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Wikipedia:Algebraic connectivity#0 | The algebraic connectivity (also known as Fiedler value or Fiedler eigenvalue after Miroslav Fiedler) of a graph G is the second-smallest eigenvalue (counting multiple eigenvalues separately) of the Laplacian matrix of G. This eigenvalue is greater than 0 if and only if G is a connected graph. This is a corollary to the fact that the number of times 0 appears as an eigenvalue in the Laplacian is the number of connected components in the graph. The magnitude of this value reflects how well connected the overall graph is. It has been used in analyzing the robustness and synchronizability of networks. == Properties == The algebraic connectivity of undirected graphs with nonnegative weights is a ( G ) ≥ 0 {\displaystyle a(G)\geq 0} , with the inequality being strict if and only if G is connected. However, the algebraic connectivity can be negative for general directed graphs, even if G is a connected graph. Furthermore, the value of the algebraic connectivity is bounded above by the traditional (vertex) connectivity of a graph, algebraic connectivity ≤ connectivity {\displaystyle {\text{algebraic connectivity}}\leq {\text{connectivity}}} , unless the graph is complete (the algebraic connectivity of a complete graph Kn is its order n). For an undirected connected graph with nonnegative edge weights, n vertices, and diameter D, the algebraic connectivity is also known to be bounded below by 1 n D {\textstyle {\frac {1}{nD}}} , and in fact (in a result due to Brendan McKay) by 4 n D {\textstyle {\frac {4}{nD}}} . For the example graph with 6 nodes show above ( n = 6 , D = 3 {\textstyle n=6,D=3} ), these bounds would be calculated as: 4 / 18 = 0.222 ≤ algebraic connectivity 0.722 ≤ connectivity 1. {\displaystyle 4/18=0.222\leq {\text{algebraic connectivity 0.722}}\leq {\text{connectivity 1.}}} Unlike the traditional form of graph connectivity, defined by local configurations whose removal would disconnect the graph, the algebraic connectivity is dependent on the global number of vertices, as well as the way in which vertices are connected. In random graphs, the algebraic connectivity decreases with the number of vertices, and increases with the average degree. The exact definition of the algebraic connectivity depends on the type of Laplacian used. Fan Chung has developed an extensive theory using a rescaled version of the Laplacian, eliminating the dependence on the number of vertices, so that the bounds are somewhat different. In models of synchronization on networks, such as the Kuramoto model, the Laplacian matrix arises naturally, so the algebraic connectivity gives an indication of how easily the network will synchronize. Other measures, such as the average distance (characteristic path length) can also be used, and in fact the algebraic connectivity is closely related to the (reciprocal of the) average distance. The algebraic connectivity also relates to other connectivity attributes, such as the isoperimetric number, which is bounded below by half the algebraic connectivity. == Fiedler vector == The original theory related to algebraic connectivity was produced by Miroslav Fiedler. In his honor the eigenvector associated with the algebraic connectivity has been named the Fiedler vector. The Fiedler vector can be used to partition a graph. === Partitioning a graph using the Fiedler vector === For the example graph in the introductory section, the Fiedler vector is ( 0.415 0.309 0.069 − 0.221 0.221 − 0.794 ) {\textstyle {\begin{pmatrix}0.415&0.309&0.069&-0.221&0.221&-0.794\end{pmatrix}}} . The negative values are associated with the poorly connected vertex 6, and the neighbouring articulation point, vertex 4; while the positive values are associated with the other vertices. The signs of the values in the Fiedler vector can therefore be used to partition this graph into two components: { 1 , 2 , 3 , 5 } , { 4 , 6 } {\textstyle \{1,2,3,5\},\{4,6\}} . Alternatively, the value of 0.069 (which is close to zero) can be placed in a class of its own, partitioning the graph into three components: { 1 , 2 , 5 } , { 3 } , { 4 , 6 } {\textstyle \{1,2,5\},\{3\},\{4,6\}} or moved to the other partition { 1 , 2 , 5 } , { 3 , 4 , 6 } {\textstyle \{1,2,5\},\{3,4,6\}} , as pictured. The squared values of the components of the Fiedler vector, summing up to one since the vector is normalized, can be interpreted as probabilities of the corresponding data points to be assigned to the sign-based partition. == See also == Connectivity (graph theory) Graph property == References == |
Wikipedia:Algebraic expression#0 | In mathematics, an algebraic expression is an expression built up from constants (usually, algebraic numbers), variables, and the basic algebraic operations: addition (+), subtraction (-), multiplication (×), division (÷), whole number powers, and roots (fractional powers).. For example, 3 x 2 − 2 x y + c {\displaystyle 3x^{2}-2xy+c} is an algebraic expression. Since taking the square root is the same as raising to the power 1/2, the following is also an algebraic expression: 1 − x 2 1 + x 2 {\displaystyle {\sqrt {\frac {1-x^{2}}{1+x^{2}}}}} An algebraic equation is an equation involving polynomials, for which algebraic expressions may be solutions. If you restrict your set of constants to be numbers, any algebraic expression can be called an arithmetic expression. However, algebraic expressions can be used on more abstract objects such as in Abstract algebra. If you restrict your constants to integers, the set of numbers that can be described with an algebraic expression are called Algebraic numbers. By contrast, transcendental numbers like π and e are not algebraic, since they are not derived from integer constants and algebraic operations. Usually, π is constructed as a geometric relationship, and the definition of e requires an infinite number of algebraic operations. More generally, expressions which are algebraically independent from their constants and/or variables are called transcendental. == Terminology == Algebra has its own terminology to describe parts of an expression: == Conventions == === Variables === By convention, letters at the beginning of the alphabet (e.g. a , b , c {\displaystyle a,b,c} ) are typically used to represent constants, and those toward the end of the alphabet (e.g. x , y {\displaystyle x,y} and z {\displaystyle z} ) are used to represent variables. They are usually written in italics. === Exponents === By convention, terms with the highest power (exponent), are written on the left, for example, x 2 {\displaystyle x^{2}} is written to the left of x {\displaystyle x} . When a coefficient is one, it is usually omitted (e.g. 1 x 2 {\displaystyle 1x^{2}} is written x 2 {\displaystyle x^{2}} ). Likewise when the exponent (power) is one, (e.g. 3 x 1 {\displaystyle 3x^{1}} is written 3 x {\displaystyle 3x} ), and, when the exponent is zero, the result is always 1 (e.g. 3 x 0 {\displaystyle 3x^{0}} is written 3 {\displaystyle 3} , since x 0 {\displaystyle x^{0}} is always 1 {\displaystyle 1} ). == In roots of polynomials == The roots of a polynomial expression of degree n, or equivalently the solutions of a polynomial equation, can always be written as algebraic expressions if n < 5 (see quadratic formula, cubic function, and quartic equation). Such a solution of an equation is called an algebraic solution. But the Abel–Ruffini theorem states that algebraic solutions do not exist for all such equations (just for some of them) if n ≥ {\displaystyle \geq } 5. == Rational expressions == Given two polynomials P ( x ) {\displaystyle P(x)} and Q ( x ) {\displaystyle Q(x)} , their quotient is called a rational expression or simply rational fraction. A rational expression P ( x ) Q ( x ) {\textstyle {\frac {P(x)}{Q(x)}}} is called proper if deg P ( x ) < deg Q ( x ) {\displaystyle \deg P(x)<\deg Q(x)} , and improper otherwise. For example, the fraction 2 x x 2 − 1 {\displaystyle {\tfrac {2x}{x^{2}-1}}} is proper, and the fractions x 3 + x 2 + 1 x 2 − 5 x + 6 {\displaystyle {\tfrac {x^{3}+x^{2}+1}{x^{2}-5x+6}}} and x 2 − x + 1 5 x 2 + 3 {\displaystyle {\tfrac {x^{2}-x+1}{5x^{2}+3}}} are improper. Any improper rational fraction can be expressed as the sum of a polynomial (possibly constant) and a proper rational fraction. In the first example of an improper fraction one has x 3 + x 2 + 1 x 2 − 5 x + 6 = ( x + 6 ) + 24 x − 35 x 2 − 5 x + 6 , {\displaystyle {\frac {x^{3}+x^{2}+1}{x^{2}-5x+6}}=(x+6)+{\frac {24x-35}{x^{2}-5x+6}},} where the second term is a proper rational fraction. The sum of two proper rational fractions is a proper rational fraction as well. The reverse process of expressing a proper rational fraction as the sum of two or more fractions is called resolving it into partial fractions. For example, 2 x x 2 − 1 = 1 x − 1 + 1 x + 1 . {\displaystyle {\frac {2x}{x^{2}-1}}={\frac {1}{x-1}}+{\frac {1}{x+1}}.} Here, the two terms on the right are called partial fractions. === Irrational fraction === An irrational fraction is one that contains the variable under a fractional exponent. An example of an irrational fraction is x 1 / 2 − 1 3 a x 1 / 3 − x 1 / 2 . {\displaystyle {\frac {x^{1/2}-{\tfrac {1}{3}}a}{x^{1/3}-x^{1/2}}}.} The process of transforming an irrational fraction to a rational fraction is known as rationalization. Every irrational fraction in which the radicals are monomials may be rationalized by finding the least common multiple of the indices of the roots, and substituting the variable for another variable with the least common multiple as exponent. In the example given, the least common multiple is 6, hence we can substitute x = z 6 {\displaystyle x=z^{6}} to obtain z 3 − 1 3 a z 2 − z 3 . {\displaystyle {\frac {z^{3}-{\tfrac {1}{3}}a}{z^{2}-z^{3}}}.} == Algebraic and other mathematical expressions == The table below summarizes how algebraic expressions compare with several other types of mathematical expressions by the type of elements they may contain, according to common but not universal conventions. A rational algebraic expression (or rational expression) is an algebraic expression that can be written as a quotient of polynomials, such as x2 + 4x + 4. An irrational algebraic expression is one that is not rational, such as √x + 4. == See also == Algebraic function Analytical expression Closed-form expression Expression (mathematics) Precalculus Term (logic) == Notes == == References == James, Robert Clarke; James, Glenn (1992). Mathematics dictionary. Springer. p. 8. ISBN 9780412990410. == External links == Weisstein, Eric W. "Algebraic Expression". MathWorld. |
Wikipedia:Algebraic fraction#0 | In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are 3 x x 2 + 2 x − 3 {\displaystyle {\frac {3x}{x^{2}+2x-3}}} and x + 2 x 2 − 3 {\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}}} . Algebraic fractions are subject to the same laws as arithmetic fractions. A rational fraction is an algebraic fraction whose numerator and denominator are both polynomials. Thus 3 x x 2 + 2 x − 3 {\displaystyle {\frac {3x}{x^{2}+2x-3}}} is a rational fraction, but not x + 2 x 2 − 3 , {\displaystyle {\frac {\sqrt {x+2}}{x^{2}-3}},} because the numerator contains a square root function. == Terminology == In the algebraic fraction a b {\displaystyle {\tfrac {a}{b}}} , the dividend a is called the numerator and the divisor b is called the denominator. The numerator and denominator are called the terms of the algebraic fraction. A complex fraction is a fraction whose numerator or denominator, or both, contains a fraction. A simple fraction contains no fraction either in its numerator or its denominator. A fraction is in lowest terms if the only factor common to the numerator and the denominator is 1. An expression which is not in fractional form is an integral expression. An integral expression can always be written in fractional form by giving it the denominator 1. A mixed expression is the algebraic sum of one or more integral expressions and one or more fractional terms. == Rational fractions == If the expressions a and b are polynomials, the algebraic fraction is called a rational algebraic fraction or simply rational fraction. Rational fractions are also known as rational expressions. A rational fraction f ( x ) g ( x ) {\displaystyle {\tfrac {f(x)}{g(x)}}} is called proper if deg f ( x ) < deg g ( x ) {\displaystyle \deg f(x)<\deg g(x)} , and improper otherwise. For example, the rational fraction 2 x x 2 − 1 {\displaystyle {\tfrac {2x}{x^{2}-1}}} is proper, and the rational fractions x 3 + x 2 + 1 x 2 − 5 x + 6 {\displaystyle {\tfrac {x^{3}+x^{2}+1}{x^{2}-5x+6}}} and x 2 − x + 1 5 x 2 + 3 {\displaystyle {\tfrac {x^{2}-x+1}{5x^{2}+3}}} are improper. Any improper rational fraction can be expressed as the sum of a polynomial (possibly constant) and a proper rational fraction. In the first example of an improper fraction one has x 3 + x 2 + 1 x 2 − 5 x + 6 = ( x + 6 ) + 24 x − 35 x 2 − 5 x + 6 , {\displaystyle {\frac {x^{3}+x^{2}+1}{x^{2}-5x+6}}=(x+6)+{\frac {24x-35}{x^{2}-5x+6}},} where the second term is a proper rational fraction. The sum of two proper rational fractions is a proper rational fraction as well. The reverse process of expressing a proper rational fraction as the sum of two or more fractions is called resolving it into partial fractions. For example, 2 x x 2 − 1 = 1 x − 1 + 1 x + 1 . {\displaystyle {\frac {2x}{x^{2}-1}}={\frac {1}{x-1}}+{\frac {1}{x+1}}.} Here, the two terms on the right are called partial fractions. == Irrational fractions == An irrational fraction is one that contains the variable under a fractional exponent. An example of an irrational fraction is x 1 / 2 − 1 3 a x 1 / 3 − x 1 / 2 . {\displaystyle {\frac {x^{1/2}-{\tfrac {1}{3}}a}{x^{1/3}-x^{1/2}}}.} The process of transforming an irrational fraction to a rational fraction is known as rationalization. Every irrational fraction in which the radicals are monomials may be rationalized by finding the least common multiple of the indices of the roots, and substituting the variable for another variable with the least common multiple as exponent. In the example given, the least common multiple is 6, hence we can substitute x = z 6 {\displaystyle x=z^{6}} to obtain z 3 − 1 3 a z 2 − z 3 . {\displaystyle {\frac {z^{3}-{\tfrac {1}{3}}a}{z^{2}-z^{3}}}.} == See also == Partial fraction decomposition == References == Brink, Raymond W. (1951). "IV. Fractions". College Algebra. |
Wikipedia:Algebraic function#0 | In mathematics, an algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are: f ( x ) = 1 / x {\displaystyle f(x)=1/x} f ( x ) = x {\displaystyle f(x)={\sqrt {x}}} f ( x ) = 1 + x 3 x 3 / 7 − 7 x 1 / 3 {\displaystyle f(x)={\frac {\sqrt {1+x^{3}}}{x^{3/7}-{\sqrt {7}}x^{1/3}}}} Some algebraic functions, however, cannot be expressed by such finite expressions (this is the Abel–Ruffini theorem). This is the case, for example, for the Bring radical, which is the function implicitly defined by f ( x ) 5 + f ( x ) + x = 0 {\displaystyle f(x)^{5}+f(x)+x=0} . In more precise terms, an algebraic function of degree n in one variable x is a function y = f ( x ) , {\displaystyle y=f(x),} that is continuous in its domain and satisfies a polynomial equation of positive degree a n ( x ) y n + a n − 1 ( x ) y n − 1 + ⋯ + a 0 ( x ) = 0 {\displaystyle a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+\cdots +a_{0}(x)=0} where the coefficients ai(x) are polynomial functions of x, with integer coefficients. It can be shown that the same class of functions is obtained if algebraic numbers are accepted for the coefficients of the ai(x)'s. If transcendental numbers occur in the coefficients the function is, in general, not algebraic, but it is algebraic over the field generated by these coefficients. The value of an algebraic function at a rational number, and more generally, at an algebraic number is always an algebraic number. Sometimes, coefficients a i ( x ) {\displaystyle a_{i}(x)} that are polynomial over a ring R are considered, and one then talks about "functions algebraic over R". A function which is not algebraic is called a transcendental function, as it is for example the case of exp x , tan x , ln x , Γ ( x ) {\displaystyle \exp x,\tan x,\ln x,\Gamma (x)} . A composition of transcendental functions can give an algebraic function: f ( x ) = cos arcsin x = 1 − x 2 {\displaystyle f(x)=\cos \arcsin x={\sqrt {1-x^{2}}}} . As a polynomial equation of degree n has up to n roots (and exactly n roots over an algebraically closed field, such as the complex numbers), a polynomial equation does not implicitly define a single function, but up to n functions, sometimes also called branches. Consider for example the equation of the unit circle: y 2 + x 2 = 1. {\displaystyle y^{2}+x^{2}=1.\,} This determines y, except only up to an overall sign; accordingly, it has two branches: y = ± 1 − x 2 . {\displaystyle y=\pm {\sqrt {1-x^{2}}}.\,} An algebraic function in m variables is similarly defined as a function y = f ( x 1 , … , x m ) {\displaystyle y=f(x_{1},\dots ,x_{m})} which solves a polynomial equation in m + 1 variables: p ( y , x 1 , x 2 , … , x m ) = 0. {\displaystyle p(y,x_{1},x_{2},\dots ,x_{m})=0.} It is normally assumed that p should be an irreducible polynomial. The existence of an algebraic function is then guaranteed by the implicit function theorem. Formally, an algebraic function in m variables over the field K is an element of the algebraic closure of the field of rational functions K(x1, ..., xm). == Algebraic functions in one variable == === Introduction and overview === The informal definition of an algebraic function provides a number of clues about their properties. To gain an intuitive understanding, it may be helpful to regard algebraic functions as functions which can be formed by the usual algebraic operations: addition, multiplication, division, and taking an nth root. This is something of an oversimplification; because of the fundamental theorem of Galois theory, algebraic functions need not be expressible by radicals. First, note that any polynomial function y = p ( x ) {\displaystyle y=p(x)} is an algebraic function, since it is simply the solution y to the equation y − p ( x ) = 0. {\displaystyle y-p(x)=0.\,} More generally, any rational function y = p ( x ) q ( x ) {\displaystyle y={\frac {p(x)}{q(x)}}} is algebraic, being the solution to q ( x ) y − p ( x ) = 0. {\displaystyle q(x)y-p(x)=0.} Moreover, the nth root of any polynomial y = p ( x ) n {\textstyle y={\sqrt[{n}]{p(x)}}} is an algebraic function, solving the equation y n − p ( x ) = 0. {\displaystyle y^{n}-p(x)=0.} Surprisingly, the inverse function of an algebraic function is an algebraic function. For supposing that y is a solution to a n ( x ) y n + ⋯ + a 0 ( x ) = 0 , {\displaystyle a_{n}(x)y^{n}+\cdots +a_{0}(x)=0,} for each value of x, then x is also a solution of this equation for each value of y. Indeed, interchanging the roles of x and y and gathering terms, b m ( y ) x m + b m − 1 ( y ) x m − 1 + ⋯ + b 0 ( y ) = 0. {\displaystyle b_{m}(y)x^{m}+b_{m-1}(y)x^{m-1}+\cdots +b_{0}(y)=0.} Writing x as a function of y gives the inverse function, also an algebraic function. However, not every function has an inverse. For example, y = x2 fails the horizontal line test: it fails to be one-to-one. The inverse is the algebraic "function" x = ± y {\displaystyle x=\pm {\sqrt {y}}} . Another way to understand this, is that the set of branches of the polynomial equation defining our algebraic function is the graph of an algebraic curve. === The role of complex numbers === From an algebraic perspective, complex numbers enter quite naturally into the study of algebraic functions. First of all, by the fundamental theorem of algebra, the complex numbers are an algebraically closed field. Hence any polynomial relation p(y, x) = 0 is guaranteed to have at least one solution (and in general a number of solutions not exceeding the degree of p in y) for y at each point x, provided we allow y to assume complex as well as real values. Thus, problems to do with the domain of an algebraic function can safely be minimized. Furthermore, even if one is ultimately interested in real algebraic functions, there may be no means to express the function in terms of addition, multiplication, division and taking nth roots without resorting to complex numbers (see casus irreducibilis). For example, consider the algebraic function determined by the equation y 3 − x y + 1 = 0. {\displaystyle y^{3}-xy+1=0.\,} Using the cubic formula, we get y = − 2 x − 108 + 12 81 − 12 x 3 3 + − 108 + 12 81 − 12 x 3 3 6 . {\displaystyle y=-{\frac {2x}{\sqrt[{3}]{-108+12{\sqrt {81-12x^{3}}}}}}+{\frac {\sqrt[{3}]{-108+12{\sqrt {81-12x^{3}}}}}{6}}.} For x ≤ 3 4 3 , {\displaystyle x\leq {\frac {3}{\sqrt[{3}]{4}}},} the square root is real and the cubic root is thus well defined, providing the unique real root. On the other hand, for x > 3 4 3 , {\displaystyle x>{\frac {3}{\sqrt[{3}]{4}}},} the square root is not real, and one has to choose, for the square root, either non-real square root. Thus the cubic root has to be chosen among three non-real numbers. If the same choices are done in the two terms of the formula, the three choices for the cubic root provide the three branches shown, in the accompanying image. It may be proven that there is no way to express this function in terms of nth roots using real numbers only, even though the resulting function is real-valued on the domain of the graph shown. On a more significant theoretical level, using complex numbers allows one to use the powerful techniques of complex analysis to discuss algebraic functions. In particular, the argument principle can be used to show that any algebraic function is in fact an analytic function, at least in the multiple-valued sense. Formally, let p(x, y) be a complex polynomial in the complex variables x and y. Suppose that x0 ∈ C is such that the polynomial p(x0, y) of y has n distinct zeros. We shall show that the algebraic function is analytic in a neighborhood of x0. Choose a system of n non-overlapping discs Δi containing each of these zeros. Then by the argument principle 1 2 π i ∮ ∂ Δ i p y ( x 0 , y ) p ( x 0 , y ) d y = 1. {\displaystyle {\frac {1}{2\pi i}}\oint _{\partial \Delta _{i}}{\frac {p_{y}(x_{0},y)}{p(x_{0},y)}}\,dy=1.} By continuity, this also holds for all x in a neighborhood of x0. In particular, p(x, y) has only one root in Δi, given by the residue theorem: f i ( x ) = 1 2 π i ∮ ∂ Δ i y p y ( x , y ) p ( x , y ) d y {\displaystyle f_{i}(x)={\frac {1}{2\pi i}}\oint _{\partial \Delta _{i}}y{\frac {p_{y}(x,y)}{p(x,y)}}\,dy} which is an analytic function. === Monodromy === Note that the foregoing proof of analyticity derived an expression for a system of n different function elements fi (x), provided that x is not a critical point of p(x, y). A critical point is a point where the number of distinct zeros is smaller than the degree of p, and this occurs only where the highest degree term of p or the discriminant vanish. Hence there are only finitely many such points c1, ..., cm. A close analysis of the properties of the function elements fi near the critical points can be used to show that the monodromy cover is ramified over the critical points (and possibly the point at infinity). Thus the holomorphic extension of the fi has at worst algebraic poles and ordinary algebraic branchings over the critical points. Note that, away from the critical points, we have p ( x , y ) = a n ( x ) ( y − f 1 ( x ) ) ( y − f 2 ( x ) ) ⋯ ( y − f n ( x ) ) {\displaystyle p(x,y)=a_{n}(x)(y-f_{1}(x))(y-f_{2}(x))\cdots (y-f_{n}(x))} since the fi are by definition the distinct zeros of p. The monodromy group acts by permuting the factors, and thus forms the monodromy representation of the Galois group of p. (The monodromy action on the universal covering space is related but different notion in the theory of Riemann surfaces.) == History == The ideas surrounding algebraic functions go back at least as far as René Descartes. The first discussion of algebraic functions appears to have been in Edward Waring's 1794 An Essay on the Principles of Human Knowledge in which he writes: let a quantity denoting the ordinate, be an algebraic function of the abscissa x, by the common methods of division and extraction of roots, reduce it into an infinite series ascending or descending according to the dimensions of x, and then find the integral of each of the resulting terms. == See also == Algebraic expression Analytic function Complex function Elementary function Function (mathematics) Generalized function List of special functions and eponyms List of types of functions Polynomial Rational function Special functions Transcendental function == References == Ahlfors, Lars (1979). Complex Analysis. McGraw Hill. van der Waerden, B.L. (1931). Modern Algebra, Volume II. Springer. == External links == Definition of "Algebraic function" in the Encyclopedia of Math Weisstein, Eric W. "Algebraic Function". MathWorld. Algebraic Function at PlanetMath. Definition of "Algebraic function" Archived 2020-10-26 at the Wayback Machine in David J. Darling's Internet Encyclopedia of Science |
Wikipedia:Algebraic graph theory#0 | Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs. This is in contrast to geometric, combinatoric, or algorithmic approaches. There are three main branches of algebraic graph theory, involving the use of linear algebra, the use of group theory, and the study of graph invariants. == Branches of algebraic graph theory == === Using linear algebra === The first branch of algebraic graph theory involves the study of graphs in connection with linear algebra. Especially, it studies the spectrum of the adjacency matrix, or the Laplacian matrix of a graph (this part of algebraic graph theory is also called spectral graph theory). For the Petersen graph, for example, the spectrum of the adjacency matrix is (−2, −2, −2, −2, 1, 1, 1, 1, 1, 3). Several theorems relate properties of the spectrum to other graph properties. As a simple example, a connected graph with diameter D will have at least D+1 distinct values in its spectrum. Aspects of graph spectra have been used in analysing the synchronizability of networks. === Using group theory === The second branch of algebraic graph theory involves the study of graphs in connection to group theory, particularly automorphism groups and geometric group theory. The focus is placed on various families of graphs based on symmetry (such as symmetric graphs, vertex-transitive graphs, edge-transitive graphs, distance-transitive graphs, distance-regular graphs, and strongly regular graphs), and on the inclusion relationships between these families. Certain of such categories of graphs are sparse enough that lists of graphs can be drawn up. By Frucht's theorem, all groups can be represented as the automorphism group of a connected graph (indeed, of a cubic graph). Another connection with group theory is that, given any group, symmetrical graphs known as Cayley graphs can be generated, and these have properties related to the structure of the group. This second branch of algebraic graph theory is related to the first, since the symmetry properties of a graph are reflected in its spectrum. In particular, the spectrum of a highly symmetrical graph, such as the Petersen graph, has few distinct values (the Petersen graph has 3, which is the minimum possible, given its diameter). For Cayley graphs, the spectrum can be related directly to the structure of the group, in particular to its irreducible characters. === Studying graph invariants === Finally, the third branch of algebraic graph theory concerns algebraic properties of invariants of graphs, and especially the chromatic polynomial, the Tutte polynomial and knot invariants. The chromatic polynomial of a graph, for example, counts the number of its proper vertex colorings. For the Petersen graph, this polynomial is t ( t − 1 ) ( t − 2 ) ( t 7 − 12 t 6 + 67 t 5 − 230 t 4 + 529 t 3 − 814 t 2 + 775 t − 352 ) {\displaystyle t(t-1)(t-2)(t^{7}-12t^{6}+67t^{5}-230t^{4}+529t^{3}-814t^{2}+775t-352)} . In particular, this means that the Petersen graph cannot be properly colored with one or two colors, but can be colored in 120 different ways with 3 colors. Much work in this area of algebraic graph theory was motivated by attempts to prove the four color theorem. However, there are still many open problems, such as characterizing graphs which have the same chromatic polynomial, and determining which polynomials are chromatic. == See also == Spectral graph theory Algebraic combinatorics Algebraic connectivity Dulmage–Mendelsohn decomposition Graph property Adjacency matrix == References == == External links == Media related to Algebraic graph theory at Wikimedia Commons |
Wikipedia:Algebraic independence#0 | In abstract algebra, a subset S {\displaystyle S} of a field L {\displaystyle L} is algebraically independent over a subfield K {\displaystyle K} if the elements of S {\displaystyle S} do not satisfy any non-trivial polynomial equation with coefficients in K {\displaystyle K} . In particular, a one element set { α } {\displaystyle \{\alpha \}} is algebraically independent over K {\displaystyle K} if and only if α {\displaystyle \alpha } is transcendental over K {\displaystyle K} . In general, all the elements of an algebraically independent set S {\displaystyle S} over K {\displaystyle K} are by necessity transcendental over K {\displaystyle K} , and over all of the field extensions over K {\displaystyle K} generated by the remaining elements of S {\displaystyle S} . == Example == The real numbers π {\displaystyle {\sqrt {\pi }}} and 2 π + 1 {\displaystyle 2\pi +1} are transcendental numbers: they are not the roots of any nontrivial polynomial whose coefficients are rational numbers. Thus, the sets { π } {\displaystyle \{{\sqrt {\pi }}\}} and { 2 π + 1 } {\displaystyle \{2\pi +1\}} are both algebraically independent over the rational numbers. However, the set { π , 2 π + 1 } {\displaystyle \{{\sqrt {\pi }},2\pi +1\}} is not algebraically independent over the rational numbers Q {\displaystyle \mathbb {Q} } , because the nontrivial polynomial P ( x , y ) = 2 x 2 − y + 1 {\displaystyle P(x,y)=2x^{2}-y+1} is zero when x = π {\displaystyle x={\sqrt {\pi }}} and y = 2 π + 1 {\displaystyle y=2\pi +1} . == Algebraic independence of known constants == Although π and e are transcendental, it is not known whether { π , e } {\displaystyle \{\pi ,e\}} is algebraically independent over Q {\displaystyle \mathbb {Q} } . In fact, it is not even known whether π + e {\displaystyle \pi +e} is irrational. Nesterenko proved in 1996 that: the numbers π {\displaystyle \pi } , e π {\displaystyle e^{\pi }} , and Γ ( 1 / 4 ) {\displaystyle \Gamma (1/4)} , where Γ {\displaystyle \Gamma } is the gamma function, are algebraically independent over Q {\displaystyle \mathbb {Q} } ; the numbers e π 3 {\displaystyle e^{\pi {\sqrt {3}}}} and Γ ( 1 / 3 ) {\displaystyle \Gamma (1/3)} are algebraically independent over Q {\displaystyle \mathbb {Q} } ; for all positive integers n {\displaystyle n} , the number e π n {\displaystyle e^{\pi {\sqrt {n}}}} is algebraically independent over Q {\displaystyle \mathbb {Q} } . == Results and open problems == The Lindemann–Weierstrass theorem can often be used to prove that some sets are algebraically independent over Q {\displaystyle \mathbb {Q} } . It states that whenever α 1 , … , α n {\displaystyle \alpha _{1},\ldots ,\alpha _{n}} are algebraic numbers that are linearly independent over Q {\displaystyle \mathbb {Q} } , then e α 1 , … , e α n {\displaystyle e^{\alpha _{1}},\ldots ,e^{\alpha _{n}}} are also algebraically independent over Q {\displaystyle \mathbb {Q} } . The Schanuel conjecture would establish the algebraic independence of many numbers, including π and e, but remains unproven: Let { z 1 , . . . , z n } {\displaystyle \{z_{1},...,z_{n}\}} be any set of n {\displaystyle n} complex numbers that are linearly independent over Q {\displaystyle \mathbb {Q} } . The field extension Q ( z 1 , . . . , z n , e z 1 , . . . , e z n ) {\displaystyle \mathbb {Q} (z_{1},...,z_{n},e^{z_{1}},...,e^{z_{n}})} has transcendence degree at least n {\displaystyle n} over Q {\displaystyle \mathbb {Q} } . == Algebraic matroids == Given a field extension L / K {\displaystyle L/K} that is not algebraic, Zorn's lemma can be used to show that there always exists a maximal algebraically independent subset of L {\displaystyle L} over K {\displaystyle K} . Further, all the maximal algebraically independent subsets have the same cardinality, known as the transcendence degree of the extension. For every finite set S {\displaystyle S} of elements of L {\displaystyle L} , the algebraically independent subsets of S {\displaystyle S} satisfy the axioms that define the independent sets of a matroid. In this matroid, the rank of a set of elements is its transcendence degree, and the flat generated by a set T {\displaystyle T} of elements is the intersection of L {\displaystyle L} with the field K [ T ] {\displaystyle K[T]} . A matroid that can be generated in this way is called an algebraic matroid. No good characterization of algebraic matroids is known, but certain matroids are known to be non-algebraic; the smallest is the Vámos matroid. Many finite matroids may be represented by a matrix over a field K {\displaystyle K} , in which the matroid elements correspond to matrix columns, and a set of elements is independent if the corresponding set of columns is linearly independent. Every matroid with a linear representation of this type may also be represented as an algebraic matroid, by choosing an indeterminate for each row of the matrix, and by using the matrix coefficients within each column to assign each matroid element a linear combination of these transcendentals. The converse is false: not every algebraic matroid has a linear representation. == See also == Linear independence Transcendental number Lindemann-Weierstrass theorem Schanuel's conjecture == References == == External links == Chen, Johnny. "Algebraically Independent". MathWorld. |
Wikipedia:Algebraic interior#0 | In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic. Interior algebras form a variety of modal algebras. == Definition == An interior algebra is an algebraic structure with the signature ⟨S, ·, +, ′, 0, 1, I⟩ where ⟨S, ·, +, ′, 0, 1⟩ is a Boolean algebra and postfix I designates a unary operator, the interior operator, satisfying the identities: xI ≤ x xII = xI (xy)I = xIyI 1I = 1 xI is called the interior of x. The dual of the interior operator is the closure operator C defined by xC = ((x′)I)′. xC is called the closure of x. By the principle of duality, the closure operator satisfies the identities: xC ≥ x xCC = xC (x + y)C = xC + yC 0C = 0 If the closure operator is taken as primitive, the interior operator can be defined as xI = ((x′)C)′. Thus the theory of interior algebras may be formulated using the closure operator instead of the interior operator, in which case one considers closure algebras of the form ⟨S, ·, +, ′, 0, 1, C⟩, where ⟨S, ·, +, ′, 0, 1⟩ is again a Boolean algebra and C satisfies the above identities for the closure operator. Closure and interior algebras form dual pairs, and are paradigmatic instances of "Boolean algebras with operators." The early literature on this subject (mainly Polish topology) invoked closure operators, but the interior operator formulation eventually became the norm following the work of Wim Blok. == Open and closed elements == Elements of an interior algebra satisfying the condition xI = x are called open. The complements of open elements are called closed and are characterized by the condition xC = x. An interior of an element is always open and the closure of an element is always closed. Interiors of closed elements are called regular open and closures of open elements are called regular closed. Elements that are both open and closed are called clopen. 0 and 1 are clopen. An interior algebra is called Boolean if all its elements are open (and hence clopen). Boolean interior algebras can be identified with ordinary Boolean algebras as their interior and closure operators provide no meaningful additional structure. A special case is the class of trivial interior algebras, which are the single element interior algebras characterized by the identity 0 = 1. == Morphisms of interior algebras == === Homomorphisms === Interior algebras, by virtue of being algebraic structures, have homomorphisms. Given two interior algebras A and B, a map f : A → B is an interior algebra homomorphism if and only if f is a homomorphism between the underlying Boolean algebras of A and B, that also preserves interiors and closures. Hence: f(xI) = f(x)I; f(xC) = f(x)C. === Topomorphisms === Topomorphisms are another important, and more general, class of morphisms between interior algebras. A map f : A → B is a topomorphism if and only if f is a homomorphism between the Boolean algebras underlying A and B, that also preserves the open and closed elements of A. Hence: If x is open in A, then f(x) is open in B; If x is closed in A, then f(x) is closed in B. (Such morphisms have also been called stable homomorphisms and closure algebra semi-homomorphisms.) Every interior algebra homomorphism is a topomorphism, but not every topomorphism is an interior algebra homomorphism. === Boolean homomorphisms === Early research often considered mappings between interior algebras that were homomorphisms of the underlying Boolean algebras but that did not necessarily preserve the interior or closure operator. Such mappings were called Boolean homomorphisms. (The terms closure homomorphism or topological homomorphism were used in the case where these were preserved, but this terminology is now redundant as the standard definition of a homomorphism in universal algebra requires that it preserves all operations.) Applications involving countably complete interior algebras (in which countable meets and joins always exist, also called σ-complete) typically made use of countably complete Boolean homomorphisms also called Boolean σ-homomorphisms—these preserve countable meets and joins. === Continuous morphisms === The earliest generalization of continuity to interior algebras was Sikorski's, based on the inverse image map of a continuous map. This is a Boolean homomorphism, preserves unions of sequences and includes the closure of an inverse image in the inverse image of the closure. Sikorski thus defined a continuous homomorphism as a Boolean σ-homomorphism f between two σ-complete interior algebras such that f(x)C ≤ f(xC). This definition had several difficulties: The construction acts contravariantly producing a dual of a continuous map rather than a generalization. On the one hand σ-completeness is too weak to characterize inverse image maps (completeness is required), on the other hand it is too restrictive for a generalization. (Sikorski remarked on using non-σ-complete homomorphisms but included σ-completeness in his axioms for closure algebras.) Later J. Schmid defined a continuous homomorphism or continuous morphism for interior algebras as a Boolean homomorphism f between two interior algebras satisfying f(xC) ≤ f(x)C. This generalizes the forward image map of a continuous map—the image of a closure is contained in the closure of the image. This construction is covariant but not suitable for category theoretic applications as it only allows construction of continuous morphisms from continuous maps in the case of bijections. (C. Naturman returned to Sikorski's approach while dropping σ-completeness to produce topomorphisms as defined above. In this terminology, Sikorski's original "continuous homomorphisms" are σ-complete topomorphisms between σ-complete interior algebras.) == Relationships to other areas of mathematics == === Topology === Given a topological space X = ⟨X, T⟩ one can form the power set Boolean algebra of X: ⟨P(X), ∩, ∪, ′, ø, X⟩ and extend it to an interior algebra A(X) = ⟨P(X), ∩, ∪, ′, ø, X, I⟩, where I is the usual topological interior operator. For all S ⊆ X it is defined by SI = ∪ {O | O ⊆ S and O is open in X} For all S ⊆ X the corresponding closure operator is given by SC = ∩ {C | S ⊆ C and C is closed in X} SI is the largest open subset of S and SC is the smallest closed superset of S in X. The open, closed, regular open, regular closed and clopen elements of the interior algebra A(X) are just the open, closed, regular open, regular closed and clopen subsets of X respectively in the usual topological sense. Every complete atomic interior algebra is isomorphic to an interior algebra of the form A(X) for some topological space X. Moreover, every interior algebra can be embedded in such an interior algebra giving a representation of an interior algebra as a topological field of sets. The properties of the structure A(X) are the very motivation for the definition of interior algebras. Because of this intimate connection with topology, interior algebras have also been called topo-Boolean algebras or topological Boolean algebras. Given a continuous map between two topological spaces f : X → Y we can define a complete topomorphism A(f) : A(Y) → A(X) by A(f)(S) = f−1[S] for all subsets S of Y. Every complete topomorphism between two complete atomic interior algebras can be derived in this way. If Top is the category of topological spaces and continuous maps and Cit is the category of complete atomic interior algebras and complete topomorphisms then Top and Cit are dually isomorphic and A : Top → Cit is a contravariant functor that is a dual isomorphism of categories. A(f) is a homomorphism if and only if f is a continuous open map. Under this dual isomorphism of categories many natural topological properties correspond to algebraic properties, in particular connectedness properties correspond to irreducibility properties: X is empty if and only if A(X) is trivial X is indiscrete if and only if A(X) is simple X is discrete if and only if A(X) is Boolean X is almost discrete if and only if A(X) is semisimple X is finitely generated (Alexandrov) if and only if A(X) is operator complete i.e. its interior and closure operators distribute over arbitrary meets and joins respectively X is connected if and only if A(X) is directly indecomposable X is ultraconnected if and only if A(X) is finitely subdirectly irreducible X is compact ultra-connected if and only if A(X) is subdirectly irreducible ==== Generalized topology ==== The modern formulation of topological spaces in terms of topologies of open subsets, motivates an alternative formulation of interior algebras: A generalized topological space is an algebraic structure of the form ⟨B, ·, +, ′, 0, 1, T⟩ where ⟨B, ·, +, ′, 0, 1⟩ is a Boolean algebra as usual, and T is a unary relation on B (subset of B) such that: 0,1 ∈ T T is closed under arbitrary joins (i.e. if a join of an arbitrary subset of T exists then it will be in T) T is closed under finite meets For every element b of B, the join Σ{a ∈T | a ≤ b} exists T is said to be a generalized topology in the Boolean algebra. Given an interior algebra its open elements form a generalized topology. Conversely given a generalized topological space ⟨B, ·, +, ′, 0, 1, T⟩ we can define an interior operator on B by bI = Σ{a ∈T | a ≤ b} thereby producing an interior algebra whose open elements are precisely T. Thus generalized topological spaces are equivalent to interior algebras. Considering interior algebras to be generalized topological spaces, topomorphisms are then the standard homomorphisms of Boolean algebras with added relations, so that standard results from universal algebra apply. ==== Neighbourhood functions and neighbourhood lattices ==== The topological concept of neighbourhoods can be generalized to interior algebras: An element y of an interior algebra is said to be a neighbourhood of an element x if x ≤ yI. The set of neighbourhoods of x is denoted by N(x) and forms a filter. This leads to another formulation of interior algebras: A neighbourhood function on a Boolean algebra is a mapping N from its underlying set B to its set of filters, such that: For all x ∈ B, max{y ∈ B | x ∈ N(y)} exists For all x,y ∈ B, x ∈ N(y) if and only if there is a z ∈ B such that y ≤ z ≤ x and z ∈ N(z). The mapping N of elements of an interior algebra to their filters of neighbourhoods is a neighbourhood function on the underlying Boolean algebra of the interior algebra. Moreover, given a neighbourhood function N on a Boolean algebra with underlying set B, we can define an interior operator by xI = max{y ∈ B | x ∈ N(y)} thereby obtaining an interior algebra. N ( x ) {\displaystyle N(x)} will then be precisely the filter of neighbourhoods of x in this interior algebra. Thus interior algebras are equivalent to Boolean algebras with specified neighbourhood functions. In terms of neighbourhood functions, the open elements are precisely those elements x such that x ∈ N(x). In terms of open elements x ∈ N(y) if and only if there is an open element z such that y ≤ z ≤ x. Neighbourhood functions may be defined more generally on (meet)-semilattices producing the structures known as neighbourhood (semi)lattices. Interior algebras may thus be viewed as precisely the Boolean neighbourhood lattices i.e. those neighbourhood lattices whose underlying semilattice forms a Boolean algebra. === Modal logic === Given a theory (set of formal sentences) M in the modal logic S4, we can form its Lindenbaum–Tarski algebra: L(M) = ⟨M / ~, ∧, ∨, ¬, F, T, □⟩ where ~ is the equivalence relation on sentences in M given by p ~ q if and only if p and q are logically equivalent in M, and M / ~ is the set of equivalence classes under this relation. Then L(M) is an interior algebra. The interior operator in this case corresponds to the modal operator □ (necessarily), while the closure operator corresponds to ◊ (possibly). This construction is a special case of a more general result for modal algebras and modal logic. The open elements of L(M) correspond to sentences that are only true if they are necessarily true, while the closed elements correspond to those that are only false if they are necessarily false. Because of their relation to S4, interior algebras are sometimes called S4 algebras or Lewis algebras, after the logician C. I. Lewis, who first proposed the modal logics S4 and S5. === Preorders === Since interior algebras are (normal) Boolean algebras with operators, they can be represented by fields of sets on appropriate relational structures. In particular, since they are modal algebras, they can be represented as fields of sets on a set with a single binary relation, called a Kripke frame. The Kripke frames corresponding to interior algebras are precisely the preordered sets. Preordered sets (also called S4-frames) provide the Kripke semantics of the modal logic S4, and the connection between interior algebras and preorders is deeply related to their connection with modal logic. Given a preordered set X = ⟨X, «⟩ we can construct an interior algebra B(X) = ⟨P(X), ∩, ∪, ′, ø, X, I⟩ from the power set Boolean algebra of X where the interior operator I is given by SI = {x ∈ X | for all y ∈ X, x « y implies y ∈ S} for all S ⊆ X. The corresponding closure operator is given by SC = {x ∈ X | there exists a y ∈ S with y « x} for all S ⊆ X. SI is the set of all worlds inaccessible from worlds outside S, and SC is the set of all worlds accessible from some world in S. Every interior algebra can be embedded in an interior algebra of the form B(X) for some preordered set X giving the above-mentioned representation as a field of sets (a preorder field). This construction and representation theorem is a special case of the more general result for modal algebras and Kripke frames. In this regard, interior algebras are particularly interesting because of their connection to topology. The construction provides the preordered set X with a topology, the Alexandrov topology, producing a topological space T(X) whose open sets are: {O ⊆ X | for all x ∈ O and all y ∈ X, x « y implies y ∈ O}. The corresponding closed sets are: {C ⊆ X | for all x ∈ C and all y ∈ X, y « x implies y ∈ C}. In other words, the open sets are the ones whose worlds are inaccessible from outside (the up-sets), and the closed sets are the ones for which every outside world is inaccessible from inside (the down-sets). Moreover, B(X) = A(T(X)). === Monadic Boolean algebras === Any monadic Boolean algebra can be considered to be an interior algebra where the interior operator is the universal quantifier and the closure operator is the existential quantifier. The monadic Boolean algebras are then precisely the variety of interior algebras satisfying the identity xIC = xI. In other words, they are precisely the interior algebras in which every open element is closed or equivalently, in which every closed element is open. Moreover, such interior algebras are precisely the semisimple interior algebras. They are also the interior algebras corresponding to the modal logic S5, and so have also been called S5 algebras. In the relationship between preordered sets and interior algebras they correspond to the case where the preorder is an equivalence relation, reflecting the fact that such preordered sets provide the Kripke semantics for S5. This also reflects the relationship between the monadic logic of quantification (for which monadic Boolean algebras provide an algebraic description) and S5 where the modal operators □ (necessarily) and ◊ (possibly) can be interpreted in the Kripke semantics using monadic universal and existential quantification, respectively, without reference to an accessibility relation. === Heyting algebras === The open elements of an interior algebra form a Heyting algebra and the closed elements form a dual Heyting algebra. The regular open elements and regular closed elements correspond to the pseudo-complemented elements and dual pseudo-complemented elements of these algebras respectively and thus form Boolean algebras. The clopen elements correspond to the complemented elements and form a common subalgebra of these Boolean algebras as well as of the interior algebra itself. Every Heyting algebra can be represented as the open elements of an interior algebra and the latter may be chosen to be an interior algebra generated by its open elements—such interior algebras correspond one-to-one with Heyting algebras (up to isomorphism) being the free Boolean extensions of the latter. Heyting algebras play the same role for intuitionistic logic that interior algebras play for the modal logic S4 and Boolean algebras play for propositional logic. The relation between Heyting algebras and interior algebras reflects the relationship between intuitionistic logic and S4, in which one can interpret theories of intuitionistic logic as S4 theories closed under necessity. The one-to-one correspondence between Heyting algebras and interior algebras generated by their open elements reflects the correspondence between extensions of intuitionistic logic and normal extensions of the modal logic S4.Grz. === Derivative algebras === Given an interior algebra A, the closure operator obeys the axioms of the derivative operator, D. Hence we can form a derivative algebra D(A) with the same underlying Boolean algebra as A by using the closure operator as a derivative operator. Thus interior algebras are derivative algebras. From this perspective, they are precisely the variety of derivative algebras satisfying the identity xD ≥ x. Derivative algebras provide the appropriate algebraic semantics for the modal logic wK4. Hence derivative algebras stand to topological derived sets and wK4 as interior/closure algebras stand to topological interiors/closures and S4. Given a derivative algebra V with derivative operator D, we can form an interior algebra I(V) with the same underlying Boolean algebra as V, with interior and closure operators defined by xI = x·x ′ D ′ and xC = x + xD, respectively. Thus every derivative algebra can be regarded as an interior algebra. Moreover, given an interior algebra A, we have I(D(A)) = A. However, D(I(V)) = V does not necessarily hold for every derivative algebra V. == Stone duality and representation for interior algebras == Stone duality provides a category theoretic duality between Boolean algebras and a class of topological spaces known as Boolean spaces. Building on nascent ideas of relational semantics (later formalized by Kripke) and a result of R. S. Pierce, Jónsson, Tarski and G. Hansoul extended Stone duality to Boolean algebras with operators by equipping Boolean spaces with relations that correspond to the operators via a power set construction. In the case of interior algebras the interior (or closure) operator corresponds to a pre-order on the Boolean space. Homomorphisms between interior algebras correspond to a class of continuous maps between the Boolean spaces known as pseudo-epimorphisms or p-morphisms for short. This generalization of Stone duality to interior algebras based on the Jónsson–Tarski representation was investigated by Leo Esakia and is also known as the Esakia duality for S4-algebras (interior algebras) and is closely related to the Esakia duality for Heyting algebras. Whereas the Jónsson–Tarski generalization of Stone duality applies to Boolean algebras with operators in general, the connection between interior algebras and topology allows for another method of generalizing Stone duality that is unique to interior algebras. An intermediate step in the development of Stone duality is Stone's representation theorem, which represents a Boolean algebra as a field of sets. The Stone topology of the corresponding Boolean space is then generated using the field of sets as a topological basis. Building on the topological semantics introduced by Tang Tsao-Chen for Lewis's modal logic, McKinsey and Tarski showed that by generating a topology equivalent to using only the complexes that correspond to open elements as a basis, a representation of an interior algebra is obtained as a topological field of sets—a field of sets on a topological space that is closed with respect to taking interiors or closures. By equipping topological fields of sets with appropriate morphisms known as field maps, C. Naturman showed that this approach can be formalized as a category theoretic Stone duality in which the usual Stone duality for Boolean algebras corresponds to the case of interior algebras having redundant interior operator (Boolean interior algebras). The pre-order obtained in the Jónsson–Tarski approach corresponds to the accessibility relation in the Kripke semantics for an S4 theory, while the intermediate field of sets corresponds to a representation of the Lindenbaum–Tarski algebra for the theory using the sets of possible worlds in the Kripke semantics in which sentences of the theory hold. Moving from the field of sets to a Boolean space somewhat obfuscates this connection. By treating fields of sets on pre-orders as a category in its own right this deep connection can be formulated as a category theoretic duality that generalizes Stone representation without topology. R. Goldblatt had shown that with restrictions to appropriate homomorphisms such a duality can be formulated for arbitrary modal algebras and Kripke frames. Naturman showed that in the case of interior algebras this duality applies to more general topomorphisms and can be factored via a category theoretic functor through the duality with topological fields of sets. The latter represent the Lindenbaum–Tarski algebra using sets of points satisfying sentences of the S4 theory in the topological semantics. The pre-order can be obtained as the specialization pre-order of the McKinsey–Tarski topology. The Esakia duality can be recovered via a functor that replaces the field of sets with the Boolean space it generates. Via a functor that instead replaces the pre-order with its corresponding Alexandrov topology, an alternative representation of the interior algebra as a field of sets is obtained where the topology is the Alexandrov bico-reflection of the McKinsey–Tarski topology. The approach of formulating a topological duality for interior algebras using both the Stone topology of the Jónsson–Tarski approach and the Alexandrov topology of the pre-order to form a bi-topological space has been investigated by G. Bezhanishvili, R.Mines, and P.J. Morandi. The McKinsey–Tarski topology of an interior algebra is the intersection of the former two topologies. == Metamathematics == Grzegorczyk proved the first-order theory of closure algebras undecidable. Naturman demonstrated that the theory is hereditarily undecidable (all its subtheories are undecidable) and demonstrated an infinite chain of elementary classes of interior algebras with hereditarily undecidable theories. == Notes == == References == Blok, W.A., 1976, Varieties of interior algebras, Ph.D. thesis, University of Amsterdam. Esakia, L., 2004, "Intuitionistic logic and modality via topology," Annals of Pure and Applied Logic 127: 155-70. McKinsey, J.C.C. and Alfred Tarski, 1944, "The Algebra of Topology," Annals of Mathematics 45: 141-91. Naturman, C.A., 1991, Interior Algebras and Topology, Ph.D. thesis, University of Cape Town Department of Mathematics. Bezhanishvili, G., Mines, R. and Morandi, P.J., 2008, Topo-canonical completions of closure algebras and Heyting algebras, Algebra Universalis 58: 1-34. Schmid, J., 1973, On the compactification of closure algebras, Fundamenta Mathematicae 79: 33-48 Sikorski R., 1955, Closure homomorphisms and interior mappings, Fundamenta Mathematicae 41: 12-20 |
Wikipedia:Algebraic operation#0 | In mathematics, a basic algebraic operation is any one of the common operations of elementary algebra, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power). These operations may be performed on numbers, in which case they are often called arithmetic operations. They may also be performed, in a similar way, on variables, algebraic expressions, and more generally, on elements of algebraic structures, such as groups and fields. An algebraic operation may also be defined more generally as a function from a Cartesian power of a given set to the same set. The term algebraic operation may also be used for operations that may be defined by compounding basic algebraic operations, such as the dot product. In calculus and mathematical analysis, algebraic operation is also used for the operations that may be defined by purely algebraic methods. For example, exponentiation with an integer or rational exponent is an algebraic operation, but not the general exponentiation with a real or complex exponent. Also, the derivative is an operation on numerical functions and algebraic expressions that is not algebraic. == Notation == Multiplication symbols are usually omitted, and implied, when there is no operator between two variables or terms, or when a coefficient is used. For example, 3 × x2 is written as 3x2, and 2 × x × y is written as 2xy. Sometimes, multiplication symbols are replaced with either a dot or center-dot, so that x × y is written as either x . y or x · y. Plain text, programming languages, and calculators also use a single asterisk to represent the multiplication symbol, and it must be explicitly used; for example, 3x is written as 3 * x. Rather than using the ambiguous division sign (÷), division is usually represented with a vinculum, a horizontal line, as in 3/x + 1. In plain text and programming languages, a slash (also called a solidus) is used, e.g. 3 / (x + 1). Exponents are usually formatted using superscripts, as in x2. In plain text, the TeX mark-up language, and some programming languages such as MATLAB and Julia, the caret symbol, ^, represents exponents, so x2 is written as x ^ 2. In programming languages such as Ada, Fortran, Perl, Python and Ruby, a double asterisk is used, so x2 is written as x ** 2. The plus–minus sign, ±, is used as a shorthand notation for two expressions written as one, representing one expression with a plus sign, the other with a minus sign. For example, y = x ± 1 represents the two equations y = x + 1 and y = x − 1. Sometimes, it is used for denoting a positive-or-negative term such as ±x. == Arithmetic vs algebraic operations == Algebraic operations work in the same way as arithmetic operations, as can be seen in the table below. Note: the use of the letters a {\displaystyle a} and b {\displaystyle b} is arbitrary, and the examples would have been equally valid if x {\displaystyle x} and y {\displaystyle y} were used. == Properties of arithmetic and algebraic operations == == See also == Algebraic expression Algebraic function Elementary algebra Factoring a quadratic expression Order of operations == Notes == == References == |
Wikipedia:Algebraic representation#0 | In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint functors page); there is no essential difference between modules for the resulting unital ring, in which the identity acts by the identity mapping, and representations of the algebra. == Examples == === Linear complex structure === One of the simplest non-trivial examples is a linear complex structure, which is a representation of the complex numbers C, thought of as an associative algebra over the real numbers R. This algebra is realized concretely as C = R [ x ] / ( x 2 + 1 ) , {\displaystyle \mathbb {C} =\mathbb {R} [x]/(x^{2}+1),} which corresponds to i2 = −1. Then a representation of C is a real vector space V, together with an action of C on V (a map C → E n d ( V ) {\displaystyle \mathbb {C} \to \mathrm {End} (V)} ). Concretely, this is just an action of i , as this generates the algebra, and the operator representing i (the image of i in End(V)) is denoted J to avoid confusion with the identity matrix I. === Polynomial algebras === Another important basic class of examples are representations of polynomial algebras, the free commutative algebras – these form a central object of study in commutative algebra and its geometric counterpart, algebraic geometry. A representation of a polynomial algebra in k variables over the field K is concretely a K-vector space with k commuting operators, and is often denoted K [ T 1 , … , T k ] , {\displaystyle K[T_{1},\dots ,T_{k}],} meaning the representation of the abstract algebra K [ x 1 , … , x k ] {\displaystyle K[x_{1},\dots ,x_{k}]} where x i ↦ T i . {\displaystyle x_{i}\mapsto T_{i}.} A basic result about such representations is that, over an algebraically closed field, the representing matrices are simultaneously triangularisable. Even the case of representations of the polynomial algebra in a single variable are of interest – this is denoted by K [ T ] {\displaystyle K[T]} and is used in understanding the structure of a single linear operator on a finite-dimensional vector space. Specifically, applying the structure theorem for finitely generated modules over a principal ideal domain to this algebra yields as corollaries the various canonical forms of matrices, such as Jordan canonical form. In some approaches to noncommutative geometry, the free noncommutative algebra (polynomials in non-commuting variables) plays a similar role, but the analysis is much more difficult. == Weights == Eigenvalues and eigenvectors can be generalized to algebra representations. The generalization of an eigenvalue of an algebra representation is, rather than a single scalar, a one-dimensional representation λ : A → R {\displaystyle \lambda \colon A\to R} (i.e., an algebra homomorphism from the algebra to its underlying ring: a linear functional that is also multiplicative). This is known as a weight, and the analog of an eigenvector and eigenspace are called weight vector and weight space. The case of the eigenvalue of a single operator corresponds to the algebra R [ T ] , {\displaystyle R[T],} and a map of algebras R [ T ] → R {\displaystyle R[T]\to R} is determined by which scalar it maps the generator T to. A weight vector for an algebra representation is a vector such that any element of the algebra maps this vector to a multiple of itself – a one-dimensional submodule (subrepresentation). As the pairing A × M → M {\displaystyle A\times M\to M} is bilinear, "which multiple" is an A-linear functional of A (an algebra map A → R), namely the weight. In symbols, a weight vector is a vector m ∈ M {\displaystyle m\in M} such that a m = λ ( a ) m {\displaystyle am=\lambda (a)m} for all elements a ∈ A , {\displaystyle a\in A,} for some linear functional λ {\displaystyle \lambda } – note that on the left, multiplication is the algebra action, while on the right, multiplication is scalar multiplication. Because a weight is a map to a commutative ring, the map factors through the abelianization of the algebra A {\displaystyle {\mathcal {A}}} – equivalently, it vanishes on the derived algebra – in terms of matrices, if v {\displaystyle v} is a common eigenvector of operators T {\displaystyle T} and U {\displaystyle U} , then T U v = U T v {\displaystyle TUv=UTv} (because in both cases it is just multiplication by scalars), so common eigenvectors of an algebra must be in the set on which the algebra acts commutatively (which is annihilated by the derived algebra). Thus of central interest are the free commutative algebras, namely the polynomial algebras. In this particularly simple and important case of the polynomial algebra F [ T 1 , … , T k ] {\displaystyle \mathbf {F} [T_{1},\dots ,T_{k}]} in a set of commuting matrices, a weight vector of this algebra is a simultaneous eigenvector of the matrices, while a weight of this algebra is simply a k {\displaystyle k} -tuple of scalars λ = ( λ 1 , … , λ k ) {\displaystyle \lambda =(\lambda _{1},\dots ,\lambda _{k})} corresponding to the eigenvalue of each matrix, and hence geometrically to a point in k {\displaystyle k} -space. These weights – in particularly their geometry – are of central importance in understanding the representation theory of Lie algebras, specifically the finite-dimensional representations of semisimple Lie algebras. As an application of this geometry, given an algebra that is a quotient of a polynomial algebra on k {\displaystyle k} generators, it corresponds geometrically to an algebraic variety in k {\displaystyle k} -dimensional space, and the weight must fall on the variety – i.e., it satisfies the defining equations for the variety. This generalizes the fact that eigenvalues satisfy the characteristic polynomial of a matrix in one variable. == See also == Representation theory Intertwiner Representation theory of Hopf algebras Lie algebra representation Schur’s lemma Jacobson density theorem Double commutant theorem == Notes == == References == |
Wikipedia:Algebraic signal processing#0 | Algebraic signal processing (ASP) is an emerging area of theoretical signal processing (SP). In the algebraic theory of signal processing, a set of filters is treated as an (abstract) algebra, a set of signals is treated as a module or vector space, and convolution is treated as an algebra representation. The advantage of algebraic signal processing is its generality and portability. == History == In the original formulation of algebraic signal processing by Puschel and Moura, the signals are collected in an A {\displaystyle {\mathcal {A}}} -module for some algebra A {\displaystyle {\mathcal {A}}} of filters, and filtering is given by the action of A {\displaystyle {\mathcal {A}}} on the A {\displaystyle {\mathcal {A}}} -module. == Definitions == Let K {\displaystyle K} be a field, for instance the complex numbers, and A {\displaystyle {\mathcal {A}}} be a K {\displaystyle K} -algebra (i.e. a vector space over K {\displaystyle K} with a binary operation ∗ : A ⊗ A → A {\displaystyle \ast :{\mathcal {A}}\otimes {\mathcal {A}}\to {\mathcal {A}}} that is linear in both arguments) treated as a set of filters. Suppose M {\displaystyle {\mathcal {M}}} is a vector space representing a set signals. A representation of A {\displaystyle {\mathcal {A}}} consists of an algebra homomorphism ρ : A → E n d ( M ) {\displaystyle \rho :{\mathcal {A}}\to \mathrm {End} ({\mathcal {M}})} where E n d ( M ) {\displaystyle \mathrm {End} ({\mathcal {M}})} is the algebra of linear transformations T : M → M {\displaystyle T:{\mathcal {M}}\to {\mathcal {M}}} with composition (equivalent, in the finite-dimensional case, to matrix multiplication). For convenience, we write ρ a {\displaystyle \rho _{a}} for the endomorphism ρ ( a ) {\displaystyle \rho (a)} . To be an algebra homomorphism, ρ {\displaystyle \rho } must not only be a linear transformation, but also satisfy the property ρ a ∗ b = ρ b ∘ ρ a ∀ a , b ∈ A {\displaystyle \rho _{a\ast b}=\rho _{b}\circ \rho _{a}\quad \forall a,b\in {\mathcal {A}}} Given a signal x ∈ M {\displaystyle x\in {\mathcal {M}}} , convolution of the signal by a filter a ∈ A {\displaystyle a\in {\mathcal {A}}} yields a new signal ρ a ( x ) {\displaystyle \rho _{a}(x)} . Some additional terminology is needed from the representation theory of algebras. A subset G ⊆ A {\displaystyle {\mathcal {G}}\subseteq {\mathcal {A}}} is said to generate the algebra if every element of A {\displaystyle {\mathcal {A}}} can be represented as polynomials in the elements of A {\displaystyle {\mathcal {A}}} . The image of a generator g ∈ G {\displaystyle g\in {\mathcal {G}}} is called a shift operator. In all practically all examples, convolutions are formed as polynomials in E n d ( M ) {\displaystyle \mathrm {End} ({\mathcal {M}})} generated by shift operators. However, this is not necessarily the case for a representation of an arbitrary algebra. == Examples == === Discrete Signal Processing === In discrete signal processing (DSP), the signal space is the set of complex-valued functions M = L 2 ( Z ) {\displaystyle {\mathcal {M}}={\mathcal {L}}^{2}(\mathbb {Z} )} with bounded energy (i.e. square-integrable functions). This means the infinite series ∑ n = − ∞ ∞ | ( x ) n | < ∞ {\displaystyle \sum _{n=-\infty }^{\infty }|(x)_{n}|<\infty } where | ⋅ | {\displaystyle |\cdot |} is the modulus of a complex number. The shift operator is given by the linear endomorphism ( S x ) n = ( x ) n − 1 {\displaystyle (Sx)_{n}=(x)_{n-1}} . The filter space is the algebra of polynomials with complex coefficients A = C [ z − 1 , z ] {\displaystyle {\mathcal {A}}=\mathbb {C} [z^{-1},z]} and convolution is given by ρ h = ∑ k = − ∞ ∞ h k S k {\displaystyle \rho _{h}=\sum _{k=-\infty }^{\infty }h_{k}S^{k}} where h ( t ) = ∑ k = − ∞ ∞ h k z k {\displaystyle h(t)=\sum _{k=-\infty }^{\infty }h_{k}z^{k}} is an element of the algebra. Filtering a signal by h {\displaystyle h} , then yields ( y ) n = ∑ k = − ∞ ∞ h k x n − k {\displaystyle (y)_{n}=\sum _{k=-\infty }^{\infty }h_{k}x_{n-k}} because ( S k x ) n = ( x ) n − k {\displaystyle (S^{k}x)_{n}=(x)_{n-k}} . === Graph Signal Processing === A weighted graph is an undirected graph G = ( V , E ) {\displaystyle {\mathcal {G}}=({\mathcal {V}},{\mathcal {E}})} with pseudometric on the node set V {\displaystyle {\mathcal {V}}} written a i j {\displaystyle a_{ij}} . A graph signal is simply a real-valued function on the set of nodes of the graph. In graph neural networks, graph signals are sometimes called features. The signal space is the set of all graph signals M = R V {\displaystyle {\mathcal {M}}=\mathbb {R} ^{\mathcal {V}}} where V {\displaystyle {\mathcal {V}}} is a set of n = | V | {\displaystyle n=|{\mathcal {V}}|} nodes in G = ( V , E ) {\displaystyle {\mathcal {G}}=({\mathcal {V}},{\mathcal {E}})} . The filter algebra is the algebra of polynomials in one indeterminate A = R [ t ] {\displaystyle {\mathcal {A}}=\mathbb {R} [t]} . There a few possible choices for a graph shift operator (GSO). The (un)normalized weighted adjacency matrix of [ A ] i j = a i j {\displaystyle [A]_{ij}=a_{ij}} is a popular choice, as well as the (un)normalized graph Laplacian [ L ] i j = { ∑ j = 1 n a i j i = j − a i j i ≠ j {\displaystyle [L]_{ij}={\begin{cases}\sum _{j=1}^{n}a_{ij}&i=j\\-a_{ij}&i\neq j\end{cases}}} . The choice is dependent on performance and design considerations. If S {\displaystyle S} is the GSO, then a graph convolution is the linear transformation ρ h = ∑ k = 0 ∞ h k S k {\displaystyle \rho _{h}=\sum _{k=0}^{\infty }h_{k}S^{k}} for some h ( t ) = ∑ k = 0 ∞ h k z k {\displaystyle h(t)=\sum _{k=0}^{\infty }h_{k}z^{k}} , and convolution of a graph signal x : V → R {\displaystyle \mathbf {x} :{\mathcal {V}}\to \mathbb {R} } by a filter h ( t ) {\displaystyle h(t)} yields a new graph signal y = ( ∑ k = 0 ∞ h k S k ) ⋅ x {\displaystyle \mathbf {y} =\left(\sum _{k=0}^{\infty }h_{k}S^{k}\right)\cdot \mathbf {x} } . === Other Examples === Other mathematical objects with their own proposed signal-processing frameworks are algebraic signal models. These objects include including quivers, graphons, semilattices, finite groups, and Lie groups, and others. == Intertwining Maps == In the framework of representation theory, relationships between two representations of the same algebra are described with intertwining maps which in the context of signal processing translates to transformations of signals that respect the algebra structure. Suppose ρ : A → E n d ( M ) {\displaystyle \rho :{\mathcal {A}}\to \mathrm {End} ({\mathcal {M}})} and ρ ′ : A → E n d ( M ′ ) {\displaystyle \rho ':{\mathcal {A}}\to \mathrm {End} ({\mathcal {M}}')} are two different representations of A {\displaystyle {\mathcal {A}}} . An intertwining map is a linear transformation α : M → M ′ {\displaystyle \alpha :{\mathcal {M}}\to {\mathcal {M}}'} such that α ∘ ρ a = ρ a ′ ∘ α ∀ a ∈ A {\displaystyle \alpha \circ \rho _{a}=\rho '_{a}\circ \alpha \quad \forall a\in {\mathcal {A}}} Intuitively, this means that filtering a signal by a {\displaystyle a} then transforming it with α {\displaystyle \alpha } is equivalent to first transforming a signal with α {\displaystyle \alpha } , then filtering by a {\displaystyle a} . The z transform is a prototypical example of an intertwining map. == Algebraic Neural Networks == Inspired by a recent perspective that popular graph neural networks (GNNs) architectures are in fact convolutional neural networks (CNNs), recent work has been focused on developing novel neural network architectures from the algebraic point-of-view. An algebraic neural network is a composition of algebraic convolutions, possibly with multiple features and feature aggregations, and nonlinearities. == References == == External links == Smart Project: Algebraic Theory of Signal Processing at the Department of Electrical and Computer Engineering at Carnegie Mellon University. Lecture 12: "Algebraic Neural Networks," University of Pennsylvania (ESE 514). |
Wikipedia:Algebraic structure#0 | In mathematics, an algebraic structure or algebraic system consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A (typically binary operations such as addition and multiplication), and a finite set of identities (known as axioms) that these operations must satisfy. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a vector space involves a second structure called a field, and an operation called scalar multiplication between elements of the field (called scalars), and elements of the vector space (called vectors). Abstract algebra is the name that is commonly given to the study of algebraic structures. The general theory of algebraic structures has been formalized in universal algebra. Category theory is another formalization that includes also other mathematical structures and functions between structures of the same type (homomorphisms). In universal algebra, an algebraic structure is called an algebra; this term may be ambiguous, since, in other contexts, an algebra is an algebraic structure that is a vector space over a field or a module over a commutative ring. The collection of all structures of a given type (same operations and same laws) is called a variety in universal algebra; this term is also used with a completely different meaning in algebraic geometry, as an abbreviation of algebraic variety. In category theory, the collection of all structures of a given type and homomorphisms between them form a concrete category. == Introduction == Addition and multiplication are prototypical examples of operations that combine two elements of a set to produce a third element of the same set. These operations obey several algebraic laws. For example, a + (b + c) = (a + b) + c and a(bc) = (ab)c are associative laws, and a + b = b + a and ab = ba are commutative laws. Many systems studied by mathematicians have operations that obey some, but not necessarily all, of the laws of ordinary arithmetic. For example, the possible moves of an object in three-dimensional space can be combined by performing a first move of the object, and then a second move from its new position. Such moves, formally called rigid motions, obey the associative law, but fail to satisfy the commutative law. Sets with one or more operations that obey specific laws are called algebraic structures. When a new problem involves the same laws as such an algebraic structure, all the results that have been proved using only the laws of the structure can be directly applied to the new problem. In full generality, algebraic structures may involve an arbitrary collection of operations, including operations that combine more than two elements (higher arity operations) and operations that take only one argument (unary operations) or even zero arguments (nullary operations). The examples listed below are by no means a complete list, but include the most common structures taught in undergraduate courses. == Common axioms == === Equational axioms === An axiom of an algebraic structure often has the form of an identity, that is, an equation such that the two sides of the equals sign are expressions that involve operations of the algebraic structure and variables. If the variables in the identity are replaced by arbitrary elements of the algebraic structure, the equality must remain true. Here are some common examples. Commutativity An operation ∗ {\displaystyle *} is commutative if x ∗ y = y ∗ x {\displaystyle x*y=y*x} for every x and y in the algebraic structure. Associativity An operation ∗ {\displaystyle *} is associative if ( x ∗ y ) ∗ z = x ∗ ( y ∗ z ) {\displaystyle (x*y)*z=x*(y*z)} for every x, y and z in the algebraic structure. Left distributivity An operation ∗ {\displaystyle *} is left distributive with respect to another operation + {\displaystyle +} if x ∗ ( y + z ) = ( x ∗ y ) + ( x ∗ z ) {\displaystyle x*(y+z)=(x*y)+(x*z)} for every x, y and z in the algebraic structure (the second operation is denoted here as + {\displaystyle +} , because the second operation is addition in many common examples). Right distributivity An operation ∗ {\displaystyle *} is right distributive with respect to another operation + {\displaystyle +} if ( y + z ) ∗ x = ( y ∗ x ) + ( z ∗ x ) {\displaystyle (y+z)*x=(y*x)+(z*x)} for every x, y and z in the algebraic structure. Distributivity An operation ∗ {\displaystyle *} is distributive with respect to another operation + {\displaystyle +} if it is both left distributive and right distributive. If the operation ∗ {\displaystyle *} is commutative, left and right distributivity are both equivalent to distributivity. === Existential axioms === Some common axioms contain an existential clause. In general, such a clause can be avoided by introducing further operations, and replacing the existential clause by an identity involving the new operation. More precisely, let us consider an axiom of the form "for all X there is y such that f ( X , y ) = g ( X , y ) {\displaystyle f(X,y)=g(X,y)} ", where X is a k-tuple of variables. Choosing a specific value of y for each value of X defines a function φ : X ↦ y , {\displaystyle \varphi :X\mapsto y,} which can be viewed as an operation of arity k, and the axiom becomes the identity f ( X , φ ( X ) ) = g ( X , φ ( X ) ) . {\displaystyle f(X,\varphi (X))=g(X,\varphi (X)).} The introduction of such auxiliary operation complicates slightly the statement of an axiom, but has some advantages. Given a specific algebraic structure, the proof that an existential axiom is satisfied consists generally of the definition of the auxiliary function, completed with straightforward verifications. Also, when computing in an algebraic structure, one generally uses explicitly the auxiliary operations. For example, in the case of numbers, the additive inverse is provided by the unary minus operation x ↦ − x . {\displaystyle x\mapsto -x.} Also, in universal algebra, a variety is a class of algebraic structures that share the same operations, and the same axioms, with the condition that all axioms are identities. What precedes shows that existential axioms of the above form are accepted in the definition of a variety. Here are some of the most common existential axioms. Identity element A binary operation ∗ {\displaystyle *} has an identity element if there is an element e such that x ∗ e = x and e ∗ x = x {\displaystyle x*e=x\quad {\text{and}}\quad e*x=x} for all x in the structure. Here, the auxiliary operation is the operation of arity zero that has e as its result. Inverse element Given a binary operation ∗ {\displaystyle *} that has an identity element e, an element x is invertible if it has an inverse element, that is, if there exists an element inv ( x ) {\displaystyle \operatorname {inv} (x)} such that inv ( x ) ∗ x = e and x ∗ inv ( x ) = e . {\displaystyle \operatorname {inv} (x)*x=e\quad {\text{and}}\quad x*\operatorname {inv} (x)=e.} For example, a group is an algebraic structure with a binary operation that is associative, has an identity element, and for which all elements are invertible. === Non-equational axioms === The axioms of an algebraic structure can be any first-order formula, that is a formula involving logical connectives (such as "and", "or" and "not"), and logical quantifiers ( ∀ , ∃ {\displaystyle \forall ,\exists } ) that apply to elements (not to subsets) of the structure. Such a typical axiom is inversion in fields. This axiom cannot be reduced to axioms of preceding types. (it follows that fields do not form a variety in the sense of universal algebra.) It can be stated: "Every nonzero element of a field is invertible;" or, equivalently: the structure has a unary operation inv such that ∀ x , x = 0 or x ⋅ inv ( x ) = 1. {\displaystyle \forall x,\quad x=0\quad {\text{or}}\quad x\cdot \operatorname {inv} (x)=1.} The operation inv can be viewed either as a partial operation that is not defined for x = 0; or as an ordinary function whose value at 0 is arbitrary and must not be used. == Common algebraic structures == === One set with operations === Simple structures: no binary operation: Set: a degenerate algebraic structure S having no operations. Group-like structures: one binary operation. The binary operation can be indicated by any symbol, or with no symbol (juxtaposition) as is done for ordinary multiplication of real numbers. Group: a monoid with a unary operation (inverse), giving rise to inverse elements. Abelian group: a group whose binary operation is commutative. Ring-like structures or Ringoids: two binary operations, often called addition and multiplication, with multiplication distributing over addition. Ring: a semiring whose additive monoid is an abelian group. Division ring: a nontrivial ring in which division by nonzero elements is defined. Commutative ring: a ring in which the multiplication operation is commutative. Field: a commutative division ring (i.e. a commutative ring which contains a multiplicative inverse for every nonzero element). Lattice structures: two or more binary operations, including operations called meet and join, connected by the absorption law. Complete lattice: a lattice in which arbitrary meet and joins exist. Bounded lattice: a lattice with a greatest element and least element. Distributive lattice: a lattice in which each of meet and join distributes over the other. A power set under union and intersection forms a distributive lattice. Boolean algebra: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation. === Two sets with operations === Module: an abelian group M and a ring R acting as operators on M. The members of R are sometimes called scalars, and the binary operation of scalar multiplication is a function R × M → M, which satisfies several axioms. Counting the ring operations these systems have at least three operations. Vector space: a module where the ring R is a field or, in some contexts, a division ring. Algebra over a field: a module over a field, which also carries a multiplication operation that is compatible with the module structure. This includes distributivity over addition and linearity with respect to multiplication. Inner product space: a field F and vector space V with a definite bilinear form V × V → F. == Hybrid structures == Algebraic structures can also coexist with added structure of non-algebraic nature, such as partial order or a topology. The added structure must be compatible, in some sense, with the algebraic structure. Topological group: a group with a topology compatible with the group operation. Lie group: a topological group with a compatible smooth manifold structure. Ordered groups, ordered rings and ordered fields: each type of structure with a compatible partial order. Archimedean group: a linearly ordered group for which the Archimedean property holds. Topological vector space: a vector space whose M has a compatible topology. Normed vector space: a vector space with a compatible norm. If such a space is complete (as a metric space) then it is called a Banach space. Hilbert space: an inner product space over the real or complex numbers whose inner product gives rise to a Banach space structure. Vertex operator algebra Von Neumann algebra: a *-algebra of operators on a Hilbert space equipped with the weak operator topology. == Universal algebra == Algebraic structures are defined through different configurations of axioms. Universal algebra abstractly studies such objects. One major dichotomy is between structures that are axiomatized entirely by identities and structures that are not. If all axioms defining a class of algebras are identities, then this class is a variety (not to be confused with algebraic varieties of algebraic geometry). Identities are equations formulated using only the operations the structure allows, and variables that are tacitly universally quantified over the relevant universe. Identities contain no connectives, existentially quantified variables, or relations of any kind other than the allowed operations. The study of varieties is an important part of universal algebra. An algebraic structure in a variety may be understood as the quotient algebra of term algebra (also called "absolutely free algebra") divided by the equivalence relations generated by a set of identities. So, a collection of functions with given signatures generate a free algebra, the term algebra T. Given a set of equational identities (the axioms), one may consider their symmetric, transitive closure E. The quotient algebra T/E is then the algebraic structure or variety. Thus, for example, groups have a signature containing two operators: the multiplication operator m, taking two arguments, and the inverse operator i, taking one argument, and the identity element e, a constant, which may be considered an operator that takes zero arguments. Given a (countable) set of variables x, y, z, etc. the term algebra is the collection of all possible terms involving m, i, e and the variables; so for example, m(i(x), m(x, m(y,e))) would be an element of the term algebra. One of the axioms defining a group is the identity m(x, i(x)) = e; another is m(x,e) = x. The axioms can be represented as trees. These equations induce equivalence classes on the free algebra; the quotient algebra then has the algebraic structure of a group. Some structures do not form varieties, because either: It is necessary that 0 ≠ 1, 0 being the additive identity element and 1 being a multiplicative identity element, but this is a nonidentity; Structures such as fields have some axioms that hold only for nonzero members of S. For an algebraic structure to be a variety, its operations must be defined for all members of S; there can be no partial operations. Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g., fields and division rings. Structures with nonidentities present challenges varieties do not. For example, the direct product of two fields is not a field, because ( 1 , 0 ) ⋅ ( 0 , 1 ) = ( 0 , 0 ) {\displaystyle (1,0)\cdot (0,1)=(0,0)} , but fields do not have zero divisors. == Category theory == Category theory is another tool for studying algebraic structures (see, for example, Mac Lane 1998). A category is a collection of objects with associated morphisms. Every algebraic structure has its own notion of homomorphism, namely any function compatible with the operation(s) defining the structure. In this way, every algebraic structure gives rise to a category. For example, the category of groups has all groups as objects and all group homomorphisms as morphisms. This concrete category may be seen as a category of sets with added category-theoretic structure. Likewise, the category of topological groups (whose morphisms are the continuous group homomorphisms) is a category of topological spaces with extra structure. A forgetful functor between categories of algebraic structures "forgets" a part of a structure. There are various concepts in category theory that try to capture the algebraic character of a context, for instance algebraic category essentially algebraic category presentable category locally presentable category monadic functors and categories universal property. == Different meanings of "structure" == In a slight abuse of notation, the word "structure" can also refer to just the operations on a structure, instead of the underlying set itself. For example, the sentence, "We have defined a ring structure on the set A {\displaystyle A} ", means that we have defined ring operations on the set A {\displaystyle A} . For another example, the group ( Z , + ) {\displaystyle (\mathbb {Z} ,+)} can be seen as a set Z {\displaystyle \mathbb {Z} } that is equipped with an algebraic structure, namely the operation + {\displaystyle +} . == See also == Free object Mathematical structure Signature (logic) Structure (mathematical logic) == Notes == == References == Mac Lane, Saunders; Birkhoff, Garrett (1999), Algebra (2nd ed.), AMS Chelsea, ISBN 978-0-8218-1646-2 Michel, Anthony N.; Herget, Charles J. (1993), Applied Algebra and Functional Analysis, New York: Dover Publications, ISBN 978-0-486-67598-5 Burris, Stanley N.; Sankappanavar, H. P. (1981), A Course in Universal Algebra, Berlin, New York: Springer-Verlag, ISBN 978-3-540-90578-3 Category theory Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98403-2 Taylor, Paul (1999), Practical foundations of mathematics, Cambridge University Press, ISBN 978-0-521-63107-5 == External links == Jipsen's algebra structures. Includes many structures not mentioned here. Mathworld page on abstract algebra. Stanford Encyclopedia of Philosophy: Algebra by Vaughan Pratt. |
Wikipedia:Algebraic topology (object)#0 | Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. == Main branches == Below are some of the main areas studied in algebraic topology: === Homotopy groups === In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. === Homology === In algebraic topology and abstract algebra, homology (in part from Greek ὁμός homos "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group. === Cohomology === In homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign "quantities" to the chains of homology theory. === Manifolds === A manifold is a topological space that near each point resembles Euclidean space. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot be embedded in three dimensions, but can be embedded in four dimensions. Typically, results in algebraic topology focus on global, non-differentiable aspects of manifolds; for example Poincaré duality. === Knot theory === Knot theory is the study of mathematical knots. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined so that it cannot be undone. In precise mathematical language, a knot is an embedding of a circle in three-dimensional Euclidean space, R 3 {\displaystyle \mathbb {R} ^{3}} . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R 3 {\displaystyle \mathbb {R} ^{3}} upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself. === Complexes === A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. A CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). == Method of algebraic invariants == An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones (the modern standard tool for such construction is the CW complex). In the 1920s and 1930s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraic groups, which led to the change of name to algebraic topology. The combinatorial topology name is still sometimes used to emphasize an algorithmic approach based on decomposition of spaces. In the algebraic approach, one finds a correspondence between spaces and groups that respects the relation of homeomorphism (or more general homotopy) of spaces. This allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure, often making these statements easier to prove. Two major ways in which this can be done are through fundamental groups, or more generally homotopy theory, and through homology and cohomology groups. The fundamental groups give us basic information about the structure of a topological space, but they are often nonabelian and can be difficult to work with. The fundamental group of a (finite) simplicial complex does have a finite presentation. Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated. Finitely generated abelian groups are completely classified and are particularly easy to work with. == Setting in category theory == In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural transformation originated here. Fundamental groups and homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms also correspond—a continuous mapping of spaces induces a group homomorphism on the associated groups, and these homomorphisms can be used to show non-existence (or, much more deeply, existence) of mappings. One of the first mathematicians to work with different types of cohomology was Georges de Rham. One can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf cohomology to investigate the solvability of differential equations defined on the manifold in question. De Rham showed that all of these approaches were interrelated and that, for a closed, oriented manifold, the Betti numbers derived through simplicial homology were the same Betti numbers as those derived through de Rham cohomology. This was extended in the 1950s, when Samuel Eilenberg and Norman Steenrod generalized this approach. They defined homology and cohomology as functors equipped with natural transformations subject to certain axioms (e.g., a weak equivalence of spaces passes to an isomorphism of homology groups), verified that all existing (co)homology theories satisfied these axioms, and then proved that such an axiomatization uniquely characterized the theory. == Applications == Classic applications of algebraic topology include: The Brouwer fixed point theorem: every continuous map from the unit n-disk to itself has a fixed point. The free rank of the nth homology group of a simplicial complex is the nth Betti number, which allows one to calculate the Euler–Poincaré characteristic. One can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf cohomology to investigate the solvability of differential equations defined on the manifold in question. A manifold is orientable when the top-dimensional integral homology group is the integers, and is non-orientable when it is 0. The n-sphere admits a nowhere-vanishing continuous unit vector field if and only if n is odd. (For n = 2, this is sometimes called the "hairy ball theorem".) The Borsuk–Ulam theorem: any continuous map from the n-sphere to Euclidean n-space identifies at least one pair of antipodal points. Any subgroup of a free group is free. This result is quite interesting, because the statement is purely algebraic yet the simplest known proof is topological. Namely, any free group G may be realized as the fundamental group of a graph X. The main theorem on covering spaces tells us that every subgroup H of G is the fundamental group of some covering space Y of X; but every such Y is again a graph. Therefore, its fundamental group H is free. On the other hand, this type of application is also handled more simply by the use of covering morphisms of groupoids, and that technique has yielded subgroup theorems not yet proved by methods of algebraic topology; see Higgins (1971). Topological combinatorics. == Notable people == == Important theorems == == See also == == Notes == == References == Allegretti, Dylan G. L. (2008), Simplicial Sets and van Kampen's Theorem (Discusses generalized versions of van Kampen's theorem applied to topological spaces and simplicial sets). Bredon, Glen E. (1993), Topology and Geometry, Graduate Texts in Mathematics, vol. 139, Springer, ISBN 0-387-97926-3. Brown, R. (2007), Higher dimensional group theory, archived from the original on 2016-05-14, retrieved 2022-08-17 (Gives a broad view of higher-dimensional van Kampen theorems involving multiple groupoids). Brown, R.; Razak, A. (1984), "A van Kampen theorem for unions of non-connected spaces", Arch. Math., 42: 85–88, doi:10.1007/BF01198133, S2CID 122228464. "Gives a general theorem on the fundamental groupoid with a set of base points of a space which is the union of open sets." Brown, R.; Hardie, K.; Kamps, H.; Porter, T. (2002), "The homotopy double groupoid of a Hausdorff space", Theory Appl. Categories, 10 (2): 71–93. Brown, R.; Higgins, P.J. (1978), "On the connection between the second relative homotopy groups of some related spaces", Proc. London Math. Soc., S3-36 (2): 193–212, doi:10.1112/plms/s3-36.2.193. "The first 2-dimensional version of van Kampen's theorem." Brown, Ronald; Higgins, Philip J.; Sivera, Rafael (2011), Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids, European Mathematical Society Tracts in Mathematics, vol. 15, European Mathematical Society, arXiv:math/0407275, ISBN 978-3-03719-083-8, archived from the original on 2009-06-04 This provides a homotopy theoretic approach to basic algebraic topology, without needing a basis in singular homology, or the method of simplicial approximation. It contains a lot of material on crossed modules. Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: Addison-Wesley, ISBN 0-201-01984-1 Greenberg, Marvin J.; Harper, John R. (1981), Algebraic Topology: A First Course, Revised edition, Mathematics Lecture Note Series, Westview/Perseus, ISBN 9780805335576. A functorial, algebraic approach originally by Greenberg with geometric flavoring added by Harper. Hatcher, Allen (2002), Algebraic Topology, Cambridge: Cambridge University Press, ISBN 0-521-79540-0. A modern, geometrically flavoured introduction to algebraic topology. Higgins, Philip J. (1971), Notes on categories and groupoids, Van Nostrand Reinhold, ISBN 9780442034061 Maunder, C. R. F. (1970), "Algebraic Topology", Nature, 227 (5259), London: Van Nostrand Reinhold: 756, Bibcode:1970Natur.227..756F, doi:10.1038/227756a0, ISBN 0-486-69131-4. tom Dieck, Tammo (2008), Algebraic Topology, EMS Textbooks in Mathematics, European Mathematical Society, ISBN 978-3-03719-048-7 van Kampen, Egbert (1933), "On the connection between the fundamental groups of some related spaces", American Journal of Mathematics, 55 (1): 261–7, JSTOR 51000091 == Further reading == Hatcher, Allen (2002). Algebraic topology. Cambridge University Press. ISBN 0-521-79160-X. and ISBN 0-521-79540-0. "Algebraic topology", Encyclopedia of Mathematics, EMS Press, 2001 [1994] May JP (1999). A Concise Course in Algebraic Topology (PDF). University of Chicago Press. Archived (PDF) from the original on 2022-10-09. Retrieved 2008-09-27. Section 2.7 provides a category-theoretic presentation of the theorem as a colimit in the category of groupoids. |
Wikipedia:Algebrator#0 | Algebrator (also called Softmath) is a computer algebra system (CAS), originally known as Edusym and developed beginning in 1988. This is a CAS specifically geared towards algebra education. Beside the computation results, it shows step by step the solution process and context sensitive explanations. == See also == List of computer algebra systems == References == == External links == Official website |
Wikipedia:Algebroid function#0 | In mathematics, an algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are: f ( x ) = 1 / x {\displaystyle f(x)=1/x} f ( x ) = x {\displaystyle f(x)={\sqrt {x}}} f ( x ) = 1 + x 3 x 3 / 7 − 7 x 1 / 3 {\displaystyle f(x)={\frac {\sqrt {1+x^{3}}}{x^{3/7}-{\sqrt {7}}x^{1/3}}}} Some algebraic functions, however, cannot be expressed by such finite expressions (this is the Abel–Ruffini theorem). This is the case, for example, for the Bring radical, which is the function implicitly defined by f ( x ) 5 + f ( x ) + x = 0 {\displaystyle f(x)^{5}+f(x)+x=0} . In more precise terms, an algebraic function of degree n in one variable x is a function y = f ( x ) , {\displaystyle y=f(x),} that is continuous in its domain and satisfies a polynomial equation of positive degree a n ( x ) y n + a n − 1 ( x ) y n − 1 + ⋯ + a 0 ( x ) = 0 {\displaystyle a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+\cdots +a_{0}(x)=0} where the coefficients ai(x) are polynomial functions of x, with integer coefficients. It can be shown that the same class of functions is obtained if algebraic numbers are accepted for the coefficients of the ai(x)'s. If transcendental numbers occur in the coefficients the function is, in general, not algebraic, but it is algebraic over the field generated by these coefficients. The value of an algebraic function at a rational number, and more generally, at an algebraic number is always an algebraic number. Sometimes, coefficients a i ( x ) {\displaystyle a_{i}(x)} that are polynomial over a ring R are considered, and one then talks about "functions algebraic over R". A function which is not algebraic is called a transcendental function, as it is for example the case of exp x , tan x , ln x , Γ ( x ) {\displaystyle \exp x,\tan x,\ln x,\Gamma (x)} . A composition of transcendental functions can give an algebraic function: f ( x ) = cos arcsin x = 1 − x 2 {\displaystyle f(x)=\cos \arcsin x={\sqrt {1-x^{2}}}} . As a polynomial equation of degree n has up to n roots (and exactly n roots over an algebraically closed field, such as the complex numbers), a polynomial equation does not implicitly define a single function, but up to n functions, sometimes also called branches. Consider for example the equation of the unit circle: y 2 + x 2 = 1. {\displaystyle y^{2}+x^{2}=1.\,} This determines y, except only up to an overall sign; accordingly, it has two branches: y = ± 1 − x 2 . {\displaystyle y=\pm {\sqrt {1-x^{2}}}.\,} An algebraic function in m variables is similarly defined as a function y = f ( x 1 , … , x m ) {\displaystyle y=f(x_{1},\dots ,x_{m})} which solves a polynomial equation in m + 1 variables: p ( y , x 1 , x 2 , … , x m ) = 0. {\displaystyle p(y,x_{1},x_{2},\dots ,x_{m})=0.} It is normally assumed that p should be an irreducible polynomial. The existence of an algebraic function is then guaranteed by the implicit function theorem. Formally, an algebraic function in m variables over the field K is an element of the algebraic closure of the field of rational functions K(x1, ..., xm). == Algebraic functions in one variable == === Introduction and overview === The informal definition of an algebraic function provides a number of clues about their properties. To gain an intuitive understanding, it may be helpful to regard algebraic functions as functions which can be formed by the usual algebraic operations: addition, multiplication, division, and taking an nth root. This is something of an oversimplification; because of the fundamental theorem of Galois theory, algebraic functions need not be expressible by radicals. First, note that any polynomial function y = p ( x ) {\displaystyle y=p(x)} is an algebraic function, since it is simply the solution y to the equation y − p ( x ) = 0. {\displaystyle y-p(x)=0.\,} More generally, any rational function y = p ( x ) q ( x ) {\displaystyle y={\frac {p(x)}{q(x)}}} is algebraic, being the solution to q ( x ) y − p ( x ) = 0. {\displaystyle q(x)y-p(x)=0.} Moreover, the nth root of any polynomial y = p ( x ) n {\textstyle y={\sqrt[{n}]{p(x)}}} is an algebraic function, solving the equation y n − p ( x ) = 0. {\displaystyle y^{n}-p(x)=0.} Surprisingly, the inverse function of an algebraic function is an algebraic function. For supposing that y is a solution to a n ( x ) y n + ⋯ + a 0 ( x ) = 0 , {\displaystyle a_{n}(x)y^{n}+\cdots +a_{0}(x)=0,} for each value of x, then x is also a solution of this equation for each value of y. Indeed, interchanging the roles of x and y and gathering terms, b m ( y ) x m + b m − 1 ( y ) x m − 1 + ⋯ + b 0 ( y ) = 0. {\displaystyle b_{m}(y)x^{m}+b_{m-1}(y)x^{m-1}+\cdots +b_{0}(y)=0.} Writing x as a function of y gives the inverse function, also an algebraic function. However, not every function has an inverse. For example, y = x2 fails the horizontal line test: it fails to be one-to-one. The inverse is the algebraic "function" x = ± y {\displaystyle x=\pm {\sqrt {y}}} . Another way to understand this, is that the set of branches of the polynomial equation defining our algebraic function is the graph of an algebraic curve. === The role of complex numbers === From an algebraic perspective, complex numbers enter quite naturally into the study of algebraic functions. First of all, by the fundamental theorem of algebra, the complex numbers are an algebraically closed field. Hence any polynomial relation p(y, x) = 0 is guaranteed to have at least one solution (and in general a number of solutions not exceeding the degree of p in y) for y at each point x, provided we allow y to assume complex as well as real values. Thus, problems to do with the domain of an algebraic function can safely be minimized. Furthermore, even if one is ultimately interested in real algebraic functions, there may be no means to express the function in terms of addition, multiplication, division and taking nth roots without resorting to complex numbers (see casus irreducibilis). For example, consider the algebraic function determined by the equation y 3 − x y + 1 = 0. {\displaystyle y^{3}-xy+1=0.\,} Using the cubic formula, we get y = − 2 x − 108 + 12 81 − 12 x 3 3 + − 108 + 12 81 − 12 x 3 3 6 . {\displaystyle y=-{\frac {2x}{\sqrt[{3}]{-108+12{\sqrt {81-12x^{3}}}}}}+{\frac {\sqrt[{3}]{-108+12{\sqrt {81-12x^{3}}}}}{6}}.} For x ≤ 3 4 3 , {\displaystyle x\leq {\frac {3}{\sqrt[{3}]{4}}},} the square root is real and the cubic root is thus well defined, providing the unique real root. On the other hand, for x > 3 4 3 , {\displaystyle x>{\frac {3}{\sqrt[{3}]{4}}},} the square root is not real, and one has to choose, for the square root, either non-real square root. Thus the cubic root has to be chosen among three non-real numbers. If the same choices are done in the two terms of the formula, the three choices for the cubic root provide the three branches shown, in the accompanying image. It may be proven that there is no way to express this function in terms of nth roots using real numbers only, even though the resulting function is real-valued on the domain of the graph shown. On a more significant theoretical level, using complex numbers allows one to use the powerful techniques of complex analysis to discuss algebraic functions. In particular, the argument principle can be used to show that any algebraic function is in fact an analytic function, at least in the multiple-valued sense. Formally, let p(x, y) be a complex polynomial in the complex variables x and y. Suppose that x0 ∈ C is such that the polynomial p(x0, y) of y has n distinct zeros. We shall show that the algebraic function is analytic in a neighborhood of x0. Choose a system of n non-overlapping discs Δi containing each of these zeros. Then by the argument principle 1 2 π i ∮ ∂ Δ i p y ( x 0 , y ) p ( x 0 , y ) d y = 1. {\displaystyle {\frac {1}{2\pi i}}\oint _{\partial \Delta _{i}}{\frac {p_{y}(x_{0},y)}{p(x_{0},y)}}\,dy=1.} By continuity, this also holds for all x in a neighborhood of x0. In particular, p(x, y) has only one root in Δi, given by the residue theorem: f i ( x ) = 1 2 π i ∮ ∂ Δ i y p y ( x , y ) p ( x , y ) d y {\displaystyle f_{i}(x)={\frac {1}{2\pi i}}\oint _{\partial \Delta _{i}}y{\frac {p_{y}(x,y)}{p(x,y)}}\,dy} which is an analytic function. === Monodromy === Note that the foregoing proof of analyticity derived an expression for a system of n different function elements fi (x), provided that x is not a critical point of p(x, y). A critical point is a point where the number of distinct zeros is smaller than the degree of p, and this occurs only where the highest degree term of p or the discriminant vanish. Hence there are only finitely many such points c1, ..., cm. A close analysis of the properties of the function elements fi near the critical points can be used to show that the monodromy cover is ramified over the critical points (and possibly the point at infinity). Thus the holomorphic extension of the fi has at worst algebraic poles and ordinary algebraic branchings over the critical points. Note that, away from the critical points, we have p ( x , y ) = a n ( x ) ( y − f 1 ( x ) ) ( y − f 2 ( x ) ) ⋯ ( y − f n ( x ) ) {\displaystyle p(x,y)=a_{n}(x)(y-f_{1}(x))(y-f_{2}(x))\cdots (y-f_{n}(x))} since the fi are by definition the distinct zeros of p. The monodromy group acts by permuting the factors, and thus forms the monodromy representation of the Galois group of p. (The monodromy action on the universal covering space is related but different notion in the theory of Riemann surfaces.) == History == The ideas surrounding algebraic functions go back at least as far as René Descartes. The first discussion of algebraic functions appears to have been in Edward Waring's 1794 An Essay on the Principles of Human Knowledge in which he writes: let a quantity denoting the ordinate, be an algebraic function of the abscissa x, by the common methods of division and extraction of roots, reduce it into an infinite series ascending or descending according to the dimensions of x, and then find the integral of each of the resulting terms. == See also == Algebraic expression Analytic function Complex function Elementary function Function (mathematics) Generalized function List of special functions and eponyms List of types of functions Polynomial Rational function Special functions Transcendental function == References == Ahlfors, Lars (1979). Complex Analysis. McGraw Hill. van der Waerden, B.L. (1931). Modern Algebra, Volume II. Springer. == External links == Definition of "Algebraic function" in the Encyclopedia of Math Weisstein, Eric W. "Algebraic Function". MathWorld. Algebraic Function at PlanetMath. Definition of "Algebraic function" Archived 2020-10-26 at the Wayback Machine in David J. Darling's Internet Encyclopedia of Science |
Wikipedia:Alhazen's problem#0 | Alhazen's problem, also known as Alhazen's billiard problem, is a mathematical problem in geometrical optics first formulated by Ptolemy in 150 AD. It is named for the 11th-century Arab mathematician Alhazen (Ibn al-Haytham), who presented a geometric solution in his Book of Optics. The algebraic solution involves quartic equations and was found in 1965 by Jack M. Elkin. == Geometric formulation == The problem comprises drawing lines from two points, meeting at a third point on the circumference of a circle and making equal angles with the normal at that point (specular reflection). Thus, its main application in optics is to "Find the point on a spherical convex mirror at which a ray of light coming from a given point must strike in order to be reflected to another point." This leads to an equation of the fourth degree. Alhazen himself never used this algebraic rewriting of the problem. == Alhazen's solution == Ibn al-Haytham solved the problem using conic sections and a geometric proof. == Algebraic solution == Later mathematicians such as Christiaan Huygens, James Gregory, Guillaume de l'Hôpital, Isaac Barrow, and many others attempted to find an algebraic solution to the problem, using various methods, including analytic methods of geometry and derivation by complex numbers. An algebraic solution to the problem was finally found first in 1965 by Jack M. Elkin (an actuary), by means of a quartic polynomial. Other solutions were rediscovered later: in 1989, by Harald Riede; in 1990 (submitted in 1988), by Miller and Vegh; and in 1992, by John D. Smith and also by Jörg Waldvogel. In 1997, the Oxford mathematician Peter M. Neumann proved that there is no ruler-and-compass construction for the general solution of Alhazen's problem (although in 1965 Elkin had already provided a counterexample to Euclidean construction). == Generalization == Researchers have extended Alhazen's problem to general rotationally symmetric quadric mirrors, including hyperbolic, parabolic and elliptical mirrors. They showed that the mirror reflection point can be computed by solving an eighth-degree equation in the most general case. If the camera (eye) is placed on the axis of the mirror, the degree of the equation reduces to six. Alhazen's problem can also be extended to multiple refractions from a spherical ball. Given a light source and a spherical ball of certain refractive index, the closest point on the spherical ball where the light is refracted to the eye of the observer can be obtained by solving a tenth-degree equation. == References == |
Wikipedia:Ali H. Nayfeh#0 | Ali Hasan Nayfeh (Arabic: علي نايفة) (21 December 1933 – 27 March 2017) was a Palestinian-Jordanian mathematician, mechanical engineer and physicist. He is regarded as the most influential scholar and scientist in the area of applied nonlinear dynamics in mechanics and engineering. He was the inaugural winner of the Thomas K. Caughey Dynamics Award, and was awarded the Benjamin Franklin Medal in mechanical engineering. His pioneering work in nonlinear dynamics has been influential in the construction and maintenance of machines and structures that are common in daily life, such as ships, cranes, bridges, buildings, skyscrapers, jet engines, rocket engines, aircraft and spacecraft. == Biography == Ali Hasan Nayfeh was born on 21 December 1933, in the neighborhood of Shweikeh in Tulkarem city, in Mandatory Palestine. He was born to an illiterate and poor Palestinian family, who encouraged him to pursue education. He worked as a mathematics teacher at towns and villages in Jordan for ten years, up until he won a scholarship at the age of 26 to study at Stanford University in the United States. Nayfeh received his B.S. with great distinction in engineering science (1962) and his M.S. (1963) and PhD (1964) in aeronautics and astronautics from Stanford University. He was a University Distinguished Professor of engineering at Virginia Tech since 1976. He was a volunteer at the University of Jordan. He was the editor-in-chief of Nonlinear Dynamics. He was editor-in-chief of the Journal of Vibration and Control from 1995 until his resignation in May 2014. He held honorary doctorates from Marine Technical University (Russia), Technical University of Munich (Germany), and Politechnika Szczecińska (Poland). He died on 27 March 2017. == Contributions == In a career spanning four decades, he made important contributions to a number of fields, including perturbation techniques, nonlinear oscillations, aerodynamics, flight mechanics, acoustics, ship motions, hydrodynamic stability, nonlinear waves, structural dynamics, experimental dynamics, linear and nonlinear control, and micromechanics, and fluid dynamics. He authored over a thousand publications, which have collectively been cited at least 43,364 times by other scholars, as of 2017. His contributions have had a significant influence on the lives of many people. His contributions in nonlinear dynamics have had an impact on numerous practical applications, including devices, structures and systems that are common in daily life. His pioneering work in nonlinear dynamics, mainly focused on finding stability and predictability in what may appear to be chaos, has been influential and widely adopted by various industries for the construction and maintenance of safe, reliable machines and structures, such as ships, cranes, bridges, buildings, skyscrapers, jet engines, rocket engines, aircraft and spacecraft. == Professional memberships == Nayfeh was a fellow of the American Physical Society, the American Institute of Aeronautics and Astronautics, the American Society of Mechanical Engineers, the Society of Design and Process Science, and the American Academy of Mechanics. == Awards == Nayfeh received the Pendray Aerospace Literature Award from the American Institute of Aeronautics and Astronautics in 1995; the J. P. Den Hartog Award from the American Society of Mechanical Engineers in 1997; the Frank J. Maher Award for Excellence in Engineering Education in 1997; the Lyapunov Award from the American Society of Mechanical Engineers in 2005; the Virginia Academy of Science's Life Achievement in Science Award in 2005; the Gold Medal of Honor from the Academy of Trans-Disciplinary Learning and Advanced Studies in 2007; and the Thomas K. Caughey Dynamics Award in 2008. In 2014, Nayfeh was awarded the Benjamin Franklin Medal in mechanical engineering. Nayfeh gives his name to two awards offered by the International Society of Nonlinear Dynamics (NODYS), the Ali H. Nayfeh Senior Award and the Ali H. Nayfeh Early Career Award, to recognize senior and young researchers who have made significant contributions to the field of nonlinear dynamics. == Publications == === Books === Ali H. Nayfeh, Dean T. Mook (1979). Nonlinear Oscillations. John Wiley & Sons. Ali H. Nayfeh (1993). The Method of Normal Forms. John Wiley & Sons. Ali H. Nayfeh (1993). Introduction to Perturbation Techniques. John Wiley & Sons. Ali H. Nayfeh, Balakumar Balachandran (1995). Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods. John Wiley & Sons. Bibcode:1995anda.book.....N. Ali H. Nayfeh (2000). Perturbation methods. John Wiley & Sons. Ali H. Nayfeh (2000). Nonlinear Interactions: Analytical, Computational, and Experimental Methods. John Wiley & Sons. Ali H. Nayfeh, P. Frank Pai (2004). Linear and Nonlinear Structural Mechanics. John Wiley & Sons. == References == == External links == Official website Ali H. Nayfeh at the Mathematics Genealogy Project |
Wikipedia:Ali Qushji#0 | Ala al-Dīn Ali ibn Muhammed (1403 – 18 December 1474), Persian: علاءالدین علی بن محمد سمرقندی known as Ali Qushji (Ottoman Turkish : علی قوشچی, kuşçu – falconer in Turkish; Latin: Ali Kushgii) was a Timurid theologian, jurist, astronomer, mathematician and physicist, who settled in the Ottoman Empire some time before 1472. As a disciple of Ulugh Beg, he is best known for the development of astronomical physics independent from natural philosophy, and for providing empirical evidence for the Earth's rotation in his treatise, Concerning the Supposed Dependence of Astronomy upon Philosophy. In addition to his contributions to Ulugh Beg's famous work Zij-i-Sultani and to the founding of Sahn-ı Seman Medrese, one of the first centers for the study of various traditional Islamic sciences in the Ottoman Empire, Ali Kuşçu was also the author of several scientific works and textbooks on astronomy. == Biography == === Early life and works === Ali Kuşçu was born in 1403 in the city of Samarkand, in present-day Uzbekistan. His full name at birth was Ala al-Dīn Ali ibn Muhammed al-Qushji. The last name Qushji derived from the Turkish term kuşçu—"falconer"—due to the fact that Ali's father Muhammad was the royal falconer of Ulugh Beg. Sources consider him Turkic or Persian. He attended the courses of Qazi zadeh Rumi, Ghiyāth al-Dīn Jamshīd Kāshānī and Muin al-Dīn Kashi. He moved to Kerman, Iran (Persia), where he conducted some research on storms in the Oman sea. He completed Hall-e Eshkal-i Ghammar (Explanations of the Periods of the Moon) and Sharh-e Tajrid in Kirman. He moved to Herat and taught Molla Cami about astronomy (1423). After professing in Herat for a while, he returned to Samarkand. There he presented his work on the Moon to Ulugh Beg, who found it so fascinating that he read the entire work while standing up. Ulugh Beg assigned him to Ulugh Beg Observatory, which was called Samarkand Observatory at that time. Qushji worked there until Ulugh Beg was assassinated. After Ulugh Beg's death, Ali Kuşçu went to Herat, Tashkent, and finally Tabriz where, around 1470, the Ak Koyunlu ruler Uzun Hasan sent him as a delegate to the Ottoman Sultan Mehmed II. At that time Husayn Bayqarah had come to reign in Herat but Qushji preferred Constantinople over Herat because of Sultan Mehmed's attitude toward scientists and intellectuals. === Constantinople era === When he came to Constantinople (present-day Istanbul), his grandson Ghutb al-Dīn Muhammed had a son Mirim Çelebi who would be a great mathematician and astronomer in the future. Ali Kuşçu composed "risalah dar hay’at" in Persian for Mehmed II at Constantinople in 1470. Also he wrote "Sharh e resalye Fathiyeh", "resalye Mohammadiye" in Constantinople, which are in Arabic on the topic of mathematics. He then finished "Sharh e tejrid" on Nasir al-Din al-Tusi's "Tejrid al-kalam". That work is called "Sharh e Jadid" in scientific community. == Contributions to astronomy == Qushji improved on Nasir al-Din al-Tusi's planetary model and presented an alternative planetary model for Mercury. He was also one of the astronomers that were part of Ulugh Beg's team of researchers working at the Samarqand observatory and contributed towards the Zij-i-Sultani compiled there. In addition to his contributions to Zij, Ali Kuşçu wrote nine works in astronomy, two of them in Persian and seven in Arabic. A Latin translation of two of Qushji's works, the Tract on Arithmetic and Tract on Astronomy, was published by John Greaves in 1650. === Concerning the Supposed Dependence of Astronomy upon Philosophy === Qushji's most important astronomical work is Concerning the Supposed Dependence of Astronomy upon Philosophy. Under the influence of Islamic theologians who opposed the interference of Aristotelianism in astronomy, Qushji rejected Aristotelian physics and completely separated natural philosophy from Islamic astronomy, allowing astronomy to become a purely empirical and mathematical science. This allowed him to explore alternatives to the Aristotelian notion of a stationary Earth, as he explored the idea of a moving Earth instead (though Emilie Savage-Smith asserts that no Islamic astronomers proposed a heliocentric universe). He found empirical evidence for the Earth's rotation through his observation on comets and concluded, on the basis of empirical evidence rather than speculative philosophy, that the moving Earth theory is just as likely to be true as the stationary Earth theory. His predecessor al-Tusi had previously realized that "the monoformity of falling bodies, and the uniformity of celestial motions," both moved "in a single way", though he still relied on Aristotelian physics to provide "certain principles that only the natural philosophers could provide the astronomer." Qushji took this concept further and proposed that "the astronomer had no need for Aristotelian physics and in fact should establish his own physical principles independently of the natural philosophers." Alongside his rejection of Aristotle's concept of a stationary Earth, Qushji suggested that there was no need for astronomers to follow the Aristotelian notion of the heavenly bodies moving in uniform circular motion. Qushji's work was an important step away from Aristotelian physics and towards an independent astronomical physics. This is considered to be a "conceptual revolution" that had no precedent in European astronomy prior to the Copernican Revolution in the 16th century. Qushji's view on the Earth's motion was similar to the later views of Nicolaus Copernicus on this issue, though it is uncertain whether the former had any influence on the latter. However, it is likely that they both may have arrived at similar conclusions due to using the earlier work of Nasir al-Din al-Tusi as a basis. This is more of a possibility considering "the remarkable coincidence between a passage in De revolutionibus (I.8) and one in Ṭūsī’s Tadhkira (II.1[6]) in which Copernicus follows Ṭūsī’s objection to Ptolemy’s "proofs" of the Earth's immobility." == His works == === Astronomy === Sharḥ e Zîj e Ulugh Beg (In Persian) Risāla fī Halle Eshkale Moadeleye Ghamar lil-Masir (Arabic) Risāla fī aṣl al-Hâric yumkin fī al-sufliyyeyn (Arabic) Sharḥ ʿalā al-tuḥfat al-shāhiyya fī al-hayāt (Arabic) Risāla dar elm-i ḥeyāt (In Persian) Al-Fatḥīya fī ʿilm al-hayʾa (in Arabic) Risāla fi Hall-e Eshkal-i Ghammar (in Persian) Concerning the Supposed Dependence of Astronomy upon Philosophy (Arabic) === Mathematics === Risāla al-muḥhammadiyya fi-ḥisāb (In Arabic) Risāla dār ʿilm al-ḥisāb: Suleymaniye (Arabic) === Kalam and Fiqh === Sharh Tajrid al-I'tiqad Hashiye ale't-Telvîh Unkud-üz-Zevahir fi Nazm-al-Javaher === Mechanics === Tazkare fi Âlâti'r-Ruhâniyye === Linguistics === Sharh Risâleti'l-Vadiyye El-Ifsâh El-Unkûdu'z-Zevâhir fî Nazmi'l-Javâher Sharh e'Sh-Shâfiye Resale fî Beyâni Vadi'l-Mufredât Fâ'ide li-Tahkîki Lâmi't-Ta'rîf Resale mâ Ene Kultu Resale fî'l-Hamd Resale fî Ilmi'l-Me'ânî Resale fî Bahsi'l-Mufred Resale fî'l-Fenni's-Sânî min Ilmihal-Beyân Tafsir e-Bakara ve Âli Imrân Risâle fî'l-İstişâre Mahbub-al-Hamail fi kashf-al-mesail Tajrid-al-Kalam == Notes == Yavuz Unat, Ali Kuşçu, Kaynak Yayınları, 2010. == References == Ragep, F. Jamil (2001a), "Tusi and Copernicus: The Earth's Motion in Context", Science in Context, 14 (1–2), Cambridge University Press: 145–163, doi:10.1017/s0269889701000060, S2CID 145372613 == External links == Fazlıoğlu, İhsan (2007), "Qūshjī: Abū al-Qāsim ʿAlāʾ al-Dīn ʿAlī ibn Muḥammad Qushči-zāde", in Thomas Hockey; et al. (eds.), The Biographical Encyclopedia of Astronomers, New York: Springer, pp. 946–8, ISBN 978-0-387-31022-0. (PDF version) |
Wikipedia:Alice Christine Stickland#0 | Alice Christine Stickland (16 March 1906 – 16 April 1987) was an applied mathematician and astrophysics engineer with interests in radar and radiowave propagation. She worked on long-wave propagation, short-wave propagation and the ionosphere. She was also a supported of the International Council for Science’s Committee on Space Research (COSPAR) and the Girl Guides’ Association. == Early life == Alice Christine Stickland was born in Camberwell, London, on 16 March 1906. Her father was a publisher's clerk. == Education == Stickland studied mathematics at King's College, London, and graduated with a BSc in 1927. She then went on to study privately while working at the in Department of Scientific and Industrial Research at the Radio Research Station (RRS) in Ditton Park. Stickland first studied for an MSc in mathematical physics in 1929 and then being awarded a PhD in mathematical physics from University of London in 1943. Her dissertation title was "The Propagation of the Magnetic Field of the Electron Magnetic Wave along the Ground and in the Lower Atmosphere." == Career == Stickland worked as an Assistant Grade II scientific civil servant at the Radio Research Station between 1928 and 1947. She worked with radar pioneers, including with Robert Watson-Watt, on long-wave propagation, Reginald Smith-Rose on short-wave propagation, and Edward Appleton on the properties of the ionosphere. Stickland, along with Smith-Rose, read a paper titled "Ultra-Short Wave Propagation - Comparison Between Theory and Experimental data" at the Institution of Electrical Engineers. The paper described the results of field intensity measurements obtained between 1937 and 1939 using the Post Office radio-telephone link between Guernsey and Chaldon. She was a member of the Institute of Physics and the Physical Society. She officially retired in 1968, but continued to work as General Editor of the Annals of the International Years of the Quiet Sun (1964-65), and with the International Council for Science’s Committee on Space Research (COSPAR). She was heavily involved in the Girl Guides’ Association. == Death and legacy == Stickland died on 16 April 1987 in Northwood, Middlesex, England. The new Stickland room at the headquarters of the Institute of Physics in London was named in her honour. == Selected publications == Ultra-Short Wave Propagation - Comparison Between Theory and Experimental data - Dr. R. L. Smith-Rose, Miss A. C. Stickland == References == |
Wikipedia:Alice Roth#0 | Alice Roth (6 February 1905 – 22 July 1977) was a Swiss mathematician who invented the Swiss cheese set and made significant contributions to approximation theory. She was born, lived and died in Bern, Switzerland. == Life == Alice attended the Höhere Töchterschule of Zürich, a municipal school for higher education for girls. After graduation in 1924 she studied mathematics, physics and astronomy at ETH Zurich under George Pólya. She graduated with a diploma in 1930. Her Master's thesis was titled "Extension of Weierstrass's Approximation Theorem to the complex plane and to an infinite interval". After that, she was a teacher at multiple high schools for girls in the Zurich area while continuing working with Pólya at ETH. In 1938 she became the second woman to graduate with a PhD from ETH. Her PhD Thesis was titled "Properties of approximations and radial limits of meromorphic and entire functions" and was so well regarded that it received a monetary prize and the ETH silver medal. Her supervisors were George Pólya and Heinz Hopf. From 1940 she was mathematics and physics teacher at Humboldtianum in Bern, a private school. It was only after her retirement in 1971 that she returned to mathematical research, again in the area of complex approximation. She published three papers on her own, as well as a shared paper with Paul Gauthier of the University of Montreal and Harvard University professor Joseph L. Walsh. In 1975, at the age of 70, she was invited to give a public lecture at the University of Montreal. In 1976 she was diagnosed with cancer, and she died the next year. == Contribution to mathematics == One of the main results of Roth's 1938 thesis was an example of a compact set on which not every continuous function can by approximated uniformly by rational functions. This set, now known as the "Swiss cheese," was forgotten and independently rediscovered in 1952 in Russia by Mergelyan, and proper credit was restored by 1969. The following excerpt by her former student, Peter Wilker, appeared in an obituary he wrote after her death: "In Switzerland, as elsewhere, women mathematicians are few and far between.... Alice Roth's dissertation was awarded a medal from the ETH, and appeared shortly after its completion in a Swiss mathematical journal....One year later war broke out, the world had other worries than mathematics, and Alice Roth's work was simply forgotten. So completely forgotten that around 1950 a Russian mathematician re-discovered similar results without having the slightest idea that a young Swiss woman mathematician had published the same ideas more than a decade before he did. However, her priority was recognized." Roth developed other important results during her brief return to research at the end of her life: "Roth's past as well as future work was to have a strong and lasting influence on mathematicians working in this area [rational approximation theory]. Her Swiss cheese has been modified (to an entire variety of cheeses).... Roth's Fusion Lemma, which appeared in her 1976 paper...influenced a new generation of mathematicians worldwide." == Lecture series and movie == ETH Zürich's Department of Mathematics now sponsors the annual Alice Roth Lecture Series to honor women with outstanding achievements in mathematics. The inaugural lecture was delivered in March 2022 by number theorist and later Fields medalist Maryna Viazovska, who spoke on "Fourier interpolation pairs and their applications". The Spring 2023 lecture will be given by harmonic analyst Gigliola Staffilani. ETH Zürich has also produced an 8 minute documentary movie about Alice Roth's life and work. == References == == External links == Alice Roth portrait, a video from ETH Zurich department of Mathematics. |
Wikipedia:Alicia Alva Mantari#0 | Alicia Katherine Alva Mantari is a Peruvian specialist in biomedical informatics and telemedicine. She holds a master's degree with focus on global health and has been actively involved in telemedicine projects since 2008, leading tele-diagnosis systems projects for diseases such as tuberculosis and melanoma. For ten years, she was a member of the Bioinformatics and Molecular Biology Laboratory at Cayetano Heredia University (Universidad Peruana Cayetano Heredia; UPCH) . Alva has been involved in national research projects focused on health technology, including initiatives to combat COVID-19 and studies on heavy metal contamination. Her work has contributed to the advancement of telemedicine in Peru, improving healthcare accessibility in remote areas. == Biography == === Early years and education === Alicia Alva studied in Alcides Spelucín Vega School in Callao, where she participated in mathematics competitions and developed an interest in science. Later on, she studied mathematics at the National University of Engineering (Universidad Nacional de Ingeniería; UNI), where she developed analytical and computational thinking. During her master's studies, she participated on a project to develop an algorithm for tuberculosis detection. Her programming studies led her to be a member of the molecular biology laboratory at UPCH. She earned a master's degree in biomedical informatics at UPCH, in collaboration with the University of Washington through a QUIPU program scholarship. === Career === As a researcher at the University of Sciences and Humanities, she was one of the winners of CONCYTEC funding in 2020 with her project SAMAYCOV, a portable device designed to assess the risk of pneumonia in patients suspected of having COVID-19 by detecting sounds. It converts these sounds into electrical signals, which are then processed by a Python-based program. In 2022, Alva developed Soft-Warmi, an automated software for diagnosing bacterial vaginosis (BV). This research was part of her master's thesis at UPCH, in collaboration with the University of Washington. The main objective of the project was to improve the accuracy and accessibility of BV diagnosis using pattern recognition algorithms to analyze microscopic images of vaginal smears. In 2024, she participated in the creation of Yanadevn, an early dengue diagnostic system developed by researchers at the University of Sciences and Humanities in collaboration with UPCH and regional hospitals. The device uses CRISPR/Cas13 technology to detect the presence of dengue RNA in blood. == Research and publications == She co-authored the study "Implementation of a telediagnostic system for tuberculosis and determination of multi-drug resistance based on the MODS method in Trujillo, Peru". It focused on developing a remote diagnostic system for tuberculosis and multidrug resistance (MDR) using the MODS method. It optimized an image recognition algorithm to detect Mycobacterium tuberculosis in digital MODS culture images from the CENEX-Trujillo laboratory. She also participated in the development of Mathematical algorithms for the automatic recognition of intestinal parasites, which was a tool for detecting intestinal parasites in microscopic images of fecal smears. The study implemented an image processing algorithm in SCILAB, capable of identifying Taenia sp., Trichuris trichiura, Diphyllobothrium latum, and Fasciola hepatica. == Awards and recognitions == She won the "Special Projects: COVID-19 Response" 2020 competition by Concytec for the SAMAYCOV project, a device designed to assess COVID-19 risk related to lung damage. She won the II National Innotec Competition 2010, for the development of a tele-diagnosis system for multidrug-resistant tuberculosis (TB-MDR), capable of providing results in less than 15 seconds. == External links == Publications by Alicia Alva Mantari at ResearchGate. Raise of Awareness for Blood Donation Campaigns on University Campuses in Lima, Peru in the IEEE Xplore digital library. MELapp: Mobile application that detects skin cancer . TVPerú Noticias. == References == |
Wikipedia:Alicja Derkowska#0 | Alicja Maria Derkowska (born 1940) is a Polish social activist, mathematician and educator. == Biography == Alicja was born in Sosnowiec and received her doctoral degree in theoretical mathematics from the University of Lodz. She remained there to teach and conduct research until 1975 when, for health reasons, she and her husband moved to Nowy Sącz, a small rural city where both could be instructors at the teacher's Training College. In her new hometown, Alicja co-founded the regional branch of the Polish Mathematical Society, which she headed for the maximum term of six years. It was the first time a scientific society was established in a town without a university. She became involved in the activities of the first "Solidarity" movement. After martial law was imposed, she and her husband lost their jobs, and until the end of the 1980s they ran a private store. At the same time, she cooperated with the clandestine activities of the political opposition. Alicja was also an important member of Solidarity when it was still underground as a member of the Solidarity Education Board, which planned reforms to the Polish educational system. After Poland's return to democracy in 1989, Alicja did not want to teach for the state again, and began in earnest to establish a truly independent, non-state, non-religious, private school in Nowy Sacz. According to Ashoka, Alicja's educational plan incorporated the teaching of the "habit of democracy" to young students in Poland. She "designed and implemented an innovative school model in which students are trained in democratic procedures, both theoretically and in practice... Her plan to popularize her ideas rests on two strategies: first, she runs a teacher training program through which teachers from all over the region are trained to organize lessons and school life around principles, values, and activities that promote civic awareness and democratic behavior. Second, she has organized a multi-national exchange program for students and teachers that allows both to gain exposure to other cultural and linguistic environments and build inter-cultural friendships and ties." In 1988, she was among the founders of the Malopolska Educational Society (MTO), which she headed for several years and became a board member. MTO focused on promoting innovative teaching methods and democratizing school management. The organization was awarded, among other prizes, in the 2005 "Pro Publico Bono" Competition for the Best Civic Initiative. It was also involved in the creation of a network of schools in the Balkans. In the 1990s Alicja Derkowska was active in the Citizens' Movement for Democratic Action, the Democratic Union and the Freedom Union. She was part of the governing body of the World Movement for Democracy, and became a fellow of Ashoka. == Honors == Derkowska has received numerous awards for her work in educational reform. In 2004, she was awarded the POLCUL Foundation award by the Jerzy Boniecki Foundation in Australia. In 2004, she received the “For the Future of the Children of Europe” prize from the Future of Europe Association in Hungary. In 2010, she received the Polish National Education Commission Medal for outstanding contributions to education and upbringing given by the Ministry of Education. In 2011, President Bronisław Komorowski awarded her the Officer's Cross of the Order of Polonia Restituta. == References == |
Wikipedia:Alida Rossander#0 | Alida Rossander (1843-1909) was a Swedish educator, mathematician, women's rights activist and bank clerk official. In 1864, she became the first female bank clerk official in Sweden. She and her sister Jenny Rossander were students of the pioneering Lärokurs för fruntimmer in 1859, were among the first teachers employed when it was transformed to the Högre lärarinneseminariet in 1861, and were fired by Jane Miller Thengberg when the school was given an organized structure in 1864, and in 1865 they became the founders and managers of the Rossander Course. == References == == Further reading == Alida Rossander at Svenskt kvinnobiografiskt lexikon |
Wikipedia:Aline Huke Frink#0 | Aline Huke Frink (March 2, 1904 – March 14, 2000) was an American mathematician, and a professor on the faculty of the Pennsylvania State University from 1930 to 1969. == Early life and education == Aline Huke was born in Torrington, Connecticut and raised in Massachusetts, the daughter of Allen Johnson Huke and Mary Evelyn Feustel Huke. Her father was a businessman, and her mother was a schoolteacher. Huke earned a bachelor's degree in mathematics from Mount Holyoke College in 1924. She trained as a teacher at the New York State Teachers' College in Albany, and completed a master's degree and doctorate at the University of Chicago. Her doctoral advisor was Gilbert Ames Bliss, who oversaw her 1930 dissertation, "An Historical and Critical Study of the Fundamental Lemma in the Calculus of Variations". She also studied with David Widder at Bryn Mawr College. == Career == Frink taught at a high school in Cobleskill, New York from 1924 to 1926. She taught mathematics at Mount Holyoke College from 1929 to 1930, and at the Pennsylvania State University, part time after she married and full-time after 1947, two years after her last child was born. She held the rank of assistant professor until 1952, associate professor until 1962, and became professor emeritus when she retired in 1969. In addition to her teaching, Frink translated a Russian-language mathematical text, Calculus of Variations by Naum Akhiezer, published in 1962. Her mathematical research was published in Bulletin of the American Mathematical Society. She was a charter member of the Women's Scientific Club at Penn State, along with Pauline Gracia Beery Mack, Mary Louisa Willard, and Teresa Cohen; the club became a chapter of Sigma Delta Epsilon (now Graduate Women in Science). == Personal life == Aline Huke married fellow mathematician Orrin Frink in 1931. They had four children together. She was widowed when Orrin Frink died in 1988, and she died in 2000, in Kennebunkport, Maine, aged 96 years. Since 2006, there has been an endowed scholarship in Penn State's mathematics department, named for Aline Huke Frink and Orrin Frink. == References == |
Wikipedia:Alison Harcourt#0 | Alison Grant Harcourt (née Doig; born 24 November 1929) is an Australian mathematician and statistician most well-known for co-defining the branch and bound algorithm along with Ailsa Land whilst carrying out research at the London School of Economics. She was also part of the team which developed a poverty line as part of the Henderson Inquiry into poverty in Australia and helped to introduce the double randomisation method of ordering candidates used in Australian elections. == Early life and education == Harcourt was born Alison Doig in Colac, Victoria, in 1929. Her father was Keith Doig, a physician and Australian rules footballer who received the Military Cross during World War I. Her mother, Louie Grant, was of Scottish descent and was sister to physicist Sir Kerr Grant. She was schooled at Colac West State School, Colac High School and Fintona Girls' School. After her schooling, she enrolled at the University of Melbourne, gaining a Bachelor of Arts with a major in mathematics, and then a Bachelor of Science majoring in physics. While specialising in statistics undertaking a Master of Arts degree, she developed a technique for integer linear programming. == London School of Economics == On the basis of her work in linear programming, she started work at the London School of Economics (LSE) in the late 1950s. In 1960, Doig and fellow LSE mathematician Ailsa Land, published a landmark paper in the economics journal Econometrica ("An Automatic Method for Solving Discrete Programming Problems"), which outlined a branch and bound optimisation algorithm for solving NP-hard problems. The algorithm is the backbone idea behind all modern Integer programming solvers such as Gurobi, Cplex. == University of Melbourne == In 1963, Doig returned to Melbourne, where she took up a position as a senior lecturer in statistics at the University of Melbourne. In the mid-1960s, she joined a team headed by the sociologist Ronald Henderson which was attempting to quantify the extent of poverty in Australia. The team developed the Henderson Poverty Line in 1973, which was the disposable income required to support the basic needs of a family of two adults and two dependent children. The techniques developed by the Henderson team have been used by the Melbourne Institute of Applied Economic and Social Research to regularly update the poverty line for Australia since 1979. In 1970, Harcourt took study leave in Sweden, where she co-authored two papers on theoretical chemistry—"A simple demonstration of Hund’s Rule for the helium 2S and 2P States" and "Wavefunctions for 4-electron 3-centre bonding"—with her husband, the chemist Richard Harcourt. In 1975, following the dismissal of the Whitlam government, Harcourt and fellow statistician Malcolm Clark noticed irregularities in the distribution of party ordering on the Senate ballot papers for the 1975 federal election which was determined by drawing envelopes from a box, with Coalition parties holding one of the first two positions in every state. Harcourt and Clark made a submission to the Joint Select Committee on Electoral Reform, which resulted in a 1984 amendment to the Commonwealth Electoral Act to introduce a more rigorous double randomisation method. Harcourt and Clark published a paper about their analysis and recommendations for the Australian & New Zealand Journal of Statistics in 1991. Harcourt retired as an academic from the University of Melbourne in 1994, but continues to work there as a sessional tutor in statistics. In October 2018 Harcourt was named as 2019 Senior Victorian Australian of the Year. In early December 2018, the University of Melbourne awarded Harcourt with an honorary Doctor of Science degree. In June 2019, Harcourt was made an Officer of the Order of Australia in recognition of her "distinguished service to mathematics and computer science through pioneering research and development of integer linear programming". == References == |
Wikipedia:Alison Ramage#0 | Alison Ramage is a British applied mathematician and numerical analyst specialising in preconditioning methods for numerical linear algebra, and their applications to the numerical solution of partial differential equations. She is a reader in the Department of Mathematics and Statistics at the University of Strathclyde. == Education and career == Ramage is a graduate of the University of St Andrews. She completed her PhD in 1991 at the University of Bristol. Her dissertation, Preconditioned Conjugate Gradient Methods for Galerkin Finite Element Equations, was supervised by Andrew Wathen, and involved preconditioners for the Galerkin method. She is a member of the board of trustees of the Society for Industrial and Applied Mathematics. She currently serves as Chair of the Society for Industrial and Applied Mathematics (SIAM) Board of Trustees. == Research == Ramage's thesis involved preconditioners for the Galerkin method, and applications to the numerical solution of partial differential equations in computational fluid dynamics, including those for incompressible flow and advection–diffusion equations. Subsequently, she has applied her research to other areas including geotechnical engineering, financial mathematics, modeling liquid crystals, weather forecasting, and sensor networks. == References == == External links == Home page Alison Ramage publications indexed by Google Scholar |
Wikipedia:Alison Tomlin#0 | Alison Sarah Tomlin is a British physical chemist and applied mathematician whose research involves building detailed mathematical models of combustion, including uncertainty quantification for those models. She is a professor in the School of Chemical and Process Engineering at the University of Leeds, where she heads the Clean Combustion Research Group. == Education and career == Tomlin was a student of mathematics and of the history of science and philosophy of science at the University of Leeds, where she earned a combined bachelor's degree in those topics in 1987. She continued at Leeds as a graduate student in physical chemistry, completing her dissertation Bifurcation analysis for non-linear chemical kinetics in 1990. After earning her doctorate, and performing post-doctoral research at Leeds and Princeton University, she returned to Leeds as a lecturer in the Department of Fuel and Energy in 1994. == Book == With Tamás Turányi, Tomlin is coauthor of the book Analysis of Kinetic Reaction Mechanisms (Springer, 2014). == Recognition == A paper coauthored by Tomlin won the 1992 Sugden Award of The Combustion Institute. Tomlin was elected to the inaugural 2018 class of Fellows of The Combustion Institute, "for innovative research on the development and application of mechanism reduction, sensitivity analysis and uncertainty quantification in combustion models". == References == == External links == Alison Tomlin publications indexed by Google Scholar Alison Tomlin on Twitter |
Wikipedia:Allan Sly (mathematician)#0 | Allan Murray Sly is an Australian mathematician and statistician specializing in probability theory. He is a professor of mathematics at Princeton University and was awarded a MacArthur Fellowship in 2018. == Education and career == Sly was a member of the Australian team at the 1999 and 2000 International Mathematical Olympiads, earning an honourable mention and a silver medal respectively. He attended Radford College, where he was dux of the year in 2000. He then studied at Australian National University, winning the University Medal in 2004, earning a bachelor's degree, and in 2006 earning a M.Phil. He completed his Ph.D. in 2009 at the University of California, Berkeley. His dissertation, Spatial and Temporal Mixing of Gibbs Measures, was supervised by Elchanan Mossel. After postdoctoral study at Microsoft Research, he joined the statistics faculty at Berkeley in 2011, and moved to Princeton University as a professor of mathematics in 2016. == Contributions == Sly's work has included research on finding clusters in networks, the use of information percolation to analyze the "cutoff" phenomenon in which Markov chains exhibit a sharp transition to their stationary distribution, embeddings of random sequences, and phase transitions for random instances of the satisfiability problem. == Recognition == Sly won a Sloan Research Fellowship in 2012 and was awarded the Rollo Davidson Prize in 2013. He was named a MacArthur Fellow in 2018 for "applying probability theory to resolve long-standing problems in statistical physics and computer science". He was the winner of the 2019 Loeve prize. == References == == External links == Home page |
Wikipedia:Allegory (mathematics)#0 | In the mathematical field of category theory, an allegory is a category that has some of the structure of the category Rel of sets and binary relations between them. Allegories can be used as an abstraction of categories of relations, and in this sense the theory of allegories is a generalization of relation algebra to relations between different sorts. Allegories are also useful in defining and investigating certain constructions in category theory, such as exact completions. In this article we adopt the convention that morphisms compose from right to left, so RS means "first do S, then do R". == Definition == An allegory is a category in which every morphism R : X → Y {\displaystyle R\colon X\to Y} is associated with an anti-involution, i.e. a morphism R ∘ : Y → X {\displaystyle R^{\circ }\colon Y\to X} with R ∘ ∘ = R {\displaystyle R^{\circ \circ }=R} and ( R S ) ∘ = S ∘ R ∘ ; {\displaystyle (RS)^{\circ }=S^{\circ }R^{\circ }{\text{;}}} and every pair of morphisms R , S : X → Y {\displaystyle R,S\colon X\to Y} with common domain/codomain is associated with an intersection, i.e. a morphism R ∩ S : X → Y {\displaystyle R\cap S\colon X\to Y} all such that intersections are idempotent: R ∩ R = R , {\displaystyle R\cap R=R,} commutative: R ∩ S = S ∩ R , {\displaystyle R\cap S=S\cap R,} and associative: ( R ∩ S ) ∩ T = R ∩ ( S ∩ T ) ; {\displaystyle (R\cap S)\cap T=R\cap (S\cap T);} anti-involution distributes over intersection: ( R ∩ S ) ∘ = R ∘ ∩ S ∘ ; {\displaystyle (R\cap S)^{\circ }=R^{\circ }\cap S^{\circ };} composition is semi-distributive over intersection: R ( S ∩ T ) ⊆ R S ∩ R T {\displaystyle R(S\cap T)\subseteq RS\cap RT} and ( R ∩ S ) T ⊆ R T ∩ S T ; {\displaystyle (R\cap S)T\subseteq RT\cap ST;} and the modularity law is satisfied: R S ∩ T ⊆ ( R ∩ T S ∘ ) S . {\displaystyle RS\cap T\subseteq (R\cap TS^{\circ })S.} Here, we are abbreviating using the order defined by the intersection: R ⊆ S {\displaystyle R\subseteq S} means R = R ∩ S . {\displaystyle R=R\cap S.} A first example of an allegory is the category of sets and relations. The objects of this allegory are sets, and a morphism X → Y {\displaystyle X\to Y} is a binary relation between X and Y. Composition of morphisms is composition of relations, and the anti-involution of R {\displaystyle R} is the converse relation R ∘ {\displaystyle R^{\circ }} : y R ∘ x {\displaystyle yR^{\circ }x} if and only if x R y {\displaystyle xRy} . Intersection of morphisms is (set-theoretic) intersection of relations. == Regular categories and allegories == === Allegories of relations in regular categories === In a category C, a relation between objects X and Y is a span of morphisms X ← R → Y {\displaystyle X\gets R\to Y} that is jointly monic. Two such spans X ← S → Y {\displaystyle X\gets S\to Y} and X ← T → Y {\displaystyle X\gets T\to Y} are considered equivalent when there is an isomorphism between S and T that make everything commute; strictly speaking, relations are only defined up to equivalence (one may formalise this either by using equivalence classes or by using bicategories). If the category C has products, a relation between X and Y is the same thing as a monomorphism into X × Y (or an equivalence class of such). In the presence of pullbacks and a proper factorization system, one can define the composition of relations. The composition X ← R → Y ← S → Z {\displaystyle X\gets R\to Y\gets S\to Z} is found by first pulling back the cospan R → Y ← S {\displaystyle R\to Y\gets S} and then taking the jointly-monic image of the resulting span X ← R ← ∙ → S → Z . {\displaystyle X\gets R\gets \bullet \to S\to Z.} Composition of relations will be associative if the factorization system is appropriately stable. In this case, one can consider a category Rel(C), with the same objects as C, but where morphisms are relations between the objects. The identity relations are the diagonals X → X × X . {\displaystyle X\to X\times X.} A regular category (a category with finite limits and images in which covers are stable under pullback) has a stable regular epi/mono factorization system. The category of relations for a regular category is always an allegory. Anti-involution is defined by turning the source/target of the relation around, and intersections are intersections of subobjects, computed by pullback. === Maps in allegories, and tabulations === A morphism R in an allegory A is called a map if it is entire ( 1 ⊆ R ∘ R ) {\displaystyle (1\subseteq R^{\circ }R)} and deterministic ( R R ∘ ⊆ 1 ) . {\displaystyle (RR^{\circ }\subseteq 1).} Another way of saying this is that a map is a morphism that has a right adjoint in A when A is considered, using the local order structure, as a 2-category. Maps in an allegory are closed under identity and composition. Thus, there is a subcategory Map(A) of A with the same objects but only the maps as morphisms. For a regular category C, there is an isomorphism of categories C ≅ Map ( Rel ( C ) ) . {\displaystyle C\cong \operatorname {Map} (\operatorname {Rel} (C)).} In particular, a morphism in Map(Rel(Set)) is just an ordinary set function. In an allegory, a morphism R : X → Y {\displaystyle R\colon X\to Y} is tabulated by a pair of maps f : Z → X {\displaystyle f\colon Z\to X} and g : Z → Y {\displaystyle g\colon Z\to Y} if g f ∘ = R {\displaystyle gf^{\circ }=R} and f ∘ f ∩ g ∘ g = 1. {\displaystyle f^{\circ }f\cap g^{\circ }g=1.} An allegory is called tabular if every morphism has a tabulation. For a regular category C, the allegory Rel(C) is always tabular. On the other hand, for any tabular allegory A, the category Map(A) of maps is a locally regular category: it has pullbacks, equalizers, and images that are stable under pullback. This is enough to study relations in Map(A), and in this setting, A ≅ Rel ( Map ( A ) ) . {\displaystyle A\cong \operatorname {Rel} (\operatorname {Map} (A)).} === Unital allegories and regular categories of maps === A unit in an allegory is an object U for which the identity is the largest morphism U → U , {\displaystyle U\to U,} and such that from every other object, there is an entire relation to U. An allegory with a unit is called unital. Given a tabular allegory A, the category Map(A) is a regular category (it has a terminal object) if and only if A is unital. === More sophisticated kinds of allegory === Additional properties of allegories can be axiomatized. Distributive allegories have a union-like operation that is suitably well-behaved, and division allegories have a generalization of the division operation of relation algebra. Power allegories are distributive division allegories with additional powerset-like structure. The connection between allegories and regular categories can be developed into a connection between power allegories and toposes. == References == |
Wikipedia:Alma Johanna Ruubel#0 | Alma Johanna Ruubel (28 September 1899 – 21 January 1990) was an Estonian mathematician and professor engaged in the development of curvilinear representational geometry. == Life and work == Alma Johanna Ruubel was born in Õisu Parish (present-day Viljandi Parish), Viljandi County, and grew up on the Peebu farm in the Ōisu municipality of Viljandimaa. Her father, Juhan Ruubel, was a master builder and joiner and her mother Ann (née Mankin) was a housewife. In 1909, she entered the three-grade Russian-language Peebu school in Õisu Parish and in 1912, the Viljandi Estonian Educational Society's Estonian-language girls' school from which she graduated in 1916. She continued her studies at the Russian-language Viljandi Girls' High School, which was converted into a seven-grade school during World War I and the German occupation in 1918. In order to continue her studies, Ruubel, together with other girls, formed a group of students of the eighth-grade course, which invited separate teachers. With the end of the German occupation, Viljandi City Girls' High School started working in Viljandi, and Alma's group was included in the eighth grade of the school there. Upon its completion in 1919, Ruubel was allowed to enter a university. She entered the summer teacher preparation courses organized by the University of Tartu (TRÜ), and in August 1919 she was already selected as a mathematics teacher at the Pärnu Commercial School. From the next school year, she worked as a teacher in her old school in Viljandi. Ruubel also continued her studies in the courses held at the university and finished them in 1921. === University student === She officially entered the University of Tartu in 1926. At first, she earned money by giving tutoring lessons, but in 1929 Professor Gerhard Rägo invited her to become an assistant at the Institute of Mathematics, even though she was only a third-year student. With this, Ruubel became the first female lecturer in mathematics at the University of Tartu. In 1932, she graduated cum laude. In 1935, she wanted to start research related to numerical and graphical methods. Professor Rägo recommended to her John Couch Adams's method of numerical integration of differential equations and Richard von Mises's recently published error estimation approach to that method. Rägo considered her finished research suitable for submission as a master's thesis. Ruubel received her master's degree in mathematics in 1936. Her thesis, "The JC Adams Method for the Numerical Integration of Ordinary Differential Equations," was recertified in 1946 and she was awarded a Ph.D. in Physics and Mathematics. === Teacher === After graduating from the university, Ruubel continued to work there, first as a senior teacher and later as an associate professor (appointed in 1949). She taught theoretical mechanics, applied mathematics, computational and graphical methods, probability theory, analytical, differential and representational geometry, and higher mathematics. She also worked in leading positions, being the head of the department, the vice-dean and dean of the Faculty of Mathematics and Natural Sciences, the supervisor of the postgraduate course and the head of the methodological council. In 1952, the department of visual geometry of the Estonian Academy of Agriculture was opened, and Ruubel was invited to head it. This allowed her to focus on her subject. At the same time, she continued to work at TRÜ as an associate professor until 1955. Later, in 1969, she was invited to be a member of the Scientific Council of the Faculty of Mathematics of TRÜ for another three years. Ruubel's work at the university led her to establish a new branch of mathematics, namely curvilinear representational geometry. She spoke at many scientific conferences in Moscow, Leningrad, Tallinn as well as Tartu to introduce it and publish brochures and articles. After the Department of Visual Geometry and Graphics was merged with other departments, she worked until her retirement in 1968 as an Associate Professor of the Combined Department and headed the Section of Visual Geometry and Graphics. As a pensioner, she gave lectures from time to time and participated in the organization of educational methodological work until 1973. Alma Ruubel died in 1990 In Tartu on the same day as mathematician John Couch Adams. She is buried in the Vana-Jaani cemetery in Tartu. == Research == Ruubel's areas of research: curvilinear projection methods, their properties and applications, graphical and mechanical integration of differential equations and application of projection methods in visual geometry. She was one of the founders of the curvilinear projection direction. She published articles in cooperation with professor Sinaiida Riives, who was her colleague in the department of visual geometry of the Estonian Academy of Agriculture. == Selected works == Orthogonal circular projection. Tartu, 1958 Complex drawings of curve projections. Tartu, 1961 Generalized axonometry. Tartu, 1967 School mathematics 1. Tartu, 1979. == References == |
Wikipedia:Almagest#0 | The Almagest ( AL-mə-jest) is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths, written by Claudius Ptolemy (c. AD 100 – c. 170) in Koine Greek. One of the most influential scientific texts in history, it canonized a geocentric model of the Universe that was accepted for more than 1,200 years from its origin in Hellenistic Alexandria, in the medieval Byzantine and Islamic worlds, and in Western Europe through the Middle Ages and early Renaissance until Copernicus. It is also a key source of information about ancient Greek astronomy. Ptolemy set up a public inscription at Canopus, Egypt, in 147 or 148. N. T. Hamilton found that the version of Ptolemy's models set out in the Canopic Inscription was earlier than the version in the Almagest. Hence the Almagest could not have been completed before about 150, a quarter-century after Ptolemy began observing. == Names == The name comes from Arabic اَلْمَجِسْطِيّ al-majisṭī, with اَلـ al- meaning 'the' and majisṭī being a corruption of Greek μεγίστη megístē 'greatest'. The Arabic name was popularized by a Latin translation known as Almagestum made in the 12th century from an Arabic translation, which would endure until original Greek copies resurfaced in the 15th century. The work was originally called Μαθηματικὴ Σύνταξις (Mathēmatikḕ Sýntaxis) in Koine Greek, as also in Modern Greek (primarily), and was known as Syntaxis Mathematica in Latin. The treatise was later called Ἡ Μεγάλη Σύνταξις (Hē Megálē Sýntaxis), "The Great Treatise" (Latin: Magna Syntaxis), and the superlative form of this (Greek: μεγίστη megístē, 'greatest') lies behind the Arabic name from which the English name Almagest derives. In the study of medieval Hebrew texts, the Almagest is sometimes referred to as "Ptolemy's Book of Elections", to emphasize parallelism with Abraham ibn Ezra's manuscript of the same name. == History == Written possibly around 150 CE, the text survives as copies, the oldest being from the 9th century when Arabic scholars started to translate the text, which in turn have survived in copies from the 11th and 13th century. == Contents == === The Syntaxis Mathematica books === The Syntaxis Mathematica consists of thirteen sections, called books. As with many medieval manuscripts that were handcopied or, particularly, printed in the early years of printing, there were considerable differences between various editions of the same text, as the process of transcription was highly personal. An example illustrating how the Syntaxis was organized is given below; it is a Latin edition printed in 1515 at Venice by Petrus Lichtenstein. Book I contains an outline of Aristotle's cosmology: on the spherical form of the heavens, with the spherical Earth lying motionless as the center, with the fixed stars and the various planets revolving around the Earth. Then follows an explanation of chords with table of chords; observations of the obliquity of the ecliptic (the apparent path of the Sun through the stars); and an introduction to spherical trigonometry. Book II covers problems associated with the daily motion attributed to the heavens, namely risings and settings of celestial objects, the length of daylight, the determination of latitude, the points at which the Sun is vertical, the shadows of the gnomon at the equinoxes and solstices, and other observations that change with the observer's position. There is also a study of the angles made by the ecliptic with the vertical, with tables. Book III covers the length of the year, and the motion of the Sun. Ptolemy explains Hipparchus' discovery of the precession of the equinoxes and begins explaining the theory of epicycles. Books IV and V cover the motion of the Moon, lunar parallax, the motion of the lunar apogee, and the sizes and distances of the Sun and Moon relative to the Earth. Book VI covers solar and lunar eclipses. Books VII and VIII cover the motions of the fixed stars, including precession of the equinoxes. They also contain a star catalogue of 1022 stars, described by their positions in the constellations, together with ecliptic longitude and latitude. Book IX addresses general issues associated with creating models for the five naked eye planets, and the motion of Mercury. Book X covers the motions of Venus and Mars. Book XI covers the motions of Jupiter and Saturn. Book XII covers stations and retrograde motion, which occurs when planets appear to pause, then briefly reverse their motion against the background of the zodiac. Ptolemy understood these terms to apply to Mercury and Venus as well as the outer planets. Book XIII covers motion in latitude, that is, the deviation of planets from the ecliptic. The final topic of this chapter also covers how to determine when a planet first becomes visible after being hidden by the glare of the sun, as well as the last time it is seen before being hidden by the sun's glare. === Ptolemy's cosmos === The cosmology of the Syntaxis includes five main points, each of which is the subject of a chapter in Book I. What follows is a close paraphrase of Ptolemy's own words from Toomer's translation. The celestial realm is spherical, and moves as a sphere. The Earth is a sphere. The Earth is at the center of the cosmos. The Earth, in relation to the distance of the fixed stars, has no appreciable size and must be treated as a mathematical point. The Earth does not move. === The star catalogue === The layout of the catalogue has always been tabular. Ptolemy writes explicitly that the coordinates are given as (ecliptical) "longitudes" and "latitudes", which are given in columns, so this has probably always been the case. It is significant that Ptolemy chooses the ecliptical coordinate system because of his knowledge of precession, which distinguishes him from all his predecessors. Hipparchus' celestial globe had an ecliptic drawn in, but the coordinates were equatorial. Since Hipparchus' star catalogue has not survived in its original form, but was absorbed into the Almagest star catalogue (and heavily revised in the 265 years in between), the Almagest star catalogue is the oldest one in which complete tables of coordinates and magnitudes have come down to us. As mentioned, Ptolemy includes a star catalog containing 1022 stars. He says that he "observed as many stars as it was possible to perceive, even to the sixth magnitude". The ecliptic longitudes are given in terms of a zodiac sign and a number of degrees and fractions of a degree. The zodiac signs each represent exactly 30°, starting with Aries representing longitude 0° to 30°. The degrees are added to the lower limit of the 30-degree range to obtain the longitude. Unlike the situation with the zodiac of modern-day astrology, most of the stars of a given zodiac constellation in the catalog fall in the 30-degree range designated by the same name (the so-called 'zodiac sign'). The ecliptic longitudes are about 26° lower than those of AD 2000 (the J2000 epoch). Ptolemy says that the ecliptic longitudes are for the beginning of the reign of Antoninus Pius (138 AD) and that he found that the longitudes had increased by 2° 40′ since the time of Hipparchus which was 265 years earlier (Alm. VII, 2). But calculations show that his ecliptic longitudes correspond more closely to around the middle of the first century CE (+48 to +58). Since Tycho Brahe found this offset, astronomers and historians investigated this problem and suggested several causes: that all coordinates were calculated from Hipparchus' observations, whereby the precession constant, which was known too inaccurately at the time, led to a summation error (Delambre 1817); that the data had in fact been observed a century earlier by Menelaus of Alexandria (Björnbo 1901); that the difference is a sum of individual errors of various kinds, including calibration with outdated solar data; that Ptolemy's instrument was wrongly calibrated and had a systematic offset. Subtracting the systematic error leaves other errors that cannot be explained by precession. Of these errors, about 18 to 20 are also found in Hipparchus' star catalogue (which can only be reconstructed incompletely). From this it can be concluded that a subset of star coordinates in the Almagest can indeed be traced back to Hipparchus, but not that the complete star catalogue was simply "copied". Rather, Hipparchus' major errors are no longer present in the Almagest and, on the other hand, Hipparchus' star catalogue had some stars that are entirely absent from the Almagest. It can be concluded that Hipparchus' star catalogue, while forming the basis, has been reobserved and revised. ==== Errors in the coordinates ==== The figure he used is based on Hipparchus' own estimate for precession, which was 1° in 100 years, instead of the correct 1° in 72 years. Dating attempts through proper motion of the stars also appear to date the actual observation to Hipparchus' time instead of Ptolemy. Many of the longitudes and latitudes have been corrupted in the various manuscripts. Most of these errors can be explained by similarities in the symbols used for different numbers. For example, the Greek letters Α and Δ were used to mean 1 and 4 respectively, but because these look similar copyists sometimes wrote the wrong one. In Arabic manuscripts, there was confusion between for example 3 and 8 (ج and ح). (At least one translator also introduced errors. Gerard of Cremona, who translated an Arabic manuscript into Latin around 1175, put 300° for the latitude of several stars. He had apparently learned from Moors, who used the letter س (sin) for 300 (like the Hebrew ש (shin)), but the manuscript he was translating came from the East, where س was used for 60, like the Hebrew ס (samekh).) Even without the errors introduced by copyists, and even accounting for the fact that the longitudes are more appropriate for 58 AD than for 137 AD, the latitudes and longitudes are not fully accurate, with errors as great as large fractions of a degree. Some errors may be due to atmospheric refraction causing stars that are low in the sky to appear higher than where they really are. A series of stars in Centaurus are off by a couple of degrees, including the star we call Alpha Centauri. These were probably measured by a different person or persons from the others, and in an inaccurate way. ==== Constellations in the star catalogue ==== The star catalogue contains 48 constellations, which have different surface areas and numbers of stars. In Book VIII, Chapter 3, Ptolemy writes that the constellations should be outlined on a globe, but it is unclear exactly how he means this: should surrounding polygons be drawn or should the figures be sketched or even line figures be drawn? This is not stated. Although no line figures have survived from antiquity, the figures can be reconstructed on the basis of the descriptions in the star catalogue: The exact celestial coordinates of the figures' heads, feet, arms, wings and other body parts are recorded. It is therefore possible to draw the stick figures in the modern sense so that they fit the description in the Almagest. These constellations form the basis for the modern constellations that were formally adopted by the International Astronomical Union in 1922, with official boundaries that were agreed in 1928. Of the stars in the catalogue, 108 (just over 10%) were classified by Ptolemy as 'unformed', by which he meant lying outside the recognized constellation figures. These were later absorbed into their surrounding constellations or in some cases used to form new constellations. === Ptolemy's planetary model === In Almagest, Ptolemy assigned the following order to the planetary spheres, beginning with the innermost: Later, in his "Planetary Hypothesis", he concludes that Mercury is the second closest planet. Other classical writers suggested different sequences. Plato (c. 427 – c. 347 BC) placed the Sun second in order after the Moon. Martianus Capella (5th century AD) put Mercury and Venus in motion around the Sun. Ptolemy's authority was preferred by most medieval Islamic and late medieval European astronomers. Ptolemy inherited from his Greek predecessors a geometrical toolbox and a partial set of models for predicting where the planets would appear in the sky. Apollonius of Perga (c. 262 – c. 190 BC) had introduced the deferent and epicycle and the eccentric deferent to astronomy. Hipparchus (2nd century BC) had crafted mathematical models of the motion of the Sun and Moon. Hipparchus had some knowledge of Mesopotamian astronomy, and he felt that Greek models should match those of the Babylonians in accuracy. He was unable to create accurate models for the remaining five planets. The Syntaxis adopted Hipparchus' solar model, which consisted of a simple eccentric deferent. For the Moon, Ptolemy began with Hipparchus' epicycle-on-deferent, then added a device that historians of astronomy refer to as a "crank mechanism": he succeeded in creating models for the other planets, where Hipparchus had failed, by introducing a third device called the equant. Ptolemy wrote the Syntaxis as a textbook of mathematical astronomy. It explained geometrical models of the planets based on combinations of circles, which could be used to predict the motions of celestial objects. In a later book, the Planetary Hypotheses, Ptolemy explained how to transform his geometrical models into three-dimensional spheres or partial spheres. In contrast to the mathematical Syntaxis, the Planetary Hypotheses is sometimes described as a book of cosmology. == Influence == Ptolemy's comprehensive treatise of mathematical astronomy superseded most older texts of Greek astronomy. Much of what we know about the work of astronomers like Hipparchus comes from references in the Syntaxis. The book was circulated among astronomers, and also among philosophers who are interested in astronomy. The Almagest, however, was not translated into Latin in ancient times and had little influence on popular literature. The first translations into Arabic were made in the 9th century, with two separate efforts, one sponsored by the caliph Al-Ma'mun, who received a copy as a condition of peace with the Byzantine emperor. Sahl ibn Bishr is thought to be the first Arabic translator. No Latin translation was made before the 12th century. Henry Aristippus made the first Latin translation directly from a Greek copy, but it was not as influential as a later translation into Latin made in Spain by the Italian scholar Gerard of Cremona from the Arabic (finished in 1175). Gerard translated the Arabic text while working at the Toledo School of Translators, although he was unable to translate many technical terms such as the Arabic Abrachir for Hipparchus. In the 13th century a Spanish version was produced, which was later translated under the patronage of Alfonso X. In the 15th century, a Greek version appeared in Western Europe. The German astronomer Johannes Müller (known as Regiomontanus, after his birthplace of Königsberg in Lower Frankonia) made an abridged Latin version at the instigation of the Greek churchman Cardinal Bessarion. Around the same time, George of Trebizond made a full translation accompanied by a commentary that was as long as the original text. George's translation, done under the patronage of Pope Nicholas V, was intended to supplant the old translation. The new translation was a great improvement; the new commentary was not, and aroused criticism. The Pope declined the dedication of George's work, and Regiomontanus's translation had the upper hand for over 100 years. During the 16th century, Guillaume Postel, who had been on an embassy to the Ottoman Empire, brought back Arabic disputations of the Almagest, such as the works of al-Kharaqī, Muntahā al-idrāk fī taqāsīm al-aflāk ("The Ultimate Grasp of the Divisions of Spheres", 1138–39). Commentaries on the Syntaxis were written by Theon of Alexandria (extant), Pappus of Alexandria (only fragments survive), and Ammonius Hermiae (lost). == Modern assessment == Under the scrutiny of modern scholarship, and the cross-checking of observations contained in the Almagest against figures produced through backwards extrapolation, various patterns of errors have emerged within the work. A prominent example is Ptolemy's use of measurements said to have been taken at noon, but which systematically produce readings that are off by half an hour, as if the observations were taken at 12:30pm. However, an explanation for this error was found in 1969. The overall quality of Claudius Ptolemy's scholarship and place as "one of the most outstanding scientists of antiquity" has been challenged by several modern writers, most prominently by Robert R. Newton in the 1977 book The Crime of Claudius Ptolemy, which asserted that the scholar fabricated his observations to fit his theories. Newton accused Ptolemy of systematically inventing data or doctoring the data of earlier astronomers, and labelled him "the most successful fraud in the history of science". One striking error noted by Newton was an autumn equinox said to have been observed by Ptolemy and "measured with the greatest care" at 2pm on 25 September 132, when the equinox should have been observed at 9:54am the day prior. Herbert Lewis, who had reworked some of Ptolemy's calculations, agreed with Newton that "Ptolemy was an outrageous fraud", and that "all those results capable of statistical analysis point beyond question towards fraud and against accidental error". Although some have described the charges laid by Newton as "erudite and imposing", others have disagreed with the findings. Bernard R. Goldstein wrote, "Unfortunately, Newton’s arguments in support of these charges are marred by all manner of distortions, misunderstandings, and excesses of rhetoric due to an intensely polemical style." Owen Gingerich, while agreeing that the Almagest contains "some remarkably fishy numbers", including in the matter of the 30-hour displaced equinox, which he noted aligned perfectly with predictions made by Hipparchus 278 years earlier, rejected the qualification of fraud. John Phillips Britton, Visiting Fellow at Yale University, wrote of R.R. Newton, "I think that his main conclusion with respect to Ptolemy’s stature and achievements as an astronomer is simply wrong, and that the Almagest should be seen as a great, if not indeed the first, scientific treatise." He continued, "Newton’s work does focus critical attention on the many difficulties and inconsistencies apparent in the fine structure of the Almagest. In particular, his conclusion that the Almagest is not a historical account of how Ptolemy actually derived his models and parameters is essentially the same as mine, although our reasons for this conclusion and our inferences from it differ radically." == Modern editions == The Almagest under the Latin title Syntaxis mathematica, was edited by J. L. Heiberg in Claudii Ptolemaei opera quae exstant omnia, vols. 1.1 and 1.2 (1898, 1903). Three translations of the Almagest into English have been published. The first, by R. Catesby Taliaferro of St. John's College in Annapolis, Maryland, was included in volume 16 of the Great Books of the Western World in 1952. The second, by G. J. Toomer, Ptolemy's Almagest in 1984, with a second edition in 1998. The third was a partial translation by Bruce M. Perry in The Almagest: Introduction to the Mathematics of the Heavens in 2014. A direct French translation from the Greek text was published in two volumes in 1813 and 1816 by Nicholas Halma, including detailed historical comments in a 69-page preface. It has been described as "suffer[ing] from excessive literalness, particularly where the text is difficult" by Toomer, and as "very faulty" by Serge Jodra. The scanned books are available in full at the Gallica French National library. == Gallery == == See also == Abū al-Wafā' Būzjānī (who also wrote an Almagest) Book of Fixed Stars Star cartography Euclid's Elements == References == === Notes === === Citations === === Sources === ==== Books ==== ==== Journals and magazines ==== ==== Websites ==== == Further reading == Evans, James (1998). The History and Practice of Ancient Astronomy. Oxford University Press. ISBN 978-0-19-509539-5. Neugebauer, Otto (1948). "Mathematical Methods in Ancient Astronomy". Bulletin of the American Mathematical Society. 54 (11): 1013–1041. doi:10.1090/S0002-9904-1948-09089-9 (inactive 1 May 2025).{{cite journal}}: CS1 maint: DOI inactive as of May 2025 (link) Neugebauer, Otto (1975). A History of Ancient Mathematical Astronomy. Berlin: Springer. Part 1, Part 2, Part 3. ISBN 3-540-06995-X. Hanson, Norwood Russell (1960). "The Mathematical Power of Epicyclical Astronomy". Isis. 51 (2): 150–158. doi:10.1086/348869. JSTOR 226846. Ridpath, Ian (2018) [1998]. Star Tales (Rev. ed.). Cambridge: Lutterworth Press. ISBN 978-0-7188-9478-8. Pedersen, Olaf (1993) [1974]. Early Physics and Astronomy: A Historical Introduction (2nd ed.). Cambridge University Press. ISBN 978-0-521-40899-8. Pedersen, Olaf (2011) [1974]. A Survey of the Almagest (2nd ed.). Berlin: Springer. ISBN 978-1-4939-3922-0. Ptolemaeus, Claudius. Syntaxis mathematica (in Greek). OCLC 767751182. Retrieved 10 April 2023 – via MDZ/München, Bayerische Staatsbibliothek. Shank, Michael H. (2009). "Islamic Science and the Making of European Renaissance by George Saliba". Aestimatio (Book review). 6: 63–72. == External links == Syntaxis Mathematica (Almagest), original Greek, edited by Johan Ludvig Heiberg, 1898. Syntaxis mathematica in J.L. Heiberg's edition (1898–1903) Ptolemy's De Analemmate. PDF scans of Heiberg's Greek edition, now in the public domain (Koine Greek) Toomer's English translation Duckworth, 1984. Ptolemy. Almagest. Latin translation from the Arabic by Gerard of Cremona. Digitized version of manuscript made in Northern Italy c. 1200–1225 held by the State Library of Victoria. University of Vienna: Almagestum (1515) PDFs of different resolutions. Edition of Petrus Liechtenstein, Latin translation of Gerard of Cremona. Almagest Ephemeris Calculator by Robert Van Gent. Positions for any date, translation of dates in calendars used by Ptolemy. Extensive list of references and articles. Online luni-solar and planetary ephemeris calculator based on the Almagest A podcast discussion by Prof. M Heath and Dr A. Chapman of a recent re-discovery of a 14th-century manuscript in the university of Leeds Library Star catalog in ASCII (in Latin) Animation of Ptolemy's model of the universe by Andre Rehak (YouTube) (in French) French translation, with some diagrams animated, interactive, or random (in Hebrew) Maimonides explaining why you need to learn Almagest first to understand science |
Wikipedia:Almost commutative ring#0 | In algebra, a filtered ring A is said to be almost commutative if the associated graded ring gr A = ⊕ A i / A i − 1 {\displaystyle \operatorname {gr} A=\oplus A_{i}/{A_{i-1}}} is commutative. Basic examples of almost commutative rings involve differential operators. For example, the enveloping algebra of a complex Lie algebra is almost commutative by the PBW theorem. Similarly, a Weyl algebra is almost commutative. == See also == Ore condition Gelfand–Kirillov dimension == References == Victor Ginzburg, Lectures on D-modules |
Wikipedia:Almut Burchard#0 | Almut Burchard is a mathematician whose research interests include functional analysis, probability theory, and their applications in network calculus for the stochastic analysis of communication networks. Educated in Germany and the US, she has worked in the US and Canada, where she is a professor of mathematics at the University of Toronto. == Education and career == Burchard earned a diploma in mathematics (the German equivalent of a master's degree) in 1989 from Heidelberg University, with research supervised by Willi Jäger on mathematical modeling of chemical reactions. She shifted to pure mathematics for her doctoral work on the Riesz rearrangement inequality at Georgia Tech, supervised by Michael Loss, completing her Ph.D. in 1994. She was a faculty member in the Princeton University department of mathematics from 1994 to 1998, and at the University of Virginia from 1998 to 2005. She has held her present position at the University of Toronto since 2005. == Recognition == In 2021, the Fields Institute listed Burchard as a Fellow, recognizing her "energetic support" of the Fields Institute's, in helping develop an online program on fluid dynamics, and in mentorship of undergraduates. == References == == External links == Home page Almut Burchard publications indexed by Google Scholar |
Wikipedia:Alona Ben-Tal#0 | Alona Ben-Tal (Hebrew: אלונה בן טל) is an Israeli and New Zealand applied mathematician who works as an associate professor and deputy head of school in the School of Natural and Computational Sciences at Massey University. Her research concerns dynamical systems and the mathematical modeling of human and bird breathing and of electrical power systems. == Education and career == Ben-Tal originally studied mechanical engineering at the Technion – Israel Institute of Technology, earning a bachelor's degree there in 1991 and a master's degree in 1994. After working in industry for three years, she moved with her family to New Zealand and returned to graduate study in mathematics, completing a Ph.D. in 2001 at the University of Auckland with the dissertation A Study of Symmetric Forced Oscillators supervised by Vivien Kirk, Graeme Wake and Geoff Nicholls. After she completed her doctorate, she held positions at the University of Auckland as a fixed-term lecturer in mathematics, and then as a NZ Science & Technology Post-doctoral Fellow in the Bioengineering Institute, before moving to Massey University as a lecturer in 2005. == Contributions == In her work on human breathing, Ben-Tal has studied respiratory sinus arrhythmia, the phenomenon that the heart rate speeds up while inhaling and slows down while exhaling. Initially hypothesising that this variability would improve the rate of gas exchange in the lungs, her research found that instead it saves effort by the heart while maintaining even levels of blood oxygenation. In birds, Ben-Tal has studied the one-way nature of certain air passages in bird lungs, and the ability of birds to change the speed of airflow through these passages. Her research found that, in some circumstances, birds spend less time inhaling than they do exhaling. == Recognition == Ben-Tal was named a fellow of the New Zealand Mathematical Society in 2016. == References == == External links == Alona Ben-Tal publications indexed by Google Scholar |
Wikipedia:Alon–Boppana bound#0 | In spectral graph theory, the Alon–Boppana bound provides a lower bound on the second-largest eigenvalue of the adjacency matrix of a d {\displaystyle d} -regular graph, meaning a graph in which every vertex has degree d {\displaystyle d} . The reason for the interest in the second-largest eigenvalue is that the largest eigenvalue is guaranteed to be d {\displaystyle d} due to d {\displaystyle d} -regularity, with the all-ones vector being the associated eigenvector. The graphs that come close to meeting this bound are Ramanujan graphs, which are examples of the best possible expander graphs. Its discoverers are Noga Alon and Ravi Boppana. == Theorem statement == Let G {\displaystyle G} be a d {\displaystyle d} -regular graph on n {\displaystyle n} vertices with diameter m {\displaystyle m} , and let A {\displaystyle A} be its adjacency matrix. Let λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{n}} be its eigenvalues. Then λ 2 ≥ 2 d − 1 − 2 d − 1 − 1 ⌊ m / 2 ⌋ . {\displaystyle \lambda _{2}\geq 2{\sqrt {d-1}}-{\frac {2{\sqrt {d-1}}-1}{\lfloor m/2\rfloor }}.} The above statement is the original one proved by Noga Alon. Some slightly weaker variants exist to improve the ease of proof or improve intuition. Two of these are shown in the proofs below. == Intuition == The intuition for the number 2 d − 1 {\displaystyle 2{\sqrt {d-1}}} comes from considering the infinite d {\displaystyle d} -regular tree. This graph is a universal cover of d {\displaystyle d} -regular graphs, and it has spectral radius 2 d − 1 . {\displaystyle 2{\sqrt {d-1}}.} == Saturation == A graph that essentially saturates the Alon–Boppana bound is called a Ramanujan graph. More precisely, a Ramanujan graph is a d {\displaystyle d} -regular graph such that | λ 2 | , | λ n | ≤ 2 d − 1 . {\displaystyle |\lambda _{2}|,|\lambda _{n}|\leq 2{\sqrt {d-1}}.} A theorem by Friedman shows that, for every d {\displaystyle d} and ϵ > 0 {\displaystyle \epsilon >0} and for sufficiently large n {\displaystyle n} , a random d {\displaystyle d} -regular graph G {\displaystyle G} on n {\displaystyle n} vertices satisfies max { | λ 2 | , | λ n | } < 2 d − 1 + ϵ {\displaystyle \max\{|\lambda _{2}|,|\lambda _{n}|\}<2{\sqrt {d-1}}+\epsilon } with high probability. This means that a random n {\displaystyle n} -vertex d {\displaystyle d} -regular graph is typically "almost Ramanujan." == First proof (slightly weaker statement) == We will prove a slightly weaker statement, namely dropping the specificity on the second term and simply asserting λ 2 ≥ 2 d − 1 − o ( 1 ) . {\displaystyle \lambda _{2}\geq 2{\sqrt {d-1}}-o(1).} Here, the o ( 1 ) {\displaystyle o(1)} term refers to the asymptotic behavior as n {\displaystyle n} grows without bound while d {\displaystyle d} remains fixed. Let the vertex set be V . {\displaystyle V.} By the min-max theorem, it suffices to construct a nonzero vector z ∈ R | V | {\displaystyle z\in \mathbb {R} ^{|V|}} such that z T 1 = 0 {\displaystyle z^{\text{T}}\mathbf {1} =0} and z T A z z T z ≥ 2 d − 1 − o ( 1 ) . {\displaystyle {\frac {z^{\text{T}}Az}{z^{\text{T}}z}}\geq 2{\sqrt {d-1}}-o(1).} Pick some value r ∈ N . {\displaystyle r\in \mathbb {N} .} For each vertex in V , {\displaystyle V,} define a vector f ( v ) ∈ R | V | {\displaystyle f(v)\in \mathbb {R} ^{|V|}} as follows. Each component will be indexed by a vertex u {\displaystyle u} in the graph. For each u , {\displaystyle u,} if the distance between u {\displaystyle u} and v {\displaystyle v} is k , {\displaystyle k,} then the u {\displaystyle u} -component of f ( v ) {\displaystyle f(v)} is f ( v ) u = w k = ( d − 1 ) − k / 2 {\displaystyle f(v)_{u}=w_{k}=(d-1)^{-k/2}} if k ≤ r − 1 {\displaystyle k\leq r-1} and 0 {\displaystyle 0} if k ≥ r . {\displaystyle k\geq r.} We claim that any such vector x = f ( v ) {\displaystyle x=f(v)} satisfies x T A x x T x ≥ 2 d − 1 ( 1 − 1 2 r ) . {\displaystyle {\frac {x^{\text{T}}Ax}{x^{\text{T}}x}}\geq 2{\sqrt {d-1}}\left(1-{\frac {1}{2r}}\right).} To prove this, let V k {\displaystyle V_{k}} denote the set of all vertices that have a distance of exactly k {\displaystyle k} from v . {\displaystyle v.} First, note that x T x = ∑ k = 0 r − 1 | V k | w k 2 . {\displaystyle x^{\text{T}}x=\sum _{k=0}^{r-1}|V_{k}|w_{k}^{2}.} Second, note that x T A x = ∑ u ∈ V x u ∑ u ′ ∈ N ( u ) x u ′ ≥ ∑ k = 0 r − 1 | V k | w k [ w k − 1 + ( d − 1 ) w k + 1 ] − ( d − 1 ) | V r − 1 | w r − 1 w r , {\displaystyle x^{\text{T}}Ax=\sum _{u\in V}x_{u}\sum _{u'\in N(u)}x_{u'}\geq \sum _{k=0}^{r-1}|V_{k}|w_{k}\left[w_{k-1}+(d-1)w_{k+1}\right]-(d-1)|V_{r-1}|w_{r-1}w_{r},} where the last term on the right comes from a possible overcounting of terms in the initial expression. The above then implies x T A x ≥ 2 d − 1 ( ∑ k = 0 r − 1 | V k | w k 2 − 1 2 | V r − 1 | w r − 1 2 ) , {\displaystyle x^{\text{T}}Ax\geq 2{\sqrt {d-1}}\left(\sum _{k=0}^{r-1}|V_{k}|w_{k}^{2}-{\frac {1}{2}}|V_{r-1}|w_{r-1}^{2}\right),} which, when combined with the fact that | V k + 1 | ≤ ( d − 1 ) | V k | {\displaystyle |V_{k+1}|\leq (d-1)|V_{k}|} for any k , {\displaystyle k,} yields x T A x ≥ 2 d − 1 ( 1 − 1 2 r ) ∑ k = 0 r − 1 | V k | w k 2 . {\displaystyle x^{\text{T}}Ax\geq 2{\sqrt {d-1}}\left(1-{\frac {1}{2r}}\right)\sum _{k=0}^{r-1}|V_{k}|w_{k}^{2}.} The combination of the above results proves the desired inequality. For convenience, define the ( r − 1 ) {\displaystyle (r-1)} -ball of a vertex v {\displaystyle v} to be the set of vertices with a distance of at most r − 1 {\displaystyle r-1} from v . {\displaystyle v.} Notice that the entry of f ( v ) {\displaystyle f(v)} corresponding to a vertex u {\displaystyle u} is nonzero if and only if u {\displaystyle u} lies in the ( r − 1 ) {\displaystyle (r-1)} -ball of x . {\displaystyle x.} The number of vertices within distance k {\displaystyle k} of a given vertex is at most 1 + d + d ( d − 1 ) + d ( d − 1 ) 2 + ⋯ + d ( d − 1 ) k − 1 = d k + 1. {\displaystyle 1+d+d(d-1)+d(d-1)^{2}+\cdots +d(d-1)^{k-1}=d^{k}+1.} Therefore, if n ≥ d 2 r − 1 + 2 , {\displaystyle n\geq d^{2r-1}+2,} then there exist vertices u , v {\displaystyle u,v} with distance at least 2 r . {\displaystyle 2r.} Let x = f ( v ) {\displaystyle x=f(v)} and y = f ( u ) . {\displaystyle y=f(u).} It then follows that x T y = 0 , {\displaystyle x^{\text{T}}y=0,} because there is no vertex that lies in the ( r − 1 ) {\displaystyle (r-1)} -balls of both x {\displaystyle x} and y . {\displaystyle y.} It is also true that x T A y = 0 , {\displaystyle x^{\text{T}}Ay=0,} because no vertex in the ( r − 1 ) {\displaystyle (r-1)} -ball of x {\displaystyle x} can be adjacent to a vertex in the ( r − 1 ) {\displaystyle (r-1)} -ball of y . {\displaystyle y.} Now, there exists some constant c {\displaystyle c} such that z = x − c y {\displaystyle z=x-cy} satisfies z T 1 = 0. {\displaystyle z^{\text{T}}\mathbf {1} =0.} Then, since x T y = x T A y = 0 , {\displaystyle x^{\text{T}}y=x^{\text{T}}Ay=0,} z T A z = x T A x + c 2 y T A y ≥ 2 d − 1 ( 1 − 1 2 r ) ( x T x + c 2 y T y ) = 2 d − 1 ( 1 − 1 2 r ) z T z . {\displaystyle z^{\text{T}}Az=x^{\text{T}}Ax+c^{2}y^{\text{T}}Ay\geq 2{\sqrt {d-1}}\left(1-{\frac {1}{2r}}\right)(x^{\text{T}}x+c^{2}y^{\text{T}}y)=2{\sqrt {d-1}}\left(1-{\frac {1}{2r}}\right)z^{\text{T}}z.} Finally, letting r {\displaystyle r} grow without bound while ensuring that n ≥ d 2 r − 1 + 2 {\displaystyle n\geq d^{2r-1}+2} (this can be done by letting r {\displaystyle r} grow sublogarithmically as a function of n {\displaystyle n} ) makes the error term o ( 1 ) {\displaystyle o(1)} in n . {\displaystyle n.} == Second proof (slightly modified statement) == This proof will demonstrate a slightly modified result, but it provides better intuition for the source of the number 2 d − 1 . {\displaystyle 2{\sqrt {d-1}}.} Rather than showing that λ 2 ≥ 2 d − 1 − o ( 1 ) , {\displaystyle \lambda _{2}\geq 2{\sqrt {d-1}}-o(1),} we will show that λ = max ( | λ 2 | , | λ n | ) ≥ 2 d − 1 − o ( 1 ) . {\displaystyle \lambda =\max(|\lambda _{2}|,|\lambda _{n}|)\geq 2{\sqrt {d-1}}-o(1).} First, pick some value k ∈ N . {\displaystyle k\in \mathbb {N} .} Notice that the number of closed walks of length 2 k {\displaystyle 2k} is tr A 2 k = ∑ i = 1 n λ i 2 k ≤ d 2 k + n λ 2 k . {\displaystyle \operatorname {tr} A^{2k}=\sum _{i=1}^{n}\lambda _{i}^{2k}\leq d^{2k}+n\lambda ^{2k}.} However, it is also true that the number of closed walks of length 2 k {\displaystyle 2k} starting at a fixed vertex v {\displaystyle v} in a d {\displaystyle d} -regular graph is at least the number of such walks in an infinite d {\displaystyle d} -regular tree, because an infinite d {\displaystyle d} -regular tree can be used to cover the graph. By the definition of the Catalan numbers, this number is at least C k ( d − 1 ) k , {\displaystyle C_{k}(d-1)^{k},} where C k = 1 k + 1 ( 2 k k ) {\displaystyle C_{k}={\frac {1}{k+1}}{\binom {2k}{k}}} is the k th {\displaystyle k^{\text{th}}} Catalan number. It follows that tr A 2 k ≥ n 1 k + 1 ( 2 k k ) ( d − 1 ) k {\displaystyle \operatorname {tr} A^{2k}\geq n{\frac {1}{k+1}}{\binom {2k}{k}}(d-1)^{k}} ⟹ λ 2 k ≥ 1 k + 1 ( 2 k k ) ( d − 1 ) k − d 2 k n . {\displaystyle \implies \lambda ^{2k}\geq {\frac {1}{k+1}}{\binom {2k}{k}}(d-1)^{k}-{\frac {d^{2k}}{n}}.} Letting n {\displaystyle n} grow without bound and letting k {\displaystyle k} grow without bound but sublogarithmically in n {\displaystyle n} yields λ ≥ 2 d − 1 − o ( 1 ) . {\displaystyle \lambda \geq 2{\sqrt {d-1}}-o(1).} == References == |
Wikipedia:Alp Eden#0 | Osman Alp Eden (born 1958) is a Turkish mathematician, scientist and professor of mathematics. He is a retired member of the Boğaziçi University Mathematics Department in Istanbul, Turkey. == Education == Alp Eden was born in Istanbul in 1958. He finished the high school Robert College of Istanbul in 1976. He graduated from Boğaziçi University Civil Engineering and Mathematics departments in 1981. He received his PhD in Mathematics under the supervision of Ciprian Ilie Foiaș at the Indiana University Bloomington in the United States in 1989. == Academic career == He worked in Arizona State University in 1989–1992. Then he became a full time member of the Department of Mathematics, at Boğaziçi University, Istanbul. In years he served as the chair of the department and the vice-dean of the Faculty of Arts and Sciences. He retired in 2015 and moved to İzmir. He is one of the founders of the Masters program in Financial Engineering at Boğaziçi University. Between 2006-2016 he was a member of the Steering Committee of the Istanbul Center for Mathematical Sciences (IMBM). He served as the Chief Editor of the Turkish Journal of Mathematics. After retirement he served as the Editor of the popular mathematics journal Matematik Dünyası published by the Turkish Mathematical Society (TMD). He has been an active researcher, having published more than 50 scientific manuscripts with more than 600 citations. In 1995 he was awarded the Science Award for Young Scientists given by the Scientific and Technological Research Council of Turkey (TUBITAK). He was awarded the Boğaziçi University Award for Excellence in Research in 1998 (for young scientists) and again in 2009. A one-day PDE workshop was held in his honor in 2015 by TMD. There is a conjecture in the field of dynamical systems named after Alp Eden. == Research areas == His research interests include non-linear PDEs, dynamical systems, finance mathematics and mathematical modelling. === Representative scientific publications === Book: Eden, A.; Foias, C.; Nicolaenko, B.; Temam, R. Exponential attractors for dissipative evolution equations. RAM: Research in Applied Mathematics, 37. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. Eden, A.; Kalantarov, V. K. The convective Cahn-Hilliard equation. Appl. Math. Lett. 20 (2007), no. 4, 455–461. Eden, A.; Milani, A. J. "Exponential attractors for extensible beam equations". Nonlinearity 6 (1993), no. 3, 457–479. Eden, A.; Milani, A. J.; Nicolaenko, B. Finite-dimensional exponential attractors for semilinear wave equations with damping. J. Math. Anal. Appl. 169 (1992), no. 2, 408–419. Eden, A.; Michaux, B.; Rakotoson, J.-M. Doubly nonlinear parabolic-type equations as dynamical systems. J. Dynam. Differential Equations 3 (1991), no. 1, 87–131. == References == == External links == Alp Eden publications indexed by Google Scholar |
Wikipedia:Alpha centrality#0 | In graph theory, the Katz centrality or alpha centrality of a node is a measure of centrality in a network. It was introduced by Leo Katz in 1953 and is used to measure the relative degree of influence of an actor (or node) within a social network. Unlike typical centrality measures which consider only the shortest path (the geodesic) between a pair of actors, Katz centrality measures influence by taking into account the total number of walks between a pair of actors. It is similar to Google's PageRank and to the eigenvector centrality. == Measurement == Katz centrality computes the relative influence of a node within a network by measuring the number of the immediate neighbors (first degree nodes) and also all other nodes in the network that connect to the node under consideration through these immediate neighbors. Connections made with distant neighbors are, however, penalized by an attenuation factor α {\displaystyle \alpha } . Each path or connection between a pair of nodes is assigned a weight determined by α {\displaystyle \alpha } and the distance between nodes as α d {\displaystyle \alpha ^{d}} . For example, in the figure on the right, assume that John's centrality is being measured and that α = 0.5 {\displaystyle \alpha =0.5} . The weight assigned to each link that connects John with his immediate neighbors Jane and Bob will be ( 0.5 ) 1 = 0.5 {\displaystyle (0.5)^{1}=0.5} . Since Jose connects to John indirectly through Bob, the weight assigned to this connection (composed of two links) will be ( 0.5 ) 2 = 0.25 {\displaystyle (0.5)^{2}=0.25} . Similarly, the weight assigned to the connection between Agneta and John through Aziz and Jane will be ( 0.5 ) 3 = 0.125 {\displaystyle (0.5)^{3}=0.125} and the weight assigned to the connection between Agneta and John through Diego, Jose and Bob will be ( 0.5 ) 4 = 0.0625 {\displaystyle (0.5)^{4}=0.0625} . == Mathematical formulation == Let A be the adjacency matrix of a network under consideration. Elements ( a i j ) {\displaystyle (a_{ij})} of A are variables that take a value 1 if a node i is connected to node j and 0 otherwise. The powers of A indicate the presence (or absence) of links between two nodes through intermediaries. For instance, in matrix A 3 {\displaystyle A^{3}} , if element ( a 2 , 12 ) = 1 {\displaystyle (a_{2,12})=1} , it indicates that node 2 and node 12 are connected through some walk of length 3. If C K a t z ( i ) {\displaystyle C_{\mathrm {Katz} }(i)} denotes Katz centrality of a node i, then, given a value α ∈ ( 0 , 1 ) {\displaystyle \alpha \in (0,1)} , mathematically: C K a t z ( i ) = ∑ k = 1 ∞ ∑ j = 1 n α k ( A k ) j i {\displaystyle C_{\mathrm {Katz} }(i)=\sum _{k=1}^{\infty }\sum _{j=1}^{n}\alpha ^{k}(A^{k})_{ji}} Note that the above definition uses the fact that the element at location ( i , j ) {\displaystyle (i,j)} of A k {\displaystyle A^{k}} reflects the total number of k {\displaystyle k} degree connections between nodes i {\displaystyle i} and j {\displaystyle j} . The value of the attenuation factor α {\displaystyle \alpha } has to be chosen such that it is smaller than the reciprocal of the absolute value of the largest eigenvalue of A. In this case the following expression can be used to calculate Katz centrality: C → K a t z = ( ( I − α A T ) − 1 − I ) I → {\displaystyle {\overrightarrow {C}}_{\mathrm {Katz} }=((I-\alpha A^{T})^{-1}-I){\overrightarrow {I}}} Here I {\displaystyle I} is the identity matrix, I → {\displaystyle {\overrightarrow {I}}} is a vector of size n (n is the number of nodes) consisting of ones. A T {\displaystyle A^{T}} denotes the transposed matrix of A and ( I − α A T ) − 1 {\displaystyle (I-\alpha A^{T})^{-1}} denotes matrix inversion of the term ( I − α A T ) {\displaystyle (I-\alpha A^{T})} . An extension of this framework allows for the walks to be computed in a dynamical setting. By taking a time dependent series of network adjacency snapshots of the transient edges, the dependency for walks to contribute towards a cumulative effect is presented. The arrow of time is preserved so that the contribution of activity is asymmetric in the direction of information propagation. Network producing data of the form: { A [ k ] ∈ R N × N } for k = 0 , 1 , 2 , … , M , {\displaystyle \left\{A^{[k]}\in \mathbb {R} ^{N\times N}\right\}\qquad {\text{for}}\quad k=0,1,2,\ldots ,M,} representing the adjacency matrix at each time t k {\displaystyle t_{k}} . Hence: ( A [ k ] ) i j = { 1 there is an edge from node i to node j at time t k 0 otherwise {\displaystyle \left(A^{[k]}\right)_{ij}={\begin{cases}1&{\text{there is an edge from node }}i{\text{ to node }}j{\text{ at time }}t_{k}\\0&{\text{otherwise}}\end{cases}}} The time points t 0 < t 1 < ⋯ < t M {\displaystyle t_{0}<t_{1}<\cdots <t_{M}} are ordered but not necessarily equally spaced. Q ∈ R N × N {\displaystyle Q\in \mathbb {R} ^{N\times N}} for which ( Q ) i j {\displaystyle (Q)_{ij}} is a weighted count of the number of dynamic walks of length w {\displaystyle w} from node i {\displaystyle i} to node j {\displaystyle j} . The form for the dynamic communicability between participating nodes is: Q = ( I − α A [ 0 ] ) − 1 ⋯ ( I − α A [ M ] ) − 1 . {\displaystyle {\mathcal {Q}}=\left(I-\alpha A^{[0]}\right)^{-1}\cdots \left(I-\alpha A^{[M]}\right)^{-1}.} This can be normalized via: Q ^ [ k ] = Q ^ [ k − 1 ] ( I − α A [ k ] ) − 1 ‖ Q ^ [ k − 1 ] ( I − α A [ k ] ) − 1 ‖ . {\displaystyle {\hat {\mathcal {Q}}}^{[k]}={\frac {{\hat {\mathcal {Q}}}^{[k-1]}\left(I-\alpha A^{[k]}\right)^{-1}}{\left\|{\hat {\mathcal {Q}}}^{[k-1]}\left(I-\alpha A^{[k]}\right)^{-1}\right\|}}.} Therefore, centrality measures that quantify how effectively node n {\displaystyle n} can 'broadcast' and 'receive' dynamic messages across the network: C n b r o a d c a s t := ∑ k = 1 N Q n k a n d C n r e c e i v e := ∑ k = 1 N Q k n {\displaystyle C_{n}^{\mathrm {broadcast} }:=\sum _{k=1}^{N}{\mathcal {Q}}_{nk}\quad \mathrm {and} \quad C_{n}^{\mathrm {receive} }:=\sum _{k=1}^{N}{\mathcal {Q}}_{kn}} . === Alpha centrality === Given a graph with adjacency matrix A i , j {\displaystyle A_{i,j}} , Katz centrality is defined as follows: x → = ( I − α A T ) − 1 e → − e → {\displaystyle {\vec {x}}=(I-\alpha A^{T})^{-1}{\vec {e}}-{\vec {e}}\,} where e j {\displaystyle e_{j}} is the external importance given to node j {\displaystyle j} , and α {\displaystyle \alpha } is a nonnegative attenuation factor which must be smaller than the inverse of the spectral radius of A {\displaystyle A} . The original definition by Katz used a constant vector e → {\displaystyle {\vec {e}}} . Hubbell introduced the usage of a general e → {\displaystyle {\vec {e}}} . Half a century later, Bonacich and Lloyd defined alpha centrality as: x → = ( I − α A T ) − 1 e → {\displaystyle {\vec {x}}=(I-\alpha A^{T})^{-1}{\vec {e}}\,} which is essentially identical to Katz centrality. More precisely, the score of a node j {\displaystyle j} differs exactly by e j {\displaystyle e_{j}} , so if e → {\displaystyle {\vec {e}}} is constant the order induced on the nodes is identical. == Applications == Katz centrality can be used to compute centrality in directed networks such as citation networks and the World Wide Web. Katz centrality is more suitable in the analysis of directed acyclic graphs where traditionally used measures like eigenvector centrality are rendered useless. Katz centrality can also be used in estimating the relative status or influence of actors in a social network. The work presented in shows the case study of applying a dynamic version of the Katz centrality to data from Twitter and focuses on particular brands which have stable discussion leaders. The application allows for a comparison of the methodology with that of human experts in the field and how the results are in agreement with a panel of social media experts. In neuroscience, it is found that Katz centrality correlates with the relative firing rate of neurons in a neural network. The temporal extension of the Katz centrality is applied to fMRI data obtained from a musical learning experiment in where data is collected from the subjects before and after the learning process. The results show that the changes to the network structure over the musical exposure created in each session a quantification of the cross communicability that produced clusters in line with the success of learning. A generalized form of Katz centrality can be used as an intuitive ranking system for sports teams, such as in college football. Alpha centrality is implemented in igraph library for network analysis and visualization. == References == |
Wikipedia:Alphasyllabic numeral system#0 | Alphasyllabic numeral systems are a type of numeral systems, developed mostly in India starting around 500 AD. Based on various alphasyllabic scripts, in this type of numeral systems glyphs of the numerals are not abstract signs, but syllables of a script, and numerals are represented with these syllable-signs. On the basic principle of these systems, numeric values of the syllables are defined by the consonants and vowels which constitute them, so that consonants and vowels are - or are not in some systems in case of vowels - ordered to numeric values. While there are many hundreds of possible syllables in a script, and since in alphasyllabic numeral systems several syllables receive the same numeric value, so the mapping is not injective. == Alphasyllabaries == The basic principle of the Indian alphasyllabaries is a set of 33 consonant-signs, which are combined with a set of about 20 diacritic marks that indicate vowels of the brahmi scripts, these produce a set of signs for syllables; unmarked consonant-signs denote the syllable with the inherent vowel ’a’. == Indian alphasyllabic numeration == Starting around 500 AD, Indian astronomers and astrologers began to use this new principle for numeration with assigning numeral values to the phonetic signs of various Indian alphasyllabic scripts – the brahmi scripts. Earlier 20th-century scholars supposed that the Indian grammarian Pāṇini used alphasyllabic numerals already in the 7th century BC. Since there is no direct evidence for any alphasyllabic numeration in India until about 510 AD, recently this theory is not supported. These systems, known collectively as varnasankhya systems, were considered to be distinct from other Indian systems – i.e. brahmi or kharosthi numerals - that had abstract numeral-signs. Alike the alphabetic systems of Europe and the Middle East, these systems used phonetic signs of a script for numeration, but they were more flexible than those. Three significant systems of them: Āryabhaṭa numeration, katapayadi system, and the aksharapalli numerals. Alphasyllabic numeration are very important for understanding Indian astronomy, astrology, and numerology, since Indian astronomical texts were written in Sanskrit verse, which had strict metrical form. These systems had the advantage of being able to give any word a numerical value, and to find many words corresponding to one given number. This made possible the construction of various mnemonics to aid scholars and students, and would have served a prosodic function. == Structure == Structure of the Indian alphasyllabic numeration systems differs basically from one another. Though in each of the systems consonants and vowels are ordered to numeric values, thereby each syllable has a numeric value, but on the base of each system's own rules. In various systems the V, CV, CCV syllables receive different values, and the methods, how the numbers are represented by these syllables, are quite different. Āryabhaṭa numeration system operates on the additive principle, so that the number's value, which is represented in it, is computed as the sum of each syllable's numeric value. In his mapping, the consonants are ordered from 1 to 25, then by tens from 30 to 100. Each successive vowel is ordered to the different exponent of 100. In Āryabhaṭa numeration’s the diacritic signs, which mark vowels, multiply the value of the syllable’s consonant by the given power of 100. Direction of his script is right to left, which reflects the order of the Sanskrit lexical numerals. In katapayadi system, syllables have the numeric values only from 0 to 9. To each V, CV and CCV syllable is given a value between 0 and 9. In this way each number between 0 and 9 are ordered to several syllables. Unlike Aryabhata's system, changing the vowel in the syllable doesn’t change the syllable’s numerical value. The number’s value, which is represented in this way, is given as positional number with one syllable on each position. Direction of this script is right to left. In aksharapalli system, syllables were assigned the numerical values 1–9, 10–90, but never as high as 1000. According to S. Chrisomalis there was never a single regular system for correlating signs with numeral values in this system. It was used widely for paginating books, aksharapalli numerals were written in the margins from top to bottom. == Systems == Āryabhaṭa numeration Katapayadi system Aksharapalli Tuubhyara == References == == Sources == Stephen Chrisomalis (2010). Numerical Notation: A Comparati-ve History. Cambridge University Press. ISBN 9780521878180. Retrieved 2019-07-05. Datta, Bibhutibhusan; Singh, Avadhesh Narayan (1962) [1935]. History of Hindu Mathematics. Bombay: Asia Publishing House. Georges Ifrah: The Universal History of Numbers. From Prehistory to the Invention of the Computer. John Wiley & Sons, New York, 2000, ISBN 0-471-39340-1. == See also == Alphabetic numeral system Bhutasamkhya system |
Wikipedia:Alternating multilinear map#0 | In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (for example, a bilinear form or a multilinear form) that is zero whenever any pair of its arguments is equal. This generalizes directly to a module over a commutative ring. The notion of alternatization (or alternatisation) is used to derive an alternating multilinear map from any multilinear map of which all arguments belong to the same space. == Definition == Let R {\displaystyle R} be a commutative ring and V {\displaystyle V} , W {\displaystyle W} be modules over R {\displaystyle R} . A multilinear map of the form f : V n → W {\displaystyle f:V^{n}\to W} is said to be alternating if it satisfies the following equivalent conditions: whenever there exists 1 ≤ i ≤ n − 1 {\textstyle 1\leq i\leq n-1} such that x i = x i + 1 {\displaystyle x_{i}=x_{i+1}} then f ( x 1 , … , x n ) = 0 {\displaystyle f(x_{1},\ldots ,x_{n})=0} . whenever there exists 1 ≤ i ≠ j ≤ n {\textstyle 1\leq i\neq j\leq n} such that x i = x j {\displaystyle x_{i}=x_{j}} then f ( x 1 , … , x n ) = 0 {\displaystyle f(x_{1},\ldots ,x_{n})=0} . == Vector spaces == Let V , W {\displaystyle V,W} be vector spaces over the same field. Then a multilinear map of the form f : V n → W {\displaystyle f:V^{n}\to W} is alternating if it satisfies the following condition: if x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are linearly dependent then f ( x 1 , … , x n ) = 0 {\displaystyle f(x_{1},\ldots ,x_{n})=0} . == Example == In a Lie algebra, the Lie bracket is an alternating bilinear map. The determinant of a matrix is a multilinear alternating map of the rows or columns of the matrix. == Properties == If any component x i {\displaystyle x_{i}} of an alternating multilinear map is replaced by x i + c x j {\displaystyle x_{i}+cx_{j}} for any j ≠ i {\displaystyle j\neq i} and c {\displaystyle c} in the base ring R {\displaystyle R} , then the value of that map is not changed. Every alternating multilinear map is antisymmetric, meaning that f ( … , x i , x i + 1 , … ) = − f ( … , x i + 1 , x i , … ) for any 1 ≤ i ≤ n − 1 , {\displaystyle f(\dots ,x_{i},x_{i+1},\dots )=-f(\dots ,x_{i+1},x_{i},\dots )\quad {\text{ for any }}1\leq i\leq n-1,} or equivalently, f ( x σ ( 1 ) , … , x σ ( n ) ) = ( sgn σ ) f ( x 1 , … , x n ) for any σ ∈ S n , {\displaystyle f(x_{\sigma (1)},\dots ,x_{\sigma (n)})=(\operatorname {sgn} \sigma )f(x_{1},\dots ,x_{n})\quad {\text{ for any }}\sigma \in \mathrm {S} _{n},} where S n {\displaystyle \mathrm {S} _{n}} denotes the permutation group of degree n {\displaystyle n} and sgn σ {\displaystyle \operatorname {sgn} \sigma } is the sign of σ {\displaystyle \sigma } . If n ! {\displaystyle n!} is a unit in the base ring R {\displaystyle R} , then every antisymmetric n {\displaystyle n} -multilinear form is alternating. == Alternatization == Given a multilinear map of the form f : V n → W , {\displaystyle f:V^{n}\to W,} the alternating multilinear map g : V n → W {\displaystyle g:V^{n}\to W} defined by g ( x 1 , … , x n ) := ∑ σ ∈ S n sgn ( σ ) f ( x σ ( 1 ) , … , x σ ( n ) ) {\displaystyle g(x_{1},\ldots ,x_{n})\mathrel {:=} \sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )f(x_{\sigma (1)},\ldots ,x_{\sigma (n)})} is said to be the alternatization of f {\displaystyle f} . Properties The alternatization of an n {\displaystyle n} -multilinear alternating map is n ! {\displaystyle n!} times itself. The alternatization of a symmetric map is zero. The alternatization of a bilinear map is bilinear. Most notably, the alternatization of any cocycle is bilinear. This fact plays a crucial role in identifying the second cohomology group of a lattice with the group of alternating bilinear forms on a lattice. == See also == Alternating algebra Bilinear map Exterior algebra § Alternating multilinear forms Map (mathematics) Multilinear algebra Multilinear map Multilinear form Symmetrization == Notes == == References == Bourbaki, N. (2007). Eléments de mathématique. Vol. Algèbre Chapitres 1 à 3 (reprint ed.). Springer. Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley. Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Vol. 211 (revised 3rd ed.). Springer. ISBN 978-0-387-95385-4. OCLC 48176673. Rotman, Joseph J. (1995). An Introduction to the Theory of Groups. Graduate Texts in Mathematics. Vol. 148 (4th ed.). Springer. ISBN 0-387-94285-8. OCLC 30028913. Tu, Loring W. (2011). An Introduction to Manifolds. Springer-Verlag New York. ISBN 978-1-4419-7400-6. |
Wikipedia:Alternating polynomial#0 | In algebra, an alternating polynomial is a polynomial f ( x 1 , … , x n ) {\displaystyle f(x_{1},\dots ,x_{n})} such that if one switches any two of the variables, the polynomial changes sign: f ( x 1 , … , x j , … , x i , … , x n ) = − f ( x 1 , … , x i , … , x j , … , x n ) . {\displaystyle f(x_{1},\dots ,x_{j},\dots ,x_{i},\dots ,x_{n})=-f(x_{1},\dots ,x_{i},\dots ,x_{j},\dots ,x_{n}).} Equivalently, if one permutes the variables, the polynomial changes in value by the sign of the permutation: f ( x σ ( 1 ) , … , x σ ( n ) ) = s g n ( σ ) f ( x 1 , … , x n ) . {\displaystyle f\left(x_{\sigma (1)},\dots ,x_{\sigma (n)}\right)=\mathrm {sgn} (\sigma )f(x_{1},\dots ,x_{n}).} More generally, a polynomial f ( x 1 , … , x n , y 1 , … , y t ) {\displaystyle f(x_{1},\dots ,x_{n},y_{1},\dots ,y_{t})} is said to be alternating in x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} if it changes sign if one switches any two of the x i {\displaystyle x_{i}} , leaving the y j {\displaystyle y_{j}} fixed. == Relation to symmetric polynomials == Products of symmetric and alternating polynomials (in the same variables x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} ) behave thus: the product of two symmetric polynomials is symmetric, the product of a symmetric polynomial and an alternating polynomial is alternating, and the product of two alternating polynomials is symmetric. This is exactly the addition table for parity, with "symmetric" corresponding to "even" and "alternating" corresponding to "odd". Thus, the direct sum of the spaces of symmetric and alternating polynomials forms a superalgebra (a Z 2 {\displaystyle \mathbf {Z} _{2}} -graded algebra), where the symmetric polynomials are the even part, and the alternating polynomials are the odd part. This grading is unrelated to the grading of polynomials by degree. In particular, alternating polynomials form a module over the algebra of symmetric polynomials (the odd part of a superalgebra is a module over the even part); in fact it is a free module of rank 1, with the Vandermonde polynomial in n variables as generator. If the characteristic of the coefficient ring is 2, there is no difference between the two concepts: the alternating polynomials are precisely the symmetric polynomials. == Vandermonde polynomial == The basic alternating polynomial is the Vandermonde polynomial: v n = ∏ 1 ≤ i < j ≤ n ( x j − x i ) . {\displaystyle v_{n}=\prod _{1\leq i<j\leq n}(x_{j}-x_{i}).} This is clearly alternating, as switching two variables changes the sign of one term and does not change the others. The alternating polynomials are exactly the Vandermonde polynomial times a symmetric polynomial: a = v n ⋅ s {\displaystyle a=v_{n}\cdot s} where s {\displaystyle s} is symmetric. This is because: v n {\displaystyle v_{n}} is a factor of every alternating polynomial: ( x j − x i ) {\displaystyle (x_{j}-x_{i})} is a factor of every alternating polynomial, as if x i = x j {\displaystyle x_{i}=x_{j}} , the polynomial is zero (since switching them does not change the polynomial, we get f ( x 1 , … , x i , … , x j , … , x n ) = f ( x 1 , … , x j , … , x i , … , x n ) = − f ( x 1 , … , x i , … , x j , … , x n ) , {\displaystyle f(x_{1},\dots ,x_{i},\dots ,x_{j},\dots ,x_{n})=f(x_{1},\dots ,x_{j},\dots ,x_{i},\dots ,x_{n})=-f(x_{1},\dots ,x_{i},\dots ,x_{j},\dots ,x_{n}),} so ( x j − x i ) {\displaystyle (x_{j}-x_{i})} is a factor), and thus v n {\displaystyle v_{n}} is a factor. an alternating polynomial times a symmetric polynomial is an alternating polynomial; thus all multiples of v n {\displaystyle v_{n}} are alternating polynomials Conversely, the ratio of two alternating polynomials is a symmetric function, possibly rational (not necessarily a polynomial), though the ratio of an alternating polynomial over the Vandermonde polynomial is a polynomial. Schur polynomials are defined in this way, as an alternating polynomial divided by the Vandermonde polynomial. === Ring structure === Thus, denoting the ring of symmetric polynomials by Λn, the ring of symmetric and alternating polynomials is Λ n [ v n ] {\displaystyle \Lambda _{n}[v_{n}]} , or more precisely Λ n [ v n ] / ⟨ v n 2 − Δ ⟩ {\displaystyle \Lambda _{n}[v_{n}]/\langle v_{n}^{2}-\Delta \rangle } , where Δ = v n 2 {\displaystyle \Delta =v_{n}^{2}} is a symmetric polynomial, the discriminant. That is, the ring of symmetric and alternating polynomials is a quadratic extension of the ring of symmetric polynomials, where one has adjoined a square root of the discriminant. Alternatively, it is: R [ e 1 , … , e n , v n ] / ⟨ v n 2 − Δ ⟩ . {\displaystyle R[e_{1},\dots ,e_{n},v_{n}]/\langle v_{n}^{2}-\Delta \rangle .} If 2 is not invertible, the situation is somewhat different, and one must use a different polynomial W n {\displaystyle W_{n}} , and obtains a different relation; see Romagny. == Representation theory == From the perspective of representation theory, the symmetric and alternating polynomials are subrepresentations of the action of the symmetric group on n letters on the polynomial ring in n variables. (Formally, the symmetric group acts on n letters, and thus acts on derived objects, particularly free objects on n letters, such as the ring of polynomials.) The symmetric group has two 1-dimensional representations: the trivial representation and the sign representation. The symmetric polynomials are the trivial representation, and the alternating polynomials are the sign representation. Formally, the scalar span of any symmetric (resp., alternating) polynomial is a trivial (resp., sign) representation of the symmetric group, and multiplying the polynomials tensors the representations. In characteristic 2, these are not distinct representations, and the analysis is more complicated. If n > 2 {\displaystyle n>2} , there are also other subrepresentations of the action of the symmetric group on the ring of polynomials, as discussed in representation theory of the symmetric group. == Unstable == Alternating polynomials are an unstable phenomenon: the ring of symmetric polynomials in n variables can be obtained from the ring of symmetric polynomials in arbitrarily many variables by evaluating all variables above x n {\displaystyle x_{n}} to zero: symmetric polynomials are thus stable or compatibly defined. However, this is not the case for alternating polynomials, in particular the Vandermonde polynomial. == See also == Symmetric polynomial Euler class == Notes == == References == Giambruno, Antonio; Zaicev, Mikhail (2005). Polynomial Identities and Asymptotic Methods. Vol. 122. American Mathematical Society. ISBN 978-0-8218-3829-7. The fundamental theorem of alternating functions, by Matthieu Romagny, September 15, 2005 |
Wikipedia:Alternativity#0 | In abstract algebra, alternativity is a property of a binary operation. A magma G is said to be left alternative if ( x x ) y = x ( x y ) {\displaystyle (xx)y=x(xy)} for all x , y ∈ G {\displaystyle x,y\in G} and right alternative if y ( x x ) = ( y x ) x {\displaystyle y(xx)=(yx)x} for all x , y ∈ G {\displaystyle x,y\in G} . A magma that is both left and right alternative is said to be alternative (flexible). Any associative magma (that is, a semigroup) is alternative. More generally, a magma in which every pair of elements generates an associative submagma must be alternative. The converse, however, is not true, in contrast to the situation in alternative algebras. == Examples == Examples of alternative algebras include: Any Semigroup is associative and therefore alternative. Moufang loops are alternative and flexible but not associative. See Moufang loop § Examples for more examples. Octonion multiplication is alternative and flexible. More generally Cayley-Dickson algebra over a commutative ring is alternative. == See also == Flexible algebra Power associativity == References == |
Wikipedia:Amanda Chetwynd#0 | Amanda G. Chetwynd is a British mathematician and statistician specializing in combinatorics and spatial statistics. She is Professor of Mathematics and Statistics and Provost for Student Experience, Colleges and the Library at Lancaster University, and a Principal Fellow of the Higher Education Academy. == Education and research == Chetwynd earned a Ph.D. from the Open University in 1985. Her dissertation, Edge-colourings of graphs, was jointly supervised by Anthony Hilton and Robin Wilson. She did postdoctoral research at the Stockholm University before joining Lancaster University. Her research interests include graph theory, edge coloring, and latin squares in combinatorics, as well as geographical clustering in medical statistics. == Recognition and service == In 2003, Chetwynd won a National Teaching Fellowship recognizing her teaching excellence. She was vice president of the London Mathematical Society in 2005, at a time when university study of mathematics was shrinking, and as vice president encouraged the UK government to counter the decline by providing more funds for mathematics education. == Books == With Peter Diggle, Chetwynd is the author of the books Discrete Mathematics (Modular Mathematics series, Arnold, 1995) and Statistics and Scientific Method: An Introduction for Students and Researchers (Oxford University Press, 2011). With Bob Burn she is the author of A Cascade of Numbers: An Introduction to Number Theory (Arnold, 1995). == References == == External links == Home page Amanda Chetwynd publications indexed by Google Scholar |
Wikipedia:Amanda Montejano#0 | Amanda Montejano Cantoral is a Mexican mathematician specializing in combinatorics, and particularly in the application of graph coloring to geometric graphs. She is a professor at the Juriquilla campus of the National Autonomous University of Mexico, in the Multidisciplinary Unit of Teaching and Research of the Faculty of Sciences. == Education and career == Montejano graduated from the National Autonomous University of Mexico in 2004, and earned a doctorate in applied mathematics at the Polytechnic University of Catalonia in Spain in 2009. Her doctoral dissertation, Colored combinatorial structures: homomorphisms and counting, was supervised by Oriol Serra Albó. She was a postdoctoral researcher at the National Autonomous University of Mexico, in the Center for Applied Physics and Advanced Technology, before taking her present position in the Multidisciplinary Unit of Teaching and Research. == Recognition == Montejano is a member of the Mexican Academy of Sciences. == References == == External links == Home page Amanda Montejano publications indexed by Google Scholar |
Wikipedia:Amel Ben Abda#0 | Amel Ben Abda is a professor of mathematics at the National Engineering School of Tunis. She was the first person in Tunisia to earn a PhD in applied mathematics. She is the Tunisian representative of the steering committee of the International Laboratory for Computer Sciences and Applied Mathematics on the advisory board of the Tunisian Woman Mathematician Association. == Early life and education == Ben Abda studied applied mathematics at the National Engineering School of Tunis, graduating in 1988. After her degree, she worked in the Preparatory Institute for Engineering Studies of Nabeul. She went on to earn the first PhD in Applied Mathematics in Tunisia in 1993. == Research and career == In the field of applied mathematics, Ben Abda has worked on a "reciprocity gap" method that can be used as an indication of defects in materials. She also works on the problem of reconstructing boundary conditions from incomplete data. In 1993, Ben Abda joined the Preparatory Institute for Scientific and Technical Studies, where she was promoted to assistant professor that same year. In 1999 she joined the National Engineering School of Tunis. She defended Tunisia's first habilitation in 1998. She is responsible for the inverse theorems team at the Laboratory of Mathematical Modelling and Numeric in Engineering Sciences (LAMSIN). She is the Tunisian representative of the steering committee of the International Laboratory for Computer Sciences and Applied Mathematics. She is on the advisory board of the Tunisian Woman Mathematician Association (TWMA). The TWMA give an annual award for the best PhD thesis in mathematics. In 2018 she was selected as one of OkayAfrica's Top 100 Women. == References == |
Wikipedia:American Mathematical Society#0 | The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. == History == The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the American Journal of Mathematics. The result was the Bulletin of the American Mathematical Society, with Fiske as editor-in-chief. The de facto journal, as intended, was influential in increasing membership. The popularity of the Bulletin soon led to the launches of the Transactions of the American Mathematical Society and Proceedings of the American Mathematical Society, which were also de facto journals. In 1891, Charlotte Scott of Britain became the first woman to join the AMS, then called the New York Mathematical Society. The society reorganized under its present name (American Mathematical Society) and became a national society in 1894, and that year Scott became the first woman on the first Council of the society. In 1927 Anna Pell-Wheeler became the first woman to present a lecture at the society's Colloquium. In 1951 there was a southeastern sectional meeting of the Mathematical Association of America in Nashville. The citation delivered at the 2007 MAA awards presentation, where Lee Lorch received a standing ovation, recorded that: "Lee Lorch, the chair of the mathematics department at Fisk University, and three Black colleagues, Evelyn Boyd (now Granville), Walter Brown, and H. M. Holloway came to the meeting and were able to attend the scientific sessions. However, the organizer for the closing banquet refused to honor the reservations of these four mathematicians. (Letters in Science, August 10, 1951, pp. 161–162 spell out the details). Lorch and his colleagues wrote to the governing bodies of the AMS [American Mathematical Society] and MAA seeking bylaws against discrimination. Bylaws were not changed, but non-discriminatory policies were established and have been strictly observed since then." Also in 1951, the American Mathematical Society's headquarters moved from New York City to Providence, Rhode Island. The society later added an office in Ann Arbor, Michigan in 1965 and an office in Washington, D.C. in 1992. In 1954 the society called for the creation of a new teaching degree, a Doctor of Arts in Mathematics, similar to a PhD but without a research thesis. In the 1970s, as reported in "A Brief History of the Association for Women in Mathematics: The Presidents' Perspectives" by Lenore Blum, "In those years the AMS was governed by what could only be called an 'old boys network,' closed to all but those in the inner circle." Mary W. Gray challenged that situation by "sitting in on the Council meeting in Atlantic City. When she was told she had to leave, she refused saying she would wait until the police came. (Mary relates the story somewhat differently: When she was told she had to leave, she responded she could find no rules in the by-laws restricting attendance at Council meetings. She was then told it was by 'gentlemen's agreement.' Naturally Mary replied 'Well, obviously I'm no gentleman.') After that time, Council meetings were open to observers and the process of democratization of the Society had begun." Also, in 1971 the AMS established its Joint Committee on Women in the Mathematical Sciences (JCW), which later became a joint committee of multiple scholarly societies. Julia Robinson was the first female president of the American Mathematical Society (1983–1984), but was unable to complete her term as she was suffering from leukemia. In 1988, the Journal of the American Mathematical Society was created, as the flagship journal of the AMS. == AMS and mathematical research == The American Mathematical Society plays a significant role in advancing mathematical research by fostering collaboration, supporting early-career researchers, and maintaining influential publications and databases. === Research collaborations and support === The AMS facilitates collaboration among mathematicians through a variety of programs aimed at different career stages. The Mathematical Research Communities, established in 2008, provides early-career researchers with opportunities to engage in intensive research workshops, collaborate with peers, and receive mentoring from senior mathematicians. These programs often lead to the formation of long-term research groups that contribute to emerging fields in mathematics. In addition, the AMS supports Research Experiences for Undergraduates through advocacy and funding partnerships, ensuring that undergraduate students are exposed to high-level mathematical research. The AMS also offers travel grants and fellowships to encourage participation in international conferences and collaborative research projects. == Influence on policy and education == === Advocacy for mathematics funding === The AMS advocates for federal funding for mathematical research. It collaborates with organizations such as the National Science Foundation and the National Academy of Sciences to promote funding initiatives. The AMS is also a member of the Joint Policy Board for Mathematics, which works with policymakers to emphasize the role of mathematics in technological advancements and national security. In partnership with the American Association for the Advancement of Science, the AMS has contributed to discussions on STEM workforce development and the applications of mathematics in areas such as cybersecurity and data science. The society has supported initiatives for stable funding in mathematical research, citing its importance in economic growth and scientific development. == Meetings == The AMS, along with more than a dozen other organizations, holds the largest annual research mathematics meeting in the world, the Joint Mathematics Meeting, in early January. The 2019 Joint Mathematics Meeting in Baltimore drew approximately 6,000 attendees. Each of the four regional sections of the AMS (Central, Eastern, Southeastern, and Western) holds meetings in the spring and fall of each year. The society also co-sponsors meetings with other international mathematical societies. == Fellows == The AMS selects an annual class of Fellows who have made outstanding contributions to the advancement of mathematics. == Publications == The AMS publishes Mathematical Reviews, a database of reviews of mathematical publications, various journals, and books. In 1997 the AMS acquired Chelsea Publishing Company, which it uses as an imprint. In 2017, the AMS acquired MAA Press, the book publishing program of the Mathematical Association of America. The AMS has continued to publish books under the MAA Press imprint. Journals: General Bulletin of the American Mathematical Society — published quarterly Communications of the American Mathematical Society — online only Electronic Research Announcements of the American Mathematical Society — online only Journal of the American Mathematical Society — published quarterly Memoirs of the American Mathematical Society — published six times per year Notices of the American Mathematical Society — published monthly, one of the most widely read mathematical periodicals Proceedings of the American Mathematical Society — published monthly Transactions of the American Mathematical Society — published monthly Subject-specific Conformal Geometry and Dynamics — online only Journal of Algebraic Geometry – published quarterly Mathematics of Computation — published quarterly Mathematical Surveys and Monographs Representation Theory — online only Translation Journals St. Petersburg Mathematical Journal Theory of Probability and Mathematical Statistics Transactions of the Moscow Mathematical Society Sugaku Expositions Proceedings and Collections: Advances in Soviet Mathematics American Mathematical Society Translations AMS/IP Studies in Advanced Mathematics Centre de Recherches Mathématiques (CRM) Proceedings & Lecture Notes Contemporary Mathematics IMACS: Series in Discrete Mathematics and Theoretical Computer Science Fields Institute Communications Proceedings of Symposia in Applied Mathematics Proceedings of Symposia in Pure Mathematics == Prizes == Some prizes are awarded jointly with other mathematical organizations. See specific articles for details. Bôcher Memorial Prize Cole Prize David P. Robbins Prize Fulkerson Prize Leroy P. Steele Prizes Morgan Prize Norbert Wiener Prize in Applied Mathematics Oswald Veblen Prize in Geometry == Outreach == The AMS creates outreach materials aimed at middle school, high school, and college students. These include: Posters about mathematicians and mathematics Mathematical Moments: posters and interviews about applications of math to science and society Math in the Media: a monthly rundown of news articles that mention math, paired with classroom activities on the relevant math concepts. == Typesetting == The AMS was an early advocate of the typesetting program TeX, requiring that contributions be written in it and producing its own packages AMS-TeX and AMS-LaTeX. TeX and LaTeX are now ubiquitous in mathematical publishing. == Presidents == The AMS is led by the president, who is elected for a two-year term, and cannot serve for two consecutive terms. The current president is Ravi Vakil, who took office in February 2025. === 1888–1900 === John Howard Van Amringe (New York Mathematical Society) (1888–1890) Emory McClintock (New York Mathematical Society) (1891–94) George Hill (1895–96) Simon Newcomb (1897–98) Robert Woodward (1899–1900) === 1901–1950 === Eliakim Moore (1901–02) Thomas Fiske (1903–04) William Osgood (1905–06) Henry White (1907–08) Maxime Bôcher (1909–10) Henry Fine (1911–12) Edward Van Vleck (1913–14) Ernest Brown (1915–16) Leonard Dickson (1917–18) Frank Morley (1919–20) Gilbert Bliss (1921–22) Oswald Veblen (1923–24) George Birkhoff (1925–26) Virgil Snyder (1927–28) Earle Raymond Hedrick (1929–30) Luther Eisenhart (1931–32) Arthur Byron Coble (1933–34) Solomon Lefschetz (1935–36) Robert Moore (1937–38) Griffith C. Evans (1939–40) Marston Morse (1941–42) Marshall Stone (1943–44) Theophil Hildebrandt (1945–46) Einar Hille (1947–48) Joseph L. Walsh (1949–50) === 1951–2000 === John von Neumann (1951–52) Gordon Whyburn (1953–54) Raymond Wilder (1955–56) Richard Brauer (1957–58) Edward McShane (1959–60) Deane Montgomery (1961–62) Joseph Doob (1963–64) Abraham Albert (1965–66) Charles B. Morrey Jr. (1967–68) Oscar Zariski (1969–70) Nathan Jacobson (1971–72) Saunders Mac Lane (1973–74) Lipman Bers (1975–76) R. H. Bing (1977–78) Peter Lax (1979–80) Andrew Gleason (1981–82) Julia Robinson (1983–84) Irving Kaplansky (1985–86) George Mostow (1987–88) William Browder (1989–90) Michael Artin (1991–92) Ronald Graham (1993–94) Cathleen Morawetz (1995–96) Arthur Jaffe (1997–98) Felix Browder (1999–2000) === 2001–present === Hyman Bass (2001–02) David Eisenbud (2003–04) James Arthur (2005–06) James Glimm (2007–08) George E. Andrews (2009–10) Eric M. Friedlander (2011–12) David Vogan (2013–14) Robert L. Bryant (2015–16) Ken Ribet (2017–18) Jill Pipher (2019–20) Ruth Charney (2021–22) Bryna Kra (2023–24) Ravi Vakil (2025–present) == Executive directors == The AMS has an executive director who sits at the helm of the organization, steering it, managing its operations, and carrying out its mission according to the strategic direction of the board of trustees. Holbrook MacNeille (1949–1954) John Curtiss (1954–1959) Gordon Walker (1959–1977) William LeVeque (1977–1988) William Jaco (1988–1995) John H. Ewing (1995–2009) Donald McClure (2009–2016) Catherine Roberts (2016–2023) John Meier (2024-) == See also == Canadian Mathematical Society Mathematical Association of America European Mathematical Society London Mathematical Society List of mathematical societies == References == == External links == Official website MacTutor: The American Mathematical Society This article incorporates material from American Mathematical Society on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. |
Wikipedia:Amir Dembo#0 | Amir Dembo (Hebrew: אמיר דמבו; born October 25, 1958, Haifa) is an Israeli-American mathematician, specializing in probability theory. He was elected a member of the National Academy of Sciences in 2022, and of the American Academy of Arts and Sciences in 2023. == Biography == Dembo received his bachelor's degree in electrical engineering in 1980 from the Technion. He obtained in 1986 his doctorate in electrical engineering under the supervision of David Malah with the thesis "Design of Digital FIR Filter Arrays". He joined Stanford University as Assistant Professor of Statistics and Mathematics in 1990, and is currently the Marjorie Mhoon Fair Professor in Quantitative Science there. His research deals with probability theory and stochastic processes, the theory of large deviations, the spectral theory of random matrices, random walks, and interacting particle systems. He was Invited Speaker with the talk Simple random covering, disconnection, late and favorite points at the ICM in Madrid in 2006. Dembo is a fellow of the Institute of Mathematical Statistics. His doctoral students include Scott Sheffield and Jason P. Miller. == Selected publications == === Articles === with Yuval Peres, Jay Rosen and Ofer Zeitouni: Dembo, Amir; Peres, Yuval; Rosen, Jay; Zeitouni, Ofer (2001). "Thick points for planar Brownian motion and the Erdős-Taylor conjecture on random walk". Acta Mathematica. 186 (2): 239–270. doi:10.1007/BF02401841. with Bjorn Poonen, Qi-Man Shao and Ofer Zeitouni: Dembo, Amir; Poonen, Bjorn; Shao, Qi-Man; Zeitouni, Ofer (2002). "Random polynomials having few or no real zeros". J. Amer. Math. Soc. 15 (4): 857–892. arXiv:math/0006113. doi:10.1090/S0894-0347-02-00386-7. with Yuval Peres, Jay Rosen and Ofer Zeitouni: Dembo, Amir; Peres, Yuval; Rosen, Jay; Zeitouni, Ofer (2002). "Thick points for intersections of planar sample paths". Trans. Amer. Math. Soc. 354 (12): 4969–5003. doi:10.1090/S0002-9947-02-03080-5. === Books === with Ofer Zeitouni: Large Deviations Techniques and Applications, Springer, Dembo, Amir; Zeitouni, Ofer (2009). 2nd edition, corrected printing of 1998 edition. ISBN 9783642033117; pbk{{cite book}}: CS1 maint: postscript (link) == Sources == Zhan Shi: Problèmes de recouvrement et points exceptionnels pour la marche aléatoire et le mouvement brownien, d'après Dembo, Peres, Rosen, Zeitouni, Seminaire Bourbaki, No. 951, 2005 == References == == External links == Amir Dembo's home page, Stanford University |
Wikipedia:Amitsur–Levitzki theorem#0 | In algebra, the Amitsur–Levitzki theorem states that the algebra of n × n matrices over a commutative ring satisfies a certain identity of degree 2n. It was proved by Amitsur and Levitsky (1950). In particular matrix rings are polynomial identity rings such that the smallest identity they satisfy has degree exactly 2n. == Statement == The standard polynomial of degree n is S n ( x 1 , … , x n ) = ∑ σ ∈ S n sgn ( σ ) x σ ( 1 ) ⋯ x σ ( n ) {\displaystyle S_{n}(x_{1},\dots ,x_{n})=\sum _{\sigma \in S_{n}}{\text{sgn}}(\sigma )x_{\sigma (1)}\cdots x_{\sigma (n)}} in non-commuting variables x1, ..., xn, where the sum is taken over all n! elements of the symmetric group Sn. The Amitsur–Levitzki theorem states that for n × n matrices A1, ..., A2n whose entries are taken from a commutative ring then S 2 n ( A 1 , … , A 2 n ) = 0. {\displaystyle S_{2n}(A_{1},\dots ,A_{2n})=0.} == Proofs == Amitsur and Levitzki (1950) gave the first proof. Kostant (1958) deduced the Amitsur–Levitzki theorem from the Koszul–Samelson theorem about primitive cohomology of Lie algebras. Swan (1963) and Swan (1969) gave a simple combinatorial proof as follows. By linearity it is enough to prove the theorem when each matrix has only one nonzero entry, which is 1. In this case each matrix can be encoded as a directed edge of a graph with n vertices. So all matrices together give a graph on n vertices with 2n directed edges. The identity holds provided that for any two vertices A and B of the graph, the number of odd Eulerian paths from A to B is the same as the number of even ones. (Here a path is called odd or even depending on whether its edges taken in order give an odd or even permutation of the 2n edges.) Swan showed that this was the case provided the number of edges in the graph is at least 2n, thus proving the Amitsur–Levitzki theorem. Razmyslov (1974) gave a proof related to the Cayley–Hamilton theorem. Rosset (1976) gave a short proof using the exterior algebra of a vector space of dimension 2n. Procesi (2015) gave another proof, showing that the Amitsur–Levitzki theorem is the Cayley–Hamilton identity for the generic Grassman matrix. == References == Amitsur, A. S.; Levitzki, Jakob (1950), "Minimal identities for algebras" (PDF), Proceedings of the American Mathematical Society, 1 (4): 449–463, doi:10.1090/S0002-9939-1950-0036751-9, ISSN 0002-9939, JSTOR 2032312, MR 0036751 Amitsur, A. S.; Levitzki, Jakob (1951), "Remarks on Minimal identities for algebras" (PDF), Proceedings of the American Mathematical Society, 2 (2): 320–327, doi:10.2307/2032509, ISSN 0002-9939, JSTOR 2032509 Formanek, E. (2001) [1994], "Amitsur–Levitzki theorem", Encyclopedia of Mathematics, EMS Press Formanek, Edward (1991), The polynomial identities and invariants of n×n matrices, Regional Conference Series in Mathematics, vol. 78, Providence, RI: American Mathematical Society, ISBN 0-8218-0730-7, Zbl 0714.16001 Kostant, Bertram (1958), "A theorem of Frobenius, a theorem of Amitsur–Levitski and cohomology theory", J. Math. Mech., 7 (2): 237–264, doi:10.1512/iumj.1958.7.07019, MR 0092755 Razmyslov, Ju. P. (1974), "Identities with trace in full matrix algebras over a field of characteristic zero", Mathematics of the USSR-Izvestiya, 8 (4): 727, doi:10.1070/IM1974v008n04ABEH002126, ISSN 0373-2436, MR 0506414 Rosset, Shmuel (1976), "A new proof of the Amitsur–Levitski identity", Israel Journal of Mathematics, 23 (2): 187–188, doi:10.1007/BF02756797, ISSN 0021-2172, MR 0401804, S2CID 121625182 Swan, Richard G. (1963), "An application of graph theory to algebra" (PDF), Proceedings of the American Mathematical Society, 14 (3): 367–373, doi:10.2307/2033801, ISSN 0002-9939, JSTOR 2033801, MR 0149468 Swan, Richard G. (1969), "Correction to "An application of graph theory to algebra"" (PDF), Proceedings of the American Mathematical Society, 21 (2): 379–380, doi:10.2307/2037008, ISSN 0002-9939, JSTOR 2037008, MR 0255439 Procesi, Claudio (2015), "On the theorem of Amitsur—Levitzki", Israel Journal of Mathematics, 207: 151–154, arXiv:1308.2421, Bibcode:2013arXiv1308.2421P, doi:10.1007/s11856-014-1118-8 |
Wikipedia:Amplitwist#0 | In mathematics, the amplitwist is a concept created by Tristan Needham in the book Visual Complex Analysis (1997) to represent the derivative of a complex function visually. == Definition == The amplitwist associated with a given function is its derivative in the complex plane. More formally, it is a complex number z {\displaystyle z} such that in an infinitesimally small neighborhood of a point a {\displaystyle a} in the complex plane, f ( ξ ) = z ξ {\displaystyle f(\xi )=z\xi } for an infinitesimally small vector ξ {\displaystyle \xi } . The complex number z {\displaystyle z} is defined to be the derivative of f {\displaystyle f} at a {\displaystyle a} . == Uses == The concept of an amplitwist is used primarily in complex analysis to offer a way of visualizing the derivative of a complex-valued function as a local amplification and twist of vectors at a point in the complex plane. == Examples == Define the function f ( z ) = z 3 {\displaystyle f(z)=z^{3}} . Consider the derivative of the function at the point e i π 4 {\displaystyle e^{i{\frac {\pi }{4}}}} . Since the derivative of f ( z ) {\displaystyle f(z)} is 3 z 2 {\displaystyle 3z^{2}} , we can say that for an infinitesimal vector γ {\displaystyle \gamma } at e i π 4 {\displaystyle e^{i{\frac {\pi }{4}}}} , f ( γ ) = 3 ( e i π 4 ) 2 γ = 3 e i π 2 γ {\displaystyle f(\gamma )=3(e^{i{\frac {\pi }{4}}})^{2}\gamma =3e^{i{\frac {\pi }{2}}}\gamma } . == References == |
Wikipedia:Amy Roth McDuffie#0 | Amy Roth McDuffie is an American scholar of mathematics education and a professor in the College of Education at Washington State University. == Education and career == McDuffie majored in mathematics at Franklin & Marshall College, graduating in 1987. Next, she studied education at Johns Hopkins University, receiving a secondary mathematics teaching certificate in 1989 and a master's degree in technology for education in 1991. She completed a Ph.D. in mathematics education at the University of Maryland, College Park in 1998. She joined Washington State University Tri-Cities as an instructor in 1998, became a tenure-track assistant professor in 2000, was tenured as an associate professor in 2006, and was promoted to full professor in 2013. In 2017 she moved to the main (Pullman) campus of Washington State University. From 2015 to 2019, she was associate dean for research and external funding for the university's College of Education. In 2021 she began a two-year term on the Mathematics Standing Committee of the National Assessment of Educational Progress. == Books == McDuffie is the coeditor of edited volumes including: Using Research to Improve Instruction (2014) Mathematical Modeling and Modeling Mathematics (2016) Transforming Mathematics Teacher Education: An Equity-Based Approach (2019) == Recognition == In Spring 2025 Washington State University gave McDuffie their Chosen Coug award. == References == == External links == Home page Amy Roth McDuffie publications indexed by Google Scholar |
Wikipedia:Amèle El Mahdi#0 | Amèle El Mahdi, born in 1956 in Blida, is an Algerian professor of mathematics and writer. She lived in many of the cities in southern Algeria, which inspired many of her writings. She has written for the Algerian newspaper El Watan. == Literary works == The Beauty and the Poet, Algiers, Casbah Editions, 2012, 187 p. Yamsel, son of the Ahaggar, Algiers, Casbah Editions, 2014, 275 p. Tin Hinan, My Queen, Algiers, Casbah Editions, 2014, 141 p. Grandma's Beautiful Stories, Algiers, Casbah Editions, 2015 Under the flag of Raïs, Algiers, Casbah Editions, 2016 An African odyssey. The tragedy of illegal migration, Algiers, Casbah Editions, 2018, 172 p. A collective work entitled Hiziya My Love with texts by 14 novelists and poets (Amar Achour, Nassira Belloula, Maïssa Bey, Aicha Bouabaci, Slimane Djouadi, Saléha Imekraz, Abdelmadjid Kaouah, Azzedine Menasra, Miloud Khaizar, Fouzia Laradi, Amèle El Mahdi, Arezki Metref, Lazhari Labter, Smail Yabrir), Hibr Editions. Oasis: yesterday's images, today's views; a collective work supported by the publication program of the French Institute of Algeria. Chihab Editions, 2018 == References == |
Wikipedia:Ana Carpio#0 | Ana María Carpio Rodríguez is a Spanish applied mathematician whose research has included inverse problems, the propagation of dislocations in crystals, fluid dynamics, reaction–diffusion systems, and cancer metastasis. She is a professor of applied mathematics at the Complutense University of Madrid. == Education and career == Carpio studied mathematics and numerical analysis at the University of the Basque Country, earning bachelor's and master's degrees in 1988. Next, she went to Pierre and Marie Curie University in France, where she earned a Diplome d'Etudes Approfondies in 1989. She continued there for a Ph.D. in 1993, directed by Alain Haraux. Her doctoral dissertation there was Etude de quelques problèmes d'équations aux dérivées partielles nonlinéaires, and concerned partial differential equations. In the same year, the Complutense University of Madrid recognized her with a Ph.D., for the same dissertation (in Spanish), listing Enrique Zuazua as her advisor. She completed a Spanish habilitation in 2004. She was appointed as an assistant professor at the Complutense University of Madrid in 1992, associate professor in 1994, and full professor in 2006. From 1996 to 1997 she traveled to the University of Oxford for postdoctoral research at the Oxford Centre for Industrial and Applied Mathematics. Since 2007 she has also been affiliated with the Gregorio Millán Barbany University Institute for Modelling and Simulation in Fluodynamics, Nanoscience and Industrial Mathematics of Charles III University of Madrid. == Recognition == Carpio was the inaugural winner of the SEMA Young Researcher Award of the Spanish Society of Applied Mathematics, in 1998. == References == == External links == Home page Ana Carpio publications indexed by Google Scholar |
Wikipedia:Analysis and Applications#0 | The Journal of Mathematical Analysis and Applications is an academic journal in mathematics, specializing in mathematical analysis and related topics in applied mathematics. It was founded in 1960 by Richard Bellman, as part of a series of new journals on areas of mathematics published by Academic Press, and is now published by Elsevier. For most years since 2003 it has been ranked by SCImago Journal Rank as among the top 25% of journals in its topic areas. == References == |
Wikipedia:Analysis on fractals#0 | Analysis on fractals or calculus on fractals is a generalization of calculus on smooth manifolds to calculus on fractals. The theory describes dynamical phenomena which occur on objects modelled by fractals. It studies questions such as "how does heat diffuse in a fractal?" and "How does a fractal vibrate?" In the smooth case the operator that occurs most often in the equations modelling these questions is the Laplacian, so the starting point for the theory of analysis on fractals is to define a Laplacian on fractals. This turns out not to be a full differential operator in the usual sense but has many of the desired properties. There are a number of approaches to defining the Laplacian: probabilistic, analytical or measure theoretic. == See also == Time scale calculus for dynamic equations on a cantor set. Differential geometry Discrete differential geometry == References == Christoph Bandt; Siegfried Graf; Martina Zähle (2000). Fractal Geometry and Stochastics II. Birkhäuser. ISBN 978-3-7643-6215-7. Jun Kigami (2001). Analysis on Fractals. Cambridge University Press. ISBN 978-0-521-79321-6. Robert S. Strichartz (2006). Differential Equations on Fractals. Princeton. ISBN 978-0-691-12542-8. Pavel Exner; Jonathan P. Keating; Peter Kuchment; Toshikazu Sunada & Alexander Teplyaev (2008). Analysis on graphs and its applications: Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, January 8-June 29, 2007. AMS Bookstore. ISBN 978-0-8218-4471-7. == External links == Analysis on Fractals, Robert S. Strichartz - Article in Notices of the AMS University of Connecticut - Analysis on fractals Research projects Calculus on fractal subsets of real line - I: formulation |
Wikipedia:Anand Pillay#0 | Anand Pillay (born 7 May 1951) is a British mathematician and logician working in model theory and its applications in algebra and number theory. == Biography == Pillay studied as an undergraduate at the University of Oxford, obtaining a Bachelor in Mathematics and Philosophy in 1973 at Balliol College. At the University of London, he received his master's degree in mathematics in 1974 and his PhD in 1978 with Wilfrid Hodges at Bedford College, titled Gaifman Operations, Minimal Models, and the Number of Countable Models. In 1978, he was a Royal Society Fellow and visiting scientist at CNRS at Paris Diderot University. After teaching at the University of Manchester starting in 1981 and at McGill University in Canada, he joined the University of Notre Dame as an assistant professor in 1983, where he became an associate professor in 1986 and a full professor in 1988. From 1996 to 2006, he was Swanlund Professor at the University of Illinois Urbana-Champaign, where he is now Professor Emeritus. Since 2005, he has been the Chair of Mathematical Logic at the University of Leeds. He also held positions as a visiting scholar at the Fields Institute in Toronto, at the Mathematical Sciences Research Institute in Berkeley, and at the Isaac Newton Institute in Cambridge. == Career == Pillay's dissertation work concerned the number of countable models of countable theories; under the influence of the Paris school of model theory, he also worked on stability theory. Later, he dealt with applications of model theory in other areas of mathematics, including Nash manifolds and groups, algebraic theory of differential equations and differential algebra, classification of compact complex manifolds, and diophantine geometry. Pillay was an invited speaker at the International Congress of Mathematicians in Zürich in 1994. In 2009 he was invited to present the Tarski Lectures, titled Compact Spaces, Definability, and Measures, in Model Theory. His three lectures were titled "The Logic Topology", "Lie Groups from Nonstandard Models", and "Measures and Domination". In 2001, he received the Humboldt Foundation's research award, and was also a Humboldt Fellow at the University of Kiel in 1988 and at the University of Freiburg in 1992. In 2011, he gave the Gödel Lecture. He is a Fellow of the American Mathematical Society. == Selected works == An introduction to stability theory (Oxford Logic Guides 8). Clarendon Press, Oxford 1983, ISBN 0-19-853186-9. Geometric Stability Theory (Oxford Logic Guides 32). Clarendon Press, Oxford 1996, ISBN 0-19-853437-X. with David Marker and Margit Messmer: Model theory of fields (Lecture Notes in Logic 5). Springer, Berlin. 1996, ISBN 3-540-60741-2. Model Theory and Diophantine Geometry. In: Bulletin of the American Mathematical Society. Vol. 34, No. 4, 1997, pp. 405–422, doi:10.1090/S0273-0979-97-00730-1. Model Theory. In: Notices of the American Mathematical Society. Vol. 47, No. 11, 2000, pp. 1373–1381. with Deidre Haskell and Charles Steinhorn (ed.): Model theory, algebra and geometry (Mathematical Sciences Research Institute Publications 39). Cambridge University Press, Cambridge, 2000, ISBN 0-521-78068-3. == References == == External links == Homepage at University of Notre Dame Short Biography |
Wikipedia:Anania Shirakatsi#0 | Anania Shirakatsi (Old Armenian: Անանիա Շիրակացի, Anania Širakac’i, anglicized: Ananias of Shirak) was a 7th-century Armenian polymath and natural philosopher, author of extant works covering mathematics, astronomy, geography, chronology, and other fields. Little is known for certain of his life outside of his own writings, but he is considered the father of the exact and natural sciences in Armenia—the first Armenian mathematician, astronomer, and cosmographer. A part of the Armenian Hellenizing School and one of the few secular scholars in medieval Armenia, Anania was educated primarily by Tychicus, in Trebizond. He composed science textbooks and the first known geographic work in classical Armenian (Ashkharhatsuyts), which provides detailed information about Greater Armenia, Persia and the Caucasus (Georgia and Caucasian Albania). In mathematics, his accomplishments include the earliest known table of results of the four basic operations, the earliest known collection of recreational math puzzles and problems, and the earliest book of math problems in Armenian. He also devised a system of mathematical notation based on the Armenian alphabet, although he was the only writer known to have used it. == Name == His name is usually anglicized as Ananias of Shirak (Širak). Anania is the Armenian variant of the biblical name Ananias, itself the Greek version of the Hebrew Hananiah. The second part of his name denotes his place of origin, the region of Shirak, though it may have become a sort of surname. In some manuscripts, he is called Shirakuni (Շիրակունի) and Shirakavants’i (Շիրակաւանցի). == Life == === Background === Anania Shirakatsi lived in the 7th century. The dates of his birth and death have not been definitively established. Robert H. Hewsen noted in 1968 that Anania is widely believed to have been born between 595 and 600; a quarter-century later he settled on c. 610 as a birthdate and 685 as the year he died. Agop Jack Hacikyan et al. place his birth in early 600s but agree on 685. Sen Arevshatyan, James R. Russell, Edward G. Mathews, and Theo van Lint also concur with 610–685, while Greenwood suggests c. 600–670. Vardanyan places his death in the early 690s. Anania is the only classical Armenian scholar to have written an autobiography. It is a brief text, characterized as "somewhat self-congratulatory" and "more a statement of academic pedigree" than autobiography. It was probably written as the preface to one of his scholarly works, possibly the K’nnikon. He was the son of Yovhannes and was born in the village of Anania/Aneank’ (Անեանք) or in the town of Shirakavan (Yerazgavors), in the canton of Shirak, in the central Armenian province of Ayrarat. Aneank' may be connected to the later city of Ani, the Bagratid Armenian capital. Anania probably came from a noble family. Since his name is sometimes spelled as Shirakuni (Շիրակունի), Hewsen argued that he may have belonged to the house of the Kamsarakan or Arsharuni princes of Shirak and Arsharunik’, respectively. Greenwood suggests that it is more likely that Anania came from the lesser nobility in Shirak, who served the house of Kamsarakan. Broutian describes his father as a "minor Armenian nobleman." Vardanyan believes he either came from the Kamsarakan family or that they were his patrons. Anania is traditionally thought to have been buried in the village of Anavank’; however, the tradition probably originated from the name of the village. === Education === Anania received his early education at the local Armenian schools, possibly at Dprevank’ monastery, where he studied sacred texts and earlier Armenian authors. Due to the lack of teachers and books in Armenia, he decided to travel to the Byzantine Empire to study mathematics. After first traveling to Theodosiopolis, then to the Byzantine-controlled province of Fourth Armenia (probably Martyropolis), where he studied under the mathematician Christosatur for six months. He then left to find a better teacher and learned about Tychicus, who was based at the monastery (or martyrium) of Saint Eugenios in Trebizond. Redgate placed this in the 620s. Anania devoted a significant part of his autobiography to Tychicus (born c. 560), with whom he spent eight years in the 620s or 630s. Tychicus had studied the Armenian language and its literature while serving in the Byzantine army in Armenia. Wounded by the Persians, he retired from the military and later studied in Alexandria, Rome, and Constantinople. Tychicus later returned to his native Trebizond, where he established a school c. 615. Tychicus taught many students from Constantinople (including from the imperial court) and was renowned among Byzantine kings. He provided Anania special attention and taught him what Anania called a "perfect knowledge of mathematics". In Tychicus's vast library, Anania found "everything, exoteric and esoteric", including sacred and secular Greek authors, including works on the sciences, medicine, chronology, and history. Russell believed his library may have included Pythagorean and alchemical books. Anania considered Tychicus to have been "predestined by God for the introduction of science into Armenia." === Educator and scientist === Anania himself established a school in Armenia upon his return. That school, the first in Armenia to teach quadrivium, is presumed to have been located in his native Shirak. He was disappointed with the laziness of his students and their departure after learning the basics. Anania complained about Armenians' lack of interest in mathematics, writing that they "love neither learning, nor knowledge." Nicholas Adontz considered it an exaggeration, "if not an absolute slander, to deny the Armenian innate love of investigation." The 12th-century chronicler Samuel of Ani listed five of Shirakatsi's students, who are otherwise unknown. Anania financed his research in several fields with the money he earned teaching. == Relationship with the church == Thomson wrote that as a lay scholar, Anania was a "rarity in early Armenia." Hewsen termed him the only lay classical Armenian author besides Grigor Magistros, adding that he had a close relationship with the Armenian Church. Hacikyan et al. describe Anania as a "devout Christian and well versed in the Bible" who "made some attempts to reconcile science and Scripture." In his later years, Anania may have been a monk in the Armenian Church. This is based on his religious discourses and attempts to date the feasts of the church. John A. C. Greppin doubts that Anania was ever in any religious order. Several scholars consider him a church ideologist akin to Cosmas Indicopleustes, whom he actually criticized. Hewsen noted that some of Anania's "more revolutionary ideas" were suppressed by the Armenian Church after his death. Greppin suggested that Anania, a largely secular author, had fallen into a "bad clerical odor." S. Peter Cowe disagreed with Ashot G. Abrahamian's hypothesis that his name was "censored in the Middle Ages because of ecclesiastical disapproval" and argued that it is "more applicable to Soviet practice than that of the relatively tolerant Armenian and other eastern churches." Soviet historians represented him as a founder of irreligious and anti-clerical thought in Armenia, who pioneered double-truth theory. Vazgen Chaloyan called him a "progressive representative of the feudal period of Armenian science." Gevorg Khrlopian went as far as to argue that Anania was an enemy of the Armenian Church and fought against its obscurantism. Hewsen opposed this view, suggesting that, instead, he was an "independent thinker of sorts." == Philosophy == Anania is considered by modern scholars to be a representative of the Hellenizing School since many of his works were based on classical Greek sources. He was the first Armenian scholar to have "imported a set of scientific notions, and examples of their applications, from the Greek-speaking schools" into Armenia. He was well versed in Greek literature, and the influence of Greek syntax is evident in his works. Anania was also knowledgeable about native Armenian and Iranian cultural traditions; several of his works provide important information on late Sassanian Iran. James R. Russell describes him as an alchemist and a Pythagorean who "does not usually rely on mythology to explain natural phenomena." Anania accepted the importance of experience, observation, rational practice and theory, and was influenced by the ideas of the 5th-century Neoplatonist philosopher Davit Anhaght (the Invincible), and Greek philosophers Thales of Miletus, Hippocrates, Democritus, Plato, Aristotle, Zeno of Citium, Epicurus, Ptolemy, Pappus of Alexandria, and Cosmas Indicopleustes. Aristotle's On the Heavens had a significant influence on Anania's thought. According to Gevorg Khrlopian, Anania was heavily influenced by Yeghishe's An Interpretation of Creation, the anonymous Interpretation of the Categories of Aristotle, and the works of Davit Anhaght, who had established Neoplatonism in Armenian thought. Anania was also the first Armenian scholar to quote Philo of Alexandria by name. Anania was the last known lay scholar in Christian Armenia until Grigor Magistros Pahlavuni in the 11th century. He advocated rationalism in studying nature and attacked superstitious beliefs and astrology as the "babblings of the foolish." He adopted the classical theory of four elements, which considered all matter to be composed of four elements: fire, air, water, and earth. He believed that while God directly created these elements, He did not interfere with the "natural course of the development of things." He asserted that the creation, existence, and decay of natural bodies and phenomena occurred through the union of these elements—without the interference of God. Both living and non-living matter came into existence from a synthesis of the four elements. Anania accepted that the Earth is round, describing it as "like an egg with a spherical yolk (the globe) surrounded by a layer of white (the atmosphere) and covered with a hard shell (the sky)." He accurately explained solar and lunar eclipses, the phases of the Moon, and the structure of the Milky Way, describing the latter as a "mass of dense but faintly luminous stars." Anania also correctly attributed tides to the influence of the Moon. He described the topmost sphere as the aether (arp’i), the source of light and heat (through the Sun). == Works == Anania was a polymath and natural philosopher. About 40 works in various disciplines have been attributed to Anania, but only half are extant. They include studies and translations in mathematics, astronomy, cosmology, geography, chronology, and meteorology. Many of his works are believed to have been part of the K’nnikon (Քննիկոն, from "canon", Greek: Kanonikon), completed circa 666, and used as the standard science textbook in medieval Armenia. Artashes Matevosyan termed it the first secular Armenian science textbook, while Valentina Calzolari described it as a "monumental encyclopedic work." Greenwood argued that the K’nnikon was a "fluid compilation, whose contents fluctuated over time, reflecting the interests and resources of different teachers and practitioners." Modern scholars have praised Anania's writing as concise, simple, and to the point, retaining the reader's attention and citing examples to illustrate his point. === Mathematical === Anania was primarily devoted to mathematics, which he considered the "mother of all knowledge." His mathematical books were used as textbooks in Armenia. Of Anania's several mathematical works, the most important is the book of arithmetic (Hamaroghut’iun, Համարողութիւն; or T’vabanut’iun, Թւաբանութիւն), a comprehensive collection of tables on the four basic operations. It is the earliest extant known work of its kind. The operations reach up to a total of 80 million, which is the highest number. A possible theoretical part is believed lost. Problems and Solutions (alternatively translated as On Questions and Answers), a collection of 24 arithmetical problems and their solutions, is based on the application of fractions; it is the earliest such work in Armenian. Many of its problems allude to real-world situations: six connect to the princely house of Shirak, the Kamsarakans, and at least three to Iran. Greenwood calls the problems "a rich source for seventh-century history whose value has not been sufficiently recognized." The third work, probably an appendix of the book of arithmetic, is titled Xraxč̣anakank’ (Խրախճանականք), literally "things for festive occasions". It has been translated into English as Mathematical Pastimes, Fun with Arithmetic or Problems for Amusement. It also contains 24 problems "intended for mathematical entertainment in social gatherings." According to Mathews this may be the oldest extant text of its kind. ==== Numerical notation ==== For his mathematical works, Anania developed a unique numerical notation based on 12 letters of the Armenian alphabet. For the units, he used the first nine letters of the Armenian script (Ա, Բ, Գ, Դ, Ե, Զ, Է, Ը, Թ), similar to the standard traditional Armenian numerical system. The letters used for 10, 100, and 1000 were also identical to the traditional Armenian system (Ժ, Ճ, Ռ), but all other numbers up to 10,000 were written using these 12 letters. For instance, 50 would be written ԵԺ (5×10) and not Ծ as in the standard system. Thus, the notation is multiplicative-additive as opposed to the ciphered-additive standard system and requires knowing 12 letters, instead of 36, to write numbers less than 10,000. Numbers greater than that could be written using multiplicative combinations of just 2 or 3 signs, but using all 36 letters. Stephen Chrisomalis believes this system was created by Anania since it only occurs in his works and is not found in Greek, Syriac, Hebrew, or any other alphabetic numeral system. Allen Shaw has argued it was just a variant of the Armenian numerals designed specifically for the representation of large numbers. No other writer used it. === Astronomical === One of Anania's most significant works is the Cosmology (Տիեզերագիտութիւն, Tiezeragitut’iun). Abrahamian's version is composed of ten chapters, with an introduction titled "In the Fulfillment of a Promise", implying a patron. It covers the sun, the moon, celestial spheres, constellations, the Milky Way, and meteorological changes. Works used for the parts of the Cosmology include the Bible (mostly the Pentateuch and Psalms) and works by the Church Fathers. Anania cites the work of Basil of Caesarea, Gregory the Illuminator, and Amphilochius (perhaps, of Iconium). Some chapters of the work, such as "On Clouds" (also called "On the Sky" or "Concerning the Skies"), are largely based on Basil's Hexameron. Anania also repeats the classical Greek notions in the fields of astronomy, physics or meteorology. Pambakian wrote about the significance of the Cosmology: Another of Anania's astronomical works, Tables of the Motions of the Moon (Խորանք ընթացիք լուսոյ, Xorank’ ënt’ac’ik’ lusoy), is based on the works of Meton of Athens and his own observations. ==== Perpetual calendar ==== In 667 Anania was invited by Catholicos Anastas I of Akori (r. 661/2–667) to the Armenian Church's central seat at Dvin to establish a fixed calendar of the movable and immovable feasts of the Armenian Church. The result was a perpetual calendar based on a 532-year cycle (ՇԼԲ բոլորակ), combining the solar cycle and the lunar cycle since they coincide every 532 years. It was first proposed by Victorius of Aquitaine in 457 and adopted by the Church of Alexandria. Anania's calendar was never implemented by the Armenian Church; Hovhannes Draskhanakerttsi believes that Anastas's death prevented a church council from ratifying it. Krikor H. Maksoudian described the endeavor a "serious attempt", which was not adopted due to circumstances. === Geographical === The Ashkharhatsuyts (Classical Armenian: Աշխարհացոյց, Ašxarhac’oyc’, lit. "showing the world") is an anonymously published world map, believed to have been written sometime between 610 and 636, "probably shortly before AD 636", i.e. the Muslim Arab invasions and conquests. It lacks a drawn map, but contains "earliest surviving detailed description of Armenian lands." Its authorship has been disputed in the modern period; formerly believed to have been the work of Movses Khorenatsi, most scholars now attribute it to Anania. Hewsen calls it "one of the most valuable works to come down to us from Armenian antiquity." The Armenian Geography—as it is alternatively known—has been especially important for research into the history and geography of Greater Armenia, the Caucasus (Georgia and Caucasian Albania) and the Sasanian Empire, which are all described in detail. The information on Armenia is not found elsewhere in historical sources, as it is the only known Armenian geographical work prior to the 13th century. Cowe described it as "the only one of its kind in Armenia until the introduction of modern cartography in the seventeenth century." Christina Maranci described it as the "earliest surviving description of the entire known world in the Armenian language." The Ashkharhatsuyts has survived in long and short recensions. According to the scholarly consensus, the long recension was the original. For the description of Europe, North Africa and Asia (all the known world from Spain to China), it largely uses Greek sources, namely the now lost geography of Pappus of Alexandria (4th century), which in turn, is based on the Geography of Ptolemy (2nd century). According to Hewsen, it is the "last work based on ancient geographical knowledge written before the Renaissance." Edmond Schütz called it an "outstanding work of medieval sciences, a rich post-Ptolemaid heredity." It was one of the earliest secular Armenian works to be published (in 1668 by Voskan Yerevantsi). It has been translated into four languages: English, Latin (both 1736), French (1819), and Russian (1877). In 1877, Kerovbe Patkanian first attributed it to Anania as the most probable author. Another geographical work of Anania, The Itinerary (Մղոնաչափք, Mghonach’ap’k’), may have been a part of the Ashkharhatsuyts. It presents six routes from Dvin, Armenia's capital at that time, to the major settlements in different directions, with distances in miles (մղոն, mghon), referring to the Arabic mile of 1,917.6 metres (6,291 ft), according to Hakob Manandian. === Chronology === Anania's major chronological work, the Chronicle, listed important events in order of their occurrence. Written between 686 and 690, it is composed of two parts: a universal chronicle, utilizing the lost works of Annianus of Alexandria and the lost Roman imperial sequence from Eusebius's Chronographia, and an ecclesiastical history from a miaphysite perspective, which records the six ecumenical councils. Another chronological work, known as the Calendar (Tomar), included texts and tables about the calendars of 15 peoples: Armenians, Hebrews, Arabs, Macedonians, Romans, Syrians, Greeks, Egyptians, Ethiopians, Athenians, Bythanians, Cappadocians, Georgians, Caucasian Albanians, and Persians. The calendars of the Armenians, Romans, Hebrews, Syrians, Greeks, and Egyptians contain texts, while those of other peoples only have the names of months and their length. === Other === Anania wrote several books on weights and measures. He extensively used the work of Epiphanius of Salamis to present the system of weights used by the Greeks, Jews, and Syrians, and his own knowledge as well as other sources for those of the Armenians and Persians. Anania wrote several works on precious stones, music, and the known languages of the world. Anania's discourses on Christmas/Epiphany and Easter are discussions on the dates of the two feasts. In the first, he uses a lost work he ascribes to Polycarp of Smyrna and insists that the Armenian custom of celebrating Christmas and the Epiphany on the same date is truer to the holidays' intent than celebrating them separately as is common elsewhere in the Christian world. == Traditions and legends == Anania also wrote on herbal medicine, though none of his medical writings have survived. He is traditionally credited with the discovery of the miraculous flower called hamasp’iwṙ or hamasp’iur (համասփիւռ). One 16th century manuscript mentions that he dealt with its therapeutic properties. It has been identified by modern scholars as Silene latifolia (white campion). He is credited with discovering the plant in Dzoghakert (near modern Taşburun, Iğdır, Turkey) and using it medically. The authorship of the "Book of the Six Thousand" (Vec‘ hazareak), the "most important Armenian magical text of the Middle Ages", has traditionally been attributed to Anania. According to a later legend, Anania taught alchemy to the king of Venice. == Influence in the Middle Ages == Anania laid the foundations of the exact sciences in Armenia and greatly influenced many Armenian scholars who came after him. Hovhannes Imastaser (Sarkavag) and other medieval scholars extensively cited and incorporated Anania's works. In a 1037 letter, Grigor Magistros, a scholar from the Pahlavuni noble family, asked Catholicos Petros Getadardz for Anania's manuscripts of his K’nnikon, which were locked up at the catholicosate for centuries. Grigor used these as a textbook at his school at Sanahin Monastery. Anania may have also influenced Byzantine scholars of Armenian origin, namely Leo the Mathematician in the 9th century and Nicholas Artabasdos Rhabdas in the 14th century. == Reemergence in the modern period == In the printed age, passing references to Anania were made as early as 1742 by Paghtasar Dpir, but it was not until the latter half of the 19th century that Anania and his work became a subject of scholarly study. In 1877 Kerovbe Patkanian published a collection of Anania's works in the original classical Armenian at St Petersburg University. Titled Sundry Studies (Մնացորդք բանից, Mnats’ordk’ banits’), it is the first-ever print publication of his works. Galust Ter-Mkrtchian published a number of Anania's works in 1896. The Russian Academy of Sciences published Joseph Orbeli's Russian translation of Anania's Problems and Solutions in 1918. Systematic study and publication of his works began in the Soviet period. Ashot G. Abrahamian, who began his research at the Matenadaran in the 1930s, first published one of Anania's arithmetical texts in 1939, followed by a complete compilation of Anania's work in 1944. One critic objected to his compilation for attributing disputed works to Anania. Abrahamian and Garegin Petrosian published an updated edition in 1979. Some criticism persisted: Varag Arakelian noted a number of errors in translations from classical Armenian and concluded that a new translation of his works was needed. Another Soviet scholar, Suren T. Eremian, studied the Geography. He insisted on Anania's authorship and published his research in 1963. The first translation of Anania's work into a European language was done by the British Orientalist Frederick Cornwallis Conybeare, who translated into English Anania's On Christmas, in 1896, and On Easter and Anania's autobiography, in 1897. Lemerle noted that Conybeare translated Anania's autobiography from a Russian translation, and it contains numerous serious errors. Renewed interest in Anania's work emerged in the West since the 1950s, with a series of English articles in the Armenian Review. A French translation of his autobiography appeared in 1964 by Haïg Berbérian. Robert H. Hewsen authored an introductory article on Anania's life and scholarship in 1968. Hewsen dedicated his monumental atlas of Armenia (2001) to Anania, whom he called "Armenia's First Scientist." == Modern assessment == Hailed as a polymath, Anania is considered by modern scholars as the "father of the exact sciences in Armenia." Modern historians consider him as the greatest scientist of medieval Armenia and, possibly, all Armenian history, up to the 20th century astrophysicist Viktor Ambartsumian. He is widely regarded as the founder of the natural sciences in the country. He was the first classical Armenian scholar to study mathematics and several scientific subjects, such as cosmography and chronology. Nicholas Adontz argued that Anania "occupied the same position in Armenian education as Leo [the Mathematician] did in Byzantine education. He was the first to sow the seeds of science among the Armenians." Hacikyan et al. wrote in The Heritage of Armenian Literature: "Shirakatsi was an educator and an organizer of ideas and materials rather than an original thinker. He was often in the forefront of scientific thinking, but at other times he repeated the accepted theories of his time." Suren Yeremian named Anania, along with historian Movses Khorenatsi and philosopher David the Invincible, one of the prominent thinkers of the "great cultural flourishing" in Armenia of the fifth to seventh centuries, when Hellenistic traditions were still strong and continued to bear the influence of the secular thinking of the pre-Christian times. Sen Arevshatyan argued that the "Christian spirit" is more prominent in Anania than in earlier figures of the Hellenizing School as he aims to bridge Hellenistic science and church doctrine. "Nevertheless, despite his inconsistencies, he emerges as the greatest representative of natural knowledge and natural philosophy in early feudal Armenia." In Soviet reference books, Anania was often mentioned as the earliest astronomer among its peoples. Greenwood argues that studying Anania and his works "resonated with twentieth-century political beliefs and offered a suitable subject for academic research in ways that works on medieval theology or Biblical exegesis did not. Anania came to be projected as a national hero from the distant Armenian past, linking and affirming past and present identities." == Tributes == Statuettes of Anania were created by Nikolai Nikoghosyan (c. 1945) and Yervand Kochar (gypsum, 1952). Ara Sargsyan crafted a plaquette of him in 1957. Anania was one of six medieval scholars whose statue was erected in front of the Matenadaran, the museum-institute of Armenian manuscripts in Yerevan, in the 1960s. Sculpted by Grigor Badalyan in basalt, it was erected in 1963. A bust of Anania by Badalyan stands inside the Matenadaran. A statue, sculpted by Aram Gharibyan, was erected in the front yard of Yerevan State University in 1999. Another basalt statue of Anania by Samvel Petrosyan was erected in Gyumri in 2009. A crater on the Moon was named Shirakatsi in 1979. The Anania Shirakatsy Lyceum, an International Baccalaureate school in Yerevan, was established in 1990. In 1993 the Medal of Anania Shirakatsi, a state award, was established. It is awarded for "significant activities, inventions, and discoveries in the spheres of economy, engineering, architecture, science, and technology." In 2005 the Central Bank of Armenia issued a commemorative coin, while HayPost issued a stamp dedicated to him. A short 2016 documentary by Armenian Public TV. == References == Notes Citations == Bibliography == == Further reading == Mahé, Jean-Pierre (1987). "Quadrivium et cursus d'études au VIIe siècle en Arménie et dans le monde byzantin d'après le "K'nnikon" d'Anania Širakac'i". Travaux et Mémoires (in French). 10. Centre de recherche d'Histoire et Civilisation de Byzance: 159–206. == External links == Media related to Anania Shirakatsi at Wikimedia Commons Quotations related to Anania Shirakatsi at Wikiquote |
Wikipedia:Anastasia Stavrova#0 | Anastasia Konstantinovna Stavrova (Russian: Анастасия Константиновна Ставрова) is a Russian mathematician specializing in algebraic groups, non-associative algebra, and algebraic K-theory. She is a researcher in the Chebyshev Laboratory at Saint Petersburg State University. == Education and career == Stavrova earned a specialist degree in mathematics at Saint Petersburg State University in 2005. After traveling to the University of Leiden and University of Padua for a master's degree, which she completed in 2007, she returned to Saint Petersburg State University for her doctoral studies. Her 2009 dissertation, Structure of Isotropic Reductive Groups, was supervised by Nikolai Vavilov. She returned to Saint Petersburg State University as a researcher, after doing postdoctoral research from 2010 to 2012 at Ludwig Maximilian University of Munich and the University of Duisburg-Essen in Germany, and in 2013 as the Jerrold E. Marsden Postdoctoral Fellow at the Fields Institute in Canada. == Recognition == Stavrova won the Young Mathematician Prize of the Saint Petersburg Mathematical Society in 2009, and the Young Russian Mathematics Scholarship in 2016. In 2018, she won the G. de B. Robinson Award of the Canadian Mathematical Society. == References == == External links == Anastasia Stavrova publications indexed by Google Scholar |
Wikipedia:Anatol Slissenko#0 | Anatol Slissenko (a former transliteration: Slisenko; Russian: Анатоль Олесьевич Слисенко), a Soviet, Russian and French mathematician and computer scientist, was born on August 15, 1941, in Siberia, where his father served as the commander of a regiment of military topography. In 1950 his parents moved to Leningrad. == Research == Source: In 1958 A.O. Slissenko entered the Faculty of Mathematics and Mechanics of Leningrad (now Saint Petersburg) State University. There he started his research in constructive mathematics (recursive analysis) under the supervision of Nikolai Shanin. Having graduated from the university in 1963, he became a researcher at the Leningrad Department of Steklov Mathematical Institute of the Academy of Sciences of the USSR (nowadays it is called slightly differently. He defended his PhD dissertation in constructive mathematics 1967, his superviser was Nikolai Shanin. In 1981 he defended his DSc dissertation at Steklov Mathematical Institute in Moscow. In 1963–1966 he continued his research in constructive mathematics and at the same time he participated in the development and implementation of Shanin's algorithm for automatic theorem proving in classical propositional logic. Then he gradually began a research in algorithmics and computational complexity. In paper he discusses how to define 'computational' complexity of an individual problem, a topic that influenced his subsequent papers on entropic convergence. In he gave an unexpected solution to the problem of recognizing palindromes by multi-head Turing machines, namely, he proved that a 6-headed Turing machine with one tape can recognize palindromes in real time; the widespread expectation was that it was impossible. His very long proof was later simplified by Zvi Galil who used some new results that were not known when was being written (see also. Another major result of Slissenko was a real-time algorithm that solved a large variety of string-matching problems (including finding of all periodicities in a compact form). The algorithm can be formalized as LRAM (address machine), a random access machine with registers whose length is bounded by the logarithm of time complexity, introduced in and also described in. This model allows the use of various operations, including multiplication, that without this bound on the length can give unrealistically fast algorithms. Besides that LRAM has a very dense time complexity hierarchy. During 1981–1992 Slissenko was the head of a laboratory in St. Petersburg Institute for Informatics and Automation of the Russian Academy of Sciences. He worked there on applications, however it was a time of agony of the Soviet Union, and there were no noteworthy publications in this field. In he introduced a class of graph grammars (called Slisenko (Slissenko) grammars in ) that generate graphs for which the existence of Hamiltonian cycles can be decided in polynomial time. The graphs generated by these grammars are of bounded tree-width. The paper was his first one where he introduced an entropy to evaluate the quality of inference systems. From 1993 until 2009 he was a professor at the University Paris-East Créteil (UPEC: Université Paris-Est-Créteil; the previous name was Paris-12 University) , Faculty of Sciences and Technology, Department of Informatics. He worked on the complexity of Markov decision processes, on algorithms constructing shortest paths amidst semi-algebraic and other obstacles and on the verification of timed systems. Paper describes a polytime algorithm for constructing a shortest path touching skew straight lines in 3-dimensional space that solves a known open problem. A model checking algorithm for a rather powerful logic with an operator of probability was developed in; it was a first result for this kind of logics. Various topics of verification were investigated for timed models. In a powerful logic for the specification of hard real time systems, named FOTL (First Order Timed Logic), was introduced and decidable classes were presented. More general decidable classes were investigated in. Entropic convergence of algorithms was introduced in; it was applied to knowledge bases evaluation. In he noticed that one can slightly reformulate P≠NP problem in such a way that it remains practically interesting but its independence from arithmetics implies that P≠NP. Slissenko was an invited speaker at many conferences, in particular at the International Congress of Mathematicians in 1983, in Warsaw, Poland. He collaborated with Nikolai Shanin, S.Maslov, G.Mints and V.Orevkov on automatic theorem proving, with D.Beauquier, Dima Grigoriev, D.Burago, A.Rabinovich and others on various topics related to algorithmics. == Teaching and Organizational Activity == A.O.Slissenko was a part-time professor in Leningrad Polytechnical Institute in 1981–1987, and in 1988–1992 he was a part-time professor in the faculty of Mathematics and Mechanics of Leningrad State University where he was the head of the Department of Computer Science whose creation he initiated (the student teams of the Department were world champions of ACM International Collegiate Programming Contest four times). In 1993–2009 he was a full professor at the University Paris-East Creteil at the Faculty of Science and Technology, Department of Informatics. Since 2009 he has remained a professor emeritus at this university. He had also been the head (and, in a way, the founder) of Laboratory for Algorithmics Complexity and Logic (LACL) from 1997 until 2007. In all these positions he contributed to modernizing the curriculum and research. In 1967 Slissenko organized together with Grigori Tseitin (1936–2022) and Robert I.Freidson (1942–2018) the Leningrad Seminar on Computational Complexity that first had its meetings in the Leningrad State University, and then in the Leningrad Department of Steklov Mathematical Institute (at that time Slissenko became its formal head). The seminar. played an important role in the development of this field in the Soviet Union. The seminar functioned until 1992, date on which after the collapse of the Soviet Union (and of its research system) the main part of its participants left the country and found jobs in USA, France, UK. Historical information on the Soviet computer science can be found in, and some remarks are in. == References == == External links == Anatol Slissenko at the Mathematics Genealogy Project Anatol Slissenko home page Publications of Anatol Slissenko Publications of Anatol Slissenko in Russian |
Wikipedia:Anatolii Goldberg#0 | Anatolii Asirovich Goldberg (Russian: Анатолий Асирович Гольдберг, Ukrainian: Анатолій Асірович Гольдберг, Hebrew: אנטולי גולדברג; April 2, 1930, in Kyiv – October 11, 2008, in Netanya) was a Soviet and Israeli mathematician working in complex analysis. His main area of research was the theory of entire and meromorphic functions. == Life and work == Goldberg received his PhD in 1955 from Lviv University under the direction of Lev Volkovyski. He worked as a docent in Uzhhorod National University (1955–1963), then in Lviv University (1963–1997), where he became a full professor in 1965, and in Bar Ilan University (1997–2008). Goldberg, jointly with Iossif Ostrovskii and Boris Levin, was awarded the State Prize of Ukraine in 1992. Among his main achievements are: construction of meromorphic functions with infinitely many deficient values, solution of the inverse problem of Nevanlinna theory for finitely many deficient values, development of the integral with respect to a semi-additive measure. He authored a book Goldberg & Ostrovskii (2008) and over 150 research papers. Several things are named after him: Goldberg's examples, Goldberg's constants, and Goldberg's conjecture. == Selected publications == Goldberg, A. A.; Ostrovskii, I. V. (1970). Distribution of values of meromorphic functions (in Russian). Moscow: Nauka. MR 0280720., translated as Goldberg, A. A.; Ostrovskii, I. V. (2008). Distribution of values of meromorphic functions. Providence, RI: Amer. Math. Soc. ISBN 978-0-8218-4265-2. MR 2435270. == References == == External links == "McTutor history of mathematics archive". {{cite journal}}: Cite journal requires |journal= (help) Anatolii Goldberg at the Mathematics Genealogy Project Eremenko, A.; Ostrovskii, I.; Sodin, M. (1998). "Anatolii Asirovich Gol'dberg" (PDF). Complex Variables, Theory and Application. 37 (1–4): 1–51. CiteSeerX 10.1.1.299.355. doi:10.1080/17476939808815121. hdl:11693/48936. The International conference on complex analysis and related topics dedicated to the 90-th anniversary of Anatolii Asirovich Goldberg (1930-2008) |
Wikipedia:Anatoliy Skorokhod#0 | Anatoliy Volodymyrovych Skorokhod (Ukrainian: Анато́лій Володи́мирович Скорохо́д; September 10, 1930 – January 3, 2011) was a Soviet and Ukrainian mathematician. Skorokhod is well-known for his comprehensive treatise on the theory of stochastic processes which he co-authored with Gikhman. == Career == Skorokhod worked at Kyiv University from 1956 to 1964. He was subsequently at the Institute of Mathematics of the National Academy of Sciences of Ukraine from 1964 until 2002. Since 1993, he had been a professor at Michigan State University in the US, and a member of the American Academy of Arts and Sciences. He was an academician of the National Academy of Sciences of Ukraine from 1985 up to his death in 2011. His scientific works are on the theory of: stochastic differential equations, limit theorems of random processes, distributions in infinite-dimensional spaces, statistics of random processes and Markov processes. Skorokhod authored over 450 scientific works, including more than 40 monographs and books. Many terms and concepts have his name, specifically: Skorokhod space (Skorokhod space) Skorokhod integral Skorokhod problem Skorokhod's embedding theorem Skorokhod's representation theorem == Selected works == with I. I. Gikhman: Introduction to the theory of random processes, W. B. Saunders 1969, Dover 1996 with I. I. Gikhman: Stochastic Differential Equations, Springer Verlag 1972 with I. I. Gikhman: Controlled stochastic processes, Springer Verlag 1979 with I. I. Gikhman: The Theory of Stochastic Processes, Springer Verlag, 3 vols., 2004–2007 Random processes with independent increments, Kluwer 1991 Asymptotic methods in the theory of stochastic differential equations , American Mathematical Society 1989 Random linear operators, Reidel 1984 Studies in the theory of random processes, Dover 1982 Stochastic equations for complex systems, Reidel/Kluwer 1988 ISBN 90-277-2408-3 Stochastische Differentialgleichungen, Berlin, Akademie Verlag 1971 Integration in Hilbert Space, Springer Verlag 1974 with Yu. V. Prokhorov: Basic principles and applications of probability theory, Springer Verlag 2005 with Frank C. Hoppensteadt, Habib Salehi: Random perturbation methods with applications in science and engineering, Springer Verlag 2002 == See also == List of Ukrainian mathematicians == Notes == == External links == Anatoliy Skorokhod at the Mathematics Genealogy Project "Skorokhod, Anatoliy Volodymyrovych Collection: Interviews conducted February 20, 1992 and in 2004". Cornell University Library. Retrieved 10 March 2011. "A short sketch of life and research of A.V. Skorokhod". Dept. of Mathematics, Kyiv University. Retrieved 12 July 2011. "Anatolii Skorokhod". Dept. of Probability, Kyiv University. "MacTutor Skorokhod biography". Archived from the original on 2015-11-25. Retrieved 2015-11-25. |
Wikipedia:Anatoly Karatsuba#0 | Anatoly Alexeyevich Karatsuba (his first name often spelled Anatolii) (Russian: Анато́лий Алексе́евич Карацу́ба; Grozny, Soviet Union, 31 January 1937 – Moscow, Russia, 28 September 2008) was a Russian mathematician working in the field of analytic number theory, p-adic numbers and Dirichlet series. For most of his student and professional life he was associated with the Faculty of Mechanics and Mathematics of Moscow State University, defending a D.Sc. there entitled "The method of trigonometric sums and intermediate value theorems" in 1966. He later held a position at the Steklov Institute of Mathematics of the Academy of Sciences. His textbook Foundations of Analytic Number Theory went to two editions, 1975 and 1983. The Karatsuba algorithm is the earliest known divide and conquer algorithm for multiplication and lives on as a special case of its direct generalization, the Toom–Cook algorithm. The main research works of Anatoly Karatsuba were published in more than 160 research papers and monographs. His daughter, Yekaterina Karatsuba, also a mathematician, constructed the FEE method. == Work on informatics == As a student of Lomonosov Moscow State University, Karatsuba attended the seminar of Andrey Kolmogorov and found solutions to two problems set up by Kolmogorov. This was essential for the development of automata theory and started a new branch in Mathematics, the theory of fast algorithms. === Automata === In the paper of Edward F. Moore, ( n ; m ; p ) {\displaystyle (n;m;p)} , an automaton (or a machine) S {\displaystyle S} , is defined as a device with n {\displaystyle n} states, m {\displaystyle m} input symbols and p {\displaystyle p} output symbols. Nine theorems on the structure of S {\displaystyle S} and experiments with S {\displaystyle S} are proved. Later such S {\displaystyle S} machines got the name of Moore machines. At the end of the paper, in the chapter «New problems», Moore formulates the problem of improving the estimates which he obtained in Theorems 8 and 9: Theorem 8 (Moore). Given an arbitrary ( n ; m ; p ) {\displaystyle (n;m;p)} machine S {\displaystyle S} , such that every two states can be distinguished from each other, there exists an experiment of length n ( n − 1 ) / 2 {\displaystyle n(n-1)/2} that identifies the state of S {\displaystyle S} at the end of this experiment. In 1957 Karatsuba proved two theorems which completely solved the Moore problem on improving the estimate of the length of experiment in his Theorem 8. Theorem A (Karatsuba). If S {\displaystyle S} is a ( n ; m ; p ) {\displaystyle (n;m;p)} machine such that each two its states can be distinguished from each other then there exists a ramified experiment of length at most ( n − 1 ) ( n − 2 ) / 2 + 1 {\displaystyle (n-1)(n-2)/2+1} , by means of which one can find the state S {\displaystyle S} at the end of the experiment. Theorem B (Karatsuba). There exists a ( n ; m ; p ) {\displaystyle (n;m;p)} machine, every states of which can be distinguished from each other, such that the length of the shortest experiment finding the state of the machine at the end of the experiment, is equal to ( n − 1 ) ( n − 2 ) / 2 + 1 {\displaystyle (n-1)(n-2)/2+1} . These two theorems were proved by Karatsuba in his 4th year as a basis of his 4th year project; the corresponding paper was submitted to the journal "Uspekhi Mat. Nauk" on December 17, 1958 and published in June 1960. Up to this day (2011) this result of Karatsuba that later acquired the title "the Moore-Karatsuba theorem", remains the only precise (the only precise non-linear order of the estimate) non-linear result both in the automata theory and in the similar problems of the theory of complexity of computations. == Work on number theory == The main research works of A. A. Karatsuba were published in more than 160 research papers and monographs. === The p-adic method === A.A.Karatsuba constructed a new p {\displaystyle p} -adic method in the theory of trigonometric sums. The estimates of so-called L {\displaystyle L} -sums of the form S = ∑ x = 1 P e 2 π i ( a 1 x / p n + ⋯ + a n x n / p ) , ( a s , p ) = 1 , 1 ≤ s ≤ n , {\displaystyle S=\sum _{x=1}^{P}e^{2\pi i(a_{1}x/p^{n}+\cdots +a_{n}x^{n}/p)},\quad (a_{s},p)=1,\quad 1\leq s\leq n,} led to the new bounds for zeros of the Dirichlet L {\displaystyle L} -series modulo a power of a prime number, to the asymptotic formula for the number of Waring congruence of the form x 1 n + ⋯ + x t n ≡ N ( mod p k ) , 1 ≤ x s ≤ P , 1 ≤ s ≤ n , P < p k , {\displaystyle x_{1}^{n}+\dots +x_{t}^{n}\equiv N{\pmod {p^{k}}},\quad 1\leq x_{s}\leq P,\quad 1\leq s\leq n,\quad P<p^{k},} to a solution of the problem of distribution of fractional parts of a polynomial with integer coefficients modulo p k {\displaystyle p^{k}} . A.A. Karatsuba was the first to realize in the p {\displaystyle p} -adic form the «embedding principle» of Euler-Vinogradov and to compute a p {\displaystyle p} -adic analog of Vinogradov u {\displaystyle u} -numbers when estimating the number of solutions of a congruence of the Waring type. Assume that : x 1 n + ⋯ + x t n ≡ N ( mod Q ) , 1 ≤ x s ≤ P , 1 ≤ s ≤ t , ( 1 ) {\displaystyle x_{1}^{n}+\dots +x_{t}^{n}\equiv N{\pmod {Q}},\quad 1\leq x_{s}\leq P,\quad 1\leq s\leq t,\quad (1)} and moreover : P r ≤ Q < P r + 1 , 1 ≤ r ≤ 1 12 n , Q = p k , k ≥ 4 ( r + 1 ) n , {\displaystyle P^{r}\leq Q<P^{r+1},\quad 1\leq r\leq {\frac {1}{12}}{\sqrt {n}},\quad Q=p^{k},\quad k\geq 4(r+1)n,} where p {\displaystyle p} is a prime number. Karatsuba proved that in that case for any natural number n ≥ 144 {\displaystyle n\geq 144} there exists a p 0 = p 0 ( n ) {\displaystyle p_{0}=p_{0}(n)} such that for any p 0 > p 0 ( n ) {\displaystyle p_{0}>p_{0}(n)} every natural number N {\displaystyle N} can be represented in the form (1) for t ≥ 20 r + 1 {\displaystyle t\geq 20r+1} , and for t < r {\displaystyle t<r} there exist N {\displaystyle N} such that the congruence (1) has no solutions. This new approach, found by Karatsuba, led to a new p {\displaystyle p} -adic proof of the Vinogradov mean value theorem, which plays the central part in the Vinogradov's method of trigonometric sums. Another component of the p {\displaystyle p} -adic method of A.A. Karatsuba is the transition from incomplete systems of equations to complete ones at the expense of the local p {\displaystyle p} -adic change of unknowns. Let r {\displaystyle r} be an arbitrary natural number, 1 ≤ r ≤ n {\displaystyle 1\leq r\leq n} . Determine an integer t {\displaystyle t} by the inequalities m t ≤ r ≤ m t + 1 {\displaystyle m_{t}\leq r\leq m_{t+1}} . Consider the system of equations { x 1 m 1 + ⋯ + x k m 1 = y 1 m 1 + ⋯ + y k m 1 … … … … … … … … x 1 m s + ⋯ + x k m s = y 1 m s + ⋯ + y k m s x 1 n + ⋯ + x k n = y 1 n + ⋯ + y k n {\displaystyle {\begin{cases}x_{1}^{m_{1}}+\dots +x_{k}^{m_{1}}=y_{1}^{m_{1}}+\dots +y_{k}^{m_{1}}\\\dots \dots \dots \dots \dots \dots \dots \dots \\x_{1}^{m_{s}}+\dots +x_{k}^{m_{s}}=y_{1}^{m_{s}}+\dots +y_{k}^{m_{s}}\\x_{1}^{n}+\dots +x_{k}^{n}=y_{1}^{n}+\dots +y_{k}^{n}\end{cases}}} 1 ≤ x 1 , … , x k , y 1 , … , y k ≤ P , 1 ≤ m 1 < m 2 < ⋯ < m s < m s + 1 = n . {\displaystyle 1\leq x_{1},\dots ,x_{k},y_{1},\dots ,y_{k}\leq P,\quad 1\leq m_{1}<m_{2}<\dots <m_{s}<m_{s+1}=n.} Karatsuba proved that the number of solutions I k {\displaystyle I_{k}} of this system of equations for k ≥ 6 r n log n {\displaystyle k\geq 6rn\log n} satisfies the estimate I k ≪ P 2 k − δ , δ = m 1 + ⋯ + m t + ( s − t + 1 ) r . {\displaystyle I_{k}\ll P^{2k-\delta },\quad \delta =m_{1}+\dots +m_{t}+(s-t+1)r.} For incomplete systems of equations, in which the variables run through numbers with small prime divisors, Karatsuba applied multiplicative translation of variables. This led to an essentially new estimate of trigonometric sums and a new mean value theorem for such systems of equations. === The Hua Luogeng problem on the convergency exponent of the singular integral in the Terry problem === p {\displaystyle p} -adic method of A.A.Karatsuba includes the techniques of estimating the measure of the set of points with small values of functions in terms of the values of their parameters (coefficients etc.) and, conversely, the techniques of estimating those parameters in terms of the measure of this set in the real and p {\displaystyle p} -adic metrics. This side of Karatsuba's method manifested itself especially clear in estimating trigonometric integrals, which led to the solution of the problem of Hua Luogeng. In 1979 Karatsuba, together with his students G.I. Arkhipov and V.N. Chubarikov obtained a complete solution of the Hua Luogeng problem of finding the exponent of convergency of the integral: ϑ 0 = ∫ − ∞ + ∞ ⋯ ∫ − ∞ + ∞ | ∫ 0 1 e 2 π i ( α n x n + ⋯ + α 1 x ) d x | 2 k d α n … d α 1 , {\displaystyle \vartheta _{0}=\int \limits _{-\infty }^{+\infty }\cdots \int \limits _{-\infty }^{+\infty }{\biggl |}\int \limits _{0}^{1}e^{2\pi i(\alpha _{n}x^{n}+\cdots +\alpha _{1}x)}dx{\biggr |}^{2k}d\alpha _{n}\ldots d\alpha _{1},} where n ≥ 2 {\displaystyle n\geq 2} is a fixed number. In this case, the exponent of convergency means the value γ {\displaystyle \gamma } , such that ϑ 0 {\displaystyle \vartheta _{0}} converges for 2 k > γ + ε {\displaystyle 2k>\gamma +\varepsilon } and diverges for 2 k < γ − ε {\displaystyle 2k<\gamma -\varepsilon } , where ε > 0 {\displaystyle \varepsilon >0} is arbitrarily small. It was shown that the integral ϑ 0 {\displaystyle \vartheta _{0}} converges for 2 k > 1 2 ( n 2 + n ) + 1 {\displaystyle 2k>{\tfrac {1}{2}}(n^{2}+n)+1} and diverges for 2 k ≤ 1 2 ( n 2 + n ) + 1 {\displaystyle 2k\leq {\tfrac {1}{2}}(n^{2}+n)+1} . At the same time, the similar problem for the integral was solved: ϑ 1 = ∫ − ∞ + ∞ ⋯ ∫ − ∞ + ∞ | ∫ 0 1 e 2 π i ( α n x n + α m x m + ⋯ + α r x r ) d x | 2 k d α n d α m … d α r , {\displaystyle \vartheta _{1}=\int _{-\infty }^{+\infty }\cdots \int _{-\infty }^{+\infty }{\biggl |}\int _{0}^{1}e^{2\pi i(\alpha _{n}x^{n}+\alpha _{m}x^{m}+\cdots +\alpha _{r}x^{r})}dx{\biggr |}^{2k}d\alpha _{n}d\alpha _{m}\ldots d\alpha _{r},} where n , m , … , r {\displaystyle n,m,\ldots ,r} are integers, satisfying the conditions : 1 ≤ r < … < m < n , r + … + m + n < 1 2 ( n 2 + n ) . {\displaystyle 1\leq r<\ldots <m<n,\quad r+\ldots +m+n<{\tfrac {1}{2}}(n^{2}+n).} Karatsuba and his students proved that the integral ϑ 1 {\displaystyle \vartheta _{1}} converges, if 2 k > n + m + … + r {\displaystyle 2k>n+m+\ldots +r} and diverges, if 2 k ≤ n + m + … + r {\displaystyle 2k\leq n+m+\ldots +r} . The integrals ϑ 0 {\displaystyle \vartheta _{0}} and ϑ 1 {\displaystyle \vartheta _{1}} arise in the studying of the so-called Prouhet–Tarry–Escott problem. Karatsuba and his students obtained a series of new results connected with the multi-dimensional analog of the Tarry problem. In particular, they proved that if F {\displaystyle F} is a polynomial in r {\displaystyle r} variables ( r ≥ 2 {\displaystyle r\geq 2} ) of the form : F ( x 1 , … , x r ) = ∑ ν 1 = 0 n 1 ⋯ ∑ ν r = 0 n r α ( ν 1 , … , ν r ) x 1 ν 1 … x r ν r , {\displaystyle F(x_{1},\ldots ,x_{r})\,=\,\sum \limits _{\nu _{1}=0}^{n_{1}}\cdots \sum \limits _{\nu _{r}=0}^{n_{r}}\alpha (\nu _{1},\ldots ,\nu _{r})x_{1}^{\nu _{1}}\ldots x_{r}^{\nu _{r}},} with the zero free term, m = ( n 1 + 1 ) … ( n r + 1 ) − 1 {\displaystyle m=(n_{1}+1)\ldots (n_{r}+1)-1} , α ¯ {\displaystyle {\bar {\alpha }}} is the m {\displaystyle m} -dimensional vector, consisting of the coefficients of F {\displaystyle F} , then the integral : ϑ 2 = ∫ − ∞ + ∞ ⋯ ∫ − ∞ + ∞ | ∫ 0 1 ⋯ ∫ 0 1 e 2 π i F ( x 1 , … , x r ) d x 1 … d x r | 2 k d α ¯ {\displaystyle \vartheta _{2}=\int \limits _{-\infty }^{+\infty }\cdots \int \limits _{-\infty }^{+\infty }{\biggl |}\int \limits _{0}^{1}\cdots \int \limits _{0}^{1}e^{2\pi iF(x_{1},\ldots ,x_{r})}dx_{1}\ldots dx_{r}{\biggr |}^{2k}d{\bar {\alpha }}} converges for 2 k > m n {\displaystyle 2k>mn} , where n {\displaystyle n} is the highest of the numbers n 1 , … , n r {\displaystyle n_{1},\ldots ,n_{r}} . This result, being not a final one, generated a new area in the theory of trigonometric integrals, connected with improving the bounds of the exponent of convergency ϑ 2 {\displaystyle \vartheta _{2}} (I. A. Ikromov, M. A. Chahkiev and others). === Multiple trigonometric sums === In 1966–1980, Karatsuba developed (with participation of his students G.I. Arkhipov and V.N. Chubarikov) the theory of multiple Hermann Weyl trigonometric sums, that is, the sums of the form S = S ( A ) = ∑ x 1 = 1 P 1 … ∑ x r = 1 P r e 2 π i F ( x 1 , … , x r ) {\displaystyle S=S(A)=\sum _{x_{1}=1}^{P_{1}}\dots \sum _{x_{r}=1}^{P_{r}}e^{2\pi iF(x_{1},\dots ,x_{r})}} , where F ( x 1 , … , x r ) = ∑ t 1 = 1 n 1 … ∑ t r = 1 n r α ( t 1 , … , t r ) x 1 t 1 … x r t r {\displaystyle F(x_{1},\dots ,x_{r})=\sum _{t_{1}=1}^{n_{1}}\dots \sum _{t_{r}=1}^{n_{r}}\alpha (t_{1},\dots ,t_{r})x_{1}^{t_{1}}\dots x_{r}^{t_{r}}} , A {\displaystyle A} is a system of real coefficients α ( t 1 , … , t r ) {\displaystyle \alpha (t_{1},\dots ,t_{r})} . The central point of that theory, as in the theory of the Vinogradov trigonometric sums, is the following mean value theorem. Let n 1 , … , n r , P 1 , … , P r {\displaystyle n_{1},\dots ,n_{r},P_{1},\dots ,P_{r}} be natural numbers, P 1 = min ( P 1 , … , P r ) {\displaystyle P_{1}=\min(P_{1},\dots ,P_{r})} , m = ( n 1 + 1 ) … ( n r + 1 ) {\displaystyle m=(n_{1}+1)\dots (n_{r}+1)} . Furthermore, let Ω {\displaystyle \Omega } be the m {\displaystyle m} -dimensional cube of the form :: 0 ≤ α ( t 1 , … , t r ) < 1 {\displaystyle 0\leq \alpha (t_{1},\dots ,t_{r})<1} , 0 ≤ t 1 ≤ n 1 , … , 0 ≤ t r ≤ n r {\displaystyle 0\leq t_{1}\leq n_{1},\dots ,0\leq t_{r}\leq n_{r}} , in the euclidean space : and :: J = J ( P 1 , … , P r ; n 1 , … , n r ; K , r ) = ∫ … ∫ Ω | S ( A ) | 2 K d A {\displaystyle J=J(P_{1},\dots ,P_{r};n_{1},\dots ,n_{r};K,r)={\underset {\Omega }{\int \dots \int }}|S(A)|^{2K}dA} . : Then for any τ ≥ 0 {\displaystyle \tau \geq 0} and K ≥ K τ = m τ {\displaystyle K\geq K_{\tau }=m\tau } the value J {\displaystyle J} can be estimated as follows J ≤ K τ 2 m τ ϰ 4 ϰ 2 Δ ( τ ) 2 8 m ϰ τ ( P 1 … P r ) 2 K P − ϰ Δ ( τ ) {\displaystyle J\leq K_{\tau }^{2m\tau }\varkappa ^{4\varkappa ^{2}\Delta (\tau )}2^{8m\varkappa \tau }(P_{1}\dots P_{r})^{2K}P^{-\varkappa \Delta (\tau )}} , : where ϰ = n 1 ν 1 + ⋯ + n r ν r {\displaystyle \varkappa =n_{1}\nu _{1}+\dots +n_{r}\nu _{r}} , γ ϰ = 1 {\displaystyle \gamma \varkappa =1} , Δ ( τ ) = m 2 ( 1 − ( 1 − γ ) τ ) {\displaystyle \Delta (\tau )={\frac {m}{2}}(1-(1-\gamma )^{\tau })} , P = ( P 1 n 1 … P r n r ) γ {\displaystyle P=(P_{1}^{n_{1}}\dots P_{r}^{n_{r}})^{\gamma }} , and the natural numbers ν 1 , … , ν r {\displaystyle \nu _{1},\dots ,\nu _{r}} are such that: :: − 1 < P s P 1 − ν s ≤ 0 {\displaystyle -1<{\frac {P_{s}}{P_{1}}}-\nu _{s}\leq 0} , s = 1 , … , r {\displaystyle s=1,\dots ,r} . The mean value theorem and the lemma on the multiplicity of intersection of multi-dimensional parallelepipeds form the basis of the estimate of a multiple trigonometric sum, that was obtained by Karatsuba (two-dimensional case was derived by G.I. Arkhipov). Denoting by Q 0 {\displaystyle Q_{0}} the least common multiple of the numbers q ( t 1 , … , t r ) {\displaystyle q(t_{1},\dots ,t_{r})} with the condition t 1 + … t r ≥ 1 {\displaystyle t_{1}+\dots t_{r}\geq 1} , for Q 0 ≥ P 1 / 6 {\displaystyle Q_{0}\geq P^{1/6}} the estimate holds | S ( A ) | ≤ ( 5 n 2 n ) r ν ( Q 0 ) ( τ ( Q 0 ) ) r − 1 P 1 … P r Q − 0.1 μ + 2 8 r ( r μ − 1 ) r − 1 P 1 … P r P − 0.05 μ {\displaystyle |S(A)|\leq (5n^{2n})^{r\nu (Q_{0})}(\tau (Q_{0}))^{r-1}P_{1}\dots P_{r}Q^{-0.1\mu }+2^{8r}(r\mu ^{-1})^{r-1}P_{1}\dots P_{r}P^{-0.05\mu }} , where τ ( Q ) {\displaystyle \tau (Q)} is the number of divisors of the integer Q {\displaystyle Q} , and ν ( Q ) {\displaystyle \nu (Q)} is the number of distinct prime divisors of the number Q {\displaystyle Q} . === The estimate of the Hardy function in the Waring problem === Applying his p {\displaystyle p} -adic form of the Hardy-Littlewood-Ramanujan-Vinogradov method to estimating trigonometric sums, in which the summation is taken over numbers with small prime divisors, Karatsuba obtained a new estimate of the well known Hardy function G ( n ) {\displaystyle G(n)} in the Waring's problem (for n ≥ 400 {\displaystyle n\geq 400} ): G ( n ) < 2 n log n + 2 n log log n + 12 n . {\displaystyle \!G(n)<2n\log n+2n\log \log n+12n.} === Multi-dimensional analog of the Waring problem === In his subsequent investigation of the Waring problem Karatsuba obtained the following two-dimensional generalization of that problem: Consider the system of equations x 1 n − i y 1 i + ⋯ + x k n − i y k i = N i {\displaystyle x_{1}^{n-i}y_{1}^{i}+\dots +x_{k}^{n-i}y_{k}^{i}=N_{i}} , i = 0 , 1 , … , n {\displaystyle i=0,1,\dots ,n} , where N i {\displaystyle N_{i}} are given positive integers with the same order or growth, N 0 → + ∞ {\displaystyle N_{0}\to +\infty } , and x ϰ , y ϰ {\displaystyle x_{\varkappa },y_{\varkappa }} are unknowns, which are also positive integers. This system has solutions, if k > c n 2 log n {\displaystyle k>cn^{2}\log n} , and if k < c 1 n 2 {\displaystyle k<c_{1}n^{2}} , then there exist such N i {\displaystyle N_{i}} , that the system has no solutions. === The Artin problem of local representation of zero by a form === Emil Artin had posed the problem on the p {\displaystyle p} -adic representation of zero by a form of arbitrary degree d. Artin initially conjectured a result, which would now be described as the p-adic field being a C2 field; in other words non-trivial representation of zero would occur if the number of variables was at least d2. This was shown not to be the case by an example of Guy Terjanian. Karatsuba showed that, in order to have a non-trivial representation of zero by a form, the number of variables should grow faster than polynomially in the degree d; this number in fact should have an almost exponential growth, depending on the degree. Karatsuba and his student Arkhipov proved, that for any natural number r {\displaystyle r} there exists n 0 = n 0 ( r ) {\displaystyle n_{0}=n_{0}(r)} , such that for any n ≥ n 0 {\displaystyle n\geq n_{0}} there is a form with integral coefficients F ( x 1 , … , x k ) {\displaystyle F(x_{1},\dots ,x_{k})} of degree smaller than n {\displaystyle n} , the number of variables of which is k {\displaystyle k} , k ≥ 2 u {\displaystyle k\geq 2^{u}} , u = n ( log 2 n ) ( log 2 log 2 n ) … ( log 2 … log 2 n ) ⏟ r ( log 2 … log 2 n ) 3 ⏟ r + 1 {\displaystyle u={\frac {n}{(\log _{2}n)(\log _{2}\log _{2}n)\dots \underbrace {(\log _{2}\dots \log _{2}n)} _{r}\underbrace {(\log _{2}\dots \log _{2}n)^{3}} _{r+1}}}} which has only trivial representation of zero in the 2-adic numbers. They also obtained a similar result for any odd prime modulus p {\displaystyle p} . === Estimates of short Kloosterman sums === Karatsuba developed (1993—1999) a new method of estimating short Kloosterman sums, that is, trigonometric sums of the form ∑ n ∈ A exp ( 2 π i a n ∗ + b n m ) , {\displaystyle \sum \limits _{n\in A}\exp {{\biggl (}2\pi i\,{\frac {an^{*}+bn}{m}}{\biggr )}},} where n {\displaystyle n} runs through a set A {\displaystyle A} of numbers, coprime to m {\displaystyle m} , the number of elements ‖ A ‖ {\displaystyle \|A\|} in which is essentially smaller than m {\displaystyle m} , and the symbol n ∗ {\displaystyle n^{*}} denotes the congruence class, inverse to n {\displaystyle n} modulo m {\displaystyle m} : n n ∗ ≡ 1 ( mod m ) {\displaystyle nn^{*}\equiv 1(\mod m)} . Up to the early 1990s, the estimates of this type were known, mainly, for sums in which the number of summands was higher than m {\displaystyle {\sqrt {m}}} (H. D. Kloosterman, I. M. Vinogradov, H. Salié, L. Carlitz, S. Uchiyama, A. Weil). The only exception was the special moduli of the form m = p α {\displaystyle m=p^{\alpha }} , where p {\displaystyle p} is a fixed prime and the exponent α {\displaystyle \alpha } increases to infinity (this case was studied by A. G. Postnikov by means of the method of Vinogradov). Karatsuba's method makes it possible to estimate Kloosterman sums where the number of summands does not exceed m ε , {\displaystyle m^{\varepsilon },} and in some cases even exp { ( ln m ) 2 / 3 + ε } , {\displaystyle \exp {\{(\ln m)^{2/3+\varepsilon }\}},} where ε > 0 {\displaystyle \varepsilon >0} is an arbitrarily small fixed number. The final paper of Karatsuba on this subject was published posthumously. Various aspects of the method of Karatsuba have found applications in the following problems of analytic number theory: finding asymptotics of the sums of fractional parts of the form : ∑ n ≤ x ′ { a n ∗ + b n m } , ∑ p ≤ x ′ { a p ∗ + b p m } , {\displaystyle {\sum _{n\leq x}}'{\biggl \{}{\frac {an^{*}+bn}{m}}{\biggr \}},{\sum _{p\leq x}}'{\biggl \{}{\frac {ap^{*}+bp}{m}}{\biggr \}},} : where n {\displaystyle n} runs, one after another, through the integers satisfying the condition ( n , m ) = 1 {\displaystyle (n,m)=1} , and p {\displaystyle p} runs through the primes that do not divide the module m {\displaystyle m} (Karatsuba); finding a lower bound for the number of solutions of inequalities of the form : α < { a n ∗ + b n m } ≤ β {\displaystyle \alpha <{\biggl \{}{\frac {an^{*}+bn}{m}}{\biggr \}}\leq \beta } : in the integers n {\displaystyle n} , 1 ≤ n ≤ x {\displaystyle 1\leq n\leq x} , coprime to m {\displaystyle m} , x < m {\displaystyle x<{\sqrt {m}}} (Karatsuba); the precision of approximation of an arbitrary real number in the segment [ 0 , 1 ] {\displaystyle [0,1]} by fractional parts of the form : { a n ∗ + b n m } , {\displaystyle {\biggl \{}{\frac {an^{*}+bn}{m}}{\biggr \}},} : where 1 ≤ n ≤ x {\displaystyle 1\leq n\leq x} , ( n , m ) = 1 {\displaystyle (n,m)=1} , x < m {\displaystyle x<{\sqrt {m}}} (Karatsuba); a more precise constant c {\displaystyle c} in the Brun–Titchmarsh theorem : π ( x ; q , l ) < c x φ ( q ) ln 2 x q , {\displaystyle \pi (x;q,l)<{\frac {cx}{\varphi (q)\ln {\frac {2x}{q}}}},} : where π ( x ; q , l ) {\displaystyle \pi (x;q,l)} is the number of primes p {\displaystyle p} , not exceeding x {\displaystyle x} and belonging to the arithmetic progression p ≡ l ( mod q ) {\displaystyle p\equiv l{\pmod {q}}} (J. Friedlander, H. Iwaniec); a lower bound for the greatest prime divisor of the product of numbers of the form : n 3 + 2 {\displaystyle n^{3}+2} , N < n ≤ 2 N {\displaystyle N<n\leq 2N} (D. R. Heath-Brown); proving that there are infinitely many primes of the form: a 2 + b 4 {\displaystyle a^{2}+b^{4}} (J. Friedlander, H. Iwaniec); combinatorial properties of the set of numbers : n ∗ ( mod m ) {\displaystyle n^{*}{\pmod {m}}} 1 ≤ n ≤ m ε {\displaystyle 1\leq n\leq m^{\varepsilon }} (A. A. Glibichuk). === The Riemann zeta function === ==== The Selberg zeroes ==== In 1984 Karatsuba proved, that for a fixed ε {\displaystyle \varepsilon } satisfying the condition 0 < ε < 0.001 {\displaystyle 0<\varepsilon <0.001} , a sufficiently large T {\displaystyle T} and H = T a + ε {\displaystyle H=T^{a+\varepsilon }} , a = 27 82 = 1 3 − 1 246 {\displaystyle a={\tfrac {27}{82}}={\tfrac {1}{3}}-{\tfrac {1}{246}}} , the interval ( T , T + H ) {\displaystyle (T,T+H)} contains at least c H ln T {\displaystyle cH\ln T} real zeros of the Riemann zeta function ζ ( 1 2 + i t ) {\displaystyle \zeta {\Bigl (}{\tfrac {1}{2}}+it{\Bigr )}} . The special case H ≥ T 1 / 2 + ε {\displaystyle H\geq T^{1/2+\varepsilon }} was proven by Atle Selberg earlier in 1942. The estimates of Atle Selberg and Karatsuba can not be improved in respect of the order of growth as T → + ∞ {\displaystyle T\to +\infty } . ==== Distribution of zeros of the Riemann zeta function on the short intervals of the critical line ==== Karatsuba also obtained a number of results about the distribution of zeros of ζ ( s ) {\displaystyle \zeta (s)} on «short» intervals of the critical line. He proved that an analog of the Selberg conjecture holds for «almost all» intervals ( T , T + H ] {\displaystyle (T,T+H]} , H = T ε {\displaystyle H=T^{\varepsilon }} , where ε {\displaystyle \varepsilon } is an arbitrarily small fixed positive number. Karatsuba developed (1992) a new approach to investigating zeros of the Riemann zeta-function on «supershort» intervals of the critical line, that is, on the intervals ( T , T + H ] {\displaystyle (T,T+H]} , the length H {\displaystyle H} of which grows slower than any, even arbitrarily small degree T {\displaystyle T} . In particular, he proved that for any given numbers ε {\displaystyle \varepsilon } , ε 1 {\displaystyle \varepsilon _{1}} satisfying the conditions 0 < ε , ε 1 < 1 {\displaystyle 0<\varepsilon ,\varepsilon _{1}<1} almost all intervals ( T , T + H ] {\displaystyle (T,T+H]} for H ≥ exp { ( ln T ) ε } {\displaystyle H\geq \exp {\{(\ln T)^{\varepsilon }\}}} contain at least H ( ln T ) 1 − ε 1 {\displaystyle H(\ln T)^{1-\varepsilon _{1}}} zeros of the function ζ ( 1 2 + i t ) {\displaystyle \zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}} . This estimate is quite close to the one that follows from the Riemann hypothesis. ==== Zeros of linear combinations of Dirichlet L-series ==== Karatsuba developed a new method of investigating zeros of functions which can be represented as linear combinations of Dirichlet L {\displaystyle L} -series. The simplest example of a function of that type is the Davenport-Heilbronn function, defined by the equality f ( s ) = 1 2 ( 1 − i κ ) L ( s , χ ) + 1 2 ( 1 + i κ ) L ( s , χ ¯ ) , {\displaystyle f(s)={\tfrac {1}{2}}(1-i\kappa )L(s,\chi )+{\tfrac {1}{2}}(1\,+\,i\kappa )L(s,{\bar {\chi }}),} where χ {\displaystyle \chi } is a non-principal character modulo 5 {\displaystyle 5} ( χ ( 1 ) = 1 {\displaystyle \chi (1)=1} , χ ( 2 ) = i {\displaystyle \chi (2)=i} , χ ( 3 ) = − i {\displaystyle \chi (3)=-i} , χ ( 4 ) = − 1 {\displaystyle \chi (4)=-1} , χ ( 5 ) = 0 {\displaystyle \chi (5)=0} , χ ( n + 5 ) = χ ( n ) {\displaystyle \chi (n+5)=\chi (n)} for any n {\displaystyle n} ), κ = 10 − 2 5 − 2 5 − 1 . {\displaystyle \kappa ={\frac {{\sqrt {10-2{\sqrt {5}}}}-2}{{\sqrt {5}}-1}}.} For f ( s ) {\displaystyle f(s)} Riemann hypothesis is not true, however, the critical line R e s = 1 2 {\displaystyle Re\ s={\tfrac {1}{2}}} contains, nevertheless, abnormally many zeros. Karatsuba proved (1989) that the interval ( T , T + H ] {\displaystyle (T,T+H]} , H = T 27 / 82 + ε {\displaystyle H=T^{27/82+\varepsilon }} , contains at least H ( ln T ) 1 / 2 e − c ln ln T {\displaystyle H(\ln T)^{1/2}e^{-c{\sqrt {\ln \ln T}}}} zeros of the function f ( 1 2 + i t ) {\displaystyle f{\bigl (}{\tfrac {1}{2}}+it{\bigr )}} . Similar results were obtained by Karatsuba also for linear combinations containing arbitrary (finite) number of summands; the degree exponent 1 2 {\displaystyle {\tfrac {1}{2}}} is here replaced by a smaller number β {\displaystyle \beta } , that depends only on the form of the linear combination. ==== The boundary of zeros of the zeta function and the multi-dimensional problem of Dirichlet divisors ==== To Karatsuba belongs a new breakthrough result in the multi-dimensional problem of Dirichlet divisors, which is connected with finding the number D k ( x ) {\displaystyle D_{k}(x)} of solutions of the inequality x 1 ∗ … ∗ x k ≤ x {\displaystyle x_{1}*\ldots *x_{k}\leq x} in the natural numbers x 1 , … , x k {\displaystyle x_{1},\ldots ,x_{k}} as x → + ∞ {\displaystyle x\to +\infty } . For D k ( x ) {\displaystyle D_{k}(x)} there is an asymptotic formula of the form D k ( x ) = x P k − 1 ( ln x ) + R k ( x ) {\displaystyle D_{k}(x)=xP_{k-1}(\ln x)+R_{k}(x)} , where P k − 1 ( u ) {\displaystyle P_{k-1}(u)} is a polynomial of degree ( k − 1 ) {\displaystyle (k-1)} , the coefficients of which depend on k {\displaystyle k} and can be found explicitly and R k ( x ) {\displaystyle R_{k}(x)} is the remainder term, all known estimates of which (up to 1960) were of the form | R k ( x ) | ≤ x 1 − α ( k ) ( c ln x ) k {\displaystyle |R_{k}(x)|\leq x^{1-\alpha (k)}(c\ln x)^{k}} , where α = 1 a k + b {\displaystyle \alpha ={\frac {1}{ak+b}}} , a , b , c {\displaystyle a,b,c} are some absolute positive constants. Karatsuba obtained a more precise estimate of R k ( x ) {\displaystyle R_{k}(x)} , in which the value α ( k ) {\displaystyle \alpha (k)} was of order k − 2 / 3 {\displaystyle k^{-2/3}} and was decreasing much slower than α ( k ) {\displaystyle \alpha (k)} in the previous estimates. Karatsuba's estimate is uniform in x {\displaystyle x} and k {\displaystyle k} ; in particular, the value k {\displaystyle k} may grow as x {\displaystyle x} grows (as some power of the logarithm of x {\displaystyle x} ). (A similar looking, but weaker result was obtained in 1960 by a German mathematician Richert, whose paper remained unknown to Soviet mathematicians at least until the mid-seventies.) Proof of the estimate of R k ( x ) {\displaystyle R_{k}(x)} is based on a series of claims, essentially equivalent to the theorem on the boundary of zeros of the Riemann zeta function, obtained by the method of Vinogradov, that is, the theorem claiming that ζ ( s ) {\displaystyle \zeta (s)} has no zeros in the region R e s ≥ 1 − c ( ln | t | ) 2 / 3 ( ln ln | t | ) 1 / 3 , | t | > 10 {\displaystyle Re\ s\geq 1-{\frac {c}{(\ln |t|)^{2/3}(\ln \ln |t|)^{1/3}}},\quad |t|>10} . Karatsuba found (2000) the backward relation of estimates of the values R k ( x ) {\displaystyle R_{k}(x)} with the behaviour of ζ ( s ) {\displaystyle \zeta (s)} near the line R e s = 1 {\displaystyle Re\ s=1} . In particular, he proved that if α ( y ) {\displaystyle \alpha (y)} is an arbitrary non-increasing function satisfying the condition 1 / y ≤ α ( y ) ≤ 1 / 2 {\displaystyle 1/y\leq \alpha (y)\leq 1/2} , such that for all k ≥ 2 {\displaystyle k\geq 2} the estimate | R k ( x ) | ≤ x 1 − α ( k ) ( c ln x ) k {\displaystyle |R_{k}(x)|\leq x^{1-\alpha (k)}(c\ln x)^{k}} holds, then ζ ( s ) {\displaystyle \zeta (s)} has no zeros in the region R e s ≥ 1 − c 1 α ( ln | t | ) ln ln | t | , | t | ≥ e 2 {\displaystyle Re\ s\geq 1-c_{1}\,{\frac {\alpha (\ln |t|)}{\ln \ln |t|}},\quad |t|\geq e^{2}} ( c , c 1 {\displaystyle c,c_{1}} are some absolute constants). ==== Estimates from below of the maximum of the modulus of the zeta function in small regions of the critical domain and on small intervals of the critical line ==== Karatsuba introduced and studied the functions F ( T ; H ) {\displaystyle F(T;H)} and G ( s 0 ; Δ ) {\displaystyle G(s_{0};\Delta )} , defined by the equalities F ( T ; H ) = max | t − T | ≤ H | ζ ( 1 2 + i t ) | , G ( s 0 ; Δ ) = max | s − s 0 | ≤ Δ | ζ ( s ) | . {\displaystyle F(T;H)=\max _{|t-T|\leq H}{\bigl |}\zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}{\bigr |},\quad G(s_{0};\Delta )=\max _{|s-s_{0}|\leq \Delta }|\zeta (s)|.} Here T {\displaystyle T} is a sufficiently large positive number, 0 < H ≪ ln ln T {\displaystyle 0<H\ll \ln \ln T} , s 0 = σ 0 + i T {\displaystyle s_{0}=\sigma _{0}+iT} , 1 2 ≤ σ 0 ≤ 1 {\displaystyle {\tfrac {1}{2}}\leq \sigma _{0}\leq 1} , 0 < Δ < 1 3 {\displaystyle 0<\Delta <{\tfrac {1}{3}}} . Estimating the values F {\displaystyle F} and G {\displaystyle G} from below shows, how large (in modulus) values ζ ( s ) {\displaystyle \zeta (s)} can take on short intervals of the critical line or in small neighborhoods of points lying in the critical strip 0 ≤ R e s ≤ 1 {\displaystyle 0\leq Re\ s\leq 1} . The case H ≫ ln ln T {\displaystyle H\gg \ln \ln T} was studied earlier by Ramachandra; the case Δ > c {\displaystyle \Delta >c} , where c {\displaystyle c} is a sufficiently large constant, is trivial. Karatsuba proved, in particular, that if the values H {\displaystyle H} and Δ {\displaystyle \Delta } exceed certain sufficiently small constants, then the estimates F ( T ; H ) ≥ T − c 1 , G ( s 0 ; Δ ) ≥ T − c 2 , {\displaystyle F(T;H)\geq T^{-c_{1}},\quad G(s_{0};\Delta )\geq T^{-c_{2}},} hold, where c 1 , c 2 {\displaystyle c_{1},c_{2}} are certain absolute constants. ==== Behaviour of the argument of the zeta-function on the critical line ==== Karatsuba obtained a number of new results related to the behaviour of the function S ( t ) = 1 π arg ζ ( 1 2 + i t ) {\displaystyle S(t)={\frac {1}{\pi }}\arg {\zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}}} , which is called the argument of Riemann zeta function on the critical line (here arg ζ ( 1 2 + i t ) {\displaystyle \arg {\zeta {\bigl (}{\tfrac {1}{2}}+it{\bigr )}}} is the increment of an arbitrary continuous branch of arg ζ ( s ) {\displaystyle \arg \zeta (s)} along the broken line joining the points 2 , 2 + i t {\displaystyle 2,2+it} and 1 2 + i t {\displaystyle {\tfrac {1}{2}}+it} ). Among those results are the mean value theorems for the function S ( t ) {\displaystyle S(t)} and its first integral S 1 ( t ) = ∫ 0 t S ( u ) d u {\displaystyle S_{1}(t)=\int _{0}^{t}S(u)du} on intervals of the real line, and also the theorem claiming that every interval ( T , T + H ] {\displaystyle (T,T+H]} for H ≥ T 27 / 82 + ε {\displaystyle H\geq T^{27/82+\varepsilon }} contains at least H ( ln T ) 1 / 3 e − c ln ln T {\displaystyle H(\ln T)^{1/3}e^{-c{\sqrt {\ln \ln T}}}} points where the function S ( t ) {\displaystyle S(t)} changes sign. Earlier similar results were obtained by Atle Selberg for the case H ≥ T 1 / 2 + ε {\displaystyle H\geq T^{1/2+\varepsilon }} . === The Dirichlet characters === ==== Estimates of short sums of characters in finite fields ==== In the end of the sixties Karatsuba, estimating short sums of Dirichlet characters, developed a new method, making it possible to obtain non-trivial estimates of short sums of characters in finite fields. Let n ≥ 2 {\displaystyle n\geq 2} be a fixed integer, F ( x ) = x n + a n − 1 x n − 1 + … + a 1 x + a 0 {\displaystyle F(x)=x^{n}+a_{n-1}x^{n-1}+\ldots +a_{1}x+a_{0}} a polynomial, irreducible over the field Q {\displaystyle \mathbb {Q} } of rational numbers, θ {\displaystyle \theta } a root of the equation F ( θ ) = 0 {\displaystyle F(\theta )=0} , Q ( θ ) {\displaystyle \mathbb {Q} (\theta )} the corresponding extension of the field Q {\displaystyle \mathbb {Q} } , ω 1 , … , ω n {\displaystyle \omega _{1},\ldots ,\omega _{n}} a basis of Q ( θ ) {\displaystyle \mathbb {Q} (\theta )} , ω 1 = 1 {\displaystyle \omega _{1}=1} , ω 2 = θ {\displaystyle \omega _{2}=\theta } , ω 3 = θ 2 , … , ω n = θ n − 1 {\displaystyle \omega _{3}=\theta ^{2},\ldots ,\omega _{n}=\theta ^{n-1}} . Furthermore, let p {\displaystyle p} be a sufficiently large prime, such that F ( x ) {\displaystyle F(x)} is irreducible modulo p {\displaystyle p} , G F ( p n ) {\displaystyle \mathrm {GF} (p^{n})} the Galois field with a basis ω 1 , ω 2 , … , ω n {\displaystyle \omega _{1},\omega _{2},\ldots ,\omega _{n}} , χ {\displaystyle \chi } a non-principal Dirichlet character of the field G F ( p n ) {\displaystyle \mathrm {GF} (p^{n})} . Finally, let ν 1 , … , ν n {\displaystyle \nu _{1},\ldots ,\nu _{n}} be some nonnegative integers, D ( X ) {\displaystyle D(X)} the set of elements x ¯ {\displaystyle {\bar {x}}} of the Galois field G F ( p n ) {\displaystyle \mathrm {GF} (p^{n})} , x ¯ = x 1 ω 1 + … + x n ω n {\displaystyle {\bar {x}}=x_{1}\omega _{1}+\ldots +x_{n}\omega _{n}} , such that for any i {\displaystyle i} , 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} , the following inequalities hold: ν i < x i < ν i + X {\displaystyle \nu _{i}<x_{i}<\nu _{i}+X} . Karatsuba proved that for any fixed k {\displaystyle k} , k ≥ n + 1 {\displaystyle k\geq n+1} , and arbitrary X {\displaystyle X} satisfying the condition p 1 4 + 1 4 k ≤ X ≤ p 1 2 + 1 4 k {\displaystyle p^{{\frac {1}{4}}+{\frac {1}{4k}}}\leq X\leq p^{{\frac {1}{2}}+{\frac {1}{4k}}}} the following estimate holds: | ∑ x ¯ ∈ D ( X ) χ ( x ¯ ) | ≤ c ( X 1 − 1 k p 1 4 k + 1 4 k 2 ) n ( ln p ) γ , {\displaystyle {\biggl |}\sum \limits _{{\bar {x}}\in D(X)}\chi ({\bar {x}}){\biggr |}\leq c{\Bigl (}X^{1-{\frac {1}{k}}}p^{{\frac {1}{4k}}+{\frac {1}{4k^{2}}}}{\Bigr )}^{\!\!n}(\ln p)^{\gamma },} where γ = 1 k ( 2 n + 1 − 1 ) {\displaystyle \gamma ={\frac {1}{k}}(2^{n+1}-1)} , and the constant c {\displaystyle c} depends only on n {\displaystyle n} and the basis ω 1 , … , ω n {\displaystyle \omega _{1},\ldots ,\omega _{n}} . ==== Estimates of linear sums of characters over shifted prime numbers ==== Karatsuba developed a number of new tools, which, combined with the Vinogradov method of estimating sums with prime numbers, enabled him to obtain in 1970 an estimate of the sum of values of a non-principal character modulo a prime q {\displaystyle q} on a sequence of shifted prime numbers, namely, an estimate of the form | ∑ p ≤ N χ ( p + k ) | ≤ c N q − ε 2 1024 , {\displaystyle {\biggl |}\sum \limits _{p\leq N}\chi (p+k){\biggr |}\leq cNq^{-{\frac {\varepsilon ^{2}}{1024}}},} where k {\displaystyle k} is an integer satisfying the condition k ≢ 0 ( mod q ) {\displaystyle k\not \equiv 0(\mod q)} , ε {\displaystyle \varepsilon } an arbitrarily small fixed number, N ≥ q 1 / 2 + ε {\displaystyle N\geq q^{1/2+\varepsilon }} , and the constant c {\displaystyle c} depends on ε {\displaystyle \varepsilon } only. This claim is considerably stronger than the estimate of Vinogradov, which is non-trivial for N ≥ q 3 / 4 + ε {\displaystyle N\geq q^{3/4+\varepsilon }} . In 1971 speaking at the International conference on number theory on the occasion of the 80th birthday of Ivan Matveyevich Vinogradov, Academician Yuri Linnik noted the following: «Of a great importance are the investigations carried out by Vinogradov in the area of asymptotics of Dirichlet character on shifted primes ∑ p ≤ N χ ( p + k ) {\displaystyle \sum \limits _{p\leq N}\chi (p+k)} , which give a decreased power compared to N {\displaystyle N} compared to N ≥ q 3 / 4 + ε {\displaystyle N\geq q^{3/4+\varepsilon }} , ε > 0 {\displaystyle \varepsilon >0} , where q {\displaystyle q} is the modulus of the character. This estimate is of crucial importance, as it is so deep that gives more than the extended Riemann hypothesis, and, it seems, in that directions is a deeper fact than that conjecture (if the conjecture is true). Recently this estimate was improved by A.A.Karatsuba». This result was extended by Karatsuba to the case when p {\displaystyle p} runs through the primes in an arithmetic progression, the increment of which grows with the modulus q {\displaystyle q} . ==== Estimates of sums of characters on polynomials with a prime argument ==== Karatsuba found a number of estimates of sums of Dirichlet characters in polynomials of degree two for the case when the argument of the polynomial runs through a short sequence of subsequent primes. Let, for instance, q {\displaystyle q} be a sufficiently high prime, f ( x ) = ( x − a ) ( x − b ) {\displaystyle f(x)=(x-a)(x-b)} , where a {\displaystyle a} and b {\displaystyle b} are integers, satisfying the condition a b ( a − b ) ≢ 0 ( mod q ) {\displaystyle ab(a-b)\not \equiv 0(\mod q)} , and let ( n q ) {\displaystyle \left({\frac {n}{q}}\right)} denote the Legendre symbol, then for any fixed ε {\displaystyle \varepsilon } with the condition 0 < ε < 1 2 {\displaystyle 0<\varepsilon <{\tfrac {1}{2}}} and N > q 3 / 4 + ε {\displaystyle N>q^{3/4+\varepsilon }} for the sum S N {\displaystyle S_{N}} , S N = ∑ p ≤ N ( f ( p ) q ) , {\displaystyle S_{N}=\sum \limits _{p\leq N}{\biggl (}{\frac {f(p)}{q}}{\biggr )},} the following estimate holds: | S N | ≤ c π ( N ) q − ε 2 100 {\displaystyle |S_{N}|\leq c\pi (N)q^{-{\frac {\varepsilon ^{2}}{100}}}} (here p {\displaystyle p} runs through subsequent primes, π ( N ) {\displaystyle \pi (N)} is the number of primes not exceeding N {\displaystyle N} , and c {\displaystyle c} is a constant, depending on ε {\displaystyle \varepsilon } only). A similar estimate was obtained by Karatsuba also for the case when p {\displaystyle p} runs through a sequence of primes in an arithmetic progression, the increment of which may grow together with the modulus q {\displaystyle q} . Karatsuba conjectured that the non-trivial estimate of the sum S N {\displaystyle S_{N}} for N {\displaystyle N} , which are "small" compared to q {\displaystyle q} , remains true in the case when f ( x ) {\displaystyle f(x)} is replaced by an arbitrary polynomial of degree n {\displaystyle n} , which is not a square modulo q {\displaystyle q} . This conjecture is still open. ==== Lower bounds for sums of characters in polynomials ==== Karatsuba constructed an infinite sequence of primes p {\displaystyle p} and a sequence of polynomials f ( x ) {\displaystyle f(x)} of degree n {\displaystyle n} with integer coefficients, such that f ( x ) {\displaystyle f(x)} is not a full square modulo p {\displaystyle p} , 4 ( p − 1 ) ln p ≤ n ≤ 8 ( p − 1 ) ln p , {\displaystyle {\frac {4(p-1)}{\ln p}}\leq n\leq {\frac {8(p-1)}{\ln p}},} and such that ∑ x = 1 p ( f ( x ) p ) = p . {\displaystyle \sum \limits _{x=1}^{p}\left({\frac {f(x)}{p}}\right)=p.} In other words, for any x {\displaystyle x} the value f ( x ) {\displaystyle f(x)} turns out to be a quadratic residues modulo p {\displaystyle p} . This result shows that André Weil's estimate | ∑ x = 1 p ( f ( x ) p ) | ≤ ( n − 1 ) p {\displaystyle {\biggl |}\sum \limits _{x=1}^{p}\left({\frac {f(x)}{p}}\right){\biggr |}\leq (n-1){\sqrt {p}}} cannot be essentially improved and the right hand side of the latter inequality cannot be replaced by say the value C n p {\displaystyle C{\sqrt {n}}{\sqrt {p}}} , where C {\displaystyle C} is an absolute constant. ==== Sums of characters on additive sequences ==== Karatsuba found a new method, making it possible to obtain rather precise estimates of sums of values of non-principal Dirichlet characters on additive sequences, that is, on sequences consisting of numbers of the form x + y {\displaystyle x+y} , where the variables x {\displaystyle x} and y {\displaystyle y} runs through some sets A {\displaystyle A} and B {\displaystyle B} independently of each other. The most characteristic example of that kind is the following claim which is applied in solving a wide class of problems, connected with summing up values of Dirichlet characters. Let ε {\displaystyle \varepsilon } be an arbitrarily small fixed number, 0 < ε < 1 2 {\displaystyle 0<\varepsilon <{\tfrac {1}{2}}} , q {\displaystyle q} a sufficiently large prime, χ {\displaystyle \chi } a non-principal character modulo q {\displaystyle q} . Furthermore, let A {\displaystyle A} and B {\displaystyle B} be arbitrary subsets of the complete system of congruence classes modulo q {\displaystyle q} , satisfying only the conditions ‖ A ‖ > q ε {\displaystyle \|A\|>q^{\varepsilon }} , ‖ B ‖ > q 1 / 2 + ε {\displaystyle \|B\|>q^{1/2+\varepsilon }} . Then the following estimate holds: | ∑ x ∈ A ∑ y ∈ B χ ( x + y ) | ≤ c ‖ A ‖ ⋅ ‖ B ‖ q − ε 2 20 , c = c ( ε ) > 0. {\displaystyle {\biggl |}\sum \limits _{x\in A}\sum \limits _{y\in B}\chi (x+y){\biggr |}\leq c\|A\|\cdot \|B\|q^{-{\frac {\varepsilon ^{2}}{20}}},\quad c=c(\varepsilon )>0.} Karatsuba's method makes it possible to obtain non-trivial estimates of that sort in certain other cases when the conditions for the sets A {\displaystyle A} and B {\displaystyle B} , formulated above, are replaced by different ones, for example: ‖ A ‖ > q ε {\displaystyle \|A\|>q^{\varepsilon }} , ‖ A ‖ ⋅ ‖ B ‖ > q 1 / 2 + ε . {\displaystyle {\sqrt {\|A\|}}\cdot \|B\|>q^{1/2+\varepsilon }.} In the case when A {\displaystyle A} and B {\displaystyle B} are the sets of primes in intervals ( 1 , X ] {\displaystyle (1,X]} , ( 1 , Y ] {\displaystyle (1,Y]} respectively, where X ≥ q 1 / 4 + ε {\displaystyle X\geq q^{1/4+\varepsilon }} , Y ≥ q 1 / 4 + ε {\displaystyle Y\geq q^{1/4+\varepsilon }} , an estimate of the form | ∑ p ≤ X ∑ p ′ ≤ Y χ ( p + p ′ ) | ≤ c π ( X ) π ( Y ) q − c 1 ε 2 , {\displaystyle {\biggl |}\sum \limits _{p\leq X}\sum \limits _{p'\leq Y}\chi (p+p'){\biggr |}\leq c\pi (X)\pi (Y)q^{-c_{1}\varepsilon ^{2}},} holds, where π ( Z ) {\displaystyle \pi (Z)} is the number of primes, not exceeding Z {\displaystyle Z} , c = c ( ε ) > 0 {\displaystyle c=c(\varepsilon )>0} , and c 1 {\displaystyle c_{1}} is some absolute constant. ==== Distribution of power congruence classes and primitive roots in sparse sequences ==== Karatsuba obtained (2000) non-trivial estimates of sums of values of Dirichlet characters "with weights", that is, sums of components of the form χ ( n ) f ( n ) {\displaystyle \chi (n)f(n)} , where f ( n ) {\displaystyle f(n)} is a function of natural argument. Estimates of that sort are applied in solving a wide class of problems of number theory, connected with distribution of power congruence classes, also primitive roots in certain sequences. Let k ≥ 2 {\displaystyle k\geq 2} be an integer, q {\displaystyle q} a sufficiently large prime, ( a , q ) = 1 {\displaystyle (a,q)=1} , | a | ≤ q {\displaystyle |a|\leq {\sqrt {q}}} , N ≥ q 1 2 − 1 2 ( k + 1 ) + ε {\displaystyle N\geq q^{{\frac {1}{2}}-{\frac {1}{2(k+1)}}+\varepsilon }} , where 0 < ε < min { 0.01 , 2 3 ( k + 1 ) } {\displaystyle 0<\varepsilon <\min {\{0.01,{\tfrac {2}{3(k+1)}}\}}} , and set, finally, D k ( x ) = ∑ x 1 ∗ … ∗ x k ≤ x 1 = ∑ n ≤ x τ k ( n ) {\displaystyle D_{k}(x)=\sum \limits _{x_{1}*\ldots *x_{k}\leq x}1=\sum \limits _{n\leq x}\tau _{k}(n)} (for an asymptotic expression for D k ( x ) {\displaystyle D_{k}(x)} , see above, in the section on the multi-dimensional problem of Dirichlet divisors). For the sums V 1 ( x ) {\displaystyle V_{1}(x)} and V 2 ( x ) {\displaystyle V_{2}(x)} of the values τ k ( n ) {\displaystyle \tau _{k}(n)} , extended on the values n ≤ x {\displaystyle n\leq x} , for which the numbers ( n + a ) {\displaystyle (n+a)} are quadratic residues (respectively, non-residues) modulo q {\displaystyle q} , Karatsuba obtained asymptotic formulas of the form V 1 ( x ) = 1 2 D k ( x ) + O ( x q − 0.01 ε 2 ) , V 2 ( x ) = 1 2 D k ( x ) + O ( x q − 0.01 ε 2 ) {\displaystyle V_{1}(x)={\tfrac {1}{2}}D_{k}(x)+O{\bigl (}xq^{-0.01\varepsilon ^{2}}{\bigr )},\quad V_{2}(x)={\tfrac {1}{2}}D_{k}(x)+O{\bigl (}xq^{-0.01\varepsilon ^{2}}{\bigr )}} . Similarly, for the sum V ( x ) {\displaystyle V(x)} of values τ k ( n ) {\displaystyle \tau _{k}(n)} , taken over all n ≤ x {\displaystyle n\leq x} , for which ( n + a ) {\displaystyle (n+a)} is a primitive root modulo q {\displaystyle q} , one gets an asymptotic expression of the form V ( x ) = ( 1 − 1 p 1 ) … ( 1 − 1 p s ) D k ( x ) + O ( x q − 0.01 ε 2 ) {\displaystyle V(x)=\left(1-{\frac {1}{p_{1}}}\right)\ldots \left(1-{\frac {1}{p_{s}}}\right)D_{k}(x)+O{\bigl (}xq^{-0.01\varepsilon ^{2}}{\bigr )}} , where p 1 , … , p s {\displaystyle p_{1},\ldots ,p_{s}} are all prime divisors of the number q − 1 {\displaystyle q-1} . Karatsuba applied his method also to the problems of distribution of power residues (non-residues) in the sequences of shifted primes p + a {\displaystyle p+a} , of the integers of the type x 2 + y 2 + a {\displaystyle x^{2}+y^{2}+a} and some others. == Late work == In his later years, apart from his research in number theory (see Karatsuba phenomenon), Karatsuba studied certain problems of theoretical physics, in particular in the area of quantum field theory. Applying his ATS theorem and some other number-theoretic approaches, he obtained new results in the Jaynes–Cummings model in quantum optics. == Awards and titles == 1981: P.L.Tchebyshev Prize of Soviet Academy of Sciences 1999: Distinguished Scientist of Russia 2001: I.M.Vinogradov Prize of Russian Academy of Sciences == See also == ATS theorem Karatsuba algorithm Moore machine == References == G. I. Archipov; V. N. Chubarikov (1997). "On the mathematical works of professor A. A. Karatsuba". Proc. Steklov Inst. Math. 218. == External links == Anatoly Karatsuba at the Mathematics Genealogy Project "Karatsuba Anatolii Alexeevitch (personal home page)". Archived from the original on 6 October 2008. Retrieved November 17, 2008. List of Research Works at Steklov Institute of Mathematics |
Wikipedia:Anatoly Samoilenko#0 | Anatoly Mykhailovych Samoilenko (Ukrainian: Анато́лій Миха́йлович Само́йленко) (2 January 1938 – 4 December 2020) was a Ukrainian mathematician, an Academician of the National Academy of Sciences of Ukraine (since 1995), the Director of the Institute of Mathematics of the National Academy of Sciences of Ukraine (since 1988). == Biography == Anatoly Samoilenko was born in 1938 in the village of Potiivka, Radomyshl district, Zhytomyr region. In 1955, he entered the Geologic Department at the Shevchenko Kyiv State University. In 1960, Samoilenko graduated from the Department of Mechanics and Mathematics at the Shevchenko Kyiv State University with mathematics specialization. At the same time, his first scientific works were published. In 1963, after the graduation from the postgraduate courses at the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR, Samoilenko defended his candidate-degree thesis "Application of Asymptotic Methods to the Investigation of Nonlinear Differential Equations with Irregular Right-Hand Side" and began his work at the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR under the supervision of Academician Yu. A. Mitropolskiy. In few years of diligent research work, Samoilenko became one of the leading experts in the qualitative theory of differential equations. In 1967, based on the results of his research in the theory of multifrequency oscillations, he defended his doctoral-degree thesis "Some Problems of the Theory of Periodic and Quasiperiodic Systems", the official opponents of which were V. I. Arnold and D. V. Anosov. In 1965–1974, Samoilenko worked as a senior research fellow at the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR and gave lectures at the Shevchenko Kyiv State University. In 1974, he obtained the professor degree. In 1978, he was elected to become a Corresponding Member of the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR. His monograph brought him worldwide recognition. This monograph was written by Samoilenko together with his teachers, Academicians N. N. Bogolyubov and Mitropolskiy. Thirty six years later, Samoilenko reminisced, "In Kyiv, at the Institute of Mathematics, great scientists were my teachers... In many fields of science, they were 'trendsetters' on the scale of the Soviet Union. It is very important for a young scientist to belong to a serious scientific school. Probably, only in this case he has a chance to obtain results at the world level. The atmosphere of a good scientific school itself stimulates a young scientist to carry out his research work at the cutting edge of modern science. And if he suddenly opens a new direction in science, then his name immediately gains recognition". In 1974–1987, Samoilenko headed the Chair of Integral and Differential Equations of the Department of Mechanics and Mathematics at the Shevchenko Kyiv State University. These years were marked by especially high scientific activity of the chair. Based on results of the research in the theory of differential equations with delay performed at that time, the monograph of Mitropolskiy, Samoilenko, and D. I. Martynyuk was published. At the same time, Samoilenko, together with his disciple M. O. Perestyuk, published the well-known monograph devoted to the theory of impulsive differential equations. These monographs (especially their English translations) are frequently cited in scientific literature. Since 1987, Samoilenko has headed the Department of Ordinary Differential Equations at the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR (at present, the Department of Differential Equations and Theory of Oscillations at the Institute of Mathematics of the National Academy of Sciences of Ukraine), and since 1988 he has been the Director of the Institute of Mathematics of the National Academy of Sciences of Ukraine. The beginning of this fruitful creative period was marked by the fundamental monograph devoted to the qualitative theory of invariant manifolds of dynamical systems. This monograph served as a foundation for the construction of the general perturbation theory of invariant tori of nonlinear dynamical systems on a torus. The English version of this monograph is also well known. Three years later, the monograph of Samoilenko (in coauthorship with Mitropol'skii and V. L. Kulyk) was published. In this monograph, in particular, the method of Lyapunov functions was used for the investigation of dichotomies in linear differential systems of the general form. The results of many-year investigations of constructive methods in the theory of boundary-valued problems for ordinary differential equations carried out by Samoilenko together with M. Ronto are presented in monographs. Constructive algorithms for finding solutions of boundary-value problems with different classes of multipoint boundary conditions were developed by Samoilenko, V. M. Laptyns'kyi, and K. Kenzhebaev; the obtained results are presented in monograph. Complex classes of resonance boundary-value problems whose linear pan cannot be described by Fredholm operators of index zero were investigated by Samoilenko, together with O. A. Boichuk and V. F. Zhuravlev, in monographs. The monograph of Samoilenko and Yu. V. Teplins'kyi is devoted to the theory of countable systems of ordinary differential equations. The monographs of Samoilenko and R. I. Petryshyn cover a broad class of qualitative problems in the theory of nonlinear dynamical systems on a torus. Samoilenko is the author of about 400 scientific works, including 30 monographs and 15 textbooks, most of which have been translated into foreign languages. His monographs made an important contribution to mathematical science and education. According to MathSciNet, the scientific papers of Samoilenko were cited 336 times by 208 authors. The scientific interests of Samoilenko covered a broad range of important problems in the qualitative theory of differential equations, nonlinear mechanics, and the theory of nonlinear oscillations. His deep results in the theory of multifrequency oscillations, perturbation theory of toroidal manifolds, asymptotic methods of nonlinear mechanics, theory of impulsive systems, theory of differential equations with delay, and theory of boundary-value problems were highly appreciated in Ukraine and abroad. Academician Samoilenko was the founder of a scientific school in the theory of multifrequency oscillations and theory of impulsive systems recognized by the international mathematical community. His successful many-year guidance of the Institute of Mathematics of the Ukrainian National Academy of Sciences furthered the rapid development of mathematics in Ukraine and the continuation of the best traditions of the world-known Bogolyubov – Krylov – Mitropolskiy Kyiv scientific school. The worldwide recognition of Samoilenko's mathematical results is illustrated by notions well known in the mathematical literature such as the Samoilenko numerical-analytic method and the Samoilenko – Green function (the kernel of an integral operator related to the problem of an invariant torus of a dynamical system). Samoilenko gave much attention to training scientists of the highest qualification. For many years, he had given lectures at the Shevchenko Kyiv National University and the "Kyiv Polytechnic Institute" National Technical University and guided the scientific work of postgraduate and doctoral students. Despite the extremely busy schedule of his work as the Director of the Institute of Mathematics of the Ukrainian National Academy of Sciences for about 20 last years (since 2006, he was the Academician-Secretary of the Department of Mathematics at the National Academy of Sciences of Ukraine), Samoilenko found time for organizational and public activities. In particular, Samoilenko was the President of the "Foundation for Support of the Development of Mathematical Sciences" All-Ukrainian charity organization. Many young talents from the "small homeland" of Samoilenko (Malynshchyna) are grateful to him for founding and heading the charity foundation for support of the development of gifted children and youth. Samoilenko found and taught many nonordinary scientists. He created an international scientific school in differential equations. Among his disciples, there are 33 doctors and 82 candidates of physical and mathematical sciences, who are now researchers of prestigious scientific institutions, professors, heads of chairs, deans, and rectors (scientific researchers, pedagogs, and administrators of various levels). For example, Samoilenko's alma mater (the Department of Mechanics and Mathematics at the Shevchenko Kyiv National University) has been headed for many years by his disciples (Professors M. O. Perestyuk and I. O. Parasyuk). Among other well-known scientists belonging to Samoilenko's mathematical school, one may mention Professor Kenzhebaev, the rector of the Zhubanov Aktobe University, one of the most reputable universities in Kazakhstan, and Academician M. Ilolov, the President of the Tajik Academy of Sciences. Samoilenko was a member of the Ukrainian Mathematical Society, the American Mathematical Society, and the editorial boards of numerous Ukrainian and foreign mathematical journals, among which there are Differential Equations, Reports of the National Academy of Sciences of Ukraine, In the world of mathematics, Nonlinear Mathematical Physics, Memoirs on Differential Equations and Mathematical Physics, and Miskolc Mathematical Notes. He is an editor-in-chief of the Ukrainian Mathematical Journal, Nonlinear Oscillations journal and the Ukrainian Mathematical Bulletin. Samoilenko was a full member of the National Academy of Sciences of Ukraine (since 1995) and the European Academy of Sciences (since 2002). He was a Foreign Member of the Tajik Academy of Sciences (since 2011). Samoilenko was awarded Order of Friendship of Peoples (1984), and Order of Merit of degree III (2003), Order of Prince Yaroslav the Wise of degree V (2008), a Diploma of the Presidium of the Supreme Soviet of Ukraine (1987), and the titles of an Honored Scientist of Ukraine (1998) and a Soros Professor (1998). He was also awarded the State Prize of Ukraine in the Field of Science and Engineering (1985 and 1996), State Prize of Ukraine in the field of education (2012), Ostrovsky Prize (1968), Krylov Prize (1981), Bogolyubov Prize (1998), Lavrentyev Prize (2000), Ostrogradsky Prize (2004) and Mitropolskiy Prize (2010). == References == == External links == O'Connor, John J.; Robertson, Edmund F., "Anatoly Samoilenko", MacTutor History of Mathematics Archive, University of St Andrews A.A. Boichuk, A.G. Mazko, A.A. Martynyuk and M.O. Perestyuk. Academician A.M. Samoilenko. On His 75th Birthday. Nonlinear Dynamics and Systems Theory, 13 (2) (2013), 107–113. Personal site. Google Scholar Citations. |
Wikipedia:Anatoly Styopin#0 | Anatoly Mikhailovich Styopin (Russian: Анатолий Михайлович Стёпин (may be transliterated as Stepin), 20 July 1940 – 7 November 2020) was a Soviet-Russian mathematician, specializing in dynamical systems and ergodic theory. == Education and career == Stepin was born in Moscow on 20 July 1940. In 1965 he graduated from the Mechanics and Mathematics Faculty of Moscow State University. There he received in 1968 his Ph.D. under Felix Berezin with thesis "Применение метода аппроксимации динамических систем периодическими в спектральной теории" (Application of the method of approximation of dynamical systems by periodic spectral theory) and in 1986 his Russian doctorate (Doctor Nauk) with thesis "Спектральные и метрические свойства динамических систем и групп преобразований" (Spectral and metric properties of dynamical systems and groups of transformations). In 1970 he was an Invited Speaker at the ICM in Nice. In 1993 he was awarded the academic title of Professor in Mathematics. Since 1993, he has taught at the department of the theory of functions and functional analysis of the Mechanics and Mathematics Faculty of Moscow State University. In 2009 he was awarded the title of Honorary Professor of Moscow State University. His doctoral students include Rostislav Grigorchuk and Yiangdong Ye. On 7 November 2020, Stepin died at the age of 80. == Awards == Kolmogorov Prize (together with Boris Markovich Gurevich and Valery lustinovich Oseledets) for 2009 — for their series of works ergodic theory and related topics Award of the Moscow Mathematical Society == References == == External links == "Стёпин Анатолий Михайлович (Летопись Московского университета)". letopis.msu.ru. "К юбилею Анатолия Михайловича Степина". msu.ru. "Персоналии: Степин Анатолий Михайлович". mathnet.ru. "Степин Анатолий Михайлович - пользователь, сотрудник". istina.msu.ru. |
Wikipedia:Anchor losses#0 | Anchor losses are a type of damping commonly highlighted in micro-resonators. They refer to the phenomenon where energy is dissipated as mechanical waves from the resonator attenuate into the substrate. == Introduction == In physical systems, damping is the loss of energy of an oscillating system by dissipation. In the field of micro-electro-mechanicals, the damping is usually measured by a dimensionless parameter Q factor (Quality factor). A higher Q factor indicates lower damping and reduced energy dissipation, which is desirable for micro-resonators as it leads to lower energy consumption, better accuracy and efficiency, and reduced noise. Several factors contribute to the damping of micro-electro-mechanical resonators, including fluid damping and solid damping. Anchor losses are a type of solid damping observed in resonators operating in various environments. When a resonator is fixed to a substrate, either directly or via other structures such as tethers, mechanical waves propagate into the substrate through these connections. The wave traveling through a perfectly elastic solid would have a constant energy and an isolated perfectly elastic solid once set into vibration would continue to vibrate indefinitely. Actual materials do not show such behavior and dissipation will happen due to some imperfection of elasticity within the body. In typical micro-resonators, the substrate dimensions are significantly larger than those of the resonator itself. Consequently, it can be approximated that all waves entering the substrate will attenuate without reflecting back to the resonator. In other words, the energy carried by the waves will dissipate, leading to damping. This phenomenon is referred to as anchor losses. == Estimation of anchor losses == === Analytical estimation === Standard theories of structural mechanics permit the expression of concentrated forces and couples exerted by the structure on the support.These generally include a constant component (due, for instance, to pre-stresses or initial deformation) and a sinusoidal varying contribution. Some researchers have investigated some simple geometries following this idea, and one example is the anchor losses of a cantilaver beam connected to a 3-D semi-infinite region: Q a n c h o r ≈ C ( L W ) ( L H ) 4 {\displaystyle Q_{anchor}\approx C\left({\frac {L}{W}}\right)\left({\frac {L}{H}}\right)^{4}} where L is the length of the beam, H is the in-plane (curvature plane) thickness, W is the out-of-plane thickness, C is a constant depending on the Poisson's coefficient, with C = 3.45 for ν = 0.25, C = 3.23 for ν = 0.3, C = 3.175 for ν = 0.33. === Numerical estimation === Due to the complexity of geometries and the anisotropy or inhomogeneities of materials, usually it is difficult to use analytical method to estimate the anchor losses of some devices. Numerical methods are more widely applied for this issue. An artificial boundary or an artificial absorbing layer is applied to the numerical model to prevent the wave reflection. One such method is the perfectly matched layer, initially developed for electromagnetic wave transmission and later adapted for solid mechanics. Perfectly matched layers act as special elements where wave attenuation occurs through a complex coordinate transformation, ensuring all waves entering the layer are absorbed, thus simulating anchor losses. To determine the Q factor from a Finite Element Method model with perfectly matched layers, two common approaches are used: Using the complex eigenfrequency from a modal analysis: Q = R e ( ω ) 2 I m ( ω ) {\displaystyle Q={\frac {Re(\omega )}{2Im(\omega )}}} where R e ( ω ) {\textstyle Re(\omega )} and I m ( ω ) {\textstyle Im(\omega )} is the real and imaginary part of the complex eigenfrequency. Generating the frequency response from a frequency domain analysis and applying methods such as the half-bandwidth method to calculate the Q factor. == Methods to mitigate anchor losses == Anchor losses are highly dependent on the geometry of the resonator. How to anchor the resonator or the size of the tether has a strong effect on the anchor losses. Some common methods to eliminate anchor losses are summarized as followings. === Anchor at nodal points === A common method is to fix the resonator at the nodal points, where the motion amplitude is minimum. From the definitions of anchor losses, now the wave magnitude into the substrate will be minimized and less energy will dissipate. However, this method may not apply to certain resonators, in which the nodal points are not around the resonator edges, causing difficulty in tether designs. === Quater wavelength tethers === Quarter wavelength tether is an effective approach to minimize the energy loss through these tethers. Similar to the theory used for transmission lines, quarter wavelength tether is assumed as the best acoustic isolation, since the complete in phase reflection occurs as the tether length equals to a quarter acoustic wavelength, or λ/4. Therefore, there is hardly any energy dissipation to the substrate through the tethers. However, quarter-wavelength design results in extremely long tether structures, usually in tens to hundreds of micrometers, which is counter to minimization and leads to a decrease in the mechanical stability of the devices. === Material-mismatched support === The resonator structure and anchoring stem are made with different materials. The acoustic impedance mismatch between these two suppresses the energy from the resonator to the stem, thus reducing anchor losses and allowing high Q factor. === Acoustic reflection cavity === The basic mechanism is to reflect back a portion of the elastic waves at the anchor boundary due to the discontinuity in the acoustic impedance caused by the acoustic cavity (the etching trenches). === Phonon crystal tether and metamaterial === Phonon crystal tether is a promising way to restrain the acoustic wave propagation in the supporting tethers, since they can arouse complete band gaps in which the transmissions of the elastic waves are prohibited. Thus, the vibration energy is retained in the resonator body, reducing the anchor losses into the substrate. Besides the phonon crystal tether, some other kinds of metamaterial could be applied to the anchor and surrounding regions to prohibit the wave transmission. A key drawback of this method is the challenge to the fabrication process. === Optimized anchor geometry === Anchor losses are highly sensitive to the geometry of the anchors. Features such as fillets, curvature, sidewall inclination, and other detailed geometric aspects can affect anchor losses. By carefully optimizing these geometric configurations, anchor losses can be significantly reduced. == See also == Dynamical systems theory Finite element method Finite-difference time-domain method Micro-Electro-Mechanical Systems Resonator Infinite element method == References == == External links == How to Model Different Types of Damping in COMSOL Multiphysics® Effect of Perfectly Matched Layers (PML) in FDTD Simulations Notes on Perfectly Matched Layers (PMLs) |
Wikipedia:Ancient Egyptian mathematics#0 | Ancient Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt c. 3000 to c. 300 BCE, from the Old Kingdom of Egypt until roughly the beginning of Hellenistic Egypt. The ancient Egyptians utilized a numeral system for counting and solving written mathematical problems, often involving multiplication and fractions. Evidence for Egyptian mathematics is limited to a scarce amount of surviving sources written on papyrus. From these texts it is known that ancient Egyptians understood concepts of geometry, such as determining the surface area and volume of three-dimensional shapes useful for architectural engineering, and algebra, such as the false position method and quadratic equations. == Overview == Written evidence of the use of mathematics dates back to at least 3200 BC with the ivory labels found in Tomb U-j at Abydos. These labels appear to have been used as tags for grave goods and some are inscribed with numbers. Further evidence of the use of the base 10 number system can be found on the Narmer Macehead which depicts offerings of 400,000 oxen, 1,422,000 goats and 120,000 prisoners. Archaeological evidence has suggested that the Ancient Egyptian counting system had origins in Sub-Saharan Africa. Also, fractal geometry designs which are widespread among Sub-Saharan African cultures are also found in Egyptian architecture and cosmological signs. The evidence of the use of mathematics in the Old Kingdom (c. 2690–2180 BC) is scarce, but can be deduced from inscriptions on a wall near a mastaba in Meidum which gives guidelines for the slope of the mastaba. The lines in the diagram are spaced at a distance of one cubit and show the use of that unit of measurement. The earliest true mathematical documents date to the 12th Dynasty (c. 1990–1800 BC). The Moscow Mathematical Papyrus, the Egyptian Mathematical Leather Roll, the Lahun Mathematical Papyri which are a part of the much larger collection of Kahun Papyri and the Berlin Papyrus 6619 all date to this period. The Rhind Mathematical Papyrus which dates to the Second Intermediate Period (c. 1650 BC) is said to be based on an older mathematical text from the 12th dynasty. The Moscow Mathematical Papyrus and Rhind Mathematical Papyrus are so called mathematical problem texts. They consist of a collection of problems with solutions. These texts may have been written by a teacher or a student engaged in solving typical mathematics problems. An interesting feature of ancient Egyptian mathematics is the use of unit fractions. The Egyptians used some special notation for fractions such as 1/2, 1/3 and 2/3 and in some texts for 3/4, but other fractions were all written as unit fractions of the form 1/n or sums of such unit fractions. Scribes used tables to help them work with these fractions. The Egyptian Mathematical Leather Roll for instance is a table of unit fractions which are expressed as sums of other unit fractions. The Rhind Mathematical Papyrus and some of the other texts contain 2/n tables. These tables allowed the scribes to rewrite any fraction of the form 1/n as a sum of unit fractions. During the New Kingdom (c. 1550–1070 BC) mathematical problems are mentioned in the literary Papyrus Anastasi I, and the Papyrus Wilbour from the time of Ramesses III records land measurements. In the workers village of Deir el-Medina several ostraca have been found that record volumes of dirt removed while quarrying the tombs. == Sources == Current understanding of ancient Egyptian mathematics is impeded by the paucity of available sources. The sources that do exist include the following texts (which are generally dated to the Middle Kingdom and Second Intermediate Period): The Moscow Mathematical Papyrus The Egyptian Mathematical Leather Roll The Lahun Mathematical Papyri The Berlin Papyrus 6619, written around 1800 BC The Akhmim Wooden Tablet The Reisner Papyrus, dated to the early Twelfth dynasty of Egypt and found in Nag el-Deir, the ancient town of Thinis The Rhind Mathematical Papyrus (RMP), dated from the Second Intermediate Period (c. 1650 BC), but its author, Ahmes, identifies it as a copy of a now lost Middle Kingdom papyrus. The RMP is the largest mathematical text. From the New Kingdom there are a handful of mathematical texts and inscriptions related to computations: The Papyrus Anastasi I, a literary text written as a (fictional) letter written by a scribe named Hori and addressed to a scribe named Amenemope. A segment of the letter describes several mathematical problems. Ostracon Senmut 153, a text written in hieratic Ostracon Turin 57170, a text written in hieratic Ostraca from Deir el-Medina contain computations. Ostracon IFAO 1206 for instance shows the calculation of volumes, presumably related to the quarrying of a tomb. According to Étienne Gilson, Abraham "taught the Egyptians arythmetic and astronomy". == Numerals == Ancient Egyptian texts could be written in either hieroglyphs or in hieratic. In either representation the number system was always given in base 10. The number 1 was depicted by a simple stroke, the number 2 was represented by two strokes, etc. The numbers 10, 100, 1000, 10,000 and 100,000 had their own hieroglyphs. Number 10 is a hobble for cattle, number 100 is represented by a coiled rope, the number 1000 is represented by a lotus flower, the number 10,000 is represented by a finger, the number 100,000 is represented by a frog, and a million was represented by a god with his hands raised in adoration. Egyptian numerals date back to the Predynastic period. Ivory labels from Abydos record the use of this number system. It is also common to see the numerals in offering scenes to indicate the number of items offered. The king's daughter Neferetiabet is shown with an offering of 1000 oxen, bread, beer, etc. The Egyptian number system was additive. Large numbers were represented by collections of the glyphs and the value was obtained by simply adding the individual numbers together. The Egyptians almost exclusively used fractions of the form 1/n. One notable exception is the fraction 2/3, which is frequently found in the mathematical texts. Very rarely a special glyph was used to denote 3/4. The fraction 1/2 was represented by a glyph that may have depicted a piece of linen folded in two. The fraction 2/3 was represented by the glyph for a mouth with 2 (different sized) strokes. The rest of the fractions were always represented by a mouth super-imposed over a number. == Notation == Steps of calculations were written in sentences in Egyptian languages. (e.g. "Multiply 10 times 100; it becomes 1000.") In Rhind Papyrus Problem 28, the hieroglyphs (D54, D55), symbols for feet, were used to mean "to add" and "to subtract." These were presumably shorthands for meaning "to go in" and "to go out." == Multiplication and division == Egyptian multiplication was done by a repeated doubling of the number to be multiplied (the multiplicand), and choosing which of the doublings to add together (essentially a form of binary arithmetic), a method that links to the Old Kingdom. The multiplicand was written next to figure 1; the multiplicand was then added to itself, and the result written next to the number 2. The process was continued until the doublings gave a number greater than half of the multiplier. Then the doubled numbers (1, 2, etc.) would be repeatedly subtracted from the multiplier to select which of the results of the existing calculations should be added together to create the answer. As a shortcut for larger numbers, the multiplicand can also be immediately multiplied by 10, 100, 1000, 10000, etc. For example, Problem 69 on the Rhind Papyrus (RMP) provides the following illustration, as if Hieroglyphic symbols were used (rather than the RMP's actual hieratic script). The denotes the intermediate results that are added together to produce the final answer. The table above can also be used to divide 1120 by 80. We would solve this problem by finding the quotient (80) as the sum of those multipliers of 80 that add up to 1120. In this example that would yield a quotient of 10 + 4 = 14. A more complicated example of the division algorithm is provided by Problem 66. A total of 3200 ro of fat are to be distributed evenly over 365 days. First the scribe would double 365 repeatedly until the largest possible multiple of 365 is reached, which is smaller than 3200. In this case 8 times 365 is 2920 and further addition of multiples of 365 would clearly give a value greater than 3200. Next it is noted that 2/3 + 1/10 + 1/2190 times 365 gives us the value of 280 we need. Hence we find that 3200 divided by 365 must equal 8 + 2/3 + 1/10 + 1/2190. == Algebra == Egyptian algebra problems appear in both the Rhind mathematical papyrus and the Moscow mathematical papyrus as well as several other sources. Aha problems involve finding unknown quantities (referred to as Aha) if the sum of the quantity and part(s) of it are given. The Rhind Mathematical Papyrus also contains four of these type of problems. Problems 1, 19, and 25 of the Moscow Papyrus are Aha problems. For instance problem 19 asks one to calculate a quantity taken 1+1/2 times and added to 4 to make 10. In other words, in modern mathematical notation we are asked to solve the linear equation: 3 2 × x + 4 = 10. {\displaystyle {\frac {3}{2}}\times x+4=10.\ } Solving these Aha problems involves a technique called method of false position. The technique is also called the method of false assumption. The scribe would substitute an initial guess of the answer into the problem. The solution using the false assumption would be proportional to the actual answer, and the scribe would find the answer by using this ratio. The mathematical writings show that the scribes used (least) common multiples to turn problems with fractions into problems using integers. In this connection red auxiliary numbers are written next to the fractions. The use of the Horus eye fractions shows some (rudimentary) knowledge of geometrical progression. Knowledge of arithmetic progressions is also evident from the mathematical sources. === Quadratic equations === The ancient Egyptians were the first civilization to develop and solve second-degree (quadratic) equations. This information is found in the Berlin Papyrus fragment. Additionally, the Egyptians solve first-degree algebraic equations found in Rhind Mathematical Papyrus. == Geometry == There are only a limited number of problems from ancient Egypt that concern geometry. Geometric problems appear in both the Moscow Mathematical Papyrus (MMP) and in the Rhind Mathematical Papyrus (RMP). The examples demonstrate that the Ancient Egyptians knew how to compute areas of several geometric shapes and the volumes of cylinders and pyramids. Area: Triangles: The scribes record problems computing the area of a triangle (RMP and MMP). Rectangles: Problems regarding the area of a rectangular plot of land appear in the RMP and the MMP. A similar problem appears in the Lahun Mathematical Papyri in London. Circles: Problem 48 of the RMP compares the area of a circle (approximated by an octagon) and its circumscribing square. This problem's result is used in problem 50, where the scribe finds the area of a round field of diameter 9 khet. Hemisphere: Problem 10 in the MMP finds the area of a hemisphere. Volumes: Cylindrical (cylinder): Several problems compute the volume of cylindrical granaries (RMP 41–43), while problem 60 RMP seems to concern a pillar or a cone instead of a pyramid. It is rather small and steep, with a seked (reciprocal of slope) of four palms (per cubit). In section IV.3 of the Lahun Mathematical Papyri the volume of a granary with a circular base is found using the same procedure as RMP 43. Rectangular (Cuboid): Several problems in the Moscow Mathematical Papyrus (problem 14) and in the Rhind Mathematical Papyrus (numbers 44, 45, 46) compute the volume of a rectangular granary. Truncated pyramid (frustum) Frustum: The volume of a truncated pyramid is computed in MMP 14. === The Seqed === Problem 56 of the RMP indicates an understanding of the idea of geometric similarity. This problem discusses the ratio run/rise, also known as the seqed. Such a formula would be needed for building pyramids. In the next problem (Problem 57), the height of a pyramid is calculated from the base length and the seked (Egyptian for the reciprocal of the slope), while problem 58 gives the length of the base and the height and uses these measurements to compute the seqed. In Problem 59 part 1 computes the seqed, while the second part may be a computation to check the answer: If you construct a pyramid with base side 12 [cubits] and with a seqed of 5 palms 1 finger; what is its altitude? == See also == Red auxiliary number History of mathematics History of geometry Egyptian hieroglyphics and Transliteration of Ancient Egyptian Ancient Egyptian units of measurement and technology Mathematics and architecture == References == == Further reading == == External links == Egyptian Arithmetic Introduction to Early Mathematics |
Wikipedia:Ancient Egyptian multiplication#0 | In mathematics, ancient Egyptian multiplication (also known as Egyptian multiplication, Ethiopian multiplication, Russian multiplication, or peasant multiplication), one of two multiplication methods used by scribes, is a systematic method for multiplying two numbers that does not require the multiplication table, only the ability to multiply and divide by 2, and to add. It decomposes one of the multiplicands (preferably the smaller) into a set of numbers of powers of two and then creates a table of doublings of the second multiplicand by every value of the set which is summed up to give result of multiplication. This method may be called mediation and duplation, where mediation means halving one number and duplation means doubling the other number. It is still used in some areas. The second Egyptian multiplication and division technique was known from the hieratic Moscow and Rhind Mathematical Papyri written in the seventeenth century B.C. by the scribe Ahmes. Although in ancient Egypt the concept of base 2 did not exist, the algorithm is essentially the same algorithm as long multiplication after the multiplier and multiplicand are converted to binary. The method as interpreted by conversion to binary is therefore still in wide use today as implemented by binary multiplier circuits in modern computer processors. == Method == The ancient Egyptians had laid out tables of a great number of powers of two, rather than recalculating them each time. To decompose a number, they identified the powers of two which make it up. The Egyptians knew empirically that a given power of two would only appear once in a number. For the decomposition, they proceeded methodically; they would initially find the largest power of two less than or equal to the number in question, subtract it out and repeat until nothing remained. (The Egyptians did not make use of the number zero in mathematics.) After the decomposition of the first multiplicand, the person would construct a table of powers of two times the second multiplicand (generally the smaller) from one up to the largest power of two found during the decomposition. The result is obtained by adding the numbers from the second column for which the corresponding power of two makes up part of the decomposition of the first multiplicand. Because mathematically speaking, multiplication of natural numbers is just "exponentiation in the additive monoid", this multiplication method can also be recognised as a special case of the Square and multiply algorithm for exponentiation. === Example === 25 × 7 = ? Decomposition of the number 25: The largest power of two is 16 and the second multiplicand is 7. As 25 = 16 + 8 + 1, the corresponding multiples of 7 are added to get 25 × 7 = 112 + 56 + 7 = 175. == Russian peasant multiplication == In the Russian peasant method, the powers of two in the decomposition of the multiplicand are found by writing it on the left and progressively halving the left column, discarding any remainder, until the value is 1 (or −1, in which case the eventual sum is negated), while doubling the right column as before. Lines with even numbers on the left column are struck out, and the remaining numbers on the right are added together. === Example === 238 × 13 = ? == See also == Egyptian fraction Egyptian mathematics Multiplication algorithms Binary numeral system == References == === Other sources === Boyer, Carl B. (1968) A History of Mathematics. New York: John Wiley. Brown, Kevin S. (1995) The Akhmin Papyrus 1995 --- Egyptian Unit Fractions. Bruckheimer, Maxim, and Y. Salomon (1977) "Some Comments on R. J. Gillings' Analysis of the 2/n Table in the Rhind Papyrus," Historia Mathematica 4: 445–52. Bruins, Evert M. (1953) Fontes matheseos: hoofdpunten van het prae-Griekse en Griekse wiskundig denken. Leiden: E. J. Brill. ------- (1957) "Platon et la table égyptienne 2/n," Janus 46: 253–63. Bruins, Evert M (1981) "Egyptian Arithmetic," Janus 68: 33–52. ------- (1981) "Reducible and Trivial Decompositions Concerning Egyptian Arithmetics," Janus 68: 281–97. Burton, David M. (2003) History of Mathematics: An Introduction. Boston Wm. C. Brown. Chace, Arnold Buffum, et al. (1927) The Rhind Mathematical Papyrus. Oberlin: Mathematical Association of America. Cooke, Roger (1997) The History of Mathematics. A Brief Course. New York, John Wiley & Sons. Couchoud, Sylvia. "Mathématiques égyptiennes". Recherches sur les connaissances mathématiques de l'Egypte pharaonique., Paris, Le Léopard d'Or, 1993. Daressy, Georges. "Akhmim Wood Tablets", Le Caire Imprimerie de l'Institut Francais d'Archeologie Orientale, 1901, 95–96. Eves, Howard (1961) An Introduction to the History of Mathematics. New York, Holt, Rinehard & Winston. Fowler, David H. (1999) The mathematics of Plato's Academy: a new reconstruction. Oxford Univ. Press. Gardiner, Alan H. (1957) Egyptian Grammar being an Introduction to the Study of Hieroglyphs. Oxford University Press. Gardner, Milo (2002) "The Egyptian Mathematical Leather Roll, Attested Short Term and Long Term" in History of the Mathematical Sciences, Ivor Grattan-Guinness, B.C. Yadav (eds), New Delhi, Hindustan Book Agency:119-34. -------- "Mathematical Roll of Egypt" in Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Springer, Nov. 2005. Gillings, Richard J. (1962) "The Egyptian Mathematical Leather Roll," Australian Journal of Science 24: 339–44. Reprinted in his (1972) Mathematics in the Time of the Pharaohs. MIT Press. Reprinted by Dover Publications, 1982. -------- (1974) "The Recto of the Rhind Mathematical Papyrus: How Did the Ancient Egyptian Scribe Prepare It?" Archive for History of Exact Sciences 12: 291–98. -------- (1979) "The Recto of the RMP and the EMLR," Historia Mathematica, Toronto 6 (1979), 442–447. -------- (1981) "The Egyptian Mathematical Leather Role–Line 8. How Did the Scribe Do it?" Historia Mathematica: 456–57. Glanville, S.R.K. "The Mathematical Leather Roll in the British Museum" Journal of Egyptian Archaeology 13, London (1927): 232–8 Griffith, Francis Llewelyn. The Petrie Papyri. Hieratic Papyri from Kahun and Gurob (Principally of the Middle Kingdom), Vols. 1, 2. Bernard Quaritch, London, 1898. Gunn, Battiscombe George. Review of The Rhind Mathematical Papyrus by T. E. Peet. The Journal of Egyptian Archaeology 12 London, (1926): 123–137. Hultsch, F. Die Elemente der Aegyptischen Theihungsrechmun 8, Übersicht über die Lehre von den Zerlegangen, (1895):167-71. Imhausen, Annette. "Egyptian Mathematical Texts and their Contexts", Science in Context 16, Cambridge (UK), (2003): 367–389. Joseph, George Gheverghese. The Crest of the Peacock/the non-European Roots of Mathematics, Princeton, Princeton University Press, 2000 Klee, Victor, and Wagon, Stan. Old and New Unsolved Problems in Plane Geometry and Number Theory, Mathematical Association of America, 1991. Knorr, Wilbur R. "Techniques of Fractions in Ancient Egypt and Greece". Historia Mathematica 9 Berlin, (1982): 133–171. Legon, John A.R. "A Kahun Mathematical Fragment". Discussions in Egyptology, 24 Oxford, (1992). Lüneburg, H. (1993) "Zerlgung von Bruchen in Stammbruche" Leonardi Pisani Liber Abbaci oder Lesevergnügen eines Mathematikers, Wissenschaftsverlag, Mannheim: 81=85. Neugebauer, Otto (1969) [1957]. The Exact Sciences in Antiquity (2 ed.). Dover Publications. ISBN 978-0-486-22332-2. Robins, Gay and Charles Shute, The Rhind Mathematical Papyrus: an Ancient Egyptian Text, London, British Museum Press, 1987. Roero, C. S. "Egyptian mathematics" Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences" I. Grattan-Guinness (ed), London, (1994): 30–45. Sarton, George. Introduction to the History of Science, Vol I, New York, Williams & Son, 1927 Scott, A. and Hall, H.R., "Laboratory Notes: Egyptian Mathematical Leather Roll of the Seventeenth Century BC", British Museum Quarterly, Vol 2, London, (1927): 56. Sylvester, J. J. "On a Point in the Theory of Vulgar Fractions": American Journal of Mathematics, 3 Baltimore (1880): 332–335, 388–389. Vogel, Kurt. "Erweitert die Lederolle unserer Kenntniss ägyptischer Mathematik Archiv für Geschichte der Mathematik, V 2, Julius Schuster, Berlin (1929): 386-407 van der Waerden, Bartel Leendert. Science Awakening, New York, 1963 Hana Vymazalova, The Wooden Tablets from Cairo:The Use of the Grain Unit HK3T in Ancient Egypt, Archiv Orientalai, Charles U Prague, 2002. == External links == RMP 2/n table The Ahmes code Egyptian Mathematical Leather Roll The first LCM method Red Auxiliary numbers Egyptian fraction Math forum and two ways to calculate 2/7 New and Old classifications of Ahmes Papyrus Russian Peasant Multiplication The Russian Peasant Algorithm (pdf file) Peasant Multiplication from cut-the-knot Egyptian Multiplication by Ken Caviness, The Wolfram Demonstrations Project. Russian Peasant Multiplication at The Daily WTF Michael S. Schneider explains how the Ancient Egyptians (and Chinese) and modern computers multiply and divide Russian Multiplication - Numberphile |
Wikipedia:Ancient Greek mathematics#0 | Ancient Greek mathematics refers to the history of mathematical ideas and texts in Ancient Greece during Classical and Late antiquity, mostly from the 5th century BC to the 6th century AD. Greek mathematicians lived in cities spread around the shores of the ancient Mediterranean, from Anatolia to Italy and North Africa, but were united by Greek culture and the Greek language. The development of mathematics as a theoretical discipline and the use of deductive reasoning in proofs is an important difference between Greek mathematics and those of preceding civilizations, such as Ancient Egypt and Babylonia. The early history of Greek mathematics is obscure, and traditional narratives of mathematical theorems found before the fifth century BC are regarded as later inventions. It is now generally accepted that treatises of deductive mathematics written in Greek began circulating around the mid-fifth century BC, but the earliest complete work on the subject is the Elements, written during the Hellenistic period. The works of renown mathematicians Archimedes and Apollonius, as well as of the astronomer Hipparchus, also belong to this period. In the Imperial Roman era, Ptolemy used trigonometry to determine the positions of stars in the sky, while Nicomachus and other ancient philosophers revived ancient number theory and harmonics. In Late antiquity, Pappus of Alexandria wrote his Collection, summarizing the work of his predecessors, while Diophantus' Arithmetica dealt with the solution of arithmetic problems by way of pre-modern algebra. Later authors such as Theon of Alexandria, his daughter Hypatia, and Eutocius of Ascalon wrote commentaries on the authors making up the ancient Greek mathematical corpus. The works of ancient Greek mathematicians were copied in the medieval Byzantine period and translated into Arabic and Latin, where they exerted influence on mathematics in the Islamic world and in Medieval Europe. During the Renaissance, the texts of Euclid, Archimedes, Apollonius, and Pappus in particular went on to influence the development of early modern mathematics. Some problems in Ancient Greek mathematics were solved only in the modern era by mathematicians such as Gauss, and attempts to prove or disprove Euclid's parallel line postulate spurred the development of non-Euclidean geometry. == Etymology == Greek mathēmatikē (Ancient Greek: μαθηματική) derives from the Ancient Greek: μάθημα, romanized: máthēma, Attic Greek: [má.tʰɛː.ma] Koinē Greek: [ˈma.θi.ma], from the verb manthano, "I learn". Strictly speaking, a máthēma could be any branch of learning, or anything learnt; however, since antiquity certain mathēmata were granted special status: arithmetic, geometry, astronomy, and harmonics. Arithmetic, which dealt with numbers, included not only basic operations of addition, subtraction, multiplication, and division, but also what we would now consider algebra and number theory. Geometry (lit. "land mensuration") included not only plane and solid geometry and the theory of conic sections, but also optics. Astronomy dealt with phenomena related to the stars and the five planets, and fostered the development of astronomical models and trigonometry. Harmonics dealt primarily with the theory of music scales by way of means and ratios. These four mathēmata would later become the medieval quadrivium. == Origins == The origins of Greek mathematics are not well documented. The earliest known written treatises on Ancient Greek mathematics, starting with Hippocrates of Chios in the 5th century BC, have been lost, and the early history of mathematics must be reconstructed from information passed down through later authors, beginning in the mid-4th century BC. Much of the knowledge about Ancient Greek mathematics in this period is thanks to references by Plato, Aristotle, and from quotations of Eudemus of Rhodes' histories of geometry and arithmetic by later authors. These references provide near-contemporary accounts for many mathematicians active in the 4th century BC. Euclid's Elements is also believed to contain many theorems that are attributed to mathematicians in the preceding centuries. === Bronze Age === The earliest advanced civilizations in Greece were the Minoan and later Mycenaean (1500-1200 BC) civilizations, both of which flourished in the second half of the Bronze Age. While these civilizations possessed writing, and many Linear B documents written in Mycenaean Greek have been deciphered, no mathematical writings have yet been discovered. After the Bronze Age collapse, there was a significant population decline and loss of writing. When Greek writing re-emerge in the 7th century BC, it was based on an entirely new system derived from the Phoenician alphabet, with papyrus from Ancient Egyptian being the preferred medium. Mathematical writings from two preceding Bronze Age civilizations are extant, in the form of Babylonian cuneiform tablets and Egyptian mathematical papyri. Unlike later Greek mathematics, the mathematics in these extant texts are primarily focused on land mensuration and accounting; although problems in Babylonian and Egyptian mathematics went beyond purely utilitarian aims, including constructing artificial scenarios involving the solution of quadratic equations, there are no signs of explicit theoretical concerns. Though no direct evidence of transmission is available, it is generally thought that Babylonian and Egyptian mathematics had an influence on the younger Greek culture, possibly through an oral tradition of mathematical problems over the course of centuries. === Archaic period === Later traditions in Ancient Greece attribute the origin of Greek mathematics to either Thales of Miletus, one of the legendary Seven Sages of Greece, or to Pythagoras of Samos, both of whom are said to have visited Egypt and Babylon and learned mathematics there. However, modern scholarship tends to be skeptical of such claims as neither Thales or Pythagoras left any writings behind that were available in the Classical period. Additionally, widespread literacy and the scribal culture that would have supported the transmission of mathematical treatises did not emerge fully until the 5th century; the oral literature of their time was primarily focused on public speeches and recitations of poetry. The standard view among historians is that the discoveries Thales and Pythagoras are credited with, such as Thales' Theorem, the Pythagorean theorem, and the Platonic solids, are the product of attributions by much later authors. === Classical Greece === While Ancient Greek literature in the Archaic period was primarily orally transmitted, the earliest traces of Greek mathematicians who wrote dedicated mathematical treatises do not appear until the second half of the fifth century BC. According to Eudemus, Hippocrates of Chios was the first known author to write a book of Elements in the tradition later continued by Euclid. Hippocrates wrote a now lost mathematical treatise that is described by Eudemus which provides a description of the Lune of Hippocrates. According to the testimony of Eudemus, Hippocrates studied an astronomer named Oenopides, also from Chios. Although the remainder of the writings of these mathematicians are lost, later mathematicians associated with Chios include Andron and Zenodotus, who may be associated with a "school of Oenopides" mentioned by Proclus. Some of the later Pythagoreans who lived in the 5th century may have made significant contributions to mathematics. Aristotle, one of the earliest authors to associate Pythagoreanism with mathematics, even refused to attribute anything specifically to Pythagoras, who he seems to have regarded as a fictional person, and only discussed the work of the Pythagoreans as a group. Early traditions about the Pythagoreans discuss a taboo on publishing their doctrines; many of the stories of early Pythagoreans are apocryphal, including stories that Hippasus was drowned for publishing the discovery of the dodecagon, or that another Pythagorean philosopher was exiled for sharing the discovery of irrational numbers. However, more concrete accounts of mathematical work start to appear beginning with Philolaus of Croton, who associated together arithmetic, geometry, astronomy, and harmonics. Fragments of Philolaus' work are preserved in quotations from later authors. In the 5th century BC, many other philosophers also made claims about mathematics that have been recorded; Antiphon claimed to be able to construct a rectilinear figure with the same area as a given circle, while Hippias is credited with a method for squaring a circle with a neusis construction. Protagoras and Democritus debated the tangency of a line with a circle; Protagoras argued that it was impossible for a line to intersect a circle at a single point, whereas Democritus stated that this was impossible. Democritus also asserted, apparently without proof, that the area of a cone was 1/3 the area of a cylinder with the same base, a result which was later proved by Eudoxus of Cnidus. ==== Mathematics in the time of Plato ==== While Plato was not a mathematician himself, numerous early mathematicians, including Archytas, Theaetetus, and Eudoxus, were associated with Plato or with his Academy, and Plato mentions mathematics in several of his dialogues, including the Meno, the Theaetetus, the Republic, and the Timaeus. Archytas, a Pythagorean philosopher from Tarentum, was a friend of Plato who made several mathematical discoveries. Archytas is often credited with books VII to IX in the Elements, which deal with the Euclidean algorithm, prime numbers, mean ratios, and perfect numbers. Archytas solved the problem of doubling the cube, now known to be impossible with only a compass and a straightedge, with an alternative method, systematized the Pythagorean means, and made contributions to optics and mechanics. Theaetetus, who figures as a character in the Platonic dialogue named after him, where he is working on a problem given to him by Theodorus of Cyrene to demonstrate that the square roots of several numbers from 3 to 17 are irrational, a construction now known as the Spiral of Theodorus. Theaetetus is traditionally credited with much of the work contained in Books X of Euclid's Elements, concerned with incommensurable magnitudes, Book XIII, which outlines the construction of the regular polyhedra. Although some of the regular polyhedra were certainly known prior to Theaetetus, he is credited with the systematic construction of them, and the proof that only five of them exist. Another mathematician associated with Plato's academy is Eudoxus of Cnidus, developed a theory of proportion in book V of the Elements. Archimedes also credits Eudoxus of Cnidus with two propositions in book XII of Euclid's Elements, proving that the volume of a cone is one-third the volume of a cylinder with the same base, which use an early form of calculus known as the method of exhaustion. This method is also used by Archimedes himself in order to find an approximation to π (Measurement of the Circle) and to prove that the area enclosed by a parabola and a straight line is 4/3 times the area of a triangle with equal base and height (Quadrature of the Parabola). Eudoxus also developed an astronomical calendar, now lost, that remains partially preserved by an imitation in poetic form called Phaenomena by Aratus. Eudoxus seems to have founded a school of mathematics in Cyzicus, where one of Eudoxus' students, Menaechmus went on to develop a theory of conic sections. == Hellenistic and early Roman period == Ancient Greek mathematics reached its acme during the Hellenistic era and early Roman periods following Alexander the Great's conquest of the Eastern Mediterranean, Egypt, Mesopotamia, the Iranian plateau, Central Asia, and parts of India, leading to the spread of the Greek language and culture across these regions. Koine Greek became the lingua franca of scholarship throughout the Hellenistic world, and the mathematics of the Classical period merged with Egyptian and Babylonian mathematics to give rise to Hellenistic mathematics, and several centers of learning appeared during the Hellenistic period, of which the most important one was the Musaeum in Alexandria, in Ptolemaic Egypt. Although few in number, Hellenistic mathematicians actively communicated with each other via letters; who were then responsible for distributing publication consisted of passing and copying someone's work among colleagues. Working at the Library of Alexandria, Euclid collected many previous mathematical results and theorems in the Elements, a compilation of many of the works of his predecessors that would become a canon of geometry and elementary number theory for many centuries. Archimedes, building on the work in Elements, used the method of exhaustion to approximate Pi (Measurement of a Circle), measured the surface area and volume of a sphere (On the Sphere and Cylinder), devised a mechanical method for developing solutions to mathematical problems using the law of the lever, (Method of Mechanical Theorems), and a developed method for representing very large numbers in order to show that the number of grains of sand filling the universe was not uncountable.(The Sand-Reckoner), Apollonius of Perga, in his extant work Conics, refined and developed the theory of conic sections first outlined by Menaechmus, Euclid, and Conon of Samos. Trigonometry was developed around the time of Hipparchus, an early astronomer, and both trigonometry and astronomy further developed by Ptolemy in his Almagest. === Construction problems === Much of the extant literature on hellenistic mathematics deals with three construction problems: Doubling the Cube, Angle trisection, and Squaring the circle, all of which are now known to be impossible with only a straight edge and a compass, however, many attempts were made using neusis constructions including the Cissoid of Diocles, Quadratrix, and the Conchoid of Nicomedes. The constructions regular polygons and polyhedra had already been known by the time of the publication of Euclid's elements. Archimedes extended this in a now lost work by constructing the semiregular polyhedra, also sometimes known as Archimedean solids. A work transmitted as "Book XIV" of Euclid's Elements, likely written a few centuries later by Hypsicles, provides a historical development after Theatetus; Aristaeus the Elder's comparison of five figures and Apollonius of Perga's Comparison of the Dodecahedron and the Icosahedron. Another book, transmitted as "Book XV" of Euclid's elements, which was compiled in the 6th century CE, provides further developments. Many of the works on the solution of construction problems became part of a standard curriculum of works which were studied during the Hellenistic period: Data and Porisms by Euclid, several works by Apollonius of Perga including Cutting off a ratio, Cutting off an area, Determinate section, Tangencies, and Neusis, and several works dealing with loci, including Plane Loci and Conics by Apollonius, Solid Loci by Aristaeus the Elder, Loci on a Surface by Euclid, and On Means by Eratosthenes of Cyrene. All of these works other than Data, Conics Books I to VII, and Cutting off a ratio are lost. However, a rough outline of the contents of can be obtained in Book 7 of the Collection of Pappus of Alexandria, who provides brief epitomes of each of the works, along with lemmas for Cutting off an area, Determinate section, Tangencies, Porisms, Neusis, Plane Loci, and Book VIII of the Conics. The study of optics in Ancient Greece was also considered a part of geometry. An extant work on Catoptrics is dubiously attributed to Euclid, Archimedes is known to have written a now lost work on catoptrics, and another work, On Burning Mirrors, by Diocles is extant in an arabic translation. === Astronomy === The Little Astronomy, a collection of shorter works from the 4th–2nd century BC, mostly with astronomical relevance, have survived because they were bundled together as an astronomy curriculum beginning in the 2nd century AD and transmitted as a group: Theodosius's Spherics, Autolycus's On the Moving Sphere, Euclid's Optics and Phaenomena, Theodosius's On Habitations and On Days and Nights, Aristarchus's On the Sizes and Distances, Autolycus's On Risings and Settings, and Hypsicles's On Ascensions. These works are all extant in Vaticanus gr. 204, which also contains Apollonius's Conics books I-IV and the commentary by Eutocius, and Euclid's Catoptrics and his Data with an introduction by Marinus of Neapolis. This collection was translated into Arabic with a few additions such as Euclid's Data, Menelaus's Spherics (which only survives in Arabic), and various works by Archimedes as the Middle Books, intermediate between Euclid's Elements and Ptolemy's Almagest. Around the 2nd century BC, the works of Babylonian astronomers became available to Ancient Greek mathematicians. The development of trigonometry as a synthesis of Babylonian and Greek methods in mathematical astronomy is commonly attributed to Hipparchus, who made extensive astronomical observations and wrote several mathematical treatises; however, all of Hipparchus's works have been lost with the exception of his Commentary on the Phaenomena of Eudoxus and Aratus, a critical commentary on a lost treatise by Eudoxus and a popular poem based on it by Aratus about astronomical phenomena, which was preserved bundled among other commentary on Aratus's poem. In the 2nd century AD, Claudius Ptolemy compiled the observations of Hipparchus and other astronomers and wrote a work now called the Almagest explaining the motions of the stars and planets according to a geocentric model, and calculated out chord tables to a higher degree of precision than had been done previously, along with an instruction manual, Handy Tables. === Arithmetic === Building on the works of the earlier Pythagoreans, Nicomachus of Gerasa wrote an Introduction to Arithmetic which would go on to receive later commentary in Neopythagoreanism. The continuing influence of Platonism in mathematics is shown by another extant work, Mathematics Useful For Understanding Plato, by Theon of Smyrna, written around the same time. Diophantus wrote on polygonal numbers and a work in pre-modern algebra (Arithmetica), === Applied mathematics === Much of the work represented by authors such as Euclid, Archimedes, Apollonius, Hipparchus, and Diophantus was of a very advanced level and rarely mastered outside a small circle. Ancient Greek mathematics was not limited to theoretical works but was also used in other activities, such as business transactions and in land mensuration, as evidenced by extant texts where computational procedures and practical considerations took more of a central role. Examples of applied mathematics around this time include the construction of analogue computers like the Antikythera mechanism, the accurate measurement of the circumference of the Earth by Eratosthenes, and the mathematical and mechanical works of Heron. == Mathematics in late antiquity == The mathematicians in the later Roman era from the 4th century onward generally had few notable original works, however, they are distinguished for their commentaries and expositions on the works of earlier mathematicians. These commentaries have preserved valuable extracts from works which have perished, or historical allusions which, in the absence of original documents, are precious because of their rarity. === Pappus' Collection === Pappus of Alexandria compiled a canon of results of earlier mathematics in the Collection in eight books, of which part of book II and books III through VII are extant in Greek and book VIII is extant in Arabic. The collection attempts to sum up the whole of Ancient Greek mathematics up to that time as interpreted by Pappus: Book III is framed as a letter to Pandrosion, a mathematican in Athens, and discusses three construction problems and attempts to solve them: Doubling the Cube, Angle trisection, and Squaring the Circle. Book IV discusses classical geometry, which Pappus divides into plane geometry, Line geometry, and Solid geometry, and includes a discussion of Archimedes' construction of the Arbelos, otherwise only known via a Pseudo-archimedean work, Book of Lemmas. Book V discusses isoperimetric figures, summarizing otherwise lost works by Zenodotus and Archimedes on isoperimetric plane figures and solid figures, respectively. Book VI deals with astronomy, providing commentary on some of the works of the Little Astronomy corpus. Book VII deals with analysis, providing epitomes and lemmas from otherwise lost works. Book VIII deals with mechanics. The Greek version breaks off in the middle of a sentence discussing Hero of Alexandria, but a complete edition of the book survives in Arabic. === Commentaries === The commentary tradition, which had begun during the Hellenistic period, continued into late antiquity. The first known commentary on the Elements was written by Hero of Alexandria, who likely set the format for future commentaries. Serenus of Antinoöpolis wrote a lost commentary on the Conics of Apollonius, along with two works that survive, Section of a Cylinder and Section of a Cone, expanding on specific subjects in the Conics. Pappus wrote a commentary on Book X of the elements, dealing with incommensurable magnitudes. Heliodorus of Larissa wrote a summary of the Optics. Many of the late antique commentators were associated with Neoplatonist philosophy; Porphyry of Tyre, a student of Plotinus, the founder of Neoplatonism, wrote a commentary on Ptolemy's Harmonics. Iamblichus, who was himself a student of Porphyry, wrote a commentary on Nicomachus' Introduction to Arithmetic. In Alexandria in the 4th century, Theon of Alexandria wrote commentaries on the writings of Ptolemy, including a commentary on the Almagest and two commentaries on the Handy Tables, one of which is more of an instruction manual ("Little Commentary"), and another with a much more detailed exposition and derivations ("Great Commentary"). Hypatia, Theon's daughter, also wrote a commentary on Diophantus' Arithmetica and a commentary on the Conics of Apollonius, which have not survived. In the 5th century, in Athens, Proclus wrote a commentary on Euclid's elements, which the first book survives. Proclus' contemporary, Domninus of Larissa, wrote a summary of Nicomachus' Introduction to Arithmetic, while Marinus of Neapolis, Proclus' successor, wrote an Introduction to Euclid's Data. Meanwhile in Alexandria, Ammonius Hermiae, John Philoponus and Simplicius of Cilicia wrote commentaries on the works of Aristotle that preserve information on earlier mathematicians and philosophers. Eutocius of Ascalon,(c. 480–540 AD) another student of Ammonius, wrote commentaries that are extant on Apollonius' Conics along with some treatises of Archimedes: On the Sphere and Cylinder, Measurement of a Circle, and On Balancing Planes (though the authorship of the last one is disputed). In Rome, Boethius, seeking to preserve Ancient Greek philosophical, translated works on the quadrivium into Latin, deriving much of his work on Arithmetic and Harmonics from Nicomachus. After the closure of the Neoplatonic schools by the emperor Justinian in 529 AD, the institution of mathematics as a formal enterprise entered a decline. However, two mathematicians connected to the Neoplatonic tradition were commissioned to build the Hagia Sophia: Anthemius of Tralles and Isidore of Miletus. Anthemius constructed many advanced mechanisms and wrote a work On Surprising Mechanisms which treats "burning mirrors" and skeptically attempts to explain the function of Archimedes' heat ray. Isidore, who continued the project of the Hagia Sophia after Anthemius' death, also supervised the revision of Eutocius' commentaries of Archimedes. From someone in Isidore's circle we also have a work on polyhedra that is transmitted pseudepigraphically as Book XV of Euclid's Elements. == Reception and legacy == The majority of mathematical treatises written in Ancient Greek, along with the discoveries made within them, have been lost; around 30% of the works known from references to them are extant. Authors whose works survive in Greek manuscripts include: Euclid, Autolycus of Pitane, Archimedes, Aristarchus of Samos, Philo of Byzantium, Biton of Pergamon, Apollonius of Perga, Hipparchus, Theodosius of Bithynia, Hypsicles, Athenaeus Mechanicus, Geminus, Hero of Alexandria, Apollodorus of Damascus, Theon of Smyrna, Cleomedes, Nicomachus, Ptolemy, Cleonides, Gaudentius, Anatolius of Laodicea, Aristides Quintilian, Porphyry, Diophantus, Alypius, Heliodorus of Larissa, Pappus of Alexandria, Serenus of Antinoöpolis, Theon of Alexandria, Proclus, Marinus of Neapolis, Domninus of Larissa, Anthemius of Tralles, and Eutocius. The earliest surviving papyrus to record a Greek mathematical text is P. Hib. i 27, which contains a parapegma of Eudoxus' astronomical calendar, along with several ostraca from the 3rd century BC that deal with propositions XIII.10 and XIII.16 of Euclid's Elements. A papyrus recovered from Herculaneum contains an essay by the Epicurean philosopher Demetrius Lacon on Euclid's Elements. Most of the oldest extant manuscripts for each text date from the 9th century onward, copies of works written during and before the Hellenistic period. The two major sources of manuscripts are Byzantine-era codices, copied some 500 to 1500 years after their originals, and Arabic translations of Greek works; what has survived reflects the preferences of readers in late antiquity along with the interests of mathematicians in the Byzantine empire and the medieval Islamic world who preserved and copied them. Despite the lack of original manuscripts, the dates for some Greek mathematicians are more certain than the dates of surviving Babylonian or Egyptian sources because a number of overlapping chronologies exist, though many dates remain uncertain. === Byzantine mathematics === With the closure of the Neoplatonist schools in the 6th century, Greek mathematics declined in the medieval Byzantine period, although many works were preserved in medieval manuscript transmission and translated into first Syriac and Arabic, and later into Latin. The transition to miniscule manuscript in the 9th century, however, many works that were not copied during this time period were lost, although a few uncial manuscripts do survive. Many surviving works are derived from only a single manuscript; such as Pappus' Collection and Books I-IV of the Conics. Many of the surviving manuscripts originate from two scholars in this period in the circle of Photios I, Leo the Mathematician and Arethas of Caesarea. Scholia written in the margins of Euclid's elements that have been copied throughout multiple extant manuscripts that were also written by Arethas, derived from Proclus' commentary along with many commentaries on Euclid which are now lost. The works of Archimedes survived in three different recensions in manuscripts from the 9th and 10th centuries; two of which are now lost after being copied, the third of which, the Archimedes Palimpsest, was only rediscovered in 1906. In the later Byzantine period, George Pachymeres wrote a summary of the quadrivium, and Maximus Planudes wrote scholia on the first two books of Diophantus. === Medieval Islamic mathematics === Numerous mathematical treatises were translated into Arabic in the 9th century; many works that are only extent today in Arabic translation, and there is evidence for several more that have since been lost. Medieval Islamic scientists such as Alhazen developed the ideas of the Ancient Greek geometry into advanced theories in optics and astronomy, and Diophantus' Arithmetica was synthezied with the works of Al-Khwarizmi and works from Indian mathematics to develop a theory of algebra. The following works are extant only in Arabic translations: Apollonius, Conics books V to VII, Cutting Off of a Ratio Archimedes, Book of Lemmas Diocles, On Burning Mirrors Diophantus, Arithmetica books IV to VII Euclid, On Divisions of Figures, On Weights Menelaus, Sphaerica Hero, Catoptrica, Mechanica Pappus, Commentary on Euclid's Elements book X, Collection Book VIII Ptolemy, Planisphaerium, Additionally, the work Optics by Ptolemy only survives in a Latin translations of the Arabic translation of a Greek original. === In Latin Medieval Europe === The works derived from Ancient Greek mathematical writings that had been written in late antiquity by Boethius and Martianus Capella had formed the basis of early medieval quadrivium of arithmetic, geometry, astronomy, and music. In the 12th century the original works of Ancient Greek mathematics were translated into Latin first from Arabic by Gerard of Cremona, and then from the original Greek a century later by William of Moerbeke. === Renaissance === The publication of Greek mathematical works increased their audience; Pappus's collection was published in 1588, Diophantus in 1621. Diophantus would go on to influence Pierre de Fermat's work on number theory; Fermat scribbled his famous note about Fermat's Last theorem in his copy of Arithmetica. Descartes, working through the Problem of Apollonius from his edition of Pappus, proved what is now called Descartes' theorem and laid the foundations for Analytic geometry. === Modern mathematics === Ancient Greek mathematics constitutes an important period in the history of mathematics: fundamental in respect of geometry and for the idea of formal proof. Greek mathematicians also contributed to number theory, mathematical astronomy, combinatorics, mathematical physics, and, at times, approached ideas close to the integral calculus. Richard Dedekind acknowledged Eudoxus's theory of proportion as an inspiration for the Dedekind cut, a method of contructing the real numbers. == See also == Timeline of ancient Greek mathematicians List of Greek mathematicians Music of ancient Greece – Musical traditions of ancient Greece == Notes == == References == Acerbi, Fabio (2018), "Hellenistic Mathematics", in Keyser, Paul T; Scarborough, John (eds.), Oxford Handbook of Science and Medicine in the Classical World, pp. 268–292, doi:10.1093/oxfordhb/9780199734146.013.69, ISBN 978-0-19-973414-6, retrieved 2021-05-26 Boyer, Carl B. (1991), A History of Mathematics (3rd ed.), John Wiley & Sons, Inc., ISBN 978-0-471-54397-8 Cameron, A. (1990), "Isidore of Miletus and Hypatia: On the Editing of Mathematical Texts", Greek, Roman, and Byzantine Studies, 31 (1): 103–127 Fowler, D. H. (1999), The Mathematics of Plato's Academy (2nd ed.), Clarendon Press Høyrup, J. (1990), "Sub-scientific mathematics: Undercurrents and missing links in the mathematical technology of the Hellenistic and Roman world" (PDF) (Unpublished manuscript, written for Aufstieg und Niedergang der römischen Welt) Knorr, Wilbur R. (1986), The Ancient Tradition of Geometric Problems Knorr, Wilbur R. (1996), "The method of indivisibles in Ancient Geometry", Vita Mathematica, MAA Press, pp. 67–86 Mansfeld, J. (2016), Prolegomena Mathematica: From Apollonius of Perga to the Late Neoplatonism. With an Appendix on Pappus and the History of Platonism, Brill, ISBN 978-90-04-32105-2 Netz, Reviel (2022), A New History of Greek Mathematics, Cambridge University Press, ISBN 978-1-108-83384-4 Netz, Reviel (2014), "The problem of Pythagorean mathematics", in Huffman, Carl A. (ed.), A History of Pythagoreanism, Cambridge University Press, pp. 167–184, doi:10.1017/CBO9781139028172.009, ISBN 978-1-107-01439-8 Schofield, Malcolm (2014), "Archytas", in Huffman, Carl A. (ed.), A History of Pythagoreanism, Cambridge University Press, pp. 69–87, doi:10.1017/CBO9781139028172.009, ISBN 978-1-107-01439-8 == Further reading == A. Barker, Porphyry’s Commentary on Ptolemy’s Harmonics A. Barker, Greek Musical Writings, Vol. 2: Harmonic and Acoustic Theory A. Bernard, “Ancient Rhetoric and Greek Mathematics: A Response to a Modern Historiographical Dilemma,” I. Bodnár, Oenopides of Chius: A Survey of the Modern Literature with a Collection of the Ancient Testimonia Burton, David M. (1997), The History of Mathematics: An Introduction (3rd ed.), The McGraw-Hill Companies, Inc., ISBN 978-0-07-009465-9 M. F. Burnyeat, “Plato on Why Mathematics Is Good for the Soul,” Proceedings of the British Academy 2000 M. F. Burnyeat, “The Philosophical Sense of Theaetetus’ Mathematics,” 1978 L. Corry, A Brief History of Number S. Cuomo, Pappus of Alexandria and the Mathematics of Late Antiquity Christianidis, Jean, ed. (2004), Classics in the History of Greek Mathematics, Dordrecht: Kluwer, ISBN 978-1-4020-0081-2 Cooke, Roger (1997), The History of Mathematics: A Brief Course, Wiley-Interscience, ISBN 978-0-471-18082-1 Derbyshire, John (2006), Unknown Quantity: A Real And Imaginary History of Algebra, Joseph Henry Press, ISBN 978-0-309-09657-7 E. J. Dijksterhuis, Archimedes M. N. Fried, and S. Unguru, Apollonius of Perga’s Conica: Text, Context, Subtext Heath, Thomas Little (1981) [First published 1921], A History of Greek Mathematics, Dover publications, ISBN 978-0-486-24073-2 Heath, Thomas Little (2003) [First published 1931], A Manual of Greek Mathematics, Dover publications, ISBN 978-0-486-43231-1 Huffman, Archytas Huffman, Philolaus A. Jones, A Portable Cosmos R. W. Knorr, The Evolution of the Euclidean Elements, 1975 H. Mendell, “Reflections on Eudoxus, Callippus and Their Curves: Hippopedes and Callippopedes,” I. Mueller, Philosophy of Mathematics and Deductive Structure in Euclid’s Elements Netz, “Eudemus of Rhodes, Hippocrates of Chios and the Earliest Form of a Greek Mathematical Text,” R. Netz, Ludic Proof: Greek Mathematics and the Alexandrian Aesthetics R. Netz, The Shaping of Deduction in Greek Mathematics O. Pedersen, A Survey of the Almagest: With Annotation and New Commentary by Alexander Jones D. N. Sedley, “Epicurus and the Mathematicians of Cyzicus,” M. Sialaros, J. Christianidis, and A. Megremi (eds.), “On Mathemata: Commenting on Greek and Arabic Mathematical Texts,” Sing, Robert; Berkel, Tazuko Angela van; Osborne, Robin (2022), Numbers and numeracy in the Greek polis, Brill, ISBN 978-90-04-46721-7 Stillwell, John (2004), Mathematics and its History (2nd ed.), Springer Science + Business Media Inc., ISBN 978-0-387-95336-6 Szabó, Árpád; Szabó, Árpád (1978), The Beginnings of Greek Mathematics, Budapest: Akadémiai Kiadó, ISBN 978-963-05-1416-3 S. Unguru, “On the Need to Rewrite the History of Greek Mathematics,” Archive for History of Exact Sciences 15 (1975): 67-114 G. Vlastos, “Elenchus and Mathematics: A Turning-Point in Plato’s Philosophical Development,” I. Yavetz, “On the Homocentric Spheres of Eudoxus,” Archive for History of Exact Sciences == External links == Vatican Exhibit History of Mathematics MacTutor archive of History of Mathematics |
Wikipedia:Ancient Mesopotamian units of measurement#0 | Ancient Mesopotamian units of measurement originated in the loosely organized city-states of Early Dynastic Sumer. Each city, kingdom and trade guild had its own standards until the formation of the Akkadian Empire when Sargon of Akkad issued a common standard. This standard was improved by Naram-Sin, but fell into disuse after the Akkadian Empire dissolved. The standard of Naram-Sin was readopted in the Ur III period by the Nanše Hymn which reduced a plethora of multiple standards to a few agreed upon common groupings. Successors to Sumerian civilization including the Babylonians, Assyrians, and Persians continued to use these groupings. Akkado-Sumerian metrology has been reconstructed by applying statistical methods to compare Sumerian architecture, architectural plans, and issued official standards such as Statue B of Gudea and the bronze cubit of Nippur. == Archaic system == The systems that would later become the classical standard for Mesopotamia were developed in parallel with writing in Sumer during Late Uruk Period (c. 3500–3000). Studies of protocuneiform indicate twelve separate counting systems used in Uruk IV-III. Seven of these were also used in the contemporary Proto-Elamite writing system. The bisexagesimal systems went out of use after the Early Dynastic I/II period. Sexagesimal System S used to count slaves, animals, fish, wooden objects, stone objects, containers. Sexagesimal System S' used to count dead animals, certain types of beer Bisexagesimal System B used to count cereal, bread, fish, milk products Bisexagesimal System B* used to count rations GAN2 System G used to count field measurement ŠE system Š used to count barley by volume ŠE system Š' used to count malt by volume ŠE system Š" used to count wheat by volume ŠE System Š* used to count barley groats EN System E used to count weight U4 System U used to count calendrics DUGb System Db used to count milk by volume DUGc System Db used to count beer by volume In Early Dynastic Sumer (c. 2900–2300 BCE) metrology and mathematics were indistinguishable and treated as a single scribal discipline. The idea of an abstract number did not yet exist, thus all quantities were written as metrological symbols and never as numerals followed by a unit symbol. For example there was a symbol for one-sheep and another for one-day but no symbol for one. About 600 of these metrological symbols exist, for this reason archaic Sumerian metrology is complex and not fully understood. In general however, length, volume, and mass are derived from a theoretical standard cube, called 'gur (also spelled kor in some literature)', filled with barley, wheat, water, or oil. However, because of the different specific gravities of these substances combined with dual numerical bases (sexagesimal or decimal), multiple sizes of the gur-cube were used without consensus. The different gur-cubes are related by proportion, based on the water gur-cube, according to four basic coefficients and their cubic roots. These coefficients are given as: Komma = 80⁄81 correction when planning rations with a 360-day year Leimma = 24⁄25 conversion from decimal to a sexagesimal number system Diesis = 15⁄16 Euboic = 5⁄6 One official government standard of measurement of the archaic system was the Cubit of Nippur (2650 BCE). It is a Euboic Mana + 1 Diesis (432 grams). This standard is the main reference used by archaeologists to reconstruct the system. == Classical system == A major improvement came in 2150 BCE during the Akkadian Empire under the reign of Naram-Sin when the competing systems were unified by a single official standard, the royal gur-cube. His reform is considered the first standardized system of measure in Mesopotamia. The royal gur-cube (Cuneiform: LU2.GAL.GUR, 𒈚𒄥; Akkadian: šarru kurru) was a theoretical cuboid of water approximately 6 m × 6 m × 0.5 m from which all other units could be derived. The Neo-Sumerians continued use of the royal gur-cube as indicated by the Letter of Nanse issued in 2000 BCE by Gudea. Use of the same standard continued through the Neo-Babylonian Empire, Neo-Assyrian Empire, and Achaemenid Empire. === Length === Units of length are prefixed by the logogram DU (𒁺) a convention of the archaic period counting system from which it was evolved. Basic length was used in architecture and field division. Distance units were geodectic as distinguished from non-geodectic basic length units. Sumerian geodesy divided latitude into seven zones between equator and pole. === Area === The GAN2 system G counting system evolved into area measurements. A special unit measuring brick quantity by area was called the brick-garden (Cuneiform: SIG.SAR 𒊬𒋞; Sumerian: šeg12-sar; Akkadian: libittu-mūšaru) which held 720 bricks. === Capacity or volume === Capacity was measured by either the ŠE system Š for dry capacity or the ŠE system Š* for wet capacity. A sila was about 1 liter. === Mass or weight === Mass was measured by the EN system E Values below are an average of weight artifacts from Ur and Nippur. The ± value represents 1 standard deviation. All values have been rounded to second digit of the standard deviation. === Time === In the Archaic System time notation was written in the U4 System U. Multiple lunisolar calendars existed; however the civil calendar from the holy city of Nippur (Ur III period) was adopted by Babylon as their civil calendar. The calendar of Nippur dates to 3500 BCE and was itself based on older astronomical knowledge of an uncertain origin. The main astronomical cycles used to construct the calendar were the month, year, and day. == Relationship to other metrologies == The Classical Mesopotamian system formed the basis for Elamite, Hebrew, Urartian, Hurrian, Hittite, Ugaritic, Phoenician, Babylonian, Assyrian, Persian, Arabic, and Islamic metrologies. The Classical Mesopotamian System also has a proportional relationship, by virtue of standardized commerce, to Bronze Age Harappan and Egyptian metrologies. == See also == Assyrian lion weights Babylonian mathematics Historical weights and measures Weights and measures == References == === Citations === === Bibliography === Conder, Claude Reignier (1908). The Rise of Man. University of Michigan: J. Murray. pp. 368. hittite metrology. Melville, Duncan J (2006-06-06). "Old Babylonian Weights and Measures". Archived from the original on 13 May 2008. Retrieved 2008-06-28. Powell, Marvin A (1995). "Metrology and Mathematics in Ancient Mesopotamia". In Sasson, Jack M. (ed.). Civilizations of the Ancient Near East. Vol. III. New York, NY: Charles Scribner’s Sons. pp. 3024. ISBN 0-684-19279-9. Ronan, Colin Alistair (2008). "Measurement of time and types of calendars » Standard units and cycles". Encyclopædia Britannica Online. Archived from the original on 25 June 2008. Retrieved 2008-06-28. Whitrow, G.J. (1988). Time in History: Views of Time from Prehistory to the Present Day. New York: Oxford University Press. pp. 217. ISBN 0-19-285211-6. == Further reading == Katz, Victor, J (2007). The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 712. ISBN 978-0-691-11485-9.{{cite book}}: CS1 maint: multiple names: authors list (link) Nissen, Hans Jörg; Peter Damerow; Robert K. Englund; Paul Larsen (1993). Archaic Bookkeeping: Early Writing and Techniques of Economic Administration. University of Chicago Press. p. 169. ISBN 0-226-58659-6. Robson, Eleanor (1999). Mesopotamian Mathematics, 2100–1600 BC: Technical Constants in Bureaucracy. Oxford University Press. ISBN 0-19-815246-9. Sarton, George (1993). Ancient science through the golden age of Greece. Courier Dover Publications. p. 646. ISBN 0-486-27495-0. == External links == An online calculator [1] Archived 2017-02-13 at the Wayback Machine Robson, Eleanor (2007). "Digital Corpus of Cuneiform Mathematical Texts". Archived from the original on 2009-01-16. Retrieved 2008-08-13. Aleff, H. Peter (2008). "Auspicious latitudes". Retrieved 2008-08-13. Kreidik, L. G.; T. S. Kortneva; G. P. Shpenkov (2005). "4. Fundamental periods of the World and ancient metrology". Journal of Theoretical Dialectics-Physics-Mathematics. Dialectical Academy, Russia-Belarus. Retrieved 2009-08-20. |
Wikipedia:Anders C. Hansen#0 | Anders C. Hansen is a Norwegian mathematician, who is currently a Professor of Mathematics at University of Cambridge, where he is the head of the Applied Functional and Harmonic Analysis group, and also Professor II at the University of Oslo. He works in functional analysis, harmonic analysis (applied), foundations of mathematics (computational), data science and numerical analysis . == Education == Hansen studied mathematics at the University of Cambridge, University of California, Berkeley and the Norwegian University of Science and Technology, where he was awarded a PhD (2008), a MA (2005) and a BA (2002) respectively. == Career and research == He was a von Kármán instructor at California Institute of Technology from 2008 to 2009, held a junior research fellowship at Homerton College, Cambridge from 2009 to 2012, and held a Marie Skłodowska-Curie Actions fellowship at the University of Vienna in 2012. Since 2012, he has held a Royal Society University Research Fellowship (URF) at the University of Cambridge, where he is now a professor at the Faculty of Mathematics, University of Cambridge and a Bye-Fellow of Peterhouse. Among other results, he has established the Solvability Complexity Index (SCI) and its following classification hierarchy. It is linked to Steve Smale's question on the existence of iterative convergent algorithms for polynomial root finding answered by Curt McMullen and Peter Doyle, as well as Alan Turing's work and the Arithmetical hierarchy. === Awards and honours === In 2017, he was awarded the Leverhulme Prize for having "solved very hard problems and opened new directions in areas of great impact in applied analysis [...] Notably, by introducing the Solvability Complexity Index he has made a major contribution to the advancement of Smale’s programme on the foundation of computational mathematics". In 2018, he was awarded the IMA Prize in Mathematics and its Applications for having "made a transformative impact on the mathematical sciences and their applications [...] in particular, for his development of the Solvability Complexity Index and its corresponding classification hierarchy". In 2019, he was awarded the Whitehead Prize of the London Mathematical Society for having "contributed fundamentally to the mathematics of data, sampling theory, computational harmonic analysis and compressed sensing" and "especially his development of the Solvability Complexity Index and its corresponding classification hierarchy ". == Selected publications == Research articles Hansen, Anders (12 July 2010). "On the Solvability Complexity Index, the 𝑛-pseudospectrum and approximations of spectra of operators". Journal of the American Mathematical Society. 24 (1). American Mathematical Society (AMS): 81–124. doi:10.1090/s0894-0347-2010-00676-5. ISSN 0894-0347. Antun, Vegard; Renna, Francesco; Poon, Clarice; Adcock, Ben; Hansen, Anders C. (11 May 2020). "On instabilities of deep learning in image reconstruction and the potential costs of AI". Proceedings of the National Academy of Sciences. 117 (48). Proceedings of the National Academy of Sciences: 30088–30095. arXiv:1902.05300. Bibcode:2020PNAS..11730088A. doi:10.1073/pnas.1907377117. ISSN 0027-8424. PMC 7720232. PMID 32393633. Colbrook, Matthew J.; Antun, Vegard; Hansen, Anders C. (16 March 2022). "The difficulty of computing stable and accurate neural networks: On the barriers of deep learning and Smale's 18th problem". Proceedings of the National Academy of Sciences. 119 (12): e2107151119. Bibcode:2022PNAS..11907151C. doi:10.1073/pnas.2107151119. ISSN 0027-8424. PMC 8944871. PMID 35294283. S2CID 247499099. ADCOCK, BEN; HANSEN, ANDERS C.; POON, CLARICE; ROMAN, BOGDAN (2017). "Breaking the Coherence Barrier: A New Theory for Compressed Sensing". Forum of Mathematics, Sigma. 5. Cambridge University Press (CUP). arXiv:1302.0561. doi:10.1017/fms.2016.32. ISSN 2050-5094. S2CID 263901. Adcock, Ben; Hansen, Anders C. (20 August 2015). "Generalized Sampling and Infinite-Dimensional Compressed Sensing". Foundations of Computational Mathematics. 16 (5). Springer Science and Business Media LLC: 1263–1323. doi:10.1007/s10208-015-9276-6. ISSN 1615-3375. S2CID 223542. Colbrook, Matthew J.; Roman, Bogdan; Hansen, Anders C. (28 June 2019). "How to Compute Spectra with Error Control". Physical Review Letters. 122 (25). American Physical Society (APS): 250201. Bibcode:2019PhRvL.122y0201C. doi:10.1103/physrevlett.122.250201. ISSN 0031-9007. PMID 31347861. S2CID 198463498. Research expository highlights A. Bastounis, A. C. Hansen, D. Higham, I. Tyukin and V. Vlacic: "Deep Learning: What Could Go Wrong?", SIAM News (October 2021). V. Antun, N. Gottschling, A. C. Hansen and B. Adcock, "Deep Learning in Scientific Computing: Understanding the Instability Mystery", SIAM News (March 2021). A. Bastounis, B. Adcock and A. C. Hansen, "From Global to Local: Getting More from Compressed Sensing", SIAM News (October 2017). Books Adcock, Ben; Hansen, Anders C. (2021). Compressive imaging : structure, sampling, learning. Cambridge, United Kingdom: Cambridge University Press. ISBN 978-1-108-37744-7. OCLC 1260468467. == References == |
Wikipedia:Anders Hald#0 | Anders Hjorth Hald (3 July 1913 – 11 November 2007) was a Danish statistician. He was a professor at the University of Copenhagen from 1960 to 1982. While a professor, he did research in industrial quality control and other areas, and also authored textbooks. After retirement, he made important contributions to the history of statistics. Hald was a Fellow of the American Statistical Association, a Member of the Royal Danish Academy of Science and Letters, a Member of the Institute of Mathematical Statistics, an Honorary Fellow of the Royal Statistical Society, and a Corresponding Fellow of the Royal Society of Edinburgh. == Bibliography == A History of Parametric Statistical Inference from Bernoulli to Fisher, 1713–1935. New York: Springer. 2007. ISBN 978-0-387-46408-4. A History of Probability and Statistics and Their Applications before 1750. Hoboken, NJ: Wiley. 2003. ISBN 0-471-47129-1. A History of Mathematical Statistics from 1750 to 1930. New York: Wiley. 1998. ISBN 0-471-17912-4. Statistical theory of sampling inspection by attributes. London: Academic Press. 1981. ISBN 0-12-318350-2. Statistical Theory with Engineering Applications. New York: Wiley. 1952. Statistical Tables and Formulas. New York: Wiley. 1952. "T. N. Thiele's contributions to statistics", International Statistical Review, 49, number 1 (1981): 1–20. (Reprinted in Steffen L. Lauritzen, ed. (2002). Thiele: Pioneer in Statistics. Oxford University Press.) "The early history of the cumulants and the Gram–Charlier series", International Statistical Review, 68, number 2 (2000): 137–153. (Reprinted in Steffen L. Lauritzen, ed. (2002). Thiele: Pioneer in Statistics. Oxford University Press.) === In Danish === Statistiske Metoder, 1949 Statistisk Kvalitetskontrol, 1954 Statistiske Metoder i Arbejdsstudierteknikken, 1955 Elementær Lærebog i Statistisk Kvalitetskontrol, 1956 == References == == External links == There is a photograph at Anders Hald on the Portraits of Statisticians page. There is a bibliography of Hald's writings on the history of statistics at Anders Hald (1913–2007): Writings on the History of Probability and Statistics in the Electronic Journal for History of Probability and Statistics. |
Wikipedia:Anders Planman#0 | Anders Planman (1724 – 25 April 1803) was a Finnish astronomer, professor of physics and mathematician. He was one of the first people to make systematical astronomical observations in Finland. == Life == Planman was born in Hattula. He came from a Swedish-speaking Finnish family and his father was a lieutenant. He studied at the Royal Academy of Turku from 1744 to 1754 and then continued his studies at Uppsala University. In 1756 he received the grade of docent in astronomy. In 1763 he was appointed professor of physics in Turku and retained the position until 1801, when he quit due to poor health. For three years he was also head of the academy. Because the tenureship as a professor was sometimes without a salary, Planman had also been ordained and worked as parish priest in Nousiainen and Paimio. From 1767 he was a member of the Royal Swedish Academy of Sciences. He was also a member of the Royal Society of Sciences in Uppsala. == Work == Planman was one of the first individuals to make systematic astronomical observations in Finland. He made his most important observations during the transits of Venus in 1761 and 1769. The Royal Swedish Academy of Sciences provided money for expeditions to the north of Finland (Lapland) to make the observations. The overarching aim of these expeditions was to contribute to the measurement of the solar parallax. By participating in the expeditions and also by working with data supplied to him the academy in Stockholm, he developed a new method to calculate the parallax. His own observations in 1761 were not very precise but those made by him in 1769 were considered some of the most exact in Europe. He took an active part in the then on-going debate on whether or not Venus had an atmosphere. During the expedition in 1761, he also calculated the correct longitude for six locations on the way, among them Kajaani, Mikkeli and Hämeenlinna. He could do this partly because he used surveying tools for his astronomical observations; these were some of the most exact contemporary measuring tools. Planman has presented in his research in 1753 that in the future it will be possible for people to take off into the air with the "help of a controlled machine". The minor planet 2639 Planman is named after Anders Planman. == References == |
Wikipedia:Anders Szepessy#0 | Anders Szepessy (born 1960) is a Swedish mathematician. Szepessy received his PhD in 1989 from Chalmers University of Technology with thesis Convergence of the streamline diffusion finite element method for conservation laws under the supervision of Claes Johnson. Szepessy is now a professor of mathematics and numerical analysis at KTH Royal Institute of Technology. His research area is applied mathematics, especially partial differential equations. Szepessy was an invited speaker at the International Congress of Mathematicians in 2006 in Madrid. He was elected a member of the Royal Swedish Academy of Sciences in 2007. == Selected publications == Johnson, Claes; Szepessy, Anders (1987). "On the convergence of a finite element method for a nonlinear hyperbolic conservation law". Mathematics of Computation. 49 (180): 427. doi:10.1090/S0025-5718-1987-0906180-5. Szepessy, Anders (1989). "An existence result for scalar conservation laws using measure valued solutions". Communications in Partial Differential Equations. 14 (10): 1329–1350. doi:10.1080/03605308908820657. Szepessy, Anders (1989). "Measure-valued solutions of scalar conservation laws with boundary conditions". Archive for Rational Mechanics and Analysis. 107 (2): 181–193. Bibcode:1989ArRMA.107..181S. doi:10.1007/BF00286499. S2CID 120515809. Szepessy, Anders (1989). "Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions". Mathematics of Computation. 53 (188): 527–545. Bibcode:1989MaCom..53..527S. doi:10.1090/S0025-5718-1989-0979941-6. Johnson, Claes; Szepessy, Anders; Hansbo, Peter (1990). "On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws". Mathematics of Computation. 54 (189): 107. Bibcode:1990MaCom..54..107J. doi:10.1090/S0025-5718-1990-0995210-0. Hansbo, Peter; Szepessy, Anders (1990). "A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations". Computer Methods in Applied Mechanics and Engineering. 84 (2): 175–192. Bibcode:1990CMAME..84..175H. doi:10.1016/0045-7825(90)90116-4. Szepessy, Anders; Xin, Zhouping (1993). "Nonlinear stability of viscous shock waves". Archive for Rational Mechanics and Analysis. 122 (1): 53–103. Bibcode:1993ArRMA.122...53S. doi:10.1007/BF01816555. S2CID 122130129. Goodman, Jonathan; Szepessy, Anders; Zumbrun, Kevin (1994). "A Remark on the Stability of Viscous Shock Waves". SIAM Journal on Mathematical Analysis. 25 (6): 1463–1467. doi:10.1137/S0036141092239648. ISSN 0036-1410. Johnson, Claes; Szepessy, Anders (1995). "Adaptive finite element methods for conservation laws based on a posteriori error estimates". Communications on Pure and Applied Mathematics. 48 (3): 199–234. doi:10.1002/cpa.3160480302. Jaffre, J.; Johnson, C.; Szepessy, A. (1995). "Convergence of the Discontinuous Galerkin Finite Element Method for Hyperbolic Conservation Laws". Mathematical Models and Methods in Applied Sciences. 05 (3): 367–386. doi:10.1142/S021820259500022X. Szepessy, Anders; Zumbrun, Kevin (1996). "Stability of rarefaction waves in viscous media". Archive for Rational Mechanics and Analysis. 133 (3): 249–298. doi:10.1007/BF00380894. S2CID 18558122. Szepessy, Anders; Tempone, Raúl; Zouraris, Georgios E. (2001). "Adaptive weak approximation of stochastic differential equations". Communications on Pure and Applied Mathematics. 54 (10): 1169–1214. doi:10.1002/cpa.10000. ISSN 0010-3640. S2CID 7182000. == References == |
Wikipedia:Anderson function#0 | Anderson functions describe the projection of a magnetic dipole field in a given direction at points along an arbitrary line. They are useful in the study of magnetic anomaly detection, with historical applications in submarine hunting and underwater mine detection. They approximately describe the signal detected by a total field sensor as the sensor passes by a target (assuming the targets signature is small compared to the Earth's magnetic field). == Definition == The magnetic field from a magnetic dipole along a given line, and in any given direction can be described by the following basis functions: θ i − 1 ( θ 2 + 1 ) 5 2 , for i = 1 , 2 , 3 {\displaystyle {\frac {\theta ^{~i-1}}{(\theta ^{2}+1)^{\frac {5}{2}}}},{\text{ for }}i=1,2,3} which are known as Anderson functions. Definitions: m → {\displaystyle {\vec {m}}} is the dipole's strength and direction B → E {\displaystyle {\vec {B}}_{E}} is the projected direction (often the Earth's magnetic field in a region) x {\displaystyle x} is the position along the line v ^ {\displaystyle {\hat {v}}} points in the direction of the line r → {\displaystyle {\vec {r}}} is a vector from the dipole to the point of closest approach (CPA) of the line θ = x / r {\displaystyle \theta =x/r} , a dimensionless quantity for simplification The total magnetic field along the line is given by B ( θ ) = μ 0 4 π | m | r 3 ( A 1 ( θ 2 + 1 ) 5 2 + A 2 θ 1 ( θ 2 + 1 ) 5 2 + A 3 θ 2 ( θ 2 + 1 ) 5 2 ) {\displaystyle B(\theta )={\frac {\mu _{0}}{4\pi }}{\frac {|m|}{r^{3}}}\left({\frac {A_{1}}{(\theta ^{2}+1)^{\frac {5}{2}}}}+{\frac {A_{2}\theta ^{1}}{(\theta ^{2}+1)^{\frac {5}{2}}}}+{\frac {A_{3}\theta ^{2}}{(\theta ^{2}+1)^{\frac {5}{2}}}}\right)} where μ 0 {\displaystyle \mu _{0}} is the magnetic constant, and A 1 , 2 , 3 {\displaystyle A_{1,2,3}} are the Anderson coefficients, which depend on the geometry of the system. These are A 1 = 3 ( m ^ ⋅ r ^ ) ( r ^ ⋅ B ^ E ) − ( m ^ ⋅ B ^ E ) A 2 = 3 ( m ^ ⋅ r ^ ) ( v ^ ⋅ B ^ E ) + 3 ( m ^ ⋅ v ^ ) ( r ^ ⋅ B ^ E ) A 3 = 3 ( m ^ ⋅ v ^ ) ( v ^ ⋅ B ^ E ) − ( m ^ ⋅ B ^ E ) {\displaystyle {\begin{aligned}A_{1}&=3({\hat {m}}\cdot {\hat {r}})({\hat {r}}\cdot {\hat {B}}_{E})-~~({\hat {m}}\cdot {\hat {B}}_{E})\\A_{2}&=3({\hat {m}}\cdot {\hat {r}})({\hat {v}}\cdot {\hat {B}}_{E})+3({\hat {m}}\cdot {\hat {v}})({\hat {r}}\cdot {\hat {B}}_{E})\\A_{3}&=3({\hat {m}}\cdot {\hat {v}})({\hat {v}}\cdot {\hat {B}}_{E})-~~({\hat {m}}\cdot {\hat {B}}_{E})\end{aligned}}} where m ^ , r ^ , {\displaystyle {\hat {m}},{\hat {r}},} and B ^ E {\displaystyle {\hat {B}}_{E}} are unit vectors (given by m → | m → | , r → | r → | , {\displaystyle {\frac {\vec {m}}{|{\vec {m}}|}},{\frac {\vec {r}}{|{\vec {r}}|}},} and B → E | B → E | {\displaystyle {\frac {{\vec {B}}_{E}}{|{\vec {B}}_{E}|}}} , respectively). Note, the antisymmetric portion of the function is represented by the second function. Correspondingly, the sign of A 2 {\displaystyle A_{2}} depends on how v → {\displaystyle {\vec {v}}} is defined (e.g. direction is 'forward'). == Total field measurements == The total field measurement resulting from a dipole field B → D {\displaystyle {\vec {B}}_{D}} in the presence of a background field B → E {\displaystyle {\vec {B}}_{E}} (such as earth magnetic field) is | B | = ( B → D + B → E ) ⋅ ( B → D + B → E ) = | B E | 1 + 2 B → D ⋅ B → E | B E | 2 + B → D 2 | B E | 2 ≈ | B E | + B → D ⋅ B → E | B E | , | B E | ≫ | B D | . {\displaystyle {\begin{aligned}|B|&={\sqrt {({\vec {B}}_{D}+{\vec {B}}_{E})\cdot ({\vec {B}}_{D}+{\vec {B}}_{E})}}\\&=|B_{E}|{\sqrt {1+{\frac {2{\vec {B}}_{D}\cdot {\vec {B}}_{E}}{|B_{E}|^{2}}}+{\frac {{\vec {B}}_{D}^{2}}{|B_{E}|^{2}}}}}\\&\approx |B_{E}|+{\frac {{\vec {B}}_{D}\cdot {\vec {B}}_{E}}{|B_{E}|}},&|B_{E}|\gg |B_{D}|.\end{aligned}}} The last line is an approximation that is accurate if the background field is much larger than contributions from the dipole. In such a case the total field reduces to the sum of the background field, and the projection of the dipole field onto the background field. This means that the total field can be accurately described as an Anderson function with an offset. == References == |
Wikipedia:Andre Punt#0 | André Eric Punt (born February 1965) is a South African fisheries scientist and mathematician, best known for his work on fisheries stock assessment. He received the K. Radway Allen Award in 1999 for his contributions to fisheries science. == Early years and education == André Punt was born in February 1965 in Cape Town, South Africa. He attended the University of Cape Town, where he received a BSc (with Honours) in Computer Science in 1986, an MSc in Applied Mathematics in 1988, and a PhD in Applied Mathematics in 1991. As a PhD student in the late 1980s, Punt and his colleague Doug Butterworth competed in an informal competition against other international research groups to develop computer simulations that could guide quota decisions for whale harvests. Punt and his collaborators went on to produce simulations and formulae that could account for uncertainties in whale abundance data. These approaches were subsequently applied to stock assessments for South Africa's hake, sardine, anchovy and West Coast rock lobster fisheries. == Career == In 1992, after completing his doctoral studies, Punt joined the School of Fisheries at the University of Washington as a research associate. In 1994, he moved to Australia to work as a resource modeller in the CSIRO's Division of Marine Research. Here, his work on stock assessments was influential for Australian fisheries. In 1999, the Australian Society for Fish Biology awarded Punt its K. Radway Allen Award in recognition of his scientific contributions. He later rejoined the University of Washington, in the School of Aquatic and Fishery Sciences where he has received numerous distinguished teaching awards. Punt is known for his international collaborations, particularly his work with the International Whaling Commission and the International Commission for the Conservation of Atlantic Tunas. == References == |
Wikipedia:Andreas Seeger#0 | Seeger is the surname of various people. == Etymology == Seeger is one of the variant forms of Seagar, a surname of Middle English origin based on the given name Segar, which was formed from Old English sæ ("sea") and gar ("spear"). == Seeger family of musicians == Charles Louis Seeger, Sr. (1860–1943), American businessman Charles Louis Seeger, Jr. (1886–1979), American musicologist, composer, and teacher Pete Seeger (1919–2014), one of the preeminent American folk and protest singers of the 20th century Toshi Seeger (1922–2013), filmmaker and environmental activist Mika Seeger American ceramic artist Tao Rodríguez-Seeger (b. 1972), a contemporary American folk musician Ruth Crawford Seeger (1901–1953), a modernist composer and an American folk music specialist Mike Seeger (1933–2009), American folk musician and folklorist Peggy Seeger (born 1935), American folk singer and songwriter; wife of Scottish folk singer Ewan MacColl and stepmother of singer Kirsty MacColl Alan Seeger (1888–1916), American poet Elizabeth Seeger (1889–1973), teacher at Dalton School and author == Others == Al Seeger (b. 1980), American boxer Andreas Seeger, German mathematician Britta Seeger (b. 1969), German business executive Charles M. Seeger (b. 1948), American author and attorney Christopher A. Seeger (b. 1960), American attorney Daniel Seeger (b. 1934), defendant in a case on conscription of pacifists Frank Seeger (b. 1972), German sports shooter Fritz Seeger (b. 1912), Swiss sprinter Hal Seeger (1917–2005), American animated cartoon producer and director Harald Seeger (1922–2015), German footballer and manager Helmut Seeger (b. 1932), German sports shooter Hermann Seeger (1857–1945), German painter Louis Seeger (1798–1865), German equestrian Matthew Seeger (b. 1957), American professor and university dean Melanie Seeger (b. 1977), German race walker Petra Seeger, German documentary film director and producer Raymond Seeger (1906–1992), American physicist Ruth Taubert Seeger (1924–2014), American athlete and coach Stanley J. Seeger (1930–2011), American art collector Stefan Seeger (born 1962), German chemist and professor Steven C. Seeger (b. 1971), American judge == Legal cases == "Seeger" may also reference either of two legal cases involving the individuals below: United States v. Seeger, regarding Daniel Seeger's conscription == Schools == Seeger Memorial Junior-Senior High School in West Lebanon, Indiana Elizabeth Seeger School in Greenwich Village, established in 1971 by five teachers from Dalton School == See also == Seger Seager Segger (surname) Object names Seeger ring, alternative name for a circlip The closest modern named object that corresponds to a seaspear is a harpoon. == References == |
Wikipedia:Andreas Speiser#0 | Andreas Speiser (June 10, 1885 – October 12, 1970) was a Swiss mathematician and philosopher of science. == Life and work == Speiser studied in Göttingen, starting in 1904, notably with David Hilbert, Felix Klein, Hermann Minkowski. In 1917 he became full-time professor at the University of Zurich but later relocated in Basel. During 1924/25 he was president of the Swiss Mathematical Association. Speiser worked on number theory, group theory, and the theory of Riemann surfaces. He organized the translation of Leonard Dickson's seminal 1923 book Algebras and Their Arithmetics (Algebren und ihre Zahlentheorie, 1927), which was heavily influenced by the work on the theory of algebras done by the schools of Emmy Noether and Helmut Hasse. Speiser also added an appendix on ideal theory to Dickson's book. Speiser's book Theorie der Gruppen endlicher Ordnung is a classic, richly illustrated work on group theory. In this book, there are group theoretical applications in Galois theory, elementary number theory, and Platonic solids, as well as extensive studies of ornaments, such as those that Speiser studied on a 1928 trip to Egypt. Speiser also worked on the history of mathematics and was the chief editor for the Euler Commission's edition of Leonhard Euler's Opera Omnia and the editor of the works of Johann Heinrich Lambert. As a philosopher Speiser was chiefly concerned with Plato and wrote a commentary on the Parmenides Dialogue, but he was also an expert of the philosophies of Plotinus and Hegel. Speiser's doctoral students include J. J. Burckhardt. == Writings == Die Theorie der Gruppen von endlicher Ordnung – mit Anwendungen auf algebraische Zahlen und Gleichungen sowie auf die Kristallographie. Springer 1923 (1st edition), 1927 (2nd edition), 1937 (3rd edition); Birkhäuser 1956. Klassische Stücke der Mathematik. Orell Füssli 1925 (mit Abdruck von Quellen, u.a. auch Dante, Rousseau). Leonhard Euler und die Deutsche Philosophie. Orell Füssli 1934. Leonhard Euler. In: Große Schweizer. Atlantis Verlag, Zürich 1939, 1940, S.1-6. Die mathematische Denkweise. Rascher 1932, Birkhäuser 1945, 1952. Leonhard Euler. Vortrag gehalten an der Generalversammlung des S.I.A. in Basel am 11. September 1949. Schweizerische Bauzeitung, Jg.67, Nr.48. 26. November 1949, Zürich. Elemente der Philosophie und Mathematik. Birkhäuser 1952. Die Geistige Arbeit. Birkhäuser 1955 (Vorträge). Ein Parmenideskommentar – Studien zur Platonischen Dialektik. Koehler, Leipzig, Stuttgart, 1937, 1959. Ueber Riemannsche Flächen. Comm.Math.Helvetici (CMH), Bd.2, 1930, S.284-293. Zur Theorie der Substitutionsgruppen. Mathematische Annalen, Bd. 75, 1914, S.443-448. Zahlentheoretische Sätze aus der Gruppentheorie. Math.Zeitschrift Bd.5, 1919, S. 1-6. Naturphilosophische Untersuchungen von Euler und Riemann. Crelle Journal Bd. 157, 1927, S.105-114. Zahlentheorie in rationalen Algebren. CMH, Bd.8, 1936, S.391-406. Riemann'sche Flächen vom hyperbolischen Typus. CMH Bd.10, 1937, S.232-242. Geometrisches zur Riemannschen Zetafunktion. Mathematische Annalen Bd.110, 1934, S.514-521. Einteilung der sämtlichen Werke Leonhard Eulers. CMH Bd.20, 1947, S. 288-318. == See also == Hilbert–Speiser theorem Jordan–Schur theorem == References == Martin Eichler, Nachruf in den Verhandlungen der Schweizer Naturforschenden Gesellschaft, Bd.150, 1970, S.325 J. J. Burckhardt, Nachruf in Vierteljahresschrift der Naturforschenden Gesellschaft Bd.115, 1970, 471 J. J. Burckhardt: Die Mathematik an der Universität Zurich 1916-1950 unter den Professoren R. Fueter, A. Speiser und P. Finsler, Basel, 1980 == External links == Media related to Andreas Speiser (mathematician) at Wikimedia Commons |
Wikipedia:Andreas Stöberl#0 | Andreas Stöberl (c. 1464 in Pleiskirchen near Altötting – September 3, 1515 in Vienna), better known by his latinised name Andreas Stiborius (Boius), was a German humanist astronomer, mathematician, and theologian working mainly at the University of Vienna. == Life == Stöberl studied from 1479 on at the University of Ingolstadt, where he became a magister in 1484, and subsequently a member of the Faculty of Arts. At Ingolstadt, he met and became a friend of Conrad Celtis, an eminent advocate of humanism who lectured there between 1492 and 1497. When Celtis moved to Vienna in 1497, Stöberl followed his mentor. Stiborius was a member of the Sodalitas Litterarum Danubiana, a circle of humanists founded by Celtis. In 1502 he became one of two professors for mathematics (the other was Johannes Stabius, his friend from Ingolstadt) at the Collegium poetarum et mathematicorum, founded on Celtis' initiative by emperor Maximilian I the year before, as a part of the University of Vienna. At the Collegium, he taught courses in astronomy and astrology, as he did later at the University, where he got a chair at the Collegium ducale in 1503. Stiborius was a gifted teacher and well-liked by his students. In 1507 or 1508 he became a canon at St. Stephen's, and until his death in 1515 in Vienna he was also parish priest in Stockerau, where he was buried. == Works == At Vienna, Stiborius worked with Georg Tannstetter, who came to Vienna from Ingolstadt in Autumn 1502. Together they became the most prominent exponents of the "Second Viennese School of Mathematics" (the first having been the circle around Johann von Gmunden, Georg von Peuerbach, and Regiomontanus). Tannstetter, in his Viri Mathematici names both Stabius and Stiborius as his teachers. As editor, Stiborius published an edition of Robert Grosseteste's Libellus Linconiensis de Phisicis lineis, angulis et figuris, per quas omnes acciones naturales complentur in 1503. For Tannstetter's edition Tabulae Eclypsium..., which was published in 1514 and contained tables of eclipses of Georg von Peuerbach and the primi mobilis tables of Regiomontanus, Stiborius wrote two prefaces. In preparation of the 10th session of the 5th council of the Lateran, Pope Leo X requested in October 1514 from various rulers to have their scientists offer proposals on the calendar reform. Emperor Maximilian gave the task to Stiborius and Tannstetter in Vienna, and to Johannes Stöffler at Tübingen. Stiborius and Tannstetter proposed to omit one leap year every 134 years, and to drop the 19-year metonic cycle used by the Church to calculate the Easter date. Instead of the metonic cycle, they proposed to simply use the true astronomic calculation for the full moon dates to determine Paschal Full Moon. Furthermore, they pointed out that the true astronomic March equinox and full moons, on which the whole calculation of the Easter date and thus other Church holidays was based, would occur at different times, sometimes even different dates in places at different longitudes around the globe, leading to Church holidays falling on different days in different places. They recommended to use universally the equinox at the Meridian of Jerusalem or Rome. Tannstetter and Stiborius's calendar reform proposal was published as Super requisitione sanctissimi Leonis Papae X. et divi Maximiliani Imp. p.f. Aug. De Romani Calendarii correctione Consilium in Florentissimo studio Viennensi Anustriae conscriptum et aeditum ca. 1515 by the printer Johannes Singriener in Vienna. As it turned out, the whole topic of the calendar reform was not even discussed at the fifth Lateran Council. Tannstetter gives in his Viri Mathematici a list of books in Stiborius's library, and also a list of works written by the latter himself. He mentions a five-volume Opus Umbrarum ("Work of Shadows"), in which Stiborius treated various astronomical and mathematical topics such as cartographic projections, the theory and use of the astrolabe, including the saphea, the construction of sundials, and others. The work was the basis of his lectures in Vienna; it appears never to have been published though. A partial copy made in 1500 of these lecture notes has survived. == Legacy == The lunar crater Stiborius is named after him. == Footnotes == == References == == Literature == Schöner, Christoph: Mathematik und Astronomie an der Universität Ingolstadt im 15. und 16. Jahrhundert, Ludovico Maximilianea. Forschungen; Vol. 13, Berlin : Duncker und Humblot, 1994. ISBN 3-428-08118-8. In German. |
Wikipedia:Andrei Bolibrukh#0 | Andrei Andreevich Bolibrukh (Russian: Андрей Андреевич Болибрух) (30 January 1950 – 11 November 2003) was a Soviet and Russian mathematician. He was known for his work on ordinary differential equations especially Hilbert's twenty-first problem (Riemann–Hilbert problem). Bolibrukh was the author of about a hundred research articles on theory of ordinary differential equations including Riemann–Hilbert problem and Fuchsian system. == Work == Bolibrukh was born on 30 January 1950 in Moscow and studied at the 45th Physics-Mathematics School in Saint Petersburg. After receiving his mathematical education at the Lomonosov Moscow State University, with Mikhail Mikhailovich Postnikov and Alexey Chernavskii as thesis advisers, he started working on the proof of the existence of linear differential equations having a prescribed monodromic group. He applied modern methods of complex analytic geometry to classical problems about ordinary differential equations and was an expert on Hilbert's twenty-first problem. In 1989, Bolibrukh produced his famous counterexamples which invalidated the Josip Plemelj's 1908 solution of Hilbert's twenty-first problem. Bolibrukh dedicated much of his efforts to the Riemann-Hilbert problem in order to find full necessary and sufficient conditions for given monodromy data to be those of a Fuchsian system. During his short career he served as Deputy Director of the Steklov Institute of Mathematics and professor at the Moscow Institute of Physics and Technology. == Honours and awards == In 1994 Bolibrukh was elected to the Russian Academy of Sciences. He was an Invited Speaker at the ICM in Zürich in 1994. He was awarded the Lyapunov Prize from the Academy of Sciences, Russia in 1995. In 2001 Bolibrukh received the State Prize of the Russian Federation. == References == == External links == Andrei Bolibrukh at the Mathematics Genealogy Project Andrei Bolibrukh. Web page at the Russian Academy of Sciences. |
Wikipedia:Andrei Krylov (mathematician)#0 | Andrei Sergeyevich Krylov (Russian: Андре́й Се́рджевич Крыло́в) (born 1956) is a Russian mathematician, specialist in mathematical methods of image processing, and Professor, Dr. Sc., a professor at the Faculty of Computer Science at the Moscow State University. He defended his thesis Mathematical modeling and computer analysis of liquid metal systems for his degree for Doctor of Physical and Mathematical Sciences (2009). He is the author of three books and more than 140 scientific articles. == References == == Bibliography == Evgeny Grigoriev (2010). Faculty of Computational Mathematics and Cybernetics: History and Modernity: A Biographical Directory (1 500 экз ed.). Moscow: Publishing house of Moscow University. pp. 56–57. ISBN 978-5-211-05838-5. == External links == Laboratory of Mathematical Methods of Image Processing(in English) MSU CMC(in Russian) Scientific works of Andrei Krylov Scientific works of Andrei Krylov(in English) |
Wikipedia:Andrei Roiter#0 | Andrei Vladimirovich Roiter (Russian: Андрей Владимирович Ройтер; Ukrainian: Андрій Володимирович Ройтер, November 30, 1937, Dnipro – July 26, 2006, Riga, Latvia) was a Ukrainian mathematician, specializing in algebra. A. V. Roiter's father was the Ukrainian physical chemist V. A. Roiter, a leading expert on catalysis. In 1955 Andrei V. Roiter matriculated at Taras Shevchenko National University of Kyiv, where he met a fellow mathematics major Lyudmyla Nazarova. In 1958 he and Nazarova transferred to Saint Petersburg State University (then named Leningrad State University). They married and began a lifelong collaboration on representation theory. He received in 1960 his Diploma (M.S.) and in 1963 his Candidate of Sciences degree (PhD). His PhD thesis was supervised by Dmitry Konstantinovich Faddeev, who also supervised Ludmila Nazarova's PhD. A. V. Roiter was hired in 1961 as a researcher at the Institute of Mathematics of the Academy of Sciences of Ukraine, where he worked until his death in 2006 and since 1991 was Head of the Department of Algebra. He received his Doctor of Sciences degree (habilitation) in 1969. In 1978 he was an invited speaker at the International Congress of Mathematicians in Helsinki. In his first published paper, Roiter in 1960 proved an important result that eventually led several other mathematicians to establish that a finite group G {\displaystyle G} has finitely many non-isomorphic indecomposable integral representations if and only if, for each prime p, its Sylow p-subgroup is cyclic of order at most p2. In a 1966 paper he proved an important theorem in the theory of the integral representation of rings. In a famous 1968 paper he proved the first Brauer-Thrall conjecture. Roiter proved the first Brauer-Thrall conjecture for finite-dimensional algebras; his paper never mentioned Artin algebras, but his techniques work for Artin algebras as well. There is an important line of research inspired by the paper and started by Maurice Auslander and Sverre Olaf Smalø in a 1980 paper. Auslander and Smalø's paper and its follow-ups by several researchers introduced, among other things, covariantly and contravariantly finite subcategories of the category of finitely generated modules over an Artin algebra, which led to the theory of almost split sequences in subcategories. According to Auslander and Smalø: ... it is perhaps surprising that the original impetus for our work did not come from the theory of hereditary artin algebras or those stably equivalent to hereditary artin algebras. Rather, the research came from an effort to explain a much older result of Gabriel and Roiter ... concerning artin algebras of finite representation type in terms of the technics and ideas developed by Auslander and Reiten in connection with almost split sequences and irreducible morphisms ... Roiter did important research on p-adic representations, especially his 1967 paper with Yuriy Drozd and Vladimir V. Kirichenko on hereditary and Bass orders and the Drozd-Roiter criterion for a commutative order to have finitely many non-isomorphic indecomposable representations. An important tool in this research was his theory of divisibility of modules. In 1972 Nazarova and Roiter introduced representations of partially ordered sets, an important class of matrix problems with many applications in mathematics, such as the representation theory of finite dimensional algebras. (In 2005 they with M. N. Smirnova proved a theorem about antimonotone quadratic forms and partially ordered sets.) Also in the 1970s Roiter in three papers, two of which were joint work with Mark Kleiner, introduced representations of bocses, a very large class of matrix problems. The monograph by Roiter and P. Gabriel (with a contribution by Bernhard Keller), published by Springer in 1992 in English translation, is important for its influence on the theory of representations of finite-dimensional algebras and the theory of matrix problems. There is a 1997 reprint of the English translation. In the years shortly before his death, Roiter did research on representations in Hilbert spaces. In two papers, he with his wife and Stanislav A. Kruglyak introduced the notion of locally scalar representations of quivers (i.e. directed multigraphs) in Hilbert spaces. In their 2006 paper they constructed for such representations Coxeter functors analogous to Bernstein-Gelfand-Ponomarev functors and applied the new functors to the study of locally scalar representations. In particular, they proved that a graph has only finitely many indecomposable locally scalar representations (up to unitary isomorphism) if and only if it is a Dynkin graph. Their result is analogous to that of Gabriel for the “usual” representations of quivers. In 1961 Roiter started in Kyiv a seminar on the theory of representations. The seminar became the foundation of the highly esteemed Kyiv school of the representation theory. He was the supervisor for 13 Candidate of Sciences degrees (PhDs). In 2007 A. V. Roiter was posthumously awarded the State Prize of Ukraine in Science and Technology for his research on representation theory. == References == Conde, Teresa (2020). "Gabriel–Roiter Measure, Representation Dimension and Rejective Chains". The Quarterly Journal of Mathematics. 71 (2): 619–635. arXiv:1903.05555. doi:10.1093/qmathj/haz062. Külshammer, Julian (2016). "In the bocs seat: Quasi-hereditary algebras and representation type". arXiv:1601.03899 [math.RT]. |
Wikipedia:Andrei Zelevinsky#0 | Andrei Vladlenovich Zelevinsky (Андрей Владленович Зелевинский; 30 January 1953 – 10 April 2013) was a Russian-American mathematician who made important contributions to algebra, combinatorics, and representation theory, among other areas. == Biography == Zelevinsky graduated in 1969 from the Moscow Mathematical School No. 2. After winning a silver medal as a member of the USSR team at the International Mathematical Olympiad he was admitted without examination to the mathematics department of Moscow State University where he obtained his PhD in 1978 under the mentorship of Joseph Bernstein, Alexandre Kirillov and Israel Gelfand. He worked in the mathematical laboratory of Vladimir Keilis-Borok at the Institute of Earth Science (1977–85), and at the Council for Cybernetics of the Soviet Academy of Sciences (1985–90). In the early 1980s, at a great personal risk, he taught at the Jewish People's University, an unofficial organization offering first-class mathematics education to talented students denied admission to Moscow State University's math department. In 1990–91, Zelevinsky was a visiting professor at Cornell University, and from 1991 until his death was on faculty at Northeastern University, Boston. With his wife, Galina, he had a son and a daughter; he also had several grandchildren. Zelevinsky is a relative of the physicists Vladimir Zelevinsky and Tanya Zelevinsky. == Research == Zelevinsky's most notable achievement is the discovery (with Sergey Fomin) of cluster algebras. His other contributions include: Bernstein–Zelevinsky classification of representations of p-adic groups; introduction (jointly with Israel Gelfand and Mikhail Kapranov) of A-systems of hypergeometric equations (also known as GKZ-systems) and development of the theory of hyperdeterminants; generalization of the Littlewood–Richardson rule and Robinson–Schensted correspondence using the combinatorics of "pictures"; work (jointly with Arkady Berenstein and Sergey Fomin) on total positivity; work (with Sergey Fomin) on the Laurent phenomenon, including its applications to Somos sequences. == Awards and recognition == Invited lecture at the International Congress of Mathematicians (Berlin, 1998) Humboldt Research Award (2004) Fellow (2012) of the American Mathematical Society University Distinguished Professorship (2013) at Northeastern University Steele Prize for Seminal Contribution to Research (2018) == References == == External links == Home page of Andrei Zelevinsky (including CV) Conference in memory of Andrei Zelevinsky Publications of Andrei Zelevinsky (in Russian) Publications of Andrei Zelevinsky (in English) Research Focus: Andrei Zelevinsky's Cluster Algebras Live journal run by Andrei Zelevinsky from 2007 to 2013 Andrei Zelevinsky at the Mathematics Genealogy Project |
Wikipedia:Andrej Dujella#0 | Andrej Dujella (born May 21, 1966 in Pula) is a Croatian professor of mathematics at the University of Zagreb and a fellow of the Croatian Academy of Sciences and Arts. == Life == Born in Pula, a native of Zadar, Dujella took part in the International Mathematical Olympiad, where he won a bronze medal in 1984. He received his M.Sc. and Ph.D. in mathematics from the University of Zagreb with a dissertation titled "Generalized Diophantine–Davenport problem". His main area of research is number theory, in particular Diophantine equations, elliptic curves, and applications of number theory in cryptography. Dujella is author of the monograph "Number Theory" (translated from Croatian). Dujella presently serves as the editor-in-chief of Rad-HAZU (Mathematical Section), a mathematics journal published by the Croatian Academy of Sciences and Arts (HAZU). Dujella's main contribution to number theory is in connection to Diophantine m-tuples. Dujella has shown that there exists no Diophantine 6-tuple and that there exist at most a finite number of Diophantine 5-tuples. He applied Diophantine tuples to construct elliptic curves with high rank. In 1998, Dujella and Attila Pethő introduced congruence method to obtain lower bound for number of Diophantine 5-tuples. In 2017, Dujella received an honorary doctorate from the University of Debrecen. == References == |
Wikipedia:Andrej Pazman#0 | Andrej Pázman (born 1938) is a Slovak mathematician working in the area of optimum experimental design and in the theory of nonlinear statistical models. He is an elected fellow of the International Statistical Institute (2004), of the Learned Society of SAS (2004) and also a member of the Royal Statistical Society (1992). He wrote also several books, three of them are monographs published in English. Today there are all presented by Springer. He obtained the Price of the Slovak Literary Fund for the Nonlinear statistical models (1994). In 2004, he obtained the WU Best Paper Award der Stadt Wien (together with W.G. Mueller), in 2008, the Golden Medal of the rector of CU Bratislava, in 2014, the Golden Medal of SAS and also the Price for Science and Technology from the Minister of Education of Slovakia. He is currently a professor (professor emeritus from 2016) at Comenius University Bratislava. == Biography == Born 6 December 1938 in Prague, studied physics and mathematics at Comenius University (CU). In 1964 he obtained his PhD in statistics (in measurement theory) from the Institute of Measurement of the Slovak Academy of Sciences (SAS), jointly advised by Juraj Bolf and Jiří Nedoma. He left this institute in 1981 to enter in the Mathematical Institute of SAS until 1991. In the years 1966–69 he has been researcher in statistics for physics in the Joint Institutes for Nuclear Research in Dubna, Russia. In 1992 he became full professor in probability and statistics at the Faculty of Mathematics, Physics and Informatics of Comenius University, head of the department 1992–1998, head of the Section of Mathematics 1999–2002. He has been appointed as invited professor at WU Wien (1995), at Universität Augsburg (1998–1999) and at TU Wien (2000), each time for one semester. He published more than 100 research papers. Main results: Hilbert space methods in experimental design, probability density of the non-linear MLE under finite samples, a differential-geometric analysis of the uniqueness of the non-linear MLE, the method of virtual noise for design under correlated observations, various methods for the design in non-linear models. He educated 9 PhD students. == Bibliography == Andrej Pázman (1986). Foundations of optimum experimental design. Vol. 14. Springer. Andrej Pázman (1993). Nonlinear statistical models. Kluwer Acad. Publ., Dordrecht. ISBN 9789401724500. Pronzato Luc and Andrej Pázman (2013). Design of Experiments in Nonlinear Models. Springer, New York. === List of other publications === http://dblp.uni-trier.de/pers/hd/p/P=aacute=zman:Andrej http://www.iam.fmph.uniba.sk/ospm/Pazman/ppazman.htm == References == == External links == Comenius University website |
Wikipedia:Andres Luure#0 | Andres Luure (born 22 May 1959, in Tallinn) is an Estonian philosopher and translator, and a researcher at Tallinn University. Luure graduated from the Moscow State University in 1983, majoring in mathematics. In 1998, he successfully defended his MA thesis titled "A combinatorial model of referring" from the Tallinn Pedagogical University. In 2006, he successfully defended his Ph. D. thesis titled "Duality and sextets: a new structure of categories" in semiotics at the University of Tartu. Luure has translated philosophical works into Estonian, including titles by Ludwig Wittgenstein, Jürgen Habermas and Gilbert Ryle. == Recognition == In 2008, Luure was recognised by Estonian Volunteering Development Centre as the Volunteer of the Year for his contribution to the Estonian Wikipedia. He received the Order of the White Star, 5th Class in 2013 for his contribution to the Estonian Wikipedia. == References == |
Wikipedia:Andrew Blake (computer scientist)#0 | Andrew Blake (born 12 March 1956) is a British scientist, former laboratory director of Microsoft Research Cambridge and Microsoft Distinguished Scientist, former director of the Alan Turing Institute, Chair of the Samsung AI Centre in Cambridge, honorary professor at the University of Cambridge, Fellow of Clare Hall, Cambridge, and a leading researcher in computer vision. == Education == Blake was educated at Rugby School and graduated in 1977 from Trinity College, Cambridge with a Bachelor of Arts degree in Mathematics and Electrical Sciences. After a year as a Kennedy Scholar at Massachusetts Institute of Technology (MIT) and two years in the defence electronics industry, he studied at the University of Edinburgh for a PhD, which was awarded in 1983 and supervised by Donald Michie. == Career and research == Until 1987 he was on the faculty of the department of Computer Science at the University of Edinburgh, as a Royal Society University Research Fellow. From 1987 to 1999, he was on the academic staff of the Department of Engineering Science in the University of Oxford, where he became a professor in 1996, and was a Royal Society Senior Research Fellow for 1998-9. In 1999 he moved to Microsoft Research Cambridge as senior research scientist, where he founded the Computer Vision Group. In 2008 he became a deputy managing director at the lab, before becoming laboratory director in 2010. From 2015-2018 he was director at the Alan Turing Institute. Since 2018 he has been the inaugural chair of the Samsung AI Centre in Cambridge. == Honours and awards == Blake was elected Fellow of the Royal Academy of Engineering (FREng) in 1998, Fellow of the Royal Society (FRS) in 2005 and an IEEE Fellow in 2008. In 2006 the Royal Academy of Engineering awarded Andrew its Silver Medal. He has twice won the prize of the European Conference on Computer Vision, with Roberto Cipolla in 1992 and with M. Isard in 1996, and was awarded the IEEE David Marr Prize (jointly with Kentaro Toyama for their paper on Probabilistic Tracking with Exemplars in a Metric Space) in 2001. In 2007 he was awarded the Mountbatten Medal by the Institution of Engineering and Technology (IET). In 2009 he was awarded the Institute of Electrical and Electronics Engineers (IEEE) Computer Vision Distinguished Researcher Award. In 2010 Blake was elected to the council of the Royal Society. In 2011, he and colleagues at Microsoft Research received the Royal Academy of Engineering MacRobert gold medal for their machine learning contribution to Microsoft Kinect human motion-capture. In 2012 he was elected to the board of the EPSRC and also received an honorary degree of Doctor of Science from the University of Edinburgh. In 2013 he was awarded an honorary degree of Doctor of Engineering from the University of Sheffield. In 2014, Blake gave the Josiah Willard Gibbs lecture at the Joint Mathematics Meetings. == Publications == Markov Random Fields for Vision and Image Processing. 2011. MIT Press. (Ed.) Active Vision. 1992. MIT Press. Visual Reconstruction. 1987. MIT Press. == References == |
Wikipedia:Andrew C. Berry#0 | Andrew Campbell Berry (November 23, 1906 – January 13, 1998) was an American mathematician. The Berry–Esseen theorem is named after him. Berry was born in Somerville, Massachusetts, US on November 23, 1906. He spent eight years (1921–1929) at Harvard University, receiving his A.B. degree in 1925, A.M. degree in 1926, and a Ph.D. in 1929. After two years at Brown University and Princeton University on a National Research Fellowship, he joined the faculty of Columbia University in 1931, where he was assistant professor from 1935 to 1941. In 1941, he joined Lawrence University as associate professor. In 1944, during World War II, the university "loaned" him to the 5th and 13th Air Forces in the Pacific War. There he worked as an operations analyst, including "development of an improved gunsight for waist-gunners on B-24 aircraft". He received the Medal of Freedom in 1946 for his actions during the Battle of Guadalcanal. Berry died in Appleton, Wisconsin on January 13, 1998. == References == |
Wikipedia:Andrew Crowther Hurley#0 | Andrew Crowther Hurley (1926–1988) was a quantum chemist and mathematician who was elected a Fellow of the Australian Academy of Science in 1972. He was a student of the University of Melbourne and obtained First Class Honours B Sc in Theoretical Physics and theory of Statistics. He received his Bachelor of Arts (Honours) in 1947 and his Bachelor of Science in 1948. He received his Master of Arts degree in March 1949 for his thesis 'Finite Rotation Groups and Crystal Classes in Four Dimensions', receiving First Class Honours and first place. In 1950 he moved to Trinity College, Cambridge, where he studied for his PhD in theoretical physics under the supervision of Paul Dirac. After one term, he transferred to the Department of Theoretical Chemistry under the supervision of John Lennard-Jones, and also interacted with S. Francis Boys, George G. Hall and John Pople. He worked on Moffitt's method of atoms in molecules and introduced the method of 'intra-atomic correlation correction' using the rather poor computational facilities available, which limited the calculations to small molecules. In 1953 he joined the Commonwealth Scientific and Industrial Research Organisation, Chemical Physics Section, and remained there until his death in 1988. He was an expert on group theory and its uses in quantum chemistry. In 1963, he wrote a monograph on the 'Electronic Theory of Small Molecules' for the series 'Theoretical Chemistry' published by Academic Press. == References == |
Wikipedia:Andrew Masondo#0 | Lieutenant General Andrew Masondo, born Andrew Mandla Lekoto Masondo (27 October 1936 – 20 April 2008) was a South African mathematician, political prisoner, a former general in the South African National Defence Force (SANDF), and a national commissar of the African National Congress's military wing, Umkhonto weSizwe, == Early life and education == Andrew Masondo was born on 27 October 1936 in Sophiatown, Johannesburg, to Alois Emmanuel Mathanjane Masondo, and Elsie Seraka Masondo. He was raised in a working class African family who believed in the value of education. After completing Grade 12 in 1954, Masondo went to Fort Hare University and majored in physics and mathematics. He completed his BSc in 1957 and, in the following year, became one of the first two black students to complete the BSc (Honours) degree in applied mathematics at the University of the Witwatersrand. In 1959, these two students completed the one-year University Education Diploma at Fort Hare, again the first black students to do so. By 1960, Masondo was lecturing pure and applied mathematics at the University of Fort Hare. His wide reading and cultural interaction with other scholars as well as the township community helped to develop an individual in touch with diverse groups of people. == Military career and activism == In 1953, Andrew Masondo joined the African National Congress (ANC). In 1962/3, he held several posts within the structure of the ANC in the Eastern Cape, mainly as a rural area organiser but also in the higher command structure. In 1963, as the command director of the underground Umkhonto we Sizwe (MK) movement, the military wing of the ANC, he began to take part in sabotage activities in South Africa, cutting electricity pylons in the vicinity of Alice, armed with a number of devices, including a saw and an old rifle he had found buried in a garden. === Awards === Decoration for Merit (Gold) (DMG) Military Merit Medal (MMM) Operational Medal for Southern Africa South Africa Service Medal Unitas (Unity) Medal Service Medal (Gold) Service Medal (Silver) Service Medal (Bronze) == Imprisonment == Masondo was arrested in 1963 and sentenced to twelve years' imprisonment on Robben Island, where he was later joined by top Rivonia trialists such as Govan Mbeki, Walter Sisulu, and Nelson Mandela. In 1964, he was sentenced to an additional three years, two of which would be served concurrently with his original sentence, bringing the total to thirteen years. In prison, Masondo again turned to education. Through UNISA, he completed second-year mathematical statistics and third-year mathematics for the second time and obtained his BSc (Honours) in Mathematical Statistics. In 1975, he registered for a BCom in statistics, but did not complete the course after losing his study privileges when he intervened to defend Walter Sisulu and Govan Mbeki in a case of alleged insubordination against a white man. == Release from prison and exile == Andrew Masondo was released from prison in 1976 and placed under house arrest. With the help of Oliver Tambo and his wife, he escaped to Swaziland in June 1976. From there, he went to Mozambique and Tanzania. He received military training, including a commander's course and a course in guerilla warfare, in the Soviet Union. In Angola, he became a national commissar of the MK, as well as a member of the ANC's national executive and Revolutionary Council. From 1978 to 1990, he lectured at the Solomon Mahlangu Freedom College in Tanzania, becoming the principal of the college. After serving as an MK ambassador and underground commander in Uganda, he returned to Angola in 1994, to deal with the repatriation of MK soldiers in exile, returning to a new, democratic South Africa. == Later career and retirement == Gen Masondo attended a Joint Staff Course at the Defence College, after the South African National Defence Force (SANDF) was established in April 1994, and subsequently promoted to the rank of major general. From 1994 until his retirement on 31 October 2001 with the rank of lieutenant-general, he held many posts - Chairman of the Integration Committee, Chief Director of Equal Opportunity, Chief Director of Corporate Communication, and Chief of the Service Corps. After his retirement, General Masondo remained active in the corporate, education, and heritage spheres of South African life. He served on numerous boards and committees in the fields of education, museums, indigenous knowledge, traditional healing, reconciliation and medicine. == Death == Masondo was diagnosed with kidney failure in 2005, he died on Sunday 20 April 2008 in the One Military Hospital in Tshwane at the age of 71. He is survived by five children and six grandchildren. == References == |
Wikipedia:Andrew Pullan#0 | Andrew John Pullan (1963 – 7 March 2012) was a New Zealand mathematician specialising in bio-electrical modelling. He was a fellow of the Royal Society of New Zealand. == Academic career == After attending Aorere College in Māngere, Pullan received a scholarship to the University of Auckland where he studied mathematics before moving to the engineering school to work on biomedical engineering finite-element models of the heart and models of electrical activity in the gastrointestinal tract. His 1988 doctoral thesis was titled Quasilinearised infiltration and the boundary element method with Professor Ian Collins as his supervisor. He was appointed head of department from 2008 to 2010. He died in March 2012 of metastatic melanoma. == Selected works == New advances in gastrointestinal motility research Mathematically modelling the electrical activity of the heart : from cell to body surface and back again == References == == External links == Google scholar Blog |
Wikipedia:Andrew Soward#0 | Andrew Michael Soward (born 20 October 1943) is a British fluid dynamicist. He is an emeritus professor at the Department of Mathematics of the University of Exeter. == Education == Soward was educated at Queens' College, Cambridge. He earned his PhD in 1969, under the supervision of Keith Moffatt. == Research == Soward is known for his work on magnetohydrodynamics (MHD) and especially dynamo theory, and also for his contributions to linear and nonlinear stability theory. He used asymptotic analysis to solve a number of outstanding problems in applied mathematics. By a new pseudo-Lagrangian technique for studying lightly damped fluid systems, he elucidated previously inexplicable features of Braginskii's geodynamo. Soward has provided explicit examples of steady fast dynamo action, thus disproving a conjecture that such dynamos did not exist. He identified new rotating modes of nonlinear convection in rotating systems, and in collaboration with Steven Childress, established an MHD dynamo model in a rapidly rotating Bénard layer; he also gave the first demonstration that situations exist where oscillatory MHD dynamos generate magnetic fields more readily than steady flows can. He collaborated with Eric Priest to provide the first mathematically consistent account of the Petschek mechanism of magnetic field line reconnection. Soward also gave the first complete solution of the Stefan (freezing) problem in cylindrical geometry; with C.A. Jones, he provided the first completely correct solution of the spherical Taylor problem. == Awards and honours == Soward was elected a Fellow of the Royal Society (FRS) in 1991. == References == |
Wikipedia:Andrew Vázsonyi#0 | Andrew Vázsonyi (1916–2003), also known as Endre Weiszfeld and Zepartzatt Gozinto) was a Hungarian mathematician and operations researcher. He is known for Weiszfeld's algorithm for minimizing the sum of distances to a set of points, and for founding The Institute of Management Sciences. == Biography == Endre Weiszfeld was born on November 4, 1916, the middle son of a Jewish family in Budapest, where his father was the owner of a shoe store. At age 14, he met and befriended Paul Erdős (his elder by three years), and at age 16, he began working on the geometric median problem for which he would later publish a solution. He studied at the Pázmány Péter Catholic University in Budapest, from which he earned a doctorate in 1936. His thesis, on higher-dimensional surfaces, was supervised by Lipót Fejér. Because of increasing discrimination against Jews in the 1930s and following the lead of his cousin, politician Vilmos Vázsonyi, he changed his name in 1937 to Andrew Vázsonyi. The name comes from that of his father's native town, Nagyvázsony. During this period, Vázsonyi studied graph theory, working with Erdős on finding necessary and sufficient conditions for an infinite graph to have an Euler tour. In 1938, Vázsonyi was invited by Otto Szász to escape Europe and work with Szász at the University of Cincinnati, but was only able to obtain a one-year student visa. Instead, he traveled to Paris, and finally succeeded in traveling to the US in April 1940, two months before France's fall to the Nazis. He spent a year at a Quaker workshop at Haverford, Pennsylvania, and in 1941 began graduate studies in mechanical engineering at Harvard University, studying there under Richard von Mises with the support of a Gordon McKay Fellowship. He earned an M.S. in 1942 and continued to work at Harvard for Howard Wilson Emmons, studying the design of supersonic aircraft. While at Harvard, he met and married Baroness Laura Vladimirovna Saparova, a musician and immigrant from Georgia whom he had met at Harvard's International Club. In 1945, Vázsonyi took US citizenship and left Harvard, working as an engineer for the Elliott Company in Jeannette, Pennsylvania. From there, he moved to southern California, where he worked on missile design for North American Aviation. He moved to the U.S. Naval Ordnance Test Station in 1948, where he headed their missile guidance and control division, and in 1953 moved again to Hughes Aircraft. At Hughes, his interests shifted from aeronautics to management science. He began working on computerization of Hughes' payroll and production lines, and on diagramming parts requirements. His alias "Zepartzatt Gozinto" began during this period, when he visited the RAND Corporation and, during a presentation there, made a joke that was misinterpreted by attendee George Dantzig. Through the 1950s and 1960s, Vázsonyi continued to work on management science problems at several other companies, including the Ramo-Wooldridge Corporation, Roe Alderson, and a second stint at NAA. In 1970, Vázsonyi joined the School of Management at the University of Southern California, but he did not get tenure there, and in 1973 he moved to the Graduate School of Business at the University of Rochester. In the late 1970s, threatened with forced retirement at Rochester as he neared age 65, he moved again to St. Mary's University, Texas. He retired in 1987, but continued to teach as an emeritus professor at the University of San Francisco. Vázsonyi died on November 13, 2003, in Santa Rosa, California. In 2009, a memorial collection of research articles was published in his honor. == Contributions == === Weiszfeld's algorithm === The geometric median of a set of points in the Euclidean plane is the point (not necessarily in the given set) that minimizes the sum of distances to the given points; the solution for three points was first given by Evangelista Torricelli, after being challenged with it by Pierre de Fermat in the 17th century. An algorithm for the more general problem with an arbitrarily large number of points, published by Weiszfeld in 1937, solves this problem numerically using a hill climbing procedure that repeatedly finds a point improving the sum of distances until no more improvements can be made. Each step of this algorithm assigns weights to the points, inversely proportional to the distances to the current solution, and then finds the weighted average of the points, which is the point that minimizes the sum of the squares of the weighted distances. The algorithm has been frequently rediscovered, and although other methods for finding the geometric median are known, Weiszfeld's algorithm is still frequently used due to its simplicity and rapid convergence. === Kruskal's tree theorem === Kruskal's tree theorem states that, in every infinite set of finite trees, there exists a pair of trees one of which is homeomorphically embedded into the other; another way of stating the same fact is that the homeomorphisms of trees form a well-quasi-ordering. In his 1960 paper giving the first proof of this result, Joseph Kruskal credits it to a conjecture of Vázsonyi. The Robertson–Seymour theorem greatly generalizes this result from trees to graphs. === TIMS and DSI === While working in the aerospace industry, Vázsonyi attended meetings of the Operations Research Society of America, but found it to be too remote from the business interests of his employers. In 1953, with William W. Cooper and Mel Salveson, Vázsonyi founded The Institute of Management Sciences; Cooper became the first president of the new society, and Vázsonyi became the first past president (without ever having been president). ORSA and TIMS later merged in 1995 to form the Institute for Operations Research and the Management Sciences. Vázsonyi also helped found the Decision Sciences Institute, and became a fellow of it. === Books === As well as his 2002 autobiography, Which Door Has the Cadillac: Adventures of a Real-Life Mathematician, Vázsonyi was the author of several technical books, including: Scientific programming in business and industry (Wiley, 1963) Problem solving by digital computers with PL/I programming (Prentice-Hall, 1970) Finite mathematics: quantitative analysis for management (Wiley, 1977) Introduction to data processing (R. D. Irwin, 1980) == References == == External resources == Biography of Andrew Vazsonyi from the Institute for Operations Research and the Management Sciences |
Wikipedia:Andrews–Curtis conjecture#0 | In mathematics, the Andrews–Curtis conjecture states that every balanced presentation (i.e. a presentation with the same number of generators and relations) of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on the relators together with conjugations of relators, named after James J. Andrews and Morton L. Curtis who proposed it in 1965. It is difficult to verify whether the conjecture holds for a given balanced presentation or not. It is widely believed that the Andrews–Curtis conjecture is false. While there are no counterexamples known, there are numerous potential counterexamples. It is known that the Zeeman conjecture on collapsibility implies the Andrews–Curtis conjecture. == References == Andrews, J. J.; Curtis, M. L. (1965), "Free groups and handlebodies", Proceedings of the American Mathematical Society, 16 (2), American Mathematical Society: 192–195, doi:10.2307/2033843, JSTOR 2033843, MR 0173241 "Low-dimensional topology, problems in", Encyclopedia of Mathematics, EMS Press, 2001 [1994] |
Wikipedia:Andrey Muchnik#0 | Andrey Albertovich Muchnik (February 24, 1958 - March 18, 2007) was a Soviet and Russian mathematician who practiced mathematical logic. He was awarded the A. N. Kolmogorov Prize in 2006. == Biography == Andrey Muchnik was born on February 24, 1958, in the Soviet Union. His parents were Albert Abramovich Muchnik and Nadezhda Mitrofanovna Ermolaeva. Both of his parents were mathematicians and students of P. S. Novikov, a Soviet mathematician. Muchnik's father, Albert Muchnik, solved Post's problem regarding the existence of a non-trivial enumerable degree of Turing reducibility. Andrey Muchnik began his academic journey at Moscow State University. Muchnik began working as a mathematician at the seminar of Evgenii Landis and Yulij Ilyashenko for third year students of the Faculty of Mechanics and Mathematics of Lomonosov Moscow State University. In his second year, he published his first work on differential equations under the guidance of Ilyashenko. Starting in his third year, he specialized in definability theory at the Department of Mathematical Logic, under the supervision of Alexei Semenov. In 1981, he completed his diploma on the solution to a problem posed by Michael Rabin at the International Congress of Mathematicians in Nice. The problem involved eliminating transfinite induction in the proof of Rabin's theorem on the solvability of the monadic theory of infinite trees. Later on, Muchnik applied his approach to prove a generalization of Rabin's theorem, which had been announced by Shelah and Stupp. Using the original idea of Alfred Tarski, he introduced in the notion of self-definability to derive a proof of the Cobham-Semenov theorem. He earned his Ph.D. in 2001. Subsequently, he worked at the Institute of New Technologies and the Scientific Council of the USSR Academy of Sciences in the field of cybernetics. He eventually became one of the leaders of the Kolmogorov seminar at Moscow State University. Muchnik also contributed results to the field of algorithmic information theory. Many of his results and collaborations were published after his death. == Awards == Andrey Muchnik was awarded the A.N. Kolmogorov Prize (along with Alexei Semenov, 2006) for his work in the field of mathematics and for the series of works "On the Refinement of A.N. Kolmogorov, Related to the Theory of Chance.” == References == == External links == Muchnik, Andrei Albertovich on the official website of the Russian Academy of Sciences Persons: Muchnik Andrey Albertovich. mathnet.ru. Retrieved 2016-3-15. "С. И. Адян, А. Л. Семёнов, В. А. Успенский, «Андрей Альбертович Мучник (некролог)», УМН, 62:4(376) (2007), 140–144". mathnet.ru. Retrieved 2016-03-15. |
Wikipedia:Andrey Piontkovsky#0 | Andrey Andreyevich Piontkovsky (Russian: Андре́й Андре́евич Пионтко́вский, born 30 June 1940) is a Russian-Georgian scientist and political writer and analyst, a member of International PEN Club. He is a former member of the Russian Opposition Coordination Council. == Biography == === Mathematics career === He graduated from the Mathematics Department of Moscow State University and has published more than a hundred scientific papers on applied mathematics. Piontkovsky is a member of the American Mathematical Society. Early in his career he wrote on the terminal control problem, and nonstationary nonlinear systems. Later papers were devoted to the mathematical modelling of strategic stability in the Cold War dual opponent system. === Polemical career === In his article published on 11 January 2000 in Sovetskaya Rossiya and placed on the Yabloko website on the same day, was the first to use the term "putinism" which he had defined as "the highest and final stage of bandit capitalism in Russia, the stage where, as one half-forgotten classic said, the bourgeoisie throws the flag of the democratic freedoms and the human rights overboard; and also as a war, "consolidation" of the nation on the ground of hatred against some ethnic group, attack on freedom of speech and information brainwashing, isolation from the outside world and further economic degradation". In the same article, Piontkovsky stated that putinism is the terminal shot to the head of Russia, and also he compared Yeltsin to Hindenburg who in 1932 gave Hitler power over Germany. Piontkovsky was an executive director of the Strategic Studies Center (Moscow) think tank that has been closed since 2006. He contributed regularly to Novaya Gazeta, The Moscow Times, The Russia Journal and the online journals Grani.ru and Transitions Online. He is also a regular political commentator for the BBC World Service and Radio Liberty in Moscow. He was an outspoken critic of Putin's "managed" democracy in Russia and, as such, has described Russia as a "soft totalitarian regime" and "hybrid fascism." Piontkovsky is the author of several books on the Putin presidency in Russia, including his most recent book, Another Look Into Putin's Soul. Piontkovsky is one of the 34 first signatories of the online anti-Putin manifesto "Putin Must Go", published on 10 March 2010. In his subsequent articles he has repeatedly stressed its importance and urged citizens to sign it. On 26 June 2013, Piontkovsky commented the case of Edward Snowden by saying, "If Pushkov dares to draw a parallel between Snowden and Soviet dissidents, I must respond that none of them had anything to do with Soviet special services and none of them pledged not to betray state and departmental secrets." Piontkovsky compared the Crimean speech of Vladimir Putin in 2014 to Hitler's speech on Sudetenland in 1939. He described Putin as using "the same arguments and vision of history" and beyond that, that this speech played a key role in starting the war in Donbas. In 2016 he published an article "Бомба, готовая взорваться" ("A bomb that is ready to explode") about Russian-Chechen ethnic conflict. When the General Prosecutor Office found his article "extremist" and started criminal prosecution Piontkovsky at last left Russia on 19 February 2016. ==== 2014 condemnation of fascism ==== Piontkovsky adduces Igor Girkin's name among those of like-minded persons and says, "The authentic high-principled Hitlerites, true Aryans Dugin, Prokhanov, Prosvirnin, Kholmogorov, Girkin, Prilepin are a marginalized minority in Russia." Piontkovsky adds, "Putin has stolen the ideology of the Russian Reich from the domestic Hitlerites, he has preventively burned them down, using their help to do so, hundreds of their most active supporters in the furnace of the Ukrainian Vendée." In his interview with Radio Liberty, Piontkovsky says that maybe the meaning of the operation conducted by Putin is to reveal all these potential passionate leaders of social revolt, send them to Ukraine and burn them in the furnace of the Ukrainian Vendée. === Exile in America === In an interview published on 11 February 2022 Piontkovsky argues that the ideology of Rashism is in many ways similar to German fascism (Nazism), while in an interview published on 11 August 2021 he opines that the speeches and policies of Putin are similar to the ideas of Hitler. Piontkovsky has written extensively on political affairs for the Jamestown Foundation, the Kyiv Post, and Project Syndicate. His byline includes mention that he is a visiting fellow at the Hudson Institute. == Some works == In English "Modern-day Rasputin". The Moscow Times. 12 November 1997. Piontkovsky, Andrei; Tsygichko, Vitali (August 1998). "Russia and NATO after Paris and Madrid: a perspective from Moscow". Contemporary Security Policy. 19 (2): 121–125. doi:10.1080/13523269808404196. Piontkovsky, Andrei (17–23 January 2000). "Stasi for president" (PDF). The Russia Journal. 3 (1): 5. Archived from the original on 13 October 2015. Retrieved 13 October 2015. "Obseqiousness toward Putin". The Washington Times. 29 September 2005. Another look into Putin's soul. Washington, DC: Hudson Institute. 2006. ISBN 1558131515. East or West? Russia's identity crisis in foreign policy (PDF). London: Foreign Policy Centre. 2006. ISBN 1903558786. Archived from the original (PDF) on 30 September 2006. Retrieved 13 October 2015. Russian Identity. Washington, DC: Hudson Institute. 2008. ISBN 978-1558131620. Andrei Piontkovsky (April 2009). "The dying mutant". Journal of Democracy. 20 (2): 52–55. doi:10.1353/jod.0.0074. S2CID 153474174. "Putinism may be fading". The Moscow Times. 7 April 2010. "The Caucasus dark circle". The Moscow Times. 1 June 2011. "The Russian spring has begun. The Putin regime will never recover legitimacy, but financial interests mean it will hang on as long as it can". The Wall Street Journal. 14 December 2011. "From protest to nausea". The Moscow Times. 2 February 2012. "The 4 stages of putinism". The Moscow Times. 6 March 2013. "Putin fears democracy in Ukraine". The Moscow Times. 1 June 2014. In Russian Gelovani, Viktor; Yegorov, Vsevolod; Mitrophanov, Viktor; Piontkovsky, Andrey (1974). Решение одной задачи управления для глобальной динамической модели Форрестера [The solution of one task for Forrester’s world dynamics model] (in Russian) (56). The Keldysh Institute of Applied Mathematics of the USSR Academy of Sciences. {{cite journal}}: Cite journal requires |journal= (help) Yurchenko, Valentin; Gelovani, Viktor; Piontkovsky, Andrey (1975). О задаче управления в глобальной модели World-3 [On the task of governing in world model WORLD-3] (in Russian). Moscow: The Institute for Problems of Management of the USSR Academy of Sciences. Yegorov, Vsevolod; Kallistov, Yuri; Mitrophanov, Viktor; Piontkovsky, Andrey (1980). Математические модели глобального развития [Mathematical models of world development] (in Russian). Leningrad: Гидрометеоиздат. Исследование стратегической стабильности методами математического моделирования [The study of strategic stability by the methods of mathematical modeling] (in Russian). Moscow: The Institute of System Analysis of the USSR Academy of Sciences. 1988. Gelovani, Viktor; Piontkovsky, Andrey; Yemeliyanov, Stanislav (1997). Эволюция концепций стратегической стабильности (Ядерное оружие в XX и XXI веке) [Evolution of conceptions of strategical stability (Nuclear weapons in the 20th and 21st centuries)] (in Russian). Moscow: РАЙМС. ISBN 5876640840. Путинизм как высшая и заключительная стадия бандитского капитализма в России [Putinism as the highest and final stage of bandit capitalism in Russia] (in Russian). Yabloko. 11 January 2000. {{cite magazine}}: Cite magazine requires |magazine= (help) За Родину! За Абрамовича! Огонь! [For motherland! For Abramovich! Fire!] (PDF) (in Russian). Moscow: ЭПИцентр. 2005. ISBN 589069099X. Archived (PDF) from the original on 11 May 2005. Нелюбимая страна [Unloved country] (in Russian). Moscow: Yabloko. 2006. ISBN 5856910613. Третий путь …к рабству [The third way …to slavery] (PDF) (in Russian). Moscow: M.Graphics Publishing. 2014 [2010]. ISBN 978-1934881422. Archived (PDF) from the original on 19 July 2014. Захар Прилепин как зеркало путинского фашизма [Zakhar Prilepin as a mirror of Putin's fascism] (in Russian). Чёртова дюжина Путина: Хроники последних лет [The baker's dozen of Putin. The chronicles of last years] (in Russian). Moscow: Алгоритм. 2014. ISBN 978-5443806396. Искушение Владимира Путина [Vladimir Putin’s temptation] (in Russian). Moscow: Алгоритм. 2013. ISBN 978-5443803296. His articles in The Jamestown Foundation His articles in Project Syndicate The Law of the Nerd, English translation from grani.ru His articles in grani.ru (Russian) Putin's Russia as a Revisionist Power == Video == "Vladimir Putin and Russia's increasingly aggressive nuclear threat (with the video of Piontkovsky's speech)". Hudson Institute. 1 October 2014. А. Пионтковский: "Путину нужна большая война" (A.Piontkovsky, "Putin needs a big war", in Russian with English subtitles, 17 November 2015, 22 min) on YouTube А. Пионтковский: "Жёсткий анализ ситуации" (A.Piontkovsky, "Harsh situation analysis", in Russian with English subtitles, 9 December 2015, 15 min) on YouTube == References == |
Wikipedia:Andrey Voronenko#0 | Andrey Voronenko (Russian: Андре́й Анато́льевич Вороне́нко) (born 1972) is a Russian mathematician, Professor, Dr.Sc., a professor at the Faculty of Computer Science at the Moscow State University. He defended the thesis «Methods for representing discrete functions in problems of calculating, testing and recognizing properties» for the degree of Doctor of Physical and Mathematical Sciences (2008). Was awarded the title of Professor (2009). Author of 16 books and more than 80 scientific articles. == References == == Literature == Evgeny Grigoriev (2010). Faculty of Computational Mathematics and Cybernetics: History and Modernity: A Biographical Directory (1 500 экз ed.). Moscow: Publishing house of Moscow University. pp. 375–376. ISBN 978-5-211-05838-5. == External links == MSU CMC(in Russian) Scientific works of Andrey Voronenko Scientific works of Andrey Voronenko(in English) |
Wikipedia:Andries Brouwer#0 | Andries Evert Brouwer (born 1951) is a Dutch mathematician and computer programmer, Professor Emeritus at Eindhoven University of Technology (TU/e). He is known as the creator of the greatly expanded 1984 to 1985 versions of the roguelike computer game Hack that formed the basis for NetHack. He is also a Linux kernel hacker. He is sometimes referred to by the handle aeb. == Biography == Born in Amsterdam, Brouwer attended the gymnasium, and obtained his MSc in mathematics at the University of Amsterdam in 1971. In 1976 he received his Ph.D. in mathematics from Vrije Universiteit with a thesis entitled "Treelike Spaces and Related Topological Spaces", under the supervision of Maarten Maurice and Pieter Baayen, both of whom were in turn students of Johannes de Groot. In 2004 he received an honorary doctorate from Aalborg University. After graduation Brouwer started his academic career at the Mathematisch Centrum, later Centrum Wiskunde & Informatica. From 1986 to 2012 he was Professor at Eindhoven University of Technology (TU/e). == Work == Brouwer's varied research interests include several branches of discrete mathematics, particularly graph theory, finite geometry and coding theory. He has published dozens of papers in graph theory and other areas of combinatorics, many of them in collaboration with other researchers. His co-authors include at least 9 of the co-authors of Paul Erdős, giving him an Erdős number of 2. === Hack === In December 1984, while at the Centrum Wiskunde & Informatica (CWI), he made the first public release of Hack on Usenet. Hack was an implementation of Rogue originally written in 1982 by Jay Fenlason and a few others, but Brouwer heavily modified and expanded it. He distributed a total of four versions of Hack between December 1984 and July 1985. The source code was released as free software, and it was widely copied, played, and ported to multiple computer platforms. When Mike Stephenson brought together a large development team via Usenet to produce an enhanced version in 1987 incorporating changes from many of the Hack derivatives, they respected Brouwer's wishes by renaming their game NetHack, as Brouwer might "...eventually release a new version of his own." === Linux kernel === Brouwer has also been involved with the development of Unix-like computer operating systems based on the Linux kernel. He was previously the maintainer of the man pager program man and the maintainer of the Linux man-pages project (from 1995 to 2004), and he is a kernel maintainer in the areas of disk geometry and partition handling. Brouwer also serves as specialist in security aspects of Unix and Linux for EiPSI (Eindhoven Institute for the Protection of Systems and Information), TU/e's information security research institute. == Selected publications == Brouwer, Andries; Arjeh Cohen; Arnold Neumaier (August 1989). Distance Regular Graphs. Ergebnisse der Mathematik und ihrer Grenzgebiete 3.19. Springer-Verlag. ISBN 0-387-50619-5. Brouwer, Andries; Haemers, Willem (16 December 2011). Spectra of Graphs. Springer. ISBN 978-1-4614-1938-9. == References == == External links == Brouwer's University home page hack(6) – Linux Games Manual Brouwer's Hack page at CWI |
Wikipedia:Andries Mac Leod#0 | Andries Hugo Donald Mac Leod (10 August 1891 – 28 March 1977) was a Belgian-Swedish philosopher and mathematician. Andries Mac Leod was born in Ledeberg, a suburb of Ghent, as a son of Julius Mac Leod, a botanist and professor at Ghent University, and of Fanny Mac Leod born Maertens, who was translator from English into Dutch of two books by Kropotkin. While Mac Leod was attending the atheneum in Ghent, he already got interested in philosophy and he was one of the founders of a Wijsgerige Kring (philosophical circle) there. One of the other members of this circle was Marcel Minnaert, with whom he maintained a lifelong friendship. Mac Leod studied mathematics and physics at Ghent University, where he obtained his doctorate in July 1914 by submitting a thesis on a problem in fluid mechanics (which appeared later as Mac Leod 1923). Immediately afterwards he travelled for holidays to Lapland in Sweden. There he learnt that the German army had invaded Belgium on 4 August 1914. He decided to stay in Sweden. He got a job in the large mathematical library of Gösta Mittag-Leffler in Djursholm near Stockholm. He also attended the seminars of the philosopher Adolph Phalén at Uppsala University. He was deeply impressed by Phalén's work. In 1921 Mac Leod returned to Belgium, where he became a high school teacher in mathematics and physics, first in Diest and later in Ghent. In 1922 he married with Gunhild Sahlén, whom he had met during his stay in Sweden. In that year also his first book appeared, an introduction to non-Euclidean geometry (Mac Leod 1922). In 1927 his first philosophical monograph (Mac Leod 1927) was published. In February 1939 he moved with his wife to Sweden, where he would stay during the rest of his life, devoting all his time to philosophical research on fundamental ontological and epistemological questions. Notably he published two lengthy philosophical monographs in Swedish (Mac Leod 1960, Mac Leod 1972). In 1960 he got an honorary doctorate at Uppsala University and a few years later he was honoured with a Festschrift (Henschen-Dahlquist 1966). == Publications == Mac Leod, A. (1922), Introduction à la géometrie non-euclidienne, Paris: J. Hermann, p. 433, JFM 48.0633.03 Mac Leod, A. (1923), "Over een geval van wenteling eener ideale vloeistof, waarbij negatieve drukkingen optreden" [On a case of rotation of an ideal fluid, where negative pressures occur], Wis- en Natuurkundig Tijdschrift (in Dutch), 2: 22–60 Mac Leod, A. (1927), Sur diverses questions se présentant dans l'étude du concept de réalité, Paris: J. Hermann, p. 239 Mac Leod, A. (1960), Beskaffenhet och innehåll av ett medvetande [The nature and content of a consciousness] (in Swedish), Uppsala: Kungl. Humanistiska Vetenskaps-Samfundet, p. 569 Mac Leod, A. (1972), Verklighet och negation [Reality and negation] (in Swedish), Stockholm and Uppsala: Almqvist & Wiksell, p. 1324 == References == Buning, L. (1977), "Van en over wijlen Dr. Andries Mac Leod (1891–1977)" (in Dutch), Wetenschappelijke tijdingen 36, nr. 3: 129–144 Dahlquist, T. (1980), "Andries Mac Leod (1891–1977) In memoriam", Theoria 46: 1–4 Henschen-Dahlquist, A.-M., ed. (1980), "Writings of Andries Mac Leod", Theoria 46: 5–6 Henschen-Dahlquist, A.-M., ed. (1966) (in Swedish), Analyser och argument. Filosofiska uppsatser tillägnade Andries MacLeod (Analyses and arguments. Philosophical essays dedicated to Andries Mac Leod), Uppsala Philosophical Studies, vol. 4, Department of Philosophy, University of Uppsala |
Wikipedia:Andrzej Grzegorczyk#0 | Andrzej Grzegorczyk ([ˈandʐɛj ɡʐɛˈɡɔrt͡ʂɨk]; 22 August 1922 – 20 March 2014) was a Polish logician, mathematician, philosopher and ethicist. He was noted for his work in computability, mathematical logic and the foundations of mathematics. == Family == In 1953, Grzegorczyk married Renata Maria Grzegorczykowa, a Polish philologist and expert in polonist linguistics. They had a daughter and a son. Grzegorczyk died of natural causes in Warsaw on 20 March 2014 at the age of 91. His body is buried in the Cemetery of Pruszków. == See also == List of Polish people == Sources == Odintsov, Sergei Pavlovich (2018): Larisa Maksimova on Implication, Interpolation, and Definability. Springer International Publishing, Cham Golińska-Pilarek, Joanna; Huuskonen, Taneli (2017): Grzegorczyk's non-Fregean logics and their formal properties. In Urbaniak, Rafał; Payette, Gillman (editors) (2017): Applications of Formal Philosophy: The Road Less Travelled. Springer International Publishing, Cham, Chapter 12, pp. 243–263 Brożek, Anna; Stadler, Friedrich; Woleński, Jan (editors) (2017): The Significance of the Lvov–Warsaw School in the European Culture. Springer International Publishing, Cham Majewska, Lucyna (2017): Этическое творчество Анджея Гжегорчика (Andrzej Grzegorczyk's works in ethics) Archived 1 March 2018 at the Wayback Machine. ВЕЧЕ: Журнал русской философии и культуры, Volume 29, pp. 285–295 (Publishing House of Saint Petersburg State University, Saint Petersburg) Śliwerski, Bogusław (2016): O kluczowej dla pedagogiki twórczości filozofa Andrzeja Grzegorczyka. Blog Pedagog, 4 January 2016 Niwiński, Damian (2016): Contribution of Warsaw Logicians to Computational Logic. Axioms, Volume 5, Issue 16, 8 pages Golińska-Pilarek, Joanna (2016): On the Minimal Non-Fregean Grzegorczyk Logic: To the Memory of Andrzej Grzegorczyk. Studia Logica: An International Journal for Symbolic Logic, Volume 104, Issue 2, pp. 209–234 Hirnyy, Oleg (June 2016). "Andrzej Grzegorczyk as a Philosopher of Education". Filosofiya Osvity / Philosophy of Education. 18 (1). Institute of Higher Education NAES of Ukraine: 242–256. doi:10.31874/2309-1606-2016-18-1-242-256. Visser, Albert (2016): The Second Incompleteness Theorem: Reflections and Rumination. In Horsten, Leon; Welch, Philip (editors) (2016): Gödel's Disjunction: The Scope and Limits of Mathematical Knowledge. Oxford University Press, Oxford, pp. 67–91 Huuskonen, Taneli (2015): Grzegorczyk's Logics: Part I Archived 10 March 2018 at the Wayback Machine. Formalized Mathematics, Volume 23, Issue 3, pp. 177–187 Huuskonen, Taneli (2015): Polish Notation Archived 11 March 2018 at the Wayback Machine. Formalized Mathematics, Volume 23, Issue 3, pp. 161–176 Biłat, Andrzej (2015): Non-Fregean Logics of Analytic Equivalence (II). Bulletin of the Section of Logic, Volume 44, Issue 1–2, pp. 69–79 Kładoczny, Piotr (2015): The Release and Rehabilitation of Victims of Stalinist Terror in Poland. In McDermott, Kevin; Stibbe, Matthew (editors) (2015): De-Stalinising Eastern Europe: The Rehabilitation of Stalin's Victims after 1953. Palgrave Macmillan, Basingstoke, pp. 67–86 Góralski, Andrzej (editor) (2015): Andrzej Grzegorczyk – Człowiek i dzieło. Biblioteka Dialogu. Universitas Rediviva, Warsaw Woleński, Jan; Marek, Victor Witold (2015): Logic in Poland after 1945 (until 1975). European Review, Volume 23, pp. 159–197 Jankowska, Małgorzata (2015): Życie to wyzwanie: Pamięci Profesora Andrzeja Grzegorczyka. Kwartalnik Filozoficzny, Volume 43, Issue 1, pp. 5–13 Pelc, Jerzy (2015): Od wydawcy: Pożegnanie ze "Studiami Semiotycznymi". Studia Semiotyczne, Volume 28–29, pp. 5–30 Krajewski, Stanisław (2015): Andrzej Grzegorczyk (1922–2014) Archived 8 March 2018 at the Wayback Machine. Studia Semiotyczne, Volume 28–29, pp. 63–88 Krajewski, Stanisław (2014): Andrzej Grzegorczyk (1922–2014). Wiadomości Matematyczne, Volume 50, Issue 1, pp. 171–173 Jankowska, Małgorzata (2014): Filozoficzne dekalogi – tekst dedykowany pamięci profesora Andrzeja Grzegorczyka (1922–2014). Zeszyty Naukowe Centrum Badań im. Edyty Stein, Number 12: Wobec Samotności, Wydawnictwo Naukowe UAM, Poznań, pp. 251–265 Trela, Grzegorz (2014): Logika – sprawa ludzka: Wspomnienie o profesorze Andrzeju Grzegorczyku (1922–2014). Argument, Volume 4, Number 2, pp. 491–498 Maksimova, Larisa Lvovna (2014): The Lyndon property and uniform interpolation over the Grzegorczyk logic. Siberian Mathematical Journal, Volume 55, Number 1, pp. 118–124 (translated from the Russian version) Avigad, Jeremy; Brattka, Vasco (2014): Computability and Analysis: The Legacy of Alan Turing. In Downey, Rod (editor) (2014): Turing's Legacy: Developments from Turing's Ideas in Logic. Cambridge University Press, Cambridge, pp. 1–47 Duchliński, Piotr (2014): W stronę aporetycznej filozofii klasycznej: Konfrontacja tomizmu egzystencjalnego z wybranymi koncepcjami filozofii współczesnej. Akademia Ignatianum, Wydawnictwo WAM, Kraków Murawski, Roman (2014): The Philosophy of Mathematics and Logic in the 1920s and 1930s in Poland. Birkhäuser, Basel Urbaniak, Rafał (2014): Leśniewski's Systems of Logic and Foundations of Mathematics. Springer International Publishing, Cham Kamiński, Łukasz; Waligóra, Grzegorz (editors) (2014): Kryptonim "Pegaz". Służba Bezpieczeństwa wobec Towarzystwa Kursów Naukowych 1978–1980 Archived 21 April 2015 at the Wayback Machine. Instytut Pamięci Narodowej – Komisja Ścigania Zbrodni przeciwko Narodowi Polskiemu, Warsaw Kamiński, Łukasz; Waligóra, Grzegorz (editors) (2014): Kryptonim Wasale: Służba bezpieczeństwa wobec studenckich komitetów Solidarności 1977–1980 Archived 21 April 2015 at the Wayback Machine. Instytut Pamięci Narodowej – Komisja Ścigania Zbrodni przeciwko Narodowi Polskiemu, Warsaw Feferman, Solomon (2013): About and around Computing over the Reals. In Copeland, Brian Jack; Posy, Carl; Shagrir, Oron (editors) (2013): Computability: Turing, Gödel, Church, and Beyond. MIT Press, Cambridge, Massachusetts, pp. 55–76 Mints, Grigori; Olkhovikov, Grigory; Urquhart, Alasdair (2013): Failure of Interpolation in Constant Domain Intuitionistic Logic. Journal of Symbolic Logic, Volume 78, Issue 3, pp. 937–950 Trzęsicki, Kazimierz; Krajewski, Stanisław; Woleński, Jan (editors) (2012): Papers on Logic and Rationality: Festschrift in Honour of Andrzej Grzegorczyk. Studies in Logic, Grammar and Rhetoric, Volume 27, Issue 40. University of Białystok, Białystok Mikołajczuk, Agnieszka (2012): O życiu zawodowym i dokonaniach naukowych Profesor Renaty Grzegorczykowej. Etnolingwistyka, Volume 24, pp. 7–10 Tavana, Nazanin Roshandel; Weihrauch, Klaus (2011): Turing machines on represented sets, a model of computation for analysis. Logical Methods in Computer Science, Volume 7, Issue 2, pp. 1–21 Resnick, Rebecca Abigail (2011): Finding the best model for continuous computation Archived 14 November 2016 at the Wayback Machine. Senior Thesis, Harvard University, Cambridge Woleński, Jan (2011): Jews in Polish Philosophy. Shofar: An Interdisciplinary Journal of Jewish Studies, Volume 29, Number 3, pp. 68–82 Murawski, Roman (2011): Logos and Mathema: Studies in the Philosophy of Mathematics and History of Logic. Peter Lang, Frankfurt am Main Murawski, Roman (2010): Essays in the Philosophy and History of Logic and Mathematics. Rodopi, Amsterdam Čačić, Vedran; Pudlák, Pavel; Restall, Greg; Urquhart, Alasdair; Visser, Albert (2010): Decorated linear order types and the theory of concatenation. 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Dissertation presented for the degree of Philosophiae Doctor (PhD), Department of Mathematics, University of Oslo, November 2009, supervised by Lars Kristiansen Ehrenfeucht, Andrzej; Marek, Victor Witold; Srebrny, Marian (editors) (2008): Andrzej Mostowski and Foundational Studies. IOS Press, Amsterdam Krajewski, Stanisław; Marek, Victor Witold; Mirkowska, Grażyna; Salwicki, Andrzej; Woleński, Jan (editors) (2008): Topics in Logic, Philosophy and Foundations of Mathematics and Computer Science: In Recognition of Professor Andrzej Grzegorczyk. Fundamenta Informaticae, Volume 81, Issue 1–3. IOS Press, Amsterdam Krajewski, Stanisław (2008): Andrzej Grzegorczyk – logika i religia, samotność i solidarność. Wiadomości Matematyczne, Volume 44, Number 01, pp. 53–59 Matuszewski, Roman; Zalewska, Anna (editors) (2007): From Insight to Proof: Festschrift in Honour of Andrzej Trybulec. Studies of Logic, Grammar, and Rhetoric, Volume 10, Issue 23 Švejdar, Vítězslav (2007): An interpretation of Robinson's Arithmetic in its Grzegorczyk's weaker variant. Fundamenta Informaticae, Volume 81, Issue 1–3, pp. 347–354 Maksimova, Larisa Lvovna (2006): Projective Beth Property in Extensions of Grzegorczyk Logic. Studia Logica: An International Journal for Symbolic Logic, Volume 83, pp. 365–391 Eisler, Jerzy Krzysztof (2006): Polski rok 1968 Archived 25 April 2015 at the Wayback Machine. Instytut Pamięci Narodowej – Komisja Ścigania Zbrodni przeciwko Narodowi Polskiemu, Warszawa Jadacki, Jacek Juliusz (2006): The Lvov–Warsaw School and Its Influence on Polish Philosophy of the Second Half of the 20th Century. In Jadacki, Jacek Juliusz and Paśniczek, Jacek (editors): The Lvov–Warsaw School – The New Generation. Rodopi, Amsterdam, pp. 41–83 Trzęsicki, Kazimierz (2006): Wkład logików polskich w światową informatykę. 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Republished in The Polish Foreign Affairs Digest: Quarterly, Volume 3, Number 2, Issue 7, pp. 29–37 Słowik, Zdzisław (2002): O duchu Europy i jej powołaniu: Rozmowa z profesorem Andrzejem Grzegorczykiem. Res Humana, Number 4, Issue 59, pp. 24–28 Chmurzyński, Jerzy Andrzej (2002): Searching Europe's Destination. Dialogue and Universalism, Number 6-7, pp. 133–144 Ciesielski, Remigiusz Tadeusz (2002): Sens Europy. Kultura Współczesna: Teorie, Interpretacje, Praktyka, Issue 3–4, pp. 111–117 Fiorentini, Camillo; Miglioli, Pierangelo (1999): A Cut-free Sequent Calculus for the Logic of Constant Domains with a Limited Amount of Duplications. Logic Journal of the IGPL, Volume 7, Issue 6, pp. 733–753 Wiśniewski, Ryszard; Tyburski, Włodzimierz (editors) (1999): Polska filozofia analityczna: W kręgu szkoły lwowsko-warszawskiej. Wydawnictwo Naukowe Uniwersytetu Mikołaja Kopernika, Toruń Murawski, Roman (1999): Recursive Functions and Metamathematics: Problems of Completeness and Decidability, Gödel's theorems. Kluwer Academic Publishers, Dordrecht Woleński, Jan; Köhler, Eckehart (editors) (1999): Alfred Tarski and the Vienna Circle: Austro–Polish Connections in Logical Empiricism. Kluwer Academic Publishers, Dordrecht Kijania-Placek, Katarzyna; Woleński, Jan (editors) (1998):The Lvov–Warsaw School and Contemporary philosophy. Kluwer Academic Publishers, Dordrecht Srzednicki, Jan Tadeusz Jerzy; Stachniak, Zbigniew (editors) (1998): Leśniewski's Systems Protothetic. Kluwer Academic Publishers, Dordrecht Weihrauch, Klaus (1997): A Foundation for Computable Analysis. In Freksa, Christian; Jantzen, Matthias; Valk, Rüdiger (editors) (1997): Foundations of Computer Science: Potential-Theory-Cognition. Springer, Heidelberg, pp. 185–199 Kaczor, Anna (1997): Perspektywy edukacji. Przegląd Humanistyczny, Volume 41, Number 3, Issue 342, pp. 188–193 Jadacki, Jacek Juliusz (1996): The Conceptual System of the Lvov–Warsaw School. Axiomathes, Number 3, pp. 325–333 Błaszczyk, Jolanta (1994): Dziewięć Lat Warszawskiej Premiery Literackiej (Nine years of the Warsaw Literary Award). Bibliotekarz, Issue 3, pp. 25–27 Mardaev, Sergei Il'ich (1993): Least fixed points in Grzegorczyk's logic and in the intuitionistic propositional logic. Algebra and Logic, Volume 32, Issue 5, pp. 279–288 Boolos, George (1993): The Logic of Provability. Cambridge University Press, Cambridge Woleński, Jan (editor) (1990): Philosophical Logic in Poland. Kluwer Academic Publishers, Dordrecht Woleński, Jan (editor) (1990): Kotarbiński: Logic, Semantics and Ontology. Kluwer Academic Publishers, Dordrecht Woleński, Jan (1989): Logic and Philosophy in the Lvov–Warsaw School. Kluwer Academic Publishers, Dordrecht Maryniarczyk, Andrzej (1989): Filozofia naukowa czy kolejny zabobon?. Ethos: Kwartalnik Instytutu Jana Pawła II KUL, Volume 2, Number 8, pp. 332–337 Srzednicki, Jan Tadeusz Jerzy; Rickey, Vincent Frederick; Czelakowski, Janusz (editors) (1984): Leśniewski's Systems: Ontology and Mereology. Martinus Nijhoff Publishers, The Hague & Ossolineum: Publishing House of the Polish Academy of Sciences, Wrocław Gabbay, Dov; Guenthner, Franz (editors) (1984): Handbook of Philosophical Logic, Volume II: Extensions of Classical Logic. D. Reidel, Dordrecht Paris, Jeffrey Bruce; Wilkie, Alex James (1984): Δ 0 {\displaystyle \Delta _{0}} sets and induction. In Guzicki, Wojciech; Marek, Wiktor Witold; Pelc, Andrzej; Rauszer, Cecylia (editors) (1984): Open Days in Model Theory and Set Theory: Proceedings of a Conference held in September 1981 at Jadwisin, near Warsaw, Poland. University of Leeds, Leeds, pp. 237–248 Cichoń, Eugeniusz Adam; Wainer, Stanley Scott (1983): The Slow-Growing and the Grzegorczyk Hierarchies. Journal of Symbolic Logic, Volume 48, Issue 2, pp. 399–408 Boolos, George (1980): Provability in arithmetic and a schema of Grzegorczyk. Fundamenta Mathematicae, Volume 60, Number 1, pp. 41–45 Boolos, George (1979): The Unprovability of Consistency: An Essay in Modal Logic. Cambridge University Press, Cambridge Hass, Ludwik (1979): Wolnomularze i loże wolnomularskie Płocka (1803–1821). Rocznik Mazowiecki, Volume 7, pp. 69–126 Tatarkiewicz, Teresa; Tatarkiewicz, Władysław (1979): Wspomnienia. Państwowy Instytut Wydawniczy, Warsaw Mostowski, Andrzej (1979): Thirty Years of Foundational Studies: Lectures on the Development of Mathematical Logic and the Study of the Foundations of Mathematics in 1930–1964. In Kuratowski, Kazimierz; Marek, Wiktor Witold; Pacholski, Leszek; Rasiowa, Helena; Ryll-Nardzewski, Czesław; Zbierski, Paweł (editors): Andrzej Mostowski: Foundational Studies, Selected Works, Volume I. North-Holland, Amsterdam & PWN–Polish Scientific Publishers, Warsaw, pp. 1–176 Chomsky, Noam (1977): The Right to Help: Noam Chomsky and Andrzej Grzegorczyk. The New York Review of Books, 4 August 1977. Ptaczek, Józef (editor) (1976): Memoriał 59, inne dokumenty protestu oraz list otwarty prof. dr Edwarda Lipińskiego do Gierka. Wydawnictwo Komitetu Głównego P.P.S. w Niemczech, Munich Wainer, Stanley Scott (1972): Ordinal Recursion, and a Refinement of the Extended Grzegorczyk Hierarchy. Journal of Symbolic Logic, Volume 37, Issue 2, pp. 281–292 Segerberg, Karl Krister (1971): An Essay in Classical Modal Logic Archived 25 February 2018 at the Wayback Machine. PhD dissertation under Dana Stewart Scott, Stanford University. Filosofiska Studier utgivna av Filosofiska Föreningen och Filosofiska Institutionen vid Uppsala Universitet, Number 13, Uppsala, 1971 Klemke, Dieter (1971): Ein Henkin-Beweis für die Vollständigkeit eines Kalküls relativ zur Grzegorczyk-Semantik. Archiv für Mathematische Logik und Grundlagenforschung, Volume 14, pp. 148–161 Görnemann, Sabine (1971): A logic stronger than intuitionism. Journal of Symbolic Logic, Volume 36, Issue 2, pp. 249–261 Klemke, Dieter (1970): Ein vollständiger Kalkül für die Folgerungsbeziehung der Grzegorczyk-Semantik. Dissertation, Albert-Ludwigs-Universität Freiburg im Breisgau, Naturwissenschaftlich-Mathematische Fakultät Görnemann, Sabine (1969): Über eine Verschärfung der intuitionistischen Logik. Proefschrift, Technische Hochschule Hannover, Fakultät für Mathematik und Naturwissenschaft Gabbay, Dov (1969): Montague Type Semantics for Non-Classical Logics I. U.S. Air Force Office of Scientific Research, contract No. F 61052-68-C-0036, Report No. 4 Kostanecki, Stanisław (1969): Mirosław Zdziarski (1892–1939). Rocznik Mazowiecki, Volume 2, pp. 309–339 Starnawski, Jerzy (1969): Piotr Grzegorczyk (17 listopada 1894 – 20 maja 1968). Pamiętnik Literacki: Czasopismo kwartalne poświęcone historii i krytyce literatury polskiej, Volume 60, Issue 2, pp. 409–415 Addison, John West; Henkin, Leon; Tarski, Alfred (editors) (1965): The Theory of Models: Proceedings of the 1963 International Symposium at Berkeley. North-Holland, Amsterdam Jordan, Zbigniew Antoni (1963): Philosophy and Ideology: The Development of Philosophy and Marxism-Leninism in Poland since the Second World War. D. Reidel, Dordrecht Luschei, Eugene Charles (1962): The Logical Systems of Lesniewski. North-Holland, Amsterdam Czeżowski, Tadeusz (editor) (1960): Charisteria: Rozprawy filozoficzne złożone w darze Władysławowi Tatarkiewiczowi w siedemdziesiątą rocznicę urodzin. Państwowe Wydawnictwo Naukowe, Warsaw Załęski, Stanisław (1908): O masonii w Polsce od roku 1738 do 1822: na źródłach wyłącznie masońskich. Druk W.L. Anczyca i Spółki, Kraków Wójcicki, Kazimierz Władysław (1858): Cmentarz Powązkowski oraz cmentarze katolickie i innych wyznań pod Warszawą i w okolicach tegoż miasta, Tom III. Drukarnia S. Orgelbranda, Warsaw == External links == Andrzej Grzegorczyk profile at Mathematics Genealogy Project Andrzej Grzegorczyk's Warsaw Uprising biogram (in Polish) Andrzej Grzegorczyk's obituary (in Polish) by Polish Mathematical Society Andrzej Grzegorczyk's profile by Calculemus 2012 Interview with Andrzej Grzegorczyk (in Polish) Andrzej Grzegorczyk's 90th birthday at the Warsaw branch of the Polish Philosophical Society Andrzej Grzegorczyk's lecture on 17 May 2011, the University of Opole, Poland 2006 Appeal of the Polish Science Representatives to the Minister of Environment Jan Szyszko in Defense of the Tatra Mountains (in Polish) |
Wikipedia:Andrzej Swierniak#0 | Andrzej Piotr Świerniak (born February 22, 1950, in Wałbrzych) is a Polish mathematician, specializing in bioinformatics and control theory. == Biography == In 1972 he obtained a master's degree in automation engineering at the Faculty of Automation of the Silesian University of Technology, and in 1975 he received a master's degree in mathematics at the University of Silesia in Katowice. In 1978, at the Faculty of Automatic Control and Computer Science of the Silesian University of Technology, he received his doctoral degree technical sciences. There, based on scientific achievements and his habilitation monograph, he was awarded in 1988 his Habilitation (higher doctoral degree). In 1996 he became a professor of technical sciences. He became a full professor at the Silesian University of Technology at the Faculty of Automatic Control, Electronics and Computer Science and the director of the Institute of Automatic Control at this department. He became a member of the Automation and Robotics Committee of the Polish Academy of Sciences, the Committee of Biocybernetics and Biomedical Engineering of the Polish Academy of Sciences, the Central Commission for Degrees and Titles (Section VI - Technical Sciences), the Committee of the Scientific Research Committee (Electronics, Automation and Robotics, Information Technology and Telecommunications, T-11 ) and the Appeals Team of the Science Council at the Ministry of Science and Higher Education; Science Council; Appeal Team. In 2019 he became a member of the Rada Doskonałości Naukowej (Council of Scientific Excellence), the nation's highest distinction in the discipline of biomedical engineering. In 2012 he was elected a Fellow of the American Mathematical Society. == Awards and honors == Golden Cross of Merit (1993) Medal of Merit for the Silesian University of Technology (1997) Knight's Cross of the Order of Polonia Restituta (1999) Silver Badge of the Meritorious SEP (2002) Medal of the National Education Committee (2003) Honorary Medal of Prof. Obrąpalski (2005) PTETiS Gold Badge (2008) Officer's Cross of the Order of Polonia Restituta (2011) == Selected publications == Swierniak, A.; Polanski, A.; Kimmel, M. (1996). "Optimal control problems arising in cell-cycle-specific cancer chemotherapy". Cell Proliferation. 29 (3): 117–139. doi:10.1046/j.1365-2184.1996.00995.x. ISSN 0960-7722. PMID 8652742. S2CID 1413722. Tarnawski, Rafal; Fowler, Jack; Skladowski, Krzysztof; Świerniak, Andrzej; Suwiński, Rafal; Maciejewski, Boguslaw; Wygoda, Andrzej (2002). "How fast is repopulation of tumor cells during the treatment gap?". International Journal of Radiation Oncology, Biology, Physics. 54 (1): 229–236. doi:10.1016/S0360-3016(02)02936-X. ISSN 0360-3016. PMID 12182996. with Barbara Jarząb, Małgorzata Wiench, Krzysztof Fujarewicz, Krzysztof Simek, Michał Jarząb, Małgorzata Oczko-Wojciechowska, Jan Włoch et al.: "Gene expression profile of papillary thyroid cancer: sources of variability and diagnostic implications." Cancer Research 65, no. 4 (2005): 1587-1597. == References == |
Wikipedia:András P. Huhn#0 | András P Huhn (Szeged, 26 January 1947 – Szeged, 6 June 1985) was a Hungarian mathematician. Huhn's theorem on the representation of distributive semilattices is named after him. == References == O'Connor, John J.; Robertson, Edmund F., "András P. Huhn", MacTutor History of Mathematics Archive, University of St Andrews == External links == A Tribute to Andras Huhn by E. Tamás Schmidt |
Wikipedia:András Sebő#0 | András Sebő (born 24 April 1954) is a Hungarian-French mathematician working in the areas of combinatorial optimization and discrete mathematics. Sebő is a French National Centre for Scientific Research (CNRS) Director of Research and the former head of the Combinatorial Optimization. group in Laboratory G-SCOP, affiliated with the University of Grenoble and the CNRS. == Biography == Sebő received his Ph.D. in 1984 from Faculty of Sciences of the Eötvös Loránd University and he obtained the Candidate's Degree from the Hungarian Academy of Sciences in 1989, advised by András Frank. From 1979 through 1988, Sebő was a Research Assistant and Research Fellow at The Computer and Automation Research Institute, Hungarian Academy of Sciences in Budapest. He moved to the University of Grenoble in 1988, where he advanced to his current position of CNRS Director of Research. He has held visiting positions at leading mathematical centers, including the Research Institute for Discrete Mathematics in Bonn, Germany (1988-89 as an Alexander von Humboldt Foundation Fellow and 1992-93 as the John von Neumann Professor), DIMACS (1989), University of Waterloo Faculty of Mathematics (multiple years), and the Hausdorff Center for Mathematics (2015). He is also one of seven honorary members of the Egerváry Research Group on Combinatorial Optimization. == Research work == In 2012, Sebő and Jens Vygen developed a 7/5-approximation algorithm for the graph version of the traveling salesman problem; currently the best-known approximation, improving on the widely cited 1.5-epsilon result of Gharan, Saberi, and Singh. In 2013, Sebő found also an 8/5-approximation algorithm for the path version of the TSP. A scientific conference in honor of Sebő was held April 24–25, 2014 in Grenoble, France. == References == == External links == András Sebő publications indexed by Microsoft Academic András Sebő at DBLP Bibliography Server András Sebő's publications indexed by the Scopus bibliographic database. (subscription required) |
Wikipedia:André Darré#0 | André (Andrew) Darré (1750–1833) was a French priest and academic. He was one of the four exiles from France, the others being professors Francois Anglade, Louis-Gilles Delahogue, and Pierre-Justin Delort, sometimes called the French "founding fathers" of Maynooth College in Ireland. == Life == A native of the small town of Montaut, Auch near Toulouse in Gascony, France, he was born on 5 February 1750. Darré studied philosophy and theology at the University of Toulouse, was ordained a priest in Auch in 1774, and became a professor of philosophy in Toulouse. Darré was exiled following the French Revolution and moved to Ireland in 1793, where he was appointed Professor of Logic, Metaphysics and Ethics in 1795 at the newly established Royal College of St. Patrick, Maynooth, and Professor of Mathematics and Natural Philosophy from 1801 to 1813. He succeeded fellow French exile, the Rev. Pierre-Justin Delort, who had returned to France in 1801, as Chair of Natural Philosophy and Mathematics. During the Irish rebellion of 1803, which occurred close to the college, Darre helped negotiate the surrender of the local rebels. He returned to France in 1813 (and was succeeded in the Chair of Natural Philosophy and Mathematics by his former pupil Cornelius Denvir), and was serving as canon of Sainte-Marie d'Auch when he died in 1833. == Publications == A copy of his work is stored in the Russell Library, Maynooth. Elements of Geometry, with both Plane and Spherical Trigonometry. Designed for the use of the Students at the R. C. College, Maynooth, by André Darré, printed and published by H. Fitzpatrick of 4, Capel Street, Dublin 1813. == Legacy == His textbook was used by his successors in Maynooth Professors Denvir and Callan, and Rev. Dr. Callan produced a revised edition in 1844. The Darré Exhibitions prize, named in his honour, is awarded to students based on their leaving certificate results, on entry to first-year Mathematics at Maynooth University. == References == |
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