+� = (|Z2⟩ + ��¯Z2 � )/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The polarized state is |0⟩ ≡ |0000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (a) For a two-site unit cell K = 2 and µ = 0, the |Z2⟩ state lies on a periodic trajectory identified in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We also illustrate the trajectory of |0⟩ state, which is predicted by TDVP to evolve to ��Z+� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' however, this point lies within a region of high leakage where the TDVP dynamics does not accurately describe the quantum evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This is consistent with the |0⟩ state thermalizing at µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (b) Taking K = 1 we focus on the evolution of the |0⟩ state trajectory as µ is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For µ = 0, the trajectory is periodic but passes through a region of high leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' When µ ̸= 0, the trajectory shrinks, whilst gradually exiting the high leakage area, and QMBS dynamics starts to emerge in the full system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In this regime, the QMBS dynamics can be seen as an oscillation between |0⟩ and a new state, |¯0(µ)⟩, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Finally, in the extreme µ → ±∞ limit, the orbit shrinks to a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' limit of N → ∞ to obtain the equations of motions for the θ and φ angles, see Appendix B for K = 1 and Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 53 and 67 for some K = 2 and K = 3 examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Integrating this system of differential equations yields the trajectory in M taken by |ψMPS({θ, φ})⟩ during the course of quan- tum evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 2 shows a pictorial representation of the manifold and the projection of exact dynamics into it, for the cases of interest in the PXP model perturbed by the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Importantly, beyond equations of motion, it is possible to estimate “quantum leakage”: the difference between exact quantum evolution and its projection into the man- ifold [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Quantum leakage, γ, is defined as the instan- taneous rate at which the exact wave function leaves M: γ2 = lim N→∞ 1 N ����iH |ψMPS(x)⟩ + � j ˙xj∂xj |ψMPS(x)⟩ ���� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (10) Red regions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 2 indicate areas of large γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In these high-leakage regions, the instantaneous TDVP dynamics is expected to poorly capture the exact dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Con- sequently, trajectories passing through such regions will generally be of limited accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' On the other hand, as first noted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 53, the special property of the PXP phase space is that it has regions of remarkably low leak- age, such as the region traversed by the semiclassical or- bit associated with the |Z2⟩ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 2(a) where the orbit is sketched, lying within a re- gion of low leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Note that, in general, there can exist multiple periodic orbits within the same manifold [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' TDVP interpretation of the dynamical phase diagram Much of the PXP dynamical phase diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1 can be understood by considering the trajectory of the polarized state in the TDVP manifold introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 2(b) sketches this trajectory for three different values of the chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Within TDVP, a periodic orbit exists even for µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' However, the orbit passes through the superposition of the two N´eel states, |Z+⟩, which is located in the high-leakage region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The TDVP dynamics is therefore not a good approximation in this case, which accounts for the absence of revivals observed in the full quantum dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The addition of a finite chemical potential µ contracts the trajectory and pushes it into a low-leakage region, as shown in the middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 2(b), effectively allow- ing the revivals from the polarized state to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' As we will explain in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' IV, in this intermediate range of µ, the ground state of HPXP(µ) occupies an antipodal posi- tion on the orbit, corresponding to a chemical-potential dependent state we label |¯0(µ)⟩, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (7) for unit cell size K = 1: |¯0(µ)⟩ = |ψMPS(θmax, φmax)⟩ , (11) with angles (θmax, φmax) denoting the antipodal point in the TDVP orbit of the initial polarized state, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' As µ has the effect of deforming the trajec- tory, the antipodal angles also depend on µ, as will be specified in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (16) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Note that the sign of µ has no effect on the deformation of the particular orbit dis- cussed here, as we explain in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Finally, in the extreme limit µ → ±∞, the trajectory is restricted to the vicinity of the initial state and the dynamics is effectively frozen, as shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' SCARRING IN GAPPED REGIMES OF THE PHASE DIAGRAM In this section we focus on regions (1), (2), and (3) of the phase diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1, in particular for the values of the chemical potential away from the EPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Based on the o←n o←n μ≠0 μ IH8 K=2 K=1 iHt (a) [2) (b) Z2/ (()" 0>Z+> Z+> z+> /z+> 10) 10)00) 10)* OJ Z2)6 discussion of TDVP in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' III and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 2, the origin of regions (1) and (3) can be understood by examining the form of the PXP ground state in the presence of chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' When µi → −∞, excitations are favored and the ground state is (for PBCs) a superposition of the two N´eel states, |Z+⟩ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' By contrast, µi → ∞ pe- nalizes excitations, therefore the ground state is the po- larized state |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The superposition state |Z+⟩ is known to display revivals when quenched to µf = 0 [37], while the polarized state revives when quenched with µf ̸= 0 as shown more recently in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 24 and 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' By continu- ity, these limiting cases explain the mechanism behind revivals in regions (1) and (3) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In the remainder of this section, we focus on the more interesting region (2) where the pre-quench initial state is an entangled state with low overlap on both |0⟩ and |Z2⟩ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Scarring in region (2) of the phase diagram We focus on region (2) of the phase diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1 and pick (µ∗ i , µ∗ f ) = (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='76, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='60) as an illustrative point in this region, marked by the cross in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1(a)-(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' QMBS dynamics at this point was first noted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 24 and here we will characterize it in detail and explain its origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The evolution of fidelity and overlap with the po- larized and N´eel state are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 3(a), where per- sistent fidelity revivals can be observed while the overlap with |Z2⟩ remains negligible throughout the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Curiously, while the initial state at µ∗ i has low overlap with |0⟩, the evolved state does develop a relatively high overlap with |0⟩ state, approximately half way between the main revival peaks – see the green line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This is reminiscent of the |Z2⟩ state, which in the pure PXP model undergoes state transfer to ��¯Z2 � at half the revival period [53], implying that the ground state of HPXP(µ∗ i ) is related to the polarized state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Another tell-tale signature of QMBS is a slower growth of entanglement entropy, SE(t), for special initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The entanglement entropy is defined as the von Neu- mann entropy of the reduced density matrix, ρA = TrB|ψ(t)⟩⟨ψ(t)|, obtained by tracing out degrees of free- dom belonging to one half of the chain (denoted B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We plot the dynamics of SE(t) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Compared to both |Z+⟩ and a random product state, |σRandom⟩, the entropy growth from the ground state of HPXP(µ∗ i ) is strongly suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Moreover, for the latter state, we observe clear oscillations in the time series of SE(t), rem- iniscent of entropy dynamics in the PXP model in the absence of chemical potential [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We emphasize that the special point (µ∗ i , µ∗ f ) is repre- sentative of the entire region (2) in the phase diagram, where similar QMBS phenomenology is numerically ob- served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In the remainder of this section, we use TDVP to garner a further understanding of this QMBS regime from a semiclassical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 | (t)| 2 | | (t)|0 | | (t)| (0) | 0 2 4 6 8 10 12 14 t 1 2 3 S(t) | random(t) | + | (t) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Dynamics of quantum fidelity and entanglement entropy, following a global quench of the chemical potential, µ∗ i = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='76 → µ∗ f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6, corresponding to the point marked by the cross in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Quantum fidelity for the initial state |ψ(0)⟩ defined as the ground state of the PXP model with µ∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Also shown is the projection of the time-evolved state on the |Z2⟩ and |0⟩ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' While the overlap with the |Z2⟩ state is low throughout the evolution, the overlap with |0⟩ reaches relatively high values between the main revival peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (b) Growth of entanglement entropy, SE(t), for the same initial state |ψ(0)⟩ as in (a), as well as for a random state |σRandom⟩ and ��Z+� state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The initial state |ψ(0)⟩ has strongly sup- pressed entanglement growth compared to the other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Data is for system size N = 28 obtained using exact diago- nalization with PBCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' TDVP analysis of scarring in region (2) Before we apply TDVP to extract the semiclassical de- scription of the dynamics in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 3, we need to make sure that the PXP ground state in the presence of chemical potential is represented within the manifold spanned by states in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In a recent work [68], a method of “op- timal steering” has been devised to smoothly prepare a class of PXP ground states based on the minimization of quantum leakage along the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' To show that the detuned PXP ground states are captured in the TDVP manifold, here we follow a simpler approach of optimiz- ing the overlap |⟨ψMPS({x})|ψ(µi)⟩|2, where |ψ(µi)⟩ is the ground state of the PXP model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For a unit cell size K = 1, we performed exhaustive numerical sampling at system size N = 20 and found that most states belonging to the TDVP manifold (> 90% of them) can be approximated with better than 98% accuracy by a ground state of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' As a side note, we mention that in order to prepare the states in the TDVP manifold with unit cell K ≥ 2, we need to make two modifications to the preparation procedure: (i) we need to allow chemical potential to be different for different atoms within the unit cell;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (ii) we need to include a unit-cell modulated pulse in the z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' As explained in Appendix D, after these generalizations, one can also successfully pre- pare TDVP states with K ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' While we do not have a general proof, this provides a numerical confirmation of the representability of the ground states of the PXP 7 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (a) Phase space portrait of quantum dynamics within the K = 1 TDVP manifold for the PXP model with µf=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Grey shading indicates quantum leakage (darker regions represent larger leakage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The trajectory of the |0⟩ state for the given value of µf is highlighted in red, while colored symbols indicate the location of the PXP ground states corresponding to various µi indicated on the color bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The ground states with µi≈−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='76 can be seen to lie close to the point which is antipodal to the |0⟩ state in its trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' With changing µf, this trajectory either expands or compresses, meaning all ground states will lie on this antipodal point for some µf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (b) In region (2) of the phase diagram, for a given µf, |¯0(µf)⟩ state is well-approximated by a detuned PXP ground state with some µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Color bar shows the highest overlap between the |¯0(µf)⟩ state, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (16) for a range of fixed µf ∈ [−5, 5], and the family of ground states of HPXP(µi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Dashed lines denote the EPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For negative chemical potentials, especially relevant for region (1) of the phase diagram, the mapping requires an additional phase pulse, as described in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (c) Matching the detuned PXP ground state with a |¯0⟩ state becomes progressively more difficult at the critical point (dashed line) as system size N is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In contrast to panel (b), here we fix the PXP ground state at µi and vary µf to find the optimal |¯0(µf)⟩ state with the highest overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' All plots are obtained using exact diagonalization with PBCs and system size N = 20 in panels (a)-(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' model with a suitably-defined generalization of the chem- ical potential within the TDVP manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Having established that our pre-quench ground state at arbitrary µi can be approximately mapped to an MPS state in the K = 1 TDVP manifold for some variational parameters (θ, φ), we now proceed to describe the dy- namics from this initial state using the classical dynam- ical system defined by (θ(t), φ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (9), one can derive the TDVP equations of motion for K = 1 and arbitrary chemical potential µ (see Appendix B for details): ˙θ = − cos θ cos φ � 1 + sin2 θ � , (12) ˙φ = µ + sin φ sin θ � 1 − 4 sin2 θ − sin4 θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (13) Unlike the special case µ = 0, where φ variables can be set to zero in the flow-invariant subspace [53], for general values of µ one must consider both θ and φ variables simultaneously [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Integrating Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (12)-(13), we plot the phase space θ, φ portrait for the chemical potential value µf = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The greyscale background indicates the quan- tum leakage at any given point in the manifold, γ2 = sin6θ 1 + sin2θ, (14) which only depends on θ variable (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' By integrating the equations of motion for µf = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6, start- ing from the polarized state |ψMPS(0, 0)⟩, we obtain the trajectory plotted in red color in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Generally, for any |µf| ̸= 0, the polarized state has a periodic or- bit within TDVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' When µf is large, the orbit is pinned around θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Decreasing |µf| stretches out the orbit un- til the maximal point in the trajectory eventually tends towards the |Z+⟩ superposition state, (θ, φ) ≡ (π/2, π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Due to the presence of a quantum leakage gradient, the |Z+⟩ point is not reached for any finite time, consis- tent with the lack of revivals from the polarized state in full quantum dynamics for sufficiently small values of µf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Thus, we conclude that the orbit corresponding to the cross in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1(a) is a compromise between two com- peting effects: the orbit is sufficiently stretched so that it has nontrivial dynamics, while at the same time, by be- ing not stretched too much, it can avoid the large leakage in the vicinity of |Z+⟩ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' To verify this picture across the entire region (2), we study the projection of the PXP ground state at µi, |GS(µi)⟩, to the TDVP manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We numerically max- imize the overlap |⟨ψMPS(θ, φ)|GS(µi)⟩|2, with the MPS state given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We plot the resulting (θ, φ) phase space coordinates for a variety of µi in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 4(a), where the colored dots correspond to the ground states from our phase diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' As expected, some of the ground states are “distant” from |Z+⟩ or |0⟩ but tend towards either in their respective limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' All suc- cessfully optimized ground states lie on the same φ plane in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1(a), such that the deformation of the trajectory means they will correspond to some maximum point µf on the polarized state trajectory, denoted by the state |¯0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' By analogy with the N´eel state, whose translation partner – the anti-N´eel state – displays identical scarring behavior [19], here we have a similar relation between |0⟩ and |¯0(µf)⟩ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The main difference with the anti-N´eel state is that |¯0⟩ state depends on the value of µf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (b) (a) (c) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 3+ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='00 》 (m)0l → N =14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5 N =16 4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='8 N=18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0似 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='50 S N =20 2 1 N=22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5 一 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6 N =24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='00 0- 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5 5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 0 5 5 i ui8 To substantiate this further, we analytically derive the phase-space coordinates corresponding to |¯0(µf)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (12), we see that the turning point in the gradient of θ along the trajectory is governed by cos φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' A sign flip therefore must occur when φ = ±π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Because energy is exactly conserved along a TDVP trajectory, |¯0(µf)⟩ must have the same energy as the polarized state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For states belonging to K = 1 TDVP manifold, the energy density is given by E(θ, φ)/N = sin θ 1 + sin2 θ � µf sin θ + 2 cos2 θ sin φ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (15) For the polarized state, E(0, 0) = 0 and, setting φmax = π/2, allows us to determine the θmax coordinate of the |¯0(µf)⟩ turning point: sin θmax = � |µf| − � µ2 f + 16 � /4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (16) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 4(b) we test the overlap of the state |¯0(µf)⟩, with the MPS angles given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (16), against the family of ground states of HPXP(µi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We scan through a set of values µf ∈ [−5, 5] and, for each µf, plot the maximum overlap obtained by maximizing over µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Although µf < 0 is not particularly relevant for region (2), we note that the optimization fails there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This, however, can be fixed by including an additional phase pulse, as explained in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 4(b) to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1(a), we see a striking correspondence between the successful optimiza- tion and region (2) in the phase diagram, which confirms that the QMBS phenomena in region (2) are indeed as- sociated with |¯0(µf)⟩ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 4(c) we study the system size scal- ing of the mapping between the PXP ground state with chemical potential and states in the TDVP manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We scan for the maximal overlap of the ground state at some µi with the set of all |¯0(µf)⟩ states in the in- terval µf ∈ [−20, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Remarkably, for the vast majority of region (2) when µi > 0, we see a near perfect over- lap between the ground state and |¯0(µf)⟩, independent of system size – suggesting that the TDVP state captures well the PXP ground state in region (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Nevertheless, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 4(c) we also observe a breakdown of the mapping at the EPT point µi = µc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This is expected since the ground state at the critical point develops a diverging entangle- ment entropy and the χ = 2 MPS approximation must deteriorate as system size is increased, since an area-law state cannot capture the critical ground state in the ther- modynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This naturally leads to the question: is the observed scarring in the critical ground state an artefact of finite size and what is its origin?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' INTERPLAY BETWEEN SCARRING AND CRITICALITY We now focus on the nature of QMBS regime when quenching from the critical ground state at µi = µc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' De- spite the complexity of this state, we find robust signa- tures of ergodicity breaking in the area between regions (1) and (2) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For example, by fixing µi = µc and scanning µf to determine the largest δF, we find the most robust revivals occur at µf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='633 – a point that was marked by the diamond in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This turns out to be one of the best reviving points in all of regions (1), (2) and (3), including the |Z2⟩ and |0⟩ initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' As discussed above, the TDVP semiclassical formalism is not well-suited for describing this case as it cannot capture the diverging entanglement entropy of the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This immediately raises the question if the ob- served QMBS behavior is a finite size effect and whether one should rather expect a sharp boundary between re- gions (1) and (2) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1 in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' To probe the robustness of QMBS revivals in the ther- modynamic limit we simulated the quench dynamics µi = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='31 → µf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6 in large systems up to N = 401 using the MPS method [54] in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The fidelity, plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 5(a), demonstrates that revivals exist in all accessi- ble system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The fidelity is not an intensive quantity, therefore it is generically expected to decay in the N → ∞ limit, as indeed can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Thus, to compare different system sizes, we take the fidelity at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 N =21 N =31 N =41 N =51 N =101 N =151 N =201 N =251 N=301 N=401 0 2 4 6 8 10 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='4 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='02 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Fidelity and entanglement entropy dynamics for the quench from the critical ground state with µi = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='31 to µf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (a) Fidelity revivals persist up to the largest system size N = 401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' While the fidelity decays with N, the fidelity density of the first revival peak, − log(F1)/N, plotted against inverse system size, 1/N, extrapolates to a value close to 0 (inset), indicating non-ergodic behavior in the thermo- dynamic limit at a finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (b) Dynamics of the half-chain entanglement entropy SE(t) for the same quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We scale the entropy by the critical value given by the Cardy-Calabrese formula with central charge c = 1/2 [69], which collapses the data to 1 at t = 0 (inset shows the unscaled entropy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The growth of entropy is seen to be linear, with pronounced os- cillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Data is obtained by MPS simulations with OBCs, bond dimension χ = 300, and time step δt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 9 the first revival peak F1 and plot its density, − log(F1)/N against 1/N, in inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This serves as an in- dicator of ergodicity breaking at a finite time that can be properly scaled to the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For a random state in the constrained Hilbert space of the PXP model, we expect − log(F1)/N to asymptotically approach log � (1 + √ 5)/2 � ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Contrary to this ex- pectation, the fidelity density in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 5(a) continues to decrease as N → ∞, signaling non-ergodicity in the ther- modynamic limit at a finite time t ∼ 5/Ω, well beyond the initial relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 5(b) we observe a slow growth of entangle- ment entropy following the same quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In contrast to previous QMBS cases in the literature, where the sys- tem was initialized in a product state with zero entropy, such as |Z2⟩, here we start from a critical ground state whose entropy is expected to diverge logarithmically with system size according to the Cardy-Calabrese formula, Scrit = (c/6) log(N/π) [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The universal prefactor is determined by the central charge c of the conformal field theory, which is c = 1/2 for our critical point in the Ising universality class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Scaling the data by Scrit indeed yields a good collapse at time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' At later times, the en- tropy grows linearly with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' On top of linear growth, we observe prominent oscillations that are typically found in QMBS systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=', the |Z2⟩ initial state in the PXP model [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The amplitude of these oscillations is roughly independent of system size, as can be seen in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' At much later times, which are inaccessible to MPS methods, we expect the entropy to saturate to a value proportional to the volume of the subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Apart from the diverging entropy of the initial state, the overall picture from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 5 is broadly similar to pre- vious studies of QMBS dynamics [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' What remains to be explained is why the critical ground state is poised to- wards QMBS-like dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' To identify the microscopic origin of this robust ergodicity breaking in the vicinity of µf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='633, we plot the overlap of the initial criti- cal ground state with the eigenstates of the post-quench Hamiltonian in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The overlap exhibits clear tow- ers of eigenstates which are emblematic of QMBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' While these features are present throughout the spectrum, the dominant contributions to the initial state come from low-energy eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In order to approximate their characteristics, we can treat them as magnons with a given momentum k on top of the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For µf = 0, this has been shown to give a good approximation of scarred states even at relatively high energies when us- ing magnons with momentum k = π [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Similarly, we find this to be true in our case near µf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6, where much of the low-energy spectrum can be approximately recon- structed from pairs of non-interacting magnons with mo- menta k and −k, see the dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 6 and in- set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Note that the PXP model is gapped for µf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='633, hence the ground state and the first tower in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 6 are separated by a finite energy that is independent of N in sufficiently large systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' A detailed analysis of the magnon dispersion as a func- Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Overlap between the ground state at the critical point µi = µc = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='31 and the eigenstates of the PXP model with µf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The color indicates the density of data- points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The red dashed lines indicate multiples of the energy of a k = π excitation on top of the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This matches well with the scarred towers in the relevant part of the spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The inset shows the first set of excited states, with the grey dashed lines indicating the expected energy for non- interacting pairs of excitations with momenta k and −k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Due to the flatness of the band near k = π and k = 0, the lines are denser near the scarred states, leading to sharper towers and better revivals (see further analysis of the magnon dispersion in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 7 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Data is obtained by exact diagonalization for system size N = 28 with PBCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' tion of chemical potential is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The dispersion relation for several values of µf is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For µf < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6, the single-magnon band merges with the two-magnon continuum, causing the downward slope near k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Near µf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6, the band becomes remarkably flat for small k, coinciding with the one- magnon and two-magnon bands barely touching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' At that point, the energies of the first excited states at k = 0 are well approximated by twice the energies of the single- magnon states, indicating that they correspond to a pair of two non-interacting magnons with momenta k and −k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 7(b) and the inset of panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This simple picture of non-interacting excitations al- lows us to predict the energies of the low-energy excited states based solely on the dispersion relation of the single- magnon states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In particular, the flatness of the band near k = 0 and k = π means that the eigenstates near the scarred ones have approximately the same energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This implies that the towers of states will be sharper, and that the effective energy spacing, which determines the dynamics at intermediate times, is the spacing be- tween the towers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In turn, the fact that the magnons are very weakly interacting means that the spacing between these towers will be approximately equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In summary, we showed that QMBS in the critical ini- tial state can persist due to (i) the post-quench Hamil- tonian HPXP(µf) having a gapped spectrum with a suffi- ciently flat band of the low-lying magnon excitations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (ii) the magnons are weakly interacting and their multiplets (GS-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='31|E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='633)I2 10 10-7 10-10 10-13 10-16 10 10 20 E10 (a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5 3 2 1 0 6 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 0 1 0 1 1/2 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (a) Dispersion relation of the low-lying excitations of the PXP model for several values of the chemical potential µf, shown in different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' When µf ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6, the dispersion becomes visibly flat near both k = 0 and k = π momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Inset shows the difference between the actual energies of the first excited states in the spectrum and their approximation by a pair of two non-interacting excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For all momenta k, the best agreement between the approximation and exact energy is attained at µf ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (b) Low energy spectrum of the PXP model with µf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6 – the value with the best revivals when quenching from the critical ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The ground state and first excited states are indicated, along with energies corresponding to a non-interacting pair of excitations with momenta k and −k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In this instance, we see the approximate excitations and exact energy levels lie close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Data is obtained by exact diagonalization for system size N = 24 with PBCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' give rise to regularly spaced QMBS-like towers in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' While this scenario is reminiscent of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 71, where quantum revivals in some non-integrable models were related to the low-lying quasiparticle states, in our case the chemical potential needs to be finely tuned to a value µf ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6 to meet the conditions (i)-(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Indeed, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1, varying µf around this value leads to a sharp decay of QMBS revivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In contrast to the PXP model with µi = 0 and the |Z2⟩ initial state, the QMBS eigenstates in the µi = µc case are clearly skewed towards the low-energy part of the spectrum, however this allows the QMBS revivals to persist in large systems, despite the highly entangled initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' EXPERIMENTAL PROTOCOL Finally, in this section we address the experimental ob- servation of the phase diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The key step is the preparation of the PXP ground state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The protocol below is directly applicable to Rydberg atom arrays [22], however it can also be adapted to ultracold bosons in a tilted optical lattice, where the chemical po- tential µ maps to the energy mismatch between the Hub- bard interaction and electric field which induces a tilt potential [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Ground state preparation is accomplished via a “ramp- ing” procedure utilized in related experiments [21, 46, 48, 72, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This assumes fine control of the chemical poten- tial that is varied in time, µ = µ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Taking the chemical potential very large, µ → ±∞, one can prepare |0⟩ and |Z2⟩ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Starting in one of these states, one can then ramp to a desired ground state in the interior of our phase diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1 by evolving with a time-dependent PXP Hamiltonian, HPXP(µ(t)), where µ(t) is appropri- ately parameterized for an adiabatic evolution, as spec- ified below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The adiabaticity implies that the ramping will not be able to prepare the critical ground state af- ter a finite time in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Therefore, with finite resources, we can only hope to approach the critical point from different gapped regions of the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We start the ramp either in |Z2⟩ or |0⟩, depend- ing on whether we are in a ordered (µ < µc) or disordered (µ > µc) phase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Specifically, we make use of the following ramp µ(t) = A (t − B)2 − A (t − C)2 + µc, (17) where A, B, and C are tunable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' One par- ticularly successful choice was found to be A = ∓40, when ramping from |0⟩ or |Z2⟩, respectively, B = 30, and C = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' An example of this ramping curve is plotted in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 8(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We include µc due to the need for a much slower ramp as the gap between the ground state and first excited state closes in the vicinity of the EPT point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' After specifying the ramp and the initial state, we evolve by the PXP Hamiltonian in the presence of chemical potential, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (17), until some time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The evolution time is determined by numerically min- imizing 1 − |⟨ψ(t)|GS(µtarget)⟩|2, where |GS(µtarget)⟩ is the state we are trying to prepare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 8(a) illustrates the success of the ramping proce- dure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For system sizes ranging from N = 6 to N = 14, we have ramped to prepare the ground states from µ = ±6, in increments δµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5, towards the critical point, µc = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 8(b) shows the time that the ramp took for each ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We see the ramp time is insensitive to system size in gapped regions of the phase diagram, while it sharply increases near µc and exhibits strong fluctua- tions with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For fixed ramp parameters, we expect it will take an infinite amount of time to prepare the critical ground state in the N → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='00 =6 =8 =10 =12 =14 0 5 10 15 4 2 6 0 2 4 6 6 4 2 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 0 30 60 0 60 =51 =75 =101 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (a) The success of preparing the PXP ground state at chemical potential µ by ramping the chemical potential ac- cording to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The total ramp time is varied for each point to maximize the overlap, which is plotted on the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For µ > µc, the initial state is |0⟩ (square symbols), while for µ < µc we start the ramp in the |Z2⟩ state (triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Separate optimizations were performed for different system sizes N, shown in the same plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Black dashed line (in all the panels) denotes the critical point µc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Inset: using the optimal parameters and average ramping time determined in smaller sizes in the main panel, we prepare the ground states for the same values of µ in much larger system sizes N = 51, 75, 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The preparation in this case was done using the MPS method with time step δt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='025 and maximum bond dimension χ = 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' While in the gapped phases the preparation remains successful, there is a visible drop near the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (b) Total ramp time tramp returned by the optimizations in the main panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Inset shows the ramping curve µ(t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We observe an increase of the ramp time and strong finite-size fluctuations at the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The data in the main panels (a) and (b) was computed using exact diagonalization in k=0 momentum and p=+1 inversion symmetry sector with PBCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Finally, to verify our preparation scheme in large sys- tems, we repeated the preparation of the detuned PXP ground states for system sizes of N = 51, 75 and 101 using MPS simulations with bond dimension χ = 128 and the the ramping protocol in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (17), with the same A, B, C parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 8(a) demonstrates that the ramping continues to successfully reproduce the desired ground state with high fidelity, with the exception of the critical point where we see a clear drop in overlap with the target state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This suggests the ramping procedure is a viable method for generating desired ground states even in large systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' With this in hand, along with the al- ready existing capabilities to quench with a detuned PXP Hamiltonian and conduct measurements of local observ- ables [21, 22], all the tools are, in principle, available to reconstruct the dynamical phase diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In particular, local fidelity measurements [24] can be used to approximate the numerically computed global fidelity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This would allow to experimentally verify the persistence of QMBS across the phase diagram and its robustness near the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' CONCLUSIONS AND DISCUSSION We have mapped out the dynamical phase diagram of the PXP model, based on ergodicity breaking in its dy- namics following the global quench of the chemical po- tential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We have demonstrated the existence of extended regions which harbor QMBS phenomena, either associ- ated with the previously studied initial conditions, such as |Z2⟩ and |0⟩, or with new entangled states such as |¯0(µ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The mechanisms giving rise to these QMBS phe- nomena, in particular the underlying periodic trajecto- ries, were identified within the TDVP framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We have analyzed in detail the robustness of QMBS when the system is tuned to the EPT point, arguing that this does not provide an obstacle for QMBS, provided that the post-quench Hamiltonian is tuned in such a way that the low-lying quasiparticle excitations are weakly inter- acting and possess a flat energy-momentum dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This enables different QMBS regions in the dynamical phase diagram to connect smoothly, bridging across the EPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Finally, we have also outlined an adiabatic prepa- ration scheme that allows to map out the same phase diagram in experiments on Rydberg atoms and ultracold bosons in tilted optical lattices, both of which have re- cently realized the PXP model in the presence of a tun- able chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In light of these experiments, our discussion of the phase diagram above was restricted to finite times, however in Appendix F we discuss the corresponding phase diagram for time t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We note that the existence of a continuous family of QMBS states, tunable by the chemical potential, is of independent in- terest in quantum-enhanced metrology, for which QMBS states were shown to be advantageous [74–76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' One motivation behind this work is the open prob- lem of identifying all initial conditions associated with QMBS for a given model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For the pure PXP model it had originally appeared that only the |Z2⟩ state is special in this regard [21], however, more recent explorations of the chemical potential [24] have revealed that the latter can also stabilize QMBS from a different initial state, |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In this paper, we have shown that these two product states share the semiclassical description and belong to a larger family, which also includes some other weakly-entangled states such as |¯0(µ)⟩ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' While we have numerically related these initial states and their quench dynamics, it is not obvious how to relate them at the level of a spectrum-generating su(2) algebra, which has provided an elegant description of revivals from the |Z2⟩ state in the pure PXP model [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Moreover, our present investi- gation focused on the dynamics with periodicity K = 1 and it would be interesting to extend it to K ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For ex- ample, it is known that |Z3⟩ = |100100100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 100⟩ state also exhibits revivals in the pure PXP model model [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' However, this state necessitates a TDVP description with K = 3 unit cell, which already gives rise to an intricate phase space at the semiclassical level [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' It would be interesting to understand the dynamical phase diagram associated with such states that have larger unit cells, 12 either in the PXP model or analogous models for larger Rydberg blockade radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Finally, our results for the initial state at the critical point suggest that QMBS dynamics is not necessarily as- sociated with preparing the system in a product state or even an area-law entangled state, but in principle allows for highly-entangled initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In this case, QMBS dynamics is more strongly temperature-dependent, as the initial state has dominant support on the relatively low- lying energy eigenstates of the post-quench Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The key ingredient for making this work was to suppress the interaction between quasiparticles and flatten their energy dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' It would be interesting to understand how to engineer such conditions in other models and thereby realize similar dynamics from highly-entangled initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by the Leverhulme Trust Re- search Leadership Award RL-2019-015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' acknowl- edges support by EPSRC grant EP/R513258/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' ac- knowledges funding provided by the Institute of Physics Belgrade, through the grant by the Ministry of Educa- tion, Science, and Technological Development of the Re- public of Serbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Part of the numerical simulations were performed at the Scientific Computing Laboratory, Na- tional Center of Excellence for the Study of Complex Systems, Institute of Physics Belgrade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Statement of compliance with EPSRC policy framework on research data: This publication is theoretical work that does not require supporting research data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' acknowledges sup- port by the Erwin Schr¨odinger International Institute for Mathematics and Physics (ESI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' acknowledges funding from the European Research Council (ERC) un- der the European Union’s Horizon 2020 research and innovation programm (Grant Agreement no 948141) — ERC Starting Grant SimUcQuam, and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foun- dation) under Germany’s Excellence Strategy – EXC- 2111 – 390814868.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Appendix A: Other regions of the phase diagram Several regions of the phase diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1 exhibit fidelity revivals that have a simple origin that can be understood without invoking QMBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Here we explain in more detail these regions labeled (4), (5), (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' It is useful to consider the Inverse Participation Ratio (IPR), one of the traditional measures of ergodicity of the eigenfunctions introduced in the context of Anderson localization [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The IPR is defined as IPR = 1 � E | ⟨E|ψ⟩ |4 , (A1) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Logarithm (base 10) of the IPR of the ground state of HPXP(µi) with respect to the eigenstates of HPXP(µf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' All the labels have the same meaning as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Data is obtained using exact diagonalization in the sector with k = 0 momentum and p = +1 inversion symmetry for size N = 26 with PBCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' and it intuitively tells us about how many basis states |E⟩ the state |ψ⟩ has support on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For example, if |ψ⟩ is a basis state, its IPR will be 1, while if |ψ⟩ is homogeneously spread over the entire Hilbert space, the IPR will be equal to the Hilbert space dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Note that IPR is a basis- dependent quantity and, in our case, we have a natural choice of eigenstates |E⟩ of HPXP(µf) as the basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The log of IPR for µi ground states with respect to µf eigenstates is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This allows us to fur- ther distinguish between different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For conven- tional |Z2⟩ scarring we expect the IPR to be on the order of system size N, since the |Z2⟩ state has high overlap with a band of N + 1 scarred eigenstates of HPXP(0) but low overlap with the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This is evidenced in region (1) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' On the other hand, the band of scarred eigenstates associated with |0⟩ state in the detuned PXP model is “tilted” to one edge of the spectrum, so we ex- pect the IPR to be smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In general, the regions with high IPR are expected to be ergodic, while the least inter- esting regimes are characterized by very low IPR, such as around the µi = µf diagonal and in regions (5) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The IPR is not as low in parts of regions (4) and (7) visible in this figure, but it decreases with increasing |µi| and |µf| as the ground state of HPXP(µi) approaches an eigenstate of HPXP(µf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Large |µf| leads to fragmentation of the Hilbert space, which can effectively trap the initial state in a simple oscillating superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' For example, region (4) [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=', µi > 0, −µf ≫ 1] roughly corresponds to the polarized state in the strongly detuned regime, since the initial ground state has significant overlap with |0⟩ for µi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In the µi → ∞ limit, it is expected to become the exact mirror image of region (3), given that the polarized state has the same dynamics for ±µf (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Simi- 6 (7) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5 4 (2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 2 (3) X (1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5 0 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 (6) 4 (4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5 (5) 9- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 i13 larly, region (7) [µi < 0, µf ≫ 1] has a simple explanation in terms of |Z+⟩ state in the strongly detuned regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The origin of revivals in region (5) [µf < µi < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='3] is perhaps not immediately obvious, since the initial state in that case does not have high overlap with one of the previously studied states such as |0⟩ or |Z+⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We now briefly investigate this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The fidelity and the aver- age number of excitations after quenching from µi = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5 to µf = −6 can be seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 10(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The quenched state maintains high overlap with the |Z+⟩ state, with peaks in the middle between the fidelity re- vivals, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 10(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This situation is reminiscent of the |¯0⟩ state in region (2), which periodically evolves to |0⟩ and back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Although it oscillates, the overlap with |Z+⟩ never drops to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In contrast, the overlap with |0⟩ is constantly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 10(b) we also see that the av- erage occupation is remarkably stable, fluctuating only slightly around ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' As explained above for regions (4) and (7), such behavior arises due to the fact that in the large-µ limit the Hilbert space becomes fragmented and the initial state has support on a small number of eigenstates that are disconnected from the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' This can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 10(c), which shows the overlap of the ini- tial state and the eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The fragmentation and high overlap with the ground state are apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Fur- ther evidence comes from the inverse participation ratio (IPR), which we find to be very low in this region, indi- cating overlap with only a small number of eigenstates, as will be shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Finally, region (6) [µi < µf < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='3] has a similar phenomenology to its mirroring region (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Dynamics and eigenstate properties of the PXP model quenched from µi = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5 to µf = −6, corresponding to region (5) of the phase diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (a) Fidelity of the initial state |ψ(0)⟩, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=', the ground state of HPXP(−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5), as well as the overlap with both the polarized state |0⟩, and superposition state ��Z+� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (b) The average number of excita- tions remains nearly constant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (c) The overlap of the initial state with eigenstates of HPXP(−6) reveals fragmenta- tion and large projection on the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Data obtained by exact diagonalization for N = 28 with PBCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' In summary, we have argued that regions (4), (7) and part of (5) correspond to regimes where µf has a large absolute value, leading to a simple oscillatory dynamics due to Hilbert space fragmentation, while in regions (5) and (6), µf ≈ µi causes the initial state to be close to an eigenstate of the post-quench Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Appendix B: Derivation of TDVP equations of motion and quantum leakage In this section we first derive the TDVP equations of motion and then compute the instantaneous leakage rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' These derivations follow Appendices A and C of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Equations of motion The TDVP equations of motion can be derived as the saddle point equations for the following Lagrangian [62, 64]: L = i 2 � ⟨ψMPS| ˙ψMPS⟩ − ⟨ ˙ψMPS|ψMPS⟩ � − ⟨ψMPS |H| ψMPS⟩, (B1) where it will be convenient to split our Hamiltonian into two terms, H = HPXP+Hµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Unlike Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' 60, we restrict to K = 1 which greatly simplifies the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' Through- out this section we will consider mixed MPS transfer ma- trices, denoted by T B C = � σ ¯Bσ ⊗ Cσ, (B2) where B and C are arbitrary MPS tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' The MPS transfer matrix for the PXP ansatz chosen in the main text takes the form T A A = T = � � � � cos2 θ 0 0 1 cos θ sin θ 0 0 0 cos θ sin θ 0 0 0 sin2 θ 0 0 0 � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (B3) The dominant left and right eigenvalues of the transfer matrix are equal to 1, and the corresponding eigenvectors are |R) = � � � � 1 cos θ sin θ cos θ sin θ sin2 θ � � � � , (L| = � 1 0 0 1 � , (B4) which obey (L|R) = 1 + sin2 θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' We also introduce the following shorthand for a 3-site local Hamiltonian term contracted with MPS tensors on every site: H = HA,A,A A,A,A = � σi ¯Aσ1 ¯Aσ2 ¯Aσ3hσ1,σ2,σ3 σ4,σ5,σ6Aσ4Aσ5Aσ6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content=' (B5) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='0 a <(t) /b(0)> <2b(t)|Z2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E2T4oBgHgl3EQfEAaS/content/2301.03631v1.pdf'} +page_content='5 1000 K) have water abundances close to 10−4 been +⋆ We dedicate this manuscript to Tom Phillips, who was an inspira- +tion to all of us. Tables with the data used for Figures 2 and B.3 are avail- +able at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) +or via http://cdsarc.u-strasbg.fr/viz-bin/cat/J/A+A/XXX/XX +derived. For O2, low gas-phase abundances have been suggested +by early Submillimeter Wave Astronomy Satellite (SWAS) and +Odin observations (Goldsmith et al. 2000; Larsson et al. 2007). +Herschel pushed the limits even lower, with O2 detections re- +ported only in the Orion shock (Goldsmith et al. 2011; Chen +et al. 2014) and in the ρ Ophiuchi cloud (Liseau et al. 2012; +Larsson & Liseau 2017), where an abundance of 5 × 10−8 with +respect to H2 has been derived. In other sources, limits as low as +6 × 10−9 (3σ) were reported (NGC 1333 IRAS4A; Yıldız et al. +2013). +A combined analysis of water vapor, water ice, and O2 limits +in cold clouds thus indicates that a large fraction of the oxy- +gen is unaccounted for. A number of possible explanations have +been hypothesized (van Dishoeck et al. 2021). Within the sim- +ple water chemistry models, the only solution is to have a short +pre-stellar stage of only 0.1 Myr to prevent all oxygen from be- +ing turned into water. An alternative is for dense cores to have +a small fraction of large grains (> 1 µm), which prevents more +than 50% of the water ice from being observed through infrared +absorption spectroscopy. However, this solution does not apply +to hot cores and shocks, where the large icy grains should have +sublimated and where a large fraction of oxygen is also missing. +Another option is, therefore, that oxygen is in some refractory +form called unidentified depleted oxygen (UDO), which con- +sists of material that does not vaporize or atomize even in strong +shocks (up to 1000 K). +A schematic overview of the oxygen budget in diffuse and +translucent clouds is shown in Fig. 3 of Whittet (2010). In dif- +fuse clouds, most of the oxygen is in atomic form, with silicates +and oxides contributing up to about 100 pm (out of a total of 575 +Article number, page 1 of 7 +arXiv:2301.03651v1 [astro-ph.GA] 9 Jan 2023 + +A&A proofs: manuscript no. ms_final +ppm relative to H nuclei). The UDO wedge starts at hydrogen +nucleus densities of about 0.1 cm−3 and increases to about 150 +ppm (∼ 25% of the total oxygen) at densities of about 7 cm−3, +corresponding to the effective observational limit on depletion +studies in the UV. In higher-density clouds (nH ∼ 1000 cm−3), +gas-phase CO is expected to contribute at about 50 ppm, with +ices, and silicates and oxides contributing at about 100 ppm each. +The atomic oxygen contribution is predicted to be small, leaving +about 300 ppm in the form of UDO (∼ 50% of the total oxygen). +This denser gas is not accessible in the UV due to extinction, but +can be studied using far-infrared spectroscopy. +Atomic oxygen is an important part of the oxygen budget, +and a number of studies aimed at constraining its abundance +have been carried out. The [O i] 63 µm fine-structure line was ob- +served in the diffuse ISM using the Infrared Space Observatory +(ISO) Long Wavelength Spectrometer (LWS) by means of ab- +sorption spectroscopy against bright background dust continuum +sources. Such studies were limited by the relatively low spectral +resolution of the LWS instrument (10 km s−1 in the Fabry-Perot +mode with the maximum entropy deconvolution). However, in +some sources, the foreground absorption is well separated in ve- +locity space from the background source, allowing a determina- +tion of the oxygen abundance in the foreground clouds (Vastel et +al. 2000; Lis et al. 2001; Vastel et al. 2002). +The [O i] 63 µm fine-structure line emission has been widely +used as a tracer of star formation both in Galactic sources (Liseau +et al. 1999; Oberst et al. 2001; Karska et al. 2014) and in exter- +nal galaxies (Malhotra et al. 2001; Dale et al. 2004; González- +Alfonso et al. 2012; Farrah et al. 2013). Comparison of the +63 µm and 145 µm line intensities (Stacey et al. +1983) and +63 µm studies with higher spectral resolution (Kraemer et al. +1998; Boreiko & Betz 1996; Leurini et al. 2015; Schneider et +al. 2018; Mookerjea et al. 2019) suggest that the lower-lying +63 µm line observed in emission is optically thick. Goldsmith +et al. (2021) find that approximately half of the 12 sources ob- +served with the German REceiver for Astronomy at Terahertz +Frequencies (GREAT) instrument on the Stratospheric Obser- +vatory for Infrared Astronomy (SOFIA) showed clear evidence +of self-absorption profiles, indicating the presence of large col- +umn densities of low-excitation atomic oxygen with N(O0 +le) = +2 − 7 × 1018 cm−2. Much of this is in regions that would typi- +cally be assumed to be totally molecular, but which in fact have +X(Oo) ≃ 10−5. The low-excitation foreground gas can be studied +by means of absorption spectroscopy toward bright background +dust continuum sources. +Sgr B2 is one of the brightest far-infrared continuum sources +in the Galaxy, and thus an excellent target for absorption studies. +The differential rotation of the Milky Way allows spectral fea- +tures from gas clouds at different galactocentric radii to be sepa- +rated in velocity (Greaves & Williams 1994; Fig. 4). Even at the +limited spectral resolution of the ISO LWS, the Sgr B2(M) [O i] +spectrum could be decomposed into three foreground velocity +components (Fig. 3 of Lis et al. 2001), and the atomic oxygen +column density was shown to be correlated with the CO column +density, as expected if the two species are well mixed spatially. +An average atomic oxygen abundance of 2.7 × 10−4 with respect +to hydrogen nuclei was derived in the molecular phase (Lis et al. +2001). +The ISO study was limited by the spectral resolution of the +LWS instrument, which resulted in blending of multiple ve- +locity components. The GREAT instrument on SOFIA offers +tremendous improvements in sensitivity and spectral resolution +at 63 µm over ISO LWS (see, e.g., Wiesemeyer et al. 2016; +Goldsmith et al. 2021 for velocity-resolved [O i] observations of +other sources). In the present paper, we revisit the atomic oxygen +abundance in the foreground clouds on the sightline toward Sgr +B2 using archival SOFIA observations. The high spectral res- +olution of GREAT allows, for the first time, the [O i] emission +from individual line-of-sight clouds to be separated, velocity in- +tervals affected by saturated absorption to be correctly masked, +and accurate atomic oxygen column densities and abundances to +be derived. +2. Observations +We used publicly available SOFIA/GREAT (Heyminck et al. +2012) observations of the 63 µm fine-structure [O i] line from the +NASA Infrared Processing and Analysis Center (IPAC) SOFIA +Science Archive1. The data downloaded from the archive were +re-reduced using the latest version of the GREAT pipeline (see +Appendix A). The [O i] spectrum is centered at the position of +Sgr B2M, 17h47m20.16s; −28d23′04.5′′ (J2000). +Figure 1 (upper panel) shows the final [O i] 63 µm spectrum +divided by the continuum, resampled to 1 km s−1 spectral res- +olution. Local standard of rest (LSR) velocities between −120 +and +40 km s−1 correspond to the foreground gas, while those +greater than 40 km s−1 correspond to the envelope of the Sgr B2 +cloud. The higher gas densities present in this component make +the excitation and the resulting column densities uncertain. Con- +sequently, we excluded these velocities from the analysis. Ve- +locities between −6 and 0 km s−1, where the [O i] spectrum is +contaminated by telluric absorption, were also excluded. To es- +timate the noise level in the [O i] absorption region, we split the +data into two independent subsets with comparable integration +times. The lower histogram shows the difference spectrum be- +tween the two subsets divided by 2, which is a measure of the +uncertainty in the final [O i] spectrum. The difference spectrum +is flat over most velocities. The rms computed in the −130 to 40 +km s−1 velocity range is 0.0193, and we used this value as the +uncertainty of the [O i] line-to-continuum ratio in the analysis. +The rms increases toward the edges, where the two local oscil- +lator (LO) settings used in the observations do not fully overlap, +resulting in an effectively shorter integration time. In addition, a +higher rms is seen in the difference spectrum at positive veloc- +ities, where the foreground absorption may be contaminated by +wings of the [O i] emission from the Sgr B2 envelope. The full +width at half maximum (FWHM) SOFIA beam size at the [O i] +frequency is ∼ 6.6′′. +We used archival Herschel Heterodyne Instrument for the +Far-Infrared (HIFI) (de Graauw et al. 2010) observations of hy- +drogen fluoride (HF) to determine H2 column densities in the +various velocity components on the line of sight toward Sgr B2. +Two independent data sets were used in the analysis, which al- +lows an accurate quantification of the instrumental uncertain- +ties. Both data sets, reduced using the latest HIFI instrument +pipeline, were downloaded from the European Space Agency +Herschel Science Archive2 and imported into the Institut de +radioastronomie millimétrique (IRAM) Gildas3 software pack- +age for subsequent analysis. The first observation is a Band 5A +spectral scan of Sgr B2(M), centered at the same position as +the O i spectrum (OBSID 1342204739). The second observation +is a 4′ long north-south strip taken in the double-beam-switch +(DBS) observing mode (OBSID 1342205881). The DBS refer- +ence beams lie approximately 3′ to the east and west (i.e., per- +1 https://irsa.ipac.caltech.edu/Missions/sofia.html; AOR ID 03_0088. +2 https://archives.esac.esa.int/hsa/whsa/ +3 https://www.iram.fr/IRAMFR/GILDAS/ +Article number, page 2 of 7 + +Lis et al.: Atomic oxygen +Fig. 1. Spectra of [O i] and HF absorption toward Sgr B2(M) divided +by the corresponding continua. (Upper) SOFIA/GREAT [O i] spectrum. +The lower histogram shows the difference between two independent +data subsets divided by 2, which is a measure of the uncertainty in the +[O i] spectrum. Gray areas show velocities excluded from the analysis. +This includes the Sgr B2 envelope at velocities greater than 40 km s−1 +and the region between -6 and 0 km s−1, where the [O i] spectrum is con- +taminated by telluric absorption. (Lower) Average Herschel/HIFI spec- +trum of HF. The lower histogram shows the difference between the two +independent observations divided by 2, which is a measure of the un- +certainty in the average spectrum. +pendicular to the roughly north–south elongation of Sgr B2). +Two spectra in the strip closest to Sgr B2(M) (< 10′′ offsets) +were averaged with uniform weighting to produce the final spec- +trum used in the analysis. We used spectra taken with the HIFI +wide band spectrometer, which provided a spectral resolution +of 1.1 MHz over a 4 GHz intermediate frequency (IF) band- +width. The FWHM HIFI beam size at the HF frequency is ∼ +18′′, about three times larger than the SOFIA beam size at the +[O i] frequency. However, the foreground clouds are expected to +be extended on such angular scales and to fully cover the back- +ground continuum source. This conclusion is supported by the +good agreement between the two independent HF spectra taken +at positions offset by about half of the HIFI beam. +Figure 1 (lower panel) shows the final HF spectrum, an +equally weighted average of the two instrumental polarizations +and the two independent observations, resampled to a 1 km s−1 +velocity resolution. The lower histogram shows the difference +between the spectral scan and DBS observations divided by 2, +which is a measure of the uncertainty in the final HF spectrum. +The difference spectrum is very flat and shows no residuals, even +at positive velocities, where the background absorption may po- +tentially be contaminated by the Sgr B2 envelope emission (see +the HF emission wing at velocities greater than 85 km s−1). The +rms computed in the −130 to 40 km s−1 velocity range is 0.0145, +and we used this value as the uncertainty of the HF line-to- +continuum ratio in the analysis. +3. Results +To derive the oxygen optical depths and the corresponding col- +umn densities in the individual channels, we followed estab- +lished procedures commonly used in the analysis of HIFI ob- +servations of light hydrides (e.g., Neufeld et al. 2010; Lis et al. +2010; Monje et al. 2011). We first derived the optical depth +of the [O i] and HF lines (τ = −ln[1 − TL/TC], where TL/TC is +the line-to-continuum ratio), assuming that the foreground ab- +sorption completely covers the continuum source and that all +oxygen atoms are in the ground state. The spiral arm clouds on +the line of sight toward Sgr B2 have moderate densities, up to a +few times 104 cm−3 (Greaves & Williams 1994). This is lower +than the critical density for the excitation of the 63 µm O i line +(5.0 × 105 cm−3 for collisions with H2 and 7.8 × 105 cm−3 for +collisions with H; Lique et al. 2018; Goldsmith 2019); the as- +sumption that the entire population is in the ground state is thus +well justified. Figure B.1 shows the [O i] optical depths and the +corresponding column density as a function of velocity. +To determine the oxygen abundance, H2 and H column den- +sities in the line-of-sight clouds are required. Because of its +unique thermochemistry, HF has been shown to be an excellent +tracer of H2 (e.g., Phillips et al. 2010; Neufeld et al. 2010; +Sonnentrucker et al. 2010; Monje et al. 2011). The HF abun- +dance with respect to H2 in diffuse or translucent clouds, deter- +mined from a comparison with CH observations, is in the range +(1.1 − 1.6) × 10−8, with an average of (1.4 ± 0.17) × 10−8 (mul- +tiple velocity components toward W51, W49N, and NGC6334I; +Sonnentrucker et al. 2010; Emprechtinger et al. 2012). We used +this value to convert the HF column density to the H2 column +density (Fig. B.2). To characterize the atomic gas component on +the line of sight toward Sgr B2, we used the H i column densities +of Winkel et al. (2017). Figure B.3 shows the total hydrogen +nucleus column density as a function of velocity, along with the +molecular and atomic contributions. +Figure 2 shows the gas-phase atomic oxygen abundance as +a function of velocity, derived from the observations of the +63 µm line. The average abundance with respect to hydrogen +nuclei, computed over the −120 to +40 km s−1 velocity range, +is (2.51 ± 0.69) × 10−4. The ±1σ dispersion of the individual +measurements computed from the ensemble of 120 independent +velocity channels is shown. Black error bars are the formal 1σ +uncertainties of the individual measurements, computed by com- +bining in quadrature corresponding uncertainties in the column +densities of the atomic oxygen, atomic, and molecular hydro- +gen. They are typically smaller than the ensemble dispersion, +suggesting the presence of variations in the local atomic oxygen +abundance among different velocity components. +4. Discussion +The average atomic oxygen abundance toward Sgr B2 derived +here, (2.51 ± 0.69) × 10−4 with respect to hydrogen nuclei, is +in excellent agreement with the ISO value of 2.7 × 10−4 (Lis +et al. 2001), which was based on 13CO column density esti- +mates for the molecular gas component. Figure 3 shows a nor- +malized histogram of the O0 abundances in the 120 individual +velocity channels. The histogram is non-Gaussian and shows a +Article number, page 3 of 7 + +A&A proofs: manuscript no. ms_final +Fig. 2. Atomic oxygen abundance relative to hydrogen nuclei as a func- +tion of velocity. The mean value of the 120 individual channels within +the −120 and 40 km s−1 velocity range is 2.51×10−4, and the dispersion +of the individual channels is 0.65 × 10−4. The horizontal dotted lines +mark the mean value and ±1σ dispersion computed from the ensemble +of individual measurements. The black vertical error bars mark ±1σ un- +certainties in the individual channels, as described in the text. Color bars +mark velocity ranges corresponding to the 3 kpc, 5 kpc, Sagittarius, and +Scutum arms (red, magenta, blue, and yellow, respectively). Velocities +corresponding to the Galactic center gas are marked in green and those +of the local gas in gray. Channels with saturated absorption are masked. +An electronic table with the data used for this figure is available at the +CDS. +narrow peak around 2.25 × 10−4 and a broader shoulder around +3.15 × 10−4, comparable to the values of 3.1 − 3.5 × 10−4 derived +by Wiesemeyer et al. (2016) toward W31C, G34.26, and W49N. +The origin of the variations in the derived atomic oxygen abun- +dance among different velocity components can be investigated +further by using independent observations of additional molec- +ular tracers, including the oxygen ions, CH as a proxy for H2 +(Gerin et al. 2010), argonium as a proxy for purely atomic gas +(Schilke et al. 2014), and ammonia as a tracer of high-density +gas; such analysis is beyond the scope of the present paper. +As a reference, the cosmic standard abundance of oxygen is +(5.75 ± 0.4) × 10−4, as measured in a representative sample of +unevolved early B-type stars in nearby OB associations (Przy- +billa et al. 2008). The latest solar photospheric abundance is +4.57 × 10−4 (Asplund et al. 2005), significantly lower than the +earlier Grevesse & Sauval (1998) value of 6.76 × 10−4. Car- +tledge et al. (2004) presented a comprehensive analysis of high- +resolution Hubble Space Telescope observations of O i and H i +Ly α UV absorption along 36 sight lines that probe a variety of +Galactic disk environments. They derive an average O/H ratio of +3.90 × 10−4 in the low-density warm gas that should be least af- +fected by depletion. Sight lines of higher mean density are char- +acterized by a lower average O/H ratio of 2.84×10−4. Taking the +higher value as a reference for the atomic oxygen abundance in +the ISM gas, our average abundance of 2.51×10−4 on the line of +sight toward Sgr B2 corresponds to about 35% gas-phase oxygen +depletion. CO will also contribute to the oxygen budget4 with a +typical abundance of 1 × 10−4 with respect to H2 in the molecu- +lar gas, which is dominant at most velocities on this line of sight +4 Other oxygen-bearing gas-phase species contribute at a much lower +level, e.g., the H2O fractional abundance with respect to H2 is in the +range 3 − 7 × 10−7 (Neufeld et al. 2000; Lis et al. 2010). +Fig. 3. Normalized probability density function (PDF) of the O0 abun- +dances in 120 individual velocity channels toward Sgr B2. The vertical +dashed black line shows the mean abundance, the gray shaded area ±1σ +departures from the mean, and the black arrow the corresponding ap- +proximate gas-phase oxygen content, including atomic oxygen and CO. +The vertical red line shows the cosmic standard abundance (Przybilla +et al. 2008) and the magenta line the latest solar abundance (Asplund +et al. 2005), with the magenta arrow pointing toward the earlier value +of Grevesse & Sauval (1998). The blue line is the average UV-derived +ISM abundance in the low-density warm gas that is least affected by de- +pletion, with the blue arrow showing the corresponding value for higher +mean density site lines (Cartledge et al. 2004). +(Fig. B.3). Adding the two contributions, we derive an estimate +of the total gas-phase O0 + CO oxygen content toward Sgr B2 +of ∼ 3 × 10−4 with respect to hydrogen nuclei, about 25% lower +than the O/H value derived in the low-density warm gas from the +UV measurements. +Figure 4 shows the good correlation between the atomic oxy- +gen and total hydrogen nucleus column densities (Pearson’s cor- +relation coefficient 0.85). The average abundance is 2.51 × 10−4. +Error bars mark the formal 1σ uncertainties of the individual +channels. We note that points with the highest O0 column den- +sities are located on average above the best-fit line. However, +owing to the nonlinear dependence of the opacity on the line- +to-continuum ratio at such high column densities, these points +have large error bars and the result may not be significant. If we +exclude points with the atomic oxygen column densities above +3 × 1017 cm−2, the resulting average atomic oxygen abundance +is lower by only 2%. +5. Conclusions +We have presented an analysis of archival SOFIA/GREAT ob- +servations of the [O i] 63 µm absorption toward the Sagittar- +ius B2(M) continuum source in the Galactic center. The high +spectral resolution of the GREAT instrument allows, for the first +time, the [O i] absorption from individual line-of-sight clouds to +be separated, velocity intervals affected by saturated absorption +and telluric absorption to be masked, and accurate atomic oxy- +gen column densities and abundances to be derived. The atomic +oxygen column density in the foreground spiral arm clouds to- +ward Sgr B2 is well correlated with the total hydrogen column +density, as determined from HF and H i observations, with an +average abundance of (2.51 ± 0.69) × 10−4 with respect to H +nuclei in individual 1 km s−1 velocity channels. This value is in +good agreement with the earlier ISO measurements on the same +Article number, page 4 of 7 + +Lis et al.: Atomic oxygen +Fig. 4. Atomic oxygen column density as a function of total hydrogen +nucleus column density. Error bars are 1σ. The dotted line corresponds +to a fractional abundance of 2.51 × 10−4. +line of sight, and about 35% lower than the average O/H ratio of +3.90 × 10−4 in low-density warm gas derived from UV measure- +ments (Cartledge et al. 2004). If we add a typical gas-phase CO +content at 1×10−4 with respect to H2 in the molecular gas, which +is dominant at most velocities on this line of sight (Fig. B.3), the +total gas-phase oxygen content (O0 + CO) on the line of sight +toward Sgr B2 is ∼ 3 × 10−4, or 300 ppm with respect to H, +about 25% lower than the low-density warm ISM oxygen abun- +dance derived from UV measurements (Cartledge et al. 2004). +With silicates and oxides contributing another 100 ppm (Whittet +2010), the remaining oxygen fraction is about 175 ppm, which +can be viewed as an upper limit for UDO on the line of sight +toward Sagittarius B2. However, the expected ice contribution in +this density regime is about 125 ppm (Whittet 2010), leaving +little room for UDO. +Acknowledgements. Based on observations made with the NASA/DLR Strato- +spheric Observatory for Infrared Astronomy (SOFIA). SOFIA is jointly oper- +ated by the Universities Space Research Association, Inc. (USRA), under NASA +contract NAS2-97001, and the Deutsches SOFIA Institut (DSI) under DLR con- +tract 50 OK 0901 to the University of Stuttgart. GREAT is a development by +the MPI für Radioastronomie and the KOSMA/Universität zu Köln, in cooper- +ation with the DLR Institut für Optische Sensorsysteme, financed by the par- +ticipating institutes, by the German Aerospace Center (DLR) under grants 50 +OK 1102, 1103 and 1104, and within the Collaborative Research Centre 956, +funded by the Deutsche Forschungsgemeinschaft (DFG). Part of this research +was carried out at the Jet Propulsion Laboratory, California Institute of Technol- +ogy, under a contract with the National Aeronautics and Space Administration +(80NM0018D0004). We thank B. Winkel for providing us with the H i column +densities used in the analysis and an anonymous referee for helpful comments +regarding the overall oxygen budget. D. C. 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M., et al. 2017, A&A, 600, A2 +Yıldız, U. A., Acharyya, K., Goldsmith, P. F., et al. 2013, A&A, 558, A58 +Article number, page 5 of 7 + +A&A proofs: manuscript no. ms_final +Appendix A: SOFIA data reduction +The data were collected on 2015 July 19, on the southern hemi- +sphere deployment of SOFIA’s Cycle 1, at 11.3 to 11.5 km alti- +tude, under a precipitable water vapor column of typically 6 µm +at zenith. The high-frequency channel of GREAT was tuned to +the [O i] line, alternating between the lower and upper sideband, +so as to synthesize a sightline velocity interval from −200 to ++135 km s−1. In the velocity interval considered here, the median +single-sideband system temperatures at zenith were 2100 and +2400 K for lower-sideband and upper-sideband tuning, respec- +tively. Atmospheric and Galactic backgrounds were removed by +chopping to a reference position at 160′′ on both sides of Sgr +B2(M), at a position angle of 30◦ (east to south), that is to say, +perpendicular to the object’s elongation. The FWHM beam size +of 6.6′′ was measured from cross-scans on Mars. +The spectra were calibrated to forward-beam brightness tem- +peratures (with a 97% forward efficiency) against loads at am- +bient and cold temperatures, and then to main-beam brightness +temperatures using a 67% main beam efficiency. The 63 µm [OI] +transition is located in the wing of a broad water vapor ab- +sorption feature; the applied transmission correction was derived +from modeling the measured atmospheric total power emission +received in the signal and image bands. In order to minimize the +impact of mixer gain drifts, only the off-target spectra immedi- +ately following a calibration load measurement were used (i.e., +the correction was determined scan-wise and then scaled to the +current elevation of each recorded spectrum). +The continuum emission of Sgr B2(M) was obtained from +a dedicated double-sideband calibration. While the signal-to- +image band gain ratio may deviate from unity (a standard de- +viation of 5%), the overall reliability of the calibration scheme +can be monitored with two tests: First, the saturation of the +[OI] line at the systemic velocity defines the zero level for the +single-sideband calibration. Second, the mesospheric [OI] line +serves as a “beacon” that undergoes the same attenuation in the +stratosphere as the astronomical signal. With these precautions, +the calibrations and, consequently, the continuum levels of the +lower- and upper-sideband tunings were brought into agreement. +The spectra, s, of Sgr B2(M) in the lower- and upper-sideband +tuning are thus identical within the radiometric noise. The dif- +ference between the actually measured spectra (yL and yU for +lower- and upper-sideband tuning, respectively) displays a well- +defined linear baseline below 15 km s−1. Instabilities that arise at +velocities above this baseline are from the upper-sideband tun- +ing and can be ignored there thanks to the redundancy with the +lower-sideband tuning. The spectra from the two tunings can +then be expressed as yL = s + aLν, yU = s + aUν , where ν is +the frequency in the rest frame of Sgr B2(M). Thanks to the lin- +ear baseline fit to the difference spectrum, only two parameters +(offset and slope) remained to be optimized, which was done in +a way that ensures equal continuum levels on both sides of the +line-free portion of the spectra and reproduces the saturated ab- +sorption at systemic velocity. The good agreement between the +two tunings in the overlapping velocity interval, from −110 to ++15 km s−1, is taken as an assessment of the data processing al- +gorithm. +Appendix B: Atomic oxygen and HF column +densities +We converted the [O i] optical depth to the atomic oxygen col- +umn density assuming that the absorbing gas covers the back- +Fig. B.1. [O i] optical depth (left axis) and atomic oxygen column den- +sity (right axis) in 1 km s−1 channels as a function of velocity. Error bars +are 1σ. +Fig. B.2. HF optical depth (left axis) and the H nucleus column den- +sity in the molecular component (right axis) in 1 km s−1 channels as a +function of velocity. Error bars are 1σ. +ground continuum source and the entire population is in the +lower state (e.g., Neufeld et al. 2010): +� +τdv = Aulguλ3 +8πgl +N(O0) = 5.365 × 10−18N(O0) cm2 km s−1, +(B.1) +where Aul = 8.91−5 s−1 is the spontaneous radiative decay rate, +gu = 3 and gl = 5 are the degeneracies of the upper and lower +levels, and λ = 63.184 µm is the transition wavelength. Fig- +ure B.1 shows the [O i] optical depth (left vertical scale) and the +resulting atomic oxygen column density (right vertical scale) in +1 km s−1 velocity channels as a function of LSR velocity. +The corresponding formula for HF is +� +τdv = Aulguλ3 +8πgl +N(HF) = 4.157 × 10−13N(HF) cm2 km s−1, +(B.2) +with Aul = 2.42−2 s−1, gu = 3, gl = 1, and λ = 243.2444 µm. +Figure B.2 shows the HF optical depth and the resulting column +Article number, page 6 of 7 + +Lis et al.: Atomic oxygen +Fig. B.3. Total hydrogen nucleus column density toward Sgr B2(M) as +a function of velocity (black squares). The molecular, 2 × N(H2), and +atomic, N(H i), components are shown in cyan and green, respectively. +Error bars are ±1σ. An electronic table with the data used for this figure +is available at the CDS. +density in 1 km s−1 velocity channels as a function of LSR veloc- +ity. Figure B.3 shows the total hydrogen nucleus column density +as a function of velocity, along with the molecular and atomic +contributions, as described in the text. +Article number, page 7 of 7 + diff --git a/c9E2T4oBgHgl3EQfawcz/content/tmp_files/2301.03877v1.pdf.txt b/c9E2T4oBgHgl3EQfawcz/content/tmp_files/2301.03877v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c53508aab7d5efed5544f249890b6a7d4ea26fb0 --- /dev/null +++ b/c9E2T4oBgHgl3EQfawcz/content/tmp_files/2301.03877v1.pdf.txt @@ -0,0 +1,870 @@ +arXiv:2301.03877v1 [math.FA] 10 Jan 2023 +NUMERICAL RADIUS INEQUALITIES OF BOUNDED LINEAR +OPERATORS AND (α, β)-NORMAL OPERATORS +PINTU BHUNIA +Abstract. We obtain various upper bounds for the numerical radius w(T ) +of a bounded linear operator T defined on a complex Hilbert space H, by +developing the upper bounds for the α-norm of T , which is defined as ∥T ∥α = +sup +�� +α|⟨T x, x⟩|2 + (1 − α)∥T x∥2 : x ∈ H, ∥x∥ = 1 +� +for 0 ≤ α ≤ 1. Further, +we prove that +w(T ) +≤ +�� +min +α∈[0,1] ∥α|T | + (1 − α)|T ∗|∥ +� +∥T ∥ +≤ +∥T ∥. +For 0 ≤ α ≤ 1 ≤ β, the operator T is called (α, β)-normal if α2T ∗T ≤ T T ∗ ≤ +β2T ∗T holds. Note that every invertible operator is an (α, β)-normal operator +for suitable values of α and β. Among other lower bound for the numerical +radius of an (α, β)-normal operator T , we show that +w(T ) +≥ +� +max +� +1 + α2, 1 + 1 +β2 +� ∥T ∥2 +4 ++ |∥ℜ(T )∥2 − ∥ℑ(T )∥2| +2 +≥ +max +�� +1 + α2, +� +1 + 1 +β2 +� ∥T ∥ +2 +> +∥T ∥ +2 , +where ℜ(T ) and ℑ(T ) are the real part and imaginary part of T , respectively. +1. Introduction +Let B(H) denote the C∗-algebra of all bounded linear operators on a complex +Hilbert space H, with inner product ⟨·, ·⟩. Let T ∈ B(H). We write |T| = (T ∗T)1/2 +and |T ∗| = (TT ∗)1/2, where T ∗ denotes the adjoint of T. The numerical radius of +T, denoted by w(T), is defined as +w(T) = sup{|⟨Tx, x⟩| : x ∈ H, ∥x∥ = 1}. +2020 Mathematics Subject Classification. 47A12, 47A30, 15A60. +Key words and phrases. Numerical radius, operator norm, (α, β)-normal operator, Bounded +linear operator. +The author would like to sincerely acknowledge Prof. Kallol Paul for his valuable comments +on this paper. +The author also would like to thank SERB, Govt. of India for the financial support in the +form of National Post Doctoral Fellowship (N-PDF, File No. PDF/2022/000325) under the +mentorship of Prof. Apoorva Khare. +1 + +2 +P. BHUNIA +Note that, the numerical radius defines a norm on B(H), and it is equivalent to +the operator norm, satisfies +1 +2∥T∥ ≤ w(T) ≤ ∥T∥, for every T ∈ B(H). +(1.1) +Various refinements of the inequalities in (1.1) and related results have been +discussed in a recent published book [2]. Also, the reader can see the papers +[1, 6, 16] and references therein. For 0 ≤ α ≤ 1, the α-norm of T (see [4, 5, 20]) +is given by +∥T∥α = sup +�� +α|⟨Tx, x⟩|2 + (1 − α)∥Tx∥2 : x ∈ H, ∥x∥ = 1 +� +. +This α-norm is also a norm on B(H), and it satisfies +w(T) ≤ ∥T∥α ≤ ∥T∥, for every T ∈ B(H) and for all α ∈ [0, 1]. +(1.2) +The inequality (1.2) together with (1.1) implies that the α-norm is equivalent +to both the operator norm and the numerical radius norm. For more details of +the α-norm, the reader can see [20]. Next, we turn our attention to study the +(α, β)-normal operators (see [11]), where 0 ≤ α ≤ 1 ≤ β. An operator T ∈ B(H) +is said to be (α, β)-normal operator if T satisfies α2T ∗T ≤ TT ∗ ≤ β2T ∗T. This is +equivalent to α2⟨T ∗Tx, x⟩ ≤ ⟨TT ∗x, x⟩ ≤ β2⟨T ∗Tx, x⟩ for all x ∈ H, that is, +α∥Tx∥ ≤ ∥T ∗x∥ ≤ β∥Tx∥ for all x ∈ H. +Every normal and hyponormal operators are (α, β)-normal for suitable values of +α, β. For normal operators α = β = 1 and for hyponormal operators β = 1. +Also, there exist operators which are neither normal nor hyponormal. As for +example, the matrix +� +1 +0 +1 +1 +� +is an (α, β)-normal with α = +� +3− +√ +5 +2 +and β = +� +3+ +√ +5 +2 +. However, the matrix is neither normal nor hyponormal. If T is an (α, β)- +normal operator, then both, T mazorizes T ∗ and T ∗ mazorizes T. According to +Douglas (Mazorization lemma) [12], an operator T ∈ B(H) mazorizes an operator +S ∈ B(H) if one of the following equivalent statements holds: +(i) Ran(T) ⊆ Ran(S), where Ran(T) (Ran(S)) denotes the range space of T (S). +(ii) TT ∗ ≤ µ2SS∗ for some µ ≥ 0. +Therefore, T is an (α, β)-normal operator if and only if Ran(T) = Ran(T ∗). So, +every invertible operator is an (α, β)-normal operator for suitable values of α, β. +In this paper, we develop upper bounds for the α-norm of bounded linear op- +erators and consequentially we obtain bounds for the numerical radius, which +improve on the existing ones. Further, we develop lower bounds for the numeri- +cal radius of the (α, β)-normal operators, which improve on the classical bound +w(T) ≥ ∥T∥ +2 . +2. +The α-norm and numerical radius inequalities +We obtain new upper bounds for the numerical radius of bounded linear operators, +by developing the bounds of α-norm. First, we recall some necessary inequalities + +3 +from the literature. Kato [15] generalized the Cauchy-Schwarz inequality, namely, +|⟨Tx, y⟩|2 ≤ +� +|T|2α x, x +� � +|T ∗|2(1−α) y, y +� +∀x, y ∈ H, ∀α ∈ [0, 1]. +(2.1) +In particular, for α = 1/2 (see [13, pp. 75-76]), +|⟨Tx, y⟩| ≤ ⟨|T|x, x⟩1/2 ⟨|T ∗|y, y⟩1/2 ∀x, y ∈ H. +(2.2) +In [18, Theorem 1], Kittaneh generalized the Kato’s inequality, which is as follows: +If f, g : [0, ∞] → [0, ∞] are continuous functions satisfying f(t)g(t) = t, ∀t ≥ 0, +then +|⟨Tx, y⟩|2 ≤ +� +f 2(|T|)x, x +� � +g2(|T ∗|)y, y +� +∀x, y ∈ H. +(2.3) +Next, we recall the H¨older–McCarthy inequality (see [21, p. 20]) which states +that if T ∈ B(H) is positive, then +⟨Tx, x⟩r ≤ ⟨T rx, x⟩ +∀r ≥ 1 ∀x ∈ H, ∥x∥ = 1. +(2.4) +The inequality (2.4) is reversed when 0 ≤ r ≤ 1. The Cartesian decomposition +of T ∈ B(H) is given by T = ℜ(T) + iℑ(T), where ℜ(T) = +1 +2(T + T ∗) and +ℑ(T) = 1 +2i(T − T ∗). Now, we are in a position to prove our first result. +Theorem 2.1. If T ∈ B(H), then +∥T∥2 +α ≤ +���α +4 +� +f 4(|T|) + g4(|T ∗|) +� ++ (1 − α)|T|2��� + α +2 +��ℜ +� +f 2(|T|)g2(|T ∗|) +��� , +where f and g are as in (2.3). +Proof. Let x ∈ H with ∥x∥ = 1. Then, it follows from (2.3) that +|⟨Tx, x⟩|2 +≤ +� +f 2(|T|)x, x +� � +g2(|T ∗|)x, x +� += +�� +f 2(|T|)x, x +�1/2 � +g2(|T ∗|)x, x +�1/2�2 +≤ +1 +4 +�� +f 2(|T|)x, x +� ++ +� +g2(|T ∗|)x, x +��2 += +1 +4 +�� +f 2(|T|) + g2(|T ∗|) +� +x, x +�2 +≤ +1 +4 +�� +f 2(|T|) + g2(|T ∗|) +�2 x, x +� +(by using (2.4) ) += +1 +4 +�� +f 4(|T|) + g4(|T ∗|) +� +x, x +� ++ 1 +2 +� +ℜ +� +f 2(|T|)g2(|T ∗|) +� +x, x +� +. +Thus, we have +∥T∥2 +α += +sup +∥x∥=1 +� +α |⟨Tx, x⟩|2 + (1 − α)∥Tx∥2� +≤ +sup +∥x∥=1 +��α +4 +� +f 4(|T|) + g4(|T ∗|) +� ++ (1 − α)|T|2� +x, x +� ++α +2 sup +∥x∥=1 +��� +ℜ +� +f 2(|T|)g2(|T ∗|) +� +x, x +��� += +���α +4 +� +f 4(|T|) + g4(|T ∗|) +� ++ (1 − α)|T|2��� + α +2 +��ℜ +� +f 2(|T|)g2(|T ∗|) +��� , + +4 +P. BHUNIA +as required. +□ +Now, as a consequence of Theorem 2.1, we get the following corollary. +Corollary 2.2. If T ∈ B(H), then +∥T∥2 +α +≤ +���� +� +1 − 3α +4 +� +|T|2 + α +4 |T ∗|2 +���� + α +2 ∥ℜ(|T||T ∗|)∥ +(2.5) +and +∥T∥2 +α +≤ +���� +� +1 − 3α +4 +� +|T ∗|2 + α +4 |T|2 +���� + α +2 ∥ℜ(|T||T ∗|)∥ . +(2.6) +Further, +w(T) ≤ min{γ, δ}, +(2.7) +where +γ = min +α∈[0,1] +����� +� +1 − 3α +4 +� +|T|2 + α +4 |T ∗|2 +���� + α +2 ∥ℜ(|T||T ∗|)∥ +�1/2 +and +δ = min +α∈[0,1] +����� +� +1 − 3α +4 +� +|T ∗|2 + α +4 |T|2 +���� + α +2 ∥ℜ(|T||T ∗|)∥ +�1/2 +. +Proof. The inequality (2.5) follows from Theorem 2.1 by taking f(t) = g(t) = t1/2. +The inequality (2.6) follows from (2.5) by replacing T by T ∗ and using the equality +∥T∥α = ∥T ∗∥α (see [20, Proposition 2.6]). The inequality (2.7) follows from the +fact that w(T) ≤ minα∈[0,1] ∥T∥α. +□ +Remark 2.3. Recently, Bhunia and Paul obtained in [3, The inequality (2.3)] +and [8, Corollary 2.6] that w2(T) ≤ 1 +4 ∥|T|2 + |T ∗|2∥+ 1 +2 ∥ℜ(|T||T ∗|)∥ and w2(T) ≤ +1 +4 ∥|T|2 + |T ∗|2∥ + 1 +2w(|T||T ∗|), respectively. Clearly, we see that +min{γ2, δ2} +≤ +1 +4 +��|T|2 + |T ∗|2�� + 1 +2 ∥ℜ(|T||T ∗|)∥ +≤ +1 +4 +��|T|2 + |T ∗|2�� + 1 +2w(|T||T ∗|). +Thus, we would like to remark that the inequality (2.7) refines both the inequali- +ties in [3, The inequality (2.3)] and [8, Corollary 2.6]. To show proper refinement, +we consider a matrix T = + + +0 +1 +0 +0 +0 +2 +0 +0 +0 + +. Then, by elementary calculations we see +that γ = +� +28 +13 and δ = 3 +2, and so +min{γ2, δ2} = 28 +13 < 9 +4 = 1 +4 +��|T|2 + |T ∗|2�� + 1 +2 ∥ℜ(|T||T ∗|)∥ . +Another bound for the α-norm reads as follows: +Theorem 2.4. If T ∈ B(H), then +∥T∥2 +α +≤ +min +���α|T|2 + (1 − α)|T ∗|2�� , +��α|T ∗|2 + (1 − α)|T|2��� +. + +5 +Proof. Let x ∈ H with ∥x∥ = 1. By using Cauchy-Schwarz inequality, we get +∥T∥2 +α += +sup +∥x∥=1 +� +α |⟨Tx, x⟩|2 + (1 − α)∥Tx∥2� += +sup +∥x∥=1 +� +α |⟨x, T ∗x⟩|2 + (1 − α)∥Tx∥2� +≤ +sup +∥x∥=1 +� +α∥T ∗x∥2 + (1 − α)∥Tx∥2� += +sup +∥x∥=1 +� +α⟨|T ∗|2x, x⟩ + (1 − α)⟨|T|2x, x⟩ +� += +sup +∥x∥=1 +⟨(α|T ∗|2 + (1 − α)|T|2)x, x⟩ += +∥α|T ∗|2 + (1 − α)|T|2∥. +(2.8) +Replacing T by T ∗ in (2.8) and using ∥T∥α = ∥T ∗∥α, we get +∥T∥2 +α +≤ +∥α|T|2 + (1 − α)|T ∗|2∥. +(2.9) +Therefore, the required inequality follows by combining (2.8) and (2.9). +□ +Now, the following upper bound for the numerical radius follows easily from +Theorem 2.4 together with the first inequality in (1.2): +w2(T) +≤ +min +α∈[0,1] +��α|T|2 + (1 − α)|T ∗|2�� . +(2.10) +Note that, the bound (2.10) is also obtained in [7]. To obtain our next result we +need the Buzano’s inequality [10], namely, if x, y, e ∈ H with ∥e∥ = 1, then +2|⟨x, e⟩⟨e, y⟩| +≤ +∥x∥∥y∥ + |⟨x, y⟩|. +(2.11) +By employing the Buzano’s inequality (2.11), we prove the following theorem. +Theorem 2.5. If T ∈ B(H), then +∥T∥2 +α +≤ +α +4 w2(|T| + i|T ∗|) + α +4 w(|T||T ∗|) + +���� +� +1 − 7α +8 +� +|T|2 + α +8 |T ∗|2 +���� . +Proof. Let x ∈ H with ∥x∥ = 1. Then, from the inequality (2.2), we get +|⟨Tx, x⟩|2 +≤ +⟨|T|x, x⟩⟨|T ∗|x, x⟩ +≤ +1 +4 (⟨|T|x, x⟩ + ⟨|T ∗|x, x⟩)2 += +1 +4(⟨|T|x, x⟩2 + ⟨|T ∗|x, x⟩2) + 1 +2⟨|T|x, x⟩⟨|T ∗|x, x⟩ += +1 +4|⟨|T|x, x⟩ + i⟨|T ∗|x, x⟩|2 + 1 +2⟨|T ∗|x, x⟩⟨x, |T|x⟩ +≤ +1 +4|⟨(|T| + i|T ∗|)x, x⟩|2 + 1 +4∥|T ∗|x∥ ∥|T|x∥ + 1 +4|⟨|T ∗|x, |T|x⟩| (by (2.11)) +≤ +1 +4|⟨(|T| + i|T ∗|)x, x⟩|2 + 1 +8(∥|T ∗|x∥2 + ∥|T|x∥2) + 1 +4|⟨|T||T ∗|x, x⟩|. + +6 +P. BHUNIA +Hence, +α|⟨Tx, x⟩|2 + (1 − α)∥Tx∥2 +≤ +α +4 |⟨(|T| + i|T ∗|)x, x⟩|2 + α +4 |⟨|T||T ∗|x, x⟩| + +��� +1 − 7α +8 +� +|T|2 + α +8 |T ∗|2 +� +x, x +� +≤ +α +4 w2(|T| + i|T ∗|) + α +4 w(|T||T ∗|) + +���� +� +1 − 7α +8 +� +|T|2 + α +8 |T ∗|2 +���� . +Therefore, taking supremum over ∥x∥ = 1, we get the required inequality. +□ +Replacing T by T ∗ in Theorem 2.5, and then using ∥T∥α = ∥T ∗∥α, we also get +the following inequality: +Corollary 2.6. If T ∈ B(H), then +∥T∥2 +α +≤ +α +4 w2(|T| + i|T ∗|) + α +4 w(|T||T ∗|) + +���� +� +1 − 7α +8 +� +|T ∗|2 + α +8 |T|2 +���� . +As an application of the bounds for the α-norm, obtained in Theorem 2.5 and +Corollary 2.6, we have a new upper bound for the numerical radius, which is +given below. +Corollary 2.7. If T ∈ B(H), then +w2(T) +≤ +α +4 w2(|T| + i|T ∗|) + α +4 w(|T||T ∗|) ++ min +����� +� +1 − 7α +8 +� +|T ∗|2 + α +8 |T|2 +���� , +���� +� +1 − 7α +8 +� +|T|2 + α +8 |T ∗|2 +���� +� +, +for all α ∈ [0, 1]. +In particular, for α = 1 we have (see also in [14]), +w2(T) +≤ +1 +4w2(|T| + i|T ∗|) + 1 +4w(|T||T ∗|) + 1 +8 +��|T ∗|2 + |T|2�� +(2.12) +≤ +1 +2 ∥T ∗T + TT ∗∥ . +Therefore, Corollary 2.7 generalizes and improves on the inequality (2.12) as well +as the well-known inequality w2(T) ≤ 1 +2 ∥T ∗T + TT ∗∥, obtained by Kittaneh [16]. +Next, we note that every operator T ∈ B(H) satisfies the following numeri- +cal radius inequality (see [17]): w(T) ≤ 1 +2∥|T| + |T ∗|∥. In [7], it is proved that +w2(T) ≤ ∥α|T|2 + (1 − α)|T ∗|2∥ , ∀α ∈ [0, 1]. Now, it is natural to ask whether +the following inequality +w(T) ≤ ∥α|T| + (1 − α)|T ∗|∥ , ∀α ∈ [0, 1] +holds or not. In this connection we have the following result. +Theorem 2.8. If T ∈ B(H), then +w2(T) +≤ +(∥α|T| + (1 − α)|T ∗|∥) ∥T∥, +for all α ∈ [0, 1]. + +7 +Proof. Let x ∈ H with ∥x∥ = 1. Then, it follows from the inequality (2.2) that +|⟨Tx, x⟩|2 +≤ +⟨|T|x, x⟩ ⟨|T ∗|x, x⟩ += +⟨|T|x, x⟩α⟨|T ∗|x, x⟩1−α⟨|T|x, x⟩1−α⟨|T ∗|x, x⟩α +≤ +(α⟨|T|x, x⟩ + (1 − α)⟨|T ∗|x, x⟩)((1 − α)⟨|T|x, x⟩ + α⟨|T ∗|x, x⟩) += +⟨(α|T| + (1 − α)|T ∗|)x, x⟩⟨(α|T ∗| + (1 − α)|T|)x, x⟩ +≤ +(∥α|T| + (1 − α)|T ∗|∥) (∥α|T ∗| + (1 − α)|T|∥) +≤ +(∥α|T| + (1 − α)|T ∗|∥) ∥T∥. +Therefore, taking supremum over ∥x∥ = 1, we get +w2(T) +≤ +(∥α|T| + (1 − α)|T ∗|∥) ∥T∥, +as required. +□ +Remark 2.9. Since Theorem 2.8 holds for all α ∈ [0, 1], we can write that +inequality in the following form: +w(T) +≤ +�� +min +α∈[0,1] ∥α|T| + (1 − α)|T ∗|∥ +� +∥T∥. +(2.13) +Note that, the bound (2.13) is a considerable refinement of the well-known bound +w(T) ≤ ∥T∥. To show proper refinement, we take the same matrix as in Remark +2.3. Then, by elementary calculations we see that +�� +min +α∈[0,1] ∥α|T| + (1 − α)|T ∗|∥ +� +∥T∥ = +� +4 +3 × 2 < 2 = ∥T∥. +3. Numerical radius of (α, β)-normal operators +In this section, we develop lower bounds for the numerical radius of the (α, β)- +normal operators. We observe that for T ∈ B(H), +w(T) ≥ max{∥ℜ(T)∥, ∥ℑ(T)∥}, +(3.1) +the proof of which follows from the Cartesian decomposition of T. The first lower +bound of this section reads as follows: +Theorem 3.1. Let T be an (α, β)-normal operator. Then +w2(T) ≥ max +� +1 + α2, 1 + 1 +β2 +� ∥T∥2 +4 ++ |∥ℜ(T)∥2 − ∥ℑ(T)∥2| +2 +. + +8 +P. BHUNIA +Proof. From the inequality (3.1), we have +w2(T) +≥ +∥ℜ(T)∥2 + ∥ℑ(T)∥2 +2 ++ |∥ℜ(T)∥2 − ∥ℑ(T)∥2| +2 +≥ +∥(ℜ(T))2 + (ℑ(T))2∥ +2 ++ |∥ℜ(T)∥2 − ∥ℑ(T)∥2| +2 += +∥|T|2 + |T ∗|2∥ +4 ++ |∥ℜ(T)∥2 − ∥ℑ(T)∥2| +2 +≥ +∥|T|2 + α2|T|2∥ +4 ++ |∥ℜ(T)∥2 − ∥ℑ(T)∥2| +2 +(since, |T ∗|2 ≥ α2|T|2) += +(1 + α2)∥T∥2 +4 ++ |∥ℜ(T)∥2 − ∥ℑ(T)∥2| +2 +. +(3.2) +Similarly, we also have +w2(T) +≥ +� +1 + +1 +β2 +� +∥T∥2 +4 ++ |∥ℜ(T)∥2 − ∥ℑ(T)∥2| +2 +. +(3.3) +Therefore, the required inequality follows by combining the inequalities in (3.2) +and (3.3). +□ +Remark 3.2. Clearly, max +� +1 + α2, 1 + +1 +β2 +� +> 1. Therefore, for an (α, β)-normal +operator T, +w(T) +≥ +� +max +� +1 + α2, 1 + 1 +β2 +� ∥T∥2 +4 ++ |∥ℜ(T)∥2 − ∥ℑ(T)∥2| +2 +≥ +� +max +� +1 + α2, 1 + 1 +β2 +� ∥T∥2 +4 += +max +�√ +1 + α2, +� +1 + 1 +β2 +� ∥T∥ +2 +(3.4) +> +∥T∥ +2 . +Therefore, we would like to remark that for the (α, β)-normal operators, Theorem +3.1 gives stronger bound than the well-known bound w(T) ≥ ∥T∥ +2 . Note that the +inequality (3.4) is also developed in [19, Theorem 3.1]. +To obtain another improvement we first note the following numerical radius +inequality (see [9]): For every T ∈ B(H), +w(T) ≥ 1 +√ +2 +max {∥ℜ(T) + ℑ(T)∥, ∥ℜ(T) − ℑ(T)∥} . +(3.5) +By using the inequality (3.5) and using the similar arguments as Theorem 3.1, we +also prove the following lower bound for the numerical radius of the (α, β)-normal +operators. + +9 +Theorem 3.3. Let T be an (α, β)-normal operator. Then +w2(T) ≥ max +� +1 + α2, 1 + 1 +β2 +� ∥T∥2 +4 ++ |∥ℜ(T) + ℑ(T)∥2 − ∥ℜ(T) − ℑ(T)∥2| +4 +. +Remark 3.4. Clearly, we see that for an (α, β)-normal operator T, +w(T) +≥ +� +max +� +1 + α2, 1 + 1 +β2 +� ∥T∥2 +4 ++ |∥ℜ(T) + ℑ(T)∥2 − ∥ℜ(T) − ℑ(T)∥2| +4 +≥ +� +max +� +1 + α2, 1 + 1 +β2 +� ∥T∥2 +4 += +max +�√ +1 + α2, +� +1 + 1 +β2 +� ∥T∥ +2 +> +∥T∥ +2 . +Therefore, for the (α, β)-normal operators, Theorem 3.3 gives stronger bound +than the well-known bound w(T) ≥ ∥T∥ +2 . +At the end, we would like to remark that if T ∈ B(H) is an invertible operator, +then w(T) > ∥T∥ +2 . This implies that when w(T) = ∥T∥ +2 , then T must be a non- +invertible operator. +References +1. S. Bag, P. Bhunia, and K. Paul; Bounds of numerical radius of bounded linear operator +using t-Aluthge transform, Math. Inequal. Appl. 23 (2020), no. 3, 991–1004. +2. P. Bhunia, S.S. Dragomir, M.S. Moslehian, and K. Paul; Lectures on numerical radius +inequalities, Infosys Science Foundation Series, Infosys Science Foundation Series in Math- +ematical Sciences, Springer Cham, (2022). https://doi.org/10.1007/978-3-031-13670-2 +3. P. Bhunia, and K. Paul; Refinement of numerical radius inequalities of complex Hilbert +space operators, Acta Sci. Math. (Szeged) (2022), to appear. +4. P. Bhunia, A. Bhanja, D. Sain, and K. Paul; Numerical radius inequalities of operator +matrices from a new norm on B(H), Miskolc Math. Notes (2022), to appear. +5. P. Bhunia, A. Sen, and K. Paul; New semi-norm of semi-Hilbertian space operators and its +application, J. Convex Anal. 29 (2022), no. 4, 1149–1160. +6. P. Bhunia, and K. Paul; Development of inequalities and characterization of equality con- +ditions for the numerical radius, Linear Algebra Appl. 630 (2021), 306–315. +7. P. Bhunia, and K. Paul; Proper improvement of well-known numerical radius inequalities +and their applications, Results Math. 76 (2021), no. 4, Paper No. 177, 12 pp. +8. P. Bhunia, K. Paul; New upper bounds for the numerical radius of Hilbert space operators, +Bull. Sci. Math. 167 (2021), Paper No. 102959, 11 pp. +9. P. Bhunia, S. Jana, and K. Paul; Refined inequalities for the numerical radius of Hilbert +space operators, (2021). https://arxiv.org/abs/2106.13949 +10. M.L. Buzano; Generalizzatione della disuguaglianza di Cauchy-Schwarz, Rend. Semin. Mat. +Univ. Politech. Torino 31(1971/73) (1974) 405–409. +11. S.S. Dragomir, and M.S. Moslehian; Some inequalities for (α, β)-normal operators in Hilbert +spaces, Facta Univ. Ser. Math. Inform. 23 (2008), 39–47. +12. R.G. Duglas; On majorization, factorization, and range inclusion of operators on Hilbert +space, Proc. Amer. Math. Soc. 17 (1966), 413–415. +13. P.R. Halmos; A Hilbert space problems book, Springer Verlag, New York, 1982. + +10 +P. BHUNIA +14. S. Jana, P. Bhunia, and K. Paul; Numerical radius inequalities and estimation of zeros of +polynomials, (2023). https://arxiv.org/abs/2301.03159 +15. T. Kato; Notes on some inequalities for linear operators, Math. Ann. 125 (1952), 208–212. +16. F. Kittaneh; Numerical radius inequalities for Hilbert space operators, Studia Math. 168 +(2005), no. 1, 73–80. +17. F. Kittaneh; Numerical radius inequality and an estimate for the numerical radius of the +Frobenius companion matrix, Studia Math. 158 (2003), no. 1, 11–17. +18. F. Kittaneh; Notes on some inequalities for Hilbert space operators, Publ. RIMS Kyoto +Univ. 24 (1988) 283–293. +19. M. Sababheh, and H.R. Moradi; New orders among Hilbert space operators, (2022). +https://arxiv.org/abs/2212.06382v1 +20. D. Sain, P. Bhunia, A. Bhanja, and K. Paul; On a new norm on B(H) and its application +to numerical radius inequalities, Ann. Funct. Anal. 12 (2021), no. 4, Paper No. 51, 25 pp. +21. B. Simon; Trace ideals and their applications, Camrbidge University Press, 1979. +Department of Mathematics, Indian Institute of Science, Bengaluru 560012, +Karnataka, India +Email address: pintubhunia5206@gmail.com + diff --git a/c9E2T4oBgHgl3EQfawcz/content/tmp_files/load_file.txt b/c9E2T4oBgHgl3EQfawcz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2b6e83256719d31cd6686b205dc619264d38cc82 --- /dev/null +++ b/c9E2T4oBgHgl3EQfawcz/content/tmp_files/load_file.txt @@ -0,0 +1,396 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf,len=395 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='03877v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='FA] 10 Jan 2023 NUMERICAL RADIUS INEQUALITIES OF BOUNDED LINEAR OPERATORS AND (α, β)-NORMAL OPERATORS PINTU BHUNIA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' We obtain various upper bounds for the numerical radius w(T ) of a bounded linear operator T defined on a complex Hilbert space H, by developing the upper bounds for the α-norm of T , which is defined as ∥T ∥α = sup �� α|⟨T x, x⟩|2 + (1 − α)∥T x∥2 : x ∈ H, ∥x∥ = 1 � for 0 ≤ α ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Further, we prove that w(T ) ≤ �� min α∈[0,1] ∥α|T | + (1 − α)|T ∗|∥ � ∥T ∥ ≤ ∥T ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' For 0 ≤ α ≤ 1 ≤ β, the operator T is called (α, β)-normal if α2T ∗T ≤ T T ∗ ≤ β2T ∗T holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Note that every invertible operator is an (α, β)-normal operator for suitable values of α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Among other lower bound for the numerical radius of an (α, β)-normal operator T , we show that w(T ) ≥ � max � 1 + α2, 1 + 1 β2 � ∥T ∥2 4 + |∥ℜ(T )∥2 − ∥ℑ(T )∥2| 2 ≥ max �� 1 + α2, � 1 + 1 β2 � ∥T ∥ 2 > ∥T ∥ 2 , where ℜ(T ) and ℑ(T ) are the real part and imaginary part of T , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Introduction Let B(H) denote the C∗-algebra of all bounded linear operators on a complex Hilbert space H, with inner product ⟨·, ·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Let T ∈ B(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' We write |T| = (T ∗T)1/2 and |T ∗| = (TT ∗)1/2, where T ∗ denotes the adjoint of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' The numerical radius of T, denoted by w(T), is defined as w(T) = sup{|⟨Tx, x⟩| : x ∈ H, ∥x∥ = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 47A12, 47A30, 15A60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Numerical radius, operator norm, (α, β)-normal operator, Bounded linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' The author would like to sincerely acknowledge Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Kallol Paul for his valuable comments on this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' The author also would like to thank SERB, Govt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' of India for the financial support in the form of National Post Doctoral Fellowship (N-PDF, File No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' PDF/2022/000325) under the mentorship of Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Apoorva Khare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 1 2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' BHUNIA Note that, the numerical radius defines a norm on B(H), and it is equivalent to the operator norm, satisfies 1 2∥T∥ ≤ w(T) ≤ ∥T∥, for every T ∈ B(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='1) Various refinements of the inequalities in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='1) and related results have been discussed in a recent published book [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Also, the reader can see the papers [1, 6, 16] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' For 0 ≤ α ≤ 1, the α-norm of T (see [4, 5, 20]) is given by ∥T∥α = sup �� α|⟨Tx, x⟩|2 + (1 − α)∥Tx∥2 : x ∈ H, ∥x∥ = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' This α-norm is also a norm on B(H), and it satisfies w(T) ≤ ∥T∥α ≤ ∥T∥, for every T ∈ B(H) and for all α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='2) The inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='2) together with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='1) implies that the α-norm is equivalent to both the operator norm and the numerical radius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' For more details of the α-norm, the reader can see [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Next, we turn our attention to study the (α, β)-normal operators (see [11]), where 0 ≤ α ≤ 1 ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' An operator T ∈ B(H) is said to be (α, β)-normal operator if T satisfies α2T ∗T ≤ TT ∗ ≤ β2T ∗T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' This is equivalent to α2⟨T ∗Tx, x⟩ ≤ ⟨TT ∗x, x⟩ ≤ β2⟨T ∗Tx, x⟩ for all x ∈ H, that is, α∥Tx∥ ≤ ∥T ∗x∥ ≤ β∥Tx∥ for all x ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Every normal and hyponormal operators are (α, β)-normal for suitable values of α, β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' For normal operators α = β = 1 and for hyponormal operators β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Also, there exist operators which are neither normal nor hyponormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' As for example, the matrix � 1 0 1 1 � is an (α, β)-normal with α = � 3− √ 5 2 and β = � 3+ √ 5 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' However, the matrix is neither normal nor hyponormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' If T is an (α, β)- normal operator, then both, T mazorizes T ∗ and T ∗ mazorizes T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' According to Douglas (Mazorization lemma) [12], an operator T ∈ B(H) mazorizes an operator S ∈ B(H) if one of the following equivalent statements holds: (i) Ran(T) ⊆ Ran(S), where Ran(T) (Ran(S)) denotes the range space of T (S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' (ii) TT ∗ ≤ µ2SS∗ for some µ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Therefore, T is an (α, β)-normal operator if and only if Ran(T) = Ran(T ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' So, every invertible operator is an (α, β)-normal operator for suitable values of α, β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' In this paper, we develop upper bounds for the α-norm of bounded linear op- erators and consequentially we obtain bounds for the numerical radius, which improve on the existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Further, we develop lower bounds for the numeri- cal radius of the (α, β)-normal operators, which improve on the classical bound w(T) ≥ ∥T∥ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' The α-norm and numerical radius inequalities We obtain new upper bounds for the numerical radius of bounded linear operators, by developing the bounds of α-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' First, we recall some necessary inequalities 3 from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Kato [15] generalized the Cauchy-Schwarz inequality, namely, |⟨Tx, y⟩|2 ≤ � |T|2α x, x � � |T ∗|2(1−α) y, y � ∀x, y ∈ H, ∀α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='1) In particular, for α = 1/2 (see [13, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 75-76]), |⟨Tx, y⟩| ≤ ⟨|T|x, x⟩1/2 ⟨|T ∗|y, y⟩1/2 ∀x, y ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='2) In [18, Theorem 1], Kittaneh generalized the Kato’s inequality, which is as follows: If f, g : [0, ∞] → [0, ∞] are continuous functions satisfying f(t)g(t) = t, ∀t ≥ 0, then |⟨Tx, y⟩|2 ≤ � f 2(|T|)x, x � � g2(|T ∗|)y, y � ∀x, y ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='3) Next, we recall the H¨older–McCarthy inequality (see [21, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 20]) which states that if T ∈ B(H) is positive, then ⟨Tx, x⟩r ≤ ⟨T rx, x⟩ ∀r ≥ 1 ∀x ∈ H, ∥x∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='4) The inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='4) is reversed when 0 ≤ r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' The Cartesian decomposition of T ∈ B(H) is given by T = ℜ(T) + iℑ(T), where ℜ(T) = 1 2(T + T ∗) and ℑ(T) = 1 2i(T − T ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Now, we are in a position to prove our first result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' If T ∈ B(H), then ∥T∥2 α ≤ ���α 4 � f 4(|T|) + g4(|T ∗|) � + (1 − α)|T|2��� + α 2 ��ℜ � f 2(|T|)g2(|T ∗|) ��� , where f and g are as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Let x ∈ H with ∥x∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Then, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='3) that |⟨Tx, x⟩|2 ≤ � f 2(|T|)x, x � � g2(|T ∗|)x, x � = �� f 2(|T|)x, x �1/2 � g2(|T ∗|)x, x �1/2�2 ≤ 1 4 �� f 2(|T|)x, x � + � g2(|T ∗|)x, x ��2 = 1 4 �� f 2(|T|) + g2(|T ∗|) � x, x �2 ≤ 1 4 �� f 2(|T|) + g2(|T ∗|) �2 x, x � (by using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='4) ) = 1 4 �� f 4(|T|) + g4(|T ∗|) � x, x � + 1 2 � ℜ � f 2(|T|)g2(|T ∗|) � x, x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Thus, we have ∥T∥2 α = sup ∥x∥=1 � α |⟨Tx, x⟩|2 + (1 − α)∥Tx∥2� ≤ sup ∥x∥=1 ��α 4 � f 4(|T|) + g4(|T ∗|) � + (1 − α)|T|2� x, x � +α 2 sup ∥x∥=1 ��� ℜ � f 2(|T|)g2(|T ∗|) � x, x ��� = ���α 4 � f 4(|T|) + g4(|T ∗|) � + (1 − α)|T|2��� + α 2 ��ℜ � f 2(|T|)g2(|T ∗|) ��� , 4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' BHUNIA as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' □ Now, as a consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='1, we get the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' If T ∈ B(H), then ∥T∥2 α ≤ ���� � 1 − 3α 4 � |T|2 + α 4 |T ∗|2 ���� + α 2 ∥ℜ(|T||T ∗|)∥ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='5) and ∥T∥2 α ≤ ���� � 1 − 3α 4 � |T ∗|2 + α 4 |T|2 ���� + α 2 ∥ℜ(|T||T ∗|)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='6) Further, w(T) ≤ min{γ, δ}, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='7) where γ = min α∈[0,1] ����� � 1 − 3α 4 � |T|2 + α 4 |T ∗|2 ���� + α 2 ∥ℜ(|T||T ∗|)∥ �1/2 and δ = min α∈[0,1] ����� � 1 − 3α 4 � |T ∗|2 + α 4 |T|2 ���� + α 2 ∥ℜ(|T||T ∗|)∥ �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' The inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='5) follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='1 by taking f(t) = g(t) = t1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' The inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='6) follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='5) by replacing T by T ∗ and using the equality ∥T∥α = ∥T ∗∥α (see [20, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' The inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='7) follows from the fact that w(T) ≤ minα∈[0,1] ∥T∥α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Recently, Bhunia and Paul obtained in [3, The inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='3)] and [8, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='6] that w2(T) ≤ 1 4 ∥|T|2 + |T ∗|2∥+ 1 2 ∥ℜ(|T||T ∗|)∥ and w2(T) ≤ 1 4 ∥|T|2 + |T ∗|2∥ + 1 2w(|T||T ∗|), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Clearly, we see that min{γ2, δ2} ≤ 1 4 ��|T|2 + |T ∗|2�� + 1 2 ∥ℜ(|T||T ∗|)∥ ≤ 1 4 ��|T|2 + |T ∗|2�� + 1 2w(|T||T ∗|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Thus, we would like to remark that the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='7) refines both the inequali- ties in [3, The inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='3)] and [8, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' To show proper refinement, we consider a matrix T = \uf8eb \uf8ed 0 1 0 0 0 2 0 0 0 \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Then, by elementary calculations we see that γ = � 28 13 and δ = 3 2, and so min{γ2, δ2} = 28 13 < 9 4 = 1 4 ��|T|2 + |T ∗|2�� + 1 2 ∥ℜ(|T||T ∗|)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Another bound for the α-norm reads as follows: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' If T ∈ B(H), then ∥T∥2 α ≤ min ���α|T|2 + (1 − α)|T ∗|2�� , ��α|T ∗|2 + (1 − α)|T|2��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Let x ∈ H with ∥x∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' By using Cauchy-Schwarz inequality, we get ∥T∥2 α = sup ∥x∥=1 � α |⟨Tx, x⟩|2 + (1 − α)∥Tx∥2� = sup ∥x∥=1 � α |⟨x, T ∗x⟩|2 + (1 − α)∥Tx∥2� ≤ sup ∥x∥=1 � α∥T ∗x∥2 + (1 − α)∥Tx∥2� = sup ∥x∥=1 � α⟨|T ∗|2x, x⟩ + (1 − α)⟨|T|2x, x⟩ � = sup ∥x∥=1 ⟨(α|T ∗|2 + (1 − α)|T|2)x, x⟩ = ∥α|T ∗|2 + (1 − α)|T|2∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='8) Replacing T by T ∗ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='8) and using ∥T∥α = ∥T ∗∥α, we get ∥T∥2 α ≤ ∥α|T|2 + (1 − α)|T ∗|2∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='9) Therefore, the required inequality follows by combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' □ Now, the following upper bound for the numerical radius follows easily from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='4 together with the first inequality in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='2): w2(T) ≤ min α∈[0,1] ��α|T|2 + (1 − α)|T ∗|2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='10) Note that, the bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='10) is also obtained in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' To obtain our next result we need the Buzano’s inequality [10], namely, if x, y, e ∈ H with ∥e∥ = 1, then 2|⟨x, e⟩⟨e, y⟩| ≤ ∥x∥∥y∥ + |⟨x, y⟩|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='11) By employing the Buzano’s inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='11), we prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' If T ∈ B(H), then ∥T∥2 α ≤ α 4 w2(|T| + i|T ∗|) + α 4 w(|T||T ∗|) + ���� � 1 − 7α 8 � |T|2 + α 8 |T ∗|2 ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Let x ∈ H with ∥x∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Then, from the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='2), we get |⟨Tx, x⟩|2 ≤ ⟨|T|x, x⟩⟨|T ∗|x, x⟩ ≤ 1 4 (⟨|T|x, x⟩ + ⟨|T ∗|x, x⟩)2 = 1 4(⟨|T|x, x⟩2 + ⟨|T ∗|x, x⟩2) + 1 2⟨|T|x, x⟩⟨|T ∗|x, x⟩ = 1 4|⟨|T|x, x⟩ + i⟨|T ∗|x, x⟩|2 + 1 2⟨|T ∗|x, x⟩⟨x, |T|x⟩ ≤ 1 4|⟨(|T| + i|T ∗|)x, x⟩|2 + 1 4∥|T ∗|x∥ ∥|T|x∥ + 1 4|⟨|T ∗|x, |T|x⟩| (by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='11)) ≤ 1 4|⟨(|T| + i|T ∗|)x, x⟩|2 + 1 8(∥|T ∗|x∥2 + ∥|T|x∥2) + 1 4|⟨|T||T ∗|x, x⟩|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 6 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' BHUNIA Hence, α|⟨Tx, x⟩|2 + (1 − α)∥Tx∥2 ≤ α 4 |⟨(|T| + i|T ∗|)x, x⟩|2 + α 4 |⟨|T||T ∗|x, x⟩| + ��� 1 − 7α 8 � |T|2 + α 8 |T ∗|2 � x, x � ≤ α 4 w2(|T| + i|T ∗|) + α 4 w(|T||T ∗|) + ���� � 1 − 7α 8 � |T|2 + α 8 |T ∗|2 ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Therefore, taking supremum over ∥x∥ = 1, we get the required inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' □ Replacing T by T ∗ in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='5, and then using ∥T∥α = ∥T ∗∥α, we also get the following inequality: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' If T ∈ B(H), then ∥T∥2 α ≤ α 4 w2(|T| + i|T ∗|) + α 4 w(|T||T ∗|) + ���� � 1 − 7α 8 � |T ∗|2 + α 8 |T|2 ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' As an application of the bounds for the α-norm, obtained in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='5 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='6, we have a new upper bound for the numerical radius, which is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' If T ∈ B(H), then w2(T) ≤ α 4 w2(|T| + i|T ∗|) + α 4 w(|T||T ∗|) + min ����� � 1 − 7α 8 � |T ∗|2 + α 8 |T|2 ���� , ���� � 1 − 7α 8 � |T|2 + α 8 |T ∗|2 ���� � , for all α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' In particular, for α = 1 we have (see also in [14]), w2(T) ≤ 1 4w2(|T| + i|T ∗|) + 1 4w(|T||T ∗|) + 1 8 ��|T ∗|2 + |T|2�� (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='12) ≤ 1 2 ∥T ∗T + TT ∗∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Therefore, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='7 generalizes and improves on the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='12) as well as the well-known inequality w2(T) ≤ 1 2 ∥T ∗T + TT ∗∥, obtained by Kittaneh [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Next, we note that every operator T ∈ B(H) satisfies the following numeri- cal radius inequality (see [17]): w(T) ≤ 1 2∥|T| + |T ∗|∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' In [7], it is proved that w2(T) ≤ ∥α|T|2 + (1 − α)|T ∗|2∥ , ∀α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Now, it is natural to ask whether the following inequality w(T) ≤ ∥α|T| + (1 − α)|T ∗|∥ , ∀α ∈ [0, 1] holds or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' In this connection we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' If T ∈ B(H), then w2(T) ≤ (∥α|T| + (1 − α)|T ∗|∥) ∥T∥, for all α ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Let x ∈ H with ∥x∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Then, it follows from the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='2) that |⟨Tx, x⟩|2 ≤ ⟨|T|x, x⟩ ⟨|T ∗|x, x⟩ = ⟨|T|x, x⟩α⟨|T ∗|x, x⟩1−α⟨|T|x, x⟩1−α⟨|T ∗|x, x⟩α ≤ (α⟨|T|x, x⟩ + (1 − α)⟨|T ∗|x, x⟩)((1 − α)⟨|T|x, x⟩ + α⟨|T ∗|x, x⟩) = ⟨(α|T| + (1 − α)|T ∗|)x, x⟩⟨(α|T ∗| + (1 − α)|T|)x, x⟩ ≤ (∥α|T| + (1 − α)|T ∗|∥) (∥α|T ∗| + (1 − α)|T|∥) ≤ (∥α|T| + (1 − α)|T ∗|∥) ∥T∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Therefore, taking supremum over ∥x∥ = 1, we get w2(T) ≤ (∥α|T| + (1 − α)|T ∗|∥) ∥T∥, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Since Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='8 holds for all α ∈ [0, 1], we can write that inequality in the following form: w(T) ≤ �� min α∈[0,1] ∥α|T| + (1 − α)|T ∗|∥ � ∥T∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='13) Note that, the bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='13) is a considerable refinement of the well-known bound w(T) ≤ ∥T∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' To show proper refinement, we take the same matrix as in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Then, by elementary calculations we see that �� min α∈[0,1] ∥α|T| + (1 − α)|T ∗|∥ � ∥T∥ = � 4 3 × 2 < 2 = ∥T∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Numerical radius of (α, β)-normal operators In this section, we develop lower bounds for the numerical radius of the (α, β)- normal operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' We observe that for T ∈ B(H), w(T) ≥ max{∥ℜ(T)∥, ∥ℑ(T)∥}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='1) the proof of which follows from the Cartesian decomposition of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' The first lower bound of this section reads as follows: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Let T be an (α, β)-normal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Then w2(T) ≥ max � 1 + α2, 1 + 1 β2 � ∥T∥2 4 + |∥ℜ(T)∥2 − ∥ℑ(T)∥2| 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 8 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' BHUNIA Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' From the inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='1), we have w2(T) ≥ ∥ℜ(T)∥2 + ∥ℑ(T)∥2 2 + |∥ℜ(T)∥2 − ∥ℑ(T)∥2| 2 ≥ ∥(ℜ(T))2 + (ℑ(T))2∥ 2 + |∥ℜ(T)∥2 − ∥ℑ(T)∥2| 2 = ∥|T|2 + |T ∗|2∥ 4 + |∥ℜ(T)∥2 − ∥ℑ(T)∥2| 2 ≥ ∥|T|2 + α2|T|2∥ 4 + |∥ℜ(T)∥2 − ∥ℑ(T)∥2| 2 (since, |T ∗|2 ≥ α2|T|2) = (1 + α2)∥T∥2 4 + |∥ℜ(T)∥2 − ∥ℑ(T)∥2| 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='2) Similarly, we also have w2(T) ≥ � 1 + 1 β2 � ∥T∥2 4 + |∥ℜ(T)∥2 − ∥ℑ(T)∥2| 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='3) Therefore, the required inequality follows by combining the inequalities in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Clearly, max � 1 + α2, 1 + 1 β2 � > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Therefore, for an (α, β)-normal operator T, w(T) ≥ � max � 1 + α2, 1 + 1 β2 � ∥T∥2 4 + |∥ℜ(T)∥2 − ∥ℑ(T)∥2| 2 ≥ � max � 1 + α2, 1 + 1 β2 � ∥T∥2 4 = max �√ 1 + α2, � 1 + 1 β2 � ∥T∥ 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='4) > ∥T∥ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Therefore, we would like to remark that for the (α, β)-normal operators, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='1 gives stronger bound than the well-known bound w(T) ≥ ∥T∥ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Note that the inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='4) is also developed in [19, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' To obtain another improvement we first note the following numerical radius inequality (see [9]): For every T ∈ B(H), w(T) ≥ 1 √ 2 max {∥ℜ(T) + ℑ(T)∥, ∥ℜ(T) − ℑ(T)∥} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='5) By using the inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='5) and using the similar arguments as Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='1, we also prove the following lower bound for the numerical radius of the (α, β)-normal operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 9 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Let T be an (α, β)-normal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Then w2(T) ≥ max � 1 + α2, 1 + 1 β2 � ∥T∥2 4 + |∥ℜ(T) + ℑ(T)∥2 − ∥ℜ(T) − ℑ(T)∥2| 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Clearly, we see that for an (α, β)-normal operator T, w(T) ≥ � max � 1 + α2, 1 + 1 β2 � ∥T∥2 4 + |∥ℜ(T) + ℑ(T)∥2 − ∥ℜ(T) − ℑ(T)∥2| 4 ≥ � max � 1 + α2, 1 + 1 β2 � ∥T∥2 4 = max �√ 1 + α2, � 1 + 1 β2 � ∥T∥ 2 > ∥T∥ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Therefore, for the (α, β)-normal operators, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='3 gives stronger bound than the well-known bound w(T) ≥ ∥T∥ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' At the end, we would like to remark that if T ∈ B(H) is an invertible operator, then w(T) > ∥T∥ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' This implies that when w(T) = ∥T∥ 2 , then T must be a non- invertible operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Bag, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Bhunia, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Paul;' metadata={'source': 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Bhunia, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Paul;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Refinement of numerical radius inequalities of complex Hilbert space operators, Acta Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' (Szeged) (2022), to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Bhunia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Bhanja, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Sain, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Paul;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Numerical radius inequalities of operator matrices from a new norm on B(H), Miskolc Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Notes (2022), to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Bhunia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Sen, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Paul;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' New semi-norm of semi-Hilbertian space operators and its application, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Convex Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 29 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 4, 1149–1160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Bhunia, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Paul;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Development of inequalities and characterization of equality con- ditions for the numerical radius, Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 630 (2021), 306–315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Bhunia, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Paul;' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' New orders among Hilbert space operators, (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='org/abs/2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='06382v1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Sain, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Bhunia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Bhanja, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Paul;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' On a new norm on B(H) and its application to numerical radius inequalities, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Anal.' metadata={'source': 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applications, Camrbidge University Press, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content=' Department of Mathematics, Indian Institute of Science, Bengaluru 560012, Karnataka, India Email address: pintubhunia5206@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E2T4oBgHgl3EQfawcz/content/2301.03877v1.pdf'} diff --git a/dtAyT4oBgHgl3EQfwvmK/vector_store/index.faiss b/dtAyT4oBgHgl3EQfwvmK/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..3dc4b472144b974e4eaaf2ed891d244703be0654 --- /dev/null +++ b/dtAyT4oBgHgl3EQfwvmK/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1813bdce3259962fa44ad5bc113a579ef6833b00379d01ec719b5a2676a4213e +size 9502765 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b/eNE2T4oBgHgl3EQfbQcn/content/2301.03882v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4019a52f19ed30027310bdc3478251b1b37102e1f7ac0c0d6aaa7092e3ed5e32 +size 2763276 diff --git a/eNE5T4oBgHgl3EQfgg9f/content/tmp_files/2301.05634v1.pdf.txt b/eNE5T4oBgHgl3EQfgg9f/content/tmp_files/2301.05634v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3763d9e7f1b217fa516b1af08be87b8e9279b7cc --- /dev/null +++ b/eNE5T4oBgHgl3EQfgg9f/content/tmp_files/2301.05634v1.pdf.txt @@ -0,0 +1,1442 @@ +arXiv:2301.05634v1 [hep-ph] 13 Jan 2023 +The quark orbital angular momentum of ground state octet baryons +Jing-Feng Li,1 Cheng Chen,2, 3 Gang Li,4, ∗ Chun-Sheng An,1, † Cheng-Rong Deng,1, ‡ and Ju-Jun Xie2, 3, 5, § +1School of Physical Science and Technology, Southwest University, Chongqing 400715, China +2Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China +3School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 101408, China +4College of Physics and Engineering, Qufu Normal University, Qufu 273165, China +5Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical +Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China +(Dated: January 16, 2023) +Here we study the quark orbital angular momentum of the ground octet baryons employing an +extended chiral constituent quark model, within which the baryon wave functions are taken to be +superposition of the traditional qqq and the qqqq¯q higher Fock components. +Coupling between +the two configurations is estimated using the 3P0 quark-antiquark creation mechanism, and the +corresponding coupling strength is determined by fitting the sea flavor asymmetry of the nucleon. +The obtained numerical results show that the quark angular momentum of the nucleon, Σ, Λ and Ξ +hyperons are in the range 0.10-0.30. In addition, the quark angular momentum of all the hyperons +are a little bit smaller than that of the nucleon. And the octet baryons spin fractions taken by the +intrinsic quark orbital angular momentum could be up to 60% in present model. +I. +INTRODUCTION +In the traditional constituent quark models, +the +ground octet baryons are composed of three quarks that +are in their S-wave, thus the total spin of the baryons +are contributed from only the spin of their quark content, +namely, the SU(2) mixed symmetric spin configuration +of the three-quark system with total spin S = 1/2 should +account for the total spin of corresponding baryons com- +pletely [1, 2]. +However, in the late 1980s, the famous +European muon collaboration (EMC) found that the +quark spin could take only a small fraction of the proton +spin [3, 4], which raised up the proton spin crisis. +Since then intensively experimental measurements on +the spin structure of proton have been taken [5–11], and, +on the theoretical side, the spin decomposition of proton +in gauge theory was proposed in Refs. [12–15]. Accord- +ingly, contributions of the quarks, the gluons and orbital +angular momentum (OAM) to the total proton spin have +been investigated by lattice QCD [16–23], chiral pertur- +bation theory [24–27], as well as other theoretical ap- +proaches [28–39], for recent reviews on status of the ex- +perimental and theoretical investigations on nucleon spin +structure, see Refs. [40–46]. +Based on these previous theoretical studies, one may +conclude that the intrinsic sea content in baryons should +play important roles in the static properties of baryons. +On the other hand, the experimentally observed sea +∗Electronic address: gli@qfnu.edu.cn +†Electronic address: ancs@swu.edu.cn +‡Electronic address: crdeng@swu.edu.cn +§Electronic address: xiejujun@impcas.ac.cn +flavor asymmetry of proton [47, 48] also reveals the +non-perturbative effects of the intrinsic sea content in +baryons. +Constructively, the proton can be depicted by a meson- +cloud picture that the proton is explained to be a neutron +core surrounding by pion meson cloud. Furthermore, one +can also consider effects of the KΛ, KΣ, and π∆ compo- +nents in proton [49–52]. Within the meson cloud model, +spin of proton could be directly decomposed into the spin +and orbital angular momentum of quarks, since the me- +son clouds should be in their P-wave states relative to +the baryon cores. And the sea flavor asymmetry may be +described by the πN and π∆ components with appro- +priate probabilities in proton. Within this picture, one +can also explain the intrinsic strange-antistrange quark +asymmetry in proton if the KΛ and KΣ components are +taken into account [52]. +Straightforwardly, the meson +cloud picture for nucleon can be extended to the other +octet baryons [53]. +Alternatively, the extended chiral constituent quark +model (EχCQM), in which the higher Fock compo- +nents in the baryon’s wave function are assumed to +be compact pentaquark configurations, was proposed to +study the strangeness magnetic moment [54], strangeness +form factor [55] and strangeness spin of proton [32], +and further developed to investigate the intrinsic sea +content and meson-baryon sigma terms of the octet +baryons [56–58]. In addition, the experimental data for +decays of baryon resonances such as ∆(1232), P11(1440), +S11(1535), S11(1650), D13(1520) and D13(1700) [59–64] +can be also well reproduced by EχCQM. Very recently, +the EχCQM has been applied to study the quark OAM +of the proton and flavor-dependent axial charges of the +ground state octet baryons [65–67]. It’s shown that the +singlet axial charge g(0) +A , the isovector axial charge g(3) +A + +2 +and the SU(3) octet axial charge g(8) +A of the octet baryons +obtained in the EχCQM, are consistent with predictions +by lattice QCD and chiral perturbation theory, if the +model parameters are fixed by fitting the data for ¯d − ¯u +asymmetry of proton [48]. +As an direct extension of +Ref. [65], here we investigate the quark OAM of the +ground octet baryons, and analyze the spin decompo- +sition of the corresponding baryons in the EχCQM. +The present manuscript is organized as follows. +In +Sec. II, we present the framework which includes the +EχCQM and the formalism for calculations on the OAM +in corresponding model, the explicit numerical results +and discussions are given in Sec. III. Finally, we give a +brief summary of present work in Sec. IV. +II. +FRAMEWORK +To investigate the quark orbital angular momentum of +the ground octet baryons, here we employ the EχCQM, +in which the wave functions of baryons can be expressed +in a general form as: +|B⟩ = +1 +√ +N +� +|qqq⟩ + +� +i +Cq +i |qqq(q¯q), i⟩ +� +, +(1) +where the first term just represents the wave functions +for the traditional three-quark components of the octet +baryons, while the second term denotes the wave func- +tions for the compact five-quark components, with the +sum over i runs over all the possible five-quark configu- +rations with a q¯q (q = u, d, s) pair that may form con- +siderable higher Fock components in the octet baryons. +Cq +i / +√ +N are the corresponding probability amplitudes for +the five-quark components with N being a normalization +constant. The seventeen possible qqq(q¯q) configurations +with i = 1 · · · 17 are shown in Table I, where the flavor, +spin, color and orbital wave functions for the four-quark +subsystem are denoted by the Young tableaux of the S4 +permutation group. +As we can see in Table I, spin wave functions of the +four-quark subsystem in configurations with i = 1 · · · 7 +are [22]S, which leads to the total spin S4 = 0 for the +four-quark subsystem, thus one only need to consider the +combination of the quark (antiquark) OAM and spin of +the antiquark to get the total spin of a given baryon. And +a general form for the wave functions of these configura- +tions can be expressed as +|B, i = 1 · · · 7⟩5q = +� +ijkln +� +ab +� +m¯sz +C +1 +2 ,↑ +1,m; 1 +2 ,¯szC[14] +[31]k +χF S;[211]¯k +CC +[31]k +χF S +[O]i +χ;[F S]j +F SC +[F S]j +F S +[F ]l +F ;[22]n +SC[23]C +a,b +|[211] +¯k +C(a)⟩|[11]C,¯q(b)⟩|I, I3⟩[F ]l +F +|1, m⟩[O]i +χ|[22]n +S⟩|¯χ, ¯sz⟩φ({⃗rq}), +(2) +with the coefficients C[··· ] +[··· ][··· ] represent the CG coefficients +of the S4 permutation group, and C +1 +2 ,↑ +1,m; 1 +2 ,¯sz the CG co- +efficients for the combination of the quark (antiquark) +OAM and spin of the antiquark to form a spin |1/2, +1/2⟩ +baryon state. And the explicit flavor, orbital, spin and +color wave functions for the presently considered five- +quark configurations have been given in Refs. [67]. +While for the five-quark configurations with i = 8 · · · 17 +shown in Table I, the spin wave function of the four-quark +subsystem is [31]S which results in the total spin S4 = 1 +for the four-quark subsystem. Accordingly, we have to +take into account the combination of the spin of both the +four-quark and the antiquark and the quark (antiquark) +OAM to get the total spin of the octet baryon. +One +should note that combination of the spin for four-quark +subsystem S4 = 1 and the quark OAM L = 1 will lead to +J = L⊕S4 = 0, 1 or 2, and the former two J those could +form the total spin SB = 1/2 of the presently studied +octet baryons, when combine to the spin of the antiquark +S¯q = 1/2 are applicable. Hereafter, we denote these two +cases of wave functions as Set I and Set II, respectively. +The general forms for wave functions of the five-quark +configurations with i = 8 · · · 17 in these two cases are +then given by +|B, i = 8 · · · 17⟩I +5q = +� +ijkln +� +ab +� +msz +C00 +1,m;1,szC[14] +[31]k +χF S;[211]¯k +CC +[31]k +χF S +[O]χ +i ;[F S]j +FSC +[F S]j +F S +[F ]l +F ;[31]n +SC[23]C +a,b +|[211] +¯k +C(a)⟩|[11]C,¯q(b)⟩|I, I3⟩[F ]l +F +|1, m⟩[O]i +χ|[31]n +S, sz⟩|¯χ, ¯sz⟩φ({⃗rq}) , +(3) + +3 +TABLE I: The orbital-flavor-spin configurations for the four-quark subsystem of the five-quark configurations those may exist as higher +Fock components in ground octet baryons. +i +1 +2 +3 +4 +5 +Config. +[31]χ[4]F S[22]F [22]S +[31]χ[31]F S[211]F [22]S +[31]χ[31]F S[31]F1[22]S +[31]χ[31]F S[31]F2[22]S +[4]χ[31]F S[211]F [22]S +i +6 +7 +8 +9 +10 +Config. +[4]χ[31]F S[31]F1 [22]S +[4]χ[31]F S[31]F2 [22]S +[31]χ[4]FS[31]F1 [31]S +[31]χ[4]F S[31]F2[31]S +[31]χ[31]FS[211]F [31]S +i +11 +12 +13 +14 +15 +Config. +[31]χ[31]F S[22]F [31]S +[31]χ[31]F S[31]F1[31]S +[31]χ[31]F S[31]F2[31]S +[4]χ[31]F S[211]F [31]S +[4]χ[31]FS[22]F [31]S +i +16 +17 +Config. +[4]χ[31]F S[31]F1 [31]S +[4]χ[31]F S[31]F2 [31]S +and +|B, i = 8 · · · 17⟩II +5q = +� +ijkln +� +ab +� +Jz¯sz +� +msz +C +1 +2 , 1 +2 +1,Jz; 1 +2 ,¯szC1,Jz +1,m;1,szC[14] +[31]k +χF S;[211]¯k +CC +[31]k +χF S +[O]iχ;[F S]j +F SC +[F S]j +F S +[F ]l +F ;[31]n +SC[23]C +a,b +|[211] +¯k +C(a)⟩|[11]C,¯q(b)⟩ +|I, I3⟩[F ]l +F |1, m⟩[O]χ +i |[31]S +n, sz⟩|¯χ, ¯sz⟩φ({⃗rq}) . +(4) +Next we turn to the coefficients Cq +i in Eq. (1). Gen- +erally, to get the probability amplitude for a five-quark +component in a given baryon, one has to evaluate energy +of the five-quark configuration Eq +i and its coupling to the +three-quark component in the corresponding baryon. +In present work, the energy Eq +i for a given five-quark +configuration is estimated using the chiral constituent +quark model, within which the hyperfine interaction be- +tween quarks is [2] +Hhyp = − +� +i 20,000 nT, dur- +ing a CGF orbit. The minimum field strength was recorded +during one crossing of the magnetodisk (Bunce et al., 2007). +Multiple crossings or close approaches to the magnetodisk +are evident from the periodic decreases of the magnetic field +strength during the outbound portion of the pass. The az- +imuthal component, which stayed within [−30, 50] nT while +the background field varied by 4 orders of magnitude, re- +vealed many dynamical features of Saturn’s magnetosphere. +A northern-hemisphere high-latitude crossing of the mag- +netic field lines connecting Saturn and Enceladus occurred +during this orbit, see feature labelled as Enceladus flux tube +crossing on 02 Sep. 2017, where a strong negative Bφ (∼ +−30 nT) was observed. The sign of Bφ there is consistent +with a bend-back of Saturn’s field lines resulting from the in- +teraction of flowing magnetospheric plasma with Enceladus +(Dougherty et al., 2006), and the amplitude of the associ- +ated field-aligned currents (FACs) has been estimated to be +∼ 200 nA m−2 (Sulaiman et al., 2018). The expected auro- +ral FACs (Hunt et al., 2020) and the ubiquitous planetary +period oscillations (PPOs) (Provan et al., 2018, 2019a) are +also evident in the measured Bφ component. Provan et al. +(2019a) showed that the dual modulations by the northern +and southern PPOs are present throughout Saturn’s inner- +most magnetosphere in all three magnetic field components. +Moreover, the measured Bφ revealed a new feature in Sat- +urn’s innermost magnetosphere: a low-latitude FAC system +located near the magnetic shells connecting Saturn and its +tenuous D-ring (Dougherty et al., 2018; Khurana et al., 2018; +Provan et al., 2019b; Hunt et al., 2019). +5.3.1 Discovery of a low-latitude field-aligned +current system +Unexpected Bφ perturbations on the order of 20 nT were +consistently observed around periapses along CGF orbits +(Dougherty et al., 2018). When mapped along Saturn’s back- +ground magnetic field lines, these Bφ perturbations map to +the tenuous D-ring of Saturn and its planetward cavity (see +Figs. 5.3 & 5.4). The sharpness of the spatial variation lead- +ing to the center peak is a strong indicator that these Bφ +signatures are of magnetospheric-ionospheric origin, instead +of a deep interior origin. The predominantly positive nature +of the low-latitude Bφ perturbations (see Fig. 5.4) further +points to their toroidal nature, which must be associated +with local meridional electric currents. + +4 +Cao, Dougherty, Hunt, Bunce, Christensen, Khurana, & Kivelson +100 +101 +102 +103 +104 +|B| [nT] +Aug 30 +Aug 31 +Sep 01 +Sep 02 +Sep 03 +Sep 04 +Sep 05 +Sep 06 +UTC Time, Year =2017 +B [nT] +Enceladus fluxtube crossing +magnetodisk crossing +40 nT +400 nT +10,000 nT +Auroral FACs +Planetary Period Oscillations (PPOs) +Auroral FACs +Low-latitude +(Intra-D ring) FACs +-30 +-20 +-10 +0 +10 +20 +30 +40 +50 +10:00:00 11:00:00 12:00:00 13:00:00 14:00:00 15:00:00 16:00:00 17:00:00 +UTC Time, Sep 02 2017 +-50 +-40 +-30 +-20 +-10 +0 +10 +20 +30 +40 +50 +B [nT] +Enceladus fluxtube crossing +Southern +Auroral FACs +Northern +Auroral FACs +Low-latitude +(Intra-D ring) FACs +A +B +C +Figure 5.2 Magnetic field strength and the azimuthal +component measured during a 6.5-day Cassini Grand Finale +orbit. Panel A shows the total field strength on a log-scale, +while Panel B shows the azimuthal field component Bφ on a +linear scale. Panel C shows a zoom-in of Bφ for a few hours +around the periapsis. Figure from Cao et al. (2020). +The electric current system associated with positive low- +latitude Bφ is inter-hemispheric, with field-aligned currents +J∥ flowing from the north to the south in the magnetosphere +and return Pedersen currents JP in the ionosphere flow- +ing south to north as illustrated in Fig. 5.3B. A negative +Bφ would correspond to a reversed current loop. Khurana +et al. (2018) further proposed that such a low-latitude elec- +tric current system is driven by zonal wind shear between +the two “ends” of Saturn’s magnetic field embedded in the +ionosphere. Khurana et al. (2018) estimated the Joule dissi- +pation associated with this low-latitude current system and +found it to be ∼ 2 × 1011 W, assuming a height-integrated +conductance ∼ 9 S (siemens). This dissipation rate is simi- +lar to the estimated heating rate from solar extreme ultra- +Figure 5.3 Panel A shows the measured azimuthal magnetic +field, Bφ, around the periapsis of the first Cassini Grand Finale +orbit (Rev 271) color-coded along the trajectory of the orbit. +The high-latitude auroral FACs and the low-latitude intra-D +ring FACs are both evident from the measured Bφ. Panel B +shows a sketch of the magnetospheric-ionospheric current +system consistent with the observed positive Bφ in Saturn’s +innermost magnetosphere. JP is the Pedersen current while J∥ +is the field-aligned current. Panel A is from Dougherty et al. +(2018) while Panel B is from Khurana et al. (2018). +violet radiation at Saturn, but is not sufficient to explain +the higher-than-expected temperature in Saturn’s equato- +rial thermosphere (M¨uller-Wodarg et al., 2006). +Hunt et al. (2019) estimated the electric current density +associated with these low-latitude Bφ perturbations. Assum- +ing that the azimuthal extent of the current system is much +wider than that traversed by the spacecraft, they inferred +that the low-latitude ionospheric meridional currents are ∼ +0.5 - 1.5 MA (Mega Ampere) per radian of azimuth (see +Fig. 5.5 for two examples), similar in intensity to the auro- +ral region current (e.g., Hunt et al., 2020). Hunt et al. (2019) +then computed the current density of the associated FACs, + +2 +Bo [nT] +Auroral +20 +FACs +1.5 +15 +10 +0.5 +5 +S +C +B +A +R +0 +0 +N +-0.5 +-5 +-1 +-10 +Auroral +-1.5 +-15 +FACs +-20 +-2 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +0 +4 +A +Cassini's trajectory +300 +150 +0 +Field lines +-150 +-300 +m/s +BSaturn’s Magnetic Field at Unprecedented Detail Achieved by Cassini’s Close Encounters +5 +-30 +-15 +0 +15 +30 +BÁ [nT] +D +D +C +C +B +B +A +A +a +Untraversed +Field Region +-30 +-15 +0 +15 +30 +¢BÁ [nT] +D +D +C +C +B +B +A +A +b +-50 +54 +2.25 +-40 +46 +1.65 +-30 +37 +1.34 +-20 +28 +1.16 +-10 +20 +1.06 +0 +10 +1.03 +10 +0 +1.03 +20 +-10 +1.07 +30 +-22 +1.19 +40 +-33 +1.43 +50 +-45 +1.9 +-2.5 +0.0 +2.5 +5.0 +­ [rad s¡1] +µi [°] +µc [°] +½eq[RS] +D +D +C +C +B +B +A +A +c 1e-6 +Figure 5.4 The measured azimuthal magnetic field, Bφ, along +the Cassini Grand Finale orbits. The horizontal axis is labelled +with the mapping of the magnetic field line traversed by the +spacecraft to the ionosphere and the ring plane: θi is the +ionospheric latitude of the field line footprint in the same +hemisphere as the measurement, θc is the ionospheric latitude of +the conjugate footprint in the opposite hemisphere along the +field line, ρeq is the (cylindrical) radial distance in the +equatorial/ring plane of the field line from the center of Saturn. +The top panel shows the total observed Bφ while the middle +panel shows de-trended field ∆Bφ where a fourth order +polynomial fit has been subtracted from the data to isolate the +intra-D ring features. In both panels, colored traces represent +different CGF orbits while the black traces represent the +running median from all orbits. The bottom panel shows the +angular velocity profile of the 1-bar zonal winds at Saturn as a +function of θi, referenced to an assumed planetary rotation +period of 10 hr 34 min 13 sec. Figure from Agiwal et al. (2021). +and found it to be ∼ 5 - 10 nA m−2 (Fig. 5.5), over an order +of magnitude smaller than that of typical auroral FACs. +Provan et al. (2019b) showed that the measured low- +latitude Bφ is generally symmetric about the field-parallel +point, where the spacecraft trajectory is tangent to the back- +ground field line. This symmetry is consistent with field per- +turbations associated with inter-hemispheric FACs (also see +Fig. 3 in Khurana et al., 2018). Provan et al. (2019b) further +examined the morphology of the low-latitude Bφ signature +and showed that they can be categorized into four different +groups: ∼ 35% cases/orbits feature a single positive central +peak ∼ 20 - 40 nT (category A), ∼ 30% cases feature two +or three weaker positive peaks ∼ 10 - 20 nT (category B), +∼ 15% cases feature rather irregular central positive peaks +in regions well inside the magnetic shell connected to the D- +ring inner boundary (category C), and ∼ 20% cases feature +unique features including two with ∼ 20 - 30 nT negative +fields and two with < 10 nT fields (category U). Provan et al. +(2019b) tried to correlate these different categories with the +spacecraft altitude, local time, PPO phase, and the orbital +phase of the D-68 ringlet but found no convincing correla- +tion. +Agiwal et al. (2021) explored the possibility of temporal +variations in the thermospheric zonal winds as the origin +of the variability observed in the low-latitude Bφ. We will +discuss more about this work in section 5.4.1. +5.3.2 Discovery of Alfv´en waves planetward of the +D-ring +Southwood et al. (2021) examined the shorter timescale vari- +ations in the azimuthal magnetic field measured along the + +6 +Cao, Dougherty, Hunt, Bunce, Christensen, Khurana, & Kivelson +Figure 5.5 Ionospheric meridional current Im and field-aligned +current density j∥ i associated with the low-latitude Bφ observed +along Cassini Rev 287 (top) and Rev 292 (bottom). The vertical +lines indicate magnetic mapping to ring boundaries: the pair of +red dashed lines correspond to the outer boundary of the +A-ring, while the solid (dashed) pair of blue lines correspond to +the outer (inner) boundaries of the D-ring. Figure from Hunt +et al. (2019). +CGF orbits. They showed that Bφ oscillations with a typ- +ical amplitude of a few nT and a typical time scale of a +few minutes prevail in the innermost magnetosphere plan- +etward of the D-ring, regardless of the morphology of the +background Bφ structure. They applied cubic spline fittings +with a 5-min window and a 10-min window successively to +the measured Bφ, which extracted two bands of Bφ oscil- +lations (middle two panels of Fig. 5.6). This procedure also +isolated the large-scale background Bφ, as shown in the bot- +tom panel of Fig. 5.6. It can be seen from Fig. 5.6 that both +the large-scale background Bφ and the oscillatory Bφ are +generally confined within the magnetic shell connecting to +the outer edge of Saturn’s D-ring. The background Bφ fea- +tures a typical amplitude of 20 nT, while both bands of the +oscillatory Bφ have amplitudes on the order of 5 nT or less. +When examining the peaks (anti-nodes) and zeros (nodes) +of the Bφ oscillations with respect to the local background +planetary field (Br & Bθ), Southwood et al. (2021) noticed +that the nodes of Bφ oscillations from both bands lie close +to the proxy magnetic equator (defined as where the radial +component of the field vanishes Br = 0). For the 1-5 min +oscillations, 15 orbits show a strong clustering of a Bφ node +within 5 secs of T = 45 sec from Br = 0; the remaining 7 +orbits show a clustering of nodes between T = −12 and +16 +sec from Br = 0 (Fig. 5.7). This feature is consistent with +the odd-mode (wave number n = 1, 3, 5, etc.) of a standing +Alfv´en wave (assuming similar ionospheric electrical conduc- +tivity in the northern hemisphere and southern hemisphere). +For the 5-10 min oscillations, the picture is somewhat more +complex: 9 orbits display a magnetic node between T = −6 +and 35 sec on either side of the proxy magnetic equator +(Br = 0), another 9 orbits display a magnetic node between +T = −30 and −40 sec from the proxy magnetic equator. +There are four additional orbits with mixed behaviors, in- +cluding one with a Bφ anti-node (maximum) near the proxy +magnetic equator and two with relatively flat Bφ oscillations +near Br = 0. +Southwood et al. (2021) propose that these few minutes +Bφ oscillations planetward of Saturn’s D-ring are standing +(transverse) Alfv´en waves. The fact that 1) both oscillation +bands feature a magnetic node near the proxy magnetic +equator (Br = 0) and 2) the (temporal) spectral content +of these oscillations remains relatively simple despite the +up to a factor of two change in local field line length led +Southwood et al. (2021) to suggest that these are local field +line resonances being pumped by global magnetospheric cav- +ity modes, a coupling originally suggested by Kivelson and +Southwood (1986) for the 1 - 10 min ultra-low frequency +(ULF) magnetic pulsations in Earth’s magnetosphere. +5.3.3 The internal magnetic field: extreme +axisymmetry and high-degree magnetic moments +The small amplitude of the azimuthal component (Bφ +∼ 0.1% |B| near CGF periapses) and the identifiable +magnetospheric-ionospheric +features +in +this +component +serve as direct evidence for the extreme axisymmetry of +Saturn’s internal magnetic field. Quantitative analyses con- +firm this first impression. Two different analyses have been +performed with the CGF magnetic field measurements to +quantify the amount of non-axisymmetry in Saturn’s inter- +nal magnetic field (Cao et al., 2020): longitudinal variation +in Saturn’s magnetic equator positions and Gauss coeffi- +cients inversion with all three field components. The two +analyses yielded similar upper limits on large-scale non- +axisymmetry in Saturn’s internal magnetic field: the dipole +tilt at Saturn must be smaller than 0.007◦ (25.2 arcsecs), +while the non-axisymmetric contribution to the spherical + +Rev 287 +2.5 (a) +2.0 +1.5 +rad' +/MA +1.0 +_m +0.5 +0.0 +-0.5 +15 +(b) +10 +5 +UAA +-5 +-10 +15 +30 +60 +90 +120 +150 +Co-latitude [deg] +Rev 292 +1.5 +1.0 +0.5 +/MA rad' +0.0 +-0.5 +-1.0 +(b) +5日 +/nA +-5 E +-10 +-15 +20 +30 +60 +90 +120 +150 +Co-latitude [deg]Saturn’s Magnetic Field at Unprecedented Detail Achieved by Cassini’s Close Encounters +7 +Figure 5.6 Oscillations in the azimuthal magnetic field +measured along 22 Cassini Grand Finale orbits as identified by +Southwood et al. (2021). The bottom panel shows the +background Bφ, the middle two panels show the Bφ oscillations +in the 1 - 5 min band and the 5 - 10 min band, and the top +panel shows the magnetic mapping of the spacecraft position to +the ring plane in which the mapping to the D-ring is highlighted +with the grey shade. The data is organized in the time frame in +which T = 0 represents the time when the spacecraft was on the +innermost magnetic shell (L-shell). L-shell describes the set of +field lines which cross the equator at the radial distance defined +by the numerical value of L. Figure from Southwood et al. +(2021). +Figure 5.7 Azimuthal magnetic field oscillations in the 1-5 min +band as a function of time from the crossing of the proxy +magnetic equator (where the background radial field, Br = 0). +Orbit Rev numbers are indicated on each panel. The top panel +displays 15 CGF orbits during which a magnetic node occurred +between T = 40 and 50 sec (vertical shaded bar) from Br = 0. +The lower panel displays 7 CGF orbits that show a clustering of +nodes between T= −12 and +16 sec (vertical shaded bar). +Figure from Southwood et al. (2021). + +8 +Cao, Dougherty, Hunt, Bunce, Christensen, Khurana, & Kivelson +harmonic (SH) degree 2 & 3 magnetic moments are less +than 0.15% (Cao et al., 2020). +Although extremely axisymmetric, Saturn’s internal mag- +netic field displays surprisingly rich (spatial) spectral con- +tent, corresponding to an axisymmetric magnetic field on +many different length-scales (Dougherty et al., 2018; Cao +et al., 2020). Two different mathematical representations +of the internal magnetic field were adopted by Cao et al. +(2020): the Gauss coefficients representation and the Green’s +function representation. The Green’s function relates the +vector magnetic field at the point of observation to its ra- +dial component at a reference surface inside the planet. For +technical details of the two different representations, the in- +terested readers can refer to the Appendices A & B of Cao +et al. (2020) and references therein. +Utilizing the Green’s function, Cao et al. (2020) computed +the sensitivity of the magnetic field measurements along the +CGF orbits to the axisymmetric field at Saturn’s “dynamo +surface”, taken to be the a = 0.75 RS, c = 0.6993 RS iso- +baric surface. Here a and c represent the equatorial and +polar radii of the isobaric surface, respectively. The shape +of the isobaric surface was determined from interior struc- +tural models constrained by Cassini gravity measurements +(Iess et al., 2019; Militzer et al., 2019). The choice for the +depth of “dynamo surface” is guided by an estimation of +where the local magnetic Reynolds number reaches order +unity (Dougherty et al., 2018; Cao et al., 2020). Cao et al. +(2020) showed that magnetic field measurements along the +CGF orbits are sensitive to Saturn’s large-scale (SH degree +n ≤ 3) axisymmetric magnetic field at depth up to very +high latitude (e.g., ±80◦) but likely are not very sensitive to +Saturn’s small-scale (n > 3) axisymmetric magnetic field be- +yond ±60◦ latitude. These sensitivity characteristics result +from 1) the particular trajectory of CGF orbits with closest +approach near the equator and 2) the highly axisymmetric +nature of Saturn’s internal magnetic field. +After determining the magnetodisk field (Bunce et al., +2007) contributions orbit-by-orbit with CGF magnetic field +measurements slightly away from the periapsis and removing +them from the measurements, Cao et al. (2020) performed +inversion analysis on the CGF magnetic field data to extract +features of Saturn’s internal magnetic field with both the +Gauss coefficients and the Green’s function representations. +Inversion analysis with the Gauss coefficients representation +revealed that axisymmetric Gauss coefficients up to at least +SH degree-9 are needed to give a reasonable match to the +measurements (with Root-Mean-Square residual < 5 nT). +Cao et al. (2020) noted that, on the dynamo surface, the +contribution of the internal magnetic field corresponding to +the degrees 4 - 9 axisymmetric Gauss coefficients is substan- +tially larger beyond ±60◦ latitude than at lower latitudes +(see Fig. 12 in Cao et al., 2020). The Green’s function sensi- +tivity analysis of the data does not support this result. Cao +et al. (2020) employed the technique of regularized inversion, +which introduces a damping parameter, γ, that regularizes +the behavior of the model while simultaneously fitting the +measurements (for technical details, see section 5.1.2 in Cao +et al., 2020). +Fig. 5.8 shows characteristic properties of a few selected +models, both in spectral space (the amplitude of g0n) and in +real space (∆Br/|B| on the “dynamo surface”), as a func- +tion of the damping parameter γ which sets the relative +importance of model constraints. It can be seen that models +tend to feature 1) large amplitude of ∆Br at high-latitude +when the damping is weak, 2) significantly reduced ampli- +tude at high latitude when some damping is allowed, and 3) +a substantially worse fit to the data when the damping is +strong (see the RMS residual values in the legend in Figure +5.8). Fig. 5.8 further shows that the small-scale field struc- +ture within ±60◦ latitude is well-resolved regardless of the +field behavior at higher latitudes: there are eight alternat- +ing latitudinal bands of radial magnetic fluxes between ±60◦ +latitude (see also Fig. 5.9B). The model that features a well- +behaved field at the “dynamo surface” and a good match to +the data, corresponding to γ = 0.03 (thick red traces in Fig. +5.8), was selected as the preferred model and was named the +Cassini 11+ model (see Table 5.1 for the coefficients). +Cao et al. (2020) also performed inversion analysis on the +CGF magnetic field measurements with the Green’s function +representation. Compared to the Cassini 11+ field model, +the Green’s function analysis returned an almost identical +small-scale axisymmetric magnetic field within ±60◦ lati- +tude on the “dynamo surface”, but (as expected) produced +different field behaviors at higher latitudes (see their Fig. +15). +With a robust understanding of which features are well- +resolved by the CGF data, Cao et al. (2020) examined the +characteristics of Saturn’s internal magnetic field at the “dy- +namo surface” (the a = 0.75 RS, c = 0.6993 RS isobaric +surface). As shown in Fig. 5.9A, Saturn’s large-scale (SH +degrees 1 - 3) magnetic field is relatively weak in the equa- +torial region: Br remains less than 1/3 of its peak value +within ±40◦ latitude (panel A). As shown in Fig. 5.9B, +Saturn’s small-scale (beyond SH degree-3) magnetic field +features eight alternating zonal bands between ±60◦ lati- +tude (panel B), with typical amplitude ∼ 5 - 10% of the +background large-scale field (see Fig. 5.8B). Like that of the +large-scale field, the polarity of the eight alternating mag- +netic bands displays predominant anti-symmetry with re- +spect to the equator (dipole-octupole-like), while their am- +plitudes are stronger at northern mid-latitudes. Cao et al. +(2020) further noted that the number of alternating mag- +netic bands at the a = 0.75 isobaric surface coincides with +the number of alternating zonal wind bands if one projects +the observed surface wind along the spin-axis to the same +depth. Both the small-scale magnetic bands and the pro- +jected surface zonal winds feature typical latitudinal widths +of ∼ 15◦. +It is interesting to examine the pattern of the Gauss co- +efficients of Saturn’s internal field (see Fig. 5.8A and Ta- +ble 5.1). Overall, there is a dramatic drop in amplitude be- +yond degree-3: g0 +4 drops by more than an order-of-magnitude +compared to g0 +2 and g0 +3, and all higher-degree moments are +smaller than g0 +4. The degree-5 moment, g0 +5, seems to be +strongly suppressed: it is almost another order-of-magnitude +smaller compared to g0 +4. However, the degree-7 moment, g0 +7, +bounces back by a factor 5 - 7 compared to g0 +5. g0 +8 to g0 +11 + +Saturn’s Magnetic Field at Unprecedented Detail Achieved by Cassini’s Close Encounters +9 +10 +11 +12 +13 +14 +Spherical Harmonic Degree +10-1 +100 +101 +102 +103 +104 +105 +=3 +10-3,RMS=4.58 nT +=1 +10-3,RMS=4.66 nT +=3 +10-2,RMS=4.73 nT +=0.1,RMS=5.32 nT +=0.2,RMS=7.64 nT +|gn +0| [nT] +−80 +−60 +−40 +−20 +0 +20 +40 +60 +80 +Latitude [deg] +B +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 + Br/|B| @ a=0.75 RS +A +5 +6 +7 +8 +9 +1 +2 +3 +4 +Figure 5.8 The axisymmetric Gauss coefficients g0 +n versus SH +degree (panel A) and the small-scale (g0 +n for n > 3) internal +magnetic field of Saturn versus latitude (panel B) derived from +Cassini Grand Finale magnetic field measurements with +regularized inversion. The damping parameter γ and the RMS +residual associated with each solution are shown in the legend in +panel A. In panel B, the small-scale magnetic field, ∆Br, +corresponds to g0 +n with n > 3. It is normalized with respect to +the strength of the background field, |B|, corresponding to g0 +n +with n ≤ 3. Figure from Cao et al. (2020). +are comparable to g0 +5 and g0 +6. Although an overall decay- +ing (with SH degree) trend is present, 1) the strong dip in +g0 +5 and 2) the seemingly separate slopes for the low-degree +moments and the high-degree moments are unexpected. +Cao et al. (2020) attempted to extract an electromagnetic +induction signal from Saturn’s interior, with the orbit-to- +orbit varying magnetodisk BZ field as the external sound- +ing signal. They solved for the orbit-to-orbit varying internal +dipole ∆g0 +1 after removing the Cassini 11+ model, and then +compared those values to the orbit-to-orbit varying magne- +todisk field ∆BZ. Although the expected induction signal is +within 1σ of a formal inversion analysis of ∆g0 +1 versus ∆BZ, +the large scatter in the data precluded a definitive constraint +on the induction depth inside Saturn. +Br @ a=0.75 RS +Br @ a=0.75 RS + [nT] +A +B +60° N +60° S +40° S +40° N +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +105 +-8000 +-6000 +-4000 +-2000 +0 +2000 +4000 +6000 +8000 + [nT] +Figure 5.9 Saturn’s large scale (g0 +n for n ≤ 3) and small scale +(g0 +n for n > 3) radial magnetic field at the a = 0.75 RS, +c = 0.6993 RS isobaric surface according to the Cassini 11+ +field model. Saturn’s large scale radial magnetic field at this +depth features a relatively weak equatorial region, Br remains +less than 50,000 nT (<1/3 of its peak value) between ±40◦. +Saturn’s small-scale magnetic field at this depth features eight +alternating bands between ±60◦, with typical amplitude of ∼ +5%–10% of the background field. Figure from Cao et al. (2020). +[nT] +Cassini 11 +Cassini 11+ +g0 +1 +21140 +21141 +g0 +2 +1581 +1583 +g0 +3 +2260 +2262 +g0 +4 +91 +95 +g0 +5 +12.6 +10.3 +g0 +6 +17.2 +17.4 +g0 +7 +−59.6 +−68.8 +g0 +8 +−10.5 +−15.5 +g0 +9 +−12.9 +−24.2 +g0 +10 +15 +9.0 +g0 +11 +18 +11.3 +g0 +12 +−2.8 +g0 +13 +−2.4 +g0 +14 +−0.8 +Table 5.1. Gauss coefficients of the Cassini 11 model +(Dougherty et al., 2018) and the Cassini 11+ model (Cao +et al., 2020) for Saturn. These coefficients refer to a sur- +face radius RS=60268 km. The Cassini 11 model was con- +structed from magnetic field measurements from the first ten +Cassini Grand Finale orbits while the Cassini 11+ model +was constructed using data from all 22.5 CGF orbits. + +10 +Cao, Dougherty, Hunt, Bunce, Christensen, Khurana, & Kivelson +Figure 5.10 Data-model comparisons between the observed +Bφ along two CGF orbits and a kinematic ionospheric zonal +shear model with different assumed zonal flow profiles. Panels a +& c display Bφ, with the solid grey traces showing the +de-trended observations, dashed black traces showing the model +associated with the baseline zonal flow profile, and red dashed +traces showing the model associated with the perturbed zonal +flow profile. Panels b & d display the angular velocity profile, +with the black traces showing the baseline model, and the solid +red traces showing the perturbed model. It can be seen from the +data-model comparison for Rev 292 (panels c & d) that one only +needs to reduce the amplitude of the zonal flow in the southern +hemisphere (instead of reversing the wind) to flip the sign of the +low-latitude Bφ. Figure from Agiwal et al. (2021). +5.4 Implications and open questions +Here we briefly discuss the implications and open questions +for Saturn’s innermost magnetosphere, ionosphere, and in- +terior that are closely related to the magnetic field investiga- +tion. As will become clear, our understanding of the physical +mechanisms behind many of the observed phenomena is still +in a preliminary stage: most of our interpretations are kine- +matic. A fully dynamic understanding of the Saturn system +has not been obtained. +5.4.1 Saturn’s equatorial ionosphere: zonal shear +and atmospheric-wave-induced temporal +variability? +As introduced in section 5.3.1, a considerable amount of +orbit-to-orbit variability has been observed in the large-scale +low-latitude Bφ peak along the Cassini Grand Finale orbits +(e.g., see Fig. 5.4). Although most of the CGF orbits feature +positive large-scale low-latitude Bφ around the periapses, a +few orbits (e.g. Revs 286 & 292) feature strong negative low- +latitude Bφ (see the categorization by Provan et al. 2019b). +As explained by Khurana et al. (2018), due to the northward +shift of Saturn’s magnetic equator, a north-south symmetric +eastward zonal flow in Saturn’s ionosphere could naturally +lead to a low-latitude positive Bφ (see Fig. 5.3B). +M¨uller-Wodarg et al. (2019) and Brown et al. (2020) re- +ported atmospheric waves in Saturn’s thermosphere with +Cassini Ion Neutral Mass Spectrometer (INMS) and Ultra- +violet Imaging Spectrograph (UVIS) measurements during +the Cassini Grand Finale. These atmospheric waves could +assist the vertical transport of the 1-bar wind structure to +higher altitude, but could also introduce temporal variability +to the higher altitude winds. It is thus natural to consider +time variability in the zonal shear in Saturn’s ionosphere +as the cause of the observed variability in the large-scale +Bφ around the CGF periapses. Agiwal et al. (2021) inves- +tigated quantitatively the amount of variation in the iono- +spheric zonal wind shear needed to account for the observed +variability in the low-latitude Bφ. +The technical starting point of Agiwal et al. (2021) is the +rigorously derived formula relating the steady-state zonal +shear in the ionosphere to the meridional current in the iono- +sphere (under a thin-shell approximation of the ionosphere) +and Bφ along the field line, first presented in Appendix A of +Provan et al. (2019b). The steady-state meridional current, +Im, associated with the differential ionospheric wind drag is +shown to be +Im = +ΣP NΣP S (ΩnS − ΩnN) +� +ΣP N +|biS| +ρ2 +iSB2 +iS + ΣP S +|biN| +ρ2 +iNB2 +iN +�, +(5.1) +where ΣP N,S are the height-integrated Pedersen conduc- +tivities in the northern and southern ionospheres, ΩnN,S +are the angular velocities of the neutral gas in the iono- +spheres, BiN,S are the planetary poloidal fields in the two +ionospheres, while biN,S are the poloidal field components +normal to the ionosphere. Assuming axisymmetry, the asso- +ciated azimuthal magnetic field at any point along the field +line is +Bφ = µ0Im +ρ +, +(5.2) +where µ0 is the permeability of free-space, and ρ is the cylin- +drical radius from the magnetic axis (also the spin-axis) of +Saturn. It should be immediately clear from Eqs. 5.1 & 5.2 +that 1) the difference between the angular velocities at the +two ends of a field line, ΩnS − ΩnN, determines the sign of +Bφ, while 2) the amplitude of the height-integrated Pedersen +conductivity modulates the amplitude of the Bφ. These for- +mulae also highlight the degenerate nature of the amplitude +of the zonal wind shear and the ionospheric Pedersen con- +ductivity if the magnetic field is the only observable quan- +tity. Bφ would remain the same if we increase the zonal wind +shear by a factor 2, while decreasing the height-integrated +ionospheric Pedersen conductivity in both hemispheres by +the same factor. + +Rev 271 +a +40 +A +B +B +A +20 +[nT] +0 +0 +B +-20 +MAG +Bs1 +B? +(s=1.4 +b +F1e-6 +[s pe] +2.5 +0.0 +2m +-2.5 +[。]'0 +50 +40 +30 +20 +10 +-10 +0 +-20 +-30 +-40 +-50 +Rev 292 +c +40 +C +B +B +C +D +A +A +D +20 +0 +0 +B +-20 +B*(s =0.2) +d +1e-6 +2.5 +0.0 +-2.5 +[。]'0 +50 +30 +10 +-50 +40 +20 +0 +-10 +-20 +-30 +-40Saturn’s Magnetic Field at Unprecedented Detail Achieved by Cassini’s Close Encounters +11 +Vriesema et al. (2020) investigated the observed low- +latitude Bφ at Saturn with a height resolved ionospheric- +thermospheric electrodynamics model. The Vriesema et al. +(2020) model also assumes steady-state and axisymmetry, +and is a kinematic model. Agiwal et al. (2021) demonstrated +the equivalence of the Provan et al. (2019b) formulation +(Eqs. 5.1 & 5.2) and the Vriesema et al. (2020) formulation +in the thin ionospheric current layer approximation. +With an ionospheric conductivity model (M¨uller-Wodarg +et al., 2006; M¨uller-Wodarg et al., 2019) evaluated under +northern summer conditions at Saturn, Agiwal et al. (2021) +adopted the latitudinal profile of the 1-bar atmospheric +zonal winds but decreased their amplitude by a factor of two +as the starting point of Saturn’s low-latitude thermospheric +wind. By systematically perturbing the zonal winds in both +hemispheres, they showed that variability in the equato- +rial thermospheric wind up to 350 m/s can account for the +observed variability in the low-latitude Bφ along CGF or- +bits. Fig. 5.10 showcases two examples from Agiwal et al. +(2021). The data-model comparison for Rev 292 shows that +one only needs to reduce the amplitude of the zonal flow +in the southern hemisphere (solid red traces in panel d), +instead of reversing the flow direction, to flip the sign of +Bφ around the periapses of the CGF orbits. Future inves- +tigations that model the lower atmosphere-thermosphere- +ionosphere-inner magnetosphere interactions with the rele- +vant dynamical processes such as gravity waves are needed +to evaluate whether such variability in the thermospheric +wind is plausible at Saturn. +Furthermore, will there be a strong seasonal dependence +of the thermospheric wind shear and the low-latitude field- +aligned current system at Saturn? Will the low-latitude Bφ +flip to mainly negative when the season switches to northern +winter at Saturn? +5.4.2 Saturn’s interior: a tale of two dynamos? +The fact that Saturn’s internal magnetic field is extremely +axisymmetric and yet full of structures in the latitudinal +direction (e.g., see Fig. 5.9) is intriguing. Dougherty et al. +(2018) and Cao et al. (2020) proposed that two spatially +separate dynamos, one in the deep metallic hydrogen layer +and the other in the semi-conducting layer, are responsi- +ble for this intriguing magnetic field behavior (e.g., see Fig. +5.3A). The deep dynamo is responsible for generating the +large-scale dipolar background field, while the shallow sec- +ondary dynamo (Cao and Stevenson, 2017b) is responsible +for generating the small-scale latitudinally banded magnetic +perturbations. +Gastine et al. (2014) observed localized secondary dynamo +action at low-latitudes in their 3D Jovian dynamo simula- +tions which they attributed to interaction with deep zonal +winds. Cao and Stevenson (2017b) proposed the possibil- +ity of a global secondary dynamo inside Jupiter and Saturn, +and illustrated this process with mean-field electrodynamics +(Moffatt and Dormy, 2019). Three key ingredients of their +envisioned secondary dynamo inside giant planets are: 1) +the background magnetic field, B0, generated by the deep +dynamo; 2) differential rotation (zonal flows) in the semi- +conducting layer which generates the toroidal magnetic field, +BT, via the ω−effect (this process bears similarity to the +proposed process of zonal wind shear in the ionosphere gen- +erating the low-latitude Bφ); and 3) small-scale helical con- +vection in the semi-conducting layer providing the critical +α−effect that generates the externally observable poloidal +magnetic field, ∆BP, from BT. Both BT and ∆BP are +expected to be spatially correlated with the zonal flows in +the semi-conducting layer. Furthermore, Cao and Stevenson +(2017b) pointed out that the ω−effect and the α−effect in +the secondary dynamo may operate at different depths: BT +resulting from the ω−effect can be generated at a relatively +shallow depth and then diffuse several scale-heights down- +ward due to the rapidly increasing electrical conductivity as +a function of depth in the semi-conducting layer. +Galanti and Kaspi (2020) analyzed Saturn’s gravity mo- +ments and magnetic moments jointly, aiming to construct +a deep differential rotation profile for Saturn. Their analy- +sis of Saturn’s dynamic gravity moments was based on the +Cassini radio tracking observations (Iess et al., 2019) and the +diagnostic thermal wind (TW) relation (Kaspi, 2013; Cao +and Stevenson, 2017a). Their analysis of Saturn’s magnetic +moments was based on the Cassini 11 model (Dougherty +et al., 2018) and the (kinematic) mean-field electrodynamics +(MFED) model of Cao and Stevenson (2017b). They showed +that the same deep zonal wind profile, which is only slightly +modified from the observed surface zonal winds around ±30◦ +latitude, and penetrating to about 7500 km below the 1- +bar level can account for both the dynamic gravity field +and the small-scale magnetic bands at Saturn. This is an +encouraging result, and calls for future dynamic investiga- +tions, in which the physical processes that control the depth +of rapid zonal flows inside Saturn (and Jupiter) need to be +self-consistently modeled. +5.4.3 Saturn’s extremely axisymmetric magnetic +field: electromagnetic filtering or a double-diffusive +dynamo in the diffusive core? +The axisymmetry of Saturn’s internal magnetic field has +long been a puzzle. Here we briefly explain the challenge +and then describe the proposed kinematic solution and some +dynamical tests with 3D numerical dynamo simulations. We +conclude this subsection with a discussion of the new chal- +lenges raised by the inference of a large diffusive core inside +Saturn (Mankovich and Fuller, 2021). +From a theoretical point of view, Cowling’s theorem +(Cowling, 1933) precludes the possibility of maintaining a +purely axisymmetric magnetic field in a steady-state via +MHD dynamos. However, Cowling’s theorem itself does not +place any lower limit on the amount of non-axisymmetry +necessary to maintain dynamo action. When examining +the observational evidence in the solar system, Earth and +Jupiter feature a modest amount of non-axisymmetry with +dipole tilts ∼ 10◦, Uranus and Neptune feature a significant +amount of non-axisymmetry with dipole tilts ∼ 50◦ (e.g., see +Table 7.2 in Kivelson and Bagenal, 2014). For Mercury and +Ganymede, the total amount of non-axisymmetry in their +large-scale internal magnetic field are less clear at this stage + +12 +Cao, Dougherty, Hunt, Bunce, Christensen, Khurana, & Kivelson +while their dipole tilts have been estimated to be on the +order of 1◦. +The most widely invoked explanation for Saturn’s very +axisymmetric magnetic field is the one proposed by Steven- +son (1980, 1982). The essence of this mechanism is electro- +magnetic (EM) filtering: if there exists a passive, electrically +conducting layer on top of a regular dynamo, and if this fil- +ter layer is rotating at a different angular speed compared +to that of the deep dynamo, then any non-axisymmetric +magnetic field from the deep dynamo would appear as a +time-varying field to this layer and thus be electromagnetic +filtered while the axisymmetric part of the magnetic field +would appear as time-stationary and pass through. For Sat- +urn in particular, Stevenson (1980) proposed that helium +rain-out would create a stably stratified layer (with no large- +scale overturning motion) on top of Saturn’s deep dynamo. +One should remember that both the qualitative descrip- +tion above and the quantitative model presented in Steven- +son (1982) are kinematic: no dynamical feedback between +the deep dynamo and the stably stratified layer has been +considered. Within this kinematic framework, Cao et al. +(2020) showed that a 0.007◦ dipole tilt requires a stable +layer thicker than 2500 km if the deep dynamo features a +Jupiter-like ∼ 10◦ dipole tilt. This estimation represents a +lower limit on the thickness of the stable layer, as the dy- +namical feedback from the non-axisymmetric magnetic field +on the flows in the stable layer is expected to reduce the +filtering efficiency. +3D numerical dynamo models have also been constructed +to reproduce the highly axisymmetric magnetic field of Sat- +urn. Christensen and Wicht (2008); Stanley and Moham- +madi (2008); Stanley (2010); Christensen (2018); Gastine +et al. (2020); Yan and Stanley (2021) employed numerical +MHD models under the Boussinesq approximation to inves- +tigate the effects of a stable layer on top of a convective +dynamo. In these models, background density and electrical +conductivity are assumed to be constant while the Ohmic +and viscous dissipation are assumed to be negligible. While +these assumptions are approximately valid for the cores of +terrestrial planets, they are quite different from the condi- +tions inside giant planets, which feature large variations in +background density, temperature, and electrical conductiv- +ity as a function of depth (French et al., 2012). +These simplified models illustrate many dynamical pro- +cesses involved in the interaction between the convective +dynamo and the stable layer. The electromagnetic filtering +effect proposed by Stevenson (1982) was indeed observed in +many of these studies (e.g. Christensen and Wicht, 2008; +Stanley, 2010; Gastine et al., 2020). However, the situation +is more complicated than the kinematic picture. Convection +eddies can penetrate into the stably stratified layer (Gastine +et al., 2020), horizontal circulation can erase certain types +of thermal heterogeneity imposed from above (Christensen, +2018), certain types of zonal flow in the stable layer could +destabilize the dynamo-generated magnetic field instead of +axisymmetrizing it (Stanley and Mohammadi, 2008). In one +of the latest 3D Boussinesq modeling effort, Yan and Stan- +ley (2021) achieved a dipole tilt ∼ 0.066◦ with a thick sta- +bly stratified layer (∼ 0.28 RS) combined with a particular +type of heterogeneous heat flux variation on top of the stable +layer. This is still about one order of magnitude larger than +the latest observational upper bound at Saturn (Dougherty +et al., 2018; Cao et al., 2020). Part of this discrepancy could +result from the relatively low magnetic Reynolds number as- +sociated with differential rotation in the stable layer in the +numerical simulations compared to the realistic values inside +Saturn. See section 4.4.3 in Christensen et al. (2018) for a +more detailed discussion of this point. +More advanced models, where the molecular envelope and +the deep dynamo with smooth transition in material prop- +erties were simulated simultaneously, have also been con- +structed under the anelastic approximation (Jones, 2014; +Dietrich and Jones, 2018; Gastine and Wicht, 2021). Most +of the anelastic dynamo models were constructed without +any stable layer (Duarte et al., 2013; Jones, 2014; Dietrich +and Jones, 2018), the resulting magnetic field in the dipolar +branch features a modest amount of non-axisymmetry sim- +ilar to that observed at Jupiter. In a survey of the effects +of relative thickness of the molecular envelope, Dietrich and +Jones (2018) observed an interesting oscillatory dynamo in +which the magnetic field flips its polarity regularly and re- +sembles Saturn’s main magnetic field qualitatively during +about a quarter of every cycle. The dipole tilts during such +times are typically 1-2◦. +Three recent studies implemented stable layers in anelas- +tic models applicable to Jupiter and Saturn. Dietrich and +Wicht (2018) constructed a hydrodynamic model for Sat- +urn with a sandwiched stable layer and analyzed the depth +and mechanism of penetrative convection. Christensen et al. +(2020) investigated how a stable layer might work with the +dynamo-generated magnetic field to truncate deep zonal +flows in a 2.5D (axisymmetric) set-up. Gastine and Wicht +(2021) constructed one of the first 3D MHD model for gas +giants with a stably stratified layer between 0.82 RP and +0.86 RP near the molecular-metallic transition. The resul- +tant surface magnetic field are dipole-dominant with appre- +ciable amount of non-axisymmetry (see their Fig. 9), similar +to that of Jupiter. +The latest inference of a large stably stratified diffusive +core inside Saturn (Mankovich and Fuller, 2021) raised new +challenges for understanding the origin of Saturn’s axisym- +metric magnetic field. A diffusive core extending out to 0.6 +RS (see Fig. 5.1BC) leaves very little space for a traditional +convective dynamo inside Saturn. Can an MHD dynamo op- +erate in a stably stratified diffusive core? Small-scale dou- +ble diffusive or oscillatory convection are expected to exist +in the stably stratified layer(s) inside Saturn. Could these +small-scale motions produce the necessary α−effect to main- +tain an MHD dynamo and generate a highly axisymmetric +magnetic field? +5.5 Outlook for future exploration of Saturn +and other giant planets +It is appropriate to end this data-driven chapter with an out- +look for future exploration of Saturn and other giant planets, + +Saturn’s Magnetic Field at Unprecedented Detail Achieved by Cassini’s Close Encounters +13 +focusing on aspects related to magnetic field investigations. +In the Saturn system, other than the intriguing moons Ence- +ladus and Titan, four areas near Saturn that have not been +visited in-situ are 1) the low altitude region near the poles +of Saturn, 2) regions inside the orbit of F-ring at local time +sectors away from noon, 3) the equatorial region immedi- +ately above and below the rings, and 4) the atmosphere of +Saturn. Magnetic field measurements in these four areas will +help 1) resolve Saturn’s small-scale magnetic field, includ- +ing any local non-axisymmetry, near the poles, 2) map out +EM coupling between Saturn’s northern and southern iono- +spheres as well as between Saturn and its rings at different +local time sectors, 3) clarify the existence of a “ring iono- +sphere”, and 4) distinguish the ionospheric magnetic field +and the internal magnetic field from the deep interior. +Comparing different planets is an important step to dif- +ferentiate common processes from special realizations. Af- +ter the Juno mapping campaign at Jupiter, we shall have +a much better knowledge of the similarities and differences +between Jupiter and Saturn. Orbital missions to Uranus and +Neptune are needed to complete our mapping of the Solar +System giant planets. The single Voyager 2 flyby at Uranus +and Neptune informed us that Uranus and Neptune possess +fundamentally different magnetic fields compared to those +at Jupiter and Saturn (see a recent review by Soderlund +and Stanley, 2020). However, meaningful theoretical inter- +pretations of the Uranus and Neptune systems will require +significantly more constraints from in situ observations. +The ever expanding categories of exoplanets offers many +giant planets for investigation, some of which function in ex- +treme environments (e.g., the hot Jupiters). Measuring their +surface composition, wind pattern, and magnetic field can +provide unique observational constraints in vastly different +settings compared to giant planets in the solar system. As +in many other circumstances, studying the extremes could +help us understand the “norm”. Last but not least, measure- +ments and theoretical studies at the giant planets also help +us better understand planet Earth, as many processes are +common despite the differences in the parameter regimes in +which they function. +Acknowledgement +H.C. is funded by the NASA Cassini Data Analysis Program +(Grant Number 80NSSC21K1128). H.C.’s visit to Imperial +College London was funded by the Royal Society, UK grant +RP 180014. Work at Imperial College London was funded by +Science and Technology Facilities Council (STFC), UK con- +solidated grant ST/N000692/1. M.K.D. is funded by Royal +Society, UK Research Professorship RP140004. Work at the +University of Leicester was funded by STFC, UK consol- +idated grant ST/N000749/1. E.J.B. was supported by a +Royal Society Wolfson Research Merit Award. +References +Agiwal, O., Cao, H., Cowley, S. W. H., et al. 2021. 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Recipe for a Saturn-like dynamo. +AGU Advances, 2(2), e00318. + diff --git a/fNE0T4oBgHgl3EQf5wLJ/content/tmp_files/load_file.txt b/fNE0T4oBgHgl3EQf5wLJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b4e22f03cc79888236e00db6d8f13262d173e1c5 --- /dev/null +++ b/fNE0T4oBgHgl3EQf5wLJ/content/tmp_files/load_file.txt @@ -0,0 +1,1238 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf,len=1237 +page_content='5 Saturn’s Magnetic Field at Unprecedented Detail Achieved by Cassini’s Close Encounters H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' CAO1,2,3, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='K.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' KHURANA1, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' KIVELSON1 1 Department of Earth, Planetary, and Space Sciences, Univer- sity of California, Los Angeles, Charles Young Drive East, Los Angeles, CA 90095, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 2 Department of Earth and Planetary Sciences, Harvard Univer- sity, 20 Oxford Street, Cambridge, MA 02138, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 3 Physics Department, The Blackett Laboratory, Imperial Col- lege London, London, SW7 2AZ, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 4 Department of Physics and Astronomy, University of Leicester, Leicester, LE1 7RH, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5 Max Planck Institute for Solar System Research, Justus-von- Liebig-Weg 3, 37077 G¨ottingen, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Copyright Notice The Chapter, “Saturn’s Magnetic Field at Unprecedented Detail Achieved by Cassini’s Close Encounters”, is to be published by Cambridge University Press as part of a multi-volume work edited by Kevin Baines, Michael Flasar, Norbert Krupp, and Thomas Stallard, entitled “Cassini at Saturn: The Grand Finale” (‘the Volume’) © in the Chapter, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Cao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Dougherty, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Hunt, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Bunce, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Christensen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Khurana and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Kivelson, © in the Volume, Cambridge University Press NB: The copy of the Chapter, as displayed on this website, is a draft, pre-publication copy only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The final, published version of the Chapter will be available to purchase through Cambridge University Press and other standard distribution channels as part of the wider, edited Volume, once published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' This draft copy is made available for personal use only and must not be sold or re-distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Abstract The last 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 orbits of the Cassini mission brought the spacecraft to less than 3000 km from Saturn’s 1-bar sur- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' These close encounters offered an unprecedented view of Saturn’s magnetic field, including contributions from the internal dynamo, the ionosphere, and the magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' In this chapter, we highlight the new picture of Saturn’s magnetic field from the Cassini mission including the per- sistent yet time-varying low-latitude field-aligned currents, Alfv´en waves planetward of the D-ring, extreme axisymme- try, and high-degree magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' We then discuss the implications and new questions raised for Saturn’s innermost magnetosphere, equatorial ionosphere, and interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' We con- clude this chapter with an outlook for the future exploration of Saturn and other giant planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1 Introduction Magnetic field investigations have played central roles in our multi-disciplinary exploration of the Saturn system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', Dougherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The intrinsic magnetic field, origi- nating in the deep interior, determines key properties of the electromagnetic environment around planet Saturn and its rings and moons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The Grand Finale phase of the Cassini mission (Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' to Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 2017) brought the spacecraft to ex- treme proximity to Saturn, its trajectory passing through the gap between its upper atmosphere and its innermost ring, a region never visited before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The Cassini Grand Fi- nale (CGF) orbits provided an unprecedented opportunity to decipher Saturn’s interior, ionosphere, and the innermost magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' In the conventional picture, the bulk interior of Saturn is qualitatively divided into four layers based on composi- tion or material properties (Militzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2019): a central rock-icy core, a metallic hydrogen layer (Weir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 1996), a helium rain layer (Morales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Brygoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2021), and the outer molecular hydrogen layer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Given our current understanding of material properties in the relevant pressure-temperature ranges, the transition be- tween each adjacent pair of layers likely is smooth (Weir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Wilson and Militzer, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Employing ring seis- mology in addition to gravity, Mankovich and Fuller (2021) showed that a diffusive, stably-stratified core could extend to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='6 Saturn radii (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1BC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The origin (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', giant im- pact versus core erosion) and dynamical consequences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', flow characteristics and magnetic field generation) of such an extended diffusive core inside Saturn are active areas of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Convective motion, either in the form of overturning convection or double diffusive convection (Garaud, 2018), is primarily driven by the cooling of the planet and the subsequent phase-separation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', helium rain) (Stevenson, 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Magnetic field generation is a natural outcome of rapidly rotating overturning convection in electrically con- ducting fluids, a process commonly referred to as magne- tohydrodynamic (MHD) dynamo action (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', Moffatt and Dormy, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Whether double diffusive convection can sup- port planetary-scale magnetic field generation still needs to be clarified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Measuring the characteristics of the intrinsic magnetic field, including its time variations, would provide key observational constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='02756v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='EP] 7 Jan 2023 2 Cao, Dougherty, Hunt, Bunce, Christensen, Khurana, & Kivelson Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1 Pictures of Saturn’s bulk interior structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Panel A shows a classical four-layer model of Saturn’s interior structure constructed to match Saturn’s gravity field: a molecular hydrogen layer, a helium rain layer, a metallic hydrogen layer, and a rock-ice core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Panels B & C depict a new picture of Saturn’s interior with a large diffuse core extending to about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='6 Saturn radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' In this new picture, the heavy element (Z) abundance features a gradual transition from the center of Saturn to about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='6 Saturn radii (panel B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The diffuse core of Saturn is stably stratified, featuring Brunt-V¨ais¨al¨a frequencies N as high as twice that of the natural frequency of Saturn ωdyn = (GMS/R3 S)1/2 where G is the gravitational constant, MS is the mass of Saturn, and RS is the radius of Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Panel A is from Militzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2019), while Panels B & C are from Mankovich and Fuller (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Another common feature of rapidly rotating convection is the emergence of differential rotation, also referred to as zonal flows or jets, in which different parts of the fluid fea- ture different mean angular velocities with respect to the spin-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The zonal flows on Saturn’s surface feature equa- torial super-rotation that are about 4% faster than the bulk rotation, and alternating bands of super-rotation and sub- rotation in the off-equatorial region (Read et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Deep zonal flows will inevitably interact with the planetary mag- netic field at depths with even modest electrical conductivity (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Cao and Stevenson, 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' For example, the Lorentz force associated with the interaction could play a role in truncating the zonal flow (Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Gastine and Wicht, 2021) and the zonal flow could modify the magnetic field (Cao and Stevenson, 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The bulk characteristics of Saturn’s ionosphere have been constrained with radio occultation and ground-based H+ 3 emission measurements (Kliore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' These measurements revealed a highly dynamic iono- sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' In addition to the expected strong H+ 3 emission at auroral latitudes, the mid-to-low latitude H+ 3 emission at Saturn shows distinct patterns correlated with the spatial structure of the rings when mapped along magnetic field lines (O’Donoghue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' This led to the suggestion of material in-fall from the rings along magnetic field lines to the ionosphere of Saturn, a phenomenon sometimes re- ferred to as “ring rain” (Connerney and Waite, 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The material exchange and electromagnetic interaction between Saturn and its rings likely are defining factors of the inner- most magnetosphere at Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Moreover, these interactions could shape the long term evolution of Saturn’s rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' In section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2 we present a brief overview of pre-Cassini Grand Finale knowledge of Saturn’s magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' In sec- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='3, we summarize the new picture of Saturn’s magnetic field revealed by the Cassini Grand Finale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' In section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='4, we discuss the implications for Saturn’s innermost magne- tosphere, ionosphere, and interior as well as open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' In section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5, we offer our outlook for future exploration of Saturn and other giant planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2 A brief overview of pre-Cassini Grand Finale knowledge of Saturn’s magnetic field Saturn’s magnetic field was discovered during the Pioneer 11 Saturn flyby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Together with data from the subsequent two flybys by Voyager 1 & 2 , these measurements revealed that Saturn’s large-scale internal magnetic field is dominated by the axial dipole, with ∼10% contributions from axial quadrupole and octupole magnetic moments when evalu- ated on the 1-bar surface (Davis and Smith, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Saturn’s surface magnetic field strength is slightly weaker than that of the Earth and almost one order of magnitude weaker than that of Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' This relatively weak surface field at Saturn was a surprise given the relatively strong surface heat flux measured at Saturn (Ingersoll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' A bigger surprise of Saturn’s internal magnetic field was the seeming lack of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='0 Molecular hydrogen B (helium depleted) ¥－2 3 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 log Likelihood Helium rain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='0 Metallic hydrogen (helium rich) 2 Linear Wdyn Smooth Rock-ice 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='0 core r/Rs ASaturn’s Magnetic Field at Unprecedented Detail Achieved by Cassini’s Close Encounters 3 departure from symmetry around the spin-axis, an unex- pected feature based on a naive interpretation of Cowling’s theorem (Cowling, 1933) which precludes a purely axisym- metric magnetic field from being maintained by dynamo ac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The sparse spatial coverage of these three flybys did not yield a stringent upper-limit on the departure from ax- isymmetry in Saturn’s internal magnetic field: a dipole tilt on the order of 1◦ was still permitted by these early mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The Cassini magnetometer measurements in the first few years offered reasonably good latitude-longitude coverage for investigating non-axisymmetry in Saturn’s internal mag- netic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Sterenborg and Bloxham (2010) analyzed the Cassini magnetometer data between Jun 2004 and May 2008 within 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='9 RS and placed an upper bound on the order- 1 non-axisymmetric magnetic moments to be less than ∼ 5% of the total axisymmetric moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2011) restricted their analysis to measurements inside the mag- netic shell connecting Saturn and the orbit of Enceladus be- tween 2005 and 2010 and placed stringent constraints on the low-degree non-axisymmetry in Saturn’s internal magnetic field: the dipole tilt must be smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='06◦ while the non-axisymmetric quadrupolar moments are less than 6% of the axisymmetric quadrupolar moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The non-detection of internal magnetic non-axisymmetry also implies that the rotation rate of the deep interior of Saturn needs to be deter- mined from other means such as gravity, shape, wind shear, and ring seismology (Militzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Mankovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' One important measure of a planetary internal magnetic field is its change with time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', Holme and Olsen, 2006), commonly referred to as the magnetic secular variation (SV), which offers a window into the internal dynamo flows and waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2011) compared their low-degree model of Saturn’s internal magnetic field built from the pre- Grand Finale Cassini measurements to the SPV (Saturn Pi- oneer Voyager) model (Davis and Smith, 1990) built from the Pioneer 11, Voyager 1 & Voyager 2 measurements, and found that the differences in Saturn’s low-degree internal magnetic moments between the Pioneer-Voyager era and the early Cassini era are very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' If interpreting the difference (with their uncertainty ranges) as the linear secular varia- tion rate, then Saturn’s magnetic SV rate is on the order of 1 nT/year or less which is much smaller than the order 10 nT/year secular variation rate of the recent geomagnetic field (Finlay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='3 A new picture of Saturn’s magnetic field from the Cassini Grand Finale measurements Following a gravity assist from Titan, the Cassini space- craft embarked upon the Grand Finale phase of its journey, which consisted of 22 high inclination orbits with periapses in the gap between Saturn and its innermost ring, before descending into the atmosphere of Saturn on 15 Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The periapsis altitudes of the Cassini Grand Finale (CGF) orbits ranged between 3911 km and 1444 km from Saturn’s 1-bar surface (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', see Table 1 in Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2020), signifi- cantly lower than the altitudes of all previous Cassini orbits and Pioneer-Voyager flybys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' For comparison, the periapsis altitudes of Cassini SOI and Pioneer 11 were ∼ 20,000 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' In an inertial frame, the periapses of the CGF orbits were near local noon while the apoapses were near midnight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The inclination of the CGF orbits, with respect to the rotational equator of Saturn, was ∼ 62◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2 displays the mea- sured magnetic field strength |B| and the azimuthal compo- nent, Bφ, along a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5-day CGF orbit (Rev 291), which serves as an illustration of the bulk features of Saturn’s magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' It can be seen that the magnetic field strength varied by 4 orders of magnitude, from < 2 nT to > 20,000 nT, dur- ing a CGF orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The minimum field strength was recorded during one crossing of the magnetodisk (Bunce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Multiple crossings or close approaches to the magnetodisk are evident from the periodic decreases of the magnetic field strength during the outbound portion of the pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The az- imuthal component, which stayed within [−30, 50] nT while the background field varied by 4 orders of magnitude, re- vealed many dynamical features of Saturn’s magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' A northern-hemisphere high-latitude crossing of the mag- netic field lines connecting Saturn and Enceladus occurred during this orbit, see feature labelled as Enceladus flux tube crossing on 02 Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 2017, where a strong negative Bφ (∼ −30 nT) was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The sign of Bφ there is consistent with a bend-back of Saturn’s field lines resulting from the in- teraction of flowing magnetospheric plasma with Enceladus (Dougherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2006), and the amplitude of the associ- ated field-aligned currents (FACs) has been estimated to be ∼ 200 nA m−2 (Sulaiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The expected auro- ral FACs (Hunt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2020) and the ubiquitous planetary period oscillations (PPOs) (Provan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2018, 2019a) are also evident in the measured Bφ component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Provan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2019a) showed that the dual modulations by the northern and southern PPOs are present throughout Saturn’s inner- most magnetosphere in all three magnetic field components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Moreover, the measured Bφ revealed a new feature in Sat- urn’s innermost magnetosphere: a low-latitude FAC system located near the magnetic shells connecting Saturn and its tenuous D-ring (Dougherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Khurana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Provan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Hunt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1 Discovery of a low-latitude field-aligned current system Unexpected Bφ perturbations on the order of 20 nT were consistently observed around periapses along CGF orbits (Dougherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' When mapped along Saturn’s back- ground magnetic field lines, these Bφ perturbations map to the tenuous D-ring of Saturn and its planetward cavity (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='3 & 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The sharpness of the spatial variation lead- ing to the center peak is a strong indicator that these Bφ signatures are of magnetospheric-ionospheric origin, instead of a deep interior origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The predominantly positive nature of the low-latitude Bφ perturbations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='4) further points to their toroidal nature, which must be associated with local meridional electric currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 4 Cao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Dougherty,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Hunt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Bunce,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Christensen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Khurana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' & Kivelson 100 101 102 103 104 |B| [nT] Aug 30 Aug 31 Sep 01 Sep 02 Sep 03 Sep 04 Sep 05 Sep 06 UTC Time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Year =2017 B [nT] Enceladus fluxtube crossing magnetodisk crossing 40 nT 400 nT 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='000 nT Auroral FACs Planetary Period Oscillations (PPOs) Auroral FACs Low-latitude (Intra-D ring) FACs 30 20 10 0 10 20 30 40 50 10:00:00 11:00:00 12:00:00 13:00:00 14:00:00 15:00:00 16:00:00 17:00:00 UTC Time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Sep 02 2017 50 40 30 20 10 0 10 20 30 40 50 B [nT] Enceladus fluxtube crossing Southern Auroral FACs Northern Auroral FACs Low-latitude (Intra-D ring) FACs A B C Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2 Magnetic field strength and the azimuthal component measured during a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5-day Cassini Grand Finale orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Panel A shows the total field strength on a log-scale, while Panel B shows the azimuthal field component Bφ on a linear scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Panel C shows a zoom-in of Bφ for a few hours around the periapsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Figure from Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The electric current system associated with positive low- latitude Bφ is inter-hemispheric, with field-aligned currents J∥ flowing from the north to the south in the magnetosphere and return Pedersen currents JP in the ionosphere flow- ing south to north as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' A negative Bφ would correspond to a reversed current loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Khurana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2018) further proposed that such a low-latitude elec- tric current system is driven by zonal wind shear between the two “ends” of Saturn’s magnetic field embedded in the ionosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Khurana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2018) estimated the Joule dissi- pation associated with this low-latitude current system and found it to be ∼ 2 × 1011 W, assuming a height-integrated conductance ∼ 9 S (siemens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' This dissipation rate is simi- lar to the estimated heating rate from solar extreme ultra- Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='3 Panel A shows the measured azimuthal magnetic field, Bφ, around the periapsis of the first Cassini Grand Finale orbit (Rev 271) color-coded along the trajectory of the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The high-latitude auroral FACs and the low-latitude intra-D ring FACs are both evident from the measured Bφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Panel B shows a sketch of the magnetospheric-ionospheric current system consistent with the observed positive Bφ in Saturn’s innermost magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' JP is the Pedersen current while J∥ is the field-aligned current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Panel A is from Dougherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2018) while Panel B is from Khurana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' violet radiation at Saturn, but is not sufficient to explain the higher-than-expected temperature in Saturn’s equato- rial thermosphere (M¨uller-Wodarg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Hunt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2019) estimated the electric current density associated with these low-latitude Bφ perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Assum- ing that the azimuthal extent of the current system is much wider than that traversed by the spacecraft, they inferred that the low-latitude ionospheric meridional currents are ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 MA (Mega Ampere) per radian of azimuth (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 for two examples), similar in intensity to the auro- ral region current (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', Hunt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Hunt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2019) then computed the current density of the associated FACs, 2 Bo [nT] Auroral 20 FACs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 15 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 5 S C B A R 0 0 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 5 1 10 Auroral 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 15 FACs 20 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content="5 0 4 A Cassini's trajectory 300 150 0 Field lines 150 300 m/s BSaturn’s Magnetic Field at Unprecedented Detail Achieved by Cassini’s Close Encounters 5 30 15 0 15 30 BÁ [nT] D D C C B B A A a Untraversed Field Region 30 15 0 15 30 ¢BÁ [nT] D D C C B B A A b 50 54 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='25 40 46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='65 30 37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='34 20 28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='16 10 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='06 0 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='03 10 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='03 20 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='07 30 22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='19 40 33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='43 50 45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='0 \xad [rad s¡1] µi [°] µc [°] ½eq[RS] D D C C B B A A c 1e-6 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='4 The measured azimuthal magnetic field, Bφ, along the Cassini Grand Finale orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The horizontal axis is labelled with the mapping of the magnetic field line traversed by the spacecraft to the ionosphere and the ring plane: θi is the ionospheric latitude of the field line footprint in the same hemisphere as the measurement, θc is the ionospheric latitude of the conjugate footprint in the opposite hemisphere along the field line, ρeq is the (cylindrical) radial distance in the equatorial/ring plane of the field line from the center of Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The top panel shows the total observed Bφ while the middle panel shows de-trended field ∆Bφ where a fourth order polynomial fit has been subtracted from the data to isolate the intra-D ring features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' In both panels, colored traces represent different CGF orbits while the black traces represent the running median from all orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The bottom panel shows the angular velocity profile of the 1-bar zonal winds at Saturn as a function of θi, referenced to an assumed planetary rotation period of 10 hr 34 min 13 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Figure from Agiwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' and found it to be ∼ 5 - 10 nA m−2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5), over an order of magnitude smaller than that of typical auroral FACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Provan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2019b) showed that the measured low- latitude Bφ is generally symmetric about the field-parallel point, where the spacecraft trajectory is tangent to the back- ground field line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' This symmetry is consistent with field per- turbations associated with inter-hemispheric FACs (also see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 3 in Khurana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Provan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2019b) further examined the morphology of the low-latitude Bφ signature and showed that they can be categorized into four different groups: ∼ 35% cases/orbits feature a single positive central peak ∼ 20 - 40 nT (category A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' ∼ 30% cases feature two or three weaker positive peaks ∼ 10 - 20 nT (category B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' ∼ 15% cases feature rather irregular central positive peaks in regions well inside the magnetic shell connected to the D- ring inner boundary (category C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' and ∼ 20% cases feature unique features including two with ∼ 20 - 30 nT negative fields and two with < 10 nT fields (category U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Provan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2019b) tried to correlate these different categories with the spacecraft altitude, local time, PPO phase, and the orbital phase of the D-68 ringlet but found no convincing correla- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Agiwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2021) explored the possibility of temporal variations in the thermospheric zonal winds as the origin of the variability observed in the low-latitude Bφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' We will discuss more about this work in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2 Discovery of Alfv´en waves planetward of the D-ring Southwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2021) examined the shorter timescale vari- ations in the azimuthal magnetic field measured along the 6 Cao, Dougherty, Hunt, Bunce, Christensen, Khurana, & Kivelson Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 Ionospheric meridional current Im and field-aligned current density j∥ i associated with the low-latitude Bφ observed along Cassini Rev 287 (top) and Rev 292 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The vertical lines indicate magnetic mapping to ring boundaries: the pair of red dashed lines correspond to the outer boundary of the A-ring, while the solid (dashed) pair of blue lines correspond to the outer (inner) boundaries of the D-ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Figure from Hunt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' CGF orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' They showed that Bφ oscillations with a typ- ical amplitude of a few nT and a typical time scale of a few minutes prevail in the innermost magnetosphere plan- etward of the D-ring, regardless of the morphology of the background Bφ structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' They applied cubic spline fittings with a 5-min window and a 10-min window successively to the measured Bφ, which extracted two bands of Bφ oscil- lations (middle two panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' This procedure also isolated the large-scale background Bφ, as shown in the bot- tom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='6 that both the large-scale background Bφ and the oscillatory Bφ are generally confined within the magnetic shell connecting to the outer edge of Saturn’s D-ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The background Bφ fea- tures a typical amplitude of 20 nT, while both bands of the oscillatory Bφ have amplitudes on the order of 5 nT or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' When examining the peaks (anti-nodes) and zeros (nodes) of the Bφ oscillations with respect to the local background planetary field (Br & Bθ), Southwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2021) noticed that the nodes of Bφ oscillations from both bands lie close to the proxy magnetic equator (defined as where the radial component of the field vanishes Br = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' For the 1-5 min oscillations, 15 orbits show a strong clustering of a Bφ node within 5 secs of T = 45 sec from Br = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' the remaining 7 orbits show a clustering of nodes between T = −12 and +16 sec from Br = 0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' This feature is consistent with the odd-mode (wave number n = 1, 3, 5, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=') of a standing Alfv´en wave (assuming similar ionospheric electrical conduc- tivity in the northern hemisphere and southern hemisphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' For the 5-10 min oscillations, the picture is somewhat more complex: 9 orbits display a magnetic node between T = −6 and 35 sec on either side of the proxy magnetic equator (Br = 0), another 9 orbits display a magnetic node between T = −30 and −40 sec from the proxy magnetic equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' There are four additional orbits with mixed behaviors, in- cluding one with a Bφ anti-node (maximum) near the proxy magnetic equator and two with relatively flat Bφ oscillations near Br = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Southwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2021) propose that these few minutes Bφ oscillations planetward of Saturn’s D-ring are standing (transverse) Alfv´en waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The fact that 1) both oscillation bands feature a magnetic node near the proxy magnetic equator (Br = 0) and 2) the (temporal) spectral content of these oscillations remains relatively simple despite the up to a factor of two change in local field line length led Southwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2021) to suggest that these are local field line resonances being pumped by global magnetospheric cav- ity modes, a coupling originally suggested by Kivelson and Southwood (1986) for the 1 - 10 min ultra-low frequency (ULF) magnetic pulsations in Earth’s magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='3 The internal magnetic field: extreme axisymmetry and high-degree magnetic moments The small amplitude of the azimuthal component (Bφ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1% |B| near CGF periapses) and the identifiable magnetospheric-ionospheric features in this component serve as direct evidence for the extreme axisymmetry of Saturn’s internal magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Quantitative analyses con- firm this first impression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Two different analyses have been performed with the CGF magnetic field measurements to quantify the amount of non-axisymmetry in Saturn’s inter- nal magnetic field (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2020): longitudinal variation in Saturn’s magnetic equator positions and Gauss coeffi- cients inversion with all three field components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The two analyses yielded similar upper limits on large-scale non- axisymmetry in Saturn’s internal magnetic field: the dipole tilt at Saturn must be smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='007◦ (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2 arcsecs), while the non-axisymmetric contribution to the spherical Rev 287 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 (a) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content="5 rad' /MA 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='0 _m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 15 (b) 10 5 UAA 5 10 15 30 60 90 120 150 Co-latitude [deg] Rev 292 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content="5 /MA rad' 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='0 (b) 5日 /nA 5 E 10 15 20 30 60 90 120 150 Co-latitude [deg]Saturn’s Magnetic Field at Unprecedented Detail Achieved by Cassini’s Close Encounters 7 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='6 Oscillations in the azimuthal magnetic field measured along 22 Cassini Grand Finale orbits as identified by Southwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The bottom panel shows the background Bφ, the middle two panels show the Bφ oscillations in the 1 - 5 min band and the 5 - 10 min band, and the top panel shows the magnetic mapping of the spacecraft position to the ring plane in which the mapping to the D-ring is highlighted with the grey shade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The data is organized in the time frame in which T = 0 represents the time when the spacecraft was on the innermost magnetic shell (L-shell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' L-shell describes the set of field lines which cross the equator at the radial distance defined by the numerical value of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Figure from Southwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='7 Azimuthal magnetic field oscillations in the 1-5 min band as a function of time from the crossing of the proxy magnetic equator (where the background radial field, Br = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Orbit Rev numbers are indicated on each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The top panel displays 15 CGF orbits during which a magnetic node occurred between T = 40 and 50 sec (vertical shaded bar) from Br = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The lower panel displays 7 CGF orbits that show a clustering of nodes between T= −12 and +16 sec (vertical shaded bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Figure from Southwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 8 Cao, Dougherty, Hunt, Bunce, Christensen, Khurana, & Kivelson harmonic (SH) degree 2 & 3 magnetic moments are less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='15% (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Although extremely axisymmetric, Saturn’s internal mag- netic field displays surprisingly rich (spatial) spectral con- tent, corresponding to an axisymmetric magnetic field on many different length-scales (Dougherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Two different mathematical representations of the internal magnetic field were adopted by Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020): the Gauss coefficients representation and the Green’s function representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The Green’s function relates the vector magnetic field at the point of observation to its ra- dial component at a reference surface inside the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' For technical details of the two different representations, the in- terested readers can refer to the Appendices A & B of Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020) and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Utilizing the Green’s function, Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020) computed the sensitivity of the magnetic field measurements along the CGF orbits to the axisymmetric field at Saturn’s “dynamo surface”, taken to be the a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='75 RS, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='6993 RS iso- baric surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Here a and c represent the equatorial and polar radii of the isobaric surface, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The shape of the isobaric surface was determined from interior struc- tural models constrained by Cassini gravity measurements (Iess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Militzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The choice for the depth of “dynamo surface” is guided by an estimation of where the local magnetic Reynolds number reaches order unity (Dougherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020) showed that magnetic field measurements along the CGF orbits are sensitive to Saturn’s large-scale (SH degree n ≤ 3) axisymmetric magnetic field at depth up to very high latitude (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', ±80◦) but likely are not very sensitive to Saturn’s small-scale (n > 3) axisymmetric magnetic field be- yond ±60◦ latitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' These sensitivity characteristics result from 1) the particular trajectory of CGF orbits with closest approach near the equator and 2) the highly axisymmetric nature of Saturn’s internal magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' After determining the magnetodisk field (Bunce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2007) contributions orbit-by-orbit with CGF magnetic field measurements slightly away from the periapsis and removing them from the measurements, Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020) performed inversion analysis on the CGF magnetic field data to extract features of Saturn’s internal magnetic field with both the Gauss coefficients and the Green’s function representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Inversion analysis with the Gauss coefficients representation revealed that axisymmetric Gauss coefficients up to at least SH degree-9 are needed to give a reasonable match to the measurements (with Root-Mean-Square residual < 5 nT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020) noted that, on the dynamo surface, the contribution of the internal magnetic field corresponding to the degrees 4 - 9 axisymmetric Gauss coefficients is substan- tially larger beyond ±60◦ latitude than at lower latitudes (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 12 in Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The Green’s function sensi- tivity analysis of the data does not support this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020) employed the technique of regularized inversion, which introduces a damping parameter, γ, that regularizes the behavior of the model while simultaneously fitting the measurements (for technical details, see section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2 in Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='8 shows characteristic properties of a few selected models, both in spectral space (the amplitude of g0n) and in real space (∆Br/|B| on the “dynamo surface”), as a func- tion of the damping parameter γ which sets the relative importance of model constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' It can be seen that models tend to feature 1) large amplitude of ∆Br at high-latitude when the damping is weak, 2) significantly reduced ampli- tude at high latitude when some damping is allowed, and 3) a substantially worse fit to the data when the damping is strong (see the RMS residual values in the legend in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='8 further shows that the small-scale field struc- ture within ±60◦ latitude is well-resolved regardless of the field behavior at higher latitudes: there are eight alternat- ing latitudinal bands of radial magnetic fluxes between ±60◦ latitude (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='9B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The model that features a well- behaved field at the “dynamo surface” and a good match to the data, corresponding to γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='03 (thick red traces in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='8), was selected as the preferred model and was named the Cassini 11+ model (see Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1 for the coefficients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020) also performed inversion analysis on the CGF magnetic field measurements with the Green’s function representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Compared to the Cassini 11+ field model, the Green’s function analysis returned an almost identical small-scale axisymmetric magnetic field within ±60◦ lati- tude on the “dynamo surface”, but (as expected) produced different field behaviors at higher latitudes (see their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' With a robust understanding of which features are well- resolved by the CGF data, Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020) examined the characteristics of Saturn’s internal magnetic field at the “dy- namo surface” (the a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='75 RS, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='6993 RS isobaric surface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='9A, Saturn’s large-scale (SH degrees 1 - 3) magnetic field is relatively weak in the equa- torial region: Br remains less than 1/3 of its peak value within ±40◦ latitude (panel A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='9B, Saturn’s small-scale (beyond SH degree-3) magnetic field features eight alternating zonal bands between ±60◦ lati- tude (panel B), with typical amplitude ∼ 5 - 10% of the background large-scale field (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='8B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Like that of the large-scale field, the polarity of the eight alternating mag- netic bands displays predominant anti-symmetry with re- spect to the equator (dipole-octupole-like), while their am- plitudes are stronger at northern mid-latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020) further noted that the number of alternating mag- netic bands at the a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='75 isobaric surface coincides with the number of alternating zonal wind bands if one projects the observed surface wind along the spin-axis to the same depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Both the small-scale magnetic bands and the pro- jected surface zonal winds feature typical latitudinal widths of ∼ 15◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' It is interesting to examine the pattern of the Gauss co- efficients of Saturn’s internal field (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='8A and Ta- ble 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Overall, there is a dramatic drop in amplitude be- yond degree-3: g0 4 drops by more than an order-of-magnitude compared to g0 2 and g0 3, and all higher-degree moments are smaller than g0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The degree-5 moment, g0 5, seems to be strongly suppressed: it is almost another order-of-magnitude smaller compared to g0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' However, the degree-7 moment, g0 7, bounces back by a factor 5 - 7 compared to g0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' g0 8 to g0 11 Saturn’s Magnetic Field at Unprecedented Detail Achieved by Cassini’s Close Encounters 9 10 11 12 13 14 Spherical Harmonic Degree 10-1 100 101 102 103 104 105 =3 10-3,RMS=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='58 nT =1 10-3,RMS=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='66 nT =3 10-2,RMS=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='73 nT =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1,RMS=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='32 nT =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2,RMS=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='64 nT |gn 0| [nT] −80 −60 −40 −20 0 20 40 60 80 Latitude [deg] B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='3 Br/|B| @ a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='75 RS A 5 6 7 8 9 1 2 3 4 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='8 The axisymmetric Gauss coefficients g0 n versus SH degree (panel A) and the small-scale (g0 n for n > 3) internal magnetic field of Saturn versus latitude (panel B) derived from Cassini Grand Finale magnetic field measurements with regularized inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The damping parameter γ and the RMS residual associated with each solution are shown in the legend in panel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' In panel B, the small-scale magnetic field, ∆Br, corresponds to g0 n with n > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' It is normalized with respect to the strength of the background field, |B|, corresponding to g0 n with n ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Figure from Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' are comparable to g0 5 and g0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Although an overall decay- ing (with SH degree) trend is present, 1) the strong dip in g0 5 and 2) the seemingly separate slopes for the low-degree moments and the high-degree moments are unexpected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020) attempted to extract an electromagnetic induction signal from Saturn’s interior, with the orbit-to- orbit varying magnetodisk BZ field as the external sound- ing signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' They solved for the orbit-to-orbit varying internal dipole ∆g0 1 after removing the Cassini 11+ model, and then compared those values to the orbit-to-orbit varying magne- todisk field ∆BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Although the expected induction signal is within 1σ of a formal inversion analysis of ∆g0 1 versus ∆BZ, the large scatter in the data precluded a definitive constraint on the induction depth inside Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Br @ a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='75 RS Br @ a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='75 RS [nT] A B 60° N 60° S 40° S 40° N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 105 8000 6000 4000 2000 0 2000 4000 6000 8000 [nT] Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='9 Saturn’s large scale (g0 n for n ≤ 3) and small scale (g0 n for n > 3) radial magnetic field at the a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='75 RS, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='6993 RS isobaric surface according to the Cassini 11+ field model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Saturn’s large scale radial magnetic field at this depth features a relatively weak equatorial region, Br remains less than 50,000 nT (<1/3 of its peak value) between ±40◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Saturn’s small-scale magnetic field at this depth features eight alternating bands between ±60◦, with typical amplitude of ∼ 5%–10% of the background field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Figure from Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' [nT] Cassini 11 Cassini 11+ g0 1 21140 21141 g0 2 1581 1583 g0 3 2260 2262 g0 4 91 95 g0 5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='3 g0 6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='4 g0 7 −59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='6 −68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='8 g0 8 −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 −15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 g0 9 −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='9 −24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2 g0 10 15 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='0 g0 11 18 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='3 g0 12 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='8 g0 13 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='4 g0 14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='8 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Gauss coefficients of the Cassini 11 model (Dougherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2018) and the Cassini 11+ model (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2020) for Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' These coefficients refer to a sur- face radius RS=60268 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The Cassini 11 model was con- structed from magnetic field measurements from the first ten Cassini Grand Finale orbits while the Cassini 11+ model was constructed using data from all 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 CGF orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 10 Cao, Dougherty, Hunt, Bunce, Christensen, Khurana, & Kivelson Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='10 Data-model comparisons between the observed Bφ along two CGF orbits and a kinematic ionospheric zonal shear model with different assumed zonal flow profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Panels a & c display Bφ, with the solid grey traces showing the de-trended observations, dashed black traces showing the model associated with the baseline zonal flow profile, and red dashed traces showing the model associated with the perturbed zonal flow profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Panels b & d display the angular velocity profile, with the black traces showing the baseline model, and the solid red traces showing the perturbed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' It can be seen from the data-model comparison for Rev 292 (panels c & d) that one only needs to reduce the amplitude of the zonal flow in the southern hemisphere (instead of reversing the wind) to flip the sign of the low-latitude Bφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Figure from Agiwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='4 Implications and open questions Here we briefly discuss the implications and open questions for Saturn’s innermost magnetosphere, ionosphere, and in- terior that are closely related to the magnetic field investiga- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' As will become clear, our understanding of the physical mechanisms behind many of the observed phenomena is still in a preliminary stage: most of our interpretations are kine- matic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' A fully dynamic understanding of the Saturn system has not been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1 Saturn’s equatorial ionosphere: zonal shear and atmospheric-wave-induced temporal variability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' As introduced in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1, a considerable amount of orbit-to-orbit variability has been observed in the large-scale low-latitude Bφ peak along the Cassini Grand Finale orbits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Although most of the CGF orbits feature positive large-scale low-latitude Bφ around the periapses, a few orbits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Revs 286 & 292) feature strong negative low- latitude Bφ (see the categorization by Provan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' As explained by Khurana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2018), due to the northward shift of Saturn’s magnetic equator, a north-south symmetric eastward zonal flow in Saturn’s ionosphere could naturally lead to a low-latitude positive Bφ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' M¨uller-Wodarg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2019) and Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020) re- ported atmospheric waves in Saturn’s thermosphere with Cassini Ion Neutral Mass Spectrometer (INMS) and Ultra- violet Imaging Spectrograph (UVIS) measurements during the Cassini Grand Finale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' These atmospheric waves could assist the vertical transport of the 1-bar wind structure to higher altitude, but could also introduce temporal variability to the higher altitude winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' It is thus natural to consider time variability in the zonal shear in Saturn’s ionosphere as the cause of the observed variability in the large-scale Bφ around the CGF periapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Agiwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2021) inves- tigated quantitatively the amount of variation in the iono- spheric zonal wind shear needed to account for the observed variability in the low-latitude Bφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The technical starting point of Agiwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2021) is the rigorously derived formula relating the steady-state zonal shear in the ionosphere to the meridional current in the iono- sphere (under a thin-shell approximation of the ionosphere) and Bφ along the field line, first presented in Appendix A of Provan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The steady-state meridional current, Im, associated with the differential ionospheric wind drag is shown to be Im = ΣP NΣP S (ΩnS − ΩnN) � ΣP N |biS| ρ2 iSB2 iS + ΣP S |biN| ρ2 iNB2 iN �, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1) where ΣP N,S are the height-integrated Pedersen conduc- tivities in the northern and southern ionospheres, ΩnN,S are the angular velocities of the neutral gas in the iono- spheres, BiN,S are the planetary poloidal fields in the two ionospheres, while biN,S are the poloidal field components normal to the ionosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Assuming axisymmetry, the asso- ciated azimuthal magnetic field at any point along the field line is Bφ = µ0Im ρ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2) where µ0 is the permeability of free-space, and ρ is the cylin- drical radius from the magnetic axis (also the spin-axis) of Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' It should be immediately clear from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1 & 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2 that 1) the difference between the angular velocities at the two ends of a field line, ΩnS − ΩnN, determines the sign of Bφ, while 2) the amplitude of the height-integrated Pedersen conductivity modulates the amplitude of the Bφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' These for- mulae also highlight the degenerate nature of the amplitude of the zonal wind shear and the ionospheric Pedersen con- ductivity if the magnetic field is the only observable quan- tity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Bφ would remain the same if we increase the zonal wind shear by a factor 2, while decreasing the height-integrated ionospheric Pedersen conductivity in both hemispheres by the same factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Rev 271 a 40 A B B A 20 [nT] 0 0 B 20 MAG Bs1 B?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (s=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='4 b F1e-6 [s pe] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='0 2m 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content="]'0 50 40 30 20 10 10 0 20 30 40 50 Rev 292 c 40 C B B C D A A D 20 0 0 B 20 B*(s =0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2) d 1e-6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content="]'0 50 30 10 50 40 20 0 10 20 30 40Saturn’s Magnetic Field at Unprecedented Detail Achieved by Cassini’s Close Encounters 11 Vriesema et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020) investigated the observed low- latitude Bφ at Saturn with a height resolved ionospheric- thermospheric electrodynamics model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The Vriesema et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020) model also assumes steady-state and axisymmetry, and is a kinematic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Agiwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2021) demonstrated the equivalence of the Provan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2019b) formulation (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1 & 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2) and the Vriesema et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020) formulation in the thin ionospheric current layer approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' With an ionospheric conductivity model (M¨uller-Wodarg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' M¨uller-Wodarg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2019) evaluated under northern summer conditions at Saturn, Agiwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2021) adopted the latitudinal profile of the 1-bar atmospheric zonal winds but decreased their amplitude by a factor of two as the starting point of Saturn’s low-latitude thermospheric wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' By systematically perturbing the zonal winds in both hemispheres, they showed that variability in the equato- rial thermospheric wind up to 350 m/s can account for the observed variability in the low-latitude Bφ along CGF or- bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='10 showcases two examples from Agiwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The data-model comparison for Rev 292 shows that one only needs to reduce the amplitude of the zonal flow in the southern hemisphere (solid red traces in panel d), instead of reversing the flow direction, to flip the sign of Bφ around the periapses of the CGF orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Future inves- tigations that model the lower atmosphere-thermosphere- ionosphere-inner magnetosphere interactions with the rele- vant dynamical processes such as gravity waves are needed to evaluate whether such variability in the thermospheric wind is plausible at Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Furthermore, will there be a strong seasonal dependence of the thermospheric wind shear and the low-latitude field- aligned current system at Saturn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Will the low-latitude Bφ flip to mainly negative when the season switches to northern winter at Saturn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2 Saturn’s interior: a tale of two dynamos?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The fact that Saturn’s internal magnetic field is extremely axisymmetric and yet full of structures in the latitudinal direction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='9) is intriguing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Dougherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2018) and Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020) proposed that two spatially separate dynamos, one in the deep metallic hydrogen layer and the other in the semi-conducting layer, are responsi- ble for this intriguing magnetic field behavior (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='3A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The deep dynamo is responsible for generating the large-scale dipolar background field, while the shallow sec- ondary dynamo (Cao and Stevenson, 2017b) is responsible for generating the small-scale latitudinally banded magnetic perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Gastine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2014) observed localized secondary dynamo action at low-latitudes in their 3D Jovian dynamo simula- tions which they attributed to interaction with deep zonal winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Cao and Stevenson (2017b) proposed the possibil- ity of a global secondary dynamo inside Jupiter and Saturn, and illustrated this process with mean-field electrodynamics (Moffatt and Dormy, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Three key ingredients of their envisioned secondary dynamo inside giant planets are: 1) the background magnetic field, B0, generated by the deep dynamo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 2) differential rotation (zonal flows) in the semi- conducting layer which generates the toroidal magnetic field, BT, via the ω−effect (this process bears similarity to the proposed process of zonal wind shear in the ionosphere gen- erating the low-latitude Bφ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' and 3) small-scale helical con- vection in the semi-conducting layer providing the critical α−effect that generates the externally observable poloidal magnetic field, ∆BP, from BT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Both BT and ∆BP are expected to be spatially correlated with the zonal flows in the semi-conducting layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Furthermore, Cao and Stevenson (2017b) pointed out that the ω−effect and the α−effect in the secondary dynamo may operate at different depths: BT resulting from the ω−effect can be generated at a relatively shallow depth and then diffuse several scale-heights down- ward due to the rapidly increasing electrical conductivity as a function of depth in the semi-conducting layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Galanti and Kaspi (2020) analyzed Saturn’s gravity mo- ments and magnetic moments jointly, aiming to construct a deep differential rotation profile for Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Their analy- sis of Saturn’s dynamic gravity moments was based on the Cassini radio tracking observations (Iess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2019) and the diagnostic thermal wind (TW) relation (Kaspi, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Cao and Stevenson, 2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Their analysis of Saturn’s magnetic moments was based on the Cassini 11 model (Dougherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2018) and the (kinematic) mean-field electrodynamics (MFED) model of Cao and Stevenson (2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' They showed that the same deep zonal wind profile, which is only slightly modified from the observed surface zonal winds around ±30◦ latitude, and penetrating to about 7500 km below the 1- bar level can account for both the dynamic gravity field and the small-scale magnetic bands at Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' This is an encouraging result, and calls for future dynamic investiga- tions, in which the physical processes that control the depth of rapid zonal flows inside Saturn (and Jupiter) need to be self-consistently modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='3 Saturn’s extremely axisymmetric magnetic field: electromagnetic filtering or a double-diffusive dynamo in the diffusive core?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The axisymmetry of Saturn’s internal magnetic field has long been a puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Here we briefly explain the challenge and then describe the proposed kinematic solution and some dynamical tests with 3D numerical dynamo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' We conclude this subsection with a discussion of the new chal- lenges raised by the inference of a large diffusive core inside Saturn (Mankovich and Fuller, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' From a theoretical point of view, Cowling’s theorem (Cowling, 1933) precludes the possibility of maintaining a purely axisymmetric magnetic field in a steady-state via MHD dynamos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' However, Cowling’s theorem itself does not place any lower limit on the amount of non-axisymmetry necessary to maintain dynamo action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' When examining the observational evidence in the solar system, Earth and Jupiter feature a modest amount of non-axisymmetry with dipole tilts ∼ 10◦, Uranus and Neptune feature a significant amount of non-axisymmetry with dipole tilts ∼ 50◦ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', see Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='2 in Kivelson and Bagenal, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' For Mercury and Ganymede, the total amount of non-axisymmetry in their large-scale internal magnetic field are less clear at this stage 12 Cao, Dougherty, Hunt, Bunce, Christensen, Khurana, & Kivelson while their dipole tilts have been estimated to be on the order of 1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The most widely invoked explanation for Saturn’s very axisymmetric magnetic field is the one proposed by Steven- son (1980, 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The essence of this mechanism is electro- magnetic (EM) filtering: if there exists a passive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' electrically conducting layer on top of a regular dynamo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' and if this fil- ter layer is rotating at a different angular speed compared to that of the deep dynamo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' then any non-axisymmetric magnetic field from the deep dynamo would appear as a time-varying field to this layer and thus be electromagnetic filtered while the axisymmetric part of the magnetic field would appear as time-stationary and pass through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' For Sat- urn in particular, Stevenson (1980) proposed that helium rain-out would create a stably stratified layer (with no large- scale overturning motion) on top of Saturn’s deep dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' One should remember that both the qualitative descrip- tion above and the quantitative model presented in Steven- son (1982) are kinematic: no dynamical feedback between the deep dynamo and the stably stratified layer has been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Within this kinematic framework, Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020) showed that a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='007◦ dipole tilt requires a stable layer thicker than 2500 km if the deep dynamo features a Jupiter-like ∼ 10◦ dipole tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' This estimation represents a lower limit on the thickness of the stable layer, as the dy- namical feedback from the non-axisymmetric magnetic field on the flows in the stable layer is expected to reduce the filtering efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 3D numerical dynamo models have also been constructed to reproduce the highly axisymmetric magnetic field of Sat- urn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Christensen and Wicht (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Stanley and Moham- madi (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Stanley (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Christensen (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Gastine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Yan and Stanley (2021) employed numerical MHD models under the Boussinesq approximation to inves- tigate the effects of a stable layer on top of a convective dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' In these models, background density and electrical conductivity are assumed to be constant while the Ohmic and viscous dissipation are assumed to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' While these assumptions are approximately valid for the cores of terrestrial planets, they are quite different from the condi- tions inside giant planets, which feature large variations in background density, temperature, and electrical conductiv- ity as a function of depth (French et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' These simplified models illustrate many dynamical pro- cesses involved in the interaction between the convective dynamo and the stable layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The electromagnetic filtering effect proposed by Stevenson (1982) was indeed observed in many of these studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Christensen and Wicht, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Stanley, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Gastine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' However, the situation is more complicated than the kinematic picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Convection eddies can penetrate into the stably stratified layer (Gastine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2020), horizontal circulation can erase certain types of thermal heterogeneity imposed from above (Christensen, 2018), certain types of zonal flow in the stable layer could destabilize the dynamo-generated magnetic field instead of axisymmetrizing it (Stanley and Mohammadi, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' In one of the latest 3D Boussinesq modeling effort, Yan and Stan- ley (2021) achieved a dipole tilt ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='066◦ with a thick sta- bly stratified layer (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='28 RS) combined with a particular type of heterogeneous heat flux variation on top of the stable layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' This is still about one order of magnitude larger than the latest observational upper bound at Saturn (Dougherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Part of this discrepancy could result from the relatively low magnetic Reynolds number as- sociated with differential rotation in the stable layer in the numerical simulations compared to the realistic values inside Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' See section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='3 in Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2018) for a more detailed discussion of this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' More advanced models, where the molecular envelope and the deep dynamo with smooth transition in material prop- erties were simulated simultaneously, have also been con- structed under the anelastic approximation (Jones, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Dietrich and Jones, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Gastine and Wicht, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Most of the anelastic dynamo models were constructed without any stable layer (Duarte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Jones, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Dietrich and Jones, 2018), the resulting magnetic field in the dipolar branch features a modest amount of non-axisymmetry sim- ilar to that observed at Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' In a survey of the effects of relative thickness of the molecular envelope, Dietrich and Jones (2018) observed an interesting oscillatory dynamo in which the magnetic field flips its polarity regularly and re- sembles Saturn’s main magnetic field qualitatively during about a quarter of every cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The dipole tilts during such times are typically 1-2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Three recent studies implemented stable layers in anelas- tic models applicable to Jupiter and Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Dietrich and Wicht (2018) constructed a hydrodynamic model for Sat- urn with a sandwiched stable layer and analyzed the depth and mechanism of penetrative convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' (2020) investigated how a stable layer might work with the dynamo-generated magnetic field to truncate deep zonal flows in a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5D (axisymmetric) set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Gastine and Wicht (2021) constructed one of the first 3D MHD model for gas giants with a stably stratified layer between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='82 RP and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='86 RP near the molecular-metallic transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The resul- tant surface magnetic field are dipole-dominant with appre- ciable amount of non-axisymmetry (see their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 9), similar to that of Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The latest inference of a large stably stratified diffusive core inside Saturn (Mankovich and Fuller, 2021) raised new challenges for understanding the origin of Saturn’s axisym- metric magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' A diffusive core extending out to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='6 RS (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='1BC) leaves very little space for a traditional convective dynamo inside Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Can an MHD dynamo op- erate in a stably stratified diffusive core?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Small-scale dou- ble diffusive or oscillatory convection are expected to exist in the stably stratified layer(s) inside Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Could these small-scale motions produce the necessary α−effect to main- tain an MHD dynamo and generate a highly axisymmetric magnetic field?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='5 Outlook for future exploration of Saturn and other giant planets It is appropriate to end this data-driven chapter with an out- look for future exploration of Saturn and other giant planets, Saturn’s Magnetic Field at Unprecedented Detail Achieved by Cassini’s Close Encounters 13 focusing on aspects related to magnetic field investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' In the Saturn system, other than the intriguing moons Ence- ladus and Titan, four areas near Saturn that have not been visited in-situ are 1) the low altitude region near the poles of Saturn, 2) regions inside the orbit of F-ring at local time sectors away from noon, 3) the equatorial region immedi- ately above and below the rings, and 4) the atmosphere of Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Magnetic field measurements in these four areas will help 1) resolve Saturn’s small-scale magnetic field, includ- ing any local non-axisymmetry, near the poles, 2) map out EM coupling between Saturn’s northern and southern iono- spheres as well as between Saturn and its rings at different local time sectors, 3) clarify the existence of a “ring iono- sphere”, and 4) distinguish the ionospheric magnetic field and the internal magnetic field from the deep interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Comparing different planets is an important step to dif- ferentiate common processes from special realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Af- ter the Juno mapping campaign at Jupiter, we shall have a much better knowledge of the similarities and differences between Jupiter and Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Orbital missions to Uranus and Neptune are needed to complete our mapping of the Solar System giant planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The single Voyager 2 flyby at Uranus and Neptune informed us that Uranus and Neptune possess fundamentally different magnetic fields compared to those at Jupiter and Saturn (see a recent review by Soderlund and Stanley, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' However, meaningful theoretical inter- pretations of the Uranus and Neptune systems will require significantly more constraints from in situ observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' The ever expanding categories of exoplanets offers many giant planets for investigation, some of which function in ex- treme environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=', the hot Jupiters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Measuring their surface composition, wind pattern, and magnetic field can provide unique observational constraints in vastly different settings compared to giant planets in the solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' As in many other circumstances, studying the extremes could help us understand the “norm”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Last but not least, measure- ments and theoretical studies at the giant planets also help us better understand planet Earth, as many processes are common despite the differences in the parameter regimes in which they function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Acknowledgement H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' is funded by the NASA Cassini Data Analysis Program (Grant Number 80NSSC21K1128).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='C.’s visit to Imperial College London was funded by the Royal Society, UK grant RP 180014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Work at Imperial College London was funded by Science and Technology Facilities Council (STFC), UK con- solidated grant ST/N000692/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' is funded by Royal Society, UK Research Professorship RP140004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' Work at the University of Leicester was funded by STFC, UK consol- idated grant ST/N000749/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' was supported by a Royal Society Wolfson Research Merit Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNE0T4oBgHgl3EQf5wLJ/content/2301.02756v1.pdf'} +page_content=' References Agiwal, O.' metadata={'source': 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Biological Science and Medical Engineering, Beihang University +2School of Artifical Intelligence, Nanjing University +3Department of Gastroenterology, Peking Union Medical College Hospital +4Department of Pathology, Peking Union Medical College Hospital +Abstract +Pathology image analysis crucially relies on the avail- +ability and quality of annotated pathological samples, +which are very difficult to collect and need lots of human +effort. To address this issue, beyond traditional preprocess +data augmentation methods, mixing-based approaches are +effective and practical. +However, previous mixing-based +data augmentation methods do not thoroughly explore the +essential characteristics of pathology images, including +the local specificity, global distribution, and inner/outer- +sample instance relationship. +To further understand the +pathology characteristics and make up effective pseudo +samples, we propose the CellMix framework with a novel +distribution-based in-place shuffle strategy. We split the im- +ages into patches with respect to the granularity of pathol- +ogy instances and do the shuffle process across the same +batch. In this way, we generate new samples while keeping +the absolute relationship of pathology instances intact. Fur- +thermore, to deal with the perturbations and distribution- +based noise, we devise a loss-drive strategy inspired by cur- +riculum learning during the training process, making the +model fit the augmented data adaptively. It is worth men- +tioning that we are the first to explore data augmentation +techniques in the pathology image field. Experiments show +SOTA results on 7 different datasets. We conclude that this +novel instance relationship-based strategy can shed light on +general data augmentation for pathology image analysis. +The code is available at https://github.com/sagizty/CellMix. +*These authors contributed equally to this work. +Tianyi Zhang (zhangtianyi@buaa.edu.cn) +Zhiling Yan (zhilingyan724@outlook.com) +†Corresponding author. +Guanglei Zhang (guangleizhang@buaa.edu.cn) +Figure 1. +Visualization of an augmented example. +Compared +with other mixing-based methods, CellMix accurately identifies +the boundary of patches and instances. It also tells the differences +between negative and positive. +1. Introduction +Deep neural networks (DNNs) have shown impressive +performance in pathology image analysis. However, this +success crucially relies on the availability and quality of an- +notated pathology images, which are very difficult to collect +and need lots of human effort. To alleviate this problem, +data augmentation, which enlarges the amount and diver- +sity of the labeled samples with label-preserving transfor- +mations, is a feasible method to improve the generalization +of DNNs. +Data augmentation has been extensively established in +the traditional computer vision field. Beyond the traditional +preprocess data augmentation methods(including: rotate, +flip, etc.), the mixing-based approach is one of the most rep- +resentative methods. Mixing-based methods are domain- +agnostic data augmentation techniques first proposed by +arXiv:2301.11513v1 [cs.CV] 27 Jan 2023 + +Original +Mixed Sample +Negative +Positive +Negative +CellMix +Positive +Mixup +Positive +CutMix[36]. It devises a strategy that mixes some sub-regions or +features from different categories of images, as new pseudo +samples. Specifically, Mixup [36], Cutout [4], and CutMix +[35] apply the mixing process at the instance level. While +Patchup [6] trims the hidden layers and mixes the image at +the feature level. PuzzleMix [17] and SaliencyMix [31] first +devise strategy to extract the salient instance of the original +images, then establish the mixing process. +However, more than directly applying the existing +mixing-based methods to pathology image analysis, the in- +novations are needed due to the following three reasons: +1. Domain characteristics of pathology images. Accord- +ing to our observation, there are three basic and essential +pathology image characteristics in the pathology field. (1) +Local specificity. Factors such as the size of cells, the ra- +tio of the nucleus to the cytoplasm, and the distribution of +cell chromatin are important components of pathology im- +age features. They are often scattered in various parts of +the image, requiring careful observation of local areas. (2) +Global distribution characteristics. Factors such as the rel- +ative size of cells, the overall orientation of cells, and the +consistency of cell spacing need to be analyzed as a whole +for the entire image. (3) Inner/outer-sample instance rela- +tionship. In pathology images, there are different granu- +larities of pathology instances in the same sample, such as +cells, tissues, etc. Instances of the same category appear +similar, while the relationship of different instances acts as +the main feature. The relationship between instances in the +same image, which is called the inner-sample instance re- +lationship, is a discriminative element in lesion identifica- +tion. Besides, the relationship across instances in differ- +ent samples is called outer-sample instance relationship. +Once we capture the proper outer relationship, we can com- +bine the related instances into a meaningful pathology im- +age. Therefore, we need a proper mixing approach as a data +augmentation method that well models the inner-sample re- +lationship and combine instances in proper outer-sample re- +lationships to generate new images. +2. Perturbation information. Pathology images are arti- +ficially processed (stained/photographed), which inevitably +introduces noise, leading to the inconsistency in the distri- +bution of pathology images corresponding to the same la- +bel. In fact, the attention map in earlier pathology image +analysis work always highlights some unrelated regions in +the images [37], which also validates the impact of pertur- +bation information. Therefore, it is much tougher to do the +augmentation while preserving the labeling consistency. +3. Different data distribution between pathology images. +In the pathology field, due to the difference in collection de- +vices or preprocess methods during the pathology data col- +lection process, a large gap exists between images under the +same pathology task. Thus, previous mixing-based meth- +ods will intermingle a variety of different distribution infor- +mation, making the subsequent learning procedure harder. +Therefore, we need a careful training strategy to reduce the +difficulty in the learning process. +To address the three issues mentioned above, first, we +propose a CellMix framework based on an in-place shuffle +strategy to achieve the augmentation process. +Specifically, we split the images into patches and shuf- +fle them in the same position across one batch at a certain +proportion. We use the same proportion to blend the la- +bels of these batch samples. This process is called the in- +place shuffle strategy (illustrated in Figure 2). The patch +size is designed to be consistent with the instance’s gran- +ularity, which contains local specificity. In this way, the +inner/outer-sample instance relationships and global fea- +tures in pathology images remain relatively stable compared +to other methods. +Therefore, we recreate newly labeled +pathology samples via the shuffle procedure which com- +plies with the characteristics of pathology images. +Second, inspired by the idea of curriculum learning [2], +we devise a loss-drive strategy during the training process. +Due to the perturbation noise and domain gap across the +pathology images, it is a challenging task to directly use the +aforementioned augmented data in the downstream training +process. Curriculum learning imitates the learning process +of human beings. It advocates that the model should learn +from the easy samples, and gradually advance to the com- +plex samples. It is proven to be very effective in dealing +with complex tasks in numerous research [21, 24, 29, 34]. +Therefore, in our CellMix framework, we design a fix- +position ratio scheduler and a patch-size scheduler towards +the in-place shuffle process. We continuously change the +patch size and the fixed-position ratio of pathology images +to reduce the difficulty of the training task. In this way, +we can gradually enhance the modeling ability towards bi- +ased pathology images and fit the domain gap adaptively. +Furthermore, to make the training model control the sched- +ule of learning process by itself, we devise a loss-drive ratio +changing guideline. We judge the learning procedure by the +loss of our model and alter the difficulty respectively, which +is also a crucial approach to further improve the training +efficiency. +In summary, our contribution can be concluded as the +following four aspects: +1. We are the first to develop data augmentation tech- +niques in pathology images and summarize three key issues +in the augmentation process. These issues includes the char- +acteristics in the pathology field, especially the relationship +modeling and original noise in pathology images, which +should be highly considered in the augmentation process. +2. We propose a CellMix framework based on an in- +place shuffle strategy, which explicitly considers the charac- +teristics of pathology images, especially the inner/outer in- +stance relationship to do the data augmentation in a mixing- + +based manner. +3. We devise a loss-drive curriculum learning strategy +during training, making the training process adaptively to +the augmented data. +4. The experiments conclude that our CellMix frame- +work outperforms the previous SOTA method among 7 dif- +ferent datasets. +2. Related Work +2.1. Data Augmentation +Data augmentation has been widely used in deep learn- +ing models. +Mixup [36] produces an augmented im- +age by a pixel-wise weighted combination of two images. +Cutout [4] proposes to mask fixed-size square regions of +the input training images. CutMix [35] randomly masks a +rectangular-shaped region of one image and replaces it with +the corresponding position of another image. Patchup [6] +and Manifold mixup [32] extend the concept of mixup from +input space to hidden feature space. FMix [9] uses random- +shape masks sampled from Fourier space. PuzzleMix [17], +SaliencyMix [31] and Co-Mix [16] focus on image saliency +analysis. The mask tries to reveal the most salient regions of +images and maximize the saliency of the augmented image. +ResizeMix [25] directly resizes the source image to a small +patch and pastes it on another image. RandomMix [22] ran- +domly selects a mixing sample data augmentation method +from candidates, which increases the diversity of the mixed +samples. +2.2. Curriculum Learning +Bengio et al. [2] proposed a new learning paradigm +called curriculum learning (CL). In this strategy, a model is +learned by gradually including from easy to complex sam- +ples in training in order to increase the entropy of training +samples. Previous researches have proved that CL is con- +sistent with the principle in human teaching [1, 15]. It of- +ten utilizes prior knowledge provided by instructors as guid- +ance for curriculum design. It means that the predetermined +curriculum heavily relies on the quality of prior knowledge +while ignoring feedback about the learner. +To alleviate this issue, Kumar et al. [19] designed a new +formulation, called self-paced learning (SPL). SPL adds a +regularization term into the learning objective as the cur- +riculum design. SPL leaves learners the freedom to adjust to +the actual curriculum according to their learning paces. Var- +ious SPL-based applications have been proposed recently +[12, 20, 30]. +However, SPL is unable to deal with prior +knowledge. Self-paced curriculum learning (SPCL) [13], +as an “instructor-student-collaborative” learning mode, con- +siders both prior knowledge and information learned during +training , which gives inspiration to CellMix. +3. Method +In our CellMix framework for pathology image augmen- +tation, there are two main components (demonstrated in +Figure 2). First is the in-place shuffle strategy, which re- +groups the patches in pathology images and generates aug- +mented samples. Second is the loss-drive strategy, which in- +cludes a patch-size scheduler and a fix-position ratio sched- +uler driven by the model loss. This designation in curricu- +lum learning aims to make the training process more effec- +tive and is flexible to deal with the pathology noise afore- +mentioned. Besides, with the shift of patch-size and fix- +position ratio, different scales of pathology image features +are included. In this way, the curriculum learning process +also makes the augmented pathology samples more repre- +sentative. We will give a detailed description of the two +main components in the following sections. +3.1. In-place Shuffle Strategy +The in-place shuffle strategy draws lessons from the idea +of a mixing-based augmentation approach from the com- +puter vision community. It holds the hypothesis that image +labels preserve consistent visual information which can be +mixed to generate interpolative samples. Besides, our in- +place shuffle strategy highly considers the characteristics of +the pathology field. Here we will introduce the main pro- +cess of the in-place shuffle process. +Let I ∈ RW ×H×C represent a pathology image, where +W, H, C denote the width, height, and channel of the im- +ages respectively. Same with ViT, we first equally split the +image into n patches, thus the image can be represented as +I = {P0, P1, . . . , Pn}, +Pi ∈ R[p,p,C] +where Pi denotes each patch, and we assign p to denote +the patch size of Pi. Let B denote the batch in the train- +ing process. In our in-place shuffle strategy, we take β as +the fix-position ratio. We randomly select m patches in the +same position across one batch B: +m = n × β, +β ∈ [0, 1] +which are regarded as position tokens F that do not partic- +ipate in the shuffle process. And other patches are relation +tokens R, acting as patches to be shuffled. By fixing a subset +of patches unchanged, we preserve this part of inner-sample +instance relationship, which is essential in pathology mod- +eling. +Next, we do the in-place shuffle process. We shuffle the +relation tokens across each batch B while keeping the po- +sition token fixed (illustrated in the right part of Figure 2) +which can be formulated as: +Pa,i = M ⊙ PF,i + (1 − M) ⊙ PR,i, +i ∈ n + +Figure 2. Overview of CellMix containing two components: In-place shuffle strategy and Loss-drive curriculum learning process. First, +images are split into patches. Then in Fix-position in-place shuffle strategy, we randomly select position tokens that do not participate +in the shuffle process, shown in blue. Yellow patches are relation tokens to be in-place shuffled. Next, in the right part of the figure, we +shuffle relation tokens across the batch, remaining their relative positions the same in images. Soft labels are generated from ground truth +labels according to the fix-position ratio. For the Loss-drive strategy in the blue box, we designed a Patch-size scheduler and Fix-position +scheduler to adaptively guide the learning process according to the model loss. It is introduced specifically in Figure 3. +where ⊙ is element-wise multiplication and M denotes a +binary mask, serving to assign the two kinds of tokens. +M = +� +0, i ∈ R +1, i ∈ F +In this way, we augment new pathology sample Pa,i through +the shuffle process. +Meanwhile, the image labels need to be processed syn- +chronously according to the fix-position ratio f. Consider- +ing that the image label is discrete, We need to use an in- +terpolation method to derive the soft label of the enhanced +sample: +ya = fyf + (1 − f)yr, +ya ∈ R[cls], +yf, yr ∈ I[cls] +where, f ∈ (0, 1) denotes the mix-proportion, ya denotes +the soft label of augmented samples, yf,yr are one-hot vec- +tors. yf denotes the label of fix-position tokens and yr de- +notes the relation tokens. The [cls] denotes the number of +classification categories. +With our in-place shuffle strategy, the absolute relation- +ship of the tokens remains, making the embedding of fea- +tures fixed in the training process. In this way, compared +to other methods, the important relative relationships and +global features in pathology images remain stable. There- +fore, we recreate pseudo samples via the shuffle procedure +which complies with the characteristics of pathology im- +ages. +3.2. Loss-drive Curriculum Learning +Due to the influence of perturbation information of the +pathology image and the inconsistency of data distribution, +it is often difficult to directly use augmented samples for +training downstream tasks. +Inspired by the idea of cur- +riculum learning, we imitate the process of human learning +knowledge and design training courses from easy to diffi- +cult for the model. +3.2.1 +Two Schedulers in Curriculum Learning +Specifically, in the in-place shuffle strategy, different patch +sizes and fixed position ratios will affect the complexity of +generated samples and bring about different training diffi- +culties. At the same time, since pathology images actually +contain features of different levels and scales, we also hope +that the samples augmented by the shuffle process can be +representative and flexible. In our framework, we design +a patch-size scheduler and fix-position ratio scheduler, as +shown in Figure 3. +With the patch-size scheduler, we continuously reduce +the patch size in splitting the pathology images as follows: +[pn, pn−1, . . . , p1], +pn > pn−1 > · · · > p1 +During this process, we have more and more refined +modeling of the relative relationship among instances. + +Loss-drive +Position +Classification +strategy +token +loss +fixed +Patch-size +Fix- +scheduler +position +Fix-position +scheduler +In-place +Model +shuffle +Relation +token +Positiontokens +shuffled +Relationtokens +Images & labels +Images & soft-labelsFigure 3. +Structure of Loss-drive curriculum learning process. +First, we design two schedulers: A Patch-size scheduler and a +Fix-position scheduler. The former one schedules patch size from +bigger to smaller, while the latter one gradually reduces the ra- +tio of fix-position tokens (patches) in the shuffle process. Then, +Loss-drive strategies, shown in the blue box, are proposed to de- +termine how schedulers work. When the model learns well, we +increase the difficulty as the red arrow. Otherwise, with the green +arrow, the difficulty decreases. The curriculum remains the same +in Loss-hold or steps backward to a simpler one in Loss-back. +From the big-scale relationship to the small-scale relation- +ship, the augmented samples cover the relationship of dif- +ferent granularity. +In the fix-position ratio scheduler, we continuously re- +duce the fix-position ratio in the shuffle process as follows: +[fn, fn−1, . . . , f1], +fn > fn−1 > · · · > f1, +fi ∈ (0, 1) +In this way, the mixing efficiency of the in-place shuf- +fle is constantly improved, and the pathological relationship +is becoming more and more complex, making the model +adaptively enhance the modeling ability. +Furthermore, in our curriculum learning, we do not learn +the subset of total samples every epoch, but all samples with +different patch sizes and fix-position ratios. In this way, +we manage to plan the augmented pathology images to be +trained for the model from easy to difficult. This process +eliminates the influence of perturbations in the pathology +field and improves the training effect and learning efficiency +of augmented samples. At the same time, due to the change +of schedule, the model can perceive the pathological charac- +teristics across different scales. This training strategy makes +the augmentation samples obtained by the in-place shuffle +process become more representative. +3.2.2 +Loss-drive Stragtegy +On the basis of the above patch size schedule and fixed po- +sition ratio schedule, we draw on the idea of self-paced cur- +riculum learning and hope that the training model can adjust +the schedule itself to control the learning process. To this +end, we guide the learning process according to the model +loss, as shown in the right half of Figure 3. +Specifically, in the training process, let the loss of the +current iteration be l, and the learning threshold is T. When +the loss value of the model is less than a certain thresh- +old, the model has been learned well enough for the current +course, which indicates the current course is simple enough +for the model. Therefore, we increase the difficulty of the +curriculum by reducing the patch size and diminishing the +fix-position ratio. +pi → pi−1; fi → fi−1 +iff +l < T +On the contrary, when the loss l is larger than the thresh- +old T, we give two strategies to control the schedule. First, +the loss-hold strategy, which make the schedule remain un- +changed. +pi → pi; fi → fi +iff +l ≥ T +Second, the loss-back strategy, which pushes back to the +previous schedule. +pi → pi+1; fi → fi+1 +iff +l ≥ T +We will compare the two strategies in the experiments +in Section 4. With loss-drive strategy, the model can adjust +the difficulty by itself, so as to further improve the training +efficiency of augmented samples. +4. Experiment +In this section, we mainly demonstrate the effectiveness, +generalizability, and robustness of CellMix. To evaluate the +performance of CellMix, we apply it to 7 different pathol- +ogy image classification datasets: ROSE [37], pRCC [7], +WBC [18], MARS1, GS [18], Warwick [28] and NCT [14]. +The datasets are introduced in Section 4.1, covering dif- +ferent feature scales, dataset scales, and variety of pathol- +ogy tasks. As an instance regrouping strategy, CellMix is +compared with several mixing-based state-of-the-art data +augmentation variants in Section 4.2. Designed for patch- +based learning approaches, CellMix also verifies the gen- +eralizability onto a wide range of backbones (CNN-based, +ViT-based and Hybrid) in Section 4.3. Different CellMix +strategies are introduced in Section 4.4 to explore the de- +sign concepts. Interestingly, we examine the effect of cur- +riculum design and feature scale with different prior knowl- +edge in Section 4.5. Lastly, for better interpretability, we +1https://www.marsbigdata.com/competition/details?id=21078355578880 + +Patch-size Scheduler +Loss-drive +strategies +Bigger +Smaller +relationship +relationship +Loss-hold +Loss-back +Pi+1 +Pi +Pi-1 +Fix-position Ratio Scheduler +Morecomplex +epoch +Less complex +relationship +relationship +Increasing difficulty +Previousvalue +Newvalue +Relation tokens +PositiontokensDataset +Category Numbers +Image Numbers +Organ/Tissue +Field +ROSE +2 +5088 +Pancreas +Cytopathology +pRCC +2 +1417 +Kidney +Histopathology +WBC +5 +14514 +Blood +Cytopathology +MARS +3 +1770 +Stomach +Histopathology +GS +2 +700 +Stomach +Histopathology +Warwick +2 +165 +Colon & Rectum +Histopathology +NCT +9 +100000 +Colon& Rectum +Histopathology +Table 1. Dataset description. +use Grad-CAM [26] to illustrate that CellMix obtains the +most accurate attention area among models in Section 4.6. +4.1. Datasets and Implementation Details +As a general approach, many datasets are explored. The +overview can be found in Table 1. More details of datasets +and implementation refer to Appendix A and Appendix B +respectively. +4.2. Comparison with State-of-the-art Methods +In this section, we provide a comprehensive compari- +son with many state-of-the-art mixing-based augmentation +methods. We reproduce the state-of-the-art data augmen- +tation methods [4, 9, 25, 31, 35, 36] with official codes and +settings and compare them with the CellMix method. The +baseline does not use any mixing-based data augmentation +methods. +We denote the experiments as Baseline, Cut- +Mix, Cutout, FMix, Mixup, ResizeMix, SaliencyMix, and +CellMix in the experiment tables. Every online augmen- +tation method is triggered by 50% chance in the training +process. Specific details can be found in Appendix B. +Table 2 shows CellMix significantly outperforms all +other mixing-based augmentation methods. On small-size +datasets including GS and Warwick which are relatively +simple, the majority of data augmentation methods perform +well including CellMix. On medium-size datasets including +ROSE, pRCC, WBC and MARS, CellMix yields a strik- +ing performance advancement up to 4.59% in accuracy and +6.14% in F1-score compared with Baseline, which largely +outperforms other SOTA methods. +In addition, CellMix +can steadily boost the performance on large datasets such +as NCT, by 1.47% for accuracy and 2.74% for F1-score. +4.3. Generalizability +To verify the generalizability of CellMix, we examine +various baseline models divided into three sets: +CNN- +based backbones, ViT-based backbones and Hybrid back- +bones. +CNN-based backbones include VGG16, VGG19 +[27], Resnet50 [10], Xception [3] and Mobilenetv3 [11]. +ViT-based models, also considered as patch-learning-based +models are ViT-base [5] and Swin-base [23]. Conformer +[8], Crossformer [33] and ResNet50-ViT [5] are Hybrid +backbones. Unless specified otherwise, the model imple- +mentation is based on official codes and minimal changes +are made to hyperparameters of these backbones. +In Table 3, CellMix reveals great advantages on ViT- +based models, with up to 4.59% and 1.43% improvements +on ViT-base and Swin-base models respectively. CellMix +cuts input images into non-overlapping patches, which per- +fectly encounters the patch-learning-based models with- +out breaking the actual value of tokens. +The CellMix +introduces an instance (patch) regrouping strategy to en- +hance the tokens’ relationship modeling in the attention- +based approaches. +This may explain why CellMix per- +forms especially well on patch-learning-based models. For +CNN-based backbones and Hybrid backbones, the pro- +posed method consistently boosts the variants in the range +of 0.1% to 3.18%. +In short, Table 3 shows the considerable generalizability +of CellMix, considering the improvements of CellMix on +non-patch-learning-based methods, and especially striking +performance on patch-learning-based models. +4.4. Variants on Shuffle and Loss-Drive Strategy +CellMix mainly innovates with the fix-position in-place +shuffle strategy and loss-drive method. In this section, we +explore variants of both the in-place shuffle strategy and the +loss-drive method. In the in-place shuffle strategy, a set of +randomly selected patches are shuffled across the batch, re- +maining the relative positions the same in images. For the +first strategy of CellMix-group, those alternative patches are +from the same image. For the second strategy CellMix-split, +each patch is replaced separately, making extremely com- +plex instance groups. In this way, the augmented images, +containing regions from various samples are more biased to- +ward each instance rather than focusing on the same-sample +relationship of the instances. +As for the loss-drive method, we have two variants +denoted as Loss-hold and Loss-back. +Loss drive is an +“instructor-student-collaborative” learning mode consider- +ing both prior knowledge of the curriculum and feedback +from learners. The specific curriculum in the next iteration +is defined by the model performance of the current epoch. +If model learns well, we increase the difficulty of learning. +While if not, the curriculum remains the same in Loss-hold +or steps backward to a simpler one in Loss-back. There- +fore, the model can learn different feature scales in a more +comprehensive way. +As shown in Table 4, all variants of CellMix can steadily +boost the baseline performance on all benchmarks. And +CellMix-group with Loss-hold strategy, as the best vari- +ant, lifts the accuracy of ViT up to a remarkable 4.59%. +Other variants lift the baseline result up to 3.89%, 1.41% +and 1.87% for CellMix-group-Loss-back, CellMix-split- +Loss-hold and CellMix-split-Loss-back respectively. +Al- +though all variants are effective, CellMix-split may over- + +Method +ROSE +pRCC +WBC +MARS +GS +Warwick +NCT +Acc +F1 +Acc +F1 +Acc +F1 +Acc +F1 +Acc +F1 +Acc +F1 +Acc +F1 +Baseline +91.63 +87.66 +92.58 +90.05 +98.39 +99.24 +96.31 +95.85 +99.29 +99.55 +100.00 +100.00 +98.19 +96.38 +CutMix +91.93 +88.15 +95.41 +93.66 +98.80 +99.45 +96.59 +96.25 +100.00 +100.00 +100.00 +100.00 +99.61 +98.83 +Cutout +92.72 +89.58 +93.29 +91.24 +97.60 +98.44 +96.88 +96.49 +98.57 +99.11 +100.00 +100.00 +98.33 +96.11 +FMix +93.41 +90.41 +93.64 +91.35 +98.32 +98.99 +96.31 +95.79 +100.00 +100.00 +100.00 +100.00 +99.39 +98.59 +Mixup +93.80 +90.94 +93.29 +90.55 +97.42 +98.44 +96.88 +96.55 +100.00 +100.00 +97.50 +97.62 +99.46 +98.76 +ResizeMix +92.52 +88.66 +95.76 +94.34 +98.20 +98.90 +97.44 +97.12 +100.00 +100.00 +100.00 +100.00 +99.57 +98.78 +SaliencyMix +93.21 +90.18 +94.70 +93.02 +98.92 +99.55 +98.01 +97.78 +100.00 +100.00 +100.00 +100.00 +99.50 +98.76 +CellMix +94.49 +92.07 +97.17 +96.19 +99.26 +99.70 +98.86 +98.74 +100.00 +100.00 +100.00 +100.00 +99.63 +99.12 +Table 2. Accuracy and F1-score comparison with state-of-the-art mixing-based data augmentation methods on datasets including ROSE, +pRCC, WBC, GS, Warwick, MARS and NCT. All counterpart methods are evaluated with ViT-base. +Model +ROSE +pRCC +WBC +MARS +GS +Warwick +NCT +Baseline +CellMix +Baseline +CellMix +Baseline +CellMix +Baseline +CellMix +Baseline +CellMix +Baseline +CellMix +Baseline +CellMix +VGG16 +91.44 +90.85 +86.22 +86.22 +98.36 +99.03 +96.59 +98.01 +99.29 +99.29 +96.25 +97.50 +98.06 +99.55 +VGG19 +89.96 +91.63 +88.34 +87.28 +98.25 +98.94 +96.88 +97.73 +99.29 +98.57 +95.00 +97.50 +98.10 +99.55 +Resnet50 +91.14 +91.24 +90.46 +89.40 +98.92 +98.50 +97.44 +97.73 +100.00 +100.00 +100.00 +100.00 +99.19 +99.67 +Xception +90.65 +90.85 +82.69 +85.87 +98.20 +98.46 +97.44 +96.59 +96.43 +98.57 +100.00 +100.00 +99.23 +99.64 +Mobilenetv3 +87.30 +86.02 +75.27 +71.38 +97.21 +97.74 +96.02 +96.02 +96.43 +96.43 +95.00 +93.75 +98.87 +99.24 +ViT-base +91.63 +94.49 +92.58 +97.17 +98.39 +99.26 +96.31 +98.86 +99.29 +100.00 +100.00 +100.00 +98.19 +99.63 +Swin-base +92.22 +93.31 +92.23 +92.93 +98.89 +98.36 +96.59 +97.44 +97.86 +99.29 +100.00 +100.00 +97.20 +99.46 +Conformer +91.34 +92.62 +92.93 +90.11 +97.74 +98.43 +96.02 +97.44 +100.00 +100.00 +100.00 +100.00 +99.60 +99.60 +Crossformer +91.73 +92.91 +89.40 +89.40 +97.14 +98.39 +96.31 +97.44 +99.29 +99.29 +97.50 +100.00 +97.77 +99.43 +ResNet50-ViT +92.62 +92.72 +90.81 +90.11 +99.06 +98.92 +96.59 +98.01 +99.29 +99.29 +100.00 +100.00 +99.73 +99.82 +Table 3. CellMix can steadily boost a wide range of model variants including CNN-based, ViT-based, and Hybrid models on all classifica- +tion benchmarks. Note that all hyperparameters within one dataset remain the same as Section 2. +Method +ROSE +pRCC +WBC +MARS +GS +Warwick +NCT +Baseline +91.63 +92.58 +98.39 +96.31 +99.29 +100.00 +98.19 +Group-Hold +94.49 +97.17 +99.26 +98.86 +100.00 +100.00 +99.63 +Group-Back +93.60 +96.47 +99.03 +98.30 +100.00 +100.00 +99.61 +Split-Hold +92.91 +93.99 +98.82 +97.44 +100.00 +100.00 +99.57 +Split-Back +93.50 +93.64 +98.27 +98.01 +100.00 +100.00 +99.55 +Table 4. Accuracy of variants for in-place shuffle strategy and loss- +drive method. Group and Split denotes the CellMix-group and +CellMix-split strategies respectively. Hold means the proposed +Loss-hold strategy, and Back means Loss-back strategy. +complicate the augmentation images and break the absolute +relationship of instances, resulting in negative effects on the +performance. Loss-hold is better than Loss-back consid- +ering the potential over-simplified curriculum in the Loss- +back strategy. CellMix-group with Loss-hold strategy is the +default strategy denoted as CellMix unless specified. +4.5. Curriculum Variants +As mentioned in Section 3, patch size and fixed position +ratio of samples represent the difficulty of the curriculum. +With an adaptive curriculum, the model can perceive the +pathology characteristics across different scales and elimi- +nate the perturbation information of pathology images. The +prior knowledge is the range, i.e. learning space, of both +two variables. +In this section, we ablate different prior +knowledge choices to explore the effect of curriculum de- +Patch Size +ROSE +pRCC +WBC +16 +93.90 +94.70 +98.50 +32 +94.09 +94.70 +98.76 +48 +93.50 +96.11 +98.78 +64 +94.19 +95.05 +98.71 +96 +94.00 +95.05 +98.20 +128 +93.31 +95.05 +98.78 +192 +92.81 +95.05 +98.27 +Table 5. Curriculum variants for patch size fixed in the range of +16 to 192. Fix-position ratio scheduler remains the same. +sign and feature scale. We fix different initial values for +patch size from 16 to 192 as well as fix position ratio from +0.5 to 0.9, shown in Table 5 and Table 7 respectively. In ad- +dition, we choose different patch size lists including [128, +64, 32, 16] and [196, 96, 48, 16], shown in Table 6. +In Table 5, CellMix performs the best with fix-patch size +64 on ROSE and 48 on pRCC and WBC, while too small +or too large patch size shows poor performance. In fact, +instances (i.e. cells, tissues, etc.) are sparsely distributed +in the majority of pathology images. With a large patch +size, it is more likely to generate augmented samples with +misleading supervisory signals. While too small, instances +are split into pieces resulting in the loss of semantic infor- +mation. This result verifies the capability of instance-based +shuffle strategy and supports our hypothesis that local (in- +stance) specificity, such as characteristics of cells, plays an + +Figure 4. CAM visualizations on un-augmented images. The first column is the original image from ROSE, pRCC and WBC, and the +other columns are Grad-CAM results. As shown in the figure, CellMix guides the model to precisely focus on target instances and can +accurately recognize the contour of the cell area. +Patch Size +ROSE +pRCC +WBC +Even +93.70 +96.47 +98.62 +Odd +94.19 +95.76 +98.32 +Full +94.49 +97.17 +99.26 +Table 6. Curriculum variants for different patch size lists. Full, as +the default set, means patch size list [192, 128, 96, 64, 48, 32,16]. +Even is [128, 64, 32, 16], while Odd is [196, 96, 48, 16]. +important role in modeling pathology image features. +Table 6 shows CellMix performs the best with a full +patch size list. Specifically, it outperforms CellMix without +full patch size lists in a range of 0.3% to 1.41%. Shortening +the patch size list reduces learnable feature scales and the +accuracy of results, which means a strategy with more com- +prehensive feature scales can work better in relationship +modeling. In Table 7, CellMix with adaptive fix-position +ratio outperforms others with given ratios from 0.13% to +2.82%. The drop in performance of the variant without a +full patch size list and the variant without an adaptive fix- +position ratio is consistent with the perspective that different +feature scales contribute to better relationship modeling. +To further evaluate the loss-drive strategy with different +loss thresholds and patch/ratio schedulers, more detailed ab- +lation experiments are conducted in Appendix C. +4.6. CAM Analysis +We extract Grad-CAM [26] for both un-augmented im- +ages and augmented images to investigate the regions of the +input image where the model focuses to recognize an object. +Figure 4 shows that CellMix guides the model to pre- +cisely focus on the target object compared to other meth- +ods, presenting solid interpretability in cell identification. +It shows that the model fully learns the feature of cell in- +stances (local specificity) and the difference between in- +Fix Position Ratio +ROSE +pRCC +WBC +0.5 +92.91 +94.35 +97.56 +0.6 +93.90 +94.35 +98.94 +0.7 +92.13 +94.70 +97.35 +0.8 +93.31 +94.35 +98.80 +0.9 +93.41 +94.35 +99.03 +Adaptive +94.49 +97.17 +99.26 +Table 7. Curriculum variants for given fix-position ratios. For +example, 0.9 means 90% patches of one image are fixed in the in- +place shuffle process, with 10% shuffled. Adaptive, as the default +setting, represents the continuously decreasing fix-position ratio. +stances and background (global distribution). +We also visualize Grad-CAM on augmented images in +Figure 1. It can be seen that CellMix effectively focuses on +the corresponding features and precisely localizes the in- +stances in the scene. Besides, it can be seen clearly that +CellMix pays attention to different cells when making neg- +ative and positive predictions, and accurately identifies the +boundary of different patches. It proves that CellMix in- +deed learns the difference between negative and positive in- +stances and the inner/outer-sample instance relationship. +5. Conclusion +In conclusion, we introduce CellMix data augmentation +method, which aims for the characteristics of pathology im- +age analysis and forces the models to observe the relation- +ship between the instances. We propose a distribution-based +in-place shuffle strategy that can regroup patches (contain- +ing instances) in the image (bag) level, with two kinds +of tokens assigned. At the image level, effective pseudo- +samples force the models to observe the instance relation- +ship. We devise a loss-drive curriculum learning strategy +during the training process to cover the multiple potential +feature scales and alter the modeling difficulty adaptively. + +Original +Baseline(ViT) +CellMix +CutMix +Cutout +FMix +Mixup +ResizeMix +SaliencyMix +ROSE +WBCOn various pathology image classification benchmarks, +CellMix significantly outperforms all other mixing-based +augmentation methods. +CellMix boosts the performance +of ViT-base model up to 4.59% in accuracy and 6.14% in +F1-score. The inspiring SOTA results with various models +prove the effectiveness of the core idea. +6. Acknowledgement +This work was partially supported by the National Natu- +ral Science Foundation of China (No. 62271023), the Bei- +jing Natural Science Foundation (No. 7202102), and the +Fundamental Research Funds for Central Universities. +References +[1] Sumit Basu and Janara Christensen. Teaching classification +boundaries to humans. In Proceedings of the AAAI Confer- +ence on Artificial Intelligence, volume 27, pages 109–115, +2013. 3 +[2] Yoshua Bengio, Jérôme Louradour, Ronan Collobert, and Ja- +son Weston. Curriculum learning (icml). Google Scholar +Google Scholar Digital Library Digital Library, 2009. 2, 3 +[3] François Chollet. Xception: Deep learning with depthwise +separable convolutions. +In Proceedings of the IEEE con- +ference on computer vision and pattern recognition, pages +1251–1258, 2017. 6 +[4] Terrance DeVries and Graham W Taylor. 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Shuffle instances-based vision transformer for pan- +creatic cancer rose image classification. +arXiv preprint +arXiv:2208.06833, 2022. 2, 5 + diff --git a/hNFJT4oBgHgl3EQfVixV/content/tmp_files/load_file.txt b/hNFJT4oBgHgl3EQfVixV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bb1f771dace7173500cecf55037ad53e7dd2ec44 --- /dev/null +++ b/hNFJT4oBgHgl3EQfVixV/content/tmp_files/load_file.txt @@ -0,0 +1,910 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf,len=909 +page_content='CellMix: A General Instance Relationship based Method for Data Augmentation Towards Pathology Image Analysis Tianyi Zhang1* Zhiling Yan1* Chunhui Li2 Nan Ying1 Yanli Lei1 Yunlu Feng3 Yu Zhao4 Guanglei Zhang1† 1School of Biological Science and Medical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Beihang University 2School of Artifical Intelligence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Nanjing University 3Department of Gastroenterology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Peking Union Medical College Hospital 4Department of Pathology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Peking Union Medical College Hospital Abstract Pathology image analysis crucially relies on the avail- ability and quality of annotated pathological samples,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' which are very difficult to collect and need lots of human effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' To address this issue, beyond traditional preprocess data augmentation methods, mixing-based approaches are effective and practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' However, previous mixing-based data augmentation methods do not thoroughly explore the essential characteristics of pathology images, including the local specificity, global distribution, and inner/outer- sample instance relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' To further understand the pathology characteristics and make up effective pseudo samples, we propose the CellMix framework with a novel distribution-based in-place shuffle strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' We split the im- ages into patches with respect to the granularity of pathol- ogy instances and do the shuffle process across the same batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In this way, we generate new samples while keeping the absolute relationship of pathology instances intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Fur- thermore, to deal with the perturbations and distribution- based noise, we devise a loss-drive strategy inspired by cur- riculum learning during the training process, making the model fit the augmented data adaptively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' It is worth men- tioning that we are the first to explore data augmentation techniques in the pathology image field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Experiments show SOTA results on 7 different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' We conclude that this novel instance relationship-based strategy can shed light on general data augmentation for pathology image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' The code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='com/sagizty/CellMix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Tianyi Zhang (zhangtianyi@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='cn) Zhiling Yan (zhilingyan724@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='com) †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Guanglei Zhang (guangleizhang@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='cn) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Visualization of an augmented example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Compared with other mixing-based methods, CellMix accurately identifies the boundary of patches and instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' It also tells the differences between negative and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Introduction Deep neural networks (DNNs) have shown impressive performance in pathology image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' However, this success crucially relies on the availability and quality of an- notated pathology images, which are very difficult to collect and need lots of human effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' To alleviate this problem, data augmentation, which enlarges the amount and diver- sity of the labeled samples with label-preserving transfor- mations, is a feasible method to improve the generalization of DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Data augmentation has been extensively established in the traditional computer vision field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Beyond the traditional preprocess data augmentation methods(including: rotate, flip, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' ), the mixing-based approach is one of the most rep- resentative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Mixing-based methods are domain- agnostic data augmentation techniques first proposed by arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='11513v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='CV] 27 Jan 2023 Original Mixed Sample Negative Positive Negative CellMix Positive Mixup Positive CutMix[36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' It devises a strategy that mixes some sub-regions or features from different categories of images, as new pseudo samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Specifically, Mixup [36], Cutout [4], and CutMix [35] apply the mixing process at the instance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' While Patchup [6] trims the hidden layers and mixes the image at the feature level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' PuzzleMix [17] and SaliencyMix [31] first devise strategy to extract the salient instance of the original images, then establish the mixing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' However, more than directly applying the existing mixing-based methods to pathology image analysis, the in- novations are needed due to the following three reasons: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Domain characteristics of pathology images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Accord- ing to our observation, there are three basic and essential pathology image characteristics in the pathology field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' (1) Local specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Factors such as the size of cells, the ra- tio of the nucleus to the cytoplasm, and the distribution of cell chromatin are important components of pathology im- age features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' They are often scattered in various parts of the image, requiring careful observation of local areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' (2) Global distribution characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Factors such as the rel- ative size of cells, the overall orientation of cells, and the consistency of cell spacing need to be analyzed as a whole for the entire image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' (3) Inner/outer-sample instance rela- tionship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In pathology images, there are different granu- larities of pathology instances in the same sample, such as cells, tissues, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Instances of the same category appear similar, while the relationship of different instances acts as the main feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' The relationship between instances in the same image, which is called the inner-sample instance re- lationship, is a discriminative element in lesion identifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Besides, the relationship across instances in differ- ent samples is called outer-sample instance relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Once we capture the proper outer relationship, we can com- bine the related instances into a meaningful pathology im- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Therefore, we need a proper mixing approach as a data augmentation method that well models the inner-sample re- lationship and combine instances in proper outer-sample re- lationships to generate new images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Perturbation information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Pathology images are arti- ficially processed (stained/photographed), which inevitably introduces noise, leading to the inconsistency in the distri- bution of pathology images corresponding to the same la- bel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In fact, the attention map in earlier pathology image analysis work always highlights some unrelated regions in the images [37], which also validates the impact of pertur- bation information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Therefore, it is much tougher to do the augmentation while preserving the labeling consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Different data distribution between pathology images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In the pathology field, due to the difference in collection de- vices or preprocess methods during the pathology data col- lection process, a large gap exists between images under the same pathology task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Thus, previous mixing-based meth- ods will intermingle a variety of different distribution infor- mation, making the subsequent learning procedure harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Therefore, we need a careful training strategy to reduce the difficulty in the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' To address the three issues mentioned above, first, we propose a CellMix framework based on an in-place shuffle strategy to achieve the augmentation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Specifically, we split the images into patches and shuf- fle them in the same position across one batch at a certain proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' We use the same proportion to blend the la- bels of these batch samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' This process is called the in- place shuffle strategy (illustrated in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' The patch size is designed to be consistent with the instance’s gran- ularity, which contains local specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In this way, the inner/outer-sample instance relationships and global fea- tures in pathology images remain relatively stable compared to other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Therefore, we recreate newly labeled pathology samples via the shuffle procedure which com- plies with the characteristics of pathology images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Second, inspired by the idea of curriculum learning [2], we devise a loss-drive strategy during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Due to the perturbation noise and domain gap across the pathology images, it is a challenging task to directly use the aforementioned augmented data in the downstream training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Curriculum learning imitates the learning process of human beings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' It advocates that the model should learn from the easy samples, and gradually advance to the com- plex samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' It is proven to be very effective in dealing with complex tasks in numerous research [21, 24, 29, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Therefore, in our CellMix framework, we design a fix- position ratio scheduler and a patch-size scheduler towards the in-place shuffle process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' We continuously change the patch size and the fixed-position ratio of pathology images to reduce the difficulty of the training task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In this way, we can gradually enhance the modeling ability towards bi- ased pathology images and fit the domain gap adaptively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Furthermore, to make the training model control the sched- ule of learning process by itself, we devise a loss-drive ratio changing guideline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' We judge the learning procedure by the loss of our model and alter the difficulty respectively, which is also a crucial approach to further improve the training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In summary, our contribution can be concluded as the following four aspects: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' We are the first to develop data augmentation tech- niques in pathology images and summarize three key issues in the augmentation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' These issues includes the char- acteristics in the pathology field, especially the relationship modeling and original noise in pathology images, which should be highly considered in the augmentation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' We propose a CellMix framework based on an in- place shuffle strategy, which explicitly considers the charac- teristics of pathology images, especially the inner/outer in- stance relationship to do the data augmentation in a mixing- based manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' We devise a loss-drive curriculum learning strategy during training, making the training process adaptively to the augmented data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' The experiments conclude that our CellMix frame- work outperforms the previous SOTA method among 7 dif- ferent datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Data Augmentation Data augmentation has been widely used in deep learn- ing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Mixup [36] produces an augmented im- age by a pixel-wise weighted combination of two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Cutout [4] proposes to mask fixed-size square regions of the input training images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' CutMix [35] randomly masks a rectangular-shaped region of one image and replaces it with the corresponding position of another image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Patchup [6] and Manifold mixup [32] extend the concept of mixup from input space to hidden feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' FMix [9] uses random- shape masks sampled from Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' PuzzleMix [17], SaliencyMix [31] and Co-Mix [16] focus on image saliency analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' The mask tries to reveal the most salient regions of images and maximize the saliency of the augmented image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' ResizeMix [25] directly resizes the source image to a small patch and pastes it on another image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' RandomMix [22] ran- domly selects a mixing sample data augmentation method from candidates, which increases the diversity of the mixed samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Curriculum Learning Bengio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' [2] proposed a new learning paradigm called curriculum learning (CL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In this strategy, a model is learned by gradually including from easy to complex sam- ples in training in order to increase the entropy of training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Previous researches have proved that CL is con- sistent with the principle in human teaching [1, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' It of- ten utilizes prior knowledge provided by instructors as guid- ance for curriculum design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' It means that the predetermined curriculum heavily relies on the quality of prior knowledge while ignoring feedback about the learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' To alleviate this issue, Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' [19] designed a new formulation, called self-paced learning (SPL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' SPL adds a regularization term into the learning objective as the cur- riculum design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' SPL leaves learners the freedom to adjust to the actual curriculum according to their learning paces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Var- ious SPL-based applications have been proposed recently [12, 20, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' However, SPL is unable to deal with prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Self-paced curriculum learning (SPCL) [13], as an “instructor-student-collaborative” learning mode, con- siders both prior knowledge and information learned during training , which gives inspiration to CellMix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Method In our CellMix framework for pathology image augmen- tation, there are two main components (demonstrated in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' First is the in-place shuffle strategy, which re- groups the patches in pathology images and generates aug- mented samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Second is the loss-drive strategy, which in- cludes a patch-size scheduler and a fix-position ratio sched- uler driven by the model loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' This designation in curricu- lum learning aims to make the training process more effec- tive and is flexible to deal with the pathology noise afore- mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Besides, with the shift of patch-size and fix- position ratio, different scales of pathology image features are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In this way, the curriculum learning process also makes the augmented pathology samples more repre- sentative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' We will give a detailed description of the two main components in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In-place Shuffle Strategy The in-place shuffle strategy draws lessons from the idea of a mixing-based augmentation approach from the com- puter vision community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' It holds the hypothesis that image labels preserve consistent visual information which can be mixed to generate interpolative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Besides, our in- place shuffle strategy highly considers the characteristics of the pathology field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Here we will introduce the main pro- cess of the in-place shuffle process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Let I ∈ RW ×H×C represent a pathology image, where W, H, C denote the width, height, and channel of the im- ages respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Same with ViT, we first equally split the image into n patches, thus the image can be represented as I = {P0, P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' , Pn}, Pi ∈ R[p,p,C] where Pi denotes each patch, and we assign p to denote the patch size of Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Let B denote the batch in the train- ing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In our in-place shuffle strategy, we take β as the fix-position ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' We randomly select m patches in the same position across one batch B: m = n × β, β ∈ [0, 1] which are regarded as position tokens F that do not partic- ipate in the shuffle process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' And other patches are relation tokens R, acting as patches to be shuffled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' By fixing a subset of patches unchanged, we preserve this part of inner-sample instance relationship, which is essential in pathology mod- eling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Next, we do the in-place shuffle process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' We shuffle the relation tokens across each batch B while keeping the po- sition token fixed (illustrated in the right part of Figure 2) which can be formulated as: Pa,i = M ⊙ PF,i + (1 − M) ⊙ PR,i, i ∈ n Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Overview of CellMix containing two components: In-place shuffle strategy and Loss-drive curriculum learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' First, images are split into patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Then in Fix-position in-place shuffle strategy, we randomly select position tokens that do not participate in the shuffle process, shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Yellow patches are relation tokens to be in-place shuffled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Next, in the right part of the figure, we shuffle relation tokens across the batch, remaining their relative positions the same in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Soft labels are generated from ground truth labels according to the fix-position ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' For the Loss-drive strategy in the blue box, we designed a Patch-size scheduler and Fix-position scheduler to adaptively guide the learning process according to the model loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' It is introduced specifically in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' where ⊙ is element-wise multiplication and M denotes a binary mask, serving to assign the two kinds of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' M = � 0, i ∈ R 1, i ∈ F In this way, we augment new pathology sample Pa,i through the shuffle process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Meanwhile, the image labels need to be processed syn- chronously according to the fix-position ratio f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Consider- ing that the image label is discrete, We need to use an in- terpolation method to derive the soft label of the enhanced sample: ya = fyf + (1 − f)yr, ya ∈ R[cls], yf, yr ∈ I[cls] where, f ∈ (0, 1) denotes the mix-proportion, ya denotes the soft label of augmented samples, yf,yr are one-hot vec- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' yf denotes the label of fix-position tokens and yr de- notes the relation tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' The [cls] denotes the number of classification categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' With our in-place shuffle strategy, the absolute relation- ship of the tokens remains, making the embedding of fea- tures fixed in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In this way, compared to other methods, the important relative relationships and global features in pathology images remain stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' There- fore, we recreate pseudo samples via the shuffle procedure which complies with the characteristics of pathology im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Loss-drive Curriculum Learning Due to the influence of perturbation information of the pathology image and the inconsistency of data distribution, it is often difficult to directly use augmented samples for training downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Inspired by the idea of cur- riculum learning, we imitate the process of human learning knowledge and design training courses from easy to diffi- cult for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='1 Two Schedulers in Curriculum Learning Specifically, in the in-place shuffle strategy, different patch sizes and fixed position ratios will affect the complexity of generated samples and bring about different training diffi- culties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' At the same time, since pathology images actually contain features of different levels and scales, we also hope that the samples augmented by the shuffle process can be representative and flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In our framework, we design a patch-size scheduler and fix-position ratio scheduler, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' With the patch-size scheduler, we continuously reduce the patch size in splitting the pathology images as follows: [pn, pn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' , p1], pn > pn−1 > · · · > p1 During this process, we have more and more refined modeling of the relative relationship among instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Loss-drive Position Classification strategy token loss fixed Patch-size Fix- scheduler position Fix-position scheduler In-place Model shuffle Relation token Positiontokens shuffled Relationtokens Images & labels Images & soft-labelsFigure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Structure of Loss-drive curriculum learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' First, we design two schedulers: A Patch-size scheduler and a Fix-position scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' The former one schedules patch size from bigger to smaller, while the latter one gradually reduces the ra- tio of fix-position tokens (patches) in the shuffle process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Then, Loss-drive strategies, shown in the blue box, are proposed to de- termine how schedulers work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' When the model learns well, we increase the difficulty as the red arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Otherwise, with the green arrow, the difficulty decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' The curriculum remains the same in Loss-hold or steps backward to a simpler one in Loss-back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' From the big-scale relationship to the small-scale relation- ship, the augmented samples cover the relationship of dif- ferent granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In the fix-position ratio scheduler, we continuously re- duce the fix-position ratio in the shuffle process as follows: [fn, fn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' , f1], fn > fn−1 > · · · > f1, fi ∈ (0, 1) In this way, the mixing efficiency of the in-place shuf- fle is constantly improved, and the pathological relationship is becoming more and more complex, making the model adaptively enhance the modeling ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Furthermore, in our curriculum learning, we do not learn the subset of total samples every epoch, but all samples with different patch sizes and fix-position ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In this way, we manage to plan the augmented pathology images to be trained for the model from easy to difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' This process eliminates the influence of perturbations in the pathology field and improves the training effect and learning efficiency of augmented samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' At the same time, due to the change of schedule, the model can perceive the pathological charac- teristics across different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' This training strategy makes the augmentation samples obtained by the in-place shuffle process become more representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='2 Loss-drive Stragtegy On the basis of the above patch size schedule and fixed po- sition ratio schedule, we draw on the idea of self-paced cur- riculum learning and hope that the training model can adjust the schedule itself to control the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' To this end, we guide the learning process according to the model loss, as shown in the right half of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Specifically, in the training process, let the loss of the current iteration be l, and the learning threshold is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' When the loss value of the model is less than a certain thresh- old, the model has been learned well enough for the current course, which indicates the current course is simple enough for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Therefore, we increase the difficulty of the curriculum by reducing the patch size and diminishing the fix-position ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' pi → pi−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' fi → fi−1 iff l < T On the contrary, when the loss l is larger than the thresh- old T, we give two strategies to control the schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' First, the loss-hold strategy, which make the schedule remain un- changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' pi → pi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' fi → fi iff l ≥ T Second, the loss-back strategy, which pushes back to the previous schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' pi → pi+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' fi → fi+1 iff l ≥ T We will compare the two strategies in the experiments in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' With loss-drive strategy, the model can adjust the difficulty by itself, so as to further improve the training efficiency of augmented samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Experiment In this section, we mainly demonstrate the effectiveness, generalizability, and robustness of CellMix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' To evaluate the performance of CellMix, we apply it to 7 different pathol- ogy image classification datasets: ROSE [37], pRCC [7], WBC [18], MARS1, GS [18], Warwick [28] and NCT [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' The datasets are introduced in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='1, covering dif- ferent feature scales, dataset scales, and variety of pathol- ogy tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' As an instance regrouping strategy, CellMix is compared with several mixing-based state-of-the-art data augmentation variants in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Designed for patch- based learning approaches, CellMix also verifies the gen- eralizability onto a wide range of backbones (CNN-based, ViT-based and Hybrid) in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Different CellMix strategies are introduced in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='4 to explore the de- sign concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Interestingly, we examine the effect of cur- riculum design and feature scale with different prior knowl- edge in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Lastly, for better interpretability, we 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='marsbigdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='com/competition/details?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='id=21078355578880 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Patch-size Scheduler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Loss-drive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='strategies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Bigger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Smaller ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='relationship ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='relationship ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Loss-hold ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Loss-back ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Pi+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Pi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Pi-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Fix-position Ratio Scheduler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Morecomplex ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='epoch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Less complex ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='relationship ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='relationship ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Increasing difficulty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Previousvalue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Newvalue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Relation tokens ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='PositiontokensDataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Category Numbers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Image Numbers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Organ/Tissue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Field ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='ROSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='5088 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Pancreas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Cytopathology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='pRCC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='1417 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Kidney ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Histopathology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='WBC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='14514 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Blood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Cytopathology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='MARS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='1770 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Stomach ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Histopathology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='GS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Stomach ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Histopathology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Warwick ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='165 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Colon & Rectum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Histopathology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='NCT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='100000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Colon& Rectum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Histopathology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Dataset description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' use Grad-CAM [26] to illustrate that CellMix obtains the most accurate attention area among models in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Datasets and Implementation Details As a general approach, many datasets are explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' The overview can be found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' More details of datasets and implementation refer to Appendix A and Appendix B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Comparison with State-of-the-art Methods In this section, we provide a comprehensive compari- son with many state-of-the-art mixing-based augmentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' We reproduce the state-of-the-art data augmen- tation methods [4, 9, 25, 31, 35, 36] with official codes and settings and compare them with the CellMix method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' The baseline does not use any mixing-based data augmentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' We denote the experiments as Baseline, Cut- Mix, Cutout, FMix, Mixup, ResizeMix, SaliencyMix, and CellMix in the experiment tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Every online augmen- tation method is triggered by 50% chance in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Specific details can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Table 2 shows CellMix significantly outperforms all other mixing-based augmentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' On small-size datasets including GS and Warwick which are relatively simple, the majority of data augmentation methods perform well including CellMix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' On medium-size datasets including ROSE, pRCC, WBC and MARS, CellMix yields a strik- ing performance advancement up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='59% in accuracy and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='14% in F1-score compared with Baseline, which largely outperforms other SOTA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In addition, CellMix can steadily boost the performance on large datasets such as NCT, by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='47% for accuracy and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='74% for F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Generalizability To verify the generalizability of CellMix, we examine various baseline models divided into three sets: CNN- based backbones, ViT-based backbones and Hybrid back- bones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' CNN-based backbones include VGG16, VGG19 [27], Resnet50 [10], Xception [3] and Mobilenetv3 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' ViT-based models, also considered as patch-learning-based models are ViT-base [5] and Swin-base [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Conformer [8], Crossformer [33] and ResNet50-ViT [5] are Hybrid backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Unless specified otherwise, the model imple- mentation is based on official codes and minimal changes are made to hyperparameters of these backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In Table 3, CellMix reveals great advantages on ViT- based models, with up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='59% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='43% improvements on ViT-base and Swin-base models respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' CellMix cuts input images into non-overlapping patches, which per- fectly encounters the patch-learning-based models with- out breaking the actual value of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' The CellMix introduces an instance (patch) regrouping strategy to en- hance the tokens’ relationship modeling in the attention- based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' This may explain why CellMix per- forms especially well on patch-learning-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' For CNN-based backbones and Hybrid backbones, the pro- posed method consistently boosts the variants in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='1% to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='18%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In short, Table 3 shows the considerable generalizability of CellMix, considering the improvements of CellMix on non-patch-learning-based methods, and especially striking performance on patch-learning-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Variants on Shuffle and Loss-Drive Strategy CellMix mainly innovates with the fix-position in-place shuffle strategy and loss-drive method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In this section, we explore variants of both the in-place shuffle strategy and the loss-drive method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In the in-place shuffle strategy, a set of randomly selected patches are shuffled across the batch, re- maining the relative positions the same in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' For the first strategy of CellMix-group, those alternative patches are from the same image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' For the second strategy CellMix-split, each patch is replaced separately, making extremely com- plex instance groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In this way, the augmented images, containing regions from various samples are more biased to- ward each instance rather than focusing on the same-sample relationship of the instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' As for the loss-drive method, we have two variants denoted as Loss-hold and Loss-back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Loss drive is an “instructor-student-collaborative” learning mode consider- ing both prior knowledge of the curriculum and feedback from learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' The specific curriculum in the next iteration is defined by the model performance of the current epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' If model learns well, we increase the difficulty of learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' While if not, the curriculum remains the same in Loss-hold or steps backward to a simpler one in Loss-back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' There- fore, the model can learn different feature scales in a more comprehensive way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' As shown in Table 4, all variants of CellMix can steadily boost the baseline performance on all benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' And CellMix-group with Loss-hold strategy, as the best vari- ant, lifts the accuracy of ViT up to a remarkable 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='59%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Other variants lift the baseline result up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='89%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='41% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='87% for CellMix-group-Loss-back, CellMix-split- Loss-hold and CellMix-split-Loss-back respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Al- though all variants are effective, CellMix-split may over- Method ROSE pRCC WBC MARS GS Warwick NCT Acc F1 Acc F1 Acc F1 Acc F1 Acc F1 Acc F1 Acc F1 Baseline 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='63 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='66 92.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='00 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='63 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='12 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Accuracy and F1-score comparison with state-of-the-art mixing-based data augmentation methods on datasets including ROSE, pRCC, WBC, GS, Warwick, MARS and NCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' All counterpart methods are evaluated with ViT-base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Model ROSE pRCC WBC MARS GS Warwick NCT Baseline CellMix Baseline CellMix Baseline CellMix Baseline CellMix Baseline CellMix Baseline CellMix Baseline CellMix VGG16 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='44 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='85 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='22 86.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='50 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='00 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='77 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='43 ResNet50-ViT 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='62 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='72 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='81 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='11 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='06 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='92 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='59 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='01 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='29 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='29 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='00 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='73 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='82 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' CellMix can steadily boost a wide range of model variants including CNN-based, ViT-based, and Hybrid models on all classifica- tion benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Note that all hyperparameters within one dataset remain the same as Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Method ROSE pRCC WBC MARS GS Warwick NCT Baseline 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='63 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='58 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='39 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='31 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='29 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='00 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='19 Group-Hold 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='49 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='17 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='26 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='86 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='00 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='63 Group-Back 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='60 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='47 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='03 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='30 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='00 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='61 Split-Hold 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='91 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='99 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='82 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='44 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='00 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='57 Split-Back 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='50 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='64 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='27 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='01 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='00 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='00 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='55 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Accuracy of variants for in-place shuffle strategy and loss- drive method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Group and Split denotes the CellMix-group and CellMix-split strategies respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Hold means the proposed Loss-hold strategy, and Back means Loss-back strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' complicate the augmentation images and break the absolute relationship of instances, resulting in negative effects on the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Loss-hold is better than Loss-back consid- ering the potential over-simplified curriculum in the Loss- back strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' CellMix-group with Loss-hold strategy is the default strategy denoted as CellMix unless specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Curriculum Variants As mentioned in Section 3, patch size and fixed position ratio of samples represent the difficulty of the curriculum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' With an adaptive curriculum, the model can perceive the pathology characteristics across different scales and elimi- nate the perturbation information of pathology images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' The prior knowledge is the range, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' learning space, of both two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In this section, we ablate different prior knowledge choices to explore the effect of curriculum de- Patch Size ROSE pRCC WBC 16 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='90 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='70 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='50 32 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='09 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='70 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='76 48 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='50 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='11 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='78 64 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='19 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='05 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='71 96 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='00 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='05 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='20 128 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='31 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='05 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='78 192 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='81 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='05 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='27 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Curriculum variants for patch size fixed in the range of 16 to 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Fix-position ratio scheduler remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' sign and feature scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' We fix different initial values for patch size from 16 to 192 as well as fix position ratio from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='5 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='9, shown in Table 5 and Table 7 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In ad- dition, we choose different patch size lists including [128, 64, 32, 16] and [196, 96, 48, 16], shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In Table 5, CellMix performs the best with fix-patch size 64 on ROSE and 48 on pRCC and WBC, while too small or too large patch size shows poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In fact, instances (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' cells, tissues, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=') are sparsely distributed in the majority of pathology images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' With a large patch size, it is more likely to generate augmented samples with misleading supervisory signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' While too small, instances are split into pieces resulting in the loss of semantic infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' This result verifies the capability of instance-based shuffle strategy and supports our hypothesis that local (in- stance) specificity, such as characteristics of cells, plays an Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' CAM visualizations on un-augmented images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' The first column is the original image from ROSE, pRCC and WBC, and the other columns are Grad-CAM results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' As shown in the figure, CellMix guides the model to precisely focus on target instances and can accurately recognize the contour of the cell area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Patch Size ROSE pRCC WBC Even 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='70 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='47 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='62 Odd 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='19 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='76 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='32 Full 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='49 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='17 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='26 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Curriculum variants for different patch size lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Full, as the default set, means patch size list [192, 128, 96, 64, 48, 32,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Even is [128, 64, 32, 16], while Odd is [196, 96, 48, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' important role in modeling pathology image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Table 6 shows CellMix performs the best with a full patch size list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Specifically, it outperforms CellMix without full patch size lists in a range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='3% to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='41%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Shortening the patch size list reduces learnable feature scales and the accuracy of results, which means a strategy with more com- prehensive feature scales can work better in relationship modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' In Table 7, CellMix with adaptive fix-position ratio outperforms others with given ratios from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='13% to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='82%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' The drop in performance of the variant without a full patch size list and the variant without an adaptive fix- position ratio is consistent with the perspective that different feature scales contribute to better relationship modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' To further evaluate the loss-drive strategy with different loss thresholds and patch/ratio schedulers, more detailed ab- lation experiments are conducted in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' CAM Analysis We extract Grad-CAM [26] for both un-augmented im- ages and augmented images to investigate the regions of the input image where the model focuses to recognize an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Figure 4 shows that CellMix guides the model to pre- cisely focus on the target object compared to other meth- ods, presenting solid interpretability in cell identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' It shows that the model fully learns the feature of cell in- stances (local specificity) and the difference between in- Fix Position Ratio ROSE pRCC WBC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='5 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='91 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='35 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='6 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='90 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='35 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='7 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='13 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='70 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='8 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='31 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='35 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='9 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='41 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='35 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='03 Adaptive 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='49 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='17 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='26 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Curriculum variants for given fix-position ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' For example, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='9 means 90% patches of one image are fixed in the in- place shuffle process, with 10% shuffled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Adaptive, as the default setting, represents the continuously decreasing fix-position ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' stances and background (global distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' We also visualize Grad-CAM on augmented images in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' It can be seen that CellMix effectively focuses on the corresponding features and precisely localizes the in- stances in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Besides, it can be seen clearly that CellMix pays attention to different cells when making neg- ative and positive predictions, and accurately identifies the boundary of different patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' It proves that CellMix in- deed learns the difference between negative and positive in- stances and the inner/outer-sample instance relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Conclusion In conclusion, we introduce CellMix data augmentation method, which aims for the characteristics of pathology im- age analysis and forces the models to observe the relation- ship between the instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' We propose a distribution-based in-place shuffle strategy that can regroup patches (contain- ing instances) in the image (bag) level, with two kinds of tokens assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' At the image level, effective pseudo- samples force the models to observe the instance relation- ship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' We devise a loss-drive curriculum learning strategy during the training process to cover the multiple potential feature scales and alter the modeling difficulty adaptively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Original Baseline(ViT) CellMix CutMix Cutout FMix Mixup ResizeMix SaliencyMix ROSE WBCOn various pathology image classification benchmarks, CellMix significantly outperforms all other mixing-based augmentation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' CellMix boosts the performance of ViT-base model up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='59% in accuracy and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content='14% in F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' The inspiring SOTA results with various models prove the effectiveness of the core idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Acknowledgement This work was partially supported by the National Natu- ral Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 62271023), the Bei- jing Natural Science Foundation (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' 7202102), and the Fundamental Research Funds for Central Universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' References [1] Sumit Basu and Janara Christensen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNFJT4oBgHgl3EQfVixV/content/2301.11513v1.pdf'} +page_content=' Teaching classification boundaries to humans.' metadata={'source': 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0000000000000000000000000000000000000000..f8e43699958200847c94a40a1b8465a82677d5e3 --- /dev/null +++ b/iNAyT4oBgHgl3EQfxvmH/content/tmp_files/2301.00672v1.pdf.txt @@ -0,0 +1,924 @@ +Reducing the Quantum Many-electron Problem to Two Electrons with Machine Learning +LeeAnn M. Sager-Smith1 and David A. Mazziotti1, ∗ +1Department of Chemistry and The James Franck Institute, +The University of Chicago, Chicago, IL 60637 USA +(Dated: Submitted July 6, 2022; Revised August 19, 2022) +Abstract. An outstanding challenge in chemical computation is the many-electron problem where computational +methodologies scale prohibitively with system size. The energy of any molecule can be expressed as a weighted +sum of the energies of two-electron wave functions that are computable from only a two-electron calculation. +Despite the physical elegance of this extended “aufbau” principle, the determination of the distribution of +weights—geminal occupations—for general molecular systems has remained elusive. +Here we introduce a +new paradigm for electronic structure where approximate geminal-occupation distributions are “learned” via +a convolutional neural network. We show that the neural network learns the N-representability conditions, +constraints on the distribution for it to represent an N-electron system. By training on hydrocarbon isomers with +only 2-7 carbon atoms, we are able to predict the energies for isomers of octane as well as hydrocarbons with +8-15 carbons. The present work demonstrates that machine learning can be used to reduce the many-electron +problem to an effective two-electron problem, opening new opportunities for accurately predicting electronic +structure. +Introduction +For any molecular system, the Schrödinger equation can, in +theory, be solved exactly using a full configuration interac- +tion (FCI) calculation [1–3] with a complete basis set; how- +ever, in practice, the computational complexity of such an +exact approach grows factorially with system size [3], mak- +ing molecular systems with more than a few dozen electrons +intractable. +Over time, many approximate methodologies +have been introduced in an attempt to obtain “good enough” +solutions to the electronic Schrödinger equation that predict +energies within chemical accuracy (∼1 kcal/mol). +Hartree-Fock theory—a mean-field approach—yields rea- +sonable results for a wide array of molecular systems con- +taining up to a few hundred atoms [2]; however, it fails +in molecules in which the motions of electrons are signifi- +cantly correlated. Techniques which more-accurately capture +correlation energy such as many-body perturbation theory, +coupled cluster theory, complete active-space self-consistent +field theory, and others remain computationally expensive for +large system sizes [2, 4]. The so-called many-electron prob- +lem—whereby the cost of highly-accurate ab initio compu- +tational methodologies scales in a prohibitive manner with +system size—is hence an outstanding challenge in chemical +computations. +Machine learning may enable us to circumvent this +problem by allowing us to use information about smaller +molecules to treat correlation in larger systems at a reduced +cost [5]. It has been used to learn the energies of various +molecular structures [6–9], new functionals for density func- +tional theory (DFT) [10–12], inverse problems in electronic +structure theory [13, 14], and even the many-body wave func- +tion of one-dimensional spin systems [15]. However, these +areas are in their early stages and have yet to demonstrate def- +inite success in decreasing the degree of scaling with system +size. +In this Article we introduce a new paradigm for utilizing +machine learning in quantum chemistry in which we reduce +the quantum many-electron problem to a more tractable, bet- +ter scaling two-electron problem. As originally proposed by +Bopp [16, 17], the energy of a molecule of arbitrary size can +be expressed without approximation as a weighted sum of the +energies of two-electron wave functions, known as geminals. +However, despite its physical significance as an extension of +the “aufbau” principle, the distribution of weights—geminal +occupations—has remained elusive. Here, we show that the +geminal-occupation distribution can be learned with machine +learning. We use a convolutional neural network (CNN) to +learn an effective temperature in a Boltzmann-like distribu- +tion for the geminal occupations. +The effective tempera- +ture—or correlation temperature—is inversely related to the +electron correlation. The neural network, we demonstrate, +learns the N-representability of the distribution—the repre- +sentability of the distribution by an N-electron system [18– +21], which appears as a nonzero temperature. The scheme +can be viewed as a two-electron reduced density matrix (2- +RDM) theory as the geminal occupations are an integral part +of the 2-RDM. A schematic of the machine learning algo- +rithm for predicting molecular energies is shown in Fig. 1. +We apply the machine learning algorithm to hydrocar- +bon systems. Specifically, by training a convolutional neu- +ral network on all isomers of ethane through heptane, we +predict the correlation temperatures—and hence molecu- +lar energies—of all of the isomers of octane as well as all +straight-chained hydrocarbons from octane through pentade- +cane. We find that this RDM-based machine learning method +accurately recovers the correlation energy for larger hydro- +carbon systems, with the N-representability conditions be- +ing learned by the CNN framework. Our approach—which +scales as O[n6]—improves upon the exponential scaling of +traditional configuration-interaction calculations, foreshad- +owing the potential utility of this machine-learning reduced +density matrix approach to the determination of accurate +molecular energies. While polynomial-scaling levels of the- +ory such as Coupled Cluster with Single and Double Exci- +tations (CCSD) can be used to treat weakly-correlated sys- +tems such as the hydrocarbons presented in this manuscript, +arXiv:2301.00672v1 [physics.chem-ph] 29 Dec 2022 + +2 +if trained on appropriate molecular data, our convolutional +network approach may be capable of accurately recovering +correlation energy for more highly-correlated systems. +Results and Discussion +Theory. Central to our modern understanding of chemistry +is the concept of the molecular orbital. +Any molecule’s +electronic structure can be readily understood in terms of +its molecular orbitals which are filled from lowest-in-energy +to highest-in-energy by the Pauli exclusion principle. When +electrons of a molecule become strongly correlated, however, +the orbital picture withunit fillingofthe lowestorbitalsbreaks +down. Because electronic interactions are at most pairwise, +the orbital picture can in principle be replaced by an exact +two-electron (geminal) picture, which is derivable from 2- +RDM theory. +The ground- or excited-state energy of any atom or +molecule is expressible as an exact functional of the 2-RDM +(2D) [16, 18, 19, 22–59] +E = +� +2 ˆK 2D(¯1¯2; 12)d1d2 +(1) +where 2 ˆK is the reduced Hamiltonian operator +2 ˆK = −N +2 +� +ˆp2 +1 +2m + ˆp2 +2 +2m + +� +k +Zk +r1k ++ +� +k +Zk +r2k +� ++N(N − 1) +2 +1 +r12 +. +(2) +In a finite orbital basis set, the operators are expressible +as a reduced Hamiltonian matrix. Diagonalization of this +reduced Hamiltonian matrix yields a set of eigenvalues and +eigenvectors (or geminals). In the basis set of geminals, the +Hamiltonian is a diagonal matrix consisting of its eigenval- +ues, the 2-RDM has a non-negative diagonal elements which +we denote by pi, and energy is the sum over the geminal +eigenvalues of the Hamiltonian matrix ϵi weighted by the +non-negative geminal occupations pi: +E = +� +i +piϵi. +(3) +By this transformation we express the energy as a functional +of the eigenvalues of the reduced Hamiltonian ϵi, which are +readily computed at the cost of the two-electron calculation, +and the unknown geminal occupations pi (see Fig. 2). +The German chemist Bopp originally proposed approx- +imating the geminal occupation numbers by a Pauli-like +filling scheme [16, 17]. +He suggested choosing the low- +est N(N − 1)/2 to be equal to one. This approach, while +analogous to the filling of orbitals in molecular-orbital the- +ory, generates accurate energies for four-electron atoms and +ions but energies for larger molecular systems that are too low. +Coleman suggested that the filling of the geminal by two elec- +trons—or the pseudo-particle called a pairon—should follow +a fundamental probability distribution as in statistical me- +chanics [16]. He proposed a Boltzmann distribution for the +geminal occupations based on the geminal energies. While +such a distribution is not exact because the pairon pseudo- +particles obey neither the Fermi-Dirac or Bose-Einstein parti- +cle statistics, there exists a Boltzmann-like distribution given +by +pi = N(N − 1) +Z +e−ϵi/kT ∗ +(4) +and parameterized by a specific correlation temperature (T ∗) +such that the resultant approximate geminal probability dis- +tribution allows for the accurate computation of a molecule’s +energy according to Eq. (3). However, the ability to deter- +mine such a correlation temperature is currently only possible +if the geminal energies (ϵi) and geminal populations (pi) are +both known. +Here, we train a convolutional neural network (CNN) to +predict the correlation temperature for a given molecular +system consistent with its ground-state energy. The convo- +lutional neural network is trained on inputs corresponding +to both geminal energies—expressed as partition functions +given by +Z = +� +i +e−ϵi/kT +(5) +for a variety of temperatures—as well as the computed +Hartree-Fock correlation temperature (T ∗ +HF ) and with train- +ing outputs corresponding to a ∆ value representing the dif- +ference between the exact (i.e. +configuration interaction) +correlation temperature and the HF correlation temperature, +i.e., ∆ = T ∗ +EXACT − T ∗ +HF . For larger molecular systems, +we then predict the ∆ values by reading in the geminal en- +ergies and Hartree-Fock correlation temperatures for those +molecules into the trained neural network. These ∆ values +are then added to the T ∗ +HF s in order to yield the exact cor- +relation temperatures, which allows for the approximation of +the geminal probability distributions and hence the molecular +energies via Eq. (3). +In general, for two-electron reduced density matrix +methodologies, the 2-RDM must be constrained to rep- +resent the N-electron wavefunction through application +of N-representability constraints [18–21]. +Here, if N- +representability conditions are not accounted for in our +Boltzmann-like machine learning approach, the correlation +temperature would be zero, which corresponds to the lowest- +energy geminal being fully occupied by all electron pairs. +This electronic structure machine learning approach, how- +ever, maintains N-representability by learning correlation +temperatures from N-representable training data and apply- +ing this inherent “learned” N-representability to the testing +data. +See the Experimental section at the end of this document +for additional details. +Energetic Predictions for Isomers of Octane. For the eigh- +teen isomers of octane—with molecular geometries obtained + +3 +FIG. 1: Graphic demonstrating algorithm flow. For a given molecule, a trained convolutional neural network is used +to predict the Boltzmann-like correlation temperature (Tf) with the eigenfunctions of the reduced Hamiltonian (ϵj) and the +Hartree-Fock correlation temperature (Ti) as inputs. The correlation temperature (Tf) allows for the approximation of the +geminal populations (pf,j) by Eq. (4), which is sufficient for the prediction of the energy by Eq. (3). +FIG. 2: Example of geminal energies and probabilities. For (a) benzene, we can use the (b) geminal energies ϵi to learn +the (c) geminal probabilities pi—both of which are computed here from a [Ne = 6, No = 6] complete active-space self- +consistent-field (CASSCF) using the minimal Slater-type orbital basis set with six Gaussian primitive functions representing +each Slater-type orbital (STO-6G). Knowing both geminal energies and geminal populations is sufficient to determine molecular +energies via Eq. (3). +from the PubChem database [60]—, the Hartree-Fock and +CASSCF energies are computed using Dunning’s double- +zeta (cc-pVDZ) basis set with complete active-space self- +consistent-field (CASSCF) calculations employing a [Ne = +8, No = 8] active space. Utilizing a convolutional neural +network trained on hydrocarbons ranging from two to seven +carbon atoms, the correlation temperature corresponding to +the CASSCF energy is predicted for each of the octane iso- +mers and used to compute the predicted CASSCF energies +shown in Fig. 3(a). As can be seen from this figure, which +shows energy versus isomer identifier, the predicted CASSCF +energies (green circles) show good agreement with the ac- +tual CASSCF energies (black boxes), vastly improving upon +the Hartree-Fock energies (blue diamonds), and hence our +predictions capture the correlation energy in a fairly accurate +manner. +Additionally, in order to demonstrate the generality of our +reduced density matrix approach for “learning” molecular +energies, Coupled Cluster Single Double (CCSD) energies +are computed for the cc-pVDZ basis for hydrocarbons rang- +ing from two to seven carbon atoms. +The corresponding +CCSD correlation temperatures are then used to train a con- +volutional neural net, and the correlation temperature cor- +responding to the CCSD energy is then predicted for each +isomer of octane, with the resultant predicted CCSD ener- +gies shown in Fig. 3(b). Similar to the CASSCF energies +from Fig. 3(a), the CCSD predicted energies (green circles) +demonstrate good agreement with the actual CCSD energies +(black boxes) when compared to the Hartree-Fock energies +(blue diamonds). Hence, for this second level of theory, our +predictions capture correlation energies in a fairly accurate +manner. Additional predictions corresponding to CCSD cal- + +Convolutional +Neural +Net +E- +Ejpf,j +Pi,jαe-/kT +Pf,jαe-/kTr(b) +(c) +0 +(a) +-0.05 +3 +E-0.10 +p +i +2 +-0.15- +-0.20- +-0.25- +5 +10 +15 +20 +25 +30 +35 +5 +10 +15 +20 +25 +30 +35 +2.4 +culations utilizing the STO-6G basis set can be seen in the +Supporting Information. +We next explore systems composed of larger hydrocarbons +to determine whether such good agreement remains consis- +tent as system size is increased while the training data remains +the same. +Energetic Predictions for Large Hydrocarbon. For the +eight straight-chained hydrocarbons ranging from octane to +pentadecane—with molecular geometries obtained from the +PubChem database [60]—, the Hartree-Fock and CASSCF +energies are computed using Dunning’s double-zeta (cc- +pVDZ) basis set with the CASSCF calculations employing +a [Ne = 8, No = 8] active space. Utilizing a convolutional +neural network trained on hydrocarbons ranging from two to +seven carbon atoms, the correlation temperature correspond- +ing to the CASSCF energy is predicted for each of the octane +to pentadecane hydrocarbon isomers and used to compute +the predicted CASSCF energies shown in Fig. 4. As can be +seen from this figure, which shows energy per carbon versus +number of carbons, the predicted CASSCF energies (green +circles) show good agreement with the actual CASSCF ener- +gies (black boxes), vastly improving upon the Hartree-Fock +energies (blue diamonds), and hence our predictions cap- +ture the correlation energy in a fairly accurate manner. Al- +though there is a slight increase in the error as system size +is increased, it appears to be small enough that the energies +of even larger hydrocarbon isomers may be able to be pre- +dicted in an accurate manner through use of our convolutional +neural network trained on only hydrocarbons with seven or +fewer carbon atoms. Similar promising results are obtained +for predicting CASSCF energies for octane, nonane, decane, +and undecane via a convolutional neural network trained on +CASSCF calculations for hydrocarbons with two to seven +carbons that utilize a [10,10] active space and the cc-pVTZ +basis set as can be seen in the Supporting Information. +Conclusions +In this Article, we introduce a new paradigm based on a two- +electron, reduced density matrix approach for the utilization +of machine learning architecture in the prediction of accurate +correlation energies for molecular systems at reduced compu- +tational expense. By employing a Boltzmann-like distribu- +tion for two-electron geminal populations parameterized by a +correlation temperature, we train a convolutional neural net- +work on correlation temperatures corresponding to CASSCF +and CCSD calculations for smaller molecular systems in or- +der to predict CASSCF and CCSD correlation temperatures +for larger, more computationally-expensive molecular sys- +tems and hence obtain predicted CASSCF/CCSD energies. +Moreover, the N-representability conditions are inherently +maintained by our CNN framework—as evinced by nonzero +correlation temperatures. This methodology for the predic- +tion of CASSCF energies scales as O[n6] with the number of +orbitals due to the diagonalization of the reduced Hamilto- +nian, which is an improvement over the exponential scaling +of a traditional CASSCF calculation. See the Experimental +section for additional comments on computational scaling. +Demonstrating the power of this technique, we train a +convolutional neural network on small hydrocarbon sys- +tems—with the number of carbon atoms ranging from two +to seven—in order to predict CASSCF energies for larger +hydrocarbon systems—with the number of carbons ranging +from eight to fifteen. We find that our RDM-based machine +learning approach accurately recovers the correlation energy +for the larger hydrocarbon systems. Thus, our trained con- +volutional neural network allows us to predict CASSCF-like +results at significantly lower computational expense. +While the hydrocarbons involved in training and testing +this implementation of our machine-learning reduced den- +sity matrix approach do not demonstrate large degrees of +correlation, the prediction of accurate correlation energies +for larger molecular systems of the type included in the train- +ing set likely indicates that as long as the convolutional neural +network is trained on appropriate small molecules, the en- +ergies of highly-correlated, larger molecules should be able +to be obtained via our methodology. +Specifically, if one +wishes to predict the energy of a molecule which demon- +strates a fairly-large degree of correlation, smaller correlated +systems would likely be necessary to train the neural network. +Application of our machine-learning reduced density matrix +approach to highly-correlated systems is a future direction of +this research. +This work foreshadows the promise of machine learning +in molecular electronic structure calculations, demonstrat- +ing that “learning” information about less-expensive, smaller +molecular systems can be directly applied to larger typi- +cally more-expensive molecules. +Future electronic struc- +ture methodologies may even include pre-trained convolu- +tional neural networks—possibly varying with the types of +atoms, basis set, active space, functional groups, and/or de- +gree of bond saturation inherent to the molecular system of +interest—trained on FCI (or similarly expensive) correlation +temperatures. This work serves as an initial step in the real- +ization of a combined reduced-density-matrix and machine- +learning approach that may provide a real advance in de- +creasing computational expense for large, highly-correlated +electronic structure calculations. +Supporting Information +An analysis of the effect of changing the active space size +on our machine-learning reduced density matrix approach; +application of our reduced density matrix machine learn- +ing algorithm to the prediction of CASSCF energies with a +[10,10] active space and cc-pVTZ basis set; application of +our reduced density matrix machine learning algorithm to +the prediction of CCSD energies with a STO-6G basis. + +5 +(a) CASSCF, cc-pVDZ, [Ne = 8, No = 8] +(b) CCSD, cc-pvDZ +FIG. 3: Octane data. Hartree-Fock energies (HF, blue diamonds), (a) Complete Active Space Self-Consistent Field/(b) +Coupled Cluster Single Double (CASSCF/CCSD, black boxes) energies, and energy values predicted via utilization of +Convolutional Neural Networks (CNN, green circles) are shown for the series of octane isomers. As can be seen, the CNN +methodology trained on smaller hydrocarbon data fairly accurately recovers the correlation energy. Isomer labels are given +by [8.01: ‘Octane’, 8.02: ‘2-Methylheptane’, 8.03: ‘3-Methylheptane’, 8.04: ‘4-Methylheptane’, 8.05: ‘2,2-Dimethylhexane’, +8.06: ‘2,3-Dimethylhexane’, 8.07: ‘2,4-Dimethylhexane’, 8.08: ‘2,5-Dimethylhexane’, 8.09: ‘3,3-Dimethylhexane’, 8.10: +‘3,4-Dimethylhexane’, 8.11: ‘3-Ethylhexane’, 8.12: ‘2,2,3-Trimethylpentane’, 8.13: ‘2,2,4-Trimethylpentane’, 8.14: ‘2,3,3- +Trimethylpentane’, 8.15: ‘2,3,4-Trimethylpentane’, 8.16: ‘3-Ethyl-2-Methylpentane’, 8.17: ‘3-Ethyl-3-Methylpentane’, 8.18: +‘2,2,4,4-Tetramethylbutane’]. Hartree-Fock, CASSCF, and CCSD calculations are all computed here using Dunning’s double- +zeta (cc-pVDZ) basis set with the CASSCF calculations employing a [Ne = 8, No = 8] active space. +Acknowledgements +D.A.M. gratefully acknowledges support from U.S. National +Science Foundation under Grant No. 2155082 and the De- +partment of Energy, Office of Basic Energy Sciences under +Grant No. DE-SC0019215. L.M.S.-S. also acknowledges +support from the U. S. National Science Foundation under +Grant No. DGE-1746045. +References +∗ damazz@uchicago.edu + +-313.40 +CASSCF +HF +CNN-CASSCF +-313.42 +313.44 +(hartrees) +313.46 +313.48 +313.50 +313.52 +-313.54 +-313.56 +2 +4 +6 +.9 +.9 +. +.9 +.% +. +.% +.% +.0 +' +.% +.0 +Isomers-313 +CCSD +HF +CNN-CCSD +313.5 +314 +314.5 +C +-315 +8.03 +8.11 +3 +5 +8 +.0 +.9 +.% +.% +.% +.9 +.0 +.0 +Isomers6 +FIG. 4: Large hydrocarbon data. 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Anderson, Redox, transmetalation, and +stacking properties of tetrathiafulvalene-2,3,6,7-tetrathiolate +bridged tin, nickel, and palladium compounds, Chem. Sci. 11, +1066 (2019). +[58] J.-N. Boyn and D. A. Mazziotti, Accurate singlet-triplet gaps +in biradicals via the spin averaged anti-hermitian contracted +schrödinger equation, J. Chem. Phys. 154, 134103 (2021). +[59] S. Ewing and D. A. Mazziotti, Correlation-driven phenomena +in periodic molecular systems from variational two-electron + +8 +reduced density matrix theory, J. Chem. Phys. 154, 214106 +(2021). +[60] S. Kim, J. Chen, T. Cheng, A. Gindulyte, J. He, S. He, +Q. Li, B. A. Shoemaker, P. A. Thiessen, B. Yu, L. Zaslavsky, +J. Zhang, and E. E. Bolton, PubChem in 2021: new data con- +tent and improved web interfaces, Nucleic Acids Research +49, +D1388 +(2020), +https://academic.oup.com/nar/article- +pdf/49/D1/D1388/35363961/gkaa971.pdf. +[61] J. M. Montgomery and D. A. Mazziotti, Maple’s quantum +chemistry package in the chemistry classroom, J. Chem. Educ. +97, 3658 (2020). +[62] RDMChem, Maple Quantum Chemistry Package from RDM- +Chem 2022 (Maplesoft, Waterloo, Ontario, 2022). +[63] Maplesoft, Maple 2022 (Maplesoft, Waterloo, Ontario, 2022). +[64] F. Chollet, Keras 2015 (GitHub, San Francisco, CA, 2015). +Experimental +Computational Methods. +The molecular geometries for +all hydrocarbon isomers are obtained from the PubChem +database [60]. +Molecular energies are then computed for +Hartree-Fock, Complete Active Space Self-Consistent Field +(CASSCF), and Coupled Cluster Single Double (CCSD) +levels of theory through use of a Dunning’s double-zeta +(cc-pVDZ) basis set, with the CASSCF calculations em- +ploying a [Ne = 8, No = 8] active space. These calcula- +tions are accomplished via the Quantum Chemistry Toolbox +[61, 62] in the Maple computing environment [63]. Note +that while—throughout this text—the size of the active space +for the training and testing molecules is made identically +[Ne = 8, No = 8] for all CASSCF calculations, changing +active space sizes with the number of carbons yielded sim- +ilar results to those we present here. (See the Supporting +Information for additional details.) +Computation of Geminal Energies and Populations. The +reduced Hamiltonian (2K) shown in Eq. (2) is obtained by +directly computing the one electron integrals and the elec- +tron repulsion integrals via the MOIntegrals function of the +Quantum Chemistry Toolbox [61] in the Maple computing +environment [63] and then applying the appropriate conver- +sions to put it into the same orbital basis as the 2-RDM. The +geminal energies (ϵi) then correspond to the eigenvalues of +the 2K matrix. The populations (pi) of the geminals are then +obtained via the following +pi = ⟨vi|2D|vi⟩ +(6) +where vi is the eigenvector of the reduced Hamiltonian cor- +responding to the the geminal energy ϵi and where 2D is the +particle-particle reduced density matrix (2-RDM). +Convolutional Neural Network. +Model Inputs. For a given molecular system, both the gem- +inal energies (ϵi) and the Hartree-Fock correlation tempera- +ture (T ∗ +HF ) are input into the convolutional neural network. +Specifically, the geminal energies are encoded as partition +functions (Z)—computed according to Eq. (5)—for β val- +ues ranging from 0 to 20 by 0.4 where +β = 1 +kT +(7) +and where k is the Boltzmann constant. The Hartree-Fock +correlation temperature is obtained by inserting Eq. (5) into +Eq. (4) which is inserted into Eq. (3) to obtain +E(T) = N(N − 1) +� +i +eiϵi/kT +� +j +ϵje−ϵj/kT +(8) +and +then +temperature +is +optimized +via +scipy.optimize.minimize +such +that +|EHF − E(T)| +is +minimized. +Model Outputs. For a given molecular system, the output +of the convolutional neural net is a ∆ value representing +the difference between the Hartree-Fock correlation temper- +ature and the predicted CASSCF correlation temperature, +i.e., ∆ = T ∗ +CAS − T ∗ +HF . +From this output, the predicted +correlation temperature corresponding to the CASSCF cal- +culation can be computed by adding the output (∆) to the +Hartree-Fock correlation temperature (T ∗ +HF ), which can be +used—along with the known geminal energies (ϵi)—to cal- +culate the predicted CASSCF energy according to Eq. (8). +Training Data. +All hydrocarbons isomers ranging from +two to seven carbon atoms are used to train the convolu- +tional neural net. Specifically, the training set—composed +of twenty-one hydrocarbon molecules—follows: +2.01: +‘Ethane’, 3.01: +‘Propane’, 4.01: +‘Butane’, 4.02: +‘2- +Methylpropane’, 5.01: ‘Pentane’, 5.02: ‘2-Methylbutane’, +5.03: +‘2,2-Dimethylpropane’, 6.01: +‘Hexane’, 6.02: +‘2- +Methylpentane’, 6.03: +‘3-Methylpentane’, 6.04: +‘2,2- +Dimethylbutane’, 6.05: ‘2,3-Dimethylbutane’, 7.01: ‘Hep- +tane’, 7.02: ‘3-Methylhexane’, 7.03: ‘2-Methylhexane’, 7.04: +‘2,2-Dimethylpentane’, 7.05: ‘2,3-Dimethylpentane’, 7.06: +‘2,4-Dimethylpentane’, 7.07: ‘3,3-Dimethylpentane’, 7.08: +‘3-Ethylpentane’, 7.09: ‘2,2,3-Trimethylbutane’. +Testing Data. +All isomers of octane isomers as well as +nonane, decane, undecane, dodecane, tridecane, tetrade- +cane, and pentadecane are used to test the trained neural +net. +Specifically, the testing set follows: 8.01: ‘Octane’, +8.02: +‘2-Methylheptane’, 8.03: +‘3-Methylheptane’, 8.04: +‘4-Methylheptane’, 8.05: ‘2,2-Dimethylhexane’, 8.06: ‘2,3- +Dimethylhexane’, 8.07: ‘2,4-Dimethylhexane’, 8.08: ‘2,5- +Dimethylhexane’, 8.09: +‘3,3-Dimethylhexane’, 8.1: +‘3,4- +Dimethylhexane’, 8.11: +‘3-Ethylhexane’, 8.12: +‘2,2,3- +Trimethylpentane’, 8.13: +‘2,2,4-Trimethylpentane’, 8.14: +‘2,3,3-Trimethylpentane’, 8.15: +‘2,3,4-Trimethylpentane’, +8.16: +‘3-Ethyl-2-Methylpentane’, +8.17: +‘3-Ethyl-3- +Methylpentane’, 8.18: +‘2,2,4,4-Tetramethylbutane’, 9.01: +‘Nonane’, 10.01: ‘Decane’, 11.01: ‘Undecane’, 12.01: ‘Do- +decane’, 13.01: ‘Tridecane’, 14.01: ‘Tetradecane’, 15.01: +‘Pentadecane’. +CNN Specifics. The convolutional neural network is com- +posed of an input layer, five additional dense layers, and an + +9 +output layer. The input layer consists of partition functions +and the Hartree-Fock correlation temperature as specified in +the Model Inputs section, and the output layer is a dense +layer consisting of the ∆ value described in the Model Out- +puts section. +The additional dense layers have 503, 240, +100, 50, and 20 nodes, respectively. All dense nodes are +initialized via the he_uniform kernel initializer with a relu +activation function. For the training of the convolutional net, +loss is measured via mean absolute error, and the adam opti- +mizer is implemented for 30, 000 epochs. This convolutional +neural network is implemented using Keras—Python’s deep +learning API [64]. +Computational Scaling +For the testing set, scaling is dominated by the determina- +tion of the geminal energies, which are obtained via the di- +agonalization of the two-electron reduced Hamiltonian, a +computation that scales as O[r6] where r is the number of +orbitals in the active space. Thus, for a given molecule in the +testing set, computational expense for prediction of molec- +ular energies scales as O[r6]. The computational expense +of the training set is dominated by the determination of the +reference CASSCF or CCSD energies necessary to obtain +the reference correlation temperature—which are known to +scale approximately as O[N!] and O[N 6], respectively, for a +given molecule where N for CASSCF is the number of active +electrons and N for CCSD is the number of total electrons. +TOC Graphic + +CNN +一E +Pij +Pfj \ No newline at end of file diff --git a/iNAyT4oBgHgl3EQfxvmH/content/tmp_files/load_file.txt b/iNAyT4oBgHgl3EQfxvmH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..af74bbd67fcd11520471a47ef3b97ef2c0939e20 --- /dev/null +++ b/iNAyT4oBgHgl3EQfxvmH/content/tmp_files/load_file.txt @@ -0,0 +1,787 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf,len=786 +page_content='Reducing the Quantum Many-electron Problem to Two Electrons with Machine Learning LeeAnn M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Sager-Smith1 and David A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Mazziotti1, ∗ 1Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, IL 60637 USA (Dated: Submitted July 6, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Revised August 19, 2022) Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' An outstanding challenge in chemical computation is the many-electron problem where computational methodologies scale prohibitively with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The energy of any molecule can be expressed as a weighted sum of the energies of two-electron wave functions that are computable from only a two-electron calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Despite the physical elegance of this extended “aufbau” principle, the determination of the distribution of weights—geminal occupations—for general molecular systems has remained elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Here we introduce a new paradigm for electronic structure where approximate geminal-occupation distributions are “learned” via a convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' We show that the neural network learns the N-representability conditions, constraints on the distribution for it to represent an N-electron system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' By training on hydrocarbon isomers with only 2-7 carbon atoms, we are able to predict the energies for isomers of octane as well as hydrocarbons with 8-15 carbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The present work demonstrates that machine learning can be used to reduce the many-electron problem to an effective two-electron problem, opening new opportunities for accurately predicting electronic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Introduction For any molecular system, the Schrödinger equation can, in theory, be solved exactly using a full configuration interac- tion (FCI) calculation [1–3] with a complete basis set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' how- ever, in practice, the computational complexity of such an exact approach grows factorially with system size [3], mak- ing molecular systems with more than a few dozen electrons intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Over time, many approximate methodologies have been introduced in an attempt to obtain “good enough” solutions to the electronic Schrödinger equation that predict energies within chemical accuracy (∼1 kcal/mol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Hartree-Fock theory—a mean-field approach—yields rea- sonable results for a wide array of molecular systems con- taining up to a few hundred atoms [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' however, it fails in molecules in which the motions of electrons are signifi- cantly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Techniques which more-accurately capture correlation energy such as many-body perturbation theory, coupled cluster theory, complete active-space self-consistent field theory, and others remain computationally expensive for large system sizes [2, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The so-called many-electron prob- lem—whereby the cost of highly-accurate ab initio compu- tational methodologies scales in a prohibitive manner with system size—is hence an outstanding challenge in chemical computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Machine learning may enable us to circumvent this problem by allowing us to use information about smaller molecules to treat correlation in larger systems at a reduced cost [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' It has been used to learn the energies of various molecular structures [6–9], new functionals for density func- tional theory (DFT) [10–12], inverse problems in electronic structure theory [13, 14], and even the many-body wave func- tion of one-dimensional spin systems [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' However, these areas are in their early stages and have yet to demonstrate def- inite success in decreasing the degree of scaling with system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' In this Article we introduce a new paradigm for utilizing machine learning in quantum chemistry in which we reduce the quantum many-electron problem to a more tractable, bet- ter scaling two-electron problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' As originally proposed by Bopp [16, 17], the energy of a molecule of arbitrary size can be expressed without approximation as a weighted sum of the energies of two-electron wave functions, known as geminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' However, despite its physical significance as an extension of the “aufbau” principle, the distribution of weights—geminal occupations—has remained elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Here, we show that the geminal-occupation distribution can be learned with machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' We use a convolutional neural network (CNN) to learn an effective temperature in a Boltzmann-like distribu- tion for the geminal occupations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The effective tempera- ture—or correlation temperature—is inversely related to the electron correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The neural network, we demonstrate, learns the N-representability of the distribution—the repre- sentability of the distribution by an N-electron system [18– 21], which appears as a nonzero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The scheme can be viewed as a two-electron reduced density matrix (2- RDM) theory as the geminal occupations are an integral part of the 2-RDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' A schematic of the machine learning algo- rithm for predicting molecular energies is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' We apply the machine learning algorithm to hydrocar- bon systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Specifically, by training a convolutional neu- ral network on all isomers of ethane through heptane, we predict the correlation temperatures—and hence molecu- lar energies—of all of the isomers of octane as well as all straight-chained hydrocarbons from octane through pentade- cane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' We find that this RDM-based machine learning method accurately recovers the correlation energy for larger hydro- carbon systems, with the N-representability conditions be- ing learned by the CNN framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Our approach—which scales as O[n6]—improves upon the exponential scaling of traditional configuration-interaction calculations, foreshad- owing the potential utility of this machine-learning reduced density matrix approach to the determination of accurate molecular energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' While polynomial-scaling levels of the- ory such as Coupled Cluster with Single and Double Exci- tations (CCSD) can be used to treat weakly-correlated sys- tems such as the hydrocarbons presented in this manuscript, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='00672v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='chem-ph] 29 Dec 2022 2 if trained on appropriate molecular data, our convolutional network approach may be capable of accurately recovering correlation energy for more highly-correlated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Results and Discussion Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Central to our modern understanding of chemistry is the concept of the molecular orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Any molecule’s electronic structure can be readily understood in terms of its molecular orbitals which are filled from lowest-in-energy to highest-in-energy by the Pauli exclusion principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' When electrons of a molecule become strongly correlated, however, the orbital picture withunit fillingofthe lowestorbitalsbreaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Because electronic interactions are at most pairwise, the orbital picture can in principle be replaced by an exact two-electron (geminal) picture, which is derivable from 2- RDM theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The ground- or excited-state energy of any atom or molecule is expressible as an exact functional of the 2-RDM (2D) [16, 18, 19, 22–59] E = � 2 ˆK 2D(¯1¯2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' 12)d1d2 (1) where 2 ˆK is the reduced Hamiltonian operator 2 ˆK = −N 2 � ˆp2 1 2m + ˆp2 2 2m + � k Zk r1k + � k Zk r2k � +N(N − 1) 2 1 r12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' (2) In a finite orbital basis set, the operators are expressible as a reduced Hamiltonian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Diagonalization of this reduced Hamiltonian matrix yields a set of eigenvalues and eigenvectors (or geminals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' In the basis set of geminals, the Hamiltonian is a diagonal matrix consisting of its eigenval- ues, the 2-RDM has a non-negative diagonal elements which we denote by pi, and energy is the sum over the geminal eigenvalues of the Hamiltonian matrix ϵi weighted by the non-negative geminal occupations pi: E = � i piϵi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' (3) By this transformation we express the energy as a functional of the eigenvalues of the reduced Hamiltonian ϵi, which are readily computed at the cost of the two-electron calculation, and the unknown geminal occupations pi (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The German chemist Bopp originally proposed approx- imating the geminal occupation numbers by a Pauli-like filling scheme [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' He suggested choosing the low- est N(N − 1)/2 to be equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' This approach, while analogous to the filling of orbitals in molecular-orbital the- ory, generates accurate energies for four-electron atoms and ions but energies for larger molecular systems that are too low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Coleman suggested that the filling of the geminal by two elec- trons—or the pseudo-particle called a pairon—should follow a fundamental probability distribution as in statistical me- chanics [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' He proposed a Boltzmann distribution for the geminal occupations based on the geminal energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' While such a distribution is not exact because the pairon pseudo- particles obey neither the Fermi-Dirac or Bose-Einstein parti- cle statistics, there exists a Boltzmann-like distribution given by pi = N(N − 1) Z e−ϵi/kT ∗ (4) and parameterized by a specific correlation temperature (T ∗) such that the resultant approximate geminal probability dis- tribution allows for the accurate computation of a molecule’s energy according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' However, the ability to deter- mine such a correlation temperature is currently only possible if the geminal energies (ϵi) and geminal populations (pi) are both known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Here, we train a convolutional neural network (CNN) to predict the correlation temperature for a given molecular system consistent with its ground-state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The convo- lutional neural network is trained on inputs corresponding to both geminal energies—expressed as partition functions given by Z = � i e−ϵi/kT (5) for a variety of temperatures—as well as the computed Hartree-Fock correlation temperature (T ∗ HF ) and with train- ing outputs corresponding to a ∆ value representing the dif- ference between the exact (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' configuration interaction) correlation temperature and the HF correlation temperature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=', ∆ = T ∗ EXACT − T ∗ HF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' For larger molecular systems, we then predict the ∆ values by reading in the geminal en- ergies and Hartree-Fock correlation temperatures for those molecules into the trained neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' These ∆ values are then added to the T ∗ HF s in order to yield the exact cor- relation temperatures, which allows for the approximation of the geminal probability distributions and hence the molecular energies via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' In general, for two-electron reduced density matrix methodologies, the 2-RDM must be constrained to rep- resent the N-electron wavefunction through application of N-representability constraints [18–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Here, if N- representability conditions are not accounted for in our Boltzmann-like machine learning approach, the correlation temperature would be zero, which corresponds to the lowest- energy geminal being fully occupied by all electron pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' This electronic structure machine learning approach, how- ever, maintains N-representability by learning correlation temperatures from N-representable training data and apply- ing this inherent “learned” N-representability to the testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' See the Experimental section at the end of this document for additional details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Energetic Predictions for Isomers of Octane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' For the eigh- teen isomers of octane—with molecular geometries obtained 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' 1: Graphic demonstrating algorithm flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' For a given molecule, a trained convolutional neural network is used to predict the Boltzmann-like correlation temperature (Tf) with the eigenfunctions of the reduced Hamiltonian (ϵj) and the Hartree-Fock correlation temperature (Ti) as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The correlation temperature (Tf) allows for the approximation of the geminal populations (pf,j) by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' (4), which is sufficient for the prediction of the energy by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' 2: Example of geminal energies and probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' For (a) benzene, we can use the (b) geminal energies ϵi to learn the (c) geminal probabilities pi—both of which are computed here from a [Ne = 6, No = 6] complete active-space self- consistent-field (CASSCF) using the minimal Slater-type orbital basis set with six Gaussian primitive functions representing each Slater-type orbital (STO-6G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Knowing both geminal energies and geminal populations is sufficient to determine molecular energies via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' from the PubChem database [60]—, the Hartree-Fock and CASSCF energies are computed using Dunning’s double- zeta (cc-pVDZ) basis set with complete active-space self- consistent-field (CASSCF) calculations employing a [Ne = 8, No = 8] active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Utilizing a convolutional neural network trained on hydrocarbons ranging from two to seven carbon atoms, the correlation temperature corresponding to the CASSCF energy is predicted for each of the octane iso- mers and used to compute the predicted CASSCF energies shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' As can be seen from this figure, which shows energy versus isomer identifier, the predicted CASSCF energies (green circles) show good agreement with the ac- tual CASSCF energies (black boxes), vastly improving upon the Hartree-Fock energies (blue diamonds), and hence our predictions capture the correlation energy in a fairly accurate manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Additionally, in order to demonstrate the generality of our reduced density matrix approach for “learning” molecular energies, Coupled Cluster Single Double (CCSD) energies are computed for the cc-pVDZ basis for hydrocarbons rang- ing from two to seven carbon atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The corresponding CCSD correlation temperatures are then used to train a con- volutional neural net, and the correlation temperature cor- responding to the CCSD energy is then predicted for each isomer of octane, with the resultant predicted CCSD ener- gies shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Similar to the CASSCF energies from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' 3(a), the CCSD predicted energies (green circles) demonstrate good agreement with the actual CCSD energies (black boxes) when compared to the Hartree-Fock energies (blue diamonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Hence, for this second level of theory, our predictions capture correlation energies in a fairly accurate manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Additional predictions corresponding to CCSD cal- Convolutional Neural Net E- Ejpf,j Pi,jαe-/kT Pf,jαe-/kTr(b) (c) 0 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='05 3 E-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='10 p i 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='15- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='20- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='25- 5 10 15 20 25 30 35 5 10 15 20 25 30 35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='4 culations utilizing the STO-6G basis set can be seen in the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' We next explore systems composed of larger hydrocarbons to determine whether such good agreement remains consis- tent as system size is increased while the training data remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Energetic Predictions for Large Hydrocarbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' For the eight straight-chained hydrocarbons ranging from octane to pentadecane—with molecular geometries obtained from the PubChem database [60]—, the Hartree-Fock and CASSCF energies are computed using Dunning’s double-zeta (cc- pVDZ) basis set with the CASSCF calculations employing a [Ne = 8, No = 8] active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Utilizing a convolutional neural network trained on hydrocarbons ranging from two to seven carbon atoms, the correlation temperature correspond- ing to the CASSCF energy is predicted for each of the octane to pentadecane hydrocarbon isomers and used to compute the predicted CASSCF energies shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' As can be seen from this figure, which shows energy per carbon versus number of carbons, the predicted CASSCF energies (green circles) show good agreement with the actual CASSCF ener- gies (black boxes), vastly improving upon the Hartree-Fock energies (blue diamonds), and hence our predictions cap- ture the correlation energy in a fairly accurate manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Al- though there is a slight increase in the error as system size is increased, it appears to be small enough that the energies of even larger hydrocarbon isomers may be able to be pre- dicted in an accurate manner through use of our convolutional neural network trained on only hydrocarbons with seven or fewer carbon atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Similar promising results are obtained for predicting CASSCF energies for octane, nonane, decane, and undecane via a convolutional neural network trained on CASSCF calculations for hydrocarbons with two to seven carbons that utilize a [10,10] active space and the cc-pVTZ basis set as can be seen in the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Conclusions In this Article, we introduce a new paradigm based on a two- electron, reduced density matrix approach for the utilization of machine learning architecture in the prediction of accurate correlation energies for molecular systems at reduced compu- tational expense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' By employing a Boltzmann-like distribu- tion for two-electron geminal populations parameterized by a correlation temperature, we train a convolutional neural net- work on correlation temperatures corresponding to CASSCF and CCSD calculations for smaller molecular systems in or- der to predict CASSCF and CCSD correlation temperatures for larger, more computationally-expensive molecular sys- tems and hence obtain predicted CASSCF/CCSD energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Moreover, the N-representability conditions are inherently maintained by our CNN framework—as evinced by nonzero correlation temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' This methodology for the predic- tion of CASSCF energies scales as O[n6] with the number of orbitals due to the diagonalization of the reduced Hamilto- nian, which is an improvement over the exponential scaling of a traditional CASSCF calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' See the Experimental section for additional comments on computational scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Demonstrating the power of this technique, we train a convolutional neural network on small hydrocarbon sys- tems—with the number of carbon atoms ranging from two to seven—in order to predict CASSCF energies for larger hydrocarbon systems—with the number of carbons ranging from eight to fifteen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' We find that our RDM-based machine learning approach accurately recovers the correlation energy for the larger hydrocarbon systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Thus, our trained con- volutional neural network allows us to predict CASSCF-like results at significantly lower computational expense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' While the hydrocarbons involved in training and testing this implementation of our machine-learning reduced den- sity matrix approach do not demonstrate large degrees of correlation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' the prediction of accurate correlation energies for larger molecular systems of the type included in the train- ing set likely indicates that as long as the convolutional neural network is trained on appropriate small molecules,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' the en- ergies of highly-correlated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' larger molecules should be able to be obtained via our methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Specifically, if one wishes to predict the energy of a molecule which demon- strates a fairly-large degree of correlation, smaller correlated systems would likely be necessary to train the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Application of our machine-learning reduced density matrix approach to highly-correlated systems is a future direction of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' This work foreshadows the promise of machine learning in molecular electronic structure calculations, demonstrat- ing that “learning” information about less-expensive, smaller molecular systems can be directly applied to larger typi- cally more-expensive molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Future electronic struc- ture methodologies may even include pre-trained convolu- tional neural networks—possibly varying with the types of atoms, basis set, active space, functional groups, and/or de- gree of bond saturation inherent to the molecular system of interest—trained on FCI (or similarly expensive) correlation temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' This work serves as an initial step in the real- ization of a combined reduced-density-matrix and machine- learning approach that may provide a real advance in de- creasing computational expense for large, highly-correlated electronic structure calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Supporting Information An analysis of the effect of changing the active space size on our machine-learning reduced density matrix approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' application of our reduced density matrix machine learn- ing algorithm to the prediction of CASSCF energies with a [10,10] active space and cc-pVTZ basis set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' application of our reduced density matrix machine learning algorithm to the prediction of CCSD energies with a STO-6G basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' 5 (a) CASSCF, cc-pVDZ, [Ne = 8, No = 8] (b) CCSD, cc-pvDZ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' 3: Octane data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Hartree-Fock energies (HF, blue diamonds), (a) Complete Active Space Self-Consistent Field/(b) Coupled Cluster Single Double (CASSCF/CCSD, black boxes) energies, and energy values predicted via utilization of Convolutional Neural Networks (CNN, green circles) are shown for the series of octane isomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' As can be seen, the CNN methodology trained on smaller hydrocarbon data fairly accurately recovers the correlation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Isomer labels are given by [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='01: ‘Octane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='02: ‘2-Methylheptane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='03: ‘3-Methylheptane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='04: ‘4-Methylheptane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='05: ‘2,2-Dimethylhexane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='06: ‘2,3-Dimethylhexane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='07: ‘2,4-Dimethylhexane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='08: ‘2,5-Dimethylhexane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='09: ‘3,3-Dimethylhexane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='10: ‘3,4-Dimethylhexane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='11: ‘3-Ethylhexane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='12: ‘2,2,3-Trimethylpentane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='13: ‘2,2,4-Trimethylpentane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='14: ‘2,3,3- Trimethylpentane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='15: ‘2,3,4-Trimethylpentane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='16: ‘3-Ethyl-2-Methylpentane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='17: ‘3-Ethyl-3-Methylpentane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='18: ‘2,2,4,4-Tetramethylbutane’].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Hartree-Fock, CASSCF, and CCSD calculations are all computed here using Dunning’s double- zeta (cc-pVDZ) basis set with the CASSCF calculations employing a [Ne = 8, No = 8] active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Acknowledgements D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' gratefully acknowledges support from U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' 2155082 and the De- partment of Energy, Office of Basic Energy Sciences under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' DE-SC0019215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' also acknowledges support from the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' DGE-1746045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' References ∗ damazz@uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='edu 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='40 CASSCF HF CNN-CASSCF 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='42 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='44 (hartrees) 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='46 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='48 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='50 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='52 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='54 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='56 2 4 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content="0 ' ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='0 Isomers-313 CCSD HF CNN-CCSD 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='5 314 314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='5 C 315 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='03 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='11 3 5 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='% .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='0 Isomers6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' 4: Large hydrocarbon data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Hartree-Fock energies (HF, blue diamonds), Complete Active Space Self-Consistent Field energies (CASSCF, black boxes), and energy values predicted via utilization of Convolutional Neural Networks (CNN, green circles) per number of carbons are shown for the series of straight-chained hydrocarbons from octane through pentadecane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' As can be seen, the CNN methodology trained on smaller hydrocarbon data fairly accurately recovers the correlation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Isomer labels are given by [8: ‘Octane’, 9: ‘Nonane’, 10: ‘Decane’, 11: ‘Undecane’, 12: ‘Dodecane’, 13: ‘Tridecane’, 14: ‘Tetradecane’, 15: ‘Pentadecane’].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Both Hartree-Fock and CASSCF calculations are computed here using Dunning’s double-zeta (cc-pVDZ) basis set with the CASSCF calculations employing a [Ne = 8, No = 8] active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Knowles and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Handy, Unlimited full configuration interaction calculations, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Chem.' metadata={'source': 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gaps in biradicals via the spin averaged anti-hermitian contracted schrödinger equation, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' 154, 134103 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' [59] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Ewing and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Mazziotti, Correlation-driven phenomena in periodic molecular systems from variational two-electron 8 reduced density matrix theory, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' 154, 214106 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' [60] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Cheng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Gindulyte, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' He, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' He, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Shoemaker, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Thiessen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Yu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Zaslavsky, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Zhang, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Bolton, PubChem in 2021: new data con- tent and improved web interfaces, Nucleic Acids Research 49, D1388 (2020), https://academic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='oup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='com/nar/article- pdf/49/D1/D1388/35363961/gkaa971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' [61] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Montgomery and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Mazziotti, Maple’s quantum chemistry package in the chemistry classroom, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Educ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' 97, 3658 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' [62] RDMChem, Maple Quantum Chemistry Package from RDM- Chem 2022 (Maplesoft, Waterloo, Ontario, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' [63] Maplesoft, Maple 2022 (Maplesoft, Waterloo, Ontario, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' [64] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Chollet, Keras 2015 (GitHub, San Francisco, CA, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Experimental Computational Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The molecular geometries for all hydrocarbon isomers are obtained from the PubChem database [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Molecular energies are then computed for Hartree-Fock, Complete Active Space Self-Consistent Field (CASSCF), and Coupled Cluster Single Double (CCSD) levels of theory through use of a Dunning’s double-zeta (cc-pVDZ) basis set, with the CASSCF calculations em- ploying a [Ne = 8, No = 8] active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' These calcula- tions are accomplished via the Quantum Chemistry Toolbox [61, 62] in the Maple computing environment [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Note that while—throughout this text—the size of the active space for the training and testing molecules is made identically [Ne = 8, No = 8] for all CASSCF calculations, changing active space sizes with the number of carbons yielded sim- ilar results to those we present here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' (See the Supporting Information for additional details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=') Computation of Geminal Energies and Populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The reduced Hamiltonian (2K) shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' (2) is obtained by directly computing the one electron integrals and the elec- tron repulsion integrals via the MOIntegrals function of the Quantum Chemistry Toolbox [61] in the Maple computing environment [63] and then applying the appropriate conver- sions to put it into the same orbital basis as the 2-RDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The geminal energies (ϵi) then correspond to the eigenvalues of the 2K matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The populations (pi) of the geminals are then obtained via the following pi = ⟨vi|2D|vi⟩ (6) where vi is the eigenvector of the reduced Hamiltonian cor- responding to the the geminal energy ϵi and where 2D is the particle-particle reduced density matrix (2-RDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Convolutional Neural Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Model Inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' For a given molecular system, both the gem- inal energies (ϵi) and the Hartree-Fock correlation tempera- ture (T ∗ HF ) are input into the convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Specifically, the geminal energies are encoded as partition functions (Z)—computed according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' (5)—for β val- ues ranging from 0 to 20 by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='4 where β = 1 kT (7) and where k is the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The Hartree-Fock correlation temperature is obtained by inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' (5) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' (4) which is inserted into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' (3) to obtain E(T) = N(N − 1) � i eiϵi/kT � j ϵje−ϵj/kT (8) and then temperature is optimized via scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='minimize such that |EHF − E(T)| is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Model Outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' For a given molecular system, the output of the convolutional neural net is a ∆ value representing the difference between the Hartree-Fock correlation temper- ature and the predicted CASSCF correlation temperature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=', ∆ = T ∗ CAS − T ∗ HF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' From this output, the predicted correlation temperature corresponding to the CASSCF cal- culation can be computed by adding the output (∆) to the Hartree-Fock correlation temperature (T ∗ HF ), which can be used—along with the known geminal energies (ϵi)—to cal- culate the predicted CASSCF energy according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Training Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' All hydrocarbons isomers ranging from two to seven carbon atoms are used to train the convolu- tional neural net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Specifically, the training set—composed of twenty-one hydrocarbon molecules—follows: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='01: ‘Ethane’, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='01: ‘Propane’, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='01: ‘Butane’, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='02: ‘2- Methylpropane’, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='01: ‘Pentane’, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='02: ‘2-Methylbutane’, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='03: ‘2,2-Dimethylpropane’, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='01: ‘Hexane’, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='02: ‘2- Methylpentane’, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='03: ‘3-Methylpentane’, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='04: ‘2,2- Dimethylbutane’, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='05: ‘2,3-Dimethylbutane’, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='01: ‘Hep- tane’, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='02: ‘3-Methylhexane’, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='03: ‘2-Methylhexane’, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='04: ‘2,2-Dimethylpentane’, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='05: ‘2,3-Dimethylpentane’, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='06: ‘2,4-Dimethylpentane’, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='07: ‘3,3-Dimethylpentane’, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='08: ‘3-Ethylpentane’, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='09: ‘2,2,3-Trimethylbutane’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Testing Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' All isomers of octane isomers as well as nonane, decane, undecane, dodecane, tridecane, tetrade- cane, and pentadecane are used to test the trained neural net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Specifically, the testing set follows: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='01: ‘Octane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='02: ‘2-Methylheptane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='03: ‘3-Methylheptane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='04: ‘4-Methylheptane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='05: ‘2,2-Dimethylhexane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='06: ‘2,3- Dimethylhexane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='07: ‘2,4-Dimethylhexane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='08: ‘2,5- Dimethylhexane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='09: ‘3,3-Dimethylhexane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='1: ‘3,4- Dimethylhexane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='11: ‘3-Ethylhexane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='12: ‘2,2,3- Trimethylpentane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='13: ‘2,2,4-Trimethylpentane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='14: ‘2,3,3-Trimethylpentane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='15: ‘2,3,4-Trimethylpentane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='16: ‘3-Ethyl-2-Methylpentane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='17: ‘3-Ethyl-3- Methylpentane’, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='18: ‘2,2,4,4-Tetramethylbutane’, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='01: ‘Nonane’, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='01: ‘Decane’, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='01: ‘Undecane’, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='01: ‘Do- decane’, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='01: ‘Tridecane’, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='01: ‘Tetradecane’, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='01: ‘Pentadecane’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' CNN Specifics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The convolutional neural network is com- posed of an input layer, five additional dense layers, and an 9 output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The input layer consists of partition functions and the Hartree-Fock correlation temperature as specified in the Model Inputs section, and the output layer is a dense layer consisting of the ∆ value described in the Model Out- puts section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The additional dense layers have 503, 240, 100, 50, and 20 nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' All dense nodes are initialized via the he_uniform kernel initializer with a relu activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' For the training of the convolutional net, loss is measured via mean absolute error, and the adam opti- mizer is implemented for 30, 000 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' This convolutional neural network is implemented using Keras—Python’s deep learning API [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Computational Scaling For the testing set, scaling is dominated by the determina- tion of the geminal energies, which are obtained via the di- agonalization of the two-electron reduced Hamiltonian, a computation that scales as O[r6] where r is the number of orbitals in the active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' Thus, for a given molecule in the testing set, computational expense for prediction of molec- ular energies scales as O[r6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' The computational expense of the training set is dominated by the determination of the reference CASSCF or CCSD energies necessary to obtain the reference correlation temperature—which are known to scale approximately as O[N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content='] and O[N 6], respectively, for a given molecule where N for CASSCF is the number of active electrons and N for CCSD is the number of total electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} +page_content=' TOC Graphic CNN 一E Pij Pfj' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfxvmH/content/2301.00672v1.pdf'} diff --git a/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf b/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..69be0aad99e238d66a877f47cd269ddaee440f16 --- /dev/null +++ b/jNAzT4oBgHgl3EQfpP3S/content/2301.01611v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5e4e23865ce628e09ad62846104cf830ce150bdd801f6599d786a3d9b822d57b +size 118114 diff --git a/jNAzT4oBgHgl3EQfpP3S/vector_store/index.faiss b/jNAzT4oBgHgl3EQfpP3S/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..845de8dcc49fdc1b6e7f88a9ed6b1920be37cdde --- /dev/null +++ b/jNAzT4oBgHgl3EQfpP3S/vector_store/index.faiss @@ -0,0 +1,3 @@ +version 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0000000000000000000000000000000000000000..22a6b966f8d65f670fec3ee0367f4941fa7eacb2 --- /dev/null +++ b/jtFPT4oBgHgl3EQfFzRC/content/tmp_files/load_file.txt @@ -0,0 +1,1554 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf,len=1553 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='13001v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='CO] 30 Jan 2023 On the minimum size of linear sets Sam Adriaensen Vrije Universiteit Brussel Paolo Santonastaso Universit`a degli Studi della Campania “Luigi Vanvitelli” Abstract Recently, a lower bound was established on the size of linear sets in projective spaces, that intersect a hyperplane in a canonical subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' There are several constructions showing that this bound is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In this paper, we generalize this bound to linear sets meeting some subspace π in a canonical subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We also give constructions of linear sets attaining equality in this bound, both in the case that π is a hyperplane, and in the case that π is a lower dimensional subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Keywords: Projective geometry, Linear set, Subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' MSC2020: 51E20, 05B25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 1 Introduction Linear sets are certain point sets in projective spaces, generalizing the notion of a subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' They have proven themselves to be very useful in constructing interesting objects in projective spaces, such as blocking sets [26] and KM-arcs [9], and have been used to construct Hamming and rank metric codes [1, 18, 20, 23–25, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' For a survey on linear sets, we refer the reader to [13,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Given the usefulness of linear sets, their recent spurt in popularity within the field of finite geometry is far from surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' One of the most natural questions arising in the study of linear sets is establishing lower and upper bounds on their size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' There is a quite trivial upper bound on the size of linear sets, and the study of linear sets attaining equality in this bound can be traced back to a paper by Blokhuis and Lavrauw [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' However, finding good lower bounds on the size of linear sets seems to be a harder problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Yet it is an interesting endeavor, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' due to its connection with the weight distribution of linear rank metric codes [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' As a consequence of the celebrated result on the number of directions determined by a function over finite field [2, 4], Bonoli and Polverino established a lower bound on the size of certain linear sets on a projective line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' More specifically, they proved the following result (for the definitions, we refer to Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1 ([6, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If LU is an Fq-linear set of rank n on PG(1, qn), and LU contains at least one point of weight 1, then |LU| ≥ qn−1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' De Beule and Van de Voorde managed to remove the condition on the rank from this bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We note that linear sets of rank greater than n on PG(1, qn) are not interesting to study, since they necessarily contain all the points of the projective line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence it is natural to limit the study to linear sets whose rank is at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='2 ([8, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If LU is an Fq-linear set of rank k, with 1 < k ≤ n on PG(1, qn), and LU contains at least one point of weight 1, then |LU| ≥ qk−1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Using an inductive argument, they obtained a bound on the size of a linear set in a higher dimensional projective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='2, which we prove later in this paper, this is equivalent to the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 1 Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3 ([8, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let LU be an Fq-linear set of rank k > d in PG(d, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If LU meets some hyperplane Ω in a canonical Fq-subgeometry of Ω, then |LU| ≥ qk−1 + qk−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' De Beule and Van de Voorde note directly after their statement of the above result that they would like to find lower bounds on linear sets satisfying less restrictive conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Furthermore, Jena and Van de Voorde [11, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5 (B)] state that they believe the above lower bound to hold for all Fq-linear sets of rank k that span PG(d, qn), if n is prime and k ≤ d + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In this article we will generalize the above result by dropping the condition that Ω is a hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let LU be an Fq-linear set of rank k in PG(d, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that there exists some (r − 1)-space Ω, with r < k, such that LU meets Ω in a canonical Fq-subgeometry of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then |LU| ≥ qk−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−r + IΩ, where IΩ denotes the number of r-spaces through Ω, containing a point of LU \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' This yields the following recursive lower bound on the size of a linear set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let Bq,n(r, k, d) denote the minimum size of an Fq-linear set of rank k that spans PG(d, qn) and intersects an (r − 1)-space in a canonical subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then Bq,n(r, k, d) ≥ � qk−1 + Bq,n(r − 1, k − 1, d − 1) if r > 0, qk−⌊ k d+1⌋ + Bq,n � 0, k − � k d+1 � , d − 1 � if r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Note in particular that for the linear set LU in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4, this implies that |LU| ≥ qk−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−r + qk−r−⌊ k−r d−r+1⌋ + Bq,n � 0, k − r − � k − r d − r + 1 � , d − r − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In PG(1, qn), the bound of De Beule and Van de Voorde is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' For every rank k ≤ n, there exist so-called (k − 1)-clubs of rank k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' These linear sets contain (an abundance of) points of weight 1, and their size matches the bound in Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Lunardon and Polverino [15] provided the first less trivial family of linear sets of rank n reaching equality in Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Their example was extended by Jena and Van de Voorde [11] to a very large family of linear sets of general rank, attaining equality in Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' More recently, there have been other constructions of such linear sets, and partial classification results, see Napolitano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Moreover, Jena and Van de Voorde generalized their constructions to higher dimensions, to obtain linear sets attaining equality in the bound of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3, some of which also satisfy the conditions of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3 [11, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5 (B)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In this article, we study the construction by Jena and Van de Voorde in general dimension, and we provide a sufficient condition for these linear sets to satisfy the hypothesis of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We also generalize the construction of Napolitano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Furthermore, we construct linear sets in PG(d, qn) satisfying the conditions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4, and attaining equality in the corresponding bound, where n is not prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The size of these linear sets is smaller than the bound from Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3, hence this illustrates the necessity of the conditions imposed in Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3 in case n is not prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Structure of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Section 2 contains preliminary results on linear sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Section 3 contains the proof of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We also discuss some sufficient conditions on linear sets for the hypothesis of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3 to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In addition, we deduce from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4 that the rank of a linear set is determined by its size and the minimum weight of its points, and that it is spanned by its points of minimum weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In Section 4 we discuss linear sets attaining 2 equality in Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' More specifically, we show a sufficient condition for the minimum size linear sets of [11] to satisfy the hypothesis of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3, and we generalize the construction from [16] to higher dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Section 5 contains constructions of linear sets attaining equality in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Finally, Section 6 contains some concluding remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 2 Preliminaries Throughout this article, q will always denote a prime power, and Fq will denote the finite field of order q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The d-dimensional projective space over Fq will be denoted by PG(d, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If the projective space is constructed from a (d + 1)-dimensional Fq-vector space V , and we want to emphasize the underlying vector space, we might also denote the projective space as PG(V, Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We note that the number of points in PG(d, q) equals qd+1−1 q−1 = qd + qd−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Throughout the article, when working in PG(d, q) = PG(Fd+1 q , Fq), we denote the vectors of Fd+1 q as (x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , xd), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' we label the coordinate positions from 0 to d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The ith standard basis vector will be denoted as ei = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , 0, 1 ���� ith position , 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , 0), and the corresponding point in PG(d, q) will be denoted as Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1 Linear sets Let V be a (d + 1)-dimensional vector space over Fqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then V is also a (d + 1)n-dimensional vector space over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let U denote an Fq-subspace of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then LU = {⟨u⟩Fqn : u ∈ U \\ {0}} is a set of points in PG(d, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Sets of this type are called Fq-linear sets, and the Fq-dimension of U is called the rank of LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We note that if U1 and U2 are Fq-subspaces, and LU1 and LU2 are equal as point set in PG(d, qn), this need not imply that dimFq U1 = dimFq U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence, the rank of a linear set LU is generally not unambiguously defined by LU as point set in PG(d, qn), without taking into account the underlying subspace U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Given an Fqn-subspace W ≤ V , we define the weight of Ω = PG(W, Fqn) to be wLU (Ω) = dimFq(U ∩ W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Note that wLU (Ω) equals the rank of the linear set LU∩W = LU ∩ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' For each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , n}, let Ni(LU) denote the number of points in PG(d, qn) of weight i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We will simply denote this as Ni if LU is clear from context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The numbers N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , Nn are called the weight distribution of LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In addition, the weight spectrum of LU is the vector (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , it) with i1 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' ≤ it and {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , it} = {wLU(P) : P ∈ LU} = {i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , n} : Ni > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let k > 0 denote the rank of LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then the weight distribution satisfies the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' |LU| = N1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + Nn, (1) n � i=1 Ni qi − 1 q − 1 = qk − 1 q − 1 , (2) 3 |LU| ≤ qk − 1 q − 1 , (3) |LU| ≡ 1 (mod q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' (4) Let T be an Fq-subspace of V with dimFq(T) = r ≤ d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If dimFqn(⟨T⟩Fqn) = r, we will say that LT ∼= PG(T, Fq) = PG(r − 1, q) is an Fq-subgeometry of PG(V, Fqn) and r is the rank of the subgeometry LT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' When r = d + 1, we say that LT is a canonical subgeometry of PG(V, Fqn) = PG(d, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Note that each point of a subgeometry LT has weight 1 and hence |LT | = qr−1 q−1 , if LT has rank r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Regarding the linearity of a linear set, we recall these definitions explored in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='2 ([12, Definitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' An Fq-linear set LU is an Fqs-linear set if U is also an Fqs-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We say that Fqs is the maximum field of linearity of LU if s is the largest exponent such that LU is Fqs-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3 ([12, Definitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' An Fq-linear set LU has geometric field of linearity Fqs if there exists an Fqs-linear set LU′ such that LU = LU′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' An Fq-linear set LU has maximum geometric field of linearity Fqs if s is the largest integer such that LU has geometric field of linearity Fqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The maximum field of linearity and the maximum geometric field of linearity do not always coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Clearly if LU is an Fqs-linear set, it has geometric field of linearity Fqs, but the converse need not hold, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' [12, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Note that if there exists a line ℓ that is (q + 1)-secant to a linear set LU, then by (4) LU has maximum geometric field of linearity Fq, see also [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We refer to [19] and [13] for comprehensive references on linear sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='2 Subspaces of complementary weights Recently, there has been an interest in linear sets admitting subspaces of complementary weights (see below for the definition), due to their application in coding theory, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' [18,20,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Linear sets on the projective line admitting two points of complementary weights have been studied in [17] (see also [12,16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The higher dimensional analogue has been studied in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' For the sake of completeness, we state the definition and prove the structural description of such linear sets here in full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Call subspaces W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , Wm ≤qn Fd+1 qn independent if each subspace Wi intersects ⟨Wj : j ̸= i⟩Fqn trivially, or equivalently if dimFqn⟨Wi : i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , m⟩ = dimFqn W1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + dimFqn Wm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , Wm be independent subspaces in Fd+1 qn , and let LU be an Fq-linear set in PG(d, qn) of rank k, that spans the entire space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then wLU (PG(W1, Fqn)) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + wLU (PG(Wm, Fqn)) ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If equality holds, then Fd+1 qn = W1 ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' ⊕ Wm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since no Wi intersects the span of the others, it is permitted to consider the direct sum W1 ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' ⊕ Wm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then k = dimFq U ≥ dimFq(U ∩ (W1 ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' ⊕ Wm)) ≥ dimFq(U ∩ W1) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + dimFq(U ∩ Wm) = wLU(PG(W1, Fqn)) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + wLU(PG(Wm, Fqn)) If equality holds, then U ∩ (W1 ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' ⊕ Wm) = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since W1 ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' ⊕ Wm is an Fqn-subspace, and ⟨U⟩Fqn = Fd+1 qn , we get that W1 ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' ⊕ Wm = Fd+1 qn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 4 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If the subspaces W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , Wm attain equality in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5, we say that PG(W1, Fqn), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , PG(Wm, Fqn) are subspaces of complementary weight (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' LU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let LU be an Fq-linear set spanning PG(d, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then there exist subspaces Ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , Ωm in PG(d, qn) of complementary weight, with dim Ωi = di and wLU (Ωi) = ki, if and only if U is GL(d + 1, qn)-equivalent to an Fq-subspace U1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' × Um, with each Ui a ki-dimensional Fq-subspace of Fdi+1 qn satisfying ⟨Ui⟩Fqn = Fdi+1 qn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' First suppose that such subspaces Ωi = PG(Wi, Fqn) exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then there exists a map ϕ ∈ GL(d + 1, qn) such that ϕ(W1) = ⟨e0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , ed1⟩Fqn, ϕ(W2) = ⟨ed1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , ed1+d2+1⟩Fqn, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' As can be seen in the proof of the previous lemma, ϕ(U) = ϕ(U ∩ W1) ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' ⊕ ϕ(U ∩ Wm), which equals U1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' × Um, with Ui = {u ∈ Fdi+1 qn : (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , 0, u, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , 0) ∈ ϕ(U) ∩ ϕ(Wi)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Clearly, dimFq Ui = dimFq ϕ(U ∩ Wi) = dimFq U ∩ Wi = wLU(Ωi) = ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Vice versa, suppose that ϕ(U) = U1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='×Um, with each Ui a ki-dimensional Fq-subspace of Fdi+1 qn , for some ϕ ∈ GL(d+1, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then define W1 = ⟨e0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , ed1⟩Fqn, W2 = ⟨ed1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , ed1+d2+1⟩Fqn, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then clearly PG(W1, Fqn), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , PG(Wm, Fqn) are subspaces of complementary weights w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Having subspaces of complementary weights is GL(d+1, qn)-invariant, which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 3 General bounds This section is devoted to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Afterwards, we provide some sufficient conditions on linear sets for the hypothesis of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3 to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Lastly, from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4 we derive that if a linear set LU contains a point of weight 1, its rank equals ⌈logq(|LU|)⌉, and ⟨LU⟩ is spanned by the points of LU of weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1 Proof of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5 De Beule and Van de Voorde proved the following bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Result 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1 ([8, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let LU be an Fq-linear set spanning PG(d, qn) of rank k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that LU meets some hyperplane Ω in exactly qd−1 q−1 points, spanning Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then |LU| ≥ qk−1 + qk−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Note that if d = 1, this result is exactly Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We now prove that this result is equivalent to Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' This follows directly from the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let LU be an Fq-linear set in PG(d − 1, qn), with d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then LU spans PG(d − 1, qn) and satisfies |LU| = qd−1 q−1 if and only if LU is a canonical Fq-subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If LU is a canonical subgeometry, then it immediately follows that LU spans the entire space, and |LU| = qd−1 q−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' So suppose that LU spans the space, and that |LU| = qd−1 q−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We need to prove that all points of LU have weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Indeed in that case, by equations (1) and (2), LU must then have rank d, which proves that LU is a canonical subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' So suppose by way 5 of contradiction that LU has points of weight greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Note that by Equations (1) and (2), the rank of LU is some number k > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let σ = ⟨P ∈ LU : wLU(P) > 1⟩ denote the subspace of PG(d − 1, qn) spanned by points of weight greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that σ is not PG(d − 1, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then LU ̸⊆ σ, and every point in LU \\ σ is a point of weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence, there are at least qk−1 points in LU \\ σ corresponding to (necessarily distinct) points of weight 1 of LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Thus, |LU| > qk−1 > qd−1 q−1 since k > d, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence σ equals PG(d − 1, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let m = max P ∈LU wLU(P) denote the maximum weight of the points of LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then we can choose d independent points P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , Pd in LU such that wLU(P1) = m, and wLU(Pi) ≥ 2 for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5, k ≥ d � i=1 wLU(Pi) ≥ m + 2(d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , Nm denote the weight distribution of LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then by Equations (1) and (2), qm − 1 q − 1 |LU| = m � i=1 Ni qm − 1 q − 1 ≥ m � i=1 Ni qi − 1 q − 1 = qk − 1 q − 1 ≥ qm+2(d−1) − 1 q − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' This implies that qd − 1 q − 1 = |LU| ≥ qm+2(d−1) − 1 qm − 1 , which yields a contradiction if d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We will now prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Consider the r-spaces Π1, Π2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' of PG(d, qn) through Ω = PG(W, Fqn), with Πi = PG(Wi, Fqn), for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We can order the r-spaces in such a way that Πi contains a point of LU \\ Ω if and only if i ≤ IΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let ki = dimFq(U ∩ Wi) denote the rank of the Fq-linear set LU∩Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then the sets Wi ∩ U \\ W partition the vectors in U \\ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since LU intersects Ω in a canonical subgeometry, dimFq W = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' This yields qk − qr = IΩ � i=1 (qki − qr) =⇒ qk−r = 1 + IΩ � i=1 (qki−r − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' (5) Analogously, the points of Πi \\ Ω partition the points of LU \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Note that for i ≤ IΩ, we have that LU ∩ Πi = LU∩Wi is an Fq-linear set in Πi of rank ki, satisfying the hypothesis of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence, |LU| = |LU ∩ Ω| + IΩ � i=1 (|LU ∩ Πi| − |LU ∩ Ω|) ≥ qr − 1 q − 1 + IΩ � i=1 � (qki−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qki−r + 1) − qr − 1 q − 1 � 6 = qr − 1 q − 1 + IΩ � i=1 � qki−r qr − 1 q − 1 − qr − 1 q − 1 + 1 � = qr − 1 q − 1 � 1 + IΩ � i=1 (qki−r − 1) � + IΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Using Equation (5) this implies that |LU| ≥ qr − 1 q − 1 qk−r + IΩ = qk−1 + qk−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−r + IΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If one wants to apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4 to a particular linear set LU, different choices of the (r − 1)-space Ω can yield different bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In other words, IΩ need not be the same for all (r − 1)-spaces meeting LU in a canonical Fq-subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' This is illustrated in the example below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Consider the (n + 1)-dimensional Fq-subspace U = {(x, xq) : x ∈ Fqn} × Fq of F3 qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Consider the corresponding Fq-linear set LU of rank n + 1 in PG(2, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Every point of LU has weight 1, so we can apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4 with Ω any point of LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' However, for a point P ∈ LU, IP = qn−1 + 1 if P lies on the line X2 = 0, and IP = qn−1 q−1 if P does not lie on X2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' These numbers are distinct if n > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We also remark that in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4 the number IΩ of r-spaces through Ω containing a point of LU \\ Ω equals the size of a certain linear set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Consider an Fq-linear set LU in PG(V, Fqn) and a subspace Ω = PG(W, Fqn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let U denote the subspace (U + W)/W of the quotient space V/W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then the projection of LU from Ω is the Fq-linear set LU of PG(V/W, Fqn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that LU is an Fq-linear set of rank k in PG(V, Fqn) and let LU be the projection of LU from an (r − 1)-space Ω = PG(W, Fqn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then for each Fqn-subspace W ′ ≤ V through W, wLU (PG((W ′ + W)/W, Fqn)) = wLU(PG(W ′, Fqn)) − wLU (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In particular, LU has rank k − wLU (Ω), and |LU| equals the number of r-spaces in PG(V, Fqn) through Ω that contain a point of LU \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Furthermore, if LU spans PG(V, Fqn), then LU spans PG(V/W, Fqn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We can find Fq-subspaces U1, U2, U3 of U such that U1 = W ∩ U, U1 ⊕ U2 = W ′ ∩ U, U1 ⊕ U2 ⊕ U3 = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then wLU (PG((W ′ + W)/W, Fqn)) = dimFq(U ∩ ((W ′ + W)/W)) = dimFq(⟨U, W⟩Fq ∩ W ′) − dimFq W = dimFq(W ⊕ U2) − dimFq(W) = dimFq(U2) = dimFq(U1 ⊕ U2) − dimFq(U1) = wLU(PG(W ′, Fqn)) − wLU(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If we put W ′ = V , we see that LU has rank k − wLU(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' It also follows that the points of LU are in 1-1 correspondence with the (r + 1)-spaces W ′ of V with wLU(PG(W, Fqn)) > wLU(Ω), which are exactly the r-spaces through Ω in PG(V, Fqn) containing a point of LU \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 7 This tells us the following about the quantity IΩ in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In the hypothesis of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4, let LU be the projection of LU from Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then |LU| ≥ qk−1 + qk−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−r + |LU|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Moreover, LU has rank k − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Together with the following lemma, we can now prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that LU is an Fq-linear set of rank k of PG(n, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let m = min{wLU (P) : P ∈ LU} denote the minimum weight of the points of LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then there exists an Fq-linear set LU′ of rank k − m + 1 in PG(n, q) containing points of weight 1 such that LU and LU′ coincide as point sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Take a vector u ∈ U such that P = ⟨u⟩Fqn has weight m in LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then there exists a (k − m + 1)-dimensional Fq-subspace U ′ of U that intersects ⟨u⟩Fqn in a 1-dimensional subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then wLU′(P) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' It remains to show that LU and LU′ coincide as points sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The inclusion LU′ ⊆ LU is evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' On the other hand, take a non-zero v ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then, by Grassmann’s identity wLU′(⟨v⟩Fqn ) = dimFq(⟨v⟩Fqn ∩ U ′) = dimFq((⟨v⟩Fqn ∩ U) ∩ U ′) = dimFq(⟨v⟩Fqn ∩ U) + dimFq(U ′) − dimFq(⟨⟨v⟩Fqn ∩ U, U ′⟩Fq) ≥ m + (k − m + 1) − dimFq(U) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' This shows that LU ⊆ LU′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Thus, LU and LU′ coincide as point sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let LU be an Fq-linear set spanning PG(d, qn) and intersecting an (r − 1)-space PG(W, Fqn) in a canonical Fq-subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If r > 0, then LU contains a point P = ⟨u⟩Fqn ∈ PG(W, Fqn) of weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='7, |LU| ≥ qk−1 + |LU|, where LU is the projection of LU from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then LU is an Fq-linear set of rank k − 1 spanning PG(d − 1, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Furthermore, W + ⟨u⟩Fqn is an (r − 2)-space in the quotient space, containing points of complementary weights, all weights equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence, it intersects LU in a canonical Fq-subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' This proves that Bq,n(r, k, d) ≥ qk−1 + Bq,n(r − 1, k − 1, d − 1) if r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Now suppose that r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then the minimum of the minimum weight of the points of LU, which we will denote by m, is at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='8, |LU| ≥ Bq,n(1, k − m + 1, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since this lower bound on |LU| is decreasing in m, we need to find an upper bound on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' There are d + 1 independent points in LU, since LU spans PG(d, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5 then tells us that m(d+1) ≤ k, which implies m ≤ � k d + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence, |LU| ≥ Bq,n � 1, k − � k d + 1 � + 1, d � ≥ qk−⌊ k d+1⌋ + Bq,n � 0, k − � k d + 1 � , d − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In the next sections, we will investigate linear sets attaining equality in the bound of Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' To this end, we introduce some relevant terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let LU be an Fq-linear set of rank k in PG(d, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If |LU| = qk−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−d + 1, 8 we say that LU is of d-minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If there is some (r−1)-space Ω such that LU and Ω satisfy the hypothesis of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4, and |LU| = qk−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−r + IΩ ≤ qk−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−d + 1, then we say that LU is of (r, d, Ω)-minimum size, or simply of (r, d)-minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' A linear set of (d, d)-minimum size, will also be called of proper d-minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' By Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4, an (r, d)-minimum size linear set has maximum field of linearity Fq whenever r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In the next proposition, we also prove that if a linear set is of (r, d)-minimum size, it is of (r′, d)-minimum size for every r′ ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let LU be an (r, d)-minimum size Fq-linear set of rank k in PG(d, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then LU is of (r′, d)-minimum size as well, for every 0 < r′ ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' It is enough to prove the statement for r′ = r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' By hypothesis, we know that there is some (r − 1)-space Ω = PG(W, Fqn) of PG(d, qn) meeting LU in a canonical subgeometry, such that |LU| = qk−1 + qk−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−r + |LU|, (6) where LU is the Fq-linear set in PG(V/W, Fqn) = PG(d − r, qn) defined by U = U + W ⊆ V/W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let Ω′ = PG(W ′, Fqn) be an (r − 2)-space of Ω that meets LU in a canonical subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' So, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='7, we have that |LU| ≥ qk−1 + qk−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−r+1 + |LU′|, (7) where LU′ is the Fq-linear set of rank k − r + 1 in PG(V/W ′, Fqn) = PG(d − r + 1, qn) defined by U ′ = U + W ′ ⊆ V/W ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Therefore, by (6), it follows qk−r + |LU| ≥ |LU′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' On the other hand, since wLU(Ω) = r, we get wLU′(PG(W/W ′, Fqn)) = 1 and so LU′ has a point of weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Now, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='7, we get that |LU′| ≥ qk−r + ���LU′ ��� with U ′ = U/W ′ + W/W ′ ≤ (V/W ′)/W, which is equal to U = U + W ≤ V/W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence, |LU′| ≥ qk−r + |LU|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then, by (6), equality holds in (7) and so LU is of (r − 1, d, Ω′)-minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='2 Sufficient conditions to apply Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3 In this part of the section, we show some conditions on a linear set that ensure the existence of at least one hyperplane meeting the linear set in a canonical subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let k, d and r be non negative integers with r < k, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let LU be an Fq-linear set in PG(d, qn) of rank k + d − r spanning PG(d, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that there is an r-space Ω of PG(d, qn) such that wLU(Ω) = k, and Ω contains an (r−1)-space Ω′ that meets LU in a canonical Fq-subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then some hyperplane Π of PG(d, qn) meets LU in a canonical Fq-subgeometry, implying |LU| ≥ qk+d−r−1 + qk+d−r−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that PG(d, qn) = PG(V, Fqn), Ω = PG(W, Fqn), and Ω′ = PG(W ′, Fqn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Con- sider the projection of LU from Ω′, which equals the linear set LU, with U = U + W ′ ⊆ V/W ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Write P0 = W/W ′, and choose any point P1 ∈ LU \\ {P0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since LU spans PG(d, qn), we can ex- tend P0, P1 to a subset P0, P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , Pd−r of LU that spans PG(V/W ′, Fqn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Also, wLU(P0) = k−r, and the rank of LU equals k + d − 2r = (k − r) + (d − r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5, (k − r) + (d − r) ≥ d−r � i=0 wLU (Pi) = (k − r) + d−r � i=1 wLU (Pi), 9 which implies that wLU(Pi) = 1 for all i ≥ 1, and P0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , Pd−r are points of complementary weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Therefore, LU meets ⟨P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , Pd−r⟩ in a canonical subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' There is a unique Fqn-space W ′′ through W ′ such that ⟨P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , Pd−r⟩ = PG(W ′′ + W ′, Fqn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' It follows that PG(W ′′, Fqn) meets LU in a canonical subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' For linear sets on a projective line PG(1, qn) of rank n, the notion of maximum field of linearity and maximum geometric field of linearity coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Result 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='13 ([7, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3] [12, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let LU be an Fq-linear set on PG(1, qn) of rank n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Define s = minP ∈LU wLU (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then the maximum field of linearity and the maximum geometric field of linearity are both Fqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In particular, when n is prime, the above result implies the existence of a point of weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Assume that n is prime and d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let LU be an Fq-linear set in PG(d, qn) of rank n + d − 1 spanning PG(d, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that there is a line ℓ of PG(d, qn) such that wLU (ℓ) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then |LU| ≥ qn+d−2 + qn+d−3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qn−1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' On the one hand, n is prime, so the maximum field of linearity of LU ∩ ℓ is Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' On the other hand, wLU (ℓ) = n and by Result 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='13, the maximum field of linearity of LU ∩ ℓ is Fqs with s = minP ∈LU∩ℓ wLU(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence, min P ∈LU∩ℓ wLU (P) = 1, and ℓ contains a point of weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The statement now follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Also when n is a prime and the rank of the linear set is n+d−1, in order to have the bound from Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3, it is enough to impose the existence of a (d − 2)-space meeting the linear set in a canonical subgeometry of this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Assume that n is prime and d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let LU be an Fq-linear set in PG(d, qn) of rank k = n+d−1 spanning PG(d, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that some (d−2)-space meets LU in a canonical subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then |LU| ≥ qk−1 + qk−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let Ω = PG(W, Fqn) be the (d − 2)-space of PG(d, qn) meeting LU in a canonical subge- ometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='7, we know that |LU| ≥ qk−1 + qk−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−d+1 + |LU|, where U = U + W is an Fq-subspace of the quotient V/W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since LU has rank n + d − 1 and dimFq(U ∩ W) = d − 1, then LU is an Fq-linear set of rank n in PG(1, qn) spanning the whole line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Therefore, since n is a prime, by Result 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='13 it follows that LU has at least one point of weight 1 and so |LU| ≥ qn−1 + 1 and the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3 Consequences of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4 When r = 1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4 looks as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let LU be an Fq-linear set of rank k ≥ 2 in PG(d, qn), admitting at least one point of weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let I be the number of secant lines through some point of weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then |LU| ≥ qk−1 + I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In particular, this result implies that the rank of a linear set is determined by its size and the minimum weight of its points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 10 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let LU be an Fq-linear set spanning PG(d, qn), containing more than one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Denote m = minP ∈LU wLU(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then the rank of LU is the unique integer k satisfying qk−m + Bq,n(0, k − m, d − 1) ≤ |LU| ≤ qk − 1 qm − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' k = ⌈logq(|LU|)⌉ + m − 1 = ⌊logq(|LU|)⌋ + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' As in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5, we have that |LU| ≥ Bq,n(1, k − m + 1, d) ≥ qk−m + Bq,n(0, k − m, d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The lower bound follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='8 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' By Equa- tion (2), (qm − 1)|LU| = (qm − 1) n � i=m Ni ≤ n � i=m Ni(qi − 1) = qk − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Another consequence of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='16 is that any Fq-linear set is spanned by its points of minimum weight, (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' [6, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='2] for linear sets on PG(1, qn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If an Fq-linear LU spans PG(d, qn), then its points of minimum weight also span PG(d, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that LU is an Fq-linear set of rank k, spanning PG(d, qn), and denote m = minP ∈LU wLU (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='16, |LU| > qk−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Now assume that the points of weight m of LU lie in a hyperplane π = PG(W, Fqn) of PG(d, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that U1 = U ∩ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then there exits a subspace U2 of U such that U = U1 ⊕ U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Now let U ′ 1 be an Fq-subspace of U1 of codimension m − 1, and let U ′ 2 be an Fq-subspace of U2 of codimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let U ′ = U ′ 1 ⊕ U ′ 2, then LU′ and LU coincide as point sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Indeed, if P ∈ LU ∩ π, then wLU (P) = wLU1(P) ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' As in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='8, this implies that wLU′(P) = wLU′ 1(P) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If P ∈ LU \\ π, then wLU (P) ≥ m + 1, and as in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='8, wLU′(P) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' But dimq U ′ = (dimq U1 − (m − 1)) + (dimq U2 − 1) = k − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence, by Equation (3), |LU| = |LU′| ≤ qk−m−1 q−1 < qk−m, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 4 Constructions of d-minimum size linear sets 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1 Exploring the Jena-Van de Voorde construction Recently, Jena and Van de Voorde constructed d-minimum size linear sets admitting points of complementary weights, and they completely determined their weight spectrum and weight distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Recall that if λ ∈ Fqn, then the degree of λ over Fq equals the degree of the minimal polynomial of λ over Fq, or equivalently the smallest integer t such that λ ∈ Fqt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1 ([11, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that λ ∈ Fqn has degree t > 1 over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Choose positive integers k0 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' ≥ kd such that k0 + k1 ≤ t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Define JVq,n(λ, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' k0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , kd) = ⟨1, λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , λk0−1⟩Fq × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' × ⟨1, λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , λkd−1⟩Fq = {(f0(λ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , fd(λ)): fi ∈ Fq[X], deg(fi) < ki}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then LJVq,n(λ,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='k0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=',kd) is a d-minimum size Fq-linear set in PG(d, qn) of rank k0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + kd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Note that since JVq,n(λ, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' k0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , kd) is a Cartesian product of Fq-subspaces of Fqn, it indeed admits points of complementary weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Before proceeding, we make some conventions regarding polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 11 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Given two polynomials f, g ∈ Fq[X], let gcd(f, g) denote the unique monic polynomial of maximal degree that divides f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We call f and g coprime if gcd(f, g) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Furthermore, we will use the convention that the degree of the zero polynomial is −∞, so that the equality deg(f · g) = deg f + deg g still holds if f or g is the zero polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3 ([11, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Jena and Van de Voorde also determined the weight spectrum of the above linear set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' It is (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , k0) if k1 = k0, and (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , k1, k0) if k1 < k0, in which case E0 is the unique point of weight k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' They also described the weight distribution, but since it is rather involved, we omit it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' It follows from their arguments that if gcd(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , fd) = 1, then wLU � ⟨f0(λ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , fd(λ)⟩Fqn � = min 0≤i≤d{ki − deg(fi)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' (8) This makes it relatively easy to determine Ni for some large values of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' For instance, let U = JVq,n(λ, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' k0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , kd) ⊆ Fd+1 qn , and assume that k1 < k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' As stated above, E0 is the unique point of weight k0, and the second largest weight of LU is k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We can determine Nk1(LU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let m denote the number of indices j with kj = k1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' k1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' = km > km+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let P = ⟨f0(λ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , fd(λ)⟩Fqn ∈ LU, with gcd(f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , fd) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then, by (8), P has weight k1 if and only if deg(f0) ≤ k0 − k1, deg(fi) ≤ 0, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , m and fi = 0, for i > m and there exists some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , m} such that deg(fj) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then, Nk1(LU) = qk0−k1+1 qm − 1 q − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The above construction has the following consequence on the existence of d-minimum size linear sets in PG(d, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4 ([11, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' There exists a d-minimum size Fq-linear set of rank k in PG(d, qn) whenever d < k ≤ � (d + 1)n+1 2 if n is odd, (d + 1)n 2 + 1 if n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We now present a sufficient condition for the linear set of Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1 to be of proper d-minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Consider U = JVq,n(λ, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' k0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , kd) as in Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that there exist pairwise coprime polynomials g0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , gd ∈ Fq[X] such that for each i, deg gi = ki − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If k0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + kd ≤ t + d, then LU is of proper d-minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' By Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1, we know that LU is an Fq-linear set in PG(d, qn) of rank k = k0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + kd of d-minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' So it remains to prove that there exists a hyperplane of PG(d, qn) meeting LU in a canonical subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Consider the points Pi = ⟨e0 + gi(λ)ei⟩Fqn for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Clearly, P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , Pd are independent, hence they span a hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Define the polynomial G(X) = d � i=1 gi(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Note that for each i ≥ 1, the polynomial �G gi � (X) = d � j=1 j̸=i gj(X) 12 is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then the equation of the hyperplane Π = ⟨P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , Pd⟩Fqn of PG(d, qn) is G(λ)X0 = d � i=1 �G gi � (λ)Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' (9) Let k = k0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + kd denote the rank of LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then deg G = d � i=1 (ki − 1) = k − k0 − d < t, hence G(λ) ̸= 0, and Equation (9) does indeed define a hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Now take a non-zero vector v = (f0(λ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , fd(λ)) ∈ U, and suppose that ⟨v⟩Fqn ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then G(λ)f0(λ) = d � i=1 �G gi � (λ)fi(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' (10) Every term in Equation (10) is a polynomial in λ, and deg(Gf0) = deg G + deg f0 = (k − k0 − d) + deg f0 < t, deg((G/gi)fi) = deg G + deg fi − deg(gi) ≤ deg(G) < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since 1, λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , λt−1 are Fq-linearly independent, Equation (10) implies that G(X)f0(X) = d � i=1 �G gi � (X)fi(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' On the one hand, this implies that f0 is a constant polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Otherwise, the left-hand side has degree greater than deg(G), but the degree of the right-hand side is at most deg(G), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' On the other hand, for each i, gi(X) | \uf8eb \uf8edG(X)f0(X) − � 1≤j̸=i � G gj � (X)fj(X) \uf8f6 \uf8f8 = �G gi � (X)fi(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since G/gi is coprime with gi, and deg(fi) ≤ deg(gi) this is only possible if fi is a multiple of gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence, v = (α0, α1g1(λ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , αdgd(λ)), for some scalars α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , αd ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Moreover, since ⟨v⟩Fqn ∈ Π, α0 = α1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + αd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence, LU intersects Π in the linear set LW , with W = � d � i=1 αi(e0 + gi(λ)ei): αi ∈ Fq � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Therefore, LU intersects Π in a canonical subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' A sufficient condition to ensure the existence of pairwise coprime polynomials g0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , gd ∈ Fq[X] such that deg(gi) = ki − 1, is to choose the size of the ground field large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Consider U = JVq,n(λ, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' k0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , kd) as in Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1, with k0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + kd ≤ t + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Assume that d � i=0 ki − d − 1 ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then LU is of proper d-minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' By the hypothesis, we can consider d+1 subsets S0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , Sd of Fq that are pairwise disjoint and such that |Si| = ki − 1, for each i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then, we can define gi(x) = � α∈Si(x − αi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' So the assertion follows by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Another sufficient condition to ensure the existence of pairwise coprime polynomials g0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , gd ∈ Fq[X] such that deg(gi) = ki − 1, is that the gi’s are different monic irreducible polynomials over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' It is well known, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' [14, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='25], that the number of monic irreducible polynomials of degree s over the finite field Fq is given by Gauss’s formula 1 s � h|s µ(s/h)qh, where h runs over the set of all positive divisors of s and µ denotes the M¨obius function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We note the following lower bound on the number of monic irreducible polyno- mials of degree s over Fq, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' [3]: 1 s � h|s µ (s/h) qh ≥ qs − 2qs/2 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' So we get the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Consider U = JVq,n(λ, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' k0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , kd) as in Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1, with k0 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='+kd ≤ t + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' For each s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , t, suppose that |{i: ki − 1 = s}| ≤ qs − 2qs/2 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then LU is of proper d-minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Clearly, if the rank of a linear set LU obtained from Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1 is greater than n + d, then every hyperplane has weight at least d + 1 in LU, so LU cannot be of proper d-minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In case the rank exceeds n + d, we can prove that LU is of (1, d)-minimum size under some constraints on the rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let U = JVq,n(λ, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' k0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , kd) be as in Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If k0 + kd−1 + kd ≤ t + 2, then LU is a (1, d)-minimum size Fq-linear set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let U ′ = {(f0(λ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , fd−1(λ) + λkd−1−1fd(λ), fd(λ)): fi ∈ Fq[X], deg(fi) < ki}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then U ′ is GL(d + 1, qn)-equivalent to U via the Fqn-linear map ϕ : v = (v0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , vd) �→ v + vdλkd−1−1ed−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The point ⟨(0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , 0, −λkd−1−1, 1)⟩Fqn has weight 1 in LU and it is mapped to point Ed by ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' So Ed has weight 1 in LU′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We prove that |LU′| = qk−1 + |LU|, where U = U ′ + Ed ≤q Fd+1 qn /Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Note that Fd+1 qn /Ed can be identified with Fd qn and U = U ′ + Ed ≤q Fd+1 qn /Ed with U = JVq,n(λ, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' k0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , kd−2, kd−1 + kd − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' By hypothesis k0 + kd−1 + kd − 1 ≤ t + 1, and so k0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , kd−2, kd−1 + kd − 1 indeed satisfy the hypothesis of Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1 when rearranged in descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Therefore, |LU| = qk−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since |LU| = |LU′| = qk−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−d + 1 = qk−1 + |LU|, we have the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 14 The above proposition together with Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4, allows to construct (1, d)-minimum size linear sets whose ranks exceed n + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' There exist (1, d)-minimum size Fq-linear sets in PG(d, qn), d ≥ 2, of rank k, whenever d < k ≤ � dn+1 2 + 1 if n is odd, dn 2 + 2 if n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='2 Generalizing the Caserta construction In [16, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1], a construction is given of linear sets on the projective line, based on the more general framework exploited in [10] and [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In this subsection, we generalize this to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The construction starts from an Fq-linear set LU′ in PG(d, qt), and yields an Fq-linear set in PG(d, qst).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Moreover, the weight distribution of LU is completely determined by the weight distribution of LU′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that n = st with s, t > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let U ′ be an Fq-subspace of Fd+1 qt ⊆ Fd+1 qn with dimFq(U ′) = k′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let Z be an Fqt-subspace of Fqn of dimension r > 0, such that 1 /∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Define Cq,s,t(Z, U ′) := {(z + u0, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , ud): z ∈ Z, (u0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , ud) ∈ U ′} ⊆ Fd+1 qn , which we will simply denote by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then (1) the Fq-linear set LU ⊆ PG(d, qn) has rank rt + k′, (2) |LU| = qrt|LU′ \\ {E0}| + 1, (3) wLU(E0) = rt + wLU′(E0), (4) Ni(LU) = qrt(Ni(LU′) − δi,wLU′ (E0)) + δi,wLU (E0), where δi,j denotes the Kronecker symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' (1) Since Z is an Fqt-subspace of Fqn, and 1 /∈ Z, Z ∩ Fqt = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Furthermore, since U ′ is an Fq-subspace of Fd+1 qt , Z ∩ {u0 : (u0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , ud) ∈ U ′} = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence, U = (Z × {0}d) ⊕Fq U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Therefore, dimFq U = dimFq Z + dimFq U ′ = rt + k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' (3) Similarly, wLU(E0) = dimFq � Z ⊕Fq {u0 : u0e0 ∈ U ′} � = dimFq Z + dimFq({u0 : u0e0 ∈ U ′}) = rt + wLU′(E0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' (2,4) Suppose that ⟨ze0 + u⟩Fqn = ⟨z′e0 + v⟩Fqn , with z, z′ ∈ Z, and u, v ∈ U ′ \\ ⟨e0⟩Fqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then ze0 + u = α(z′e0 + v) for some α ∈ Fqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since u, v are not multiples of e0, there must exist some position j > 0 such that uj, vj ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' This implies that α = vj/uj ∈ Fqt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We also have that z + u0 = α(z′ + v0), hence z − αz′ = αv0 − u0 Recall that Z is an Fqt-subspace, and that u0, v0, α ∈ Fqt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Therefore, the left-hand side of the above equality is in Z, and the right-hand side is in Fqt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since Z ∩ Fqt = {0}, this implies that z = αz′ and therefore u = αv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 15 Vice versa, if z ∈ Z, u ∈ U ′ \\ ⟨e0⟩Fqn and αu ∈ U ′ for some α ∈ Fqt, then ⟨ze0 + u⟩Fqn = ⟨αze0 + αu⟩Fqn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' This proves that wLU (⟨z + u⟩Fqn) = dimFq{α ∈ Fqt : αu ∈ U ′} = wLU′(⟨u⟩Fqn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence, varying z, we see that every point of LU′ \\ {E0} gives rise to |Z| points of LU \\ {E0} of the same weight, and this accounts for all points of LU \\{E0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Points (2) and (4) follow directly from this observation and the fact that E0 ∈ LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We remark that LU′ is contained in LU and the weight distribution and rank of LU in the above construction only depends on the weight distribution of LU′ and wLU′(E0), but not on the specific structure of U ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In particular, if ϕ ∈ ΓL(d + 1, qt), and ϕ fixes ⟨e0⟩, then Cq,s,t(Z, U ′) and Cq,s,t(Z, ϕ(U ′)) have the same rank and weight distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Given some minor conditions, the above construction preserves the property of being (r, d)- minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let U = Cq,s,t(Z, U ′) be as in Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If LU′ is an Fq-linear set of (r, d, Ω)-minimum size, and E0 ∈ LU′ \\ Ω, then LU is also of (r, d, Ω)-minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that Ω = PG(W, Fqn), and that the rank of LU′ is k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since LU′ is of (r, d, Ω)- minimum size, LU′ meets Ω in a canonical subgeometry of Ω, and |LU′| = qk′−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk′−r + |LU′|, where U ′ := U ′ + W ≤q Fd+1 qn /W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since E0 /∈ Ω, up to GL(d + 1, qn)-equivalence, we can suppose that Ω is defined by the equations X0 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' = Xd−r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence Fd+1 qn /W can be identified with Fd−r+1 qn in an obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Now, an element z +u ∈ U belongs to W if and only if z+u0 = u1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' = ud−r = 0 if and only if z = u0 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' = ud−r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Therefore, U ∩W = U ′ ∩W and so Ω also meets LU in a canonical subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Moreover, by Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='11, U = U + W = {z + u: u ∈ U ′} ⊆ Fd+1 qn /W, has size qrt(|LU′| − 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Therefore we have |LU| = qrt(|LU′| − 1) + 1 = qrt(qk′−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk′−r + |LU′| − 1) + 1 = qk−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−r + |LU|, with k = rt + k′ the rank of LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We can apply Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='11 with U ′ as in Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1, obtaining the following families of d-minimum size linear sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Consider U ′ = JVq,t(λ, t′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' k0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , kd) where t′ | t as in Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1, and choose ϕ ∈ GL(d+1, qt) such that E0 ∈ Lϕ(U′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Now define U = Cq,s,t(Z, ϕ(U ′)) as in Construc- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='11, with Z an Fqt-subspace of rank r > 0, not containing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then LU is a d-minimum size Fq-linear set of rank k = rt + k0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + kd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Moreover, the weight spectrum of LU is \uf8f1 \uf8f2 \uf8f3 � 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , k1, k0, rt + wLϕ(U′)(E0) � if wLϕ(U′)(E0) < k0 and k1 < k0, � 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , k1, rt + wLϕ(U′)(E0) � otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The Fq-linear set Lϕ(U′) has the same weight spectrum, weight distribution, and size as LU′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' So the assertions follow by applying Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='11 and Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 16 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Using [11, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='19] and Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='11 (3,4), one could in fact also determine the weight distribution of the linear set in the above theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The above construction gives new examples of proper d-minimum size linear sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In the hypothesis of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='14, suppose that LU′ is a (d, d, Π)-minimum size Fq-linear set, with Π = PG(W, Fqn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that E0 ∈ Lϕ(U′)\\˜Π, with ˜Π = PG(ϕ(W), Fqn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then LU is a proper d-minimum size Fq-linear set in PG(d, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The linear set LU′ is of (d, d, Π)-minimum size, so the hyperplane Π = PG(W, Fqn) of PG(d, qn) meets LU′ in a canonical subgeometry of Π and |LU′| = qm−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qm−d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' It follows that ˜Π also meets Lϕ(U′) in a canonical subgeometry of ˜Π, that is Lϕ(U′) is of proper d-minimum size as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The assertion follows by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1 provides constructions of d-minimum size linear sets admitting points of complementary weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Using Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='14, it is possible to construct proper d-minimum size linear sets that do not have this property, as we will see in the next example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' This proves that in general a d-minimum size linear set need not contain independent points whose weights sum to the rank of the linear set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' So in general, as already observed in [16] for the projective line, being minimum size does not determine the weight spectrum and distribution of a linear set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Consider U ′ = JVq,6(λ, 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 2, 2, 2) as in Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then LU′ is an Fq-linear set of rank 6 in PG(2, q6) having size q5 +q4 +1 and points of weight at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Moreover, wLU′(E0) + wLU′(E1) + wLU′(E2) = 2 + 2 + 2 = 6 is equal to the rank of LU′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Define ϕ ∈ GL(3, q6) : (x, y, z) �→ (x, y − λx, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then the Fq-linear set LU′′ in PG(2, q6), with U ′′ = ϕ(U ′) = {(α0 + α1λ, β0 + β1λ − α1λ2, γ0 + γ1λ): αi, βi, γi ∈ Fq} ⊆ F3 q6 has the same rank, weight spectrum and weight distribution as LU′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Note that wLU′′(E0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Choose a 1-dimensional Fq6-subspace Z ̸= Fq6 of Fq12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='14, the Fq-linear set LU of PG(2, q12), with U = Cq,2,6(Z, U ′′) has rank 12 and size q11 + q10 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' So, it is a 2-minimum size linear set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Note that the weight spectrum of LU is (1, 2, 7), and so there do not three points of complementary weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In particular, LU cannot be obtained from Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In some cases, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='14 gives us linear sets admitting points of complementary weights, but with a different weight distribution than those of Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1, as stated in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Consider U ′ = JVq,t(λ, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' k0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , kd) and let ϕi be the linear map swapping coordinates 0 and i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , d} (with ϕ0 = id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Consider U = Cq,s,t(Z, ϕi(U ′)), 17 (1) If ki = k0, then there exists an Fq-linear set obtained from Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1 with the same weight distribution as LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' (2) If ki < k0 − 1, then there does not exist an Fq-linear set obtained from Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1 with the same weight distribution as LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Write n = st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' (1) If ki = k0, choose a primitive element µ of Fqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Consider U2 = JVq,n(µ, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' k0 + rt, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , kd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then LU and LU2 have the same weight distribution by Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3 (see also [11, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='19]) and Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='11 (3,4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' (2) Now assume that ki < k0 − 1, and suppose that there exists some Fq-subspace U3 = JVq,n(µ, t′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' k′ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , k′ d) ⊆ Fd+1 qn such that LU and LU3 have the same weight distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since LU has a unique point of weight rt + ki by Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='11 (3,4), we see that by Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3, k′ 0 = rt + ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Furthermore, the second largest weight of LU and LU3 is respectively k0 and k′ 1, hence k0 = k′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let m′ denote the number of indices j with k′ j = k0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' k′ 1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' = k′ m′ > k′ m′+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then, using Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3 (see also [11, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='19]), Nk0(LU3) = qk′ 0−k′ 1+1 qm′ − 1 q − 1 = qrt+ki−k0+1 qm′ − 1 q − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' On the other hand, by Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='11 (4) and Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3 (see also [11, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='19]), Nk0(LU) = qrtNk0(LU′) = qrtqm − 1 q − 1 , with m the number indices j with kj = k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Therefore, qrt+ki−k0+1 qm′ − 1 q − 1 = qrtqm − 1 q − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since qm−1 q−1 and qm′−1 q−1 are coprime with q, this implies that ki = k0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3 Regarding equivalence We show that the two different types of Fq-subspaces that define the d-minimum size linear sets defined in Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='14 are ΓL(d + 1, qn)-inequivalent, even if the associated linear sets have the same weight spectrum and distribution (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='18 (1)), when the dimension of Z is the maximum possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The trace function Trqn/q of Fqn over Fq, defines a non-degenerate symmetric bilinear form as follows: (a, b) ∈ Fqn × Fqn �→ Trqn/q(ab) ∈ Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence, for any subset S of Fqn we can define the orthogonal complement as S⊥ = {a ∈ Fqn : Trqn/q(ab) = 0, ∀b ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Note that if S is an Fqt-subspace of Fqn, then S⊥ is an Fqt-subspace as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Given an ordered Fq-basis B = (ξ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , ξn−1) of Fqn, there exists a unique ordered Fq-basis B∗ = (ξ∗ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , ξ∗ n−1) of Fqn such that Trqn/q(ξiξ∗ j ) = δij, for i, j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , n − 1}, called the dual basis of B, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' [14, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 18 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='19 ([17, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let λ ∈ Fqn and suppose that B = (1, λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , λn−1) is an ordered Fq-basis of Fqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let f(x) = a0 + a1x + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + an−1xn−1 + xn be the minimal polynomial of λ over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then the dual basis B∗ of B is B∗ = (δ−1γ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , δ−1γn−1), where δ = f ′(λ) and γi = �n−i j=1 λj−1ai+j, for every i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that n = (s + 1)t, with s, t > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Consider U ′ = JVq,t(µ, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' k0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , kd) as in Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1, with k0 < t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let ϕi be the linear map swapping coordinates 0 and i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , d} (with ϕ0 = id) and define U1 = Cq,s,t(Z, ϕi(U ′)), as in Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='11, with Z an Fqt-subspace of dimension s, not containing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Consider U2 = JVq,n(λ, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' h0, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , kd) as in Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1, with h0 = st+k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then the Fq-subspaces U1 are U2 are ΓL(d + 1, qn)-inequivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that ki < k0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='18, LU1 and LU2 have a distinct weight distribution, hence U1 and U2 cannot be ΓL(d + 1, qn)-equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' So suppose that ki ∈ {k0 − 1, k0} and suppose by contradiction that U1 and U2 are ΓL(d + 1, qn)-equivalent via an element ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since h0 > ki, for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , d}, the point E0 is the only point in LU1 and in LU2 of weight h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' So, we have that ϕ(U1 ∩ E0) = U2 ∩ E0, that is aSρ 1 = S2, some a ∈ F∗ qn and ρ ∈ Aut(Fqn), with S1 = Z ⊕ ⟨1, µ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , µk0−1⟩Fq and S2 = ⟨1, λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , λh0−1⟩Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In particular, we have that aZρ ⊆ S2 and so (aZρ)⊥ ⊇ S⊥ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Note that dimFqt(aZρ) = dimFqt(Z) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' This implies that dimFq((aZρ)⊥) = n − st = t and hence (aZρ)⊥ is an Fqt- subspace of Fqn of dimension one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Consider the ordered Fq basis B = (1, λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , λn−1) of Fqn and its dual basis B∗ = (λ∗ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , λ∗ n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' So we have that S⊥ 2 = ⟨λ∗ h0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , λ∗ n−1⟩Fq and since k0 < t−1, we have that h0 < n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='19 it follows that λ∗ n−2 = δ−1(an−1 + λ), and λ∗ n−1 = δ−1, where f(x) = a0+a1x+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='+an−1xn−1+xn is the minimal polynomial of λ over Fq and δ = f ′(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Now, since λ∗ n−2, λ∗ n−1 ∈ (aZρ)⊥ and since (aZρ)⊥ has dimension one over Fqt, it follows λ∗ n−2 λ∗ n−1 = an−1 + λ ∈ Fqt, that is λ ∈ Fqt, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 5 Below the De Beule-Van de Voorde bound In this section, we will provide constructions of linear sets LU that in PG(d, qn), with d > 2, that are of (r, d)-minimum size but not of d-minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' They have maximum geometric field of linearity Fq, and admit two subspaces of complementary weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' For our aims, we will suppose that one of these subspaces intersects LU in a linear set with greater field of linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' This gives us the following constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let n = st, with s, t > 1, and suppose that 19 U1 is a k1-dimensional Fqt-subspace of Fd1+1 qn , U2 is a k2-dimensional Fq-subspace of Fd2+1 qt ⊆ Fd2+1 qn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Define U = U1 × U2, and d = d1 + d2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then LU is an Fq-linear set of PG(d, qn) of rank k1t + k2, with |LU| = |LU1| + qk1t|LU2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Moreover, its weight distribution satisfies Ni(LU) = Ni(LU1) + qk1tNi(LU2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Take a vector u ∈ U1 and v ∈ U2 with (u, v) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then wLU(⟨(u, v)⟩Fqn ) = dimFq{α ∈ Fqn : α(u, v) ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Evidently, α(u, v) ∈ U if and only if αu ∈ U1 and αv ∈ U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If v ̸= 0, then αv ∈ U2 implies that α ∈ Fqt, and since U1 is an Fqt-subspace, αu is automatically in U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Therefore, every point ⟨v⟩Fqn of LU2 gives rise to the qk1t points {⟨(u, v)⟩Fqn : u ∈ U1} of LU with the same weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If v = 0, then we just need that αu ∈ U1, hence in this way, every point of LU1 gives rise to one point of LU of the same weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since this accounts for all points of LU, the statement of the theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Using the above theorem, we are able to obtain constructions of linear sets in PG(d, qn), with d ≥ 3, having maximum geometric field of linearity Fq that are (r, d)-minimum size with 2 ≤ r < d and that are not d-minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let n = st, with s, t > 1, and suppose that U1 is a k1-dimensional Fqt-subspace of Fd1+1 qn , with k1 ≤ d1s, U2 is a k2-dimensional Fq-subspace of Fd2+1 qt , such that LU2 is a proper d2-minimum size Fq-linear set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Define U = U1 × U2, d = d1 + d2 + 1, and k = k1t + k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then LU is a (d2, d)-minimum size Fq-linear set of size |LU| = qk−1 + qk−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−d2 + qk1t + |LU1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence, LU is not d-minimum size if k2 ≥ d2+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Furthermore, if d2 ≥ 2, then Fq is the maximum geometric field of linearity of LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The Fq-linear set LU2 ⊆ PG(d2, qn) is of proper d2-minimum size, and so its size is |LU2| = qk2−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk2−d2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1, LU has rank k = k1t + k2 and size |LU1| + qk1t(qk2−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk2−d2 + 1) = |LU1| + qk−1 + qk−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−d2 + qk1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Moreover there exists a (d2 − 1)-space Γ = PG(W, Fqn) of PG(d2, qn), with W ⊆ Fd2+1 qn , meeting LU2 in a canonical subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Now, let W ′ = {0}d1+1 × W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then W ′ defines a d2-space of PG(d, qn) meeting LU in a canonical subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Identifying Fd+1 qn /W with Fd1+1 qn we have that U = U1 + W ≤q Fd+1 qn /W with U = U1 × U ′, where U ′ is an Fq-subspace of Fqn of dimension k2 − d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' So again, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1, we have |LU| = qk1t + |LU1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Moreover, by (3), |LU1| ≤ q(k1−1)t + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qt + 1, and since k2 > d2 + 1 it follows that |LU1| < qk−d2−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−d + 1 − qk1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' This implies that LU is not of d-minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Finally, the assertion on the geometric field of linearity follows from Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 20 By the above corollary and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5, we get the following construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let n = st, with s, t > 1, and suppose that U1 = JVqt,n(λ, n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' l0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , ld1), and denote k1 = l0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + ld1, U2 = JVq,t(µ, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' m0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , md2), and denote k2 = m0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + md2, with LU2 satisfying the condition of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Define U = U1 × U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then LU is a (d2, d)- minimum size Fq-linear set, but not of d-minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Moreover, if d2 ≥ 2, then Fq is the maximum geometric field of linearity of LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Note that |LU1| = q(k1−1)t + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + q(k1−d1)t + 1 < qk−d2−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−d + 1 − qk1t, and then the assertion follows by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Other examples of (d2, d)-minimum size linear set can be obtained by using as LU1 or LU2 in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='2 the minimum size linear sets constructed in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' It is natural to consider PG(d, qn), n not prime, and wonder what the maximal value of d2 is such that the above corollary implies the existence of an Fq-linear set in PG(d, qn) that is of (d2, d)-minimum size, but not of d-minimum size, and has maximum geometric field of linearity Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' So let t be the largest proper divisor of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Note that t ≥ √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We want to construct a set U2 = JVq,t(µ, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' m0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' , md2) with d2 maximal, such that it satisfies the conditions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence, there must exist pairwise coprime polynomials gi of degree mi − 1 such that (m0 − 1) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + (md2 − 1) ≤ t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let δ(x) denote the maximum number of distinct monic irreducible polynomials over Fq such that the sum of their degrees is smaller than x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then for any m ≥ 1, δ(qm) ≥ qm−1 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Indeed, consider the minimal polynomials of the elements of F∗ qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since every element of F∗ qm is the root of a unique such polynomial, their degrees sum to qm − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Furthermore, the maximum degree equals m, so there are at least qm−1 m such polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence, to answer the original question, asymptotically, d2 = Ω(t/ logq(t)) = Ω(√n/ logq(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We conclude this subsection with examples of linear sets of (1, 2)-minimum size that are not of (2, 2)-minimum size and have maximum geometric field of linearity Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let n = st, with s > 1, and t > 2 prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose that the smallest prime that divides s is at least t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let U1 be a k1-dimensional Fqt-subspace of Fqn, U2 = JVq,t(µ, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' m0, m1), with t = m0 + m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Define U = U1 × U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then LU is a (1, 2)-minimum size Fq-linear set, but not of 2-minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Moreover, Fq is the maximum geometric field of linearity of LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1, LU is an Fq-linear set of PG(2, qn) of rank (k1 + 1)t having size |LU| = q(k1+1)t−1 + qk1t + 1, that is not of 2-minimum size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since LU2 has a point of weight 1, there exists ϕ ∈ GL(2, qt), such that the Fq-linear set LU′, with U ′ = U1×ϕ(U2) has E2 as a point of weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Hence F3 qn/E2 can be identified with F2 qn in an obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Clearly, LU and LU′ are GL(3, qn)-equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In this way, U ′/E2 can be identified as an Fq-subspace U = U1 × U ′ 2, where U ′ 2 is an (t − 1)-dimensional Fq-subspace of Fqt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Again, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='1, we have that |LU| = qk1t + 1 and hence LU is a (1, 2)-minimum size Fq-linear set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Suppose now, that LU = LW for some Fqr-linear set LW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If r < t, then by our hypothesis, r is coprime with s and t, hence r is coprime with n = st, and Fqr is not a subfield of Fqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Therefore, r ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Let ℓ be the line of PG(2, qn) having equation X0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Then qt−1 + 1 = |LU2| = |ℓ ∩ LU| = |ℓ ∩ LW |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since ℓ∩LW is an Fqr-linear set we have that |ℓ∩LW| ≥ qr +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' So t−1 ≥ r, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' 21 6 Conclusion De Beule and Van de Voorde [8] provided a lower bound on the size of an Fq-linear set LU in PG(d, qn) that intersects some hyperplane Ω in a canonical subgeometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' In this paper, we generalized their result by allowing Ω to be an (r − 1)-space, with 1 ≤ r ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Our bound looks like |LU| ≥ qk−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−r + |LU|, where k is the rank of LU, and LU is the projection of LU from Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Unfortunately, the bound still depends on the size of the linear set LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' This raises the question what the minimum size of LU is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Taking r as large as possible, we may assume that LU does not have any points of weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We presented a recursive lower bound on the size of linear sets without points of weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Since we recursively use a rather naive upper bound on the minimum weight of the points of LU, one should not expect this bound to be tight, except for some particular cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Open problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Find a good lower bound on the size of an Fq-linear set of rank k−r spanning PG(d − r, qn), containing no points of weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' The rest of the paper was concerned with finding examples of linear sets that attain equality in the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We note that all constructions but the one of Jena and Van de Voorde [11] use a subfield in between Fq and Fqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' So it looks like the most restrictive setting to study linear sets of minimum size is the case where such a subfield does not exist, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' the case where n is prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' We reiterate that for n prime, Jena and Van de Voorde [11, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='5 (B)] stated their belief that the bound of Result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='3 is still correct with less restrictive conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Open problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Is it true that if n is prime, all Fq-linear sets LU in PG(d, qn) of rank k ≤ d + n satisfy |LU| ≥ qk−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' + qk−d + 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' If so, can we classify the linear sets attaining equality in this bound?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Acknowledgment We would like to thank Jan De Beule, Olga Polverino and Ferdinando Zullo for fruitful discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Paolo Santonastaso is very grateful for the hospitality of the Department of Mathematics and Data Science, Vrije Universiteit Brussel, Brussels, Belgium, where he was a visiting PhD student for 2 months during the preparation of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Paolo Santonastaso was supported by the project “VALERE: VAnviteLli pEr la RicErca” of the University of Campania “Luigi Vanvitelli” and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' References [1] G.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Finite Fields and Their Appli- cations, 87:102153, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content=' Sam Adriaensen Vrije Universiteit Brussel Department of Mathematics and Data Science Pleinlaan 2, 1050 Elsene, Belgium sam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='adriaensen@vub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='be Paolo Santonastaso Universit`a degli Studi della Campania “Luigi Vanvitelli” Dipartimento di Matematica e Fisica Viale Lincoln, 5, I– 81100 Caserta, Italy paolo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='santonastaso@unicampania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} +page_content='it 24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFPT4oBgHgl3EQfFzRC/content/2301.13001v1.pdf'} diff --git a/kdFAT4oBgHgl3EQfbB3C/content/2301.08555v1.pdf b/kdFAT4oBgHgl3EQfbB3C/content/2301.08555v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..04c1818971a946107b15285b67d9182e666ad124 --- /dev/null +++ b/kdFAT4oBgHgl3EQfbB3C/content/2301.08555v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1c3d8d8d03c92aadc8f78ab1efecac5b4266af2e3a77aa5b35ad72ba40d662d5 +size 9367026 diff --git a/kdFAT4oBgHgl3EQfbB3C/vector_store/index.pkl b/kdFAT4oBgHgl3EQfbB3C/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..87873cdf19106ef7e74412b9027713143a694b5a --- /dev/null +++ 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+It has been recently +demonstrated that the arrangement of two levitated nanoparticles naturally realizes the tunable +nonreciprocal dipole-dipole interaction. Motivated by this development, we here propose and ana- +lyze an array of levitated nanoparticles as an ideal platform to study non-Hermitian physics in a +highly controlled manner. We employ the non-Bloch band theory to determine the continuum bands +of the proposed setup and investigate the non-Hermitian skin effect therein. In particular, we point +out that the levitated nanoparticle array exhibits rich dynamical phases, including the dynamically +unstable phase and the unconventional critical phase where the spectral singularity persists over +a broad region of the controllable parameters. +We also show that the long-range nature of the +dipole-dipole interaction gives rise to the unique self-crossing point of the continuum band. +A levitated nanoparticle is a laser trapped nanoscale +dielectric particle smaller than wavelength of light [1]. +Recent experimental developments have allowed one +to cool a levitated nanoparticle to ultracold tempera- +tures [2–8] and offered unique opportunities to study +quantum mechanics of mesoscopic objects [9–20]. Addi- +tionally, previous studies demonstrated the potential of a +levitated nanoparticle to explore various fields of physics, +such as nonequilibrium physics [21–27] and quantum +sensing [28–37]. Remarkably, recent experimental stud- +ies have shown the possibility of extending these sys- +tems to multi-nanoparticle setups [38–43]. In particular, +Ref. [43] has reported a realization of an on-demand as- +sembly of levitated nanoparticles, in which optical tweez- +ers are used to trap and arrange the nanoparticles one by +one. +On another front, recent years have witnessed remark- +able advances in our understandings of non-Hermitian +systems, i.e., a class of nonequilibrium systems that can +be effectively described by a non-Hermitian operator [44]. +While non-Hermitian physics has been widely investi- +gated in several fields of quantum science, such as ul- +tracold atoms [45–49] and photonics [50–53], its idea has +also found numerous applications in classical systems re- +alized in optics [54–57], mechanics [58–61], and electrical +circuits [62–65]. These previous studies uncovered rich +non-Hermitian phenomena that have no counterparts +to Hermitian systems. For instance, a one-dimensional +(1D) tight-binding model with asymmetric hopping am- +plitudes exhibits the non-Hermitian skin effect [66–68], +where the bulk eigenstates are localized at open bound- +aries of the system, leading to the extreme boundary sen- +sitivity of the eigenvalue. +In this Letter, we propose and analyze a 1D levitated +nanoparticle array as an ideal platform to study previ- +ously unexplored regimes of non-Hermitian physics in a +highly controlled manner. A prominent feature here is +that there exists the tunable nonreciprocal dipole-dipole +interaction between levitated nanoparticles, which is in- +duced by the nonreciprocal interference originating from +phase difference between the trapping lasers [41]. The +proposed system then realizes a 1D tight-binding model +with arbitrarily tunable asymmetric hopping amplitudes +that have possibly negative signs and long-range depen- +dence. This high controllability allows one to explore the +whole parameter region of non-Hermitian systems, thus +opening the possibility to uncover the full potential of +non-Hermitian phenomena. In this respect, the proposed +setup should be contrasted to the previous non-Hermitian +platforms, where studies were restricted to severely lim- +ited parameter regions of the models due to the difficul- +ties of realizing tunable nonreciprocal interactions. +To determine the continuum bands and the dynamical +phase diagram of the levitated nanoparticle array, we in- +voke the non-Bloch band theory [66, 69–73], a recently +developed powerful tool to investigate models featuring +the non-Hermitian skin effect. The non-Bloch band the- +ory allows for calculating the asymptotic eigenvalues of +the systems with open boundary conditions in the limit +of a large system size. This makes contrast to the con- +ventional Bloch band theory, where the energy band re- +produces the eigenvalues under periodic boundary con- +ditions. +On the basis of this theoretical framework, we find that +the levitated nanoparticle array exhibits rich dynamical +phases, including the unconventional critical phase and +the dynamically unstable phase. In the former, a remark- +able feature is that the non-Hermitian degeneracy of the +continuum bands known as the spectral singularity ap- +pears without fine-tuning and persists over a broad region +of the parameters. The key ingredients of the latter are +negative interparticle couplings, which were difficult to +realize in the existing non-Hermitian platforms. More- +over, the proposed system can naturally realize the long- +range hopping amplitudes originating from the dipole- +dipole interaction provided that the particle distance is +arXiv:2301.05439v1 [physics.optics] 13 Jan 2023 + +2 +FIG. 1. Schematic figure of the levitated nanoparticle array. +The distance between the nearest-neighbor particles is d0, and +the mass of all the particles is m. All the trapping lasers have +the power P and the wavelength λ. We set the phase of the +nth trapping laser in the focal plane to be φ + n∆φ. +judiciously controlled. We show that this long-range na- +ture leads to the unique self-crossing point of the contin- +uum band, which corresponds to the singularity of the +generalized Brillouin zone. +To be concrete, we consider a 1D array of the trapped +levitated nanoparticles as shown in Fig. 1. The particles +are equally spaced at the interval d0, and all the particles +have the mass m. Let λ and P denote the wavelength +and the power of all the trapping lasers, respectively. +Furthermore, we assume that the motion of the particles +along the plane perpendicular to the optical axis is frozen. +The interaction between the two particles arises due +to the interference between the scattered electromagnetic +field and the trapping laser. Since the scattered field ac- +quires the phase kd0 during the propagation, the phase +difference between the trapping lasers at the positions +of the particles leads to the constructive and destruc- +tive interference depending on the propagation direction +of the scattered field. It is this spatial asymmetry that +renders the interparticle coupling nonreciprocal. Due to +the long-range nature of this nonreciprocal dipole-dipole +interaction, it is in general necessary to incorporate the +couplings that reach up to Nth neighbor particles. Al- +together, in the vicinity of the focal plane, the linearized +equation of motion of the nth particle along the z axis is +given by +m¨zn + mγ ˙zn = − +� +mΩ2 + 2 +N +� +l=1 +Kl +� +zn ++ +N +� +l=1 +�� +Kl + ¯Kl +� +zn−l + +� +Kl − ¯Kl +� +zn+l +� +. +(1) +Here, Ω is an intrinsic mechanical frequency of the par- +ticle proportional to +√ +P, γ is a friction coefficient, and +Kl and ¯ +Kl are the coupling strengths given by +� +� +� +� +� +� +� +Kl = +G +lk0d0 +cos (lk0d0) cos (l∆φ) , +¯Kl = +G +lk0d0 +sin (lk0d0) sin (l∆φ) , +(2) +where G has the dimension of a spring constant and is +proportional to P, ∆φ is the optical phase difference be- +tween the neighbor trapping lasers in the focal plane +[Fig. 1], and k0 (= 2π/λ) is a wavenumber of the trap- +ping laser. One can infer from Eq. (2) that the couplings +are long-range because the dipole-dipole interaction is +proportional to the inverse of the distance between the +particles. +We note that the sign of the coupling con- +stants Kl and ¯Kl can be controlled by changing the phase +difference. We provide the derivation of Eq. (1) in the +Supplementary Material [74]. +In general, continuum bands of non-Hermitian tight- +binding models can be obtained by invoking the non- +Bloch band theory, which reproduces the asymptotic +eigenvalues under open boundary conditions in the ther- +modynamic limit. +Specifically, the continuum band is +calculated from the generalized Brillouin zone spanned +by β ≡ eik for a complex Bloch wavenumber k. +We +here apply the non-Bloch band theory to the levitated +nanoparticle array; throughout this paper, we assume +|KN| ̸= +�� ¯KN +��. Substituting zn = ψneiωt to Eq. (1), we +have the real-space eigenequation as follows: +1 +m +N +� +l=1 +�� +Kl − ¯Kl +� +ψn+l +� +Kl + ¯Kl +� +ψn−l +� ++ +� +ω2 − iγω − Ω2 − 2 +m +N +� +l=1 +Kl +� +ψn = 0. +(3) +Importantly, an ansatz of Eq. (3) can be taken as +ψn = +2N +� +j=1 +(βj)n φ(j), +(4) +where βj (= β) is the solution of the characteristic equa- +tion given by +1 +m +N +� +l=1 +�� +Kl − ¯Kl +� +βl + +� +Kl + ¯Kl +� +β−l� ++ +� +ω2 − iγω − Ω2 − 2 +m +N +� +l=1 +Kl +� += 0. +(5) +We note that Eq. (5) is an algebraic equation for β of +2Nth degrees. The main result of the non-Bloch band +theory is that the condition for the generalized Brillouin +zone is obtained from the 2N solutions, and it is given +by +|βN| = |βN+1| +(6) + +m +m +m +do +do +r +2 +nth +(n-1)th +(n+1)th +Φ+ (n- 1) △d +Φ+nAd +Φ+ (n+ 1) △d3 +with |β1| ≤ · · · ≤ |β2N|. The trajectories of βN and βN+1 +form the generalized Brillouin zone on the complex plane, +which reveals the essential features of non-Hermitian sys- +tems (see, e.g., Refs. [75–78]). Then, we can calculate the +continuum bands by combining Eq. (5) with the gener- +alized Brillouin zone. We note that when ¯Kl = 0, the +generalized Brillouin zone reduces to a unit circle, which +means that the Bloch wavenumber becomes real. +We start our analysis from the levitated nanoparticle +array with the nearest-neighbor interaction, which cor- +responds to N = 1 in Eq. (1); in the following, we as- +sume γ > 2Ω for the sake of concreteness. From Eq. (6), +the generalized Brillouin zone can be given by the cir- +cle with the radius r = +���� +K1 + ¯K1 +� +/ +� +K1 − ¯K1 +���. We +then have the analytical form of the continuum bands as +follows: +ω± = i +2γ ± +� +Ω2 + 2 +m +� +K1 − +� +K2 +1 − ¯K2 +1 cos θ +� +− γ2 +4 , +(7) +where +θ +is +a +real +number. +Since +each +eigen- +mode contributes to the dynamics through the factor +e−Im(ω±)eiRe(ω±), we can show the dynamical phase di- +agram of the system depending on K1/m and ¯K1/m in +Fig. 2(a). Figures. 2(b)–(d) and (e)–(g) plot the evolu- +tions of the continuum bands along the black and white +arrows indicated in Fig. 2(a), respectively. +In the blue-shaded regions of Fig. 2(a), all the particles +oscillate with the attenuation because Re (ω±) ̸= 0 and +Im (ω±) > 0 [Fig. 2(b)]. In contrast, in the red-shaded re- +gions, their motion monotonically vanishes without oscil- +lations because Re (ω±) = 0 and Im (ω±) > 0 [Fig. 2(d)]. +For these reasons, we term the former (latter) the dy- +namical phase as the underdamped (overdamped) phase. +Remarkably, we find the broad green-shaded region +where the two branches ω+ and ω− coalesce at Re (ω±) = +0 [Fig. 2(c)], leading to the crossover dynamics between +the above two dynamical phases. Such a degenerate point +unique to non-Hermitian bands is known as the spectral +singularity. We shall term this intermediate regime as +the critical phase in the sense that the overdamped be- +havior eventually sets in after the initial underdamped +oscillations. Importantly, the emergence of this critical +phase is unique to the present setup with open bound- +aries because the spectral singularity disappears under +periodic boundary conditions. Indeed, after replacing β +by eik (k ∈ R) in Eq. (5), one obtains the band, which re- +produces the eigenvalues under periodic boundary condi- +tions, and it is qualitatively different from Eq. (7). Thus, +the transient phenomena discussed here is supported by +the non-Hermitian degeneracy, and the non-Hermitian +skin effect plays an essential role to realize the aforemen- +tioned critical phase. We note that, as shown in Fig. 2(f), +the spectral singularity also appears along the green ver- +tical lines in Fig. 2(a). However, one would need fine- +FIG. 2. +Dynamical phase diagram and continuum bands of +the levitated nanoparticle array. +(a) Dynamical phase dia- +gram exhibiting the underdamped, critical, overdamped, and +dynamically unstable phases shown in the blue, green, red, +and gray-shaded regions, respectively. The spectral singular- +ity (SS) appears in the green-shaded regions. On the black +dashed lines, |K1| = +�� ¯K1 +�� is satisfied. We set the parameters +to be Ω = 1 and γ = 5. (b)–(g) Evolutions of the continuum +bands along the arrows in (a). The magenta and cyan express +ω+ and ω−, respectively. The numerical values in each panel +specify +� +K1, ¯K1 +� +. +tuning of the parameters in this case as indicated by +Figs. 2(e)–(g), where the two continuum bands are re- +combined across the green line. +In addition to the above phases, we also find the dy- + +10 +(a) +(b)" +5 +(c) +(d) +K1 +0 +m +(g) +(f) +-5 +(e) +-105 +_10 +-5 +0 +5 +10 +K1 +m +(b) (6,5.5) +(c) (4,3.5) +(d) (0.8,0.5) +5 +5 +5 +SS +3 +3 +Im(w) +Im(w) +1F +1 +1 +0 +0 +0 +-3 +3 +-3 +0 +3 +-3 +0 +3 +Re(w) +Re(w) +Re(w) +(e) (5,-6) +(f) (21/8,-29/8) +(g) (0.8,-1.8) +5 r +5 +5 +SS +3 +Im(w) +1 +入 +1 +0L +0 +0 +-3 +0 +3 +-3 +0 +3 +-3 +0 +3 +Re(w) +Re(w) +Re(w)4 +namically unstable phase as indicated by the gray-shaded +region in Fig. 2(a). There, either of the hopping ampli- +tudes, K1 ± ¯K1, becomes negative, and the oscillation +amplitudes diverges in the long-time limit because the +imaginary part of the eigenvalues can take negative val- +ues. Physically, this instability originates from the fact +that the negative hopping amplitudes cause the force that +increasingly keeps away the particles from their equilib- +rium positions. We emphasize that the dynamically un- +stable phase discussed here is rather difficult to realize in +the previous non-Hermitian systems due to the lack of the +ability to implement tunable negative coupling strengths. +It is worthwhile to mention that, in finite-size systems, +the boundary between the overdamped and dynamically +unstable phases can be slightly modified [74]. +We next investigate how the long-range nature of the +couplings can affect the continuum band and the cor- +responding generalized Brillouin zone of the levitated +nanoparticle array; in the following, we neglect the fric- +tion for the sake of simplicity. +To this end, we as- +sume that the interaction reaches up to the next-nearest- +neighbor particles, which corresponds to N += 2 in +Eq. (1). In Fig. 3, we plot the continuum bands with the +positive branch of the square root and the corresponding +generalized Brillouin zone at different ∆φ. We note that +the black dashed curves in Figs. 3(d)–(f) indicate the +conventional Brillouin zone formed by β ≡ eik (k ∈ R). +One can see from Figs. 3(d) and (f) that the general- +ized Brillouin zone with N = 2 forms a skewed closed +curve with the cusps, at which it becomes indifferen- +tiable, while the generalized Brillouin zone with N = 1 is +merely a circle. Importantly, the cusps correspond to the +self-crossing points of the continuum band [Fig. 3(a) and +(c)] [79]. Thus, the long-range nature of the nonrecip- +rocal interaction can lead to these unconventional band +structures. +Meanwhile, at ∆φ = π/2, the generalized +Brillouin zone becomes the unit circle independently of +N as shown in Fig. 3(e), where the non-Hermitian skin +effect disappears. Accordingly, there are no self-crossing +points of the continuum band as shown in Fig. 3(b). +In summary, we propose and analyze the levitated +nanoparticle array as an ideal platform to study new +realms of non-Hermitian physics in a highly controlled +manner. We show that the system exhibits the uncon- +ventional critical phase, where the spectral singularity +originating from the non-Hermitian skin effect persists +over a broad region of the controllable parameters. We +also point out that the tunable dipole-dipole nonrecipro- +cal interaction in the proposed setup allows for extremely +nonreciprocal hopping amplitudes with possibly negative +signs, which result in the dynamical instability. We fi- +nally reveal that the long-range nature of the nonrecip- +rocal couplings further enriches the non-Hermitian band +structures, leading to the cusps of the generalized Bril- +louin zone and the self-crossing points of the continuum +band. +FIG. 3. Continuum bands and generalized Brillouin zones of +the levitated nanoparticle array at different ∆φ. (a)–(c) The +continuum bands with the positive branch of the square root, +and (d)–(f) the corresponding generalized Brillouin zones are +shown. The red (blue) curves indicate the results for N = 1 +(N = 2). In (d)–(f), the black dashed curve expresses the +conventional Brillouin zone spanned by β ≡ eik (k ∈ R). The +system parameters are set to be λ = 1.064 × 10−6 m, d0 = +10−5 m, Ω = 10 s−1, and G/ (mkd0) = 1 s−2. +Several open questions remain for future studies. First, +in a levitated nanoparticle array, the continuum bands +can be experimentally observed by measuring the power +spectral density. Hence, it should be possible to directly +observe the spectral singularity and the correspondence +between the cusps of the generalized Brillouin zone and +the self-crossing points of the continuum band. +Second, besides the model discussed here, a levitated +nanoparticle array allows one to realize various non- +Hermitian tight-binding models with arbitrary param- +eters, thanks to its high controllability. +For instance, +it provides an ideal setup to realize the non-Hermitian +Su-Schrieffer-Heeger model [66], where rich phenomena +including a topological phase transition are expected to + +(a) △Φ = π /10 +(d) △Φ = π /10 +0.05 +1 +Im(β) +(3) +-0.05 +9.8 +9.9 +10 +10.1 +-1 +0 +Re(w) +Re(β) +(b) △Φ = /2 +(e) △= /2 +0.05 +1 +(m)wl +(d)ul +0 +0 +-0.05 +9.8 +9.9 +10 +10.1 +0 +Re(w) +Re(β) +(c) △ = 3π /5 +(f) +△Φ = 3元/5 +0.05 +Im(w) +Im(β) +0 +-1 +-0.05F +9.8 +9.9 +10 +10.1 +0 +Re(w) +Re(β)5 +be observed. +Third, it merits further study to examine nonlinear ef- +fects, which should play a crucial role in the dynamically +unstable phase. 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Jackson, Classical electrodynamics (Wiley, New +York, 1999). + +Supplementary Material for “Non-Hermitian Physics of Levitated Nanoparticle +Array” +Kazuki Yokomizo1 and Yuto Ashida1, 2 +1Department of Physics, The University of Tokyo, +7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan +2Institute for Physics of Intelligence, University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan +EQUATION OF MOTION +We derive the equation of motion of a single levitated +nanoparticle, coupled two levitated nanoparticles, and an +array of levitated nanoparticles as shown in Figs. S1(a), +(b), and (c), respectively. We assume that the motion of +the particles along the plane perpendicular to the optical +axis is frozen. +Single levitated nanoparticle +We first focus on the motion of a single levitated +nanoparticle along the z axis [Fig. S1(a)]. +Let P and +λ denote the power and the wavelength of the trapping +laser, respectively. +In this system, the subwavelength- +FIG. S1. Schematic figure of a single levitated nanoparticle, +coupled two levitated nanoparticles, and an array of leviated +nanoparticles. In (a), (b), and (c), the mass of the particle is +m, and the trapping lasers have the power P and the wave- +length λ. In (b) and (c), the distance between the particles is +d0. In (c), we set the phase of the nth trapping laser in the +focal plane to be φ + n∆φ. +sized particle with the refractive index n is surrounded +by a medium with the refractive index n′, and it can be +regarded as a point dipole with the electric dipole mo- +ment given by +p (r, t) = αE (r, t) . +(S1) +Here, E (r, t) expresses the electric field of the trapping +laser, and α is the polarizability of the particle. +The +latter can explicitly be written as +α = 3ε0n′2V n2 +r − 1 +n2r + 2, +(S2) +where c is the speed of light in vacuum, V is the volume +of the particle, and nr ≡ n/n′ is the relative refractive +index of the particle. The electric field then acts on the +point dipole, and the particle feels the gradient force [1] +given by +Fgrad (r) = +�1 +2Re [(αE∗ (r, t) · ∇) E (r, t)] +� +T += +α +2cε0n′ ∇I (r) , +(S3) +where ⟨· · · ⟩T means time average, and I (r) is the spatial +profile of the intensity of the trapping laser. We note that +the trapping laser is described by the Gaussian beam, of +which the electric field is given by +E (r, t) = E0 +w0 +w (z) +× exp +� +− x2 + y2 +{w (z)}2 + ik x2 + y2 +2R (z) − iζ (z) +� +×ei(k0z−ωt), +(S4) +where +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +w (z) = w0 +� +1 + +� z +z0 +�2 +, +R (z) = z +� +1 + +�z0 +z +�2� +, +ζ (z) = tan−1 +� z +z0 +� +. +(S5) +Here, k0 (= 2π/λ) is the wavenumber of the trapping +laser, w0 is the beam waist, and z0 = πw2 +0/λ is the +arXiv:2301.05439v1 [physics.optics] 13 Jan 2023 + +(b) +(a) +m +m +m +op +之4 +2 +Particle 1 +Particle 2 +(c) +m +m +m +do +do +r +nth +(n-1)th +(n+1)th +Φ+(n- 1)△d +b+nAd +Φ+(n+1)b2 +Rayleigh length. The combination of Eqs. (S3) and (S4) +leads to the z component of the gradient force given by +Fgrad,z (z) = − +Pα +2πcε0n′w2 +0z2 +0 +z +� +1 + (z/z0)2�2 . +(S6) +In the vicinity of the focal plane of the trapping laser, +the linearized equation of the motion of the levitated +nanoparticle is obtained as +m¨z + mγ ˙z = −mΩ2z, +(S7) +where +Ω2 = +Pα +2πcε0n′mw2 +0z2 +0 +, +(S8) +and γ is a friction coefficient. We remark that the lin- +earized motion of the particle exhibits the harmonic os- +cillation because the gradient force plays a role as the +restoring force. +Coupled two levitated nanoparticles +We next consider the equation of motion of the coupled +two levitated nanoparticles [Fig. S1(b)]. We assume that +the polarization direction of the particles points to the y +axis. Then, in addition to the gradient force, both par- +ticles feel the interaction caused by the dipole radiation, +which is called the optical binding force [2]. +First of all, we shall explain the mechanism of the op- +tical binding force. Importantly, there exist two contri- +butions to the optical binding force. +The first contri- +bution results from the combination of the electric field +scattered from one particle to another particle and the +electric dipole induced by the trapping laser. The second +contribution originates from the acting of the trapping +laser on the electric dipole induced by the scattered field. +Specifically, for particle 1, the sum of these contributions +leads to the form of the optical binding force as follows: +F 2→1 +bind (r1, r2) += +�1 +2Re +� +(αE∗ (r1, t) · ∇r1) E2→1 +sca (r1, r2, t) +� ++1 +2Re +�� +αE2→1∗ +sca +(r1, r2, t) · ∇r1 +� +E (r1, t) +�� +T +. +(S9) +Here, E2→1 +sca (r1, r2, t) is the electric field scattered from +particle 2 to particle 1, and its form is written as +E2→1 +sca (r1, r2, t) = G (r1 − r2) αE (r2, t) , +(S10) +where G (r) called the Green’s tensor is the electric field +propagator between the two dipoles [3], and it is given +by +G (r) = eik0r +4πε0 +�3r ⊗ r − r2 +r5 +(1 − ik0r) + k2 +0 +r2 − r ⊗ r +r3 +� +. +(S11) +By using +Re +� +(∇r1 × E (r1, t)) × E2→1∗ +sca +(r1, r2, t) +� +−Re +� +E2→1 +sca (r1, r2, t) × (∇r1 × E∗ (r1, r2, t)) +� += Re +� +(E∗ (r1, t) · ∇r1) E2→1 +sca (r1, r2, t) +� ++Re +�� +E2→1∗ +sca +(r1, r2, t) · ∇r1 +� +E (r1, t) +� +−Re +� +∇r1 +� +E∗ (r1, t) · E2→1 +sca (r1, r2, t) +�� +(S12) +and ∇r1 × E (r1, t) = 0, we can rewrite Eq. (S9) to a +brief form given by +F 2→1 +bind (r1, r2) += +�1 +2∇r1Re [αE∗ (r1, t) G (r1 − r2) αE (r2, t)] +� +T +. +(S13) +In the following, we calculate the z component of the +optical binding force. In the far-field regime with k0d0 ≫ +1, the dominant contribution from the Green’s tensor +to the optical binding force is the term proportional to +1/r. Furthermore, in the vicinity of the focal plane, the +z dependence of the electric field is approximated as +E (rj, t) ≈ E0 exp +� +iφj + i +� +k0 − 1 +z0 +� +zj − iωt +� +(S14) +for j = 1, 2, where φj expresses the optical phase at +the focal plane. Substituting Eqs. (S11) and (S14) into +Eq. (S13), we can get the optical binding force along the +z axis as follows: +F 2→1 +bind,z (z1, z2) ≈ Pα2k3 +0 (k0 − 1/z0) +2π2cε2 +0n′w2 +0 +× sin +� +k0d0 − ∆φ − +� +k0 − 1 +z0 +� +(z1 − z2) +� +, +(S15) +where ∆φ ≡ φ1 − φ2. +We can similarly obtain the optical binding force for +particle 2 as follows: +F 1→2 +bind (r1, r2) += +�1 +2∇r2Re [αE∗ (r2, t) G (r2 − r1) αE (r1, t)] +� +T +. +(S16) +Therefore, the explicit form of the optical binding force +along the z axis is derived as +F 1→2 +bind,z (z1, z2) ≈ Pα2k3 +0 (k0 − 1/z0) +2π2cε2 +0n′w2 +0 +× sin +� +k0d0 + ∆φ + +� +k0 − 1 +z0 +� +(z1 − z2) +� +. +(S17) + +3 +Remarkably, the interaction between these two par- +ticles becomes nonreciprocal because F 2→1 +bind,z (z1, z2) ̸= +−F 1→2 +bind,z (z2, z1). The key ingredient of this nonreciproc- +ity is the interference between the trapping laser and +the scattered field. Let Φj (j = 1, 2) denote the optical +phase of the trapping laser at the position of the par- +ticle. The interference depends on the local phase dif- +ference ∆Φ ≡ Φ1 − Φ2 and the phase accumulation kd0 +which the scattered field acquires during the propagation. +Specifically, while the contribution of the interference is +kd0 − ∆Φ within the propagation of the scattered field +from particle 1 to particle 2, it becomes kd0 + ∆Φ in the +opposite case. +As a result, the interaction originating +from the interference becomes spatially asymmetric. +We can now obtain the linearized equation of motion +of the coupled levitated nanoparticles as follows: +� +m¨z1 + mγ ˙z1 = − +� +mΩ2 + K + ¯K +� +z1 + +� +K + ¯K +� +z2, +m¨z2 + mγ ˙z2 = − +� +mΩ2 + K − ¯K +� +z2 + +� +K − ¯K +� +z1, +(S18) +where Ω is the intrinsic mechanical frequency given by +Eq. (S8). The coupling constants are given by +� +� +� +� +� +� +� +K = +G +k0d0 +cos (k0d0) cos (∆φ) , +¯K = +G +k0d0 +sin (k0d0) sin (∆φ) , +(S19) +and +G = Pα2k3 +0 (k0 − 1/z0)2 +2π2cε2 +0n′w2 +0 +. +(S20) +Levitated nanoparticle array +We finally study the arrangement of the multiple lev- +itated nanoparticles at equal interval d0 [Fig. S1(c)] as +considered in the main text. In this system, the dipole- +dipole interaction among the several particles arises from +the multiple scattering of the trapping lasers. Neverthe- +less, the dominant contribution to the dynamics of the +system comes from the interaction between two parti- +cles. Thus, we neglect higher-order scattering processes. +This corresponds to approximating the optical binding +force up to O +� +|p|2� +. +We shall explain how one can derive the equation of +motion of the levitated nanoparticle array. We set the +optical phase of the nth trapping laser in the focal plane +to be φ + n∆φ. For the nth and n + lth particles, the +phase difference between the trapping lasers is l∆φ, and +the distance between the particles is ld0. Hence, the in- +teraction between these particles can be obtained by the +same procedure as explained above. +Due to the long- +range nature of the dipole-dipole interaction, it is neces- +sary to incorporate the couplings that reach up to Nth +neighbor particles. Then, the equation of motion of the +system is written as +m¨zn + mγ ˙zn = − +� +mΩ2 + 2 +N +� +l=1 +Kl +� +zn ++ +N +� +l=1 +�� +Kl + ¯Kl +� +zn−l + +� +Kl − ¯Kl +� +zn+l +� +, +(S21) +and +� +� +� +� +� +� +� +Kl = +G +lk0d0 +cos (lk0d0) cos (l∆φ) , +¯Kl = +G +lk0d0 +sin (lk0d0) sin (l∆φ) , +(S22) +where the constant G is given in Eq. (S20). +ARRAY OF FINITE LEVITATED +NANOPARTICLES +We investigate the array of a finite number of levi- +tated nanoparticles as shown in Fig. S2. +We assume +that this system includes only the interaction between the +nearest-neighbor particles. Furthermore, at both bound- +aries of the system, we arrange the deeply trapped lev- +itated nanoparticles, of which the motion in all the di- +rections is frozen, and we set the optical phase of the +trapping laser in the focal plane at the left and right +boundaries to be φ−∆φ and φ+(L + 1) ∆φ, respectively. +In this case, these particles are coupled with the system +so that they impose the fixed end boundary conditions +on the system. +Eigenvalue +We first show a way to calculate the eigenvalues of the +system described by +m¨zn + mγ ˙zn = − +� +mΩ2 + 2K +� +zn ++ +� +K + ¯K +� +zn−l + +� +K − ¯K +� +zn+l +(S23) +with the fixed boundary condition given by z0 = zL+1 = +0. In this equation, the coupling constants K and ¯K are +given by Eq. (S19). In the following, we suppose |K| ̸= +�� ¯K +��. By assuming zn = ψneiωt, Eq. (S23) is rewritten +into +1 +m +�� +K − ¯K +� +ψn+l +� +K + ¯K +� +ψn−l +� ++ +� +ω2 − iγω − Ω2 − 2 +mK +� +ψn = 0. +(S24) + +4 +FIG. S2. Schematic figure of the array of the L levitated nanoparticles. The distance between the nearest-neighbor particles +is d0, and the mass of all the particles is m. The trapping lasers have the power P and the wavelength λ. We set the phase +of the nth trapping laser in the focal plane to be φ + n∆φ. At both boundaries of the system, we arrange the two levitated +nanoparticles frozen in the motion in all the directions. The trapping lasers at the left and right boundaries have the optical +phase φ − ∆φ and φ + L∆φ in the focal plane, respectively. +From a general theory of a difference equation, we can +take +ψn = +2 +� +j=1 +(βj)n φ(j) +(S25) +as an ansatz of Eq. (S24). Here, βj (= β) is the solution +of the characteristic equation given by +1 +m +�� +K − ¯K +� +β + +� +K + ¯K +� +β−1� ++ +� +ω2 − iγω − Ω2 − 2 +mK +� += 0. +(S26) +We note that Eq. (S26) is a quadratic equation for β. +The boundary conditions, ψ0 = ψL+1 = 0, tell us the +condition that the combination coefficients φ(1) and φ(2) +take nonzero values, and it is written as +�β1 +β2 +�L+1 += 1. +(S27) +Then, we can obtain the explicit form of β1 and β2 from +Eqs. (S26) and (S27). When +� +K + ¯K +� � +K − ¯K +� +> 0, the +Vieta’s formula of Eq. (S26) gives +β1 = reiθl, β2 = re−iθl, +(S28) +where +r+ = +� +K + ¯K +K − ¯K , +(S29) +and +θl = +πl +N + 1 (l = 1, . . . , N) . +(S30) +Hence, the eigenvalues of the system can be calculated +as +ω> +l,± = i +2γ ± +� +Ω2 + 2 +m +� +K − +� +K2 − ¯K2 cos θl +� +− γ2 +4 . +(S31) +Similarly, when +� +K + ¯K +� � +K − ¯K +� +< 0, we obtain the +form of β1 and β2 as follows: +β1 = −ir′eiθl, β2 = −ir′e−iθl. +(S32) +Here, +r− = +����� +K + ¯K +K − ¯K +����, +(S33) +and θl is given by Eq. (S30). The eigenvalue of the system +in this case is written as +ω< +l,± = i +2γ± +� +Ω2 + 2 +m +� +K − i +���K2 − ¯K2�� cos θl +� +− γ2 +4 . +(S34) +Finite-size effect +In the main text, we investigate the levitated nanopar- +ticle array in the thermodynamic limit, L → ∞, and +obtain the dynamical phase diagram and the continuum +bands of the levitated nanoparticle array. With γ > 2Ω, +we recall that the phase diagram was obtained as shown +in Fig. S3. +We here discuss how finite-size effects can +affect the band structures in the critical phase and the +boundaries between the underdamped and dynamically +unstable phases. + +m +1st +Lth +nth +pvu+p +Φ+(L+ 1)Ad5 +FIG. S3. Dynamical phase diagram, continuum bands, and +eigenvalues of the levitated nanoparticle array. (a) Dynam- +ical phase diagram with Ω = 1 and γ = 5, which is the +same as Fig. 2(a) in the main text. +The blue, green, red, +and gray-shaded regions indicate the underdamped, critical, +overdamped, and the dynamically unstable phases, respec- +tively. (b) Continuum bands ˜ω+ (magenta) and ˜ω− (cyan), +and eigenvalues ω> +l,+ (red) and ω> +l,− (blue). (c) Continuum +band ˜ω+ (black), and eigenvalues ω< +l,+ (red). In (b) and (c), +we set L = 8 and choose the values of +� +K, ¯K +� +at the white +and black stars in (a), respectively. +To do so, we recall that the continuum bands of the +infinite-size system is given by +˜ω± = i +2γ + +� +Ω2 + 2 +m +� +K − +� +K2 − ¯K2 cos θ +� +− γ2 +4 . +(S35) +First, we consider the parameters indicated by the black +star in Fig. S3(a) and show the corresponding continuum +bands ˜ω± and the eigenvalues ω> +l,± in Fig. S3(b). One can +see that there is no degeneracy between ω> +l,+ and ω> +l,− in +a strict sense, simply because θl takes only the discrete +values determined by Eq. (S30). Nevertheless, we empha- +size that the eigenvectors around the spectral singularity, +where ˜ω+ touches ˜ω−, still exhibit the strong nonorthog- +onality which is a hallmark of the non-Hermitian degen- +eracy. Next, we consider the parameters indicated by the +white star in Fig. S3(a) and show the corresponding con- +tinuum band ˜ω+ and the eigenvalues ω< +l,+ in Fig. S3(c). +It is found that the finite-size system does not exhibit the +dynamical instability because the imaginary parts of all +the discrete eigenvalues become positive. In this sense, +the finite-size effect can slightly modify the boundary +between the dynamically unstable phase and the other +phases. One can also infer from Fig. S3(c) that a larger +system size would be favorable to observe the dynamical +instability. Nevertheless, we emphasize that the phase +diagram of a finite-size system still remains qualitatively +the same as in the result obtained in the thermodynamic +limit. +[S1] Y. Harada and T. Asakura, Opt. Commun. 124, 529 +(1996). +[S2] F. Dapasse and J.-M. Vigoureux, J. Phys. D 27, 914 +(1994). +[S3] J. D. Jackson, Classical electrodynamics (Wiley, New +York, 1999). + +10 +(a) +★ (c) +5 +(b) +K +0 +m +-5 +10 +-5 +10 +0 +5 +10 +K +m +(b)(K, K) = (5,3) +(c)(K, K) = (3, 7.5 +5 +3 +3 +(3) +3) +1 +1 +0 +0 +-0.5 +-3 +0 +3 +0 +1 +2 +3 +Re(w) +Re(w) \ No newline at end of file diff --git a/l9E5T4oBgHgl3EQfHQ75/content/tmp_files/load_file.txt b/l9E5T4oBgHgl3EQfHQ75/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ee6d076032e816de221e81476b17ae1c37a7ef6a --- /dev/null +++ b/l9E5T4oBgHgl3EQfHQ75/content/tmp_files/load_file.txt @@ -0,0 +1,1091 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf,len=1090 +page_content='Non-Hermitian Physics of Levitated Nanoparticle Array Kazuki Yokomizo1 and Yuto Ashida1, 2 1Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan 2Institute for Physics of Intelligence, University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan The ability to control levitated nanoparticles allows one to explore various fields of physics, in- cluding quantum optics, quantum metrology, and nonequilibrium physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' It has been recently demonstrated that the arrangement of two levitated nanoparticles naturally realizes the tunable nonreciprocal dipole-dipole interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Motivated by this development, we here propose and ana- lyze an array of levitated nanoparticles as an ideal platform to study non-Hermitian physics in a highly controlled manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We employ the non-Bloch band theory to determine the continuum bands of the proposed setup and investigate the non-Hermitian skin effect therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In particular, we point out that the levitated nanoparticle array exhibits rich dynamical phases, including the dynamically unstable phase and the unconventional critical phase where the spectral singularity persists over a broad region of the controllable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We also show that the long-range nature of the dipole-dipole interaction gives rise to the unique self-crossing point of the continuum band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' A levitated nanoparticle is a laser trapped nanoscale dielectric particle smaller than wavelength of light [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Recent experimental developments have allowed one to cool a levitated nanoparticle to ultracold tempera- tures [2–8] and offered unique opportunities to study quantum mechanics of mesoscopic objects [9–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Addi- tionally, previous studies demonstrated the potential of a levitated nanoparticle to explore various fields of physics, such as nonequilibrium physics [21–27] and quantum sensing [28–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Remarkably, recent experimental stud- ies have shown the possibility of extending these sys- tems to multi-nanoparticle setups [38–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In particular, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' [43] has reported a realization of an on-demand as- sembly of levitated nanoparticles, in which optical tweez- ers are used to trap and arrange the nanoparticles one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' On another front, recent years have witnessed remark- able advances in our understandings of non-Hermitian systems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=', a class of nonequilibrium systems that can be effectively described by a non-Hermitian operator [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' While non-Hermitian physics has been widely investi- gated in several fields of quantum science, such as ul- tracold atoms [45–49] and photonics [50–53], its idea has also found numerous applications in classical systems re- alized in optics [54–57], mechanics [58–61], and electrical circuits [62–65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' These previous studies uncovered rich non-Hermitian phenomena that have no counterparts to Hermitian systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' For instance, a one-dimensional (1D) tight-binding model with asymmetric hopping am- plitudes exhibits the non-Hermitian skin effect [66–68], where the bulk eigenstates are localized at open bound- aries of the system, leading to the extreme boundary sen- sitivity of the eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In this Letter, we propose and analyze a 1D levitated nanoparticle array as an ideal platform to study previ- ously unexplored regimes of non-Hermitian physics in a highly controlled manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' A prominent feature here is that there exists the tunable nonreciprocal dipole-dipole interaction between levitated nanoparticles, which is in- duced by the nonreciprocal interference originating from phase difference between the trapping lasers [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The proposed system then realizes a 1D tight-binding model with arbitrarily tunable asymmetric hopping amplitudes that have possibly negative signs and long-range depen- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' This high controllability allows one to explore the whole parameter region of non-Hermitian systems, thus opening the possibility to uncover the full potential of non-Hermitian phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In this respect, the proposed setup should be contrasted to the previous non-Hermitian platforms, where studies were restricted to severely lim- ited parameter regions of the models due to the difficul- ties of realizing tunable nonreciprocal interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' To determine the continuum bands and the dynamical phase diagram of the levitated nanoparticle array, we in- voke the non-Bloch band theory [66, 69–73], a recently developed powerful tool to investigate models featuring the non-Hermitian skin effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The non-Bloch band the- ory allows for calculating the asymptotic eigenvalues of the systems with open boundary conditions in the limit of a large system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' This makes contrast to the con- ventional Bloch band theory, where the energy band re- produces the eigenvalues under periodic boundary con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' On the basis of this theoretical framework, we find that the levitated nanoparticle array exhibits rich dynamical phases, including the unconventional critical phase and the dynamically unstable phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In the former, a remark- able feature is that the non-Hermitian degeneracy of the continuum bands known as the spectral singularity ap- pears without fine-tuning and persists over a broad region of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The key ingredients of the latter are negative interparticle couplings, which were difficult to realize in the existing non-Hermitian platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' More- over, the proposed system can naturally realize the long- range hopping amplitudes originating from the dipole- dipole interaction provided that the particle distance is arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='05439v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='optics] 13 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Schematic figure of the levitated nanoparticle array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The distance between the nearest-neighbor particles is d0, and the mass of all the particles is m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' All the trapping lasers have the power P and the wavelength λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We set the phase of the nth trapping laser in the focal plane to be φ + n∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' judiciously controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We show that this long-range na- ture leads to the unique self-crossing point of the contin- uum band, which corresponds to the singularity of the generalized Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' To be concrete, we consider a 1D array of the trapped levitated nanoparticles as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The particles are equally spaced at the interval d0, and all the particles have the mass m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Let λ and P denote the wavelength and the power of all the trapping lasers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Furthermore, we assume that the motion of the particles along the plane perpendicular to the optical axis is frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The interaction between the two particles arises due to the interference between the scattered electromagnetic field and the trapping laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Since the scattered field ac- quires the phase kd0 during the propagation, the phase difference between the trapping lasers at the positions of the particles leads to the constructive and destruc- tive interference depending on the propagation direction of the scattered field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' It is this spatial asymmetry that renders the interparticle coupling nonreciprocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Due to the long-range nature of this nonreciprocal dipole-dipole interaction, it is in general necessary to incorporate the couplings that reach up to Nth neighbor particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Al- together, in the vicinity of the focal plane, the linearized equation of motion of the nth particle along the z axis is given by m¨zn + mγ ˙zn = − � mΩ2 + 2 N � l=1 Kl � zn + N � l=1 �� Kl + ¯Kl � zn−l + � Kl − ¯Kl � zn+l � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (1) Here, Ω is an intrinsic mechanical frequency of the par- ticle proportional to √ P, γ is a friction coefficient, and Kl and ¯ Kl are the coupling strengths given by � � � � � � � Kl = G lk0d0 cos (lk0d0) cos (l∆φ) , ¯Kl = G lk0d0 sin (lk0d0) sin (l∆φ) , (2) where G has the dimension of a spring constant and is proportional to P, ∆φ is the optical phase difference be- tween the neighbor trapping lasers in the focal plane [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 1], and k0 (= 2π/λ) is a wavenumber of the trap- ping laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' One can infer from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (2) that the couplings are long-range because the dipole-dipole interaction is proportional to the inverse of the distance between the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We note that the sign of the coupling con- stants Kl and ¯Kl can be controlled by changing the phase difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We provide the derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (1) in the Supplementary Material [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In general, continuum bands of non-Hermitian tight- binding models can be obtained by invoking the non- Bloch band theory, which reproduces the asymptotic eigenvalues under open boundary conditions in the ther- modynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Specifically, the continuum band is calculated from the generalized Brillouin zone spanned by β ≡ eik for a complex Bloch wavenumber k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We here apply the non-Bloch band theory to the levitated nanoparticle array;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' throughout this paper, we assume |KN| ̸= �� ¯KN ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Substituting zn = ψneiωt to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (1), we have the real-space eigenequation as follows: 1 m N � l=1 �� Kl − ¯Kl � ψn+l � Kl + ¯Kl � ψn−l � + � ω2 − iγω − Ω2 − 2 m N � l=1 Kl � ψn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (3) Importantly, an ansatz of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (3) can be taken as ψn = 2N � j=1 (βj)n φ(j), (4) where βj (= β) is the solution of the characteristic equa- tion given by 1 m N � l=1 �� Kl − ¯Kl � βl + � Kl + ¯Kl � β−l� + � ω2 − iγω − Ω2 − 2 m N � l=1 Kl � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (5) We note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (5) is an algebraic equation for β of 2Nth degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The main result of the non-Bloch band theory is that the condition for the generalized Brillouin zone is obtained from the 2N solutions, and it is given by |βN| = |βN+1| (6) m m m do do r 2 nth (n-1)th (n+1)th Φ+ (n- 1) △d Φ+nAd Φ+ (n+ 1) △d3 with |β1| ≤ · · · ≤ |β2N|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The trajectories of βN and βN+1 form the generalized Brillouin zone on the complex plane, which reveals the essential features of non-Hermitian sys- tems (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' [75–78]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Then, we can calculate the continuum bands by combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (5) with the gener- alized Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We note that when ¯Kl = 0, the generalized Brillouin zone reduces to a unit circle, which means that the Bloch wavenumber becomes real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We start our analysis from the levitated nanoparticle array with the nearest-neighbor interaction, which cor- responds to N = 1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' in the following, we as- sume γ > 2Ω for the sake of concreteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (6), the generalized Brillouin zone can be given by the cir- cle with the radius r = ���� K1 + ¯K1 � / � K1 − ¯K1 ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We then have the analytical form of the continuum bands as follows: ω± = i 2γ ± � Ω2 + 2 m � K1 − � K2 1 − ¯K2 1 cos θ � − γ2 4 , (7) where θ is a real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Since each eigen- mode contributes to the dynamics through the factor e−Im(ω±)eiRe(ω±), we can show the dynamical phase di- agram of the system depending on K1/m and ¯K1/m in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 2(b)–(d) and (e)–(g) plot the evolu- tions of the continuum bands along the black and white arrows indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 2(a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In the blue-shaded regions of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 2(a), all the particles oscillate with the attenuation because Re (ω±) ̸= 0 and Im (ω±) > 0 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 2(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In contrast, in the red-shaded re- gions, their motion monotonically vanishes without oscil- lations because Re (ω±) = 0 and Im (ω±) > 0 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 2(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' For these reasons, we term the former (latter) the dy- namical phase as the underdamped (overdamped) phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Remarkably, we find the broad green-shaded region where the two branches ω+ and ω− coalesce at Re (ω±) = 0 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 2(c)], leading to the crossover dynamics between the above two dynamical phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Such a degenerate point unique to non-Hermitian bands is known as the spectral singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We shall term this intermediate regime as the critical phase in the sense that the overdamped be- havior eventually sets in after the initial underdamped oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Importantly, the emergence of this critical phase is unique to the present setup with open bound- aries because the spectral singularity disappears under periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Indeed, after replacing β by eik (k ∈ R) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (5), one obtains the band, which re- produces the eigenvalues under periodic boundary condi- tions, and it is qualitatively different from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Thus, the transient phenomena discussed here is supported by the non-Hermitian degeneracy, and the non-Hermitian skin effect plays an essential role to realize the aforemen- tioned critical phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We note that, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 2(f), the spectral singularity also appears along the green ver- tical lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' However, one would need fine- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Dynamical phase diagram and continuum bands of the levitated nanoparticle array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (a) Dynamical phase dia- gram exhibiting the underdamped, critical, overdamped, and dynamically unstable phases shown in the blue, green, red, and gray-shaded regions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The spectral singular- ity (SS) appears in the green-shaded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' On the black dashed lines, |K1| = �� ¯K1 �� is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We set the parameters to be Ω = 1 and γ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (b)–(g) Evolutions of the continuum bands along the arrows in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The magenta and cyan express ω+ and ω−, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The numerical values in each panel specify � K1, ¯K1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' tuning of the parameters in this case as indicated by Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 2(e)–(g), where the two continuum bands are re- combined across the green line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In addition to the above phases, we also find the dy- 10 (a) (b)" 5 (c) (d) K1 0 m (g) (f) 5 (e) 105 _10 5 0 5 10 K1 m (b) (6,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='5) (c) (4,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='5) (d) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='8,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='5) 5 5 5 SS 3 3 Im(w) Im(w) 1F 1 1 0 0 0 3 3 3 0 3 3 0 3 Re(w) Re(w) Re(w) (e) (5,-6) (f) (21/8,-29/8) (g) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='8,-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='8) 5 r 5 5 SS 3 Im(w) 1 入 1 0L 0 0 3 0 3 3 0 3 3 0 3 Re(w) Re(w) Re(w)4 namically unstable phase as indicated by the gray-shaded region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' There, either of the hopping ampli- tudes, K1 ± ¯K1, becomes negative, and the oscillation amplitudes diverges in the long-time limit because the imaginary part of the eigenvalues can take negative val- ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Physically, this instability originates from the fact that the negative hopping amplitudes cause the force that increasingly keeps away the particles from their equilib- rium positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We emphasize that the dynamically un- stable phase discussed here is rather difficult to realize in the previous non-Hermitian systems due to the lack of the ability to implement tunable negative coupling strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' It is worthwhile to mention that, in finite-size systems, the boundary between the overdamped and dynamically unstable phases can be slightly modified [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We next investigate how the long-range nature of the couplings can affect the continuum band and the cor- responding generalized Brillouin zone of the levitated nanoparticle array;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' in the following, we neglect the fric- tion for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' To this end, we as- sume that the interaction reaches up to the next-nearest- neighbor particles, which corresponds to N = 2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 3, we plot the continuum bands with the positive branch of the square root and the corresponding generalized Brillouin zone at different ∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We note that the black dashed curves in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 3(d)–(f) indicate the conventional Brillouin zone formed by β ≡ eik (k ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' One can see from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 3(d) and (f) that the general- ized Brillouin zone with N = 2 forms a skewed closed curve with the cusps, at which it becomes indifferen- tiable, while the generalized Brillouin zone with N = 1 is merely a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Importantly, the cusps correspond to the self-crossing points of the continuum band [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 3(a) and (c)] [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Thus, the long-range nature of the nonrecip- rocal interaction can lead to these unconventional band structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Meanwhile, at ∆φ = π/2, the generalized Brillouin zone becomes the unit circle independently of N as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 3(e), where the non-Hermitian skin effect disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Accordingly, there are no self-crossing points of the continuum band as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In summary, we propose and analyze the levitated nanoparticle array as an ideal platform to study new realms of non-Hermitian physics in a highly controlled manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We show that the system exhibits the uncon- ventional critical phase, where the spectral singularity originating from the non-Hermitian skin effect persists over a broad region of the controllable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We also point out that the tunable dipole-dipole nonrecipro- cal interaction in the proposed setup allows for extremely nonreciprocal hopping amplitudes with possibly negative signs, which result in the dynamical instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We fi- nally reveal that the long-range nature of the nonrecip- rocal couplings further enriches the non-Hermitian band structures, leading to the cusps of the generalized Bril- louin zone and the self-crossing points of the continuum band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Continuum bands and generalized Brillouin zones of the levitated nanoparticle array at different ∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (a)–(c) The continuum bands with the positive branch of the square root, and (d)–(f) the corresponding generalized Brillouin zones are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The red (blue) curves indicate the results for N = 1 (N = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In (d)–(f), the black dashed curve expresses the conventional Brillouin zone spanned by β ≡ eik (k ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The system parameters are set to be λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='064 × 10−6 m, d0 = 10−5 m, Ω = 10 s−1, and G/ (mkd0) = 1 s−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Several open questions remain for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' First, in a levitated nanoparticle array, the continuum bands can be experimentally observed by measuring the power spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Hence, it should be possible to directly observe the spectral singularity and the correspondence between the cusps of the generalized Brillouin zone and the self-crossing points of the continuum band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Second, besides the model discussed here, a levitated nanoparticle array allows one to realize various non- Hermitian tight-binding models with arbitrary param- eters, thanks to its high controllability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' For instance, it provides an ideal setup to realize the non-Hermitian Su-Schrieffer-Heeger model [66], where rich phenomena including a topological phase transition are expected to (a) △Φ = π /10 (d) △Φ = π /10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='05 1 Im(β) (3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='9 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='1 1 0 Re(w) Re(β) (b) △Φ = /2 (e) △= /2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='05 1 (m)wl (d)ul 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='9 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='1 0 Re(w) Re(β) (c) △ = 3π /5 (f) △Φ = 3元/5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='05 Im(w) Im(β) 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='05F 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='9 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='1 0 Re(w) Re(β)5 be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Third, it merits further study to examine nonlinear ef- fects, which should play a crucial role in the dynamically unstable phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' While the asymmetric interaction can exponentially amplify the oscillation strengths in short- time regimes, we expect that this amplification is even- tually balanced by the nonlinear suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Finally, it is interesting to extend the present analysis to quan- tum regimes of the levitated nanoparticle array, which should be within the experimental reach in view of re- cent developments of cooling levitated nanoparticles to ultracold temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In particular, understanding a role of quantum correlation and coherence between levi- tated nanoparticles in the array remains as an intriguing open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' was supported by JSPS KAKENHI through Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' JP21J01409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' acknowledges support from the Japan Society for the Promotion of Science through Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' JP19K23424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Ashkin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' A 99, 051401 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Kamba, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Kiuchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Yotsuya, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Aikawa, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' A 103, L051701 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Kamba, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Shimizu, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Aikawa, Opt.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Ye, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Painter, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Kimble, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Zoller, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 107, 1005 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' [10] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' F.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Juan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Quidant, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Kiesel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Aspelmeyer, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Cirac, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 107, 020405 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' [15] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Tebbenjohanns, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Frimmer, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Jain, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Windey, and L.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Hebestreit, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Diehl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Frimmer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Windey, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Tebbenjohanns, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Novotny, Phys.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Li, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Zhang, arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='03641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' [44] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Ashida, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Gong, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Gad- way, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Yi, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Yan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 124, 070402 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Dutt, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Wojcik, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Vuˇckovi´c, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Fan, Science 371, 1240 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Murakami, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Research 2, 043045 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' [80] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Dapasse and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Vigoureux, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' D 27, 914 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' [81] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Harada and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Asakura, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 124, 529 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' [82] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Jackson, Classical electrodynamics (Wiley, New York, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Supplementary Material for “Non-Hermitian Physics of Levitated Nanoparticle Array” Kazuki Yokomizo1 and Yuto Ashida1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 2 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 7-3-1 Hongo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Bunkyo-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 113-0033,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Japan 2Institute for Physics of Intelligence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 7-3-1 Hongo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Tokyo 113-0033,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Japan EQUATION OF MOTION We derive the equation of motion of a single levitated nanoparticle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' coupled two levitated nanoparticles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' and an array of levitated nanoparticles as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' S1(a), (b), and (c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We assume that the motion of the particles along the plane perpendicular to the optical axis is frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Single levitated nanoparticle We first focus on the motion of a single levitated nanoparticle along the z axis [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' S1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Let P and λ denote the power and the wavelength of the trapping laser, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In this system, the subwavelength- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Schematic figure of a single levitated nanoparticle, coupled two levitated nanoparticles, and an array of leviated nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In (a), (b), and (c), the mass of the particle is m, and the trapping lasers have the power P and the wave- length λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In (b) and (c), the distance between the particles is d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In (c), we set the phase of the nth trapping laser in the focal plane to be φ + n∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' sized particle with the refractive index n is surrounded by a medium with the refractive index n′, and it can be regarded as a point dipole with the electric dipole mo- ment given by p (r, t) = αE (r, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S1) Here, E (r, t) expresses the electric field of the trapping laser, and α is the polarizability of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The latter can explicitly be written as α = 3ε0n′2V n2 r − 1 n2r + 2, (S2) where c is the speed of light in vacuum, V is the volume of the particle, and nr ≡ n/n′ is the relative refractive index of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The electric field then acts on the point dipole, and the particle feels the gradient force [1] given by Fgrad (r) = �1 2Re [(αE∗ (r, t) · ∇) E (r, t)] � T = α 2cε0n′ ∇I (r) , (S3) where ⟨· · · ⟩T means time average, and I (r) is the spatial profile of the intensity of the trapping laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We note that the trapping laser is described by the Gaussian beam, of which the electric field is given by E (r, t) = E0 w0 w (z) × exp � − x2 + y2 {w (z)}2 + ik x2 + y2 2R (z) − iζ (z) � ×ei(k0z−ωt), (S4) where � � � � � � � � � � � � � � � � � � � w (z) = w0 � 1 + � z z0 �2 , R (z) = z � 1 + �z0 z �2� , ζ (z) = tan−1 � z z0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S5) Here, k0 (= 2π/λ) is the wavenumber of the trapping laser, w0 is the beam waist, and z0 = πw2 0/λ is the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='05439v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='optics] 13 Jan 2023 (b) (a) m m m op 之4 2 Particle 1 Particle 2 (c) m m m do do r nth (n-1)th (n+1)th Φ+(n- 1)△d b+nAd Φ+(n+1)b2 Rayleigh length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The combination of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S3) and (S4) leads to the z component of the gradient force given by Fgrad,z (z) = − Pα 2πcε0n′w2 0z2 0 z � 1 + (z/z0)2�2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S6) In the vicinity of the focal plane of the trapping laser, the linearized equation of the motion of the levitated nanoparticle is obtained as m¨z + mγ ˙z = −mΩ2z, (S7) where Ω2 = Pα 2πcε0n′mw2 0z2 0 , (S8) and γ is a friction coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We remark that the lin- earized motion of the particle exhibits the harmonic os- cillation because the gradient force plays a role as the restoring force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Coupled two levitated nanoparticles We next consider the equation of motion of the coupled two levitated nanoparticles [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' S1(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We assume that the polarization direction of the particles points to the y axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Then, in addition to the gradient force, both par- ticles feel the interaction caused by the dipole radiation, which is called the optical binding force [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' First of all, we shall explain the mechanism of the op- tical binding force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Importantly, there exist two contri- butions to the optical binding force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The first contri- bution results from the combination of the electric field scattered from one particle to another particle and the electric dipole induced by the trapping laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The second contribution originates from the acting of the trapping laser on the electric dipole induced by the scattered field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Specifically, for particle 1, the sum of these contributions leads to the form of the optical binding force as follows: F 2→1 bind (r1, r2) = �1 2Re � (αE∗ (r1, t) · ∇r1) E2→1 sca (r1, r2, t) � +1 2Re �� αE2→1∗ sca (r1, r2, t) · ∇r1 � E (r1, t) �� T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S9) Here, E2→1 sca (r1, r2, t) is the electric field scattered from particle 2 to particle 1, and its form is written as E2→1 sca (r1, r2, t) = G (r1 − r2) αE (r2, t) , (S10) where G (r) called the Green’s tensor is the electric field propagator between the two dipoles [3], and it is given by G (r) = eik0r 4πε0 �3r ⊗ r − r2 r5 (1 − ik0r) + k2 0 r2 − r ⊗ r r3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S11) By using Re � (∇r1 × E (r1, t)) × E2→1∗ sca (r1, r2, t) � −Re � E2→1 sca (r1, r2, t) × (∇r1 × E∗ (r1, r2, t)) � = Re � (E∗ (r1, t) · ∇r1) E2→1 sca (r1, r2, t) � +Re �� E2→1∗ sca (r1, r2, t) · ∇r1 � E (r1, t) � −Re � ∇r1 � E∗ (r1, t) · E2→1 sca (r1, r2, t) �� (S12) and ∇r1 × E (r1, t) = 0, we can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S9) to a brief form given by F 2→1 bind (r1, r2) = �1 2∇r1Re [αE∗ (r1, t) G (r1 − r2) αE (r2, t)] � T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S13) In the following, we calculate the z component of the optical binding force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In the far-field regime with k0d0 ≫ 1, the dominant contribution from the Green’s tensor to the optical binding force is the term proportional to 1/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Furthermore, in the vicinity of the focal plane, the z dependence of the electric field is approximated as E (rj, t) ≈ E0 exp � iφj + i � k0 − 1 z0 � zj − iωt � (S14) for j = 1, 2, where φj expresses the optical phase at the focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S11) and (S14) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S13), we can get the optical binding force along the z axis as follows: F 2→1 bind,z (z1, z2) ≈ Pα2k3 0 (k0 − 1/z0) 2π2cε2 0n′w2 0 × sin � k0d0 − ∆φ − � k0 − 1 z0 � (z1 − z2) � , (S15) where ∆φ ≡ φ1 − φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We can similarly obtain the optical binding force for particle 2 as follows: F 1→2 bind (r1, r2) = �1 2∇r2Re [αE∗ (r2, t) G (r2 − r1) αE (r1, t)] � T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S16) Therefore, the explicit form of the optical binding force along the z axis is derived as F 1→2 bind,z (z1, z2) ≈ Pα2k3 0 (k0 − 1/z0) 2π2cε2 0n′w2 0 × sin � k0d0 + ∆φ + � k0 − 1 z0 � (z1 − z2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S17) 3 Remarkably, the interaction between these two par- ticles becomes nonreciprocal because F 2→1 bind,z (z1, z2) ̸= −F 1→2 bind,z (z2, z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The key ingredient of this nonreciproc- ity is the interference between the trapping laser and the scattered field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Let Φj (j = 1, 2) denote the optical phase of the trapping laser at the position of the par- ticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The interference depends on the local phase dif- ference ∆Φ ≡ Φ1 − Φ2 and the phase accumulation kd0 which the scattered field acquires during the propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Specifically, while the contribution of the interference is kd0 − ∆Φ within the propagation of the scattered field from particle 1 to particle 2, it becomes kd0 + ∆Φ in the opposite case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' As a result, the interaction originating from the interference becomes spatially asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We can now obtain the linearized equation of motion of the coupled levitated nanoparticles as follows: � m¨z1 + mγ ˙z1 = − � mΩ2 + K + ¯K � z1 + � K + ¯K � z2, m¨z2 + mγ ˙z2 = − � mΩ2 + K − ¯K � z2 + � K − ¯K � z1, (S18) where Ω is the intrinsic mechanical frequency given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The coupling constants are given by � � � � � � � K = G k0d0 cos (k0d0) cos (∆φ) , ¯K = G k0d0 sin (k0d0) sin (∆φ) , (S19) and G = Pα2k3 0 (k0 − 1/z0)2 2π2cε2 0n′w2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S20) Levitated nanoparticle array We finally study the arrangement of the multiple lev- itated nanoparticles at equal interval d0 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' S1(c)] as considered in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In this system, the dipole- dipole interaction among the several particles arises from the multiple scattering of the trapping lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Neverthe- less, the dominant contribution to the dynamics of the system comes from the interaction between two parti- cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Thus, we neglect higher-order scattering processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' This corresponds to approximating the optical binding force up to O � |p|2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We shall explain how one can derive the equation of motion of the levitated nanoparticle array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We set the optical phase of the nth trapping laser in the focal plane to be φ + n∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' For the nth and n + lth particles, the phase difference between the trapping lasers is l∆φ, and the distance between the particles is ld0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Hence, the in- teraction between these particles can be obtained by the same procedure as explained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Due to the long- range nature of the dipole-dipole interaction, it is neces- sary to incorporate the couplings that reach up to Nth neighbor particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Then, the equation of motion of the system is written as m¨zn + mγ ˙zn = − � mΩ2 + 2 N � l=1 Kl � zn + N � l=1 �� Kl + ¯Kl � zn−l + � Kl − ¯Kl � zn+l � , (S21) and � � � � � � � Kl = G lk0d0 cos (lk0d0) cos (l∆φ) , ¯Kl = G lk0d0 sin (lk0d0) sin (l∆φ) , (S22) where the constant G is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' ARRAY OF FINITE LEVITATED NANOPARTICLES We investigate the array of a finite number of levi- tated nanoparticles as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We assume that this system includes only the interaction between the nearest-neighbor particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Furthermore, at both bound- aries of the system, we arrange the deeply trapped lev- itated nanoparticles, of which the motion in all the di- rections is frozen, and we set the optical phase of the trapping laser in the focal plane at the left and right boundaries to be φ−∆φ and φ+(L + 1) ∆φ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In this case, these particles are coupled with the system so that they impose the fixed end boundary conditions on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Eigenvalue We first show a way to calculate the eigenvalues of the system described by m¨zn + mγ ˙zn = − � mΩ2 + 2K � zn + � K + ¯K � zn−l + � K − ¯K � zn+l (S23) with the fixed boundary condition given by z0 = zL+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In this equation, the coupling constants K and ¯K are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In the following, we suppose |K| ̸= �� ¯K ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' By assuming zn = ψneiωt, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S23) is rewritten into 1 m �� K − ¯K � ψn+l � K + ¯K � ψn−l � + � ω2 − iγω − Ω2 − 2 mK � ψn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S24) 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Schematic figure of the array of the L levitated nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The distance between the nearest-neighbor particles is d0, and the mass of all the particles is m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The trapping lasers have the power P and the wavelength λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We set the phase of the nth trapping laser in the focal plane to be φ + n∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' At both boundaries of the system, we arrange the two levitated nanoparticles frozen in the motion in all the directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The trapping lasers at the left and right boundaries have the optical phase φ − ∆φ and φ + L∆φ in the focal plane, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' From a general theory of a difference equation, we can take ψn = 2 � j=1 (βj)n φ(j) (S25) as an ansatz of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Here, βj (= β) is the solution of the characteristic equation given by 1 m �� K − ¯K � β + � K + ¯K � β−1� + � ω2 − iγω − Ω2 − 2 mK � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S26) We note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S26) is a quadratic equation for β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The boundary conditions, ψ0 = ψL+1 = 0, tell us the condition that the combination coefficients φ(1) and φ(2) take nonzero values, and it is written as �β1 β2 �L+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S27) Then, we can obtain the explicit form of β1 and β2 from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S26) and (S27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' When � K + ¯K � � K − ¯K � > 0, the Vieta’s formula of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S26) gives β1 = reiθl, β2 = re−iθl, (S28) where r+ = � K + ¯K K − ¯K , (S29) and θl = πl N + 1 (l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' , N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S30) Hence, the eigenvalues of the system can be calculated as ω> l,± = i 2γ ± � Ω2 + 2 m � K − � K2 − ¯K2 cos θl � − γ2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S31) Similarly, when � K + ¯K � � K − ¯K � < 0, we obtain the form of β1 and β2 as follows: β1 = −ir′eiθl, β2 = −ir′e−iθl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S32) Here, r− = ����� K + ¯K K − ¯K ����, (S33) and θl is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The eigenvalue of the system in this case is written as ω< l,± = i 2γ± � Ω2 + 2 m � K − i ���K2 − ¯K2�� cos θl � − γ2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S34) Finite-size effect In the main text, we investigate the levitated nanopar- ticle array in the thermodynamic limit, L → ∞, and obtain the dynamical phase diagram and the continuum bands of the levitated nanoparticle array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' With γ > 2Ω, we recall that the phase diagram was obtained as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' We here discuss how finite-size effects can affect the band structures in the critical phase and the boundaries between the underdamped and dynamically unstable phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' m 1st Lth nth pvu+p Φ+(L+ 1)Ad5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Dynamical phase diagram, continuum bands, and eigenvalues of the levitated nanoparticle array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (a) Dynam- ical phase diagram with Ω = 1 and γ = 5, which is the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 2(a) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' The blue, green, red, and gray-shaded regions indicate the underdamped, critical, overdamped, and the dynamically unstable phases, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (b) Continuum bands ˜ω+ (magenta) and ˜ω− (cyan), and eigenvalues ω> l,+ (red) and ω> l,− (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (c) Continuum band ˜ω+ (black), and eigenvalues ω< l,+ (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In (b) and (c), we set L = 8 and choose the values of � K, ¯K � at the white and black stars in (a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' To do so, we recall that the continuum bands of the infinite-size system is given by ˜ω± = i 2γ + � Ω2 + 2 m � K − � K2 − ¯K2 cos θ � − γ2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S35) First, we consider the parameters indicated by the black star in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' S3(a) and show the corresponding continuum bands ˜ω± and the eigenvalues ω> l,± in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' S3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' One can see that there is no degeneracy between ω> l,+ and ω> l,− in a strict sense, simply because θl takes only the discrete values determined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' (S30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Nevertheless, we empha- size that the eigenvectors around the spectral singularity, where ˜ω+ touches ˜ω−, still exhibit the strong nonorthog- onality which is a hallmark of the non-Hermitian degen- eracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Next, we consider the parameters indicated by the white star in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' S3(a) and show the corresponding con- tinuum band ˜ω+ and the eigenvalues ω< l,+ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' S3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' It is found that the finite-size system does not exhibit the dynamical instability because the imaginary parts of all the discrete eigenvalues become positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' In this sense, the finite-size effect can slightly modify the boundary between the dynamically unstable phase and the other phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' One can also infer from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' S3(c) that a larger system size would be favorable to observe the dynamical instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Nevertheless, we emphasize that the phase diagram of a finite-size system still remains qualitatively the same as in the result obtained in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' [S1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Harada and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Asakura, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 124, 529 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' [S2] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Dapasse and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Vigoureux, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' D 27, 914 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' [S3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' Jackson, Classical electrodynamics (Wiley, New York, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content=' 10 (a) ★ (c) 5 (b) K 0 m 5 10 5 10 0 5 10 K m (b)(K, K) = (5,3) (c)(K, K) = (3, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='5 5 3 3 (3) 3) 1 1 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} +page_content='5 3 0 3 0 1 2 3 Re(w) Re(w)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9E5T4oBgHgl3EQfHQ75/content/2301.05439v1.pdf'} diff --git a/ltE0T4oBgHgl3EQfZAA9/vector_store/index.pkl b/ltE0T4oBgHgl3EQfZAA9/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..e9b8c61371f0031de1b5fd0f153de53a0c33271c --- /dev/null +++ b/ltE0T4oBgHgl3EQfZAA9/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3b04488c779406d023f43d6c7635ba2b1c10966b75ea4084d091e5e25dc8b26c +size 812398 diff --git a/mNE0T4oBgHgl3EQfZADU/content/tmp_files/2301.02316v1.pdf.txt b/mNE0T4oBgHgl3EQfZADU/content/tmp_files/2301.02316v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4f441a905ec518b8a13c0e9ffe5571dc31abd73f --- /dev/null +++ b/mNE0T4oBgHgl3EQfZADU/content/tmp_files/2301.02316v1.pdf.txt @@ -0,0 +1,1527 @@ +Neural Network Adaptive Control with Long Short-Term Memory +Emirhan Inanc, Yigit Gurses, Abdullah Habboush, Yildiray Yildiz and Anuradha M. Annaswamy +Abstract— In this study, we propose a novel adaptive control +architecture, which provides dramatically better transient re- +sponse performance compared to conventional adaptive control +methods. What makes this architecture unique is the synergistic +employment of a traditional, Adaptive Neural Network (ANN) +controller and a Long Short-Term Memory (LSTM) network. +LSTM structures, unlike the standard feed-forward neural +networks, can take advantage of the dependencies in an input +sequence, which can contain critical information that can +help predict uncertainty. Through a novel training method +we introduced, the LSTM network learns to compensate for +the deficiencies of the ANN controller during sudden changes +in plant dynamics. This substantially improves the transient +response of the system and allows the controller to quickly react +to unexpected events. Through careful simulation studies, we +demonstrate that this architecture can improve the estimation +accuracy on a diverse set of uncertainties for an indefinite +time span. We also provide an analysis of the contributions +of the ANN controller and LSTM network to the control input, +identifying their individual roles in compensating low and high- +frequency error dynamics. This analysis provides insight into +why and how the LSTM augmentation improves the system’s +transient response. The stability of the overall system is also +shown via a rigorous Lyapunov analysis. +I. INTRODUCTION +Although Neural Networks (NN) is a fairly old concept +[1], cheap and fast parallel computing unlocked their po- +tential and lead to their current predominance in artificial +intelligence and machine learning [2]. Computer vision and +natural language processing are examples of fields that +benefited greatly from these developments [3], [4]. Deep +learning research produced derivatives of recurrent neural +networks (RNN) such as long short-term memory (LSTM) +[5] and gated recurrent unit [6], which are powerful structures +for processing sequential data [7]. Reinforcement learning +(RL) is another concept in machine learning that made +advancements through applications of deep learning [8]. In +RL, there exists a state that is updated with respect to an +action selected by an agent that makes its choices based on +observations to maximize its cumulative reward. A similarity +can be drawn between control systems and RL where states, +control inputs and feedback connections are parallel to states, +actions, and observations, respectively. There are studies that +take advantage of this fact to design RL-based controllers [9], +[10]. +E. Inanc, A. Habboush and Y. Yildiz are with the Department of +Mechanical Engineering, and Y. Gurses is with the Department of +Computer Engineering, Bilkent University, Cankaya, Ankara 06800, Turkey. +A.M. Annaswamy is with the Department of Mechanical Engineering, +Massachusetts Institute of Technology, Cambridge, MA, 02139. (e- +mails: +emirhan.inanc@bilkent.edu.tr, +yigit.gurses@ug.bilkent.edu.tr, +a.habboush@bilkent.edu.tr, yyildiz@bilkent.edu.tr, aanna@mit.edu). +The potential shown by earlier applications provides an +incentive to utilize NN in adaptive control. The literature on +this topic is extensive and well established [11], [12], [13], +[14], [15]. Using an online-tuned feed-forward NN controller +with adaptive update laws is proven to be stable and can +successfully compensate the uncertainties [14]. However, if +there are sudden changes in the uncertainty, the transient +response can be oscillatory, which results in poor perfor- +mance. This problem is addressed in [16] with the addition +of external memory similar to a neural turing machine’s +(NTM) [17]. However, unlike the common practice in NTMs, +a feed-forward network, instead of an RNN, is used in [17]. +Feed-forward networks that have access to only the current +state cannot take full advantage of the dependencies in a +sequence. Instead, using a recurrent structure with internal +memory can increase the capability of sequence estimation +and thus improve performance [18]. +To address the above mentioned issues, we propose a +novel control architecture that consists of an Adaptive Neural +Network (ANN) controller and an LSTM. The purpose of +the LSTM is to take advantage of the long and short- +term dependencies in the input sequence to improve the +transient response of the controller. Specifically, we train the +LSTM in such a way that it learns to compensate for the +inadequacies of the ANN controller in response to sudden +and unexpected uncertainty variations. Offline training in +closed-loop systems is challenging since the trained element +affects the system dynamics. This closed loop structure sim- +ilarly exists in RL applications and is addressed by previous +studies [19], [20]. Inspired by these methods, we train the +LSTM in a closed-loop setting to predict and compensate +for the undesired transient error dynamics. We demonstrate +via simulations that thanks to its predictive nature, LSTM +offsets high-frequency errors, and thus complements the +ANN controller, where the latter helps with handling the +low-frequency dynamics. +To summarize, the contribution of this paper is a novel +adaptive control framework that provides enhanced transient +performance compared to conventional approaches. This is +achieved by making the traditional ANN controller work in +collaboration with an LSTM network that is trained in the +closed-loop system. +In Section II, we describe the formulation of the ANN +controller. In Section III, the proposed LSTM augmentation +and the training method are explained, together with a +stability analysis. Simulation results are given in Section IV +and a summary is given in Section V. +arXiv:2301.02316v1 [eess.SY] 5 Jan 2023 + +II. PROBLEM FORMULATION +Consider the following plant dynamics +˙xp(t) =Apxp(t) + Bp(u(t) + f(xp(t))), +(1a) +yp(t) =CT +p xp(t), +(1b) +where xp(t) ∈ Rnp is the accessible state vector, u(t) ∈ Rm +is the plant control input, Ap ∈ Rnp×np is a known system +matrix, f(xp(t)) : Rnp −→ Rm is a state-dependent, possibly +nonlinear, matched uncertainty, Bp ∈ Rnp×m is a known +control input matrix, and Cp ∈ Rnp×s is a known output +matrix, and yp(t) ∈ Rs is the plant output. Furthermore, it +is assumed that the pair (Ap, Bp) is controllable. +Assumption +1: +The +unknown +matched +uncertainty +f(xp(t)) : Rnp −→ Rm is continuous on a known compact +set Sp ≜ {xp(t) : ∥xp(t)∥ ≤ bx} ⊂ Rnp, where bx is a +known positive constant. +Remark 1: The proposed method also applies for an +unknown state matrix Ap: Suppose that the plant dynamics +are given by +˙xp(t) = Auxp(t) + Bp(u(t) + fu(xp(t))), +(2) +where Au +∈ Rnp×np is an unknown state matrix and +fu(xp(t)) is a state-dependent, possibly nonlinear, matched +uncertainty. The dynamics (2) can be written in the form of +(1) and for a known matrix Ap, by defining f(xp(t)) ≜ +Kuxp(t) + fu(xp(t)), given that the matching condition +Ap = Au − BpKu is satisfied for some feedback gain +Ku ∈ Rm×np. This allows us to proceed assuming Ap is +known, without loss of generality. +To achieve command following of a bounded reference +input r(t) ∈ Rs, an error-integral state xe(t) ∈ Rs is defined +as +˙xe(t) = r(t) − yp(t), +(3) +which, when augmented with (1), yields the augmented +dynamics +˙x(t) = Ax(t) + Bmr(t) + B(u(t) + f(xp(t))), +(4a) +˙y(t) = CT x(t), +(4b) +where x(t) ≜ [xp(t)T , xe(t)T ]T ∈ Rn is the augmented +state, whose dimension is n ≜ np + s, and the system +matrices A, Bm, B and C are +A ≜ +� Ap +0np×s +−CT +p +0s×s +� +∈ Rn×n, +(5a) +Bm ≜ +�0np×s +−Is×s +� +∈ Rn×s, +(5b) +B ≜ +� Bp +0s×m +� +∈ Rn×m, +(5c) +C ≜ +�−CT +p +0s×s +�T ∈ Rn×s. +(5d) +A baseline state feedback controller is designed as +ubl(t) = −Kx(t), +(6) +where K ∈ Rm×n is selected such that Am ≜ A − BK is +Hurwitz. Then, a reference model is defined as +˙xm(t) = Amxm(t) + Bmr(t), +(7) +where xm(t) ∈ Rn is the state vector of the reference model. +Although the matched uncertainty f(xp(t)) is of unknown +structure, Assumption 1 implies that it can be approximated +by a multi-layer neural network in the following form +f(xp(t)) = W T ¯σ(V T ¯xp(t)) + ε(xp(t)), +(8) +such that +∥ε(xp(t))∥ ≤ εN, +∀x ∈ S, +(9) +respectively, where the NN reconstruction error vector +ε(xp(t)) is unknown, but bounded by a known constant +εN > 0 in the domain of interest Sp ⊂ Rnp. Such a bound +depends on the number of hidden neurons nh and the weight +matrices of the neural network W and V . The following +assumption is necessary and is standard in the literature. +Assumption 2: The unknown ideal weights W and V are +bounded by known positive constants such that ∥V ∥F ≤ VM +and ∥W∥F ≤ WM. +The input to the NN and the output of the hidden layer is +given by +¯xp(t) ≜ +� +xp(t)T +1 +�T ∈ Rnp+1, +(10a) +¯σ(V T ¯xp(t)) ≜ +� +σ(V T ¯xp(t)) +1 +�T ∈ Rnh+1, +(10b) +where the nonlinear activation function σ(.) : Rnh +→ +Rnh can be either sigmoid or tanh. Note that xp(t) and +σ(V T ¯xp(t)) are augmented with unity elements to account +for hidden and outer layers biases, respectively. For the +convenience of notation, we drop the overbar notation and +write (8) as +f(xp(t)) = W T σ(V T xp(t)) + ε(xp(t)), +(11) +where the dimensions of the weight matrices are loosely +defined as W ∈ Rnh×m and V ∈ Rnp×nh. +By defining the state tracking error as +e(t) ≜ x(t) − xm(t), +(12) +the control objective is to make e(t) converge to zero while +keeping all system signals bounded, with minimal transients. +To achieve this, we use the control input +u(t) = ubl(t) + uad(t) + ulstm(t) + v(t), +(13) +where ubl(t) is the baseline controller (6) that achieves the +reference model dynamics (7) in the absence of uncertainty, +uad(t) is an adaptive neural network (ANN) controller de- +signed to compensate for the matched uncertainty f(xp(t)), +and ulstm is the output of a long short-term memory (LSTM) +network. Furthermore, v(t) is a robustifying term, which is +defined subsequently. +In this paper, we focus our attention on enhancing the +transient response of the closed-loop system. Particularly, +we propose the synergistic employment of a traditional +ANN controller and an LSTM network to obtain superior +performance compared to conventional approaches. In what +follows, we detail each control term in (13). + +A. Adaptive Neural Network Controller +The adaptive neural network (ANN) controller is de- +signed to compensate for the matched nonlinear uncertainty +f(xp(t)), and is expressed as, +uad(t) = − ˆf(xp(t)), +(14) +where ˆf(xp(t)) represents an estimate of f(xp(t)). Using +(11), we define the estimate ˆf(xp(t)) as +ˆf(xp(t)) = ˆW T (t)σ( ˆV T (t)xp(t)), +(15) +where ˆV (t) ∈ Rnh×n is the input-to-hidden-layer weight +matrix and ˆW(t) ∈ Rm×nh is the hidden-to-output-layer +weight matrix. These weights serve as estimates for the +unknown ideal weights V and W, respectively. Through +proper use of Taylor series-based arguments [14], the weights +are updated using the adaptive laws +˙ˆW = F(ˆσ − ˆσ′ ˆV T xp)eT PB − Fκ∥e∥ ˆW, +(16a) +˙ˆV = GxpeT PB ˆW T ˆσ′ − Gκ∥e∥ ˆV, +(16b) +where F ∈ Rnh×nh and G ∈ Rnp×np are symmetric positive +definite matrices, serving as learning rates, κ > 0 is a scalar +gain, and ˆσ and ˆσ′ are defined as +ˆσ ≜ σ( ˆV T (t)xp(t)), +ˆσ′ ≜ dσ(z) +dz +���� +z= ˆV +T (t)xp(t) +. +(17) +Further, P +∈ Rn×n is the symmetric positive definite +solution of the Lyapunov equation +AT +mP + PAm = −Q, +(18) +for some symmetric positive definite matrix Q ∈ Rn×n. +B. Robustifying Term +The usage of the baseline controller, ANN controller, and +the robustifying term is sufficient to solve the adaptive con- +trol problem, as demonstrated in [14] for nonlinear robot arm +dynamics. However, for a generic multi-input multi-output +(MIMO) dynamical system with matched uncertainties (1), a +difficulty arises from the control input matrix B. A subsidiary +contribution of this paper is a modified robustifying term +v(t), which achieves the same stability results in [14], but +for generic MIMO linear systems. +The robustifying term, v(t) (see (13)) is used to provide +robustness against disturbances that arise from the high-order +Taylor series terms [14]. For the considered MIMO linear +plant dynamics, we propose a modified robustifying term +which is defined as +v(t) = +� +0, +if ∥BT Pe∥ = 0 +− BT P e +∥BT P e∥kz∥e∥(∥ ˆZ∥F + ZM), +otherwise +, +(19) +with +kz > C2, +(20) +where ZM ≜ +� +W 2 +M + V 2 +M, and matrices B and P are +defined in the previous subsection, whereas e is given in +(12). Furthermore, C2 ≜ 2Cσ′ZM, where Cσ′ is a known +upper bound for the absolute value of the sigmoid or tanh +activation function derivatives, i.e. ||σ′(.)|| ≤ Cσ′, where +σ′(z) = dσ(z)/dz. +In order not to diverge from the main contribution of the +paper, we defer the derivation of the ANN controller and the +proposed modified robustifying term to the Appendix. The +following section introduces the main focus of this paper. +III. LSTM NETWORK DESIGN +We propose an LSTM network that works in coordination +with the ANN controller to compensate for the uncertainties +in the system, with better transients compared to conven- +tional approaches. Combined usage of LSTM and ANN is +not arbitrary. Indeed, in this section, we precisely define the +separate roles of LSTM and ANN in the closed-loop system. +The overall control architecture is given in Fig. 1. By taking +the state tracking error e(t) = x(t) − xm(t) as the input, the +LSTM network computes the control input ulstm(t), which +is fed to the plant to compensate for the ANN controller +deficiencies. The motivation behind the LSTM network is to +improve the transient response of the system and achieve +faster convergence by taking advantage of the sequence +prediction capabilities of the LSTM architecture. +Remark 2: LSTM network takes a sequence as an input +and computes a sequence as the output. Hence, the sampling +time of the network arises as a design parameter. The +particular selection of the sampling time of the network is +application dependent. A proper selection would match the +sampling time at which the ANN is implemented, or it can be +made larger to reduce the computational load. Nevertheless, +the exact sampling time of the network is irrelevant in +the proposed architecture since, as shown subsequently, the +LSTM network can be trained offline to perform under a +given sampling time. +Remark 3: The LSTM input sequence en is generated from +the continuous-time signal e(t) as +en = e(nT), +n = 0, 1, 2, . . . +(21) +where T > 0 is the sampling time of the LSTM. Further- +more, the output sequence of the LSTM un +lstm is converted +to the continuous-time signal ulstm(t) which accounts for +the control input of the LSTM, where +ulstm(t) = un +lstm, +for nT ≤ t < (n + 1)T, +n = 0, 1, 2, . . . +(22) +LSTM is a Recurrent Neural Network (RNN), which, +unlike a conventional neural network structure, can learn the +long-term dependencies of a sequence. This is accomplished +with an additional hidden state, a cell state, and gates that +update the states in a systematic way. One cell of LSTM can +be seen in Fig. 2. +In Fig. 2, en +norm is the normalized version of the input +sequence en, hn is the hidden state, and cn is the cell +state. R, W, and b are the recurrent weight matrices, input +weight matrices, and bias vectors, respectively. σg denotes +the gate activation function (sigmoid) and σc denotes the + +Plant +Reference +Model +Basel�ne +Controller +LSTM +Controller +D�sturbance +ANN +Controller +r(t) +u(t) +ulstm(t) +uad(t) +ubl(t) +x(t) +e(t) +f(x) +Fig. 1: Block diagram of the proposed control architecture. +𝑒𝑛𝑜𝑟𝑚 +𝑛 +ℎ𝑛−1 +𝑐𝑛−1 +𝑐𝑛 +ℎ𝑛 +σg +σg +σg +Input Gate +Forget Gate +Output Gate +σc +σc +𝑜𝑢𝑡𝑓 +𝑛 +ҧ𝑐𝑛 +𝑜𝑢𝑡𝑖 +𝑛 +𝑜𝑢𝑡𝑜𝑛 +Cell Candidate +Fig. 2: Detailed sketch of an LSTM cell +state activation function (tanh). Subscripts f, g, i, and o +denote forget gate, cell candidate, input gate, and output gate, +respectively. outn +f , ¯cn, outn +i , and outn +o denote the output of +the gate operations. The superscript n denotes the current +time step and n − 1 is the previous time step. +a) Forget Gate: The forget gate decides the relevant +information that the cell state should take from the hidden +state and the input. This is achieved with a sigmoid activation +function. +outn +f = σg(Wfen +norm + Rfhn−1 + bf). +(23) +b) Input Gate: The new information that is going to be +added to the cell state is determined by the input gate and +cell candidate. First, the hyperbolic tangent tanh activation +function is used to determine which information is going to +be added to the cell state. Then, a sigmoid function is used +to determine how much of this information is going to be +added. This process is given as +¯cn = σc(Wgen +norm + Rghn−1 + bg), +(24) +outn +i = σg(Wien +norm + Rihn−1 + bi). +(25) +c) Output Gate: The output gate decides how much of the +cell state is going to be a part of the hidden state, using a +sigmoid function as +outn +o = σg(Woen +norm + Rohn−1 + bo). +(26) +The cell state c is first erased using the forget gate outf, +then new information is added through the input gate outi +and cell candidate ¯c. This is achieved with the calculation +cn = outn +f ⊙ cn−1 + outn +i ⊙ ¯cn. +(27) +The gates mentioned above are used to determine the hidden +state h and the cell state c for the next time step. The hidden +state is updated using the output gate outo and cell state c +that is mapped between -1 and 1 using a tanh function as +hn = outn +o ⊙ σc(cn). +(28) +A +fully +connected +layer +is +utilized +to +obtain +an +m−dimensional output to match the dimension of the +control input as +un +lstm = Wfchn + bfc, +(29) +where Wfc and bfc are the weight matrix and bias vector +of the fully connected layer, respectively. Then, the control +input ulstm(t) is constructed from the LSTM output as +shown in (22). +The ability to learn the dependencies within sequences +helps the LSTM to predict the evolution of an unknown +function. It is noted that in a typical ANN implementation, +the speed of the response can be increased by increasing +the adaptation rates, which may cause undesired oscillations. +With LSTM augmentation, faster convergence is achieved +without the need for large learning rates. The training and +implementation details of LSTM are explained in the follow- +ing subsections. +A. Training Method +Since the LSTM network operates in a closed loop system, +the system states and the “truth”, which is the variable it is +trying to estimate, are affected by the LSTM output (see Fig. +1). As the training progresses, the training data itself evolves +with time, which allows the data to be used only once to train +the network. +Our main philosophy to use LSTM is to compensate for +the inadequacies of the conventional ANN controller: We +know that ANN controllers are successful in compensating +for the uncertainties asymptotically. However, adjusting their +transient characteristics is not a trivial task. Instead of +increasing the learning rates, LSTM provides better transients +by actually predicting the type of transients and compensat- +ing accordingly. To achieve this, the goal for the LSTM is +set to predict the estimation error of the ANN controller and +compensate for this error. The estimation error of the ANN, +or the deviation of the ANN controller input (14), ˆf(xp(t)), +from the actual uncertainty, f(xp(t)), is given as +˜f(xp(t)) ≜ f(xp(t)) − ˆf(xp(t)). +(30) + +The “truth” for the LSTM, or the desired signal for the LSTM +to produce, is selected as +y(t) = − ˜f(xp(t)). +(31) +Therefore, LSTM provides a signal to the system that is an +estimate of this truth, expressed as +ulstm(t) = +ˆ +y(t). +(32) +During training, an uncertainty ftrain(xp(t)) is intro- +duced to the closed loop system, and LSTM is expected +to learn ˜ftrain(xp(t)) = ftrain(xp(t)) − ˆf(xp(t)), where +− ˆf(xp(t)) is the control signal produced by ANN. Apart +from ˜ftrain(xp(t)), LSTM uses the state-tracking error e(t), +defined in (12), as its input (see Fig. 1). Every iteration of +training affects the sequence that LSTM is trained on. This +dynamic nature of training helps LSTM generalize to a set of +functions that are not used in the training set. This is further +emphasized in the Simulations Section. +Since there is no initial data set, the data needs to be +collected by running the simulation with the initial LSTM +weights. Since weight updates of the LSTM network also +affect the system dynamics, new data needs to be collected +after each episode of training. Before giving the LSTM +output ulstm to the system, a gain “klstm” scales ulstm. The +role of this gain is to avoid an untrained LSTM network +output from negatively affecting the system at the beginning +of the training. klstm starts from 0 and approaches 1 as the +number of training iterations increases, then stays at 1 for the +rest of the training. The training loop can be seen in Fig. 3. +LSTM weights are +initialized and +klstm is set to zero +System is +simulated +e and y are +obtained +LSTM weights are +updated +klstm is increased +Fig. 3: Flow chart of the training process +B. Normalization +The inputs of the LSTM network are the components of +the tracking error vector e, defined in (12). In order to make +sure that these components are on a similar scale, normaliza- +tion needs to be performed. However, since an initial training +data set does not exist, data needs to be collected first in order +to acquire the parameters for normalization. For collecting +the normalization parameters only, the simulation is run with +ftrain (see explanations after (32)), z number of times. A +gain is uniformly sampled from [0, 1] in every simulation run, +to scale ftrain. This random scaling is used only during the +normalization parameter collection, not during training.The +LSTM network output ulstm is not injected to the system +during this process. Within the collected data, the minimum, +emin +i +, and the maximum, emax +i +, values, where ”i” is used +to refer the ith component, of each component of the error +vector, are utilized to obtain the normalized input parameters +as +enormi = (ei − emin +i +)/(emax +i +− emin +i +). +(33) +C. Stability Analysis +The following theorem displays the stability properties of +the overall architecture. +Theorem 1: Consider the uncertain plant dynamics (1), +subject to Assumptions 1 and 2, and the reference model +(7). Let the control input be defined by (13), which consists +of the baseline controller (6), the ANN controller defined +by (14), (15) and (16), the robustifying term given in (19) +and (20), and the LSTM controller explained in Section III. +Then, given that xp(0) ∈ Sp (see Assumption 1), the solution +(e(t), ˜W(t), ˜V (t)) is uniformly ultimately bounded (UUB) +and converges to a predefined compact set, where ˜V (t) ≜ +V − ˆV (t) and ˜W(t) ≜ W − ˆW(t). +Proof: The proof is deferred to the Appendix II. +IV. SIMULATIONS +In this section, the performance of the proposed control +framework is examined using the short-period longitudinal +flight dynamics. The short-period dynamics is given as [21] +� ˙α +˙q +� += +� +Zα +mU +1 + Zq +mU +Mα +Iy +Mq +Iy +� �α +q +� ++ +� +Zδ +mU +Mδ +Iy +� +(u + f(xp)), (34) +where α is the angle of attack (deg), q is the pitch rate +(deg/s), u is the elevator deflection (deg), and f(xp) is a state +dependent matched uncertainty. Elevator magnitude and rate +saturation limits are set as +17/ − 23 (deg) and +37/ − 37 +(deg/s) [22]. The commanded input is the pitch rate. Zα, Zq, +Mα, Mq, Zδ and Mδ are the stability and control derivatives. +The system matrices for a B-747 aircraft flying at the +speed of 274 m/s at 6000 m altitude are given as [16] +Ap = +�−0.32 +0.86 +−0.93 +−0.43 +� +, Bp = +�−0.02 +−1.16 +� +, Cp = +�0 1�T . +(35) +The augmented state vector is +x(t) = +�x1(t) +x2(t) +x3(t)�T , +(36) +where x1 is the angle of attack (rad), x2 is the pitch rate +(rad/s) and x3 is the error-integral state (3). The input to the +LSTM network is, +inlstm(t) = +�enorm1(t) +enorm2(t) +enorm3(t)�T , +(37) +where enorm1, enorm2, and enorm3 are the components of +the normalized version of the state tracking error vector (33). +The baseline controller (6) is an LQR controller with +cost matrices QLQR = I and RLQR = 1. The nonlinear +uncertainty, ftrain (see explanations after (32)) that the Long +Short-Term Memory (LSTM) network (29) is trained on is +defined as + +ftrain(x) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +0.1x1x2 +if 0 ≤ t < 5 +exp(x2) +if 5 ≤ t < 10 +2x1x2 +if 10 ≤ t < 15 +−0.1 cos(x1) +if 15 ≤ t < 20 +0.5(x1x2) +if 20 ≤ t < 25 +0.1x1x2 +if 25 ≤ t < 30 +−x1x2(sin(5x1x2) + 5 sin(x2)) +if 30 ≤ t < 35 +x1x2(3 sin(2x1x2) + 2x1) +if 35 ≤ t < 45 +−x1x2(2(tan(2x1x2) + x2 +2) +if 45 ≤ t < 55 +x1x2(x1 + x2) +if 55 ≤ t ≤ 60 +. +(38) +The LSTM network is trained in the presence of an ANN +controller (see Section II-A) that has a hidden layer consist- +ing of four neurons. The learning rates in (16a) and (16b) +are set as F = G = 10, and the Lyapunov matrix Q in (18) +is set as Q = I. The outer weights ( ˆW) and bias (ˆbw) are +initialized to zero. The inner weights ( ˆV ) and biases (ˆbv) are +initialized randomly between 0 and 1. In the simulations, the +robustifying gain kz in (20) and scalar gain κ in (16a) and +(16b) are set to 0. +The LSTM network contains one hidden layer with 128 +neurons. The number of neurons in the input layer is 3 due to +the number of state-tracking error components given as input +to the network (37). Weights are initialized using Xavier +Initialization [23]. The network is trained with stochastic +gradient descent, and uses the Adam optimizer with a learn- +ing rate of 0.001, an L2 regularization factor of 0.0001, a +gradient decay factor of 0.9, and a squared gradient decay +factor of 0.999 [24]. The gradient clipping method is used +with a threshold value of 1. The simulation step time is set to +0.01 s. The minibatch size is taken as 1. To cover both low +and high-valued uncertainties, ftrain is scaled by a parameter +that took the alternating values of 0.2 and 2. +To obtain the normalization constants given in (33), the +system is simulated 1000 times (z = 1000), with the gain +kf, (see Section III-B). +The loss function is chosen as Mean Squared Error (MSE) +given as +L(y, ˆy) = 1 +N +N +� +i=0 +(y − ˆy)2, +(39) +where y and ˆy are defined in (31) and (32),and N is the +number of data in the data set. +The proposed control framework is tested using the un- +Fig. 4: Tracking performances with and without LSTM +augmentation, in the presence of small uncertainty. +certainty defined as +ftest(x) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +0 +if 0 ≤ t < 2 +−0.1 exp(x1x2) +if 2 ≤ t < 8 +0.5x2 +2 +if 8 ≤ t < 12 +0.05 exp(x1 + 2x2) +if 12 ≤ t < 20 +−0.1 sin(0.05x1x2) +if 20 ≤ t < 28 +0.1(x2 +1 + x3 +2) +if 28 ≤ t < 34 +−0.1 +� +cos(x2)2 +if 34 ≤ t < 40 +−0.2 sin(x1x2) +if 40 ≤ t < 49 +0.1(x1x2)2 +if 49 ≤ t < 53 +0.5x2 +if 53 ≤ t < 60 +0.1(|x2|x1) +if 60 ≤ t < 67 +−0.5 sin(x2) +if 67 ≤ t < 79 +0.01 exp(x1) +if 79 ≤ t < 86 +−0.4x2 +1 +if 86 ≤ t < 98 +0.2x2 +2 +if 98 ≤ t ≤ 110 +, (40) +where the function is chosen to have different types of +sub-functions with different time intervals, compared to the +training uncertainty (38). +A. Controller performance in the presence of small uncer- +tainty +In this section, the effect of the proposed LSTM aug- +mentation is examined using low-value uncertainties. For +this purpose, ftest is scaled by 0.1 during the tests. Then, +the tracking performance, tracking error, and control inputs +of the closed-loop system are compared with and without +LSTM augmentation. +Figures 4 and 5 show the tracking and tracking error +curves, respectively. While the error plots demonstrate that +LSTM augmentation dramatically reduces both the error +magnitudes and the transient oscillations, the error values are +small enough to be ignored compared to the absolute values +of the states. Therefore, one can conclude that in this scenario + +Fig. 5: Tracking errors with and without LSTM augmenta- +tion, in the presence of small uncertainty. +8 +9 +10 +11 +12 +-14 +-13 +-12 +-11 +-10 +Zoomed +8 +9 +10 +11 +12 +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +Zoomed +Fig. 6: Control inputs in the presence of small uncertainty. +Top: Total control input. Middle: The contributions of indi- +vidual control inputs in the absence of LSTM. Bottom: The +contributions of individual control inputs in the presence of +LSTM. +LSTM augmented and not-augmented cases show similar +performances. In Fig. 6, the control inputs are presented. This +figure shows that although LSTM augmentation does not +affect the magnitude of the total control input in a meaningful +manner, it provides damping to the oscillations by providing +small but fast compensation to the introduced uncertainties. +As we discussed above, since the ANN controller is already +successful in compensating for the uncertainties, in this case, +LSTM contribution is not prominent. +B. Controller performance in the presence of large uncer- +tainty +In this section, the effect of the proposed LSTM augmenta- +tion is examined in the presence of high-valued uncertainties, +which is obtained by using the ftest as is. +Fig. 7: Tracking performances with and without LSTM +augmentation, in the presence of large uncertainty. +Fig. 8: Tracking errors with and without LSTM augmenta- +tion, in the presence of large uncertainty. +Figures 7 and 8 demonstrate that LSTM augmentation +substantially improves the transient response of the system, +especially in pitch rate tracking, which is the output of +interest (see (35)). It is noted that for this case, the same +ANN controller and the LSTM network are used as in the +small uncertainty case. Individual control inputs are shown +in Fig. 9. In this case, unlike the low-uncertainty case, the +LSTM augmentation makes the total control signal observ- +ably more agile, which is the main reason why excessive +oscillations are prevented. This shows the LSTM network +uses its memory to predict high-frequency changes in the +system. It is noted that in this large uncertainty case the +total control input is saturated. This is evident from Fig. +9, where the middle and the bottom sub-figures show the +control signals created by the individual components of the +overall controller, while the top sub-figure shows the total +control input after saturation. The figure shows that the +LSTM network produces large and very fast compensation, + +which causes saturation. Here, the rate saturation can be +problematic since it creates some oscillations in the total +control signal (see the zoomed section, blue line, in the top +sub-figure). Although the oscillations could be acceptable +since they are short-lived, we believe that this issue can also +be tackled by training the LSTM network with saturation +information. We explain this solution in the next section. +C. Addressing Saturation +To handle the saturation issue, we modified LSTM training +by a) introducing the saturation limits to the plant dynamics +during training, and b) informing the LSTM network when- +ever the control signal rate-saturates. The latter is achieved +by providing additional input to the LSTM network as +ur(t) = +� +� +� +� +� +0.1, +if rate saturation is positive +−0.1, +if rate saturation is negative, +0, +otherwise +(41) +8 +9 +10 +11 +12 +-20 +-15 +-10 +-5 +Zoomed +8 +9 +10 +11 +12 +-80 +-40 +0 +40 +Zoomed +Fig. 9: Control inputs in the presence of high uncertainty. +Top: Total control input. Middle: The contributions of indi- +vidual control inputs in the absence of LSTM. Bottom: The +contributions of individual control inputs in the presence of +LSTM. +The number of neurons in the input layer of the LSTM +network is increased to 4 since the modified input to the +LSTM network is +inlstm(t) = +�enorm1(t) +enorm2(t) +enorm3(t) +ur(t)�T . +(42) +Figure 10 shows the LSTM network contribution and the +total control inputs, with and without rate-limit information +during training. It is seen that the LSTM network uses the +rate-limit information to provide a smoother compensation. +Figures 11 and 12 show the tracking and tracking error +curves, respectively, when the LSTM network is trained +using the rate-limit information. It is seen that the perfor- +mance of the proposed controller remains similar for the case +8 +9 +10 +11 +12 +-20 +-15 +-10 +-5 +Zoomed +8 +9 +10 +11 +12 +-80 +-60 +-40 +-20 +0 +20 +40 +Zoomed +Fig. 10: Control signals with an without rate-limit-informed +LSTM. Top: Total control input. Bottom: LSTM network +contribution to the total control input. +when LSTM is not trained using the saturation information, +although the pitch rate shows initial small jumps at the +instances of uncertainty switches. +Fig. 11: Tracking performances with LSTM trained without +rate information and rate-limit-informed LSTM +7.5 +8 +8.5 +9 +9.5 +-0.14 +-0.12 +-0.1 +-0.08 +-0.06 +-0.04 +-0.02 +Zoomed +7.5 +8 +8.5 +9 +9.5 +-0.40 +-0.20 +0.00 +Zoomed +Fig. 12: Tracking errors with LSTM trained without rate +information and rate-limit-informed LSTM +V. SUMMARY +In this work, we propose a Long Short-Term Memory +(LSTM) augmented adaptive neural network (ANN) con- + +trol structure to improve the transient response of adaptive +closed-loop control systems. We demonstrate that thanks +to its time-series prediction capabilities, LSTM helps the +ANN controller compensate for the uncertainties in a more +agile fashion, resulting in dramatically improved tracking +performance. +REFERENCES +[1] B. 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Khalil, Nonlinear Systems, 3rd ed. +Upper Saddle River, NJ: +Prentice Hall, 2002. +APPENDIX I +ERROR DYNAMICS AND USEFUL REMARKS +In this section, the derivation of the ANN controller +adaptive laws (16) and the robustifying term (19) is shown. +We start by substituting the control input (13) and the +baseline controller (6) into (4), which yields +˙x(t) =Amx(t) + Bmr(t) ++ B(uad(t) + ulstm(t) + v(t) + f(xp(t))). +(43) +Using (12), (43) and (7), the state-tracking error dynamics +can be written as +˙e(t) = Ame(t) + B(uad(t) + ulstm(t) + v(t) + f(xp(t))), +(44) +which, by using the NN approximation property (11), can be +written as +˙e(t) =Ame(t) + B(uad(t) + W T σ(V T xp(t)) ++ ulstm(t) + v(t) + ε(xp(t))). +(45) +The ANN controller uad(t) is chosen to be the output of +a multi-layer NN, as defined in (14) and (15). Substituting +(14) and (15) into (45), yields +˙e(t) =Ame(t) + B(− ˆW T (t)σ( ˆV T (t)xp(t)) + W T σ(V T xp(t)) ++ ulstm(t) + v(t) + ε(xp(t))), +(46) +where ˆV (t) ∈ Rnh×n and ˆW(t) ∈ Rm×nh are adaptive NN +weights serving as estimates for the unknown ideal weights +V and W, respectively. The weight estimation errors are +defined as +˜W(t) =W − ˆW(t), +(47a) +˜V (t) =V − ˆV (t). +(47b) +Throughout the remainder of this section, we drop the time +dependency notation for simplicity. Further, we denote σ ≜ +σ(V T xp(t)) and ˆσ ≜ σ( ˆV T xp(t)). Then, (46) can be written +as +˙e = Ame + B(− ˆW T ˆσ + W T σ + ulstm + v + ε(xp)). +(48) +Using (47a) in (48) yields +˙e = Ame + B( ˜W T ˆσ + W T (σ − ˆσ) + ulstm + v + ε(xp)). +(49) + +Remark 4: Since r(t) is bounded and Am is Hurwitz, it +follows from (7) that +∥xm(t)∥ ≤ Cm, +(50) +where Cm > 0 is a known constant, which depends on the +user-defined reference input r(t). Therefore, it follows from +(12), and the fact that x(t) ≜ [xp(t)T , xe(t)T ]T ∈ Rn, that +∥xp(t)∥ ≤ ∥x(t)∥ ≤ Cm + ∥e(t)∥, +(51) +Remark 5: From the Taylor series expansion +σ(V T xp) = σ( ˆV T xp) + ˆσ′ ˜V T xp + O( ˜V T xp)2, +(52) +one can write +σ(V T xp) − σ( ˆV T xp) = ˆσ′ ˜V T xp + O( ˜V T xp)2, +(53) +where ˆσ′ is defined in (17) and O( ˜V T xp)2 denote the higher +order terms in the series. Furthermore, since +O( ˜V T xp)2 = σ(V T xp) − σ( ˆV T xp) − ˆσ′ ˜V T xp, +(54) +then, for sigmoid and tanh activation functions, ∥σ(.)∥ ≤ Cσ +and ∥σ′(.)∥ ≤ Cσ′ for known constants Cσ > 0 and Cσ′ > +0. Hence, using (51), the higher-order terms in the Taylor +series are bounded by +∥O( ˜V T xp)2∥ ≤ 2Cσ + Cσ′∥ ˜V ∥F Cm + Cσ′∥ ˜V ∥F ∥e∥. +(55) +Substituting (53) for (σ − ˆσ) in (49), yields +˙e =Ame + B( ˜W T ˆσ + W T ˆσ′ ˜V T xp ++ W T O( ˜V T xp)2 + ulstm + v + ε(xp)), +(56) +which, by using (47a) an (47b), can be written as +˙e =Ame + B( ˜W T (ˆσ − ˆσ′ ˆV T xp) + ˆW T ˆσ′ ˜V T xp +w + ulstm + v), +(57) +where +w ≜ ˜W T ˆσ′V T xp + W T O( ˜V T xp)2 + ε(xp). +(58) +Remark 6: Let +Z ≜ +� +W +0nh×nh +0np×m +V +� +∈ R(nh+np)×(m+nh). +(59) +Using (59) and assumption 2, one can write +∥Z∥2 +F = ∥W∥2 +F + ∥V ∥2 +F ≤ W 2 +M + V 2 +M, +(60) +and hence, ∥Z∥F +≤ ZM, where ZM +≜ +� +W 2 +M + V 2 +M. +Furthermore, it follows from (60) that ∥W∥2 +F ≤ ZM and +∥V ∥2 +F ≤ ZM. Similarly, by defining +ˆZ(t) ≜ +� ˆW(t) +0nh×nh +0np×m +ˆV (t) +� +∈ R(nh+np)×(m+nh), +(61) +one can show that ∥ ˜W(t)∥F ≤ ∥ ˜Z(t)∥F and ∥ ˜V (t)∥F ≤ +∥ ˜Z(t)∥F , where ˜Z(t) ≜ Z − ˆZ(t). +Remark 7: Using (9), (55) and Remark 6, one can write +∥w∥ ≤∥ ˜Z∥F Cσ′ZM(Cm + ∥e∥) ++ ZM(2Cσ + Cσ′∥ ˜Z∥F Cm + Cσ′∥ ˜Z∥F ∥e∥) + εN, +(62) +or +∥w∥ ≤C0 + C1∥ ˜Z∥F + C2∥ ˜Z∥F ∥e∥, +(63) +where +C0 ≜2CσZM + εN, +(64a) +C1 ≜2Cσ′CmZM, +(64b) +C2 ≜2Cσ′ZM. +(64c) +Remark 8: Sigmoid and tanh activation functions help +define a bound on the LSTM output: We can write (28) as +hn = +� +� +� +σg1(· · · )σc1(· · · ) +... +σgnh(· · · )σgnh(· · · ) +� +� +� ∈ Rnh, +(65) +where nh is the number of neurons in the last hidden layer. +Then, +∥hn∥2 = +nh +� +i +σ2 +gi(· · · )σ2 +c i(· · · ) ≤ nh. +(66) +Hence, +∥hn∥ ≤ √nh. +(67) +The bound on the LSTM output can be defined as +∥un +lstm∥ ≤ ∥Wfc∥F +√nh + ∥bfc∥ ≜ ¯ulstm, +(68) +which by using (22) implies that ∥ulstm(t)∥ ≤ ¯ulstm. +APPENDIX II +PROOF OF THEOREM 1: +Let the approximation property (11) hold on a known +compact set Sp ≜ {xp : ∥xp∥ ≤ bx} as stated in Assumption +1, for some bx > Cm. Defining +S ≜ {x : ∥x∥ ≤ bx}, +bx > Cm, +(69) +and using the fact that ∥xp∥ ≤ ∥x∥, x ∈ S implies that +xp ∈ Sp. Consider the compact set +Se ≜ {e : ∥e∥ ≤ bx − Cm}, +(70) +which imply that once e ∈ Se, then xp ∈ Sp. Let e(0) ∈ Se. +Then, xp(0) ∈ Sp and (11) holds. The proof proceeds by +showing that e(t) ∈ Se, ∀t ≥ 0. +Consider the Lyapunov function +V = eT Pe + tr{ ˜W T F −1 ˜W} + tr{ ˜V T G−1 ˜V }. +(71) +Differentiating (71) along the trajectory (57) yields, +˙V = ˙eT Pe + eT P ˙e + 2tr{ ˜W T F −1 ˙˜W} + 2tr{ ˜V T G−1 ˙˜V } +=eT AT +mPe + eT PAme + 2eT PB ˜W T (ˆσ − ˆσ′ ˆV T xp) ++ 2eT PB ˆW T ˆσ′ ˜V T xp + 2eT PB(w + ulstm + v) ++ 2tr{ ˜W T F −1 ˙˜W} + 2tr{ ˜V T G−1 ˙˜V }. +(72) + +Using (18) and tr{A1A2} = tr{A2A1} yields +˙V = − eT Qe + 2tr{ ˜W T (ˆσ − ˆσ′ ˆV T xp)eT PB} ++ 2tr{ ˜V T xpeT PB ˆW T ˆσ′} + 2eT PB(w + ulstm + v) ++ 2tr{ ˜W T F −1 ˙˜W} + 2tr{ ˜V T G−1 ˙˜V }. +(73) +It follows from (47) that ˙˜W = − ˙ˆW. Substituting (16) into +(73) yields +˙V = − eT Qe + 2eT PB(w + ulstm + v) ++ 2tr{ ˜W T κ∥e∥ ˆW} + 2tr{ ˜V T κ∥e∥ ˆV }. +(74) +Using (47), (59), and (61), one can write +˙V = − eT Qe + 2eT PB(w + ulstm + v) ++ 2κ∥e∥tr{ ˜W T (W − ˜W)} + 2κ∥e∥tr{ ˜V T (V − ˜V )} += − eT Qe + 2eT PB(w + ulstm + v) ++ 2κ∥e∥tr{ ˜ZT (Z − ˜Z)}. +(75) +Since +tr{ ˜ZT (Z − ˜Z)} =tr{ ˜ZT Z} − ∥ ˜Z∥2 +F +≤ ∥ ˜Z∥F (ZM − ∥ ˜Z∥F ), +(76) +it follows from (75) that +˙V ≤ − λmin(Q)∥e∥2 + 2∥BT Pe∥(∥w∥ + ∥ulstm∥) ++ 2eT PBv + 2κ∥e∥∥ ˜Z∥F (ZM − ∥ ˜Z∥F ), +(77) +which, by using (63) and Remark 8, can be written as +˙V ≤ − λmin(Q)∥e∥2 + 2∥BT Pe∥(C0 + ¯ulstm) ++ 2C1∥BT Pe∥∥ ˜Z∥F + 2C2∥BT Pe∥∥e∥∥ ˜Z∥F ++ 2eT PBv + 2κ∥e∥∥ ˜Z∥F (ZM − ∥ ˜Z∥F ). +(78) +The proposed robustifying term (19) is designed to cancel +the 4th term of (78). Two cases follow from using (19) in +(78): a) ∥BT Pe∥ = 0 and b) ∥BT Pe∥ ̸= 0. +Case a) ∥BT Pe∥ = 0: In this case, v = 0, and (78) +reduces to +˙V ≤ − λmin(Q)∥e∥2 + 2κ∥e∥∥ ˜Z∥F (ZM − ∥ ˜Z∥F ) += − ∥e∥(λmin(Q)∥e∥ + 2κ∥ ˜Z∥F (∥ ˜Z∥F − ZM)) += − ∥e∥ +� +λmin(Q)∥e∥ + 2κ(∥ ˜Z∥F − ZM +2 )2 − κZ2 +M +2 +� +, +(79) +which implies that ˙V < 0 if either +∥e∥ > +κZ2 +M +2λmin(Q), +(80) +or +∥ ˜Z∥F > ZM. +(81) +Case b) ∥BT Pe∥ ̸= 0: Then, by using ∥ ˜Z∥F ≤ ∥ ˆZ∥F + +ZM, and substituting (19) into (78), it follows that +˙V ≤ − λmin(Q)∥e∥2 + 2∥BT Pe∥(C0 + ¯ulstm) ++ 2C1∥BT Pe∥∥ ˜Z∥F ++ 2C2∥BT Pe∥∥e∥(∥ ˆZ∥F + ZM) ++ 2eT PBv + 2κ∥e∥∥ ˜Z∥F (ZM − ∥ ˜Z∥F ) += − λmin(Q)∥e∥2 + 2∥BT Pe∥(C0 + ¯ulstm) ++ 2C1∥BT Pe∥∥ ˜Z∥F + 2κ∥e∥∥ ˜Z∥F (ZM − ∥ ˜Z∥F ) ++ 2C2∥BT Pe∥∥e∥(∥ ˆZ∥F + ZM) +− 2kz∥BT Pe∥∥e∥(∥ ˆZ∥F + ZM). +(82) +Since kz > C2, (82) reduces to +˙V ≤ − λmin(Q)∥e∥2 + 2∥BT Pe∥(C0 + ¯ulstm) ++ 2C1∥BT Pe∥∥ ˜Z∥F + 2κ∥e∥∥ ˜Z∥F (ZM − ∥ ˜Z∥F ) +≤ − λmin(Q)∥e∥2 + 2∥BT P∥F ∥e∥(C0 + ¯ulstm) ++ 2C1∥BT P∥F ∥e∥∥ ˜Z∥F + 2κ∥e∥∥ ˜Z∥F (ZM − ∥ ˜Z∥F ) += − ∥e∥ +� +λmin(Q)∥e∥ − 2∥BT P∥F (C0 + ¯ulstm) +− 2C1∥BT P∥F ∥ ˜Z∥F − 2κ∥ ˜Z∥F (ZM − ∥ ˜Z∥F ) +� += − ∥e∥ +� +λmin(Q)∥e∥ + 2κ(∥ ˜Z∥F − q1 +2 )2 − κq2 +1 +2 − 2q0 +� +, +(83) +where +q0 +≜ +∥BT P∥F (C0 + ¯ulstm) +and +q1 +≜ +C1∥BT P∥F /κ + ZM. Therefore, it follows that ˙V < 0 if +either +∥e∥ > κq2 +1 + 4q0 +2λmin(Q), +(84) +or +∥ ˜Z∥F > q1 +2 + +� +q2 +1 +4 + q0 +κ . +(85) +Hence, using the fact that the bounds given in (84) and +(85) are bigger than the bounds in (80) and (81), and by +selecting the Lyapunov matrix Q such that +λmin(Q) > κq2 +1 + 4q0 +2(bx + Cm), +(86) +it can be shown that the solution (e(t), ˜Z(t)) is uniformly ul- +timately bounded (UUB) [25], and converges to the compact +set +E ≜ +� +(e, ˜Z) :∥e∥ ≤ κq2 +1 + 4q0 +2λmin(Q) < bx + Cm +and ∥ ˜Z∥F ≤ q1 +2 + +� +q2 +1 +4 + q0 +κ +� +. +(87) +This implies that, since e(0) ∈ Se and +˙V +< 0 on the +boundary of Se, e(t) stays within Se for all t ≥ 0. Therefore +xp(t) ∈ Sp for all t ≥ 0. +Since the reference input r(t) is bounded, xm(t) is +bounded. Therefore, the boundedness of e(t) = x(t)−xm(t) +implies that x(t), and therefore all system signals, are +bounded. +■ + diff --git a/mNE0T4oBgHgl3EQfZADU/content/tmp_files/load_file.txt b/mNE0T4oBgHgl3EQfZADU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c2ff2c08a9eff2d2be8e1494535ad7282d695675 --- /dev/null +++ b/mNE0T4oBgHgl3EQfZADU/content/tmp_files/load_file.txt @@ -0,0 +1,590 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf,len=589 +page_content='Neural Network Adaptive Control with Long Short-Term Memory Emirhan Inanc, Yigit Gurses, Abdullah Habboush, Yildiray Yildiz and Anuradha M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Annaswamy Abstract— In this study, we propose a novel adaptive control architecture, which provides dramatically better transient re- sponse performance compared to conventional adaptive control methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' What makes this architecture unique is the synergistic employment of a traditional, Adaptive Neural Network (ANN) controller and a Long Short-Term Memory (LSTM) network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' LSTM structures, unlike the standard feed-forward neural networks, can take advantage of the dependencies in an input sequence, which can contain critical information that can help predict uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Through a novel training method we introduced, the LSTM network learns to compensate for the deficiencies of the ANN controller during sudden changes in plant dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' This substantially improves the transient response of the system and allows the controller to quickly react to unexpected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Through careful simulation studies, we demonstrate that this architecture can improve the estimation accuracy on a diverse set of uncertainties for an indefinite time span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' We also provide an analysis of the contributions of the ANN controller and LSTM network to the control input, identifying their individual roles in compensating low and high- frequency error dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' This analysis provides insight into why and how the LSTM augmentation improves the system’s transient response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The stability of the overall system is also shown via a rigorous Lyapunov analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' INTRODUCTION Although Neural Networks (NN) is a fairly old concept [1], cheap and fast parallel computing unlocked their po- tential and lead to their current predominance in artificial intelligence and machine learning [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Computer vision and natural language processing are examples of fields that benefited greatly from these developments [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Deep learning research produced derivatives of recurrent neural networks (RNN) such as long short-term memory (LSTM) [5] and gated recurrent unit [6], which are powerful structures for processing sequential data [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Reinforcement learning (RL) is another concept in machine learning that made advancements through applications of deep learning [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' In RL, there exists a state that is updated with respect to an action selected by an agent that makes its choices based on observations to maximize its cumulative reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' A similarity can be drawn between control systems and RL where states, control inputs and feedback connections are parallel to states, actions, and observations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' There are studies that take advantage of this fact to design RL-based controllers [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Inanc, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Habboush and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Yildiz are with the Department of Mechanical Engineering, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Gurses is with the Department of Computer Engineering, Bilkent University, Cankaya, Ankara 06800, Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Annaswamy is with the Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (e- mails: emirhan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='inanc@bilkent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='tr, yigit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='gurses@ug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='bilkent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='tr, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='habboush@bilkent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='tr, yyildiz@bilkent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='tr, aanna@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The potential shown by earlier applications provides an incentive to utilize NN in adaptive control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The literature on this topic is extensive and well established [11], [12], [13], [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Using an online-tuned feed-forward NN controller with adaptive update laws is proven to be stable and can successfully compensate the uncertainties [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' However, if there are sudden changes in the uncertainty, the transient response can be oscillatory, which results in poor perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' This problem is addressed in [16] with the addition of external memory similar to a neural turing machine’s (NTM) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' However, unlike the common practice in NTMs, a feed-forward network, instead of an RNN, is used in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Feed-forward networks that have access to only the current state cannot take full advantage of the dependencies in a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Instead, using a recurrent structure with internal memory can increase the capability of sequence estimation and thus improve performance [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' To address the above mentioned issues, we propose a novel control architecture that consists of an Adaptive Neural Network (ANN) controller and an LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The purpose of the LSTM is to take advantage of the long and short- term dependencies in the input sequence to improve the transient response of the controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Specifically, we train the LSTM in such a way that it learns to compensate for the inadequacies of the ANN controller in response to sudden and unexpected uncertainty variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Offline training in closed-loop systems is challenging since the trained element affects the system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' This closed loop structure sim- ilarly exists in RL applications and is addressed by previous studies [19], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Inspired by these methods, we train the LSTM in a closed-loop setting to predict and compensate for the undesired transient error dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' We demonstrate via simulations that thanks to its predictive nature, LSTM offsets high-frequency errors, and thus complements the ANN controller, where the latter helps with handling the low-frequency dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' To summarize, the contribution of this paper is a novel adaptive control framework that provides enhanced transient performance compared to conventional approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' This is achieved by making the traditional ANN controller work in collaboration with an LSTM network that is trained in the closed-loop system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' In Section II, we describe the formulation of the ANN controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' In Section III, the proposed LSTM augmentation and the training method are explained, together with a stability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Simulation results are given in Section IV and a summary is given in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='02316v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='SY] 5 Jan 2023 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' PROBLEM FORMULATION Consider the following plant dynamics ˙xp(t) =Apxp(t) + Bp(u(t) + f(xp(t))), (1a) yp(t) =CT p xp(t), (1b) where xp(t) ∈ Rnp is the accessible state vector, u(t) ∈ Rm is the plant control input, Ap ∈ Rnp×np is a known system matrix, f(xp(t)) : Rnp −→ Rm is a state-dependent, possibly nonlinear, matched uncertainty, Bp ∈ Rnp×m is a known control input matrix, and Cp ∈ Rnp×s is a known output matrix, and yp(t) ∈ Rs is the plant output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Furthermore, it is assumed that the pair (Ap, Bp) is controllable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Assumption 1: The unknown matched uncertainty f(xp(t)) : Rnp −→ Rm is continuous on a known compact set Sp ≜ {xp(t) : ∥xp(t)∥ ≤ bx} ⊂ Rnp, where bx is a known positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Remark 1: The proposed method also applies for an unknown state matrix Ap: Suppose that the plant dynamics are given by ˙xp(t) = Auxp(t) + Bp(u(t) + fu(xp(t))), (2) where Au ∈ Rnp×np is an unknown state matrix and fu(xp(t)) is a state-dependent, possibly nonlinear, matched uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The dynamics (2) can be written in the form of (1) and for a known matrix Ap, by defining f(xp(t)) ≜ Kuxp(t) + fu(xp(t)), given that the matching condition Ap = Au − BpKu is satisfied for some feedback gain Ku ∈ Rm×np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' This allows us to proceed assuming Ap is known, without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' To achieve command following of a bounded reference input r(t) ∈ Rs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' an error-integral state xe(t) ∈ Rs is defined as ˙xe(t) = r(t) − yp(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (3) which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' when augmented with (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' yields the augmented dynamics ˙x(t) = Ax(t) + Bmr(t) + B(u(t) + f(xp(t))),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (4a) ˙y(t) = CT x(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (4b) where x(t) ≜ [xp(t)T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' xe(t)T ]T ∈ Rn is the augmented state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' whose dimension is n ≜ np + s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' and the system matrices A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Bm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' B and C are A ≜ � Ap 0np×s −CT p 0s×s � ∈ Rn×n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (5a) Bm ≜ �0np×s −Is×s � ∈ Rn×s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (5b) B ≜ � Bp 0s×m � ∈ Rn×m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (5c) C ≜ �−CT p 0s×s �T ∈ Rn×s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (5d) A baseline state feedback controller is designed as ubl(t) = −Kx(t), (6) where K ∈ Rm×n is selected such that Am ≜ A − BK is Hurwitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Then, a reference model is defined as ˙xm(t) = Amxm(t) + Bmr(t), (7) where xm(t) ∈ Rn is the state vector of the reference model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Although the matched uncertainty f(xp(t)) is of unknown structure, Assumption 1 implies that it can be approximated by a multi-layer neural network in the following form f(xp(t)) = W T ¯σ(V T ¯xp(t)) + ε(xp(t)), (8) such that ∥ε(xp(t))∥ ≤ εN, ∀x ∈ S, (9) respectively, where the NN reconstruction error vector ε(xp(t)) is unknown, but bounded by a known constant εN > 0 in the domain of interest Sp ⊂ Rnp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Such a bound depends on the number of hidden neurons nh and the weight matrices of the neural network W and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The following assumption is necessary and is standard in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Assumption 2: The unknown ideal weights W and V are bounded by known positive constants such that ∥V ∥F ≤ VM and ∥W∥F ≤ WM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The input to the NN and the output of the hidden layer is given by ¯xp(t) ≜ � xp(t)T 1 �T ∈ Rnp+1, (10a) ¯σ(V T ¯xp(t)) ≜ � σ(V T ¯xp(t)) 1 �T ∈ Rnh+1, (10b) where the nonlinear activation function σ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=') : Rnh → Rnh can be either sigmoid or tanh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Note that xp(t) and σ(V T ¯xp(t)) are augmented with unity elements to account for hidden and outer layers biases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' For the convenience of notation, we drop the overbar notation and write (8) as f(xp(t)) = W T σ(V T xp(t)) + ε(xp(t)), (11) where the dimensions of the weight matrices are loosely defined as W ∈ Rnh×m and V ∈ Rnp×nh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' By defining the state tracking error as e(t) ≜ x(t) − xm(t), (12) the control objective is to make e(t) converge to zero while keeping all system signals bounded, with minimal transients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' To achieve this, we use the control input u(t) = ubl(t) + uad(t) + ulstm(t) + v(t), (13) where ubl(t) is the baseline controller (6) that achieves the reference model dynamics (7) in the absence of uncertainty, uad(t) is an adaptive neural network (ANN) controller de- signed to compensate for the matched uncertainty f(xp(t)), and ulstm is the output of a long short-term memory (LSTM) network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Furthermore, v(t) is a robustifying term, which is defined subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' In this paper, we focus our attention on enhancing the transient response of the closed-loop system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Particularly, we propose the synergistic employment of a traditional ANN controller and an LSTM network to obtain superior performance compared to conventional approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' In what follows, we detail each control term in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Adaptive Neural Network Controller The adaptive neural network (ANN) controller is de- signed to compensate for the matched nonlinear uncertainty f(xp(t)), and is expressed as, uad(t) = − ˆf(xp(t)), (14) where ˆf(xp(t)) represents an estimate of f(xp(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Using (11), we define the estimate ˆf(xp(t)) as ˆf(xp(t)) = ˆW T (t)σ( ˆV T (t)xp(t)), (15) where ˆV (t) ∈ Rnh×n is the input-to-hidden-layer weight matrix and ˆW(t) ∈ Rm×nh is the hidden-to-output-layer weight matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' These weights serve as estimates for the unknown ideal weights V and W, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Through proper use of Taylor series-based arguments [14], the weights are updated using the adaptive laws ˙ˆW = F(ˆσ − ˆσ′ ˆV T xp)eT PB − Fκ∥e∥ ˆW, (16a) ˙ˆV = GxpeT PB ˆW T ˆσ′ − Gκ∥e∥ ˆV, (16b) where F ∈ Rnh×nh and G ∈ Rnp×np are symmetric positive definite matrices, serving as learning rates, κ > 0 is a scalar gain, and ˆσ and ˆσ′ are defined as ˆσ ≜ σ( ˆV T (t)xp(t)), ˆσ′ ≜ dσ(z) dz ���� z= ˆV T (t)xp(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (17) Further, P ∈ Rn×n is the symmetric positive definite solution of the Lyapunov equation AT mP + PAm = −Q, (18) for some symmetric positive definite matrix Q ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Robustifying Term The usage of the baseline controller, ANN controller, and the robustifying term is sufficient to solve the adaptive con- trol problem, as demonstrated in [14] for nonlinear robot arm dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' However, for a generic multi-input multi-output (MIMO) dynamical system with matched uncertainties (1), a difficulty arises from the control input matrix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' A subsidiary contribution of this paper is a modified robustifying term v(t), which achieves the same stability results in [14], but for generic MIMO linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The robustifying term, v(t) (see (13)) is used to provide robustness against disturbances that arise from the high-order Taylor series terms [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' For the considered MIMO linear plant dynamics, we propose a modified robustifying term which is defined as v(t) = � 0, if ∥BT Pe∥ = 0 − BT P e ∥BT P e∥kz∥e∥(∥ ˆZ∥F + ZM), otherwise , (19) with kz > C2, (20) where ZM ≜ � W 2 M + V 2 M, and matrices B and P are defined in the previous subsection, whereas e is given in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Furthermore, C2 ≜ 2Cσ′ZM, where Cσ′ is a known upper bound for the absolute value of the sigmoid or tanh activation function derivatives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' ||σ′(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' )|| ≤ Cσ′, where σ′(z) = dσ(z)/dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' In order not to diverge from the main contribution of the paper, we defer the derivation of the ANN controller and the proposed modified robustifying term to the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The following section introduces the main focus of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' LSTM NETWORK DESIGN We propose an LSTM network that works in coordination with the ANN controller to compensate for the uncertainties in the system, with better transients compared to conven- tional approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Combined usage of LSTM and ANN is not arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Indeed, in this section, we precisely define the separate roles of LSTM and ANN in the closed-loop system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The overall control architecture is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' By taking the state tracking error e(t) = x(t) − xm(t) as the input, the LSTM network computes the control input ulstm(t), which is fed to the plant to compensate for the ANN controller deficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The motivation behind the LSTM network is to improve the transient response of the system and achieve faster convergence by taking advantage of the sequence prediction capabilities of the LSTM architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Remark 2: LSTM network takes a sequence as an input and computes a sequence as the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Hence, the sampling time of the network arises as a design parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The particular selection of the sampling time of the network is application dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' A proper selection would match the sampling time at which the ANN is implemented, or it can be made larger to reduce the computational load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Nevertheless, the exact sampling time of the network is irrelevant in the proposed architecture since, as shown subsequently, the LSTM network can be trained offline to perform under a given sampling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Remark 3: The LSTM input sequence en is generated from the continuous-time signal e(t) as en = e(nT), n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (21) where T > 0 is the sampling time of the LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Further- more, the output sequence of the LSTM un lstm is converted to the continuous-time signal ulstm(t) which accounts for the control input of the LSTM, where ulstm(t) = un lstm, for nT ≤ t < (n + 1)T, n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (22) LSTM is a Recurrent Neural Network (RNN), which, unlike a conventional neural network structure, can learn the long-term dependencies of a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' This is accomplished with an additional hidden state, a cell state, and gates that update the states in a systematic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' One cell of LSTM can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 2, en norm is the normalized version of the input sequence en, hn is the hidden state, and cn is the cell state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' R, W, and b are the recurrent weight matrices, input weight matrices, and bias vectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' σg denotes the gate activation function (sigmoid) and σc denotes the Plant Reference Model Basel�ne Controller LSTM Controller D�sturbance ANN Controller r(t) u(t) ulstm(t) uad(t) ubl(t) x(t) e(t) f(x) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 1: Block diagram of the proposed control architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 𝑒𝑛𝑜𝑟𝑚 𝑛 ℎ𝑛−1 𝑐𝑛−1 𝑐𝑛 ℎ𝑛 σg σg σg Input Gate Forget Gate Output Gate σc σc 𝑜𝑢𝑡𝑓 𝑛 ҧ𝑐𝑛 𝑜𝑢𝑡𝑖 𝑛 𝑜𝑢𝑡𝑜𝑛 Cell Candidate Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 2: Detailed sketch of an LSTM cell state activation function (tanh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Subscripts f, g, i, and o denote forget gate, cell candidate, input gate, and output gate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' outn f , ¯cn, outn i , and outn o denote the output of the gate operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The superscript n denotes the current time step and n − 1 is the previous time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' a) Forget Gate: The forget gate decides the relevant information that the cell state should take from the hidden state and the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' This is achieved with a sigmoid activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' outn f = σg(Wfen norm + Rfhn−1 + bf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (23) b) Input Gate: The new information that is going to be added to the cell state is determined by the input gate and cell candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' First, the hyperbolic tangent tanh activation function is used to determine which information is going to be added to the cell state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Then, a sigmoid function is used to determine how much of this information is going to be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' This process is given as ¯cn = σc(Wgen norm + Rghn−1 + bg), (24) outn i = σg(Wien norm + Rihn−1 + bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (25) c) Output Gate: The output gate decides how much of the cell state is going to be a part of the hidden state, using a sigmoid function as outn o = σg(Woen norm + Rohn−1 + bo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (26) The cell state c is first erased using the forget gate outf, then new information is added through the input gate outi and cell candidate ¯c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' This is achieved with the calculation cn = outn f ⊙ cn−1 + outn i ⊙ ¯cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (27) The gates mentioned above are used to determine the hidden state h and the cell state c for the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The hidden state is updated using the output gate outo and cell state c that is mapped between -1 and 1 using a tanh function as hn = outn o ⊙ σc(cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (28) A fully connected layer is utilized to obtain an m−dimensional output to match the dimension of the control input as un lstm = Wfchn + bfc, (29) where Wfc and bfc are the weight matrix and bias vector of the fully connected layer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Then, the control input ulstm(t) is constructed from the LSTM output as shown in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The ability to learn the dependencies within sequences helps the LSTM to predict the evolution of an unknown function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' It is noted that in a typical ANN implementation, the speed of the response can be increased by increasing the adaptation rates, which may cause undesired oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' With LSTM augmentation, faster convergence is achieved without the need for large learning rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The training and implementation details of LSTM are explained in the follow- ing subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Training Method Since the LSTM network operates in a closed loop system, the system states and the “truth”, which is the variable it is trying to estimate, are affected by the LSTM output (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' As the training progresses, the training data itself evolves with time, which allows the data to be used only once to train the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Our main philosophy to use LSTM is to compensate for the inadequacies of the conventional ANN controller: We know that ANN controllers are successful in compensating for the uncertainties asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' However, adjusting their transient characteristics is not a trivial task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Instead of increasing the learning rates, LSTM provides better transients by actually predicting the type of transients and compensat- ing accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' To achieve this, the goal for the LSTM is set to predict the estimation error of the ANN controller and compensate for this error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The estimation error of the ANN, or the deviation of the ANN controller input (14), ˆf(xp(t)), from the actual uncertainty, f(xp(t)), is given as ˜f(xp(t)) ≜ f(xp(t)) − ˆf(xp(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (30) The “truth” for the LSTM, or the desired signal for the LSTM to produce, is selected as y(t) = − ˜f(xp(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (31) Therefore, LSTM provides a signal to the system that is an estimate of this truth, expressed as ulstm(t) = ˆ y(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (32) During training, an uncertainty ftrain(xp(t)) is intro- duced to the closed loop system, and LSTM is expected to learn ˜ftrain(xp(t)) = ftrain(xp(t)) − ˆf(xp(t)), where − ˆf(xp(t)) is the control signal produced by ANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Apart from ˜ftrain(xp(t)), LSTM uses the state-tracking error e(t), defined in (12), as its input (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Every iteration of training affects the sequence that LSTM is trained on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' This dynamic nature of training helps LSTM generalize to a set of functions that are not used in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' This is further emphasized in the Simulations Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Since there is no initial data set, the data needs to be collected by running the simulation with the initial LSTM weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Since weight updates of the LSTM network also affect the system dynamics, new data needs to be collected after each episode of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Before giving the LSTM output ulstm to the system, a gain “klstm” scales ulstm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The role of this gain is to avoid an untrained LSTM network output from negatively affecting the system at the beginning of the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' klstm starts from 0 and approaches 1 as the number of training iterations increases, then stays at 1 for the rest of the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The training loop can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' LSTM weights are initialized and klstm is set to zero System is simulated e and y are obtained LSTM weights are updated klstm is increased Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 3: Flow chart of the training process B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Normalization The inputs of the LSTM network are the components of the tracking error vector e, defined in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' In order to make sure that these components are on a similar scale, normaliza- tion needs to be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' However, since an initial training data set does not exist, data needs to be collected first in order to acquire the parameters for normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' For collecting the normalization parameters only, the simulation is run with ftrain (see explanations after (32)), z number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' A gain is uniformly sampled from [0, 1] in every simulation run, to scale ftrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' This random scaling is used only during the normalization parameter collection, not during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='The LSTM network output ulstm is not injected to the system during this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Within the collected data, the minimum, emin i , and the maximum, emax i , values, where ”i” is used to refer the ith component, of each component of the error vector, are utilized to obtain the normalized input parameters as enormi = (ei − emin i )/(emax i − emin i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (33) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Stability Analysis The following theorem displays the stability properties of the overall architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Theorem 1: Consider the uncertain plant dynamics (1), subject to Assumptions 1 and 2, and the reference model (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Let the control input be defined by (13), which consists of the baseline controller (6), the ANN controller defined by (14), (15) and (16), the robustifying term given in (19) and (20), and the LSTM controller explained in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Then, given that xp(0) ∈ Sp (see Assumption 1), the solution (e(t), ˜W(t), ˜V (t)) is uniformly ultimately bounded (UUB) and converges to a predefined compact set, where ˜V (t) ≜ V − ˆV (t) and ˜W(t) ≜ W − ˆW(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Proof: The proof is deferred to the Appendix II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' SIMULATIONS In this section, the performance of the proposed control framework is examined using the short-period longitudinal flight dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The short-period dynamics is given as [21] � ˙α ˙q � = � Zα mU 1 + Zq mU Mα Iy Mq Iy � �α q � + � Zδ mU Mδ Iy � (u + f(xp)), (34) where α is the angle of attack (deg), q is the pitch rate (deg/s), u is the elevator deflection (deg), and f(xp) is a state dependent matched uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Elevator magnitude and rate saturation limits are set as +17/ − 23 (deg) and +37/ − 37 (deg/s) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The commanded input is the pitch rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Zα, Zq, Mα, Mq, Zδ and Mδ are the stability and control derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The system matrices for a B-747 aircraft flying at the speed of 274 m/s at 6000 m altitude are given as [16] Ap = �−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='86 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='93 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='43 � , Bp = �−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='02 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='16 � , Cp = �0 1�T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (35) The augmented state vector is x(t) = �x1(t) x2(t) x3(t)�T , (36) where x1 is the angle of attack (rad), x2 is the pitch rate (rad/s) and x3 is the error-integral state (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The input to the LSTM network is, inlstm(t) = �enorm1(t) enorm2(t) enorm3(t)�T , (37) where enorm1, enorm2, and enorm3 are the components of the normalized version of the state tracking error vector (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The baseline controller (6) is an LQR controller with cost matrices QLQR = I and RLQR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The nonlinear uncertainty, ftrain (see explanations after (32)) that the Long Short-Term Memory (LSTM) network (29) is trained on is defined as ftrain(x) = � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='1x1x2 if 0 ≤ t < 5 exp(x2) if 5 ≤ t < 10 2x1x2 if 10 ≤ t < 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='1 cos(x1) if 15 ≤ t < 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='5(x1x2) if 20 ≤ t < 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='1x1x2 if 25 ≤ t < 30 −x1x2(sin(5x1x2) + 5 sin(x2)) if 30 ≤ t < 35 x1x2(3 sin(2x1x2) + 2x1) if 35 ≤ t < 45 −x1x2(2(tan(2x1x2) + x2 2) if 45 ≤ t < 55 x1x2(x1 + x2) if 55 ≤ t ≤ 60 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (38) The LSTM network is trained in the presence of an ANN controller (see Section II-A) that has a hidden layer consist- ing of four neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The learning rates in (16a) and (16b) are set as F = G = 10, and the Lyapunov matrix Q in (18) is set as Q = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The outer weights ( ˆW) and bias (ˆbw) are initialized to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The inner weights ( ˆV ) and biases (ˆbv) are initialized randomly between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' In the simulations, the robustifying gain kz in (20) and scalar gain κ in (16a) and (16b) are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The LSTM network contains one hidden layer with 128 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The number of neurons in the input layer is 3 due to the number of state-tracking error components given as input to the network (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Weights are initialized using Xavier Initialization [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The network is trained with stochastic gradient descent, and uses the Adam optimizer with a learn- ing rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='001, an L2 regularization factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='0001, a gradient decay factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='9, and a squared gradient decay factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='999 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The gradient clipping method is used with a threshold value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The simulation step time is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='01 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The minibatch size is taken as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' To cover both low and high-valued uncertainties, ftrain is scaled by a parameter that took the alternating values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' To obtain the normalization constants given in (33), the system is simulated 1000 times (z = 1000), with the gain kf, (see Section III-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The loss function is chosen as Mean Squared Error (MSE) given as L(y, ˆy) = 1 N N � i=0 (y − ˆy)2, (39) where y and ˆy are defined in (31) and (32),and N is the number of data in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The proposed control framework is tested using the un- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 4: Tracking performances with and without LSTM augmentation, in the presence of small uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' certainty defined as ftest(x) = � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 0 if 0 ≤ t < 2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='1 exp(x1x2) if 2 ≤ t < 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='5x2 2 if 8 ≤ t < 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='05 exp(x1 + 2x2) if 12 ≤ t < 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='1 sin(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='05x1x2) if 20 ≤ t < 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='1(x2 1 + x3 2) if 28 ≤ t < 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='1 � cos(x2)2 if 34 ≤ t < 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='2 sin(x1x2) if 40 ≤ t < 49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='1(x1x2)2 if 49 ≤ t < 53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='5x2 if 53 ≤ t < 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='1(|x2|x1) if 60 ≤ t < 67 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='5 sin(x2) if 67 ≤ t < 79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='01 exp(x1) if 79 ≤ t < 86 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='4x2 1 if 86 ≤ t < 98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='2x2 2 if 98 ≤ t ≤ 110 , (40) where the function is chosen to have different types of sub-functions with different time intervals, compared to the training uncertainty (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Controller performance in the presence of small uncer- tainty In this section, the effect of the proposed LSTM aug- mentation is examined using low-value uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' For this purpose, ftest is scaled by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='1 during the tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Then, the tracking performance, tracking error, and control inputs of the closed-loop system are compared with and without LSTM augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Figures 4 and 5 show the tracking and tracking error curves, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' While the error plots demonstrate that LSTM augmentation dramatically reduces both the error magnitudes and the transient oscillations, the error values are small enough to be ignored compared to the absolute values of the states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Therefore, one can conclude that in this scenario Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 5: Tracking errors with and without LSTM augmenta- tion, in the presence of small uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 8 9 10 11 12 14 13 12 11 10 Zoomed 8 9 10 11 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='4 Zoomed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 6: Control inputs in the presence of small uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Top: Total control input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Middle: The contributions of indi- vidual control inputs in the absence of LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Bottom: The contributions of individual control inputs in the presence of LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' LSTM augmented and not-augmented cases show similar performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 6, the control inputs are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' This figure shows that although LSTM augmentation does not affect the magnitude of the total control input in a meaningful manner, it provides damping to the oscillations by providing small but fast compensation to the introduced uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' As we discussed above, since the ANN controller is already successful in compensating for the uncertainties, in this case, LSTM contribution is not prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Controller performance in the presence of large uncer- tainty In this section, the effect of the proposed LSTM augmenta- tion is examined in the presence of high-valued uncertainties, which is obtained by using the ftest as is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 7: Tracking performances with and without LSTM augmentation, in the presence of large uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 8: Tracking errors with and without LSTM augmenta- tion, in the presence of large uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Figures 7 and 8 demonstrate that LSTM augmentation substantially improves the transient response of the system, especially in pitch rate tracking, which is the output of interest (see (35)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' It is noted that for this case, the same ANN controller and the LSTM network are used as in the small uncertainty case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Individual control inputs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' In this case, unlike the low-uncertainty case, the LSTM augmentation makes the total control signal observ- ably more agile, which is the main reason why excessive oscillations are prevented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' This shows the LSTM network uses its memory to predict high-frequency changes in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' It is noted that in this large uncertainty case the total control input is saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' This is evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 9, where the middle and the bottom sub-figures show the control signals created by the individual components of the overall controller, while the top sub-figure shows the total control input after saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The figure shows that the LSTM network produces large and very fast compensation, which causes saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Here, the rate saturation can be problematic since it creates some oscillations in the total control signal (see the zoomed section, blue line, in the top sub-figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Although the oscillations could be acceptable since they are short-lived, we believe that this issue can also be tackled by training the LSTM network with saturation information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' We explain this solution in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Addressing Saturation To handle the saturation issue, we modified LSTM training by a) introducing the saturation limits to the plant dynamics during training, and b) informing the LSTM network when- ever the control signal rate-saturates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The latter is achieved by providing additional input to the LSTM network as ur(t) = � � � � � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='1, if rate saturation is positive −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='1, if rate saturation is negative, 0, otherwise (41) 8 9 10 11 12 20 15 10 5 Zoomed 8 9 10 11 12 80 40 0 40 Zoomed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 9: Control inputs in the presence of high uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Top: Total control input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Middle: The contributions of indi- vidual control inputs in the absence of LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Bottom: The contributions of individual control inputs in the presence of LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The number of neurons in the input layer of the LSTM network is increased to 4 since the modified input to the LSTM network is inlstm(t) = �enorm1(t) enorm2(t) enorm3(t) ur(t)�T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (42) Figure 10 shows the LSTM network contribution and the total control inputs, with and without rate-limit information during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' It is seen that the LSTM network uses the rate-limit information to provide a smoother compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Figures 11 and 12 show the tracking and tracking error curves, respectively, when the LSTM network is trained using the rate-limit information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' It is seen that the perfor- mance of the proposed controller remains similar for the case 8 9 10 11 12 20 15 10 5 Zoomed 8 9 10 11 12 80 60 40 20 0 20 40 Zoomed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 10: Control signals with an without rate-limit-informed LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Top: Total control input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Bottom: LSTM network contribution to the total control input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' when LSTM is not trained using the saturation information, although the pitch rate shows initial small jumps at the instances of uncertainty switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 11: Tracking performances with LSTM trained without rate information and rate-limit-informed LSTM 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='5 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='02 Zoomed 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='5 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='00 Zoomed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 12: Tracking errors with LSTM trained without rate information and rate-limit-informed LSTM V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' SUMMARY In this work, we propose a Long Short-Term Memory (LSTM) augmented adaptive neural network (ANN) con- trol structure to improve the transient response of adaptive closed-loop control systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' We demonstrate that thanks to its time-series prediction capabilities, LSTM helps the ANN controller compensate for the uncertainties in a more agile fashion, resulting in dramatically improved tracking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' REFERENCES [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Macukow, “Neural networks–state of art, brief 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' 249–256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Kingma and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Ba, “Adam: A method for stochastic optimiza- tion,” arXiv preprint arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='6980, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' [25] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Khalil, Nonlinear Systems, 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Upper Saddle River, NJ: Prentice Hall, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' APPENDIX I ERROR DYNAMICS AND USEFUL REMARKS In this section, the derivation of the ANN controller adaptive laws (16) and the robustifying term (19) is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' We start by substituting the control input (13) and the baseline controller (6) into (4), which yields ˙x(t) =Amx(t) + Bmr(t) + B(uad(t) + ulstm(t) + v(t) + f(xp(t))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (43) Using (12), (43) and (7), the state-tracking error dynamics can be written as ˙e(t) = Ame(t) + B(uad(t) + ulstm(t) + v(t) + f(xp(t))), (44) which, by using the NN approximation property (11), can be written as ˙e(t) =Ame(t) + B(uad(t) + W T σ(V T xp(t)) + ulstm(t) + v(t) + ε(xp(t))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (45) The ANN controller uad(t) is chosen to be the output of a multi-layer NN, as defined in (14) and (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Substituting (14) and (15) into (45), yields ˙e(t) =Ame(t) + B(− ˆW T (t)σ( ˆV T (t)xp(t)) + W T σ(V T xp(t)) + ulstm(t) + v(t) + ε(xp(t))), (46) where ˆV (t) ∈ Rnh×n and ˆW(t) ∈ Rm×nh are adaptive NN weights serving as estimates for the unknown ideal weights V and W, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The weight estimation errors are defined as ˜W(t) =W − ˆW(t), (47a) ˜V (t) =V − ˆV (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (47b) Throughout the remainder of this section, we drop the time dependency notation for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Further, we denote σ ≜ σ(V T xp(t)) and ˆσ ≜ σ( ˆV T xp(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Then, (46) can be written as ˙e = Ame + B(− ˆW T ˆσ + W T σ + ulstm + v + ε(xp)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (48) Using (47a) in (48) yields ˙e = Ame + B( ˜W T ˆσ + W T (σ − ˆσ) + ulstm + v + ε(xp)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (49) Remark 4: Since r(t) is bounded and Am is Hurwitz, it follows from (7) that ∥xm(t)∥ ≤ Cm, (50) where Cm > 0 is a known constant, which depends on the user-defined reference input r(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Therefore, it follows from (12), and the fact that x(t) ≜ [xp(t)T , xe(t)T ]T ∈ Rn, that ∥xp(t)∥ ≤ ∥x(t)∥ ≤ Cm + ∥e(t)∥, (51) Remark 5: From the Taylor series expansion σ(V T xp) = σ( ˆV T xp) + ˆσ′ ˜V T xp + O( ˜V T xp)2, (52) one can write σ(V T xp) − σ( ˆV T xp) = ˆσ′ ˜V T xp + O( ˜V T xp)2, (53) where ˆσ′ is defined in (17) and O( ˜V T xp)2 denote the higher order terms in the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Furthermore, since O( ˜V T xp)2 = σ(V T xp) − σ( ˆV T xp) − ˆσ′ ˜V T xp, (54) then, for sigmoid and tanh activation functions, ∥σ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' )∥ ≤ Cσ and ∥σ′(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' )∥ ≤ Cσ′ for known constants Cσ > 0 and Cσ′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Hence, using (51), the higher-order terms in the Taylor series are bounded by ∥O( ˜V T xp)2∥ ≤ 2Cσ + Cσ′∥ ˜V ∥F Cm + Cσ′∥ ˜V ∥F ∥e∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (55) Substituting (53) for (σ − ˆσ) in (49), yields ˙e =Ame + B( ˜W T ˆσ + W T ˆσ′ ˜V T xp + W T O( ˜V T xp)2 + ulstm + v + ε(xp)), (56) which, by using (47a) an (47b), can be written as ˙e =Ame + B( ˜W T (ˆσ − ˆσ′ ˆV T xp) + ˆW T ˆσ′ ˜V T xp w + ulstm + v), (57) where w ≜ ˜W T ˆσ′V T xp + W T O( ˜V T xp)2 + ε(xp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (58) Remark 6: Let Z ≜ � W 0nh×nh 0np×m V � ∈ R(nh+np)×(m+nh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (59) Using (59) and assumption 2, one can write ∥Z∥2 F = ∥W∥2 F + ∥V ∥2 F ≤ W 2 M + V 2 M, (60) and hence, ∥Z∥F ≤ ZM, where ZM ≜ � W 2 M + V 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Furthermore, it follows from (60) that ∥W∥2 F ≤ ZM and ∥V ∥2 F ≤ ZM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Similarly, by defining ˆZ(t) ≜ � ˆW(t) 0nh×nh 0np×m ˆV (t) � ∈ R(nh+np)×(m+nh), (61) one can show that ∥ ˜W(t)∥F ≤ ∥ ˜Z(t)∥F and ∥ ˜V (t)∥F ≤ ∥ ˜Z(t)∥F , where ˜Z(t) ≜ Z − ˆZ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Remark 7: Using (9), (55) and Remark 6, one can write ∥w∥ ≤∥ ˜Z∥F Cσ′ZM(Cm + ∥e∥) + ZM(2Cσ + Cσ′∥ ˜Z∥F Cm + Cσ′∥ ˜Z∥F ∥e∥) + εN, (62) or ∥w∥ ≤C0 + C1∥ ˜Z∥F + C2∥ ˜Z∥F ∥e∥, (63) where C0 ≜2CσZM + εN, (64a) C1 ≜2Cσ′CmZM, (64b) C2 ≜2Cσ′ZM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (64c) Remark 8: Sigmoid and tanh activation functions help define a bound on the LSTM output: We can write (28) as hn = � � � σg1(· · · )σc1(· · · ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' σgnh(· · · )σgnh(· · · ) � � � ∈ Rnh, (65) where nh is the number of neurons in the last hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Then, ∥hn∥2 = nh � i σ2 gi(· · · )σ2 c i(· · · ) ≤ nh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (66) Hence, ∥hn∥ ≤ √nh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (67) The bound on the LSTM output can be defined as ∥un lstm∥ ≤ ∥Wfc∥F √nh + ∥bfc∥ ≜ ¯ulstm, (68) which by using (22) implies that ∥ulstm(t)∥ ≤ ¯ulstm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' APPENDIX II PROOF OF THEOREM 1: Let the approximation property (11) hold on a known compact set Sp ≜ {xp : ∥xp∥ ≤ bx} as stated in Assumption 1, for some bx > Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Defining S ≜ {x : ∥x∥ ≤ bx}, bx > Cm, (69) and using the fact that ∥xp∥ ≤ ∥x∥, x ∈ S implies that xp ∈ Sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Consider the compact set Se ≜ {e : ∥e∥ ≤ bx − Cm}, (70) which imply that once e ∈ Se, then xp ∈ Sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Let e(0) ∈ Se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Then, xp(0) ∈ Sp and (11) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' The proof proceeds by showing that e(t) ∈ Se, ∀t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Consider the Lyapunov function V = eT Pe + tr{ ˜W T F −1 ˜W} + tr{ ˜V T G−1 ˜V }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (71) Differentiating (71) along the trajectory (57) yields, ˙V = ˙eT Pe + eT P ˙e + 2tr{ ˜W T F −1 ˙˜W} + 2tr{ ˜V T G−1 ˙˜V } =eT AT mPe + eT PAme + 2eT PB ˜W T (ˆσ − ˆσ′ ˆV T xp) + 2eT PB ˆW T ˆσ′ ˜V T xp + 2eT PB(w + ulstm + v) + 2tr{ ˜W T F −1 ˙˜W} + 2tr{ ˜V T G−1 ˙˜V }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (72) Using (18) and tr{A1A2} = tr{A2A1} yields ˙V = − eT Qe + 2tr{ ˜W T (ˆσ − ˆσ′ ˆV T xp)eT PB} + 2tr{ ˜V T xpeT PB ˆW T ˆσ′} + 2eT PB(w + ulstm + v) + 2tr{ ˜W T F −1 ˙˜W} + 2tr{ ˜V T G−1 ˙˜V }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (73) It follows from (47) that ˙˜W = − ˙ˆW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Substituting (16) into (73) yields ˙V = − eT Qe + 2eT PB(w + ulstm + v) + 2tr{ ˜W T κ∥e∥ ˆW} + 2tr{ ˜V T κ∥e∥ ˆV }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (74) Using (47), (59), and (61), one can write ˙V = − eT Qe + 2eT PB(w + ulstm + v) + 2κ∥e∥tr{ ˜W T (W − ˜W)} + 2κ∥e∥tr{ ˜V T (V − ˜V )} = − eT Qe + 2eT PB(w + ulstm + v) + 2κ∥e∥tr{ ˜ZT (Z − ˜Z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (75) Since tr{ ˜ZT (Z − ˜Z)} =tr{ ˜ZT Z} − ∥ ˜Z∥2 F ≤ ∥ ˜Z∥F (ZM − ∥ ˜Z∥F ), (76) it follows from (75) that ˙V ≤ − λmin(Q)∥e∥2 + 2∥BT Pe∥(∥w∥ + ∥ulstm∥) + 2eT PBv + 2κ∥e∥∥ ˜Z∥F (ZM − ∥ ˜Z∥F ), (77) which, by using (63) and Remark 8, can be written as ˙V ≤ − λmin(Q)∥e∥2 + 2∥BT Pe∥(C0 + ¯ulstm) + 2C1∥BT Pe∥∥ ˜Z∥F + 2C2∥BT Pe∥∥e∥∥ ˜Z∥F + 2eT PBv + 2κ∥e∥∥ ˜Z∥F (ZM − ∥ ˜Z∥F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (78) The proposed robustifying term (19) is designed to cancel the 4th term of (78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Two cases follow from using (19) in (78): a) ∥BT Pe∥ = 0 and b) ∥BT Pe∥ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Case a) ∥BT Pe∥ = 0: In this case, v = 0, and (78) reduces to ˙V ≤ − λmin(Q)∥e∥2 + 2κ∥e∥∥ ˜Z∥F (ZM − ∥ ˜Z∥F ) = − ∥e∥(λmin(Q)∥e∥ + 2κ∥ ˜Z∥F (∥ ˜Z∥F − ZM)) = − ∥e∥ � λmin(Q)∥e∥ + 2κ(∥ ˜Z∥F − ZM 2 )2 − κZ2 M 2 � , (79) which implies that ˙V < 0 if either ∥e∥ > κZ2 M 2λmin(Q), (80) or ∥ ˜Z∥F > ZM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (81) Case b) ∥BT Pe∥ ̸= 0: Then, by using ∥ ˜Z∥F ≤ ∥ ˆZ∥F + ZM, and substituting (19) into (78), it follows that ˙V ≤ − λmin(Q)∥e∥2 + 2∥BT Pe∥(C0 + ¯ulstm) + 2C1∥BT Pe∥∥ ˜Z∥F + 2C2∥BT Pe∥∥e∥(∥ ˆZ∥F + ZM) + 2eT PBv + 2κ∥e∥∥ ˜Z∥F (ZM − ∥ ˜Z∥F ) = − λmin(Q)∥e∥2 + 2∥BT Pe∥(C0 + ¯ulstm) + 2C1∥BT Pe∥∥ ˜Z∥F + 2κ∥e∥∥ ˜Z∥F (ZM − ∥ ˜Z∥F ) + 2C2∥BT Pe∥∥e∥(∥ ˆZ∥F + ZM) − 2kz∥BT Pe∥∥e∥(∥ ˆZ∥F + ZM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (82) Since kz > C2, (82) reduces to ˙V ≤ − λmin(Q)∥e∥2 + 2∥BT Pe∥(C0 + ¯ulstm) + 2C1∥BT Pe∥∥ ˜Z∥F + 2κ∥e∥∥ ˜Z∥F (ZM − ∥ ˜Z∥F ) ≤ − λmin(Q)∥e∥2 + 2∥BT P∥F ∥e∥(C0 + ¯ulstm) + 2C1∥BT P∥F ∥e∥∥ ˜Z∥F + 2κ∥e∥∥ ˜Z∥F (ZM − ∥ ˜Z∥F ) = − ∥e∥ � λmin(Q)∥e∥ − 2∥BT P∥F (C0 + ¯ulstm) − 2C1∥BT P∥F ∥ ˜Z∥F − 2κ∥ ˜Z∥F (ZM − ∥ ˜Z∥F ) � = − ∥e∥ � λmin(Q)∥e∥ + 2κ(∥ ˜Z∥F − q1 2 )2 − κq2 1 2 − 2q0 � , (83) where q0 ≜ ∥BT P∥F (C0 + ¯ulstm) and q1 ≜ C1∥BT P∥F /κ + ZM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Therefore, it follows that ˙V < 0 if either ∥e∥ > κq2 1 + 4q0 2λmin(Q), (84) or ∥ ˜Z∥F > q1 2 + � q2 1 4 + q0 κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (85) Hence, using the fact that the bounds given in (84) and (85) are bigger than the bounds in (80) and (81), and by selecting the Lyapunov matrix Q such that λmin(Q) > κq2 1 + 4q0 2(bx + Cm), (86) it can be shown that the solution (e(t), ˜Z(t)) is uniformly ul- timately bounded (UUB) [25], and converges to the compact set E ≜ � (e, ˜Z) :∥e∥ ≤ κq2 1 + 4q0 2λmin(Q) < bx + Cm and ∥ ˜Z∥F ≤ q1 2 + � q2 1 4 + q0 κ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' (87) This implies that, since e(0) ∈ Se and ˙V < 0 on the boundary of Se, e(t) stays within Se for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Therefore xp(t) ∈ Sp for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Since the reference input r(t) is bounded, xm(t) is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' Therefore, the boundedness of e(t) = x(t)−xm(t) implies that x(t), and therefore all system signals, are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} +page_content=' ■' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE0T4oBgHgl3EQfZADU/content/2301.02316v1.pdf'} diff --git a/mNE5T4oBgHgl3EQfHg55/content/tmp_files/2301.05440v1.pdf.txt b/mNE5T4oBgHgl3EQfHg55/content/tmp_files/2301.05440v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f07e9fefb2be440aec02ca6050704d3cd227eef8 --- /dev/null +++ b/mNE5T4oBgHgl3EQfHg55/content/tmp_files/2301.05440v1.pdf.txt @@ -0,0 +1,1176 @@ +NeuralNetworks +Learnable Heterogeneous Convolution: Learning both topology and +strength +Rongzhen Zhao, Zhenzhi Wu ∗, Qikun Zhang +Lynxi Technologies, Beijing 100097, China +a r t i c l e +i n f o +Article history: +Received 8 August 2020 +Received in revised form 10 March 2021 +Accepted 29 March 2021 +Available online 20 April 2021 +Keywords: +Convolution neural network +Efficiency & performance +Learning topology & strength +Fine-grained but structural +Hardware acceleration +a b s t r a c t +Existing convolution techniques in artificial neural networks suffer from huge computation complexity, +while the biological neural network works in a much more powerful yet efficient way. Inspired +by the biological plasticity of dendritic topology and synaptic strength, our method, Learnable +Heterogeneous Convolution, realizes joint learning of kernel shape and weights, which unifies existing +handcrafted convolution techniques in a data-driven way. A model based on our method can converge +with structural sparse weights and then be accelerated by devices of high parallelism. In the +experiments, our method either reduces VGG16/19 and ResNet34/50 computation by nearly 5× on +CIFAR10 and 2× on ImageNet without harming the performance, where the weights are compressed +by 10× and 4× respectively; or improves the accuracy by up to 1.0% on CIFAR10 and 0.5% on +ImageNet with slightly higher efficiency. The code will be available on www.github.com/Genera1Z/ +LearnableHeterogeneousConvolution. +1. Introduction +Convolution neural networks (CNNs) are showing their supe- +rior performance in vision tasks like classification, detection and +segmentation, but their advantages in practice are impeded by +their heavy computation. +To improve CNN efficiency, researchers have developed many +recipes: (1) compressing existing models with pruning (Liu et al., +2019; Zhou, Zhang, Wang, & Tian, 2019), quantization (Cao et al., +2019; Gong et al., 2019) or knowledge distillation (Jin et al., 2019; +Peng et al., 2019); (2) designing efficient network structures +either by hand (Hu, Shen, & Sun, 2018) and Ma, Zhang, Zheng, and +Sun (2018) or by automatic search (Chen, Xie, Wu, & Tian, 2019; +Howard et al., 2019); (3) exploiting efficient convolution oper- +ators, which are either smaller (Simonyan & Zisserman, 2014), +or factorized (Szegedy, Ioffe, Vanhoucke, & Alemi, 2016; Szegedy, +Vanhoucke, Ioffe, Shlens, & Wojna, 2016), or sparse (Huang, Liu, +M.L.V., & Weinberger, 2018; Sun, Li, Liu, & Wang, 2018). On the +other hand, (4) the neural network in a brain, which features +sparse activation and high dynamics (Holtmaat et al., 2005; Stet- +tler, Yamahachi, Li, Denk, & Gilbert, 2006), works in a much more +complex and powerful yet efficient way (Merolla et al., 2014). +∗ Corresponding author. +E-mail addresses: +rongzhen.zhao@lynxi.com (R. Zhao), +zhenzhi.wu@lynxi.com (Z. Wu), qikun.zhang@lynxi.com (Q. Zhang). +For recipe (1), a large model of high performance must be +available firstly, and the pruning-retrain iteration is very time- +consuming; for recipe (2), it is hard to design an excellent struc- +ture by hand, and also too costly to search one even with tens +of GPUs and days. So we choose to combine recipes (3) and (4), +namely, sparsifying convolutions by learning from the brain. +A biological neuron connects from multiple predecessor neu- +ron via dendrites, and to a subsequent neuron via an axon. The +connections are plastic both in topology and strength, through +forming/losing/developing +the +synapses +on +dendritic +spines +(Bhatt, Zhang, & Gan, 2009; Harms & Dunaevsky, 2007). To make +an analogy (Beysolow II, 2017), like Fig. 1, for the co kernels +in a standard convolution layer, each kernel of shape (k, k, ci) +belongs to a neuron, which is shared at different spatial positions +of feature maps; for the ci slices in a kernel, each kernel slice +of shape (k, k) is a dendrite; for the k ∗ k elements in a slice, +each element is a synapse. But unlike biological neurons, in a +standard convolution layer, only the strength of kernel elements, +i.e., weights is learnable, while the topology of kernel slices is not, +let alone the number of slices and kernels. Such differences are +where the CNN can learn from the brain. +Improved convolution techniques like Group/Depth/Point- +Wise Convolution (GWC, DWC, PWC) can be seen as sparsifica- +tion of neurons in a layer or dendrites in a neuron, which are, +however, predefined instead of learnt (Alex, Sutskever, & Hinton, +2012; Sandler, Howard, Zhu, Zhmoginov, & Chen, 2018); works +like SeeSaw and DeepR introduce in learnability (Guillaume, +David, Wolfgang, & Robert, 2018; Zhang, 2019). Other works like + +Fig. 1. A convolution layer (green, the biases and activation are ignored) be- +tween input feature maps (orange) and output features (blue). For the standard +convolution, 1⃝ the number of kernels in a layer is fixed co, 2⃝ the number of +slices in each kernel is fixed ci, 3⃝ the shape of each kernel slice (topology) is +fixed (k, k), and only 4⃝ the value of the k∗k elements (strength) in each kernel +slice can be learnt. Our method makes 1⃝∼ 4⃝ all learnable. (For interpretation +of the references to color in this figure legend, the reader is referred to the web +version of this article.) +Inception and HetConv sparsify the dendritic topology square to a +row, line or dot, which are also not learnable (Singh, Verma, Rai, +& Namboodiri, 2019; Szegedy et al., 2015); works like L0 training +hit the mark by a fluke, but the weights learnt is non-structural, +which is not friendly for hardware to accelerate (Christos, Max, & +Diederik, 2018). See Section 2 for detailed reviewing. +Our method, Learnable Heterogeneous Convolution (LHC), a +seamless replacement to the standard convolution, is proposed +to break those limitations by integrating the plasticity in both +the synaptic strength, i.e., learnable weights, which is originally +possessed by the standard convolution, and additionally the den- +dritic topology, i.e., learnable kernel slice shapes. With the latter +as a unit, the number of slices in a neuron, and the number of +neurons in a layer all become learnable. +Comprehensive experiments demonstrate the superiority of +our method. At parallelism = 512, suppose batch size is 1, we can +either reduce VGG/ResNet computation by nearly 5× on CIFAR10 +and 2× on ImageNet without harming the performance, where +the weights are compressed by 10× and 4× respectively; or +improve the accuracy by up to 1.0% on CIFAR10 and 0.5% on +ImageNet with slightly higher efficiency. The extra costs are no +more than slightly longer training time. +Our contributions are: +(1) LHC is proposed to realize dual-plasticity of strength and +topology in convolution, which is fine-grainedly yet structurally +sparse thus fits hardware acceleration; +(2) It sparsifies convolutions in full back-propagation, instead +of undifferentiable ways used by most pruning methods such as +cutting of weights that have least magnitude; +(3) It requires negligible extra costs at training and can greatly +improve CNN models’ efficiency at inference stage even with +some performance gain. +The remaining content is organized as follows: related works +are reviewed in Section 2; the proposal is detailed in Section 3; +how our method unifies various convolution techniques is ana- +lyzed in Section 4; experiments are presented in Section 5; more +advanced analyses are discussed in Section 6; the conclusion is +drawn in Section 7. +Given a convolution layer, the notations list in Table 1. +Table 1 +Notations used in this article. +Shape of the kernels +(k, k, ci, co) +Shape of input feature maps or ‘‘feat in’’ +(b, hi, wi, ci) +Shape of output feature maps or ‘‘feat out’’ +(b, ho, wo, co) +Batch dimension of feature maps +b +Number of input/output channels +ci, co +Height/width of input/output features +hi, wi, ho, wo +Spatial size of a kernel +k ∗ k, (k, k) +Topology constraint of input/output channels +cgi, cgo +Tera Operations Per Second +TOPS +Floating Point Operation(s) +FLOP(s) +Multiply Accumulate (unit) +MAC +2. Related works +Here the sparse convolution techniques for improving CNN +efficiency are reviewed, and keypoints where the convolution can +imitate the brain are inducted. +Structural Sparse Convolutions: Handcrafted +Convolution can be sparsified in spatial dimensions. A 2D +convolution kernel is factorized into two perpendicular 1D ones +in Szegedy, Ioffe, Vanhoucke, and Alemi (2016) and Szegedy, +Vanhoucke, Ioffe, Shlens, and Wojna (2016). Kernels are designed +into incremental sizes in Tan and Le (2019). Slices of a 3*3 kernel +are replaced by 1*1 sizes at intervals in Singh et al. (2019) shown +in Fig. 8. +Sparsification can also be taken in channels. In a GWC (SIfre & +Mallat, 2014), the input channels are grouped and each output +channel is correlated with one of the groups. Features among +different groups are further exchanged in Sun et al. (2018), Xie +et al. (2018) and Zhang, Qi, Xiao, and Wang (2017). Shifting +or negation saves kernels/channels either, like in Shang, Sohn, +Almeida, and Lee (2016) and Yan, Li, Li, Zuo, and Shan (2018). +Dimensions of space and channel can be considered together. +Like in Wang, Xu, Chunjing, Xu, and Tao (2018), standard kernels +are turned into secondary kernels, which are degressive in space +and partially connected in channel. +Neuron number in a layer, dendrite number in a neuron, and +dendritic topology are all handcrafted in such methods; only the +synaptic strength is learnable. +Structural Sparse Convolutions: Learnt +Compared with the above, works thinking in this way bring +in learnability of sparsity. But the techniques employed are still +restricted to GWC, DWC and PWC. +Works like Huang et al. (2018) require too many iterations, +where connections between input and output channels of less +importance are cut off progressively while training to get a de- +sired group number. Some realize learnability under the guidance +of Singular Value Decomposition like Peng et al. (2018), which is +built of GWC and PWC in a way similar to Howard et al. (2017). +Others like Zhang (2019) realize unevenly grouped GWC layers in +a model by fusing network architecture search into training. +Only sparsity in channel dimension is considered by them. +Namely, besides the synaptic strength, they do take into account +the plasticity of neurons number in a layer or dendrite number +in a neuron, but overlook the dendritic topology. +By the way, Verelst and Tuytelaars (2020) is worth learning +from, which generates feature masks to skip computations cor- +responding to zeros, even if it sparsifies activations instead of +weights. +Non-Structural Sparse Convolutions: Learnt +Such works are similar to the pruning methods (Liu et al., +2019; Zhou et al., 2019), except that the sparsity is gained dur- +ing rather than after training. Since non-structural sparsity goes +against hardware acceleration (Deng, Li, Han, Shi, & Xie, 2020), +recent works are not that many. + +Fig. 2. Kernels of uncommon shapes designed handcrafted by SWF (Yin et al., +2019) to address specific image contents, which inspired our work. +L0 regularization are often used to train models full of ze- +ros, like Christos et al. (2018), enabling joint optimization of +weights’ value and topology via non-negative random gates of +hard concrete distribution. Inspired by dynamic connections in +the brain Bhatt et al. (2009) and Harms and Dunaevsky (2007), +the rewiring mechanism is proposed by Guillaume et al. (2018) +to enable simultaneous learning of connections and weights for +constrained hardware resources. +Their implementation of plasticity in dendritic topology, +namely, neuron number in a layer or dendrite number in a +neuron, are good references. +Our inspiration: making it possible in a convolution layer to +(a) learn the dendritic topology and synaptic strength at the +same time, and (b) realize fine-grained yet structural sparsity for +hardware acceleration. +3. Proposed method +How LHC integrates the plasticity in dendritic topology and +synaptic strength is elaborated here. The structural sparsifica- +tion and hardware acceleration of LHC-based models are also +described. +3.1. Prototype +Convolution Kernels of Uncommon Shapes +For either traditional or CNN-based algorithms, it is quite +common to use convolution kernels of square shapes. However, +SWF (Yin, Gong, & Qiu, 2019), a latest work, broke fresh ground +and is equal in force compared with CNN-based algorithms. The +key is that kernels of uncommon shapes, shown in Fig. 2, were +meticulously designed for different image patterns to gain excel- +lent feature extraction capability. +Enlightened by this, we empirically design 15 rigid kernel +shapes, shown in Fig. 3(a). Among them, shape ⟨1⟩1, ⟨2⟩1 and +⟨6⟩1 are common in use; shape ⟨3⟩1 and ⟨3⟩3 are already used in +GoogLeNet Inceptions; shape group ⟨4⟩ and ⟨5⟩ inherit from SWF. +Shape group ⟨2⟩∼⟨5⟩ are designed to sparsify a convolution layer +in spatial dimensions, while group ⟨1⟩ is to reduce redundancy in +channel dimension. +The theoretical foundation for such a design is that CNNs +are able to extract visual patterns of different levels like dot, +edge, curve and plane at different layers (Goodfellow, Bengio, & +Courville, 2016). Accordingly, very sparse shape groups ⟨2⟩ and +⟨3⟩ are designed to address simple patterns like dots and edges; +shape groups ⟨4⟩∼⟨6⟩, not that sparse, are to handle complex +patterns like curves and planes. +Further on, we relieve the constraints of the aforementioned +rigid shapes, and for a 3*3 kernel, we provide 2 +3∗3 = 512 free +shapes for an LHC layer to choose, shown in Fig. 3(b). Obviously, +the rigid shapes are a sub-set of the free shapes. +An LHC layer equipped with the rigid shapes is called LHCR, +and LHC with the free shapes is LHCF. In LHC, these shapes are +formed as masks consisting of 0/1 elements, and are paired with +corresponding effect factors to guide the learning of the shapes +of kernel slices. Details are presented in Section 3.3. +Fig. 3. Two sets of uncommon shapes for LHC. (a) 15 rigid shapes: designed +empirically in six groups; digits on the top and right are their indexes, e.g., ⟨3⟩4. +(b) 23×3 = 512 free shapes: generated by arbitrarily setting each element to 0 or +1; digits on every top-right corner are their serial indexes, e.g., 502. The former +is a sub-set of the latter. +To make a comparison, LHCR implies priori knowledge of +those rigid shapes while LHCF has more freedom in sparsification, +which is verified by experiments in Section 5. +Learnable Heterogeneous Convolution +For a Learnable Heterogeneous Convolution (LHC) layer, its +kernel slices are learnt to be these uncommon shapes, rather than +pre-defined. +Suppose a convolution layer with ci input channels and co +output channels. With the aforementioned uncommon shapes, +the kernels can be shaped (1) kernel-by-kernel (KbK), where +shapes are the same within a kernel and different among kernels, +or (2) slice-by-slice (SbS), where shapes are different both within +a kernel and among kernels. Refer to Fig. 4 the top two rows. +Obviously, SbS is more flexible than KbK and thus can elim- +inate the convolution redundancy in both spatial and channel +dimensions better; but the computation graph of SbS is too frag- +mented for the hardware to accelerate. So we split the difference: +only one shape combination is allowed for every adjacent cgi +slices and every adjacent cgo kernels. Then we can learn a struc- +tural sparse LHC layer, which supports parallelism up to cgi∗cgo ≡ +64 ∗ 8. Refer to the bottom two rows of Fig. 4. Besides, these +constraints happen to be a kind of regularization, benefiting the +performance. See Section 6 for details. +3.2. Computation reduction +To facilitate discussion, we take the 3*3 convolution as an +example, and use the indexes in Fig. 3 to represent shapes. +Following the aforementioned setting, the computation quan- +tity or the total number of multiplication–addition of a standard +convolution layer is: +CSTD = ho × wo × ci × co × ∥s511∥0 += ho × wo × ci × co × 9 +(1) +where s511 the 511th shape in Fig. 3 and its L0 norm is 9. + +Fig. 4. Topology of LHC Kernels. 1st row: LHC-KbK, where kernel slices have +the same shapes within a kernel and different shapes among kernels; 2nd row: +LHC-SbS, where slices have different shapes both within and among kernels; +3rd row: LHCR with topology constraints cgi and cgo, where slices are shaped +into those rigid shapes and every cgi ∗ cgo slices have the same shape; 4th row: +LHCF, where slices are shaped into those free shapes. +Similarly, the computation quantity of an LHC layer is: +CLHC = ho × wo × ( +co/cgo +∑ +y=1 +( +ci/cgi +∑ +x=1 +∥sx,y∥0 × cgi) × cgo) += ho × wo × cgi × cgo × +ci/cgi×co/cgo +∑ +s=1 +ns +(2) +where cgi and cgo are the aforementioned topology constraints; +sx,y is the shape of the xth slice in the yth kernel; ns is L0 norm +of the sth shape in all ci/cgi × co/cgo positions, i.e., ∥sx,y∥0 = ns. +So, the computation reduction is: +∆C = ho × wo × (ci × co × 9 +− cgi × cgo × +ci/cgi×co/cgo +∑ +s=1 +ns) +(3) +when ns = 9, ∆C = 0, where all shapes in LHC is s511, i.e., the +standard convolution; when ns = 0, ∆C = CSTD, where all shapes +in LHC is s0, which means all features are redundant and thus can +only be approximated. +Besides, existing convolution techniques can be unified by +LHC. See Section 4 for details. +3.3. Learnability +At the training stage, the only difference between LHC and a +standard convolution is the construction of the kernels. As shown +in Fig. 5, the masks composed of zeros and ones are constructed +first, then multiplied with the kernels, where elements multiplied +with zeros in the masks are deactivated during both forward +and back-propagation; finally, the masked kernels are convolved +with the input features, and the output features are obtained. +The masks and kernels realize plasticity in dendritic topology and +synaptic strength respectively. +Guide the Learning of Topology +For a convolution layer, its computation quantity is propor- +tional to the density of its kernels; for a CNN model, its com- +putation quantity is positively correlated with its global density, +Fig. 5. Training principle of LHC. Left box: (1) every cgi ∗ cgo kernel slices are +equipped with a set of effect factors, each of which points to a shape belonging +to the rigid shapes or the free; (2) through a differentiable step function, a mask +slice is got then tiled into masks partial of shape (k, k, cgi, cgo). Center box: do +(2) for every cgi∗cgo slices in the kernels, then the masks of shape (k, k, ci, co) are +constructed. Right box: (3) multiply the masks with the kernels element-wisely; +(4) do the convolution with the masked kernels and the input feature maps of +shape (b, hi, wi, ci), finally get the output features of shape (b, ho, wo, co). (5) +In the backward propagation, the shapes get different gradients according to +their contributions. (6) As training goes on, the advantage of different shapes is +accumulated in the effect factors, and the most suitable shapes gradually win +out. Note: in the left box, purple stars are where gradients pass during back +propagation; purple stars of the other two boxes are omitted; in the right box, +the biases and activation are omitted for simplicity. +of which the maximum is 1, namely, no sparsification, and the +minimum limit is 0. +Given a model with L LHC layers, we can set a global density +target and calculate mask regularization loss; then by minimizing +it, we can guide the model to converge to a state that costs much +less computation: +lmask = |dt − +∑L +l=1 ∥Ml∥1 +∑L +l=1 size(Ml) +| +(4) +where lmask is the mask regularization loss; dt is the density +target; Ml is the masks in the lth LHC layer. +Here, L1 norm is used instead of L0, because (1) Ml only +consists of zeros and ones, which means L0 is equal to L1; and +(2) L1 norm is differentiable. +Hence, optimizing this model turns into simultaneously min- +imizing the mask regularization loss and the task loss, e.g., cate- +gorical cross-entropy loss: +min +M1,...ML +α × lmask + ltask +(5) +s.t. +cgi = C1, cgo = C2 +(6) +where α is a positive constant; ltask is the task loss; cgi and cgo are +the aforementioned topology constraints for structural sparsity, +and their value C1 ≡ 64 and C2 ≡ 8 typically. +Now the point is how to construct the Ml. +Given the kernels of a layer and the topology constraints cgi +and cgo, to construct the masks, we have ci/cgi × co/cgo positions +to determine. In other words, for each of these positions, we need +to calculate a mask slice, a k∗k matrix, using the aforementioned +shapes that are either rigid or free. +Enroll Shapes into Competition +To calculate a mask slice, we hope those shapes illustrated in +Fig. 3 are enrolled in all together so that they compete with one +another along with the training steps, letting the fittest eventually +win out. +For LHCR, at each of these positions, a 15D vector e called +effect factors is used to represent the effects of those 15 rigid + +shapes. So the mask slice for this position is: +m = +15 +∑ +i=1 +step(ei, e) × si +(7) +step(ei, e) = +{ +1 +ei = max(e) +0 +ei < max(e) +(8) +∇step(ei, e) ≡ +{ +1 +|ei − mean(e)| < 1 +c +others +(9) +where ei is the ith element in e and si is the ith of those 15 rigid +shapes; c is a small positive constant, empirically set to 0.1; step +is differentiable and ensures the summation of effect factors be +one. softmax is not used because it is heavy to calculate and can +hardly make one of the shapes win out. +For LHCF, equip each of these positions with a 3*3 matrix e as +the effect factors, which represent the effects of those 512 free +shapes for this position. So the mask slice for this position is: +m = step(e) +(10) +step(ei,j) = +{ +1 +ei,j > 0 +0 +ei,j ≤ 0 +(11) +∇step(ei,j) ≡ +{ +1 +|ei,j| < 1 +c +others +(12) +where i, j indexes the elements in matrix e; c is a small positive +constant, empirically set to 0.1. +Both of the ∇step functions are designed to have grad = 1 +and grad = 0.1 segments, so that under Xavier-like initialization, +which is widely adopted, the shape of each mask slice is actively +learnable at the early stage of training and becomes more stable +but not absolutely stable at the late stage of training. Refer to +Section 6 for more details. +Via the aforementioned formulas, a mask slice m is calcu- +lated, then tiled/repeated in the dimensions of input and output +channel cgi and cgo times respectively, then get the masks partial +Mpart of shape (3, 3, cgi, cgo) for this position. By iterating all those +ci/cgi × co/cgo positions, the masks M of shape (3, 3, ci, co) are +finally constructed. +Smoothing the Training Process +To smooth the training process, two warm-up tricks are em- +ployed. +Algorithm 1 Mask Enabling Warm-Up +Set the number of epochs nwarm for warm-up +1. calculate the incremental unit per epoch δ = 1/nwarm +2. on the ith epoch begins: enable every mask at probability pi = +δ ∗ (i − 1) +3. conduct this training epoch +4. turn to step 2 until nwarm reached +Algorithm 2 Mask Regularizing Warm-Up +Set the number of epochs nwarm, the weight for mask regulariza- +tion αt, target density dt +1. calculate the incremental unit per epoch δ = αt/nwarm +2. conduct a training epoch, and keep the task loss ltask +3. calculate the upper limit and the scale factor of this loss: lmax = +abs(1.0 − dt), f = ltask/lmax +4. on epoch i begins: update the weight α = f ∗ δ ∗ (i − 1) +5. conduct this training epoch +6. turn to step 4 until nwarm is reached +Algorithm 1: mask enabling warm-up, enables the masks in +LHC layers randomly with increasing possibilities at the beginning +epochs, to eliminate non-convergence issues. +Algorithm 2: mask regularizing warm-up, increases the weight +of rmask gradually at the starting epochs, to let the CNN model +learn strength first and then topology. +Extra Cost Analysis +Compared with the standard convolution, the extra weights, +i.e., the effect factors, brought in by an LHC layer during training +is 3 ∗ 3 ∗ ci ∗ co/(cgi ∗ cgo)/(3 ∗ 3 ∗ ci ∗ co) = 1/(cgi ∗ cgo). Suppose +cgi = 64 and cgo = 8, 0.1953% of extra storage is paid for the +topology plasticity. +The extra computation, i.e., the construction of the masks, +brought in by an LHC layer during training is 3∗3∗ci∗co/(ho∗wo∗ +3 ∗ 3 ∗ ci ∗ co) = 1/(ho ∗ wo). Suppose a 224*224 input image and +a VGG16 model, the computation increases by about 0.3856%. +So at the training stage, the cost of our method is nearly the +same as that of standard convolution. See Section 5 for more +details. +3.4. Acceleration +At the inference stage, CNN models built of LHCs can be easily +accelerated by hardware of high parallelism, which consumes +much less memory and computation. +Given a MAC array of parallelism = 512, i.e., 512 multiplica- +tion and addition units. To maximize hardware utilization, two +groups of 512 numbers must be fed to them in one clock pe- +riod, which means such numbers must be stored continuously +respectively. This can be achieved under ‘‘structural sparsity’’, but +cannot in fragmented computation graphs. This is the key point +in implementing the optimal hardware acceleration. +Efficient Hardware Implementation +As mentioned above, the calculation of convolution is the only +part influenced by LHC, so the following discussion focuses on this +part. +To realize high parallel computing of a standard convolution +layer, sufficient units of multipliers and adders to form a MAC +array are needed. Similarly, to make the most of the structural +sparsity of LHC layers, it is needed that +(1) In the weight buffer, only valid weights of LHC are loaded +from external storage before runtime, and are stored in an inten- +sive way. The memory consumption of an LHC layer is just about +50%∼1% of that of a standard convolution layer, so bandwidth and +computation at runtime are saved. +(2) In the I/O buffers, input and output feature maps are stored +at runtime, just like that of standard convolution. +Data Management and Parallel Computing +Here explains how all invalid weights and invalid computation +are avoided and how high parallelism is achieved. +As shown in Fig. 6, as well as Fig. 7, the input/output features +are organized in a cgi-aligned /cgo-aligned manner, i.e., the width +of the IO buffers is cgi/cgo; yet the weights are organized in a +cgi ∗ cgo-aligned manner. +Given input features of shape (hi, wi, ci), for every pixel of +shape (ci, ) in it, every cgi adjacent elements are stored as a row +in the input buffer. At each convolution step, elements of shape +(3, 3, ci) in current sliding window are copied to the window +buffer and are stored in the same way. +Given weights of shape (3, 3, ci, co), for each kernel slice group +of shape (3, 3, cgi, cgo) in it, every cgi ∗cgo elements are stored as a +row in the weight buffer, with full-zero cgi ∗ cgo-length segments +being skipped. At every convolution step, feature elements in the +buffer window that correspond to these full-zero segments are +skipped according to the discontinuous addresses provided by +ALUT. + +Fig. 6. Parallel processing of LHC inference on hardware — how all invalid +weights and computation are avoided, and how high parallelism is reached. +Input buffer: input features are stored in cgi-aligned manner. Window buffer: +elements in current sliding window are copied from the input buffer. Weight +buffer: weights are stored in cgi ∗cgo-aligned manner, with full-zero segments of +length cgi ∗cgo being skipped. AGU: address generation unit, counts addresses for +the weights buffer incrementally. ALUT: address look-up table, stores addresses +calculated in advance for the window buffer. At every convolution step, every +row of cgi∗cgo weights in the weight buffer, and the corresponding row of length +cgi in the window buffer, are sent to the MAC array; then the cgo results are +sent to cgo output registers, where the value accumulates 3 ∗ 3 ∗ ci/cgi times +to get the final cgo results in the output features. With topology constraints +cgi ≡ 64 and cgo ≡ 8, a 512-way parallel processing is achieved. If taking the +batch dimension b (not drawn for simplicity) into account, the parallelism can +be further increased by b times. +Since employing different storing ways, the window buffer +and input buffer need different fetch addresses. Data in the +weights buffer are fetched via the incremental address counted +by AGU (address generation unit); data in the window buffer +are fetched via the pre-defined address saved in ALUT (address +lookup table). +With the addresses from AGU and ALUT, every row of length +cgi ∗cgo in the weight buffer, and the corresponding row of length +cgi in the window buffer, are sent to the MAC array to do parallel +multiplication–addition; then the cgo results are sent to cgo output +registers to accumulate 3 ∗ 3 ∗ ci/cgi times; finally the output +element at location (x, y) in the corresponding channel of the +output features is got. By iterating all the sliding windows and +all the kernels, the full output features are got. +Under topology constraints cgi and cgo, for a kernel, every cgi +adjacent kernel slices have the identical topology, and for a layer, +every cgo adjacent kernels have the identical topology. So at each +clock cycle, cgi∗cgo elements can be fed to the MAC array. Suppose +cgi = 64 and cgo = 8, parallelism = 512 is reached. This means there +are 1024 multiplication and addition operations per clock, and the +peak compute power reaches 1TOPS at 1 GHz clock frequency, +which meets the compute requirement of most terminal devices +specifically designed for CNN acceleration (Andri, Karunaratne, +Cavigelli, & Benini, 2020; Moons, Bankman, Yang, Murmann, & +Verhelst, 2018). +Of course the parallelism can be further improved if taking the +batch dimension into account. Given batch size b, the parallelism +can be further increased to b ∗ cgi ∗ cgo. +Extra Cost Analysis +Compared with the standard convolution, extra components +are required to accelerate an LHC layer during inference, i.e., ALUT +for indexing sparse weights and AGU for skipping invalid feature +elements. +For ALUT, every cgi∗cgo weights need an address index, and the +overall density of all LHC layers is no greater than 20%, so extra +memory consumption is just k∗ k∗ ci/cgi ∗ co/cgo/(k∗ k∗ ci ∗ co) = +1/cgi/cgo = 0.1953% of that of the standard convolution. Let alone +80+% of space is already saved by our method. +Fig. 7. Time scheduling of LHC vs traditional convolution. (a) The timing of an LHC layer, where its structural sparsity is fully utilized and thus all redundant clocks +and other computational resources are saved. (b) The timing of a standard convolution layer, where sparsity (gray parts) is not considered. Each column is cgi × cgo +parallel operations such as read/write, multiply–add and accumulation. Refer to Fig. 6 for the corresponding hardware logic. (For interpretation of the references to +color in this figure legend, the reader is referred to the web version of this article.) + +Fig. 8. The unification of various convolution techniques. GWC, DWC and +HetConv can all be taken as LHCs that use some certain shapes. +Fig. 9. Rebuilding GoogLeNet InceptionV4-A/B/C with LHC. The shapes used by +LHCs are the ones used in the original Inceptions. (For interpretation of the +references to color in this figure legend, the reader is referred to the web version +of this article.) +For AGU, the implementation just requires negligible resources +compared with the whole. +So at the inference stage, a large portion of computational re- +sources can be saved, even though a little extra cost is introduced +in. +4. Unification +With the topology plasticity, various convolution techniques +discussed in Section 2 can be unified by LHC. As shown in Figs. 8 +and 9, representative ones like GWC, DWC, HetConv and Incep- +tion are used as examples. +The shape of the xth slice in the yth kernel of a convolution +layer is notated as sx,y. +Group-Wise Convolution +Group-Wise Convolution or GWC (Alex et al., 2012), can be +viewed as an LHC using shapes ⟨1⟩1 and ⟨6⟩1 only, illustrated +in Fig. 8 the first row. Under the settings of Section 3.3, when +the following conditions are satisfied, an LHC degenerates into a +GWC: +sx,y = +⎧ +⎨ +⎩ +s511 +( +kcgi < x ≤ (k + 1)cgi +kcgo < y ≤ (k + 1)cgo +) +s0 +others +(13) +where cgi = ci/ngroup, cgo = co/ngroup, k = 0, 1, . . . , ngroup-1. +Depth-Wise Convolution +Depth-Wise Convolution or DWC (Sandler et al., 2018), illus- +trated in Fig. 8 the second row, a special case of GWC, is an LHC +using shapes ⟨1⟩1 and ⟨6⟩1 only. When the following conditions +are satisfied, an LHC degenerates into a DWC: +sx,y = +⎧ +⎨ +⎩ +s511 +( +kcgi < x ≤ (k + 1)cgi +kcgo < y ≤ (k + 1)cgo +) +s0 +others +(14) +where cgi = 1, cgo = co/ci, and k = 0, 1, . . . , ci-1. +HetConv +HetConv (Singh et al., 2019) is an LHC using shapes ⟨2⟩1 +and ⟨6⟩1 only, illustrated in Fig. 8 3rd row. When the following +conditions are satisfied, given p as the number of shape ⟨6⟩1 in a +kernel, LHC degenerates into HetConv: +sx,y = +{ +s511 +(x + y − 1)%p = 0 +s1 +others +(15) +where cgi = 1, cgo = 1, and p is a hyper parameter selected from +{1, 2, . . . , ci}. +Inception +GoogLeNet Inceptions (Szegedy, Ioffe, Vanhoucke, & Alemi, +2016) can be rebuilt with LHC as shown in Fig. 9. Each dashed +purple rectangle is equivalent to an LHC layer. For Inception-A, +the 2nd and 3rd rectangles are GWCs where the groups are not +evenly divided, and thus can be replaced by LHC. In Inception-B/C, +orthogonal 1D convolution pairs are employed to approximate +standard 2D convolutions, either serially or parallelly, where the +former amounts to double non-linearity and cannot be equivalent +to one LHC layer. But for simplicity, they are all replaced by LHC, +then a unified Inception shown in the bottom right of Fig. 9 is got. +Others +Other convolution techniques designed for efficiency can also +be unified. PWC can be viewed as standard convolution of kernel +size 1*1; MixConv (Tan & Le, 2019) is a GWC variant where kernel +sizes are different; SeeSaw (Zhang, 2019), IGC (Sun et al., 2018; +Xie et al., 2018; Zhang et al., 2017), etc., are unevenly grouped +GWCs. +5. Experiments +Here experiments are presented to compare LHC with other +convolution techniques that are either widely used or achieve +state-of-the-art results. +5.1. Experiment settings +Listed in Table 2 are the experiment items. +Networks and Convolution Techniques +The comparison of LHC, standard convolution, HetConv, +FPGM (He et al., 2019) and Taylor (Molchanov et al., 2019) +are conducted on VGG16/19 and ResNet34/50. Among them, +HetConv represents techniques of structural sparse convolution; +and FPGM/Taylor represent techniques of non-structural sparse +convolution. +The comparison of LHC, Inception, GWC+channel-shuffle and +DWC+PWC are conducted on GoogLeNet InceptionV4, Shuffle- +NetV1 and MobileNetV1 respectively. +Switchover of Different Convolution Techniques +For VGG16/19 and ResNet34/50, all convolution layers except +the 1st are replaced by LHCR/LHCF and HetConv. +For GoogLeNet, Inceptions are rebuilt by LHCR in the way +described in Fig. 9. Shapes used in LHCR are ⟨1⟩1, ⟨2⟩1, ⟨6⟩1, ⟨3⟩1 +and ⟨3⟩3. +For ShuffleNet, all GWC+channel-shuffle structures are re- +placed by LHCR, where shapes used are ⟨1⟩1 and ⟨6⟩1. + +Inception-A +Inception-B +concat +concat +c7x1 +c3x3 +c1x7 +c7x1 +c7x1 +c1x1 +c3x3 +c3x3 +c1x1 +c1x7 +c1x7 +pool +c1x1 +c1x1 +c1x1 +pool +c1x1 +c1x1 +c1x1 +feat in +feat in +Inception-C +Inception-LHC +concat +concat +c7x1c1x7 +LHC +c3x1 +c1x1 +c3x1c1x3 +c1x1 +LHC +c1x3 +pool +c1x1 +c1x1 +c1x1 +pool +LHC +feat in +feat inTable 2 +Experiment items: candidate convolution techniques on mainstream networks. +Origin +GWC (Alex +et al., 2012) +DWC (Sandler +et al., 2018) +Inception (Szegedy, +Ioffe, Vanhoucke, & +Alemi, 2016) +HetConv (Singh +et al., 2019) +LHC +(ours) +FPGM (He, Liu, +Wang, Hu, & +Yang, 2019) +Taylor (Molchanov, +Mallya, Tyree, Frosio, +& Kautz, 2019) +VGG16 (Simonyan & +Zisserman, 2014) +✓ +✓ +✓ +✓ +✓ +VGG19 (Simonyan & +Zisserman, 2014) +✓ +✓ +✓ +✓ +✓ +ResNet34 (He, Zhang, +Ren, & Sun, 2016) +✓ +✓ +✓ +✓ +✓ +ResNet50 (He et al., +2016) +✓ +✓ +✓ +✓ +✓ +GoogLeNet (Szegedy, +Ioffe, Vanhoucke, & +Alemi, 2016) +– +✓ +✓ +ShuffleNet (Zhang, Zhou, +Lin, & Sun, 2018) +– +✓ +✓ +MobileNet (Howard +et al., 2017) +– +✓ +✓ +Fig. 10. Curves of computation reduction vs accuracy of different networks built of different techniques on different datasets. The x-axis ∆flops is computation +reduction in percentage, and the y-axis acc is classification accuracy in percentage. (For interpretation of the references to color in this figure legend, the reader is +referred to the web version of this article.) +For MobileNet, all DWC+PWC structures are replaced by LHCR, +where shapes used are ⟨1⟩1, ⟨2⟩1 and ⟨6⟩1. +Tasks for Making Evaluations +The tasks for evaluating different convolution techniques are +the image classification, on two widely recognized datasets, CI- +FAR10 and ImageNet. +Training +procedures +are +designed +to +be +identical: +on +CIFAR10/ImageNet, models are to be trained 200/100 epochs, +with early stopping patience 40/20, using SGD with initial learn- +ing rate 1e−2/1e−3 and decay factor 0.1. Data augmentations +include random flipping, translation, rotation, hue, saturation, +brightness and contrast. +Specifically, for networks built of LHC, +(1) Hyper-parameter dt, is set to {invalid, 0.2, 0.1, 0.05, 0.01} +for CIFAR10, and {invalid, 0.25, 0.1} for ImageNet. Here invalid +means no density target is set, and thus the network can explore +the most suitable shapes and converge to the best performance. +(2) Hyper-parameter cgi and cgo are set to 64 and 8 respectively +as parallelism ≥ 512 is a typical value of hardware accelera- +tion (Andri et al., 2020; Moons et al., 2018). +For FPGM and Taylor, experiments are carried out by following +their official procedures. +5.2. Result analysis +Results are presented in computation reduction vs classifica- +tion accuracy, as shown in Fig. 10. +Fig. 11. Hardware resource saving with LHC. Clocks of 5× can be saved, which +either benefits improving the throughput or reducing energy consumption. +Memory of 10× can also be saved, which dramatically reduces the on-chip +memory block resource. This model VGG16 is trained on CIFAR10 under +constraints of parallelism = 512 and no obvious accuracy loss. +LHC vs Standard Conv, HetConv, FPGM & Taylor +Shown in Fig. 10 are the results of LHC and other convolution +techniques. The x-axis is the computation reduction in percent- +age, and y-axis is the classification accuracy. The more a line to +the top right corner, the better it is. +By comparing LHCR and LHCF, LHCR works slightly better if +∆flops is smaller, while LHCF is a bit superior if ∆flops is larger. +The former can be attributed to the priori knowledge implied in + +Fig. 12. Shape distribution of the 3rd/7th/11th layer of VGG16 built of LHCR and LHCF respectively. Left: LHCR; right: LHCF. The x-axis is the serial index of all 512 +shapes; y-axis is the ratio a shape takes up in an LHC layer. +the design of those rigid shapes, and the latter is likely due to the +full freedom of those free shapes in dropping redundant weights. +By comparing LHC with HetConv/FPGM/Taylor, the networks +built of LHC always achieve better accuracy at any computation +reduction ratio. Given the same computation reduction, ours ac- +curacy surpasses others by 2.0% on CIFAR10 or 1.0% on ImageNet. +For example, our method can improve VGG16 computation ef- +ficiency by nearly 5× on CIFAR10 or 2× on ImageNet without +harming its performance. +Interestingly, there is always appropriate ∆flops value, at +which networks built of LHC surprisingly outperform the original, +i.e., the horizontal dashed blue line in every sub-figure. This +rarely happens with existing techniques. Specifically, our method +improves top1 classification accuracy by 1.0% on CIFAR10 or 0.5% +on ImageNet at most. Thus, our method is also a powerful training +technique. +The parameter reduction is directly up to the dt we set, +namely, dt ∈ {invalid, 0.2, 0.1, 0.05, 0.01} for CIFAR10, and dt ∈ +{invalid, 0.25, 0.1} for ImageNet. Without harming the accuracy, +our method can compress VGG/ResNet weights by about 10× on +CIFAR10, or 4× on ImageNet. +We also conducted the hardware acceleration simulation, +mentioned in Section 3.4, on a VGG16 model, which is trained +on CIFAR10 under dt = 0.1 and achieves ∆flops = 78% and acc += 91.45% (almost the same as the original performance). The +number of clocks and memory rows (each row contains 3∗3∗cgi∗ +cgo weights) required during the inference of an input image of +shape 32*32 are drawn in Fig. 11. In hardware implementation, +the number of clocks and memory rows can also be saved by 5× +and 10×. With less requirements in clock and memory, the power +dissipation of hardware is accordingly reduced. +LHC vs GWC+Channel-Shuffle, DWC+PWC & Inception +Here +∆flops +are +evaluated +when +ShuffleNet/MobileNet/ +GoogLeNet and their LHC variants have similar accuracy. +The GoogLeNet model, with its Inceptions rebuilt by LHC, +achieves computation reduction ∆flops = 24.39%, and parame- +ter compression ∆param = 64.56%; the MobileNet model, with +its DWC+PWC modules replaced, achieves ∆flops = 15.68% and +∆params = 18.82%; and the ShuffleNet model, with its GWC+ +channel-shuffle modules replaced, achieves ∆flops = 5.36% and +∆params = 11.73%. +Clearly, the efficiency of GoogLeNet/ShuffleNet/MobileNet, ei- +ther well-designed or light-weight, can be further improved by +LHC, which means the efficiency gain contributed by our layer- +level method even beats that of beyond-layer-level methods. +Interestingly, the computation reduction amplitude of our +method on ShuffleNet is obviously less than on the other two, +which verifies the necessity of information exchange provided by +channel-shuffle. +6. Discussions +Here we discuss why LHC works. The VGG16 models trained +on ImageNet with dt = invalid are chosen, since other models +have similar results. +What LHC Learns +The core difference between LHC and standard convolution is +the dendritic topology plasticity, i.e., learning masks of suitable +shapes for current task in a data-driven manner. We count the +number of shapes learnt by the 3rd/7th/11th LHC layer, as shown +in Fig. 12. The x-axis is the serial indexes of all 512 shapes drawn +in Fig. 3, and the y-axis is the ratio a shape takes up in an LHC +layer. +By comparing the shape distribution of different LHCR/LHCF +layers, we find that: (1) low layers prefer denser shapes and +mainly have no shape ⟨1⟩1, which is totally sparse; (2) high layers +have the least amount of denser shapes but the most amount +of shape ⟨1⟩1; (3) middle layers takes the intermediate shape +distribution. +This means that enough number of kernels in low layers is +necessary for extracting sufficient basic patterns, yet after the +non-linearity of a layer upon a layer, the patterns in the feature +maps become increasingly abstract and sparse, and thus just need +sparser kernels to extract. How LHC Converges +We penetrate into the training process by inspecting how the +shapes/masks in LHC layers evolve along with the training epochs. +We calculate the masks correlation like Guillaume et al. (2018) +of an LHC layer between two adjacent epochs, i.e., the mean of + +Fig. 13. Evolution of masks along with the training epochs of the 3rd/7th/11th +LHCF layer. The x-axis is the epoch number; y-axis is the masks correlation. Mask +correlation grows as training goes, which means LHC masks generally evolve +from dynamic states to more static states. +Fig. 14. The spectrums of the 3rd/7th/11th LHC layers and corresponding +standard convolution layers from two models that are also used in Fig. 11. The +x-axis is spectrum components, and y-axis is their amplitudes. The more the +convolution layers’ spectrum of a CNN model approximates uniform distribution, +the better the model’s performance will be. Apparently, LHC layers’ spectrums +are always more uniformly distributed than that of standard convolutions, +which intuitively explains why under appropriate conditions models based +on our method have performance better than models based on the standard +convolution. +element-wise-logical-and. We choose the 3rd/7th/11th layers of +VGG16 built of LHCF to visualize in Fig. 13; LHCR has similar +results. The x-axis is the number of epochs and the y-axis is the +correlation of masks between two epochs. +The correlation of masks in each layer increases as the training +epochs carry forward, but is always less than 100%, even if the +model converges enough. This means our differentiable step func- +tion keeps the dendritic topology updating all the time (of course +so is the synaptic strength), just like the highly dynamic biological +neural networks in the brain (Holtmaat et al., 2005; Stettler et al., +2006). +Besides, the correlation of lower layers is larger than the +higher, which is also as expected, because the lower layers learn +the patterns that are more task-agnostic and thus requires more +stable dendritic topology. +Why LHC Excels +As pointed out by OCNN (Wang, Chen, Chakraborty, & Yu, +2020), the spectrum of a convolution layer’s DBT (doubly block +Toeplitz) matrix reasonably indicates how well the CNN model’s +capacity is utilized. The more uniformly the spectrums distribute, +the better the model performances. Clearly drawn in Fig. 14, our +method’s spectrums are always closer to uniform distribution +than the original. Combined with Fig. 13, we may assure that it +is the changing masks that force the model to utilize its capacity +better. +Now we know about the weights in a model built of LHC: +(1) there are a lot of zeros, up to 80+%, which can be taken as a +case of L0/L1 regularization training, i.e., strength regularization; +(2) these zeros distribute in a structural way in the kernels, +which further provides another kind of regularization, i.e., topol- +ogy regularization; +(3) the position of these zeros changes along with the training +steps, which can be seen as the dropout of the weights. +Hence, it is not difficult to understand why our method sur- +passes existing methods. +7. Conclusion +Our method LHC integrates the plasticity of both dendritic +topology and synaptic strength, making both the kernel shapes +and the weights learnable during training. CNN models built of +LHC can be greatly sparsified structurally and can be acceler- +ated in high parallelism. Experiments against existing convolution +techniques show that LHC always achieves better performance at +any computation reduction ratio; and experiments of rebuilding +typical network structures show that LHC can improve their +efficiency even further. Our method also achieves better CNN +performance if taken as a training technique. +CRediT authorship contribution statement +Rongzhen Zhao: Conceptualization, Methodology, Software. +Zhenzhi Wu: Methodology. Qikun Zhang: Software. +Declaration of competing interest +The authors declare that they have no known competing finan- +cial interests or personal relationships that could have appeared +to influence the work reported in this paper. +Acknowledgments +This work was supported by ‘‘Science and Technology Inno- +vation 2030 - New Generation of Artificial Intelligence’’, China +project (2020AAA0109100) and Beijing Science and Technology +Plan, China (Z191100007519009). +References +Alex, K., Sutskever, I., & Hinton, G. E. (2012). 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In IEEE international conference on computer vision. + diff --git a/mNE5T4oBgHgl3EQfHg55/content/tmp_files/load_file.txt b/mNE5T4oBgHgl3EQfHg55/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d1e8bdf696836f4ba71d56112edc7cc60d5122e8 --- /dev/null +++ b/mNE5T4oBgHgl3EQfHg55/content/tmp_files/load_file.txt @@ -0,0 +1,993 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf,len=992 +page_content='NeuralNetworks Learnable Heterogeneous Convolution: Learning both topology and strength Rongzhen Zhao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Zhenzhi Wu ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Qikun Zhang Lynxi Technologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Beijing 100097,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' China a r t i c l e i n f o Article history: Received 8 August 2020 Received in revised form 10 March 2021 Accepted 29 March 2021 Available online 20 April 2021 Keywords: Convolution neural network Efficiency & performance Learning topology & strength Fine-grained but structural Hardware acceleration a b s t r a c t Existing convolution techniques in artificial neural networks suffer from huge computation complexity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' while the biological neural network works in a much more powerful yet efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Inspired by the biological plasticity of dendritic topology and synaptic strength, our method, Learnable Heterogeneous Convolution, realizes joint learning of kernel shape and weights, which unifies existing handcrafted convolution techniques in a data-driven way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' A model based on our method can converge with structural sparse weights and then be accelerated by devices of high parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' In the experiments, our method either reduces VGG16/19 and ResNet34/50 computation by nearly 5× on CIFAR10 and 2× on ImageNet without harming the performance, where the weights are compressed by 10× and 4× respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' or improves the accuracy by up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='0% on CIFAR10 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='5% on ImageNet with slightly higher efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The code will be available on www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='com/Genera1Z/ LearnableHeterogeneousConvolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Introduction Convolution neural networks (CNNs) are showing their supe- rior performance in vision tasks like classification, detection and segmentation, but their advantages in practice are impeded by their heavy computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' To improve CNN efficiency, researchers have developed many recipes: (1) compressing existing models with pruning (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Zhou, Zhang, Wang, & Tian, 2019), quantization (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2019) or knowledge distillation (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (2) designing efficient network structures either by hand (Hu, Shen, & Sun, 2018) and Ma, Zhang, Zheng, and Sun (2018) or by automatic search (Chen, Xie, Wu, & Tian, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (3) exploiting efficient convolution oper- ators, which are either smaller (Simonyan & Zisserman, 2014), or factorized (Szegedy, Ioffe, Vanhoucke, & Alemi, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Szegedy, Vanhoucke, Ioffe, Shlens, & Wojna, 2016), or sparse (Huang, Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', & Weinberger, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Sun, Li, Liu, & Wang, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' On the other hand, (4) the neural network in a brain, which features sparse activation and high dynamics (Holtmaat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Stet- tler, Yamahachi, Li, Denk, & Gilbert, 2006), works in a much more complex and powerful yet efficient way (Merolla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' ∗ Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' E-mail addresses: rongzhen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='zhao@lynxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='com (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Zhao), zhenzhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='wu@lynxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='com (Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Wu), qikun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='zhang@lynxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='com (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Zhang).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' For recipe (1), a large model of high performance must be available firstly, and the pruning-retrain iteration is very time- consuming;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' for recipe (2), it is hard to design an excellent struc- ture by hand, and also too costly to search one even with tens of GPUs and days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' So we choose to combine recipes (3) and (4), namely, sparsifying convolutions by learning from the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' A biological neuron connects from multiple predecessor neu- ron via dendrites, and to a subsequent neuron via an axon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The connections are plastic both in topology and strength, through forming/losing/developing the synapses on dendritic spines (Bhatt, Zhang, & Gan, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Harms & Dunaevsky, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' To make an analogy (Beysolow II, 2017), like Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 1, for the co kernels in a standard convolution layer, each kernel of shape (k, k, ci) belongs to a neuron, which is shared at different spatial positions of feature maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' for the ci slices in a kernel, each kernel slice of shape (k, k) is a dendrite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' for the k ∗ k elements in a slice, each element is a synapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' But unlike biological neurons, in a standard convolution layer, only the strength of kernel elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', weights is learnable, while the topology of kernel slices is not, let alone the number of slices and kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Such differences are where the CNN can learn from the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Improved convolution techniques like Group/Depth/Point- Wise Convolution (GWC, DWC, PWC) can be seen as sparsifica- tion of neurons in a layer or dendrites in a neuron, which are, however, predefined instead of learnt (Alex, Sutskever, & Hinton, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Sandler, Howard, Zhu, Zhmoginov, & Chen, 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' works like SeeSaw and DeepR introduce in learnability (Guillaume, David, Wolfgang, & Robert, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Zhang, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Other works like Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' A convolution layer (green, the biases and activation are ignored) be- tween input feature maps (orange) and output features (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' For the standard convolution, 1⃝ the number of kernels in a layer is fixed co, 2⃝ the number of slices in each kernel is fixed ci, 3⃝ the shape of each kernel slice (topology) is fixed (k, k), and only 4⃝ the value of the k∗k elements (strength) in each kernel slice can be learnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Our method makes 1⃝∼ 4⃝ all learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=') Inception and HetConv sparsify the dendritic topology square to a row, line or dot, which are also not learnable (Singh, Verma, Rai, & Namboodiri, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' works like L0 training hit the mark by a fluke, but the weights learnt is non-structural, which is not friendly for hardware to accelerate (Christos, Max, & Diederik, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' See Section 2 for detailed reviewing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Our method, Learnable Heterogeneous Convolution (LHC), a seamless replacement to the standard convolution, is proposed to break those limitations by integrating the plasticity in both the synaptic strength, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', learnable weights, which is originally possessed by the standard convolution, and additionally the den- dritic topology, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', learnable kernel slice shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' With the latter as a unit, the number of slices in a neuron, and the number of neurons in a layer all become learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Comprehensive experiments demonstrate the superiority of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' At parallelism = 512, suppose batch size is 1, we can either reduce VGG/ResNet computation by nearly 5× on CIFAR10 and 2× on ImageNet without harming the performance, where the weights are compressed by 10× and 4× respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' or improve the accuracy by up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='0% on CIFAR10 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='5% on ImageNet with slightly higher efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The extra costs are no more than slightly longer training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Our contributions are: (1) LHC is proposed to realize dual-plasticity of strength and topology in convolution, which is fine-grainedly yet structurally sparse thus fits hardware acceleration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (2) It sparsifies convolutions in full back-propagation, instead of undifferentiable ways used by most pruning methods such as cutting of weights that have least magnitude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (3) It requires negligible extra costs at training and can greatly improve CNN models’ efficiency at inference stage even with some performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The remaining content is organized as follows: related works are reviewed in Section 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' the proposal is detailed in Section 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' how our method unifies various convolution techniques is ana- lyzed in Section 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' experiments are presented in Section 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' more advanced analyses are discussed in Section 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' the conclusion is drawn in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Given a convolution layer, the notations list in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Table 1 Notations used in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Shape of the kernels (k, k, ci, co) Shape of input feature maps or ‘‘feat in’’ (b, hi, wi, ci) Shape of output feature maps or ‘‘feat out’’ (b, ho, wo, co) Batch dimension of feature maps b Number of input/output channels ci, co Height/width of input/output features hi, wi, ho, wo Spatial size of a kernel k ∗ k, (k, k) Topology constraint of input/output channels cgi, cgo Tera Operations Per Second TOPS Floating Point Operation(s) FLOP(s) Multiply Accumulate (unit) MAC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Related works Here the sparse convolution techniques for improving CNN efficiency are reviewed, and keypoints where the convolution can imitate the brain are inducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Structural Sparse Convolutions: Handcrafted Convolution can be sparsified in spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' A 2D convolution kernel is factorized into two perpendicular 1D ones in Szegedy, Ioffe, Vanhoucke, and Alemi (2016) and Szegedy, Vanhoucke, Ioffe, Shlens, and Wojna (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Kernels are designed into incremental sizes in Tan and Le (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Slices of a 3*3 kernel are replaced by 1*1 sizes at intervals in Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (2019) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Sparsification can also be taken in channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' In a GWC (SIfre & Mallat, 2014), the input channels are grouped and each output channel is correlated with one of the groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Features among different groups are further exchanged in Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (2018), Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (2018) and Zhang, Qi, Xiao, and Wang (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Shifting or negation saves kernels/channels either, like in Shang, Sohn, Almeida, and Lee (2016) and Yan, Li, Li, Zuo, and Shan (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Dimensions of space and channel can be considered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Like in Wang, Xu, Chunjing, Xu, and Tao (2018), standard kernels are turned into secondary kernels, which are degressive in space and partially connected in channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Neuron number in a layer, dendrite number in a neuron, and dendritic topology are all handcrafted in such methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' only the synaptic strength is learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Structural Sparse Convolutions: Learnt Compared with the above, works thinking in this way bring in learnability of sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' But the techniques employed are still restricted to GWC, DWC and PWC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Works like Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (2018) require too many iterations, where connections between input and output channels of less importance are cut off progressively while training to get a de- sired group number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Some realize learnability under the guidance of Singular Value Decomposition like Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (2018), which is built of GWC and PWC in a way similar to Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Others like Zhang (2019) realize unevenly grouped GWC layers in a model by fusing network architecture search into training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Only sparsity in channel dimension is considered by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Namely, besides the synaptic strength, they do take into account the plasticity of neurons number in a layer or dendrite number in a neuron, but overlook the dendritic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' By the way, Verelst and Tuytelaars (2020) is worth learning from, which generates feature masks to skip computations cor- responding to zeros, even if it sparsifies activations instead of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Non-Structural Sparse Convolutions: Learnt Such works are similar to the pruning methods (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2019), except that the sparsity is gained dur- ing rather than after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Since non-structural sparsity goes against hardware acceleration (Deng, Li, Han, Shi, & Xie, 2020), recent works are not that many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Kernels of uncommon shapes designed handcrafted by SWF (Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2019) to address specific image contents, which inspired our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' L0 regularization are often used to train models full of ze- ros, like Christos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (2018), enabling joint optimization of weights’ value and topology via non-negative random gates of hard concrete distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Inspired by dynamic connections in the brain Bhatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (2009) and Harms and Dunaevsky (2007), the rewiring mechanism is proposed by Guillaume et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (2018) to enable simultaneous learning of connections and weights for constrained hardware resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Their implementation of plasticity in dendritic topology, namely, neuron number in a layer or dendrite number in a neuron, are good references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Our inspiration: making it possible in a convolution layer to (a) learn the dendritic topology and synaptic strength at the same time, and (b) realize fine-grained yet structural sparsity for hardware acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Proposed method How LHC integrates the plasticity in dendritic topology and synaptic strength is elaborated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The structural sparsifica- tion and hardware acceleration of LHC-based models are also described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Prototype Convolution Kernels of Uncommon Shapes For either traditional or CNN-based algorithms, it is quite common to use convolution kernels of square shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' However, SWF (Yin, Gong, & Qiu, 2019), a latest work, broke fresh ground and is equal in force compared with CNN-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The key is that kernels of uncommon shapes, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 2, were meticulously designed for different image patterns to gain excel- lent feature extraction capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Enlightened by this, we empirically design 15 rigid kernel shapes, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Among them, shape ⟨1⟩1, ⟨2⟩1 and ⟨6⟩1 are common in use;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' shape ⟨3⟩1 and ⟨3⟩3 are already used in GoogLeNet Inceptions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' shape group ⟨4⟩ and ⟨5⟩ inherit from SWF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Shape group ⟨2⟩∼⟨5⟩ are designed to sparsify a convolution layer in spatial dimensions, while group ⟨1⟩ is to reduce redundancy in channel dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The theoretical foundation for such a design is that CNNs are able to extract visual patterns of different levels like dot, edge, curve and plane at different layers (Goodfellow, Bengio, & Courville, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Accordingly, very sparse shape groups ⟨2⟩ and ⟨3⟩ are designed to address simple patterns like dots and edges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' shape groups ⟨4⟩∼⟨6⟩, not that sparse, are to handle complex patterns like curves and planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Further on, we relieve the constraints of the aforementioned rigid shapes, and for a 3*3 kernel, we provide 2 3∗3 = 512 free shapes for an LHC layer to choose, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Obviously, the rigid shapes are a sub-set of the free shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' An LHC layer equipped with the rigid shapes is called LHCR, and LHC with the free shapes is LHCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' In LHC, these shapes are formed as masks consisting of 0/1 elements, and are paired with corresponding effect factors to guide the learning of the shapes of kernel slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Details are presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Two sets of uncommon shapes for LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (a) 15 rigid shapes: designed empirically in six groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' digits on the top and right are their indexes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', ⟨3⟩4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (b) 23×3 = 512 free shapes: generated by arbitrarily setting each element to 0 or 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' digits on every top-right corner are their serial indexes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The former is a sub-set of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' To make a comparison, LHCR implies priori knowledge of those rigid shapes while LHCF has more freedom in sparsification, which is verified by experiments in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Learnable Heterogeneous Convolution For a Learnable Heterogeneous Convolution (LHC) layer, its kernel slices are learnt to be these uncommon shapes, rather than pre-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Suppose a convolution layer with ci input channels and co output channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' With the aforementioned uncommon shapes, the kernels can be shaped (1) kernel-by-kernel (KbK), where shapes are the same within a kernel and different among kernels, or (2) slice-by-slice (SbS), where shapes are different both within a kernel and among kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 4 the top two rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Obviously, SbS is more flexible than KbK and thus can elim- inate the convolution redundancy in both spatial and channel dimensions better;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' but the computation graph of SbS is too frag- mented for the hardware to accelerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' So we split the difference: only one shape combination is allowed for every adjacent cgi slices and every adjacent cgo kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Then we can learn a struc- tural sparse LHC layer, which supports parallelism up to cgi∗cgo ≡ 64 ∗ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Refer to the bottom two rows of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Besides, these constraints happen to be a kind of regularization, benefiting the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' See Section 6 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Computation reduction To facilitate discussion, we take the 3*3 convolution as an example, and use the indexes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 3 to represent shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Following the aforementioned setting, the computation quan- tity or the total number of multiplication–addition of a standard convolution layer is: CSTD = ho × wo × ci × co × ∥s511∥0 = ho × wo × ci × co × 9 (1) where s511 the 511th shape in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 3 and its L0 norm is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Topology of LHC Kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 1st row: LHC-KbK, where kernel slices have the same shapes within a kernel and different shapes among kernels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 2nd row: LHC-SbS, where slices have different shapes both within and among kernels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 3rd row: LHCR with topology constraints cgi and cgo, where slices are shaped into those rigid shapes and every cgi ∗ cgo slices have the same shape;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 4th row: LHCF, where slices are shaped into those free shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Similarly, the computation quantity of an LHC layer is: CLHC = ho × wo × ( co/cgo ∑ y=1 ( ci/cgi ∑ x=1 ∥sx,y∥0 × cgi) × cgo) = ho × wo × cgi × cgo × ci/cgi×co/cgo ∑ s=1 ns (2) where cgi and cgo are the aforementioned topology constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' sx,y is the shape of the xth slice in the yth kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' ns is L0 norm of the sth shape in all ci/cgi × co/cgo positions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', ∥sx,y∥0 = ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' So, the computation reduction is: ∆C = ho × wo × (ci × co × 9 − cgi × cgo × ci/cgi×co/cgo ∑ s=1 ns) (3) when ns = 9, ∆C = 0, where all shapes in LHC is s511, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', the standard convolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' when ns = 0, ∆C = CSTD, where all shapes in LHC is s0, which means all features are redundant and thus can only be approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Besides, existing convolution techniques can be unified by LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' See Section 4 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Learnability At the training stage, the only difference between LHC and a standard convolution is the construction of the kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 5, the masks composed of zeros and ones are constructed first, then multiplied with the kernels, where elements multiplied with zeros in the masks are deactivated during both forward and back-propagation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' finally, the masked kernels are convolved with the input features, and the output features are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The masks and kernels realize plasticity in dendritic topology and synaptic strength respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Guide the Learning of Topology For a convolution layer, its computation quantity is propor- tional to the density of its kernels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' for a CNN model, its com- putation quantity is positively correlated with its global density, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Training principle of LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Left box: (1) every cgi ∗ cgo kernel slices are equipped with a set of effect factors, each of which points to a shape belonging to the rigid shapes or the free;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (2) through a differentiable step function, a mask slice is got then tiled into masks partial of shape (k, k, cgi, cgo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Center box: do (2) for every cgi∗cgo slices in the kernels, then the masks of shape (k, k, ci, co) are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Right box: (3) multiply the masks with the kernels element-wisely;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (4) do the convolution with the masked kernels and the input feature maps of shape (b, hi, wi, ci), finally get the output features of shape (b, ho, wo, co).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (5) In the backward propagation, the shapes get different gradients according to their contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (6) As training goes on, the advantage of different shapes is accumulated in the effect factors, and the most suitable shapes gradually win out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Note: in the left box, purple stars are where gradients pass during back propagation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' purple stars of the other two boxes are omitted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' in the right box, the biases and activation are omitted for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' of which the maximum is 1, namely, no sparsification, and the minimum limit is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Given a model with L LHC layers, we can set a global density target and calculate mask regularization loss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' then by minimizing it, we can guide the model to converge to a state that costs much less computation: lmask = |dt − ∑L l=1 ∥Ml∥1 ∑L l=1 size(Ml) | (4) where lmask is the mask regularization loss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' dt is the density target;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Ml is the masks in the lth LHC layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Here, L1 norm is used instead of L0, because (1) Ml only consists of zeros and ones, which means L0 is equal to L1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' and (2) L1 norm is differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Hence, optimizing this model turns into simultaneously min- imizing the mask regularization loss and the task loss, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', cate- gorical cross-entropy loss: min M1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='ML α × lmask + ltask (5) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' cgi = C1, cgo = C2 (6) where α is a positive constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' ltask is the task loss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' cgi and cgo are the aforementioned topology constraints for structural sparsity, and their value C1 ≡ 64 and C2 ≡ 8 typically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Now the point is how to construct the Ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Given the kernels of a layer and the topology constraints cgi and cgo, to construct the masks, we have ci/cgi × co/cgo positions to determine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' In other words, for each of these positions, we need to calculate a mask slice, a k∗k matrix, using the aforementioned shapes that are either rigid or free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Enroll Shapes into Competition To calculate a mask slice, we hope those shapes illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 3 are enrolled in all together so that they compete with one another along with the training steps, letting the fittest eventually win out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' For LHCR, at each of these positions, a 15D vector e called effect factors is used to represent the effects of those 15 rigid shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' So the mask slice for this position is: m = 15 ∑ i=1 step(ei, e) × si (7) step(ei, e) = { 1 ei = max(e) 0 ei < max(e) (8) ∇step(ei, e) ≡ { 1 |ei − mean(e)| < 1 c others (9) where ei is the ith element in e and si is the ith of those 15 rigid shapes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' c is a small positive constant, empirically set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' step is differentiable and ensures the summation of effect factors be one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' softmax is not used because it is heavy to calculate and can hardly make one of the shapes win out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' For LHCF, equip each of these positions with a 3*3 matrix e as the effect factors, which represent the effects of those 512 free shapes for this position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' So the mask slice for this position is: m = step(e) (10) step(ei,j) = { 1 ei,j > 0 0 ei,j ≤ 0 (11) ∇step(ei,j) ≡ { 1 |ei,j| < 1 c others (12) where i, j indexes the elements in matrix e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' c is a small positive constant, empirically set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Both of the ∇step functions are designed to have grad = 1 and grad = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='1 segments, so that under Xavier-like initialization, which is widely adopted, the shape of each mask slice is actively learnable at the early stage of training and becomes more stable but not absolutely stable at the late stage of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Refer to Section 6 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Via the aforementioned formulas, a mask slice m is calcu- lated, then tiled/repeated in the dimensions of input and output channel cgi and cgo times respectively, then get the masks partial Mpart of shape (3, 3, cgi, cgo) for this position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' By iterating all those ci/cgi × co/cgo positions, the masks M of shape (3, 3, ci, co) are finally constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Smoothing the Training Process To smooth the training process, two warm-up tricks are em- ployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Algorithm 1 Mask Enabling Warm-Up Set the number of epochs nwarm for warm-up 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' calculate the incremental unit per epoch δ = 1/nwarm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' on the ith epoch begins: enable every mask at probability pi = δ ∗ (i − 1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' conduct this training epoch 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' turn to step 2 until nwarm reached Algorithm 2 Mask Regularizing Warm-Up Set the number of epochs nwarm, the weight for mask regulariza- tion αt, target density dt 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' calculate the incremental unit per epoch δ = αt/nwarm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' conduct a training epoch, and keep the task loss ltask 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' calculate the upper limit and the scale factor of this loss: lmax = abs(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='0 − dt), f = ltask/lmax 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' on epoch i begins: update the weight α = f ∗ δ ∗ (i − 1) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' conduct this training epoch 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' turn to step 4 until nwarm is reached Algorithm 1: mask enabling warm-up, enables the masks in LHC layers randomly with increasing possibilities at the beginning epochs, to eliminate non-convergence issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Algorithm 2: mask regularizing warm-up, increases the weight of rmask gradually at the starting epochs, to let the CNN model learn strength first and then topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Extra Cost Analysis Compared with the standard convolution, the extra weights, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', the effect factors, brought in by an LHC layer during training is 3 ∗ 3 ∗ ci ∗ co/(cgi ∗ cgo)/(3 ∗ 3 ∗ ci ∗ co) = 1/(cgi ∗ cgo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Suppose cgi = 64 and cgo = 8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='1953% of extra storage is paid for the topology plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The extra computation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', the construction of the masks, brought in by an LHC layer during training is 3∗3∗ci∗co/(ho∗wo∗ 3 ∗ 3 ∗ ci ∗ co) = 1/(ho ∗ wo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Suppose a 224*224 input image and a VGG16 model, the computation increases by about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='3856%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' So at the training stage, the cost of our method is nearly the same as that of standard convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' See Section 5 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Acceleration At the inference stage, CNN models built of LHCs can be easily accelerated by hardware of high parallelism, which consumes much less memory and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Given a MAC array of parallelism = 512, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 512 multiplica- tion and addition units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' To maximize hardware utilization, two groups of 512 numbers must be fed to them in one clock pe- riod, which means such numbers must be stored continuously respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' This can be achieved under ‘‘structural sparsity’’, but cannot in fragmented computation graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' This is the key point in implementing the optimal hardware acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Efficient Hardware Implementation As mentioned above, the calculation of convolution is the only part influenced by LHC, so the following discussion focuses on this part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' To realize high parallel computing of a standard convolution layer, sufficient units of multipliers and adders to form a MAC array are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Similarly, to make the most of the structural sparsity of LHC layers, it is needed that (1) In the weight buffer, only valid weights of LHC are loaded from external storage before runtime, and are stored in an inten- sive way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The memory consumption of an LHC layer is just about 50%∼1% of that of a standard convolution layer, so bandwidth and computation at runtime are saved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (2) In the I/O buffers, input and output feature maps are stored at runtime, just like that of standard convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Data Management and Parallel Computing Here explains how all invalid weights and invalid computation are avoided and how high parallelism is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 6, as well as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 7, the input/output features are organized in a cgi-aligned /cgo-aligned manner, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', the width of the IO buffers is cgi/cgo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' yet the weights are organized in a cgi ∗ cgo-aligned manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Given input features of shape (hi, wi, ci), for every pixel of shape (ci, ) in it, every cgi adjacent elements are stored as a row in the input buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' At each convolution step, elements of shape (3, 3, ci) in current sliding window are copied to the window buffer and are stored in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Given weights of shape (3, 3, ci, co), for each kernel slice group of shape (3, 3, cgi, cgo) in it, every cgi ∗cgo elements are stored as a row in the weight buffer, with full-zero cgi ∗ cgo-length segments being skipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' At every convolution step, feature elements in the buffer window that correspond to these full-zero segments are skipped according to the discontinuous addresses provided by ALUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Parallel processing of LHC inference on hardware — how all invalid weights and computation are avoided, and how high parallelism is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Input buffer: input features are stored in cgi-aligned manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Window buffer: elements in current sliding window are copied from the input buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Weight buffer: weights are stored in cgi ∗cgo-aligned manner, with full-zero segments of length cgi ∗cgo being skipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' AGU: address generation unit, counts addresses for the weights buffer incrementally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' ALUT: address look-up table, stores addresses calculated in advance for the window buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' At every convolution step, every row of cgi∗cgo weights in the weight buffer, and the corresponding row of length cgi in the window buffer, are sent to the MAC array;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' then the cgo results are sent to cgo output registers, where the value accumulates 3 ∗ 3 ∗ ci/cgi times to get the final cgo results in the output features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' With topology constraints cgi ≡ 64 and cgo ≡ 8, a 512-way parallel processing is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' If taking the batch dimension b (not drawn for simplicity) into account, the parallelism can be further increased by b times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Since employing different storing ways, the window buffer and input buffer need different fetch addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Data in the weights buffer are fetched via the incremental address counted by AGU (address generation unit);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' data in the window buffer are fetched via the pre-defined address saved in ALUT (address lookup table).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' With the addresses from AGU and ALUT, every row of length cgi ∗cgo in the weight buffer, and the corresponding row of length cgi in the window buffer, are sent to the MAC array to do parallel multiplication–addition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' then the cgo results are sent to cgo output registers to accumulate 3 ∗ 3 ∗ ci/cgi times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' finally the output element at location (x, y) in the corresponding channel of the output features is got.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' By iterating all the sliding windows and all the kernels, the full output features are got.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Under topology constraints cgi and cgo, for a kernel, every cgi adjacent kernel slices have the identical topology, and for a layer, every cgo adjacent kernels have the identical topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' So at each clock cycle, cgi∗cgo elements can be fed to the MAC array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Suppose cgi = 64 and cgo = 8, parallelism = 512 is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' This means there are 1024 multiplication and addition operations per clock, and the peak compute power reaches 1TOPS at 1 GHz clock frequency, which meets the compute requirement of most terminal devices specifically designed for CNN acceleration (Andri, Karunaratne, Cavigelli, & Benini, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Moons, Bankman, Yang, Murmann, & Verhelst, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Of course the parallelism can be further improved if taking the batch dimension into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Given batch size b, the parallelism can be further increased to b ∗ cgi ∗ cgo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Extra Cost Analysis Compared with the standard convolution, extra components are required to accelerate an LHC layer during inference, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', ALUT for indexing sparse weights and AGU for skipping invalid feature elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' For ALUT, every cgi∗cgo weights need an address index, and the overall density of all LHC layers is no greater than 20%, so extra memory consumption is just k∗ k∗ ci/cgi ∗ co/cgo/(k∗ k∗ ci ∗ co) = 1/cgi/cgo = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='1953% of that of the standard convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Let alone 80+% of space is already saved by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Time scheduling of LHC vs traditional convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (a) The timing of an LHC layer, where its structural sparsity is fully utilized and thus all redundant clocks and other computational resources are saved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (b) The timing of a standard convolution layer, where sparsity (gray parts) is not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Each column is cgi × cgo parallel operations such as read/write, multiply–add and accumulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 6 for the corresponding hardware logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=') Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The unification of various convolution techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' GWC, DWC and HetConv can all be taken as LHCs that use some certain shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Rebuilding GoogLeNet InceptionV4-A/B/C with LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The shapes used by LHCs are the ones used in the original Inceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=') For AGU, the implementation just requires negligible resources compared with the whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' So at the inference stage, a large portion of computational re- sources can be saved, even though a little extra cost is introduced in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Unification With the topology plasticity, various convolution techniques discussed in Section 2 can be unified by LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' As shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 8 and 9, representative ones like GWC, DWC, HetConv and Incep- tion are used as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The shape of the xth slice in the yth kernel of a convolution layer is notated as sx,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Group-Wise Convolution Group-Wise Convolution or GWC (Alex et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2012), can be viewed as an LHC using shapes ⟨1⟩1 and ⟨6⟩1 only, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 8 the first row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Under the settings of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='3, when the following conditions are satisfied, an LHC degenerates into a GWC: sx,y = ⎧ ⎨ ⎩ s511 ( kcgi < x ≤ (k + 1)cgi kcgo < y ≤ (k + 1)cgo ) s0 others (13) where cgi = ci/ngroup, cgo = co/ngroup, k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' , ngroup-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Depth-Wise Convolution Depth-Wise Convolution or DWC (Sandler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2018), illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 8 the second row, a special case of GWC, is an LHC using shapes ⟨1⟩1 and ⟨6⟩1 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' When the following conditions are satisfied, an LHC degenerates into a DWC: sx,y = ⎧ ⎨ ⎩ s511 ( kcgi < x ≤ (k + 1)cgi kcgo < y ≤ (k + 1)cgo ) s0 others (14) where cgi = 1, cgo = co/ci, and k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' , ci-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' HetConv HetConv (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2019) is an LHC using shapes ⟨2⟩1 and ⟨6⟩1 only, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 8 3rd row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' When the following conditions are satisfied, given p as the number of shape ⟨6⟩1 in a kernel, LHC degenerates into HetConv: sx,y = { s511 (x + y − 1)%p = 0 s1 others (15) where cgi = 1, cgo = 1, and p is a hyper parameter selected from {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' , ci}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Inception GoogLeNet Inceptions (Szegedy, Ioffe, Vanhoucke, & Alemi, 2016) can be rebuilt with LHC as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Each dashed purple rectangle is equivalent to an LHC layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' For Inception-A, the 2nd and 3rd rectangles are GWCs where the groups are not evenly divided, and thus can be replaced by LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' In Inception-B/C, orthogonal 1D convolution pairs are employed to approximate standard 2D convolutions, either serially or parallelly, where the former amounts to double non-linearity and cannot be equivalent to one LHC layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' But for simplicity, they are all replaced by LHC, then a unified Inception shown in the bottom right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 9 is got.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Others Other convolution techniques designed for efficiency can also be unified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' PWC can be viewed as standard convolution of kernel size 1*1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' MixConv (Tan & Le, 2019) is a GWC variant where kernel sizes are different;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' SeeSaw (Zhang, 2019), IGC (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2017), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', are unevenly grouped GWCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Experiments Here experiments are presented to compare LHC with other convolution techniques that are either widely used or achieve state-of-the-art results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Experiment settings Listed in Table 2 are the experiment items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Networks and Convolution Techniques The comparison of LHC, standard convolution, HetConv, FPGM (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2019) and Taylor (Molchanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2019) are conducted on VGG16/19 and ResNet34/50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Among them, HetConv represents techniques of structural sparse convolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' and FPGM/Taylor represent techniques of non-structural sparse convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The comparison of LHC, Inception, GWC+channel-shuffle and DWC+PWC are conducted on GoogLeNet InceptionV4, Shuffle- NetV1 and MobileNetV1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Switchover of Different Convolution Techniques For VGG16/19 and ResNet34/50, all convolution layers except the 1st are replaced by LHCR/LHCF and HetConv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' For GoogLeNet, Inceptions are rebuilt by LHCR in the way described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Shapes used in LHCR are ⟨1⟩1, ⟨2⟩1, ⟨6⟩1, ⟨3⟩1 and ⟨3⟩3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' For ShuffleNet, all GWC+channel-shuffle structures are re- placed by LHCR, where shapes used are ⟨1⟩1 and ⟨6⟩1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Inception-A Inception-B concat concat c7x1 c3x3 c1x7 c7x1 c7x1 c1x1 c3x3 c3x3 c1x1 c1x7 c1x7 pool c1x1 c1x1 c1x1 pool c1x1 c1x1 c1x1 feat in feat in Inception-C Inception-LHC concat concat c7x1c1x7 LHC c3x1 c1x1 c3x1c1x3 c1x1 LHC c1x3 pool c1x1 c1x1 c1x1 pool LHC feat in feat inTable 2 Experiment items: candidate convolution techniques on mainstream networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Origin GWC (Alex et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2012) DWC (Sandler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2018) Inception (Szegedy, Ioffe, Vanhoucke, & Alemi, 2016) HetConv (Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2019) LHC (ours) FPGM (He, Liu, Wang, Hu, & Yang, 2019) Taylor (Molchanov, Mallya, Tyree, Frosio, & Kautz, 2019) VGG16 (Simonyan & Zisserman, 2014) ✓ ✓ ✓ ✓ ✓ VGG19 (Simonyan & Zisserman, 2014) ✓ ✓ ✓ ✓ ✓ ResNet34 (He, Zhang, Ren, & Sun, 2016) ✓ ✓ ✓ ✓ ✓ ResNet50 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2016) ✓ ✓ ✓ ✓ ✓ GoogLeNet (Szegedy, Ioffe, Vanhoucke, & Alemi, 2016) – ✓ ✓ ShuffleNet (Zhang, Zhou, Lin, & Sun, 2018) – ✓ ✓ MobileNet (Howard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2017) – ✓ ✓ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Curves of computation reduction vs accuracy of different networks built of different techniques on different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The x-axis ∆flops is computation reduction in percentage, and the y-axis acc is classification accuracy in percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=') For MobileNet, all DWC+PWC structures are replaced by LHCR, where shapes used are ⟨1⟩1, ⟨2⟩1 and ⟨6⟩1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Tasks for Making Evaluations The tasks for evaluating different convolution techniques are the image classification, on two widely recognized datasets, CI- FAR10 and ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Training procedures are designed to be identical: on CIFAR10/ImageNet, models are to be trained 200/100 epochs, with early stopping patience 40/20, using SGD with initial learn- ing rate 1e−2/1e−3 and decay factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Data augmentations include random flipping, translation, rotation, hue, saturation, brightness and contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Specifically, for networks built of LHC, (1) Hyper-parameter dt, is set to {invalid, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='01} for CIFAR10, and {invalid, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='1} for ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Here invalid means no density target is set, and thus the network can explore the most suitable shapes and converge to the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (2) Hyper-parameter cgi and cgo are set to 64 and 8 respectively as parallelism ≥ 512 is a typical value of hardware accelera- tion (Andri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Moons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' For FPGM and Taylor, experiments are carried out by following their official procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Result analysis Results are presented in computation reduction vs classifica- tion accuracy, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Hardware resource saving with LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Clocks of 5× can be saved, which either benefits improving the throughput or reducing energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Memory of 10× can also be saved, which dramatically reduces the on-chip memory block resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' This model VGG16 is trained on CIFAR10 under constraints of parallelism = 512 and no obvious accuracy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' LHC vs Standard Conv, HetConv, FPGM & Taylor Shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 10 are the results of LHC and other convolution techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The x-axis is the computation reduction in percent- age, and y-axis is the classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The more a line to the top right corner, the better it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' By comparing LHCR and LHCF, LHCR works slightly better if ∆flops is smaller, while LHCF is a bit superior if ∆flops is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The former can be attributed to the priori knowledge implied in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Shape distribution of the 3rd/7th/11th layer of VGG16 built of LHCR and LHCF respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Left: LHCR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' right: LHCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The x-axis is the serial index of all 512 shapes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' y-axis is the ratio a shape takes up in an LHC layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' the design of those rigid shapes, and the latter is likely due to the full freedom of those free shapes in dropping redundant weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' By comparing LHC with HetConv/FPGM/Taylor, the networks built of LHC always achieve better accuracy at any computation reduction ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Given the same computation reduction, ours ac- curacy surpasses others by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='0% on CIFAR10 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='0% on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' For example, our method can improve VGG16 computation ef- ficiency by nearly 5× on CIFAR10 or 2× on ImageNet without harming its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Interestingly, there is always appropriate ∆flops value, at which networks built of LHC surprisingly outperform the original, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', the horizontal dashed blue line in every sub-figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' This rarely happens with existing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Specifically, our method improves top1 classification accuracy by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='0% on CIFAR10 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='5% on ImageNet at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Thus, our method is also a powerful training technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The parameter reduction is directly up to the dt we set, namely, dt ∈ {invalid, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='01} for CIFAR10, and dt ∈ {invalid, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='1} for ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Without harming the accuracy, our method can compress VGG/ResNet weights by about 10× on CIFAR10, or 4× on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' We also conducted the hardware acceleration simulation, mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='4, on a VGG16 model, which is trained on CIFAR10 under dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='1 and achieves ∆flops = 78% and acc = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='45% (almost the same as the original performance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The number of clocks and memory rows (each row contains 3∗3∗cgi∗ cgo weights) required during the inference of an input image of shape 32*32 are drawn in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' In hardware implementation, the number of clocks and memory rows can also be saved by 5× and 10×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' With less requirements in clock and memory, the power dissipation of hardware is accordingly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' LHC vs GWC+Channel-Shuffle, DWC+PWC & Inception Here ∆flops are evaluated when ShuffleNet/MobileNet/ GoogLeNet and their LHC variants have similar accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The GoogLeNet model, with its Inceptions rebuilt by LHC, achieves computation reduction ∆flops = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='39%, and parame- ter compression ∆param = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='56%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' the MobileNet model, with its DWC+PWC modules replaced, achieves ∆flops = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='68% and ∆params = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='82%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' and the ShuffleNet model, with its GWC+ channel-shuffle modules replaced, achieves ∆flops = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='36% and ∆params = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='73%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Clearly, the efficiency of GoogLeNet/ShuffleNet/MobileNet, ei- ther well-designed or light-weight, can be further improved by LHC, which means the efficiency gain contributed by our layer- level method even beats that of beyond-layer-level methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Interestingly, the computation reduction amplitude of our method on ShuffleNet is obviously less than on the other two, which verifies the necessity of information exchange provided by channel-shuffle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Discussions Here we discuss why LHC works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The VGG16 models trained on ImageNet with dt = invalid are chosen, since other models have similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' What LHC Learns The core difference between LHC and standard convolution is the dendritic topology plasticity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', learning masks of suitable shapes for current task in a data-driven manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' We count the number of shapes learnt by the 3rd/7th/11th LHC layer, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The x-axis is the serial indexes of all 512 shapes drawn in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 3, and the y-axis is the ratio a shape takes up in an LHC layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' By comparing the shape distribution of different LHCR/LHCF layers, we find that: (1) low layers prefer denser shapes and mainly have no shape ⟨1⟩1, which is totally sparse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (2) high layers have the least amount of denser shapes but the most amount of shape ⟨1⟩1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (3) middle layers takes the intermediate shape distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' This means that enough number of kernels in low layers is necessary for extracting sufficient basic patterns, yet after the non-linearity of a layer upon a layer, the patterns in the feature maps become increasingly abstract and sparse, and thus just need sparser kernels to extract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' How LHC Converges We penetrate into the training process by inspecting how the shapes/masks in LHC layers evolve along with the training epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' We calculate the masks correlation like Guillaume et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (2018) of an LHC layer between two adjacent epochs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', the mean of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Evolution of masks along with the training epochs of the 3rd/7th/11th LHCF layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The x-axis is the epoch number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' y-axis is the masks correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Mask correlation grows as training goes, which means LHC masks generally evolve from dynamic states to more static states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The spectrums of the 3rd/7th/11th LHC layers and corresponding standard convolution layers from two models that are also used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The x-axis is spectrum components, and y-axis is their amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The more the convolution layers’ spectrum of a CNN model approximates uniform distribution, the better the model’s performance will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Apparently, LHC layers’ spectrums are always more uniformly distributed than that of standard convolutions, which intuitively explains why under appropriate conditions models based on our method have performance better than models based on the standard convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' element-wise-logical-and.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' We choose the 3rd/7th/11th layers of VGG16 built of LHCF to visualize in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' LHCR has similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The x-axis is the number of epochs and the y-axis is the correlation of masks between two epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The correlation of masks in each layer increases as the training epochs carry forward, but is always less than 100%, even if the model converges enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' This means our differentiable step func- tion keeps the dendritic topology updating all the time (of course so is the synaptic strength), just like the highly dynamic biological neural networks in the brain (Holtmaat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Stettler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Besides, the correlation of lower layers is larger than the higher, which is also as expected, because the lower layers learn the patterns that are more task-agnostic and thus requires more stable dendritic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Why LHC Excels As pointed out by OCNN (Wang, Chen, Chakraborty, & Yu, 2020), the spectrum of a convolution layer’s DBT (doubly block Toeplitz) matrix reasonably indicates how well the CNN model’s capacity is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' The more uniformly the spectrums distribute, the better the model performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Clearly drawn in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 14, our method’s spectrums are always closer to uniform distribution than the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Combined with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 13, we may assure that it is the changing masks that force the model to utilize its capacity better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Now we know about the weights in a model built of LHC: (1) there are a lot of zeros, up to 80+%, which can be taken as a case of L0/L1 regularization training, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', strength regularization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (2) these zeros distribute in a structural way in the kernels, which further provides another kind of regularization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', topol- ogy regularization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (3) the position of these zeros changes along with the training steps, which can be seen as the dropout of the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Hence, it is not difficult to understand why our method sur- passes existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Conclusion Our method LHC integrates the plasticity of both dendritic topology and synaptic strength, making both the kernel shapes and the weights learnable during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' CNN models built of LHC can be greatly sparsified structurally and can be acceler- ated in high parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Experiments against existing convolution techniques show that LHC always achieves better performance at any computation reduction ratio;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' and experiments of rebuilding typical network structures show that LHC can improve their efficiency even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Our method also achieves better CNN performance if taken as a training technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' CRediT authorship contribution statement Rongzhen Zhao: Conceptualization, Methodology, Software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Zhenzhi Wu: Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Qikun Zhang: Software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Declaration of competing interest The authors declare that they have no known competing finan- cial interests or personal relationships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Acknowledgments This work was supported by ‘‘Science and Technology Inno- vation 2030 - New Generation of Artificial Intelligence’’, China project (2020AAA0109100) and Beijing Science and Technology Plan, China (Z191100007519009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' References Alex, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' In IEEE conference on computer vision and pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=', & Tian, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' Accelerate CNN via recursive Bayesian pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} +page_content=' In IEEE international conference on computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfHg55/content/2301.05440v1.pdf'} diff --git a/ndAzT4oBgHgl3EQfqf0O/vector_store/index.faiss b/ndAzT4oBgHgl3EQfqf0O/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..8d8f273f566fea6f99fef1834eaba2214569985d --- /dev/null +++ b/ndAzT4oBgHgl3EQfqf0O/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:252389e97e906b422e9bbe6a6b369f7781126bfb06deabd6205b4838e1385f7e +size 3735597 diff --git a/ndE3T4oBgHgl3EQf7Asa/content/2301.04794v1.pdf 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Hocky∗,† +†Department of Chemistry and Simons Center for Computational Physical Chemistry, New York University, +New York, NY 10003 +‡Department of Chemistry, Oklahoma State University, Stillwater, OK 74078 +E-mail: martin.mccullagh@okstate.edu; hockyg@nyu.edu +Abstract +In this work, we demonstrate that Linear Discriminant +Analysis (LDA) applied to atomic positions in two dif- +ferent states of a biomolecule produces a good reaction +coordinate between those two states. Atomic coordi- +nates of a macromolecule are a direct representation of +a macromolecular configuration, and yet they have not +been used in enhanced sampling studies due to a lack +of rotational and translational invariance. We resolve +this issue using the technique of our prior work, where +a molecular configuration is considered a member of +an equivalence class in size-and-shape space, which is +the set of all configurations that can be translated and +rotated to a single point within a reference multivariate +Gaussian distribution characterizing a single molecu- +lar state. The reaction coordinates produced by LDA +applied to positions are shown to be good reaction co- +ordinates both in terms of characterizing the transition +between two states of a system within a long molecular +dynamics (MD) simulation, and also ones that allow +us to readily produce free energy estimates along that +reaction coordinate using enhanced sampling MD tech- +niques. +1 +Introduction +A large class of enhanced sampling techniques work by +biasing a system to explore along a low dimensional set +of collective variables (CVs).1 These methods allow us, +in principle, to use the known bias applied to reconstruct +the free energy landscape in that low dimensional space. +In practice, the choice of the CVs is crucial, with an +ideal set of CVs allowing the system to explore all rele- +vant states within available simulation time.1 Recently, +extensive effort has been invested in using a variety of +machine learning approaches, from very simple to very +sophisticated, to determine optimal coordinates for sam- +pling from molecular dynamics (MD) simulation data +(Refs. 2–21 provide a representative but not exhaustive +sample). +One commonly encountered challenge is to compute +the free energy path of a transition between two states +along a linear dimension that chemists term a reaction +coordinate (RC). For a macromolecule such as a pro- +tein, the two states could be configurations for which +we have known structures (e.g. the PDB structure of +a protein solved with and without a bound ligand), or +processes for which one state is known and the other can +be at least qualitatively defined (e.g. folding/unfolding +or binding/unbinding). If a long MD trajectory contain- +ing multiple transitions between these states is available, +then reaction coordinates could be trained based on the +idea that we want to enhance sampling along the slow- +est modes in the system.4,10,13,14,22,23 However, having +this data is rare, in which case one can try iterative en- +hanced sampling and learning reaction coordinates with +the goal of maximizing the number of transitions be- +tween the two states in a fixed amount of simulation +time.4,5,11,13,15,24 +An alternative approach which has shown some suc- +cess is to train reaction coordinates based on short sim- +ulations within the two states, and use a method that +produces a coordinate representing the difference be- +tween the two sets of data. Linear dimensionality re- +duction techniques such as Principal Component Anal- +ysis (PCA) and Linear Discriminant Analysis (LDA) +are the simplest approaches to combining a large set of +variables defining a system of interest and producing +a small set of CVs that characterize the available data. +While PCA, which produces coordinates that capture +the most variance in the data, has been used for explo- +1 +arXiv:2301.03479v1 [physics.chem-ph] 9 Jan 2023 + +ration in enhanced sampling simulations, LDA seems +to hold more promise as an RC since it is a supervised +approach whose goal is to maximally separate different +labeled classes of data (i.e. reactants and products). We +describe LDA in full detail in the next section. In one +study, Mendels et al.6 produced a modified approach +to LDA termed harmonic LDA (HLDA, because the +covariance matrices in the two different states of inter- +est are combined by a harmonic average rather than a +simple sum), and in that work and subsequent ones,7,9 +combined it with Metadynamics (MetaD) to effectively +enhance sampling between two states in several different +systems. Later, a neural network was used to combine +features before training LDA vectors to produce the +reaction coordinate.16 +In prior examples of reaction coordinate design for +free energy sampling of biomolecules that we are aware +of, the input features to the method were internal co- +ordinates, or a function of internal coordinates, for the +molecule(s) of interest—for example, distances, angles, +and dihedrals. Often, these could be CVs based not on +atomic positions directly, but on coarse-grained (CG) +representations of the biomolecule, such as the distance +between the centers of masses (COMs) of two different +domains, or the distance between the COM of a ligand +and certain atoms in its binding pocket. This is not sur- +prising, because these often correspond to our physical +intuition about what the biomolecular reaction is. More- +over, internal coordinates are invariant to translation and +rotation of the molecule, and thus bias forces applied +to these coordinates do not depend on the position or +orientation of the molecule. +Recently, we presented atomic coordinates as an alter- +native set of features to use in the context of clustering +biomolecular data.25 Atomic coordinates of a subset of +atoms, or of beads corresponding to a CG representation +of a molecule, offer an alternative to internal coordi- +nates with the advantage that there is little choice in +selecting the features to use. Using a protein as an +example, we need only make the standard choice be- +tween Cα atoms, backbone, all heavy atoms, and so +on. Moreover, only 3N − 6 atomic coordinates essen- +tially describe the state of a biomolecular system with N +important atoms (but ignoring contributions of solvent, +salt, etc.), whereas use of internal coordinates often re- +sults in an over-determined set of features, such as all +O(N2) pairs of distances. In Ref. 25 we developed a +procedure for clustering molecular configurations into a +Gaussian mixture model (GMM) using atomic positions +that overcomes challenges of orientational dependence +that prevented their use earlier, as described below. Be- +cause a Gaussian mixture model in positions is a natural +way to coarse-grain a free energy landscape,25–28 with +locally harmonic bins around metastable states, the re- +sulting clustering is a physically appealing definition of +the “states” one’s molecule can adopt. +However, our Gaussian mixture model still relies on +a very high (3N − 6) dimensional representation of our +molecule. Given that the output of our clustering algo- +rithm is a set of states each defined by a multivariate +Gaussian distribution, LDA is a natural approach to pro- +duce a low dimensional representation of our data with +large separation between states. In this work, we first +apply LDA to the folded and unfolded states determined +from shapeGMM clustering of a long unbiased MD tra- +jectory of a fast-folding protein, and demonstrate that it +produces a physically reasonable ordering of states from +folded to unfolded. We then show that this coordinate +is a “good” reaction coordinate because the position of +the barrier separating folded and unfolded is very close +to the location where the system is equally likely to +proceed to folded or unfolded (in terms of a committor +function to be defined below). We implement this posi- +tion LDA coordinate in the PLUMED sampling library, +and demonstrate that biased sampling along this coordi- +nate can accelerate transitions between the folded and +unfolded states, and produce a qualitatively similar free +energy surface as compared to the unbiased trajectory in +3% of the simulation time, without any additional tuning +of the CV. Finally, we train a position LDA coordinate +on an achiral helical system where data is only avail- +able in the left and right handed states, and show that +this coordinate also allows us to readily sample between +the two states, even though no information about the +transition was provided during training. +2 +Theory and Methods +2.1 +Molecules in Size-and-Shape Space +Consistent with our previous work on structural align- +ment and clustering,25 we consider structures from a +molecular dynamics (MD) simulation to be associated +with Gaussian distributions in atomic positions. Struc- +tures are represented by N particles (a subset of atoms) +using a vector x of dimension N × 3 which is a member +of an equivalence class, +[xi] = {xiRi + 1N⃗ξT +i : ⃗ξi ∈ R3, Ri ∈ SO(3)}, +(1) +where ⃗ξi is a translation in R3, Ri is a rotation R3 → R3, +and 1N is the N × 1 vector of ones. [xi] is a point in +size-and-shape space29 which has dimension 3N − 6 +and is defined as S Σ3 +N = R3N/G where G = R3 × SO(3) +2 + +is the group of all rigid-body transformations for each +frame with elements g = (⃗ξ, R). +Within the shapeGMM framework, the probability +density of particle positions is assumed to be a Gaussian +mixture, +P(xi) = +K +� +j=1 +φ jN(xigi,j | µ j, Σ j), +(2) +where N(xigi,j | µ j, Σ j) is the jth normalized, multi- +variate Gaussian with mean µ j, covariance matrix Σ j, +and weight φ j (the weights are normalized such that +�K +j=1 φj = 1). gi,j is the element of G that minimizes +the Mahalanobis distance between xi and µ j. Iterative +determination of gi,j and µ j is performed in a Maximum +Likelihood procedure.25 +In the current work, we will only consider LDA coor- +dinates learned using data from only two states. Addi- +tionally, we will only consider “weighted” alignment of +particle positions, which equates to using a Kronecker +product covariance (where Σj = ΣN ⊗ I3, for ΣN the +N × N covariance of particle positions) in defining the +Mahalanobis distance between frame and average struc- +ture as described in detail in Ref. 25. +2.2 +Dimensionality Reduction using Linear +Discriminant Analysis on Particle Posi- +tions +We propose to use LDA directly on aligned particle po- +sitions as a reaction coordinate. LDA for two states pro- +duces the linear model with the maximal inter-average +variance while minimizing intra-cluster variance.30 This +is achieved by first computing the within-cluster scatter +matrix, +Sw = +K +� +i=1 +� +j∈Ni +(x j − µi)(x j − µi)T, +(3) +and the between-cluster scatter matrix, +Sb = +K +� +i=1 +(µi − µ)(µi − µ)T, +(4) +where µi is the average structure of cluster i and µ is +the global average. The simultaneous minimization of +within-cluster scatter and maximization of between clus- +ter scatter can be achieved by finding the transformation +G that maximizes the quantity +Tr +� +(GTSwG)−1GTSbG +� +. +(5) +This maximization can be achieved through an eigen- +value/eigenvector decomposition but such a procedure +is only applicable when Sw is non-singular. The LDA +method was reformulated in terms of the generalized +singular value decomposition (SVD)31 extending the +applicability of the method to singular Sw matrices such +as those encountered when using particle positions. +In addition to employing the SVD solution of to the +LDA approach, care must be taken in how particle posi- +tions are aligned when performing LDA. This is evident +when one considers the scatter matrices in Equation 3 +and Equation 4. The values and null spaces of these +scatter matrices will depend on the specific alignment +procedure chosen. There are three obvious choices for +structural alignment prior to LDA: (1) alignment of each +frame to its respective cluster mean/covariance, (2) align- +ment to one cluster or another, and (3) alignment to a +global average. The first choice will lead to scatter ma- +trices with different null spaces for each cluster making +their addition in Equation 3 unsatisfactory. Alignment +options (2) and (3) will both yield consistent null spaces +making the choice of one over the other not immediately +obvious. While there may be reasons to selection op- +tion (2), we have chosen to move forward with global +alignments for the systems studied here. +The result of an LDA procedure will be a set of K − 1 +vectors {v}K−1 of coefficients that best separate the data. +These vectors are similar in nature to the eigenvectors +from a principle component analysis, a procedure more +familiar to the bio-simulation field. +2.3 +Biasing a linear combination of positions +The value of the LDA coordinate after this procedure is +a dot product of the vector v with the atomic coordinates +x − µ. When computing this value on the fly within an +MD simulation, we need to consider the value of [x(t)], +the equivalence class of the position at time t, translated +and rotated to a reference {µ, Σ}. +Therefore, to compute the value of the LDA coordi- +nate l, we first translate x(t) by ⃗ξ(t) = 1 +N +�N +i=1 ⃗xi(t) − +1 +N +�N +i=1 ⃗µi(t), the difference in the geometric mean of the +current frame and that of the reference configuration. +Then, we compute R(t), the rotation matrix which mini- +mizes the Mahalanobis difference between x(t) −⃗ξ and +µ, for a given Σ, as described in Ref. 25. Finally, we +compute +l(x) = v · �R · (x(t) − ⃗ξ(t)) − µ� +(6) +By definition, l(µ) = 0. +To apply bias forces to this coordinate, we must be +able to compute ∇l(x(t)). Because of the inclusion of +3 + +the optimal rotation process by SVD, it is non-trivial +to compute this analytically, and we instead compute +derivatives numerically. +2.4 +Enhanced sampling with OPES-MetaD +Enhanced sampling simulations on LDA coordinates +were performed using Well-tempered Metadynam- +ics (WT-MetaD), and On the Fly Probability En- +hanced Sampling-Metadynamics (OPES-MetaD) as im- +plemented in PLUMED.32–35 +WT-MetaD works by adding a bias formed from a his- +tory dependent sum of progressively shrinking Gaussian +hills.36,37 The bias at time t for CV value Qi is given by +the expression +V(Qi, t) = +� +τ 0, the topological phase cor- +responds to the case where we have an even number of +sites and δ > 0. A signature of the topological phase +is the existence for open conditions of quasi-degenerate +zero energy eigenstates [21] in which a single particle is +in a coherent superposition between the two edges of the +chain - see e.g [22, 23] for details on the non-interacting +case. +The Fourier transform for open boundaries is given by +cj = i +� +2 +N + 1 +N +� +n=1 +˜cn sin +� πjn +N + 1 +� +. +(2) +For δ = 0, this rotation diagonalizes the problem, i.e we +have Hδ=0 = � +n εn˜c† +n˜cn with εn ≡ −2t cos +� +πn +N+1 +� +. +The continuum limit is obtained by introducing the +lattice spacing a and defining the position x ≡ ja and +arXiv:2301.00472v1 [cond-mat.str-el] 1 Jan 2023 + +2 +the momentum k ≡ +2πn +2a(N+1). The size of the sytem is +taken to be L ≡ a(N +1) so that k = πn +L . The continuous +fermionic field is Ψ(x) ≡ +1 +√acj. The boundary conditions +for Ψ are obtained by extending the discrete formula (2) +to site 0 and site N +1 : Ψ(0) = 0 and Ψ(x = a(N +1) = +L) = 0. +Following the usual bosonization procedure [17, 24], +we split Ψ into a left and a right moving fields ΨR/L by +expanding around the Fermi energy εkF : +Ψ(x) = ΨR(x) + ΨL(x), +(3) +ΨR(x) = +� +L +2 eikF x +� ∞ +−∞ +dk +π ˜ck+kF eikx, +(4) +ΨL(x) = −ΨR(−x). +(5) +For convenience, we also define the “slow” fields R(x) ≡ +ΨR(x)e−ikF x, +L(x) ≡ eikF xΨL(x). Importantly, be- +cause of open boundaries, the left and right movers are +not independent as is encapsulated by Eq.(2), see e.g [25– +27] for previous discussions of open boudaries bosoniza- +tion. +In the continuum, R(x) taken alone can be thought +of as a field living on a space of size 2L with periodic +boundary conditions. In the bosonized language, it can +then be reexpressed as +R(x) = FR +1 +√ +2παeiφR(x), +(6) +φR(x) ≡ πx +L NR + +� +n>0 +1 +√n(anei nπx +L + a† +ne−i nπx +L )e−α πn +2L . +(7) +where FR is the Klein factor associated to the right-mover +with F † +RFR = FRF † +R = 1, NR the particle number op- +erator associated to the right movers, (an, a† +n)n>0 a set +of bosonic modes indexed by n and α a regularization +parameter. +The expression of the bosonic field associ- +ated to the left-movers can be readily deduced from (5). +FL = −FR, φL(x) = φR(−x). The conjugated fields φ, θ +are customarily defined as +φ(x) ≡ −φR(x) + φL(x) +2 +, += −πx +L NR − i +� +n>0 +1 +√n(an − a† +n) sin +�nπx +L +� +e−α πn +2L , +(8) +θ(x) ≡ φR(x) + φL(x) +2 +, += +� +n>0 +1 +√n(an + a† +n) cos +�nπx +L +� +e−α πn +2L . +(9) +The particle density operator ρ, which counts the number +of particle above the Fermi sea, is deduced from φ(x) +through the relation +ρ(x) = − 1 +π ∂xφ(x). +(10) +In the remaining, as we want to characterize the 0 energy +modes, we will work at half-filling. +It is important to +notice that the definition of the half-filling depends on the +total number of sites. Let NF ∈ N label the last occupied +state of the Fermi sea. For N even, we have NF ≡ N +2 . +The corresponding momentum is kF = +πN/2 +a(N+1) ≈ +π +2a − +π +2aN ≈ +π +2a − π +2L to first order in 1/L. For N odd, the two +possible definitions of the half-filled state are NF = N±1 +2 +with corresponding Fermi momenta kF = +π +2a − +π +2L(1 ∓ +1). +In the remaining of the paper, we will chose the +convention NF = +N−1 +2 +for odd number of sites. +The +Fermi sea state with all modes filled up to NF and empty +above will be referred to as the vacuum state. +We show in the SM [28] that H can be expressed in +terms of the bosonic fields as +H = +� L +0 +dx +�vF +2π +� +: (∂xφ)2 : + : (∂xθ)2 : +� +(11) ++ +δ +L sin πx +L +: cos +� +2φ(x) + +�π +a − 2kF − π +L +� +x +� +: +� +where :: denotes normal-ordering of the fermionic modes +with respect to the vacuum and vF ≡ 2ta sin(kF a) the +Fermi velocity. +Formula (11) constitutes one of the +main results of this paper. We see that the SSH in the +bosonized language is almost equivalent to a sine-Gordon +Hamiltonian except for the spatial dependence of the +prefactor in front of the sinus. To derive (11), we dis- +carded constant terms and fast-varying modes ∝ e2ikF x, +so the bosonic field describes modes with slow spatial +variation with respect to the lattice spacing. +Since +bosonization is a theory describing low energy excita- +tions, we also expect this expression to be valid for δ +t ≪ 1. +We now turn to the computation of ρ using the clas- +sical Euler-Lagrange (EL) equations of motion. +Let +Z ≡ tr(e−βH) and Π the conjugated field to φ, Π ≡ 1 +π∂xθ. +In the imaginary time formalism, we have +Z = +� +DφDΠe +� β +0 +� L +0 dτdxL, +(12) +with L the Lagrangian density : +� +dxL = +� +dxiΠ∂τφ−H. +The EL equations ∂L +∂φ = +∂ +∂τ +∂L +∂∂τ φ + +∂ +∂x +∂L +∂∂xφ yields +1 +vF +∂2 +τφ + vF ∂2 +xφ += − +2πδ +L sin πx +L +sin +� +2φ(x) + +�π +a − 2kF − π +L +� +x +� +. +(13) +In the 0 temperature limit, all the weight of the probabil- +ity measure will be contained in the stationary solution +∂τφ = 0. +Introducing the natural rescaling y ≡ x/L, +ϕ(y) ≡ φ(yL), we get the L-independent equation +∂2 +yϕ(y) = −∆sin(2ϕ(y) + ϵπy) +sin(πy) +. +(14) + +3 +where we introduced ∆ = 2πδL +vF +≈ πδ(N+1) +t +and ϵ = 0, +for an even number of sites where kF = +π +2a − +π +2L and +ϵ = 1 for an odd number of sites using the convention +kF = +π +2a − π +L. The boundary conditions for ϕ are read +from Eq.(8): At y = 0 we have ϕ = 0 and at y = 1 we +have ϕ = −πnR with nR the number of particles created +on top of the vacuum. +Let us make some remarks here. First, note that ∆ is +an adimensioned parameter that fully characterizes the +solution of the EL equations. Note that ∆ scales linearly +in N and +δ +t , so increasing the system size has exactly +the same effect as increasing the ratio δ +t . Interestingly, +going from the odd to the even case is equivalent to shift +ϕ by π +2 y which can be interpreted as substracting half a +particle to the system. +To the best of our knowledge, (14) has no known an- +alytical solution and we have to resort to numerics. To +assert the validity of our approach, we compare numerical +solutions of (14) with exact diagonalization (ED) results +performed on the discrete Hamiltonian (1) in the zero +temperature ground state. The ED results show fast os- +cillations on the scale of the lattice spacing ∝ 1/kF that +we do not see from the solutions of the EL equations of +motion since we precisely discarded these terms. Coarse- +graining over the fast oscillations gives a smoothly vary- +ing density profile on the scale of the total system size. +We observe excellent agreement between the ED and the +EL solutions - see Fig.1. We also checked that the agree- +ment holds both for δ positive or negative, for an even +or an odd number of sites and for different values of nR +[28]. +For the even case and nR = 0 the solution of the EL +equations is simply φ = 0 so that ρ(x) = 0, which is con- +sistent with the particle-hole symmetry of the model. For +δ > 0, fixing nR = 1 amounts to populate the first mode +above the vacuum state, i.e the Majorana edge mode. +From (10,14), we can thus deduce the density profile of +this edge mode. We plot on Fig.1 the numerical solu- +tion of (14) and indeed see a concentration of the density +at the boundaries. There is a nice interpretation from +Eq.(11). The term proportional to δ wants to lock the +field in the minima of the cos term. For N even and δ > 0, +this corresponds to φ = − π +2 [π]. To match the boundary +condition, the field φ must jump from 0 to −π/2 and then +from −π/2 to −π. These jumps translates in concentra- +tion on the edges for the density ρ(x) = − 1 +π∂xφ(x). The +stiffness of the jumps of φ is controlled by ∆ and deter- +mines how much the mode is concentrated at the edges. +Eq.(14) is consistent with the exponential localization of +the edge mode near the boundary. Indeed, for small y, +one crude approximation of Eq.(14) at first order in y is +given by ∂y(∂yϕ) ≈ − 2∆ +π ∂yϕ. Using additionally that +the total particle number is 1, i.e +1 +π +� 1 +0 dy∂yϕ(y) = 1, +leads to the following ansatz ϕA when ∆ ≫ 1 : +Figure 1. Comparison between the results of the discrete ED, +the bosonization result and the exponential fit for ∆ = 20, +N = 500. The uniform vacuum density ρj = 1/2 has been +substracted. The light-blue curve corresponds to exact dis- +crete ED result which show fast oscillations at the scale of +the lattice spacing. The green dashed curve represents the +same data coarse-grained over 2 sites. The red curve is the +density profile obtained by solving the EL equation (14) from +the bosonized theory and using ρj = 1 +2 − 1 +π ∂xφ(x = ja) and +φ(x = ja) = φ′(y). Lastly, the dashed blue line is obtained +from the exponential ansatz (15). Note that the latter two +appear superposed on this plot. +∂yϕA(y) = −∆cosh( ∆ +π (2y − 1)) +sinh( ∆ +π ) +. +(15) +This exponential ansatz dictates the expression for the +typical localization length of the edge mode +ℓ ≡ πL +2∆ = vF +4δ . +(16) +For the free SSH, it is known -see e.g [22, 23]- that ℓ ∝ +a +ln t+δ +t−δ ≈ at +2δ for δ +t ≪ 1, which is consistent with our result +since vF ≈ 2ta at half-filling. Remark that being in the +δ +t ≪ 1 regime automatically implies that ℓ ≫ a. +We now turn to interactions. We consider a nearest +neighbor interacting term of the form +HI := V +N−1 +� +j=1 +(nj − 1/2) (nj+1 − 1/2) , +(17) +with nj := c† +jcj the particle number operator. From now +on, we will work exclusively, in the topological phase, i.e +δ > 0, N even and NF = N +2 . We show in the SM [28] that +in the presence of this interacting term, the bosonization +procedure leads for the total Hamiltonian to +H = +� L +0 +dx +� 1 +2π +� u +K : (∂xφ)2 :I +uK : (∂xθ)2 :I +� ++ δ +� 2 +aπ +�1−K +1 +� +L sin πx +L +�K : cos(2φ) :I +� +. +(18) + +0.025 +Discrete ED +Coarse-grained ED +Bosonization +0.020 +Exp Ansatz +0.015 +p +0.010 +0.005 +0.000 +0 +100 +200 +300 +400 +500 +j4 +Figure 2. a Plots of relation (22) for the localization length in the interacting case as a function of K for different values of +δ. We see that the effect of interactions on the edge mode strongly depends on the value of δ. b Comparison between the +solution of the EL equations of motion (19) in the interacting case with DMRG simulations. The DMRG results for attractive +(repulsive) interactions have been coarse-grained once (twice) over two sites. Only half of the solution is shown for better +readibility. We took N = 100, δ = 0.05, V = 1.6, K = 0.7 for the repulsive case and V = −0.77, K = 1.4 for the attractive +case. In this case, attractive (repulsive) interactions delocalize (localize) the edge mode. c Same than b with parameters value +N = 26, δ = 0.2, V = 1.6, K = 0.7 for the repulsive case and V = −0.77, K = 1.4 for the attractive case. We see that, in +comparison to case b, the qualitative effect of interactions are swaped. +With uK ≡ vF and u +K ≡ vF + 4V a +π . Since the free part of +the Hamiltonian has been rescaled by interactions, nor- +mal ordering needs to be done with respect to the new +“squeezed” vacuum which we denote by ::I. The K depen- +dence of the prefactors of the cos terms is a direct conse- +quence of that. In principle, since we are at half-filling, +there should also be a cos(4φ) term [28]. For simplifica- +tion, we will neglect this contribution in the present work +as it is irrelevant in the RG sense if the interactions are +not too repulsive, i.e if K > 1/2. Eq. (18) is the second +crucial result of the paper. The EL equations of motion +in the presence of interactions become +∂2 +yϕ(y) = −2 +�L +a +�2 �aπ +2L +�K +K2 δ +t +sin(2ϕ) +(sin(πy))K . +(19) +A comparison of numerical solutions of (19) and density- +matrix renormalization group (DMRG) simulations is +shown on Fig.2-b for δ = 0.05, N = 100 and V = 1.6, +K = 1.4 for the attractive case and V = −0.77, K = 1.4 +for the repulsive case. We see that the EL equations of +motion predicts the correct density profile. For these pa- +rameters, we see that attractive interactions delocalize +the edge mode into the bulk while repulsive interaction +localize it further. +For K < 2, we can give an estimation of the localiza- +tion length by expanding (19) for y ≪ 1. This gives +∂2 +yϕ(y) = − +�2L +a +�2−K +K2 δ +t +ϕ(y) +yK +(20) +Imposing ϕ(y) = 0, the solution to this equation are of +the form +φ(x ≪ L) = +� x +LJ +1 +|2−K| +��2x +a +� 2−K +2 +�δ +t +� 1 +2 +2K +|2 − K| +� +(21) +where Jα is the Bessel function of the first kind - see [28] +for the proof. The precise shape of the Bessel function +depends on α but, as one can easily verify, the position +of the first maximum of Jα scales linearly with α. Thus, +we define the localization length to be the value ℓI such +that ϕ(ℓI) = +2 +|2−K| where the factor 2 has been put in +order to match with the localization length of the free +case. Following this definition, we obtain that +ℓI = a +2 +� +t +δK2 +� +1 +2−K +. +(22) +The localization length diverges at K = 2 if δ +t < 1 +4. This +gives a rough criteria for having a localized mode in the +attractive regime K > 1. Importantly, note that ℓI is +not, in general, a monotonic function of K see Fig-2-a. +Interestingly, one consequence of this is that attractive or +repulsive interaction do not systematically delocalize or +localize the edge mode, their effect can change depending +on the value of δ. This is illustrated on Fig.2-c where we +took δ = 0.2. Contrary to the previous case shown on +Fig.2-b , we see that the effects of attractive interactions +is to localize the edge mode further to the boundary and +the other way around for the repulsive ones. +Conclusion - In this paper, we derived the bosonized +theory of the interacting SSH model with open bound- +aries. +Importantly, our study offers quantitative argu- +ments to determine the effects of interactions on the + +20.0 +6 +Bosonization attractive +Bosonization attractive +0.16 +0.05 +0.04 +17.5 +DMRG attractive +DMRG attractive +0.1 +Bosonization repulsive +Bosonization repulsive +15.0 +0.2 +DMRG repulsive +DMRG repulsive +0.25 +0.12 +0.03 +12.5 +0.4 +p +Q +0.02 +0.08 +7.5 +5.0 +0.01 +0.04 +2.5 +0.00 +0.0 +0.00 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +10 +20 +30 +40 +50 +0 +2 +4 +6 +8 +10 +12 +K +j +j +a +b +c5 +edge mode and pave the way to study other interact- +ing topological models such as the spin 1 antiferromag- +netic Heisenberg chain [29, 30], the AKLT model [31] or +the Kitaev fermionic chain [32]. One of our remarkable +results for the SSH model is that attractive (repulsive) +interactions do not systematically delocalize or localize +the edge mode, but this behavior is strongly dependent +on the value of the staggering parameter δ. +We focused on the mean density profile of the edge +mode but it would be interesting as a future direction to +understand the interplay between interactions and quan- +tum correlations of the edges. +Another notable point +is that we systematically discarded the Umklapp term in +our study. Nevertheless, in the strongly repulsive regime, +it is expected to play a role, leading to possibly interest- +ing new phenomena. +Acknowledgements The DMRG simulations pre- +sented in this paper were done using the TeNPy pack- +age for tensor network calculations with python [33]. T.J +thanks Aashish Clerk for interesting discussions and com- +ments on this work. 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Notes , 5 (2018). +SUPPLEMENTAL MATERIAL +Abacus +For the reader’s convenience, we first recall the expression of the fermionic fields in terms of bosonic modes as well +as various useful relations they satisfy. +The continuous fermionic field Ψ is split between a right-moving field and a left-moving one : +Ψ(x) = ΨR(x) + ΨL(x), +(23) +ΨR(x) = +� +L +2 eikF x +� ∞ +−∞ +dk +π ˜ck+kF eikx, +(24) +ΨL(x) = −ΨR(−x). +(25) +The main difference with bosonization on an infinite system size is that the left mover is defined from the right mover +for open boundaries. The slow-mode is defined as +R(x) ≡ FR +1 +√ +2παeiφR(x), +(26) +φR(x) ≡ πx +L NR + +� +n>0 +1 +√n(anei nπx +L + a† +ne−i nπx +L )e−α πn +2L . +(27) +One key observation is that R(x) can be interpreted as a field with periodic boundary conditions on a system with +size 2L. +The corresponding left mode is +L(x) = FL +1 +√ +2παeiφL(x), +(28) +FL = −FR, +(29) +φL(x) = −πx +L NR + +� +n>0 +1 +√n(ane−i nπx +L + a† +nei nπx +L )e−α πn +2L . +(30) +and the canonical fields φ and θ are defined as +φ(x) := −φR(x) + φL(x) +2 +, += −πx +L NR − i +� +n>0 +1 +√n(an − a† +n) sin +�nπx +L +� +e−α πn +2L , +(31) +θ(x) := φR(x) + φL(x) +2 +, += +� +n>0 +1 +√n(an + a† +n) cos +�nπx +L +� +e−α πn +2L . +(32) +It will turn out to be useful to divide the φR operator into an annihilation part ϕR and a creation part with respect +to the vacuum state : +φR = ϕR + ϕR†, +(33) +ϕR = πx +L NR + +� +n>0 +1 +√nanei nπx +L e−α πn +2L , +(34) +ϕR† = +� +n>0 +1 +√na† +ne−i 2nπx +2L e−α πn +2L . +(35) + +7 +These operators have commutation relation +[ϕR, ϕR†] = − log(1 − e−α π +L ). +(36) +Using Baker-Campbell-Hausdorff relation eA+B = eAeBe− 1 +2 [A,B] when [[A, B], B] = [[A, B], A] = 0 and e− 1 +2 [ϕR,ϕR†] = +� +1 − e−α π +L ≈ � απ +L , we can express R(x) in terms of normal-ordered operators (Normal-ordering denoted by :: here +refers to the normal order of bosons with respect to the Fermi sea vacuum state denoted |Ω⟩) : +R(x) = FR +1 +√ +2L +: eiφR(x) : . +(37) +The normal ordered version of the left field is +L(x) = −FR +1 +√ +2L +: eiφR(−x) : . +(38) +A useful relation allowing us to deal with product of normal ordered field is +: eiaφR(x) :: eibφR(y) := e−ab⟨Ω|φR(x)φR(y)|Ω⟩ : eiaφR(x)+ibφR(y) : +(39) +and an explicit calculation gets us +e−ab⟨Ω|φR(x)φR(y)|Ω⟩ = (1 − e− π +L (α+i(y−x)))ab. +(40) +Similarly, for the field φ : +: eiaφ(x) :: eibφ(y) :=: eiaφ(x)+ibφ(y) : +� +� +sin2 � +π(x+y) +2L +� +sin2 � +π(x−y) +2L +� +� +� +−ab/4 +. +(41) +Finally, another useful relation for normal-ordering is : +eibφ(x) =: eibφ(x) : e− b2 +2 ⟨Ω|φ2(x)|Ω⟩, +(42) +=: eibφ(x) : +� 2L +πα sin +�πx +L +��− b2 +4 +. +(43) +Derivation of the bosonized form of the SSH Hamiltonian +In this part, we show how to derive the bosonized Hamiltonians (11) and (18) of the main text. +The discrete Hamiltonian of the fermionic interacting SSH with open boundaries was given by +H = +N−1 +� +j=1 +� +−t − δ(−1)j� � +c† +jcj+1 + h.c +� ++ V +� +nj − 1 +2 +� � +nj+1 − 1 +2 +� +. +(44) +We split it in three parts that we will treat separately. H = H0 + H1 + HI with H0 the tight-binding term, H1 the +staggered part and HI the interacting part +H0 = −t +N−1 +� +j=1 +� +c† +jcj+1 + h.c. +� +, +(45) +H1 = −δ(−1)j +N−1 +� +j=1 +� +c† +jcj+1 + h.c. +� +, +(46) +HI = V +N−1 +� +j=1 +� +nj − 1 +2 +� � +nj+1 − 1 +2 +� +. +(47) + +8 +Bosonization of the tight-binding part H0 +Although the bosonization of the tight-binding chain is a standard textbook derivation, we redo it here because we +are dealing with open boundaries. Because of this, the left and right movers are not independent fields and this may +give rise to differences with the standard infinite system case. +Let us write H0 in terms of the continuous fermionic field : +H0 = −t +N−1 +� +j=1 +� +c† +jcj+1 + h.c. +� +(48) += −t +� L +0 +dx +� +Ψ†(x)Ψ(x + a) + h.c +� +, +(49) +≈ − t +2 +� L +−L +dx +� +eikF aR†(x)R(x + a) + e−ikF aL†(x)L(x + a) + h.c +� +(50) +where going from the second line to the third we “unfolded” the fields on [−L, L] by making use of R(x) = −L(−x) +and discarded fast oscillating terms proportional to e±i2kF x, Ψ† +RΨL and Ψ† +LΨR. As usual in continuous field theory, +the a → 0 limit must be regularized. To this end, we denote by :: R†(x)R(x+a) :: the point-splitting procedure which +consists in substracting the infinite vacuum contribution to our operator +:: R†(x)R(x + a) :: = lim +a→0[R†(x)R(x + a) − ⟨Ω| R†(x)R(x + a) |Ω⟩]. +(51) +Using Eq.(39) we get +R†(x)R(x + a) = 1 +2L : e−iφR(x)+iφR(x+a) : (1 − e− π +L (α+ia))−1, +(52) +⟨Ω| R†(x)R(x + a) |Ω⟩ = 1 +2L(1 − e− π +L (α+ia))−1. +(53) +:: R†(x)R(x + a) :: = +1 +2iπa : +� +ia∂xφR(x) − 1 +2 +� +a∂xφR(x) +�2 + ia2 +2 ∂2 +xφR(x) +� +: +(54) +Similarly, for the left moving field, we have +:: L†(x)L(x + a) :: = +1 +2iπa : +� +ia∂xφR(−x) + 1 +2 +� +a∂xφR(−x) +�2 − ia2 +2 ∂2 +xφR(−x) +� +: +(55) +When integrating these fields over [−L, L], using the periodicity of φR on that interval, the terms ∂xφR and ∂2 +xφR +vanish. Now combining with the h.c. we get +− t +2 +� L +−L +dx +� +eikF a :: R†(x)R(x + a) :: +h.c. +� += −i ta +8π +� L +−L +dx +� +eikF a : +� +∂xφR(x) +�2 : −e−ikF a : +� +∂xφR(x) +�2 : +� +, +(56) += 2ta sin(kF a) +� +�� +� +=vF +1 +8π +� L +−L +dx : +� +∂xφR(x) +�2 : . +(57) +And similarly for the left field +− t +2 +� L +−L +dx +� +e−ikF a :: L†(x)L(x + a) :: +h.c. +� += vF +8π +� L +−L +dx : +� +∂xφL(x) +�2 : . +Using φR(x) = φL(−x) we can fold back everything on the interval [0, L]. Using additionally that (∂xφR)2+(∂xφL)2 = +2((∂xφ)2 + (∂xθ)2), we end up with +H0 = +� L +0 +dxvF +2π +� +: (∂xφ)2 : + : (∂xθ)2 : +� +. +(58) + +9 +Bosonization of the staggered part H1. +We now turn to the bosonization of the staggered part H1. The philosophy is essentially the same as before except +that, because of the (−1)j prefactor, the slow-varying terms are going to be the cross terms Ψ† +RΨL and Ψ† +LΨR and +the fast varying ones, that we will discard, Ψ† +RΨR and Ψ† +LΨL. +H1 = −δ(−1)j +N−1 +� +j=1 +� +c† +jcj+1 + h.c. +� +, +(59) += −δ +� L +0 +dxei πx +a � +Ψ†(x)Ψ(x + a) + h.c +� +, +(60) +≈ −δ +� L +0 +dx +� +ei πx +a e−i2kF xe−ikF aR†(x)L(x + a) + e−i πx +a e2ikF xeikF aL†(x)R(x + a) + h.c +� +(61) +where going from the second to the third line, we ignored the fast oscillating terms and made use of the fact that +(−1)j = ei πx +a += e−i πx +a +for x = ja. Since we are half-filling 2kF x ≈ πx +a which will cancel the (−1)j contribution. +However, as is discussed in the main text, the expression of kF has subleading correction in 1/L depending on the +parity of the number of sites. For N even, kF ≈ +π +2a − +π +2L to first order in 1/L. For N odd, the two possible Fermi +momenta are kF ≈ +π +2a − π +2L(1 ∓ 1) depending on the convention. We will use the convention kF = +π +2a for odd sites in +the following. It is important to keep track of 1/L term since x itself ranges from 0 to L. Thus, terms proportional +to x/L are in general not negligible in the thermodynamic limit. +To regularize this expression, we again need to consider the point-splitting of H1. We focus on the first term : +:: R†(x)L(x + a) :: = lim +a→0[R†(x)L(x + a) − ⟨R†(x)L(x + a)⟩], +(62) +R†(x)L(x + a) = − 1 +2L : e−iφR(x) :: eiφR(−x−a) : . +(63) +Using Eq.(39) we get +R†(x)L(x + a) = − 1 +2L : e−iφR(x)+iφR(−x−a) : (1 − e− iπ(−2x−a) +L +)−1, +(64) += − +i +4L sin π(x+a/2) +L +e− iπ(x+a/2) +L +: e2iφ(x)+ia∂xφL(x) : +(65) +which in the limit a → 0 becomes +R†(x)L(x + a) = − ie− iπx +L +4L sin πx +L +: e2iφ(x) : +(66) +and +:: R†(x)L(x + a) ::= − ie− iπx +L +4L sin πx +L +(: e2iφ(x) : −1). +(67) +Similarly +:: L†(x)R(x + a) :: = +ie +iπx +L +4L sin πx +L +(: e−2iφ(x) : −1) +(68) +which leads to +H1 = −δ +� L +0 +dx +1 +4iL sin πx +L +� +ei(−kF a+x( π +a −2kF − π +L ))(: e2iφ(x) : −1) − e−i(−kF a+x( π +a −2kF − π +L ))(: e−2iφ(x) : −1) + h.c +� +, +(69) += −δ +� L +0 +dx +1 +L sin πx +L +� +: sin +� +2φ(x) + (π +a − 2kF − π +L)x − kF a +� +: − sin +� +(π +a − 2kF − π +L)x − kF a +�� +, +(70) += δ +� L +0 +dx +1 +L sin πx +L +� +: cos +� +2φ(x) + (π +a − 2kF − π +L)x +� +: − cos +� +(π +a − 2kF − π +L)x +�� +. +(71) + +10 +In the last line, we simplified the expression by making the approximation kF a ≈ π +2 . Note that such approximation +cannot be done for the kF x term. The part independent from φ(x) is precisely the contribution of the vacuum energy. +For the rest of the derivation, we will focus on the case of an even number of sites for which kF = +π +2a − +π +2L so that +H1 = δ +� L +0 +dx +1 +L sin πx +L +(: cos (2φ(x)) : −1) . +(72) +Bosonization of the interacting part HI. +We now derive the bosonization of the interacting term +HI ≡ V +N−1 +� +j=1 +� +nj − 1 +2 +� � +nj+1 − 1 +2 +� +(73) +with nj the particle number operator c† +jcj. Recall +c† +jcj = a(R†(x)R(x) + L†(x)L(x) + R†(x)L(x)e−2ikF x + L†(x)R(x)e2ikF x). +The point-split expressions we previously found were +:: R†(x)R(x) ::= 1 +2π +� +: ∂xφR : +� +, +(74) +:: L†(x)L(x) ::= − 1 +2π +� +: ∂xφL : +� +, +(75) +:: R†(x)L(x) ::= − i +4L +e−i πx +L +sin πx +L +� +: e2iφ : −1 +� +, +(76) +:: L†(x)R(x) ::= +i +4L +ei πx +L +sin πx +L +� +: e−2iφ : −1 +� +. +(77) +So that +:: R†(x)R(x) :: + :: L†(x)L(x) :: = − 1 +π ∂xφ, +(78) +:: R†(x)L(x)e−2ikF x :: + :: L†(x)R(x)e2ikF x :: = +i +4L +e−i πx +a +sin πx +L +� +: e2iφ : − : e−2iφ : +� +, +(79) += −e−i πx +a +πα +(sin(2φ)) . +(80) +Where we used the formula eibφ(x) =: eibφ(x) : +� 2L +πα sin +� πx +L +��− b2 +4 . Thus, +HI =aV +� L +0 +dx +� +− 1 +π ∂xφ(x) − e−i πx +a +πα +(sin(2φ(x))) +� � +− 1 +π ∂xφ(x + a) − e−i π(x+a) +a +πα +(sin(2φ(x + a))) +� +. +(81) +Terms proportional to e±ikF x in this product will be discarded as they are oscillating rapidly. We will also discard +any constant term whenever they arise and systematically take the limit a → 0 when it is unambiguous. Note that, +because we are at half-filling, we have to keep the Umklapp terms proportional to e±i4kF x. +HI =aV +� L +0 +dx +� 1 +π2 (∂xφ)2 − +1 +(πα)2 (sin(2φ(x)) sin(2φ(x + a))) +� +, +(82) +=aV +� L +0 +dx +� 1 +π2 (∂xφ)2 + +1 +2(πα)2 (cos(4φ(x)) − cos(2a∂xφ)) +� +, +(83) +≈aV +� L +0 +dx +� 2 +π2 (∂xφ)2 + cos(4φ(x)) +2(πa)2 +� +(84) +where in the last line we identified the cut-off α and the lattice spacing a. The normal-ordered version is +HI = +� x +0 +dx +� +2aV +π2 +: (∂xφ)2 : + +V π2a3 +25 � +L sin πx +L +�4 : cos(4φ) : +� +. +(85) + +11 +Putting everything together and normal-ordering with the new vacuum +Putting everything together, we finally arrive for the total Hamiltonian to +H = +� L +0 +dx +� +1 +2π +� u +K : (∂xφ)2 : +uK : (∂xθ)2 : +� ++ +δ +L sin πx +L +: cos(2φ) : + +V π2a3 +25 � +L sin πx +L +�4 : cos(4φ) : +� +, +(86) +with uK ≡ vF and +u +K = vF + 4V a +π . The non-normal ordered version is +H = +� L +0 +dx +� 1 +2π +� u +K (∂xφ)2 + uK (∂xθ)2� ++ 2δ +aπ cos(2φ) + +V +2aπ2 cos(4φ) +� +. +(87) +With the interactions, the free part of the Hamiltonian has been renormalized. We have to redefine the normal- +ordering with respect to this new free Hamiltonian in order to get the correct path integral formulation. Let’s call +the new normal order ::I. We introduce the rescaled fields ˜φ = +√ +Kφ and ˜θ = +1 +√ +K θ. This transformation preserves +the commutation relation between the field operators. The free part of the Hamiltonian in terms of these fields is +expressed as +� L +0 +dx +2π u +� +(∂x ˜φ)2 + (∂x˜θ)2� +. +(88) +We see that it has the same form than a genuine free Hamiltonian whose Fermi velocity would be given by u. In +particular, let |ΩI⟩ be the new vacuum. We have : +⟨ΩI| ˜φ2 |ΩI⟩ = ⟨Ω| φ2 |Ω⟩ = 1 +2 log +� 2L +πα sin(πx +L ) +� +, +(89) +so that +eiζφ =: eiζφ :I e− ζ2 +2 K⟨ΩI| ˜φ2|ΩI⟩, +(90) +=: eiζφ :I +� 2L +πα sin(πx +L ) +�− Kζ2 +4 +. +(91) +The normal-ordered Hamiltonian thus becomes : +H = +� L +0 +dx +� +1 +2π +� u +K : (∂xφ)2 :I +uK : (∂xθ)2 :I +� ++ 2δ +aπ +�πa +2 +�K +1 +� +L sin πx +L +�K : cos(2φ) :I +(92) ++ +V +2aπ2 +1 +� 2L +πα sin( πx +L ) +�4K : cos(4φ) :I +� +. +(93) +Note that in the main text, we neglected the Umklapp contribution which is the cos(4φ) term. +Bessel functions +In this section, we show that the differential equation +∂2 +xh = −a h +xb +(94) +with a, b real and positive admits as solution functions of the form +h(x) ≡ √xf(|2−b|)−1 +�√a +2 +|2 − b|x +2−b +2 +� +(95) +where fα is a Bessel function of the first or second kind, i.e a function satisfying the following relation : +y2f ′′ +α(y) + yf ′ +α(y) + (y2 − α2)fα(y) = 0. +(96) + +12 +Let y = µxν and h(x) = √xf (y = µxν) with µ ≥ 0 and ν free variables for now. The previous equation translates +for h into the relation +h(x)x−1/2 +� 1 +4ν2 − α2 +� ++ 1 +ν2 h′′(x)x3/2 + µ2x2ν−1/2h(x) = 0. +(97) +Fixing α = +1 +|2ν|, we end up with +h′′(x) = −ν2µ2x2(ν−1)h(x). +(98) +For h to be solution of (94), we must have +ν = 2 − b +2 +, +(99) +|µ| = √a +2 +|2 − b| +(100) +which ends our proof. +Additional plots for the free case +For the reader’s convenience, we provide in this section additional plots of the numerical solution of Eq.(14) of the +main text as well as comparison with ED result. We vary the sign of δ to go from the topological to the trivial phase, +the parity of the number of sites N and the number of excitations nR. For each plot, we substracted the average +particle number occupation of the vacuum mode for the ED plots. + +13 +Figure 3. Additional plots comparing the solutions of the bosonization EL equations and the ED results. All the plots are done +with |∆| = 7 and t = 1 Top-Left : First excited state of the trivial phase N = 500, δ < 0, nR = 1. Top-Right : Second +excited state of the topological state N = 500, δ > 0, nR = 2. Bottom-Left : Odd number of sites N = 501, δ > 0, nR = 1. +Bottom-Right : Odd number of sites N = 501, δ < 0, nR = 1. + +0.012 +0.0030 +0.010 +0.0025 +0.008 +0.0020 +Q +0.006 +0.0015 +0.004 +0.0010 +0.0005 +0.002 +DiscreteED +DiscreteED +Coarse-grained ED +Coarse-grained ED +0.0000 +Bosonization +0.000 +Bosonization +0 +100 +200 +300 +400 +500 +0 +100 +200 +300 +400 +500 +0.010 +0.010 +DiscreteED +DiscreteED +Coarse-grained ED +Coarse-grained ED +Bosonization +Bosonization +0.008 +0.008 +0.006 +0.006 +Q +0.004 +0.004 +0.002 +0.002 +wwwwwwwww +100 +200 +300 +500 +0 +100 +200 +300 +0 +400 +400 +500 +j \ No newline at end of file diff --git a/sdAyT4oBgHgl3EQfmfiR/content/tmp_files/load_file.txt b/sdAyT4oBgHgl3EQfmfiR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..86015e528a387855c059586b1091a47e6e884f90 --- /dev/null +++ b/sdAyT4oBgHgl3EQfmfiR/content/tmp_files/load_file.txt @@ -0,0 +1,574 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf,len=573 +page_content='Bosonization of the interacting Su-Schrieffer-Heeger model Tony Jin,1, 2, ∗ Paola Ruggiero,3 and Thierry Giamarchi1 1DQMP, University of Geneva, Quai Ernest-Ansermet 24, CH-1211 Geneva, Switzerland 2Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA 3King’s College London, Strand, WC2R 2LS London, United Kingdom We derive the bosonization of the interacting fermionic Su-Schrieffer-Heeger (SSH) with open boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We use the classical Euler-Lagrange equations of motions of the bosonized theory to compute the density profile of the Majorana edge mode and observe excellent agreement with numerical results, notably the localization of the mode near the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Remarkably, we find that repulsive or attractive interactions do not systematically localize or delocalize the edge mode but their effects depend on the value of the staggering parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We provide quantitative predictions of these effects on the localization length of the edge mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Topological concepts have become a central part of contemporary condensed matter physics [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The un- derstanding of the geometrical and topological objects underpinning band theory [2–5] has fostered intense ac- tivities in diverse areas of condensed matter physics such as the study of the quantum Hall effect [6, 7], spin-orbit induced topological band insulators [8–10], topological quantum computing [11] and so on and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' One current limitation of topological band theory is its restriction to non-interacting systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Interactions will in general spoil the band structure, rendering usual clas- sification schemes inoperative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' There, novel phenomena may be expected, the most famous example being the fractional quantum Hall effect [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' One of the simplest model capturing the key fea- tures of topological insulators is the Su-Shrieffer-Heeger (SSH) model [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The fermionic SSH model consists of a 1D tight-binding model with alternating bond value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Depending on whether the first bond is weak (strong) the model is either in the topological (trivial) phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' For open boundaries, one possible characterization of the topological phase is the presence of two-fold, quasi- degenerate, zero energy, Majorana edge modes that are exponentially localized at the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Remarkably, these edge modes have been observed and characterized experimentally in one-dimensional optical lattices [14] and artificial spin chains simulated with Rydberg atoms [15] in ultracold atoms setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Although the SSH is now considered a textbook model for topological insulators, the inclusion of interactions in this model remains to this day an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' On the other hand, a powerful technique that was developped in the previous decades to treat interacting fermionic systems in 1D is bosonization [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' One trademark prowess of bosonization is to map interacting spinless fermions in 1D to a free bosonic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Remarkably this technique has been successfully applied to study the ef- fects of interactions on Majorana modes in superconduct- ing wires [18–20] but so far, to the best of our knowledge, the SSH model has escaped from a similar treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' In particular, the spatial localization of the edge modes has not been described within the bosonization language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' In this paper, we fill this gap by deriving the bosonized theory of the interacting SSH model with open bound- aries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We use the classical Euler-Lagrange (EL) equa- tions of motion of the bosonized theory to compute the density profile of the Majorana edge mode and observe excellent agreement with numerical results, notably the localization of the mode near the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Remark- ably, we find that repulsive or attractive interactions do not systematically localize or delocalize the edge mode but their effects depend on the value of the staggering parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We provide quantitative predictions of these effects on the localization length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Our results pave the way to a generalization to other interacting topological models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We begin by discussing the bosonization procedure in the absence of interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Let (cj, c† j)j∈[1,N] be the usual fermionic creation and annihilation operators as- sociated to site j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The discrete free SSH Hamiltonian on N sites in 1D is given by H = N−1 � j=1 � −t − δ(−1)j� � c† jcj+1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='c � , (1) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='e we have a tight binding chain with alternating values for the bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Fixing t > 0, the topological phase cor- responds to the case where we have an even number of sites and δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' A signature of the topological phase is the existence for open conditions of quasi-degenerate zero energy eigenstates [21] in which a single particle is in a coherent superposition between the two edges of the chain - see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='g [22, 23] for details on the non-interacting case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The Fourier transform for open boundaries is given by cj = i � 2 N + 1 N � n=1 ˜cn sin � πjn N + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (2) For δ = 0, this rotation diagonalizes the problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='e we have Hδ=0 = � n εn˜c† n˜cn with εn ≡ −2t cos � πn N+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The continuum limit is obtained by introducing the lattice spacing a and defining the position x ≡ ja and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='00472v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='str-el] 1 Jan 2023 2 the momentum k ≡ 2πn 2a(N+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The size of the sytem is taken to be L ≡ a(N +1) so that k = πn L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The continuous fermionic field is Ψ(x) ≡ 1 √acj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The boundary conditions for Ψ are obtained by extending the discrete formula (2) to site 0 and site N +1 : Ψ(0) = 0 and Ψ(x = a(N +1) = L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Following the usual bosonization procedure [17, 24], we split Ψ into a left and a right moving fields ΨR/L by expanding around the Fermi energy εkF : Ψ(x) = ΨR(x) + ΨL(x), (3) ΨR(x) = � L 2 eikF x � ∞ −∞ dk π ˜ck+kF eikx, (4) ΨL(x) = −ΨR(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (5) For convenience, we also define the “slow” fields R(x) ≡ ΨR(x)e−ikF x, L(x) ≡ eikF xΨL(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Importantly, be- cause of open boundaries, the left and right movers are not independent as is encapsulated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (2), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='g [25– 27] for previous discussions of open boudaries bosoniza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' In the continuum, R(x) taken alone can be thought of as a field living on a space of size 2L with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' In the bosonized language, it can then be reexpressed as R(x) = FR 1 √ 2παeiφR(x), (6) φR(x) ≡ πx L NR + � n>0 1 √n(anei nπx L + a† ne−i nπx L )e−α πn 2L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (7) where FR is the Klein factor associated to the right-mover with F † RFR = FRF † R = 1, NR the particle number op- erator associated to the right movers, (an, a† n)n>0 a set of bosonic modes indexed by n and α a regularization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The expression of the bosonic field associ- ated to the left-movers can be readily deduced from (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' FL = −FR, φL(x) = φR(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The conjugated fields φ, θ are customarily defined as φ(x) ≡ −φR(x) + φL(x) 2 , = −πx L NR − i � n>0 1 √n(an − a† n) sin �nπx L � e−α πn 2L , (8) θ(x) ≡ φR(x) + φL(x) 2 , = � n>0 1 √n(an + a† n) cos �nπx L � e−α πn 2L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (9) The particle density operator ρ, which counts the number of particle above the Fermi sea, is deduced from φ(x) through the relation ρ(x) = − 1 π ∂xφ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (10) In the remaining, as we want to characterize the 0 energy modes, we will work at half-filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' It is important to notice that the definition of the half-filling depends on the total number of sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Let NF ∈ N label the last occupied state of the Fermi sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' For N even, we have NF ≡ N 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The corresponding momentum is kF = πN/2 a(N+1) ≈ π 2a − π 2aN ≈ π 2a − π 2L to first order in 1/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' For N odd, the two possible definitions of the half-filled state are NF = N±1 2 with corresponding Fermi momenta kF = π 2a − π 2L(1 ∓ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' In the remaining of the paper, we will chose the convention NF = N−1 2 for odd number of sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The Fermi sea state with all modes filled up to NF and empty above will be referred to as the vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We show in the SM [28] that H can be expressed in terms of the bosonic fields as H = � L 0 dx �vF 2π � : (∂xφ)2 : + : (∂xθ)2 : � (11) + δ L sin πx L : cos � 2φ(x) + �π a − 2kF − π L � x � : � where :: denotes normal-ordering of the fermionic modes with respect to the vacuum and vF ≡ 2ta sin(kF a) the Fermi velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Formula (11) constitutes one of the main results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We see that the SSH in the bosonized language is almost equivalent to a sine-Gordon Hamiltonian except for the spatial dependence of the prefactor in front of the sinus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' To derive (11), we dis- carded constant terms and fast-varying modes ∝ e2ikF x, so the bosonic field describes modes with slow spatial variation with respect to the lattice spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Since bosonization is a theory describing low energy excita- tions, we also expect this expression to be valid for δ t ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We now turn to the computation of ρ using the clas- sical Euler-Lagrange (EL) equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Let Z ≡ tr(e−βH) and Π the conjugated field to φ, Π ≡ 1 π∂xθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' In the imaginary time formalism, we have Z = � DφDΠe � β 0 � L 0 dτdxL, (12) with L the Lagrangian density : � dxL = � dxiΠ∂τφ−H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The EL equations ∂L ∂φ = ∂ ∂τ ∂L ∂∂τ φ + ∂ ∂x ∂L ∂∂xφ yields 1 vF ∂2 τφ + vF ∂2 xφ = − 2πδ L sin πx L sin � 2φ(x) + �π a − 2kF − π L � x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (13) In the 0 temperature limit, all the weight of the probabil- ity measure will be contained in the stationary solution ∂τφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Introducing the natural rescaling y ≡ x/L, ϕ(y) ≡ φ(yL), we get the L-independent equation ∂2 yϕ(y) = −∆sin(2ϕ(y) + ϵπy) sin(πy) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (14) 3 where we introduced ∆ = 2πδL vF ≈ πδ(N+1) t and ϵ = 0, for an even number of sites where kF = π 2a − π 2L and ϵ = 1 for an odd number of sites using the convention kF = π 2a − π L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The boundary conditions for ϕ are read from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (8): At y = 0 we have ϕ = 0 and at y = 1 we have ϕ = −πnR with nR the number of particles created on top of the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Let us make some remarks here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' First, note that ∆ is an adimensioned parameter that fully characterizes the solution of the EL equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Note that ∆ scales linearly in N and δ t , so increasing the system size has exactly the same effect as increasing the ratio δ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Interestingly, going from the odd to the even case is equivalent to shift ϕ by π 2 y which can be interpreted as substracting half a particle to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' To the best of our knowledge, (14) has no known an- alytical solution and we have to resort to numerics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' To assert the validity of our approach, we compare numerical solutions of (14) with exact diagonalization (ED) results performed on the discrete Hamiltonian (1) in the zero temperature ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The ED results show fast os- cillations on the scale of the lattice spacing ∝ 1/kF that we do not see from the solutions of the EL equations of motion since we precisely discarded these terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Coarse- graining over the fast oscillations gives a smoothly vary- ing density profile on the scale of the total system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We observe excellent agreement between the ED and the EL solutions - see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We also checked that the agree- ment holds both for δ positive or negative, for an even or an odd number of sites and for different values of nR [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' For the even case and nR = 0 the solution of the EL equations is simply φ = 0 so that ρ(x) = 0, which is con- sistent with the particle-hole symmetry of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' For δ > 0, fixing nR = 1 amounts to populate the first mode above the vacuum state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='e the Majorana edge mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' From (10,14), we can thus deduce the density profile of this edge mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We plot on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='1 the numerical solu- tion of (14) and indeed see a concentration of the density at the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' There is a nice interpretation from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='(11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The term proportional to δ wants to lock the field in the minima of the cos term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' For N even and δ > 0, this corresponds to φ = − π 2 [π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' To match the boundary condition, the field φ must jump from 0 to −π/2 and then from −π/2 to −π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' These jumps translates in concentra- tion on the edges for the density ρ(x) = − 1 π∂xφ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The stiffness of the jumps of φ is controlled by ∆ and deter- mines how much the mode is concentrated at the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (14) is consistent with the exponential localization of the edge mode near the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Indeed, for small y, one crude approximation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (14) at first order in y is given by ∂y(∂yϕ) ≈ − 2∆ π ∂yϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Using additionally that the total particle number is 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='e 1 π � 1 0 dy∂yϕ(y) = 1, leads to the following ansatz ϕA when ∆ ≫ 1 : Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Comparison between the results of the discrete ED, the bosonization result and the exponential fit for ∆ = 20, N = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The uniform vacuum density ρj = 1/2 has been substracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The light-blue curve corresponds to exact dis- crete ED result which show fast oscillations at the scale of the lattice spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The green dashed curve represents the same data coarse-grained over 2 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The red curve is the density profile obtained by solving the EL equation (14) from the bosonized theory and using ρj = 1 2 − 1 π ∂xφ(x = ja) and φ(x = ja) = φ′(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Lastly, the dashed blue line is obtained from the exponential ansatz (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Note that the latter two appear superposed on this plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' ∂yϕA(y) = −∆cosh( ∆ π (2y − 1)) sinh( ∆ π ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (15) This exponential ansatz dictates the expression for the typical localization length of the edge mode ℓ ≡ πL 2∆ = vF 4δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (16) For the free SSH, it is known -see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='g [22, 23]- that ℓ ∝ a ln t+δ t−δ ≈ at 2δ for δ t ≪ 1, which is consistent with our result since vF ≈ 2ta at half-filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Remark that being in the δ t ≪ 1 regime automatically implies that ℓ ≫ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We now turn to interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We consider a nearest neighbor interacting term of the form HI := V N−1 � j=1 (nj − 1/2) (nj+1 − 1/2) , (17) with nj := c† jcj the particle number operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' From now on, we will work exclusively, in the topological phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='e δ > 0, N even and NF = N 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We show in the SM [28] that in the presence of this interacting term, the bosonization procedure leads for the total Hamiltonian to H = � L 0 dx � 1 2π � u K : (∂xφ)2 :I +uK : (∂xθ)2 :I � + δ � 2 aπ �1−K 1 � L sin πx L �K : cos(2φ) :I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (18) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='025 Discrete ED Coarse-grained ED Bosonization 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='020 Exp Ansatz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='015 p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='000 0 100 200 300 400 500 j4 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' a Plots of relation (22) for the localization length in the interacting case as a function of K for different values of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We see that the effect of interactions on the edge mode strongly depends on the value of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' b Comparison between the solution of the EL equations of motion (19) in the interacting case with DMRG simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The DMRG results for attractive (repulsive) interactions have been coarse-grained once (twice) over two sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Only half of the solution is shown for better readibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We took N = 100, δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='05, V = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='6, K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='7 for the repulsive case and V = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='77, K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='4 for the attractive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' In this case, attractive (repulsive) interactions delocalize (localize) the edge mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' c Same than b with parameters value N = 26, δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='2, V = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='6, K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='7 for the repulsive case and V = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='77, K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='4 for the attractive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We see that, in comparison to case b, the qualitative effect of interactions are swaped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' With uK ≡ vF and u K ≡ vF + 4V a π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Since the free part of the Hamiltonian has been rescaled by interactions, nor- mal ordering needs to be done with respect to the new “squeezed” vacuum which we denote by ::I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The K depen- dence of the prefactors of the cos terms is a direct conse- quence of that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' In principle, since we are at half-filling, there should also be a cos(4φ) term [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' For simplifica- tion, we will neglect this contribution in the present work as it is irrelevant in the RG sense if the interactions are not too repulsive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='e if K > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (18) is the second crucial result of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The EL equations of motion in the presence of interactions become ∂2 yϕ(y) = −2 �L a �2 �aπ 2L �K K2 δ t sin(2ϕ) (sin(πy))K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (19) A comparison of numerical solutions of (19) and density- matrix renormalization group (DMRG) simulations is shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='2-b for δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='05, N = 100 and V = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='6, K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='4 for the attractive case and V = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='77, K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='4 for the repulsive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We see that the EL equations of motion predicts the correct density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' For these pa- rameters, we see that attractive interactions delocalize the edge mode into the bulk while repulsive interaction localize it further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' For K < 2, we can give an estimation of the localiza- tion length by expanding (19) for y ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' This gives ∂2 yϕ(y) = − �2L a �2−K K2 δ t ϕ(y) yK (20) Imposing ϕ(y) = 0, the solution to this equation are of the form φ(x ≪ L) = � x LJ 1 |2−K| ��2x a � 2−K 2 �δ t � 1 2 2K |2 − K| � (21) where Jα is the Bessel function of the first kind - see [28] for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The precise shape of the Bessel function depends on α but, as one can easily verify, the position of the first maximum of Jα scales linearly with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Thus, we define the localization length to be the value ℓI such that ϕ(ℓI) = 2 |2−K| where the factor 2 has been put in order to match with the localization length of the free case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Following this definition, we obtain that ℓI = a 2 � t δK2 � 1 2−K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (22) The localization length diverges at K = 2 if δ t < 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' This gives a rough criteria for having a localized mode in the attractive regime K > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Importantly, note that ℓI is not, in general, a monotonic function of K see Fig-2-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Interestingly, one consequence of this is that attractive or repulsive interaction do not systematically delocalize or localize the edge mode, their effect can change depending on the value of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' This is illustrated on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='2-c where we took δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Contrary to the previous case shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='2-b , we see that the effects of attractive interactions is to localize the edge mode further to the boundary and the other way around for the repulsive ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Conclusion - In this paper, we derived the bosonized theory of the interacting SSH model with open bound- aries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Importantly, our study offers quantitative argu- ments to determine the effects of interactions on the 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='0 6 Bosonization attractive Bosonization attractive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='04 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='5 DMRG attractive DMRG attractive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='1 Bosonization repulsive Bosonization repulsive 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='2 DMRG repulsive DMRG repulsive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='03 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='4 p Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='08 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='0 0 10 20 30 40 50 0 2 4 6 8 10 12 K j j a b c5 edge mode and pave the way to study other interact- ing topological models such as the spin 1 antiferromag- netic Heisenberg chain [29, 30], the AKLT model [31] or the Kitaev fermionic chain [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' One of our remarkable results for the SSH model is that attractive (repulsive) interactions do not systematically delocalize or localize the edge mode, but this behavior is strongly dependent on the value of the staggering parameter δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We focused on the mean density profile of the edge mode but it would be interesting as a future direction to understand the interplay between interactions and quan- tum correlations of the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Another notable point is that we systematically discarded the Umklapp term in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Nevertheless, in the strongly repulsive regime, it is expected to play a role, leading to possibly interest- ing new phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Acknowledgements The DMRG simulations pre- sented in this paper were done using the TeNPy pack- age for tensor network calculations with python [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' T.' metadata={'source': 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+page_content=' Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Notes , 5 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' SUPPLEMENTAL MATERIAL Abacus For the reader’s convenience, we first recall the expression of the fermionic fields in terms of bosonic modes as well as various useful relations they satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The continuous fermionic field Ψ is split between a right-moving field and a left-moving one : Ψ(x) = ΨR(x) + ΨL(x), (23) ΨR(x) = � L 2 eikF x � ∞ −∞ dk π ˜ck+kF eikx, (24) ΨL(x) = −ΨR(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (25) The main difference with bosonization on an infinite system size is that the left mover is defined from the right mover for open boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The slow-mode is defined as R(x) ≡ FR 1 √ 2παeiφR(x), (26) φR(x) ≡ πx L NR + � n>0 1 √n(anei nπx L + a† ne−i nπx L )e−α πn 2L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (27) One key observation is that R(x) can be interpreted as a field with periodic boundary conditions on a system with size 2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The corresponding left mode is L(x) = FL 1 √ 2παeiφL(x), (28) FL = −FR, (29) φL(x) = −πx L NR + � n>0 1 √n(ane−i nπx L + a† nei nπx L )e−α πn 2L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (30) and the canonical fields φ and θ are defined as φ(x) := −φR(x) + φL(x) 2 , = −πx L NR − i � n>0 1 √n(an − a† n) sin �nπx L � e−α πn 2L , (31) θ(x) := φR(x) + φL(x) 2 , = � n>0 1 √n(an + a† n) cos �nπx L � e−α πn 2L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (32) It will turn out to be useful to divide the φR operator into an annihilation part ϕR and a creation part with respect to the vacuum state : φR = ϕR + ϕR†, (33) ϕR = πx L NR + � n>0 1 √nanei nπx L e−α πn 2L , (34) ϕR† = � n>0 1 √na† ne−i 2nπx 2L e−α πn 2L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (35) 7 These operators have commutation relation [ϕR, ϕR†] = − log(1 − e−α π L ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (36) Using Baker-Campbell-Hausdorff relation eA+B = eAeBe− 1 2 [A,B] when [[A, B], B] = [[A, B], A] = 0 and e− 1 2 [ϕR,ϕR†] = � 1 − e−α π L ≈ � απ L , we can express R(x) in terms of normal-ordered operators (Normal-ordering denoted by :: here refers to the normal order of bosons with respect to the Fermi sea vacuum state denoted |Ω⟩) : R(x) = FR 1 √ 2L : eiφR(x) : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (37) The normal ordered version of the left field is L(x) = −FR 1 √ 2L : eiφR(−x) : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (38) A useful relation allowing us to deal with product of normal ordered field is : eiaφR(x) :: eibφR(y) := e−ab⟨Ω|φR(x)φR(y)|Ω⟩ : eiaφR(x)+ibφR(y) : (39) and an explicit calculation gets us e−ab⟨Ω|φR(x)φR(y)|Ω⟩ = (1 − e− π L (α+i(y−x)))ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (40) Similarly, for the field φ : : eiaφ(x) :: eibφ(y) :=: eiaφ(x)+ibφ(y) : � � sin2 � π(x+y) 2L � sin2 � π(x−y) 2L � � � −ab/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (41) Finally, another useful relation for normal-ordering is : eibφ(x) =: eibφ(x) : e− b2 2 ⟨Ω|φ2(x)|Ω⟩, (42) =: eibφ(x) : � 2L πα sin �πx L ��− b2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (43) Derivation of the bosonized form of the SSH Hamiltonian In this part, we show how to derive the bosonized Hamiltonians (11) and (18) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The discrete Hamiltonian of the fermionic interacting SSH with open boundaries was given by H = N−1 � j=1 � −t − δ(−1)j� � c† jcj+1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='c � + V � nj − 1 2 � � nj+1 − 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (44) We split it in three parts that we will treat separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' H = H0 + H1 + HI with H0 the tight-binding term, H1 the staggered part and HI the interacting part H0 = −t N−1 � j=1 � c† jcj+1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' � , (45) H1 = −δ(−1)j N−1 � j=1 � c† jcj+1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' � , (46) HI = V N−1 � j=1 � nj − 1 2 � � nj+1 − 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (47) 8 Bosonization of the tight-binding part H0 Although the bosonization of the tight-binding chain is a standard textbook derivation, we redo it here because we are dealing with open boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Because of this, the left and right movers are not independent fields and this may give rise to differences with the standard infinite system case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Let us write H0 in terms of the continuous fermionic field : H0 = −t N−1 � j=1 � c† jcj+1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' � (48) = −t � L 0 dx � Ψ†(x)Ψ(x + a) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='c � , (49) ≈ − t 2 � L −L dx � eikF aR†(x)R(x + a) + e−ikF aL†(x)L(x + a) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='c � (50) where going from the second line to the third we “unfolded” the fields on [−L, L] by making use of R(x) = −L(−x) and discarded fast oscillating terms proportional to e±i2kF x, Ψ† RΨL and Ψ† LΨR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' As usual in continuous field theory, the a → 0 limit must be regularized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' To this end, we denote by :: R†(x)R(x+a) :: the point-splitting procedure which consists in substracting the infinite vacuum contribution to our operator :: R†(x)R(x + a) :: = lim a→0[R†(x)R(x + a) − ⟨Ω| R†(x)R(x + a) |Ω⟩].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (51) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (39) we get R†(x)R(x + a) = 1 2L : e−iφR(x)+iφR(x+a) : (1 − e− π L (α+ia))−1, (52) ⟨Ω| R†(x)R(x + a) |Ω⟩ = 1 2L(1 − e− π L (α+ia))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (53) :: R†(x)R(x + a) :: = 1 2iπa : � ia∂xφR(x) − 1 2 � a∂xφR(x) �2 + ia2 2 ∂2 xφR(x) � : (54) Similarly, for the left moving field, we have :: L†(x)L(x + a) :: = 1 2iπa : � ia∂xφR(−x) + 1 2 � a∂xφR(−x) �2 − ia2 2 ∂2 xφR(−x) � : (55) When integrating these fields over [−L, L], using the periodicity of φR on that interval, the terms ∂xφR and ∂2 xφR vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Now combining with the h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' we get − t 2 � L −L dx � eikF a :: R†(x)R(x + a) :: +h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' � = −i ta 8π � L −L dx � eikF a : � ∂xφR(x) �2 : −e−ikF a : � ∂xφR(x) �2 : � , (56) = 2ta sin(kF a) � �� � =vF 1 8π � L −L dx : � ∂xφR(x) �2 : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (57) And similarly for the left field − t 2 � L −L dx � e−ikF a :: L†(x)L(x + a) :: +h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' � = vF 8π � L −L dx : � ∂xφL(x) �2 : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Using φR(x) = φL(−x) we can fold back everything on the interval [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Using additionally that (∂xφR)2+(∂xφL)2 = 2((∂xφ)2 + (∂xθ)2), we end up with H0 = � L 0 dxvF 2π � : (∂xφ)2 : + : (∂xθ)2 : � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (58) 9 Bosonization of the staggered part H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We now turn to the bosonization of the staggered part H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The philosophy is essentially the same as before except that, because of the (−1)j prefactor, the slow-varying terms are going to be the cross terms Ψ† RΨL and Ψ† LΨR and the fast varying ones, that we will discard, Ψ† RΨR and Ψ† LΨL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' H1 = −δ(−1)j N−1 � j=1 � c† jcj+1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' � , (59) = −δ � L 0 dxei πx a � Ψ†(x)Ψ(x + a) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='c � , (60) ≈ −δ � L 0 dx � ei πx a e−i2kF xe−ikF aR†(x)L(x + a) + e−i πx a e2ikF xeikF aL†(x)R(x + a) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='c � (61) where going from the second to the third line, we ignored the fast oscillating terms and made use of the fact that (−1)j = ei πx a = e−i πx a for x = ja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Since we are half-filling 2kF x ≈ πx a which will cancel the (−1)j contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' However, as is discussed in the main text, the expression of kF has subleading correction in 1/L depending on the parity of the number of sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' For N even, kF ≈ π 2a − π 2L to first order in 1/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' For N odd, the two possible Fermi momenta are kF ≈ π 2a − π 2L(1 ∓ 1) depending on the convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We will use the convention kF = π 2a for odd sites in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' It is important to keep track of 1/L term since x itself ranges from 0 to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Thus, terms proportional to x/L are in general not negligible in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' To regularize this expression, we again need to consider the point-splitting of H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We focus on the first term : :: R†(x)L(x + a) :: = lim a→0[R†(x)L(x + a) − ⟨R†(x)L(x + a)⟩], (62) R†(x)L(x + a) = − 1 2L : e−iφR(x) :: eiφR(−x−a) : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (63) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (39) we get R†(x)L(x + a) = − 1 2L : e−iφR(x)+iφR(−x−a) : (1 − e− iπ(−2x−a) L )−1, (64) = − i 4L sin π(x+a/2) L e− iπ(x+a/2) L : e2iφ(x)+ia∂xφL(x) : (65) which in the limit a → 0 becomes R†(x)L(x + a) = − ie− iπx L 4L sin πx L : e2iφ(x) : (66) and :: R†(x)L(x + a) ::= − ie− iπx L 4L sin πx L (: e2iφ(x) : −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (67) Similarly :: L†(x)R(x + a) :: = ie iπx L 4L sin πx L (: e−2iφ(x) : −1) (68) which leads to H1 = −δ � L 0 dx 1 4iL sin πx L � ei(−kF a+x( π a −2kF − π L ))(: e2iφ(x) : −1) − e−i(−kF a+x( π a −2kF − π L ))(: e−2iφ(x) : −1) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='c � , (69) = −δ � L 0 dx 1 L sin πx L � : sin � 2φ(x) + (π a − 2kF − π L)x − kF a � : − sin � (π a − 2kF − π L)x − kF a �� , (70) = δ � L 0 dx 1 L sin πx L � : cos � 2φ(x) + (π a − 2kF − π L)x � : − cos � (π a − 2kF − π L)x �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (71) 10 In the last line, we simplified the expression by making the approximation kF a ≈ π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Note that such approximation cannot be done for the kF x term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The part independent from φ(x) is precisely the contribution of the vacuum energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' For the rest of the derivation, we will focus on the case of an even number of sites for which kF = π 2a − π 2L so that H1 = δ � L 0 dx 1 L sin πx L (: cos (2φ(x)) : −1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (72) Bosonization of the interacting part HI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We now derive the bosonization of the interacting term HI ≡ V N−1 � j=1 � nj − 1 2 � � nj+1 − 1 2 � (73) with nj the particle number operator c† jcj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Recall c† jcj = a(R†(x)R(x) + L†(x)L(x) + R†(x)L(x)e−2ikF x + L†(x)R(x)e2ikF x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The point-split expressions we previously found were :: R†(x)R(x) ::= 1 2π � : ∂xφR : � , (74) :: L†(x)L(x) ::= − 1 2π � : ∂xφL : � , (75) :: R†(x)L(x) ::= − i 4L e−i πx L sin πx L � : e2iφ : −1 � , (76) :: L†(x)R(x) ::= i 4L ei πx L sin πx L � : e−2iφ : −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (77) So that :: R†(x)R(x) :: + :: L†(x)L(x) :: = − 1 π ∂xφ, (78) :: R†(x)L(x)e−2ikF x :: + :: L†(x)R(x)e2ikF x :: = i 4L e−i πx a sin πx L � : e2iφ : − : e−2iφ : � , (79) = −e−i πx a πα (sin(2φ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (80) Where we used the formula eibφ(x) =: eibφ(x) : � 2L πα sin � πx L ��− b2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Thus, HI =aV � L 0 dx � − 1 π ∂xφ(x) − e−i πx a πα (sin(2φ(x))) � � − 1 π ∂xφ(x + a) − e−i π(x+a) a πα (sin(2φ(x + a))) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (81) Terms proportional to e±ikF x in this product will be discarded as they are oscillating rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We will also discard any constant term whenever they arise and systematically take the limit a → 0 when it is unambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Note that, because we are at half-filling, we have to keep the Umklapp terms proportional to e±i4kF x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' HI =aV � L 0 dx � 1 π2 (∂xφ)2 − 1 (πα)2 (sin(2φ(x)) sin(2φ(x + a))) � , (82) =aV � L 0 dx � 1 π2 (∂xφ)2 + 1 2(πα)2 (cos(4φ(x)) − cos(2a∂xφ)) � , (83) ≈aV � L 0 dx � 2 π2 (∂xφ)2 + cos(4φ(x)) 2(πa)2 � (84) where in the last line we identified the cut-off α and the lattice spacing a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The normal-ordered version is HI = � x 0 dx � 2aV π2 : (∂xφ)2 : + V π2a3 25 � L sin πx L �4 : cos(4φ) : � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (85) 11 Putting everything together and normal-ordering with the new vacuum Putting everything together, we finally arrive for the total Hamiltonian to H = � L 0 dx � 1 2π � u K : (∂xφ)2 : +uK : (∂xθ)2 : � + δ L sin πx L : cos(2φ) : + V π2a3 25 � L sin πx L �4 : cos(4φ) : � , (86) with uK ≡ vF and u K = vF + 4V a π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The non-normal ordered version is H = � L 0 dx � 1 2π � u K (∂xφ)2 + uK (∂xθ)2� + 2δ aπ cos(2φ) + V 2aπ2 cos(4φ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (87) With the interactions, the free part of the Hamiltonian has been renormalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We have to redefine the normal- ordering with respect to this new free Hamiltonian in order to get the correct path integral formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Let’s call the new normal order ::I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We introduce the rescaled fields ˜φ = √ Kφ and ˜θ = 1 √ K θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' This transformation preserves the commutation relation between the field operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The free part of the Hamiltonian in terms of these fields is expressed as � L 0 dx 2π u � (∂x ˜φ)2 + (∂x˜θ)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (88) We see that it has the same form than a genuine free Hamiltonian whose Fermi velocity would be given by u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' In particular, let |ΩI⟩ be the new vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We have : ⟨ΩI| ˜φ2 |ΩI⟩ = ⟨Ω| φ2 |Ω⟩ = 1 2 log � 2L πα sin(πx L ) � , (89) so that eiζφ =: eiζφ :I e− ζ2 2 K⟨ΩI| ˜φ2|ΩI⟩, (90) =: eiζφ :I � 2L πα sin(πx L ) �− Kζ2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (91) The normal-ordered Hamiltonian thus becomes : H = � L 0 dx � 1 2π � u K : (∂xφ)2 :I +uK : (∂xθ)2 :I � + 2δ aπ �πa 2 �K 1 � L sin πx L �K : cos(2φ) :I (92) + V 2aπ2 1 � 2L πα sin( πx L ) �4K : cos(4φ) :I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (93) Note that in the main text, we neglected the Umklapp contribution which is the cos(4φ) term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Bessel functions In this section, we show that the differential equation ∂2 xh = −a h xb (94) with a, b real and positive admits as solution functions of the form h(x) ≡ √xf(|2−b|)−1 �√a 2 |2 − b|x 2−b 2 � (95) where fα is a Bessel function of the first or second kind, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='e a function satisfying the following relation : y2f ′′ α(y) + yf ′ α(y) + (y2 − α2)fα(y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (96) 12 Let y = µxν and h(x) = √xf (y = µxν) with µ ≥ 0 and ν free variables for now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' The previous equation translates for h into the relation h(x)x−1/2 � 1 4ν2 − α2 � + 1 ν2 h′′(x)x3/2 + µ2x2ν−1/2h(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (97) Fixing α = 1 |2ν|, we end up with h′′(x) = −ν2µ2x2(ν−1)h(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (98) For h to be solution of (94), we must have ν = 2 − b 2 , (99) |µ| = √a 2 |2 − b| (100) which ends our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Additional plots for the free case For the reader’s convenience, we provide in this section additional plots of the numerical solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' (14) of the main text as well as comparison with ED result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' We vary the sign of δ to go from the topological to the trivial phase, the parity of the number of sites N and the number of excitations nR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' For each plot, we substracted the average particle number occupation of the vacuum mode for the ED plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' 13 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Additional plots comparing the solutions of the bosonization EL equations and the ED results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' All the plots are done with |∆| = 7 and t = 1 Top-Left : First excited state of the trivial phase N = 500, δ < 0, nR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Top-Right : Second excited state of the topological state N = 500, δ > 0, nR = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Bottom-Left : Odd number of sites N = 501, δ > 0, nR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' Bottom-Right : Odd number of sites N = 501, δ < 0, nR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='0030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='0020 Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='002 DiscreteED DiscreteED Coarse-grained ED Coarse-grained ED 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='0000 Bosonization 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='000 Bosonization 0 100 200 300 400 500 0 100 200 300 400 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='010 DiscreteED DiscreteED Coarse-grained ED Coarse-grained ED Bosonization Bosonization 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='006 Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdAyT4oBgHgl3EQfmfiR/content/2301.00472v1.pdf'} +page_content='004 0.' metadata={'source': 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Mikhail Kazantsevc, Ilya Makarova,c,d +aArtificial Intelligence Research Institute (AIRI), Moscow, Russia +bJCS MERI, Moscow, Russia +cHSE University, Moscow, Russia +dBig Data Research Center, National University of Science and Technology MISIS, +Moscow, Russia +Abstract +The paper deals with the problem of controlling the state of industrial devices +according to the readings of their sensors. The current methods rely on one +approach to feature extraction in which the prediction occurs. We proposed a +technique to build a scalable model that combines multiple different feature ex- +tractor blocks. A new model based on sequential sensor space analysis achieves +state-of-the-art results on the C-MAPSS benchmark for equipment remaining +useful life estimation. The resulting model performance was validated including +the prediction changes with scaling. +Keywords: +time-series, Remaining Useful life, Conditional monitoring, deep +learning +∗Corresponding author +Email addresses: DmitryZhev@yandex.ru (Dmitry Zhevnenko), +misha.kazantsev@gmail.com (Mikhail Kazantsev), iamakarov@misis.ru (Ilya Makarov) +Preprint submitted to Journal of Industrial Information Integration +January 13, 2023 +arXiv:2301.05029v1 [cs.LG] 10 Jan 2023 + +1. Introduction +Modern Industry 5.0 trends determine high requirements for efficient moni- +toring and state control systems due to equipment complexity increase. Indus- +trial data are characterized [1, 2] by high variability from task and device, as +well as the complexity of generating a sufficient number of anomalous states +to build large expert models, similar to other domains [3, 4]. Thus, the de- +velopment of the deep learning “conduit-based maintenance” (CBM) models is +related to domain-specific approaches for limiting data operation. The current +methods described in the Related work section are limited to a few ideas with +no possible expansion. +Mostly, researchers focus on evaluation of the methods using C-MAPSS +(NASA) dataset [5], which contains the aircraft engine state control problem +[6]. An accurate solution is not only to reduce the production costs associated +with its operation but also to avoid failures situations during flight. +We fixed the dataset preprocessing and compared current approaches and +optimization methods using the base [7, 8, 9] simple models to evaluate the +preprocessing importance. Having confirmed the preprocessing role, we describe +the conditions sufficient for the performance comparison. +We proposed a technique to build a scalable model that combines different +idea models as feature extractor blocks. We proposed a new model based on +sequential sensor space analysis and scaled it according to the proposed methods +achieving state-of-the-art (SOTA) results on the C-MAPSS dataset. +The main contributions of this work are as follows: +• We proposed a novel feature extraction approach based on sequential mod- +eling of sensor space, which achieved state-of-the-art results for the C- +MAPSS dataset. +• We developed a scalable approach to a model extension using a com- +bination of feature extractors on different operating principles and loss +functions increasing the variety of extracting features. +2 + +• We proposed regularization techniques useful to prevent the overfitting +specific to monotonic piece-wise RUL prediction problem. +The rest of the paper is organized as follows. The section “Experiment Set- +ting” presents the dataset and its problem. The section “Related work” briefly +gives the related research to the C-MAPSS dataset RUL problem. The section +“Proposed approach” presents a detailed description of the proposed prognostic +approach and used model structures. Section “Experiments and Discussion” +includes a description of the learning process, experiments on the proposed +models and their performance study. Finally, the “Conclusion” summarizes our +research and includes possible feature work. +2. Experiment Setting +The subsection includes the C-MAPSS dataset description, the physical +meaning of containing data, the preprocessing of the feature and target spaces, +and related metrics. +2.1. Dataset description +The dataset [5] includes the data generated by the “Commercial Modular +Aero-Propulsion System Simulation” (C-MAPSS). The program was used to +simulate the behavior of turbofan engine sensors under various flight conditions +until its failure. The flight conditions represent situations during ascent and +descent to a standard flight altitude of about 10 km. +The C-MAPSS dataset consists of four subsets of data, the details of which +are presented in Table 1. Each subset contains readings from ‘21’ sensors (Table +2) located in different parts of the degrading engine, which are controlled by +three characteristics of the operational settings (height, speed and acceleration +in terms of altitude, Mach number, and throttle resolver angle). +The simulation was used to study the sensors during the engine’s gradual +deterioration up to the failure. Different noises were added to modeled engine +3 + +Table 1: Brief description of four C-MAPSS subdatasets. +Subdataset +FD001 +FD002 +FD003 +FD004 +Train trajectories +100 +260 +100 +249 +Test trajectories +100 +259 +100 +248 +Train samples +20631 +53759 +24720 +61249 +Test samples +13096 +33991 +16596 +41214 +Operating conditions +1 +6 +1 +6 +Fault modes +1 +1 +2 +2 +sensor reading to approximate the numerical solution to the complicated real- +world engine behavior [5]. +Similar to most of the work in the Related work section, we used normalize +technique to the sensor readings. +2.2. Problem statement +The common assumption to target space is that if the engine is functional, it +is guaranteed to run to failure for a certain amount of time, which is determined +by the constant value N: +RUL = +� +� +� +� +� +N, +if RUL ≥ N +RUL, +if RUL < N. +(1) +N is an heuristic parameter, and in most cases for the C-MAPSS dataset +task N = 125 value is used due to previously performed empirical study. This +piece-wise RUL target space, first proposed by Heimes in [10], is widely adopted +in remaining useful life estimation tasks. The sensor readings of a failing motor +1a and the associated piece-wise RUL curve 1b presented in Fig 1. +Piece-wise RUL target function is at Fig. 1b: +One of the problems limiting the model comparison ability is the use of +different preprocessing. The severe unstable impact is caused by various pre- +processing and postprocessing based on operation condition parameters (for +4 + +Table 2: C-MAPSS dataset engine sensors +Sensor # +Description +Units +1 +Total temperature at the fan inlet +°R +2 +Total temperature at the low-pressure compressor outlet +°R +3 +Total temperature at the high-pressure compressor outlet +°R +4 +Total temperature at the low-pressure turbine outlet +°R +5 +Pressure at the fan inlet +psia +6 +Total pressure in bypass-duct +psia +7 +Total pressure at the high-pressure compressor outlet +psia +8 +Physical fan speed +rpm +9 +Physical core speed +rpm +10 +Engine pressure ratio +— +11 +Static pressure at the high-pressure compressor outlet (Ps30) +psia +12 +Ratio of fuel flow to Ps30 +pps/psi +13 +Corrected fan speed +rpm +14 +Corrected core speed +rpm +15 +Bypass ratio +— +16 +Burner fuel-air ratio +— +17 +Bleed enthalpy +— +18 +Demanded fan speed +rpm +19 +Demanded corrected fan speed +rpm +20 +High-pressure turbine coolant bleed +lbm/s +21 +Low-pressure turbine coolant bleed +lbm/s +(a) Sensor reading +(b) Piece-wise RUL +Figure 1: Normalized sensor values and corresponding piece-wise RUL for the second engine +of the fd001 subdataset +5 + +2 +Normalized value +10 +11 +12 +13 +14 +15 +2 +16 +17 +18 +50 +100 +150 +200 +250 +300 +Time steps120 - +100 - +80 - +60 +R +40 +20 - +Fo +200 +250 +300 +350 +400 +450 +Time +stepsexample, [11]). Thus, we considered only fd001 and fg003 sub-datasets with +fixed operation conditions for proper model comparison. +2.3. Objective Quality Metrics +The performance of the models on the C-MAPSS dataset was evaluated +using RMSE and Score metric: +RMSE = +� +� +� +� 1 +n +n +� +i=1 +(yi − ˆyi)2 +(2) +diff(i) = ˆyi − yi +score(i) = +� +� +� +� +� +e− diff(i) +13 +− 1, +if diff(i) < 0, +e +diff(i) +10 +− 1, +if diff(i) ≥ 0 +Score = +n +� +i=1 +score(i) +(3) +where ‘yi’ and yi are predicted and true value of RUL for engine ‘i’. +The asymmetric Score metric penalizes above the true value predictions due +to maintenance regulations. The asymmetry is achieved via the coefficients in +the denominator. These coefficients are empirically defined constants designed +by the authors of the original dataset paper [5] for the PHM2008 competition. +3. Related Work +There are two main ways for the C-MAPSS dataset RUL prediction problem: +using auxiliary health index characteristics or a natural RUL curve. +3.1. Auxiliary health characteristic +The first method is domain-based auxiliary characteristics models adopted +to map test samples to train-based latent space and to determine the relations +of it to related train vectors to RUL prediction. +One of the most common +approaches is to construct a health index (HI) curve using deep learning and +classical machine learning methods. +6 + +One of the main directions related to health index space modeling is the use +of the similarity-based curve matching technique, first presented in [12]. The +approach basis is the construction of multi-label time-series window mapping to +the latent space. The latent space construction model is based on the training +dataset, the prediction is performed by test samples mapping and distance to +the closest train sample estimation. +The article [13] uses sparse Bayesian learning (SBL) [14] to establish health +indicators. In addition, there are methods of constructing an HI index based +on classical ML clustering, proposed, for example, in [15], where a new index is +constructed using the sample distances to each cluster. +One of the most important areas of HI generation is empirically constructed +models, both in the form of classical equations [16] and deep learning approaches +[12, 17]. There are several Ways of constructing an HI index by the example of +statistical approaches, e.g., in [18] there are methods based on Dempster–Shafer +evidence theory. Nevertheless, the construction of a plausible HI curve imposes +serious requirements on the function or model to display. +The HI curve can be generated using autoencoders of various [19, 20] princi- +ples for similarity-based curve matching techniques. In [21] as a feature extrac- +tor, a convolution-based autoencoder is used to construct a latent representation +of the engine state that is maximally correlated with the function-based curve +constructed from training sample sensor values. In the article, [22] the authors +proposed a comprehensive approach using a feature extractor built by apply- +ing multilayer CNN layers to a time-sequence sliding window. The main idea +of the model is to construct a common feature space for both RUL prediction +and device state classification. Some works contain elements of semi-supervised +approaches based mainly on the use of different autoencoders [23]. +3.2. Direct RUL prediction +The second method is a direct prediction of RUL or peace-wise RUL without +using auxiliary features and reliance on domain-based knowledge based on DL +models. The development of this direction is related to the possibility of using +7 + +feature extractors and efficient optimizers and obtaining SOTA results on them. +RUL prediction methods, in contrast to the health curves, estimate degradation +processes, omitting local anomalies, for which other methods are developed and +tested on specialized datasets [24, 25]. In [26] discusses an approach to gen- +erating partial training trajectories aimed at optimizing a linear RUL curve +with a piece-wise continuous function that simulates the curve behavior in test +datasets. In addition, the classical approach is to use some constant area of +the RUL curve [26, 23] corresponding to the normal mode of the device, and +the value of this area is usually chosen empirically. This heuristic is related to +the fact that sensor readings in normal operation should not differ significantly +from each other in the absence of noise - and thus model readings should not +differ either. In addition to this assumption, the accuracy of the validation of +the constructed model is also used to determine the value. Due to the various +methods of estimation, most of the empirical estimates of RUL lie in the range +of ‘115’ to ‘160’ switching cycles [27, 28, 21, 29, 9, 30, 31, 32]. The most popu- +lar approach to solving the RUL curve prediction problem is to use structures +consisting of feature extractors and a regression head. The shape of a feature +extractor varies widely, and for a regression head, MLP or gradient boosting are +usually used. Early work in this area involves assessing the applicability of basic +models like shallow networks and Multi-Layer Perceptrons (MLPs) [33]. The +authors of [9] used LSTMs on three benchmark datasets and found them to work +best compared to MLPs and CNNs. Wu et al. [34] compared LSTMs against +vanilla RNNs and GRUs. In [35] feature extractor the authors present a series +of heterogeneous heuristics on time series parameter extraction, and the use of +LigthGBM to predict RUL and the associated curve, which the authors called +a cumulative dynamic differential health indicator. Hong et al. in [27] used +stacked CNN, LSTM, and Bi-LSTM with residual connections to create a fea- +ture extractor, for which they considered approaches to increase explainability +through dimensionality reduction and Shapley additive explanation techniques. +In [36, 37] the authors proposed the variants of feature extractors based on a +multiscale convolutional neural network and a bidirectional recurrent block with +8 + +control, respectively. +3.3. Novel sota approaches +In this section, several state-of-the-art models on various principles of oper- +ation are analyzed in terms of the proposed properties and ways to improve. As +SOTA results, we would like to highlight the three following methods. The first +principle [38, 39] involves autoencoders and variational autoencoders for feature +construction or creation of an aggregated health index device. Forecasting is +usually achieved by using an lstm-type model, in which the principles of time +transfer and information cleansing occur through stacked BiLSTM layers. As +an example approach, we can consider [38], a distinctive feature of which is the +use of KL-div and MSE RUL loss to form a continuous ordered relative to RUL +latent space, over which it is possible to perform state control by proximity simi- +lar to the similarity-based curve matching method discussed in the related work. +The second principle [40, 41, 42] work includes feature construction approaches +for further use in LSTM-based models. The use of different approaches to fea- +ture generation is so diverse that it includes not only the standard approaches. +In particular, in [40] discusses the fusion of parallel models based on Bi-LSTM, +with hand-crafted time-series features per model as input. The equipment state +control is based on piece-wise RUL prediction. +The third principle [43, 44, 45] includes the addition of any form of the +attention mechanism to the previously presented approaches. For example, in +[43] attention mechanism was used both in the sensor allocation phase and in +accounting for the influence of time series position. The state of the equipment +is supposed to be monitored based on RUL prediction. +These papers, to a greater or lesser extent, share a common conception of +cleansing and interaction of information within the input. However, each work +relies on a few ideas with no possible expansion. We want to propose a new +feature extraction approach and a method to use different models as extraction +blocks to create a joint prediction. +9 + +4. Proposed Approach +In this paper, we proposed a new approach focused on building blocks aimed +at different features extracting and fusing the results to build high-level features. +Different features extraction include selecting blocks of distinct nature and loss +functions that ensure the distinction of selected output feature vectors before +blending by the attention layer [46] into a common latent space. Thus, the pro- +posed approach includes different feature extraction blocks resulting in diverse +latent space and attention layers to form high-level feature. +Figure 2: Proposed method general structure +The general structure of the model built within the proposed approach is +shown at Fig. 2, and the neural network input goes through the following stages: +• Dataset-related preprocessing +• Preprocessed time-series is passed to the input of “Local preprocessing” +in form of attention to specific features or time points, generalization or +refinement of the sequence, and focusing on the time or feature domain. +• The resulting sequence is passed to the feature extraction block with the +skipped connection, allowing to maintain the gradient distribution regard- +less of the block structure. +• Latent space vectors at the output of the feature extraction blocks are +10 + +Feature extraction +Local preprocessing +block No 1 +Concatenation +Attention +MLP +RUL +Preprocessed +Input +Feature extraction +Local preprocessing +block No N +FC-Nnormalized and accounted for in loss, increasing the distinction between +them. +• Each normalized vector passes through additional fully connected layers +for RUL prediction so that at the expense of the general error to form a +selection of this level important features. +• The resulting different vectors are concatenated and passed through an +attention layer to compensate for unavoidable errors in the form of noise +associated with the requirement for vector difference. +• The resulting latent representation passes through the regression head in +the form of a multilayer perceptron for RUL prediction. +The described approach resulted in the following step-by-step created struc- +tures. First, we studied the effect of the chosen data preprocessing methods on +LSTM and CNN based models. Second, we evaluated the effect of input data +transposition as local preprocessing. Finally, we built three consecutive models, +increasing the number of feature extractor blocks according to the proposed +framework. +4.1. Baseline setup +We evaluated how much the use of the proposed preprocessing can affect +the simulation results. To investigate the effect of dataset based preprocessing +heuristics, we use the well-known models proposed for the CMAPSS dataset +problem as described in Related Work section. For that, we have reimplemented +models based on Long Short-Term Memory (LSTM) and Convolution Neural +Network (CNN), which have been state-of-the-art models for solving the C- +MAPSS dataset task shortly after it emerged. +Figure 3 demonstrates the general structure of LSTM-based approach. “Stacked +LSTMs” bock consists of three stacked LSTM layers with the parameters: hid- +den state and input features is ‘21’, dropout is ‘0.5’. “MLP block” consists of +11 + +Figure 3: LSTM-based general structure +two fully-connected layers (441, 126) and (126, 1) with ReLU nonlinearity and +0.5 dropout. +In turn, Convolutional Neural Network is an algorithm, usually employed +for image processing. But due to their structure and ability to capture spatial +and temporal dependencies, they are widely used for time series forecasting as +well. Figure 4 illustrates the general structure of the implemented CNN-based +approach. “Stacked CNN” includes two consecutive blocks of 1d Convolution +(kernel sizes are ‘7’ and ‘3’, out channels are ‘5’ and ‘10’, stride sizes are ‘1’ and +‘0’), ReLU non-linearity, max pooling (kernel sizes are ‘3’ and ‘3’, stride sizes are +‘2’ and ‘1’), and flatten procedure after them. “MLP block” consists of three +fully-connected layers (30, 60), (60, 120) and (120, 1) with ReLU nonlinearity +and 0.5 dropout. +Figure 4: CNN-based general structure +12 + +Preprocessed +stacked LSTMs +RUL +InputPreprocessed +stacked CNN +RUL +Input4.2. Feature approach +We investigated one feature extractor block of the proposed approach. The +effectiveness of the ”local preprocessing” block was tested for the first time +through a swap of the time and feature domains. A study for window sizes +one and three showed that domain swap leads to a 20% change in score metrics +while maintaining comparable RMSEs, so we investigated the effect by adding +that transform to the extractor block-like structure. +Figure 5: Time-feature model structure +The structure of the time-feature model (TFM) for impact evaluation studied +is shown in 5. Block “Transpose” is time and feature domains swap, “LSTM- +based extraction block” is a block of 3 stacked LSTMs with implementation of +skipped connection (feature size is equal to hidden size) with parameters dropout +‘0.5’, feature size is equal to window size ‘32’ and ‘40’ for fd001 and fd003 +subdatasets, self-attention implementation corresponds to [46]. “MLP block” +consists of two fully-connected layers (672, 256), (256, 1) with SiLU nonlinearity +[47] and ‘0.5’ dropout. +4.3. Time-feature interaction +Next, we tested how effective the proposed Interaction approach is in in- +creasing the prediction accuracy compared to using the exact single model. The +model presented in the previous paragraph was used as one of the feature ex- +tractor blocks because of its high efficiency.The first step is to test the use of +double TFM (DTFM), relying on the separation of different features through +the use of an modified cosine loss function: +cos (zi, zj) = +zizj +max (∥zi∥2∥zj∥2, ϵ) +(4) +13 + +Attention +Preprocessed +LSTM-based +MLP +Transpose +RUL +Input +extraction blockmCosineLoss = +� +(i,j),i̸=j +max (cos (zi, zj), 0) +(5) +where ϵ = 1e−7 is to avoid zero in the denominator, zk is the kth extractor +block output vector. +The use of Huber losses reduces the effect of local variance in predicting a +nonnatural monotonic RUL curve shape: +HuberLoss(i) = +� +� +� +� +� +1 +2(yi − ˆyi)2, +if |(yi − ˆyi)| < β, +β(|(yi − ˆyi)| − 1 +2β), +otherwise +(6) +where hyperparameter β = 1. +Thus, the overall loss function consists of three parts, representing the loss +on RUL predictions based on concatenated vectors, losses on RUL predictions +based on feature extractor blocks latent vectors, and differences between that +latent vectors: +L = HuberLossmodel + λ · +� +k +HuberLossk + σ · mCosineLoss +(7) +where the empirical parameters lambda and sigma depend on the number +of feature extractor blocks and the batch size, ‘k’ is the feature extractor block +index and are equal ‘0.3’ and ‘1’ correspondingly, ‘model’ is index of final ‘MLP’ +output (Fig 2). +The structure of the model for impact evaluation is shown in Fig. 6. The +model blocks are described above, except: +“Concatenation” is concatenate +feature extractor block outputs, “FC1” and “FC2” are fully-connected layers +(672, 1) with SiLU nonlinearity [47] and ‘0.5’ dropout, “MLP block” consists of +two fully-connected layers (1344, 256), (256, 1) with SiLU nonlinearity [47] and +‘0.5’ dropout. The dimensionality of the hidden vector is much larger than the +window size and applying blocks of the same nature further seems redundant. +To further improve the metric, we adapted state-of-the-art stacked SCINet +[48] as an additional future extractor (Fig. +7) to build the time-feature in- +teraction model (TFIM). The reimplemented structure includes an adaptation +14 + +Figure 6: Double time-feature model structure +of the multilayer hierarchical model (Fig. 8), where ”Hierarchical structure” +represents convolution-based affine interactions, detailed in [48]. The main pa- +rameters of the built block are as follows: hidden size is ‘16’, three levels in the +hierarchical structure, kernel size is ‘3’, and dropout is ‘0.65’. Other parameters +staing constant except “MLP” first layer change to (2016, 256). +Figure 7: Time-feature interaction model structure +Adaptation of the proposed block to the task of RUL prediction consists +in optimization of the latent layer for RUL prediction between stacked blocks +at the expense of additional skipped connections through the adapted SciNet +15 + +FC1 +Transpose +LSTM-based +extraction block N1 +Concatenation +Attention +MLP +RUL +Transpose +LsTM-based +extraction block N2 +FC2LSTM-based +Transpose +extraction block No1 +Concatenation +Attention +MLP +RUL +Preprocessed +Adapted SCINet +Input +extraction block +LSTM-based +Transpose +extraction block No2 +FC1 +FC2 +FC3Figure 8: Adapted SCINet extraction block structure +block, and using the approach for processing the output vector in accordance +with the proposed method. +5. Experiments and Discussion +5.1. Training procedure +In this section, we represented the learning process, the validation, the fea- +tures of overfitting, and the hyperparameter choice. +Due to the described dataset properties, one of the important tasks is reg- +ularization. +In addition to mentioned class imbalance, there are anomalous +functioning states in the data, realized by random outliers in the engine work- +ing regime. Thus, the importance of regularization increases. In this paper, +16 + +Preprocessed +input +SCINet +Hierarchical structure +Concatentate +Concatenate +SCINet +FC +Concatenate +Latent +representationregularization was performed by four main effects: reducing the class imbalance +during dataset formation by considering only sixth window with RUL = 125, +adding white noise to the training sample, using high dropout values, and av- +eraging several RUL predictions to reduce variance. +Thus, summarizing the proposed regularization methods for the problem +and the traditional approaches described in Related work, we fix the following +conditions: +• Selecting a sliding window of sizes ‘32’ and ‘40’ for fd001 and fd003, the +insufficient test sample (engines number ‘1’ and ‘52’ for fd001 and fd003 +accordingly) expanded to the left. +• AdamW [49] as the optimizer. +• Huber loss to all RUL prediction blocks. +• Data noise augmentation (Normal distribution sampling, σ = 0.04) . +• Averaging over five values to reduce variance in prediction. +• Averaging over five runs to reduce variance in training. +Basic models were trained with a sliding window size of ‘32’ and batches +of size ‘128’. The adaptive triangular cyclic learning rate scheduler [50] with +[1e−4; 5e−4] interval. +The optimal learning configuration for training Interaction models include +‘210’ batch size and adaptive triangular cyclic learning rate updating rule in +[2e−4, 9e−5]. +The validation includes the selection of ‘37’ (fd001) and ‘45’ (fd003) consecu- +tive switching points (to average over ‘5’ sliding windows of sizes ‘32’ and ‘40’), +which are excluded from the random location of each training sample engine +and used for five-fold cross-validation with division by engines. +This approach reduces the training datasets by about ‘5%’ due to ‘37’ and +‘45’ time points per engine exclusion and leads to a scoring decrease due to the +high diversity of trajectories in feature space. The choice of other validation +17 + +techniques leads to even more significant deterioration. However, due to the +proposed regularization method, there was a plateau with minimal metric values +and a number of epochs to reach for every model. Thus, the proposed models +were trained on the full dataset and the epoch number obtained on validation. +To obtain the final result, the predictions on the test dataset averaged over five +epochs after reaching the required epoch. On average, the training procedure +lasted over ‘120’ epochs, reaching a plateau around ‘45’ or ‘70’ depending on +the model. +Once the minimum was reached, the overfitting began. One interesting effect +that can be associated with overfitting is a decrease in prediction accuracy +as the final characteristic (the RUL prediction curve for all engine readings) +approaches the reference peace-wise form. This effect can be attributed to the +“close” trajectories [5] in the sensor reading space due to the presence of multiple +noises added during the dataset data simulation. +The non-monotonicity of the characteristic guarantees a more accurate hit in +the mean, but automatically enhances the variance of the result. The variance +impact of the model was reduced by averaging the last five predictions with the +correction for the successive reduction of the RUL parameter. +5.2. Domain preprocessing impact +This section includes the performance of the discussed preprocessing ap- +proach to well-known base models (Table 3). +Table 3 shows the comparison results of the constructed models with the +models that used other preprocessing and learning methods. These works were +considered state-of-the-art, but the use of described approaches on base models +has surpassed their results. The base models outperformed the referenced mod- +els in terms of RMSE and nonsymmetrical score constructed according to state +control requirements. Thus, this study demonstrated the importance of actual +general learning and preprocessing methods for a fair comparison. +18 + +Table 3: State-of-the-art approaches of previous years with a performance, comparable to +baselines performance from this thesis +FD001 +FD003 +RMSE +Score +RMSE +Score +MODBNE (2017) [51] +17.96 +640 +19.41 +683 +Deep LSTM (2017) [9] +16.14 +338 +16.18 +852 +BiLSTM-ED (2019) [19] +14.74 +273 +17.48 +574 +Attention-based DL Approach (2021) [52] +14.53 +322 +— +— +CNN (ours) +14.38 +332 +15.12 +495 +LSTM-baseline (ours) +14.88 +400 +14.75 +382 +5.3. Time-feature interaction experiments +Using the approach described above, we trained the models presented in the +Time-feature interaction section. The training results in comparison to state- +of-the-art results are in Table 4. +The performance of the models on the standard for demonstration engines +and model training are shown in Fig. 9b and 9a. The 35th engine is used to +show the prediction results of each of the three models considered, AE is the +absolute error between the model prediction and the true RUL value at that +point. +Each of the proposed models showed state-of-the-art results. However, when +moving from a model with one feature extractor block to a model with three +feature extractor blocks, the RMSE changed by less than ‘2%’, although the +inference time, measured by ‘300’ runs on a dummy example using the GPU +(NVIDIA A100) warm-up technique, changes from ‘0.01’ to ‘0.75’ seconds. +The standard deviation in the table is composed of the variation of the metric +value at the five plateau points, over which we average the performance, and the +variation between five consecutive runs to measure the stability of the model. +It is important to note that models with non-monotonic noisy predictions +19 + +Table 4: Recent C-MAPSS dataset state-of-the-art approaches +FD001 +FD003 +RMSE +Score +RMSE +Score +GNMR (2020) [53] +12.14 +212 +13.23 +370 +BLS-TFCN (2021) [54] +12.08 +243 +11.43 +244 +TCNN-Transformer (2021) [55] +12.31 +252 +12.32 +296 +RVE (2022) [56] +13.42 +323 +12.51 +256 +MCA-BGRU (2022) [57] +12.44 +211 +— +— +DA-TCN (2021) [58] +11.78 ± 0.29 +229 ± 8 +11.56 ± 0.61 +257 ± 58 +ADL-DNN (2022) [59] +13.05 ± 0.16 +238 ± 5 +12.59 ± 0.25 +281 ± 5 +Feature space model (ours) +11.73 ± 0.09 +215 ± 5 +11.64 ± 0.03 +191 ± 9 +Double feature space model (ours) +11.65 ± 0.07 +210 ± 6 +11.58 ± 0.03 +204 ± 6 +Time-feature interaction model (ours) +11.59 ± 0.08 +208 ± 3 +10.9 ± 0.06 +187 ± 8 +(a) Learning curves +(b) 35th test engine prediction +Figure 9: Learning curve and prediction result for 35th engine of dataset fd001 +obtained the best results. A typical example of the overfitting that occurs after +exceeding the plateau is shown in Fig. 10. Increasing the accuracy of modeling +the curve shape near-constant values significantly reduces the prediction accu- +racy in the degradation region, which leads to a decrease in the metric on the +test dataset. We attribute this process to the unnatural shape of the piece-wise +RUL curve, which forces the model to learn how to convert noisy trajectories +into a monotonically decreasing curve, creating a noticeable loss. +To check if all the features are used in the prediction, we perform a zeroing +of the part of the latent representation corresponding to each feature extractor +block 11. The presented result demonstrates that the interaction of both FSM +20 + +FSM +DFSM +16 +TFIM +15 +RMSE +14 +13 +12 +0 +10 +20 +30 +40 +50 +60 +70 +Time steps140 +FSM +DFSM +120 +TFIM +AE FSM +100 +AE DFSM +AE TFIM +80 +60 +40 +20 - +25 +75 +100 +150 +Time stepsFigure 10: Smoothed prediction of RUL engine ‘2’ by trained and overfitted models +and adapted SciNet blocks are qualitatively similar, which can be interpreted as +the extraction of close features. However, due to the use of loss on the difference +of vectors, locally both predictions are different, which suggests the extraction of +additional information. The combined information usage from all three blocks +allows us to obtain a qualitatively and quantitatively close prediction. +Figure 11: TFIM 10th engine prediction with zeroed outputs of related feature extractor block +During the comparison study of three models, we found that the use of an +additional feature extractor block primarily affects the generalizability. To con- +firm this effect, we selected test engine number ‘6’ from the fd001 subdataset, +on which the models show one of the worst results, and over-smoothed it using +the “Locally Weighted Scatterplot Smoothing” method with the frac param- +21 + +120 - +140 +overfittedTFIM +TFIM +AE overfitted TFIM +AE TFIM +41 +21 +21 +Time steps140 +FSM_1 +FSM_2 +120 +TFIM +Original +Time stepseter ‘0.11’, essentially retaining only the overall trend (Fig. 12). The engine +was chosen as an example of prediction changes corresponding to the improved +quality of the models. The figure shows that TFIM supports state predictions +with RUL = 125 in the working state and contains a clear degradation trend +in the transition region, unlike other models. A similar pattern is observed for +most of the test engines, indicating that further improvements in model quality +are possible. +Figure 12: Smoothed RUL prediction for the difficult to simulation 6th engine +6. Conclusion +In this study, we proposed the ”Interaction” approach to the state control +problem on the example of the C-MAPSS dataset. The method includes build- +ing scalable models based on blocks of distinct nature and loss functions that +ensure the distinction of selected output feature vectors. Using multiple feature +extractor blocks for feature extraction leads to scalability and interpretability +due to the simplification of block individual inference research. In the frame- +work, we have explicitly proposed a dataset preprocessing, the effectiveness of +which was measured relative to basic models. Also, overfitting for the C-MAPSS +dataset models was described as a model adaptation to the monotonic piece-wise +RUL curve prediction. +22 + +120 - +100 - +80 - +RUL +R +60 +FSM +DFSM +40 - +TFIM +AE FSM +20 - +AE DFSM +AE +: TFIM +-0 +0 +10 +20 +40 + 50 +60 +70 +Time +stepsWe proposed a new Feature space model (FSM) that demonstrated state- +of-the-art results for the C-MAPSS task in selected preprocessing and augmen- +tations approaches. The resulting model was used to form blocks of general +Interaction structure. The approach to different latent space vector formation +building allowed the doubled FSM (DFSM) significantly improve the metric. +Adding an adapted SCINet model to the structure demonstrates the possibil- +ity of extracting additional information at the expense of a block of a different +nature. The constructed time-feature interaction model (TFIM) not only im- +proves the metrics but also leads to an increase in the generalizability of the +model. 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Gryllias, Automatic multi-differential +deep learning and its application to machine remaining useful life pre- +diction, Reliability Engineering & System Safety 223 (2022) 108531. +doi:https://doi.org/10.1016/j.ress.2022.108531. +URL +https://www.sciencedirect.com/science/article/pii/ +S0951832022001855 +33 + diff --git a/tdE4T4oBgHgl3EQfWAx3/content/tmp_files/load_file.txt b/tdE4T4oBgHgl3EQfWAx3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f697be85fd32fc7add62db43e573bd4943af5b74 --- /dev/null +++ b/tdE4T4oBgHgl3EQfWAx3/content/tmp_files/load_file.txt @@ -0,0 +1,979 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf,len=978 +page_content='Interaction models for remaining useful life estimation Dmitry Zhevnenkoa,b,∗, Mikhail Kazantsevc, Ilya Makarova,c,d aArtificial Intelligence Research Institute (AIRI), Moscow, Russia bJCS MERI, Moscow, Russia cHSE University, Moscow, Russia dBig Data Research Center, National University of Science and Technology MISIS, Moscow, Russia Abstract The paper deals with the problem of controlling the state of industrial devices according to the readings of their sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The current methods rely on one approach to feature extraction in which the prediction occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' We proposed a technique to build a scalable model that combines multiple different feature ex- tractor blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' A new model based on sequential sensor space analysis achieves state-of-the-art results on the C-MAPSS benchmark for equipment remaining useful life estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The resulting model performance was validated including the prediction changes with scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Keywords: time-series, Remaining Useful life, Conditional monitoring, deep learning ∗Corresponding author Email addresses: DmitryZhev@yandex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='ru (Dmitry Zhevnenko), misha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='kazantsev@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='com (Mikhail Kazantsev), iamakarov@misis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='ru (Ilya Makarov) Preprint submitted to Journal of Industrial Information Integration January 13, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='05029v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='LG] 10 Jan 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Introduction Modern Industry 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='0 trends determine high requirements for efficient moni- toring and state control systems due to equipment complexity increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Indus- trial data are characterized [1, 2] by high variability from task and device, as well as the complexity of generating a sufficient number of anomalous states to build large expert models, similar to other domains [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Thus, the de- velopment of the deep learning “conduit-based maintenance” (CBM) models is related to domain-specific approaches for limiting data operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The current methods described in the Related work section are limited to a few ideas with no possible expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Mostly, researchers focus on evaluation of the methods using C-MAPSS (NASA) dataset [5], which contains the aircraft engine state control problem [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' An accurate solution is not only to reduce the production costs associated with its operation but also to avoid failures situations during flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' We fixed the dataset preprocessing and compared current approaches and optimization methods using the base [7, 8, 9] simple models to evaluate the preprocessing importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Having confirmed the preprocessing role, we describe the conditions sufficient for the performance comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' We proposed a technique to build a scalable model that combines different idea models as feature extractor blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' We proposed a new model based on sequential sensor space analysis and scaled it according to the proposed methods achieving state-of-the-art (SOTA) results on the C-MAPSS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The main contributions of this work are as follows: We proposed a novel feature extraction approach based on sequential mod- eling of sensor space, which achieved state-of-the-art results for the C- MAPSS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' We developed a scalable approach to a model extension using a com- bination of feature extractors on different operating principles and loss functions increasing the variety of extracting features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 2 We proposed regularization techniques useful to prevent the overfitting specific to monotonic piece-wise RUL prediction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The section “Experiment Set- ting” presents the dataset and its problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The section “Related work” briefly gives the related research to the C-MAPSS dataset RUL problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The section “Proposed approach” presents a detailed description of the proposed prognostic approach and used model structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Section “Experiments and Discussion” includes a description of the learning process, experiments on the proposed models and their performance study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Finally, the “Conclusion” summarizes our research and includes possible feature work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Experiment Setting The subsection includes the C-MAPSS dataset description, the physical meaning of containing data, the preprocessing of the feature and target spaces, and related metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Dataset description The dataset [5] includes the data generated by the “Commercial Modular Aero-Propulsion System Simulation” (C-MAPSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The program was used to simulate the behavior of turbofan engine sensors under various flight conditions until its failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The flight conditions represent situations during ascent and descent to a standard flight altitude of about 10 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The C-MAPSS dataset consists of four subsets of data, the details of which are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Each subset contains readings from ‘21’ sensors (Table 2) located in different parts of the degrading engine, which are controlled by three characteristics of the operational settings (height, speed and acceleration in terms of altitude, Mach number, and throttle resolver angle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The simulation was used to study the sensors during the engine’s gradual deterioration up to the failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Different noises were added to modeled engine 3 Table 1: Brief description of four C-MAPSS subdatasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Subdataset FD001 FD002 FD003 FD004 Train trajectories 100 260 100 249 Test trajectories 100 259 100 248 Train samples 20631 53759 24720 61249 Test samples 13096 33991 16596 41214 Operating conditions 1 6 1 6 Fault modes 1 1 2 2 sensor reading to approximate the numerical solution to the complicated real- world engine behavior [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Similar to most of the work in the Related work section, we used normalize technique to the sensor readings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Problem statement The common assumption to target space is that if the engine is functional, it is guaranteed to run to failure for a certain amount of time, which is determined by the constant value N: RUL = � � � � � N, if RUL ≥ N RUL, if RUL < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' (1) N is an heuristic parameter, and in most cases for the C-MAPSS dataset task N = 125 value is used due to previously performed empirical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' This piece-wise RUL target space, first proposed by Heimes in [10], is widely adopted in remaining useful life estimation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The sensor readings of a failing motor 1a and the associated piece-wise RUL curve 1b presented in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Piece-wise RUL target function is at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 1b: One of the problems limiting the model comparison ability is the use of different preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The severe unstable impact is caused by various pre- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='processing and postprocessing based on operation condition parameters (for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Table 2: C-MAPSS dataset engine sensors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Sensor # ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Units ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Total temperature at the fan inlet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='°R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Total temperature at the low-pressure compressor outlet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='°R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Total temperature at the high-pressure compressor outlet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='°R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Total temperature at the low-pressure turbine outlet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='°R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Pressure at the fan inlet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='psia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Total pressure in bypass-duct ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='psia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Total pressure at the high-pressure compressor outlet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='psia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Physical fan speed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='rpm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Physical core speed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='rpm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Engine pressure ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Static pressure at the high-pressure compressor outlet (Ps30) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='psia ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Ratio of fuel flow to Ps30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='pps/psi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Corrected fan speed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='rpm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Corrected core speed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='rpm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Bypass ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Burner fuel-air ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Bleed enthalpy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='— ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Demanded fan speed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='rpm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Demanded corrected fan speed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='rpm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='High-pressure turbine coolant bleed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='lbm/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Low-pressure turbine coolant bleed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='lbm/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='(a) Sensor reading ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='(b) Piece-wise RUL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Figure 1: Normalized sensor values and corresponding piece-wise RUL for the second engine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='of the fd001 subdataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Normalized value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Time steps120 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='100 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='80 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='20 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Fo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='350 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='450 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='stepsexample,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Thus, we considered only fd001 and fg003 sub-datasets with fixed operation conditions for proper model comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Objective Quality Metrics The performance of the models on the C-MAPSS dataset was evaluated using RMSE and Score metric: RMSE = � � � � 1 n n � i=1 (yi − ˆyi)2 (2) diff(i) = ˆyi − yi score(i) = � � � � � e− diff(i) 13 − 1, if diff(i) < 0, e diff(i) 10 − 1, if diff(i) ≥ 0 Score = n � i=1 score(i) (3) where ‘yi’ and yi are predicted and true value of RUL for engine ‘i’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The asymmetric Score metric penalizes above the true value predictions due to maintenance regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The asymmetry is achieved via the coefficients in the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' These coefficients are empirically defined constants designed by the authors of the original dataset paper [5] for the PHM2008 competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Related Work There are two main ways for the C-MAPSS dataset RUL prediction problem: using auxiliary health index characteristics or a natural RUL curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Auxiliary health characteristic The first method is domain-based auxiliary characteristics models adopted to map test samples to train-based latent space and to determine the relations of it to related train vectors to RUL prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' One of the most common approaches is to construct a health index (HI) curve using deep learning and classical machine learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 6 One of the main directions related to health index space modeling is the use of the similarity-based curve matching technique, first presented in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The approach basis is the construction of multi-label time-series window mapping to the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The latent space construction model is based on the training dataset, the prediction is performed by test samples mapping and distance to the closest train sample estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The article [13] uses sparse Bayesian learning (SBL) [14] to establish health indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' In addition, there are methods of constructing an HI index based on classical ML clustering, proposed, for example, in [15], where a new index is constructed using the sample distances to each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' One of the most important areas of HI generation is empirically constructed models, both in the form of classical equations [16] and deep learning approaches [12, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' There are several Ways of constructing an HI index by the example of statistical approaches, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=', in [18] there are methods based on Dempster–Shafer evidence theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Nevertheless, the construction of a plausible HI curve imposes serious requirements on the function or model to display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The HI curve can be generated using autoencoders of various [19, 20] princi- ples for similarity-based curve matching techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' In [21] as a feature extrac- tor, a convolution-based autoencoder is used to construct a latent representation of the engine state that is maximally correlated with the function-based curve constructed from training sample sensor values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' In the article, [22] the authors proposed a comprehensive approach using a feature extractor built by apply- ing multilayer CNN layers to a time-sequence sliding window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The main idea of the model is to construct a common feature space for both RUL prediction and device state classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Some works contain elements of semi-supervised approaches based mainly on the use of different autoencoders [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Direct RUL prediction The second method is a direct prediction of RUL or peace-wise RUL without using auxiliary features and reliance on domain-based knowledge based on DL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The development of this direction is related to the possibility of using 7 feature extractors and efficient optimizers and obtaining SOTA results on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' RUL prediction methods, in contrast to the health curves, estimate degradation processes, omitting local anomalies, for which other methods are developed and tested on specialized datasets [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' In [26] discusses an approach to gen- erating partial training trajectories aimed at optimizing a linear RUL curve with a piece-wise continuous function that simulates the curve behavior in test datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' In addition, the classical approach is to use some constant area of the RUL curve [26, 23] corresponding to the normal mode of the device, and the value of this area is usually chosen empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' This heuristic is related to the fact that sensor readings in normal operation should not differ significantly from each other in the absence of noise - and thus model readings should not differ either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' In addition to this assumption, the accuracy of the validation of the constructed model is also used to determine the value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Due to the various methods of estimation, most of the empirical estimates of RUL lie in the range of ‘115’ to ‘160’ switching cycles [27, 28, 21, 29, 9, 30, 31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The most popu- lar approach to solving the RUL curve prediction problem is to use structures consisting of feature extractors and a regression head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The shape of a feature extractor varies widely, and for a regression head, MLP or gradient boosting are usually used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Early work in this area involves assessing the applicability of basic models like shallow networks and Multi-Layer Perceptrons (MLPs) [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The authors of [9] used LSTMs on three benchmark datasets and found them to work best compared to MLPs and CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' [34] compared LSTMs against vanilla RNNs and GRUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' In [35] feature extractor the authors present a series of heterogeneous heuristics on time series parameter extraction, and the use of LigthGBM to predict RUL and the associated curve, which the authors called a cumulative dynamic differential health indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' in [27] used stacked CNN, LSTM, and Bi-LSTM with residual connections to create a fea- ture extractor, for which they considered approaches to increase explainability through dimensionality reduction and Shapley additive explanation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' In [36, 37] the authors proposed the variants of feature extractors based on a multiscale convolutional neural network and a bidirectional recurrent block with 8 control, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Novel sota approaches In this section, several state-of-the-art models on various principles of oper- ation are analyzed in terms of the proposed properties and ways to improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' As SOTA results, we would like to highlight the three following methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The first principle [38, 39] involves autoencoders and variational autoencoders for feature construction or creation of an aggregated health index device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Forecasting is usually achieved by using an lstm-type model, in which the principles of time transfer and information cleansing occur through stacked BiLSTM layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' As an example approach, we can consider [38], a distinctive feature of which is the use of KL-div and MSE RUL loss to form a continuous ordered relative to RUL latent space, over which it is possible to perform state control by proximity simi- lar to the similarity-based curve matching method discussed in the related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The second principle [40, 41, 42] work includes feature construction approaches for further use in LSTM-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The use of different approaches to fea- ture generation is so diverse that it includes not only the standard approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' In particular, in [40] discusses the fusion of parallel models based on Bi-LSTM, with hand-crafted time-series features per model as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The equipment state control is based on piece-wise RUL prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The third principle [43, 44, 45] includes the addition of any form of the attention mechanism to the previously presented approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' For example, in [43] attention mechanism was used both in the sensor allocation phase and in accounting for the influence of time series position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The state of the equipment is supposed to be monitored based on RUL prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' These papers, to a greater or lesser extent, share a common conception of cleansing and interaction of information within the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' However, each work relies on a few ideas with no possible expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' We want to propose a new feature extraction approach and a method to use different models as extraction blocks to create a joint prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Proposed Approach In this paper, we proposed a new approach focused on building blocks aimed at different features extracting and fusing the results to build high-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Different features extraction include selecting blocks of distinct nature and loss functions that ensure the distinction of selected output feature vectors before blending by the attention layer [46] into a common latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Thus, the pro- posed approach includes different feature extraction blocks resulting in diverse latent space and attention layers to form high-level feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Figure 2: Proposed method general structure The general structure of the model built within the proposed approach is shown at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 2, and the neural network input goes through the following stages: Dataset-related preprocessing Preprocessed time-series is passed to the input of “Local preprocessing” in form of attention to specific features or time points, generalization or refinement of the sequence, and focusing on the time or feature domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The resulting sequence is passed to the feature extraction block with the skipped connection, allowing to maintain the gradient distribution regard- less of the block structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Latent space vectors at the output of the feature extraction blocks are 10 Feature extraction Local preprocessing block No 1 Concatenation Attention MLP RUL Preprocessed Input Feature extraction Local preprocessing block No N FC-Nnormalized and accounted for in loss, increasing the distinction between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Each normalized vector passes through additional fully connected layers for RUL prediction so that at the expense of the general error to form a selection of this level important features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The resulting different vectors are concatenated and passed through an attention layer to compensate for unavoidable errors in the form of noise associated with the requirement for vector difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The resulting latent representation passes through the regression head in the form of a multilayer perceptron for RUL prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The described approach resulted in the following step-by-step created struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' First, we studied the effect of the chosen data preprocessing methods on LSTM and CNN based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Second, we evaluated the effect of input data transposition as local preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Finally, we built three consecutive models, increasing the number of feature extractor blocks according to the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Baseline setup We evaluated how much the use of the proposed preprocessing can affect the simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' To investigate the effect of dataset based preprocessing heuristics, we use the well-known models proposed for the CMAPSS dataset problem as described in Related Work section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' For that, we have reimplemented models based on Long Short-Term Memory (LSTM) and Convolution Neural Network (CNN), which have been state-of-the-art models for solving the C- MAPSS dataset task shortly after it emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Figure 3 demonstrates the general structure of LSTM-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' “Stacked LSTMs” bock consists of three stacked LSTM layers with the parameters: hid- den state and input features is ‘21’, dropout is ‘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='5’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' “MLP block” consists of 11 Figure 3: LSTM-based general structure two fully-connected layers (441, 126) and (126, 1) with ReLU nonlinearity and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='5 dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' In turn, Convolutional Neural Network is an algorithm, usually employed for image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' But due to their structure and ability to capture spatial and temporal dependencies, they are widely used for time series forecasting as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Figure 4 illustrates the general structure of the implemented CNN-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' “Stacked CNN” includes two consecutive blocks of 1d Convolution (kernel sizes are ‘7’ and ‘3’, out channels are ‘5’ and ‘10’, stride sizes are ‘1’ and ‘0’), ReLU non-linearity, max pooling (kernel sizes are ‘3’ and ‘3’, stride sizes are ‘2’ and ‘1’), and flatten procedure after them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' “MLP block” consists of three fully-connected layers (30, 60), (60, 120) and (120, 1) with ReLU nonlinearity and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='5 dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Figure 4: CNN-based general structure 12 Preprocessed stacked LSTMs RUL InputPreprocessed stacked CNN RUL Input4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Feature approach We investigated one feature extractor block of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The effectiveness of the ”local preprocessing” block was tested for the first time through a swap of the time and feature domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' A study for window sizes one and three showed that domain swap leads to a 20% change in score metrics while maintaining comparable RMSEs, so we investigated the effect by adding that transform to the extractor block-like structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Figure 5: Time-feature model structure The structure of the time-feature model (TFM) for impact evaluation studied is shown in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Block “Transpose” is time and feature domains swap, “LSTM- based extraction block” is a block of 3 stacked LSTMs with implementation of skipped connection (feature size is equal to hidden size) with parameters dropout ‘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='5’, feature size is equal to window size ‘32’ and ‘40’ for fd001 and fd003 subdatasets, self-attention implementation corresponds to [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' “MLP block” consists of two fully-connected layers (672, 256), (256, 1) with SiLU nonlinearity [47] and ‘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='5’ dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Time-feature interaction Next, we tested how effective the proposed Interaction approach is in in- creasing the prediction accuracy compared to using the exact single model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The model presented in the previous paragraph was used as one of the feature ex- tractor blocks because of its high efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='The first step is to test the use of double TFM (DTFM), relying on the separation of different features through the use of an modified cosine loss function: cos (zi, zj) = zizj max (∥zi∥2∥zj∥2, ϵ) (4) 13 Attention Preprocessed LSTM-based MLP Transpose RUL Input extraction blockmCosineLoss = � (i,j),i̸=j max (cos (zi, zj), 0) (5) where ϵ = 1e−7 is to avoid zero in the denominator, zk is the kth extractor block output vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The use of Huber losses reduces the effect of local variance in predicting a nonnatural monotonic RUL curve shape: HuberLoss(i) = � � � � � 1 2(yi − ˆyi)2, if |(yi − ˆyi)| < β, β(|(yi − ˆyi)| − 1 2β), otherwise (6) where hyperparameter β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Thus, the overall loss function consists of three parts, representing the loss on RUL predictions based on concatenated vectors, losses on RUL predictions based on feature extractor blocks latent vectors, and differences between that latent vectors: L = HuberLossmodel + λ · � k HuberLossk + σ · mCosineLoss (7) where the empirical parameters lambda and sigma depend on the number of feature extractor blocks and the batch size, ‘k’ is the feature extractor block index and are equal ‘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='3’ and ‘1’ correspondingly, ‘model’ is index of final ‘MLP’ output (Fig 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The structure of the model for impact evaluation is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The model blocks are described above, except: “Concatenation” is concatenate feature extractor block outputs, “FC1” and “FC2” are fully-connected layers (672, 1) with SiLU nonlinearity [47] and ‘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='5’ dropout, “MLP block” consists of two fully-connected layers (1344, 256), (256, 1) with SiLU nonlinearity [47] and ‘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='5’ dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The dimensionality of the hidden vector is much larger than the window size and applying blocks of the same nature further seems redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' To further improve the metric, we adapted state-of-the-art stacked SCINet [48] as an additional future extractor (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 7) to build the time-feature in- teraction model (TFIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The reimplemented structure includes an adaptation 14 Figure 6: Double time-feature model structure of the multilayer hierarchical model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 8), where ”Hierarchical structure” represents convolution-based affine interactions, detailed in [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The main pa- rameters of the built block are as follows: hidden size is ‘16’, three levels in the hierarchical structure, kernel size is ‘3’, and dropout is ‘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='65’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Other parameters staing constant except “MLP” first layer change to (2016, 256).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Figure 7: Time-feature interaction model structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Adaptation of the proposed block to the task of RUL prediction consists ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='in optimization of the latent layer for RUL prediction between stacked blocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='at the expense of additional skipped connections through the adapted SciNet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='FC1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Transpose ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='LSTM-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='extraction block N1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Concatenation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='MLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='RUL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Transpose ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='LsTM-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='extraction block N2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='FC2LSTM-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Transpose ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='extraction block No1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Concatenation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='MLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='RUL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Preprocessed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Adapted SCINet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='extraction block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='LSTM-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='Transpose ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='extraction block No2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='FC1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='FC2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='FC3Figure 8: Adapted SCINet extraction block structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='block,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' and using the approach for processing the output vector in accordance with the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Experiments and Discussion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Training procedure In this section, we represented the learning process, the validation, the fea- tures of overfitting, and the hyperparameter choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Due to the described dataset properties, one of the important tasks is reg- ularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' In addition to mentioned class imbalance, there are anomalous functioning states in the data, realized by random outliers in the engine work- ing regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Thus, the importance of regularization increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' In this paper, 16 Preprocessed input SCINet Hierarchical structure Concatentate Concatenate SCINet FC Concatenate Latent representationregularization was performed by four main effects: reducing the class imbalance during dataset formation by considering only sixth window with RUL = 125, adding white noise to the training sample, using high dropout values, and av- eraging several RUL predictions to reduce variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Thus, summarizing the proposed regularization methods for the problem and the traditional approaches described in Related work, we fix the following conditions: Selecting a sliding window of sizes ‘32’ and ‘40’ for fd001 and fd003, the insufficient test sample (engines number ‘1’ and ‘52’ for fd001 and fd003 accordingly) expanded to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' AdamW [49] as the optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Huber loss to all RUL prediction blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Data noise augmentation (Normal distribution sampling, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='04) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Averaging over five values to reduce variance in prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Averaging over five runs to reduce variance in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Basic models were trained with a sliding window size of ‘32’ and batches of size ‘128’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The adaptive triangular cyclic learning rate scheduler [50] with [1e−4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 5e−4] interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The optimal learning configuration for training Interaction models include ‘210’ batch size and adaptive triangular cyclic learning rate updating rule in [2e−4, 9e−5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The validation includes the selection of ‘37’ (fd001) and ‘45’ (fd003) consecu- tive switching points (to average over ‘5’ sliding windows of sizes ‘32’ and ‘40’), which are excluded from the random location of each training sample engine and used for five-fold cross-validation with division by engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' This approach reduces the training datasets by about ‘5%’ due to ‘37’ and ‘45’ time points per engine exclusion and leads to a scoring decrease due to the high diversity of trajectories in feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The choice of other validation 17 techniques leads to even more significant deterioration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' However, due to the proposed regularization method, there was a plateau with minimal metric values and a number of epochs to reach for every model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Thus, the proposed models were trained on the full dataset and the epoch number obtained on validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' To obtain the final result, the predictions on the test dataset averaged over five epochs after reaching the required epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' On average, the training procedure lasted over ‘120’ epochs, reaching a plateau around ‘45’ or ‘70’ depending on the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Once the minimum was reached, the overfitting began.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' One interesting effect that can be associated with overfitting is a decrease in prediction accuracy as the final characteristic (the RUL prediction curve for all engine readings) approaches the reference peace-wise form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' This effect can be attributed to the “close” trajectories [5] in the sensor reading space due to the presence of multiple noises added during the dataset data simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The non-monotonicity of the characteristic guarantees a more accurate hit in the mean, but automatically enhances the variance of the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The variance impact of the model was reduced by averaging the last five predictions with the correction for the successive reduction of the RUL parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Domain preprocessing impact This section includes the performance of the discussed preprocessing ap- proach to well-known base models (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Table 3 shows the comparison results of the constructed models with the models that used other preprocessing and learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' These works were considered state-of-the-art, but the use of described approaches on base models has surpassed their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The base models outperformed the referenced mod- els in terms of RMSE and nonsymmetrical score constructed according to state control requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Thus, this study demonstrated the importance of actual general learning and preprocessing methods for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 18 Table 3: State-of-the-art approaches of previous years with a performance, comparable to baselines performance from this thesis FD001 FD003 RMSE Score RMSE Score MODBNE (2017) [51] 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='96 640 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='41 683 Deep LSTM (2017) [9] 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='14 338 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='18 852 BiLSTM-ED (2019) [19] 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='74 273 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='48 574 Attention-based DL Approach (2021) [52] 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='53 322 — — CNN (ours) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='38 332 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='12 495 LSTM-baseline (ours) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='88 400 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='75 382 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Time-feature interaction experiments Using the approach described above, we trained the models presented in the Time-feature interaction section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The training results in comparison to state- of-the-art results are in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The performance of the models on the standard for demonstration engines and model training are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 9b and 9a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The 35th engine is used to show the prediction results of each of the three models considered, AE is the absolute error between the model prediction and the true RUL value at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Each of the proposed models showed state-of-the-art results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' However, when moving from a model with one feature extractor block to a model with three feature extractor blocks, the RMSE changed by less than ‘2%’, although the inference time, measured by ‘300’ runs on a dummy example using the GPU (NVIDIA A100) warm-up technique, changes from ‘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='01’ to ‘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='75’ seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The standard deviation in the table is composed of the variation of the metric value at the five plateau points, over which we average the performance, and the variation between five consecutive runs to measure the stability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' It is important to note that models with non-monotonic noisy predictions 19 Table 4: Recent C-MAPSS dataset state-of-the-art approaches FD001 FD003 RMSE Score RMSE Score GNMR (2020) [53] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='14 212 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='23 370 BLS-TFCN (2021) [54] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='08 243 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='43 244 TCNN-Transformer (2021) [55] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='31 252 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='32 296 RVE (2022) [56] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='42 323 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='51 256 MCA-BGRU (2022) [57] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='44 211 — — DA-TCN (2021) [58] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='29 229 ± 8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='61 257 ± 58 ADL-DNN (2022) [59] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='16 238 ± 5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='25 281 ± 5 Feature space model (ours) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='09 215 ± 5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='03 191 ± 9 Double feature space model (ours) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='07 210 ± 6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='03 204 ± 6 Time-feature interaction model (ours) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='08 208 ± 3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='06 187 ± 8 (a) Learning curves (b) 35th test engine prediction Figure 9: Learning curve and prediction result for 35th engine of dataset fd001 obtained the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' A typical example of the overfitting that occurs after exceeding the plateau is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Increasing the accuracy of modeling the curve shape near-constant values significantly reduces the prediction accu- racy in the degradation region, which leads to a decrease in the metric on the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' We attribute this process to the unnatural shape of the piece-wise RUL curve, which forces the model to learn how to convert noisy trajectories into a monotonically decreasing curve, creating a noticeable loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' To check if all the features are used in the prediction, we perform a zeroing of the part of the latent representation corresponding to each feature extractor block 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The presented result demonstrates that the interaction of both FSM 20 FSM DFSM 16 TFIM 15 RMSE 14 13 12 0 10 20 30 40 50 60 70 Time steps140 FSM DFSM 120 TFIM AE FSM 100 AE DFSM AE TFIM 80 60 40 20 - 25 75 100 150 Time stepsFigure 10: Smoothed prediction of RUL engine ‘2’ by trained and overfitted models and adapted SciNet blocks are qualitatively similar, which can be interpreted as the extraction of close features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' However, due to the use of loss on the difference of vectors, locally both predictions are different, which suggests the extraction of additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The combined information usage from all three blocks allows us to obtain a qualitatively and quantitatively close prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Figure 11: TFIM 10th engine prediction with zeroed outputs of related feature extractor block During the comparison study of three models, we found that the use of an additional feature extractor block primarily affects the generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' To con- firm this effect, we selected test engine number ‘6’ from the fd001 subdataset, on which the models show one of the worst results, and over-smoothed it using the “Locally Weighted Scatterplot Smoothing” method with the frac param- 21 120 - 140 overfittedTFIM TFIM AE overfitted TFIM AE TFIM 41 21 21 Time steps140 FSM_1 FSM_2 120 TFIM Original Time stepseter ‘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content='11’, essentially retaining only the overall trend (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The engine was chosen as an example of prediction changes corresponding to the improved quality of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The figure shows that TFIM supports state predictions with RUL = 125 in the working state and contains a clear degradation trend in the transition region, unlike other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' A similar pattern is observed for most of the test engines, indicating that further improvements in model quality are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Figure 12: Smoothed RUL prediction for the difficult to simulation 6th engine 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Conclusion In this study, we proposed the ”Interaction” approach to the state control problem on the example of the C-MAPSS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The method includes build- ing scalable models based on blocks of distinct nature and loss functions that ensure the distinction of selected output feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Using multiple feature extractor blocks for feature extraction leads to scalability and interpretability due to the simplification of block individual inference research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' In the frame- work, we have explicitly proposed a dataset preprocessing, the effectiveness of which was measured relative to basic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Also, overfitting for the C-MAPSS dataset models was described as a model adaptation to the monotonic piece-wise RUL curve prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' 22 120 - 100 - 80 - RUL R 60 FSM DFSM 40 - TFIM AE FSM 20 - AE DFSM AE : TFIM 0 0 10 20 40 50 60 70 Time stepsWe proposed a new Feature space model (FSM) that demonstrated state- of-the-art results for the C-MAPSS task in selected preprocessing and augmen- tations approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The resulting model was used to form blocks of general Interaction structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The approach to different latent space vector formation building allowed the doubled FSM (DFSM) significantly improve the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Adding an adapted SCINet model to the structure demonstrates the possibil- ity of extracting additional information at the expense of a block of a different nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' The constructed time-feature interaction model (TFIM) not only im- proves the metrics but also leads to an increase in the generalizability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' Further work on developing the approach could be to find rules for scal- ing the system in-depth with the development of regularization methods that compensate for the drawbacks of the piece-wise constant RUL form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE4T4oBgHgl3EQfWAx3/content/2301.05029v1.pdf'} +page_content=' References [1] Y.' metadata={'source': 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b/utA0T4oBgHgl3EQfLv_t/content/tmp_files/2301.02124v1.pdf.txt @@ -0,0 +1,4128 @@ +R´enyi entropies for one-dimensional quantum systems with +mixed boundary conditions +Benoit Estienne, Yacine Ikhlef, Andrei Rotaru +Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique et Hautes ´Energies, +LPTHE, F-75005 Paris, France +January 6, 2023 +Abstract +We present a general method for calculating R´enyi entropies in the ground state of a +one-dimensional critical system with mixed open boundaries, for an interval starting at +one of its ends. In the conformal field theory framework, this computation boils down to +the evaluation of the correlation function of one twist field and two boundary condition +changing operators in the cyclic orbifold. Exploiting null-vectors of the cyclic orbifold, we +derive ordinary differential equations satisfied by these correlation functions. In particular +we obtain an explicit expression for the second R´enyi entropy valid for any diagonal minimal +model, but with a particular set of mixed boundary conditions. In order to compare our +results with numerical data for the Ising and three-state Potts critical chains, we also identify +and compute the leading finite size corrections. +1 +arXiv:2301.02124v1 [cond-mat.stat-mech] 5 Jan 2023 + +Contents +1 +Introduction +4 +2 +The cyclic orbifold +7 +2.1 +The cyclic orbifold on the Riemann sphere . . . . . . . . . . . . . . . . . . . . . . +7 +2.1.1 +Symmetry algebra and operator content . . . . . . . . . . . . . . . . . . . +7 +2.1.2 +Null vectors for untwisted operators +. . . . . . . . . . . . . . . . . . . . . +9 +2.1.3 +The induction procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +2.2 +The cyclic orbifold on the upper half plane +. . . . . . . . . . . . . . . . . . . . . +11 +2.3 +Operator algebra of the cyclic orbifold BCFT . . . . . . . . . . . . . . . . . . . . +12 +2.3.1 +Calculation of boundary-boundary structure constants . . . . . . . . . . . +13 +2.3.2 +Orbifold bulk-boundary structure constants . . . . . . . . . . . . . . . . . +13 +3 +Differential equations in the Z2 and Z3 orbifold BCFT +14 +3.1 +Setup for the calculations +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +3.2 +The function ⟨Ψ12 · σ · Ψ12⟩ in a generic Z2 orbifold . . . . . . . . . . . . . . . . . +15 +3.3 +The function ⟨Ψ12 · σh · Ψ12⟩ in a generic Z2 orbifold . . . . . . . . . . . . . . . . +18 +3.4 +The function ⟨Ψ12 · σ · Ψ12⟩ in a generic Z3 orbifold . . . . . . . . . . . . . . . . . +21 +3.5 +The function ⟨Ψ12 · σ13 · Ψ12⟩ in the Z3 orbifold of the Ising model +. . . . . . . . +22 +3.6 +More hypergeometric differential equations in the Ising cyclic orbifold BCFTs +. +24 +4 +Numerical checks and finite-size corrections in quantum chains +25 +4.1 +The Ising quantum chain with mixed BC +. . . . . . . . . . . . . . . . . . . . . . +26 +4.2 +The three-state Potts quantum chain with mixed BC . . . . . . . . . . . . . . . . +29 +5 +Conclusion +36 +A Mother BCFT conventions +37 +B Computation of orbifold structure constants +38 +2 + +B.1 +Composite twist one-point structure constant in the ZN orbifold BCFT +. . . . . +38 +B.2 +Bulk-boundary structure constant in the Z2 orbifold CFT . . . . . . . . . . . . . +38 +C Orbifold Ward identities for bulk fields +39 +D R´enyi entropies for the critical Ising chain with mixed fixed BC +40 +E Hypergeometric differential equation +41 +F Fusion rules in the ZN orbifold +42 +G Derivation of differential equation in the Z3 orbifold BCFT +43 +H Numerical implementation of the Frobenius method +45 +3 + +1 +Introduction +The understanding of quantum entanglement has proved to be a research topic of continued and +central interest for physicists working in domains as diverse as high energy physics, condensed +matter theory and quantum information. Entanglement measures have turned out to be useful +diagnostic tools for tensor network algorithms, quantities of interest for the AdS/CFT corre- +spondence, and, most relevantly for the present work, a powerful tool for probing the physics +of quantum many-body systems. With respect to the latter, the study of entanglement has +proved crucial to the study of phase transitions in one dimensional quantum systems, by al- +lowing their detection and the characterization of their critical exponents and corresponding +central charge [1–4]. Important applications of entanglement are found in higher dimensions +too. We mention, for two-dimensional systems, the establishment of intrinsic topological or- +der and various anyonic quantum dimensions [5, 6] and the detection and counting of critical +Dirac fermions [7–10]. Finally, entanglement can also be used, in two [11–15] or higher dimen- +sions [16,17] to reveal gapless interface modes. +The basic setup is as follows: we consider a quantum system in a pure state |Ψ⟩, and a spatial +bipartition of said system into two complementary subregions A and B. The entanglement +between them is then encoded in the reduced density matrix ρA = TrB|Ψ⟩⟨Ψ| and it can be +quantified through entanglement measures, such as the R´enyi entanglement entropies [18–22] +Sn(A) = +1 +1 − n log TrA (ρn +A) , +(1.1) +and in particular the n → 1 case corresponding to the well-known von Neumann entropy: +S(A) = −TrA (ρA log ρA) . +(1.2) +While the focus on entanglement entropies has been mostly theoretical, in recent years experi- +mental proposals as well as actual experiments have been designed to measure them [23–28]. +For strongly correlated quantum systems, the theoretical computation of entanglement en- +tropies is a technically challenging endeavour. However, if these systems are one-dimensional +and critical, the formidable toolbox of two-dimensional Conformal Field Theory (CFT) is avail- +able to tackle such computations. The calculations of entanglement entropies through such +methods rests on two crucial insights. The first insight is that, for integer values of n, and a +subsystem A = ∪i[ui, vi] built as the union of some disjoint intervals, the moments of the reduced +density matrix TrA (ρn +A) can be expressed as the partition function of an n-sheeted Riemann +surface with conical singularities corresponding to the endpoints of the intervals [ui, vi] [2,29]. +Such partition functions have been evaluated, with significant toil, for free theories and some +special cases of interacting models [4, 30–38]. In general, however, a second insight is needed +to make progress: the replication of the spacetime of the theory can be ”exchanged” for the +replication of the target space of the CFT [39–41]. Such a construction, known in the literature +as the cyclic orbifold CFT [40], is built from the permutation symmetric product of n copies of +the original CFT (referred to as the mother CFT), by modding out the discrete subgroup Zn +of cyclic permutations. In this framework, the conical singularities of the mother CFT defined +on the replicated surface are accounted for by insertions of twist fields [39] in cyclic orbifold +correlators. Thus, by computing correlators of twist operators, one can evaluate TrA (ρn +A) for +a variety of setups. To give a few examples, one can easily adapt the twist field formalism +to encode modified initial conditions around the branch points [42], which is fitting for com- +putations of more refined entanglement measures such as the symmetry-resolved entanglement +entropy [43–48] or for explorations of entanglement in non-unitary systems [42, 49]. Arguably +4 + +the most renowned result obtained in this framework is [1,2,29,50–52] +Sn(ℓ) +∼ +ℓ→∞ +c +6 +n + 1 +n +log ℓ , +(1.3) +which gives the universal asymptotic behaviour for the ground state entanglement entropy of +an interval of length ℓ in an infinite system (with c the central charge of the critical system). +In this article, we consider the R´enyi entanglement entropy in an open system with mixed +boundary conditions, when the subregion A is a single interval touching the boundary – we take +the boundary condition (BC) at one end of the chain to be different from the BC at the other +end (see Figure 1). In the scaling limit, such an open critical system is described by a Boundary +Conformal Field Theory (BCFT), with a well understood [53–56] correspondence between the +chiral Virasoro representations and the conformal boundary conditions allowed by the theory, +and an algebra of boundary operators that interpolate between them. +ℓ +β +α +L +Figure 1: An interval of length ℓ in a 1d critical chain with mixed BC (αβ) and length L. +The more accessible setup of an interval touching one of two identical boundaries has been +thoroughly analysed using either conformal field theory methods [2, 52, 57–60] or exact free +fermion techniques [61–63]. +Such configurations are also well-handled numerically, through +density-matrix renormalization group (DMRG) techniques [64–67]. In that setup, the subsys- +tem A is at the end of a finite system with the same boundary condition α on both sides. The +computation of the R´enyi entanglement entropies rests on the evaluation of a twist one-point +function on the upper half-plane. Such a correlation function is straightforwardly fixed by con- +formal invariance, and as a consequence the entanglement entropy exhibits a simple dependence +on the interval and system sizes. Explicitly, in the case of an interval of length ℓ at the end of +a system of size L, one finds the leading universal behaviour [2]: +Sn(ℓ) ∼ c +12 +n + 1 +n +log +�2L +πa sin +�πℓ +L +�� ++ log gα , +(1.4) +where a is the lattice spacing and gα is the universal boundary entropy [68] associated to the +boundary condition α. +When one studies systems with mixed BC, at the level of the BCFT one has to introduce +boundary condition changing operators (BCCOs), and thus the corresponding correlators are +more complicated. The core idea of this framework is that the singular behaviour associated +to the change in boundary conditions, can be encoded in the form of operators placed on the +boundary, that interpolate between regions of different BC α ̸= β. Thus, to compute the R´enyi +entropy Sn in this setup, we will evaluate three-point functions with one twist operator and +two BCCO insertions. +Such setups have already been studied for the Ising and XX chains +in [57], at the level of the CFT on the replicated surface, and rely on the knowledge of relatively +simple closed form expressions for the 2n-point correlator of BCCOs on the unit disk for their +calculations. +However, such knowledge is the exception, rather than the norm, for generic +BCFTs. +5 + +In this work, we present a general method to compute such twist correlators functions +with mixed BCs. The most technically demanding part of this framework is finding Ordinary +Differential Equations (ODEs) that the correlators satisfy. +According to Cardy’s doubling +trick [53], in the half-plane geometry, the three-point functions of interest obey the same Ward +identities as a four-point conformal block with the corresponding operators, where the bulk +twist operator σ(z, ¯z) is replaced by the insertion of σ(z)σ†(¯z). Thus, in an adaptation of the +method of [42], we can derive a differential equation by combining knowledge of the null-vector +conditions obeyed by the twisted and untwisted fields under the symmetry algebra of the cyclic +orbifold [40] with the derivation of well-chosen Ward identities obtained from current insertions +in the correlators of interest. The final ingredient is the determination of a subset of the (bulk +and boundary) structure constants of the cyclic orbifold BCFT, which fix the specific linear +combination of solutions of the differential equation that gives the sought correlator. +We have illustrated this approach with a variety of BCFT setups, that share a common +assumption: in the mother CFT, the mixed boundary conditions (βα) are implemented by a +BCCO which is degenerate at level two under the Virasoro algebra. +With this restriction, in the Z2 orbifold of a generic BCFT, we have derived a second-order +and a fourth order ODE, respectively for the bare and composite1 twist correlator. Under the +same restrictions, in the Z3 orbifold of a generic BCFT, we have determined a third-order +ODE for the bare twist correlator. We have also worked out, for the case of the Z2 and Z3 +cyclic orbifolds of the Ising BCFT, a variety of lower-order ODEs. The latter calculations were +found compatible with the results of [57], and have been tested against numerical results for +the critical Ising chain, for all possible combinations of BCs, to excellent agreement. Finally, +we have also considered the Z2 orbifold of the three-state Potts model, and compared it against +lattice data for the critical three-state Potts model with states {R, G, B}, with less accurate, +but consistent results. We quote here the leading behaviour of the second R´enyi entropy of +the critical three-state Potts chain of size L for mixed fixed R and restricted GB boundary +conditions: +S(R,GB) +2 +(ℓ) ∼ cPotts +8 +log 2L +πa sin +�πℓ +L +� ++ log gR − log +� +η−2h122F1 (−8/5, −9/10; −9/5 | 1 − η) +� +(1.5) +with η = e2πiℓ/L, cPotts = 4/5 the central charge, h12 = 2/5 the scaling dimension of the BCCO, +and gR = [(5 − +√ +5)/30]1/4 the ground state degeneracy associated to the fixed R BC [69]. +The expression (1.5) is, in fact, only a particular case of a more general result obtained +in this paper, which applies to any critical system described by a BCFT based on a minimal +model M(p, p′) with mixed conformal BC (α, β) chosen such that the most relevant BCCO +interpolating between them is ψ(αβ) +1,2 +and there is no boundary operator ψ(ββ) +1,3 +allowed in the +theory. Under these conditions, the second R´enyi entropy of an interval A = [0, ℓ], in a finite +system of size L touching the boundary β is: +S(α,β) +2 +(ℓ) ∼ c +8 log 2L +πa sin +�πℓ +L +� ++ log gβ − log +� +η−2h122F1 +� +2 − 3 p +p′ , 3/2 − p +p′ ; 3 − 4 p +p′ | 1 − η +�� +(1.6) +where c, gβ and h12 generalize the notation in (1.5). +In both (1.5) and (1.6), one should keep in mind, especially for the purpose of numerical +studies, that the hypergeometric function in the third term of these equations converges inside +the unit circle centred at η = 1, which only overlaps with the subinterval Arg(η) ∈ (0, π/3) ∪ +1obtained by fusing the bare twist operator with an untwisted operator φ. +6 + +(5π/3, 2π) of the parameter space of interest Arg(η) ∈ [0, 2π]. Thus, to evaluate the expressions +for Arg(η) ∈ (π/3, 5π/3), it is necessary to analytically continue the third term to this range. +We give here the outline of the article. In Section 2, we give a review of the cyclic orbifold +construction, with a focus on its implementation on the upper-half plane. We discuss in this +section the bulk and boundary operator algebra, and show how some orbifold bulk and boundary +structure constants can be expressed in terms of mother BCFT quantities by unfolding and +factorizing arguments. We dedicate Section 3 to the derivation of ODEs for the different setups +described above. On top of the announced derivations involving orbifold Ward identities, we +also use the results on the fusion rules of the ZN cyclic orbifold of [70] and some mathematical +facts about the hypergeometric differential equation, to derive low-order differential equations +for the Ising case. Section 4 contains a comparison of our analytical results with lattice data, +for both the Ising and three-state Potts critical chains. Finally, we have relegated the more +technical derivations to the Appendix, to avoid congesting the logical flow of the paper. +2 +The cyclic orbifold +In this section, we will present the construction of the cyclic orbifold BCFT on the upper half- +plane H. After reviewing a few essential features of the ZN orbifold on the Riemann sphere, we +will discuss conformal boundary conditions, boundary operators as well as bulk-boundary and +boundary-boundary operator algebras. +2.1 +The cyclic orbifold on the Riemann sphere +To build a cyclic orbifold CFT, one starts from any mother CFT M and constructs the tensor +product theory M⊗N. Then one considers all the ZN equivalent ways of connecting the copies +of the product theory, which creates N different sectors, each with its corresponding operator +families and labelled by a ZN twist charge [k]. The spectrum of the cyclic orbifold MN is then +built as a reunion of the operator families from all the sectors [k]. +2.1.1 +Symmetry algebra and operator content +In MN, each copy a of the mother CFT carries the components of the stress-energy tensor +Ta(z), ¯Ta(¯z). We define the discrete Fourier modes of these currents as +T (r)(z) = +N−1 +� +a=0 +ωar Ta(z) , +¯T (r)(¯z) = +N−1 +� +a=0 +ωar ¯Ta(¯z) , +(2.1) +where r is considered modulo N, and we have used the notation ω = exp(2iπ/N). They satisfy +the OPEs +T (r)(z)T (s)(w) = δr+s,0 Nc/2 +(z − w)4 ++ 2T (r+s)(w) +(z − w)2 ++ ∂T (r+s)(w) +z − w ++ regz→w , +¯T (r)(¯z) ¯T (s)( ¯w) = δr+s,0 Nc/2 +(¯z − ¯w)4 ++ 2 ¯T (r+s)( ¯w) +(¯z − ¯w)2 ++ ∂ ¯T (r+s)( ¯w) +¯z − ¯w ++ reg¯z→ ¯w , +(2.2) +where the Kronecker symbols δr+s,0 are understood modulo N. The symmetric modes T (0)(z) +and ¯T (0)(¯z) are the components of the stress-energy tensor of MN with central charge Nc, +7 + +whereas the other Fourier modes T (r)(z), ¯T (r)(¯z) with r ̸= 0 should be regarded as additional +conserved currents. Altogether, these Fourier modes encode an extended conformal symmetry. +The modes associated to these currents are defined in the usual way through: +L(r) +m = +1 +2iπ +� +dz zm+1 T (r)(z) , +¯L(r) +m = +1 +2iπ +� +d¯z ¯zm+1 ¯T (r)(¯z) . +(2.3) +In the sector of twist charge [k] one has the following mode decompositions +T (r)(z) = +� +m∈−kr/N+Z +z−m−2 L(r) +m +¯T (r)(¯z) = +� +m∈+kr/N+Z +¯z−m−2 ¯L(r) +m +(2.4) +and the commutation relations +� +L(r) +m , L(s) +n +� += (m − n)L(r+s) +m+n + Nc +12 m(m2 − 1) δm+n,0 δr+s,0 , +� +¯L(r) +m , ¯L(s) +n +� += (m − n)¯L(r+s) +m+n + Nc +12 m(m2 − 1) δm+n,0 δr+s,0 . +(2.5) +Hermitian conjugation of the modes acts as: +� +L(r) +n +�† += L(−r) +−n , +� +¯L(r) +n +�† += ¯L(−r) +−n . +(2.6) +Orbifold primary operators are, by definition, annihilated by the action of all the positive +modes of OVir ⊗ OVir. Descendant operators with respect to this algebra are constructed by +the action of the negative m modes. We establish the notation for descendants of a scaling +(primary or not) operator O: +� +L(r) +m · O +� +(z, ¯z) := +1 +2iπ +� +Cz +dw (w − z)m+1 T (r)(w)O(z, ¯z) , +� +¯L(r) +m · O +� +(z, ¯z) := +1 +2iπ +� +Cz +d ¯w ( ¯w − ¯z)m+1 ¯T (r)( ¯w)O(z, ¯z) , +(2.7) +where the contour Cz encloses the point z. +It will be useful to work with the primary operator spectrum with respect to the neutral +subalgebra A ⊗ ¯A generated by the algebra elements +L(r1) +m1 . . . L(rp) +mp +and +¯L(r1) +m1 . . . ¯L(rp) +mp , +with r1 + · · · + rp = 0 +mod N . +(2.8) +One can classify all ZN-symmetric operators of MN into representations of A ⊗ ¯A. This orga- +nization, described in detail by the authors of the present work in [70], distinguishes between +three types of operators. First, we have identified the untwisted non-diagonal operators Φ[j1...jN]. +These operators are built from ZN-symmetrized combinations of products of mother CFT pri- +mary operators φj (with j = 1 referring to the identity operator 1): +Φ[j1...jN] := +1 +√ +N +N−1 +� +a=0 +(φj1+a ⊗ · · · ⊗ φjN+a) , +(2.9) +in which at least one pair satisfies ji ̸= jk. Its conformal dimension is given by h[j1...jN] = � +s hjs. +8 + +The second type of primary operators under the neutral algebra are the untwisted diagonal +fields Φ(r) +j , where the Fourier replica index r takes values in ZN . The r = 0 diagonal fields are +defined to be: +Φ(0) +j += Φj := φj ⊗ · · · ⊗ φj , +(2.10) +while for r ̸= 0, they are constructed as: +Φ(r) +j +:= +1 +2Nhj +L(r) +−1 ¯L(−r) +−1 +· Φj , +1(r) := +2 +NcL(r) +−2 ¯L(−r) +−2 +· Φ1 , +(2.11) +The conformal dimension of a diagonal operator Φ(r) +j +is then generically given by +h(r) +j += Nhj + δr,0 (1 + δj,1) +(2.12) +We should note that the diagonal operators with r = 0 and the non-diagonal operators are also +primary under OVir ⊗ OVir. +Finally, we have to consider twist operators, which come in distinct flavours. For the pur- +poses of this paper, we will mostly work with twist operators with Fourier replica index r = 0. +Thusly, just as for the diagonal fields, we will drop this specification when the context heavily +implies it, to decongest the notation. +We first consider the ubiquitous bare twist operators [2, 30, 71, 39] which are denoted in +our conventions σ[k] = σ[k] +1 , or, in light notation, σ = σ[1] and σ† = σ[−1] . We have also the +composite twist fields σ[k] +j , which can be defined through point-splitting as in [49]: +σ[k] +j (z, ¯z) := Aj lim +ϵ→0 +� +ϵ2(1−N−1)hjΦ[j,1,...,1](z + ϵ, ¯z + ¯ϵ) · σ[k](z, ¯z) +� +, +(2.13) +where the constant Aj = N−2(1−N−1)hj−1/2 ensures that non-vanishing two-point functions of +twist operators are normalized to one. If N and k are coprime, the conformal dimension of the +bare twist operator is +hσ = c +24 +� +N − 1 +N +� +, +(2.14) +while for composite twist operators one has: +hσj = hσ + hj +N . +(2.15) +Having established the primary operator spectrum of the orbifold, we will now review how the +null-vectors of the diagonal and twisted fields in MN are inferred from the ones of the mother +theory M. +2.1.2 +Null vectors for untwisted operators +Let us consider a generic mother CFT M, with central charge +c = 1 − 6(1 − g)2 +g +, +0 < g ≤ 1 . +(2.16) +The conformal dimensions of degenerate primary operators are given by the Kac formula +hrs = (r − sg)2 − (1 − g)2 +4g +, +(2.17) +9 + +where r, s are positive integers. The corresponding operator φrs is degenerate at level rs. If the +parameter g is rational, i.e. g = p/p′ with coprime p and p′, then the set of operators φrs with +1 ≤ r ≤ p − 1 and 1 ≤ s ≤ p′ − 1 generates a closed operator algebra, and the related CFT is +the minimal model Mp,p′. While we do employ this parametrization extensively, in the present +work we will consider a more generic mother CFT, and we do not assume that it is a minimal +model—unless explicitly indicated. +Consider the situation when the mother CFT includes the degenerate operator φ12, with +null-vector condition +� +L−2 − 1 +gL2 +−1 +� +φ12 = 0 . +(2.18) +In the untwisted sector of the orbifold CFT, we have +L(r) +n = +N +� +a=1 +e2iπra/N +� +1 ⊗ . . . 1 ⊗ +Ln +(a−th) +⊗ 1 ⊗ . . . 1 +� +, +n ∈ Z , +(2.19) +and the diagonal untwisted operator associated to φ12 is +Φ12 = φ12 ⊗ · · · ⊗ φ12 . +(2.20) +Using an inverse discrete Fourier transform, one easily finds, for any r ∈ ZN, +� +L(r) +−2 − 1 +Ng +N−1 +� +s=0 +L(s) +−1L(r−s) +−1 +� +· Φ12 = 0 . +(2.21) +When inserted into a correlation function, the modes L(0) +m act as linear differential operators. +The treatment of the modes L(r) +m with r ̸= 0 introduces an additional difficulty, that we will +address case by case, with the help of orbifold Ward identities. +2.1.3 +The induction procedure +The null-vectors of the mother CFT also determine the null vector conditions on twist operators +in MN, through the induction procedure [40]. +In the present work, we shall only be concerned with the twist sectors with charges [±1]. +In the notations of [70], induction can be expressed in terms of a norm-preserving, invertible +linear map Θ from the Hilbert space of the mother CFT to that of the twist sector [1], defined +by +Θ|φ⟩ = |σφ⟩ , +ΘLmΘ−1 = N +� +L(−m) +m/N − hσ δm0 +� +, +(2.22) +where φ is any primary operator in the mother CFT, and σφ is the associated composite twist +operator in the orbifold CFT. +The simplest application to null-vectors is the case of the identity: +L−1 · 1 = 0 +⇒ +L(1) +−1/N · σ = 0 . +(2.23) +For a degenerate operator at level two, applying the induction map on (2.18) yields +� +L(2) +−2/N − N +g (L(1) +−1/N)2 +� +· σ12 = 0 . +(2.24) +The corresponding null-vector conditions for the operators σ† and σ† +12 are easily obtained by +conjugation. +10 + +2.2 +The cyclic orbifold on the upper half plane +To construct the cyclic orbifold BCFT, we will work on the upper half-plane H, with the +boundary along the real axis. We parametrize H by z = x + iy with x ∈ R and y > 0, and we +impose the gluing condition on the boundary for the stress-energy tensor components: +T (0)(x) = ¯T (0)(x) +for +x ∈ R , +(2.25) +which ensures that the boundary is conformal i.e., preserves a copy of the Virasoro algebra [72]. +The ZN orbifold, however, has an extended symmetry, and we must choose if and how the +components of the additional currents T (r̸=0) are glued at the boundary. +Our usage of the +replica trick provides a clear indication for these choices: since we are considering N copies +of the same mother BCFT, we must impose the gluing condition Ta(x) = ¯Ta(x) on each of +them. By taking the Fourier transform of this relation, we find that in the orbifold CFT we are +effectively imposing: +T (r)(x) = ¯T (r)(x) +for +x ∈ R , +(2.26) +for all the discrete Fourier modes of the stress-energy tensor components defined in (2.1). This +implies that the boundary preserves a full copy of the OVir algebra. +By the same reasoning on CFT replicas, the orbifold boundary states we are interested in +correspond to having the same conformal BC on the N copies of the mother CFT. They are +simply given by |α⟩⊗N and |β⟩⊗N. +On the upper half-plane, we will set the conformal BC α on the positive real axis x > 0 and +the conformal BC β on x < 0. To implement such mixed conformal BC in a BCFT, we will have +to work with the formalism of boundary condition changing operators [53]. These operators, +restricted to live on the boundary, are placed at the points of suture of regions of different +BC. The full operator algebra of a BCFT is then formed by considering the OPEs between +both BCCOs and bulk operators, as detailed in Appendix A. For a given pair of conformal +BCs (α, β), there can be several primary BCCOs implementing the change α → β: we denote +such an operator ψ(αβ) +h +, where h specifies its conformal dimension. The most relevant BCCO +implementing α → β is simply referred to as ψ(αβ). +In the ZN orbifold CFT, we will be concerned with the calculation of correlators with +insertions of diagonal BCCOs, namely : +Ψ(αβ) +h += ψ(αβ) +h +⊗ · · · ⊗ ψ(αβ) +h +� +�� +� +N times +. +(2.27) +Then, orbifold correlators with mixed BC are obtained by inserting the most relevant diagonal +BCCO: +⟨O1(z1, ¯z1) . . . On(zn, ¯zn)⟩αβ +H = ⟨Ψ(αβ)(∞) O1(z1, ¯z1) . . . On(zn, ¯zn)Ψ(βα)(0)⟩H . +(2.28) +By Cardy’s doubling trick [72], [54], such (n + 2)-point correlators satisfy the same Ward iden- +tities as any of the (2n + 2)-point conformal blocks on the Riemann sphere C with external +operators +Φ(∞), O1(z1), O1(¯z1), . . . , On(zn), On(¯zn), Φ(0) , +(2.29) +where Oi(¯z) is the antiholomorphic counterpart of Oi(z), and Φ(z) is the holomorphic part of +the diagonal primary operator defined in (2.10), with the conformal dimension of Ψ(αβ). In +more precise terms, Oi is the operator conjugate to Oi with respect to the symmetry algebra +11 + +preserved by the boundary [73]. For ZN twist operators, conjugation acts as σi = σ† +i [39], so +that the one-twist function +⟨σi(z, ¯z)⟩(αβ) +H += ⟨Ψ(αβ)(∞)σi(z, ¯z)Ψ(αβ)(0)⟩H +(2.30) +satisfies the same Ward identities as the functions ¯z−2hσi × Fk(z/¯z), where Fk is the rescaled +conformal block: +Fk(η) = +Φk +σi(1) +Φ(0) +σ† +i (η) +Φ(∞) += ⟨Φ|σi(1)Pkσ† +i (η)|Φ⟩ , +(2.31) +Pk is the projector onto the (A ⊗ ¯A)-module of Φk, and {Φk} is the set of allowed intermediary +untwisted (diagonal or not) primary operators. In the following, it will also be necessary to +consider the conformal blocks in the channel η → 1, namely +�Fℓ(η) = +Φℓ +σi(1) +Φ(0) +Φ(∞) +σ† +i (η) += ⟨Φ|Φ(1)Pℓσ† +i (1 − η)|σi⟩ . +(2.32) +Using the Ward identities implied by the OVir algebra (discussed in detail in Section 3) for +these functions, together with the null-vectors of the previous section, will allow us to extract +an ODE that the functions (2.31–2.32) satisfy. +To understand the structure of the conformal blocks, we also need to define non-diagonal +BCCOs, paralleling (2.9): +Ψ(αβ) +[j1...jN] := +1 +√ +N +N−1 +� +ℓ=0 +(ψ(αβ) +j1+ℓ ⊗ · · · ⊗ ψ(αβ) +jN+ℓ) , +(2.33) +where the 1/ +√ +N factor ensures that the non-vanishing two-point functions of these BCCOs are +normalized to one. These operators appear in the OPE of the diagonal boundary fields Ψ(αβ) +j +, +and thus their conformal dimension determines the leading singular behaviour of the conformal +blocks. Naturally, we should now discuss the operator algebra of the ZN orbifold BCFT. +2.3 +Operator algebra of the cyclic orbifold BCFT +In the orbifold BCFT, the operator algebra consists of OPEs of three types. First, there is the +operator subalgebra of bulk operators, inherited from the ZN orbifold CFT on C. We shall not +directly use the structure constants of this subalgebra in this work, but they have been discussed +in [40, 42, 71, 74]. The second type of OPE we need to consider in the orbifold BCFT, is the +bulk-boundary OPE which encapsulates the singular behaviour of a bulk field as it approaches +a conformal boundary. In our calculations, we will only need to work with the OPEs of primary +twist operators σi(z, ¯z) as they are sent towards a conformal boundary α: +σi(x, y) ∼ +y→0 +� +j +A(α) +σi,Ψj (2y)hj−2hσi Ψ(αα) +j +(x) +(2.34) +where the sum runs over all the boundary operators Ψ(αα) +j +and their descendants under OVir, +and we have denoted the bulk-boundary structure constants by A(α) +σi,Ψj. +Finally, we need to +12 + +consider the OPEs of orbifold boundary operators. For generic diagonal BCCOs, this takes the +form +Ψ(αβ) +i1 +(x1)Ψ(βγ) +i2 +(x2) +∼ +x1→x2 +� +j +B(βγ)Ψj +Ψi1,Ψi2(x1 − x2)−hi1−hi2+hjΨ(αγ) +j +(x2) +(2.35) +with the index j running over all the orbifold BCCOs interpolating between the conformal +boundary conditions α and γ. We have denoted the boundary-boundary structure constants by +B(αβγ)Ψj +Ψi1,Ψi2 . To calculate the structure constants of the OPEs that are relevant for the present +work, we will need to use factorization and unfolding arguments for the correlator that determine +them, along the lines of [70], [75] and [71]. +2.3.1 +Calculation of boundary-boundary structure constants +Let us consider the calculation of boundary-boundary structure constants of the type B(ββα)Ψk +Ψ∗,Ψj +, +where Ψ∗ denotes a generic untwisted orbifold primary BCCO. We can express this as a three- +point function on the upper half-plane H: +B(ββα)Ψk +Ψ∗,Ψj += ⟨Ψ(αβ) +k +(∞)Ψ(ββ) +j +(1)Ψ(βα) +∗ +(0)⟩H . +(2.36) +Since there are no twist insertions in the above correlator, it just factorizes into a linear com- +bination of products of mother BCFT three-point functions. Let us first consider the case of a +diagonal BCCO, with Ψ(βα) +∗ += Ψ(βα) +i +. Then, the orbifold correlator factorizes into mother CFT +three-point functions as: +B(ββα)Ψk +Ψi,Ψj += +� +⟨ψ(αβ) +k +(∞)ψ(ββ) +j +(1)ψ(βα) +i +(0)⟩H +�N +, +(2.37) +so we find a simple expression for these coefficients, in terms of mother BCFT boundary- +boundary structure constants: +B(ββα)Ψk +Ψi,Ψj += +� +B(ββα) k +ij +�N +. +(2.38) +By similar considerations, the structure constants involving a non-diagonal BCCO Ψ(βα) +[i1...iN] can +be expressed as: +B(ββα)Ψk +Ψ[i1...iN ],Ψj = +√ +N +N +� +a=1 +B(ββα) k +iaj +(2.39) +The rest of the boundary-boundary structure constants of untwisted BCCOs can similarly be +expressed in terms of mother BCFT quantities, but we will not need them in this work. +2.3.2 +Orbifold bulk-boundary structure constants +The first bulk-boundary structure constant we need to calculate is A(α) +σ,Ψ1, where Ψ(αα) +1 +is just +the identity boundary field. This can be expressed as the one-point function on the unit disk +D: +A(α) +σ,Ψ1 = ⟨σ(0, 0)⟩α +D , +(2.40) +which is just the ratio of mother CFT partition functions: +⟨σ(0, 0)⟩α +D = +Z(α) +DN +� +Z(α) +D +�N , +(2.41) +13 + +where DN denotes the N-th covering of the unit disk with branch points at 0 and 1. As shown +in [76], we can express (2.40) in terms of the ground state degeneracy gα = ⟨0|α⟩ [69] (which is +defined as the overlap between the vacuum state |0⟩ and the boundary state |α⟩ in the mother +BCFT): +A(α) +σ,Ψ1 = g1−N +α +. +(2.42) +Using this result, we can calculate the one-point structure constants of composite twist operators +σi, by using the definition (2.13) and the relation between twist correlators on the disk D and +the mother CFT partition function on DN, which simply gives: +A(α) +σi,Ψ1 = A(α) +σ,Ψ1 Aα +φi , +(2.43) +where Aα +φi is the mother CFT one-point structure constant of φi with conformal boundary +condition α. The proof is relegated to Appendix B.1. +Extending these results to more complicated bulk-boundary structure constants A(α) +σ[k] +i +,Ψj for +generic choices of mother CFT and cyclic group ZN is not usually straightforward and depends +on our knowledge of correlation functions in the mother CFT. For example, in Appendix B.2 +we calculate the structure constant A(α) +σ,Ψ13 in the Z2 orbifold BCFT, since it can be expressed +in terms of a two-point function of boundary operators in the mother CFT. For generic N +and composite twist operator σ[k] +i , knowledge of higher-point correlators in the mother CFT is +required to compute such structure constants through the same unfolding methods. +3 +Differential equations in the Z2 and Z3 orbifold BCFT +3.1 +Setup for the calculations +We consider the case of a generic BCFT, with central charge c. The model is defined on the +upper half plane, with conformal boundary conditions α and β set on the negative ℜ(z) < 0 +and positive ℜ(z) > 0 parts of the real axis, respectively. We will work, for the entirety of +this section, under the assumption that the most relevant BCCO interpolating between these +boundary conditions is ψ(αβ) +12 +, with conformal dimension h12. This implies that the BCCO has +a null-vector at level 2. Of course, our results also apply to the case where the BCCO is ψ(αβ) +21 +, +up to changing g → 1/g. +In the ZN orbifold of this theory, we will consider one-point correlators of generic composite +twist operators σi of twist charge [k = 1], in a background with mixed BC α and β, corresponding +to the replicated boundary conditions of the mother BCFT. The change in boundary conditions +in the orbifold theory will be implemented by the diagonal BCCO Ψ(αβ) +12 +defined in (2.27), +with conformal dimension hΨ12 = Nh12. Since we will aim to compare our CFT results with +lattice data in Section 4, we will define our twist correlator on an infinite strip S of width L, +parametrized by the complex coordinate w = u + iv, with u ∈ [0, L] and v ∈ R. The conformal +boundary conditions on the u = 0 and u = L sides of the strip are set to be α and β, respectively. +We will consider correlators with a twist σi inserted at w = ℓ: +⟨σi(ℓ, ℓ)⟩αβ +S , +(3.1) +where ℓ is measured from the boundary β, in accordance with Figure 1. +14 + +This correlator is now mapped to the upper half plane, through: +w = −iL +π +ln z , +(3.2) +and expressed, using (2.28), as: +⟨σi(z, ¯z)⟩αβ +H = ⟨Ψ(αβ) +12 +(∞) σi(z, ¯z)Ψ(βα) +12 +(0)⟩H , +(3.3) +with z = exp iπℓ/L in terms of strip coordinates. Using the information about the operator +algebra of the orbifold BCFT we have presented in Section 2.3, we can write the following block +expansion for (3.1) +⟨σi(ℓ, ℓ)⟩αβ +S += J +� +ℓ +A(β) +σi,ΨℓB(ββα)Ψ12 +Ψℓ,Ψ12 +�Fℓ(η) , +(3.4) +where η = z/¯z = exp (2πiℓ/L), and J = (L¯z/π)−2hσi is the combined Jacobian associated to +the M¨obius map ζ �→ ζ/¯z that takes (0, z, ¯z, ∞) �→ (0, η, 1, ∞) and the map w �→ z from the +strip to the upper half plane. We recall that the �Fℓ’s are the conformal blocks in the channel +η → 1. +As per Cardy’s doubling argument [72], the functions ˜Fk(η) are four-point conformal blocks +(2.31) with Φ = Φ12. +To proceed, we need to determine the differential equation satisfied by these functions. To +this end, we will use a combination of the null-vector conditions and the orbifold Ward identities +derived in the Appendix. +3.2 +The function ⟨Ψ12 · σ · Ψ12⟩ in a generic Z2 orbifold +Following the general approach described above, the function ⟨Ψ12(∞)σ(z, ¯z)Ψ12(0)⟩H is given, +up to the overall factor ¯z−2hσ, by a linear combination of the conformal blocks +Fk(η) = ⟨Φ12|σ(1)Pkσ(η)|Φ12⟩ . +(3.5) +It turns out that this family of conformal blocks was already studied in [42], for the calculation +of the single-interval R´enyi entropy in the excited state |Φ12⟩ with periodic BC. Let us recall +how the derivation of the corresponding ODE goes. +We use the null-vectors at level two of the untwisted chiral state |Φ12⟩, given by: +� +L(0) +−2 − 1 +2g +� +L(0) +−1 +�2 +− 1 +2g +� +L(1) +−1 +�2� +· |Φ12⟩ ≡ 0, +� +L(1) +−2 − 1 +gL(0) +−1L(1) +−1 +� +· |Φ12⟩ ≡ 0 , +(3.6) +and the null vector at level 1/N of the bare twist operator σ: +L(1) +−1/2 · σ ≡ 0 . +(3.7) +We combine these with the orbifold Ward identity for the chiral correlator: +G(1)(w, η) = ⟨Φ12| σ(1)Pkσ(η)T (1)(w)L(1) +−1 |Φ12⟩ , +(3.8) +15 + +with (m1, m2, m3, m4) = (0, −1/2, −1/2, −1) in the notation of (C.1). This gives, after taking +into account (3.7): +� +p=0,1,2 +dp ⟨Φ12| σ(1)Pkσ(η)L(1) +−p+2L(1) +−1 |Φ12⟩ = 0 , +(3.9) +with dp calculated from the series (C.6). +By substituting the null vectors (2.18–3.7) and employing the identity (C.7), one obtains +the differential equation: +64g2η2(η − 1)2 ∂2 +ηF + 16gη(η − 1) +� +(−14g2 + 23g − 6)η + 2g(1 − 4g) +� +∂ηF ++ (3g − 2) +� ++3(5g − 6)(1 − 2g)2η2 + 12g(1 − 2g)η + 16g2(g − 1) +� +F = 0 , +(3.10) +whose Riemann scheme is given by: +0 +1 +∞ +−2h12 +−2hσ +2hσ − 2h12 +−2h12 + h13/2 +−2hσ + 2h13 +2hσ − 2h12 + h13/2 +This corresponds to the intermediary states {σ, σ13} in the channels η → 0 and η → ∞, and +{1, Φ13} in the channel η → 1. Note that, when the mother CFT is a minimal model Mp,p′, +one can check for various values of (p, p′) that these are exactly the intermediary states allowed +by the orbifold fusion given in F, and that they all have multiplicity one. +To proceed, one can define the shifted function f(η): +f(η) = (1 − η)2hση2h12F(η) , +(3.11) +and substitute in (3.10) to find that f(η) satisfies a second order hypergeometric equation (E.1) +with parameters: +a = 2 − 3g , +b = 3 +2 − 2g , +c = 3 +2 − g . +(3.12) +We can work with the basis (E.3) of solutions around η = 1 for this hypergeometric equation— +in the block expansion (3.4), this corresponds to approaching the twist operator to the boundary +β. Thus, the conformal blocks �Fℓ(η) we seek are: +�F1(η) = (1 − η)−2hση−2h122F1(a, b; a + b − c + 1 | 1 − η) , +�F13(η) = (1 − η)−2hσ+2h13η−2h122F1(c − b, c − a; c − a − b + 1 | 1 − η) , +(3.13) +and they are normalized to be of the form �Fℓ(η) ∼ (1 − η)−2hσ+h(1 + . . . ) as η → 1 , where h +is the conformal dimension of the internal operator of the conformal block in this channel. The +bulk-boundary fusion rules corresponding to the exponents are +σ +��� +β → Ψ1 + Ψ13 , +(3.14) +as σ is approached to the conformal boundary β. Substituting the blocks and the expressions +for the orbifold BCFT structure constants in (3.4) one finds the expression of the one-point +twist correlator +⟨σ(z, ¯z)⟩αβ +H = ¯z−2hσ � +A(β) +σ,Ψ1B(ββα)Ψ12 +Ψ1,Ψ12 +˜F1(η) + A(β) +σ,Ψ13B(ββα)Ψ12 +Ψ13,Ψ12 +˜F13(η) +� +, +(3.15) +16 + +where the various structure constants are expressed in terms of the mother BCFT data as +A(β) +σ,Ψ1 = A(β) +σ,Ψ13 = g−1 +β +, +B(ββα)Ψ12 +Ψ1,Ψ12 += 1 , +B(ββα)Ψ12 +Ψ13,Ψ12 += +� +B(ββα) ψ12 +ψ13ψ12 +�2 +. +(3.16) +It is interesting now to observe that, for some pairs of conformal BCs (α, β), the block +expansion (3.15) greatly simplifies because the boundary-boundary structure constant B(ββα)Ψ12 +Ψ13,Ψ12 +vanishes. At the level of the mother BCFT, this is equivalent to demanding that the operator +ψ(ββ) +13 +is not allowed in the theory. +For BCFTs based on A-series minimal models M(p, p′), this holds for any pair of mixed +conformal BCs (α, β) ≡ (φr2, φr1), labelled by bulk primary fields with 1 ≤ r < p. One can use +well-established results about fusion rules in such models [77], to check that: +φ12 ∈ φr1 × φr2 +(3.17) +so that these BCs are interpolated by ψ(αβ) +12 +and +φ13 ̸∈ φr1 × φr1 +(3.18) +which implies that ψ(ββ) +13 +is not in the boundary operator spectrum of the BCFT. +Under these conditions, the correlator in (3.15) simplifies to: +⟨σ(z, ¯z)⟩αβ +H = ηhσ � +g−1 +φr1 ˜F1(η) +� +, +(3.19) +so that one finds the second R´enyi entropy in the strip setup to be: +S(α,β) +2 +(ℓ) ∼ c +8 log 2L +πa sin +�πℓ +L +� ++ log gφr1 − log +� +η−2h122F1 (2 − 3g, 3/2 − g; 3 − 4g | 1 − η) +� +(3.20) +It is now interesting to check that the theoretical prediction for this kind of mixed BC has +the expected behaviour near the φr1 and φr2 boundaries. It is known [2] that the second R´enyi +entropy of an interval ℓ touching one of the identical boundaries of a finite system of size L is +given by: +S(α,α) +2 +([0, ℓ]) = c +8 log +�2L +π sin +�πℓ +L +�� ++ log gα +(3.21) +We then anticipate that the mixed BC result (3.20) will interpolate between S(φr1,φr1) +2 +and +S(φr2,φr2) +2 +as ℓ → 0 and ℓ → L +Furthermore, suppose we consider the difference in second R´enyi entropies between two +mixed BC setups (φr2, φr1) and (φr′2, φr′1) for the same bulk CFT. We then find the following +universal result: +∆S2 = S(φr′2,φr′1) +2 +− S(φr2,φr1) +2 += log gφr1 +gφr′1 += log gφr2 +gφr′2 +(3.22) +where the latter equation follows from the expression of gφr,s = Sφr,s,φ0/�S1,φ0 [69] in terms of +S-matrix elements of minimal models [77] - here φ0 denotes the field with the lowest conformal +dimension of the diagonal bulk CFT. +This entropy difference is thus determined, in these cases, by the identical BC expressions +(3.21) that describe the asymptotic behaviour near the boundaries of the mixed BC setup. We +illustrate these ideas graphically for the A-series minimal model M(6, 5) in Figure 2, by plotting +the second R´enyi entropies shifted by L2hσ for two mixed BC setups (φ12, φ11) and (φ32, φ31) +for the BCFTs based on M(6, 5), together with the relevant identical BC setups for each case. +17 + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +/L +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +L2h +( , +) +2 +([0, ]) +CFT mixed ( +11, +12) +CFT +11 BC +CFT +12 BC +CFT mixed ( +31, +32) +CFT +31 BC +CFT +32 BC +Figure 2: Shifted second R´enyi entropies for the mixed BC setups (φ12, φ11) and (φ32, φ31) for +the BCFTs based on M(6, 5). The difference between the two curves is constant with respect +to the interval size, and thusly fixed by their respective asymptotic behaviours around ℓ → 0 +and ℓ → L +3.3 +The function ⟨Ψ12 · σh · Ψ12⟩ in a generic Z2 orbifold +From the perspective of critical quantum chains, the result in (3.15) only determines the leading +contribution to the second R´enyi entropy. To understand finite-size corrections to this result, +we should also study the one-point function of subleading primary twist operators, namely +⟨σh(z, ¯z)⟩αβ +H . +We shall derive an ODE for the conformal blocks +Fk(η) = ⟨Φ12|σh(1)Pkσh(η)|Φ12⟩ . +(3.23) +Here we consider the case of a generic composite twist operator σh with conformal dimension +�h = hσ + h +N , +(3.24) +and thus we do not assume any null-vector condition on σh. Besides the null-vector conditions +at level two (3.6), we will need the null-vector at level three in the module of |Φ12⟩: +L(1) +−2L(1) +−1 |Φ12⟩ = +� +−L(0) +−3 + 1 +g +� +2gL(0) +−1L(0) +−2 − +� +L(0) +−1 +�3�� +|Φ12⟩ , +(3.25) +and two Ward identities obtained from: +⟨Φ12| σh(1)Pkσh(η)T (1)(z)L(1) +−1 |Φ12⟩ , +(3.26) +with (m1, m2, m3, m4) = (−1, 1/2, 1/2, −2): +a0|1 ⟨Φ12| L(1) +1 σh(1)Pkσh(η)L(1) +−1 |Φ12⟩ = +3 +� +p=0 +dp|1 ⟨Φ12| σh(1)Pkσh(η)L(1) +−2+pL(1) +−1 |Φ12⟩ +(3.27) +18 + +0 +1 +∞ +−2h12 +−2�hα +2�hα − 2h12 +−2h12 + 1 +2 +−2�hα + h13 +2�hα − 2h12 + 1 +2 +−�hα − 2h12 + �hα+b +−2�hα + 2h13 +�hα − 2h12 + �hα+b +−�hα − 2h12 + �hα−b +−2�hα + 2h13 + 2 +�hα − 2h12 + �hα−b +Table 1: Singular exponents around η = 0, 1, ∞ +and (m1, m2, m3, m4) = (−2, 1/2, 1/2, −1) : +a0|2 ⟨Φ12| L(1) +2 σh(1)Pkσh(x)L(1) +−1 |Φ12⟩ + a1|2 ⟨Φ12| L(1) +1 σh(1)Pkσh(x)L(1) +−1 |Φ12⟩ = += d0|2 ⟨Φ12| σh(1)Pkσh(η)L(1) +−1L(1) +−1 |Φ12⟩ + d1|2 ⟨Φ12| σh(1)Pkσh(η)L(1) +0 L(1) +−1 |Φ12⟩ + ++ d2|2 ⟨Φ12| σh(1)Pkσh(η)L(1) +1 L(1) +−1 |Φ12⟩ +(3.28) +Putting everything together, and applying the change of function +F(η) = η−2h12 (1 − η)4h12−2�h f(η) , +(3.29) +we obtain the fourth-order ODE +(η − 1)4η3 ∂4 +ηf + 1 +2(η − 1)3η2 [(2g + 13)η + (2g − 11)] ∂3 +ηf +− 1 +8(η − 1)2η +� +(16g�h + 6g2 − 45g − 60)η2 + (20g2 + 34g + 96)η + (16g�h + 6g2 + 3g − 36) +� +∂2 +ηf +− g +16(η − 1) +� +(48�h + 18g − 75)η3 + (16g2 − 18g − 112�h + 167)η2 ++(80�h + 16g2 − 74g − 53)η + (−16�h − 6g + 9) +� +∂ηf ++ g +8 +� +(16g2�h + 6g3 − 13g2 + 4)η2 + (−32g2�h + 12g3 + 34g2 − 64g + 24)η ++(24 + 16g2�h + 6g3 − 13g2 + 4) +� +f = 0 . +(3.30) +At this stage, it will be convenient to use the Coulomb-Gas parametrization to analyse the +local exponents of the ODE. Recall the relation between the mother CFT central charge and +the parameter g: +c = 1 − 24Q2 , +Q = 1 +2(1/b − b) , +b = √g . +(3.31) +The conformal dimensions in the mother CFT are parametrized by the vertex charge α as +hα = α(α − 2Q) , +(3.32) +and we use the shorthand notation for the conformal dimension of composite twisted operators +�hα = hσ + hα +N . +(3.33) +In this parametrization, the Riemann scheme for F(η) is given in Table 1. +19 + +These exponents correspond to the intermediary states (counted with their multiplicities): +{1, [1, φ13], Φ13, Φ13} +in the channel η → 1 , +{σh, L(1) +−1/2 · σh, σh′, σh′′} +in the channels η → 0 and η → ∞ . +(3.34) +Here, we have defined h′ = hα+b and h′′ = hα−b. Recall that the conformal blocks are labelled by +primary operators under the neutral subalgebra A, and that L(1) +−1/2 ·σh is one of these operators. +When the mother CFT is a minimal model, one can check on various examples that the orbifold +fusion rules derived in [70] from the Verlinde formula are consistent with these intermediary +states. +While an analytic solution to the differential equation is not known, one can determine the +conformal blocks Fk(η) around η = 0 and �F(η) around η = 1 numerically to arbitrary precision. +Assuming this step has been performed, all that is left is to calculate the structure constants +in the block expansion (3.4). The boundary-boundary structure constants are calculated from +(2.38) and (2.39), while the bulk-boundary structure constants can be calculated analytically +through unfolding, as shown, for some cases in Appendix B.2 and B.1. For numerical studies, +however, it is simpler to bootstrap some of the coefficients in the block expansion (3.4) rather +than to calculate all of them analytically. +To implement this method, as detailed in [78], one needs to compare the block expansions +in (3.4) with the block expansion corresponding to sending the twist field to the part of the +boundary, endowed with the α BC. The crucial point here is that the conformal blocks Fk(η) +are branched functions on C, with branch points {0, 1, ∞}, and, thus, sending the twist field to +the boundary with BC α is equivalent to crossing to the other branch of the function. This is +marked by appending the phase factor e2πi to the variable η to get: +⟨σh(z, ¯z)⟩αβ +S += JSL(2,C) +� +ℓ +A(α) +σh,ΨℓB(ααβ)Ψ12 +Ψℓ,Ψ12 +�Fℓ(e2πiη) . +(3.35) +To proceed, one needs to find the monodromy matrix X around zero for the basis �Fℓ(η), which +encodes the behaviour of the conformal blocks as the branch cut is crossed: +�Fℓ(e2πiη) = +� +m +Xℓm �Fm(η) . +(3.36) +Since the monodromy of the blocks �Fℓ around zero is non-diagonal, we can use the fusing +matrix Pij to express the blocks �Fℓ in terms of a basis of the blocks Fk, which have diagonal +monodromy around η = 0: +�Fℓ(η) = +� +k +PℓkFk(η) . +(3.37) +The blocks Fk(η) simply acquire a phase under z → e2πiz, so their monodromy matrix Y is +diagonal: +Fk(e2iπη) = +� +j +Ykj Fj(η) , +Ykj = δkj exp +� +2πi +� +−�hα − 2h12 + �hk +�� +, +(3.38) +where the exponents in the exponential above, are simply read off from the Riemann scheme. +Then, the monodromy matrix of the blocks �Fℓ(η) is found from the matrix product: +X = P · Y · P −1 , +(3.39) +20 + +which allows us to compare the block expansions in (3.4) and (3.35) to find a duality relation, +of the type presented in [78]: +A(β) +σh,ΨiB(ββα)Ψi +Ψ12,Ψ12 = +� +j +A(α) +σh,ΨjB(ααβ)Ψj +Ψ12,Ψ12 Xji . +(3.40) +Using the numerical determinations for Fk(η) and �Fℓ(η) , one can find a good estimate +for the fusing matrix Pij, and, consequently, Xij. +A more fleshed out example of how the +determination of Pij works has been relegated to the Appendix H, where this equation is used +for the case of the Z2 orbifold of the three-state Potts model BCFT. +After solving the linear system in (3.40) one can evaluate the unknown structure constants +A(β) +σh,Ψi and A(α) +σh,Ψi. At this point, we stress that (3.40) gives, at most, four constraints between +the unknown structure constants. To fully determine all these quantities, one should calculate +the remaining four structure constants through other methods. +3.4 +The function ⟨Ψ12 · σ · Ψ12⟩ in a generic Z3 orbifold +Here the relevant conformal blocks are +Fk(η) = ⟨Φ12|σ(1)Pkσ†(η)|Φ12⟩ . +(3.41) +We give the null-vectors of |Φ12⟩ at levels two and three: +L(r) +−2 |Φ12⟩ = 1 +3g +2 +� +s=0 +L(r−s) +−1 +L(s) +−1 |Φ12⟩ , +(3.42) +L(3−r) +−1 +L(r) +−2 |Φ12⟩ = 1 +3g +� +2L(0) +−1L(1) +−1L(2) +−1 + +� +L(3−r) +−1 +�3� +|Φ12⟩ , +(3.43) +for r ∈ {0, 1, 2}. We will also need the null-vectors for the out-state ⟨Φ12|, which can be obtained +by Hermitian conjugation (2.6). +To derive an ODE for the conformal blocks, we had to employ seven orbifold Ward identities, +together with six of the null-vector conditions above. To not overload the presentation of this +section with technical details, we relegate the specifics of the derivation to Appendix G. We +apply the change of function +F(η) = η−8h12/3 (1 − η)16h12/3−2hσ f(η) . +(3.44) +The function f satisfies the ODE +(η − 1)3η2 ∂3 +ηf + (η − 1)2η [(g + 3)η + (g − 3)] ∂2 +ηf ++ 2 +9(η − 1) +� +2(2 + 3g)η2 − 2(7 − 15g + 18g2)η + (4 − 3g) +� +∂ηf ++ 4 +27(1 − 6g)(2 − 3g)(η + 1) f = 0 . +(3.45) +The Riemann scheme for F is +0 +1 +∞ +− 8 +3h12 +−2hσ +2hσ − 8 +3h12 +− 8 +3h12 + 1 +3 +−2hσ + 2h13 +2hσ − 8 +3h12 + 1 +3 +−3h12 + h14 +3 +−2hσ + 3h13 +2hσ − 3h12 + h14 +3 +21 + +The local exponents correspond to the intermediary states: +{1, [1, φ13, φ13], Φ13} +in the channel η → 1 , +{σ12, L(1) +−1/3 · σ12, σ14} +in the channels η → 0 and η → ∞ . +(3.46) +In the orbifold BCFT, this translates into the following fusion rules for the twist operator with +the boundary β: +σ1 +����� +β +→ Ψ(ββ) +1 ++ Ψ(ββ) +[1,φ13,φ13] + Ψ(ββ) +13 +. +(3.47) +The analytic solutions to the differential equation (3.45) are not known, but they can be +evaluated numerically, to arbitrary precision. Then, one can use the bootstrap to determine +some relations between the unknown structure constants in the expansion (3.4), as outlined in +the previous section, and determine the rest analytically, by unfolding methods, to complete +the calculation of the mixed BC correlator of the bare twist. +We note that a fourth order differential equation that the correlator (2.28) satisfies has +already been found in [75], where it plays a role in the determination of the leading contribution +to the third R´enyi entropy of an excited state in a periodic 1D critical chain. As predicted in [75], +there is no degeneracy in the exponents in the more constraining third order differential equation +we have found here. Note that these exponents are the ones expected from the orbifold fusion +rules [70]. +3.5 +The function ⟨Ψ12 · σ13 · Ψ12⟩ in the Z3 orbifold of the Ising model +In this section, we will work with the Z3 orbifold of the Ising BCFT. The bulk primary fields +of this BCFT are φ11 ≡ 1, φ12 ≡ s and φ13 ≡ ε with hs = 1/16 and hε = 1/2. We will keep +labelling the fields by their Kac indices, to not overcomplicate the notation. +We will provide here an alternative method for finding a differential equation for the one- +point function: +⟨σ13(z, ¯z)⟩f+ +H , +(3.48) +where the orbifold conformal boundary conditions α = f and β = + correspond to setting fixed +and free BC respectively, on all the copies of the Ising mother BCFT. The diagonal BCCO +Ψ(f+) +12 +is the one interpolating between them in the orbifold, since in the Ising BCFT only the +ψ(f+) +12 +primary boundary field can change between the CBCs (+) ↔ (f) [78]. As in the previous +sections, we aim to find a differential equation satisfied by the conformal blocks +Fk(η) = ⟨Φ12|σ13(1)Pkσ† +13(η)|Φ12⟩ . +(3.49) +First, we use the fusion numbers in (F.1) to infer the dimension of the space of conformal blocks +for (3.48): +� +i +N i +σ13,σ† +13N Φ12 +i,Φ12 = 2 , +(3.50) +which means the differential equation we seek should be second order. +By using the null-vectors induced on σ13 together with the right combination of Ward +identities, one should be able to rigorously derive it. Instead, we will assume this equation +exists and is of Fuchsian type – a linear homogenous ODE whose three singular points are +22 + +regular. The latter assumption is based on the observation that the method exploited in the +previous sections relies on expressing orbifold modes L(r̸=0) +m +in terms of Virasoro generators, +whose combined differential action on correlators is well-understood in the literature [77], [79] +to be of Fuchsian type. +Now, using the fusion numbers of (F.1), the fusion rules are +σ13 × Φ12 → σ12 + L(2) +−2/3 · σ12 , +σ13 × σ† +13 → 1 + [1, φ13, φ13] , +(3.51) +so we can determine the asymptotic behaviour of the solutions around the regular singular +points η ∈ {0, 1, ∞} of the differential equation and infer the Riemann scheme: +0 +1 +∞ +−hσ13 − 3h12 + hσ12 +−2hσ13 +hσ13 − 3h12 + hσ12 +−hσ13 − 3h12 + hσ12 + 2 +3 +−2hσ13 + 2h13 +hσ13 − 3h12 + hσ12 + 2 +3 +One can readily check that the entries of this Riemann scheme sum up to one, so, by virtue +of a general theorem on Fuchsian ODEs (see [80]), there is a unique second-order Fuchsian ODE +with this set of singular exponents. If we define the shifted function f(η): +f(η) = ηhσ13+3h12−hσ12(1 − η)2hσ13F(η) , +(3.52) +we find, by the same considerations, that it should satisfy a second-order Fuchsian differential +equation with the Riemann scheme +0 +1 +∞ +0 +0 +−2hσ13 − 6h12 + 2hσ12 +2 +3 +2h13 +−2hσ13 − 6h12 + 2hσ12 + 2 +3 +This is just the canonical Riemann scheme of a hypergeometric differential equation (E.1), with +coefficients: +a = −2hσ13 − 6h12 + 2hσ12 = −2/3 , +b = −2hσ13 − 6h12 + 2hσ12 + 2 +3 = 0 , +c = 1 +3 , +(3.53) +in the conventions of Appendix E. We notice that the exponents in the η → 1 channel are +spaced by one, so we will have to deal with the degenerate exponents to arrive at a closed form +solution. To do this, we will use the basis of solutions in the η → 0 channel – given in (E.2) – +to construct a linearly independent basis of solutions around η → 1. The solutions for f can be +simplified, in this case, to: +I1(η) = 1 , +I2(η) = η2/3 , +(3.54) +which gives the conformal blocks around η → 0: +F1(η) = η−1/3(1 − η)−4/9 , +F2(η) = η1/3(1 − η)−4/9 . +(3.55) +in our normalisation convention. +23 + +To build the basis of solutions around η = 1, we look for the linear combinations ˜Fi(η) = +� +j P −1 +ij Fj(η) that have the following series expansion around η = 1 : +˜F1(η) = (1 − η)−4/9 � +1 + O[(1 − η)2] +� +, +˜F2(η) ∼ (1 − η)5/9 , +(3.56) +since the power series associated to the orbifold identity should have no (1 − η) term due to the +null-vectors L(r) +−1·1 ≡ 0, and the both solutions should have the leading coefficient normalised to +one, in our convention for the conformal blocks. With these requirements, one finds the fusing +matrix P −1 +ij +to be: +P −1 = 1 +2 +� 1 +1 +3 +−3 +� +. +(3.57) +Thus, the conformal blocks of (3.48) around η = 1 are found to be: +˜F1(η) = η−1/3 + η1/3 +2(1 − η)4/9 , +˜F2(η) = 3(η−1/3 − η1/3) +2(1 − η)4/9 +. +(3.58) +For the physical correlation function, we write +⟨σ13(z, ¯z)⟩f+ +H += ¯z−2hσ1,3 +� +A(+) +σ13,Ψ1B(++f)Ψ12 +Ψ1,Ψ12 +˜F1(η) + A(+) +σ13,[ψ1,ψ13,ψ13]B(++f)Ψ12 +[ψ1,ψ13,ψ13],Ψ12 ˜F2(η) +� +. +(3.59) +Finally, we observe that B(++f)ψ12 +ψ13ψ12 +vanishes, and hence B(++f)Ψ12 +[ψ1,ψ13,ψ13],Ψ12 = 0, so the expression +(3.59) simplifies to: +⟨σ13(z, ¯z)⟩f+ +H += 25/9 × cos(2θ/3) +(r sin θ)4/9 , +z = reiθ +(3.60) +where we have also used: +A(+) +σ13,Ψ1 = g−2 ++ , +B(++f)Ψ12 +Ψ1,Ψ12 += 1 , +(3.61) +and the value of the ground-state degeneracy for fixed BC g+ = 1/ +√ +2 in the Ising BCFT [69]. +3.6 +More hypergeometric differential equations in the Ising cyclic orbifold +BCFTs +We have managed, in Sections 3.3 and 3.4 to obtain differential equations for cyclic orbifolds of +generic mother BCFTs, but have not been able to provide analytic solutions for them. +One can, however, find second order differential equations for particular choices of Mp,p′ and +composite twist fields (for the correlators of Section 3.3), in the manner presented in Section 3.5, +which allow us to exactly determine the correlators. Since we want to compare the results of this +section with lattice data of the critical Ising spin chain with mixed BC, it will be particularly +satisfying to find such equations for the cyclic orbifolds of the Ising BCFT. +Let’s first consider the correlator: +⟨σ1,3(z, ¯z)⟩αβ +N=2 +(3.62) +in the Z2 Ising orbifold BCFT which should satisfy, up to a M¨obius map, the same differential +equation as: +⟨Φ1,2| σ1,3(1)σ1,3(η) |Φ1,2⟩ +(3.63) +24 + +The orbifold fusion rules of [40], imply that the space of conformal blocks is two-dimensional +since: +� +i +N i +σ1,3,σ1,3N Φ1,2 +i,Φ1,2 = 2 +(3.64) +By the same type of arguments and assumptions as in Section 3.5, we infer that (3.63) satisfies +a second order Fuchsian differential equation with the following Riemann scheme: +0 +1 +∞ +−hσ1,3 − 2h1,2 + hσ1 +−2hσ1,3 +hσ1,3 − 2h1,2 + hσ1,1 +−hσ1,3 − 2h1,2 + hσ1,3 + 1 +2 +−2hσ1,3 + 2h1,3 +hσ1,3 − 2h1,2 + hσ1,2 + 1 +2 +so that we eventually find the one-point twist correlator to be +⟨σ1,3(z, ¯z)⟩αβ +(N=2) = ¯z−2hσ1,3g−1 ++ ˜FN=2 +Ψ1 +(η) +(3.65) +with +˜F(N=2) +Ψ1 +(η) = +1 + η3/4 +2(1 − η)9/16η3/8 +(3.66) +Finally, we can find an exact expression for the bare twist correlator: +⟨σ1(z, ¯z)⟩αβ +N=3 +(3.67) +in the Z3 Ising orbifold BCFT since it also satisfies a second order differential equation with +Riemann scheme: +0 +1 +∞ +−hσ1 − 3h1,2 + hσ1,2 +−2hσ1 +hσ1 − 3h1,2+hσ1,2 +−hσ1 − 3h1,2 + hσ1,2 + 1 +3 +−2hσ1 +2h1,3 +hσ1 − 3hΨ1,2+hσ1,2 + 1 +3 +We find: +⟨σ1(z, ¯z)⟩αβ +N=3 = ¯z−1/9g−2 ++ ˜FN=3 +Ψ1 +(η) +(3.68) +with +˜FN=3 +Ψ1 +(η) = +1 + η1/3 +2(1 − η)1/9η1/6 +(3.69) +Other results for the Ising BCFT. +We have also obtained results specific to the Z2 and +Z3 orbifolds of the Ising BCFT with fixed mixed BC with α = + and β = −, for which the +most relevant primary BCCO is ψ(+−) +2,1 +. Since these results are not based on deriving differential +equations, it felt thematically appropriate to leave their presentation for the Appendix D. +4 +Numerical checks and finite-size corrections in quantum chains +To provide an independent appraisal of the validity of our CFT results, we have performed a +numerical analysis on the Ising and three-state Potts open quantum chains for different settings +of mixed BC. Once finite-size effects are properly accounted for, the validity of the CFT results +becomes apparent. +25 + +We should note that the R´enyi entropies in the Ising case have already been obtained, for +generic N in the work of [57], through a different approach. +We found that our analytical +calculations (for N = 2, 3) are compatible with their results. +Furthermore, by studying the finite-size corrections to their result, we manage to quanti- +tatively understand the deviation of the chain data from the leading CFT prediction in the +DMRG numerical analysis of [57], even for relatively large system sizes M ∼ 102. Thus, when +the subleading CFT contribution to the R´enyi entropy is taken into account, as our analysis +shall show, the agreement with the lattice data is excellent, even for the small system sizes +M ∼ 26 accessible to exact diagonalization. +4.1 +The Ising quantum chain with mixed BC +The Hamiltonian of the Ising quantum chain with open BC, describing M spins with generic +BC at the boundary, is given by: +Hαβ = − +M−1 +� +j=1 +sz +jsz +j+1 − h +M +� +j=1 +sx +j − hαsz +1 − hβsz +M , +(4.1) +where sx,y,z +j +denote Pauli spin operators acting non-trivially at site j, and as identity at all the +other sites. We denote the lattice spacing by a, so that the length of the chain is L = Ma. The +parameters hα, hβ denote external fields (in the z direction) acting at the boundary sites j = 1 +and j = M. The ground state of this Hamiltonian is then found by exact diagonalization (ED) +for system sizes M ≤ 26, and from it, the R´enyi entropies are extracted. +To take the scaling limit of the critical chain, we send M → ∞, a → 0 while keeping L fixed. +In this limit, criticality is achieved in the bulk for h = 1, while each boundary admits three +critical points hα, hβ ∈ {0, ±∞}. +From a CFT perspective, the scaling limit of the critical Ising chain with open boundaries +is very well understood. +It is described by the BCFT with central charge c = 1/2 and a +bulk operator spectrum consisting of three primary operators – the identity 1, energy ε and +spin operators s – and their descendants [77]. The three boundary critical points correspond +to the three conformal boundary conditions for the Ising BCFT, which, in the framework of +radial quantization on the annulus, allow the construction of the following physical boundary +states [53,77]: +|f⟩ = |1⟩⟩ − |ϵ⟩⟩ +(free BC) , +(4.2) +|±⟩ = +1 +√ +2|1⟩⟩ + 1 +√ +2|ϵ⟩⟩ ± +1 +21/4 |s⟩⟩ +(fixed BC) , +(4.3) +where |i⟩⟩ denotes the Ishibashi state [53] [81] corresponding to the primary operator i. The +physical boundary states |α⟩ are in one-to-one correspondence with the primary fields of the bulk +CFT 2: |f⟩ ↔ s and |±⟩ ↔ 1/ε. The boundary fields that interpolate between two conformal +BCs can be inferred from this correspondence, as shown in [53], [78]. Thus, the spectrum of +primary boundary fields ψ(αβ) +i +of the Ising BCFT is the one of Table 2. +On the discrete side, we are calculating the one-point correlator of the lattice twist operator +�σ(m, n), where (m, n) are square-lattice coordinates. In the scaling limit with a → 0, �σ(m, n) +2This statement is strictly true if the bulk CFT is diagonal, see [82] for a detailed discussion. +26 + +(αβ) ++ +− +f ++ +ψ1 +ψε +ψs +− +ψε +ψ1 +ψs +f +ψs +ψs +ψ1, ψε +Table 2: Boundary operator spectrum of the Ising BCFT +admits a local expansion into scaling operators of the corresponding orbifold CFT. The two +most relevant terms in this expansion are: +�σ(m, n) = A a2hσσ1(w, ¯w) + B a2hσεσε(w, ¯w) + less relevant terms , +(4.4) +with the composite twist operator σε defined in (2.13) and hσε = hσ + hε/N. The integers +(m, n) parametrize the lattice, and they are related to the continuum coordinate on the strip +as w = (m + in)a, ¯w = (m − in)a. We can take advantage of the translation invariance in the +n direction to fix the ”time” coordinate of the lattice twist operators to be n = 0. We will then +denote their continuum coordinate by ℓ = ma. +The amplitudes A and B in (4.4) are not universal quantities, so we cannot determine them +by CFT techniques. However, they are also independent of the global properties of the system +(e.g. choice of BC) so they can be found from a numerical analysis of the infinite Ising chain. +Here one can employ the free fermion techniques of [1] and the well-known analytical results +for the R´enyi entropy of an interval in an infinite system [50,2] to fit for the values of A and B, +with great accuracy. +We can now express the lattice one-point twist correlator with generic mixed BC as an +expansion of CFT correlators: +⟨�σ(m, 0)⟩αβ = Aa2hσ⟨σ(ℓ, ℓ)⟩αβ +SL + Ba2hσε⟨σε(ℓ, ℓ)⟩αβ +SL + . . . +(4.5) +Using the map (3.2), we can make the dependence on system size in (4.5) explicit: +⟨�σ(m, 0)⟩αβ = A +�M +π +�−2hσ +⟨σ(z, ¯z)⟩αβ +H + B +�M +π +�−2hσε +⟨σε(z, ¯z)⟩αβ +H + . . . +(4.6) +where z = exp(iπℓ/L), ¯z = exp(−iπℓ/L). In our computational setup, the system sizes accessi- +ble through exact diagonalization are limited to M ≤ 26 and, since twist operators are placed +between lattice sites, we have only considered even system sizes. +With system sizes of this order of magnitude, finite-size corrections are quite strong. The +most relevant corrections we have found arise from the subleading scaling of the lattice twist +operator, given in equation (4.6). The relative scaling of the subleading term with respect to +the leading one is O +� +M−2hϵ/N� +. Since we do not have access, numerically, to system sizes large +enough to suppress these corrections, we had to take into account the first two terms in the +expansion of (4.6) to find a good agreement with the lattice data. Furthermore, as the work +of [57] suggests, the finite-size effects are still important, even at the much larger system sizes +M ∼ 100 accessible through DMRG methods. We mention that such subleading contributions +to the lattice twist operator, which have been identified here from the operator spectrum of the +Z2 cyclic orbifold, have previously been understood, through the path integral formalism on the +corresponding replicated surface, under the name of “unusual corrections” [83,84]. +We give now the results in the Z2 orbifold for the correlators appearing in the expansion +(4.5), for mixed fixed BC with α = +, β = − (calculated in Appendix D) and mixed free-fixed +27 + +BC with α = + and β = f: +⟨σ1(ℓ, ℓ)⟩+− +SL = 2−5/2 +�2L +π +�−1/16 7 + cos 2πℓ +L +� +sin πℓ +L +�1/16 , +⟨σε(ℓ, ℓ)⟩+− +SL = 2−5/2 +�2L +π +�−9/16 1 − 9 cos 2πℓ +L +� +sin πℓ +L +�9/16 , +⟨σ1(ℓ, ℓ)⟩+f +SL = 21/2 +�2L +π +�−1/16 +cos πℓ +4L +� +sin πℓ +L +�1/16 , +⟨σε(ℓ, ℓ)⟩+f +SL = −21/2 +�2L +π +�−9/16 +cos 3πℓ +4L +� +sin πℓ +L +�9/16 , +(4.7) +where the interval ℓ starts at the α = + boundary. The expressions for the bare twist correlators +are in accord with the equivalent results obtained in [57]. +With this mention, we show in Figure 3 the remarkable agreement between our CFT calcu- +lations for the two terms contributing to the second R´enyi entropy Sαβ +2 += − log⟨�σ(m, 0)⟩(αβ) of +the interval [0, m] on the lattice, and the numerical results for the critical Ising chain from the +exact diagonalization of the Hamiltonian. Figure 3a illustrates the case of different (±) fixed +BC on the two sides of the chain, while Figure 3b corresponds to letting the m = 0 site free, +and applying a magnetic field at the boundary site m = M − 1. +To illustrate the large amplitude of finite-size effects, we show in Figure 4 how the CFT +prediction fares against the lattice results with and without the incorporation of the subleading +term. Even for the curve including both subleading and leading terms in (4.6), the agreement +with lattice data is not perfect close to the boundary. This can be traced to the presence of +corrections from descendants of twist operators, which introduce terms of O(M−hϵ−1) relative +to the bare twist contribution. +We can repeat the same kind of analysis for the third R´enyi entropy, related to the Z3- +orbifold one-point function by Sαβ +3 += − 1 +2 log⟨�σ(m, 0)⟩(αβ). The Ising orbifold correlators in this +case are given by: +⟨σ1(ℓ, ℓ)⟩+− +SL = 3−2 +�2L +π +�−1/9 7 + 2 cos 2πℓ +L +� +sin πℓ +L +�1/9 , +⟨σε(ℓ, ℓ)⟩+− +SL = 3−2 +�2L +π +�−4/9 1 + 8 cos 2πℓ +L +� +sin πℓ +L +�4/9 , +⟨σ1(ℓ, ℓ)⟩+f +SL = 2 +�2L +π +�−1/9 +cos πℓ +3L +� +sin πℓ +L +�1/9 , +⟨σ1(ℓ, ℓ)⟩+f +SL = 21/9 +�2L +π +�−4/9 +cos 2πℓ +3L +� +sin πℓ +L +�4/9 . +(4.8) +In Figure 5, we once again compare our CFT calculations (including both the leading and +subleading term) with the critical chain results for the third R´enyi entropy Sαβ +3 , to good agree- +ment for mixed fixed BC (Fig. 5a) and mixed free fixed BC (Fig. 5b). As for the Z2 results, +including the CFT subleading contribution to Sαβ +3 +is necessary to find a satisfying match with +the lattice results. Further finite-size corrections in this case decay as O +� +M− 2hε +3 −1� +. +28 + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +m/M +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +S+ +2 +([m/M]) +CFT +M=26 sites +(a) Fixed mixed BC +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +m/M +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +S+f +2 ([m/M]) +CFT +M=26 sites +(b) Fixed-free mixed BC +Figure 3: Plots of the second R´enyi entropy Sαβ +2 ([m/M]) in the critical Ising chain with two +types of mixed BC for a chain of size M = 26. The interval is grown from the β = + boundary. +As advertised in the beginning of the section, our results for the bare twist correlators (for all +configurations of mixed BC) are compatible with the ones of [57]. The subleading contribution +to the R´enyi entropies from the excited twist correlator is largely responsible for the mismatch +between the lattice and CFT data in the aforementioned article. Finite-size corrections of this +magnitude can be suppressed only with much larger system sizes M ∼ 103, as the authors of +the present work have shown in [60] . +4.2 +The three-state Potts quantum chain with mixed BC +A natural extension of the Ising chain, the three-state Potts model allows the spins at each site +to take one of three possible values {R, G, B}, which we can also conveniently parametrize by +third roots of unity {1, ω, ω2}, with ω = exp(2πi/3). The Hamiltonian of the three-state Potts +29 + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +m/M +0.10 +0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +S+f +2 ([m/M]) +CFT leading +CFT leading+subleading +M=26 sites +Figure 4: Comparison of the second R´enyi entropy in the critical Ising chain of size M = 26 +with mixed free fixed BC with CFT results. Inclusion of the subleading term in the expansion +4.6 is crucial for obtaining a satisfying agreement with lattice data +model, tuned to its bulk critical point [85] [69], [86] is given by: +Hαβ = −ζ +� +� +M−1 +� +j=1 +� +ZjZ† +j+1 + Z† +jZj+1 +� ++ +M−1 +� +j=2 +� +Xj + X† +j +� ++ H(α) +1 ++ H(β) +M +� +� , +(4.9) +where ζ = +√ +3 +2π3/2 is the conformal normalization factor [86] and the operators Zj and Xj act at +site j as: +Z = +� +� +1 +0 +0 +0 +ω +0 +0 +0 +ω2 +� +� , +X = +� +� +0 +1 +0 +0 +0 +1 +1 +0 +0 +� +� . +(4.10) +The terms H(α) +1 +and H(β) +M set the BCs at the ends of the chain. For the purpose of this analysis, +we will set fixed BC of type R at site 1 and restricted boundary conditions of type {G, B} at +site M – the spin at site M is forbidden from taking the value R. This is implemented through +the boundary terms: +H(R) = h +� +� +−1 +0 +0 +0 +0 +0 +0 +0 +0 +� +� , +H(GB) = h +� +� +1 +0 +0 +0 +0 +−1 +0 +−1 +0 +� +� , +(4.11) +The critical points of interest for the boundaries correspond to h = +∞. +However, for +any h > 0, the boundaries will flow towards the same critical points, up to irrelevant boundary +perturbations [87] . These are typically inconsequential for h a large positive value. Furthermore, +in our numerical analysis we can, in fact, implement |h| → ∞ by restricting the local Hilbert +spaces of the boundary sites to exclude the {G, B} and {R} configurations on the left and, +respectively, right boundary. +The scaling limit M → ∞, a → 0 (with L = Ma fixed) of this critical chain is also well +understood. It is given by the D-series BCFT M6,5 with central charge c = 4/5 and a bulk +30 + +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +m/M +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +0.200 +S+ +3 +([m/M]) +CFT +M=26 sites +(a) Fixed mixed BC +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +m/M +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +S+f +3 ([m/M]) +CFT +M=26 sites +(b) Fixed-free mixed BC +Figure 5: Plots of the third R´enyi entropy Sαβ +3 ([m/M]) in the critical Ising chain with two types +of mixed BC for a chain of size M = 26. The interval is grown from the α = + boundary. +primary operator spectrum that contains the scalar fields given in Table 3 as well as the non- +diagonal fields {φ2/5,7/5, φ7/5,2/5, φ3,0, φ0,3} whose labels indicate their respective holomorphic +and antiholomorphic conformal dimensions. One can, as shown in Table 1, assign a Z3 charge +to the scalar fields, and their respective conformal families, that is consistent with the fusion +rules between them. The † in Table 3 is, thusly, used to differentiate the fields with the same +conformal dimension, but different Z3 charge. +In the scaling limit, the fixed and restricted boundary critical points will correspond, natu- +31 + +Diagonal fields +(h, ¯h) +Z3 charge +1 +(0, 0) +0 +ε ≡ φ1,2 +( 2 +5, 2 +5) +0 +φ1,3 +( 7 +5, 7 +5) +0 +φ1,4 +(3, 3) +0 +s, s† ≡ φ3,3 +( 1 +15, 1 +15) +± 1 +ψ, ψ† ≡ φ3,4 +( 2 +3, 2 +3) +± 1 +Table 3: Spectrum of spinless primary operators in the three-state Potts CFT +rally, to the fixed and restricted3 conformal boundary states [53,88]. +|1⟩ = N[(|1⟩⟩ + |ψ⟩⟩ + |ψ†⟩⟩) + λ(|ϵ⟩⟩ + |s⟩⟩ + |s†⟩⟩)] +(fixed R) +|ψ⟩ = N[(|1⟩⟩ + ω|ψ⟩⟩ + ¯ω|ψ†⟩⟩) + λ(|ϵ⟩⟩ + ω|s⟩⟩ + ¯ω|s†⟩⟩)] +(fixed G) +|ψ†⟩ = N[(|1⟩⟩ + ¯ω|ψ⟩⟩ + ω|ψ†⟩⟩) + λ(|ϵ⟩⟩ + ¯ω|s⟩⟩ + ω|s†⟩⟩)] +(fixed B) +|ε⟩ = N[λ2(|1⟩⟩ + |ψ⟩⟩ + |ψ†⟩⟩) − λ−1(|ϵ⟩⟩ + |s⟩⟩ + |s†⟩⟩)] +(restricted GB) +|s⟩ = N[λ2(|1⟩⟩ + ω|ψ⟩⟩ + ¯ω|ψ†⟩⟩) − λ−1(|ϵ⟩⟩ + ω|s⟩⟩ + ¯ω|s†⟩⟩)] +(restricted RB) +|s†⟩ = N[λ2(|1⟩⟩ + ¯ω|ψ⟩⟩ + ω|ψ†⟩⟩) − λ−1(|ϵ⟩⟩ + ¯ω|s⟩⟩ + ω|s†⟩⟩)] +(restricted RG) , +(4.12) +where +N = +� +2 +√ +15 sin π +5 , +λ = +� +sin(2π/5) +sin(π/5) , +(4.13) +and the |i⟩⟩’s are the Ishibashi states defined in [53]. +These conformal boundary states are +labelled by the primary fields of Table 3. +Due to the Z3 symmetry of our model, we have some freedom to set which conformal bound- +ary state corresponds to the fixed boundary condition R in the chain. However, this uniquely +determines the CFT boundary state that corresponds to the restricted boundary conditions GB. +This can be understood by considering the spectrum of boundary fields that can interpolate +between these conformal BC [56], and ensuring the results are consistent with the underlying Z3 +symmetry . In our case, choosing fixed R ↔ |1⟩ forces us to assign restricted GB ↔ |ε⟩. The +most relevant boundary field interpolating between these BCs is ψ(R,GB) +1,2 +[56] with conformal +dimension hε = 2/5. +We will now compare the quantum chain data for the second R´enyi entropy in the critical +Potts chain with our correlator calculations in the Z2 orbifold of the BCFT defined above. Our +analysis will parallel the one for the Ising critical chain. We first hypothesize the form of the +local expansion (4.4) of the lattice twist operator ˆσm,n in the case of the three-state Potts model: +�σ(m, n) = A a2hσσ(w, ¯w) + B a2hσεσε(w, ¯w) + less relevant terms , +(4.14) +where hσ = 1/20, and the composite twist operator σε is built with the energy operator ε of +the Potts model so that hσε = 1/4. +Once again, we numerically estimated the parameters +A, B by a simple analysis on the critical three-state Potts critical chain with periodic boundary +conditions. Following this, the one-point lattice twist correlator with our choice of mixed BC +can be calculated from: +⟨�σ(m, 0)⟩(GB,R) = A +�M +π +�−2hσ1 +⟨σ(z, ¯z)⟩(GB,R) +H ++ B +�M +π +�−2hσε +⟨σε(z, ¯z)⟩(GB,R) +H ++ . . . +(4.15) +3In [53] they are referred to as ”mixed” BC. +32 + +The correlators in (4.15) satisfy the second order (3.10) and fourth order (3.30) ODEs with +g = 6/5. While the solutions to equation (3.10) are known exactly (3.13), one needs to solve +(3.30) numerically to find the conformal blocks in the expansion (3.4) of the excited twist +correlator ⟨σε(z, ¯z)⟩αβ +H . This is done by a standard numerical implementation of the Frobenius +method, whose details we leave for Appendix H. +As in the case of the Ising BCFT, not all the solutions of these differential equations are +needed to build the twist field correlators in (4.15). +Crucially, we note that in the three- +state Potts mother BCFT, there is no boundary operator ψ(RR) +7/5 +living on the fixed conformal +boundary of type R [56]. At the level of the operator algebra this translates into the vanishing +of the boundary-boundary structure constants B(R|R|GB),ψ1,2 +ψ7/5,ψ1,2 +as we have checked using the +results of [82]. This implies, through the relations between mother BCFT and orbifold structure +constants derived in Section 2.3, the vanishing of some of the coefficients in the block expansions +(3.4) of the correlators in (4.15). In effect, only the block corresponding to the identity operator +contributes to these expressions when the twist fields are sent to the β boundary. It corresponds +to the following fusion rules for σ1, σε: +σ1 +����� +β +→ Ψ(ββ) +1 +σε +����� +β +→ Ψ(ββ) +1 +(4.16) +Thusly, we are led to obtain expressions for the bare twist and excited twist correlators on the +UHP: +⟨σ(z, ¯z)⟩αβ = ¯z−2hσA(β) +σ,Ψ1B(ββα)Ψ12 +Ψ1,Ψ12 +˜F1(η) , +⟨σε(z, ¯z)⟩αβ = ¯z−2hσεA(β) +σε,Ψ1B(ββα)Ψ12 +Ψ1,Ψ12 +˜F(ε) +1 (η) , +(4.17) +where ˜F1(η) is given in (3.13), so in this case we can write an explicit result for the bare twist +correlator on the UHP: +⟨σ(z, ¯z)⟩αβ = A(β) +σ,Ψ1B(ββα)Ψ12 +Ψ1,Ψ12 +(1 − η)−2hση−2h12+hσ 2F1 (−8/5, −9/10; −9/5 | 1 − η) +(4.18) +For the excited twist correlator we have: +˜F(ε) +1 (η) = J1(u(η)) = (1 − u)−2hσε +∞ +� +n=0 +an(1 − u)n , +(4.19) +with the coefficients determined by the recursion relation (H.8), derived in Appendix H. +The structure constants can be expressed in terms of known quantities for the M(6, 5) +BCFT, also obtained in Appendix B: +A(β) +σε,Ψ1 = g−1 +R AR +ε , +A(β) +σ,Ψ1 = g−1 +R , +B(ββα)Ψ12 +Ψ1,Ψ12 += 1 , +(4.20) +where the ground state degeneracies gR and gGB have been found in [69]: +gR = +� +5 − +√ +5 +30 +� 1 +4 +gGB = gRλ2 +(4.21) +and the bulk-boundary structure constant AR +ε has been calculated in [88,89] to be: +AR +ε = +� +1 + +√ +5 +2 +� 3 +2 +. +(4.22) +33 + +Putting everything together, we can finally compare the lattice prediction for the second R´enyi +entropy S(R,GB) +2 += − log⟨�σ(m, 0)⟩(R,GB) with our analytic results in Figure 6. While the CFT +prediction does not satisfyingly match the lattice data at all points, we observe that the inclusion +of the subleading term gives an analytic curve that is closer to the lattice data. However, it is +not enough to make up for the severe finite-size effects. +Firstly, due to the operator content of the D-series M6,5 CFT, we expect the higher order +corrections in 4.15 to have a slower power law decay than in the case of the Ising CFT. We +conjecture that the next-to-subleading contribution to (4.15) will decay as ∼ M−2(hσε+1/2). +These corrections, we believe, arise from the combined contribution of the ⟨σφ1,3(w, ¯w)⟩R,GB +S +and ⟨L(1) +−1/2 ¯L(1) +−1/2σφ1,2(w, ¯w)⟩R,GB +S +. While the first correlator can be calculated by a repeat of +the method employed for the subleading term, the correlator involving the descendant twist +field requires the derivation of a new differential equation. Such an endeavour is beyond the +scope of this work. +Furthermore, the quantum chain sizes we can reach are diminished in the case of the three- +state Potts model, since the size of the space of states grows as ∼ 3M. This memory constraint +prevents us from reaching sizes at which higher order corrections are suppressed, using our +computational methods. This limitation can be, perhaps, bypassed through the usage of more +sophisticated numerical tools, such as DMRG or tensor network methods, to access system sizes +M for which the unknown higher-order correction terms are further suppressed. +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +m/M +0.4 +0.2 +0.0 +0.2 +0.4 +S(R, GB) +2 +([m/M]) +CFT leading +CFT leading+subleading +M=18 sites +Figure 6: Comparison of the second R´enyi entropy in the critical three-state Potts chain of size +M = 18 with mixed (R, GB) BC with CFT results. +Finally, one can use the method of Appendix H, applied this time to the third order ODE of +Section 3.4 to derive the leading CFT contribution to the S(R,GB) +3 +([0, ℓ]) R´enyi entropy. Since +in this case, we have not derived an ODE for the excited twist correlator, we have no handle +on the finite-size corrections to the lattice data, which should be even more severe for N = 3. +34 + +Instead, we have just checked that the CFT result for mixed BC interpolates between the +third R´enyi entropies for identical R and GB boundaries: +S(α,α) +3 +([0, ℓ]) = c +9 log +�2L +π sin +�πℓ +L +�� ++ log gα +(4.23) +Our expectations are met, as Figure 7 confirms. +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +/L +1.2 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +L2h +( , +) +3 +([0, ]) +CFT mixed (R,GB) +CFT fixed (R,R) +CFT restricted (GB,GB) +Figure 7: Comparison of shifted third R´enyi entropies for (R, GB), (R, R) and (GB, GB) BC. +The mixed BC curve can be seen to interpolate between the identical BC results +35 + +5 +Conclusion +In this article, we have presented a general method for calculating R´enyi entropies SN in the +ground state of a 1D critical system with mixed open boundaries, for an interval starting at +one of its ends. This required computing three-point functions of one twist operator and two +BCCOs on the upper-half plane H with mixed BCs (α, β) in the ZN cyclic orbifold. +For this purpose, we have derived ODEs satisfied by these correlation functions, by exploiting +the null-vectors of the twisted and untwisted representations of its symmetry algebra OVirN, +together with Ward identities obtained from the additional conserved currents of the theory. +We used a combination of analytical and numerical methods to find a basis of solutions (a.k.a +conformal blocks) of these ODEs. +For the examples provided in this work, we have calculated the boundary and bulk-boundary +structure constants needed to build the physical correlators as linear combinations of the blocks. +Among the setups we have analysed are the leading and subleading contributions to the one- +interval second and third R´enyi entropies of the Ising model, and the second R´enyi entropy for +the three-state Potts model. We have also derived differential equations for mixed BC twist field +correlators in the Z2 and Z3 orbifolds of generic BCFTs, and obtained an explicit expression +for the second R´enyi entropy valid for any diagonal minimal model, but with a particular set of +mixed boundary conditions. +We have compared the CFT results against critical Ising and three-state Potts spin chain +data. +Since finite size effects are quite significant for open chains, we have included both +the leading and subleading contributions to the lattice twist field correlator in our analytical +prediction. In the Ising case, the agreement was excellent for all choices of mixed BC, even +though the system sizes we could reach were limited. For the three-state Potts chain, however, +the finite size effects are even more severe, and as a consequence the matching is less satisfactory. +This could be improved by using more sophisticated numerical techniques such as DMRG [90,91] +or tensor network methods [92]. +The clearest limitation of our method, first identified in [42], is that the process for obtaining +a differential equation becomes more difficult as N is increased. We have checked using the fusion +rules in Appendix F for N > 3 that the expected order of the ODEs increases with N for generic +minimal models, which implies that more orbifold Ward identities will be needed to obtain the +ODEs. +There are several possible extensions of this work. A possibility would be to generalize the +setup for the calculation of R´enyi entropies of an interval contained in the bulk, with mixed +BC. However, in this situation, one would have to find a differential equation that a four-point +function with two twist fields and two BCCOs satisfies. Cardy’s doubling trick suggests that +such a correlator satisfies the same Ward identities as a six-point conformal block on the complex +plane, so the corresponding differential equation would be partial instead of ordinary. +36 + +Appendix +A +Mother BCFT conventions +We will define here our mother BCFT conventions on the upper-half plane H parametrized by +the coordinate z = x + iy. The boundary is aligned with the real axis. +Bulk operators in the mother CFT are denoted by φi(z, ¯z) while boundary operators are +written as ψ(ab) +j +(x). The operator algebra consists of three types of OPE, which we explicitate, +to fix the notations for the corresponding structure constants. +First, we have the bulk-bulk OPEs: +φi(z, ¯z)φj(0, 0) = +� +φk scaling op. +Ck +ijz−hi−hj+hk ¯z−¯hi−¯hj+¯hk φk(0, 0) +(A.1) +where Ck +ij are the bulk structure constants. +The second type of OPE are boundary-boundary OPEs between BCCOs interpolating +different boundary conditions: +ψ(ab) +i +(x)ψ(dc) +j +(y) = δbd +� +k +B(abc)ψk +ψiψj +(x − y)hk−hi−hjψ(ac) +k +(y) +(A.2) +for x > y. The B(abc)ψk +ψiψj +are the boundary-boundary structure constants. The Kronecker delta +formally expresses the fact that it only makes sense to consider correlations of boundary oper- +ators ordered such that their BCs change consistently with their labelling. +Finally, we consider the third kind of OPE, between the bulk and the boundary: +φi(z) = +� +k +A(a) +φi,ψk(2y)hk−∆i · ψ(aa) +k +(x) +(A.3) +with A(a) +φi,ψk the bulk-boundary structure constants. +In [93], [82] all the structure constants A(a) +φi,ψk and B(abc)ψk +ψiψj +have been determined for A-series +and D-series BCFTs, in terms of fusion matrix elements of bulk CFT four-point functions, and +the entries of the modular S matrix. Relevant for this paper are the results: +B(abc)ψk +ψiψj += Fbk +� +a +c +i +j +� +(A.4) +where the fusion matrix relates bases of conformal blocks around z = 0 and z = 1 +Ir +ia,cj(z) = +� +rs +Frs +� +a +c +i +j +� +J s +ij,ac(1 − z) +(A.5) +defined in the bulk. +We also give the expressions for the 1-point structure constants of the BCFT in terms of +S-matrix elements of the mother CFT +A(a) +φi ≡ A(a) +φi,ψI = Sai +Sa1 +� +S11 +Si1 +(A.6) +37 + +B +Computation of orbifold structure constants +B.1 +Composite twist one-point structure constant in the ZN orbifold BCFT +Assuming the one-point structure constant A(α) +σ1,ψ1 is known, let’s consider the correlator: +⟨σj(0, 0)⟩α +D = A(α) +σj,ψ1 +(B.1) +We now use (2.13) to write the LHS of (B.1) as: +⟨σj(0, 0)⟩α +D = Aj lim +ϵ→0 ϵ2(1−N−1)hj +� +Φ[j,1,...,1](ϵ, ¯ϵ)σ[k](0, 0) +�α +D +(B.2) +Substituting the definition (2.9) of non-diagonal fields, we find: +⟨σj(0, 0)⟩α +D = N−2(1−N−1)hj−1 lim +ϵ→0 ϵ2(1−N−1)hj +N−1 +� +a=0 +� +(φj+a ⊗ φ1+a ⊗ . . . φ1+a) (ϵ, ¯ϵ)σ[k](0, 0) +�α +D +(B.3) +Each correlator in the sum above can be written as: +� +(φj+a ⊗ φ1+a ⊗ . . . φ1+a) (ϵ, ¯ϵ)σ[k](0, 0) +�α +D = ZN,a +ZN +1,a +⟨φj(ϵ, ¯ϵ)⟩DN = A(α) +σ1,ψ1⟨φj(ϵ, ¯ϵ)⟩a +DN +(B.4) +where ZN,a denotes the partition function on the N-sheeted disk with conformal BC a, and +branch point at 0. Now, we can unfold the disk correlator through the conformal map w → w1/N +and substitute back in (B.3) to find: +⟨σj(0, 0)⟩α +D = A(α) +σ1,ψ1⟨φj(0, 0)⟩a +D +(B.5) +so that we finally find: +A(α) +σj,ψ1 = A(α) +σ1,ψ1Aa +φj +(B.6) +B.2 +Bulk-boundary structure constant in the Z2 orbifold CFT +In this section we compute the structure constant A(α) +σ1,Ψ1,3, which is given by the UHP correlator: +� +σ1(i/2, −i/2)Ψ(αα) +1,3 (1) +�α +H = A(α) +σ1,Ψ1,3 +(B.7) +We can map the LHS of (B.7) to the unit disk through: +z → z − i/2 +z + i/2 +(B.8) +and then use the partition function expression of the correlator (as in the previous section) to +find (after a global rotation): +� +σ1(0, 0)Ψα +1,3(−i) +� +D = ⟨σ1(0, 0)⟩α +D +� +ψα +1,3(−i)ψα +1,3(−ie2iπ) +� +D2,a +(B.9) +where D2,a is a 2-sheeted disk with branch point at 0. We unfold the correlator of boundary +fields through the map w → w1/2 to find: +� +ψα +1,3(−i)ψα +1,3(−ie2iπ) +� +D2,a = +� +2i−1/2�−2h1,3 � +ψ(aa) +1,3 (i1/2)ψ(aa) +1,3 (−i1/2) +� +D = 1 +(B.10) +38 + +so that, by putting everything together, we arrive at: +A(α) +σ1,Ψ1,3 = A(α) +σ1,Ψ1 +(B.11) +For generic N, expressing the bulk-boundary structure constant A(α) +σ1,Ψ1,3 in terms of mother +BCFT quantities depends on our ability to calculate N-point functions of boundary operators. +For N ≥ 5, this becomes difficult to solve for generic mother BCFTs. +C +Orbifold Ward identities for bulk fields +Following [42], we give here the orbifold Ward identities for 4-point bulk correlators: +∞ +� +p=0 +ap +� +O1 +���L(r) +−m1−pO2(1)O3(x, ¯x) +��� O4 +� += +∞ +� +p=0 +bp +� +O1 +��� +� +L(r) +m2+pO2 +� +(1)O3(x, ¯x) +��� O4 +� ++ +∞ +� +p=0 +cp +� +O1 +���O2(1) +� +L(r) +m3+pO3 +� +(x, ¯x) +��� O4 +� ++ +∞ +� +p=0 +dp +� +O1 +���O2(1)O3(x, ¯x)L(r) +m4+p +��� O4 +� +(C.1) +where the levels mi ∈ Z + rki/N satisfy: +m1 + m2 + m3 + m4 = −2 +(C.2) +and the coefficients ap, bp, cp and dp are defined from the Taylor series: +(1 − z)m2+1(1 − xz)m3+1 = +∞ +� +p=0 +apzp +(C.3) +(z − x)m3+1zm4+1 = +∞ +� +p=0 +bp(z − 1)p +(C.4) +(z − 1)m2+1zm4+1 = +∞ +� +p=0 +cp(z − x)p +(C.5) +(z − 1)m2+1(z − x)m3+1 = +∞ +� +p=0 +dpzp +(C.6) +A useful identity +We give here the following commutation identity [42]: +⟨O1 |O2(1)O3(x, ¯x)Ln| O4⟩ − ⟨O1 |LnO2(1)O3(x, ¯x)| O4⟩ += {(1 − xn) [x∂x + (n + 1)h3] + (h4 − h1) − n (h2 + h3)} ⟨O1 |O2(1)O3(x, ¯x)| O4⟩ +(C.7) +where O2 and O3 are primary fields and |O2⟩, |O4⟩ are generic states. This commutator identity +allows one to express insertions of Virasoro modes Ln inside a correlation function in terms of +differential operators acting on them. +39 + +D +R´enyi entropies for the critical Ising chain with mixed fixed +BC +In this section, we will derive the bare and excited twist contributions to the second and third +R´enyi entropy in the critical Ising chain with fixed mixed BC a = +, b = −. +In the Ising +BCFT, the boundary field that interpolates between the corresponding conformal BC |±⟩ is +the operator ψ(+−) +2,1 +, with conformal dimension h2,1 = 1/2. In the ZN orbifold of this theory, +the change in boundary conditions is implemented by the diagonal operator Ψ(αβ) +2,1 +defined as in +(2.27). +The essential observation for the derivation of this section is that the space of conformal +blocks is one-dimensional for the chiral correlators +� +Φ1,3|σ[−k] +j +(1)σ[k] +j (η)|Φ1,3 +� +(D.1) +with j ∈ {1, φ1,3}in the Z2 and Z3 Ising orbifold CFTs. +The result is obtained, as in the +discussion of Section 3, from the fusion rules of these theories, found in [40] and [70]. These +fusion rules also imply the leading singular behaviour of the conformal block around the points +η ∈ {0, 1, ∞}. The corresponding exponents are given in Table 4. +0 +1 +∞ +N = 2, j = 1 +−1 +− 1 +16 +− 15 +16 +N = 2, j = φ1,3 +−1 +− 9 +16 +− 7 +16 +N = 3, j = 1 +−1 +− 1 +9 +− 8 +9 +N = 3, j = φ1,3 +−1 +− 4 +9 +− 5 +9 +Table 4: Singular behaviour of the conformal block of (D.1) for different N and twist field +insertions σ[k] +j (η) +In the η → 1 channel, the exponent corresponds to the fusion +σ[k] +j +× σ[−k] +j +→ Φ1 +(D.2) +for all the chiral correlators we are considering in this section. The diagonal operator Φ1 is +defined as in (2.10). +From the exponents around η → 0 and η → 1 we can determine the generic form of the +conformal blocks for the four cases enumerated above to be: +f(N) +j +(η) = η−1(1 − η)−2hσj P(η) +(D.3) +where P(η) is a generic polynomial in η. Furthermore, taking into account the singular be- +haviour of f(N) +j +(η) around η → ∞, one can constrain its degree in all four cases to be ≤ 2, so +that we have: +f(N) +j +(η) = η−1(1 − η)−2hσj (a2 η2 + a1 η + a0) +(D.4) +Around η → 1, this function behaves as: +f(N) +j +(η) ∼ (1 − η)−2hσj � +(a2 + a1 + a0) + (a2 − a0)(1 − η) + a2(1 − η)2 + . . . +� +(D.5) +40 + +To find ai, we will need to consider the first few results in the module of Φ1 from the OPE +of twist fields in the ZN orbifold: +σ[k] +j (η)σ[−k] +j +(1) = Φ1(1) + 2hσj +Nc (1 − η)2 T (0)(1) + . . . +(D.6) +where T (0)(z) = L(0) +−2Φ1(z) is the SET of the chiral ZN orbifold CFT. The corresponding +structure constant has been determined by applying a L(0) +2 +from the left on both sides of the +OPE, and power matching in (1 − η). Finally, the term at level 1 has vanished because the null +vector L−11 ≡ 0 in the mother CFT induces the null-vectors L(r) +−1Φ1 ≡ 0 in the orbifold. +Inserting (D.6) into (D.1) one finds, the coefficients +a0 = a2 = 2hσj +Nc +a1 = 1 − 2a0 +(D.7) +with which we fix the conformal blocks for all the cases presented in Table 4. We then use the +block expansions for the mixed BC correlators to find: +� +σ[k] +j (z, ¯z) +�αβ +N = g1−N ++ +fN +j (η) +(D.8) +where we have also used, notably, the results of (2.42) for the 1-point structure constant of twist +fields. After mapping to the strip through (3.2), we find for N = 2: +⟨σ1(ℓ, ℓ)⟩+− +SL = 2−5/2 +�2L +π +�−1/16 7 + cos 2πℓ +L +� +sin πℓ +L +�1/16 +(D.9) +⟨σε(ℓ, ℓ)⟩+− +SL = 2−5/2 +�2L +π +�−9/16 1 − 9 cos 2πℓ +L +� +sin πℓ +L +�9/16 +(D.10) +and N = 3: +⟨σ1(ℓ, ℓ)⟩+− +SL = 3−2 +�2L +π +�−1/9 7 + 2 cos 2πℓ +L +� +sin πℓ +L +�1/9 +(D.11) +⟨σε(ℓ, ℓ)⟩+− +SL = 3−2 +�2L +π +�−4/9 1 + 8 cos 2πℓ +L +� +sin πℓ +L +�4/9 +(D.12) +E +Hypergeometric differential equation +The hypergeometric differential equation is canonically defined as: +η(η − 1)f′′(η) + [(a + b + 1)η − c]f′(η) + ab f(η) = 0 +(E.1) +with the Riemann scheme: +0 +1 +∞ +0 +0 +a +1 − c +c − a − b +b +41 + +The solutions are constructed using the Gauss hypergeometric function 2 F1(a, b; c | η). +Following the conventions of [94], we give a standard basis of fundamental solutions to (E.1) +around the singular point η = 0: +I1(η) = 2 F1(a, b; c | η) +I2(η) = η1−c2 F1(b − c + 1, a − c + 1; 2 − c | η) +(E.2) +and around η = 1: +J1(η) = 2 F1(a, b; a + b − c + 1 | 1 − η) +J2(η) = (1 − η)c−a−b2 F1(c − b, c − a; c − a − b + 1 | 1 − η) +(E.3) +The two bases of solutions are linearly related as +Ii(η) = +2 +� +j=1 +PijJj(η) +(E.4) +with the fusing matrix P +P = +� +Γ(c)Γ(d) +Γ(c−a)Γ(c−b) +Γ(c)Γ(−d) +Γ(a)Γ(b) +Γ(2−c)Γ(d) +Γ(1−a)Γ(1−b) +Γ(2−c)Γ(−d) +Γ(1−c+a)Γ(1−c+b) +� +(E.5) +and its inverse: +P−1 = +� +Γ(1−c)Γ(1−d) +Γ(1−c+a)Γ(1−c+b) +Γ(c−1)Γ(1−d) +Γ(a)Γ(b) +Γ(1−c)Γ(1+d) +Γ(1−a)Γ(1−b) +Γ(c−1)Γ(1+d) +Γ(c−a)Γ(c−b) +� +(E.6) +expressed in terms of Euler’s Gamma function Γ, with d = c − a − b. +F +Fusion rules in the ZN orbifold +In [70] we have found compact expressions for the fusion numbers of the ZN orbifold of a +diagonal RCFT. They are given by: +N [k1...kN] +[i1...iN],[j1...jN] = +N−1 +� +a,b=0 +Nk1 +i1+a,j1+b . . . NkN +iN+a,jN+b , +N k(r) +[i1...iN],[j1...jN] = +N−1 +� +a=0 +Nk +i1+a,j1 . . . Nk +iN+a,jN , +N k(s) +[i1...iN],j(r) = Nk +i1,j . . . Nk +iN,j , +N k(t) +i(r),j(s) = δr+s,t Nk +ij . +N [k1...kN] +i[p](r)j[q](s) = δp+q,0 +M +� +ℓ=1 +SiℓSjℓ · Sk1ℓ . . . SkNℓ +SN +1ℓ +, +N k(t) +i[p](r)j[q](s) = δp+q,0 +N +M +� +ℓ=1 +� +SiℓSjℓSN +kℓ +SN +1ℓ ++ +N−1 +� +n=1 +ωnp(r+s−t) (P−n)iℓ(Pn)jℓSkℓ +S1ℓ +� +, +N k[m](t) +i[p](r)j[q](s) = δp+q,m +N +M +� +ℓ=1 +� +SiℓSjℓSkℓ +SN +1ℓ ++ +N−1 +� +n=1 +ωn(r+s−t) (P † +pn−1)iℓ(P † +qn−1)jℓ(Pmn−1)kℓ +S1ℓ +� +. +(F.1) +42 + +where ω = exp (2πi/N), and Nk +ij, Sij are the fusion numbers and the modular S-matrix of the +mother CFT. One also needs the matrix Pn which is defined from +Pn = T −n/N · Qn · T [[−n−1]]/N , +n ∈ Z× +N , +(F.2) +where T is the modular T matrix of the mother CFT, [[−n−1]] denotes the inverse of (−n) in +Z× +N, with 0 < [[−n−1]] < N, and Qn is the matrix representing the linear action of the modular +map +τ �→ qn(τ) = nτ − (n[[−n−1]] + 1)/N +Nτ − [[−n−1]] +(F.3) +on the characters χj of the mother CFT. +G +Derivation of differential equation in the Z3 orbifold BCFT +We present in this section all the orbifold Ward identities and null-vector conditions necessary +to derive the third order differential equation (3.45). +The Ward identities +Ward 1 The correlator to integrate over is: +⟨Φ12| L(1) +1 σ1(1)T (1)(z)˜σ1(η)L(1) +−1 |Φ12⟩ +(G.1) +with (m1, m2, m3, m4) = (−1, 1/3, −1/3, −1), to find: +a0|1 ⟨Φ12| (L(1) +1 )2σ1(1)˜σ1(η)L(1) +−1 |Φ12⟩ + a1|1 ⟨Φ12| L(1) +1 L(1) +0 σ1(1)˜σ1(η)L(1) +−1 |Φ12⟩ = +d0|1 ⟨Φ12| L(1) +1 σ1(1)˜σ1(η)L(1) +−1L(1) +−1 |Φ12⟩ + d1|1 ⟨Φ12| L(1) +1 σ1(1)˜σ1(η)L(1) +0 L(1) +−1 |Φ12⟩ +(G.2) +Ward 2 The correlator to integrate over is: +⟨Φ12| σ1(1)T (1)(z)˜σ1(η)L(1) +−1L(1) +−1 |Φ12⟩ +(G.3) +with (m1, m2, m3, m4) = (−1, 1/3, −1/3, −1) to find: +a0|2 ⟨Φ12| L(1) +1 σ1(1)˜σ1(η) +� +L(1) +−1 +�2 +|Φ12⟩ = d0|2 ⟨Φ12| σ1(1)˜σ1(η) +� +L(1) +−1 +�3 +|Φ12⟩ ++d1|2 ⟨Φ12| σ1(1)˜σ1(η)L(1) +0 +� +L(1) +−1 +�2 +|Φ12⟩ + d2|2 ⟨Φ12| σ1(1)˜σ1(η)L(1) +1 +� +L(1) +−1 +�2 +|Φ12⟩ ++ d3|2 ⟨Φ12| σ1(1)˜σ1(η)L(1) +2 +� +L(1) +−1 +�2 +|Φ12⟩ +(G.4) +Ward 3 The correlator to integrate over is +⟨Φ12| L(1) +1 L(1) +1 σ1(1)T (1)(z)˜σ1(η) |Φ12⟩ +(G.5) +with (m1, m2, m3, m4) = (−1, 1/3, −1/3, −1) to find: +d0|3 ⟨Φ12| +� +L(1) +1 +�2 +σ1(1)˜σ1(η)L(1) +−1 |Φ12⟩ = a0|3 ⟨Φ12| +� +L(1) +1 +�3 +σ1(1)˜σ1(η) |Φ12⟩ ++a1|3 ⟨Φ12| +� +L(1) +1 +�2 +L(1) +0 σ1(1)˜σ1(η) |Φ12⟩ + a2|3 ⟨Φ12| +� +L(1) +1 +�2 +L(1) +−1σ1(1)˜σ1(η) |Φ12⟩ ++ a3|3 ⟨Φ12| +� +L(1) +1 +�2 +L(1) +−2σ1(1)˜σ1(η) |Φ12⟩ +(G.6) +43 + +Ward 4 The correlator to integrate over is: +⟨Φ12| σ1(1)T (2)(z)˜σ1(η)L(1) +−1 |Φ12⟩ +(G.7) +with (m1, m2, m3, m4) = (0, −1/3, 1/3, −2) so we find: +d0|4 ⟨Φ12| σ1(1)˜σ1(η)L(2) +−2L(1) +−1 |Φ12⟩ + d1|4 ⟨Φ12| σ1(1)˜σ1(η)L(2) +−1L(1) +−1 |Φ12⟩ + +d2|4 ⟨Φ12| σ1(1)˜σ1(η)L(2) +0 L(1) +−1 |Φ12⟩ + d3|4 ⟨Φ12| σ1(1)˜σ1(η)L(2) +1 L(1) +−1 |Φ12⟩ = 0 +(G.8) +Ward 5 The correlator to integrate over is: +⟨Φ12| L(1) +1 T (2)(z)σ1(1)˜σ1(η) |Φ12⟩ +(G.9) +with (m1, m2, m3, m4) = (−2, −1/3, 1/3, 0) so we find: +a0|5 ⟨Φ12| L(1) +1 L(2) +2 σ1(1)˜σ1(η) |Φ12⟩ + a1|5 ⟨Φ12| L(1) +1 L(2) +1 σ1(1)˜σ1(η) |Φ12⟩ + +a2|5 ⟨Φ12| L(1) +1 L(2) +0 σ1(1)˜σ1(η) |Φ12⟩ + a3|5 ⟨Φ12| L(1) +1 L(2) +−1σ1(1)˜σ1(η) |Φ12⟩ = 0 +(G.10) +Ward 6 The correlator to integrate over is: +⟨Φ12| σ1(1)T (2)(z)˜σ1(η)L(1) +−1 |Φ12⟩ +(G.11) +with (m1, m2, m3, m4) = (−1, −1/3, 1/3, −1) to find: +a0|6 ⟨Φ12| L(2) +1 σ1(1)˜σ1(η)L(1) +−1 |Φ12⟩ = d0|6 ⟨Φ12| σ1(1)˜σ1(η)L(2) +−1L(1) +−1 |Φ12⟩ ++d1|6 ⟨Φ12| σ1(1)˜σ1(η)L(2) +0 L(1) +−1 |Φ12⟩ + d2|6 ⟨Φ12| σ1(1)˜σ1(η)L(2) +1 L(1) +−1 |Φ12⟩ +(G.12) +Ward 7 The correlator to integrate over is: +⟨Φ12| σ1(1)T (1)(z)˜σ1(η)L(2) +−1 |Φ12⟩ +(G.13) +with (m1, m2, m3, m4) = (−1, 1/3, −1/3, −1) to find: +a0|7 ⟨Φ12| L(1) +1 σ1(1)˜σ1(η)L(2) +−1 |Φ12⟩ = d0|7 ⟨Φ12| σ1(1)˜σ1(η)L(1) +−1L(2) +−1 |Φ12⟩ ++d1|7 ⟨Φ12| σ1(1)˜σ1(η)L(1) +0 L(2) +−1 |Φ12⟩ + d2|7 ⟨Φ12| σ1(1)˜σ1(η)L(1) +1 L(2) +−1 |Φ12⟩ +(G.14) +The null-vector conditions +L(1) +−1L(2) +−1 |Φ12⟩ = 1 +2 +� +3gL(0) +−2 − +� +L(0) +−1 +�2� +|Φ12⟩ +(G.15) +⟨Φ12| L(1) +1 L(2) +1 = ⟨Φ12| 1 +2 +� +3gL(0) +2 − +� +L(0) +1 +�2� +(G.16) +2 L(0) +−1L(2) +−1L(1) +−1 |Φ12⟩ = +� +3gL(0) +−1L(0) +−2 − +� +L(0) +−1 +�3� +|Φ12⟩ +(G.17) +2 L(0) +−1L(2) +−1L(1) +−1 |Φ12⟩ = +� +3gL(1) +−1L(2) +−2 − +� +L(1) +−1 +�3� +|Φ12⟩ +(G.18) +2 ⟨Φ12| L(0) +1 L(2) +1 L(1) +1 = ⟨Φ12| +� +3gL(0) +2 L(0) +1 − +� +L(0) +1 +�3� +(G.19) +2 ⟨Φ12| L(0) +1 L(2) +1 L(1) +1 = ⟨Φ12| +� +3gL(2) +2 L(1) +1 − +� +L(1) +1 +�3� +(G.20) +By removing from this linear system of 13 equations all terms containing modes L(r) +n with r ̸= 0, +one indeed obtains (3.45). +44 + +H +Numerical implementation of the Frobenius method +We want to find a basis of solutions to the differential equation (3.30) that converge on the +entire range of interest - the unit circle |η| = 1. +The Fuchsian ODE (3.30) has singular points 0, 1, ∞. The solutions around η = 0 and η = 1 +converge on the disks |η| < 1 and |η − 1| < 1 respectively. Thus, only a portion of the unit +semicircle, namely 0 < Arg(η) < π/3, is contained in the convergence disk around η = 1. We +can circumvent this problem by observing that the solutions around η = ∞ can be convergent +on the whole unit circle |η| = 1. Even better, we can implement the change of variable +η �→ 1 + u +2u +, +∂η �→ −2u2∂u . +(H.1) +so that the new ODE, in the variable u has singular points at u = 0, 1, −1. The original unit +circle |η| = 1 is mapped to |u − 1/3| = 2/3, which is contained in the convergence disk |u| < 1. +Hence, applying the Froebenius method, and expressing the solutions around u = 1 in terms of +those around u = 0 will give the appropriate numerical evaluation of the desired values of η. +Now, as explained in [42], a convenient way of finding power series solutions around a point +u = u0 is to rewrite the differential equation (3.30) in terms of the operator θ = (u − u0)∂u, +which satisfies: +(u − u0)n∂n +u = +n−1 +� +k=0 +(θ − k) +(H.2) +Most importantly, we have that any polynomial P(θ) satisfies: +P(θ)(u − u0)r = P(r)(u − u0)r +(H.3) +For u0 = 0, we can then rewrite the equation as: +� 8 +� +i=0 +uiPi(θ) +� += 0 +(H.4) +where: +P0(θ) = 250θ4 − 125θ3 − 130θ2 − θ + 6 +P1(θ) = −θ +� +−2125θ2 + 450θ + 997 +� +− 174 +P2(θ) = −250θ4 + 125θ3 + 130θ2 − +� +750θ3 + 1250θ2 − 2415θ + 4264 +� +θ + θ − 3123 +P3(θ) = −θ +� +4250θ2 + 5925θ + 6016 +� ++ θ +� +−2125θ2 + 450θ + 997 +� +− 8511 +P4(θ) = 5 θ +� +150θ3 + 575θ2 + 93θ − 332 +� ++ θ +� +750θ3 + 1250θ2 − 2415θ + 4264 +� +− 6000 +P5(θ) = 2125 θ +� +θ2 + 3θ + 2 +� ++ θ +� +4250θ2 + 5925θ + 6016 +� +P6(θ) = −250 θ +� +θ3 + 6θ2 + 11θ + 6 +� +− 5θ +� +150θ3 + 575θ2 + 93θ − 332 +� +P7(θ) = −2125 θ +� +θ2 + 3θ + 2 +� +P8(θ) = 250 θ +� +θ3 + 6θ2 + 11θ + 6 +� +(H.5) +We now seek power series solutions around u = 0 of the form: +Ii(u) = uri +∞ +� +n=0 +anun +with +a0 = 1 +(H.6) +45 + +where the ri are the roots of the characteristic polynomial P0(r): +r1 = −3/10 +r2 = 1 +r3 = 1/5 +r4 = −2/5 +(H.7) +and are the same as the exponents around ∞ in Table 1. +By substituting the ansatz (H.6) in the differential equation and employing the identity +(H.3) we find the following recursion relations for the coefficients an of the solution Ii(u): +P0(ri + n)an = − +min{n,8} +� +i=1 +an−i Pi(ri + n − i) +a0 = 1 +(H.8) +The four series found in this way converge for |u| < 1 and can be evaluated numerically to +arbitrary precision. +We note, at this point, that the solution corresponding to r4 is unphysical, since it corre- +sponds, according to Table 1, to the presence in the operator algebra of the theory of a composite +twist field formed with a primary operator that is outside the Kac table, i.e. not present in +the M(6, 5) CFT. This suggests that the physical space of conformal blocks is actually three- +dimensional, and thus, that there should be a third order differential equation satisfied by the +excited twist correlator in this setup. +One should now repeat the above computation for the solutions Jj(u) around u = 1, since +these are the ones that appear in the block expansion (3.4) of BCFT correlators. The recursion +relation takes the same form as in (H.8), with different roots: +λ1 = −1/2 +λ2 = 9/10 +λ3 = 23/10 +λ4 = 23/10 + 2 +(H.9) +A slight complication appears in this case because two of the roots of the corresponding +characteristic polynomial differ by an integer, that is, r4 = r3 + 2. +This will lead to the +truncation of the corresponding recursion relations (H.8) for r3 because at n = 2, the coefficient +P0(r3 + 2) = 0. A good basis of solutions in this case is {J1(u), J2(u), J(k) +3 (u), J4(u)}, where: +Ji(u) = (1 − u)ri +∞ +� +i=0 +an(1 − u)n +J(k) +3 (u) = (1 − u)r3[a0 + a1(1 − u)] + kJ4(u) +(H.10) +where k is a free parameter and the value we choose for it should not change the final result +for the physical correlator. We have chosen to set it to k0 = 0.04428171795178596 and define +J3(u) = J(k0)(u). +The reason for this choice becomes apparent when one looks at our solution for the fusing +matrix M: +M = +� +� +� +� +� +� +0.207411 +0.393808 +0.152178 +0 +1.356 +−2.30281 +−0.444933 +0 +7.70383 +71.6374 +−22.7841 +0 +−8986.23 +−19156.8 +7211.61 +5800.8 +� +� +� +� +� +� +(H.11) +which relates the bases of conformal blocks around u = 1 and u = 0 as: +Ji(u) = +� +j +MijIj(u) +(H.12) +46 + +To obtain this solution, we have generated a linear system of equations for the unknown Mij +from the evaluation of the above relations at different points {ui} in the interval 0 < u < 1 +(where both sets of solutions converge). In this context, the parameter k0 was tuned so that +the block J3(u) does not depend on the unphysical solution I3(u) around u = 0. 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McClain, eds. +52 + diff --git a/utA0T4oBgHgl3EQfLv_t/content/tmp_files/load_file.txt b/utA0T4oBgHgl3EQfLv_t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eb93b50098daa4c9b53e0f2d7f98a48d88bff8a5 --- /dev/null +++ b/utA0T4oBgHgl3EQfLv_t/content/tmp_files/load_file.txt @@ -0,0 +1,1981 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf,len=1980 +page_content='R´enyi entropies for one-dimensional quantum systems with mixed boundary conditions Benoit Estienne, Yacine Ikhlef, Andrei Rotaru Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique et Hautes ´Energies, LPTHE, F-75005 Paris, France January 6, 2023 Abstract We present a general method for calculating R´enyi entropies in the ground state of a one-dimensional critical system with mixed open boundaries, for an interval starting at one of its ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In the conformal field theory framework, this computation boils down to the evaluation of the correlation function of one twist field and two boundary condition changing operators in the cyclic orbifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Exploiting null-vectors of the cyclic orbifold, we derive ordinary differential equations satisfied by these correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In particular we obtain an explicit expression for the second R´enyi entropy valid for any diagonal minimal model, but with a particular set of mixed boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In order to compare our results with numerical data for the Ising and three-state Potts critical chains, we also identify and compute the leading finite size corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='02124v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='stat-mech] 5 Jan 2023 Contents 1 Introduction 4 2 The cyclic orbifold 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 The cyclic orbifold on the Riemann sphere .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 Calculation of boundary-boundary structure constants .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 Orbifold bulk-boundary structure constants .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 13 3 Differential equations in the Z2 and Z3 orbifold BCFT 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 Setup for the calculations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5 The function ⟨Ψ12 · σ13 · Ψ12⟩ in the Z3 orbifold of the Ising model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6 More hypergeometric differential equations in the Ising cyclic orbifold BCFTs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 24 4 Numerical checks and finite-size corrections in quantum chains 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 The Ising quantum chain with mixed BC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 The three-state Potts quantum chain with mixed BC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 29 5 Conclusion 36 A Mother BCFT conventions 37 B Computation of orbifold structure constants 38 2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 Composite twist one-point structure constant in the ZN orbifold BCFT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 38 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 Bulk-boundary structure constant in the Z2 orbifold CFT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='C Orbifold Ward identities for bulk fields ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='D R´enyi entropies for the critical Ising chain with mixed fixed BC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='E Hypergeometric differential equation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='F Fusion rules in the ZN orbifold ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='G Derivation of differential equation in the Z3 orbifold BCFT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='43 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='H Numerical implementation of the Frobenius method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='Introduction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='The understanding of quantum entanglement has proved to be a research topic of continued and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='central interest for physicists working in domains as diverse as high energy physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' condensed matter theory and quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Entanglement measures have turned out to be useful diagnostic tools for tensor network algorithms, quantities of interest for the AdS/CFT corre- spondence, and, most relevantly for the present work, a powerful tool for probing the physics of quantum many-body systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' With respect to the latter, the study of entanglement has proved crucial to the study of phase transitions in one dimensional quantum systems, by al- lowing their detection and the characterization of their critical exponents and corresponding central charge [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Important applications of entanglement are found in higher dimensions too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We mention, for two-dimensional systems, the establishment of intrinsic topological or- der and various anyonic quantum dimensions [5, 6] and the detection and counting of critical Dirac fermions [7–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Finally, entanglement can also be used, in two [11–15] or higher dimen- sions [16,17] to reveal gapless interface modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The basic setup is as follows: we consider a quantum system in a pure state |Ψ⟩, and a spatial bipartition of said system into two complementary subregions A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The entanglement between them is then encoded in the reduced density matrix ρA = TrB|Ψ⟩⟨Ψ| and it can be quantified through entanglement measures, such as the R´enyi entanglement entropies [18–22] Sn(A) = 1 1 − n log TrA (ρn A) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) and in particular the n → 1 case corresponding to the well-known von Neumann entropy: S(A) = −TrA (ρA log ρA) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2) While the focus on entanglement entropies has been mostly theoretical, in recent years experi- mental proposals as well as actual experiments have been designed to measure them [23–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' For strongly correlated quantum systems, the theoretical computation of entanglement en- tropies is a technically challenging endeavour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' However, if these systems are one-dimensional and critical, the formidable toolbox of two-dimensional Conformal Field Theory (CFT) is avail- able to tackle such computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The calculations of entanglement entropies through such methods rests on two crucial insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The first insight is that, for integer values of n, and a subsystem A = ∪i[ui, vi] built as the union of some disjoint intervals, the moments of the reduced density matrix TrA (ρn A) can be expressed as the partition function of an n-sheeted Riemann surface with conical singularities corresponding to the endpoints of the intervals [ui, vi] [2,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Such partition functions have been evaluated, with significant toil, for free theories and some special cases of interacting models [4, 30–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In general, however, a second insight is needed to make progress: the replication of the spacetime of the theory can be ”exchanged” for the replication of the target space of the CFT [39–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Such a construction, known in the literature as the cyclic orbifold CFT [40], is built from the permutation symmetric product of n copies of the original CFT (referred to as the mother CFT), by modding out the discrete subgroup Zn of cyclic permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In this framework, the conical singularities of the mother CFT defined on the replicated surface are accounted for by insertions of twist fields [39] in cyclic orbifold correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Thus, by computing correlators of twist operators, one can evaluate TrA (ρn A) for a variety of setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' To give a few examples, one can easily adapt the twist field formalism to encode modified initial conditions around the branch points [42], which is fitting for com- putations of more refined entanglement measures such as the symmetry-resolved entanglement entropy [43–48] or for explorations of entanglement in non-unitary systems [42, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Arguably 4 the most renowned result obtained in this framework is [1,2,29,50–52] Sn(ℓ) ∼ ℓ→∞ c 6 n + 1 n log ℓ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3) which gives the universal asymptotic behaviour for the ground state entanglement entropy of an interval of length ℓ in an infinite system (with c the central charge of the critical system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In this article, we consider the R´enyi entanglement entropy in an open system with mixed boundary conditions, when the subregion A is a single interval touching the boundary – we take the boundary condition (BC) at one end of the chain to be different from the BC at the other end (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In the scaling limit, such an open critical system is described by a Boundary Conformal Field Theory (BCFT), with a well understood [53–56] correspondence between the chiral Virasoro representations and the conformal boundary conditions allowed by the theory, and an algebra of boundary operators that interpolate between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' ℓ β α L Figure 1: An interval of length ℓ in a 1d critical chain with mixed BC (αβ) and length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The more accessible setup of an interval touching one of two identical boundaries has been thoroughly analysed using either conformal field theory methods [2, 52, 57–60] or exact free fermion techniques [61–63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Such configurations are also well-handled numerically, through density-matrix renormalization group (DMRG) techniques [64–67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In that setup, the subsys- tem A is at the end of a finite system with the same boundary condition α on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The computation of the R´enyi entanglement entropies rests on the evaluation of a twist one-point function on the upper half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Such a correlation function is straightforwardly fixed by con- formal invariance, and as a consequence the entanglement entropy exhibits a simple dependence on the interval and system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Explicitly, in the case of an interval of length ℓ at the end of a system of size L, one finds the leading universal behaviour [2]: Sn(ℓ) ∼ c 12 n + 1 n log �2L πa sin �πℓ L �� + log gα , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) where a is the lattice spacing and gα is the universal boundary entropy [68] associated to the boundary condition α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' When one studies systems with mixed BC, at the level of the BCFT one has to introduce boundary condition changing operators (BCCOs), and thus the corresponding correlators are more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The core idea of this framework is that the singular behaviour associated to the change in boundary conditions, can be encoded in the form of operators placed on the boundary, that interpolate between regions of different BC α ̸= β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Thus, to compute the R´enyi entropy Sn in this setup, we will evaluate three-point functions with one twist operator and two BCCO insertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Such setups have already been studied for the Ising and XX chains in [57], at the level of the CFT on the replicated surface, and rely on the knowledge of relatively simple closed form expressions for the 2n-point correlator of BCCOs on the unit disk for their calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' However, such knowledge is the exception, rather than the norm, for generic BCFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 5 In this work, we present a general method to compute such twist correlators functions with mixed BCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The most technically demanding part of this framework is finding Ordinary Differential Equations (ODEs) that the correlators satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' According to Cardy’s doubling trick [53], in the half-plane geometry, the three-point functions of interest obey the same Ward identities as a four-point conformal block with the corresponding operators, where the bulk twist operator σ(z, ¯z) is replaced by the insertion of σ(z)σ†(¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Thus, in an adaptation of the method of [42], we can derive a differential equation by combining knowledge of the null-vector conditions obeyed by the twisted and untwisted fields under the symmetry algebra of the cyclic orbifold [40] with the derivation of well-chosen Ward identities obtained from current insertions in the correlators of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The final ingredient is the determination of a subset of the (bulk and boundary) structure constants of the cyclic orbifold BCFT, which fix the specific linear combination of solutions of the differential equation that gives the sought correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We have illustrated this approach with a variety of BCFT setups, that share a common assumption: in the mother CFT, the mixed boundary conditions (βα) are implemented by a BCCO which is degenerate at level two under the Virasoro algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' With this restriction, in the Z2 orbifold of a generic BCFT, we have derived a second-order and a fourth order ODE, respectively for the bare and composite1 twist correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Under the same restrictions, in the Z3 orbifold of a generic BCFT, we have determined a third-order ODE for the bare twist correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We have also worked out, for the case of the Z2 and Z3 cyclic orbifolds of the Ising BCFT, a variety of lower-order ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The latter calculations were found compatible with the results of [57], and have been tested against numerical results for the critical Ising chain, for all possible combinations of BCs, to excellent agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Finally, we have also considered the Z2 orbifold of the three-state Potts model, and compared it against lattice data for the critical three-state Potts model with states {R, G, B}, with less accurate, but consistent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We quote here the leading behaviour of the second R´enyi entropy of the critical three-state Potts chain of size L for mixed fixed R and restricted GB boundary conditions: S(R,GB) 2 (ℓ) ∼ cPotts 8 log 2L πa sin �πℓ L � + log gR − log � η−2h122F1 (−8/5, −9/10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' −9/5 | 1 − η) � (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5) with η = e2πiℓ/L, cPotts = 4/5 the central charge, h12 = 2/5 the scaling dimension of the BCCO, and gR = [(5 − √ 5)/30]1/4 the ground state degeneracy associated to the fixed R BC [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The expression (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5) is, in fact, only a particular case of a more general result obtained in this paper, which applies to any critical system described by a BCFT based on a minimal model M(p, p′) with mixed conformal BC (α, β) chosen such that the most relevant BCCO interpolating between them is ψ(αβ) 1,2 and there is no boundary operator ψ(ββ) 1,3 allowed in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Under these conditions, the second R´enyi entropy of an interval A = [0, ℓ], in a finite system of size L touching the boundary β is: S(α,β) 2 (ℓ) ∼ c 8 log 2L πa sin �πℓ L � + log gβ − log � η−2h122F1 � 2 − 3 p p′ , 3/2 − p p′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 3 − 4 p p′ | 1 − η �� (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6) where c, gβ and h12 generalize the notation in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In both (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6), one should keep in mind, especially for the purpose of numerical studies, that the hypergeometric function in the third term of these equations converges inside the unit circle centred at η = 1, which only overlaps with the subinterval Arg(η) ∈ (0, π/3) ∪ 1obtained by fusing the bare twist operator with an untwisted operator φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 6 (5π/3, 2π) of the parameter space of interest Arg(η) ∈ [0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Thus, to evaluate the expressions for Arg(η) ∈ (π/3, 5π/3), it is necessary to analytically continue the third term to this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We give here the outline of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In Section 2, we give a review of the cyclic orbifold construction, with a focus on its implementation on the upper-half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We discuss in this section the bulk and boundary operator algebra, and show how some orbifold bulk and boundary structure constants can be expressed in terms of mother BCFT quantities by unfolding and factorizing arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We dedicate Section 3 to the derivation of ODEs for the different setups described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' On top of the announced derivations involving orbifold Ward identities, we also use the results on the fusion rules of the ZN cyclic orbifold of [70] and some mathematical facts about the hypergeometric differential equation, to derive low-order differential equations for the Ising case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Section 4 contains a comparison of our analytical results with lattice data, for both the Ising and three-state Potts critical chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Finally, we have relegated the more technical derivations to the Appendix, to avoid congesting the logical flow of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 2 The cyclic orbifold In this section, we will present the construction of the cyclic orbifold BCFT on the upper half- plane H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' After reviewing a few essential features of the ZN orbifold on the Riemann sphere, we will discuss conformal boundary conditions, boundary operators as well as bulk-boundary and boundary-boundary operator algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 The cyclic orbifold on the Riemann sphere To build a cyclic orbifold CFT, one starts from any mother CFT M and constructs the tensor product theory M⊗N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Then one considers all the ZN equivalent ways of connecting the copies of the product theory, which creates N different sectors, each with its corresponding operator families and labelled by a ZN twist charge [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The spectrum of the cyclic orbifold MN is then built as a reunion of the operator families from all the sectors [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 Symmetry algebra and operator content In MN, each copy a of the mother CFT carries the components of the stress-energy tensor Ta(z), ¯Ta(¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We define the discrete Fourier modes of these currents as T (r)(z) = N−1 � a=0 ωar Ta(z) , ¯T (r)(¯z) = N−1 � a=0 ωar ¯Ta(¯z) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) where r is considered modulo N, and we have used the notation ω = exp(2iπ/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' They satisfy the OPEs T (r)(z)T (s)(w) = δr+s,0 Nc/2 (z − w)4 + 2T (r+s)(w) (z − w)2 + ∂T (r+s)(w) z − w + regz→w , ¯T (r)(¯z) ¯T (s)( ¯w) = δr+s,0 Nc/2 (¯z − ¯w)4 + 2 ¯T (r+s)( ¯w) (¯z − ¯w)2 + ∂ ¯T (r+s)( ¯w) ¯z − ¯w + reg¯z→ ¯w , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2) where the Kronecker symbols δr+s,0 are understood modulo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The symmetric modes T (0)(z) and ¯T (0)(¯z) are the components of the stress-energy tensor of MN with central charge Nc, 7 whereas the other Fourier modes T (r)(z), ¯T (r)(¯z) with r ̸= 0 should be regarded as additional conserved currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Altogether, these Fourier modes encode an extended conformal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The modes associated to these currents are defined in the usual way through: L(r) m = 1 2iπ � dz zm+1 T (r)(z) , ¯L(r) m = 1 2iπ � d¯z ¯zm+1 ¯T (r)(¯z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3) In the sector of twist charge [k] one has the following mode decompositions T (r)(z) = � m∈−kr/N+Z z−m−2 L(r) m ¯T (r)(¯z) = � m∈+kr/N+Z ¯z−m−2 ¯L(r) m (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) and the commutation relations � L(r) m , L(s) n � = (m − n)L(r+s) m+n + Nc 12 m(m2 − 1) δm+n,0 δr+s,0 , � ¯L(r) m , ¯L(s) n � = (m − n)¯L(r+s) m+n + Nc 12 m(m2 − 1) δm+n,0 δr+s,0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5) Hermitian conjugation of the modes acts as: � L(r) n �† = L(−r) −n , � ¯L(r) n �† = ¯L(−r) −n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6) Orbifold primary operators are, by definition, annihilated by the action of all the positive modes of OVir ⊗ OVir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Descendant operators with respect to this algebra are constructed by the action of the negative m modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We establish the notation for descendants of a scaling (primary or not) operator O: � L(r) m · O � (z, ¯z) := 1 2iπ � Cz dw (w − z)m+1 T (r)(w)O(z, ¯z) , � ¯L(r) m · O � (z, ¯z) := 1 2iπ � Cz d ¯w ( ¯w − ¯z)m+1 ¯T (r)( ¯w)O(z, ¯z) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7) where the contour Cz encloses the point z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' It will be useful to work with the primary operator spectrum with respect to the neutral subalgebra A ⊗ ¯A generated by the algebra elements L(r1) m1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' L(rp) mp and ¯L(r1) m1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' ¯L(rp) mp , with r1 + · · · + rp = 0 mod N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8) One can classify all ZN-symmetric operators of MN into representations of A ⊗ ¯A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This orga- nization, described in detail by the authors of the present work in [70], distinguishes between three types of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' First, we have identified the untwisted non-diagonal operators Φ[j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='jN].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' These operators are built from ZN-symmetrized combinations of products of mother CFT pri- mary operators φj (with j = 1 referring to the identity operator 1): Φ[j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='jN] := 1 √ N N−1 � a=0 (φj1+a ⊗ · · · ⊗ φjN+a) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='9) in which at least one pair satisfies ji ̸= jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Its conformal dimension is given by h[j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='jN] = � s hjs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 8 The second type of primary operators under the neutral algebra are the untwisted diagonal fields Φ(r) j , where the Fourier replica index r takes values in ZN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The r = 0 diagonal fields are defined to be: Φ(0) j = Φj := φj ⊗ · · · ⊗ φj , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='10) while for r ̸= 0, they are constructed as: Φ(r) j := 1 2Nhj L(r) −1 ¯L(−r) −1 Φj , 1(r) := 2 NcL(r) −2 ¯L(−r) −2 Φ1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='11) The conformal dimension of a diagonal operator Φ(r) j is then generically given by h(r) j = Nhj + δr,0 (1 + δj,1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='12) We should note that the diagonal operators with r = 0 and the non-diagonal operators are also primary under OVir ⊗ OVir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Finally, we have to consider twist operators, which come in distinct flavours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' For the pur- poses of this paper, we will mostly work with twist operators with Fourier replica index r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Thusly, just as for the diagonal fields, we will drop this specification when the context heavily implies it, to decongest the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We first consider the ubiquitous bare twist operators [2, 30, 71, 39] which are denoted in our conventions σ[k] = σ[k] 1 , or, in light notation, σ = σ[1] and σ† = σ[−1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We have also the composite twist fields σ[k] j , which can be defined through point-splitting as in [49]: σ[k] j (z, ¯z) := Aj lim ϵ→0 � ϵ2(1−N−1)hjΦ[j,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=',1](z + ϵ, ¯z + ¯ϵ) · σ[k](z, ¯z) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='13) where the constant Aj = N−2(1−N−1)hj−1/2 ensures that non-vanishing two-point functions of twist operators are normalized to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' If N and k are coprime, the conformal dimension of the bare twist operator is hσ = c 24 � N − 1 N � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='14) while for composite twist operators one has: hσj = hσ + hj N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='15) Having established the primary operator spectrum of the orbifold, we will now review how the null-vectors of the diagonal and twisted fields in MN are inferred from the ones of the mother theory M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 Null vectors for untwisted operators Let us consider a generic mother CFT M, with central charge c = 1 − 6(1 − g)2 g , 0 < g ≤ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='16) The conformal dimensions of degenerate primary operators are given by the Kac formula hrs = (r − sg)2 − (1 − g)2 4g , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='17) 9 where r, s are positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The corresponding operator φrs is degenerate at level rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' If the parameter g is rational, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' g = p/p′ with coprime p and p′, then the set of operators φrs with 1 ≤ r ≤ p − 1 and 1 ≤ s ≤ p′ − 1 generates a closed operator algebra, and the related CFT is the minimal model Mp,p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' While we do employ this parametrization extensively, in the present work we will consider a more generic mother CFT, and we do not assume that it is a minimal model—unless explicitly indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Consider the situation when the mother CFT includes the degenerate operator φ12, with null-vector condition � L−2 − 1 gL2 −1 � φ12 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='18) In the untwisted sector of the orbifold CFT, we have L(r) n = N � a=1 e2iπra/N � 1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 1 ⊗ Ln (a−th) ⊗ 1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 1 � , n ∈ Z , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='19) and the diagonal untwisted operator associated to φ12 is Φ12 = φ12 ⊗ · · · ⊗ φ12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='20) Using an inverse discrete Fourier transform, one easily finds, for any r ∈ ZN, � L(r) −2 − 1 Ng N−1 � s=0 L(s) −1L(r−s) −1 � Φ12 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='21) When inserted into a correlation function, the modes L(0) m act as linear differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The treatment of the modes L(r) m with r ̸= 0 introduces an additional difficulty, that we will address case by case, with the help of orbifold Ward identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3 The induction procedure The null-vectors of the mother CFT also determine the null vector conditions on twist operators in MN, through the induction procedure [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In the present work, we shall only be concerned with the twist sectors with charges [±1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In the notations of [70], induction can be expressed in terms of a norm-preserving, invertible linear map Θ from the Hilbert space of the mother CFT to that of the twist sector [1], defined by Θ|φ⟩ = |σφ⟩ , ΘLmΘ−1 = N � L(−m) m/N − hσ δm0 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='22) where φ is any primary operator in the mother CFT, and σφ is the associated composite twist operator in the orbifold CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The simplest application to null-vectors is the case of the identity: L−1 · 1 = 0 ⇒ L(1) −1/N · σ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='23) For a degenerate operator at level two, applying the induction map on (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='18) yields � L(2) −2/N − N g (L(1) −1/N)2 � σ12 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='24) The corresponding null-vector conditions for the operators σ† and σ† 12 are easily obtained by conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 The cyclic orbifold on the upper half plane To construct the cyclic orbifold BCFT, we will work on the upper half-plane H, with the boundary along the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We parametrize H by z = x + iy with x ∈ R and y > 0, and we impose the gluing condition on the boundary for the stress-energy tensor components: T (0)(x) = ¯T (0)(x) for x ∈ R , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='25) which ensures that the boundary is conformal i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=', preserves a copy of the Virasoro algebra [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The ZN orbifold, however, has an extended symmetry, and we must choose if and how the components of the additional currents T (r̸=0) are glued at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Our usage of the replica trick provides a clear indication for these choices: since we are considering N copies of the same mother BCFT, we must impose the gluing condition Ta(x) = ¯Ta(x) on each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' By taking the Fourier transform of this relation, we find that in the orbifold CFT we are effectively imposing: T (r)(x) = ¯T (r)(x) for x ∈ R , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='26) for all the discrete Fourier modes of the stress-energy tensor components defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This implies that the boundary preserves a full copy of the OVir algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' By the same reasoning on CFT replicas, the orbifold boundary states we are interested in correspond to having the same conformal BC on the N copies of the mother CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' They are simply given by |α⟩⊗N and |β⟩⊗N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' On the upper half-plane, we will set the conformal BC α on the positive real axis x > 0 and the conformal BC β on x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' To implement such mixed conformal BC in a BCFT, we will have to work with the formalism of boundary condition changing operators [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' These operators, restricted to live on the boundary, are placed at the points of suture of regions of different BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The full operator algebra of a BCFT is then formed by considering the OPEs between both BCCOs and bulk operators, as detailed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' For a given pair of conformal BCs (α, β), there can be several primary BCCOs implementing the change α → β: we denote such an operator ψ(αβ) h , where h specifies its conformal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The most relevant BCCO implementing α → β is simply referred to as ψ(αβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In the ZN orbifold CFT, we will be concerned with the calculation of correlators with insertions of diagonal BCCOs, namely : Ψ(αβ) h = ψ(αβ) h ⊗ · · · ⊗ ψ(αβ) h � �� � N times .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='27) Then, orbifold correlators with mixed BC are obtained by inserting the most relevant diagonal BCCO: ⟨O1(z1, ¯z1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' On(zn, ¯zn)⟩αβ H = ⟨Ψ(αβ)(∞) O1(z1, ¯z1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' On(zn, ¯zn)Ψ(βα)(0)⟩H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='28) By Cardy’s doubling trick [72], [54], such (n + 2)-point correlators satisfy the same Ward iden- tities as any of the (2n + 2)-point conformal blocks on the Riemann sphere C with external operators Φ(∞), O1(z1), O1(¯z1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' , On(zn), On(¯zn), Φ(0) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='29) where Oi(¯z) is the antiholomorphic counterpart of Oi(z), and Φ(z) is the holomorphic part of the diagonal primary operator defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='10), with the conformal dimension of Ψ(αβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In more precise terms, Oi is the operator conjugate to Oi with respect to the symmetry algebra 11 preserved by the boundary [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' For ZN twist operators, conjugation acts as σi = σ† i [39], so that the one-twist function ⟨σi(z, ¯z)⟩(αβ) H = ⟨Ψ(αβ)(∞)σi(z, ¯z)Ψ(αβ)(0)⟩H (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='30) satisfies the same Ward identities as the functions ¯z−2hσi × Fk(z/¯z), where Fk is the rescaled conformal block: Fk(η) = Φk σi(1) Φ(0) σ† i (η) Φ(∞) = ⟨Φ|σi(1)Pkσ† i (η)|Φ⟩ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='31) Pk is the projector onto the (A ⊗ ¯A)-module of Φk, and {Φk} is the set of allowed intermediary untwisted (diagonal or not) primary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In the following, it will also be necessary to consider the conformal blocks in the channel η → 1, namely �Fℓ(η) = Φℓ σi(1) Φ(0) Φ(∞) σ† i (η) = ⟨Φ|Φ(1)Pℓσ† i (1 − η)|σi⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='32) Using the Ward identities implied by the OVir algebra (discussed in detail in Section 3) for these functions, together with the null-vectors of the previous section, will allow us to extract an ODE that the functions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='31–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='32) satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' To understand the structure of the conformal blocks, we also need to define non-diagonal BCCOs, paralleling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='9): Ψ(αβ) [j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='jN] := 1 √ N N−1 � ℓ=0 (ψ(αβ) j1+ℓ ⊗ · · · ⊗ ψ(αβ) jN+ℓ) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='33) where the 1/ √ N factor ensures that the non-vanishing two-point functions of these BCCOs are normalized to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' These operators appear in the OPE of the diagonal boundary fields Ψ(αβ) j , and thus their conformal dimension determines the leading singular behaviour of the conformal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Naturally, we should now discuss the operator algebra of the ZN orbifold BCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3 Operator algebra of the cyclic orbifold BCFT In the orbifold BCFT, the operator algebra consists of OPEs of three types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' First, there is the operator subalgebra of bulk operators, inherited from the ZN orbifold CFT on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We shall not directly use the structure constants of this subalgebra in this work, but they have been discussed in [40, 42, 71, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The second type of OPE we need to consider in the orbifold BCFT, is the bulk-boundary OPE which encapsulates the singular behaviour of a bulk field as it approaches a conformal boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In our calculations, we will only need to work with the OPEs of primary twist operators σi(z, ¯z) as they are sent towards a conformal boundary α: σi(x, y) ∼ y→0 � j A(α) σi,Ψj (2y)hj−2hσi Ψ(αα) j (x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='34) where the sum runs over all the boundary operators Ψ(αα) j and their descendants under OVir, and we have denoted the bulk-boundary structure constants by A(α) σi,Ψj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Finally, we need to 12 consider the OPEs of orbifold boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' For generic diagonal BCCOs, this takes the form Ψ(αβ) i1 (x1)Ψ(βγ) i2 (x2) ∼ x1→x2 � j B(βγ)Ψj Ψi1,Ψi2(x1 − x2)−hi1−hi2+hjΨ(αγ) j (x2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='35) with the index j running over all the orbifold BCCOs interpolating between the conformal boundary conditions α and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We have denoted the boundary-boundary structure constants by B(αβγ)Ψj Ψi1,Ψi2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' To calculate the structure constants of the OPEs that are relevant for the present work, we will need to use factorization and unfolding arguments for the correlator that determine them, along the lines of [70], [75] and [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 Calculation of boundary-boundary structure constants Let us consider the calculation of boundary-boundary structure constants of the type B(ββα)Ψk Ψ∗,Ψj , where Ψ∗ denotes a generic untwisted orbifold primary BCCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We can express this as a three- point function on the upper half-plane H: B(ββα)Ψk Ψ∗,Ψj = ⟨Ψ(αβ) k (∞)Ψ(ββ) j (1)Ψ(βα) ∗ (0)⟩H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='36) Since there are no twist insertions in the above correlator, it just factorizes into a linear com- bination of products of mother BCFT three-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Let us first consider the case of a diagonal BCCO, with Ψ(βα) ∗ = Ψ(βα) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Then, the orbifold correlator factorizes into mother CFT three-point functions as: B(ββα)Ψk Ψi,Ψj = � ⟨ψ(αβ) k (∞)ψ(ββ) j (1)ψ(βα) i (0)⟩H �N , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='37) so we find a simple expression for these coefficients, in terms of mother BCFT boundary- boundary structure constants: B(ββα)Ψk Ψi,Ψj = � B(ββα) k ij �N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='38) By similar considerations, the structure constants involving a non-diagonal BCCO Ψ(βα) [i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='iN] can be expressed as: B(ββα)Ψk Ψ[i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='iN ],Ψj = √ N N � a=1 B(ββα) k iaj (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='39) The rest of the boundary-boundary structure constants of untwisted BCCOs can similarly be expressed in terms of mother BCFT quantities, but we will not need them in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 Orbifold bulk-boundary structure constants The first bulk-boundary structure constant we need to calculate is A(α) σ,Ψ1, where Ψ(αα) 1 is just the identity boundary field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This can be expressed as the one-point function on the unit disk D: A(α) σ,Ψ1 = ⟨σ(0, 0)⟩α D , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='40) which is just the ratio of mother CFT partition functions: ⟨σ(0, 0)⟩α D = Z(α) DN � Z(α) D �N , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='41) 13 where DN denotes the N-th covering of the unit disk with branch points at 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' As shown in [76], we can express (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='40) in terms of the ground state degeneracy gα = ⟨0|α⟩ [69] (which is defined as the overlap between the vacuum state |0⟩ and the boundary state |α⟩ in the mother BCFT): A(α) σ,Ψ1 = g1−N α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='42) Using this result, we can calculate the one-point structure constants of composite twist operators σi, by using the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='13) and the relation between twist correlators on the disk D and the mother CFT partition function on DN, which simply gives: A(α) σi,Ψ1 = A(α) σ,Ψ1 Aα φi , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='43) where Aα φi is the mother CFT one-point structure constant of φi with conformal boundary condition α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The proof is relegated to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Extending these results to more complicated bulk-boundary structure constants A(α) σ[k] i ,Ψj for generic choices of mother CFT and cyclic group ZN is not usually straightforward and depends on our knowledge of correlation functions in the mother CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' For example, in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 we calculate the structure constant A(α) σ,Ψ13 in the Z2 orbifold BCFT, since it can be expressed in terms of a two-point function of boundary operators in the mother CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' For generic N and composite twist operator σ[k] i , knowledge of higher-point correlators in the mother CFT is required to compute such structure constants through the same unfolding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 3 Differential equations in the Z2 and Z3 orbifold BCFT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 Setup for the calculations We consider the case of a generic BCFT, with central charge c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The model is defined on the upper half plane, with conformal boundary conditions α and β set on the negative ℜ(z) < 0 and positive ℜ(z) > 0 parts of the real axis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We will work, for the entirety of this section, under the assumption that the most relevant BCCO interpolating between these boundary conditions is ψ(αβ) 12 , with conformal dimension h12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This implies that the BCCO has a null-vector at level 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Of course, our results also apply to the case where the BCCO is ψ(αβ) 21 , up to changing g → 1/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In the ZN orbifold of this theory, we will consider one-point correlators of generic composite twist operators σi of twist charge [k = 1], in a background with mixed BC α and β, corresponding to the replicated boundary conditions of the mother BCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The change in boundary conditions in the orbifold theory will be implemented by the diagonal BCCO Ψ(αβ) 12 defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='27), with conformal dimension hΨ12 = Nh12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Since we will aim to compare our CFT results with lattice data in Section 4, we will define our twist correlator on an infinite strip S of width L, parametrized by the complex coordinate w = u + iv, with u ∈ [0, L] and v ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The conformal boundary conditions on the u = 0 and u = L sides of the strip are set to be α and β, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We will consider correlators with a twist σi inserted at w = ℓ: ⟨σi(ℓ, ℓ)⟩αβ S , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) where ℓ is measured from the boundary β, in accordance with Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 14 This correlator is now mapped to the upper half plane, through: w = −iL π ln z , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2) and expressed, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='28), as: ⟨σi(z, ¯z)⟩αβ H = ⟨Ψ(αβ) 12 (∞) σi(z, ¯z)Ψ(βα) 12 (0)⟩H , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3) with z = exp iπℓ/L in terms of strip coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Using the information about the operator algebra of the orbifold BCFT we have presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3, we can write the following block expansion for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) ⟨σi(ℓ, ℓ)⟩αβ S = J � ℓ A(β) σi,ΨℓB(ββα)Ψ12 Ψℓ,Ψ12 �Fℓ(η) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) where η = z/¯z = exp (2πiℓ/L), and J = (L¯z/π)−2hσi is the combined Jacobian associated to the M¨obius map ζ �→ ζ/¯z that takes (0, z, ¯z, ∞) �→ (0, η, 1, ∞) and the map w �→ z from the strip to the upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We recall that the �Fℓ’s are the conformal blocks in the channel η → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' As per Cardy’s doubling argument [72], the functions ˜Fk(η) are four-point conformal blocks (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='31) with Φ = Φ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' To proceed, we need to determine the differential equation satisfied by these functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' To this end, we will use a combination of the null-vector conditions and the orbifold Ward identities derived in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 The function ⟨Ψ12 · σ · Ψ12⟩ in a generic Z2 orbifold Following the general approach described above, the function ⟨Ψ12(∞)σ(z, ¯z)Ψ12(0)⟩H is given, up to the overall factor ¯z−2hσ, by a linear combination of the conformal blocks Fk(η) = ⟨Φ12|σ(1)Pkσ(η)|Φ12⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5) It turns out that this family of conformal blocks was already studied in [42], for the calculation of the single-interval R´enyi entropy in the excited state |Φ12⟩ with periodic BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Let us recall how the derivation of the corresponding ODE goes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We use the null-vectors at level two of the untwisted chiral state |Φ12⟩, given by: � L(0) −2 − 1 2g � L(0) −1 �2 − 1 2g � L(1) −1 �2� |Φ12⟩ ≡ 0, � L(1) −2 − 1 gL(0) −1L(1) −1 � |Φ12⟩ ≡ 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6) and the null vector at level 1/N of the bare twist operator σ: L(1) −1/2 · σ ≡ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7) We combine these with the orbifold Ward identity for the chiral correlator: G(1)(w, η) = ⟨Φ12| σ(1)Pkσ(η)T (1)(w)L(1) −1 |Φ12⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8) 15 with (m1, m2, m3, m4) = (0, −1/2, −1/2, −1) in the notation of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This gives, after taking into account (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7): � p=0,1,2 dp ⟨Φ12| σ(1)Pkσ(η)L(1) −p+2L(1) −1 |Φ12⟩ = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='9) with dp calculated from the series (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' By substituting the null vectors (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='18–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7) and employing the identity (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7), one obtains the differential equation: 64g2η2(η − 1)2 ∂2 ηF + 16gη(η − 1) � (−14g2 + 23g − 6)η + 2g(1 − 4g) � ∂ηF + (3g − 2) � +3(5g − 6)(1 − 2g)2η2 + 12g(1 − 2g)η + 16g2(g − 1) � F = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='10) whose Riemann scheme is given by: 0 1 ∞ −2h12 −2hσ 2hσ − 2h12 −2h12 + h13/2 −2hσ + 2h13 2hσ − 2h12 + h13/2 This corresponds to the intermediary states {σ, σ13} in the channels η → 0 and η → ∞, and {1, Φ13} in the channel η → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Note that, when the mother CFT is a minimal model Mp,p′, one can check for various values of (p, p′) that these are exactly the intermediary states allowed by the orbifold fusion given in F, and that they all have multiplicity one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' To proceed, one can define the shifted function f(η): f(η) = (1 − η)2hση2h12F(η) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='11) and substitute in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='10) to find that f(η) satisfies a second order hypergeometric equation (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) with parameters: a = 2 − 3g , b = 3 2 − 2g , c = 3 2 − g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='12) We can work with the basis (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3) of solutions around η = 1 for this hypergeometric equation— in the block expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4), this corresponds to approaching the twist operator to the boundary β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Thus, the conformal blocks �Fℓ(η) we seek are: �F1(η) = (1 − η)−2hση−2h122F1(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' a + b − c + 1 | 1 − η) , �F13(η) = (1 − η)−2hσ+2h13η−2h122F1(c − b, c − a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' c − a − b + 1 | 1 − η) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='13) and they are normalized to be of the form �Fℓ(η) ∼ (1 − η)−2hσ+h(1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' ) as η → 1 , where h is the conformal dimension of the internal operator of the conformal block in this channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The bulk-boundary fusion rules corresponding to the exponents are σ ��� β → Ψ1 + Ψ13 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='14) as σ is approached to the conformal boundary β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Substituting the blocks and the expressions for the orbifold BCFT structure constants in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) one finds the expression of the one-point twist correlator ⟨σ(z, ¯z)⟩αβ H = ¯z−2hσ � A(β) σ,Ψ1B(ββα)Ψ12 Ψ1,Ψ12 ˜F1(η) + A(β) σ,Ψ13B(ββα)Ψ12 Ψ13,Ψ12 ˜F13(η) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='15) 16 where the various structure constants are expressed in terms of the mother BCFT data as A(β) σ,Ψ1 = A(β) σ,Ψ13 = g−1 β , B(ββα)Ψ12 Ψ1,Ψ12 = 1 , B(ββα)Ψ12 Ψ13,Ψ12 = � B(ββα) ψ12 ψ13ψ12 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='16) It is interesting now to observe that, for some pairs of conformal BCs (α, β), the block expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='15) greatly simplifies because the boundary-boundary structure constant B(ββα)Ψ12 Ψ13,Ψ12 vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' At the level of the mother BCFT, this is equivalent to demanding that the operator ψ(ββ) 13 is not allowed in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' For BCFTs based on A-series minimal models M(p, p′), this holds for any pair of mixed conformal BCs (α, β) ≡ (φr2, φr1), labelled by bulk primary fields with 1 ≤ r < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' One can use well-established results about fusion rules in such models [77], to check that: φ12 ∈ φr1 × φr2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='17) so that these BCs are interpolated by ψ(αβ) 12 and φ13 ̸∈ φr1 × φr1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='18) which implies that ψ(ββ) 13 is not in the boundary operator spectrum of the BCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Under these conditions, the correlator in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='15) simplifies to: ⟨σ(z, ¯z)⟩αβ H = ηhσ � g−1 φr1 ˜F1(η) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='19) so that one finds the second R´enyi entropy in the strip setup to be: S(α,β) 2 (ℓ) ∼ c 8 log 2L πa sin �πℓ L � + log gφr1 − log � η−2h122F1 (2 − 3g, 3/2 − g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 3 − 4g | 1 − η) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='20) It is now interesting to check that the theoretical prediction for this kind of mixed BC has the expected behaviour near the φr1 and φr2 boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' It is known [2] that the second R´enyi entropy of an interval ℓ touching one of the identical boundaries of a finite system of size L is given by: S(α,α) 2 ([0, ℓ]) = c 8 log �2L π sin �πℓ L �� + log gα (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='21) We then anticipate that the mixed BC result (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='20) will interpolate between S(φr1,φr1) 2 and S(φr2,φr2) 2 as ℓ → 0 and ℓ → L Furthermore, suppose we consider the difference in second R´enyi entropies between two mixed BC setups (φr2, φr1) and (φr′2, φr′1) for the same bulk CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We then find the following universal result: ∆S2 = S(φr′2,φr′1) 2 − S(φr2,φr1) 2 = log gφr1 gφr′1 = log gφr2 gφr′2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='22) where the latter equation follows from the expression of gφr,s = Sφr,s,φ0/�S1,φ0 [69] in terms of S-matrix elements of minimal models [77] - here φ0 denotes the field with the lowest conformal dimension of the diagonal bulk CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This entropy difference is thus determined, in these cases, by the identical BC expressions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='21) that describe the asymptotic behaviour near the boundaries of the mixed BC setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We illustrate these ideas graphically for the A-series minimal model M(6, 5) in Figure 2, by plotting the second R´enyi entropies shifted by L2hσ for two mixed BC setups (φ12, φ11) and (φ32, φ31) for the BCFTs based on M(6, 5), together with the relevant identical BC setups for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='0 /L 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5 L2h ( , ) 2 ([0, ]) CFT mixed ( 11, 12) CFT 11 BC CFT 12 BC CFT mixed ( 31, 32) CFT 31 BC CFT 32 BC Figure 2: Shifted second R´enyi entropies for the mixed BC setups (φ12, φ11) and (φ32, φ31) for the BCFTs based on M(6, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The difference between the two curves is constant with respect to the interval size, and thusly fixed by their respective asymptotic behaviours around ℓ → 0 and ℓ → L 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3 The function ⟨Ψ12 · σh · Ψ12⟩ in a generic Z2 orbifold From the perspective of critical quantum chains, the result in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='15) only determines the leading contribution to the second R´enyi entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' To understand finite-size corrections to this result, we should also study the one-point function of subleading primary twist operators, namely ⟨σh(z, ¯z)⟩αβ H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We shall derive an ODE for the conformal blocks Fk(η) = ⟨Φ12|σh(1)Pkσh(η)|Φ12⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='23) Here we consider the case of a generic composite twist operator σh with conformal dimension �h = hσ + h N , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='24) and thus we do not assume any null-vector condition on σh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Besides the null-vector conditions at level two (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6), we will need the null-vector at level three in the module of |Φ12⟩: L(1) −2L(1) −1 |Φ12⟩ = � −L(0) −3 + 1 g � 2gL(0) −1L(0) −2 − � L(0) −1 �3�� |Φ12⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='25) and two Ward identities obtained from: ⟨Φ12| σh(1)Pkσh(η)T (1)(z)L(1) −1 |Φ12⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='26) with (m1, m2, m3, m4) = (−1, 1/2, 1/2, −2): a0|1 ⟨Φ12| L(1) 1 σh(1)Pkσh(η)L(1) −1 |Φ12⟩ = 3 � p=0 dp|1 ⟨Φ12| σh(1)Pkσh(η)L(1) −2+pL(1) −1 |Φ12⟩ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='27) 18 0 1 ∞ −2h12 −2�hα 2�hα − 2h12 −2h12 + 1 2 −2�hα + h13 2�hα − 2h12 + 1 2 −�hα − 2h12 + �hα+b −2�hα + 2h13 �hα − 2h12 + �hα+b −�hα − 2h12 + �hα−b −2�hα + 2h13 + 2 �hα − 2h12 + �hα−b Table 1: Singular exponents around η = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' ∞ and (m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' m3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' m4) = (−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' −1) : a0|2 ⟨Φ12| L(1) 2 σh(1)Pkσh(x)L(1) −1 |Φ12⟩ + a1|2 ⟨Φ12| L(1) 1 σh(1)Pkσh(x)L(1) −1 |Φ12⟩ = = d0|2 ⟨Φ12| σh(1)Pkσh(η)L(1) −1L(1) −1 |Φ12⟩ + d1|2 ⟨Φ12| σh(1)Pkσh(η)L(1) 0 L(1) −1 |Φ12⟩ + + d2|2 ⟨Φ12| σh(1)Pkσh(η)L(1) 1 L(1) −1 |Φ12⟩ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='28) Putting everything together, and applying the change of function F(η) = η−2h12 (1 − η)4h12−2�h f(η) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='29) we obtain the fourth-order ODE (η − 1)4η3 ∂4 ηf + 1 2(η − 1)3η2 [(2g + 13)η + (2g − 11)] ∂3 ηf − 1 8(η − 1)2η � (16g�h + 6g2 − 45g − 60)η2 + (20g2 + 34g + 96)η + (16g�h + 6g2 + 3g − 36) � ∂2 ηf − g 16(η − 1) � (48�h + 18g − 75)η3 + (16g2 − 18g − 112�h + 167)η2 +(80�h + 16g2 − 74g − 53)η + (−16�h − 6g + 9) � ∂ηf + g 8 � (16g2�h + 6g3 − 13g2 + 4)η2 + (−32g2�h + 12g3 + 34g2 − 64g + 24)η +(24 + 16g2�h + 6g3 − 13g2 + 4) � f = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='30) At this stage, it will be convenient to use the Coulomb-Gas parametrization to analyse the local exponents of the ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Recall the relation between the mother CFT central charge and the parameter g: c = 1 − 24Q2 , Q = 1 2(1/b − b) , b = √g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='31) The conformal dimensions in the mother CFT are parametrized by the vertex charge α as hα = α(α − 2Q) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='32) and we use the shorthand notation for the conformal dimension of composite twisted operators �hα = hσ + hα N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='33) In this parametrization, the Riemann scheme for F(η) is given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 19 These exponents correspond to the intermediary states (counted with their multiplicities): {1, [1, φ13], Φ13, Φ13} in the channel η → 1 , {σh, L(1) −1/2 · σh, σh′, σh′′} in the channels η → 0 and η → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='34) Here, we have defined h′ = hα+b and h′′ = hα−b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Recall that the conformal blocks are labelled by primary operators under the neutral subalgebra A, and that L(1) −1/2 ·σh is one of these operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' When the mother CFT is a minimal model, one can check on various examples that the orbifold fusion rules derived in [70] from the Verlinde formula are consistent with these intermediary states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' While an analytic solution to the differential equation is not known, one can determine the conformal blocks Fk(η) around η = 0 and �F(η) around η = 1 numerically to arbitrary precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Assuming this step has been performed, all that is left is to calculate the structure constants in the block expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The boundary-boundary structure constants are calculated from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='38) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='39), while the bulk-boundary structure constants can be calculated analytically through unfolding, as shown, for some cases in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' For numerical studies, however, it is simpler to bootstrap some of the coefficients in the block expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) rather than to calculate all of them analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' To implement this method, as detailed in [78], one needs to compare the block expansions in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) with the block expansion corresponding to sending the twist field to the part of the boundary, endowed with the α BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The crucial point here is that the conformal blocks Fk(η) are branched functions on C, with branch points {0, 1, ∞}, and, thus, sending the twist field to the boundary with BC α is equivalent to crossing to the other branch of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This is marked by appending the phase factor e2πi to the variable η to get: ⟨σh(z, ¯z)⟩αβ S = JSL(2,C) � ℓ A(α) σh,ΨℓB(ααβ)Ψ12 Ψℓ,Ψ12 �Fℓ(e2πiη) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='35) To proceed, one needs to find the monodromy matrix X around zero for the basis �Fℓ(η), which encodes the behaviour of the conformal blocks as the branch cut is crossed: �Fℓ(e2πiη) = � m Xℓm �Fm(η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='36) Since the monodromy of the blocks �Fℓ around zero is non-diagonal, we can use the fusing matrix Pij to express the blocks �Fℓ in terms of a basis of the blocks Fk, which have diagonal monodromy around η = 0: �Fℓ(η) = � k PℓkFk(η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='37) The blocks Fk(η) simply acquire a phase under z → e2πiz, so their monodromy matrix Y is diagonal: Fk(e2iπη) = � j Ykj Fj(η) , Ykj = δkj exp � 2πi � −�hα − 2h12 + �hk �� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='38) where the exponents in the exponential above, are simply read off from the Riemann scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Then, the monodromy matrix of the blocks �Fℓ(η) is found from the matrix product: X = P · Y · P −1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='39) 20 which allows us to compare the block expansions in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='35) to find a duality relation, of the type presented in [78]: A(β) σh,ΨiB(ββα)Ψi Ψ12,Ψ12 = � j A(α) σh,ΨjB(ααβ)Ψj Ψ12,Ψ12 Xji .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='40) Using the numerical determinations for Fk(η) and �Fℓ(η) , one can find a good estimate for the fusing matrix Pij, and, consequently, Xij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' A more fleshed out example of how the determination of Pij works has been relegated to the Appendix H, where this equation is used for the case of the Z2 orbifold of the three-state Potts model BCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' After solving the linear system in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='40) one can evaluate the unknown structure constants A(β) σh,Ψi and A(α) σh,Ψi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' At this point, we stress that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='40) gives, at most, four constraints between the unknown structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' To fully determine all these quantities, one should calculate the remaining four structure constants through other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4 The function ⟨Ψ12 · σ · Ψ12⟩ in a generic Z3 orbifold Here the relevant conformal blocks are Fk(η) = ⟨Φ12|σ(1)Pkσ†(η)|Φ12⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='41) We give the null-vectors of |Φ12⟩ at levels two and three: L(r) −2 |Φ12⟩ = 1 3g 2 � s=0 L(r−s) −1 L(s) −1 |Φ12⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='42) L(3−r) −1 L(r) −2 |Φ12⟩ = 1 3g � 2L(0) −1L(1) −1L(2) −1 + � L(3−r) −1 �3� |Φ12⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='43) for r ∈ {0, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We will also need the null-vectors for the out-state ⟨Φ12|, which can be obtained by Hermitian conjugation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' To derive an ODE for the conformal blocks, we had to employ seven orbifold Ward identities, together with six of the null-vector conditions above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' To not overload the presentation of this section with technical details, we relegate the specifics of the derivation to Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We apply the change of function F(η) = η−8h12/3 (1 − η)16h12/3−2hσ f(η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='44) The function f satisfies the ODE (η − 1)3η2 ∂3 ηf + (η − 1)2η [(g + 3)η + (g − 3)] ∂2 ηf + 2 9(η − 1) � 2(2 + 3g)η2 − 2(7 − 15g + 18g2)η + (4 − 3g) � ∂ηf + 4 27(1 − 6g)(2 − 3g)(η + 1) f = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='45) The Riemann scheme for F is 0 1 ∞ − 8 3h12 −2hσ 2hσ − 8 3h12 − 8 3h12 + 1 3 −2hσ + 2h13 2hσ − 8 3h12 + 1 3 −3h12 + h14 3 −2hσ + 3h13 2hσ − 3h12 + h14 3 21 The local exponents correspond to the intermediary states: {1, [1, φ13, φ13], Φ13} in the channel η → 1 , {σ12, L(1) −1/3 · σ12, σ14} in the channels η → 0 and η → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='46) In the orbifold BCFT, this translates into the following fusion rules for the twist operator with the boundary β: σ1 ����� β → Ψ(ββ) 1 + Ψ(ββ) [1,φ13,φ13] + Ψ(ββ) 13 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='47) The analytic solutions to the differential equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='45) are not known, but they can be evaluated numerically, to arbitrary precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Then, one can use the bootstrap to determine some relations between the unknown structure constants in the expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4), as outlined in the previous section, and determine the rest analytically, by unfolding methods, to complete the calculation of the mixed BC correlator of the bare twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We note that a fourth order differential equation that the correlator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='28) satisfies has already been found in [75], where it plays a role in the determination of the leading contribution to the third R´enyi entropy of an excited state in a periodic 1D critical chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' As predicted in [75], there is no degeneracy in the exponents in the more constraining third order differential equation we have found here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Note that these exponents are the ones expected from the orbifold fusion rules [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5 The function ⟨Ψ12 · σ13 · Ψ12⟩ in the Z3 orbifold of the Ising model In this section, we will work with the Z3 orbifold of the Ising BCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The bulk primary fields of this BCFT are φ11 ≡ 1, φ12 ≡ s and φ13 ≡ ε with hs = 1/16 and hε = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We will keep labelling the fields by their Kac indices, to not overcomplicate the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We will provide here an alternative method for finding a differential equation for the one- point function: ⟨σ13(z, ¯z)⟩f+ H , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='48) where the orbifold conformal boundary conditions α = f and β = + correspond to setting fixed and free BC respectively, on all the copies of the Ising mother BCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The diagonal BCCO Ψ(f+) 12 is the one interpolating between them in the orbifold, since in the Ising BCFT only the ψ(f+) 12 primary boundary field can change between the CBCs (+) ↔ (f) [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' As in the previous sections, we aim to find a differential equation satisfied by the conformal blocks Fk(η) = ⟨Φ12|σ13(1)Pkσ† 13(η)|Φ12⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='49) First, we use the fusion numbers in (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) to infer the dimension of the space of conformal blocks for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='48): � i N i σ13,σ† 13N Φ12 i,Φ12 = 2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='50) which means the differential equation we seek should be second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' By using the null-vectors induced on σ13 together with the right combination of Ward identities, one should be able to rigorously derive it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Instead, we will assume this equation exists and is of Fuchsian type – a linear homogenous ODE whose three singular points are 22 regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The latter assumption is based on the observation that the method exploited in the previous sections relies on expressing orbifold modes L(r̸=0) m in terms of Virasoro generators, whose combined differential action on correlators is well-understood in the literature [77], [79] to be of Fuchsian type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Now, using the fusion numbers of (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1), the fusion rules are σ13 × Φ12 → σ12 + L(2) −2/3 · σ12 , σ13 × σ† 13 → 1 + [1, φ13, φ13] , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='51) so we can determine the asymptotic behaviour of the solutions around the regular singular points η ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' ∞} of the differential equation and infer the Riemann scheme: 0 1 ∞ −hσ13 − 3h12 + hσ12 −2hσ13 hσ13 − 3h12 + hσ12 −hσ13 − 3h12 + hσ12 + 2 3 −2hσ13 + 2h13 hσ13 − 3h12 + hσ12 + 2 3 One can readily check that the entries of this Riemann scheme sum up to one,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' so,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' by virtue of a general theorem on Fuchsian ODEs (see [80]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' there is a unique second-order Fuchsian ODE with this set of singular exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' If we define the shifted function f(η): f(η) = ηhσ13+3h12−hσ12(1 − η)2hσ13F(η) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='52) we find, by the same considerations, that it should satisfy a second-order Fuchsian differential equation with the Riemann scheme 0 1 ∞ 0 0 −2hσ13 − 6h12 + 2hσ12 2 3 2h13 −2hσ13 − 6h12 + 2hσ12 + 2 3 This is just the canonical Riemann scheme of a hypergeometric differential equation (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1), with coefficients: a = −2hσ13 − 6h12 + 2hσ12 = −2/3 , b = −2hσ13 − 6h12 + 2hσ12 + 2 3 = 0 , c = 1 3 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='53) in the conventions of Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We notice that the exponents in the η → 1 channel are spaced by one, so we will have to deal with the degenerate exponents to arrive at a closed form solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' To do this, we will use the basis of solutions in the η → 0 channel – given in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2) – to construct a linearly independent basis of solutions around η → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The solutions for f can be simplified, in this case, to: I1(η) = 1 , I2(η) = η2/3 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='54) which gives the conformal blocks around η → 0: F1(η) = η−1/3(1 − η)−4/9 , F2(η) = η1/3(1 − η)−4/9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='55) in our normalisation convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 23 To build the basis of solutions around η = 1, we look for the linear combinations ˜Fi(η) = � j P −1 ij Fj(η) that have the following series expansion around η = 1 : ˜F1(η) = (1 − η)−4/9 � 1 + O[(1 − η)2] � , ˜F2(η) ∼ (1 − η)5/9 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='56) since the power series associated to the orbifold identity should have no (1 − η) term due to the null-vectors L(r) −1·1 ≡ 0, and the both solutions should have the leading coefficient normalised to one, in our convention for the conformal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' With these requirements, one finds the fusing matrix P −1 ij to be: P −1 = 1 2 � 1 1 3 −3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='57) Thus, the conformal blocks of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='48) around η = 1 are found to be: ˜F1(η) = η−1/3 + η1/3 2(1 − η)4/9 , ˜F2(η) = 3(η−1/3 − η1/3) 2(1 − η)4/9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='58) For the physical correlation function, we write ⟨σ13(z, ¯z)⟩f+ H = ¯z−2hσ1,3 � A(+) σ13,Ψ1B(++f)Ψ12 Ψ1,Ψ12 ˜F1(η) + A(+) σ13,[ψ1,ψ13,ψ13]B(++f)Ψ12 [ψ1,ψ13,ψ13],Ψ12 ˜F2(η) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='59) Finally, we observe that B(++f)ψ12 ψ13ψ12 vanishes, and hence B(++f)Ψ12 [ψ1,ψ13,ψ13],Ψ12 = 0, so the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='59) simplifies to: ⟨σ13(z, ¯z)⟩f+ H = 25/9 × cos(2θ/3) (r sin θ)4/9 , z = reiθ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='60) where we have also used: A(+) σ13,Ψ1 = g−2 + , B(++f)Ψ12 Ψ1,Ψ12 = 1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='61) and the value of the ground-state degeneracy for fixed BC g+ = 1/ √ 2 in the Ising BCFT [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6 More hypergeometric differential equations in the Ising cyclic orbifold BCFTs We have managed, in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4 to obtain differential equations for cyclic orbifolds of generic mother BCFTs, but have not been able to provide analytic solutions for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' One can, however, find second order differential equations for particular choices of Mp,p′ and composite twist fields (for the correlators of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3), in the manner presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5, which allow us to exactly determine the correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Since we want to compare the results of this section with lattice data of the critical Ising spin chain with mixed BC, it will be particularly satisfying to find such equations for the cyclic orbifolds of the Ising BCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Let’s first consider the correlator: ⟨σ1,3(z, ¯z)⟩αβ N=2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='62) in the Z2 Ising orbifold BCFT which should satisfy, up to a M¨obius map, the same differential equation as: ⟨Φ1,2| σ1,3(1)σ1,3(η) |Φ1,2⟩ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='63) 24 The orbifold fusion rules of [40], imply that the space of conformal blocks is two-dimensional since: � i N i σ1,3,σ1,3N Φ1,2 i,Φ1,2 = 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='64) By the same type of arguments and assumptions as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5, we infer that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='63) satisfies a second order Fuchsian differential equation with the following Riemann scheme: 0 1 ∞ −hσ1,3 − 2h1,2 + hσ1 −2hσ1,3 hσ1,3 − 2h1,2 + hσ1,1 −hσ1,3 − 2h1,2 + hσ1,3 + 1 2 −2hσ1,3 + 2h1,3 hσ1,3 − 2h1,2 + hσ1,2 + 1 2 so that we eventually find the one-point twist correlator to be ⟨σ1,3(z, ¯z)⟩αβ (N=2) = ¯z−2hσ1,3g−1 + ˜FN=2 Ψ1 (η) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='65) with ˜F(N=2) Ψ1 (η) = 1 + η3/4 2(1 − η)9/16η3/8 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='66) Finally, we can find an exact expression for the bare twist correlator: ⟨σ1(z, ¯z)⟩αβ N=3 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='67) in the Z3 Ising orbifold BCFT since it also satisfies a second order differential equation with Riemann scheme: 0 1 ∞ −hσ1 − 3h1,2 + hσ1,2 −2hσ1 hσ1 − 3h1,2+hσ1,2 −hσ1 − 3h1,2 + hσ1,2 + 1 3 −2hσ1 +2h1,3 hσ1 − 3hΨ1,2+hσ1,2 + 1 3 We find: ⟨σ1(z, ¯z)⟩αβ N=3 = ¯z−1/9g−2 + ˜FN=3 Ψ1 (η) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='68) with ˜FN=3 Ψ1 (η) = 1 + η1/3 2(1 − η)1/9η1/6 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='69) Other results for the Ising BCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We have also obtained results specific to the Z2 and Z3 orbifolds of the Ising BCFT with fixed mixed BC with α = + and β = −, for which the most relevant primary BCCO is ψ(+−) 2,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Since these results are not based on deriving differential equations, it felt thematically appropriate to leave their presentation for the Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 4 Numerical checks and finite-size corrections in quantum chains To provide an independent appraisal of the validity of our CFT results, we have performed a numerical analysis on the Ising and three-state Potts open quantum chains for different settings of mixed BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Once finite-size effects are properly accounted for, the validity of the CFT results becomes apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 25 We should note that the R´enyi entropies in the Ising case have already been obtained, for generic N in the work of [57], through a different approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We found that our analytical calculations (for N = 2, 3) are compatible with their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Furthermore, by studying the finite-size corrections to their result, we manage to quanti- tatively understand the deviation of the chain data from the leading CFT prediction in the DMRG numerical analysis of [57], even for relatively large system sizes M ∼ 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Thus, when the subleading CFT contribution to the R´enyi entropy is taken into account, as our analysis shall show, the agreement with the lattice data is excellent, even for the small system sizes M ∼ 26 accessible to exact diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 The Ising quantum chain with mixed BC The Hamiltonian of the Ising quantum chain with open BC, describing M spins with generic BC at the boundary, is given by: Hαβ = − M−1 � j=1 sz jsz j+1 − h M � j=1 sx j − hαsz 1 − hβsz M , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) where sx,y,z j denote Pauli spin operators acting non-trivially at site j, and as identity at all the other sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We denote the lattice spacing by a, so that the length of the chain is L = Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The parameters hα, hβ denote external fields (in the z direction) acting at the boundary sites j = 1 and j = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The ground state of this Hamiltonian is then found by exact diagonalization (ED) for system sizes M ≤ 26, and from it, the R´enyi entropies are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' To take the scaling limit of the critical chain, we send M → ∞, a → 0 while keeping L fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In this limit, criticality is achieved in the bulk for h = 1, while each boundary admits three critical points hα, hβ ∈ {0, ±∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' From a CFT perspective, the scaling limit of the critical Ising chain with open boundaries is very well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' It is described by the BCFT with central charge c = 1/2 and a bulk operator spectrum consisting of three primary operators – the identity 1, energy ε and spin operators s – and their descendants [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The three boundary critical points correspond to the three conformal boundary conditions for the Ising BCFT, which, in the framework of radial quantization on the annulus, allow the construction of the following physical boundary states [53,77]: |f⟩ = |1⟩⟩ − |ϵ⟩⟩ (free BC) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2) |±⟩ = 1 √ 2|1⟩⟩ + 1 √ 2|ϵ⟩⟩ ± 1 21/4 |s⟩⟩ (fixed BC) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3) where |i⟩⟩ denotes the Ishibashi state [53] [81] corresponding to the primary operator i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The physical boundary states |α⟩ are in one-to-one correspondence with the primary fields of the bulk CFT 2: |f⟩ ↔ s and |±⟩ ↔ 1/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The boundary fields that interpolate between two conformal BCs can be inferred from this correspondence, as shown in [53], [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Thus, the spectrum of primary boundary fields ψ(αβ) i of the Ising BCFT is the one of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' On the discrete side, we are calculating the one-point correlator of the lattice twist operator �σ(m, n), where (m, n) are square-lattice coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In the scaling limit with a → 0, �σ(m, n) 2This statement is strictly true if the bulk CFT is diagonal, see [82] for a detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 26 (αβ) + − f + ψ1 ψε ψs − ψε ψ1 ψs f ψs ψs ψ1, ψε Table 2: Boundary operator spectrum of the Ising BCFT admits a local expansion into scaling operators of the corresponding orbifold CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The two most relevant terms in this expansion are: �σ(m, n) = A a2hσσ1(w, ¯w) + B a2hσεσε(w, ¯w) + less relevant terms , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) with the composite twist operator σε defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='13) and hσε = hσ + hε/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The integers (m, n) parametrize the lattice, and they are related to the continuum coordinate on the strip as w = (m + in)a, ¯w = (m − in)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We can take advantage of the translation invariance in the n direction to fix the ”time” coordinate of the lattice twist operators to be n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We will then denote their continuum coordinate by ℓ = ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The amplitudes A and B in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) are not universal quantities, so we cannot determine them by CFT techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' However, they are also independent of the global properties of the system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' choice of BC) so they can be found from a numerical analysis of the infinite Ising chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Here one can employ the free fermion techniques of [1] and the well-known analytical results for the R´enyi entropy of an interval in an infinite system [50,2] to fit for the values of A and B, with great accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We can now express the lattice one-point twist correlator with generic mixed BC as an expansion of CFT correlators: ⟨�σ(m, 0)⟩αβ = Aa2hσ⟨σ(ℓ, ℓ)⟩αβ SL + Ba2hσε⟨σε(ℓ, ℓ)⟩αβ SL + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5) Using the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2), we can make the dependence on system size in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5) explicit: ⟨�σ(m, 0)⟩αβ = A �M π �−2hσ ⟨σ(z, ¯z)⟩αβ H + B �M π �−2hσε ⟨σε(z, ¯z)⟩αβ H + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6) where z = exp(iπℓ/L), ¯z = exp(−iπℓ/L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In our computational setup, the system sizes accessi- ble through exact diagonalization are limited to M ≤ 26 and, since twist operators are placed between lattice sites, we have only considered even system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' With system sizes of this order of magnitude, finite-size corrections are quite strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The most relevant corrections we have found arise from the subleading scaling of the lattice twist operator, given in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The relative scaling of the subleading term with respect to the leading one is O � M−2hϵ/N� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Since we do not have access, numerically, to system sizes large enough to suppress these corrections, we had to take into account the first two terms in the expansion of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6) to find a good agreement with the lattice data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Furthermore, as the work of [57] suggests, the finite-size effects are still important, even at the much larger system sizes M ∼ 100 accessible through DMRG methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We mention that such subleading contributions to the lattice twist operator, which have been identified here from the operator spectrum of the Z2 cyclic orbifold, have previously been understood, through the path integral formalism on the corresponding replicated surface, under the name of “unusual corrections” [83,84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We give now the results in the Z2 orbifold for the correlators appearing in the expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5), for mixed fixed BC with α = +, β = − (calculated in Appendix D) and mixed free-fixed 27 BC with α = + and β = f: ⟨σ1(ℓ, ℓ)⟩+− SL = 2−5/2 �2L π �−1/16 7 + cos 2πℓ L � sin πℓ L �1/16 , ⟨σε(ℓ, ℓ)⟩+− SL = 2−5/2 �2L π �−9/16 1 − 9 cos 2πℓ L � sin πℓ L �9/16 , ⟨σ1(ℓ, ℓ)⟩+f SL = 21/2 �2L π �−1/16 cos πℓ 4L � sin πℓ L �1/16 , ⟨σε(ℓ, ℓ)⟩+f SL = −21/2 �2L π �−9/16 cos 3πℓ 4L � sin πℓ L �9/16 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7) where the interval ℓ starts at the α = + boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The expressions for the bare twist correlators are in accord with the equivalent results obtained in [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' With this mention, we show in Figure 3 the remarkable agreement between our CFT calcu- lations for the two terms contributing to the second R´enyi entropy Sαβ 2 = − log⟨�σ(m, 0)⟩(αβ) of the interval [0, m] on the lattice, and the numerical results for the critical Ising chain from the exact diagonalization of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Figure 3a illustrates the case of different (±) fixed BC on the two sides of the chain, while Figure 3b corresponds to letting the m = 0 site free, and applying a magnetic field at the boundary site m = M − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' To illustrate the large amplitude of finite-size effects, we show in Figure 4 how the CFT prediction fares against the lattice results with and without the incorporation of the subleading term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Even for the curve including both subleading and leading terms in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6), the agreement with lattice data is not perfect close to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This can be traced to the presence of corrections from descendants of twist operators, which introduce terms of O(M−hϵ−1) relative to the bare twist contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We can repeat the same kind of analysis for the third R´enyi entropy, related to the Z3- orbifold one-point function by Sαβ 3 = − 1 2 log⟨�σ(m, 0)⟩(αβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The Ising orbifold correlators in this case are given by: ⟨σ1(ℓ, ℓ)⟩+− SL = 3−2 �2L π �−1/9 7 + 2 cos 2πℓ L � sin πℓ L �1/9 , ⟨σε(ℓ, ℓ)⟩+− SL = 3−2 �2L π �−4/9 1 + 8 cos 2πℓ L � sin πℓ L �4/9 , ⟨σ1(ℓ, ℓ)⟩+f SL = 2 �2L π �−1/9 cos πℓ 3L � sin πℓ L �1/9 , ⟨σ1(ℓ, ℓ)⟩+f SL = 21/9 �2L π �−4/9 cos 2πℓ 3L � sin πℓ L �4/9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8) In Figure 5, we once again compare our CFT calculations (including both the leading and subleading term) with the critical chain results for the third R´enyi entropy Sαβ 3 , to good agree- ment for mixed fixed BC (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 5a) and mixed free fixed BC (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' As for the Z2 results, including the CFT subleading contribution to Sαβ 3 is necessary to find a satisfying match with the lattice results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Further finite-size corrections in this case decay as O � M− 2hε 3 −1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='9 m/M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='25 S+ 2 ([m/M]) CFT M=26 sites (a) Fixed mixed BC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='9 m/M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='175 S+f 2 ([m/M]) CFT M=26 sites (b) Fixed-free mixed BC Figure 3: Plots of the second R´enyi entropy Sαβ 2 ([m/M]) in the critical Ising chain with two types of mixed BC for a chain of size M = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The interval is grown from the β = + boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' As advertised in the beginning of the section, our results for the bare twist correlators (for all configurations of mixed BC) are compatible with the ones of [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The subleading contribution to the R´enyi entropies from the excited twist correlator is largely responsible for the mismatch between the lattice and CFT data in the aforementioned article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Finite-size corrections of this magnitude can be suppressed only with much larger system sizes M ∼ 103, as the authors of the present work have shown in [60] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 The three-state Potts quantum chain with mixed BC A natural extension of the Ising chain, the three-state Potts model allows the spins at each site to take one of three possible values {R, G, B}, which we can also conveniently parametrize by third roots of unity {1, ω, ω2}, with ω = exp(2πi/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The Hamiltonian of the three-state Potts 29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='9 m/M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='20 S+f 2 ([m/M]) CFT leading CFT leading+subleading M=26 sites Figure 4: Comparison of the second R´enyi entropy in the critical Ising chain of size M = 26 with mixed free fixed BC with CFT results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Inclusion of the subleading term in the expansion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6 is crucial for obtaining a satisfying agreement with lattice data model, tuned to its bulk critical point [85] [69], [86] is given by: Hαβ = −ζ � � M−1 � j=1 � ZjZ† j+1 + Z† jZj+1 � + M−1 � j=2 � Xj + X† j � + H(α) 1 + H(β) M � � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='9) where ζ = √ 3 2π3/2 is the conformal normalization factor [86] and the operators Zj and Xj act at site j as: Z = � � 1 0 0 0 ω 0 0 0 ω2 � � , X = � � 0 1 0 0 0 1 1 0 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='10) The terms H(α) 1 and H(β) M set the BCs at the ends of the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' For the purpose of this analysis, we will set fixed BC of type R at site 1 and restricted boundary conditions of type {G, B} at site M – the spin at site M is forbidden from taking the value R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This is implemented through the boundary terms: H(R) = h � � −1 0 0 0 0 0 0 0 0 � � , H(GB) = h � � 1 0 0 0 0 −1 0 −1 0 � � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='11) The critical points of interest for the boundaries correspond to h = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' However, for any h > 0, the boundaries will flow towards the same critical points, up to irrelevant boundary perturbations [87] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' These are typically inconsequential for h a large positive value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Furthermore, in our numerical analysis we can, in fact, implement |h| → ∞ by restricting the local Hilbert spaces of the boundary sites to exclude the {G, B} and {R} configurations on the left and, respectively, right boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The scaling limit M → ∞, a → 0 (with L = Ma fixed) of this critical chain is also well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' It is given by the D-series BCFT M6,5 with central charge c = 4/5 and a bulk 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='9 m/M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='200 S+ 3 ([m/M]) CFT M=26 sites (a) Fixed mixed BC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='9 m/M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='14 S+f 3 ([m/M]) CFT M=26 sites (b) Fixed-free mixed BC Figure 5: Plots of the third R´enyi entropy Sαβ 3 ([m/M]) in the critical Ising chain with two types of mixed BC for a chain of size M = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The interval is grown from the α = + boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' primary operator spectrum that contains the scalar fields given in Table 3 as well as the non- diagonal fields {φ2/5,7/5, φ7/5,2/5, φ3,0, φ0,3} whose labels indicate their respective holomorphic and antiholomorphic conformal dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' One can, as shown in Table 1, assign a Z3 charge to the scalar fields, and their respective conformal families, that is consistent with the fusion rules between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The † in Table 3 is, thusly, used to differentiate the fields with the same conformal dimension, but different Z3 charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In the scaling limit, the fixed and restricted boundary critical points will correspond, natu- 31 Diagonal fields (h, ¯h) Z3 charge 1 (0, 0) 0 ε ≡ φ1,2 ( 2 5, 2 5) 0 φ1,3 ( 7 5, 7 5) 0 φ1,4 (3, 3) 0 s, s† ≡ φ3,3 ( 1 15, 1 15) ± 1 ψ, ψ† ≡ φ3,4 ( 2 3, 2 3) ± 1 Table 3: Spectrum of spinless primary operators in the three-state Potts CFT rally, to the fixed and restricted3 conformal boundary states [53,88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' |1⟩ = N[(|1⟩⟩ + |ψ⟩⟩ + |ψ†⟩⟩) + λ(|ϵ⟩⟩ + |s⟩⟩ + |s†⟩⟩)] (fixed R) |ψ⟩ = N[(|1⟩⟩ + ω|ψ⟩⟩ + ¯ω|ψ†⟩⟩) + λ(|ϵ⟩⟩ + ω|s⟩⟩ + ¯ω|s†⟩⟩)] (fixed G) |ψ†⟩ = N[(|1⟩⟩ + ¯ω|ψ⟩⟩ + ω|ψ†⟩⟩) + λ(|ϵ⟩⟩ + ¯ω|s⟩⟩ + ω|s†⟩⟩)] (fixed B) |ε⟩ = N[λ2(|1⟩⟩ + |ψ⟩⟩ + |ψ†⟩⟩) − λ−1(|ϵ⟩⟩ + |s⟩⟩ + |s†⟩⟩)] (restricted GB) |s⟩ = N[λ2(|1⟩⟩ + ω|ψ⟩⟩ + ¯ω|ψ†⟩⟩) − λ−1(|ϵ⟩⟩ + ω|s⟩⟩ + ¯ω|s†⟩⟩)] (restricted RB) |s†⟩ = N[λ2(|1⟩⟩ + ¯ω|ψ⟩⟩ + ω|ψ†⟩⟩) − λ−1(|ϵ⟩⟩ + ¯ω|s⟩⟩ + ω|s†⟩⟩)] (restricted RG) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='12) where N = � 2 √ 15 sin π 5 , λ = � sin(2π/5) sin(π/5) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='13) and the |i⟩⟩’s are the Ishibashi states defined in [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' These conformal boundary states are labelled by the primary fields of Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Due to the Z3 symmetry of our model, we have some freedom to set which conformal bound- ary state corresponds to the fixed boundary condition R in the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' However, this uniquely determines the CFT boundary state that corresponds to the restricted boundary conditions GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This can be understood by considering the spectrum of boundary fields that can interpolate between these conformal BC [56], and ensuring the results are consistent with the underlying Z3 symmetry .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In our case, choosing fixed R ↔ |1⟩ forces us to assign restricted GB ↔ |ε⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The most relevant boundary field interpolating between these BCs is ψ(R,GB) 1,2 [56] with conformal dimension hε = 2/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We will now compare the quantum chain data for the second R´enyi entropy in the critical Potts chain with our correlator calculations in the Z2 orbifold of the BCFT defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Our analysis will parallel the one for the Ising critical chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We first hypothesize the form of the local expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) of the lattice twist operator ˆσm,n in the case of the three-state Potts model: �σ(m, n) = A a2hσσ(w, ¯w) + B a2hσεσε(w, ¯w) + less relevant terms , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='14) where hσ = 1/20, and the composite twist operator σε is built with the energy operator ε of the Potts model so that hσε = 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Once again, we numerically estimated the parameters A, B by a simple analysis on the critical three-state Potts critical chain with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Following this, the one-point lattice twist correlator with our choice of mixed BC can be calculated from: ⟨�σ(m, 0)⟩(GB,R) = A �M π �−2hσ1 ⟨σ(z, ¯z)⟩(GB,R) H + B �M π �−2hσε ⟨σε(z, ¯z)⟩(GB,R) H + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='15) 3In [53] they are referred to as ”mixed” BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 32 The correlators in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='15) satisfy the second order (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='10) and fourth order (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='30) ODEs with g = 6/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' While the solutions to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='10) are known exactly (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='13), one needs to solve (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='30) numerically to find the conformal blocks in the expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) of the excited twist correlator ⟨σε(z, ¯z)⟩αβ H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This is done by a standard numerical implementation of the Frobenius method, whose details we leave for Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' As in the case of the Ising BCFT, not all the solutions of these differential equations are needed to build the twist field correlators in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Crucially, we note that in the three- state Potts mother BCFT, there is no boundary operator ψ(RR) 7/5 living on the fixed conformal boundary of type R [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' At the level of the operator algebra this translates into the vanishing of the boundary-boundary structure constants B(R|R|GB),ψ1,2 ψ7/5,ψ1,2 as we have checked using the results of [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This implies, through the relations between mother BCFT and orbifold structure constants derived in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3, the vanishing of some of the coefficients in the block expansions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) of the correlators in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In effect, only the block corresponding to the identity operator contributes to these expressions when the twist fields are sent to the β boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' It corresponds to the following fusion rules for σ1, σε: σ1 ����� β → Ψ(ββ) 1 σε ����� β → Ψ(ββ) 1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='16) Thusly, we are led to obtain expressions for the bare twist and excited twist correlators on the UHP: ⟨σ(z, ¯z)⟩αβ = ¯z−2hσA(β) σ,Ψ1B(ββα)Ψ12 Ψ1,Ψ12 ˜F1(η) , ⟨σε(z, ¯z)⟩αβ = ¯z−2hσεA(β) σε,Ψ1B(ββα)Ψ12 Ψ1,Ψ12 ˜F(ε) 1 (η) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='17) where ˜F1(η) is given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='13), so in this case we can write an explicit result for the bare twist correlator on the UHP: ⟨σ(z, ¯z)⟩αβ = A(β) σ,Ψ1B(ββα)Ψ12 Ψ1,Ψ12 (1 − η)−2hση−2h12+hσ 2F1 (−8/5, −9/10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' −9/5 | 1 − η) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='18) For the excited twist correlator we have: ˜F(ε) 1 (η) = J1(u(η)) = (1 − u)−2hσε ∞ � n=0 an(1 − u)n , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='19) with the coefficients determined by the recursion relation (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8), derived in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The structure constants can be expressed in terms of known quantities for the M(6, 5) BCFT, also obtained in Appendix B: A(β) σε,Ψ1 = g−1 R AR ε , A(β) σ,Ψ1 = g−1 R , B(ββα)Ψ12 Ψ1,Ψ12 = 1 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='20) where the ground state degeneracies gR and gGB have been found in [69]: gR = � 5 − √ 5 30 � 1 4 gGB = gRλ2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='21) and the bulk-boundary structure constant AR ε has been calculated in [88,89] to be: AR ε = � 1 + √ 5 2 � 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='22) 33 Putting everything together, we can finally compare the lattice prediction for the second R´enyi entropy S(R,GB) 2 = − log⟨�σ(m, 0)⟩(R,GB) with our analytic results in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' While the CFT prediction does not satisfyingly match the lattice data at all points, we observe that the inclusion of the subleading term gives an analytic curve that is closer to the lattice data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' However, it is not enough to make up for the severe finite-size effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Firstly, due to the operator content of the D-series M6,5 CFT, we expect the higher order corrections in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='15 to have a slower power law decay than in the case of the Ising CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We conjecture that the next-to-subleading contribution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='15) will decay as ∼ M−2(hσε+1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' These corrections, we believe, arise from the combined contribution of the ⟨σφ1,3(w, ¯w)⟩R,GB S and ⟨L(1) −1/2 ¯L(1) −1/2σφ1,2(w, ¯w)⟩R,GB S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' While the first correlator can be calculated by a repeat of the method employed for the subleading term, the correlator involving the descendant twist field requires the derivation of a new differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Such an endeavour is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Furthermore, the quantum chain sizes we can reach are diminished in the case of the three- state Potts model, since the size of the space of states grows as ∼ 3M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This memory constraint prevents us from reaching sizes at which higher order corrections are suppressed, using our computational methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This limitation can be, perhaps, bypassed through the usage of more sophisticated numerical tools, such as DMRG or tensor network methods, to access system sizes M for which the unknown higher-order correction terms are further suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='0 m/M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4 S(R, GB) 2 ([m/M]) CFT leading CFT leading+subleading M=18 sites Figure 6: Comparison of the second R´enyi entropy in the critical three-state Potts chain of size M = 18 with mixed (R, GB) BC with CFT results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Finally, one can use the method of Appendix H, applied this time to the third order ODE of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4 to derive the leading CFT contribution to the S(R,GB) 3 ([0, ℓ]) R´enyi entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Since in this case, we have not derived an ODE for the excited twist correlator, we have no handle on the finite-size corrections to the lattice data, which should be even more severe for N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 34 Instead, we have just checked that the CFT result for mixed BC interpolates between the third R´enyi entropies for identical R and GB boundaries: S(α,α) 3 ([0, ℓ]) = c 9 log �2L π sin �πℓ L �� + log gα (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='23) Our expectations are met, as Figure 7 confirms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='0 /L 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='0 L2h ( , ) 3 ([0, ]) CFT mixed (R,GB) CFT fixed (R,R) CFT restricted (GB,GB) Figure 7: Comparison of shifted third R´enyi entropies for (R, GB), (R, R) and (GB, GB) BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The mixed BC curve can be seen to interpolate between the identical BC results 35 5 Conclusion In this article, we have presented a general method for calculating R´enyi entropies SN in the ground state of a 1D critical system with mixed open boundaries, for an interval starting at one of its ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This required computing three-point functions of one twist operator and two BCCOs on the upper-half plane H with mixed BCs (α, β) in the ZN cyclic orbifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' For this purpose, we have derived ODEs satisfied by these correlation functions, by exploiting the null-vectors of the twisted and untwisted representations of its symmetry algebra OVirN, together with Ward identities obtained from the additional conserved currents of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We used a combination of analytical and numerical methods to find a basis of solutions (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='a conformal blocks) of these ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' For the examples provided in this work, we have calculated the boundary and bulk-boundary structure constants needed to build the physical correlators as linear combinations of the blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Among the setups we have analysed are the leading and subleading contributions to the one- interval second and third R´enyi entropies of the Ising model, and the second R´enyi entropy for the three-state Potts model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We have also derived differential equations for mixed BC twist field correlators in the Z2 and Z3 orbifolds of generic BCFTs, and obtained an explicit expression for the second R´enyi entropy valid for any diagonal minimal model, but with a particular set of mixed boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We have compared the CFT results against critical Ising and three-state Potts spin chain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Since finite size effects are quite significant for open chains, we have included both the leading and subleading contributions to the lattice twist field correlator in our analytical prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In the Ising case, the agreement was excellent for all choices of mixed BC, even though the system sizes we could reach were limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' For the three-state Potts chain, however, the finite size effects are even more severe, and as a consequence the matching is less satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This could be improved by using more sophisticated numerical techniques such as DMRG [90,91] or tensor network methods [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The clearest limitation of our method, first identified in [42], is that the process for obtaining a differential equation becomes more difficult as N is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We have checked using the fusion rules in Appendix F for N > 3 that the expected order of the ODEs increases with N for generic minimal models, which implies that more orbifold Ward identities will be needed to obtain the ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' There are several possible extensions of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' A possibility would be to generalize the setup for the calculation of R´enyi entropies of an interval contained in the bulk, with mixed BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' However, in this situation, one would have to find a differential equation that a four-point function with two twist fields and two BCCOs satisfies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Cardy’s doubling trick suggests that such a correlator satisfies the same Ward identities as a six-point conformal block on the complex plane, so the corresponding differential equation would be partial instead of ordinary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 36 Appendix A Mother BCFT conventions We will define here our mother BCFT conventions on the upper-half plane H parametrized by the coordinate z = x + iy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The boundary is aligned with the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Bulk operators in the mother CFT are denoted by φi(z, ¯z) while boundary operators are written as ψ(ab) j (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The operator algebra consists of three types of OPE, which we explicitate, to fix the notations for the corresponding structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' First, we have the bulk-bulk OPEs: φi(z, ¯z)φj(0, 0) = � φk scaling op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Ck ijz−hi−hj+hk ¯z−¯hi−¯hj+¯hk φk(0, 0) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) where Ck ij are the bulk structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The second type of OPE are boundary-boundary OPEs between BCCOs interpolating different boundary conditions: ψ(ab) i (x)ψ(dc) j (y) = δbd � k B(abc)ψk ψiψj (x − y)hk−hi−hjψ(ac) k (y) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2) for x > y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The B(abc)ψk ψiψj are the boundary-boundary structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The Kronecker delta formally expresses the fact that it only makes sense to consider correlations of boundary oper- ators ordered such that their BCs change consistently with their labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Finally, we consider the third kind of OPE, between the bulk and the boundary: φi(z) = � k A(a) φi,ψk(2y)hk−∆i · ψ(aa) k (x) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3) with A(a) φi,ψk the bulk-boundary structure constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In [93], [82] all the structure constants A(a) φi,ψk and B(abc)ψk ψiψj have been determined for A-series and D-series BCFTs, in terms of fusion matrix elements of bulk CFT four-point functions, and the entries of the modular S matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Relevant for this paper are the results: B(abc)ψk ψiψj = Fbk � a c i j � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) where the fusion matrix relates bases of conformal blocks around z = 0 and z = 1 Ir ia,cj(z) = � rs Frs � a c i j � J s ij,ac(1 − z) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5) defined in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We also give the expressions for the 1-point structure constants of the BCFT in terms of S-matrix elements of the mother CFT A(a) φi ≡ A(a) φi,ψI = Sai Sa1 � S11 Si1 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6) 37 B Computation of orbifold structure constants B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1 Composite twist one-point structure constant in the ZN orbifold BCFT Assuming the one-point structure constant A(α) σ1,ψ1 is known, let’s consider the correlator: ⟨σj(0, 0)⟩α D = A(α) σj,ψ1 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) We now use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='13) to write the LHS of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) as: ⟨σj(0, 0)⟩α D = Aj lim ϵ→0 ϵ2(1−N−1)hj � Φ[j,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=',1](ϵ, ¯ϵ)σ[k](0, 0) �α D (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2) Substituting the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='9) of non-diagonal fields, we find: ⟨σj(0, 0)⟩α D = N−2(1−N−1)hj−1 lim ϵ→0 ϵ2(1−N−1)hj N−1 � a=0 � (φj+a ⊗ φ1+a ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' φ1+a) (ϵ, ¯ϵ)σ[k](0, 0) �α D (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3) Each correlator in the sum above can be written as: � (φj+a ⊗ φ1+a ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' φ1+a) (ϵ, ¯ϵ)σ[k](0, 0) �α D = ZN,a ZN 1,a ⟨φj(ϵ, ¯ϵ)⟩DN = A(α) σ1,ψ1⟨φj(ϵ, ¯ϵ)⟩a DN (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) where ZN,a denotes the partition function on the N-sheeted disk with conformal BC a, and branch point at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Now, we can unfold the disk correlator through the conformal map w → w1/N and substitute back in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3) to find: ⟨σj(0, 0)⟩α D = A(α) σ1,ψ1⟨φj(0, 0)⟩a D (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5) so that we finally find: A(α) σj,ψ1 = A(α) σ1,ψ1Aa φj (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2 Bulk-boundary structure constant in the Z2 orbifold CFT In this section we compute the structure constant A(α) σ1,Ψ1,3, which is given by the UHP correlator: � σ1(i/2, −i/2)Ψ(αα) 1,3 (1) �α H = A(α) σ1,Ψ1,3 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7) We can map the LHS of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7) to the unit disk through: z → z − i/2 z + i/2 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8) and then use the partition function expression of the correlator (as in the previous section) to find (after a global rotation): � σ1(0, 0)Ψα 1,3(−i) � D = ⟨σ1(0, 0)⟩α D � ψα 1,3(−i)ψα 1,3(−ie2iπ) � D2,a (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='9) where D2,a is a 2-sheeted disk with branch point at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We unfold the correlator of boundary fields through the map w → w1/2 to find: � ψα 1,3(−i)ψα 1,3(−ie2iπ) � D2,a = � 2i−1/2�−2h1,3 � ψ(aa) 1,3 (i1/2)ψ(aa) 1,3 (−i1/2) � D = 1 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='10) 38 so that, by putting everything together, we arrive at: A(α) σ1,Ψ1,3 = A(α) σ1,Ψ1 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='11) For generic N, expressing the bulk-boundary structure constant A(α) σ1,Ψ1,3 in terms of mother BCFT quantities depends on our ability to calculate N-point functions of boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' For N ≥ 5, this becomes difficult to solve for generic mother BCFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' C Orbifold Ward identities for bulk fields Following [42], we give here the orbifold Ward identities for 4-point bulk correlators: ∞ � p=0 ap � O1 ���L(r) −m1−pO2(1)O3(x, ¯x) ��� O4 � = ∞ � p=0 bp � O1 ��� � L(r) m2+pO2 � (1)O3(x, ¯x) ��� O4 � + ∞ � p=0 cp � O1 ���O2(1) � L(r) m3+pO3 � (x, ¯x) ��� O4 � + ∞ � p=0 dp � O1 ���O2(1)O3(x, ¯x)L(r) m4+p ��� O4 � (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) where the levels mi ∈ Z + rki/N satisfy: m1 + m2 + m3 + m4 = −2 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2) and the coefficients ap, bp, cp and dp are defined from the Taylor series: (1 − z)m2+1(1 − xz)m3+1 = ∞ � p=0 apzp (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3) (z − x)m3+1zm4+1 = ∞ � p=0 bp(z − 1)p (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) (z − 1)m2+1zm4+1 = ∞ � p=0 cp(z − x)p (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5) (z − 1)m2+1(z − x)m3+1 = ∞ � p=0 dpzp (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6) A useful identity We give here the following commutation identity [42]: ⟨O1 |O2(1)O3(x, ¯x)Ln| O4⟩ − ⟨O1 |LnO2(1)O3(x, ¯x)| O4⟩ = {(1 − xn) [x∂x + (n + 1)h3] + (h4 − h1) − n (h2 + h3)} ⟨O1 |O2(1)O3(x, ¯x)| O4⟩ (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7) where O2 and O3 are primary fields and |O2⟩, |O4⟩ are generic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This commutator identity allows one to express insertions of Virasoro modes Ln inside a correlation function in terms of differential operators acting on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 39 D R´enyi entropies for the critical Ising chain with mixed fixed BC In this section, we will derive the bare and excited twist contributions to the second and third R´enyi entropy in the critical Ising chain with fixed mixed BC a = +, b = −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In the Ising BCFT, the boundary field that interpolates between the corresponding conformal BC |±⟩ is the operator ψ(+−) 2,1 , with conformal dimension h2,1 = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In the ZN orbifold of this theory, the change in boundary conditions is implemented by the diagonal operator Ψ(αβ) 2,1 defined as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The essential observation for the derivation of this section is that the space of conformal blocks is one-dimensional for the chiral correlators � Φ1,3|σ[−k] j (1)σ[k] j (η)|Φ1,3 � (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) with j ∈ {1, φ1,3}in the Z2 and Z3 Ising orbifold CFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The result is obtained, as in the discussion of Section 3, from the fusion rules of these theories, found in [40] and [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' These fusion rules also imply the leading singular behaviour of the conformal block around the points η ∈ {0, 1, ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The corresponding exponents are given in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 0 1 ∞ N = 2, j = 1 −1 − 1 16 − 15 16 N = 2, j = φ1,3 −1 − 9 16 − 7 16 N = 3, j = 1 −1 − 1 9 − 8 9 N = 3, j = φ1,3 −1 − 4 9 − 5 9 Table 4: Singular behaviour of the conformal block of (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) for different N and twist field insertions σ[k] j (η) In the η → 1 channel, the exponent corresponds to the fusion σ[k] j × σ[−k] j → Φ1 (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2) for all the chiral correlators we are considering in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The diagonal operator Φ1 is defined as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' From the exponents around η → 0 and η → 1 we can determine the generic form of the conformal blocks for the four cases enumerated above to be: f(N) j (η) = η−1(1 − η)−2hσj P(η) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3) where P(η) is a generic polynomial in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Furthermore, taking into account the singular be- haviour of f(N) j (η) around η → ∞, one can constrain its degree in all four cases to be ≤ 2, so that we have: f(N) j (η) = η−1(1 − η)−2hσj (a2 η2 + a1 η + a0) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) Around η → 1, this function behaves as: f(N) j (η) ∼ (1 − η)−2hσj � (a2 + a1 + a0) + (a2 − a0)(1 − η) + a2(1 − η)2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' � (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5) 40 To find ai, we will need to consider the first few results in the module of Φ1 from the OPE of twist fields in the ZN orbifold: σ[k] j (η)σ[−k] j (1) = Φ1(1) + 2hσj Nc (1 − η)2 T (0)(1) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6) where T (0)(z) = L(0) −2Φ1(z) is the SET of the chiral ZN orbifold CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The corresponding structure constant has been determined by applying a L(0) 2 from the left on both sides of the OPE, and power matching in (1 − η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Finally, the term at level 1 has vanished because the null vector L−11 ≡ 0 in the mother CFT induces the null-vectors L(r) −1Φ1 ≡ 0 in the orbifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Inserting (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6) into (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) one finds, the coefficients a0 = a2 = 2hσj Nc a1 = 1 − 2a0 (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7) with which we fix the conformal blocks for all the cases presented in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We then use the block expansions for the mixed BC correlators to find: � σ[k] j (z, ¯z) �αβ N = g1−N + fN j (η) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8) where we have also used, notably, the results of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='42) for the 1-point structure constant of twist fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' After mapping to the strip through (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2), we find for N = 2: ⟨σ1(ℓ, ℓ)⟩+− SL = 2−5/2 �2L π �−1/16 7 + cos 2πℓ L � sin πℓ L �1/16 (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='9) ⟨σε(ℓ, ℓ)⟩+− SL = 2−5/2 �2L π �−9/16 1 − 9 cos 2πℓ L � sin πℓ L �9/16 (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='10) and N = 3: ⟨σ1(ℓ, ℓ)⟩+− SL = 3−2 �2L π �−1/9 7 + 2 cos 2πℓ L � sin πℓ L �1/9 (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='11) ⟨σε(ℓ, ℓ)⟩+− SL = 3−2 �2L π �−4/9 1 + 8 cos 2πℓ L � sin πℓ L �4/9 (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='12) E Hypergeometric differential equation The hypergeometric differential equation is canonically defined as: η(η − 1)f′′(η) + [(a + b + 1)η − c]f′(η) + ab f(η) = 0 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) with the Riemann scheme: 0 1 ∞ 0 0 a 1 − c c − a − b b 41 The solutions are constructed using the Gauss hypergeometric function 2 F1(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' c | η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Following the conventions of [94], we give a standard basis of fundamental solutions to (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) around the singular point η = 0: I1(η) = 2 F1(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' c | η) I2(η) = η1−c2 F1(b − c + 1, a − c + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 2 − c | η) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2) and around η = 1: J1(η) = 2 F1(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' a + b − c + 1 | 1 − η) J2(η) = (1 − η)c−a−b2 F1(c − b, c − a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' c − a − b + 1 | 1 − η) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3) The two bases of solutions are linearly related as Ii(η) = 2 � j=1 PijJj(η) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) with the fusing matrix P P = � Γ(c)Γ(d) Γ(c−a)Γ(c−b) Γ(c)Γ(−d) Γ(a)Γ(b) Γ(2−c)Γ(d) Γ(1−a)Γ(1−b) Γ(2−c)Γ(−d) Γ(1−c+a)Γ(1−c+b) � (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5) and its inverse: P−1 = � Γ(1−c)Γ(1−d) Γ(1−c+a)Γ(1−c+b) Γ(c−1)Γ(1−d) Γ(a)Γ(b) Γ(1−c)Γ(1+d) Γ(1−a)Γ(1−b) Γ(c−1)Γ(1+d) Γ(c−a)Γ(c−b) � (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6) expressed in terms of Euler’s Gamma function Γ, with d = c − a − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' F Fusion rules in the ZN orbifold In [70] we have found compact expressions for the fusion numbers of the ZN orbifold of a diagonal RCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' They are given by: N [k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='kN] [i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='iN],[j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='jN] = N−1 � a,b=0 Nk1 i1+a,j1+b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' NkN iN+a,jN+b , N k(r) [i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='iN],[j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='jN] = N−1 � a=0 Nk i1+a,j1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Nk iN+a,jN , N k(s) [i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='iN],j(r) = Nk i1,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Nk iN,j , N k(t) i(r),j(s) = δr+s,t Nk ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' N [k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='kN] i[p](r)j[q](s) = δp+q,0 M � ℓ=1 SiℓSjℓ · Sk1ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' SkNℓ SN 1ℓ , N k(t) i[p](r)j[q](s) = δp+q,0 N M � ℓ=1 � SiℓSjℓSN kℓ SN 1ℓ + N−1 � n=1 ωnp(r+s−t) (P−n)iℓ(Pn)jℓSkℓ S1ℓ � , N k[m](t) i[p](r)j[q](s) = δp+q,m N M � ℓ=1 � SiℓSjℓSkℓ SN 1ℓ + N−1 � n=1 ωn(r+s−t) (P † pn−1)iℓ(P † qn−1)jℓ(Pmn−1)kℓ S1ℓ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) 42 where ω = exp (2πi/N), and Nk ij, Sij are the fusion numbers and the modular S-matrix of the mother CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' One also needs the matrix Pn which is defined from Pn = T −n/N · Qn · T [[−n−1]]/N , n ∈ Z× N , (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2) where T is the modular T matrix of the mother CFT, [[−n−1]] denotes the inverse of (−n) in Z× N, with 0 < [[−n−1]] < N, and Qn is the matrix representing the linear action of the modular map τ �→ qn(τ) = nτ − (n[[−n−1]] + 1)/N Nτ − [[−n−1]] (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3) on the characters χj of the mother CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' G Derivation of differential equation in the Z3 orbifold BCFT We present in this section all the orbifold Ward identities and null-vector conditions necessary to derive the third order differential equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The Ward identities Ward 1 The correlator to integrate over is: ⟨Φ12| L(1) 1 σ1(1)T (1)(z)˜σ1(η)L(1) −1 |Φ12⟩ (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) with (m1, m2, m3, m4) = (−1, 1/3, −1/3, −1), to find: a0|1 ⟨Φ12| (L(1) 1 )2σ1(1)˜σ1(η)L(1) −1 |Φ12⟩ + a1|1 ⟨Φ12| L(1) 1 L(1) 0 σ1(1)˜σ1(η)L(1) −1 |Φ12⟩ = d0|1 ⟨Φ12| L(1) 1 σ1(1)˜σ1(η)L(1) −1L(1) −1 |Φ12⟩ + d1|1 ⟨Φ12| L(1) 1 σ1(1)˜σ1(η)L(1) 0 L(1) −1 |Φ12⟩ (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2) Ward 2 The correlator to integrate over is: ⟨Φ12| σ1(1)T (1)(z)˜σ1(η)L(1) −1L(1) −1 |Φ12⟩ (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3) with (m1, m2, m3, m4) = (−1, 1/3, −1/3, −1) to find: a0|2 ⟨Φ12| L(1) 1 σ1(1)˜σ1(η) � L(1) −1 �2 |Φ12⟩ = d0|2 ⟨Φ12| σ1(1)˜σ1(η) � L(1) −1 �3 |Φ12⟩ +d1|2 ⟨Φ12| σ1(1)˜σ1(η)L(1) 0 � L(1) −1 �2 |Φ12⟩ + d2|2 ⟨Φ12| σ1(1)˜σ1(η)L(1) 1 � L(1) −1 �2 |Φ12⟩ + d3|2 ⟨Φ12| σ1(1)˜σ1(η)L(1) 2 � L(1) −1 �2 |Φ12⟩ (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) Ward 3 The correlator to integrate over is ⟨Φ12| L(1) 1 L(1) 1 σ1(1)T (1)(z)˜σ1(η) |Φ12⟩ (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5) with (m1, m2, m3, m4) = (−1, 1/3, −1/3, −1) to find: d0|3 ⟨Φ12| � L(1) 1 �2 σ1(1)˜σ1(η)L(1) −1 |Φ12⟩ = a0|3 ⟨Φ12| � L(1) 1 �3 σ1(1)˜σ1(η) |Φ12⟩ +a1|3 ⟨Φ12| � L(1) 1 �2 L(1) 0 σ1(1)˜σ1(η) |Φ12⟩ + a2|3 ⟨Φ12| � L(1) 1 �2 L(1) −1σ1(1)˜σ1(η) |Φ12⟩ + a3|3 ⟨Φ12| � L(1) 1 �2 L(1) −2σ1(1)˜σ1(η) |Φ12⟩ (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6) 43 Ward 4 The correlator to integrate over is: ⟨Φ12| σ1(1)T (2)(z)˜σ1(η)L(1) −1 |Φ12⟩ (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7) with (m1, m2, m3, m4) = (0, −1/3, 1/3, −2) so we find: d0|4 ⟨Φ12| σ1(1)˜σ1(η)L(2) −2L(1) −1 |Φ12⟩ + d1|4 ⟨Φ12| σ1(1)˜σ1(η)L(2) −1L(1) −1 |Φ12⟩ + d2|4 ⟨Φ12| σ1(1)˜σ1(η)L(2) 0 L(1) −1 |Φ12⟩ + d3|4 ⟨Φ12| σ1(1)˜σ1(η)L(2) 1 L(1) −1 |Φ12⟩ = 0 (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8) Ward 5 The correlator to integrate over is: ⟨Φ12| L(1) 1 T (2)(z)σ1(1)˜σ1(η) |Φ12⟩ (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='9) with (m1, m2, m3, m4) = (−2, −1/3, 1/3, 0) so we find: a0|5 ⟨Φ12| L(1) 1 L(2) 2 σ1(1)˜σ1(η) |Φ12⟩ + a1|5 ⟨Φ12| L(1) 1 L(2) 1 σ1(1)˜σ1(η) |Φ12⟩ + a2|5 ⟨Φ12| L(1) 1 L(2) 0 σ1(1)˜σ1(η) |Φ12⟩ + a3|5 ⟨Φ12| L(1) 1 L(2) −1σ1(1)˜σ1(η) |Φ12⟩ = 0 (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='10) Ward 6 The correlator to integrate over is: ⟨Φ12| σ1(1)T (2)(z)˜σ1(η)L(1) −1 |Φ12⟩ (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='11) with (m1, m2, m3, m4) = (−1, −1/3, 1/3, −1) to find: a0|6 ⟨Φ12| L(2) 1 σ1(1)˜σ1(η)L(1) −1 |Φ12⟩ = d0|6 ⟨Φ12| σ1(1)˜σ1(η)L(2) −1L(1) −1 |Φ12⟩ +d1|6 ⟨Φ12| σ1(1)˜σ1(η)L(2) 0 L(1) −1 |Φ12⟩ + d2|6 ⟨Φ12| σ1(1)˜σ1(η)L(2) 1 L(1) −1 |Φ12⟩ (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='12) Ward 7 The correlator to integrate over is: ⟨Φ12| σ1(1)T (1)(z)˜σ1(η)L(2) −1 |Φ12⟩ (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='13) with (m1, m2, m3, m4) = (−1, 1/3, −1/3, −1) to find: a0|7 ⟨Φ12| L(1) 1 σ1(1)˜σ1(η)L(2) −1 |Φ12⟩ = d0|7 ⟨Φ12| σ1(1)˜σ1(η)L(1) −1L(2) −1 |Φ12⟩ +d1|7 ⟨Φ12| σ1(1)˜σ1(η)L(1) 0 L(2) −1 |Φ12⟩ + d2|7 ⟨Φ12| σ1(1)˜σ1(η)L(1) 1 L(2) −1 |Φ12⟩ (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='14) The null-vector conditions L(1) −1L(2) −1 |Φ12⟩ = 1 2 � 3gL(0) −2 − � L(0) −1 �2� |Φ12⟩ (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='15) ⟨Φ12| L(1) 1 L(2) 1 = ⟨Φ12| 1 2 � 3gL(0) 2 − � L(0) 1 �2� (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='16) 2 L(0) −1L(2) −1L(1) −1 |Φ12⟩ = � 3gL(0) −1L(0) −2 − � L(0) −1 �3� |Φ12⟩ (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='17) 2 L(0) −1L(2) −1L(1) −1 |Φ12⟩ = � 3gL(1) −1L(2) −2 − � L(1) −1 �3� |Φ12⟩ (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='18) 2 ⟨Φ12| L(0) 1 L(2) 1 L(1) 1 = ⟨Φ12| � 3gL(0) 2 L(0) 1 − � L(0) 1 �3� (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='19) 2 ⟨Φ12| L(0) 1 L(2) 1 L(1) 1 = ⟨Φ12| � 3gL(2) 2 L(1) 1 − � L(1) 1 �3� (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='20) By removing from this linear system of 13 equations all terms containing modes L(r) n with r ̸= 0, one indeed obtains (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' 44 H Numerical implementation of the Frobenius method We want to find a basis of solutions to the differential equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='30) that converge on the entire range of interest - the unit circle |η| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The Fuchsian ODE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='30) has singular points 0, 1, ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The solutions around η = 0 and η = 1 converge on the disks |η| < 1 and |η − 1| < 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Thus, only a portion of the unit semicircle, namely 0 < Arg(η) < π/3, is contained in the convergence disk around η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We can circumvent this problem by observing that the solutions around η = ∞ can be convergent on the whole unit circle |η| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Even better, we can implement the change of variable η �→ 1 + u 2u , ∂η �→ −2u2∂u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='1) so that the new ODE, in the variable u has singular points at u = 0, 1, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The original unit circle |η| = 1 is mapped to |u − 1/3| = 2/3, which is contained in the convergence disk |u| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Hence, applying the Froebenius method, and expressing the solutions around u = 1 in terms of those around u = 0 will give the appropriate numerical evaluation of the desired values of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' Now, as explained in [42], a convenient way of finding power series solutions around a point u = u0 is to rewrite the differential equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='30) in terms of the operator θ = (u − u0)∂u, which satisfies: (u − u0)n∂n u = n−1 � k=0 (θ − k) (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='2) Most importantly, we have that any polynomial P(θ) satisfies: P(θ)(u − u0)r = P(r)(u − u0)r (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3) For u0 = 0, we can then rewrite the equation as: � 8 � i=0 uiPi(θ) � = 0 (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='where: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='P0(θ) = 250θ4 − 125θ3 − 130θ2 − θ + 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='P1(θ) = −θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='−2125θ2 + 450θ + 997 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='− 174 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='P2(θ) = −250θ4 + 125θ3 + 130θ2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='750θ3 + 1250θ2 − 2415θ + 4264 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='θ + θ − 3123 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='P3(θ) = −θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4250θ2 + 5925θ + 6016 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='+ θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='−2125θ2 + 450θ + 997 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='− 8511 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='P4(θ) = 5 θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='150θ3 + 575θ2 + 93θ − 332 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='+ θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='750θ3 + 1250θ2 − 2415θ + 4264 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='− 6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='P5(θ) = 2125 θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='θ2 + 3θ + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='+ θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4250θ2 + 5925θ + 6016 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='P6(θ) = −250 θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='θ3 + 6θ2 + 11θ + 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='− 5θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='150θ3 + 575θ2 + 93θ − 332 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='P7(θ) = −2125 θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='θ2 + 3θ + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='P8(θ) = 250 θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='θ3 + 6θ2 + 11θ + 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='(H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='5) We now seek power series solutions around u = 0 of the form: Ii(u) = uri ∞ � n=0 anun with a0 = 1 (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6) 45 where the ri are the roots of the characteristic polynomial P0(r): r1 = −3/10 r2 = 1 r3 = 1/5 r4 = −2/5 (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7) and are the same as the exponents around ∞ in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' By substituting the ansatz (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6) in the differential equation and employing the identity (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='3) we find the following recursion relations for the coefficients an of the solution Ii(u): P0(ri + n)an = − min{n,8} � i=1 an−i Pi(ri + n − i) a0 = 1 (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8) The four series found in this way converge for |u| < 1 and can be evaluated numerically to arbitrary precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We note, at this point, that the solution corresponding to r4 is unphysical, since it corre- sponds, according to Table 1, to the presence in the operator algebra of the theory of a composite twist field formed with a primary operator that is outside the Kac table, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' not present in the M(6, 5) CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This suggests that the physical space of conformal blocks is actually three- dimensional, and thus, that there should be a third order differential equation satisfied by the excited twist correlator in this setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' One should now repeat the above computation for the solutions Jj(u) around u = 1, since these are the ones that appear in the block expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='4) of BCFT correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The recursion relation takes the same form as in (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8), with different roots: λ1 = −1/2 λ2 = 9/10 λ3 = 23/10 λ4 = 23/10 + 2 (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='9) A slight complication appears in this case because two of the roots of the corresponding characteristic polynomial differ by an integer, that is, r4 = r3 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' This will lead to the truncation of the corresponding recursion relations (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8) for r3 because at n = 2, the coefficient P0(r3 + 2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' A good basis of solutions in this case is {J1(u), J2(u), J(k) 3 (u), J4(u)}, where: Ji(u) = (1 − u)ri ∞ � i=0 an(1 − u)n J(k) 3 (u) = (1 − u)r3[a0 + a1(1 − u)] + kJ4(u) (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='10) where k is a free parameter and the value we choose for it should not change the final result for the physical correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' We have chosen to set it to k0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='04428171795178596 and define J3(u) = J(k0)(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' The reason for this choice becomes apparent when one looks at our solution for the fusing matrix M: M = � � � � � � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='207411 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='393808 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='152178 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='356 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='30281 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='444933 0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='70383 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='6374 −22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='7841 0 −8986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='23 −19156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8 7211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='61 5800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='8 � � � � � � (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='11) which relates the bases of conformal blocks around u = 1 and u = 0 as: Ji(u) = � j MijIj(u) (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content='12) 46 To obtain this solution, we have generated a linear system of equations for the unknown Mij from the evaluation of the above relations at different points {ui} in the interval 0 < u < 1 (where both sets of solutions converge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utA0T4oBgHgl3EQfLv_t/content/2301.02124v1.pdf'} +page_content=' In this context, the parameter k0 was tuned so that the block J3(u) does not depend on the unphysical solution I3(u) 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index 0000000000000000000000000000000000000000..9535ba68d7f27cf49e3ffec8f3e2ea685bf462e6 --- /dev/null +++ b/w9AyT4oBgHgl3EQfOfbT/content/tmp_files/2301.00008v1.pdf.txt @@ -0,0 +1,1800 @@ +Effects of Data Geometry in Early Deep Learning +Saket Tiwari +Department of Computer Science +Brown University +Providence, RI 02906 +saket tiwari@brown.edu +George Konidaris +Department of Computer Science +Brown University +Providence, RI 02906 +Abstract +Deep neural networks can approximate functions on different types of data, from +images to graphs, with varied underlying structure. This underlying structure can +be viewed as the geometry of the data manifold. By extending recent advances +in the theoretical understanding of neural networks, we study how a randomly +initialized neural network with piece-wise linear activation splits the data manifold +into regions where the neural network behaves as a linear function. We derive +bounds on the density of boundary of linear regions and the distance to these +boundaries on the data manifold. This leads to insights into the expressivity +of randomly initialized deep neural networks on non-Euclidean data sets. We +empirically corroborate our theoretical results using a toy supervised learning +problem. Our experiments demonstrate that number of linear regions varies across +manifolds and the results hold with changing neural network architectures. We +further demonstrate how the complexity of linear regions is different on the low +dimensional manifold of images as compared to the Euclidean space, using the +MetFaces dataset. +1 +Introduction +The capacity of Deep Neural Networks (DNNs) to approximate arbitrary functions given sufficient +training data in the supervised learning setting is well known [Cybenko, 1989, Hornik et al., 1989, +Anthony and Bartlett, 1999]. Several different theoretical approaches have emerged that study the +effectiveness and pitfalls of deep learning. These studies vary in their treatment of neural networks +and the aspects they study range from convergence [Allen-Zhu et al., 2019, Goodfellow and Vinyals, +2015], generalization [Kawaguchi et al., 2017, Zhang et al., 2017, Jacot et al., 2018, Sagun et al., +2018], function complexity [Mont´ufar et al., 2014, Mhaskar and Poggio, 2016], adversarial attacks +[Szegedy et al., 2014, Goodfellow et al., 2015] to representation capacity [Arpit et al., 2017]. Some +recent theories have also been shown to closely match empirical observations [Poole et al., 2016, +Hanin and Rolnick, 2019b, Kunin et al., 2020]. +One approach to studying DNNs is to examine how the underlying structure, or geometry, of the data +interacts with learning dynamics. The manifold hypothesis states that high-dimensional real world +data typically lies on a low dimensional manifold [Tenenbaum, 1997, Carlsson et al., 2007, Fefferman +et al., 2013]. Empirical studies have shown that DNNs are highly effective in deciphering this +underlying structure by learning intermediate latent representations [Poole et al., 2016]. The ability +of DNNs to “flatten” complex data manifolds, using composition of seemingly simple piece-wise +linear functions, appears to be unique [Brahma et al., 2016, Hauser and Ray, 2017]. +DNNs with piece-wise linear activations, such as ReLU [Nair and Hinton, 2010], divide the input +space into linear regions, wherein the DNN behaves as a linear function [Mont´ufar et al., 2014]. The +density of these linear regions serves as a proxy for the DNN’s ability to interpolate a complex data +landscape and has been the subject of detailed studies [Mont´ufar et al., 2014, Telgarsky, 2015, Serra +36th Conference on Neural Information Processing Systems (NeurIPS 2022). +arXiv:2301.00008v1 [cs.LG] 29 Dec 2022 + +et al., 2018, Raghu et al., 2017]. The work by Hanin and Rolnick [2019a] on this topic stands out +because they derive bounds on the average number of linear regions and verify the tightness of these +bounds empirically for deep ReLU networks, instead of larger bounds that rarely materialize. Hanin +and Rolnick [2019a] conjecture that the number of linear regions correlates to the expressive power +of randomly initialized DNNs with piece-wise linear activations. However, they assume that the +data is uniformly sampled from the Euclidean space Rd, for some d. By combining the manifold +hypothesis with insights from Hanin and Rolnick [2019a], we are able to go further in estimating the +number of linear regions and the average distance from linear boundaries. We derive bounds on how +the geometry of the data manifold affects the aforementioned quantities. +To corroborate our theoretical bounds with empirical results, we design a toy problem where the +input data is sampled from two distinct manifolds that can be represented in a closed form. We count +the exact number of linear regions and the average distance to the boundaries of linear regions on +these two manifolds that a neural network divides the two manifolds into. We demonstrate how the +number of linear regions and average distance varies for these two distinct manifolds. These results +show that the number of linear regions on the manifold do not grow exponentially with the dimension +of input data. Our experiments do not provide estimates for theoretical constants, as in most deep +learning theory, but demonstrate that the number of linear regions change as a consequence of these +constants. We also study linear regions of deep ReLU networks for high dimensional data that lies +on a low dimensional manifold with unknown structure and how the number of linear regions vary +on and off this manifold, which is a more realistic setting. To achieve this we present experiments +performed on the manifold of natural face images. We sample data from the image manifold using +a generative adversarial network (GAN) [Goodfellow et al., 2014] trained on the curated images +of paintings. Specifically, we generate images using the pre-trained StyleGAN [Karras et al., 2019, +2020b] trained on the curated MetFaces dataset [Karras et al., 2020a]. We generate curves on the +image manifold of faces, using StyleGAN, and report how the density of linear regions varies on and +off the manifold. These results shed new light on the geometry of deep learning over structured data +sets by taking a data intrinsic approach to understanding the expressive power of DNNs. +2 +Preliminaries and Background +Our goal is to understand how the underlying structure of real world data matters for deep learning. +We first provide the mathematical background required to model this underlying structure as the +geometry of data. We then provide a summary of previous work on understanding the approximation +capacity of deep ReLU networks via the complexity of linear regions. For the details on how our +work fits into one of the two main approaches within the theory of DNNs, from the expressive power +perspective or from the learning dynamics perspective, we refer the reader to Appendix C. +2.1 +Data Manifold and Definitions +Figure 1: A 2D surface, here represented by a 2-torus, is embedded in a larger input space, R3. +Suppose each point corresponds to an image of a face on this 2-torus. We can chart two curves: +one straight line cutting across the 3D space and another curve that stays on the torus. Images +corresponding to the points on the torus will have a smoother variation in style and shape whereas +there will be images corresponding to points on the straight line that are not faces. +We use the example of the MetFaces dataset [Karras et al., 2020a] to illustrate how data lies on a low +dimensional manifold. The images in the dataset are 1028 × 1028 × 3 dimensional. By contrast, +the number of realistic dimensions along which they vary are limited, e.g. painting style, artist, size +and shape of the nose, jaw and eyes, background, clothing style; in fact, very few 1028 × 1028 × 3 +2 + +福dimensional images correspond to realistic faces. We illustrate how this affects the possible variations +in the data in Figure 1. A manifold formalises the notion of limited variations in high dimensional +data. One can imagine that there exists an unknown function f : X → Y from a low dimensional +space of variations, to a high dimensional space of the actual data points. Such a function f : X → Y , +from one open subset X ⊂ Rm, to another open subset Y ⊂ Rk, is a diffeomorphism if f is bijective, +and both f and f −1 are differentiable (or smooth). Therefore, a manifold is defined as follows. +Definition 2.1. Let k, m ∈ N0. A subset M ⊂ Rk is called a smooth m-dimensional submanifold +of Rk (or m-manifold in Rk) iff every point x ∈ M has an open neighborhood U ⊂ Rk such that +U ∩ M is diffeomorphic to an open subset Ω ⊂ Rm. A diffeomorphism (i.e. differentiable mapping), +f : U ∩ M → Ω +is called a coordinate chart of M and the inverse, +h := f −1 : Ω → U ∩ M +is called a smooth parametrization of U ∩ M. +For the MetFaces dataset example, suppose there are 10 dimensions along which the images vary. +Further assume that each variation can take a value continuously in some interval of R. Then the +smooth parametrization would map f : Ω ∩ R10 → M ∩ R1028×1028×3. This parametrization and its +inverse are unknown in general and computationally very difficult to estimate in practice. +There are similarities in how geometric elements are defined for manifolds and Euclidean spaces. +A smooth curve, on a manifold M, γ : I → M is defined from an interval I to the manifold M as +a function that is differentiable for all t ∈ I, just as for Euclidean spaces. The shortest such curve +between two points on a manifold is no longer a straight line, but is instead a geodesic. One recurring +geometric element, which is unique to manifolds and stems from the definition of smooth curves, is +that of a tangent space, defined as follows. +Definition 2.2. Let M be an m-manifold in Rk and x ∈ M be a fixed point. A vector v ∈ Rk is called +a tangent vector of M at x if there exists a smooth curve γ : I → M such that γ(0) = x, ˙γ(0) = v +where ˙γ(t) is the derivative of γ at t. The set +TxM := {˙γ(0)|γ : R → M is smoothγ(0) = x} +of tangent vectors of M at x is called the tangent space of M at x. +In simpler terms, the plane tangent to the manifold M at point x is called the tangent space and +denoted by by TxM. Consider the upper half of a 2-sphere, S2 ⊂ R3, which is a 2-manifold in R3. +The tangent space at a fixed point x ∈ S2 is the 2D plane perpendicular to the vector x and tangential +to the surface of the sphere that contains the point x. For additional background on manifolds we +refer the reader to Appendix B. +2.2 +Linear Regions of Deep ReLU Networks +The higher the density of these linear regions the more complex a function a DNN can approximate. +For example, a sin curve in the range [0, 2π] is better approximated by 4 piece-wise linear regions as +opposed to 2. To clarify this further, with the 4 “optimal” linear regions [0, π/2), [π/2, π), [π, 3π/2), +and [3π/2, 2π] a function could approximate the sin curve better than any 2 linear regions. In other +words, higher density of linear regions allows a DNN to approximate the variation in the curve better. +We define the notion of boundary of a linear regions in this section and provide an overview of +previous results. +We consider a neural network, F, which is a composition of activation functions. Inputs at each layer +are multiplied by a matrix, referred to as the weight matrix, with an additional bias vector that is +added to this product. We limit our study to ReLU activation function [Nair and Hinton, 2010], which +is piece-wise linear and one of the most popular activation functions being applied to various learning +tasks on different types of data like text, images, signals etc. We further consider DNNs that map +inputs, of dimension nin, to scalar values. Therefore, F : Rnin → R is defined as, +F(x) = WLσ(BL−1 + WL−1σ(...σ(B1 + W1x))), +(1) +where Wl ∈ Mnl×nl−1 is the weight matrix for the lth hidden layer, nl is the number of neurons in +the lth hidden layer, Bl ∈ Rnl is the vector of biases for the lth hidden layer, n0 = nin and σ : R → R +3 + +is the activation function. For a neuron z in the lth layer we denote the pre-activation of this neuron, +for given input x ∈ Rnin, as zl(x). For a neuron z in the layer l we have +z(x) = Wl−1,zσ(...σ(B1 + W1x)), +(2) +for l > 1 (for the base case l = 1 we have z(x) = W1,zx) where Wl−1,z is the row of weights, in the +weight matrix of the lth layer, Wl, corresponding to the neuron z. We use Wz to denote the weight +vector for brevity, omitting the layer index l in the subscript. We also use bz to denote the bias term +for the neuron z. +Neural networks with piece-wise linear activations are piece-wise linear on the input space [Mont´ufar +et al., 2014]. Suppose for some fixed y ∈ Rnin as x → y if we have z(x) → −bz then we observe a +discontinuity in the gradient ∇xσ(bz + Wzz(x)) at y. Intuitively, this is because x is approaching +the boundary of the linear region of the function defined by the output of z. Therefore, the boundary +of linear regions, for a feed forward neural network F, is defined as: +BF = {x|∇F(x) is not continuous at x}. +Hanin and Rolnick [2019a] argue that an important generalization for the approximation capacity +of a neural network F is the (nin − 1)−dimensional volume density of linear regions defined as +volnin−1(BF ∩ K)/volnin(K), for a bounded set K ⊂ Rnin. This quantity serves as a proxy for +density of linear regions and therefore the expressive capacity of DNNs. Intuitively, higher density of +linear boundaries means higher capacity of the DNN to approximate complex non-linear functions. +The quantity is applied to lower bound the distance between a point x ∈ K and the set BF , which is +distance(x, BF ) = +min +neurons z |z(x) − bz|/||∇z(x)||, +which measures the sensitivity over neurons at a given input. The above quantity measures how “far” +the input is from flipping any neuron from inactive to active or vice-versa. +Informally, Hanin and Rolnick [2019a] provide two main results for a randomly initialized DNN F, +with a reasonable initialisation. Firstly, they show that +E +�volnin−1(BF ∩ K) +volnin(K) +� +≈ #{ neurons}, +meaning the density of linear regions is bound above and below by some constant times the number +of neurons. Secondly, for x ∈ [0, 1]nin, +E +� +distance(x, BF ) +� +≥ C#{ neurons}−1, +where C > 0 depends on the distribution of biases and weights, in addition to other factors. In +other words, the distance to the nearest boundary is bounded above and below by a constant times +the inverse of the number of neurons. These results stand in contrast to earlier worst case bounds +that are exponential in the number of neurons. Hanin and Rolnick [2019a] also verify these results +empirically to note that the constants lie in the vicinity of 1 throughout training. +3 +Linear Regions on the Data Manifold +One important assumption in the results presented by Hanin and Rolnick [2019a] is that the input, x, +lies in a compact set K ⊂ Rnin and that volnin(K) is greater than 0. Also, the theorem pertaining to +the lower bound on average distance of x to linear boundaries the input assumes the input uniformly +distributed in [0, 1]nin. As noted earlier, high-dimensional real world datasets, like images, lie on +low dimensional manifolds, therefore both these assumptions are false in practice. This motivates us +to study the case where the data lies on some m−dimensional submanifold of Rnin, i.e. M ⊂ Rnin +where m ≪ nin. We illustrate how this constraint effects the study of linear regions in Figure 2. +As introduced by Hanin and Rolnick [2019a], we denote the “(nin − k)−dimensional piece” of +BF as BF,k. More precisely, BF,0 = ∅ and BF,k is recursively defined to be the set of points +x ∈ BF \ {BF,0 ∪ ... ∪ BF,k−1} with the added condition that in a neighbourhood of x the set BF,k +coincides with hyperplane of dimension nin − k. We provide a detailed and formal definition for BF,k +with intuition in Appendix E. In our setting, where the data lies on a manifold M, we define B′ +F,k +4 + +Figure 2: A circle is an example of a 1D manifold in a 2D Euclidean space. The effective number of +linear regions on the manifold, the upper half of the circle, are the number of linear regions on the +arc from −π to π. In the diagram above, each color in the 2D space corresponds to a linear region. +When the upper half of the circle is flattened into a 1D space we obtain a line. Each color on the line +corresponds to a linear region of the 2D space. +as BF,k ∩ M, and note that dim(B′ +F,k) = m − k (Appendix E Proposition E.4). For example, the +transverse intersection (see Definition E.3) of a plane in 3D with the 2D manifold S2 is a 1D curve in +S2 and therefore has dimension 1. Therefore, B′ +F,k is a submanifold of dimension 3 − 2 = 1. This +imposes the restriction k ≤ m, for the intersection BF,k ∩ M to have a well defined volume. +We first note that the definition of the determinant of the Jacobian, for a collection of neurons +z1, ..., zk, is different in the case when the data lies on a manifold M as opposed to in a compact set +of dimension nin in Rnin. Since the determinant of the Jacobian is the quantity we utilise in our proofs +and theorems repeatedly we will use the term Jacobian to refer to it for succinctness. Intuitively, +this follows from the Jacobian of a function being defined differently in the ambient space Rnin as +opposed to the manifold M. In case of the former it is the volume of the paralellepiped determined +by the vectors corresponding to the directions with steepest ascent along each one of the nin axes. In +case of the latter it is more complex and defined below. Let Hm be the m−dimensional Hausdorff +measure (we refer the reader to the Appendix B for background on Hausdorff measure). The Jacobian +of a function on manifold M, as defined by Krantz and Parks [2008] (Chapter 5), is as follows. +Definition 3.1. The (determinant of) Jacobian of a function H : M → Rk, where k ≤ dim(M) = +m, is defined as +JM +k,H(x) = sup +�Hk(DMH(P)) +Hk(P) +���P is a k-dimensional parallelepiped contained in TxM. +� +, +where DM : TxM → Rk is the differential map (see Appendix B) and we use DMH(P) to denote +the mapping of the set P in TxM, which is a parallelepiped, to Rk. The supremum is taken over all +parallelepipeds P. +We also say that neurons z1, ..., zk are good at x if there exists a path of neurons from z to the output +in the computational graph of F so that each neuron is activated along the path. Our three main +results that hold under the assumptions listed in Appendix A, each of which extend and improve upon +the theoretical results by Hanin and Rolnick [2019a], are: +Theorem 3.2. Given F a feed-forward ReLU network with input dimension nin, output dimension 1, +and random weights and biases. Then for any bounded measurable submanifold M ⊂ Rnin and any +k = 1, ...., m the average (m − k)−dimensional volume of BF,k inside M, +E[volm−k(BF,k ∩ M)] = +� +distinct neurons z1,...,zk in F +� +M +E[Yz1,...,zk]dvolm(x), +(3) +where Yz1,...,zk is JM +m,Hk(x)ρb1,...,bk(z1(x), ..., zk(x)), times the indicator function of the event that +zj is good at x for each j = 1, ..., k. Here the function ρbz1,...,bzk is the density of the joint distribution +of the biases bz1, ..., bzk. +This change in the formula, from Theorem 3.4 by Hanin and Rolnick [2019a], is a result of the +fact that z(x) has a different direction of steepest ascent when it is restricted to the data manifold +M, for any j. The proof is presented in Appendix E. Formula 3 also makes explicit the fact that +the data manifold has dimension m ≤ nin and therefore the m − k-dimensional volume is a more +representative measure of the linear boundaries. Equipped with Theorem 3.2, we provide a result for +the density of boundary regions on manifold M. +5 + +Theorem 3.3. For data sampled uniformly from a compact and measurable m dimensional manifold +M we have the following result for all k ≤ m: +volm−k(BF,k ∩ M) +volm(M) +≤ +� +# neurons +k +� +(2CgradCbiasCM)k, +where Cgrad depends on ||∇z(x)|| and the DNN’s architecture, CM depends on the geometry of M, +and Cbias on the distribution of biases ρb. +The constant CM is the supremum over the matrix norm of projection matrices onto the tangent +space, TxM, at any point x ∈ M. For the Euclidean space CM is always equal to 1 and therefore the +term does not appear in the work by Hanin and Rolnick [2019a], but we cannot say the same for our +setting. We refer the reader to Appendix F for the proof, further details, and interpretation. Finally, +under the added assumptions that the diameter of the manifold M is finite and M has polynomial +volume growth we provide a lower bound on the average distance to the linear boundary for points +on the manifold and how it depends on the geometry and dimensionality of the manifold. +Theorem 3.4. For any point, x, chosen randomly from M, we have: +E[distanceM(x, BF ∩ M)] ≥ +CM,κ +CgradCbiasCM#neurons, +where CM,κ depends on the scalar curvature, the input dimension and the dimensionality of the +manifold M. The function distanceM is the distance on the manifold M. +This result gives us intuition on how the density of linear regions around a point depends on the +geometry of the manifold. The constant CM,κ captures how volumes are distorted on the manifold +M as compared to the Euclidean space, for the exact definition we refer the reader to the proof in +Appendix G. For a manifold which has higher volume of a unit ball, on average, in comparison to +the Euclidean space the constant CM,κ is higher and lower when the volume of unit ball, on average, +is lower than the volume of the Euclidean space. For background on curvature of manifolds and a +proof sketch we refer the reader to the Appendices B and D, respectively. Note that the constant CM +is the same as in Theorem 3.3. Another difference to note is that we derive a lower bound on the +geodesic distance on the manifold M and not the Euclidean distance in Rk as done by Hanin and +Rolnick [2019a]. This distance better captures the distance between data points on a manifold while +incorporating the underlying structure. In other words, this distance can be understood as how much +a data point should change to reach a linear boundary while ensuring that all the individual points on +the curve, tracing this change, are “valid” data points. +3.1 +Intuition For Theoretical Results +One of the key ingredients of the proofs by Hanin and Rolnick [2019a] is the co-area formula +[Krantz and Parks, 2008]. The co-area formula is applied to get a closed form representation of the +k−dimensional volume of the region where any set of k neurons, z1, z2, ..., zk is “good” in terms +of the expectation over the Jacobian, in the Euclidean space. Instead of the co-area formula we use +the smooth co-area formula [Krantz and Parks, 2008] to get a closed form representation of the +m − k−dimensional volume of the region intersected with manifold, M, in terms of the Jacobian +defined on a manifold (Definition 3.1). The key difference between the two formulas is that in the +smooth co-area formula the Jacobian (of a function from the manifold M) is restricted to the tangent +plane. While the determinant of the “vanilla” Jacobian measures the distortion of volume around a +point in Euclidean space the determinant of the Jacobian defined as above (Definition 3.1) measures +the distortion of volume on the manifold instead for the function with the same domain, the function +that is 1 if the set of neurons are good and 0 otherwise. +The value of the Jacobian as defined in Definition 3.1 has the same volume as the projection of +the parallelepiped defined by the gradients ∇z(x) onto the tangent space (see Proposition F.1 in +Appendix). This introduces the constant CM, defined above. Essentially, the constant captures +how the magnitude of the gradients, ∇z(x), are modified upon being projected to the tangent plane. +Certain manifolds “shrink” vectors upon projection to the tangent plane more than others, on an +average, which is a function of their geometry. We illustrate how two distinct manifolds “shrink” +the gradients differently upon projection to the tangent plane as reflected in the number of linear +regions on the manifolds (see Figure 11 in the appendix) for 1D manifolds. We provide intuition +6 + +(a) +(b) +Figure 3: The tractrix (a) and circle (b) are plotted in grey and the target function is in blue. This is +for illustration purposes and does not match the actual function or domains used in our experiments. +for the curvature of a manifold in Appendix B, due to space constraints, which is used in the lower +bound for the average distance in Theorem 3.4. The constant CM,κ depends on the curvature as the +supremum of a polynomial whose coefficients depend on the curvature, with order at most nin and at +least nin − m. Note that despite this dependence on the ambient dimension, there are other geometric +constants in this polynomial (see Appendix G). Finally, we also provide a simple example as to how +this constant varies with nin and m, for a simple and contrived example, in Appendix G.1. +4 +Experiments +4.1 +Linear Regions on a 1D Curve +To empirically corroborate our theoretical results, we calculate the number of linear regions and +average distance to the linear boundary on 1D curves for regression tasks in two settings. The first is +for 1D manifolds embedded in 2D and higher dimensions and the second is for the high-dimensional +data using the MetFaces dataset. We use the same algorithm, for the toy problem and the high- +dimensional dataset, to find linear regions on 1D curves. We calculate the exact number of linear +regions for a 1D curve in the input space, x : I → Rnin where I is an interval in real numbers, by +finding the points where z(x(t)) = bz for every neuron z. The solutions thus obtained gives us +the boundaries for neurons on the curve x. We obtain these solutions by using the programmatic +activation of every neuron and using the sequential least squares programming (SLSQP) algorithm +[Kraft, 1988] to solve for |z(x(t)) − bz| = 0 for t ∈ I. In order to obtain the programmatic activation +of a neuron we construct a Deep ReLU network as defined in Equation 2. We do so for all the neurons +for a given DNN with fixed weights. +4.2 +Supervised Learning on Toy Dataset +We define two similar regression tasks where the data is sampled from two different manifolds with +different geometries. We parameterize the first task, a unit circle without its north and south poles, +by ψcircle : (−π, π) → R2 where ψcircle(θ) = (cos θ, sin θ) and θ is the angle made by the vector +from the origin to the point with respect to the x-axis. We set the target function for regression task +to be a periodic function in θ. The target is defined as z(θ) = a sin(νθ) where a is the amplitude +and ν is the frequency (Figure 3). DNNs have difficulty learning periodic functions [Ziyin et al., +2020]. The motivation behind this is to present the DNN with a challenging task where it has to +learn the underlying structure of the data. Moreover the DNN will have to split the circle into linear +regions. For the second regression task, a tractrix is parametrized by ψtractrix : R1 → R2 where +ψtractrix(y) = (y − tanh y, sech y) (see Figure 3). We assign a target function z(t) = a sin(νt). For +the purposes of our study we restrict the domain of ψtractrix to (−3, 3). We choose ν so as to ensure +that the number of peaks and troughs, 6, in the periodic target function are the same for both the +manifolds. This ensures that the domains of both the problems have length close to 6.28. Further +experimental details are in Appendix H. +The results, averaged over 20 runs, are presented in Figures 4 and 5. We note that CM is smaller +for Sphere (based on Figure 4) and the curvature is positive whilst CM is larger for tractrix and the +curvature is negative. Both of these constants (curvature and CM) contribute to the lower bound +7 + +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.0 +0.8 +0.6 +-10 +-5 +0.4 +0 +0.2 +5 +10 +0.00.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.0 +0.5 +-1.0 +0.0 +0.5 +0.0 +0.5 +0.5 +1.0 +1.0in Theorem 3.4. Similarly, we show results of number of linear regions divided by the number of +neurons upon changing architectures, consequently the number of neurons, for the two manifolds in +Figure 8, averaged over 30 runs. Note that this experiment observes the effect of CM × Cgrad, since +changing the architecture also changes Cgrad and the variation in Cgrad is quite low in magnitude as +observed empirically by Hanin and Rolnick [2019a]. The empirical observations are consistent with +our theoretical results. We observe that the number of linear regions starts off close to #neurons and +remains close throughout the training process for both the manifolds. This supports our theoretical +results (Theorem 3.3) that the constant CM, which is distinct across the two manifolds, affects the +number of linear regions throughout training. The tractrix has a higher value of CM and that is +reflected in both Figures 4 and 5. Note that its relationship is inverse to the average distance to the +boundary region, as per Theorem 3.4, and it is reflected as training progresses in Figure 5. This is +due to different “shrinking” of vectors upon being projected to the tangent space (Section 3.1). +4.3 +Varying Input Dimensions +To empirically corroborate the results of Theorems 2 and 3 we vary the dimension nin while keeping +m constant. We achieve this by counting the number of linear regions and the average distance to +boundary region on the 1D circle as we vary the input dimension in steps of 5. We draw samples of 1D +circles in Rnin by randomly choosing two perpendicular basis vectors. We then train a network with +the same architecture as the previous section on the periodic target function (a sin(νθ)) as defined +above. The results in Figure 6 shows that the quantities stay proportional to #neurons, and do not +vary as nin is increased, as predicted by our theoretical results. Our empirical study asserts how the +relevant upper and lower bounds, for the setting where data lies on a low-dimensional manifold, does +not grow exponentially with nin for the density of linear regions in a compact set of Rnin but instead +depend on the intrinsic dimension. Further details are in Appendix H. +4.4 +MetFaces: High Dimensional Dataset +Our goal with this experiment is to study how the density of linear regions varies across a low +dimensional manifold and the input space. To discover latent low dimensional underlying structure of +data we employ a GAN. Adversarial training of GANs can be effectively applied to learn a mapping +from a low dimensional latent space to high dimensional data [Goodfellow et al., 2014]. The generator +is a neural network that maps g : Rk → Rnin. We train a deep ReLU network on the MetFaces dataset +with random labels (chosen from 0, 1) with cross entropy loss. As noted by Zhang et al. [2017], +training with random labels can lead to the DNN memorizing the entire dataset. +We compare the log density of number of linear regions on a curve on the manifold with a straight line +off the manifold. We generate these curves using the data sampled by the StyleGAN by [Karras et al., +2020a]. Specifically, for each curve we sample a random pair of latent vectors: z1, z2 ∈ Rk, this +gives us the start and end point of the curve using the generator g(z1) and g(z2). We then generate +100 images to approximate a curve connecting the two images on the image manifold in a piece-wise +manner. We do so by taking 100 points on the line connecting z1 and z2 in the latent space that are +evenly spaced and generate an image from each one of them. Therefore, the ith image is generated as: +z′ +i = g(((100 − i) × z1 + i × z2)/100), using the StyleGAN generator g. We qualitatively verify the +images to ensure that they lie on the manifold of images of faces. The straight line, with two fixed +points g(z1) and g(z2), is defined as x(t) = (1 − t)g(z1) + tg(z2) with t ∈ [0, 1]. The approximated +curve on the manifold is defined as x′(t) = (1 − t)g(z′ +i) + tg(z′ +i+1) where i = floor(100t). We +then apply the method from Section 4.1 to obtain the number of linear regions on these curves. +The results are presented in Figure 9. This leads us to the key observation: the density of linear +regions is significantly lower on the data manifold and devising methods to “concentrate” these linear +regions on the manifold is a promising research direction. That could lead to increased expressivity +for the same number of parameters. We provide further experimental details in Appendix I. +5 +Discussion and Conclusions +There is significant work in both supervised and unsupervised learning settings for non-Euclidean +data [Bronstein et al., 2017]. Despite these empirical results most theoretical analysis is agnostic +to data geometry, with a few prominent exceptions [Cloninger and Klock, 2020, Shaham et al., +8 + +Figure 4: Graph of number of linear regions for +tractrix (blue) and sphere (orange). The shaded re- +gions represent one standard deviation. Note that +the number of neurons is 26 and the number of +linear regions are comparable to 26 but different +for both the manifolds throughout training. +Figure 5: Graph of distance to linear regions for +tractrix (blue) and sphere (orange). The distances +are normalized by the maximum distance on the +range, for both tractrix and sphere. The shaded +regions represent one standard deviation. +Figure 6: We observe that as the dimension nin is +increased, while keeping the manifold dimension +constant, the number of linear regions remains +proportional to number of neurons (26). +Figure 7: We observe that as the dimension nin is +increased, while keeping the manifold dimension +constant, the average distance varies very little. +Figure 8: The effects of changing the architecture +on the number of linear regions. We observe that +the value of CM effects the number of linear re- +gions proportionally. The number of hidden units +for three layer networks are in the legend along +with the data manifold. +Figure 9: We observe that the log density of num- +ber of linear regions is lower on the manifold +(blue) as compared to off the manifold (green). +This is for the MetFaces dataset. +9 + +12 +-13 +-14 +15 +-16 +-17 +-18 +0 +20 +40 +60 +80 +100 +120 +140 +EpochNumber45 +rRegions +40 +35 +Linear +Numberofi +30 +25 +20 +15 +0 +50 +100 +150 +200 +250 +300 +EpochNumber0.040 +0.035 +rest boundary +0.030 +to near +0.025 +istance +0.020 +D +0.015 +0.010 +0 +50 +100 +150 +200 +250 +300 +EpochNumber60 +50 +40 +30 + Number of i +20 +5 +10 +10 +15 +20 +25 +0 +20 +40 +60 +80 +100 +120 +140 +0 +Epochs0.006 +0.005 +0.004 + Distance to +0.003 +Average +0.002 +5 +10 +15 +0.001 +20 +25 +0.000 +0 +20 +40 +60 +80 +100 +120 +140 +Epochs2.00 +1.75 +Numberof Linear Regions/#Neurons +1.50 +1.25 +1.00 +0.75 +5, 8.Sphere +10,16.Sphere +0.50 +20,32.Sphere +5, 8,Tractrix +0.25 +10,16,Tractrix +20, 32, Tractrix +0.00 +0 +20 +40 +60 +80 +100 +120 +140 +EpochNumber2015, Schmidt-Hieber, 2019]. We incorporate the idea of data geometry into measuring the effective +approximation capacity of DNNs, deriving average bounds on the density of boundary regions +and distance from the boundary when the data is sampled from a low dimensional manifold. Our +experimental results corroborate our theoretical results. We also present insights into expressivity +of DNNs on low dimensional manfiolds for the case of high dimensional datasets. Estimating the +geometry, dimensionality and curvature, of these image manifolds accurately is a problem that +remains largely unsolved [Brehmer and Cranmer, 2020, Perraul-Joncas and Meila, 2013], which +limits our inferences on high dimensional dataset to observations that guide future research. We note +that proving a lower bound on the number of linear regions, as done by Hanin and Rolnick [2019a], +for the manifold setting remains open. Our work opens up avenues for further research that combines +model geometry and data geometry and can lead to empirical research geared towards developing +DNN architectures for high dimensional datasets that lie on a low dimensional manifold. +6 +Acknowledgements +This work was funded by L2M (DARPA Lifelong Learning Machines program under grant number +FA8750-18-2-0117), the Penn MURI (ONR under the PERISCOPE MURI Contract N00014- 17- +1-2699), and the ONR Swarm (the ONR under grant number N00014-21-1-2200). This research +was conducted using computational resources and services at the Center for Computation and +Visualization, Brown University. +We would like to thank Sam Lobel, Rafael Rodriguez Sanchez, and Akhil Bagaria for refining +our work, multiple technical discussions, and their helpful feedback on the implementation details. +We also thank Tejas Kotwal for assistance on deriving the mathematical details related to the 1D +Tractrix and sources for various citations. We thank Professor Pedro Lopes de Almeida, Nihal Nayak, +Cameron Allen and Aarushi Kalra for their valuable comments on writing and presentation of our +work. We thank all the members of the Brown robotics lab for their guidance and support at various +stages of our work. Finally, we are indebted to, and graciously thank, the numerous anonymous +reviewers for their time and labor as their valuable feedback and thoughtful engagement have shaped +and vastly refine our work. +References +Zeyuan Allen-Zhu, Y. Li, and Zhao Song. +A convergence theory for deep learning via over- +parameterization. ArXiv, abs/1811.03962, 2019. +M. Anthony and P. Bartlett. Neural network learning - theoretical foundations. In Neural Network +Learning - Theoretical Foundations, 1999. +Sanjeev Arora, Rong Ge, Behnam Neyshabur, and Yi Zhang. Stronger generalization bounds for +deep nets via a compression approach. ArXiv, abs/1802.05296, 2018. +Sanjeev Arora, Nadav Cohen, Noah Golowich, and Wei Hu. 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[N/A] +15 + +A +Assumptions +We first make explicit the assumptions on the distribution of weights and biases. +A1: The conditional distribution of any set of biases bz1, ..., bzk given all other weights and +biases has a density ρz1,...,zk(b1, ..., bk) with respect to Lebesgue measure on Rk. +A2: The joint distribution of all weights has a density with respect to Lebesgue measure on +R#weights. +A3: The data manifold M is smooth. +A4: (Only +needed +for +Theorem +3) +the +diameter +of +M +defined +by +dM += +supx,y∈M distanceM(x, y) is finite. +A5: (Only needed for Theorem 3) a geodesic ball in manifold M has polynomial volume growth +of order m. +B +Additional Background on Manifolds +We provide further background on the theory of manifolds. In this section we first provide the +background, definition and an interpretation for the scalar curvature of a manifold at a point. Every +smooth manifold is also equipped with a Riemannian metric tensor (or metric tensor in short). Given +any two vectors, v and w, in the tangent space of a point x on a manifold M, the metric tensor defines +a parallel to the dot product in Euclidean spaces. The metric tensor, at a point x, is defined by the +smooth functions gij : M → R, i, j ∈ {1, ..., k}. Where the matrix defined by +Gx = [gij(x)] = +� +�� +g11(x) +. . . +g1n(x) +... +... +... +gn1(x) +. . . +gnn(x) +� +�� +is symmetric and invertible. The inner product of u, v ∈ TxM is then defined by ⟨u, v⟩M = uT Gxv. +the inner product is symmetric, non-degenerate, and bilinear, i.e. +⟨ku, v⟩M =k⟨u, v⟩M = ⟨u, kv⟩M, +⟨u + w, v⟩M =⟨u, v⟩M + ⟨w, v⟩M, +⟨u, v⟩M =⟨v, u⟩M. +As can be seen, these properties also hold for the Euclidean inner product (with Gx = I for all x). +Let the inverse of G = [gij(x)] be denoted by [gij(x)]. Building on this definition of the metric +tensor the Ricci curvature tensor is defined as +Rij = − 1 +2 +n +� +a,b=1 +� ∂2gij +∂xa∂xb ++ ∂2gab +∂xi∂xj +− +∂2gib +∂xj∂xa +− ∂2gjb +∂xi∂xa +� +gab ++ +n +� +a,b,c,d=1 +�1 +2 +∂gac +∂xi +∂gbd +∂xj ++ ∂gic +∂xa +∂gjd +∂xb +− ∂gic +∂xa +∂gjb +∂xd +� +gabgcd +− 1 +4 +n +� +a,b,c,d=1 +�∂gjc +∂xi ++ ∂gic +∂xj +− ∂gij +∂xc +� +gabgcd. +For geometric interpretations of the above tensors we refer the reader to the work by Loveridge +[2004]. +Another quantity, from the theory of manifolds, which we utilise in our proofs and theorems, is scalar +curvature (or Ricci curvature). The curvature is a measure how much the volume of a geodisic ball +on the manifold M, e.g. S2, deviates from a d − 1 sphere in the flat space, e.g. R3. The volume on +the manifold deviates by an amount proportional to the curvature. We illustrate this idea in figure +10. We refer the reader to works by Gray [1974] and Wan [2016] for further technical details. Since +our main theorems relate to the volume of linear regions the scalar curvature plays an important role. +16 + +(a) +(b) +Figure 10: The geodesic circle on S2 (blue region in (a)) does not have the same area as the flat +circle (b), both of radius ϵ. One can imagine cutting the blue top off the sphere’s surface and trying to +“flatten” it. Such an effort will lead to failure, if the material of the sphere does not ”stretch”, since +the geodesic ball, on S2, cannot be mapped to a circle in R2 in a distance preserving manner. Thus, +the area of the two blue regions in (a) and (b) vary. This deviation in the area spanned by the two +spheres, despite their radii being the same, is proportional to the scalar curvature. +Formally, the scalar curvature of a manifold M at a point x with metric tensor [gij] and Ricci tensor +[Rij] is defined as +C = +n +� +i,j=1 +gijRij. +Another important concept is that of Hausdorff measure. Since the volumes are “distorted” on a +manifold it requires careful consideration when defining a measure and integrating using it on a +manifold. The m−dimensional Hausdorff measure, of a set S, is defined as +Hm(S) := sup +δ>0 +inf +� ∞ +� +i=1 +(diam Ui)d|S ⊆ ∪∞ +i=1Ui, diam Ui < δ +� +. +Next we introduce the definition of the differential map that is used in Definition 3.1, for the +determinant of the Jacobian. The differential map of a smooth function H from a manifold M to +a manifold S at a point x ∈ M is the smooth map dH : TxM → TxS such that the tangent vector +corresponding to any smooth curve γ : I → M at x, γ′(0) ∈ TxM, maps to the tangent vector of +H ◦ γ in TH(x)N. This is the analog of the total derivative of “vanilla calculus”. More intuitively, +the differential map captures how the function changes along different directions on N as its input +changes along different directions on M, this also has an analog to how rows of the Jacobian matrix +are viewed in calculus. In Definition 3.1 we use the specific case where the function H maps from +manifold M to the Euclidean space Rk and the tangent space of a Euclidean space is the Euclidean +space itself. Finally, a paralellepiped’s, P in TxM, mapping via the differential map gives us the +points in Rk that correspond to this set P. +C +Related Work +There have been various approaches to explain the efficacy of DNNs in approximating arbitrarily +complex functions. We briefly touch upon two such promising approaches. Broadly, the theory of +DNNs can be viewed from two lenses: expressive power [Hornik et al., 1989, Bartlett et al., 1998, +Poole et al., 2016, Raghu et al., 2017, Kawaguchi et al., 2017, Neyshabur et al., 2018, Hanin, 2019] +and learning dynamics [Saxe et al., 2014, Su et al., 2016, Smith and Le, 2018, Jacot et al., 2018, +Lee et al., 2019, Arora et al., 2019a,b]. These approaches are not independent of one another but +17 + +EEcomplementary. For example, Kawaguchi et al. [2017] argue theoretically how the family of DNNs +generalize well despite the large capacity of the function class. Neyshabur et al. [2018] provide +PAC-Bayes generalization bounds which are improved upon by Arora et al. [2018]. Hanin [2019] +shows that Deep ReLU networks of finite width can approximate any continuous, convex or smooth +functions on a unit cube. These works look at DNNs from the lens of expressive power. More recently, +there has been a surge in explaining how various algorithms arrive at these almost accurate function +approximations by applying different theoretical models of DNNs. Jacot et al. [2018] provide results +for convergence and generalization of DNNs in the infinite width limit by introducing a the neural +tangent kernel (NTK). Hanin and Nica [2020] provide finite depth and width corrections for the NTK. +Another line of work within the learning dynamics literature looks at implicit regularization that +emerge from the learning algorithm and over-parametrised DNNs [Arora et al., 2019a,b, Du et al., +2018, Liang et al., 2019]. +Researchers have begun to incorporate data geometry into the theoretical analyses of DNNs by +applying the assumption that the data lies on a general manifold. First we note the works looking +at DNNs from the lens of expressive power combined with the idea of data geometry. Shaham +et al. [2015] demonstrate that the size of the neural network depends on the curvature of the data +manifold and the complexity of the function, whilst depending weakly on the input data dimension, +for their construction of sparsely-connected 4-layer neural networks. Cloninger and Klock [2020] +show that their construction of deep ReLU nets achieve near optimal approximation rates which +depend only on the intrinsic dimensionality of the data. Chen et al. [2019] exploit the low dimensional +structure of data to enhance the function approximation capacity of Deep ReLU networks by means +of theoretical guarantees. Schmidt-Hieber [2019] shows that sparsely connected deep ReLU networks +can approximate a Holder function on a low dimensional manifold embedded in a high dimensional +space. Simultaneously, researchers have incorporated data geometry into the learning dynamics line +of work [Goldt et al., 2020, Paccolat et al., 2020, Buchanan et al., 2021, Wang et al., 2021]. Buchanan +et al. [2021] apply the NTK model to study how DNNs can separate two curves, representing the +data manifolds of two separate classes, on the unit sphere. Goldt et al. [2020] introduce the Hidden +Manifold Model for structured data sets to capture the dynamics of two-layer neural networks trained +with stochastic gradient descent. Rahaman et al. [2019] provide empirical results on which data +manifolds are learned faster. Finally, the work by Novak et al. [2018] comes the closes in studying the +number of linear regions on the data manifold. They study the change in input output Jacobian, and +as a consequence the number of linear regions, for DNNs with piece-wise linearities. They provide +empirical studies by counting the number of linear regions along lines connecting data points as a +proxy for number of linear regions on the data manifold. +Our work fits into the study of expressive power of DNNs. The number of linear regions is a +good proxy for the practical expressive power or approximation capacity of Deep ReLU networks +[Mont´ufar et al., 2014]. The results surrounding the density of linear regions make the fewest +simplifying assumptions both on the data and the architecture of the DNN. The results by Hanin and +Rolnick [2019a] bound the number of linear regions orders of magnitude tighter than previous results +by deriving bounds for the average case and not the worst case. Moreover, they demonstrate the +validity empirically in a setting with very few simplifying assumptions. We introduce the manifold +hypothesis to this setting in order to obtain tighter bounds for the first time. This introduces a toolbox +of ideas from differential geometry to analyse the approximation capacity of deep ReLU networks. +In addition to the theoretical works listed above, there has been significant empirical work that applies +DNNs to non-Euclidean data [Bronstein et al., 2017, 2021]. Here the data is assumed to be sampled +from manifolds with certain geometric properties. For example, Ganea et al. [2018] design DNNs +for data sampled from Hyperbolic spaces of arbitrary dimensionality and modify the forward and +backward passes accordingly. There have been numerous applications of modified DNNs, namely +graph convolutional networks, to graph data that incorporate the idea that graphs are discrete samples +from a smooth manifold [Henaff et al., 2015, Monti et al., 2017, Kipf and Welling, 2017], see the +survey by Wu et al. [2019] for a comprehensive review. Graph convolutional networks have also been +applied to point cloud data for applications in graphics [Qi et al., 2017, Wang et al., 2019]. +D +Proof Sketch +In this section we provide an overview of how the three main theorems are proved. Theorem 3.2 +provides an equality for measuring the volume of m − k dimensional boundary regions on the +18 + +manifold. To this effect, we introduce the idea of viewing boundary regions as submanifolds on +the data manifold instead of hyperplanes (Proposition 6). We then prove an equality between the +volume of boundary regions and the Jacobian of the neurons over the manifold. We utilise the smooth +coarea formula that, intuitively, is applied to integrate a function using level sets on a manifold. This +completes the proof for Theorem 3.2. +To prove Theorem 3.3 we first prove that the Jacobian of a function on a manifold can be denoted +using the volume of paralellepiped of vectors in the ambient space subject to a linear transform +(Proposition 8). Using this result and combining it with Theorem 3.2 we can then give an inequality +for the density of linear regions. As can be expected this volume depends on the aforementioned +projection, which in turn is related to the geometry of the manifold. +Finally, for proving Theorem 3.4 we first provide an inequality over the tubular neighbourhood of the +boundary region. We then use this result to lower bound the geodesic distance between the boundary +region and any random point on the manifold. The proof strategy follows that of Hanin and Rolnick +[2019a] but there are major deviations when it comes to accounting for the geometry of the data +manifold. To the best of our knowledge, we are utilising elements of differential topology that are +unique to machine learning when it comes to developing a theoretical understanding of DNNs. +E +Proof of Theorem 3.2 +We follow the proof strategy used by Hanin and Rolnick [2019a] but deviate from it to account for +our setting where x ∈ M. Let Sz be the set of values at which the neuron z has a discontinuity in the +differential of its output (or the neuron switches between the two linear regions of the piece-wise +linear activation σ), +Sz := {x ∈ Rnin|z(x) − bz = 0}. +We also have +O := +� +x ∈ Rnin|∀j = 1, ..., L ∃ neuron z with l(z) = j s.t. σ′(z(x) − bz) ̸= 0 +� +. +Further, +� +Sz := Sz ∩ O. +We state propositions 9 and 10 by Hanin and Rolnick [2019a] as we apply them to prove Theorem +3.2, relabeling them as needed. +Proposition E.1. (Proposition 9 by Hanin and Rolnick [2019a]) Under assumptions A1 and A2, +we have, with probability 1, +BF = +� +neurons z +� +Sz. +By extending the notion of Sz to multiple neurons we have +�Sz1,...,zk := +k� +j=1 +�Szj, +meaning that the set �Sz1,...,zk is, intuitively, the collection of inputs in Rin where the neurons +zj, j = 1, ..., k, switch between linear regions for σ and at which the output of F is affected by the +outputs of these neurons. We refer the reader to section B of the appendix in the work by Hanin and +Rolnick [2019a] for an intuitive explanation of proposition E.1. Before proceeding we provide a +formal definition and intuition for the set BF,k, +BF,k = {x|x ∈ BF \ {BF,0 ∪ ... ∪ BF,k−1} = BF,−k and for any ball of radius ϵ > 0, +B(x, ϵ) ∩ BF,−k is subset to a n − k dimensional hyperplane}. +Following the explanation provided by Hanin and Rolnick [2019a], BF,k is the nin − k dimensional +piece of BF . Suppose the boundaries of linear regions for nin = 2 are unions of polygon boundaries, +as depicted in Figure 2 of the main body of the paper, then BF,1 are all the open line segments of +these polygons and BF,2 are the end points. Next we state Proposition 10 by Hanin and Rolnick +[2019a]. +19 + +Proposition E.2. (Prosposition 10 by Hanin and Rolnick [2019a]) Fix k = 1, ..., nin, and k distinct +neurons z1, ..., zk in F. Then, with probability 1, for every x ∈ BF,k there exists a neighbourhood in +which BF,k coincides with a nin−k−dimensional hyperplane. +We now present Proposition E.4, and its proof, which incorporates the additional constraint that +x ∈ M, which is an m-dimensional manifold in Rnin. To prove the proposition we need the definition +of tranversal intersection of two manifolds [Guillemin and Pollack, 1974]. +Definition E.3. Two submanifolds, M1 and M2, of S are said to intersect transversally if at every +point of intersection their tangent spaces, at that point, together generate the tangent space of the +manifold, S, by means of linear combinations. Formally, for all x ∈ M1 ∩ M2 +TxS = TxM1 + TxM2, +if and only if M1 and M2 intersect transversally. +For example, given a 2D hyperplane, P, and the surface of a 3D sphere, S2, intersect in the ambient +space R3. We have that this intersection is transverse if and only if P is not tangent to S2. For the +case where a 2D hyperplane, ¯P, intersects with S2 at a point p but does not intersect tranversally it +coincides exactly with the tangent plane of S2 at point {p} = S2 ∩ P, i.e. TpS = P. Note that in +either case the tangent space of the 2D hyperplane P at any point of intersection is the plane itself. +Proposition E.4. Fix k = 1, ..., m and k distinct neurons z1, ..., zk in F. Then, with probability +1, for every x ∈ BF,k ∩ M there exists a neighbourhood in which BF,k coincides with an m − k +dimensional submanifold in Rin. +Proof. From Proposition E.2 we already know that BF,k is a nin − k-dimensional hyperplane in +some neighbourhood of x, with probability 1, for any x ∈ BF,k ∩ M. Let this hyperplane be denoted +by Pk. This is an n − k dimensional submanifold of Rnin. The tangent space of this hyperplane +at x is the hyperplane itself. Therefore, from assumptions A1 and A2 we have that the probability +that this hyperplane intersects the manifold M transversally with probability 1. In other words the +probability that this plane Pk contains or is contained in TxM is 0. Finally, we have the intersection, +M ∩ Hk, has dimension dim(M) + dim(Hk) − nin [Guillemin and Pollack, 1974], which is equal +to m − k. +One implication of Proposition E.4 is that for any k ≤ m the m − (k + 1) dimensional volume of +BF,k ∩ M is 0. In addition to that, Proposition E.4 implies that, with probability 1, +volm−k(BF,k) = +� +distinct neurons z1,...,zk +volm−k(�Sz1,...,zk ∩ M). +(4) +The final step in the proof of Theorem 3.2 is to prove the following result. +Proposition E.5. Let z1, ..., zk be distinct neurons in F and k ≤ m. +Then for a bounded +m−Hausdorff measurable manifold M embedded in Rnin, +E +� +volm−k +� +�Sz1,...,zk ∩ M +�� += +� +M +E +� +Yz1,...,zk(x) +� +dx, +where Yz1,...,zk(x) equals +JM +m,Hk(x)ρb1,...,bk(z1(x), ..., zk(x)), +times the indicator function of the event that zj, for j = 1, ..., k, is good at x for every j and +Hk : Rnin → Rk is such that Hk(x) = [z1(x), ..., zk(x)]T . The expectation is over the distribution +of weights and biases. +Proof. Let z1, ..., zk be distinct neurons in F and M be an m−dimensional compact Haudorff +measurable manifold. We seek to compute the mean of volm−k(�Sz1,...,zk ∩ M) over the distribution +of weights and biases. We can rewrite this expression as +� +Sz1,...,zk ∩M +1zj is good at xdvolm−k(x). +(5) +20 + +The map Hk is Lipschitz and C1 almost everywhere. We first note the smooth coarea formula +(theorem 5.3.9 by Krantz and Parks [2008]) in context of our notation. Suppose m ≥ k and +Hk : Rnin → Rk is C1 and M ⊆ Rnin is an m−dimensional C1 manifold in Rnin, then +� +M +g(x)JM +k,Hk(x)dvolm(x) = +� +Rk +� +M∩H−1 +k +(y) +g(y)dvolm−k(y)dvolk(x), +(6) +for every Hm-measurable function g where JM +k,Hk is as defined in Definition 3.1. +We denote preactivations and biases of neurons as z(x) = [z1(x), ..., zk(x)]T and bz = [bz1, ..., bzk]T . +From the notation in A1, we have that +ρbz = ρbz1,...,bzk , +is the joint conditional density of bz1, ..., bzk given all other weights and biases. The mean of the term +in equation 5 over the conditional distribution of bz1, ..., bzk, ρbz, is therefore +� +Rk bdvolk(b) +� +{z=b}∩M +1zj is good at xdvolm−k(x), +(7) +where we denote [b1, ..., bk]T as b. Thus applying the smooth co-area formula (Equation 6) to the +expression in 7 shows that the average 5 is equal to +� +M +Yz1,...,zk(x)dx. +Finally, we take the average over the remaining weights and biases and commute the expectation with +the dx integral. We can do this since the integrand is non-negative. This gives us the result: +E +� +volm−k +� +�Sz1,...,zk ∩ M +�� += +� +M +E +� +Yz1,...,zk(x) +� +dx, +(8) +as required. +Finally, taking the summation over all possible sets of distinct neurons z1, ..., zk and combining +equation 4 with Proposition E.5 completes the proof for Theorem 3.2. +F +Proof of Theorem 3.3 +To prove the upper bound in Theorem 3.3 we first show that the (determinant of) Jacobian for the +function Hk : M → Rk, Hk(x) = [z1(x), ..., zk(x)]T , as defined in 3.1 is equal to the volume of +the parallelopiped defined by the vectors φHk(∇zj(x)), for j = 1, ..., k, where φHk : Rk → TxM is +an orthogonal projection onto the orthogonal complement of the kernel of the differential DMHk. +Intuitively, this shows that with the added assumption x ∈ M in Theorem 3.3 how exactly we can +incorporate the geometry of the data manifold M into the upper bound provided by Hanin and +Rolnick [2019a] in corollary 7. +Proposition F.1. Given Hk : M → Rk such that Hk(x) = [z1(x), ..., zk(x)]T and the differential +DMHk is surjective at x then +JM +k,Hk(x) = +� +det(Gram(φHk(∇z1(x)), ..., φHk(∇zk(x)))), +(9) +where φHk : Rn → Rk is a linear map and Gram denotes the Gramian matrix. +Proof. We first define the orthogonal complement of the kernel of the differential DMHk. For a +manifold M ⊂ Rn and a fixed point x we have that TxM is a m−dimensional hyperplane. If we +choose an orthonormal basis e1, ..., en of Rn such that e1, ..., em spans TxM for a fixed x we can +denote all vectors in TxM using m coordinates corresponding to this basis. Therefore, for any +vector y ∈ Rk we can get the orthogonal projection of y onto TxM using a m × n matrix which we +denote as Px, where Pxy (matrix multiplied by a vector) represents a vector in TxM corresponding +to the basis e1, ..., em. For any manifold M in Rn and function Hk : M → Rk we have that +DMHk : TxM → Rk at a fixed point x is linear function. Therefore we can write DMHk(v) = Av +21 + +where v ∈ TxM is denoted using the aforementioned basis of TxM. This implies that A is a k × m +matrix. Therefore, the kernel of DMHk for a fixed point x ∈ M is +ker(DMHk) = +� +z|Az = 0 and z ∈ TxM +� +. +Since we can create a canonical basis for the space ker(DMHk) starting from the basis e1, ..., em in +Rn using the Gram-Schmidt process given the matrix A we have that for any y ∈ Rn we can project +it orthogonally onto ker(DMHk). The orthogonal complement of ker(DMHk) is therefore defined +by +ker(DMHk)⊥ = +� +a|a · z = 0 for all z ∈ ker(DMHk) and a ∈ TxM +� +. +Similar to the previous argument, we construct a canonical basis starting from e1, ..., em for +ker(DMHk)⊥ and therefore we can denote the orthogonal projection onto ker(DMHk)⊥ as a +linear transformation. We denote this linear projection for fixed x using φk. +We denote the basis vectors e1, ...., em as a m × n matrix E where each row i corresponds to the +vector ei. Therefore, the orthogonal projection of any vector y ∈ Rn is Ey. Now we can get the +matrix A using E∇zj(x) corresponding to each row j for j = 1, ..., m. This uses the fact that the +direction of steepest ascent on zj(x) restricted to the tangent space TxM of the manifold M is an +orthogonal projection of the direction of steepest ascent in Rn. +Finally, from lemma 5.3.5 by Guillemin and Pollack [1974] we have that +JM +k,Hk(x) = Hk(DMHk(P))/Hk(P), +for any parallelepiped P contained in (ker(DMHk))⊥. Arguing similar to the proof of lemma 5.3.5 +by Guillemin and Pollack [1974] we get that +JM +k,Hk(x) = +� +det((A)T A) = +� +det Gram(E∇z1(x), ..., E∇zk(x)), +thereby showing that φHk(y) = Ey is a linear mapping. +Although we state Proposition F.1 for neurons zj(x), j = 1, ..., k in the proof, it applies to any +function that satisfy the conditions laid out in the proposition. Equipped with Proposition F.1 we +prove Theorem 3.3. When the weights and biases of F are independent obtain an upper bound on +ρbz1,...,bzk (b1, ..., bk) as +Πk +j=1ρbzj (b1, ..., bk) ≤ +� +sup +neurons z ρbz(b) +�k += Ck +bias. +Hence, +Yz1,...,zk ≤ Ck +biasJM +k,Hk. +From Proposition 9 we have that JM +k,Hk is equal to the k-dimensional volume of the paralellopiped +spanned by φx(∇zj(x)) for j = 1, ..., k. Therefore, we have +JM +k,Hk ≤ Πk +j=1||E∇zj(x)|| ≤ ||E||kΠk +j=1||∇zj(x)||, +(10) +where ||E|| denotes the matrix norm which is defined as +||E|| = sup +� +||Ey|| +���y ∈ Rk, ||y|| = 1 +� +. +Note that E does not depend on F (or z1, ..., zk) but only on TxM or more generally the geometry of +M at any point x. From Theorem 3.2 by Hanin and Nica [2018] we have, for any fixed x, +E +� +Πk +j=1||∇zj(x)|| +� +≤ +� +Cgrad +�k +, +(11) +where, +Cgrad = sup +z +sup +x∈Rnin E[||∇z(x)||2k]1/k ≤ Ce +C �d +j=1 +1 +nj , +22 + +(a) +(b) +Figure 11: We illustrate how vectors project differently on tangent planes of two different manifolds: +circle (a) and tractrix (b). In case of the tractrix the tangents (and the projection of vectors onto them) +are on the inside of the tractrix whereas for the sphere the tangents are always on the outside of the +sphere. Since the projections of vectors onto the tangent space are an essential aspect of our proof we +end up with the term CM, which quantifies the “shrinking” of these vectors upon projection, in the +inequalities for Theorems 3.3 and 3.4. +wherein C > 0 depends only on µ and not on the architecture of F and nj is the width of the hidden +layer j. Let CM be defined as +CM := sup +� +C| there exists a set, S, of non zero m − k-dimensional Hausdorff measure +such that ||Ex|| ≥ C∀x ∈ S +� +Therefore, combining equations 11, 10 and result from Theorem 3.2 we have +E[volm−k(BF,k ∩ M)] +volm(M) +≤ +� +number of neurons +k +� +(2CgradCbiasCM)k, +where the expectation is over the distribution of weights and biases. +G +Proof of Theorem 3.4 +We first prove the following proposition +Proposition G.1. For a compact m-dimensional submanifold M in Rn, m, n ≥ 1 and m < n let +S ⊆ Rn be a compact fixed continuous piece-wise linear submanifold with finitely many pieces and +given any U > 0. Let S0 = ∅ and let Sk be the union of the interiors of all k-dimensional pieces of +S \ (S0 ∪ ... ∪ Sk−1). Denote by Tϵ the ϵ-tubuluar neighbourhood of any X ⊂ M such that +Tϵ(X) = +� +y|dM(y, X) < ϵ and y ∈ M +� +, +where ϵ ∈ (0, U), dM is the geodesic distance between the point y and set X on the manifold M, we +have +volm(Tϵ(S)) ≤ +d +� +k=n−m +volk(Sk ∩ M)ωn−kϵn−kCk,κ,U, +where Ck,κ,U > 0 is a constant that depends on the average scalar curvature κ(Sk∩M)⊥ and U, and +ωn−k is the volume of the unit ball in Rn−k. +Proof. Define d to be the maximal dimension of linear pieces in S. Let x ∈ Tϵ(X ∩ M). Suppose +x /∈ Tϵ(X ∩ M) for all k = n − m, ..., d − 1. Then the intersection of a geodesic ball of radius ϵ +around s with S is a ball inside Sd ∩ M. Using the convexity of this ball, with respect to the manifold +M [Robbin et al., 2011], there exists a point y in Sd ∩ M such that the geodesic γ : [0, 1] → M with +23 + +1.5 +1.0 +0.5 +0.0 +-0.5 - +-1.0 +1.5 + +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.50.8 +0.6 +0.4 +0.2 +0.0 +-6 +-4 +-2 +0 +2 +4 +6γ(0) = y and γ(1) = x is perpendicular to Sd ∩ M at y. Formally, TSd∩MM at y is perpendicular +to +˙ +γ(0) ∈ TM at y. Let Bϵ(N ∗(Sd ∩ M)) be the union of all the ϵ balls along the fiber of the +submanifold Sd ∩ M. Therefore, we have +volm(Tϵ(S ∩ M) ≤ volm(Bϵ(N ∗(Sd ∩ M)) + volm(Tϵ(S≤d−1 ∩ M)), +(12) +where S≤d−1 := ∪d−1 +k=0Sk. We also note that +volm(Bϵ(N ∗(Sd ∩ M)) = volm+d−n(Sd ∩ M)voln−d(Bϵ((M ∩ Sd)⊥)), +where Bϵ((M ∩ Sd)⊥) is the average volume of an ϵ ball in the submanifold of M orthogonal +to M ∩ Sd. This volume depends on the average scalar curvature, κ(M∩Sd)⊥ of the submanifold +(M ∩ Sd)⊥. As shown by Wan [2016], for a fixed point x ∈ (M ∩ Sd)⊥ +voln−d(Bϵ(x, (M ∩ Sd)⊥)) = ωn−dϵn−d� +1 − κ(x)(M∩Sd)⊥ +n − d + 2 +ϵ2 + O(ϵ4) +� +, +where ωn−d is the volume of the unit ball of dimension n − d, Bϵ(x, (M ∩ Sd)⊥) is the geodesic ball +of radius ϵ in the manifold (M ∩ Sd)⊥ centered at x and κ(M∩Sd)⊥(x) denotes the scalar curvature +at point x. Gray [1974] provides the second order expansion of the formula above. Given that +ϵ ∈ (0, U), for all k ∈ {n − m, n − m + 1, ..., d}, then we have a smallest Ck,κ,U such that +volk(Bϵ(x, (M ∩ Sk)⊥)) ≤ Ck,κ,Uϵk. +(13) +The above inequality follows from assumption A5. Using the above inequalities 12, 13 and repeating +the argument d − 1 − n + m times we get the result of the proposition. +We also note that Ck,κ,U increases monotonically with U, this also follows from the volume being +monotonically increasing and positive for ϵ > 0. Finally, we can now prove Theorem 3.4. Let x ∈ M +be uniformly chosen. Then, for all ϵ ∈ (0, U), using Markov’s inequality and Proposition G.1, we +have +E[distanceM(x, Bf ∩ M)] ≥ ϵ Pr(distanceM(x, BF ∩ M) > ϵ) += ϵ(1 − Pr(distanceM(x, BF ∩ M) <= ϵ)) +≥ ϵ(1 − +nin +� +k=nin−m +volk(Sk ∩ M)ωn−kϵnin−kCnin−k,κ,U +� +≥ ϵ(1 − +nin +� +k=nin−m +Cnin−k,κ,U(CgradCbiasCMϵ{#neurons})k� +. +Note that as we increase U the constants Cn−k,κ,U increase, although not strictly, for all k. To +find the supremum of the expression on the right hand side, of the last inequality, in ϵ ∈ (0, U) we +multiply and divide the expression by CgradCbiasCM#neurons to get the polynomial +pU(ζ) = ζ +� +1 − +nin +� +k=nin−m +Cnin−k,κ,Uζk� +, +where ζ = ϵCgradCbiasCM#neurons and ζ ∈ (0, U ′) where U ′ = UCgradCbiasCM#neurons. Let +dM be the diameter of the manifold M, defined by dM = supx,y∈M distanceM(x, y). We assume +that dM is finite. Taking the supremum over all U ∈ (0, dM] or U ′ ∈ (0, d′ +M], where d′ +M = +dMCgradCbiasCM#neurons, gives us the constant CM,κ +CM,κ = +sup +U ′∈(0,d′ +M] +{ sup +ζ∈(0,U ′) +{pU(ζ)}}. +Since dM is finite the constant above exists and is finite. We make a note on the existence of this +constant CM,κ in the absence of the constraint that the diameter of manifold M is finite. As U +increases the constants Cnin−k,κ,U also increase and are all positive. The solution for p′ +U(ζ) = +0, ζ > 0, which we denote by ζU, is unique and keeps decreasing as U increases. The uniqueness +of the solution follows from the fact that the coefficients Cnin−k,κ,U are all positive. We also note +24 + +Figure 12: We plot the optima for a simplified +polynomial as described in Section G.1. The in- +dividual plots correspond to nin increasing from +nin = 2 to nin = 30 (left to right) with m varying +from 1 to nin − 1 on the x-axis. +Figure 13: We plot the optima for a simplified +polynomial as described in Section G.1. The in- +dividual plots correspond to m increasing from +m = 1 to m = 29 (left to right) with nin varying +from m + 1 to 30 on the x-axis. +that pU(ζU) need not be equal to supζ∈(0,U ′){pU(ζ)} because ζU need not lie in (0, U ′). In all +such cases supζ∈(0,U ′){pU(ζ)} = pU(U ′). Given the polynomial pU(ζ) above if we can assert +that there exists a CU, and the corresponding CU ′, such that for all U > CU, and corresponding +U ′ > CU ′, we have supζ∈(0,U ′){pU(ζ)} = pU(ζU) < ∞ and for all 0 < U ≤ CU we have +supζ∈(0,U ′){pU(ζ)} = pU(U ′) < ∞. Therefore, CM,κ exists and is finite if the previous assertion +holds, proving this assertion is beyond the scope of our current work and particularly challenging. +Finally, taking the average over distribution of weights gives us the inequality +E[distanceM(x, Bf ∩ M)] ≥ +CM,κ +CgradCbiasCM#neurons, +where CM,κ is a constant which depends on the average scalar curvature of the manifold M. This +completes the proof of Theorem 3.4. +G.1 +Variations in Supremum +We illustrate the dependence of the the constant CM,κ on varying values of nin, m using a simple +example. We fix the coefficient of the polynomial p(ζ) to be all 1, this not always the case but we do +so to illustrate the relationship between the optima and the exponents for simplest such polynomial: +psimplified(ζ) = ζ +� +1 − +nin +� +k=nin−m +ζk� +We plot the supremums of this simplified polcynomial Csimplified = supζ∈(0,1) psimplified(ζ) for each +nin from the {2, ..., 30} and varying m in Figure 12. Similarly, we vary nin with fixed m and report +the supremums Csimplified in Figure 13. We notice that for a fixed nin the supremum decreases with m +and for a fixed m the supremum increases with nin. +We programatically calculate the supremum being reported by restricting the domain of psimplified +to (0, 1). We solve for the supremum by using the fminbound method from the scipy package +[Virtanen et al., 2020]. The function uses Brent’s method [Brent, 1971] to find the supremum. +H +Toy Supervised Learning Problems +For the two supervised learning tasks with different geometries (tractrix and sphere), we uniformly +sample 1000 data points from each 1D manifold to come up with samples of (xi, yi) pairs. We then +add Gaussian noise to y. We train a DNN with 2 hidden layers, with 10 and 16 neurons in each +layer and a single linear output neuron, for a total of 26 neurons with piece-wise linearity, using the +25 + +1.0 +0.8 +0.6 +Supremum +0.4 +0.2 +0.0 +20 +25 +30 +0 +5 +10 +15 +m1.0 +0.8 +0.6 +Supremum +0.4 +0.2 +0.0 +0 +5 +10 +15 +20 +25 +30 +nFigure 14: The test errors for the cases where data is sampled from the tractrix (blue) and the circle +(green). We see that the tractrix converges slower but the magnitude of the errors remains comparable +as training progresses across the two manifolds. +PyTorch library. The optimization is performed using the Adam optimizer [Kingma and Ba, 2015] +with a learning rate of 0.01. We ensure a reasonable fit of the model by reducing the test time mean +squared error (see Figure 14). We then calculate the exact number of linear regions on the respective +domains by finding the points where z(x(t)) = bz for every neuron z and x is on the 1D manifold. +We do this by adding neurons, z, one by one at every layer and using the SLSQP [Kraft, 1988] to +solve for |z(x(t)) − bz| = 0 in t for tractrix and |z(x(θ)) − bz| = 0 in θ for the circle. Note that +this methodology can be extended to solve for linear regions of a deep ReLU network for any 1D +curve x(.) in any dimension. We then split a linear region depending on where this solution lies +compared to previous layers. For every epoch, we then uniformly randomly sample points from the +1D manifold, by sampling directly from θ and t, to measure average distance to the nearest linear +boundaries. The experiment was run on CPUs, from training to counting of number of linear regions. +The intel cpus had access to 4 GB memory per core. A total of, approximately, 24 cpu hours were +required for all the experiments in this section. This was run on an on demand cloud instance. All +implementations are in PyTorch, except for SLQSP for which we used sklearn. +H.1 +Varying Input Dimensions +The experimental setup, hyperparameters, network architecture, target function and methods are all +the same as described for the toy supervised learning problem for the case where the geometry is a +sphere. The only difference is that the input dimension varies, nin. +I +High Dimensional Dataset +We utilise the official implementation of pretrained StyleGAN generator to generate curves of +images that lie on the manifold of face images. Specifically, for each curve we sample a random +pair of latent vectors: z1, z2 ∈ Rk, this gives us the start and end point of the curve using the +generator g(z1) and g(z2). We then generate 100 images to approximate a curve connecting the two +images on the image manifold in a piece-wise manner. We do so by taking 100 points on the line +connecting z1 and z2 in the latent space that are evenly spaced and generate an image from each one +of them. Therefore, the ith image is generated as: xi = g(((100 − i) × z1 + i × z2)/100), using the +StyleGAN generator g. We qualitatively verify the images to ensure that they lie on the manifold +of images of faces. 4 examples of these curves, sampled as above, are illustrated in the video here: +https://drive.google.com/file/d/1p9B8ATVQGQYoiMh3Q22D-jSaI0USsoNx/view?usp=sharing. +These two constructions allow us to formulate two curves in the high-dimensional setting. The +straight line, with two fixed points g(z1) and g(z2), is defined as x(t) = (1 − t)g(z1) + tg(z2) with +t ∈ [0, 1]. The approximated curve on the manifold is defined as x′(t) = (1 − t)g(zi) + tg(zi+1) +26 + +0.5 +0.4 +Test Errors +0.3 +0.2 +0.1 +0.0 +0 +50 +100 +150 +200 +250 +300 +Epoch Number(a) LR: 0.025, momentum: 0.5, BS: 64 +(b) LR: 0.005, momentum: 0.75, BS: 64 +(c) LR: 0.01, momentum: 0.75, BS: 128 +Figure 15: We report the log density of linear regions for various hyperparameters. Lr refers to the +learning rate and BS is the batch size. +where i = floor(100t). This once again gives us two curves and we solve for the zeros of +|z(x(t)) − bz| = 0 and |z(x′(t)) − bz| = 0 for t ∈ [0, 1] using SLQSP as described in Appendix H. +The neural network, used for classification in our MetFaces experiment, is feed forward with ReLU +activation. There are two hidden layers with 256 and 64 neurons in the first and second layers +respectively. We downsample the images to 128 × 128 × 3. We augment the dataset using random +horizontal flips of the images. All inputs are normalized. We use a batch size of 32. The neural +network is trained using SGD. The learning rate is 0.01 and the momentum is 0.5. The total time +required, for these experiments on MetFaces dataset, was approximately 36 GPU hours on a Titan +RTX GPU that has 24 GB memory. This was run on an on demand cloud instance. We chose +hyperparameters by trial and error, targeting a better fit for the training data for the results reported in +Figure 9 of the main body of the paper. +We report further results for density of linear regions with varying hyperparameters in Figure 15. +We also report the training and testing accuracy for the various sets of hyperparameters in Figure +16. Note that Figure 16(a) corresponds to the test and train accuracies on MetFaces reported in the +main body of the paper (Figure 9). Note all of these results are for the same architecture as described +above. +27 + +11 +-12 +13 +14 +15 +16 +0 +20 +40 +09 +80 +EpochNumber12 +-13 +-14 +15 +16 +0 +20 +40 +60 +80 +EpochNumber-11 +-12 +-13 +-14 +-15 +16 +0 +20 +40 +60 +80 +EpochNumber(a) LR: 0.01, momentum: 0.5, BS: 32 +(b) LR: 0.025, momentum: 0.5, BS: 64 +(c) LR: 0.005, momentum: 0.75, BS: 64 +(d) LR: 0.01, momentum: 0.75, BS: 128 +Figure 16: We report the test and train accuracies across 5 random seeds above. +28 + +1.0 +Train +Test +0.8 +0.6 +Accuracy +0.4 +0.2 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +Epoch1.0 +Train +Test +0.8 +Accuracy +0.6 +0.4 +0.2 +0.0 +0 +20 +40 +60 +80 +100 +Epoch1.0 +Train +Test +0.8 +Accuracy +0.6 +0.4 +0.2 +0.0 +0 +20 +40 +60 +80 +100 +Epoch1.0 +Train +Test +0.8 +Accuracy +0.6 +0.4 +0.2 +0.0 +0 +20 +40 +60 +80 +100 +Epoch \ No newline at end of file diff --git a/w9AyT4oBgHgl3EQfOfbT/content/tmp_files/load_file.txt b/w9AyT4oBgHgl3EQfOfbT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..70237dec022881b5783640b6182d2a3e3a0d6124 --- /dev/null +++ b/w9AyT4oBgHgl3EQfOfbT/content/tmp_files/load_file.txt @@ -0,0 +1,1446 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf,len=1445 +page_content='Effects of Data Geometry in Early Deep Learning Saket Tiwari Department of Computer Science Brown University Providence, RI 02906 saket tiwari@brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='edu George Konidaris Department of Computer Science Brown University Providence, RI 02906 Abstract Deep neural networks can approximate functions on different types of data, from images to graphs, with varied underlying structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This underlying structure can be viewed as the geometry of the data manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' By extending recent advances in the theoretical understanding of neural networks, we study how a randomly initialized neural network with piece-wise linear activation splits the data manifold into regions where the neural network behaves as a linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We derive bounds on the density of boundary of linear regions and the distance to these boundaries on the data manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This leads to insights into the expressivity of randomly initialized deep neural networks on non-Euclidean data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We empirically corroborate our theoretical results using a toy supervised learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Our experiments demonstrate that number of linear regions varies across manifolds and the results hold with changing neural network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We further demonstrate how the complexity of linear regions is different on the low dimensional manifold of images as compared to the Euclidean space, using the MetFaces dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 1 Introduction The capacity of Deep Neural Networks (DNNs) to approximate arbitrary functions given sufficient training data in the supervised learning setting is well known [Cybenko, 1989, Hornik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 1989, Anthony and Bartlett, 1999].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Several different theoretical approaches have emerged that study the effectiveness and pitfalls of deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' These studies vary in their treatment of neural networks and the aspects they study range from convergence [Allen-Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2019, Goodfellow and Vinyals, 2015], generalization [Kawaguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2017, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2017, Jacot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2018, Sagun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2018], function complexity [Mont´ufar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2014, Mhaskar and Poggio, 2016], adversarial attacks [Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2014, Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2015] to representation capacity [Arpit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Some recent theories have also been shown to closely match empirical observations [Poole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2016, Hanin and Rolnick, 2019b, Kunin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' One approach to studying DNNs is to examine how the underlying structure, or geometry, of the data interacts with learning dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The manifold hypothesis states that high-dimensional real world data typically lies on a low dimensional manifold [Tenenbaum, 1997, Carlsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2007, Fefferman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Empirical studies have shown that DNNs are highly effective in deciphering this underlying structure by learning intermediate latent representations [Poole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The ability of DNNs to “flatten” complex data manifolds, using composition of seemingly simple piece-wise linear functions, appears to be unique [Brahma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2016, Hauser and Ray, 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' DNNs with piece-wise linear activations, such as ReLU [Nair and Hinton, 2010], divide the input space into linear regions, wherein the DNN behaves as a linear function [Mont´ufar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The density of these linear regions serves as a proxy for the DNN’s ability to interpolate a complex data landscape and has been the subject of detailed studies [Mont´ufar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2014, Telgarsky, 2015, Serra 36th Conference on Neural Information Processing Systems (NeurIPS 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='00008v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='LG] 29 Dec 2022 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2018, Raghu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The work by Hanin and Rolnick [2019a] on this topic stands out because they derive bounds on the average number of linear regions and verify the tightness of these bounds empirically for deep ReLU networks, instead of larger bounds that rarely materialize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Hanin and Rolnick [2019a] conjecture that the number of linear regions correlates to the expressive power of randomly initialized DNNs with piece-wise linear activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' However, they assume that the data is uniformly sampled from the Euclidean space Rd, for some d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' By combining the manifold hypothesis with insights from Hanin and Rolnick [2019a], we are able to go further in estimating the number of linear regions and the average distance from linear boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We derive bounds on how the geometry of the data manifold affects the aforementioned quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' To corroborate our theoretical bounds with empirical results, we design a toy problem where the input data is sampled from two distinct manifolds that can be represented in a closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We count the exact number of linear regions and the average distance to the boundaries of linear regions on these two manifolds that a neural network divides the two manifolds into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We demonstrate how the number of linear regions and average distance varies for these two distinct manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' These results show that the number of linear regions on the manifold do not grow exponentially with the dimension of input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Our experiments do not provide estimates for theoretical constants, as in most deep learning theory, but demonstrate that the number of linear regions change as a consequence of these constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We also study linear regions of deep ReLU networks for high dimensional data that lies on a low dimensional manifold with unknown structure and how the number of linear regions vary on and off this manifold, which is a more realistic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' To achieve this we present experiments performed on the manifold of natural face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We sample data from the image manifold using a generative adversarial network (GAN) [Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2014] trained on the curated images of paintings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Specifically, we generate images using the pre-trained StyleGAN [Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2019, 2020b] trained on the curated MetFaces dataset [Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2020a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We generate curves on the image manifold of faces, using StyleGAN, and report how the density of linear regions varies on and off the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' These results shed new light on the geometry of deep learning over structured data sets by taking a data intrinsic approach to understanding the expressive power of DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 2 Preliminaries and Background Our goal is to understand how the underlying structure of real world data matters for deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We first provide the mathematical background required to model this underlying structure as the geometry of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We then provide a summary of previous work on understanding the approximation capacity of deep ReLU networks via the complexity of linear regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For the details on how our work fits into one of the two main approaches within the theory of DNNs, from the expressive power perspective or from the learning dynamics perspective, we refer the reader to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1 Data Manifold and Definitions Figure 1: A 2D surface, here represented by a 2-torus, is embedded in a larger input space, R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Suppose each point corresponds to an image of a face on this 2-torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We can chart two curves: one straight line cutting across the 3D space and another curve that stays on the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Images corresponding to the points on the torus will have a smoother variation in style and shape whereas there will be images corresponding to points on the straight line that are not faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We use the example of the MetFaces dataset [Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2020a] to illustrate how data lies on a low dimensional manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The images in the dataset are 1028 × 1028 × 3 dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' By contrast, the number of realistic dimensions along which they vary are limited, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' painting style, artist, size and shape of the nose, jaw and eyes, background, clothing style;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' in fact, very few 1028 × 1028 × 3 2 福dimensional images correspond to realistic faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We illustrate how this affects the possible variations in the data in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' A manifold formalises the notion of limited variations in high dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' One can imagine that there exists an unknown function f : X → Y from a low dimensional space of variations, to a high dimensional space of the actual data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Such a function f : X → Y , from one open subset X ⊂ Rm, to another open subset Y ⊂ Rk, is a diffeomorphism if f is bijective, and both f and f −1 are differentiable (or smooth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Therefore, a manifold is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Let k, m ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' A subset M ⊂ Rk is called a smooth m-dimensional submanifold of Rk (or m-manifold in Rk) iff every point x ∈ M has an open neighborhood U ⊂ Rk such that U ∩ M is diffeomorphic to an open subset Ω ⊂ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' A diffeomorphism (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' differentiable mapping), f : U ∩ M → Ω is called a coordinate chart of M and the inverse, h := f −1 : Ω → U ∩ M is called a smooth parametrization of U ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For the MetFaces dataset example, suppose there are 10 dimensions along which the images vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Further assume that each variation can take a value continuously in some interval of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Then the smooth parametrization would map f : Ω ∩ R10 → M ∩ R1028×1028×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This parametrization and its inverse are unknown in general and computationally very difficult to estimate in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' There are similarities in how geometric elements are defined for manifolds and Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' A smooth curve, on a manifold M, γ : I → M is defined from an interval I to the manifold M as a function that is differentiable for all t ∈ I, just as for Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The shortest such curve between two points on a manifold is no longer a straight line, but is instead a geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' One recurring geometric element, which is unique to manifolds and stems from the definition of smooth curves, is that of a tangent space, defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Let M be an m-manifold in Rk and x ∈ M be a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' A vector v ∈ Rk is called a tangent vector of M at x if there exists a smooth curve γ : I → M such that γ(0) = x, ˙γ(0) = v where ˙γ(t) is the derivative of γ at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The set TxM := {˙γ(0)|γ : R → M is smoothγ(0) = x} of tangent vectors of M at x is called the tangent space of M at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' In simpler terms, the plane tangent to the manifold M at point x is called the tangent space and denoted by by TxM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Consider the upper half of a 2-sphere, S2 ⊂ R3, which is a 2-manifold in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The tangent space at a fixed point x ∈ S2 is the 2D plane perpendicular to the vector x and tangential to the surface of the sphere that contains the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For additional background on manifolds we refer the reader to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2 Linear Regions of Deep ReLU Networks The higher the density of these linear regions the more complex a function a DNN can approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For example, a sin curve in the range [0, 2π] is better approximated by 4 piece-wise linear regions as opposed to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' To clarify this further, with the 4 “optimal” linear regions [0, π/2), [π/2, π), [π, 3π/2), and [3π/2, 2π] a function could approximate the sin curve better than any 2 linear regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' In other words, higher density of linear regions allows a DNN to approximate the variation in the curve better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We define the notion of boundary of a linear regions in this section and provide an overview of previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We consider a neural network, F, which is a composition of activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Inputs at each layer are multiplied by a matrix, referred to as the weight matrix, with an additional bias vector that is added to this product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We limit our study to ReLU activation function [Nair and Hinton, 2010], which is piece-wise linear and one of the most popular activation functions being applied to various learning tasks on different types of data like text, images, signals etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We further consider DNNs that map inputs, of dimension nin, to scalar values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Therefore, F : Rnin → R is defined as, F(x) = WLσ(BL−1 + WL−1σ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='σ(B1 + W1x))), (1) where Wl ∈ Mnl×nl−1 is the weight matrix for the lth hidden layer, nl is the number of neurons in the lth hidden layer, Bl ∈ Rnl is the vector of biases for the lth hidden layer, n0 = nin and σ : R → R 3 is the activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For a neuron z in the lth layer we denote the pre-activation of this neuron, for given input x ∈ Rnin, as zl(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For a neuron z in the layer l we have z(x) = Wl−1,zσ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='σ(B1 + W1x)), (2) for l > 1 (for the base case l = 1 we have z(x) = W1,zx) where Wl−1,z is the row of weights, in the weight matrix of the lth layer, Wl, corresponding to the neuron z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We use Wz to denote the weight vector for brevity, omitting the layer index l in the subscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We also use bz to denote the bias term for the neuron z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Neural networks with piece-wise linear activations are piece-wise linear on the input space [Mont´ufar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Suppose for some fixed y ∈ Rnin as x → y if we have z(x) → −bz then we observe a discontinuity in the gradient ∇xσ(bz + Wzz(x)) at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Intuitively, this is because x is approaching the boundary of the linear region of the function defined by the output of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Therefore, the boundary of linear regions, for a feed forward neural network F, is defined as: BF = {x|∇F(x) is not continuous at x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Hanin and Rolnick [2019a] argue that an important generalization for the approximation capacity of a neural network F is the (nin − 1)−dimensional volume density of linear regions defined as volnin−1(BF ∩ K)/volnin(K), for a bounded set K ⊂ Rnin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This quantity serves as a proxy for density of linear regions and therefore the expressive capacity of DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Intuitively, higher density of linear boundaries means higher capacity of the DNN to approximate complex non-linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The quantity is applied to lower bound the distance between a point x ∈ K and the set BF , which is distance(x, BF ) = min neurons z |z(x) − bz|/||∇z(x)||, which measures the sensitivity over neurons at a given input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The above quantity measures how “far” the input is from flipping any neuron from inactive to active or vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Informally, Hanin and Rolnick [2019a] provide two main results for a randomly initialized DNN F, with a reasonable initialisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Firstly, they show that E �volnin−1(BF ∩ K) volnin(K) � ≈ #{ neurons}, meaning the density of linear regions is bound above and below by some constant times the number of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Secondly, for x ∈ [0, 1]nin, E � distance(x, BF ) � ≥ C#{ neurons}−1, where C > 0 depends on the distribution of biases and weights, in addition to other factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' In other words, the distance to the nearest boundary is bounded above and below by a constant times the inverse of the number of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' These results stand in contrast to earlier worst case bounds that are exponential in the number of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Hanin and Rolnick [2019a] also verify these results empirically to note that the constants lie in the vicinity of 1 throughout training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 3 Linear Regions on the Data Manifold One important assumption in the results presented by Hanin and Rolnick [2019a] is that the input, x, lies in a compact set K ⊂ Rnin and that volnin(K) is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Also, the theorem pertaining to the lower bound on average distance of x to linear boundaries the input assumes the input uniformly distributed in [0, 1]nin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' As noted earlier, high-dimensional real world datasets, like images, lie on low dimensional manifolds, therefore both these assumptions are false in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This motivates us to study the case where the data lies on some m−dimensional submanifold of Rnin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' M ⊂ Rnin where m ≪ nin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We illustrate how this constraint effects the study of linear regions in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' As introduced by Hanin and Rolnick [2019a], we denote the “(nin − k)−dimensional piece” of BF as BF,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' More precisely, BF,0 = ∅ and BF,k is recursively defined to be the set of points x ∈ BF \\ {BF,0 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' ∪ BF,k−1} with the added condition that in a neighbourhood of x the set BF,k coincides with hyperplane of dimension nin − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We provide a detailed and formal definition for BF,k with intuition in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' In our setting, where the data lies on a manifold M, we define B′ F,k 4 Figure 2: A circle is an example of a 1D manifold in a 2D Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The effective number of linear regions on the manifold, the upper half of the circle, are the number of linear regions on the arc from −π to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' In the diagram above, each color in the 2D space corresponds to a linear region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' When the upper half of the circle is flattened into a 1D space we obtain a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Each color on the line corresponds to a linear region of the 2D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' as BF,k ∩ M, and note that dim(B′ F,k) = m − k (Appendix E Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For example, the transverse intersection (see Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='3) of a plane in 3D with the 2D manifold S2 is a 1D curve in S2 and therefore has dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Therefore, B′ F,k is a submanifold of dimension 3 − 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This imposes the restriction k ≤ m, for the intersection BF,k ∩ M to have a well defined volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We first note that the definition of the determinant of the Jacobian, for a collection of neurons z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', zk, is different in the case when the data lies on a manifold M as opposed to in a compact set of dimension nin in Rnin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Since the determinant of the Jacobian is the quantity we utilise in our proofs and theorems repeatedly we will use the term Jacobian to refer to it for succinctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Intuitively, this follows from the Jacobian of a function being defined differently in the ambient space Rnin as opposed to the manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' In case of the former it is the volume of the paralellepiped determined by the vectors corresponding to the directions with steepest ascent along each one of the nin axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' In case of the latter it is more complex and defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Let Hm be the m−dimensional Hausdorff measure (we refer the reader to the Appendix B for background on Hausdorff measure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The Jacobian of a function on manifold M, as defined by Krantz and Parks [2008] (Chapter 5), is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The (determinant of) Jacobian of a function H : M → Rk, where k ≤ dim(M) = m, is defined as JM k,H(x) = sup �Hk(DMH(P)) Hk(P) ���P is a k-dimensional parallelepiped contained in TxM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' � , where DM : TxM → Rk is the differential map (see Appendix B) and we use DMH(P) to denote the mapping of the set P in TxM, which is a parallelepiped, to Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The supremum is taken over all parallelepipeds P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We also say that neurons z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', zk are good at x if there exists a path of neurons from z to the output in the computational graph of F so that each neuron is activated along the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Our three main results that hold under the assumptions listed in Appendix A, each of which extend and improve upon the theoretical results by Hanin and Rolnick [2019a], are: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Given F a feed-forward ReLU network with input dimension nin, output dimension 1, and random weights and biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Then for any bounded measurable submanifold M ⊂ Rnin and any k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='., m the average (m − k)−dimensional volume of BF,k inside M, E[volm−k(BF,k ∩ M)] = � distinct neurons z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',zk in F � M E[Yz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',zk]dvolm(x), (3) where Yz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',zk is JM m,Hk(x)ρb1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',bk(z1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', zk(x)), times the indicator function of the event that zj is good at x for each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Here the function ρbz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',bzk is the density of the joint distribution of the biases bz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', bzk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This change in the formula, from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4 by Hanin and Rolnick [2019a], is a result of the fact that z(x) has a different direction of steepest ascent when it is restricted to the data manifold M, for any j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The proof is presented in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Formula 3 also makes explicit the fact that the data manifold has dimension m ≤ nin and therefore the m − k-dimensional volume is a more representative measure of the linear boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Equipped with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2, we provide a result for the density of boundary regions on manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 5 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For data sampled uniformly from a compact and measurable m dimensional manifold M we have the following result for all k ≤ m: volm−k(BF,k ∩ M) volm(M) ≤ � # neurons k � (2CgradCbiasCM)k, where Cgrad depends on ||∇z(x)|| and the DNN’s architecture, CM depends on the geometry of M, and Cbias on the distribution of biases ρb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The constant CM is the supremum over the matrix norm of projection matrices onto the tangent space, TxM, at any point x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For the Euclidean space CM is always equal to 1 and therefore the term does not appear in the work by Hanin and Rolnick [2019a], but we cannot say the same for our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We refer the reader to Appendix F for the proof, further details, and interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Finally, under the added assumptions that the diameter of the manifold M is finite and M has polynomial volume growth we provide a lower bound on the average distance to the linear boundary for points on the manifold and how it depends on the geometry and dimensionality of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For any point, x, chosen randomly from M, we have: E[distanceM(x, BF ∩ M)] ≥ CM,κ CgradCbiasCM#neurons, where CM,κ depends on the scalar curvature, the input dimension and the dimensionality of the manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The function distanceM is the distance on the manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This result gives us intuition on how the density of linear regions around a point depends on the geometry of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The constant CM,κ captures how volumes are distorted on the manifold M as compared to the Euclidean space, for the exact definition we refer the reader to the proof in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For a manifold which has higher volume of a unit ball, on average, in comparison to the Euclidean space the constant CM,κ is higher and lower when the volume of unit ball, on average, is lower than the volume of the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For background on curvature of manifolds and a proof sketch we refer the reader to the Appendices B and D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Note that the constant CM is the same as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Another difference to note is that we derive a lower bound on the geodesic distance on the manifold M and not the Euclidean distance in Rk as done by Hanin and Rolnick [2019a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This distance better captures the distance between data points on a manifold while incorporating the underlying structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' In other words, this distance can be understood as how much a data point should change to reach a linear boundary while ensuring that all the individual points on the curve, tracing this change, are “valid” data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1 Intuition For Theoretical Results One of the key ingredients of the proofs by Hanin and Rolnick [2019a] is the co-area formula [Krantz and Parks, 2008].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The co-area formula is applied to get a closed form representation of the k−dimensional volume of the region where any set of k neurons, z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', zk is “good” in terms of the expectation over the Jacobian, in the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Instead of the co-area formula we use the smooth co-area formula [Krantz and Parks, 2008] to get a closed form representation of the m − k−dimensional volume of the region intersected with manifold, M, in terms of the Jacobian defined on a manifold (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The key difference between the two formulas is that in the smooth co-area formula the Jacobian (of a function from the manifold M) is restricted to the tangent plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' While the determinant of the “vanilla” Jacobian measures the distortion of volume around a point in Euclidean space the determinant of the Jacobian defined as above (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1) measures the distortion of volume on the manifold instead for the function with the same domain, the function that is 1 if the set of neurons are good and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The value of the Jacobian as defined in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1 has the same volume as the projection of the parallelepiped defined by the gradients ∇z(x) onto the tangent space (see Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1 in Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This introduces the constant CM, defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Essentially, the constant captures how the magnitude of the gradients, ∇z(x), are modified upon being projected to the tangent plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Certain manifolds “shrink” vectors upon projection to the tangent plane more than others, on an average, which is a function of their geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We illustrate how two distinct manifolds “shrink” the gradients differently upon projection to the tangent plane as reflected in the number of linear regions on the manifolds (see Figure 11 in the appendix) for 1D manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We provide intuition 6 (a) (b) Figure 3: The tractrix (a) and circle (b) are plotted in grey and the target function is in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This is for illustration purposes and does not match the actual function or domains used in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' for the curvature of a manifold in Appendix B, due to space constraints, which is used in the lower bound for the average distance in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The constant CM,κ depends on the curvature as the supremum of a polynomial whose coefficients depend on the curvature, with order at most nin and at least nin − m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Note that despite this dependence on the ambient dimension, there are other geometric constants in this polynomial (see Appendix G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Finally, we also provide a simple example as to how this constant varies with nin and m, for a simple and contrived example, in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1 Linear Regions on a 1D Curve To empirically corroborate our theoretical results, we calculate the number of linear regions and average distance to the linear boundary on 1D curves for regression tasks in two settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The first is for 1D manifolds embedded in 2D and higher dimensions and the second is for the high-dimensional data using the MetFaces dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We use the same algorithm, for the toy problem and the high- dimensional dataset, to find linear regions on 1D curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We calculate the exact number of linear regions for a 1D curve in the input space, x : I → Rnin where I is an interval in real numbers, by finding the points where z(x(t)) = bz for every neuron z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The solutions thus obtained gives us the boundaries for neurons on the curve x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We obtain these solutions by using the programmatic activation of every neuron and using the sequential least squares programming (SLSQP) algorithm [Kraft, 1988] to solve for |z(x(t)) − bz| = 0 for t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' In order to obtain the programmatic activation of a neuron we construct a Deep ReLU network as defined in Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We do so for all the neurons for a given DNN with fixed weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2 Supervised Learning on Toy Dataset We define two similar regression tasks where the data is sampled from two different manifolds with different geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We parameterize the first task, a unit circle without its north and south poles, by ψcircle : (−π, π) → R2 where ψcircle(θ) = (cos θ, sin θ) and θ is the angle made by the vector from the origin to the point with respect to the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We set the target function for regression task to be a periodic function in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The target is defined as z(θ) = a sin(νθ) where a is the amplitude and ν is the frequency (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' DNNs have difficulty learning periodic functions [Ziyin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The motivation behind this is to present the DNN with a challenging task where it has to learn the underlying structure of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Moreover the DNN will have to split the circle into linear regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For the second regression task, a tractrix is parametrized by ψtractrix : R1 → R2 where ψtractrix(y) = (y − tanh y, sech y) (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We assign a target function z(t) = a sin(νt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For the purposes of our study we restrict the domain of ψtractrix to (−3, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We choose ν so as to ensure that the number of peaks and troughs, 6, in the periodic target function are the same for both the manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This ensures that the domains of both the problems have length close to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Further experimental details are in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The results, averaged over 20 runs, are presented in Figures 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We note that CM is smaller for Sphere (based on Figure 4) and the curvature is positive whilst CM is larger for tractrix and the curvature is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Both of these constants (curvature and CM) contribute to the lower bound 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='6 10 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Similarly, we show results of number of linear regions divided by the number of neurons upon changing architectures, consequently the number of neurons, for the two manifolds in Figure 8, averaged over 30 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Note that this experiment observes the effect of CM × Cgrad, since changing the architecture also changes Cgrad and the variation in Cgrad is quite low in magnitude as observed empirically by Hanin and Rolnick [2019a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The empirical observations are consistent with our theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We observe that the number of linear regions starts off close to #neurons and remains close throughout the training process for both the manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This supports our theoretical results (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='3) that the constant CM, which is distinct across the two manifolds, affects the number of linear regions throughout training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The tractrix has a higher value of CM and that is reflected in both Figures 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Note that its relationship is inverse to the average distance to the boundary region, as per Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4, and it is reflected as training progresses in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This is due to different “shrinking” of vectors upon being projected to the tangent space (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='3 Varying Input Dimensions To empirically corroborate the results of Theorems 2 and 3 we vary the dimension nin while keeping m constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We achieve this by counting the number of linear regions and the average distance to boundary region on the 1D circle as we vary the input dimension in steps of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We draw samples of 1D circles in Rnin by randomly choosing two perpendicular basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We then train a network with the same architecture as the previous section on the periodic target function (a sin(νθ)) as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The results in Figure 6 shows that the quantities stay proportional to #neurons, and do not vary as nin is increased, as predicted by our theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Our empirical study asserts how the relevant upper and lower bounds, for the setting where data lies on a low-dimensional manifold, does not grow exponentially with nin for the density of linear regions in a compact set of Rnin but instead depend on the intrinsic dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Further details are in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4 MetFaces: High Dimensional Dataset Our goal with this experiment is to study how the density of linear regions varies across a low dimensional manifold and the input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' To discover latent low dimensional underlying structure of data we employ a GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Adversarial training of GANs can be effectively applied to learn a mapping from a low dimensional latent space to high dimensional data [Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The generator is a neural network that maps g : Rk → Rnin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We train a deep ReLU network on the MetFaces dataset with random labels (chosen from 0, 1) with cross entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' As noted by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [2017], training with random labels can lead to the DNN memorizing the entire dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We compare the log density of number of linear regions on a curve on the manifold with a straight line off the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We generate these curves using the data sampled by the StyleGAN by [Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2020a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Specifically, for each curve we sample a random pair of latent vectors: z1, z2 ∈ Rk, this gives us the start and end point of the curve using the generator g(z1) and g(z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We then generate 100 images to approximate a curve connecting the two images on the image manifold in a piece-wise manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We do so by taking 100 points on the line connecting z1 and z2 in the latent space that are evenly spaced and generate an image from each one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Therefore, the ith image is generated as: z′ i = g(((100 − i) × z1 + i × z2)/100), using the StyleGAN generator g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We qualitatively verify the images to ensure that they lie on the manifold of images of faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The straight line, with two fixed points g(z1) and g(z2), is defined as x(t) = (1 − t)g(z1) + tg(z2) with t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The approximated curve on the manifold is defined as x′(t) = (1 − t)g(z′ i) + tg(z′ i+1) where i = floor(100t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We then apply the method from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1 to obtain the number of linear regions on these curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The results are presented in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This leads us to the key observation: the density of linear regions is significantly lower on the data manifold and devising methods to “concentrate” these linear regions on the manifold is a promising research direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' That could lead to increased expressivity for the same number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We provide further experimental details in Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 5 Discussion and Conclusions There is significant work in both supervised and unsupervised learning settings for non-Euclidean data [Bronstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Despite these empirical results most theoretical analysis is agnostic to data geometry, with a few prominent exceptions [Cloninger and Klock, 2020, Shaham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 8 Figure 4: Graph of number of linear regions for tractrix (blue) and sphere (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The shaded re- gions represent one standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Note that the number of neurons is 26 and the number of linear regions are comparable to 26 but different for both the manifolds throughout training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Figure 5: Graph of distance to linear regions for tractrix (blue) and sphere (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The distances are normalized by the maximum distance on the range, for both tractrix and sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The shaded regions represent one standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Figure 6: We observe that as the dimension nin is increased, while keeping the manifold dimension constant, the number of linear regions remains proportional to number of neurons (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Figure 7: We observe that as the dimension nin is increased, while keeping the manifold dimension constant, the average distance varies very little.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Figure 8: The effects of changing the architecture on the number of linear regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We observe that the value of CM effects the number of linear re- gions proportionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The number of hidden units for three layer networks are in the legend along with the data manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Figure 9: We observe that the log density of num- ber of linear regions is lower on the manifold (blue) as compared to off the manifold (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This is for the MetFaces dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 9 12 13 14 15 16 17 18 0 20 40 60 80 100 120 140 EpochNumber45 rRegions 40 35 Linear Numberofi 30 25 20 15 0 50 100 150 200 250 300 EpochNumber0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='035 rest boundary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='030 to near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='025 istance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='020 D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='010 0 50 100 150 200 250 300 EpochNumber60 50 40 30 Number of i 20 5 10 10 15 20 25 0 20 40 60 80 100 120 140 0 Epochs0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='004 Distance to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='003 Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='002 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='001 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='000 0 20 40 60 80 100 120 140 Epochs2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='75 Numberof Linear Regions/#Neurons 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='75 5, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='Sphere 10,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='Sphere 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='50 20,32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='Sphere 5, 8,Tractrix 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='25 10,16,Tractrix 20, 32, Tractrix 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='00 0 20 40 60 80 100 120 140 EpochNumber2015, Schmidt-Hieber, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We incorporate the idea of data geometry into measuring the effective approximation capacity of DNNs, deriving average bounds on the density of boundary regions and distance from the boundary when the data is sampled from a low dimensional manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Our experimental results corroborate our theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We also present insights into expressivity of DNNs on low dimensional manfiolds for the case of high dimensional datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Estimating the geometry, dimensionality and curvature, of these image manifolds accurately is a problem that remains largely unsolved [Brehmer and Cranmer, 2020, Perraul-Joncas and Meila, 2013], which limits our inferences on high dimensional dataset to observations that guide future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We note that proving a lower bound on the number of linear regions, as done by Hanin and Rolnick [2019a], for the manifold setting remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Our work opens up avenues for further research that combines model geometry and data geometry and can lead to empirical research geared towards developing DNN architectures for high dimensional datasets that lie on a low dimensional manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 6 Acknowledgements This work was funded by L2M (DARPA Lifelong Learning Machines program under grant number FA8750-18-2-0117), the Penn MURI (ONR under the PERISCOPE MURI Contract N00014- 17- 1-2699), and the ONR Swarm (the ONR under grant number N00014-21-1-2200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This research was conducted using computational resources and services at the Center for Computation and Visualization, Brown University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We would like to thank Sam Lobel, Rafael Rodriguez Sanchez, and Akhil Bagaria for refining our work, multiple technical discussions, and their helpful feedback on the implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We also thank Tejas Kotwal for assistance on deriving the mathematical details related to the 1D Tractrix and sources for various citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We thank Professor Pedro Lopes de Almeida, Nihal Nayak, Cameron Allen and Aarushi Kalra for their valuable comments on writing and presentation of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We thank all the members of the Brown robotics lab for their guidance and support at various stages of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Finally, we are indebted to, and graciously thank, the numerous anonymous reviewers for their time and labor as their valuable feedback and thoughtful engagement have shaped and vastly refine our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' References 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Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' A comprehensive survey on graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' IEEE Transactions on Neural Networks and Learning Systems, 32:4–24, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Bengio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Hardt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Recht, and Oriol Vinyals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Understanding deep learning requires rethinking generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' ArXiv, abs/1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='03530, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Liu Ziyin, Tilman Hartwig, and Masahito Ueda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Neural networks fail to learn periodic functions and how to fix it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' ArXiv, abs/2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='08195, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Checklist 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [Yes] (b) Did you describe the limitations of your work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [Yes] (c) Did you discuss any potential negative societal impacts of your work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [N/A] Our work is primarily theoretical with few toy experiments we do not see its applicability (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [Yes] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' If you are including theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' (a) Did you state the full set of assumptions of all theoretical results?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [Yes] See Appendix A for a list (b) Did you include complete proofs of all theoretical results?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [Yes] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' If you ran experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' (a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [Yes] See Appendix J (b) Did you specify all the training details (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', data splits, hyperparameters, how they were chosen)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [Yes] See experimental sections in the Appendix and main body (c) Did you report error bars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', with respect to the random seed after running experi- ments multiple times)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [Yes] Except for the cases where there are multiple graphs that are overlapping (Figure 6,7, 8) because it would make interpreting them difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' (d) Did you include the total amount of compute and the type of resources used (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', type of GPUs, internal cluster, or cloud provider)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [Yes] Appendix J 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' If you are using existing assets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', code, data, models) or curating/releasing new assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' (a) If your work uses existing assets, did you cite the creators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [Yes] (b) Did you mention the license of the assets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [Yes] (c) Did you include any new assets either in the supplemental material or as a URL?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [No] (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [N/A] (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [N/A] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' If you used crowdsourcing or conducted research with human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 14 (a) Did you include the full text of instructions given to participants and screenshots, if applicable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [N/A] 15 A Assumptions We first make explicit the assumptions on the distribution of weights and biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' A1: The conditional distribution of any set of biases bz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', bzk given all other weights and biases has a density ρz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',zk(b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', bk) with respect to Lebesgue measure on Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' A2: The joint distribution of all weights has a density with respect to Lebesgue measure on R#weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' A3: The data manifold M is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' A4: (Only needed for Theorem 3) the diameter of M defined by dM = supx,y∈M distanceM(x, y) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' A5: (Only needed for Theorem 3) a geodesic ball in manifold M has polynomial volume growth of order m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' B Additional Background on Manifolds We provide further background on the theory of manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' In this section we first provide the background, definition and an interpretation for the scalar curvature of a manifold at a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Every smooth manifold is also equipped with a Riemannian metric tensor (or metric tensor in short).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Given any two vectors, v and w, in the tangent space of a point x on a manifold M, the metric tensor defines a parallel to the dot product in Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The metric tensor, at a point x, is defined by the smooth functions gij : M → R, i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Where the matrix defined by Gx = [gij(x)] = � �� g11(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' g1n(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' gn1(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' gnn(x) � �� is symmetric and invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The inner product of u, v ∈ TxM is then defined by ⟨u, v⟩M = uT Gxv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' the inner product is symmetric, non-degenerate, and bilinear, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' ⟨ku, v⟩M =k⟨u, v⟩M = ⟨u, kv⟩M, ⟨u + w, v⟩M =⟨u, v⟩M + ⟨w, v⟩M, ⟨u, v⟩M =⟨v, u⟩M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' As can be seen, these properties also hold for the Euclidean inner product (with Gx = I for all x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Let the inverse of G = [gij(x)] be denoted by [gij(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Building on this definition of the metric tensor the Ricci curvature tensor is defined as Rij = − 1 2 n � a,b=1 � ∂2gij ∂xa∂xb + ∂2gab ∂xi∂xj − ∂2gib ∂xj∂xa − ∂2gjb ∂xi∂xa � gab + n � a,b,c,d=1 �1 2 ∂gac ∂xi ∂gbd ∂xj + ∂gic ∂xa ∂gjd ∂xb − ∂gic ∂xa ∂gjb ∂xd � gabgcd − 1 4 n � a,b,c,d=1 �∂gjc ∂xi + ∂gic ∂xj − ∂gij ∂xc � gabgcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For geometric interpretations of the above tensors we refer the reader to the work by Loveridge [2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Another quantity, from the theory of manifolds, which we utilise in our proofs and theorems, is scalar curvature (or Ricci curvature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The curvature is a measure how much the volume of a geodisic ball on the manifold M, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' S2, deviates from a d − 1 sphere in the flat space, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The volume on the manifold deviates by an amount proportional to the curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We illustrate this idea in figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We refer the reader to works by Gray [1974] and Wan [2016] for further technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Since our main theorems relate to the volume of linear regions the scalar curvature plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 16 (a) (b) Figure 10: The geodesic circle on S2 (blue region in (a)) does not have the same area as the flat circle (b), both of radius ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' One can imagine cutting the blue top off the sphere’s surface and trying to “flatten” it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Such an effort will lead to failure, if the material of the sphere does not ”stretch”, since the geodesic ball, on S2, cannot be mapped to a circle in R2 in a distance preserving manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Thus, the area of the two blue regions in (a) and (b) vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This deviation in the area spanned by the two spheres, despite their radii being the same, is proportional to the scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Formally, the scalar curvature of a manifold M at a point x with metric tensor [gij] and Ricci tensor [Rij] is defined as C = n � i,j=1 gijRij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Another important concept is that of Hausdorff measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Since the volumes are “distorted” on a manifold it requires careful consideration when defining a measure and integrating using it on a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The m−dimensional Hausdorff measure, of a set S, is defined as Hm(S) := sup δ>0 inf � ∞ � i=1 (diam Ui)d|S ⊆ ∪∞ i=1Ui, diam Ui < δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Next we introduce the definition of the differential map that is used in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1, for the determinant of the Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The differential map of a smooth function H from a manifold M to a manifold S at a point x ∈ M is the smooth map dH : TxM → TxS such that the tangent vector corresponding to any smooth curve γ : I → M at x, γ′(0) ∈ TxM, maps to the tangent vector of H ◦ γ in TH(x)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This is the analog of the total derivative of “vanilla calculus”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' More intuitively, the differential map captures how the function changes along different directions on N as its input changes along different directions on M, this also has an analog to how rows of the Jacobian matrix are viewed in calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' In Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1 we use the specific case where the function H maps from manifold M to the Euclidean space Rk and the tangent space of a Euclidean space is the Euclidean space itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Finally, a paralellepiped’s, P in TxM, mapping via the differential map gives us the points in Rk that correspond to this set P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' C Related Work There have been various approaches to explain the efficacy of DNNs in approximating arbitrarily complex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We briefly touch upon two such promising approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Broadly, the theory of DNNs can be viewed from two lenses: expressive power [Hornik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 1989, Bartlett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 1998, Poole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2016, Raghu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2017, Kawaguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2017, Neyshabur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2018, Hanin, 2019] and learning dynamics [Saxe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2014, Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2016, Smith and Le, 2018, Jacot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2018, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2019, Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2019a,b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' These approaches are not independent of one another but 17 EEcomplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For example, Kawaguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [2017] argue theoretically how the family of DNNs generalize well despite the large capacity of the function class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Neyshabur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [2018] provide PAC-Bayes generalization bounds which are improved upon by Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Hanin [2019] shows that Deep ReLU networks of finite width can approximate any continuous, convex or smooth functions on a unit cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' These works look at DNNs from the lens of expressive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' More recently, there has been a surge in explaining how various algorithms arrive at these almost accurate function approximations by applying different theoretical models of DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Jacot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [2018] provide results for convergence and generalization of DNNs in the infinite width limit by introducing a the neural tangent kernel (NTK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Hanin and Nica [2020] provide finite depth and width corrections for the NTK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Another line of work within the learning dynamics literature looks at implicit regularization that emerge from the learning algorithm and over-parametrised DNNs [Arora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2019a,b, Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2018, Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Researchers have begun to incorporate data geometry into the theoretical analyses of DNNs by applying the assumption that the data lies on a general manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' First we note the works looking at DNNs from the lens of expressive power combined with the idea of data geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Shaham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [2015] demonstrate that the size of the neural network depends on the curvature of the data manifold and the complexity of the function, whilst depending weakly on the input data dimension, for their construction of sparsely-connected 4-layer neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Cloninger and Klock [2020] show that their construction of deep ReLU nets achieve near optimal approximation rates which depend only on the intrinsic dimensionality of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [2019] exploit the low dimensional structure of data to enhance the function approximation capacity of Deep ReLU networks by means of theoretical guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Schmidt-Hieber [2019] shows that sparsely connected deep ReLU networks can approximate a Holder function on a low dimensional manifold embedded in a high dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Simultaneously, researchers have incorporated data geometry into the learning dynamics line of work [Goldt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2020, Paccolat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2020, Buchanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2021, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Buchanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [2021] apply the NTK model to study how DNNs can separate two curves, representing the data manifolds of two separate classes, on the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Goldt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [2020] introduce the Hidden Manifold Model for structured data sets to capture the dynamics of two-layer neural networks trained with stochastic gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Rahaman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [2019] provide empirical results on which data manifolds are learned faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Finally, the work by Novak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [2018] comes the closes in studying the number of linear regions on the data manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' They study the change in input output Jacobian, and as a consequence the number of linear regions, for DNNs with piece-wise linearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' They provide empirical studies by counting the number of linear regions along lines connecting data points as a proxy for number of linear regions on the data manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Our work fits into the study of expressive power of DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The number of linear regions is a good proxy for the practical expressive power or approximation capacity of Deep ReLU networks [Mont´ufar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The results surrounding the density of linear regions make the fewest simplifying assumptions both on the data and the architecture of the DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The results by Hanin and Rolnick [2019a] bound the number of linear regions orders of magnitude tighter than previous results by deriving bounds for the average case and not the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Moreover, they demonstrate the validity empirically in a setting with very few simplifying assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We introduce the manifold hypothesis to this setting in order to obtain tighter bounds for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This introduces a toolbox of ideas from differential geometry to analyse the approximation capacity of deep ReLU networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' In addition to the theoretical works listed above, there has been significant empirical work that applies DNNs to non-Euclidean data [Bronstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2017, 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Here the data is assumed to be sampled from manifolds with certain geometric properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For example, Ganea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [2018] design DNNs for data sampled from Hyperbolic spaces of arbitrary dimensionality and modify the forward and backward passes accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' There have been numerous applications of modified DNNs, namely graph convolutional networks, to graph data that incorporate the idea that graphs are discrete samples from a smooth manifold [Henaff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2015, Monti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2017, Kipf and Welling, 2017], see the survey by Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' [2019] for a comprehensive review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Graph convolutional networks have also been applied to point cloud data for applications in graphics [Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2017, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' D Proof Sketch In this section we provide an overview of how the three main theorems are proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2 provides an equality for measuring the volume of m − k dimensional boundary regions on the 18 manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' To this effect, we introduce the idea of viewing boundary regions as submanifolds on the data manifold instead of hyperplanes (Proposition 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We then prove an equality between the volume of boundary regions and the Jacobian of the neurons over the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We utilise the smooth coarea formula that, intuitively, is applied to integrate a function using level sets on a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This completes the proof for Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' To prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='3 we first prove that the Jacobian of a function on a manifold can be denoted using the volume of paralellepiped of vectors in the ambient space subject to a linear transform (Proposition 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Using this result and combining it with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2 we can then give an inequality for the density of linear regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' As can be expected this volume depends on the aforementioned projection, which in turn is related to the geometry of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Finally, for proving Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4 we first provide an inequality over the tubular neighbourhood of the boundary region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We then use this result to lower bound the geodesic distance between the boundary region and any random point on the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The proof strategy follows that of Hanin and Rolnick [2019a] but there are major deviations when it comes to accounting for the geometry of the data manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' To the best of our knowledge, we are utilising elements of differential topology that are unique to machine learning when it comes to developing a theoretical understanding of DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' E Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2 We follow the proof strategy used by Hanin and Rolnick [2019a] but deviate from it to account for our setting where x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Let Sz be the set of values at which the neuron z has a discontinuity in the differential of its output (or the neuron switches between the two linear regions of the piece-wise linear activation σ), Sz := {x ∈ Rnin|z(x) − bz = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We also have O := � x ∈ Rnin|∀j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', L ∃ neuron z with l(z) = j s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' σ′(z(x) − bz) ̸= 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Further, � Sz := Sz ∩ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We state propositions 9 and 10 by Hanin and Rolnick [2019a] as we apply them to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2, relabeling them as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' (Proposition 9 by Hanin and Rolnick [2019a]) Under assumptions A1 and A2, we have, with probability 1, BF = � neurons z � Sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' By extending the notion of Sz to multiple neurons we have �Sz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',zk := k� j=1 �Szj, meaning that the set �Sz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',zk is, intuitively, the collection of inputs in Rin where the neurons zj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', k, switch between linear regions for σ and at which the output of F is affected by the outputs of these neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We refer the reader to section B of the appendix in the work by Hanin and Rolnick [2019a] for an intuitive explanation of proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Before proceeding we provide a formal definition and intuition for the set BF,k, BF,k = {x|x ∈ BF \\ {BF,0 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' ∪ BF,k−1} = BF,−k and for any ball of radius ϵ > 0, B(x, ϵ) ∩ BF,−k is subset to a n − k dimensional hyperplane}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Following the explanation provided by Hanin and Rolnick [2019a], BF,k is the nin − k dimensional piece of BF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Suppose the boundaries of linear regions for nin = 2 are unions of polygon boundaries, as depicted in Figure 2 of the main body of the paper, then BF,1 are all the open line segments of these polygons and BF,2 are the end points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Next we state Proposition 10 by Hanin and Rolnick [2019a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 19 Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' (Prosposition 10 by Hanin and Rolnick [2019a]) Fix k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', nin, and k distinct neurons z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', zk in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Then, with probability 1, for every x ∈ BF,k there exists a neighbourhood in which BF,k coincides with a nin−k−dimensional hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We now present Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4, and its proof, which incorporates the additional constraint that x ∈ M, which is an m-dimensional manifold in Rnin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' To prove the proposition we need the definition of tranversal intersection of two manifolds [Guillemin and Pollack, 1974].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Two submanifolds, M1 and M2, of S are said to intersect transversally if at every point of intersection their tangent spaces, at that point, together generate the tangent space of the manifold, S, by means of linear combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Formally, for all x ∈ M1 ∩ M2 TxS = TxM1 + TxM2, if and only if M1 and M2 intersect transversally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For example, given a 2D hyperplane, P, and the surface of a 3D sphere, S2, intersect in the ambient space R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We have that this intersection is transverse if and only if P is not tangent to S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For the case where a 2D hyperplane, ¯P, intersects with S2 at a point p but does not intersect tranversally it coincides exactly with the tangent plane of S2 at point {p} = S2 ∩ P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' TpS = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Note that in either case the tangent space of the 2D hyperplane P at any point of intersection is the plane itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Fix k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', m and k distinct neurons z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', zk in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Then, with probability 1, for every x ∈ BF,k ∩ M there exists a neighbourhood in which BF,k coincides with an m − k dimensional submanifold in Rin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' From Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2 we already know that BF,k is a nin − k-dimensional hyperplane in some neighbourhood of x, with probability 1, for any x ∈ BF,k ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Let this hyperplane be denoted by Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This is an n − k dimensional submanifold of Rnin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The tangent space of this hyperplane at x is the hyperplane itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Therefore, from assumptions A1 and A2 we have that the probability that this hyperplane intersects the manifold M transversally with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' In other words the probability that this plane Pk contains or is contained in TxM is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Finally, we have the intersection, M ∩ Hk, has dimension dim(M) + dim(Hk) − nin [Guillemin and Pollack, 1974], which is equal to m − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' One implication of Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4 is that for any k ≤ m the m − (k + 1) dimensional volume of BF,k ∩ M is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' In addition to that, Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4 implies that, with probability 1, volm−k(BF,k) = � distinct neurons z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',zk volm−k(�Sz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',zk ∩ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' (4) The final step in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2 is to prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Let z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', zk be distinct neurons in F and k ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Then for a bounded m−Hausdorff measurable manifold M embedded in Rnin, E � volm−k � �Sz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',zk ∩ M �� = � M E � Yz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',zk(x) � dx, where Yz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',zk(x) equals JM m,Hk(x)ρb1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',bk(z1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', zk(x)), times the indicator function of the event that zj, for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', k, is good at x for every j and Hk : Rnin → Rk is such that Hk(x) = [z1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', zk(x)]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The expectation is over the distribution of weights and biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Let z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', zk be distinct neurons in F and M be an m−dimensional compact Haudorff measurable manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We seek to compute the mean of volm−k(�Sz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',zk ∩ M) over the distribution of weights and biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We can rewrite this expression as � Sz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',zk ∩M 1zj is good at xdvolm−k(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' (5) 20 The map Hk is Lipschitz and C1 almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We first note the smooth coarea formula (theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='9 by Krantz and Parks [2008]) in context of our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Suppose m ≥ k and Hk : Rnin → Rk is C1 and M ⊆ Rnin is an m−dimensional C1 manifold in Rnin, then � M g(x)JM k,Hk(x)dvolm(x) = � Rk � M∩H−1 k (y) g(y)dvolm−k(y)dvolk(x), (6) for every Hm-measurable function g where JM k,Hk is as defined in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We denote preactivations and biases of neurons as z(x) = [z1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', zk(x)]T and bz = [bz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', bzk]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' From the notation in A1, we have that ρbz = ρbz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',bzk , is the joint conditional density of bz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', bzk given all other weights and biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The mean of the term in equation 5 over the conditional distribution of bz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', bzk, ρbz, is therefore � Rk bdvolk(b) � {z=b}∩M 1zj is good at xdvolm−k(x), (7) where we denote [b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', bk]T as b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Thus applying the smooth co-area formula (Equation 6) to the expression in 7 shows that the average 5 is equal to � M Yz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',zk(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Finally, we take the average over the remaining weights and biases and commute the expectation with the dx integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We can do this since the integrand is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This gives us the result: E � volm−k � �Sz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',zk ∩ M �� = � M E � Yz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',zk(x) � dx, (8) as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Finally, taking the summation over all possible sets of distinct neurons z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', zk and combining equation 4 with Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5 completes the proof for Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' F Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='3 To prove the upper bound in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='3 we first show that the (determinant of) Jacobian for the function Hk : M → Rk, Hk(x) = [z1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', zk(x)]T , as defined in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1 is equal to the volume of the parallelopiped defined by the vectors φHk(∇zj(x)), for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', k, where φHk : Rk → TxM is an orthogonal projection onto the orthogonal complement of the kernel of the differential DMHk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Intuitively, this shows that with the added assumption x ∈ M in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='3 how exactly we can incorporate the geometry of the data manifold M into the upper bound provided by Hanin and Rolnick [2019a] in corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Given Hk : M → Rk such that Hk(x) = [z1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', zk(x)]T and the differential DMHk is surjective at x then JM k,Hk(x) = � det(Gram(φHk(∇z1(x)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', φHk(∇zk(x)))), (9) where φHk : Rn → Rk is a linear map and Gram denotes the Gramian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We first define the orthogonal complement of the kernel of the differential DMHk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For a manifold M ⊂ Rn and a fixed point x we have that TxM is a m−dimensional hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' If we choose an orthonormal basis e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', en of Rn such that e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', em spans TxM for a fixed x we can denote all vectors in TxM using m coordinates corresponding to this basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Therefore, for any vector y ∈ Rk we can get the orthogonal projection of y onto TxM using a m × n matrix which we denote as Px, where Pxy (matrix multiplied by a vector) represents a vector in TxM corresponding to the basis e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For any manifold M in Rn and function Hk : M → Rk we have that DMHk : TxM → Rk at a fixed point x is linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Therefore we can write DMHk(v) = Av 21 where v ∈ TxM is denoted using the aforementioned basis of TxM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This implies that A is a k × m matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Therefore, the kernel of DMHk for a fixed point x ∈ M is ker(DMHk) = � z|Az = 0 and z ∈ TxM � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Since we can create a canonical basis for the space ker(DMHk) starting from the basis e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', em in Rn using the Gram-Schmidt process given the matrix A we have that for any y ∈ Rn we can project it orthogonally onto ker(DMHk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The orthogonal complement of ker(DMHk) is therefore defined by ker(DMHk)⊥ = � a|a · z = 0 for all z ∈ ker(DMHk) and a ∈ TxM � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Similar to the previous argument, we construct a canonical basis starting from e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', em for ker(DMHk)⊥ and therefore we can denote the orthogonal projection onto ker(DMHk)⊥ as a linear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We denote this linear projection for fixed x using φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We denote the basis vectors e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='., em as a m × n matrix E where each row i corresponds to the vector ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Therefore, the orthogonal projection of any vector y ∈ Rn is Ey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Now we can get the matrix A using E∇zj(x) corresponding to each row j for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This uses the fact that the direction of steepest ascent on zj(x) restricted to the tangent space TxM of the manifold M is an orthogonal projection of the direction of steepest ascent in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Finally, from lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5 by Guillemin and Pollack [1974] we have that JM k,Hk(x) = Hk(DMHk(P))/Hk(P), for any parallelepiped P contained in (ker(DMHk))⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Arguing similar to the proof of lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5 by Guillemin and Pollack [1974] we get that JM k,Hk(x) = � det((A)T A) = � det Gram(E∇z1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', E∇zk(x)), thereby showing that φHk(y) = Ey is a linear mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Although we state Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1 for neurons zj(x), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', k in the proof, it applies to any function that satisfy the conditions laid out in the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Equipped with Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1 we prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' When the weights and biases of F are independent obtain an upper bound on ρbz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',bzk (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', bk) as Πk j=1ρbzj (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', bk) ≤ � sup neurons z ρbz(b) �k = Ck bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Hence, Yz1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=',zk ≤ Ck biasJM k,Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' From Proposition 9 we have that JM k,Hk is equal to the k-dimensional volume of the paralellopiped spanned by φx(∇zj(x)) for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Therefore, we have JM k,Hk ≤ Πk j=1||E∇zj(x)|| ≤ ||E||kΠk j=1||∇zj(x)||, (10) where ||E|| denotes the matrix norm which is defined as ||E|| = sup � ||Ey|| ���y ∈ Rk, ||y|| = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Note that E does not depend on F (or z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', zk) but only on TxM or more generally the geometry of M at any point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2 by Hanin and Nica [2018] we have, for any fixed x, E � Πk j=1||∇zj(x)|| � ≤ � Cgrad �k , (11) where, Cgrad = sup z sup x∈Rnin E[||∇z(x)||2k]1/k ≤ Ce C �d j=1 1 nj , 22 (a) (b) Figure 11: We illustrate how vectors project differently on tangent planes of two different manifolds: circle (a) and tractrix (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' In case of the tractrix the tangents (and the projection of vectors onto them) are on the inside of the tractrix whereas for the sphere the tangents are always on the outside of the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Since the projections of vectors onto the tangent space are an essential aspect of our proof we end up with the term CM, which quantifies the “shrinking” of these vectors upon projection, in the inequalities for Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' wherein C > 0 depends only on µ and not on the architecture of F and nj is the width of the hidden layer j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Let CM be defined as CM := sup � C| there exists a set, S, of non zero m − k-dimensional Hausdorff measure such that ||Ex|| ≥ C∀x ∈ S � Therefore, combining equations 11, 10 and result from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2 we have E[volm−k(BF,k ∩ M)] volm(M) ≤ � number of neurons k � (2CgradCbiasCM)k, where the expectation is over the distribution of weights and biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' G Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4 We first prove the following proposition Proposition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For a compact m-dimensional submanifold M in Rn, m, n ≥ 1 and m < n let S ⊆ Rn be a compact fixed continuous piece-wise linear submanifold with finitely many pieces and given any U > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Let S0 = ∅ and let Sk be the union of the interiors of all k-dimensional pieces of S \\ (S0 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' ∪ Sk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Denote by Tϵ the ϵ-tubuluar neighbourhood of any X ⊂ M such that Tϵ(X) = � y|dM(y, X) < ϵ and y ∈ M � , where ϵ ∈ (0, U), dM is the geodesic distance between the point y and set X on the manifold M, we have volm(Tϵ(S)) ≤ d � k=n−m volk(Sk ∩ M)ωn−kϵn−kCk,κ,U, where Ck,κ,U > 0 is a constant that depends on the average scalar curvature κ(Sk∩M)⊥ and U, and ωn−k is the volume of the unit ball in Rn−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Define d to be the maximal dimension of linear pieces in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Let x ∈ Tϵ(X ∩ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Suppose x /∈ Tϵ(X ∩ M) for all k = n − m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Then the intersection of a geodesic ball of radius ϵ around s with S is a ball inside Sd ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Using the convexity of this ball, with respect to the manifold M [Robbin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2011], there exists a point y in Sd ∩ M such that the geodesic γ : [0, 1] → M with 23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 6 4 2 0 2 4 6γ(0) = y and γ(1) = x is perpendicular to Sd ∩ M at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Formally, TSd∩MM at y is perpendicular to ˙ γ(0) ∈ TM at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Let Bϵ(N ∗(Sd ∩ M)) be the union of all the ϵ balls along the fiber of the submanifold Sd ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Therefore, we have volm(Tϵ(S ∩ M) ≤ volm(Bϵ(N ∗(Sd ∩ M)) + volm(Tϵ(S≤d−1 ∩ M)), (12) where S≤d−1 := ∪d−1 k=0Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We also note that volm(Bϵ(N ∗(Sd ∩ M)) = volm+d−n(Sd ∩ M)voln−d(Bϵ((M ∩ Sd)⊥)), where Bϵ((M ∩ Sd)⊥) is the average volume of an ϵ ball in the submanifold of M orthogonal to M ∩ Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This volume depends on the average scalar curvature, κ(M∩Sd)⊥ of the submanifold (M ∩ Sd)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' As shown by Wan [2016], for a fixed point x ∈ (M ∩ Sd)⊥ voln−d(Bϵ(x, (M ∩ Sd)⊥)) = ωn−dϵn−d� 1 − κ(x)(M∩Sd)⊥ n − d + 2 ϵ2 + O(ϵ4) � , where ωn−d is the volume of the unit ball of dimension n − d, Bϵ(x, (M ∩ Sd)⊥) is the geodesic ball of radius ϵ in the manifold (M ∩ Sd)⊥ centered at x and κ(M∩Sd)⊥(x) denotes the scalar curvature at point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Gray [1974] provides the second order expansion of the formula above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Given that ϵ ∈ (0, U), for all k ∈ {n − m, n − m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', d}, then we have a smallest Ck,κ,U such that volk(Bϵ(x, (M ∩ Sk)⊥)) ≤ Ck,κ,Uϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' (13) The above inequality follows from assumption A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Using the above inequalities 12, 13 and repeating the argument d − 1 − n + m times we get the result of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We also note that Ck,κ,U increases monotonically with U, this also follows from the volume being monotonically increasing and positive for ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Finally, we can now prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Let x ∈ M be uniformly chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Then, for all ϵ ∈ (0, U), using Markov’s inequality and Proposition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1, we have E[distanceM(x, Bf ∩ M)] ≥ ϵ Pr(distanceM(x, BF ∩ M) > ϵ) = ϵ(1 − Pr(distanceM(x, BF ∩ M) <= ϵ)) ≥ ϵ(1 − nin � k=nin−m volk(Sk ∩ M)ωn−kϵnin−kCnin−k,κ,U � ≥ ϵ(1 − nin � k=nin−m Cnin−k,κ,U(CgradCbiasCMϵ{#neurons})k� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Note that as we increase U the constants Cn−k,κ,U increase, although not strictly, for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' To find the supremum of the expression on the right hand side, of the last inequality, in ϵ ∈ (0, U) we multiply and divide the expression by CgradCbiasCM#neurons to get the polynomial pU(ζ) = ζ � 1 − nin � k=nin−m Cnin−k,κ,Uζk� , where ζ = ϵCgradCbiasCM#neurons and ζ ∈ (0, U ′) where U ′ = UCgradCbiasCM#neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Let dM be the diameter of the manifold M, defined by dM = supx,y∈M distanceM(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We assume that dM is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Taking the supremum over all U ∈ (0, dM] or U ′ ∈ (0, d′ M], where d′ M = dMCgradCbiasCM#neurons, gives us the constant CM,κ CM,κ = sup U ′∈(0,d′ M] { sup ζ∈(0,U ′) {pU(ζ)}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Since dM is finite the constant above exists and is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We make a note on the existence of this constant CM,κ in the absence of the constraint that the diameter of manifold M is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' As U increases the constants Cnin−k,κ,U also increase and are all positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The solution for p′ U(ζ) = 0, ζ > 0, which we denote by ζU, is unique and keeps decreasing as U increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The uniqueness of the solution follows from the fact that the coefficients Cnin−k,κ,U are all positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We also note 24 Figure 12: We plot the optima for a simplified polynomial as described in Section G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The in- dividual plots correspond to nin increasing from nin = 2 to nin = 30 (left to right) with m varying from 1 to nin − 1 on the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Figure 13: We plot the optima for a simplified polynomial as described in Section G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The in- dividual plots correspond to m increasing from m = 1 to m = 29 (left to right) with nin varying from m + 1 to 30 on the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' that pU(ζU) need not be equal to supζ∈(0,U ′){pU(ζ)} because ζU need not lie in (0, U ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' In all such cases supζ∈(0,U ′){pU(ζ)} = pU(U ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Given the polynomial pU(ζ) above if we can assert that there exists a CU, and the corresponding CU ′, such that for all U > CU, and corresponding U ′ > CU ′, we have supζ∈(0,U ′){pU(ζ)} = pU(ζU) < ∞ and for all 0 < U ≤ CU we have supζ∈(0,U ′){pU(ζ)} = pU(U ′) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Therefore, CM,κ exists and is finite if the previous assertion holds, proving this assertion is beyond the scope of our current work and particularly challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Finally, taking the average over distribution of weights gives us the inequality E[distanceM(x, Bf ∩ M)] ≥ CM,κ CgradCbiasCM#neurons, where CM,κ is a constant which depends on the average scalar curvature of the manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This completes the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1 Variations in Supremum We illustrate the dependence of the the constant CM,κ on varying values of nin, m using a simple example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We fix the coefficient of the polynomial p(ζ) to be all 1, this not always the case but we do so to illustrate the relationship between the optima and the exponents for simplest such polynomial: psimplified(ζ) = ζ � 1 − nin � k=nin−m ζk� We plot the supremums of this simplified polcynomial Csimplified = supζ∈(0,1) psimplified(ζ) for each nin from the {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 30} and varying m in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Similarly, we vary nin with fixed m and report the supremums Csimplified in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We notice that for a fixed nin the supremum decreases with m and for a fixed m the supremum increases with nin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We programatically calculate the supremum being reported by restricting the domain of psimplified to (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We solve for the supremum by using the fminbound method from the scipy package [Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The function uses Brent’s method [Brent, 1971] to find the supremum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' H Toy Supervised Learning Problems For the two supervised learning tasks with different geometries (tractrix and sphere), we uniformly sample 1000 data points from each 1D manifold to come up with samples of (xi, yi) pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We then add Gaussian noise to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We train a DNN with 2 hidden layers, with 10 and 16 neurons in each layer and a single linear output neuron, for a total of 26 neurons with piece-wise linearity, using the 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='6 Supremum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 20 25 30 0 5 10 15 m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='6 Supremum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 0 5 10 15 20 25 30 nFigure 14: The test errors for the cases where data is sampled from the tractrix (blue) and the circle (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We see that the tractrix converges slower but the magnitude of the errors remains comparable as training progresses across the two manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' PyTorch library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The optimization is performed using the Adam optimizer [Kingma and Ba, 2015] with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We ensure a reasonable fit of the model by reducing the test time mean squared error (see Figure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We then calculate the exact number of linear regions on the respective domains by finding the points where z(x(t)) = bz for every neuron z and x is on the 1D manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We do this by adding neurons, z, one by one at every layer and using the SLSQP [Kraft, 1988] to solve for |z(x(t)) − bz| = 0 in t for tractrix and |z(x(θ)) − bz| = 0 in θ for the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Note that this methodology can be extended to solve for linear regions of a deep ReLU network for any 1D curve x(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=') in any dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We then split a linear region depending on where this solution lies compared to previous layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' For every epoch, we then uniformly randomly sample points from the 1D manifold, by sampling directly from θ and t, to measure average distance to the nearest linear boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The experiment was run on CPUs, from training to counting of number of linear regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The intel cpus had access to 4 GB memory per core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' A total of, approximately, 24 cpu hours were required for all the experiments in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This was run on an on demand cloud instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' All implementations are in PyTorch, except for SLQSP for which we used sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1 Varying Input Dimensions The experimental setup, hyperparameters, network architecture, target function and methods are all the same as described for the toy supervised learning problem for the case where the geometry is a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The only difference is that the input dimension varies, nin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' I High Dimensional Dataset We utilise the official implementation of pretrained StyleGAN generator to generate curves of images that lie on the manifold of face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Specifically, for each curve we sample a random pair of latent vectors: z1, z2 ∈ Rk, this gives us the start and end point of the curve using the generator g(z1) and g(z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We then generate 100 images to approximate a curve connecting the two images on the image manifold in a piece-wise manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We do so by taking 100 points on the line connecting z1 and z2 in the latent space that are evenly spaced and generate an image from each one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Therefore, the ith image is generated as: xi = g(((100 − i) × z1 + i × z2)/100), using the StyleGAN generator g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We qualitatively verify the images to ensure that they lie on the manifold of images of faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 4 examples of these curves, sampled as above, are illustrated in the video here: https://drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='com/file/d/1p9B8ATVQGQYoiMh3Q22D-jSaI0USsoNx/view?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='usp=sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' These two constructions allow us to formulate two curves in the high-dimensional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The straight line, with two fixed points g(z1) and g(z2), is defined as x(t) = (1 − t)g(z1) + tg(z2) with t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The approximated curve on the manifold is defined as x′(t) = (1 − t)g(zi) + tg(zi+1) 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4 Test Errors 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 0 50 100 150 200 250 300 Epoch Number(a) LR: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='025, momentum: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5, BS: 64 (b) LR: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='005, momentum: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='75, BS: 64 (c) LR: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='01, momentum: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='75, BS: 128 Figure 15: We report the log density of linear regions for various hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Lr refers to the learning rate and BS is the batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' where i = floor(100t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This once again gives us two curves and we solve for the zeros of |z(x(t)) − bz| = 0 and |z(x′(t)) − bz| = 0 for t ∈ [0, 1] using SLQSP as described in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The neural network, used for classification in our MetFaces experiment, is feed forward with ReLU activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' There are two hidden layers with 256 and 64 neurons in the first and second layers respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We downsample the images to 128 × 128 × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We augment the dataset using random horizontal flips of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' All inputs are normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We use a batch size of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The neural network is trained using SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='01 and the momentum is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' The total time required, for these experiments on MetFaces dataset, was approximately 36 GPU hours on a Titan RTX GPU that has 24 GB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' This was run on an on demand cloud instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We chose hyperparameters by trial and error, targeting a better fit for the training data for the results reported in Figure 9 of the main body of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We report further results for density of linear regions with varying hyperparameters in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' We also report the training and testing accuracy for the various sets of hyperparameters in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Note that Figure 16(a) corresponds to the test and train accuracies on MetFaces reported in the main body of the paper (Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' Note all of these results are for the same architecture as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 27 11 12 13 14 15 16 0 20 40 09 80 EpochNumber12 13 14 15 16 0 20 40 60 80 EpochNumber-11 12 13 14 15 16 0 20 40 60 80 EpochNumber(a) LR: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='01, momentum: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5, BS: 32 (b) LR: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='025, momentum: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='5, BS: 64 (c) LR: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='005, momentum: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='75, BS: 64 (d) LR: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='01, momentum: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='75, BS: 128 Figure 16: We report the test and train accuracies across 5 random seeds above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content=' 28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 Train Test 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='6 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 0 20 40 60 80 100 120 140 Epoch1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 Train Test 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='8 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 0 20 40 60 80 100 Epoch1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='0 Train Test 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='8 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9AyT4oBgHgl3EQfOfbT/content/2301.00008v1.pdf'} +page_content='4 0.' metadata={'source': 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