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/*
* decaffeinate suggestions:
* DS102: Remove unnecessary code created because of implicit returns
* Full docs: https://github.com/decaffeinate/decaffeinate/blob/master/docs/suggestions.md
*/
const { fabricate } = require("@artsy/antigravity")
const sinon = require("sinon")
const routes = require("../routes")
const Backbone = require("backbone")
const { Profile } = require("../../../models/profile")
describe("#index", function () {
beforeEach(function () {
let renderStub
sinon.stub(Backbone, "sync")
routes.index(
{
params: { id: "foo" },
profile: new Profile(fabricate("profile", { owner_type: "User" })),
},
{ locals: { sd: {} }, render: (renderStub = sinon.stub()) }
)
Backbone.sync.args[0][2].success(fabricate("profile", { id: "artsy-ed" }))
this.templateName = renderStub.args[0][0]
return (this.templateOptions = renderStub.args[0][1])
})
afterEach(() => Backbone.sync.restore())
return it("renders the post page", function () {
this.templateName.should.equal("template")
return this.templateOptions.profile.get("id").should.equal("artsy-ed")
})
})
describe("#setProfile", function () {
beforeEach(() => sinon.stub(Backbone, "sync"))
afterEach(() => Backbone.sync.restore())
it("sets the profile from the vanity url and lets the next handler take it", function () {
let spy
const req = { params: { id: "foobar" } }
const res = { locals: { sd: {} } }
routes.setProfile(req, res, (spy = sinon.spy()))
Backbone.sync.args[0][2].success(fabricate("profile", { id: "foobar" }))
req.profile.get("id").should.equal("foobar")
return spy.called.should.be.ok()
})
return it("nexts if the profile already exists on the req", function () {
let next
const req = {
params: { id: "foobar" },
profile: new Profile(fabricate("profile")),
}
const res = { locals: { sd: {} } }
routes.setProfile(req, res, (next = sinon.spy()))
return next.called.should.be.ok()
})
})
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code
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सामान्य वास्तविकता और सिक्कों के वेब को फाड़ें मनोविज्ञान दुनिया
पोस्टपर्टम अवसाद सिर्फ बेबी ब्लूज़ नहीं है ग्लोब भर में कैंसर रिसर्च के राज्य रिश्ते और रिकवरी के बारे में आपको क्या पता होना चाहिए कैरियर की सफलता के भविष्य कहां हैं? द बायोस्कोसिस्को मॉडल और इसकी सीमाएं आपकी रिश्ते से परे मरम्मत क्या है? उबलते रक्त किशोर द्वारा प्रयुक्त शीर्ष पांच सामाजिक नेटवर्किंग साइटें अतीत हमेशा के बारे में है एलेन डीजेनेरस: मनोविज्ञान का क्यों हर कोई उसे प्यार करता है रचनात्मक पुनर्वास, भाग २: गंभीर सिर चोट क्रिएटिव संश्लेषण गर्मियों क्या हम सब बस साथ में आ सकते हैं? द बिग डी सिर्फ एक सुंदर चेहरा से अधिक: प्यारे फंदोम उन्मास्किंग
सामान्य वास्तविकता और सिक्कों के वेब को फाड़ें
संयोग क्या परिस्थितियों को अक्सर होता है?
संयोग आते हैं, और संयोग जाते हैं। क्या परिस्थितियों में उनकी उपस्थिति को बढ़ावा देना है?
मेरा एक परिचय खुद को प्रवाह में और फिर प्रवाह से बाहर पाता है। प्रवाह में, वह कई संयोगों का अनुभव करती है। और फिर वे रुक गए। वह नहीं जानता कि इंस या आउट ट्रिगर करता है। स्पष्टीकरण में शायद उनके व्यक्तित्व लक्षण शामिल हैं।
पिछली पोस्ट में, मैंने बताया कि व्यक्तित्व चर के एक श्रृंखला में संयोग खोजने के साथ कुछ करना है। विशेष रूप से, जिन लोगों के पास मजबूत विचार सहयोग की आसानी है यानी, जो लोग अपने दिमाग में पैटर्न के साथ पर्यावरण के आसपास पैटर्न आसानी से मेल खाते हैं वे संयोग देखने की अधिक संभावना रखते हैं। ऐसा लगता है कि मेरे इस परिचितता में कुछ समय पर विचार सहयोग की आसानी है, लेकिन दूसरों पर नहीं। ऐसा इसलिए है क्योंकि इसे न केवल व्यक्तित्व के साथ ही परिस्थितियों के साथ भी करना है। कुछ परिस्थितियां संघ की आसानी को सुविधाजनक बनाती हैं।
सामान्य वास्तविकता के वेब को फाड़ना
सामान्य वास्तविकता के वेब को फाड़ना संयोग बढ़ाता है। जब हमारे दैनिक दिनचर्या बाधित होते हैं, तो यह वेब टूटा हुआ है, जो हमारी जागरूकता में अधिक संयोग की अनुमति देता है। जन्म और मृत्यु दोनों में रूटीन बाधित होते हैं जिनमें जन्म, मृत्यु, शादी, रोमांटिक प्यार, स्नातक स्तर, नौकरी में परिवर्तन, बीमारी, तलाक, संकट और यात्रा शामिल है।
ये घटनाएं यादृच्छिकता में वृद्धि करती हैं क्योंकि वे हमें दैनिक दिनचर्या की अनुमानित बाधाओं से बाहर ले जाती हैं। कभी-कभी अराजकता उत्पन्न होती है और संकट उभरता है। संकट में अवसर हो सकता है। कम से कम कठिन परिस्थितियों में एक रास्ता खोजने के प्रयास में उद्देश्य के साथ व्यक्तिपरक को जोड़ने की हमारी प्रवृत्ति बढ़ जाती है। रोमांटिक प्यार अक्सर लोगों को प्रेमियों के बीच संबंधों की तलाश करने के लिए प्रेरित करता है, समानताएं कि एक या दोनों उम्मीदें उनके बीच घनिष्ठ संबंधों को इंगित करती हैं।
कलाकार, चित्रकार, लेखकों, और संगीतकार कभी-कभी अपनी रचनात्मक प्रेरणा को खोजने के लिए सामान्य वास्तविकता के वेब से खुद को अलग करते हैं। वे हर रोज पीछे छोड़ने की खोज की प्रक्रिया में अवशोषित हो जाते हैं। एक चित्रकार को फॉर्म और रंग के बीच पहले अनदेखा कनेक्शन के माध्यम से नए विचार मिल सकते हैं। एक लेखक कॉफी शॉप में एक टिप्पणी सुन सकता है जो कि वर्तमान विषय को पूरा करने के लिए आवश्यक चीज़ों के अनुरूप फिट बैठता है।
कुछ वातावरण दूसरों की तुलना में संयोग में समृद्ध हैं।
बस उन लोगों के संपर्क में आ रहे हैं जो संयोग के बारे में बात करते हैं और रहते हैं, दूसरों के प्रति संवेदनशीलता फैल सकते हैं।
कुछ धार्मिक समूह सभ्यता के संदेश के रूप में संयोग पर विचार करते हैं। कुछ कॉल संयोग गॉडविंक्स। इन कहानियों को दूसरों के साथ साझा करना उनके देवता के लाभ में अपनी विश्वास को मजबूत करने में मदद करता है। सामाजिक समर्थन समूह में व्यक्तियों को सहायक संयोगों को ध्यान में रखकर और उनमें शामिल प्रशंसापत्रों की पेशकश करने की अनुमति देता है।
१ ९ ६० के दशक के उत्तरार्ध में और १ ९ ७० के दशक के अंत में सैन फ्रांसिस्को के हाइट-एशबरी जिले में, टेलीफोन ध्रुवों और बुलेटिन बोर्डों पर संकेत दिए गए थे, यहां सिंक्रोनिटी स्पोकन। उस स्थान और समय के सामाजिक नियमों की लचीलापन नियमित वास्तविकता में आंसू बना रही है, अनुमति उभरने के लिए और अधिक कोइंसिडन्स्स।
नाटकों और संगीतकारों के निदेशक रिपोर्ट करते हैं कि जब किसी व्यक्ति के लिए किसी व्यक्ति की आवश्यकता होती है, तो एक व्यक्ति अक्सर दिखाता है कि कौन फिट बैठता है। एक स्क्रिप्ट की सापेक्ष सादगी से उच्च संगठन बनाने की कोशिश करने का जंगली साहस इन सहायक संयोगों को बढ़ावा देता है।
एक ज़रूरत है और इंटरनेट जवाब देता है
हम त्वरित प्रतिक्रियाओं के आदी हो गए हैं इंटरनेट खोज हमें हमारे प्रश्नों के बारे में बताते हैं। मेरे शोध से पता चलता है कि जब लोगों को इंटरनेट की आवश्यकता होती है और सर्फ करते हैं, तो जवाब आश्चर्यजनक रूप से बिना पूछे बिना दिखाई देते हैं। यह ४ सबसे आम संयोगों में से एक है।
इंटरनेट एक उच्च मात्रा अंतरंग विचारों का अवसर प्रदान करता है। अर्थपूर्ण संयोग इस तरह की स्थितियों में उभरने की अधिक संभावना है क्योंकि विचारों के अप्रत्याशित चौराहे से संयोग बनाए जाते हैं।
अन्य उच्च मात्रा अंतरण अंतरण
विचारों को छेड़छाड़ करने की एक उच्च मात्रा के लिए एक और जगह विभिन्न पृष्ठभूमि से लोगों के समूह को एक साथ रखकर और जरूरतों के साथ चुनौती देकर बनाई जा सकती है।
नाथन माईरवॉल्ड ने बस यही किया। उन्होंने उन सबसे उज्ज्वल आविष्कारकों को इकट्ठा किया जो उन्हें मिल सके और उनसे उस समय के परेशान सवालों के जवाब खोजने के लिए कहा। उनकी तीव्र बातचीत ने माइक्रोचिप्स बनाने और जेट इंजन में सुधार करने के लिए नई तकनीकें उत्पन्न की; उन्होंने जाल आस्तीन को कस्टम-टेलर करने का एक तरीका प्रस्तावित किया कि न्यूरोसर्जन का उपयोग एन्यूरियम्स की मरम्मत के लिए किया जा सकता है। समूह ने अपने पेटेंट के समूह को अस्सी मिलियन डॉलर के लिए लाइसेंस दिया है।
कभी-कभी किसी समूह में जो कुछ भी आवश्यक होता है, वह किसी के लिए संयोग के बारे में बात करना शुरू कर देता है, और फिर कहानियां बहती हैं। संयोग पर कक्षा में एक समूह को एक साथ प्राप्त करें, और अधिक संयोग दिखाई देंगे। दोस्तों के साथ अपने बारे में बात करके अपना स्वयं का संयोग-साझाकरण समूह बनाएं। मुझे बताएं कि क्या हेाता है!
चेतना के बदलते राज्य
अधिकांश मानव अस्तित्व के लिए, हमने अपनी चेतना को बदलने के तरीके खोजे हैं। क्यूं कर? नए दृष्टिकोण और अनुभवों को अनुमति देने के लिए हमारे पर्यावरण के हमारे मानक विचारों और प्रतिक्रियाओं को कम करने के लिए। ध्यान, उपवास, समूह ड्रमिंग और नृत्य और दिमाग बदलने वाली दवाओं में से प्रत्येक के पास कई संस्कृतियों और स्थानों पर उपयोग का लंबा इतिहास है। इन आत्म प्रेरित अनुभवों से एसोसिएशन की आसानी बढ़ जाती है। वे मन में और उसके पर्यावरण में अन्य विचारों के साथ कनेक्शन खोजने के लिए दिमाग में विचारों के लिए नए जैव रासायनिक संदर्भ बनाते हैं।
तलाक दवाओं धार्मिक ध्यान रोमांटिक प्यार व्यक्तित्व
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\begin{document}
\date{}
\title{$*-$open sets and $*-$ continuity in topological spaces}
\author{Aliakbar Alijani}
\address{Mollasadra Technical and Vocational College\\
Technical and Vocational University\\
Ramsar, Iran}
\email{[email protected]}
\thanks{}
\subjclass{54A10,54C10}
\begin{abstract}
In this paper, we study some properties of $*-$open and $*-$closed subsets of a space. The collection of all $*-$open subsets of a space $X$ form a topology on $X$ which is denoted by $^{*}O(X)$. We investigate the relations between topological properties of $X$ with the topology $^{*}O(X)$ and $X$. Also, we introduce the concept of a $*-$continuous map.
\end{abstract}
\maketitle
\newcommand\sfrac[2]{{#1/#2}}
\newcommand\cont{\operatorname{cont}}
\newcommand\diff{\operatorname{diff}}
\section{Introduction}
\newcommand{\stackrel}{\stackrel}
By a space, we mean a topological space. In a space $X$, the closure and the interior of a set $A$ will be denoted by $cl(A)$ and $int(A)$, respectively. If $A\subseteq Y\subseteq X$, the closure and the interior of $A$ as a subspace of $Y$ will be denoted by $cl_{Y}(A)$ and $int_{Y}(A)$, respectively. A set $A$ in $X$ is called regular open if $int(cl(A))=A$. The collection of all regular open subsets of a space $X$ denote by $R.O(X)$. A set $A$ in space $X$ is called $*-$open if for every $x\in A$, there exists $ U\in R.O(X)$ containing $x$ such that $U\subseteq A$. We denote by $^{*}O(X)$, the collection of all $*-$open subsets of a space $X$. Then, $^{*}O(X)$ is a topology on $X$ (Theorem \ref{32}). Recall that $R.O(X)$, form a base for a topology on $X$ \cite[p. 138]{B}. In fact, $^{*}O(X)$, is the topology generated by $R.O(X)$. In Section \ref{sec2}, We investigate the relations between some topological properties of $(X,^{*}O(X))$ and $X$ as follows:
\begin{itemize}
\item If $X$ is a compact space, then $(X,^{*}O(X))$ is a compact space (Lemma \ref{19}). Conversely is not true (Example \ref{20}).
\item If $X$ is a Hausdorrf, locally compact space, then $(X,^{*}O(X))$ is a Hausdorrf, locally compact space (Lemma \ref{21}). Conversely is not true (Example \ref{22}).
\item $X$ is a connected space if and only if $(X,^{*}O(X))$ is a connected space (Lemma \ref{18}).
\item If $X$ is a path connected space, then $(X,^{*}O(X))$ is a path connected space (Lemma \ref{23}). Conversely is not true (Example \ref{24}).
\item If $X$ is a locally connected space, then $(X,^{*}O(X))$ is a locally connected space (Lemma \ref{25}). Conversely is not true (Example \ref{27}).
\end{itemize}
In Section \ref{sec3}, we introduce the concept of $*-$continuous maps. A map $f:X\to Y$ is called $*-$continuous if the inverse image of every $*-$open set is $*-$open. $*-$continuous neither implies continuous nor is implied by continuous (Example \ref{28}). $*-$continuous image of connected sets are connected (Lemma \ref{29}). compactness, path connectedness and local connectedness need not to be preserved by $*-$continuous maps (Example \ref{30}).
For two spaces $X$ and $Y$, We denote by $X\times Y$, the cartesian product of $X\times Y$ with the product topology. For more information on topological spaces, see \cite{B}.
\section{$*-$open and $*-$closed sets}\label{sec2}
In this section, we introduce the concept and study some properties of $*-$open and $*-$closed sets.
\begin{definition}
Let $X$ be a space. For a set $A$ in $X$, define $^{*}int(A)$ and $^{*}cl{A}$ as follows:
$$^{*}int(A)=\{x\in A| \exists W\in R.O(X) ; x\in W\subseteq A\}$$
$$^{*}cl(A)=\{x\in X| \forall x\in W\in R.O(X); W\bigcap A\neq\emptyset\}$$
\end{definition}
\begin{definition}
A set $A$ in a space $X$ is called $*-open$ if $^{*}int(A)=A$ and $*-$closed if $^{*}cl{A}=A$.
\end{definition}
In Lemma \ref{1}, we present some properties of $^{*}int(A)$ and $^{*}cl(A)$.
\begin{lemma}\label{1}
Let $X$ be a space and $A$ an arbitrary set in $X$. Then:
\begin{enumerate}
\item $^{*}int(A)\subseteq int(A)\subseteq A\subseteq cl(A)\subseteq ^{*}cl{A}$
\item $^{*}int(A)=\bigcup\{ W\in R.O(X); W\subseteq A\}.$
\item $^{*}cl(A)=\bigcap\{W\in R.C(X); A\subseteq W\}.$
\item $^{*}int(A)$ is open and $^{*}cl(A)$ closed.
\item $^{*}cl(X-A)=X-^{*}int(A)$ and $X-^{*}cl(A)=^{*}int(X-A)$.\\
\end{enumerate}
\end{lemma}
\proof
It is clear.
\endproof
\begin{remark}
It follows from (5) of Lemma \ref{1} that a set $A$ is $*-$open if and only if $X-A$ be $*-$closed.
\end{remark}
\begin{remark}
Clearly, $*-$open sets are open. But, conversely need not be true. See Example \ref{2}.
\end{remark}
\begin{example}\label{2}
Let $\mathbb{R}$ be the set of reals and, $\tau$ be the topology on $\mathbb{R}$ generated by the union of $\tau_{1}$, the usual topology on $\mathbb{R}$, and $\tau_{2}$, the topology of countable complements on $\mathbb{R}$ \cite[Example 63]{SS}. Then, $\mathbb{R-Q}$ is open in $(\mathbb{R},\tau)$. But, $^{*}int(\mathbb{R-Q})=\emptyset$.\\
\end{example}
\begin{definition}
A space is called semi-regular if it has a basis consisting of regular open sets \cite{S}.
\end{definition}
\begin{lemma}\label{3}
Let $X$ be a semi-regular space. Then, a set $A$ in $X$ is $*-$open if and only if it is open.
\end{lemma}
\proof
Let $A$ be an open subset of $X$ and $x\in A$. Since $X$ is semi-regular, there exists a set $W\in R.O(X)$ such that $x\in W\subseteq A$. Hence, $x\in ^{*}Int(A)$ and $A$ is $*-$open.
\endproof
\begin{remark}
By (2) of Lemma \ref{1}, regular open sets are $*-$open. But, conversely need not be true. See Example \ref{4}.
\end{remark}
\begin{example}\label{4}
Since $(\mathbb{R},\tau_{1})$ is a semi-regular space, $(0,1)\bigcup(1,2)$ is $*-$open, but it is not regular open.
\end{example}
\begin{definition}
$^{*}O(X)$ will denote the class of all $*-$open sets in $X$.
\end{definition}
\begin{lemma}\label{5}
Let $(X,\tau)$ be a space. Then, $R.O(X)\subseteq ^{*}O(X)\subseteq \tau$.
\end{lemma}
\begin{lemma}\label{6}
Let $(X,\tau)$ be a semi-regular space. Then, $\tau=^{*}O(X)$.
\end{lemma}
\proof
It is clear by Lemma \ref{3}.
\endproof
\begin{remark}
In general, for a space $(X,\tau)$, $\tau\neq ^{*}O(X)$. It is sufficient to consider $(\mathbb{R},\tau)$ as in Example \ref{2}. Then, $^{*}O(X)=\tau_{1}\neq \tau$.
\end{remark}
\begin{lemma}\label{7}
Let $U$ be an open subset of $X$. Then, $Int(cl(U))$ is a regular open set.
\end{lemma}
\proof
It is clear.
\endproof
\begin{lemma}\label{8}
Let $(X,\tau)$ be a space. Then, $R.O(X,^{*}O(X))\subseteq R.O(X,\tau)$.
\end{lemma}
\proof
Let $W$ be a regular open set in $(X,^{*}O(X))$. Then,$ Int_{^{*}O(X)}(cl_{^{*}O(X)}(W))=W$. Let $x\in Int(cl(W))$. Then, $x\in U$ for some $U\in \tau$ and $U\subseteq cl(W)$. So, $x\in Int(cl(U))\subseteq cl(W)$. Since $^{*}O(X)\subseteq \tau$, $cl(W)\subseteq cl_{^{*}O(X)}(W)$. By Lemma \ref{7}, $Int(cl(U))\in ^{*}O(X)$. Hance, $x\in Int_{^{*}O(X)}(cl_{^{*}O(X)}(W))$. So, $x\in W$.
\endproof
\begin{lemma}
Let $(X,\tau)$ be a space. Then, $^{*}O(X,^{*}O(X,\tau))\subseteq^{*}O(X,\tau)$.
\end{lemma}
\proof
It is clear by Lemma \ref{8}.
\endproof
\begin{lemma}\label{9}
Let $Y$ be an open subset of $X$ and $A\subseteq Y$. Then we have:
\begin{enumerate}
\item $W\in R.O(Y)$ if and only if $W=N\bigcap Y$ where $N\in R.O(X)$.
\item $^{*}int(A)=^{*}int_{Y}(A)\bigcap ^{*}Int(Y)$
\item $^{*}cl_{Y}(A)=^{*}cl(A)\bigcap Y$
\end{enumerate}
\end{lemma}
\proof
(1) First suppose that $W\in R.O(Y)$. So, $int_{Y}(cl_{Y}(W))=W$. We have $$W=int_{Y}(cl_{Y}(W))=int(cl(U)\bigcap Y)=int(cl(U)\bigcap Y$$ Put $N=int(cl(U)$. Then, $N\in R.O(X)$ and the proof is complete. Conversely, let $W=N\bigcap Y$ where $N\in R.O(X)$. Then $$int_{Y}(cl_{Y}(W))=int(cl(N\bigcap Y)\bigcap Y)\subseteq int(cl(N))\bigcap Y=N\bigcap Y=W$$.
So, $W\in R.O(Y)$.
\endproof
\begin{corollary}\label{10}
Let $A\subseteq Y\subseteq X$ where $X$ is a space and $Y$ an open subspace. If $A\in ^{*}O(X)$, then $A\in ^{*}O(Y)$.
\end{corollary}
\proof
First recall that $^{*}Int(A)\subseteq ^{*}Int(Y)$. Since $A\in ^{*}O(X)$, $^{*}Int(A)=A$. Hence,$^{*}Int_{Y}(A)\subseteq A\subseteq ^{*}Int(Y)$. By (2) of Lemma \ref{9},$A\in ^{*}O(Y)$.
\endproof
\begin{remark}
The converse of Corollary \ref{10} need not be true. See Example \ref{11}.
\end{remark}
\begin{example}\label{11}
Let $(\mathbb{R},\tau)$ be as Example \ref{2}. Then, $\mathbb{R-Q}\in ^{*}O(\mathbb{R-Q})$. But, $\mathbb{R-Q}\notin ^{*}O(\mathbb{R-Q})$.
\end{example}
\begin{lemma}\label{12}
Let $X$ and $Y$ be two spaces,$A\in ^{*}O(X)$ and $B\in ^{*}O(Y)$. Then, $A\times B\in ^{*}O(X\times Y)$.
\end{lemma}
\proof
Let $A\in ^{*}O(X)$, $B\in ^{*}O(Y)$ and $(x,y)\in A\times B$. Then, there exists $x\in U\in R.O(X)$ and $y\in W\in R.O(Y)$ such that $U\subseteq A$ and $V\subseteq B$. Since $Int(cl(U\times V))=Int(cl(U))\times Int(cl(V))$, so $U\times V \in R.O(X\times Y)$. The proof is complete.
\endproof
\begin{remark}
It is not necessary that if $A\in ^{*}O(X\times Y)$, then $A=A_{1}\times A_{2}$ where $A_{1}\in ^{*}O(X)$ and $A_{2}\in ^{*}O(Y)$. See Example \ref{13}.
\end{remark}
\begin{example}\label{13}
Let $\mathbb{R}$ be the reals with the usual topology. By Lemma \ref{6}, $A=((0,1)\times (0,1))\cup (\mathbb{R}\times \{0\})$ is $*-$open in $\mathbb{R}\times \mathbb{R}_d$ ($\mathbb{R}_{d}$ is the reals with the discrete topology). An easy calculation shows that $A\neq A_{1}\times A_{2}$ for every $A_{1}\in ^{*}O(\mathbb{R})$ and $A_{2}\in ^{*}O(\mathbb{R}_d)$.
\end{example}
\begin{lemma}\label{14}
Let $\{A_{i};i\in I\}$ be a collection of $*-$open sets in a space $X$. Then, $\bigcup_{i}A_{i}$ is $*-$open set. If $I$ is finite, then $\bigcap_{i}A_{i}$ is $*-$open.
\end{lemma}
\proof
Let $x\in \bigcup_{i}A_{i}$. Then, $x\in A_{i}$ for some $i\in I$. So, there exists $U\in R.O(X)$ containing $x$ such that $U\subseteq A_{i}$. It is clear that $ U\subseteq \bigcup_{i}A_{i}$. Now, suppose that $x\in \bigcap_{i}A_{i}$. Then, $x\in A_{i}$ for each $i\in I$. So, there exists $U_{i}\in R.O(X)$ such that $U_{i}\subseteq A_{i}$ for each $i\in I$. Since $I$ is finite, $\bigcap_{i}U_{i}\in R.O(X)$ and $\bigcap_{i}U_{i}\subseteq \bigcap_{i}A_{i}$.
\endproof
\begin{remark}
The arbitrary intersection of $*-$open sets need not be a $*-$open set. See Example \ref{15}.
\end{remark}
\begin{example}\label{15}
In $(\mathbb{R},\tau_{1})$, it is clear that $\bigcap_{n\in \mathbb{N}}(-\frac{1}{n},\frac{1}{n})=\{0\}$ which is not $*-$open.
\end{example}
\begin{remark}
The converse of Lemma \ref{14} need not be true. See Example \ref{16}.
\end{remark}
\begin{example}\label{16}
Let $\mathbb{R}$ be the reals with the usual topology. Then, $(\mathbb{R-Q})\bigcup \mathbb{Q}=\mathbb{R}$ and $(\mathbb{R-Q})\bigcap \mathbb{Q}=\emptyset$ which are $*-$open sets. But, $\mathbb{R-Q}$ and $\mathbb{Q}$ are not $*-$open sets.
\end{example}
\begin{theorem}\label{32}
$^{*}O(X)$ is a topology on $X$.
\end{theorem}
\proof
Clearly, $\{\emptyset,X\}\subseteq ^{*}O(X)$. By Lemma \ref{14}, $^{*}O(X)$ is closed under finite intersections and arbitrary unions.
\endproof
\begin{remark}
$*-$open sets do not preserve by open continuous maps. See Example \ref{17}.
\end{remark}
\begin{example}\label{17}
Let $X$ be an infinite set and $p\in X$. Define $\tau=\{\phi\}\bigcup \{U\subseteq X;p\in U\}$. Then, $\tau$ is a topology on $X$. Consider $f:X\to X$ as follows: $f(x)=p$ for all $x\in X$. Then, $f$ is an open continuous map. But, $f(X)=\{p\}\notin ^{*}O(X)$.
\end{example}
\begin{lemma}\label{18}
Let $(X,\tau)$ be a space. Then, $(X,\tau)$ is a connected space if and only if $(X,^{*}O(X))$ be a connected space.
\end{lemma}
\proof
Let $(X,\tau)$ be connected. By Lemma \ref{5}, it is clear that $(X,^{*}O(X))$ is connected. Conversely, let $(X,^{*}O(X))$ be connected and $A$, a clopen subset of $(X,\tau)$. Then, $\{A, X-A\}\subseteq ^{*}O(X)$. So, $A$ is a clopen set in $(X,^{*}O(X))$. Since $(X,^{*}O(X))$ is connected, $A=\emptyset$ or $A=X$ and proof is complete.
\endproof
\begin{lemma}\label{19}
Let $(X,\tau)$ be compact space. Then, $(X,^{*}O(X))$ is a compact space.
\end{lemma}
\proof
It is clear by Lemma \ref{5}.
\endproof
\begin{remark}
The converse of Lemma \ref{19} need not be true. See Example \ref{20}.
\end{remark}
\begin{example}\label{20}
Let $X$ be as Example \ref{17}. Then, $^{*}O(X)=\{\emptyset,X\}$. So, $(X,^{*}O(X))$ is compact. But, $(X,\tau)$ is not compact.
\end{example}
\begin{lemma}\label{21}
Let $(X,\tau)$ be a Hausdorff, locally compact space. Then, $(X,^{*}O(X))$ is locally compact.
\end{lemma}
\proof
Let $x\in X$. Then, $U\subseteq C$ for some $x\in U\in \tau$ and compact set $C$ in $(X,\tau)$. By Lemma \ref{7}, $Int(cl(U))\in ^{*}O(X)$ and $Int(cl(U))\subseteq C$. By Lemma \ref{5}, it is clear that $C$ is compact in $(X,^{*}O(X))$.
\endproof
\begin{remark}
The converse of Lemma \ref{21} need not be true. See Example \ref{22}.
\end{remark}
\begin{example}\label{22}
Let $(\mathbb{R},\tau)$ be the same space as in Example \ref{2}. Then, $^{*}O(\mathbb{R})=\tau_{1}$. So, $(\mathbb{R},^{*}O(\mathbb{R}))$ is locally compact. But, $(\mathbb{R},\tau)$ is not locally compact.
\end{example}
\begin{lemma}\label{23}
Let $(X,\tau)$ be a path connected space. Then, $(X,^{*}O(X))$ is a path connected space.
\end{lemma}
\proof
Let $(X,\tau)$ be a path connected space and $x_{0},x_{1}\in X$. Then, there exists a continuous map $f:I\to X$ such that $f(0)=x_{0}$ and $f(1)=x_{1}$. Since the identity map $1_{X}:(X,\tau)\to (X,^{*}O(X))$ is continuous, $1_{X}of:I \to (X,^{*}O(X))$ is continuous. So, $(X,^{*}O(X))$ is a path connected.
\endproof
\begin{remark}
The converse of Lemma \ref{23} need not be true. See Example \ref{24}.
\end{remark}
\begin{example}\label{24}
Let $(\mathbb{R},\tau)$ be the same space as in Example \ref{2}. Since $^{*}O(\mathbb{R})=\tau_{1}$, $(\mathbb{R},^{*}O(\mathbb{R}))$ is path connected. Now suppose that, $x_{0},x_{1}\in \mathbb{R}$ and $f:I \to (\mathbb{R},\tau)$ be a continuous map such that $f(0)=x_{0}$ and $f(1)=x_{1}$. Then, $f(I)$ is a compact, connected subset of $(\mathbb{R},\tau)$ which is a contradiction.
\end{example}
\begin{lemma}\label{25}
Let $(X,\tau)$ be a locally connected space. Then, $(X,^{*}O(X))$ is locally connected.
\end{lemma}
\proof
Let $(X,\tau)$ be a locally connected space and $x\in U\in ^{*}O(X)$. Then, there exist $V\in R.O(X)$ containing $x$ such that $V\subseteq U$. Since $(X,\tau)$ is locally connected, there exists $ W\in \tau$ containing $x$ such that $W$ is connected and $W\subseteq V$. Since $W\subseteq Int(cl(W))\subseteq cl(W)$, $ Int(cl(W))$ is connected in $(X,\tau)$. It is clear that $Int(cl(W))$ is connected in $(X,^{*}O(X))$ and proof is complete.
\endproof
\begin{lemma}\label{26}
Let $(\mathbb{R},\tau)$ be as in Example \ref{2}. A set $A$ is connected in $(\mathbb{R},\tau)$ if and only if $A$ is connected in $(\mathbb{R},^{*}O(\mathbb{R}))$
\end{lemma}
\proof
Let $A$ be a set in $(\mathbb{R},\tau)$, $A=U_{1}\bigcup U_{2}$ where $U_{1},U_{2}\in \tau$ and $U_{1}\bigcap U_{2}=\emptyset$. Then, $U_{1},U_{2}$ are closed in $(\mathbb{R},\tau)$. Hence, $U_{1},U_{2}\in (\mathbb{R},\tau_{1})$ and proof is complete.
\endproof
\begin{remark}
The converse of Lemma \ref{25} need not be true. See Example \ref{27}.
\end{remark}
\begin{example}\label{27}
Let $(\mathbb{R},\tau)$ be the same space as in Example \ref{2}. Consider the open set $\mathbb{R-Q}$ in $(\mathbb{R},\tau)$. If $U$ be an open connected set in $(\mathbb{R},\tau)$ such that $U\subseteq \mathbb{R-Q}$, then by Lemma \ref{26}, $U\in \tau_{1}$ which is a contradiction (since $U\bigcap\mathbb{Q}\neq\emptyset$). So, $(\mathbb{R},\tau)$ is not locally connected. Recall that $^{*}O((\mathbb{R},\tau))=\tau_{1}$ and $(\mathbb{R},\tau_{1})$ is locally connected.
\end{example}
\section{$*-$continuous maps}\label{sec3}
In this section, we introduce the concept and study some properties of $*-$continuous maps.
\begin{definition}
A function $f:X\to Y$ is said to be $*-$continuous if the inverse image of every $*-$open subset of $Y$ is $*-$open subset of $X$.
\end{definition}
\begin{remark}
Recall that $*-$continuous neither implies continuous nor is implied by continuous. See Example \ref{28}.
\end{remark}
\begin{example}\label{28}
Let $(\mathbb{R},\tau)$ be as in Example \ref{2}. The identity map from $(\mathbb{R},\tau_{1})$ to $(\mathbb{R},\tau)$ is $*-$continuous. But, it is not continuous. Also, consider $f:(\mathbb{R},\tau)\to(\mathbb{R},\tau_{1})$ as follows: $f(x)=x$ if $x\in \mathbb{R-Q}$ and $f(x)=0$ if $x\in \mathbb{Q}$. Then, $f$ is continuous since if $a<b$ and $0\in (a,b)$, then $f^{-1}((a,b))=(a,b)$ and otherwise $f^{-1}((a,b))=(a,b)-\mathbb{Q}$. $f$ is not $*-$continuous, since $(0,1)$ is $*-$open in $(\mathbb{R},\tau_{1})$ but $f^{-1}((0,1))=(0,1)-\mathbb{Q}$ is not $*-$open in $(\mathbb{R},\tau)$.
\end{example}
\begin{lemma}
The projection maps $\pi_{i}:\prod_{j}X_{j}\to X_{i}$ are $*-$continuous.
\end{lemma}
\proof
For $i\in I$, let $U_{i}\in ^{*}O(X_{i})$. Then, $\pi_{i}^{-1}(U_{i})=U_{i}\times \prod_{j\neq i}X_{j}$. By Lemma \ref{12}, $\pi_{i}^{-1}(U_{i})\in ^{*}O(\prod_{i}X_{i})$.
\endproof
\begin{definition}
A map $f:X\to Y$ is called almost continuous if the inverse image of every regular open subset of $Y$ is open subset of $X$.
\end{definition}
\begin{remark}
It is clear that every $*-$continuous map is almost continuous. But, the converse need not be true. See the map $f$ of Example \ref{28}.
\end{remark}
\begin{lemma}
An open almost continuous map is $*-$continuous.
\end{lemma}
\proof
Let $f:X\to Y$ be an open almost continuous map and $y\in W\in ^{*}O(Y)$. Then, there exists $V\in R.O(Y)$ containing $y$ such that $V\subseteq W$. By \cite[Lemma 3.17]{M}, $f^{-1}(V)\in R.O(X)$ and $f^{-1}(V)\subseteq f^{-1}(W)$. So, $f^{-1}(W)\in ^{*}O(X)$.
\endproof
\begin{lemma}\label{29}
Let $f:X\to Y$ be a onto $*-$continuous map. If $X$ is connected, then $Y$ is connected.
\end{lemma}
\proof
Let $A$ be a clopen subset of $Y$. Then, $A$ is a clopen set in $(Y,^{*}O(Y))$. Hence, $f^{-1}(A)$ is a clopen set in $(X,^{*}O(X))$. By Lemma \ref{18}, $(X,^{*}O(X))$ is connected. So, $A=\emptyset$ or $A=Y$ and proof is complete.
\endproof
\begin{remark}
We know that compactness property preserved by continuous maps. But, this property do not preserve by $*-$continuous maps even though it is open. See Example \ref{30}.
\end{remark}
\begin{example}\label{30}
Let $(\mathbb{R},\tau)$ be as in Example \ref{2}. Consider the identity map $1_{\mathbb{R}}:(\mathbb{R},\tau_{1})\to (\mathbb{R},\tau)$. Then, $[0,1]$ is compact in $(\mathbb{R},\tau_{1})$, but it is not compact in $(\mathbb{R},\tau)$ ( Recall that the only compact sets in $(\mathbb{R},\tau)$ are finite).
\end{example}
\begin{remark}
Path connected and locally connected properties need not to be preserved by $*-$continuous maps. See Examples \ref{30},\ref{27} and \ref{24}.
\end{remark}
\begin{remark}
Recall that if $f:X\to Y$ is continuous, then $f(cl(A))\subseteq cl(f(A))$ for every subset $A$ of $X$. But, it is not true that $f(^{*}cl(A))\subseteq ^{*}cl(f(A))$. See Example \ref{31}.
\end{remark}
\begin{example}\label{31}
Consider the continuous map $f$ as in the Example \ref{28}. Then, $f(^{*}cl(\mathbb{Q}))=f(\mathbb{R})=(\mathbb{R-Q})\bigcup \{0\}$. But, $^{*}cl(f(\mathbb{Q}))=^{*}cl(\{0\})=\{0\}$.
\end{example}
\begin{lemma}
Let $f:X\to Y$ be a map. Then, $f$ is $*-$continuous if and only if for every subset $A$ of $X$, $f(^{*}cl(A))\subseteq ^{*}cl(f(A))$.
\end{lemma}
\proof
Let $x\in ^{*}cl(A)$ and $W$ be a regular open set in $Y$ contains $f(x)$. Then, $f^{-1}(W)$ is a $*-$open subset of $X$ contains $x$. Hence, there exists a regular open set $V$ in $X$ such that $f(x)\in f(V)\subseteq W$. Since $x\in ^{*}cl(A)$ , $V\bigcap A \neq \emptyset $. So, $\emptyset \neq f(V)\bigcap f(A)\subseteq W\bigcap f(A)$. Hence, $x\in ^{*}cl(f(A))$. Conversely, let $W\in ^{*}O(Y)$. Then, $Y-W$ is $*-$closed. By assumption , $$f(^{*}cl(f^{-1}(Y-W)))=f(^{*}cl(X-f^{-1}(W))\subseteq ^{*}cl(Y-W)=Y-W$$
Hence, $^{*}cl(X-f^{-1}(W))\subseteq X-f^{-1}(W)$. So, $f^{-1}(W)\in ^{*}O(X)$ and the proof is complete.
\endproof
\end{document}
|
math
|
\begin{document}
\color{blue}mall
\title{Affine isoperimetric inequalities on flag
manifolds}
\author{Susanna Dann\thanks{Thanks the Oberwolfach Research Institute
for Mathematics for its hospitality and support, where part of
this work was carried out.} \and Grigoris Paouris
\thanks{Supported by Simons Foundation Collaboration Grant \#527498
and NSF grant DMS-1812240.} \and Peter Pivovarov \thanks{Supported
by NSF grant DMS-1612936.}}
\maketitle
\begin{abstract}
Building on work of Furstenberg and Tzkoni, we introduce ${\bf
r}$-flag affine quermassintegrals and their dual versions. These
quantities generalize affine and dual affine quermassintegrals as
averages on flag manifolds (where the Grassmannian can be considered
as a special case). We establish affine and linear invariance
properties and extend fundamental results to this new setting. In
particular, we prove several affine isoperimetric inequalities from
convex geometry and their approximate reverse forms. We also
introduce functional forms of these quantities and establish
corresponding inequalities.
\end{abstract}
\color{blue}ection{Introduction}
Affine isoperimetric inequalities provide a rich foundation for
understanding principles in geometry and analysis that arise in the
presence of symmetries. Among the most fundamental examples is the
Blaschke-Santal\'{o} inequality \cite{Sant} on the product of volumes
of an origin-symmetric convex body $L$ in $\mathbb{R}^n$ and its polar
$L^{\circ}=\{x\in \mathbb{R}^n:\langle x, y\rangle \leq 1, \;\forall
y\in L\}$. The latter asserts that this product is maximized for
ellipsoids, i.e.,
\begin{equation}
\label{eqn:BS}
\abs{L}\abs{L^{\circ}} \leq \omega_n^2,
\end{equation}
where $\omega_n$ is the volume of the unit Euclidean ball $B_2^n$.
The Blaschke-Santal\'{o} inequality, and its version for
non-origin-symmetric bodies, is one of several equivalent forms of the
affine isoperimetric inequality; see e.g., the survey
\cite{Lutwak_survey}. Moreover, it admits numerous extensions: for
example, $L_p$ versions \cite{LZ}, generalizations from convex bodies
to functions, e.g., \cite{Ball1}, \cite{AKM}, \cite{FM} with
applications to concentration of measure \cite{AKM}, \cite{Lehec08};
further functional affine isoperimetric inequalities, e.g.,
\cite{AKSW}; stronger versions in which stochastic dominance holds
\cite{CFPP}.
Another fundamental affine isoperimetric inequality is the Petty polar
projection inequality \cite{Petty}. This concerns projection bodies,
which are special zonoids that play a fundamental role in convex
geometry, functional analysis, among other fields, e.g.,
\cite{S}, \cite{GarB}. The projection body of a convex body
$L\color{blue}ubseteq\mathbb{R}^n$ is the convex body $\Pi L$ defined by its
support function in direction $\theta\in S^{n-1}$ by $h_{\Pi
L}(\theta)=\abs{P_{\theta^{\perp}}L}$, where $P_{\theta^{\perp}}$ is
the orthogonal projection onto $\theta^{\perp}$. The Petty projection
inequality asserts that the affine-invariant quantity
$\abs{L}^{n-1}\abs{(\Pi L)^{\circ}}$ is maximized by ellipsoids, i.e.,
\begin{equation}
\label{eqn:Petty}
\abs{L}^{n-1}\abs{(\Pi L)^{\circ}}\leq
\omega_n^{n}\omega_{n-1}^{-n}.
\end{equation}
The Petty projection inequality is the geometric foundation for
Zhang's affine Sobolev inequality \cite{Zhang}. Its equivalent forms
and extensions have given rise to fundamental inequalities in
analysis, geometry and information theory, e.g., \cite{LYZ_Lp},
\cite{LYZ_Sobolev}.
The affine invariance in inequalities \eqref{eqn:BS} and
\eqref{eqn:Petty} follows from volumetric considerations. However, as
we will review below, the underlying principle goes much deeper and
extends to the family of affine quermassintegrals, of which
$\abs{L^{\circ}}$ and $\abs{(\Pi L)^{\circ}}$ are just two special
cases, up to normalization. Formally, the affine quermassintegrals are
defined for compact sets $L\color{blue}ubseteq \mathbb{R}^n$ and $1\leq k\leq n$
by
\begin{equation}
\label{aff-quer-1-def}
\Phi_{[k]}(L)=
\left(\int_{G_{n,k}}\abs{P_EL}^{-n}d\nu_{n,k}(E)\right)^{-\frac{1}{kn}},
\end{equation}
where $G_{n,k}$ is the Grassmannian manifold of $k$-dimensional linear
subspaces equipped with the Haar probability measure
$\nu_{n,k}$. Writing $\abs{L^{\circ}}$ and $\abs{(\Pi L)^{\circ}}$ in
polar coordinates shows a direct connection to $k=1$ and $k=n-1$ in
\eqref{aff-quer-1-def}, respectively. As the name suggests, they are
affine-invariant, i.e., $\Phi_{[k]}(TL)=\Phi_{[k]}(L)$ for each volume
preserving affine transformation $T$, as proved by Grinberg \cite{Gr},
extending earlier work on ellipsoids by Furstenberg-Tzkoni \cite{FT}
and Lutwak \cite{Lu1}.
The quantities $\Phi_{[k]}(L)$ are affine versions of
quermassintegrals or intrinsic volumes, which play a central role in
Brunn-Minkowski theory \cite{S}. In particular, the intrinsic volumes
$V_1(L),\ldots, V_n(L)$ of a convex body $L$ admit similar
representations through Kubota's integral recursion as
\begin{equation}
V_k(L) = c_{n,k}\int_{G_{n,k}}\abs{P_EL}d\nu_{n,k}(E),
\end{equation}
where $c_{n,k}$ is a constant that depends only on $n$ and $k$. They
enjoy many fundamental inequalities, such as
\begin{equation}
\label{eqn:Alexandrov}
V_k(L)\geq V_k(r_L B_2^n),
\end{equation}
for $k=1,\ldots,n-1$, where $r_L$ is the radius of a Euclidean ball
having the same volume as $L$. Taking $k=1$ in \eqref{eqn:Alexandrov}
corresponds to Urysohn's inequality, while $k=n-1$ is the standard
isoperimetric inequality. From Jensen's inequality one sees that
\eqref{eqn:BS} and \eqref{eqn:Petty} provide stronger affine-invariant
analogues of \eqref{eqn:Alexandrov} for $k=1$ and $k=n$, respectively.
For the intermediary values $1<k<n$, the inequalities in
\eqref{eqn:Alexandrov} are well-known consequences of
Alexandrov-Fenchel inequality, e.g., \cite{S}. On the other hand, it
is still an open problem, posed by Lutwak \cite{Lu1}, \cite[Problem
9.3]{GarB}, to determine minimizers for their affine versions,
namely, to prove that for $1<k<n-1$,
\begin{equation}
\label{eqn:Lutwak_conj}
\Phi_{[k]}(L)\geq \Phi_{[k]}(r_LB_2^n).
\end{equation}
In the last 40 years, a compelling dual theory, initiated by Lutwak in
\cite{Lut_dual}, has flourished (see, e.g., \cite{S}, \cite{GarB}). Rather than
convex bodies and projections onto lower-dimensional subspaces, this
involves star-shaped sets and intersections with subspaces. As above,
a key isoperimetric inequality lies at its foundation. The
intersection body of a star-shaped body $L$ is the star-shaped body
$IL$ with radial function $\rho_{IL} (\theta) := | L \cap
\theta^{\perp} |$. The Busemann intersection inequality \cite{Bu},
proved originally for convex bodies $L$, states that
\begin{equation}
\label{eqn:Busemann}
\abs{IL}\abs{L}^{-(n-1)}\leq \omega_{n-1}^n\omega_n^{-(n-2)}.
\end{equation}
The volume of the intersection body lies at one end-point of a
sequence of $SL_{n}$-invariant quantities that are called the dual
affine quermassintegrals. These are $SL_{n}$-invariant analogs of the
dual quermassintegrals introduced by Lutwak \cite{Lu00}. Formally, for
a compact set $L\color{blue}ubseteq \mathbb{R}^n$ and $1\leq k\leq n$, the dual
affine quermassintegrals of $L$ are defined by
\begin{equation}
\label{dual-aff-quer-1-def}
\Psi_{[k]}(L) = \left(\int_{G_{n,k}}\abs{L\cap
E}^nd\nu_{n,k}(E)\right)^{\frac{1}{kn}}.
\end{equation}
As above, Grinberg \cite{Gr}, drawing on \cite{FT}, showed that these
enjoy invariance under volume-preserving linear transformations, i.e.
$\Psi_{[k]}(TL)= \Psi_{[k]}(L)$ for $T\in SL_{n}$. They also satisfy
the following extension of \eqref{eqn:Busemann}, proved by
Busemann-Straus \cite{Bu} and Grinberg \cite{Gr}:
\begin{equation}
\label{Gr-ineq}
\Psi_{[k]}(L)\leq \Psi_{[k]}(r_L B_2^n).
\end{equation}
While the dual theory has been developed for star-shaped bodies, the
investigation of these quantities goes deeper and can be extended to
bounded Borel sets and non-negative measurable functions \cite{Gar},
\cite{DPP}. For recent developments on dual Brunn-Minkowski
theory, see \cite{S}, \cite{GarB}, \cite{HLYZ} and the references
therein.
The theory that has developed around affine and dual affine
quermassintegrals has implications outside of convex geometry. As a
sample, we mention variants of \eqref{Gr-ineq} for functions in
\cite{DPP} lead to sharp asymptotics for small-ball probabilities for
marginal densities when independence may be lacking; small-ball
probabilities for the volume of random polytopes \cite{PP2}; bounds on
marginal densities of $\log$-concave measures connected to the Slicing
Problem \cite{PaoVal}. In these applications, the main focus was on
volumetric estimates and implications for high-dimenional probability
measures. Recently, there is increasing interest in other
probabilistic aspects of Grassmannians and flag manifolds such as
topological properties of random sets in real algebraic geometry; see
\cite{BurLer} and the references therein.
\color{blue}ubsection*{Towards flag manifolds}
Given the usefulness of affine and dual affine quermassintegrals, it
is worth re-visiting the role ellipsoids have played in their
development. The work of Furstenberg-Tzkoni \cite{FT} that established
the $SL_{n}$-invariance of \eqref{dual-aff-quer-1-def} for ellipsoids
went well beyond this special case. One aspect of \cite{FT} that has
received less attention is kindred integral geometric formulas for
ellipsoids on flag manifolds. They established deeper connections to
representation of spherical functions on symmetric spaces. Unlike
affine and dual affine quermassintegrals, the corresponding notions
for convex bodies, compact sets or functions have not been
investigated in the setting of flag manifolds. Our main goal is to
initiate such a study in this paper.
Flag manifolds are natural generalizations of Grassmannians in
geometry. In convex geometry, mixed volumes admit representations in
terms of certain flag measures, e.g., \cite{HRW}. Our work goes in a
different direction and the focus here is on flag versions of
quantities like those in \eqref{aff-quer-1-def} and
\eqref{dual-aff-quer-1-def} and corresponding extremal inequalities.
We establish fundamental properties such as affine invariance and
affine inequalities. We also treat companion approximate reverse
isoperimetric inequalities, which play an important role in
high-dimensional convex geometry and probability.
\color{blue}ubsection{Main results}
We start by recalling the setting from work of Furstenberg and Tzkoni
\cite{FT}. Let $ 1\leq r \leq n-1$ and let $ {\bf r}:= ( i_{1}, i_{2},
\cdots, i_{r})$ be a strictly increasing sequence of integers, $1\leq
i_{1} < i_{2} < \cdots < i_{r} \leq n-1$. Let $ \xi_{{\bf r}} := (
F_{1}, \cdots , F_{r})$ be a (partial) flag of subspaces; i.e. $ F_{1}
\color{blue}ubset F_{2} \color{blue}ubset \cdots \color{blue}ubset F_{r}$ with each $F_{j} $ an
$i_{j}$-dimensional subspace. We denote by $ F_{{\bf r}}^{n} $ the
flag manifold (with indices ${\bf r}$) as the set of all partial flags
$\xi_{\bf r}$. $F_{{\bf r}}^{n} $ is equipped with the unique Haar
probability measure that is invariant under the action of $SO_{n}$ and
all integrations on this set in this note are meant with respect to
that measure.
In the special case when $r=1$ and $i_{1}=k$, the partial flag
manifold $F^{n}_{{\bf r}}$ is just the Grassmann manifold
$G_{n,k}$. Hence the (partial) flag-manifolds can be considered as
generalizations of Grassmannians. When $r=n-1$, so that $ {\bf r}:=
(1, 2, \ldots, n-1)$, we write $F^{n}:= F_{{\bf r}}^{n}$ for the
complete flag manifold.
We follow the convention that $i_{0} = 0$ and $i_{r+1}= n$, hence
\begin{equation}
\label{basic-id-0}
\color{blue}um_{j=1}^{r} i_{j} ( i_{j+1} - i_{j-1}) = i_{r} n.
\end{equation}
Let $L$ be a compact set in $\mathbb R^{n}$ and let $ 1\leq r \leq
n-1$ and ${\bf r}$ be a set of indices as above. We define the {\it
${\bf r}$-flag quermassintegral} of $L$ by
\begin{equation}
\label{r-flar-quer}
\Phi_{{\bf r}} (L) := \left( \int_{F_{{\bf r}}^{n}} \prod_{j=1}^{r} |
P_{F_{j}} L |^{ i_{j-1}- i_{j+1}} d\xi_{{\bf r}}
\right)^{-\frac{1}{i_{r} n}}.
\end{equation}
Similarly, we define the {\it dual ${\bf r}$-flag quermassintegral} of
$L$ by
\begin{equation}
\label{dual-r-flar-quer}
\Psi_{{\bf r}} (L) := \left( \int_{F_{{\bf r}}^{n}} \prod_{j=1}^{r} | L\cap F_{j} |^{ i_{j+1}- i_{j-1}} d\xi_{{\bf r}} \right)^{\frac{1}{i_{r} n}}.
\end{equation}
In \cite{FT}, it was shown that when $L=\cal{E}$ is an ellipsoid,
$\Psi_{{\bf r}}(\cal{E})$ is invariant under $SL_{n}$. When $r=1$, the
${\bf r}$-flag quermassintegrals are exactly the affine
quermassintegrals; similarly for the dual case. Thus the latter
quantities can be considered as extensions of the (dual) affine
quermassintegrals to flag manifolds. For complete flag manifolds, we
similarly define
\begin{equation}
\label{fullFlag}
\Psi_{{\bf F^{n}}}(L) := \left( \int_{F^{n}} \prod_{i=1}^{n-1} |L
\cap F_{i} |^{2} d \xi\right)^{\frac{1}{n(n-1)}} {\rm and } \quad
\Phi_{{\bf F^{n}}}(L) := \left( \int_{F^{n}} \prod_{i=1}^{n-1} |
P_{F_{i}} L |^{2} d \xi\right)^{\frac{1}{n(n-1)}} .
\end{equation}
Clearly, by (\ref{basic-id-0})
\begin{equation}
\label{homogen}
\Psi_{{\bf r}} ( \lambda L) = \lambda \Psi_{{\bf r}} ( L), \quad \quad
\ \Phi_{{\bf r} } ( \lambda L) = \lambda \Phi_{{\bf r} } ( L)\quad
(\lambda >0).
\end{equation}
Our first result extends the invariance results of Grinberg \cite{Gr}
that give invariance of \eqref{aff-quer-1-def} and
\eqref{dual-aff-quer-1-def} under volume-preserving affine and linear
transformations, respectively.
\begin{theorem}\label{thm-aff}
Let $L$ be a compact set in $\mathbb R^{n}$, $ 1\leq r \leq n-1$ and
$ {\bf r} := ( i_{1}, \cdots , i_{r})$ be an increasing sequence of
integers between $1$ and $n-1$. Let $A$ be an affine map that
preserves volume and $T\in SL_{n}$. Then
\begin{equation}
\label{thm-aff-1}
\Phi_{{\bf r}} ( A L ) = \Phi_{{\bf r}} ( L ) \quad {\rm and }
\quad \Psi_{{\bf r}} ( T L ) = \Psi_{{\bf r}} ( L ) .
\end{equation}
\end{theorem}
With such invariance properties, it is natural to seek extremizers of
$\Phi_{{\bf r}}(L)$ and $\Psi_{{\bf r}}(L)$, especially over convex
bodies $L \color{blue}ubseteq \mathbb{R}^n$. However, even for the Grassmannian
very few such results are known; cf. Lutwak's conjectured inequality
\eqref{eqn:Lutwak_conj}. We note, however, that inequality
\eqref{eqn:Lutwak_conj} does hold at the expense of a universal
constant, as proved by the second and third-named authors \cite{PP2}.
It is easy to construct compact sets $L \color{blue}ubseteq \mathbb{R}^n$ of a
given volume such that $\Phi_{[k]}(L)$ is arbitrarily large. This,
however, cannot happen when $L$ is convex: in \cite{DP} it was shown
that up to a logarithmic factor in the dimension $n$, $\Phi_{[k]}(L)$
does not exceed $\Phi_{[k]}(r_L B_2^n)$.
We extend the aforementioned results to the setting of ${\bf r}$-flag
quermassintegrals. In this note $c, c^{\prime}, c_{0},\cdots$
etc. will denote universal constants (not necessarily the same at each
occurrence).
\begin{theorem}\label{thm-aff-ineq}
Let $L$ be a compact set in $\mathbb R^{n}$, $ 1\leq r \leq n-1$ and
${\bf r}:= (i_{1}, \cdots , i_{r})$ an increasing sequence of integers
between $1$ and $n-1$. Then
\begin{equation}
\label{ineq-0-1}
\Psi_{{\bf r}} (L) \leq \Psi_{{\bf r}} (r_L B_2^n).
\end{equation}
If $L$ is a symmetric convex body, then
\begin{equation}
\label{ineq-0-2}
\Psi_{{\bf r}} (L) \geq \frac{ c}{\min\left\{ \color{blue}qrt{\frac{n}{i_{r}}} ,
\log{n}\right\} }\Psi_{{\bf r}} (r_L B_2^n).
\end{equation}
If $L$ is a convex body, then
\begin{equation}
\label{ineq-0-3}
\frac{1}{c} \Phi_{{\bf r}} (r_L B_2^n) \leq \Phi_{{\bf r}} (L) \leq c
\min\left\{ \color{blue}qrt{\frac{n}{i_{r}}} , \log{n}\right\} \Phi_{{\bf r}}
(r_L B_2^n) .
\end{equation}
\end{theorem}
Further drawing on \cite{FT}, we also consider variants of ${\bf
r}$-flag (dual) affine quermassintegrals involving permutations $\omega$ of
$\{1,\ldots,n\}$. We define the {\it $\omega$-flag quermassintegral}
and {\it $\omega$-flag dual quermassintegral} as follows: for every
compact set $L$ in $\mathbb R^{n}$,
\begin{equation}
\Phi_{\omega} (L) := \begin{cases} \left( \int_{{\bf F^{n}}}
\prod_{j=1}^{n-1} | P_{F_{j}}L|^{- \omega(j) + \omega(j+1)-1} d \xi
\right)^{-\frac{1}{n( n- \omega(n))}} &\ {\rm if } \ \omega(n) \neq
n,\\
\int_{{\bf F^{n}}} \prod_{j=1}^{n-1} | P_{F_{j}}L|^{-
\omega(j) + \omega(j+1)-1} d \xi &\ {\rm if } \ \omega(n) = n.
\end{cases}
\end{equation}
and
\begin{equation}
\Psi_{\omega} (L) := \begin{cases}\left( \int_{{\bf F^{n}}}
\prod_{j=1}^{n-1} | L \cap F_{j}|^{ \omega(j) - \omega(j+1)+1} d \xi
\right)^{\frac{1}{ n(n- \omega(n))}} &\ {\rm if} \ \omega(n) \neq
n,\\ \int_{{\bf F^{n}}} \prod_{j=1}^{n-1} | L \cap F_{j}|^{ \omega(j) -
\omega(j+1)+1} d \xi &\ {\rm if } \ \omega(n) = n.
\end{cases}
\end{equation}
Furstenberg and Tzkoni showed $SL_{n}$-invariance of $\Psi_{\omega}$
for ellipsoids. We investigate the extent to which this invariance
carries over to compact sets. Moreover, in the case of convex bodies
we show that such quantities cannot be too degenerate in the sense
that they admit uniform upper and lower bounds, independent of the
body. We apply V. Milman's $M$-ellipsoids \cite{Mil88}, together with
the aforementioned $SL_n$-invariance of Furstenberg-Tzkoni to
establish these bounds (see Corollary \ref{M-FT}).
In \S \ref{section:functions}, we introduce functional analogues of
the ${\bf r}$-flag dual affine quermassintegrals. We show that more
general quantities share the $SL_{n}$-invarinace properties and we
prove sharp isoperimetric inequalities. In this section, we invoke
techniques and results from our previous work \cite{DPP}. Lastly, in
\S \ref{section:functions} we also introduce a functional form of
${\bf r}$-flag affine quermassintegrals. There is much recent interest
in extending fundamental geometric inequalities from convex bodies to
certain classes of functions, e.g., \cite{KlM}, \cite{BCF},
\cite{MR}. The latter authors have studied variants of inequalities
for intrinsic volumes, or even mixed volumes, and other general
quantities; for example, they establish functional analogues of
\eqref{eqn:Alexandrov}. Of course, for functions one cannot hope for
a sharp analogue of \eqref{eqn:Lutwak_conj}, as this is open even for
affine quermassintegrals of convex bodies. On the other hand, we
establish a general functional result at the expense of a universal
constant. Invariance properties and bounds for these quantities are
treated in \S \ref{sub:func}.
\color{blue}ection{Affine invariance}
In this section we will present the proof of Theorem \ref{thm-aff}. The following proposition relates integration on a flag manifold to integration on nested Grassmannians, (see \cite{SW}
Theorem 7.1.1 on p. 267 for such a result for flags of elements consisting of two subspaces). Since we will use this fact many times throughout this paper, we include the proof. For a subspace $F \color{blue}ubset \mathbb{R}^n$, we denote by $G_{F,i}$ the Grassmannian of all $i$-dimensional subspaces contained in $F$.
\begin{proposition}
Let $ 1\leq r \leq n-1$ and $ {\bf r} := ( i_{1}, \cdots , i_{r})$ be an increasing sequence of integers between $1$ and $n-1$. For $G \in L^1(F^n_{\bf r})$,
\begin{equation}\label{basic-flags-1}
\int\limits_{F^n_{\bf r}} G (\xi_{\bf r}) d\xi_{\bf r} = \int\limits_{G_{n,i_{r}} }
\int\limits_{G_{F_{r}, i_{r-1}}} \cdots \int\limits_{G_{ F_{2}},
i_{1}} G ( F_{1}, \cdots , F_{r}) \, d F_{1}
\cdots d F_{r-1} dF_{r}.
\end{equation}
\end{proposition}
For simplicity, we have suppressed the notation to write $dF_1$ rather
than $d\mu_{G_{F_2,i_1}}(F_1)$; similarly for all other indices. This
convention will be used throughout.
\begin{proof}
Fix $i_j$. Denote by $SO(F_{j})$ the subgroup of $SO_n$ acting
transitively on $G_{F_{j},{i_{j-1}}}$. For example, if
$F_o=\mathop{\rm span}\{e_1, \dots, e_{i_j}\}$ and $E_o=\mathop{\rm span}\{e_1,
\dots, e_{i_{j-1}}\}$, then elements of $SO(F_o)$ are given by
\[ \left( \begin{array}{cc} SO_{i_j} & 0 \\ 0 & I_{n-i_j} \end{array} \right) .\]
And the stabilizer of $E_o$ in $SO(F_o)$ is
\[ \left( \begin{array}{ccc} SO_{i_{j-1}} & 0 & 0 \\ 0 & SO_{i_j - i_{j-1}} & 0 \\ 0 & 0 & I_{n-i_j} \end{array} \right) .\]
The measure $\mu_{G_{F_j},i_{j-1}}$ is invariant under $SO(F_{j})$. Further, for $g\in SO_n$ and a Borel subset $A \color{blue}ubset G_{F_j,i_{j-1}}$ we have $ \mu_{G_{g F_j,i_{j-1}}} (gA) = \mu_{G_{F_j,i_{j-1}}} (A)$.
We will show that both integrals are invariant under the action of $SO_n$. Fix $g\in SO_n$. We start with the integral on the right-hand side of (\ref{basic-flags-1}):
\begin{align*}
\int_{G_{n,i_r} } & \int_{G_{ F_r, i_{r-1} }} \cdots \int_{G_{ F_2,i_1 }} G (g^{-1} \cdot(F_{1}, \cdots, F_{r})) \, d F_1\cdots dF_{r-1} dF_r \\
&= \int_{G_{n,i_r} } \int_{G_{ g F_r, i_{r-1} }} \cdots \int_{G_{ g F_2, i_1 }} G (F_{1}, \cdots, F_{r}) \, d (g F_1)\cdots d (g F_{r-1}) d (g F_r) \\
&= \int_{G_{n,i_r} } \int_{G_{g F_r, i_{r-1} }} \cdots \int_{G_{ F_2, i_1 }} G (F_{1}, \cdots,
F_{r}) \, dE_1\cdots d (g F_{r-1}) d (g F_r) \\
&= \cdots \\
&=\int_{G_{n,i_r} } \int_{G_{ F_r, i_{r-1} }} \cdots \int_{G_{ F_2, i_1 }} G (F_{1}, \cdots, F_{r}) \, d F_1\cdots d F_{r-1} d F_r,
\end{align*}
where we have sent $(F_{1}, \cdots , F_{r}) \to g \cdot(F_{1}, \cdots, F_{r})$ and then used the invariance property
$$\mu_{G_{g F_j, i_{j-1}}} (gA) = \mu_{G_{F_j, i_{j-1}}} (A)$$
for all $r-1$ inner integrals, for the outer integral we use the $SO_n$-invariance of the measure $\mu_{G_{n,i_r}}$. Note that at each step $(F_{1}, \cdots, F_{r})$ remains an element of $F^n_{\bf r}$, this is to say that the inclusion relation is preserved. The invariance of the integral on the
left-hand side of (\ref{basic-flags-1}) is a consequence of the $SO_n$-invariance of the measure $\mu_{F^n_{\bf r}}$. The proposition now follows by the uniqueness of the $SO_n$-invariant probability measure on $F^n_{\bf r}$, see for example \S 13.3 in \cite{SW}.
\end{proof}
The following fact allows one to view an integral of a function on a
partial flag as an integral over the full flag manifold. In this case,
to avoid confusion, the subspaces of flag manifolds are indexed by
their dimension.
\begin{proposition}
Let $ 1\leq r \leq n-1$ and $ {\bf r} := ( i_{1}, \cdots , i_{r})$ be an increasing sequence of integers between $1$ and $n-1$. For a function $ G $ on the partial flag $F_{{\bf r}}^{n}$, denote by $\wtd{G}$ its trivial extension to the full flag manifold $F^{n}$, i.e., $\wtd{G}(F_1, \ldots, F_{n-1}) := G(F_{i_1}, \ldots, F_{i_r})$. Then
\begin{equation}\label{inv-proj}
\int_{F^{n}} \wtd{G}( \eta) d \eta = \int_{F_{{\bf r}}^{n}} G(\xi_{\bf r}) d \xi_{\bf r}.
\end{equation}
\end{proposition}
\begin{proof}
We ``integrate out'' the Grasmannians that do not contain subspaces
that $G$ depends on by repeatedly using the identity
$$ \int_{G_{ F_{j+1}, j}} \int_{G_{ F_j, {j-1}}} f(F_{j-1}) d F_{j-1}
d F_{j} = \int_{G_{ F_{j+1}, j-1}} f(F_{j-1}) dF_{j-1}. $$ On the
right-hand side, we integrate over the set of all $(j-1)$-dimensional
subspaces in the ambient $(j+1)$-dimensional space. On the left-hand
side we integrate over the same set of planes stepwise, we step from
one $j$-dimensional subspace in the ambient $(j+1)$-dimensional space to the next and in each such subspace we consider all $(j-1)$-dimensional subspaces. The above identity holds
since we are using probability measures on each nested Grassmannian.
Applying the latter iteratively, we get
\begin{align*}
& \int_{F^{n}} \wtd{G}( \eta) d \eta \\
&= \int_{G_{n,n-1} } \int_{G_{ F_{n-1}, n-2 }} \cdots \int_{G_{ F_2, 1 }} \wtd{G} (F_{1},
\cdots , F_{n-1}) d F_1\cdots d F_{n-2} d F_{n-1} \\
&= \int_{G_{n,n-1} } \int_{G_{ F_{n-1}, n-2 }} \cdots \int_{G_{ F_2, 1}}
G (F_{i_1}, \ldots, F_{i_r}) dF_1\cdots d F_{n-2} d F_{n-1} \\
& = \int_{G_{n,n-1} } \cdots \int_{G_{ F_{i_1+1}, i_1 }} G (F_{i_1}, \ldots, F_{i_r}) \left( \int_{G_{F_{i_1},i_1-1}} \cdots \int_{G_{F_2,1}} dF_1 \ldots
dF_{i_1-1} \right) d F_{i_1} \cdots dF_{n-1} \\
&= \int_{G_{n,n-1} } \cdots \int_{G_{ F_{i_1+1}, i_1 }} G (F_{i_1}, \ldots, F_{i_r}) d F_{i_1} \cdots dF_{n-1} \\
&= \int_{G_{n,n-1} } \cdots \left( \int_{G_{ F_{i_2}, i_2-1 }} \cdots \int_{G_{ F_{i_1+1}, i_1}} G(F_{i_1}, \ldots, F_{i_r}) d F_{i_1} \cdots dF_{i_2-1} \right)
\cdots d F_{n-1} \\
&= \int_{G_{n,n-1} } \cdots \left( \int_{G_{F_{i_2}, i_1 }} G (F_{i_1}, \ldots, F_{i_r}) d F_{i_1} \right) \cdots dF_{n-1} \\
&= \cdots \\
&= \int_{G_{n,i_r} } \int_{G_{F_{i_r}, i_{r-1} }} \cdots \int_{G_{ F_{i_2}, i_1 }} G (F_{i_1}, \cdots, F_{i_r}) d F_{i_1}\cdots dF_{i_{r-1}}d F_{i_r} \\
&=\int_{F_{{\bf r}}^{n}} G(\xi_{\bf r}) d \xi_{\bf r}.
\end{align*}
\end{proof}
We now turn to the invariance properties of the functionals
$\Phi_{{\bf r}}$ and $\Psi_{{\bf r}}$. Although self-contained proofs
are possible, they require somewhat involved machinery. Since all of the
ingredients are available in the literature \cite{FT, Gr, DPP}, we
have chosen to gather the essentials without proofs.
For readers less familiar with the relevant work, we will explain the main points
behind the affine invariance of the functionals $\Phi_{{[k]}}(K)$ and
$\Psi_{{[k]}}(K)$ along the way. There are two important changes of
variables: a `global' change of variables on the Grassmannian
$G_{n,k}$ or the flag manifold $F^n_{\bf r}$ and a `local' change of
variables on each element $F\in G_{n,k}$ or $\xi_{\bf r} \in F^n_{\bf r}$.
Let $g\in SL_n$, $F\in G_{n,k}$ and $A\color{blue}ubset F$ be a
full-dimensional Borel set, then $|g A| = |\det(g|_F)| |A|$. This
determinant of the transformation $g$ restricted to the subspace $F$,
$\det(g|_F)$, is the Jacobian in the following change of variables:
\begin{equation}\label{eq_lcv}
\int_{gF} f(g^{-1}t) dt = \int_F f(t) |\det(g|_F)| dt.
\end{equation}
Denote it as in \cite{FT} by $\color{blue}igma_k(g,F):= |\det(g|_F)| = \frac{|g
A|}{|A|}$.
For the relevant manifolds $M$ considered in this paper, denote by
$\color{blue}igma_M(g,F)$ the Jacobian determinant in the following change of
variables:
\begin{equation}\label{eq_gcv}
\int_M f(F) dF = \int_M f(gF) \color{blue}igma_M(g,F) dF .
\end{equation}
Furstenberg and Tzkoni proved in \cite{FT} that
\begin{equation}
\label{G-S-03}
\color{blue}igma_{G_{n,k}}( g, F) = \color{blue}igma_{k}^{-n} (g,F)
\end{equation}
and
\begin{equation}
\label{F-S-02}
\color{blue}igma_{F_{{\bf r}}^{n} }(g, \xi_{\bf r}) = \color{blue}igma_{i_{1}}^{-i_{2}}(g, F_1)
\color{blue}igma_{i_{2}}^{i_{1}-i_{3}}(g, F_2) \cdots \color{blue}igma_{i_{r}}^{i_{r-1}-n}(g, F_r) ,
\end{equation}
where ${\bf r}:= (i_{1}, \cdots , i_{r})$. The linear-invariance of
the dual affine quermassintergals $ \Psi_{[k]} $ now follows
immediately. Indeed, for $g\in SL_n$
\begin{align*}
\Psi^{kn}_{[k]}(gL) &=\int_{G_{n,k}} |gL\cap F|^n dF = \int_{G_{n,k}}
|gL\cap gF|^n \color{blue}igma_{G_{n,k}}(g,F) dF \\ &= \int_{G_{n,k}}
\left(\color{blue}igma_{k}(g,F) |L\cap F|\right)^n \color{blue}igma_{k}^{-n} (g,F) dF =
\Psi^{kn}_{[k]}(L),
\end{align*}
where we have used (\ref{eq_gcv}), (\ref{eq_lcv}) with $f=1_L$ and
(\ref{G-S-03}). Now we turn toward the proof of Theorem
\ref{thm-aff}. We start with the case of dual ${\bf r}$-flag
quermassintegrals.
\begin{proposition}
Let $ 1\leq r \leq n-1$ and $ {\bf r} := ( i_{1}, \cdots , i_{r})$ be an increasing sequence of integers between $1$ and $n-1$.
For every compact set $L$ in $\mathbb R^{n}$ and every $g \in SL_{n}$,
\begin{equation}
\label{sln-inv}
\Psi_{{\bf r}} (g L) = \Psi_{{\bf r}} (L) .
\end{equation}
\end{proposition}
\begin{proof}
Let us start by expressing $\color{blue}igma_{F_{{\bf r}}^{n} }(g, \xi_{\bf r})$ in terms of
sections. For this note that
$$ \color{blue}igma_{i_{j}}(g, F_{j}) = \frac{ |g( L \cap F_{j})|}{ | L\cap
F_{j}|}, $$ where as a subset of $F_{j}$ we use the section $ L \cap
F_{j}$. By (\ref{F-S-02}) with $i_0=0$ and $i_{r+1}=n$, we have
$$ \color{blue}igma_{F_{{\bf r}}^{n} } ( g, \xi_{\bf r}):= \prod_{j=1}^{r}
\color{blue}igma_{i_{j}}^{-i_{j+1}+ i_{j-1}} (g, F_{j})=\prod_{j=1}^{r}
\frac{ |L \cap F_{j}|^{ i_{j+1}- i_{j-1}} }{ |gL \cap g F_{j}|^{ i_{j+1}-i_{j-1}}} . $$
Using the change of variables (\ref{eq_gcv}) with
the above expression for $\color{blue}igma_{F_{{\bf r}}^{n} }$, yields
\begin{align*}
\Psi_{{\bf r}}^{ni_{r}} ( gL)
&= \int_{F_{\bf r}^{n}} \prod_{j=1}^{r} | gL \cap F_{j}|^{ i_{j+1}- i_{j-1}} d\xi_{\bf r} \\
&= \int_{F_{\bf r}^{n}} \prod_{j=1}^{r} | gL \cap g F_{j}|^{ i_{j+1}- i_{j-1}}
\color{blue}igma_{F_{{\bf r}}^{n}} (g,\xi_{\bf r}) d\xi_{\bf r} \\
&= \int_{F_{\bf r}^{n}} \prod_{j=1}^{r} | gL \cap g F_{j}|^{ i_{j+1}- i_{j-1}}\prod_{j=1}^{r} \frac{ | L \cap F_{j}|^{ i_{j+1}- i_{j-1}} }{ |gL \cap g F_{j}|^{i_{j+1}- i_{j-1}}} d\xi_{\bf r} \\
&=\int_{F_{\bf r}^{n}} \prod_{j=1}^{r} | L \cap F_{j}|^{ i_{j+1}- i_{j-1}} d\xi_{\bf r} \\
&=\Psi_{{\bf r}}^{ni_{r}} (L) .
\end{align*}
\noindent This proves \eqref{sln-inv}.
\end{proof}
To recall the linear-invariance of the operator $\Phi_{[k]}$, we again
follow Grinberg \cite{Gr}. Observe that for $F\in G_{n,k}$ and $g\in
SL_n$ upper-triangular with respect to the decomposition
$\mathbb{R}^n=F+F^{\perp}$, we have
$$ |P_F ( g^t L )| = |g P_F L| = |\det(g|_F)| |P_F L| = \color{blue}igma_k(g,F)
|P_F L|. $$
While for $l\in SO_n$, we have $P_F(l^t L)=P_{l F}(L)$. Since
any $g\in SL_n$ can be written as a product of a rotation and an
upper-triangular matrix, combining the two observations yields the following.
\begin{lemma}[\cite{Gr}]
\noindent Let $L$ be a compact set in $\mathbb R^{n}$, $F\in G_{n,k}$ and $g \in SL_{n}$. Then
\begin{equation}
\label{multiplier-grinberg}
| P_{F} (g^t L)| = |P_{gF} L| \color{blue}igma_{k}(g,F) .
\end{equation}
\end{lemma}
\noindent The linear-invariance of the affine quermassintergals $
\Phi_{[k]} $ can now be seen as follows: let $g\in SL_n$
$$ \Phi^{-kn}_{[k]}(g^t L)=\int_{G_{n,k}} | P_{F} (g^t L)|^{-n} dF = \int_{G_{n,k}} |P_{gF} L|^{-n} \color{blue}igma_{k}^{-n} (g,F) dF = \Phi^{-kn}_{[k]} (L), $$
where we have used (\ref{multiplier-grinberg}) and (\ref{eq_gcv}) taking into account (\ref{G-S-03}).
\begin{proposition}
Let $ 1\leq r \leq n-1$ and $ {\bf r} := ( i_{1}, \cdots , i_{r})$ be
an increasing sequence of integers between $1$ and $n-1$. Let $A$ be
an affine volume preserving map in $\mathbb R^{n}$. Then for every
compact set $L$ in $\mathbb R^{n}$,
\begin{equation}
\label{aff-inv}
\Phi_{{\bf r}} (AL) = \Phi_{{\bf r}} (L) .
\end{equation}
\end{proposition}
\begin{proof}
\noindent We will first prove the theorem in the case $A:= g\in SL_{n}$. Using (\ref{multiplier-grinberg}) for the projection onto each $F_j$, (\ref{F-S-02}) and making the change of variables (\ref{eq_gcv}), we get
\begin{align*}
\Phi^{-n i_r}_{{\bf r}} ( g^t L)
&= \int_{F_{\bf r}^{n}} \prod_{j=1}^{r} | P_{F_{j}}(g^t L)|^{- i_{j+1}+ i_{j-1}} d\xi_{\bf r} \\
&= \int_{F_{\bf r}^{n}} \prod_{j=1}^{r} \left( |P_{g F_j} L| \color{blue}igma_{i_j}(g,F_j) \right)^{- i_{j+1}+ i_{j-1}} d\xi_{\bf r} \\
&= \int_{F_{\bf r}^{n}} \prod_{j=1}^{r} |P_{g F_j} L|^{- i_{j+1}+ i_{j-1}} \prod_{j=1}^{r} \color{blue}igma^{- i_{j+1}+ i_{j-1}}_{i_j}(g,F_j) d\xi_{\bf r} \\
&= \int_{F_{\bf r}^{n}} \prod_{j=1}^{r} |P_{g F_j} L|^{- i_{j+1}+ i_{j-1}} \color{blue}igma_{F_{{\bf r}}^{n} } ( g, \xi_{\bf r}) d\xi_{\bf r} \\
&= \int_{F_{\bf r}^{n}} \prod_{j=1}^{r} | P_{F_{j}} L|^{- i_{j+1}+i_{j-1}} d\xi_{\bf r} \\
&= \Phi^{-n i_r}_{{\bf r}} (L) .
\end{align*}
\noindent The general case follows easily.
\end{proof}
\noindent The proof of Theorem \ref{thm-aff} is now complete.
\color{blue}ection{Inequalities}
\label{section:inequalities}
We start by proving an extension of the inequality of Busemann-Straus
and Grinberg \eqref{Gr-ineq} to flag manifolds.
\begin{proposition}
\label{basic-ineq}
Let $ 1 < r \leq n-1$ and $ {\bf r} := (i_{1}, \cdots, i_{r})$ be an increasing sequence of integers between $1$ and $n-1$.
Then for every compact set $L$ in $\mathbb{R}^n$,
\begin{equation}
\label{Psi-ineq}
\Psi_{{\bf r}} (L) \leq \Psi_{{\bf r}} (r_{L } B_{2}^{n})
\end{equation}
with equality if and only if $L$ is a centered ellipsoid (up to a set
of measure zero).
\end{proposition}
\begin{proof}
Inequality \eqref{Gr-ineq} implies that for every $n$, every $ E\in G_{n,m}$, any $1\leq \ell \leq m-1$ and every compact set $L \color{blue}ubseteq \mathbb R^{n}$
\begin{equation}
\label{eq-00-1}
\int_{G_{E,\ell} } | (L \cap E) \cap F |^{ m} d \mu_{G_{E, \ell}} (F)
\leq c_{m,\ell} | L \cap E|^{\ell} ,
\end{equation}
with equality iff $L \cap E $ is an ellipsoid (up to a measure $0$ set
- see \cite{Gar}). Using \eqref{basic-flags-1} and \eqref{eq-00-1} we
have that
\begin{eqnarray*}
\Psi_{\bf r}^{ i_{r} n} (L) & = &\int_{F_{\bf r}^{n}} \prod_{j=1}^{r}
|L\cap F_{{j}} |^{ i_{j+1}- i_{j-1}} d \xi_{\bf r} \\ & = & \int_{
G_{n, i_{r}} } \int_{G_{F_{r}, i_{r-1}}} \cdots \int_{ G_{F_{2},
i_{1}}} \prod_{j=1}^{r} |L\cap F_{{j}} |^{ i_{j+1}- i_{j-1}}
dF_{1} \cdots dF_{r-1} dF_{r} \\ & = & \int_{ G_{n, i_{r}} }
\int_{G_{F_{r}, i_{r-1}}} \cdots\int_{G_{F_{3}, i_{2}}}
\prod_{j=2}^{r} |L\cap F_{{j}} |^{ i_{j+1}- i_{j-1}} \times \\ & &
\;\; \times \left( \int_{ G_{F_{2}, i_{1}}} |L\cap F_{1} |^{ i_{2}}
dF_{1} \right) dF_{2} \cdots dF_{r-1} dF_{r}\\ & = & \int_{ G_{n,
i_{r}} } \int_{G_{F_{r}, i_{r-1}}} \cdots\int_{G_{F_{3}, i_{2}}}
\prod_{j=2}^{r} |L\cap F_{{j}} |^{i_{j+1}- i_{j-1}} \times \\ & & \;\;
\times \left( \int_{ G_{F_{2}, i_{1}}} |(L\cap F_{2}) \cap F_{1} |^{
i_{2}} dF_{1} \right) dF_{2} \cdots dF_{r-1} dF_{r} \\ & \leq &
c_{i_{2}, i_{1}} \int_{ G_{n, i_{r}} } \int_{G_{F_{r}, i_{r-1}}}
\cdots\int_{G_{F_{3}, i_{2}}} \prod_{j=2}^{r} |L\cap F_{{j}} |^{
i_{j+1}- i_{j-1}} \times \\ & & \;\; \times |L \cap F_{2} |^{ i_{1}}
dF_{2} \cdots dF_{r-1} dF_{r} \\ &\leq& \cdots \\ &\leq & |L|^{i_{r}}
\prod_{j=1}^{r} c_{i_{j+1}, i_{j}}.
\end{eqnarray*}
The last inequality is an equality only when $L$ is a centered
ellipsoid, up to set of measure zero; see \cite{Gar}. Since for the
Euclidean ball all inequalities in the previous chain are actually
equalities, we can compute the constants and by the linear-invariance property established by Furstenberg-Tzkoni we conclude the proof.
\end{proof}
Our next result is a type of Blaschke-Santal\'{o} and reverse
Blaschke-Santal\'{o} inequality for ${\bf r}$-flag
quermassintegrals. These inequalities concern the volume of the polar
body. For a compact set $L$ we define the {\it polar body} $L^{\circ}$
(with respect to the origin) as the convex body
$$ L^{\circ} := \{ x\in \mathbb R^{n} : \langle x, y\rangle\leq 1 ,
\ \forall y \in L \}. $$
It is straightforward to check the following
inclusion: for every compact set $L$ in $\mathbb R^{n}$ and $F\in
G_{n,k}$,
\begin{equation}\label{proj-section-inc}
P_{F} L^{\circ} \color{blue}ubseteq \left(L\cap F\right)^{\circ} .
\end{equation}
If, in addition, $L$ is convex and $0$ is in the interior of $L$,
\begin{equation}\label{proj-section}
P_{F} L^{\circ} = \left( L\cap F\right)^{\circ} .
\end{equation}
Recall that the Blaschke-Santal\'{o} inequality (for symmetric convex
bodies), e.g., \cite{GarB}, \cite{S}, states that for every symmetric
convex body $L$ in $\mathbb R^{n}$,
\begin{equation}
\label{san}
|L| | L^{\circ} | \leq |B_{2}^{n}|^{2} .
\end{equation}
Moreover \eqref{san} holds when $L$ is convex and $L^{\circ}$ is
centered \cite{S}. An approximate reverse form of this inequality is
known as the Bourgain-Milman theorem \cite{BM}: for every compact,
convex set $L$ with $ 0 \in {\rm int}(L)$,
\begin{equation}
\label{r-san}
|L| |L^{\circ} | \geq c^{n}|B_{2}^{n}|^{2}.
\end{equation}
Other proofs of this inequality include \cite{Mil88}, \cite{Kup08},
\cite{Naz12}, \cite{GPV14}.
The next proposition is the aforementioned Blaschke-Santal\'{o} and
its (approximate) reversal in the setting of ${\bf r}$-flag manifolds:
\begin{proposition}
\label{prop:BS_flag}
Let $ 1 \leq r \leq n-1$ and $ {\bf r} := ( i_{1}, \cdots, i_{r})$ be
an increasing sequence of integers between $1$ and $n-1$. Then for a
symmetric compact set $L$ in $\mathbb R^{n}$
\begin{equation}\label{S for Psi}
\Phi_{{\bf r}} (L^{\circ}) \Psi_{{\bf r}} (L) \leq \Phi_{{\bf r}}
( B_{2}^{n} ) \Psi_{{\bf r}} (B_{2}^{n}).
\end{equation}
Moreover, if $L$ is a convex body in $\mathbb R^{n}$ with $ 0 \in {\rm
int}(L)$, we have that
\begin{equation}
\label{RS for Psi}
\Phi_{{\bf r}} (L^{\circ}) \Psi_{{\bf r}} (L) \geq c \, \Phi_{{\bf r}} ( B_{2}^{n} ) \Psi_{{\bf r}} (B_{2}^{n}) ,
\end{equation}
where $c>0$ is an absolute constant - exactly the constant of the reverse Santal\'o inequality \eqref{r-san}.
\end{proposition}
\begin{proof}
First note that
$$ \Phi_{{\bf r}}^{i_{r} n} ( B_{2}^{n} ) \Psi_{{\bf r}}^{i_{r} n}
(B_{2}^{n}) = \left( \prod_{j=1}^{r} | B_{2}^{n} \cap
F_j|^{i_{j+1}- i_{j-1}} \right)^{2}. $$
Using the Blaschke-Santal\'{o} inequality \eqref{san} and
\eqref{proj-section-inc}, we have
\begin{eqnarray*}
\Psi_{{\bf r}}^{i_{r}n} (L)
&=&\int_{F_{{\bf r}}^n }\prod_{j=1}^{r}|L \cap F_j|^{ i_{j+1}- i_{j-1}} d \xi_{\bf r} \\
&\leq & \left(\prod_{j=1}^{r} | B_{2}^{n} \cap F_j|^{i_{j+1}- i_{j-1}}\right)^{2}
\int_{F_{{\bf r}}^n }\prod_{j=1}^{r} \frac{1}{ |(L \cap F_j)^{\circ}|^{ i_{j+1}- i_{j-1}} } d \xi_{\bf r} \\
& \leq & \left( \prod_{j=1}^{r} | B_{2}^{n} \cap F_j|^{i_{j+1}-i_{j-1}}\right)^{2}
\int_{F_{{\bf r}}^n}\prod_{j=1}^{r} \frac{1}{|P_{F_j}L^{\circ}|^{ i_{j+1}- i_{j-1}} } d \xi_{\bf r} \\
& = & \Phi_{{\bf r}}^{i_{r} n} ( B_{2}^{n} ) \Psi_{{\bf r}}^{i_{r} n}
(B_{2}^{n}) \Phi_{{\bf r}}^{-i_{r}n} (L^{\circ}) .
\end{eqnarray*}
On the other hand, using the reverse Blaschke-Santal\'{o} inequality \eqref{r-san}, \eqref{proj-section} and \eqref{basic-id-0} we get
\begin{eqnarray*}
\Psi_{{\bf r}}^{i_{r}n}(L)
& = &\int_{F_{\bf r}^{n}}\prod_{j=1}^{r} |L \cap F_j|^{ i_{j+1}- i_{j-1}} d \xi_{\bf r}\\
&\geq & \left( \prod_{j=1}^{r} | B_{2}^{n} \cap F_j|^{i_{j+1}- i_{j-1}} \right)^{2} c^{ \color{blue}um_{j=1}^{r} i_{j} (i_{j+1}-i_{j-1}) } \int_{F_{{\bf r}}^n }\prod_{j=1}^{r} \frac{1}{|(L \cap F_j)^{\circ}|^{ i_{j+1}- i_{j-1}} } d \xi_{\bf r} \\
& = &c^{i_{r}n} \left( \prod_{j=1}^{r} | B_{2}^{n} \cap F_j|^{i_{j+1}- i_{j-1}}\right)^{2}
\int_{F_{{\bf r}}^{n}}\prod_{j=1}^{r} \frac{1}{ | P_{F_j}L^{\circ}|^{ i_{j+1}-i_{j-1}} } d \xi_{\bf r}\\
&= & c^{i_{r}n} \Phi_{{\bf r}}^{i_{r} n} ( B_{2}^{n} )
\Psi_{{\bf r}}^{i_{r} n} (B_{2}^{n})\Phi_{{\bf r}}^{-i_{r}n}(L^{\circ}).
\end{eqnarray*}
The proof is complete.
\end{proof}
\noindent The following corollary has been proved in the case $r=1$ in \cite{PP2}.
\begin{corollary}
\noindent Let $ 1 \leq r \leq n-1$ and $ {\bf r} := ( i_{1}, \cdots, i_{r})$ be an increasing sequence of integers between $1$ and $n-1$.
Then for every convex body $L$,
\begin{equation}
\label{tilde-sym-two}
\Phi_{{\bf r}} (L) \geq c \, \Phi_{{\bf r}} (r_{L} B_{2}^{n}) ,
\end{equation}
where $c>0$ is an absolute constant.
\end{corollary}
A compact set is called {\it centered} if its centroid lies at the origin.
\begin{proof}
As $\Phi_{{\bf r}}(L)$ is translation-invariant, we may assume that
$L$ is centered. The Blaschke-Santal\'o inequality implies
$r_L r_{L^{\circ}}\leq 1$, so with \eqref{RS for Psi} and
\eqref{Psi-ineq}, we obtain
$$ \Phi_{{\bf r}} (L) \geq c \frac{ \Phi_{{\bf r} } (B_{2}^{n})
\Psi_{{\bf r}} (B_{2}^{n}) }{ \Psi_{{\bf r}}(L^{\circ})} \geq c
\frac{ \Phi_{{\bf r} } (B_{2}^{n}) \Psi_{{\bf r}} (B_{2}^{n}) }{
\Psi_{{\bf r}}( r_{L^{\circ}} B_{2}^{n})} = \frac{c}{ r_{L^{\circ}}}
\Phi_{{\bf r}} ( B_{2}^{n}) \geq c \, \Phi_{{\bf r}} (r_{L} B_{2}^{n}). $$
The proof is complete.
\end{proof}
The next proposition shows that all the quantities $\Phi_{{\bf r}}(L)$
lie between the volume-radius $r_{L}$ and the mean width $W(L)$. For a convex body $L$ in $\mathbb R^{n}$, we write
\begin{equation}\label{min-W}
W_{L} := \inf_{T\in SL_{n}} W(T L) .
\end{equation}
\begin{proposition}
Let $ 1 \leq r \leq n-1$ and $ {\bf r} := ( i_{1}, \cdots, i_{r})$ be an increasing sequence of integers between $1$ and $n-1$. Then for a convex body $L$ in $\mathbb R^{n}$
\begin{equation}\label{Bet-Ur}
c \, r_L \leq \frac{ \Phi_{{\bf r}}(L) }{ \Phi_{{\bf r}} (B_{2}^{n}) }\leq W_{L} .
\end{equation}
\end{proposition}
\begin{proof}
The left-most inequality follows from \eqref{tilde-sym-two}. Next,
since $h_{L} (x) = h_{P_{F}L}(x)$ for $x\in F$, we have
\begin{equation}
\label{fub-W}
\int_{G_{n,k}} W(P_{F} L) \, d F = W(L).
\end{equation}
We will prove the right-hand side for the case $r=2$ (the general case follows by induction on $r$). Using the Urysohn and H\"older inequalities repeatedly, we get
\begin{eqnarray*}
\Phi_{{\bf r}}(L)
& = & \left( \int_{F^n_{\bf r}} \prod_{j=1}^{2} |P_{F_{j}}L|^{-i_{j+1}+i_{j-1}} d\xi_{\bf r} \right)^{-\frac{1}{n i_2}} \\
& = & \left( \int_{G_{n,i_{2}}} \frac{1}{ |P_{F_{2}} L|^{ n-i_{1}}} \int_{G_{F_2,i_1}} \frac{1}{|P_{F_{1}}L|^{i_2}} dF_1 dF_2 \right)^{-\frac{1}{ni_{2}}} \\
&\leq& \left( \int_{G_{n,i_{2}}} \frac{1}{| P_{F_{2}} L|^{ n-i_{1}}} \int_{G_{F_{2}, i_{1}}}
\frac{1}{|P_{F_{1}}B_{2}^{n}|^{ i_{2}} W(P_{F_{1}}L)^{i_{1} i_{2}}}
dF_1 dF_2\right)^{-\frac{1}{ni_{2}}} \\
& \leq & \left( \int_{G_{n,i_{2}}} \frac{1}{| P_{F_{2}} L|^{n-i_{1}}} \frac{1}{|B_{2}^{i_{1}}|^{ i_{2}} \left(\int_{G_{F_{2}, i_{1}}} W(P_{F_{1}}L) dF_1 \right)^{i_{1}i_{2}}} dF_2 \right)^{-\frac{1}{ni_{2}}} \\
& = & \left( \int_{G_{n,i_{2}}} \frac{1}{| P_{F_{2}} L|^{ n-i_{1}}} \frac{1}{|B_{2}^{i_{1}}|^{ i_{2}} W(P_{F_{2}}L)^{i_{1}i_{2}}} dF_2 \right)^{-\frac{1}{ni_{2}}} \\
& \leq & \left( \int_{G_{n,i_{2}}} \frac{1}{ | P_{F_{2}} B_{2}^{n}|^{ n-i_{1}} W(P_{F_{2}}L)^{i_{2}(n-i_{1})}} \frac{1}{|B_{2}^{i_{1}}|^{ i_{2}} W(P_{F_{2}}L)^{i_{1}i_{2}}} dF_2
\right)^{-\frac{1}{ni_{2}}} \\
& = & \left( |B_{2}^{i_{2}}|^{n-i_{1}} |B_{2}^{i_{1}}|^{i_{2}} \right)^{\frac{1}{ ni_{2}}}\left(
\int_{G_{n,i_{2}}} W(P_{F_{2}}L)^{-n i_{2}} dF_2 \right)^{-\frac{1}{ni_{2}}} \\
& \leq & \Phi_{{\bf r}}(B^n_2) \int_{G_{n,i_{2}}} W(P_{F_{2}}L) dF_2 \\
& = &\Phi_{{\bf r}}(B^n_2) W(L).
\end{eqnarray*}
In the above argument we may replace $L$ by $TL$ with $T\in SL_n$. Since the left-hand side of this inequality remains the same for all $T$ by Theorem \ref{thm-aff}, we may take the infimum over all $T$ on the right-hand side. This completes the proof.
\end{proof}
We conclude this subsection with a discussion of inequalities of isomorphic nature. For convex bodies $L$ in $\mathbb{R}^n$, we define the Banach-Mazur distance to the Euclidean ball $B_2^n$ by $$ d_{BM}(L) := \inf\left\{ ab : a>0,b>0, \frac{1}{b} B_{2}^{n} \color{blue}ubseteq
T(L-L) \color{blue}ubseteq a B_{2}^{n}, T\in GL_{n} \right\}.$$
For symmetric convex bodies, this coincides with the standard notion of Banach-Mazur distance (for more information see, e.g., \cite{NTJ89}).
\begin{proposition}\label{prop:iso}
Let $ 1 \leq r \leq n-1$ and $ {\bf r} := ( i_{1}, \cdots, i_{r})$ be an increasing sequence of integers between $1$ and $n-1$. Then for a convex body $L$ in $\mathbb{R}^n$
\begin{equation}\label{Phi-r-upper}
\Phi_{{\bf r}}(L) \leq c \min\left\{ \color{blue}qrt{\frac{n}{i_{r}}},
\log{(1+ d_{BM}(L))} \right\} \Phi_{{\bf r}} (r_{L} B_{2}^{n}) .
\end{equation}
Moreover, if $L$ is also symmetric, then
\begin{equation}\label{Psi-r-upper}
\Psi_{{\bf r}}(L) \geq \frac{c}{\min\left\{ \color{blue}qrt{\frac{n}{i_{r}}} ,
\log{(1+ d_{BM}(L))} \right\} }\Psi_{{\bf r}} (r_{L} B_{2}^{n}) .
\end{equation}
\end{proposition}
The proof relies on several different tools. We draw on ideas from Dafnis and the second-named author \cite{DP} to exploit the affine invariance of $\Phi_{{\bf r}}(L)$, $\Psi_{{\bf r}}(L)$ by using appropriately chosen affine images of $L$. To this end, recall the following fundamental theorem which combines work of Figiel--Tomczak-Jaegermann \cite{FTJ}, Lewis \cite{Lewis}, Pisier \cite{Pi82} and Rogers-Shephard \cite{RS2} (see Theorem 1.11.5 on p. 52 in \cite{BGVV}).
\begin{theorem}\label{thm:ell}
Let $L$ be a centered convex body. Then there exists a linear map $T\in SL_{n}$ such that
\begin{equation}
\label{pisier}
W(T L) \leq c \log\{ 1+ d_{BM} (L)\} \color{blue}qrt{n}|L|^{1/n}.
\end{equation}
\end{theorem}
We will also use recent results on isotropic convex bodies. For background, the reader may consult \cite{BGVV}, we will however recall all facts that we need here. To each convex body $M\color{blue}ubseteq \mathbb{R}^n$ with unit volume, one can associate an ellipsoid $Z_{2}(M)$, called the {\it $L_{2}$-centroid body} of $M$, which is defined by its support function as
$$ h_{Z_{2}(M)} (\theta) := \left( \int_{M} | \langle x, \theta \rangle |^{2} dx \right)^{\frac{1}{2}}. $$
The {\it isotropic constant} of $M$ is defined by $L_{M} := r_{Z_{2}(M)}$. We say that $M$ is {\it isotropic} if it is centered and $Z_{2}(M) = L_{M} B_{2}^{n}$. Fix an isotropic convex body $M$ and a $k$-dimensional subspace $F$. K. Ball \cite{Ball1} proved that
\begin{equation}
\label{Ball}
|M \cap F^{\perp} |^{\frac{1}{ k}} \geq \frac{c}{L_{M}};
\end{equation}
a corresponding inequality for projections,
\begin{equation}
\label{R2-iso}
|P_{F} M| \leq \left(c \frac{n}{k} L_{M}\right)^{k},
\end{equation}
follows immediately from \eqref{Ball} and the Rogers-Shephard inequality \cite{RS2}:
$$ |P_{F} M| |M\cap F^{\perp}| \leq {n\choose k}. $$
Next, we recall a variant of $\Psi_{[k]}(M)$ studied by Dafnis and the second-named author \cite{DP}. For every $ 1\leq k \leq n-1$ and a compact set $M$ in $\mathbb R^{n}$ with $|M|=1$, we define the quantity
\begin{equation}\label{phi-tilde}
\wtd{\Phi}_{[k]}(M) := \left( \int_{G_{n,k}} |M\cap F^{\perp}|^{n}
d\mu_{G_{n,k}}(F) \right)^{\frac{1}{nk}}.
\end{equation}
In \cite{DP} it is shown that for every convex body $M$ in $\mathbb{R}^n$ of unit volume,
\begin{equation}\label{phi-tilde-bound}
\frac{c_{1}}{ L_{M}} \leq \wtd{\Phi}_{[k]}(M) \leq \wtd{ \Phi}_{[k]}(D_n) \color{blue}imeq 1 ,
\end{equation}
where $D_n$ is the Euclidean ball of volume one.
We also invoke B. Klartag's fundamental result on perturbations of
isotropic convex bodies \cite{Kl} having a well-bounded isotropic
constant.
\begin{theorem}\label{Kl}
Let $M$ be a convex body in $\mathbb R^{n}$. For every $\varepsilon\in (0,1)$ there exists a centered convex body $M_{Kl} \color{blue}ubset \mathbb R^{n}$ and a point $x\in \mathbb R^{n}$ such that
\begin{equation}
\label{Kl-1}
\frac{1}{ 1+\varepsilon} M_{Kl} \color{blue}ubseteq M + x \color{blue}ubseteq (1+\varepsilon) M_{Kl}
\end{equation}
and
\begin{equation}
\label{Kl-2}
L_{M_{Kl}} \leq \frac{c}{ \color{blue}qrt{\varepsilon}}.
\end{equation}
\end{theorem}
We are now ready to complete the proof.
\begin{proof}[Proof of Proposition \ref{prop:iso}]
By homogeneity of the operators $\Phi_{\bf r}$ and $\Psi_{\bf r}$, we can assume that $L$ has unit volume.
First we will prove the bound \eqref{Phi-r-upper} for $\bf r$-flag
affine quermassintegrals. By translation-invariance of projections, we may
further assume that $L$ is centered. Bounding $\Phi_{{\bf r}}(L)$ by
$W(L)$ according to \eqref{Bet-Ur}, using affine invariance of
$\Phi_{\bf r}$ and reverse Urysohn inequality from Theorem
\ref{thm:ell}, we get
\begin{equation}
\label{r-flag-upper-1}
\Phi_{{\bf r}} (L) \leq c\log( 1+ d_{BM}(L)) \Phi_{{\bf r}}(D_n).
\end{equation}
For the Euclidean ball $D_n$ of unit volume, for every $F\in G_{n,k}$, we have
$|P_{F} D_{n}|^{\frac{1}{k}}=| D_{n} \cap F|^{\frac{1}{k}}\color{blue}imeq \color{blue}qrt{\frac{n}{k}}$,
so
$$ \Phi_{{\bf r}} ( D_{n}) \color{blue}imeq \left( \prod_{j=1}^{r} \left(\frac{n}{ i_{j}}\right)^{i_{j} (i_{j+1}-i_{j-1}) }\right)^{\frac{1}{2i_{r} n}}. $$
The AM/GM inequality implies
$$\left( \prod_{j=1}^{r} \left( \frac{n}{ i_{j}}\right)^{i_{j} (i_{j+1}-i_{j-1}) }\right)^{\frac{1}{ 2i_{r} n}} \leq \color{blue}qrt{\frac{n}{i_{r}n} \color{blue}um_{j=1}^{r} \frac{i_{j} (i_{j+1}-j_{j-1})}{i_{j}} }\leq \color{blue}qrt{ \frac{n}{ i_{r}} }. $$
Thus
\begin{equation}
\label{est-B}
\Phi_{{\bf r}} (D_{n}) = \Psi_{{\bf r}}( D_{n}) \color{blue}imeq \left(
\prod_{j=1}^{r} \left( \frac{n}{ i_{j}}\right)^{i_{j} (
i_{j+1}-i_{j-1}) }\right)^{\frac{1}{ 2i_{r} n}} \leq \color{blue}qrt{\frac{n}{i_{r}}}.
\end{equation}
Let $K_{1} \color{blue}ubset \mathbb{R}^n$ be a centered convex body and $x\in \mathbb{R}^n$ from the conclusion of Theorem \ref{Kl} corresponding to $\varepsilon=\frac{1}{2}$. Then \eqref{Kl-1} implies $1=|L|^{1/n} \geq \frac{2}{3} | K_{1}|^{1/n}$, while \eqref{Kl-2} implies $L_{K_1} \color{blue}imeq 1$. Let $ K_{2} := \frac{K_{1}}{ |K_{1}|^{\frac{1}{n}} }$, then $L_{K_2} \color{blue}imeq L_{K_1} \color{blue}imeq 1$
and \begin{equation}
\label{phi-r-upper-1-1}
\Phi_{{\bf r}}(L)= \Phi_{{\bf r}} (L+x) \leq
\frac{3}{2} \Phi_{{\bf r}}(K_{1}) \leq \frac{9}{4} \Phi_{{\bf r}}(
K_{2}).
\end{equation}
Affine invariance of $\Phi_{\bf r}$ (Theorem \ref{thm-aff}), allows us to assume that $K_{2}$ is isotropic.
Using \eqref{R2-iso}, \eqref{est-B}, $L_{K_2} \color{blue}imeq 1$ and \eqref{est-B} one more time, we obtain
\begin{eqnarray*}
\Phi_{{\bf r}}(K_{2})
&= & \left( \int_{F_{{\bf r}}^{n}} \prod_{j=1}^{r} | P_{F_{j}} K_2|^{-i_{j+1}+i_{j-1}} d\xi_{\bf r} \right)^{-\frac{1}{i_{r} n}} \\
&\leq & \left(\prod_{j=1}^{r} \left( \frac{n}{ i_{j}}\right)^{i_{j}(i_{j+1}-i_{j-1})} \right)^{\frac{1}{i_{r}n}} (cL_{K_{2}})^{\frac{1}{i_{r}n}\color{blue}um_{j=1}^{r} i_{j}(i_{j+1}-i_{j-1}) }\\
& \leq & c \, L_{K_{2}} \Phi_{{\bf r}}(D_n)^{2} \\
&\leq & c \, \color{blue}qrt{\frac{n}{i_{r}}} \, \Phi_{{\bf r}}(D_n).
\end{eqnarray*}
By \eqref{phi-r-upper-1-1} we have $ \Phi_{{\bf r}} (L) \leq c^{\prime} \color{blue}qrt{\frac{n}{i_{r}}} \, \Phi_{{\bf r}} (D_n) $, which together with \eqref{r-flag-upper-1} gives the upper bound \eqref{Phi-r-upper}.
Applying \eqref{RS for Psi} for $L^{\circ}$, \eqref{Phi-r-upper} and the Blaschke-Santal\'{o} inequality $r_L r_{L^{\circ}}\leq 1$, we get
\begin{eqnarray*}
\Psi_{{\bf r}} (L)
&\geq & c\frac{ \Phi_{{\bf r}} (B_{2}^{n}) \Psi_{{\bf r}}(B_{2}^{n}) }{ \Phi_{{\bf r}} ( L^{\circ}) } \\
&\geq & \frac{c}{r_{L^{\circ}}} \frac{1}{ \min\left\{ \log{(1+ d_{BM}(L^{\circ}))}, \color{blue}qrt{\frac{n}{i_r}}\right\}} \frac{\Phi_{{\bf r}} (B_{2}^{n}) \Psi_{{\bf r}}(B_{2}^{n}) }{ \Phi_{{\bf r}} ( B_{2}^{n})} \\
& \geq & \frac{c}{ \min\left\{ \log{(1+ d_{BM}(L))}, \color{blue}qrt{\frac{n}{i_{r}}}\right\}} \Psi_{{\bf r}} ( r_{L} B_{2}^{n}),
\end{eqnarray*}
where we have also used the identity $d_{BM} (L^{\circ}) = d_{BM}(L)$ for symmetric convex bodies. This proves \eqref{Psi-r-upper}.
\end{proof}
\color{blue}ection{Flag manifolds and permutations}
In this section, we discuss more general quantities involving
permutations. We investigate the extent to which $SL_{n}$-invariance
properties established by Furstenberg and Tzkoni \cite{FT} carry over
from ellipsoids to compact sets. In particular, we provide an example
of a convex body for which $SL_{n}$-invariance fails. Nevertheless, we
show that for convex bodies, such quantities cannot be too degenerate
in the sense that they admit uniform upper and lower bounds,
independent of the body. The key ingredient is the notion of
$M$-ellipsoids, introduced by V. Milman \cite{Mil88}.
The next definition is motivated by the work of Furstenberg and Tzkoni
\cite{FT} for ellipsoids.
\begin{definition}
Let $\Pi_{n}$ be the set of permutations of $\{1,2,\ldots,n\}$ and
$\omega\in \Pi_{n}$. For compact sets $L$ in $\mathbb{R}^n$, we
define the {\it $\omega$-flag quermassintegral} and {\it $\omega$-flag dual
quermassintegrals} as follows: if $\omega(n) \neq n$, then
\begin{equation}
\label{perm-Psi-def}
\Psi_{\omega} (L) := \left( \int_{F^{n}} \prod_{j=1}^{n-1} | L \cap
F_{j}|^{ \omega(j) - \omega(j+1)+1} d \xi \right)^{\frac{1}{ n(n-
\omega(n))}}
\end{equation}
and
\begin{equation}
\label{perm-Phi-def-1}
\Phi_{\omega} (L) := \left( \int_{F^{n}} \prod_{j=1}^{n-1} | P_{F_{j}} L|^{- \omega(j) + \omega(j+1)-1} d \xi \right)^{-\frac{1}{n( n- \omega(n))}} .
\end{equation}
When $\omega(n)=n$, we set
\begin{equation}
\label{perm-Psi-def-2}
\Psi_{\omega} (L) := \int_{F^{n}} \prod_{j=1}^{n-1} | L \cap F_{j}|^{
\omega(j) - \omega(j+1)+1} d \xi
\end{equation}
and
\begin{equation}
\label{perm-Phi-def-2}
\Phi_{\omega} (L) := \int_{F^{n}} \prod_{j=1}^{n-1} | P_{F_{j}} L|^{-
\omega(j) + \omega(j+1)-1} d \xi.
\end{equation}
\end{definition}
\noindent Note that
\begin{equation}
\label{perm-00-01}
\color{blue}um_{j=1}^{n-1} \left( \omega(j) - \omega(j+1) +1\right) =
n-\omega(n)+ \omega(1)-1
\end{equation}and
\begin{equation}
\label{perm-00-02}
\color{blue}um_{j=1}^{n-1} j\left( \omega(j) - \omega(j+1) +1\right) = n(n-\omega(n)).
\end{equation}
Identity \eqref{perm-00-02} guarantees that $\Psi_{\omega}(L)$ and $
\Phi_{\omega}(L)$ are $1$-homogeneous when $\omega(n)\neq n$ and
$0$-homogeneous when $ \omega(n)=n$.
The following fact for $\omega$-flag dual quermassintegrals is from \cite{FT}. For $\omega$-flag quermassintegrals it follows for example by duality.
\begin{theorem}
\label{thm:FTperms}
Let ${\cal{E}}$ be an ellipsoid in $\mathbb R^{n}$ and $\omega\in
\Pi_{n}$.
\begin{equation}
\label{perm-1}
\Psi_{\omega} ({\cal{E}}) = \Psi_{\omega} (r_{\cal{E}}B_2^n) \ {\rm
and} \ \Phi_{\omega} ({\cal{E}}) = \Phi_{\omega}
(r_{\cal{E}}B_2^n).
\end{equation}
\end{theorem}
\noindent An equivalent formulation of the latter result is that for every
ellipsoid ${\cal{E}}$,
\begin{equation}
\label{perm-ell-1}
\Psi_{\omega} ({\cal{E}}) = c_{\omega} |{\cal{E}}|^{\frac{1}{n}}, \ \omega(n)
\neq n \ {\rm and } \ \Psi_{\omega} ({\cal{E}}) = c_{\omega} , \ \omega(n) =
n,
\end{equation}
where $c_{\omega} $ is a constant that depends only on $\omega$. An
analogous statement holds for $\Phi_{\omega}(\cal{E})$
(cf. \eqref{prop:BS_flag}).
The operators $\Psi_{\omega}$ and $\Phi_{\omega}$ are
generalizations of $\Psi_{{\bf r}}$ and $\Phi_{{\bf r}}$.
Indeed, let $1\leq r \leq n-1$, $1\leq i_{1}< i_{2} < \cdots < i_{r}\leq n-1$ and
$ {\bf r}:= (i_{1}, \cdots , i_{r}). $ Define $\omega $ by $\omega (1) = n-i_{1}+1$, and $\omega(t+1)= \omega(t) +1$ , for $ t\neq i_{j}$, $1\leq j \leq r$, and $\omega( i_{j}+1)= \omega(i_{j}) +1 - i_{j+1}+ i_{j-1}$ for $ 1\leq j \leq r$. Then $\omega \in \Pi_n$ with $
\omega(i_{j} ) - \omega(i_{j}+1) +1 = i_{j+1}- i_{j-1}$ for $1\leq
j\leq r$ and $ \omega ( t) - \omega( t+1) +1 = 0 , \ t\neq i_{j}$, for
$1\leq j \leq r.$ Since $\omega(n) = n-i_{r}$, for a compact set $L$ in $\mathbb R^{n}$ we have
\begin{eqnarray*}
\Psi_{\omega} (L)
&= & \left( \int_{F^{n} } \prod_{j=1}^{n-1} |L \cap F_{j}|^{\omega(j)-\omega(j+1)+1} d \xi \right)^{\frac{1}{ ni _{r}}}\\
&=& \left( \int_{F^{n} } \prod_{j=1}^{r} |L \cap F_{i_{j}}|^{i_{j+1}-i_{j-1}} d \xi \right)^{\frac{1}{ ni _{r}} } \\
&=& \Psi_{{\bf r}}(L),
\end{eqnarray*}
where, in the last equality, we have used \eqref{inv-proj}. Correspondingly,
we also have $ \Phi_{\omega}(L) = \Phi_{{\bf r}} (L)$. In particular, for this permutation $\omega$, $\Psi_{\omega}(L)$ is $SL_{n}$-invariant and $\Phi_{\omega}(L)$ is affine invariant.
As a particular case of the preceding discussion, let $r=1$, $i_{1}= k$,
$1\leq k \leq n$ and let $\omega(1)= n-k+1$ and $\omega(t+1)=
\omega(t)+1$ for $t\neq k$ and $ \omega(k+1) = \omega(k) - n +1$. Then
$\Phi_{\omega}(L) = \Phi_{[k]}(L).$
Given that $\Psi_{{\bf r}}(L)$ and $\Phi_{{\bf r}}(L)$ enjoy
invariance properties and arise as permutations, it is natural to
investigate the extent to which the invariance from Theorem
\ref{thm:FTperms} carries over to compact sets. We do not have a
complete answer. However, there are cases outside of those considered
above where the invariance holds and also counter-examples where it
fails as the next two examples show.
\begin{example} Let $n\geq 3$. Define $\omega$ by
$\omega(1)=2$, $\omega(2)=1$ and $\omega(t)=t$ for all $ 3\leq t \leq n$. Then for every symmetric compact set $L$ in $\mathbb R^{n}$,
$$ \Psi_{\omega}(L) = \frac{4}{\pi} . $$
In particular, $\Psi_{\omega}(L)$ is $SL_{n}$-invariant.
Note that our choice of the permutation $\omega$ satisfies
$$ \omega(1)-\omega(2)+1=2, \quad \omega(2)-\omega(3)+1=-1, \quad
\omega(j)-\omega(j+1)+1=0 \text{ for } 3\leq j \leq n-1,$$
or equivalently
$$ \omega(2)=\omega(1)-1, \quad \omega(3)=\omega(1)+1, \quad
\omega(j+1)=\omega(1)+(j-1) \text{ for } 3\leq j \leq n-1. $$
Since $1\leq \omega(j)\leq n$ for all $j$, it follows that
$\omega(1)=2$. Hence $\omega(1)=2, \omega(2)=1, \omega(j)=j$ for
$3\leq j \leq n$ is the unique permutation with these
properties. For an $k$-dimensional subspace $F_k$ of $\mathbb{R}^n$, denote by $S_{F_k}$ the unit sphere in $F_k$. Now, using \eqref{inv-proj}, we compute
\begin{align*}
\Psi_{\omega} (L)
&= \int_{F^{n}} \prod_{j=1}^{n-1} |L \cap F_{j}|^{\omega(j)-\omega(j+1)+1} d\xi \\
&= \int_{F^{n}} |L \cap F_1|^2 |L \cap F_2|^{-1} d\xi \\
&= \int_{G_{n,2}} |L \cap F_2|^{-1} \int_{G_{F_2,1}} |(L \cap F_2) \cap F_1|^2 dF_1 dF_2 \\
&= \int_{G_{n,2}} |L \cap F_2|^{-1} \int_{S_{F_2}} (2 \rho_{L \cap F_2}(\theta))^2 d\color{blue}igma(\theta) dF_2 \\
&= \int_{G_{n,2}} |L \cap F_2|^{-1} \frac{4}{|S_{F_2}|} \int_{S_{F_2}} \rho^2_{L \cap F_2}(\theta) d\theta \, dF_2 \\
&= \frac{4}{\pi} \int_{G_{n,2}} |L \cap F_2|^{-1} |L \cap F_2| dF_2 \\
&= \frac{4}{\pi} .
\end{align*}
\end{example}
\qed
When $n=3$, for permutations $\omega$ with $\omega(3)=3$,
$\Psi_{\omega}(L)$ are absolute constants. Moreover, the discussion
following Theorem \ref{thm:FTperms} shows that for $3$ of the
remaining $4$ permutations $\omega$ in $\Pi_3$,
$\Psi_{\omega}(L)=\Psi_{\bf r}(L)$. Altogether, for $n=3$, for 5 out
of 6 permutations $\omega$, $\Psi_{\omega}(L)$ are
$SL_n$-invariant. The next example shows that for the remaining
permutation, the invariance does not carry over for all convex bodies.
\begin{example} Let $\omega \in \Pi_3$ with $\omega(1)=1, \, \omega(2)=3$ and $\omega(3)=2$. We claim that for a centered cube $Q:=[-1,1]^3$ and the diagonal matrix $D=\mathop{\rm diag}(1,2,1/2)$, $\Phi_{\omega}(DQ)>\Phi_{\omega}(Q)$. Since $D \in SL_3$, this shows that the operator $\Phi_{\omega}$ is not invariant under volume preserving transformations.
To show this, we first note that for any convex body $L \color{blue}ubset \mathbb R^{3}$
\begin{equation}
\label{ex-1}
\Phi^{-3}_{\omega}(L) = \int_{S^{2} } \frac{ W( P_{\phi^{\perp}}L)}{ h_{\Pi L}^{2} (\phi) } d\color{blue}igma (\phi).
\end{equation}
Recall that for $\theta \in S^{n-1}$, $h_Q(\theta)=\color{blue}um_{i=1}^n |\theta_i|$ and for $g\in GL_n$, $h_{gL}(\theta) = h_L(g^{t}\theta)$.
We will also use the following facts about projection bodies (see e.g., Gardner).
The projection body of a cube is again a cube, $\Pi Q = 2 Q$ and for $g\in GL_n$,
\begin{equation}\label{ex-3}
\Pi (g L) = | {\rm det} g | \, g^{-t} \, \Pi L.
\end{equation}
Let $ A=[a_{1} \ a_{2} \ a_{3} ] \in SL_3$ with columns $a_i$. Fix $ \phi\in S^{2}$. Let $U \in O_3$ be given in column form by $ U = [ u \ v \ \phi ] $. Since $ U $ is orthogonal, $ U^{t} \phi = e_{3} $ and $ U^{t} \phi^{\perp} = {\rm span}\{ e_{1}, e_{2} \} = \mathbb R^{2} $.
Then
$$ W( P_{\phi^{\perp}} A Q)= \int_{S_{\phi^{\perp}}} h_{A Q } ( \theta) d\color{blue}igma (\theta ) = \int_{S^{1}} h_{AQ} ( U\theta ) d \color{blue}igma (\theta ) =
\int_{S^{1}} h_{Q} ( A^{t} U\theta ) d \color{blue}igma ( \theta ).$$
Thus denoting by $P$ the orthogonal projection onto $\mathbb R^{2}$, we have
$$ W( P_{\phi^{\perp}} A Q)
= \color{blue}um_{i=1}^{3} \int_{S^{1}} | \langle \theta , U^{t} A e_{i} \rangle | d\color{blue}igma( \theta )
= \color{blue}um_{i=1}^{3} \int_{S^{1}} | \langle \theta , P U^{t} A e_{i} \rangle | d\color{blue}igma( \theta )
= \frac{2}{\pi} \color{blue}um_{i=1}^{3} \| P U^{t} A e_{i} \|_{2} .$$
We have that $ A e_{i} = a_{i}$, $ U^{t} a_{i} = \left( \langle u,
a_{i} \rangle , \langle v, a_{i} \rangle, \langle \phi, a_{i} \rangle\right)^{t} $ and
$$ \| P U^{t} A e_{i} \|_{2}^{2} = \| U^{t} A e_{i} \|_{2}^{2} - \| (I-P) U^{t} A e_{i} \|_{2}^{2} = \|a_{i}\|_{2}^{2} - \langle \phi, a_{i} \rangle^{2} . $$
Therefore,
\begin{equation}
\label{ex-4}
W( P_{\phi^{\perp}} A Q)= \frac{2}{\pi} \color{blue}um_{i=1}^{3} \color{blue}qrt{ \|a_{i}\|_{2}^{2} -
\langle \phi, a_{i} \rangle^{2} } .
\end{equation}
Moreover, $ h_{\Pi (A Q) } ( \phi) = h_{\Pi Q} ( A^{-1} \phi) = 2
\color{blue}um_{i=1}^{3} | \langle A^{-1} \phi, e_{i} \rangle | $. Thus
\begin{equation}
\label{ex-5}
\Phi^{-3}_{\omega}(AQ)
= \frac{1}{2\pi} \int_{S^{2}} \frac{ \color{blue}um_{i=1}^{3} \color{blue}qrt{ \|a_{i}\|_{2}^{2} - \langle \phi, a_{i} \rangle^{2} }}{\left(\color{blue}um_{j=1}^{3} | \langle A^{-1} \phi, e_{j} \rangle \right)^{2} } d\color{blue}igma ( \phi) .
\end{equation}
Set $ A:= {\rm diag} ( d_{1}, d_{2}, d_{3})$ with $\prod_{i=1}^{3} d_{i}= 1$ and $ d_{i} >0$. Then the quantity
\begin{equation}
\label{ex-6} {\cal{A}}( d_{1}, d_{2}, d_{3}) := \int_{S^{2}} \frac{\color{blue}um_{i=1}^{3} d_{i} \color{blue}qrt{ 1- \phi_{i}^{2} }}{ \left(\color{blue}um_{j=1}^{3}\frac{|\phi_{j}|}{ d_{j}}\right)^{2} } d\color{blue}igma ( \phi)
\end{equation}
is not constant. Indeed, using MATLAB for example, one can verify that
${\cal{A}}(1,2,1/2)<{\cal{A}}(1,1,1)$.
\end{example}
\qed
In the case of convex bodies, the quantities $\Psi_{\omega}(K)$,
$\Phi_{\omega}(K)$ are uniformly bounded. We will use the following
well-known consequence of the celebrated ``existence of M-ellipsoids" by
V. Milman \cite{Mil88}.
\begin{theorem}
Let $K$ be a symmetric convex body in $\mathbb R^{n}$. Then there
exists an ellipsoid ${\cal{E}}$ such that $|{\cal{E}}|^{1/n}
\leq e^{c} |K|^{1/n}$ and for every $F\in G_{n,k}$,
\begin{equation}
\label{M-position-2}
| P_{F} {\cal{E}}| \leq | P_{F} K | \leq e^{cn} | P_{F} {\cal{E}}|
\end{equation}and
\begin{equation}
\label{M-position-3}
| {\cal{E}} \cap F| \leq | K \cap F | \leq e^{cn} | {\cal{E}} \cap F|,
\end{equation}
where $c>0$ is an absolute constant.
\end{theorem}
\begin{corollary}
\label{M-FT}
Let $\omega\in \Pi_{n}$ such that $\omega(n)\neq n$. Let $
\delta_{\omega} (j) := \omega(j) - \omega(j+1) + 1$. Set
$$I_{\omega} := \{ j\leq n : \delta_{\omega} ( j) \geq 0 \} \ {\rm and }
\ \Delta(\omega):=\frac{\min\{ \color{blue}um_{j\in I_{\omega} }
\delta_{\omega}(j) , \color{blue}um_{j\in I_{\omega}^{c} }
|\delta_{\omega}(j)|\} }{n - \omega(n)} + 1 . $$ We have that
\begin{equation}
\label{M-position-6}
e^{ -c \Delta( \omega) } c_{\omega} | K|^{\frac{1}{n}} \leq
\Psi_{\omega} (K) \leq e^{ c \Delta( \omega) } c_{\omega} |
K|^{\frac{1}{n}}
\end{equation}
and
\begin{equation}
\label{M-position-5}
e^{ - c \Delta( \omega)} c_{\omega} | K|^{\frac{1}{n}} \leq
\Phi_{\omega} (K) \leq e^{ c \Delta( \omega)} c_{\omega} |
K|^{\frac{1}{n}}
\end{equation}
where $c>0$ is an absolute constant.
\end{corollary}
\begin{proof}
Set $ \Delta_{+}(\omega) := \frac{ \color{blue}um_{j\in I_{\omega} }
\delta_{\omega}(j) }{n - \omega(n)}$ and $ \Delta_{-} := ( \omega)
\frac{ \color{blue}um_{j\in I_{\omega}^{c}} |\delta_{\omega}(j) |}{n - \omega(n)}$. Using \eqref{M-position-3} and \eqref{perm-00-01}, we have
\begin{eqnarray*}
\Psi_{\omega} (K) &:=& \left( \int_{F_{n}} \prod_{j=1}^{n-1} | K
\cap F_{j}|^{ \omega(j) - \omega(j+1)+1} d \xi \right)^{\frac{1}{
n(n- \omega(n))}}\\ &\leq & \left( \int_{F_{n}} e^{cn \color{blue}um_{j\in
I_{\omega}} \omega(j) - \omega(j+1) +1} \prod_{j=1}^{n-1} |
{\cal{E}} \cap F_{j}|^{ \omega(j) - \omega(j+1)+1} d \xi
\right)^{\frac{1}{ n(n- \omega(n))}} \\ &\leq & e^{
c\Delta_{+}(\omega) } \left( \int_{F_{n}} \prod_{j=1}^{n-1} |
{\cal{E}} \cap F_{j}|^{ \omega(j) - \omega(j+1)+1} d \xi
\right)^{\frac{1}{ n(n- \omega(n))}}\\ &=&e^{
c\Delta_{+}(\omega) }c_{\omega} |
{\cal{E}}|^{\frac{1}{n}} \\ &\leq & e^{ c (
\Delta_{+}( \omega) + 1 )}c_{\omega} | K |^{\frac{1}{ n}}.
\end{eqnarray*}
One can verify that a similar inequality with the quantity $
\Delta_{-} (\omega)$ holds as well, which leads to the right-hand side
in \eqref{M-position-6}. The proof of the other inequalities is
identical and hence is omitted.
\end{proof}
\begin{remark} A similar proposition (with the same proof) holds for the case
$\omega(n)=n$. Moreover, using Pisier's regular M-position (see
\cite{Pis89}) one can get more precise estimates.
\end{remark}
\color{blue}ection{Functional forms}
\label{section:functions}
In this section we derive functional forms of some of the previous
geometric inequalities. Note that the proofs of these functional
inequalities do not depend on the geometric inequalities. They are
much more general. The invariance of functional inequalities on flag
manifolds can be proved directly using the structure theory of
semi-simple Lie groups as was done in our previous work.
\color{blue}ubsection{Functional forms of dual ${\bf r}$-flag quermassintegrals. }
Let $ f $ be a bounded integrable function on $\mathbb{R}^{n}$. We denote by $I(f)$ the functional form of the dual ${\bf r}$-flag quermassintegral
$$ I(f) := \int_{ F_{\bf r}^n } \prod_{j=1}^{r} \| f |_{F_j}\|^{i_{j+1}-i_{j-1}} d\xi_{\bf r} . $$
\begin{theorem}\label{th_fi}
For every $ g \in SL_{n}$, $ I( g\cdot f) = I( f)$.
\end{theorem}
\begin{proof}
Starting with the left-hand side, $I( g\cdot f)$, we do a global change of variables \eqref{eq_gcv} on the flag manifold:
\begin{align*}
I( g\cdot f)
&= \int_{ F_{\bf r}^n } \prod_{j=1}^{r} \| g\cdot f |_{F_j}\|^{i_{j+1}-i_{j-1}} d\xi_{\bf r} \\
&= \int_{ F_{\bf r}^n } \prod_{j=1}^{r} \| g\cdot f |_{g\cdot F_j}\|^{i_{j+1}-i_{j-1}} \color{blue}igma_{F_{{\bf r}}^{n} }(g, \xi) d\xi_{\bf r}, \\
\end{align*}
where by \eqref{F-S-02} $\color{blue}igma_{F_{{\bf r}}^{n} }(g, \xi) = \color{blue}igma_{i_{1}}^{-i_{2}}(g, F_1) \color{blue}igma_{i_{2}}^{i_{1}-i_{3}}(g, F_2) \cdots \color{blue}igma_{i_{r}}^{i_{r-1}-n}(g, F_r) $.
Now we do $r$ local changes of variables \eqref{eq_lcv} on each nested subspace $F_j$ in the product. For each $1\leq j \leq r$, we thus have
$$ \| g\cdot f|_{g\cdot F_j}\|=\|f|_{F_j}\| \, \color{blue}igma_{i_j}(g,F_j). $$
For the product under the integral, we obtain:
$$ \prod_{j=1}^{r} \| g\cdot f |_{g\cdot F_j}\|^{i_{j+1}-i_{j-1}} = \prod_{j=1}^{r} \left( \|f|_{F_j}\| \, \color{blue}igma_{i_j}(g,F_j) \right)^{i_{j+1}-i_{j-1}} = \prod_{j=1}^{r} \|f|_{F_j}\|^{i_{j+1}-i_{j-1}} \color{blue}igma_{F_{{\bf r}}^{n} }^{-1}(g, \xi). $$
\end{proof}
It is not hard to generalize this result in several ways as was done in \cite{DPP} for functional forms of dual quermassintegrals. Instead of taking $L_1(F_j)$ norms one can take $L_{p_j}(F_j)$ norms and replace the powers $i_{j+1}-i_{j-1}$ by $\alpha_j$. As long as $\frac{\alpha_j}{p_j}=i_{j+1}-i_{j-1}$ and the integrals exist, the conclusion of the Theorem \ref{th_fi} will hold. Theorem \ref{th_fi} also generalizes to a product of $m$ functions. This allows to replace $ \| f |_{F_j}\|^{i_{j+1}-i_{j-1}} $ by $ \prod_{i=1}^m \| f_i |_{E_j} \|^{\alpha_{i,j}}_{p_{i,j}} $. For the Theorem \ref{th_fi} to hold in this case, we have to require $\color{blue}um_{i=1}^m \frac{\alpha_{i,j}}{p_{i,j}} = i_{j+1}-i_{j-1}$. Another way to generalize functional forms of dual quermassintegrals is to replace
$$\| f |_{F_j}\|^{i_{j+1}-i_{j-1}} \quad \text{ by } \quad \frac{ \| f |_{F_j} \|^{\alpha_j}_{p_j}}{{\| g |_{F_j}\|^{\beta_j}_{q_j}}} \quad \text{ with } \quad \frac{\alpha_j}{p_j} - \frac{\beta_j}{q_j} = i_{j+1}-i_{j-1}, $$
to ensure they remain invariant under volume preserving transformations. Letting $\beta_j \to \infty$ modifies the integrand to $\frac{ \| f |_{F_j} \|^{\alpha_j}_{p_j}}{{\| f |_{F_j}\|^{\beta_j}_{\infty}}}$ and the condition on the powers and norms to $\frac{\alpha_j}{p_j} = i_{j+1}-i_{j-1} $. Note that in this case the invariance holds for arbitrary powers $\beta_j$. As a particular case this proves invariance under volume preserving transformations of the integrand appearing in the next theorem. One can also take the quotient of products of functions, replacing
$$\| f |_{F_j}\|^{i_{j+1}-i_{j-1}} \quad \text{ by } \quad \frac{ \prod_{i=1}^m \| f_i |_{F_j} \|^{\alpha_{i,j}}_{p_{i,j}} }{ \prod_{l=1}^{m^{\prime}} \| g_l |_{F_j} \|^{\beta_{l,j}}_{q_{l,j}} } \quad \text{ with } \quad \color{blue}um_{i=1}^m \frac{\alpha_{i,j}}{p_{i,j}} - \color{blue}um_{l=1}^{m^{\prime}} \frac{\beta_{l,j}}{q_{l,j}}=i_{j+1}-i_{j-1}. $$
Here again we can let $q_{l,j} \rightarrow \infty$, obtaining the corresponding generalization with no restrictions on $\beta_{l,j}$.
\begin{theorem}
Let $f$ be a non-negative bounded integrable function on $\mathbb
R^n$, then
$$ \int_{F_{{\bf r}}^n} \prod_{j=1}^r
\frac{\|f|_{F_j}\|^{i_{j+1}-i_{j-1}}_1}{\|f|_{F_j}\|^{i_{j+1}-i_{j}}_{\infty}}
d \xi_{\bf r} \leq \prod_{j=1}^r
\frac{\omega_{i_j}^{i_{j+1}}}{\omega_{i_{j+1}}^{i_j}}
\|f\|_1^{i_r}. $$
\end{theorem}
\begin{proof}
The result follows by iteration of an inequality on $G_{n,k}$ for one
function from our previous work \cite{DPP}:
\begin{equation}
\label{eqn:DPP}
\int_{G_{n,k}}\frac{\|f|_{E}\|^{n}_1}{\|f|_{E}\|^{n-k}_{\infty}} dE
\leq \frac{\omega_k^{n}}{\omega_n^k}\|f\|^k_1 .
\end{equation}
\noindent
Applying the latter inequality repeatedly, we get
\begin{align*}
\int_{F_{{\bf r}}^n} &\prod_{j=1}^r
\frac{\|f|_{F_j}\|^{i_{j+1}-i_{j-1}}_1}{\|f|_{F_j}\|^{i_{j+1}-i_{j}}_{\infty}}
d \xi_{\bf r} \\ &= \int_{G_{n,i_r}} \cdots \int_{G_{F_3,i_2}}
\prod_{j=2}^r
\frac{\|f|_{F_j}\|^{i_{j+1}-i_{j-1}}_1}{\|f|_{F_j}\|^{i_{j+1}-i_{j}}_{\infty}}
\int_{G_{F_2,i_1}}
\frac{\|f|_{F_1}\|^{i_{2}}_1}{\|f|_{F_1}\|^{i_{2}-i_{1}}_{\infty}}
dF_1 dF_2 \ldots dF_r\\ &\leq
\frac{\omega_{i_1}^{i_2}}{\omega^{i_1}_{i_2}} \int_{G_{n,i_r}}
\cdots \int_{G_{F_3,i_2}} \prod_{j=2}^r
\frac{\|f|_{F_j}\|^{i_{j+1}-i_{j-1}}_1}{\|f|_{F_j}\|^{i_{j+1}-i_{j}}_{\infty}}
\|f|_{F_2}\|^{i_{1}}_1 dF_2 \ldots dF_r \\ &=
\frac{\omega_{i_1}^{i_2}}{\omega^{i_1}_{i_2}} \int_{G_{n,i_r}}
\cdots \int_{G_{F_4,i_3}} \prod_{j=3}^r
\frac{\|f|_{F_j}\|^{i_{j+1}-i_{j-1}}_1}{\|f|_{F_j}\|^{i_{j+1}-i_{j}}_{\infty}}
\int_{G_{F_3,i_2}}
\frac{\|f|_{F_2}\|^{i_{3}}_1}{\|f|_{F_2}\|^{i_{3}-i_{2}}_{\infty}}
dF_2 dF_3 \ldots \ldots dF_r \\ &\leq
\frac{\omega_{i_1}^{i_2}}{\omega^{i_1}_{i_2}}
\frac{\omega_{i_2}^{i_3}}{\omega^{i_2}_{i_3}} \int_{G_{n,i_r}}
\cdots \int_{G_{F_4,i_3}} \prod_{j=3}^r
\frac{\|f|_{F_j\|^{i_{j+1}-i_{j-1}}_1}}{\|f|_{F_j}\|^{i_{j+1}-i_{j}}_{\infty}}
\|f|_{F_3}\|^{i_{2}}_1 dF_3 \ldots dF_r\\ &=\cdots \\ &\leq \prod_{j=1}^r
\frac{\omega_{i_j}^{i_{j+1}}}{\omega_{i_{j+1}}^{i_j}} \|f\|_1^{i_r}.
\end{align*}
\end{proof}
In \cite{DPP}, more general versions of \eqref{eqn:DPP} are proved
with multiple functions and different powers. These also carry over
to extremal inequalities on flag manifolds by mimicking the previous
proof. As a sample, we mention just one statement. Let $1 \leq q \leq
i_1$ and let $f_1, \ldots, f_q$ be non-negative bounded integrable
functions on $\mathbb R^n$, then
\begin{equation}
\label{eqn:ext}
\int_{F_{{\bf r }}^n} \prod_{k=1}^q \prod_{j=2}^r
\frac{\|f_k|_{F_j}\|^{\frac{i_{j+1}-i_{j}}{i_j}}_1}{\|f_k|_{F_j}\|^{\frac{i_{j+1}-i_{j}}{i_j}}_{\infty}}
\frac{\norm{f_k|{F_1}}_1^{\frac{i_2}{i_1}}}{\norm{f_k|_{F_1}}^{\frac{i_2-i_1}{i_1}}_{\infty}}
d \xi_{\bf r} \leq \left( \prod_{j=1}^r
\frac{\omega_{i_j}^{\frac{i_{j+1}}{i_j}}}{\omega_{i_{j+1}}} \right)^q
\prod_{k=1}^q \|f_k\|_1.
\end{equation}
\color{blue}ubsection{Functional forms of the ${\bf r}$-flag quermassintegrals. }
\label{sub:func}
In this subsection we will extend the notions of ${\bf r}$-flag
quermassintegrals to functions. In particular, this will lead to
functional versions of affine quermassintegrals. This is motivated by
recent work of Bobkov, Colesanti and Fragal\'{a} \cite{BCF} and
V. Milman and Rotem \cite{MR}. The latter authors proposed and studied
a notion of quermassintegrals for $\log$-concave or even quasi-concave
functions, which we now recall.
\begin{definition}
Suppose that $f:\mathbb{R}^n\rightarrow [0,\infty)$ is
upper-semicontinuous and quasi-concave. For $ 1\leq i \leq n$, let
\begin{equation}
\label{quer-functions-1}
V_k (f) := \int_{0}^{\infty} V_k( \{ f\geq t\}) d t .
\end{equation}
\end{definition}
The above definition is consistent with the notion of projection of a
function onto a subspace as introduced by Klartag and Milman in
\cite{KlM}. Namely, let $f$ be an non-negative function $\mathbb
R^{n}\rightarrow [0,\infty]$ and $F\in G_{n,k}$. Define the orthogonal
projection of $f$ onto $F$ as the function $P_{F} f: F\rightarrow
[0,\infty]$ given by
\begin{equation}
\label{proj-def}
(P_{F} f) ( z) := \color{blue}up_{y\in
F^{\perp}} f ( z+y) .
\end{equation}
Note that if $K$ is compact and $ f:= {\bf 1}_{K}$ then $P_{F}f :=
{\bf 1}_{P_{F}(K)}$. Moreover, from the definition, one has
\begin{equation}
\label{proj-f-1}
\{ z\in F : (P_{F}f)(z) >t \} = P_{F} ( \{ x\in \mathbb R^{n}: f(x) >t\}).
\end{equation}
Assume now that $f:=\mathbb R^{n} \rightarrow [0,\infty)$ and that for
each $t>0$, the set $\{x\in \mathbb{R}^n:f(x)\geq t\}$ is
compact. For $1\leq k \leq n-1$, we define the affine
quermassintegral of $f$ by
\begin{equation}
\label{aff-quer-f}
\Phi_{[k]} (f) := \int_{0}^{\infty} \Phi_{[k]} ( \{ f\geq t\}) dt =
\int_{0}^{\infty} \left(\int_{G_{n,k}} | \{ P_{F} f\geq t\} |^{-n} d
F \right)^{\frac{1}{ nk}} dt .
\end{equation}
For $ 1\leq i_{1} < i_{2} < \cdots < i_{r} = n-1$, ${\bf r}:= (i_{1},
\cdots , i_{r})$ we define the ${\bf r}$-flag quermassintegrals of $f$
by
\begin{equation}
\label{aff-flag-quer-f}
\Phi_{{\bf r}} (f) := \int_{0}^{\infty} \Phi_{{\bf r}} ( \{ f\geq t\}) dt.
\end{equation}
For comparison, we recall that for every $f:\mathbb R^{n} \rightarrow
[0,\infty] $,
\begin{equation}
\label{L1}
\int_{\mathbb R^{n}} f (x) d x = \int_{0}^{\infty} | \{ f\geq t \}|
dt.
\end{equation} For $\lambda\in \mathbb R\color{blue}etminus \{0\}$ and $f$ as above, we write
\begin{equation}
\label{f-lambda}
f_{(\lambda)} :\mathbb R^{n} \rightarrow [0,\infty] , \ {\rm as}
\ f_{(\lambda)} ( x) := f \left(\frac{ x}{ \lambda}\right) ,
\end{equation}
and if $T\in GL_{n}$,
\begin{equation}
\label{f-T}
f\circ T :\mathbb R^{n} \rightarrow [0,\infty] , \ {\rm as} \ f\circ T ( x) := f (T^{-1} x) .
\end{equation}
\noindent Note that if $f:= {\bf 1}_{K}$, then
$$ f_{(\lambda)} (x) = {\bf 1}_{\lambda K} (x) \ {\rm and} \ f\circ T (x) = {\bf 1}_{TK} (x) . $$
Let $ f:\mathbb R^{n} \rightarrow [0, \infty]$, $\lambda >0$ and $
T\in GL_{n}$. Then
\begin{equation}
\label{b-f-lambda}
\{ f\circ T \geq t \} = T\left( \{ f\geq t \} \right) \ {\rm and }
\ \{ f_{(\lambda)} \geq t \} = \lambda \{ f \geq t \} .
\end{equation}
Then \eqref{b-f-lambda}, the $1$-homogenuity of the ${\bf r}$-flag
quermassintegrals for sets as well as the affine invariance of these
quantities imply the following.
\begin{theorem}
Let $f:\mathbb R^{n}\rightarrow [0,\infty]$, $1\leq i_{1}<\cdots <
i_{r}\leq n-1$ and ${\bf r} := (i_{1}, \cdots , i_{r})$. Let
$\lambda>0$ and $T$ be an affine volume-preserving map. Then
\begin{equation}
\Phi_{{\bf r}} (f_{(\lambda)} ) = \lambda \Phi_{{\bf r}}(f) \ {\rm and} \ \Phi_{{\bf r}} (f\circ T) = \Phi_{{\bf r}}(f) .
\end{equation}
\end{theorem}
Recall that the symmetric decreasing rearrangement of a function $f$
which is integrable (or vanishes at infinity). For a set $A\color{blue}ubseteq
\mathbb R^{n}$ with finite volume, the decreasing rearrangement
$A^{\ast}$ is defined as
$$ A^{\ast} := r_{A} B_{2}^{n}, $$ where $r_A$ is the volume-radius of
$A$. The symmetric decreasing rearrangement $f^{\ast}$ of $f$ is
defined as the radial function $f^{\ast}$ such that
$$\{f\geq t\}^{\ast} = \{ f^{\ast} \geq t \} , \ \forall t>0 . $$
Thus,
\begin{equation}
\label{basic-f-1-0}
r_{\{ f\geq t\} } B_{2}^{n} = \{ f^{\ast} \geq t \} .
\end{equation}
Using \eqref{basic-f-1-0}, \eqref{aff-flag-quer-f} and
\eqref{tilde-sym-two}, we have the following for all non-negative
quasi-concave functions $f$ on $\mathbb R^{n}$:
$$ \Phi_{{\bf r}} (f) = \int_{0}^{\infty} \Phi_{{\bf r}}(\{ f\geq t\})
d t \geq c \int_{0}^{\infty} \Phi_{{\bf r}}(r_{\{ f\geq t\}}
B_{2}^{n}) d t = $$
$$ c \int_{0}^{\infty} \Phi_{{\bf r}}(\{ f^{\ast} \geq t \} ) d t=
\Phi_{{\bf r}} (f^{\ast}) . $$
Let $f$ be a non-negative quasi-concave function on $\mathbb
R^{n}$. We define
$$ d_{BM} ( f) := \color{blue}up_{t>0}d_{BM} ( \{ f\geq t\} ). $$ The results of
\S 3 lead to the following double-sided inequality for $\Phi_{[{\bf
r}]}(f)$:
\begin{theorem}
Let $f$ be a non-negative quasi-concave function on $\mathbb R^{n}$,
$1\leq i_{1}<\cdots <i_{r} $ and let ${\bf r}:= (i_{1}, \cdots ,
i_{r})$. Then
\begin{equation}
\label{ineq-funct-quer}
c \Phi_{{\bf r}} (f^{\ast} ) \leq \Phi_{{\bf r}} ( f) \leq c^{\prime} \min\left\{ \log\{ 1+d_{BM}(f)\}, \color{blue}qrt{\frac{n}{ i_{r}}} \right\} \Phi_{{\bf r}} (f^{\ast} ) .
\end{equation}
\end{theorem}
\footnotesize
\end{document}
|
math
|
\widetilde begin{document}
\thispagestyle{empty}
\widetilde begin{center}
{\widetilde bfseries \large \textsc{Crossed products by endomorphisms and reduction of relations in relative Cuntz-Pimsner algebras}}
\widetilde bigskip
B. K. Kwa\'sniewski\footnote{Work partially supported by National Science Centre grants numbers DEC-2011/01/D/ST1/04112 and DEC-2011/01/B/ST1/03838} , \ \ A. V. Lebedev
\end{center}
\widetilde begin{abstract} Starting
from an arbitrary endomorphism $\widetilde alphapha$ of a unital $C^*$-algebra
$ A$ we construct a crossed product. It is shown that the natural
construction depends not only on the $C^*$-dynamical system
$( A,\widetilde alphapha )$ but also on the choice of an ideal $J$ orthogonal
to $\ker \widetilde alphapha $. The article gives an explicit description of
the internal structure of this crossed product and, in particular,
discusses the interrelation between relative Cuntz-Pimsner
algebras and partial isometric crossed products. We present a
canonical procedure that reduces any given $C^*$-correspondence
to the 'smallest' $C^*$-correspondence yielding the
same relative Cuntz-Pimsner algebra as the initial one. In the
context of crossed products this reduction procedure corresponds
to the reduction of $C^*$-dynamical systems and allow us to
establish a coincidence between relative Cuntz-Pimsner
algebras and crossed products introduced.
\end{abstract}
\medbreak
\textbf{Keywords:} \emph{$C^*$-algebra, endomorphism, partial
isometry, orthogonal ideal, crossed product, covariant representation, $C^*$-correspondence, relative Cuntz-Pimsner algebra, reduction}
\medbreak
{\widetilde bfseries 2000 Mathematics Subject Classification:} 47L65, 46L05,
47L30
\tableofcontents
\section*{Introduction}
The crossed product of a $C^*$-algebra $A$ by an automorphism
$\widetilde alphapha:A\to A$ is defined as a universal $C^*$-algebra
generated by a copy of $A$ and a unitary element $U$ satisfying
the relations
$$
\widetilde alphapha(a)= Ua U^*,\widetilde qquad \widetilde alphapha^{-1}(a)=U^*aU, \ \ \ \ a\in A.
$$
On one hand, algebras arising in this way (or their versions adapted to actions
of groups of automorphisms) are very well understood and became
a part of a $C^*$-folklore \cite{Pedersen}, \cite{Kadison}. On the other hand,
it is very symptomatic that, even though the first attempts on generalizing
this kind of constructions to endomorphisms go back to 1970s, articles introducing
different definitions of the related object appear almost continuously until the present-day, see,
for example, \cite{CK}, \cite{Paschke},
\cite{Stacey}, \cite{Murphy}, \cite{exel1}, \cite{exel2},
\cite{kwa}, \cite{Ant-Bakht-Leb}.
This phenomenon is caused by very fundamental problems one has to face when dealing with crossed
products by endomorphisms. Namely, one has to answer the following questions:
\widetilde begin{itemize}
\item[(i)] What relations should the element $U$ satisfy?
\item[(ii)] What should be used in place of $\widetilde alphapha^{-1}$?
\end{itemize}
It is important that in spite of substantial freedom of
choice (in answering the foregoing questions), all the above
listed papers do however have a certain nontrivial intersection.
They mostly agree, and simultaneously boast their greatest
successes, in the case when dynamics is implemented by
monomorphisms with a hereditary range. In view of the articles
\cite{Bakht-Leb}, \cite{Ant-Bakht-Leb}, \cite{kwa3}, this
coincidence is completely understood. It is shown in
\cite{Bakht-Leb} that in the case of monomorphism with hereditary
range there exists a unique non-degenerate transfer operator
$\widetilde alphapha_*$ for $(A,\widetilde alphapha)$, called by authors of
\cite{Bakht-Leb} a \emph{complete transfer operator}, and the
theory goes smooth with $\widetilde alphapha_*$ as it takes over the role
classically played by $\widetilde alphapha^{-1}$. The $C^*$-dynamical systems
of this sort will be called {\em partially reversible}.
\par
If a pair $(A,\widetilde alphapha)$ is of the above described type,
then $A$ is called a {\em coefficient algebra}.
This notion was introduced in \cite{Leb-Odz} where its investigation and
relation to the extensions of $C^*$-algebras by partial isometries was clarified.
Further in
\cite{Bakht-Leb} a certain criterion for a
$C^*$-algebra to be a coefficient algebra associated with a given
endomorphism was obtained (see also \cite{kwa3}). On the base of the results of these papers
one naturally arrives at the construction of a certain crossed
product which was implemented in \cite{Ant-Bakht-Leb}.
It was also observed in \cite{Ant-Bakht-Leb} that in the most
natural situations the coefficient algebras arise as a result of
a certain extension procedure on the initial $C^*$-algebra.
Since the crossed product is (should be) an extension of the
initial $C^*$-algebra one can consider the construction of an
appropriate coefficient algebra as one of the most important
intermediate steps in the procedure of construction of the crossed product itself (a detailed discussion of the philosophy of the arising construction is given in \cite[Section~5]{Ant-Bakht-Leb}).
It was also shown in \cite[Section~4]{Ant-Bakht-Leb} how different extension procedures lead to the most popular constructions of crossed products such as Cuntz-Krieger algebras~\cite{CK}, Paschke's crossed product~\cite{Paschke}, partial crossed product~\cite{exel1}, Exel's crossed product~\cite{exel2} and others.
The analysis of these extension procedures naturally leads to the next problem: can we extend a
$C^*$-dynamical system associated with an arbitrary endomorphism
to a {\em partially reversible} $C^*$-dynamical system? In the
{\em commutative} $C^*$-algebra situation the corresponding
procedure and the explicit description of maximal ideals of the
arising $C^*$-algebra is given in \cite{maxid}. On the base of
this construction the general construction of the crossed product
associated to an arbitrary endomorphism of a {\em commutative}
$C^*$-algebra is presented in \cite{kwa}. Further the general
construction of an extension of a $C^*$-dynamical system
associated with an arbitrary endomorphism to a partially
reversible $C^*$-dynamical system is worked out in
\cite{kwa4}. Therefore the mentioned results of
\cite{Bakht-Leb}, \cite{Ant-Bakht-Leb}, \cite{kwa4}, give us the key to
construct a general crossed product starting from a
$C^*$-dynamical system associated with an arbitrary endomorphism,
and this is one of the main themes of the present article.
\par
The most important novelties we incorporate to the theory of crossed products are
\widetilde begin{itemize}
\item[1)] an explicit description of the crossed product based on the
worked out matrix calculus presented in Section~\ref{22},
\end{itemize}
and an observation of (in a way unexpected) phenomena that
\widetilde begin{itemize}
\item[2)] in general the universal construction of the crossed product
depends not only on the algebra $A$ and an endomorphism $\widetilde alphapha$
one starts with but also on the choice of an ({\em arbitrary})
singled out ideal $J$ orthogonal to the kernel of $\widetilde alphapha$ (see
Section \ref{Crossed products-alternative1}).
\end{itemize}
So in fact we have a {\em variety} of crossed products depending
on $J$.
On the appearance of \cite{kwa-leb} B. Solel noted to the authors that the crossed product constructed
in \cite{kwa-leb} can also be modeled as a certain relative Cuntz-Pimsner algebra (Proposition \ref{universality proposition}
of the present article describes in essence the main idea of B. Solel's remark). Thus we have naturally arrived at the discussion of interrelations between relative Cuntz-Pimsner algebras and crossed products, and this was the theme of~\cite{kwa-leb1}. Since relative Cuntz-Pimsner algebras are defined by means of $C^*$-correspondences the role of the latter objects in the whole picture should be clarified and in this way we necessarily come to the analysis of the interplay: crossed products -- relative Cuntz-Pimsner algebras -- $C^*$-correspondences.
Corollary \ref{C-P-cross} of the present article states that if $X$ is a $C^*$-correspondence of a $C^*$-dynamical system $(A,\widetilde alphapha)$ and $J$ is an ideal orthogonal to the kernel of $\widetilde alphapha$ then the relative Cuntz-Pimsner algebra
$\mathcal{O}(J,X)$ and the crossed product $C^*(A,\widetilde alphapha,J)$ of the present article are canonically isomorphic. This observation in its turn causes a problem. Namely, $\mathcal{O}(J,X)$ is defined for ideals $J$ that are not
necessarily orthogonal to the kernel of $\widetilde alphapha$. Moreover, by means of $\mathcal{O}(J,X)$ one can construct crossed products seemingly different from those introduced in the present article (see, for example, Stacey's crossed product identified in Corollary \ref{corollary stacey's}). Therefore, one may guess that
$\mathcal{O}(J,X)$ is a more general object than $C^*(A,\widetilde alphapha,J)$. At
the same time it is known (see \cite[Proposition~2.21]{ms}, i.e. Proposition~\ref{injectivity of k_A} of the present article) that when
$J$ is not orthogonal to the kernel of $\widetilde alphapha$ the algebra $\mathcal{O}(J,X)$
possesses certain 'degeneracy'. All this stimulates us to take a
closer look and provide a more thorough analysis of the structure
of $\mathcal{O}(J,X)$ and its relation to $C^*(A,\widetilde alphapha,J)$, and this is one more main goal of the article.
As we show the necessary apparatus of investigation of the noted vagueness in the relation between $\mathcal{O}(J,X)$ and $C^*(A,\widetilde alphapha,J)$ is \emph{reduction}.
The general scheme of reduction procedure (taking quotients) associated with ideals in $C^*$-correspondences and the corresponding reduction in relative Cuntz-Pimsner algebras as well as the analysis of this scheme was provided in the structure theorem of \cite{fmr} (see Theorem \ref{takie tam aa} of the present article). We add \emph{canonicity} to this scheme by applying the reduction procedure to a sequence of ideals $J_n, \ n=1,2, ..., \, J_\infty$ (Definition \ref{reduction ideal for correspondences}) that are naturally generated by $J$ and $\mathcal{O}(J,X)$. Being defined in this way the canonical
procedure reduces any given $C^*$-correspondence to the 'smallest' $C^*$-correspondence yielding the same relative Cuntz-Pimsner algebra as the initial one. In the context of the crossed products this reduction procedure corresponds to the reduction of $C^*$-dynamical systems.
Using this, on the one hand, we obtain the \emph{canonical $C^*$-dynamical system} (in Section \ref{Canonical C*-dynamical systems}) and, on the other hand, eliminate the mentioned 'degeneracy' in $\mathcal{O}(J,X)$ and simultaneously establish an isomorphism between $\mathcal{O}(J,X)$ and appropriate crossed product introduced in the present article (Theorem~\ref{reduction thm} and Proposition~\ref{reducing C*-Hilbert bimodules}) obviating in this way the mentioned vagueness in their interrelations. As a byproduct we also get a refinement of Stacey's results (Example \ref{reduction of Stacey's crossed product}) and add clarity to Katsura's canonical relations (Subsection \ref{Katsura's canonical relations}).
We would also like to make a certain additional remark on the objects of the paper. It is generally agreed that a crossed product of a \emph{unital $C^*$-algebra} $A$ by a $C^*$-mapping should be a $C^*$-algebra $B$ generated by a copy (or at least a homomorphic image) of the $C^*$-algebra $A$ and an operator $U$ implementing the dynamics. On the other hand when $A$ is \emph{non-unital}, then it seems that there are various essentially different ideas of what the \emph{non-unital crossed product} $B$ should be. In particular, one can add to questions (i), (ii) (related to irreversibility of dynamics) two more still open principle problems (related to the lack of unity):
\widetilde begin{itemize}
\item[(iii)] What should $B$ be generated by? Should it be the set $A\cup AU$, cf. e.g. \cite{Pedersen}, \cite{Kadison}, \cite{Stacey}, \cite{brv}, or maybe $A\cup A_0U$ for a certain subspace $A_0$ of $A$, cf. \cite{exel1}, \cite{fmr}, if so what should $A_0$ be?
\item[(iv)] How should $U$ be related to $B$? Should it belong to the multiplier algebra $M(B)$ of $B$, cf. \cite{Pedersen}, \cite{Kadison}, \cite{Stacey}, an enveloping $W^*$-algebra $B^{**}$ of $B$, see e.g. \cite{exel1}, \cite{brv}, or maybe something else?
\end{itemize}
A way to bypass this trouble, usually adopted by most of the authors, cf. e.g. \cite{Lin-Rae}, \cite{fmr}, \cite{brv}, is to consider the non-unital crossed products only for the so-called \emph{extendible systems}, that is for systems which naturally extend from $A$ to the multiplier algebra $M(A)$. For such systems a non-unital crossed product is actually a subalgebra of a unital crossed product. In the present paper we drop the technicalities arising from consideration of extendible systems and non-trivial issues concerning non-extendible systems. \emph{We consider only unital crossed products}. Nevertheless the general $C^*$-correspondences will be considered over arbitrary (not necessarily unital) $C^*$-algebras.
The present paper is based on \cite{kwa-leb} and ~\cite{kwa-leb1},
and in essence forms their unification, refinement and development.
The paper is organized as follows.
In the first section we discuss and clarify the relations that should be used in a definition of a covariant representation. In particular, we split the class of covariant representations
into subclasses according to a certain ideal they determine arriving at the notion of a $J$-covariant representation.
In Subsection \ref{what da hell} we establish existence of such representations and introduce the corresponding crossed products as universal algebras. Section~\ref{22} presents a matrix calculus which describes the internal algebraic structure
of the crossed product serving simultaneously as its certain regular representation. This leads us in Subsection \ref{norm} to an explicit formula for the norm of elements of the crossed product introduced, thus providing us with one more its alternative definition (Definition~\ref{cr-pr-def}).
In Section \ref{isomorph} we give a series of isomorphism theorems. They provide an apparatus for verifying faithfulness of a given representation of crossed product and, in particular, establishing equivalence of different approaches to construction of
crossed products (cf. Proposition~\ref{crossed equiv}). The isomorphism theorems are discussed as on the operator algebraic level so also on the dynamical topological level exploiting topological freeness
of the arising $C^*$-dynamical systems (Theorems \ref{isomorphiasm theorem} and \ref{topolo disco-polo}). In addition we present here an overview of existing crossed product constructions and their comparison with the crossed product of the present paper. This stimulates us to undertake deeper analysis of interrelation between different approaches and in this way we pass to the next main theme of the article.
In Section~\ref{C-P}, Subsections \ref{preliminaries on C-correspondences}, \ref{The Toeplitz C*-algebra of a Hilbert bimodule},~\ref{Relative Cuntz-Pimsner algebras}, we recall the indispensable notions and objects concerning $C^*$-correspondences and Cuntz-Pimsner algebras, while
presentation of the crossed products as relative Cuntz-Pimsner algebras is given in
Subsection~\ref{Crossed prod =Relative Cuntz-Pimsner algebras}. Analysis of interrelations:
$C^*$-correspondences -- relative
Cuntz-Pimsner
algebras -- crossed products is implemented on the base of reduction. The principal results in this direction are given in Section~\ref{Reduction and canonical}. Applying the canonical reduction to $C^*$-correspondences we
eliminate the 'degeneracy' in relative Cuntz-Pimsner algebras (Theorem \ref{reduction thm}). Applying this procedure to $C^*$-dynamical systems we prove coincidence between the corresponding relative Cuntz-Pimsner algebras and the crossed products of the present article (Proposition~\ref{reducing C*-Hilbert bimodules}). Finally in Section~\ref{Canonical C*-dynamical systems} starting from a triple $(A,\widetilde alpha,J)$ we construct a $C^*$-dynamical system $(A_J,\widetilde alpha_J)$ which is cannonical in the sense that the corresponding relations defining the crossed product are 'non-degenerate' and do not include any ideal of $A$ (Theorem~\ref{canon}). This canonical construction is related to but as we argue differs from the similar result due to Katsura.
\section{Covariant representations, orthogonal ideals, crossed product}\label{Crossed products-alternative1}
\subsection{Covariant representations and ideals of covariance} \label{start}
Let $( A,\widetilde alphapha)$ be a pair consisting of a $C^*$-algebra $ A$, containing an identity and
an endomorphism $\widetilde alphapha : A \to A$ (by a homomorphism between $C^*$-algebras we always mean a $^*$-homomorphism). Throughout the paper the pair $( A,\widetilde alphapha)$ will be called
a {\em $C^*$-dynamical system}.
\widetilde begin{defn}\label{kowariant rep defn*}
Let $( A,\widetilde alphapha)$ be a $C^*$-dynamical system. A
\emph{representation} of $( A,\widetilde alphapha)$ is a triple $(\pi,U,H)$
consisting of a unital representation $\pi: A\to L(H)$ on a
Hilbert space $H$ and an operator $U\in L(H)$ satisfying the
following relation
\widetilde begin{equation}\label{covariance rel1*}
U\pi(a)U^* =\pi(\widetilde alphapha(a)),\widetilde qquad a \in A.
\end{equation}
If $\pi$ is a faithful representation of $ A$, then $(\pi,U,H)$ is
called a \emph{faithful representation}.
\end{defn}
Since $\widetilde alpha^n(1)$ is a projection for every $n$ it follows that the operator $U$ in the above definition is necessarily a \emph{power partial isometry}.
Note also that iterating \eqref{covariance rel1*} we get
\widetilde begin{equation}\label{n}
U^{n}\pi(a)U^{*n} =\pi(\widetilde alphapha^n(a)),\widetilde qquad a \in A, \, n\in \mathbb N
\end{equation}
which means that
if $(\pi,U,H)$ is a representation of $( A,\widetilde alphapha)$, then $(\pi,U^n,H)$ is a representation of $( A,\widetilde alphapha^n)$ for every $n\in \mathbb N$.
The next lemma shows that representations of $C^*$-dynamical systems possess one more important property which plays, in fact, a crucial role in the whole story.
\widetilde begin{lem}\label{iteration of representations}
If $(\pi,U,H)$ is a representation of $( A,\widetilde alphapha)$, then for every $n\in \mathbb N$ we have
\widetilde begin{equation}\label{covariance rel2*}
U^{*n}U^n \in \pi(A)'.
\end{equation}.
\end{lem}
\widetilde begin{Proof}
Let $(\pi,U,H)$ consists of a unital representation $\pi: A\to L(H)$ on a
Hilbert space $H$ and an operator $U\in L(H)$ such that \eqref{covariance rel1*} holds. In view of \eqref{n} it suffice to prove \eqref{covariance rel2*} only for $n=1$. As $U$ is a partial isometry $1-U^*U$ is a projection, and for all $a \in A$
\widetilde begin{align*}
\|U\pi(a)(1-U^*U)\|^2&=\|U\pi(a)(1-U^*U)\pi(a^*)U^*\|
\\
&=\|U\pi(a)\pi(a^*)U^* - U\pi(a)U^*U\pi(a^*)U^*\|
\\
&=\|\pi(\widetilde alpha(aa^*))-\pi(\widetilde alpha(a)\widetilde alpha(a^*))\|=0.
\end{align*}
Thus $U\pi(a)=U\pi(a)U^*U=\pi(\widetilde alpha(a))U$ and by passing to adjoints we also get $U^*\pi(\widetilde alpha(a))=\pi(a)U^*$. Using these two relations we get
$$
U^*U\pi(a)=U^*\pi(\widetilde alpha(a))U=\pi(a)U^*U,
$$
which proves the assertion.
\end{Proof}
\widetilde begin{rem}\label{remarks on representations} \widetilde begin{itemize}
\item[1.] The notion of a representation of a $C^*$-dynamical system
appears in a similar or identical form, for instance, in \cite{Stacey}, \cite{Adji_Laca_Nilsen_Raeburn}, \cite{Murphy}, \cite{exel2}, \cite{Lin-Rae}, \cite{Leb-Odz}, \cite{kwa}, \cite{kwa4}. For isometric crossed products, cf. \cite{Stacey}, \cite{Adji_Laca_Nilsen_Raeburn}, \cite{Murphy}, it is assumed that $U$ is an isometry satisfying \eqref{covariance rel1*}.
In general, cf. \cite{exel2},\cite{Lin-Rae}, \cite{Leb-Odz}, \cite{kwa}, \cite{kwa4}, definitions of representations of $C^*$-dynamical systems contained conditions \eqref{covariance rel1*} and \eqref{covariance rel2*} (for $n=1$) or a certain equivalent of \eqref{covariance rel2*}. Lemma \ref{iteration of representations} shows that \eqref{covariance rel2*} is redundant.
\item[2.] In \cite[Prop 2.2]{Leb-Odz} and \cite[Lem. 4.3]{Lin-Rae} it was shown that when \eqref{covariance rel1*} is assumed relation \eqref{covariance rel2*} is equivalent to the condition
\widetilde begin{equation}\label{covariance rel1**}
U\pi(a) =\pi(\widetilde alphapha(a))U,\widetilde qquad a \in A.
\end{equation}
In view of Lemma \ref{iteration of representations} the conditions \eqref{covariance rel2*} and \eqref{covariance rel1**} are not only equivalent in the presence of \eqref{covariance rel1*} but they actually follow from \eqref{covariance rel1*}.
\end{itemize}
\end{rem}
Relation \eqref{covariance rel2*} imply that for any representation
$(\pi,U,H)$ of $(A,\widetilde alpha)$ the set
$$
J:=\{ a\in A: U^*U \pi(a)=\pi(a)\}$$ is an ideal in $ A$ (by which we always mean a closed two-sided ideal).
This ideal will play one of the key roles in the paper. The next statement shows a certain property of $J$ which
is important for the further analysis.
\widetilde begin{prop}\label{motivation prop1}
Let $(\pi,U,H)$ be a representation of $( A,\widetilde alphapha)$ and let
\widetilde begin{equation}\label{ideals I and J}
I=\{ a\in A: (1-U^*U) \pi(a)=\pi(a)\}, \widetilde qquad J=\{ a\in A: U^*U \pi(a)=\pi(a)\}.
\end{equation}
Then
$$
I= \ker (\pi\circ\widetilde alphapha)
\widetilde qquad
\textrm{ and }\widetilde qquad I\cap J = \ker\pi.
$$
\end{prop}
\widetilde begin{Proof}
Observe that $
U (U^*U \pi(a))U^* = U \pi(a)U^*
$
and
$
U^* (U \pi(a)U^*)U =U^*U \pi(a).
$
Therefore the mappings
$
U(\cdot )U^*:U^*U \pi(a)\mapsto\pi(\widetilde alphapha(a)),
$
$
U^*(\cdot
)U:\pi(\widetilde alphapha(a))\mapsto U^*U \pi(a)
$
are each other inverses and in particular
$U^*U \pi( A)\cong \pi(\widetilde alphapha( A))$, where $U^*U \pi(a)\mapsto\pi(\widetilde alphapha(a))$. Thus
$$
\pi(\widetilde alphapha(a))=0
\mathcal Longleftrightarrow U^*U\pi(a)=0 \mathcal Longleftrightarrow
(1-U^*U)\pi(a)=\pi(a).
$$
Which means that $I$ is the kernel of $\pi\circ\widetilde alphapha$.
Clearly $\ker\pi\subset I\cap J$ and on the other hand
$$
a\in I\cap J \mathbb Rightarrow (1-U^*U) \pi(a)= U^*U \pi(a)\mathbb Rightarrow \pi(a)=0,
$$
that is $ I\cap J \subset\ker\pi$.
\end{Proof}
\widetilde begin{cor}\label{ort-id}
Let $(\pi,U,H)$ be a faithful representation of $( A,\widetilde alphapha)$, and let $I$ and $J$ be ideals in $A$ given by \eqref{ideals I and J}. Then
$$
I= {\ker}\, \widetilde alphapha \widetilde qquad and \widetilde qquad I\cap J = \{0\}.
$$
\end{cor}
Having in mind this observation we introduce the following
\widetilde begin{defn}
\label{ort}
Let $I$ and $J$ be two ideals in $ A$. We say that $J$ is \emph{orthogonal to $I$} if
$$
I\cap J =\{0\}.
$$
There exists the biggest ideal orthogonal to $I$ (in the sense
that it contains all other ideals that are orthogonal to $I$) denoted by $I^\widetilde bot$. This ideal could be defined
explicitly as $I^\widetilde bot=\{a\in A:aI=\{0\}\}$ and it is called the
\emph{annihilator} of the ideal $I$ in~$ A$.
\end{defn}
Proposition~\ref{motivation prop1} and Corollary~\ref{ort-id} make it natural to introduce the following definition.
\widetilde begin{defn}\label{kowariant rep defn 2} Let $(\pi,U,H)$ be a representation
of $( A,\widetilde alphapha)$ and let $J$ be an ideal in $ A$.
If $(\pi,U,H)$ and $J$ are such that
\widetilde begin{equation}\label{covariance rel3}
J=\{ a\in A: U^*U \pi(a)=\pi(a)\},
\end{equation}
then we will say that
$(\pi,U,H)$ is a $J$-\emph{covariant representation}. In this situation we will also say that $J$ is the
\emph{ideal of covariance for the representation} $(\pi,U,H)$. If $J = (\ker\widetilde alphapha)^\widetilde bot$, then we simply call $(\pi,U,H)$ a \emph{covariant representation}.
\end{defn}
\widetilde begin{rem}\label{remark something1} For any $C^*$-dynamical system $(A,\widetilde alpha)$ one can always
construct a faithful representation of $( A,\widetilde alphapha)$ (see e.g. the Toeplitz representation defined
in Subsection \ref{what da hell}) such that $U$ is not an isometry.
However, if $(\pi,U,H)$ is a
covariant representation of $( A,\widetilde alphapha)$, then the following implications hold true:
\widetilde begin{itemize}
\item[1)] $\widetilde alphapha$ is a monomorphism $\mathbb Rightarrow$ $U$ is an isometry,
\item[2)] $\widetilde alphapha$ is an automorphism $\mathbb Rightarrow$ $U$ is unitary.
\end{itemize}
Hence, in contrast to arbitrary representations, covariant representations as defined above involve the operators typically used in similar definitions for automorphisms and monomorphisms of $C^*$-algebras, cf. \cite{Pedersen}, \cite{Kadison}, \cite{Paschke},
\cite{Stacey}, \cite{Murphy}.
\end{rem}
In connection with the above remark and for future applications we recall the following observation, which is a part a) of \cite[Prop. 1.9]{kwa4}, and follows immediately from Proposition \ref{motivation prop1}.
\widetilde begin{prop}\label{proposition najwazniejsze}
Suppose $( A,\widetilde alphapha)$ is such that $\ker\widetilde alphapha$ is unital and let $(\pi,U,H)$ be a representation of $( A,\widetilde alphapha)$. Then the following conditions are equivalent:
\widetilde begin{itemize}
\item[i)] $(\pi,U,H)$ is a covariant representation
\item[ii)] $U^*U\in \pi( A)$
\item[iii)] $U^*U\in \pi(Z( A))$ ($Z( A)$ stands for the center of $ A$)
\item[iv)] $U^*U$ is the unit in $\pi((\ker\widetilde alphapha)^\widetilde bot)$
\end{itemize}
\end{prop}
Though we have defined $J$-covariant representations, their existence for all approriate ideals $J$ is not established yet. Henceforth we present a construction resolving this problem.
\subsection{Construction of a faithful $J$-covariant representation. Crossed product} \label{what da hell}
Let us fix a $C^*$-dynamical system $( A,\widetilde alphapha)$, an ideal
$J\subset(\ker\widetilde alphapha)^\widetilde bot$, and a faithful nondegenerate
representation $\pi: A\to L(H)$. First we define a triple
$(\widetilde{\pi},\widetilde{U}, \mathcal H)$ by the formulae
$$
\mathcal H:=\widetilde bigoplus_{n=0}^{\infty} \pi(\widetilde alphapha^n(1))H, \widetilde qquad
(\widetilde{\pi}(a)h)_n:=\pi(\widetilde alphapha^n(a))h_n,\widetilde qquad
(\widetilde{U}h)_n:= h_{n+1}.
$$
One readily sees that $(\widetilde{\pi},\widetilde{U}, \mathcal H)$ is a
faithful representation of $( A,\widetilde alphapha)$. Actually $(\widetilde{\pi},\widetilde{U}, \mathcal H)$ is $\{0\}$-covariant and having in mind the
classical associations one could call
$(\widetilde{\pi},\widetilde{U}, \mathcal H)$ a \emph{Toeplitz
representation} of $( A,\widetilde alphapha)$.
In order to obtain a $J$-covariant representation we introduce the
following algebra of operators
$$
c_0(\mathbb N,J):=\{a=\widetilde bigoplus_{n\in \mathbb N}\pi(a_n): a_n \in
\widetilde alphapha^n(1)J\widetilde alphapha^n(1),\,\,\, \lim_{n\to \infty} a_n =0\} \subset L(\mathcal H),
$$
and consider the $C^*$-algebra $C^*(c_0(\mathbb N,J),\widetilde{U})$ generated by $c_0(\mathbb N,J)$ and $\widetilde{U}$. One checks that
$$
\widetilde{U}c_0(\mathbb N,J) \widetilde{U}^*\subset c_0(\mathbb N,J) , \widetilde qquad \widetilde{U}^*c_0(\mathbb N,J) \widetilde{U}\subset c_0(\mathbb N,J), \widetilde qquad \widetilde{U}^* \widetilde{U} \in c_0(\mathbb N,J)',
$$
and hence, see \cite[Prop 2.3]{Leb-Odz}, $C^*(c_0(\mathbb N,J),\widetilde{U})$ is the closure of elements of the form
$$
\widetilde{U}^{*n}a^{(-n)} + ...+ \widetilde{U}^*a^{(-1)}+a^{(0)} + a^{(1)}\widetilde{U} + ... +a^{(n)}\widetilde{U}^n,
$$
where $ a^{(k)} \in c_0(\mathbb N,J), k=0,\pm1,...,\pm n$. Thus using the relations
$$
\widetilde{\pi}(A)c_0(\mathbb N,J)\subset c_0(\mathbb N,J), \widetilde qquad \widetilde{\pi}(A)\widetilde{U}^*=\widetilde{U}^*\widetilde{\pi}(\widetilde alpha(A)), \widetilde qquad \widetilde{U}\widetilde{\pi}(A)=\widetilde{\pi}(\widetilde alpha(A))\widetilde{U}
$$
one sees that $C^*(c_0(\mathbb N,J),\widetilde{U})$ is an ideal in $C^*(\widetilde{\pi}( A),\widetilde{U})$. Let us now take any faithful non-degenerate representation of the quotient algebra
$$
\widetilde{\pi}_J:C^*(\widetilde{\pi}( A),\widetilde{U})/C^*(c_0(\mathbb N,J),\widetilde{U})\to L(H_J)
$$
and put
$$
U_J:=\widetilde{\pi}_J([\widetilde{U}]), \widetilde qquad
\pi_J (a):=\widetilde{\pi}_J ([\widetilde{\pi}(a)]), \ a\in A,
$$
where $
[\,\cdot\,]:C^*(\widetilde{\pi}( A),\widetilde{U})\to C^*(\widetilde{\pi}( A),\widetilde{U})/C^*(c_0(\mathbb N,J),\widetilde{U})
$ is the quotient mapping.
\widetilde begin{prop}\label{regular representation proposition}
For any ideal $J\subset(\ker\widetilde alphapha)^\widetilde bot$, the triple $(\pi_J,U_J,
H_J)$ defined above is a faithful $J$-covariant representation
of $( A,\widetilde alphapha)$.
\end{prop}
\widetilde begin{Proof}
That
$(\pi_J,U_J, H_J)$ is a representation of $( A,\widetilde alphapha)$ is straightforward. What we
need to verify is that it is faithful and $J$-covariant.
To see that $\pi_J$ is
faithful let $a\in \ker \pi_J\subset c_0(\mathbb N,J)$. Then $\widetilde alphapha^{n}(a)\in J$ for all $n=0,1,2...$, and $\pi(\widetilde alphapha^n(a))\to
0$. Since $J\cap\ker\widetilde alphapha=\{0\}$ the homomorphism $\widetilde alphapha: J\to A$ is isometric,
and as $\pi$ is a faithful representation of $ A$ we
get
$$
\|a\|=\lim_{n\to \infty}\|\pi(\widetilde alphapha^n(a))\|=0.
$$
To see that $\pi_J$ is $J$-covariant note that for any $a\in A$
$$
\widetilde{U}^*\widetilde{U}\widetilde{\pi} (a) - \widetilde{\pi} (a)= \{\pi(a),0,0,... \},
$$
and so, by the definition of $c_0(\mathbb N,J)$
$$
[\widetilde{U}^*\widetilde{U}\widetilde{\pi} (a) - \widetilde{\pi} (a)] = [0]\ \mathcal Leftrightarrow
\ a\in J.
$$
\end{Proof}
As an immediate corollary we have
\widetilde begin{thm}\label{existance theorem}
Let $( A,\widetilde alphapha)$ be a $C^*$-dynamical system and $J$ be an ideal in $ A$.
Then there exists a faithful $J$-covariant representation iff \ $\ker\widetilde alphapha\cap J=\{0\}$.
\end{thm}
\widetilde begin{Proof} Sufficiency follows from Proposition~\ref{regular representation proposition}
while Corollary~\ref{ort-id} implies necessity.
\end{Proof}
The foregoing observations make the next definition of the crossed product natural.
\widetilde begin{defn}\label{crossed product defn}
Let $( A,\widetilde alphapha)$ be a $C^*$-dynamical system and $J$ an ideal in $ A$ such that $J\cap \ker \widetilde alphapha=\{0\}$. The \emph{crossed product} $
C^*( A,\widetilde alphapha,J)
$ of $ A$ by $\widetilde alphapha$ associated with $J$ is a universal $C^*$-algebra generated by the the copy of the algebra $ A$ and a partial isometry $u$ subject to relations
$$
ua u^*=\widetilde alphapha(a),\,\,\,\widetilde qquad
J=\{a\in A: u^*u a=a\}.
$$
\end{defn}
The aim of the paper is the analysis of the crossed product
introduced. It will be shown that this construction covers the
main known constructions of crossed products. In addition we
present a thorough description of the internal structure of this
crossed product (Sections~\ref{22},~\ref{isomorph}) and discuss
its relation to the relative Cuntz-Pimsner algebras
(Section~\ref{C-P}). In particular, by means of the reduction
procedure (Section~\ref{Reduction and canonical}) we show that all
relative Cuntz-Pimsner algebras are associated with orthogonal ideals.
\section{Crossed product and matrix calculus}
\label{22}
This section presents an alternative
definition of the crossed product which provides us with one more
interesting, integral point of view leading to a transparent
description of its internal structure. One can also consider the construction described here as a sort of regular representation of the crossed product.
\subsection{Matrix calculus}
\label{2}
We introduce a matrix calculus that will be an algebraic framework for our crossed product.
Let us denote by $\mathcal M( A)$ the set of infinite matrices
$\{a_{i,j}\}_{i,j\in\mathbb N} $ indexed by pairs of
natural numbers with entries $a_{i,j}$ in $ A$ such that
$$ a_{i,j}\in \widetilde alphapha^i(1) A \widetilde alphapha^j(1), \widetilde qquad i, j\in \mathbb N,$$
and there is at most finite number of $a_{i,j}$ which are non-zero.
We will take advantage of this standard matrix notation when
defining operations on $\mathcal M( A)$ and investigating a natural
homomorphism from $\mathcal M( A)$ to the covariance algebras generated by
representations of $(A, \widetilde alphapha)$. However, when calculating norms of elements
in covariance algebras, it is more handy to index the entries of
an element in $\mathcal M( A)$ by a pair consisting of a natural number
and an integer. Hence we will paralellely use two notations
concerning matrices in $\mathcal M( A)$. Namely, we presume the following
identifications
$$
a_{i,j}=a^{(j-i)}_{\min\{i,j\}},\widetilde quad i,j \in \mathbb N,\widetilde quad \widetilde qquad a_{n}^{(k)}=\widetilde begin{cases}
a_{n,k+n},& k\geq 0\\
a_{n-k,n},& k < 0
\end{cases}\widetilde quad n \in \mathbb N,\,\, k\in \mathbb Z,
$$
under which we have two equivalent matrix presentations
$$
\left(
\widetilde begin{array}{cc c c}
a_{0,0} & a_{0,1} & a_{0,2} & \cdots \\
a_{1,0} & a_{1,1} & a_{1,2} & \cdots \\
a_{2,0} & a_{2,1} & a_{2,2} & \cdots \\
\vdots & \vdots & \vdots & \ddots
\end{array}\right)
=
\left(
\widetilde begin{array}{cc c c}
a_0^{(0)} & a_0^{(1)} & a_0^{(2)} & \cdots \\
a_0^{(-1)} & a_1^{(0)} & a_1^{(1)} & \cdots \\
a_0^{(-2)} & a_1^{(-1)} & a_2^{(0)} & \cdots \\
\vdots & \vdots & \vdots & \ddots
\end{array}\right).
$$
The use of each of these conventions will always be clear from the context.
We define the
addition, multiplication by scalar, and
involution on $\mathcal M( A)$ in a natural manner. Namely, for $a=\{a_{ij}\}_{i,j\in\mathbb N}$ and $b=\{b_{ij}\}_{i,j\in\mathbb N}$ in $\mathcal M( A)$
we put
\widetilde begin{gather}\label{add1}
(a+b)_{m,n}=a_{m,n}+b_{m,n},\\[6pt]
\label{mulscal1}
(\lambda a)_{m,n}=\lambda a_{m,n}\\[6pt]
\label{invol1}
(a^*)_{m,n}=a_{n,m}^*.
\end{gather}
Moreover, we introduce a convolution multiplication '$\star$' on
$\mathcal M( A)$, which is a reflection of the operator multiplication in covariance algebras.
We set
\widetilde begin{equation}\label{star1}
a\star b= a\cdot\sum_{j=0}^\infty \mathcal Lambda^j(b)+ \sum_{j=1}^\infty \mathcal Lambda^j(a)\cdot b
\end{equation}
where $\cdot$ is the standard multiplication of matrices and mapping $\mathcal Lambda:
\mathcal M( A)
\rightarrow \mathcal M( A)$ is defined
to act as follows: $\mathcal Lambda(a)_{i,j}=\widetilde alphapha(a_{i-1,j-1})$, for $i,j> 0$, and $\mathcal Lambda(a)_{i,j}=0$ otherwise,
that is $\mathcal Lambda$ assumes the following shape
\widetilde begin{equation}\label{Lambda}
\mathcal Lambda(a)= \left(
\widetilde begin{array}{cc c c c }
0 & 0 & 0 & 0 & \cdots \\
0 & \widetilde alphapha(a_{0,0}) & \widetilde alphapha(a_{0,1}) & \widetilde alphapha(a_{0,2}) & \cdots \\
0 & \widetilde alphapha(a_{1,0}) & \widetilde alphapha(a_{1,1}) & \widetilde alphapha(a_{1,2}) & \cdots \\
0 & \widetilde alphapha(a_{2,0}) & \widetilde alphapha(a_{2,1}) & \widetilde alphapha(a_{2,2}) & \cdots \\
\vdots & \vdots & \vdots & \vdots & \ddots
\end{array}\right).
\end{equation}
\widetilde begin{prop}
The set $\mathcal M( A)$ with operations \eqref{add1},
\eqref{mulscal1}, \eqref{invol1}, \eqref{star1} becomes an algebra with involution.
\end{prop}
\widetilde begin{Proof}
The only thing we show is associativity of multiplication \eqref{star1}, the rest is straightforward.
For that purpose we note that $\mathcal Lambda$ preserves the standard matrix multiplication and thus we have
$$
a\star (b \star c)= a\star Big(b\cdot\sum_{j=0}^\infty \mathcal Lambda^j(c)+ \sum_{j=1}^\infty \mathcal Lambda^j(b)\cdot cBig)
$$
$$
=a\sum_{k=0}^\infty \mathcal Lambda^kBig( b\sum_{j=0}^\infty \mathcal Lambda^j(c)+
\sum_{j=1}^\infty \mathcal Lambda^j(b) c Big)+ \sum_{k=1}^\infty \mathcal Lambda^k(a) Big( b\sum_{j=0}^\infty \mathcal Lambda^j(c)+
\sum_{j=1}^\infty \mathcal Lambda^j(b) c
Big)$$
$$
=\sum_{k,j=0}^\infty a \mathcal Lambda^k( b)\mathcal Lambda^{j}(c) + \sum_{k=1,j=0}^\infty \mathcal Lambda^{k}(a) b\cdot \mathcal Lambda^j(c) +
\sum_{k,j=1}^\infty \mathcal Lambda^{k}(a)\mathcal Lambda^{j}(b) c
$$
$$
=Big(a\sum_{k=0}^\infty \mathcal Lambda^k(b)+ \sum_{k=1}^\infty \mathcal Lambda^k(a) bBig)\sum_{j=0}^\infty \mathcal Lambda^j(c) +
\sum_{j=1}^\infty \mathcal Lambda^j Big(a\sum_{k=0}^\infty \mathcal Lambda^k(b)+ \sum_{k=1}^\infty \mathcal Lambda^k(a) bBig) c
$$
$$
Big(a\cdot\sum_{k=0}^\infty \mathcal Lambda^k(b)+ \sum_{k=1}^\infty \mathcal Lambda^k(a)\cdot bBig)\star c= (a\star b) \star c.
$$
\end{Proof}
We embed $ A$ into $\mathcal M( A)$ by identifying an element $a\in A$ with
the matrix $\{a_{m,n}\}_{m,n\in \mathbb N}$ where $a_{0,0}=a$ and $a_{m,n}=0$
if $(m,n)\neq (0,0)$. We also define a 'partial isometry'
$u=\{u_{m,n}\}_{m,n\in \mathbb N}$ in $\mathcal M( A)$ such that
$u_{0,1}=\widetilde alphapha(1)$ and $u_{m,n}=0$ if $(m,n)\neq (0,1)$. In other
words, we adopt the following notation
\widetilde begin{equation}
\label{notation stupid} u= \left(
\widetilde begin{array}{cc c c}
0 & \widetilde alphapha(1) & 0 & \cdots \\
0 & 0 & 0 & \cdots \\
0 & 0 & 0 & \cdots \\
\vdots & \vdots & \vdots & \ddots
\end{array}\right)\widetilde quad \mathrm{and \,\,} \widetilde quad
a=\left(\widetilde begin{array}{cc c c}
a & 0 & 0 & \cdots \\
0 & 0 & 0 & \cdots \\
0 & 0 & 0 & \cdots \\
\vdots & \vdots & \vdots &\ddots
\end{array}\right), \widetilde quad \mathrm{for \,\,} a\in A.
\end{equation}
One can check that the $^*$-algebra $\mathcal M( A)$ is generated
by $u$ and $ A$. Furthermore, for every $a\in A$ we have
$$
u\star a \star u^* =\widetilde alphapha(a) \widetilde quad\textrm{ and } \widetilde quad u^*\star a \star u = \left(
\widetilde begin{array}{cc c c}
0 & 0 & 0 & \cdots \\
0 & \widetilde alphapha(1)a\widetilde alphapha(1) & 0 & \cdots \\
0 & 0 & 0 & \cdots \\
\vdots & \vdots & \vdots & \ddots
\end{array}\right).
$$
\widetilde begin{prop}\label{dense subalgera structure prop}
Let $(\pi,U,H)$ be a representation of $( A,\widetilde alphapha)$. Then there
exists a unique $^*$-homomorphism $\Psi_{(\pi, U)}$ from $\mathcal M( A)$
onto a $^*$-algebra $C_0^*(\pi( A),U)$ generated by $\pi( A)$ and
$U$, such that
$$
\Psi_{(\pi, U)}(a)=\pi(a),\widetilde quad a\in A,\widetilde qquad \Psi_{(\pi, U)}(u)=U.
$$
Moreover, $\Psi_{(\pi, U)}$ is given by the formula
\widetilde begin{equation}\label{Psi form eq}
\Psi_{(\pi, U)}( \{a_{m,n}\}_{m,n\in\mathbb N})=\sum_{m,n=0}^\infty U^{*m}\pi(a_{m,n}) U^n,
\end{equation}
and thus $C_0^*(\pi( A),U)=\left\{\sum_{m,n=0}^\infty U^{*m}\pi(a_{m,n}) U^n: \{a_{m,n}\}_{m,n\in\mathbb N}\in \mathcal M( A)\right\}$.
\end{prop}
\widetilde begin{Proof} It is clear that $\Psi_{(\pi, U)}$ has to satisfy \eqref{Psi form eq}.
Thus it is enough to check that $\Psi_{(\pi, U)}$ is a $^*$-homomorphism, and in fact we only need to show
that $\Psi_{(\pi, U)}$ is multiplicative as the rest is obvious. For that purpose
let us fix two matrices $a=\{a_{m,n}\}_{m,n\in\mathbb N},b=\{b_{m,n}\}_{m,n\in\mathbb N}\in \mathcal M( A)$. We will examine the product
$$
c_{p,r,s,t}=U^{*p}\pi(a_{p,r})U^{r} U^{*s}\pi(b_{s,t} )U^{t}
$$
Depending on the relationship between $r$ and $s$ we have two cases.
\widetilde begin{Item}{1)}
If $s \leq r$, then using equality $(U^s U^{*s})\pi(b_{s,t})=\pi(\widetilde alpha^s(1)b_{s,t})=\pi(b_{s,t}) $ and relation $U^{*r-s} U^{r-s}\in \pi(A)' $, see Lemma \ref{iteration of representations}, we get
$$
c_{p,r,s,t}=U^{*p}\pi(a_{p,r})U^{r-s} (U^s U^{*s})\pi(b_{s,t}) U^t=
U^{*p}\pi(a_{p,r})U^{r-s} U^{*r-s} U^{r-s}\pi(b_{s,t}) U^t
$$
$$
=U^{*p}\pi(a_{p,r})U^{r-s} \pi(b_{s,t})(U^{*r-s} U^{r-s}) U^t=U^{*p}\pi(a_{p,r} \widetilde alphapha^{r-s}(b_{s,t})) U^{t+r-s}
$$
Putting $r-s=j$, $r=i$, $p=m$ and $t+r-s =n$ we get
$$
c_{m,r,s,n-r+s}=c_{p,r,s,t}=U^m \pi(a_{m,i} \widetilde alphapha^{j}(b_{i-j,n-j})) U^{n}
$$
and thus
$$
\sum_{s,r \in \mathbb N \widetilde atop s \leq r} c_{m,r,s,n-r+s}=
\sum_{j=0}^\infty\sum_{i=j}^\infty U^m \pi(a_{m,i} \widetilde alphapha^{j}(b_{i-j,n-j})) U^{n}=
U^m \pi(\widetilde big(a\cdot\sum_{j=0}^\infty \mathcal Lambda^j(b)\widetilde big)_{m,n}) U^{n}
$$
\end{Item}
\widetilde begin{Item}{2)}
If $r <s$, then analogously
$$
c_{p,r,s,t}=U^{*p} \pi(a_{p,r})(U^r U^{*r}) U^{*s-r}\pi(b_{s,t}) U^l=
U^{*p}\pi( a_{p,r})(U^{*s-r}U^{r-s})U^{*r-s}\pi(b_{s,t}) U^t
$$
$$
=U^{*p} (U^{*s-r}U^{s-r})\pi(a_{p,r})U^{*s-r}\pi(b_{s,t} )U^t=U^{*p+s-r} \pi(\widetilde alphapha^{s-r}(a_{p,r})b_{s,t}) U^t
$$
Putting $s-r=j$, $r=i$, $p +s-r=m$ and $t=n$ we get
$$
c_{m-s+r,r,s,n}=c_{p,r,s,t}=U^m\pi( \widetilde alphapha^{j}(a_{m-j,i-j})b_{i,n} ) U^{n}
$$
and thus
$$
\sum_{s,r \in \mathbb N \widetilde atop r< s} c_{m-s+r,r,s,n}=
\sum_{j=1}^\infty\sum_{i=j}^\infty U^m \pi(\widetilde alphapha^{j}(a_{m-j,i-j})b_{i,n} ) U^{n}=
U^m \pi(\widetilde big(\sum_{j=1}^\infty \mathcal Lambda^j(a)\cdot b\widetilde big)_{m,n}) U^{n}
$$
\end{Item}
\noindent Using the formulas obtained in 1) and 2) we have
$$
\Psi_{(\pi, U)}(a) \Psi_{(\pi, U)}( b)= \sum_{p,r,s,t\in \mathbb N} c_{p,r,s,t}=
\sum_{p,r,s,t\in \mathbb N \widetilde atop s\leq r}c_{p,r,s,t} + \sum_{p,r,s,t\in \mathbb N \widetilde atop r<s} c_{p,r,s,t}$$
$$
=\sum_{m,r,s,n\in \mathbb N \widetilde atop s\leq r,\, n\leq r-s}c_{m,r,s,n-r+s} +
\sum_{m,r,s,n\in \mathbb N \widetilde atop r<s,\,m\leq s-r} c_{m-s+r,r,s,n}=
\sum_{m,n\in \mathbb N} U^{*m} (a\star b)_{m,n} U^n =\Psi_{(\pi, U)}(a\star b)
$$
and the proof is complete.
\end{Proof}
We now examine the structure of $\mathcal M( A)$.
We will say that a matrix $\{a_{n}^{(m)}\}_{n\in\mathbb N, m\in \mathbb Z}$ in $\mathcal M( A)$ is $k$-\emph{diagonal},
where $k$ is an integer, if it satisfies the condition
$$
a_{n}^{(m)}\neq 0 \mathcal Longrightarrow\,\, m=k.
$$
In other words $k$-diagonal matrix is the one of the form
\widetilde begin{center}\setlength{\unitlength}{1mm}
\widetilde begin{picture}(110,24)(-5,-10)
\put(-10,0){$
\left(\widetilde begin{array}{c}\widetilde begin{xy}
\widetilde xymatrix@C=-1pt@R=3pt{
& \, \, & \widetilde qquad \,\,0 \, \\
& \, & \\
& \, 0 & \\
& & \widetilde qquad
}
\end{xy}
\end{array}\right)
$ }
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\put(29,0){if $k\geq 0$, or}
\put(98,0){if $k< 0$.}
\put(61,0){$
\left(\widetilde begin{array}{c}\widetilde begin{xy}
\widetilde xymatrix@C=-1pt@R=4pt{
& \, \, & \, \\
& \, & 0 \\
& \, & \\
0 \,\, & & \widetilde qquad
}
\end{xy}
\end{array}\right)$}
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\end{picture}
\end{center}
The linear space consisting of all $k$-diagonal matrices will be denoted by $\mathcal{M}_{k}$.
These spaces will correspond to spectral subspaces, see Corollaries \ref{wniosek o spektralnych podprzestrzeniach} and \ref{spectral-sp}. We write $\mathcal M_{k}\star\mathcal{M}_{l}$ for the linear span of elements $a \star b$, $a\in \mathcal M_k$, $b\in \mathcal M_l$.
\widetilde begin{prop}
\label{**}
The spaces $\mathcal M_k$ define a $\mathbb Z$-graded algebra structure on $\mathcal M( A)$. Namely
$$
\mathcal M( A)=\widetilde bigoplus_{k\in \mathbb Z} \mathcal M_k,$$
and for every $k$, $l\in \mathbb Z$ we have the following relations
$$
\mathcal M_{k}^*=\mathcal M_{-k},\widetilde qquad\mathcal M_{k}\star\mathcal{M}_{l}\subset \mathcal M_{k+l}.
$$
In particular, $\mathcal M_{0}$ is a $^*$-algebra, $\mathcal M_{k}\star \mathcal M_{-k}$
is a self-adjoint two sided ideal in $\mathcal M_{0}$.
\end{prop}
\label{coef-alg}
\widetilde begin{Proof}
Relations $\mathcal M_k^*=\mathcal M_{-k}$, $\mathcal M_k\star \mathcal M_l\subset \mathcal M_{k+l}$ and
$(\mathcal M_k \star \mathcal M_k^*)^*=(\mathcal M_k \star \mathcal M_k^*)$ can be checked by
means of an elementary matrix calculus. Using these relations we
get
$$
\mathcal M_0 \star (\mathcal M_k \star \mathcal M_k^*)= (\mathcal M_0 \star \mathcal M_k)\star \mathcal M_k^*\subset \mathcal M_k\star \mathcal M_k^* ,
$$
$$
(\mathcal M_k\star \mathcal M_k^*)\star \mathcal M_0 = \mathcal M_k\star (\mathcal M_k^*\star
\mathcal M_0)\subset (\mathcal M_k \star \mathcal M_k^*),
$$
and thus $\mathcal M_k\star \mathcal M_k^*$ is an ideal in $\mathcal M_0$.
\end{Proof}
Proposition \ref{**} indicates in particular that $\mathcal M_0$ may be regarded as a coefficient algebra
in the sense of \cite{Leb-Odz} for $\mathcal M( A)$.
This will be shown explicitly in Proposition~\ref{nie dam rady umre}.
\widetilde begin{cor}\label{wniosek o spektralnych podprzestrzeniach}
Let $(\pi,U,H)$ be a representation of $( A,\widetilde alphapha)$ and let
$
B_k= \Psi_{(\pi, U)}(\mathcal M_k)
$
be the linear space consisting of the elements of the form
$$
\sum_{n=0}^N U^{*n} \pi(a_{n}^{(k)}) U^{n+k}, \widetilde quad \textrm{ if }\,\, k \geq 0, \widetilde quad \textrm{ or}
\widetilde quad \sum_{n=0}^N U^{*n+|k|} \pi(a_{n}^{(k)})U^{n}, \widetilde quad \textrm{ if }\,\, k < 0.
$$
Then for every $k$ and $l\in \mathbb Z$ we have the following relations
$$
B_k^*=B_{-k},\widetilde qquad B_kB_l\subset B_{k+l}
$$
In particular, $B_0$ is a $C^*$-algebra, $B_k B_k^*$ is a self-adjoint two sided ideal in $B_0$.
\end{cor}
The importance of $B_0$ was observed in \cite{Leb-Odz} and is clarified by the next proposition,
see \cite[Proposition 2.4]{Leb-Odz}.
\widetilde begin{prop}
Let $(\pi,U,H)$ be a representation of $( A,\widetilde alphapha)$ and adopt
the notation from Proposition \ref{dense subalgera structure prop} and
Corollary \ref{wniosek o spektralnych podprzestrzeniach}. Every
element $a \in C_0^*(\pi( A),U)$ can be presented in the form
$$
a= \sum_{k=1}^{\infty} U^{*k} a_{-k} + \sum_{k=0}^{\infty} a_{k}U^{*k}
$$
where $a_{-k} \in B_0\pi(\widetilde alphapha^k(1))$, $a_{k} \in \pi(\widetilde alphapha^k(1)) B_0$, $k\in \mathbb N$,
and only finite number of these coefficients are non-zero.
\end{prop}
We will now formulate a similar result concerning $\mathcal M( A)$. For
each $k\in \mathbb Z$ we define a mapping $\mathbb NN_k:\mathcal M( A)\to \mathcal M_0$, $k\in\mathbb Z$,
that carries the $k$-diagonal onto a $0$-diagonal and delete all the
remaining ones. Namely, for $a = \{ a_{n}^{(k)} \}$ we set
$$
\left[\mathbb NN_k (a)\right]_{n}^{(m)}=
\widetilde begin{cases}
a_{n}^{(k)} & \textrm{ if } m=0,\\
0 & \textrm{ otherwise },
\end{cases} \widetilde qquad \widetilde quad k \in \mathbb Z.
$$
One readily checks that for $k\geq 0$ we have $\mathbb NN_k(\mathcal M_k)
=\mathcal M_0\star \widetilde alphapha^k(1)$, $\mathbb NN_{-k}(\mathcal M_{-k})= \widetilde alphapha^k(1) \star
\mathcal M_0$. Thus the algebra $\mathcal M_0$ consists of elements
that play the role of Fourier coefficients in $\mathcal M( A)$.
\widetilde begin{prop}\label{nie dam rady umre}
Every element $a$ of $\mathcal M( A)$ is uniquely presented in the form
$$
a= \sum_{k=1}^{\infty} u^{*k}\star a_{-k} + \sum_{k=0}^{\infty} a_{k}\star u^{*k}
$$
where $u$ is given by (\ref{notation stupid}) and
$a_{-k} \in \mathcal M_0\star \widetilde alphapha^k(1)$, $a_{k}
\in \widetilde alphapha^k(1) \star \mathcal M_0$, $k\in \mathbb N$, and only finite number of
these coefficients is non-zero. Namely, $a_k=\mathbb NN_k(a)$ for $k\in
\mathbb Z$.
\end{prop}
\subsection{Norm evaluation of elements in $C_0^*(\pi( A),U)$}\label{norm}
In this subsection we gather a number of technical results
concerning norm evaluation of elements in $C_0^*(\pi( A),U)$. We
will make use of these results in the sequel.
The mappings $\mathbb NN_k:\mathcal M_k \to \mathcal M_0$ factor through $\Phi_{(\pi,U)}$ to the mappings \mbox{$N_k:B_k\to B_0$.}
\widetilde begin{prop}\label{proposition for k-diagonals}
Let $(\pi,U,H)$ be a representation of $( A,\widetilde alphapha)$ and let
$k\in \mathbb Z$. Then the norm $\|a\|$ of an element $a\in B_k$
corresponding to the matrix $\{a_{n}^{(m)}\}_{n\in \mathbb N, m\in \mathbb Z}$
in $\mathcal M_k$ is given by
$$
\lim_{n\to \infty} \max \left\{\max_{i=1,...,n}Big \|(1-U^*U)\sum_{j=0}^{i}
\pi(\widetilde alphapha^{i-j}(a_{j}^{(k)}))Big\|,\, Big\|U^*U \pi(a_{n}^{(k)})Big\| \right\}.
$$
In particular, the mapping $N_k:B_k\to B_0$ given by
$$
N_k (\Phi_{(\pi,U)}(a))=\Phi_{(\pi,U)}(\mathbb NN_k (a)), \widetilde qquad a\in \mathcal M_k,
$$
is a well defined linear isometry establishing the following isometric isomorphisms
$$
B_k\cong B_0\widetilde alphapha^k(1),\widetilde quad \textrm{ if }\,\, k\geq 0,\widetilde qquad B_{k}\cong \widetilde alphapha^{|k|}(1) B_0,\widetilde quad
\textrm{ if }\,\, k < 0.
$$
\end{prop}
\widetilde begin{Proof} Let us assume that $k \geq 0$.
Let $N$ be such that $a_{m}^{(k)}=0$ for $m>N$, that is
$$
a= \sum_{m=0}^{N} U^{*m}\pi(a_{m}^{(k)})U^{m+k}.
$$ Then, similarly as it was done in the proof of \cite[Prop. 3.1]{kwa4},
one sees that defining
$$
a_i=(1-U^*U) \pi\left(\sum_{j=0}^{i} \widetilde alphapha^{i-j}(a_{j}^{(k)})\right), \widetilde quad i=0,...,N,\widetilde qquad a_{N+1}= U^*U \pi(a_{N}^{(k)}),
$$
we have
$$
a = \widetilde big(a_0 +U^*a_1U +... + U^{*N}(a_N + a_{N+1})U^N\widetilde big) U^k
$$
and
$$
a_i\in (1-U^*U)\pi(\widetilde alphapha^i(1) A \widetilde alphapha^{i+k}(1)), \widetilde quad i=0,...,N,
\widetilde quad a_{N+1} \in U^*U\pi(\widetilde alphapha^N(1) A
\widetilde alphapha^{N+k}(1)) ,
$$
Hence it follows that
$$
U^{*i} a_i U^{i} \in (U^{*i}U^{i}-U^{*i+1}U^{i+1})\pi(A)U^kU^{*k}, \widetilde quad i=0,...,N,
$$
and
$$
U^{*N}a_{N+1}U^N \in U^{*N+1}U^{N+1}\pi( A)U^kU^{*k}
$$
These relations and the fact that $U^{*i}U^{i}-U^{*i+1}U^{i+1}$, $i=0,..,N$, and $U^{*N+1}U^{N+1}$ are
pairwise orthogonal projections lying in $\pi( A)'$ (cf. Lemma \ref{iteration of representations} or better \cite[Prop 3.6]{Leb-Odz}),
imply the following equalities
\widetilde begin{align*}
\|a\|&=\| (a_0 +U^*a_1U +... + U^{*N}(a_N+ a_{N+1}) U^N) U^kU^{*k}\|
\\
& =\|a_0 +U^*a_1U +... + U^{*N}(a_N+ a_{N+1})U^N\|
\\
&=\max \{\max_{i=0,...,N}\|U^{*i}a_iU^i\|, \|U^{*N}(a_{N+1})U^N\|\}
\\
&=\max \{\max_{i=0,...,N}\|a_i\|, \|a_{N+1}\|\}
\end{align*}
where the final equality follows from that the linear mapping
$ a \to U^{*n} a U^{n}$ is isometric on $\pi(\widetilde alpha^{n}(1)A\widetilde alpha^{n}(1))=U^nU^{*n} \pi(A)U^nU^{*n}$, $n\in \mathbb N$. Since $N$ was arbitrary (sufficiently large) this proves the assertion in the case when $k \geq 0$.
In the case of negative $k$ one may apply the part of proposition proved above to the adjoint $a^*$
of the element $a$ and thus obtain the hypotheses.
\end{Proof}
We denote by
$$
d(a, K)=\inf_{b\in K} \|a-b\|
$$
the usual distance of an element $a$ from the set $K$. The
definition of an ideal of covariance (Definition \ref{kowariant
rep defn 2}) and a known fact expressing quotient norms in terms
of projections, see for instance, \cite[Lemma 10.1.6]{Kadison}
gives us the following
\widetilde begin{cor}\label{corollary with distance}
If $(\pi,U,H)$ is a faithful $J$-covariant representation of
$( A,\widetilde alphapha)$ and $I$ denotes the kernel of $\widetilde alphapha$, then the
norm of an element $a\in B_k$ corresponding to a matrix
$\{a_{n}^{(m)}\}_{n\in \mathbb N, m\in \mathbb Z}$ in $\mathcal M_k$ is given by
$$
\|a\|=\lim_{n\to \infty} \max \left\{\max_{i=1,...,n}\widetilde big\{ d\widetilde big(\sum_{j=0}^{i}
\widetilde alphapha^{i-j}(a_{j}^{(k)}),J\widetilde big)\widetilde big\},\, d(a_{n}^{(k)},I) \right\}.
$$
In particular, if $J$ is a fixed ideal orthogonal to $I$ and
$(\pi,U,H)$ is a faithful $J$-covariant representation, then the
spaces $B_k$, $k\in\mathbb Z$ do not depend on its choice.
\end{cor}
We showed in Proposition \ref{proposition for k-diagonals} that
for an arbitrary representation $(\pi,U,H)$ of $(A,\widetilde alphapha)$, the
mappings $\mathbb NN_k$ factor through to the mappings $N_k$ acting on
spaces $B_k$. In general, however,
$\mathbb NN_k:\mathcal M( A)\to B_0$ do not factor through $\Psi_{(\pi,U)}$ to the mappings acting on
the algebra $C_0^*(\pi( A),U)$. In fact, this is the case if and only if the representation $(\pi,U,H)$
satisfies a certain property we are just about to introduce.
\widetilde begin{defn}
\label{*} We will say that a faithful representation $(\pi,U,H)$
of $( A,\widetilde alphapha)$ possesses {\em property} $(*)$ if for any $a\in
C_0^*(\pi( A),U)$ given by a matrix
$\{a_{mn}\}_{m,n\in\mathbb N}\in\mathcal M( A)$ the inequality
$$
\|\sum_{m\in\mathbb N} U^{*m}\pi(a_{m,m})U^m \| \leq \|\sum_{m,n\in\mathbb N} U^{*m}\pi(a_{m,n})U^n \|, \widetilde qquad\widetilde qquad (*)
$$
holds. In view of Corollary \ref{corollary with distance} the above equality could be equivalently stated
in the form $$
\lim_{n\to \infty} \max \left\{\max_{i=1,...,n}\widetilde big\{ d\widetilde big(\sum_{j=0}^{i}
\widetilde alphapha^{i-j}(a_{j,j}),J\widetilde big)\widetilde big\},\, d(a_{n,n},I) \right\} \leq \|a\|,\widetilde quad (*)
$$
where $I$ is the kernel of $\widetilde alphapha$ and $J$ is the ideal of
covariance of $(\pi,U,H)$.
\end{defn}
The next result, which follow immediately from \cite[Thm. 2.8]{Leb-Odz}, indicates that
under the fulfillment of property~(*) elements of $B_0$ play the
role of 'Fourier' coefficients in the algebra $C_0^*(\pi( A),U)$.
\widetilde begin{thm}
Let $(\pi,U,H)$ be a faithful representation of $(A,\widetilde alphapha)$
possessing property $(*)$ then the mappings $N_k:
C_0^*(\pi( A),U)\to B_0$, $k\in \mathbb Z$, given by formulae
\widetilde begin{equation}\label{rzuty na coefficienty}
N_k( \Phi_{(\pi,U)}(a)) = \sum_{n\in\mathbb N} U^{*n}\pi(a_{n}^{(k)})U^{n},
\end{equation}
where $\{a_{n}^{(m)}\}_{n\in \mathbb N, m\in \mathbb Z}\in \mathcal M( A)$, are well defined contractions and thus
they extend uniquely to bounded operators on $C^*(\pi( A),U)$. In particular,
every element $a \in C_0^*(\pi( A),U)$ can be uniquely presented in the form
$$
a= \sum_{k=1}^{\infty} U^{*k} a_{-k} + \sum_{k=0}^{\infty} a_{k}U^{*k}
$$
where $a_{-k} \in B_0\pi(\widetilde alphapha^k(1))$, $a_{k} \in \pi(\widetilde alphapha^k(1)) B_0$, $k\in \mathbb N$,
namely, $a_k=N_k(a)$, $k\in \mathbb Z$.
\end{thm}
Let us also recall \cite[Thm. 2.11]{Leb-Odz}.
\widetilde begin{thm}\label{3a.N}
If $(\pi,U,H)$ possesses property $(*)$, then for any element $a$ in $C_0^*(\pi( A),U)$ we have
\widetilde begin{equation}\label{be3.131}
\widetilde Vert a \widetilde Vert = \lim_{k\to\infty}
\sqrt[\leftroot{-2}\uproot{1}\scriptstyle 4k]{
\left\widetilde Vert N_0 \left[ (aa^*)^{2k}\right]\right\widetilde Vert }
\end{equation}
where $N_0$ is the mapping defined by \eqref{rzuty na coefficienty}.
\end{thm}
Using the above results one sees that in the presence of property $(*)$ the norm of an
element $a\in C_0^*(\pi( A),U)$ may be calculated only in terms of the elements
of $ A$. Indeed, as $N_0 \left[ (aa^*)^{2k}\right]$ belongs to $B_0$ one can apply Corollary
\ref{corollary with distance} to calculate $\| N_0 \left[ (aa^*)^{2k}\right]\|$ in terms of
the matrix from $\mathcal M( A)$ corresponding to $a$. However in practice, calculation of
the matrix corresponding to the element $(aa^*)^{2k}$ starting from $a$, see formula \eqref{star1},
seems to be an extremely difficult task.
\subsection{Crossed product defined by matrix calculus}
\label{cr-matr}
The foregoing observations make it now possible to give one more 'internal' definition of the crossed product. This is the aim of the present subsection.
The set $\mathcal M( A)$ with operations \eqref{add1},
\eqref{mulscal1}, \eqref{invol1}, \eqref{star1} is an algebra with involution. We define
a seminorm on $\mathcal M( A)$ that will depend on the choice of an orthogonal ideal. Let $J$ be
a fixed ideal in $ A$ having zero intersection with the kernel of $\widetilde alphapha$.
Let
$$
\|| a\||_J:=\sum_{k\in \mathbb Z} \lim_{n\to \infty} \max \left\{\max_{i=1,...,n}\widetilde big\{ d\widetilde big(\sum_{j=0}^{i}
\widetilde alphapha^{i-j}(a_{j}^{(k)}),J\widetilde big)\widetilde big\},\, d(a_{n}^{(k)},I) \right\}
$$
where $a=\{a_{n}^{(k)}\}_{n\in \mathbb N,k\in \mathbb Z}\in \mathcal M( A)$.
\widetilde begin{prop}
The function $\|| \cdot \||_J$ defined above is a seminorm on $\mathcal M( A)$ which is $^*$-invariant and submulitplicative.
\end{prop}
\widetilde begin{Proof}
Let $(\pi,U,H)$ be a faithful representation of $( A,\widetilde alphapha)$ and
$J$ be its covariance ideal. Such representation does exist by Theorem \ref{existance
theorem}. Then in view of Corollary \ref{corollary with distance}
for every $a\in \mathcal M_k$, $k\in \mathbb Z$, we have
$$
\|| a\||= \| \Phi_{(\pi,U)}(a)\|
$$
where $\Phi_{(\pi,U)}:\mathcal M( A)\to L(H)$ is the $^*$-homomorphism defined in
Proposition~\ref{dense subalgera structure prop}. Thus since every element $a\in \mathcal M( A)$ can be presented in the
form $a= \sum_{k\in \mathbb Z} a^{(k)}$ where $a^{(k)}\in \mathcal M_k$ one easily sees
that $\||| \cdot \|||$ is $^*$-invariant seminorm. To show that it is
submultiplicative take $a= \sum_{k\in \mathbb Z} a^{(k)}\in \mathcal M( A)$ and $b= \sum_{k\in \mathbb Z} b^{(k)}\in \mathcal M( A)$
such that $a^{(k)}, b^{(k)}\in \mathcal M_k$. Then
$$
\|| a \star b \|| = \||\sum_{k\in \mathbb Z} a^{(k)}\star \sum_{l\in \mathbb Z} b^{(l)}\||=\||\sum_{k\in \mathbb Z}
\sum_{l\in \mathbb Z} a^{(k)}\star b^{(l)}\||\leq \sum_{k,l\in \mathbb Z} \||a^{(k)}\star b^{(l)}\||
$$
$$
= \sum_{k,l\in \mathbb Z} \|\Phi_{(\pi,U)}(a^{(k)}\star b^{(l)})\| \leq \sum_{k,l\in \mathbb Z} \|\Phi_{(\pi,U)}(a^{(k)})\|
\cdot \| \Phi_{(\pi,U)}(b^{(l)})\|
$$
$$
= \sum_{k,l\in \mathbb Z} \||a^{(k)} \|| \cdot \||b^{(l)}\||
=\sum_{k \in\mathbb Z} \||a^{(k)}\|| \sum_{l\in \mathbb Z} \||b^{(l)}\|| =\||a\|| \cdot\|| b\||.
$$
\end{Proof}
\widetilde begin{defn}\label{cr-pr-def}
Let $( A,\widetilde alphapha)$ be a $C^*$-dynamical system and $J$ an ideal in $ A$ having zero intersection with the kernel
of $\widetilde alphapha$. The \emph{crossed product} $
C^*( A,\widetilde alphapha,J)
$ of $ A$ by $\widetilde alphapha$ associated with $J$ is the enveloping $C^*$-algebra of the quotient $^*$-algebra $\mathcal M( A)/ \|| \cdot \||_J$.
\end{defn}
Regardless of $J$, composing the quotient map with natural embedding of $ A$ into $\mathcal M( A)$
one has an embedding of $ A$ into $C^*( A,\widetilde alphapha,J)$. Moreover, denoting by $\hat{u}$
an element of $C^*( A,\widetilde alphapha,J)$ corresponding to $u\in \mathcal M( A)$ (see (\ref{notation stupid})),
one sees that $C^*( A,\widetilde alphapha,J)$ is generated by $ A$ and $\hat{u}$.
The equivalence of this definition and that introduced previously (Definition~\ref{crossed product defn}) will be established in the next section (Proposition~\ref{crossed equiv}).
\section{Isomorphism theorems and faithful representations}\label{isomorph}
Once a universal object (the crossed product) is defined it is reasonable to have its faithful representation.
This section is devoted to the description of the properties of such representations and, in particular, we establish the equivalence of two previously mentioned definitions of the crossed product. In addition we present one more alternative crossed product construction based on \cite{Ant-Bakht-Leb} approach and prove faithfulness by means of the topologically free action on the arising coefficient algebras.
\subsection{Isomorphism Theorem}\label{isomor-theor}
\widetilde begin{thm}[\textbf{Isomorphism Theorem}]\label{isomorphiasm theorem}
\label{iso} Let $J$ be an ideal in $ A$ having zero intersection
with the kernel $I$ of\, $\widetilde alphapha$ and let $(\pi_i,U_i,H_i)$,
$i=1,2$, be faithful $J$-covariant representations of
$( A,\widetilde alphapha)$ possessing property $(*)$. Then the relations
$$ \Phi(\pi_1(a)):=\pi_2(a),\widetilde quad a\in A,\widetilde qquad
\Phi(U_1):=U_2
$$
gives rise to an isomorphism between the $C^*$-algebras $C^*(\pi_1( A),U_1)$
and $C^*(\pi_2( A),U_2)$.
\end{thm}
\widetilde begin{Proof} Let $B_{0,i}$ be a $^*$-algebra consisting of elements of the form
$\sum_{n=0}^N U^{*n}_i\pi_i(a_n)U_i^n$, $i=1,2$.
In view of Corollary \ref{corollary with distance}, $\Phi$ extends to an isometric isomorphism
from $B_{0,1}$ onto $B_{0,2}$. Moreover, we have
$$
\Phi(U_1aU_1^*)=U_2(\Phi(a))U^*_2, \widetilde qquad a \in B_{0,1}.
$$
Hence the assumptions of \cite[Theorem 2.13]{Leb-Odz} are satisfied and the hypotheses follows.
\end{Proof}
\widetilde begin{cor}\label{spectral-sp}
If $(\pi,U,H)$ possesses property $(*)$, then we have a point-wise continous action
$\gamma$ of the group $S^1$ on $C^*(\pi( A),U)$ by authomorphisms
given by
$$
\gamma_z(\pi(a)):=\pi(a), \widetilde quad a\in A, \widetilde qquad \gamma_z(U) := z U, \widetilde qquad z\in S^1.
$$
Moreover the spaces $\overline{B}_k$ are the spectral subspaces corresponding to this action, that is we have
$$
\overline{B}_k=\{ a \in C^*( A,U): \gamma_z(a)= z^k a\}.
$$
In particular, the $C^*$-algebra $\overline{B}_0$ is the fixed point algebra for $\gamma$.
\end{cor}
\widetilde begin{Proof} Let $(\pi,U,H)$ be a $J$-covariant representation of $(A,\widetilde alphapha)$ possessing property $(*)$ and let $z\in S^1$. It is clear that $(\pi,zU,H)$
is also a $J$-covariant representation of $( A,\widetilde alphapha)$ and
$(\pi,zU,H)$ possesses property $(*)$. Hence by virtue of Theorem
\ref{isomorphiasm theorem}, $\gamma_z$ extends to an isomorphism of
$C^*(\pi( A),U)=C^*(\pi( A),zU)$. The remaining part of the
statement is obvious.
\end{Proof}
The next theorem is an immediate corollary of the previous statements and \cite[Thm. 2.15]{Leb-Odz}.
It is another manifestation of the fact that the elements $N_k (a)$, $k\in\mathbb Z$, should be considered as
Fourier coefficients for $a \in C^*(\pi( A),U)$.
\widetilde begin{thm}
\label{uniqueNk}
Let $(\pi,U,H)$ possess property $(*)$ and let
\widetilde begin{center}
$a \in C^*(\pi( A),U)$.
\end{center}
Then the following conditions are equivalent:
\widetilde quad\llap{$(i)$}\ \ $a=0;$
\widetilde quad\llap{$(ii)$}\ \ $N_k (a)=0$, \,$k\in\mathbb Z;$
\widetilde quad\llap{$(iii)$}\ \ $N_0 (a^*a)=0$.
\end{thm}
The results presented above give us a possibility to write out a
criterion for a representation of the crossed product to be faithful.
\widetilde begin{thm}\label{5.4} Let $C^*( A,\widetilde alphapha, J)$ be the crossed product given by
Definition~\ref{cr-pr-def} and $(\pi,U,H)$ be a faithful
$J$-covariant representations of $( A,\widetilde alphapha)$. Then the relations
\widetilde begin{equation}\label{e-fath}
(\pi\times U)(a)=\pi(a), \ a\in A;\widetilde qquad (\pi\times U)(\hat{u})=U
\end{equation}
determine in a unique way an epimorphism $\pi\times U: C^*( A,\widetilde alphapha, J) \to C^*(\pi( A),U)$. Moreover
$\pi\times U$ is an isomorphism iff $(\pi,U,H)$ possesses property $(^*)$.
\end{thm}
\subsection{Construction of faithful representations}\label{faith-covar}
Given a faithful $J$-covariant representation of $( A,\widetilde alphapha)$ one can construct a covariant representation of $( A,\widetilde alphapha)$ possessing property~$(^*)$ thus obtaining a faithful representation of $ C^*( A,\widetilde alphapha,J)$ (in view of Theorem~\ref{5.4}). Actually, we exploit the standard argument cf.~\cite{Ant-Bakht-Leb}.
\widetilde begin{prop}
For any faithful $J$-covariant representation $(\widetilde{\pi},\widetilde{U},\widetilde{H})$ of $( A,\widetilde alphapha)$ the triple $(\pi,U, { H})$ where $H :=l^2 ({\mathbb Z}, \widetilde{H})$,
\widetilde begin{equation}
(\pi (a)\widetilde xi )_n := \widetilde{\pi} (a) (\widetilde xi_n), \widetilde quad \textrm{ and }\widetilde quad
(U\widetilde xi )_n := \widetilde{U} (\widetilde xi_{n-1})\label{18e}
\end{equation}
for $a\in A$, $ \widetilde xi = \{ \widetilde xi_n \}_{n\in\mathbb Z}\in H=l^2 ({\mathbb Z}, \widetilde{H})$, is a faithful $J$-covariant representation which integrate via \eqref{e-fath} to a faithful representation $(\pi\times U)$ of $ C^*( A,\widetilde alphapha,J)$.
\end{prop}
\widetilde begin{Proof}
Routine verification shows that $(\pi,U, { H})$ is a faithful
$J$-covariant representation of $( A,\widetilde alphapha)$. By
Theorem~\ref{5.4} it suffices to verify that $(\pi,U, { H})$
possesses property $(^*)$. To this end take any $N\in \mathbb N$,
$a_{m,n}\in A$, $n,m=0,1...,N$, and note that by the explicit
form of \eqref{18e} we have
\widetilde begin{equation}\label{eq}
\widetilde Vert\sum_{m=0}^N {\widetilde{U}}^{*m}\widetilde{\pi}(a_{m,m})\widetilde{U}^m
\widetilde Vert = \widetilde Vert \sum_{m=0}^N U^{*m}\pi(a_{m,n})U^m \widetilde Vert
\end{equation}
For a given $\varepsilon > 0$ there exists a vector
$\eta \in \widetilde{H}$ such that $\widetilde Vert \eta \widetilde Vert =1$ and
\widetilde begin{equation}\label{e*}
\widetilde Vert \left(\sum_{m=0}^N {\widetilde{U}}^{*m}\widetilde{\pi}(a_{m,m})\widetilde{U}^m\right)
\eta \widetilde Vert > \widetilde Vert\sum_{m=0}^N {\widetilde{U}}^{*m}\widetilde{\pi}(a_{m,m})\widetilde{U}^m
\widetilde Vert - \varepsilon .
\end{equation}
Set $\widetilde xi = \{ \widetilde xi_n \}_{n\in
\mathbb Z} \in l^2 ({\mathbb Z}, \widetilde{H})$ by $\widetilde xi_n = \delta_{0,n}\eta $,
where $\delta_{i,j}$ is the Kronecker symbol. We have that $\widetilde Vert
\widetilde xi \widetilde Vert = 1$ and \eqref{18e} along with (\ref{eq}) and (\ref{e*}) imply
\widetilde begin{equation}\label{e21}
\widetilde Vert \left(\sum_{m=0}^N U^{*m}\pi(a_{m,n})U^m\right)\widetilde xi \widetilde Vert>
\widetilde Vert \sum_{m=0}^N U^{*m}\pi(a_{m,n})U^m \widetilde Vert
-\varepsilon
\end{equation}
Now the explicit form of \ $\left(\sum_{m,n=0}^N U^{*m}\pi(a_{m,n})U^n\right)
\widetilde xi $ \ and
\eqref{e21} imply
\[
\widetilde Vert \sum_{m,n=0}^N U^{*m}\pi(a_{m,n})U^n
\widetilde Vert^2 \ge
\widetilde Vert \left(\sum_{m,n=0}^N U^{*m}\pi(a_{m,n})U^n\right)\widetilde xi
\widetilde Vert^2\ge
\]
\widetilde begin{equation}\label{end}
\ge \widetilde Vert \left(\sum_{m=0}^N U^{*m}\pi(a_{m,n})U^m\right)\widetilde xi
\widetilde Vert^2 > \left(\widetilde Vert\sum_{m=0}^N U^{*m}\pi(a_{m,m})U^m\widetilde Vert -
\varepsilon\right)^2
\end{equation}
which by the
arbitrariness of $\varepsilon$ proves property $(^*)$ for
$(\pi,U,{ H})$. \end{Proof}
Following the foregoing construction one can also arrive at the next
\widetilde begin{prop}\label{star property*}
The crossed product $ C^*( A,\widetilde alphapha,J)$, given by Definition~\ref{crossed product defn} possesses property $(*)$, that is the algebra $ A$ is embedded in the crossed product $ C^*( A,\widetilde alphapha,J)$ and for any $N\in \mathbb N$ and $a_{m,n}\in A$, $n,m=0,1...,N$ the following inequality holds
\widetilde begin{equation}\label{*-cross}
\|\sum_{m=0}^N u^{*m}a_{m,m}u^m \| \leq \|\sum_{m,n=0}^N u^{*m}a_{m,n}u^n \|.
\end{equation}
\end{prop}
\widetilde begin{Proof}
Let $ C^*( A,\widetilde alphapha,J)$ be given by Definition~\ref{crossed product defn}. Take any faithful
representation $\overline{\pi}: C^*( A,\widetilde alphapha,J) \to L(\widetilde{H})$ of $ C^*( A,\widetilde alphapha,J)$ in a Hilbert space $\widetilde{H}$
and 'disintegrate' $\overline{\pi}$ to $(\widetilde{\pi},\widetilde{U},\widetilde{H})$, i.e. let $\widetilde{\pi}:=\overline{\pi}|_ A$
and $\widetilde{U}:=\overline{\pi}(u)$. Then $(\widetilde{\pi},\widetilde{U},\widetilde{H})$ is a faithful
$J$-covariant representation of $( A,\widetilde alphapha)$ (note that Theorem~\ref{existance theorem} and Definition~\ref{crossed product defn} imply that $\widetilde{\pi}$ is a faithful representation of $ A$). Consider the space ${ H} =l^2 ({\mathbb Z}, \widetilde{H})$ and
representation $(\pi,U,H)$ given by \eqref{18e}. By the
universality of $C^*( A,\widetilde alphapha,J)$ (Definition~\ref{crossed
product defn}) this representation give rise to a representation
of $C^*( A,\widetilde alphapha,J)$. And therefore for any any $N\in \mathbb N$ and
$a_{m,n}\in A$, $n,m=0,1...,N$ one has
$$
\|\sum_{m,n=0}^N u^{*m}a_{m,n}u^n \|\ge \widetilde Vert \sum_{m,n=0}^N U^{*m}\pi(a_{m,n})U^n
\widetilde Vert.
$$
Moreover by the explicit form of \eqref{18e} we have
$$
\|\sum_{m=0}^N u^{*m}a_{m,m}u^m \|= \widetilde Vert \sum_{m=0}^N U^{*m}\pi(a_{m,m})U^m
\widetilde Vert.
$$
Now \eqref{*-cross} follows from \eqref{end}.
\end{Proof}
The above results imply
\widetilde begin{prop}\label{crossed equiv}
The crossed products given by Definitions~\ref{crossed product defn} and~\ref{cr-pr-def} are canonically
isomorphic.
\end{prop}
\widetilde begin{Proof} Let $C^*( A,\widetilde alphapha,J)$ be the crossed product given by Definition~\ref{crossed product defn}.
Set
$$
\pi_1: A\to\hat{ A}, \ \pi_1 (a):= \hat{a}, \widetilde qquad \textrm{ and }\widetilde qquad
U_1:=\hat{u},
$$
where $\hat{a}$ and $\hat{u}$ are the universal elements, corresponding to $a\in A$ and $u$ according to Definition~\ref{crossed product defn}. Then
$$
C^*( A,\widetilde alphapha,J)=C^*(\pi_1( A),U_1).
$$
By Definition~\ref{crossed product defn} and Theorem \ref{existance theorem}, $\pi_1$ establishes an isomorphism between $ A$ and $\hat{ A}$. By Proposition~\ref{star property*} \ $C^*(\hat{ A},\hat{u})$ possesses
property~$(^*)$.
Now, let $C^*( A,\widetilde alphapha,J)$ be the crossed product given by
Definition~\ref{cr-pr-def}, and let $(\pi,U,H)$ be a faithful
$J$-covariant representation of $( A,\widetilde alphapha)$ possessing property
$(^*)$ (such representation does exist by
Remark~\ref{faith-covar}). Set $(\pi_2,U_2, H)$ by
\widetilde begin{equation}\label{e-fath*}
\pi_2(a):=(\pi\times U)(a)=\pi(a), \ a\in A;\widetilde qquad U_2:=(\pi\times U)(\hat{u})=U
\end{equation}
(cf. \eqref{e-fath}). Then by Theorem~\ref{5.4}
$$
C^*(\pi_2( A),U_2)\cong C^*( A,\widetilde alphapha,J)
$$
and $(\pi_2,U_2, H)$ possesses property $(^*)$. Thus by Theorem~\ref{isomorphiasm theorem}
$$
C^*(\pi_1( A),U_1)\cong C^*(\pi_2( A),U_2).
$$
\end{Proof}
\subsection{Topological freeness}\label{ABL=Kwa-Leb} Crossed product $C^*( A,\widetilde alphapha,J)$ can also be constructed by means of the crossed product introduced in \cite{Ant-Bakht-Leb}. Here we present this construction. It can be considered as one more alternative definition of $C^*( A,\widetilde alphapha,J)$.
Let $( A,\widetilde alphapha)$ be a $C^*$-dynamical system, $J$ be an ideal orthogonal to the kernel of $\widetilde alphapha$, and
$(\pi,U,H)$ be a faithful $J$-covariant representation of $( A,\widetilde alphapha)$. Consider the algebra
\widetilde begin{equation}\label{ext}
B:=C^*\left(\widetilde bigcup_{n\in \mathbb N}U^{*n}\pi( A)U^{n}\right).
\end{equation}
By Corollary~\ref{corollary with distance} algebra $B$ can be
described in terms of $( A, \widetilde alphapha)$ and $J$, and therefore it
does not depend on the choice of a faithful $J$-covariant
representation $(\pi,U,H)$. Routine calculation (cf.
\cite[Prop. 3.10.]{Leb-Odz}) shows that
\widetilde begin{equation}\label{B-2}
UB U^*\subset B, \widetilde qquad U^*B U\subset B, \widetilde qquad U^*U\in Z(B).
\end{equation}
Thus $B$ is a \emph{coefficient algebra} in the sense of \cite{Leb-Odz}. In particular
$$
\widetilde alphapha:B\toB, \widetilde qquad \widetilde alphapha (\cdot):= U(\cdot)U^*
$$
is an endomorphism of $B$ (identifying $\pi(A)$ with $A$ the mapping $U(\cdot)U^*$ extends the endomorphism $\widetilde alpha:A\to A$ and the notational collision disappears) and
$$
\mathcal{L}:B\toB, \widetilde qquad \mathcal{L} (\cdot):=U^*(\cdot)U
$$
is a \emph{complete transfer operator} in the sense of \cite{Bakht-Leb}. Note also that the mapping $\mathcal{L}$ is
defined uniquely by the $C^*$-dynamical system $(B,\widetilde alphapha)$ (\cite[Theorem 2.8.]{Bakht-Leb}), and therefore it is uniquely
defined by $( A,\widetilde alphapha)$ and $J$.
In particular, Proposition~\ref{star property*} and \cite[Theorem 3.5.]{Ant-Bakht-Leb} imply the following
\widetilde begin{prop}\label{crossed-ABL1} We have a natural isomorphism
\widetilde begin{equation}
\label{crossed-ABL}
C^*( A,\widetilde alphapha,J)\congB\times_\widetilde alphapha\mathbb Z,
\end{equation}
where on the right hand side stands the crossed product of \cite[Def. 2.6.]{Ant-Bakht-Leb}.
\end{prop}
\widetilde begin{rem}
Thus the alternative construction of crossed product involve essentially two steps: 1)~extending an
\emph{irreversible} system $( A,\widetilde alphapha)$ up to a \emph{reversible} system on $(B,\widetilde alphapha)$ (here $\mathcal{L}$ plays the role of the inverse to $\widetilde alphapha$),
and then 2)~attaching the crossed product from \cite{Ant-Bakht-Leb} to the extended system.
We have to stress that the general procedure of extension of $( A,\widetilde alphapha)$ up to $(B,\widetilde alphapha)$ by means of \eqref{ext} involves the ideal $J$ but 'does not see' it explicitely and namely the material of the present paper shows that by means of these orthogonal ideals
all the possible extensions are parametrised (recall in this connection the discussion in \cite[Section 5]{Ant-Bakht-Leb}).
\end{rem}
The isomorphism \eqref{crossed-ABL} and the results of
\cite{kwa-ck} give us a possibility to obtain one more isomorphism
theorem which can be written in terms of topological freeness of
the action $\widetilde alphapha$ on $B$. Indeed, it follows immediately from
\eqref{B-2} that
$$
\mathcal{L}:\widetilde alphapha(B)\to \mathcal{L}(B)\,\,\,\,\textrm{ and }\,\,\,\, \widetilde alphapha:\mathcal{L}(B)\to \widetilde alphapha(B)
$$
are mutually inverse isomorphisms $\mathcal{L}(B)=\mathcal{L}(1)B$ is an ideal in $B$
and $\widetilde alphapha(B)=\widetilde alphapha(1)B\widetilde alphapha(B)$ is a hereditary subalgebra of $B$. Therefore, cf. e.g. \cite[Thm 5.5.5]{Murphy}, we may naturally identify the spectra $\widehat{ \widetilde alphapha(B)}$ and $\widehat{\mathcal{L}(B)}$ of $\widetilde alphapha(B)$ and $\mathcal{L}(B)$ with open subsets of the spectrum $\widehat{B}$ of $B$, and then
\widetilde begin{equation}\label{identifications of spectra}
\widehat{ \widetilde alphapha(B)}=\{\pi \in \widehat{B}: \pi(\widetilde alphapha(1))\neq 0\},\widetilde qquad \widehat{ \mathcal{L}(B)}=\{\pi \in \widehat{B}: \pi(\mathcal{L}(1))\neq 0\}.
\end{equation}
Under the above identifications the dual $\widehat{\widetilde alphapha}:\widehat{ \widetilde alphapha(B)} \to \widehat{ \mathcal{L}(B)}$ to the isomorphism $\widetilde alphapha:\mathcal{L}(B)\to \widetilde alphapha(B)$ becomes a partial homeomorphism of $B$. More precisely, let $\pi:B\to L(H)$ be an irreducible representation. If $\pi(\widetilde alphapha(1))\neq 0$, then
\widetilde begin{equation}\label{homeomorphism dual to delta}
\widehat{\widetilde alphapha}(\pi)=\pi\circ \widetilde alphapha: B\to L(\widetilde alphapha(1)H)
\end{equation}
is an irreducible representation such that $\widehat{\widetilde alphapha}(\pi)(\mathcal L(1))\neq 0$. Conversely if $\pi(\mathcal{L}(1))\neq 0$, then $\widehat{\widetilde alphapha}^{-1}(\pi)=\widehat{\mathcal{L}}(\pi)$ is a unique (up to unitary equivalence) irreducible extension of the representation
$
\pi\circ \mathcal{L}: \widetilde alphapha(1)B\widetilde alphapha(1)\to L(H).
$
\widetilde begin{defn}
We say that $\widehat{\widetilde alphapha}$ where $\widehat{\widetilde alphapha}:\widehat{ \widetilde alphapha(B)} \to \widehat{ \mathcal{L}(B)}$ is a homeomorphism given by \eqref{homeomorphism dual to delta}, between the open subsets \eqref{identifications of spectra} of $\widehat{B}$, is a \emph{partial homeomorphism dual to the endomorphism }$\widetilde alphapha:B\to B$. We call the pair $(\widehat{B}, \widehat{\widetilde alphapha})$ the \emph{partial dynamical system dual to the $C^*$-dynamical system} $(B,\widetilde alphapha)$.
\end{defn}
We recall that a partial homeomorphism of a topological space, i.e. a homeomorphism between open subsets, is {\em
topologically free} if for any $n\in \mathbb N$ the set of periodic points of period $n$ has empty interior.
Applying \cite[Thm. 2.24]{kwa-ck} we arrive at the following result
\widetilde begin{thm}[\textbf{Isomorphism theorem and topologically free action}]\label{topolo disco-polo} Let $( A,\widetilde alphapha)$ be a $C^*$-dynamical system,
$J$ an ideal orthogonal to the kernel of $\widetilde alphapha$, and $(B,\widetilde alphapha)$
a partially reversible system associated to the triple $( A,\widetilde alphapha,J)$ as described above.
If the partial homeomorphism $\widehat{\widetilde alphapha}$ dual to $\widetilde alphapha:B\to B$ is topologically free,
then for any faithful $J$-covariant representations $(\pi,U,H)$ of $( A,\widetilde alphapha)$
the epimorphism $\pi\times U: C^*( A,\widetilde alphapha, J) \to C^*(\pi( A),U)$ given by \eqref{e-fath} is an isomorphism:
$$
C^*( A,\widetilde alphapha, J) \cong C^*(\pi( A),U).
$$
\end{thm}
A crucial and (in the noncommutative case) still open question is the following.
\widetilde begin{quote}
\textbf{Problem:} How to express the topological freeness of the partial reversible dynamical system $(\widehat{B},\widehat{\widetilde alphapha})$ dual to $(B,\widetilde alphapha)$ in terms of the initial $C^*$-dynamical system $( A,\widetilde alphapha)$ and the ideal $J$?
\end{quote}
\subsection{Crossed product overview}
One can see that the most popular crossed products by endomorphisms coincide with $C^*( A,\widetilde alphapha,J)$ for certain $J$. Table 1 presents the corresponding juxtaposition of the objects chosen.
\widetilde begin{table}[htb]
\widetilde begin{center}
\widetilde begin{tabular}{|c|c|c|c|} \hline
N. & endomorphism $\widetilde alphapha: A\to A$ & $J \triangleleft A$ & $C^*( A,\widetilde alphapha,J)$
\\ \hline
1. & automorphism & $J=(\ker\widetilde alphapha)^\widetilde bot= A$ & classical unitary
\\ & & & crossed product
\\ \hline
2. & monomorphism & $J=(\ker\widetilde alphapha)^\widetilde bot= A$ & isometric crossed product
\\ & & & \cite{Paschke}, \cite{Murphy}
\\ \hline
3. & $\ker\widetilde alphapha$ unital and & $J=(\ker\widetilde alphapha)^\widetilde bot$ & crossed product using
\\ & $\widetilde quad \widetilde alphapha( A)$ hereditary in $ A\widetilde quad $ & & complete transfer operator
\\ & & & \cite{Ant-Bakht-Leb}
\\ \hline
4. & $\ker\widetilde alphapha$ unital and & $J=(\ker\widetilde alphapha)^\widetilde bot$ & covariance algebra \cite{kwa}
\\ & $ A$ commutative & &
\\ \hline
5. & arbitrary & $J=\{0\}$ & partial-isometric
\\ & & & crossed product \cite{Lin-Rae}
\\ \hline
6. & arbitrary & $ \{0\}\subset J \subset (\ker\widetilde alphapha)^\widetilde bot$ & partial-isometric
\\ & & & crossed product
\\& & & $C^*( A,\widetilde alphapha,J)$
\\ & & &of the present article,
\\ \hline
\end{tabular}
\caption{Different crossed products}\label{table 1}
\end{center}
\end{table}
To see the coincidence in N.3 of Table 1 we refer the reader to
\cite[Prop. 2.6]{kwa4}. The crossed product N.6
(Definitions~\ref{crossed product defn} and~\ref{cr-pr-def}) is
the most general in the sense that it gives all the remaining ones
for an appropriate choice of $J$; namely $J=(\ker\widetilde alphapha)^\widetilde bot$ for N.1-4
and $J=\{0\}$ for N.5.
As it is shown in \cite[Section 4]{Ant-Bakht-Leb} the crossed product N.3 of Table 1 covers a lot of most popular crossed product constructions, and in particular two kinds of crossed products introduced by R. Exel in \cite{exel1} and \cite{exel2} may be obtained from the crossed product N.3.
Note also that there are a number of crossed products that at first sight are not of type $C^*( A,\widetilde alphapha,J)$
and are related to ideals that are not orthogonal to $\ker\widetilde alphapha$. As an example let us mention Stacey's (multiplicity one) crossed product \cite[Defn. 3.1]{Stacey}. For the sake of simplicity we state his definition in a unital setting, cf. \cite{Adji_Laca_Nilsen_Raeburn}.
\widetilde begin{defn}\label{Stacey}
\emph{Stacey's crossed product} for an
endomorphism $\widetilde alphapha$ of a unital $C^*$-algebra $ A$ is a unital
$C^*$-algebra $B$ together with a unital $^*$-homomorphism
$i_A:A\to B$ and an isometry $u\in B$ such that
\widetilde begin{itemize}
\item[i)] $i_ A(\widetilde alphapha(a))=u\, i_A(a)u^*$ for all $a\in A$
\item[ii)] $B$ is generated by $i_A(A)$ and $u$
\item[iii)] for every non-degenerate representation $\pi:A\to L(H)$ and an isometry $T\in L(H)$ there is a representation $\pi\times T:B\to L(H)$ such that $(\pi\times T)\circ i_ A=\pi$ and $(\pi\times T)(u)=T$.
\end{itemize}
\end{defn}
The mapping $a\to uau^*$ for an isometry $u$ is injective.
Therefore, in view of condition i), the homomorphism $i_ A$ can
not be injective unless $\widetilde alphapha$ is a monomorphism. Hence in
general $A$ does not embeds into the Stacey's crossed product.
In particular, see \cite[Prop. 2.2]{Stacey}, Stacey proved that $B$
degenerates to $\{0\}$ if and only if the inductive limit of
the inductive sequence
$ A\stackrel{\widetilde alphapha}{\rightarrow} A\stackrel{\widetilde alphapha}{\rightarrow}
...$ degenerates to zero. We will refine this result in Example \ref{reduction
of Stacey's crossed product} below.
Actually we show that Stacey's crossed product can also be presented in
the form of $C^*( A,\widetilde alphapha,J)$ for an appropriate $ A,\widetilde alphapha$ and
$J$, but to achieve this one first needs to 'reduce' the initial
$C^*$-dynamical system. The general procedure of such a reduction is
discussed in the forthcoming part of the paper where we also
analyse the relation between the crossed products
$C^*( A,\widetilde alphapha,J)$ and relative Cuntz-Pimsner algebras.
\section{Crossed products and relative Cuntz-Pimsner
algebras}\label{C-P}
In this section we start to discuss relations between the
crossed products introduced and relative Cuntz-Pimsner algebras.
This requires a description of a series of known objects
and results and we do this job in Subsections \ref{preliminaries
on C-correspondences}, \ref{The Toeplitz C*-algebra of a Hilbert
bimodule} and \ref{Relative Cuntz-Pimsner algebras}, while a
presentation of the crossed products as relative Cuntz-Pimsner
algebras is given in Subsection~\ref{Crossed prod =Relative
Cuntz-Pimsner algebras}. To begin with we recall the basic
necessary objects related to Hilbert $C^*$-modules and $C^*$-correspondences. A general
information on Hilbert $C^*$-modules can be found, for example, in \cite{lance}. The term $C^*$-correspondence was popularized, among the others, by T. Katsura \cite{katsura1}, \cite{katsura}, \cite{katsura2}.
\subsection{$C^*$-correspondences and their representations}\label{preliminaries on C-correspondences}
Let $ A$ be a (not necessarily unital) $C^*$-algebra and $X$ a
right Hilbert $ A$-module with an $ A$-valued inner product
$\langle \cdot,\cdot \rangle_ A $, see \cite[ch. 1]{lance}. We denote by
$\mathcal L(X)$ the $C^*$-algebra of adjointable operators on $X$. For
$x,y \in X$, we let ${\mathbb Theta}_{x,y}\in \mathcal L(X)$ be the
'one-dimensional operator': ${\mathbb Theta}_{x,y}(z)=x \cdot \langle
y,z\rangle_ A$, and we denote by $\mathcal K(X)$ the ideal of 'compact
operators'
in $\mathcal L(X)$
which is a closed linear span of the operators ${\mathbb Theta}_{x,y}$,
$x,y\in X$. We recall that any $C^*$-algebra $ A$ can be naturally
treated as a right Hilbert $ A$-module where $\langle a,
b\rangle_ A=a^*b$ and then $\mathcal K( A)= A$ and $\mathcal L( A)=M( A)$ is the
multiplier algebra of $ A$.
\widetilde begin{defn}
\emph{A $C^*$-correspondence $X$ over a $C^*$-algebra $ A$} is a (right) Hilbert $ A$-module equipped with a homomorphism $\phi: A \to \mathcal L(X)$.
We refer to $\phi$ as the left action of a $ A$ on $X$ and write
\widetilde begin{equation}\label{left}
a\cdot x := \phi(a)x.
\end{equation}
\end{defn}
\widetilde begin{rem}
\label{Hilbbert-bim} A $C^*$-correspondence is also sometimes
called a Hilbert bimodule, see e.g. \cite{p}, \cite{fr},
\cite{fmr}. However, there are plenty of reasons, see e.g.
\cite{aee}, \cite{katsura1}, \cite{kwa-doplicher} or \cite{kwa3}, that the term
\emph{Hilbert bimodule} should be reserved for a special sort of
$C^*$-correspondence, namely a $C^*$-correspondence $X$ with an
additional structure which is an $ A$-valued sesqui-linear form
${_A\langle} \cdot , \cdot \rangle$ such that
$$
x \cdot \langle y ,z \rangle_A = {_A\langle} x , y \rangle \cdot z, \widetilde qquad \textrm{for all}\,\,\, x,y,z\in X.
$$
Then, see \cite[Lem 3.4]{katsura1} or \cite[Prop. 1.11]{kwa-doplicher}, $X$ is both a left and a right Hilbert $ A$-module and $\|\langle x ,x \rangle_A\| = \|{_A\langle} x , x \rangle \|$.
\end{rem}
\widetilde begin{defn}
A {\em representation} $(\pi,t,B)$ of a $C^*$-correspondence
$X$ in a
$C^*$-algebra
$B$ consists of a linear map $t:X\to B$
and a homomorphism $\pi:A\to B$ such that
\widetilde begin{equation}\label{c*-corr}
t(x\cdot a) = t(x)\pi(a),\widetilde quad t(x)^*t(y)= \pi(\langle
x,y\rangle_A),\widetilde quad t(a\cdot x) = \pi(a)t(x),
\end{equation}
for $x,y\in X$ and $a\in A$. If $\pi$ is faithful (then
automatically $t$ is isometric, cf. \cite[Rem 1.1]{fr}) we say
that the representation $(\pi,t,B)$ is \emph{faithful}. If
$B=L(H)$ for a Hilbert space $H$ we say that $(\pi,t,L(H))$ is
a \emph{representation of $X$ in} $H$.
\end{defn}
\widetilde begin{rem}
The above introduced notion is called in \cite{fr}, \cite{fmr} a
\emph{Toeplitz representation} of $X$, and in
\cite{ms} it is called an \emph{isometric covariant representation} of $X$.
\end{rem}
Any representation $(\pi,t,B)$ of a $C^*$-correspondence $X$ in a $C^*$-algebra $B$
naturally give rise to a representation ${(\pi,t,B)}^{(1)}: \mathcal K(X)\to B$ of the $C^*$-algebra $\mathcal K(X)$ of 'compact operators'
on $X$ which is uniquely determined by the condition that
\widetilde begin{equation}\label{induced representation on K(X)}
{(\pi,t,B)}^{(1)}({\mathbb Theta}_{x,y}):=t(x){t(y)}^*\,\,\,\text{ for } \,\,\,x,y\in
X,
\end{equation}
see \cite[Prop.~1.6]{fr} or \cite[Prop. 3.13]{kwa-doplicher}.
Moreover the left action $\phi: A\to \mathcal L(X)$ restricted to the
ideal
$$
J(X):={\phi}^{-1}(\mathcal K(X))
$$
is a representation of $J(X)$ in $\mathcal K(X)$ and it is of a
particular interest to understand the relationship between
$\pi:J(X)\to B$ and $(\pi,t,B)^{(1)} \circ \phi:J(X)\to B$.
\widetilde begin{defn}
For any ideal $J$ contained in $J(X)$ a representation $(\pi,t,B)$ of $X$ is said to be {\em covariant on} $J$ or \emph{$J$-covariant} if
\widetilde begin{equation*}
{(\pi,t,B)}^{(1)}(\phi(a))=\pi(a)\widetilde quad\text{for all \ } a\in J.
\end{equation*}
Actually, the set $ \{a\in J(X):
{(\pi,t,B)}^{(1)}(\phi(a))=\pi(a)\}$
is the biggest ideal on which $(\pi,t,B)$ is covariant and we will call this ideal
an \emph{ideal of covariance for} $(\pi,t,B)$.
\end{defn}
\widetilde begin{rem} What we call above covariant represenations was originally called by Muhly and Solel coisometric representations, cf. \cite{ms}, \cite{fmr}. This name was motivated by specicif applications and examples, and therefore we choose, following Katsura \cite{katsura1}, \cite{katsura}, more universal (neutral) name - covariance.
\end{rem}
\widetilde begin{rem}\label{orthogonality remark}
Plainly, for \emph{$J$-covariant} representation $(\pi,t,B)$
the ideal $\ker\phi \cap J$ is contained in $\ker\pi$. Hence a
necessary condition for the existence of a faithful
\emph{$J$-covariant} representation of $X$ is that $J$ is
orthogonal to $\ker\phi$ (by Proposition \ref{injectivity of k_A}
below this condition is also sufficient).
\end{rem}
\subsection{The Fock representation }\label{The Toeplitz C*-algebra of a Hilbert bimodule}
The original idea of Pimsner \cite{p}, see also \cite{ms}, \cite{fr}, \cite{katsura}, is to construct representations of $C^*$-correspondences by a natural adaptation of the celebrated Hilbert space construction introduced by Fock. Namely, given a $C^*$-correspondence $X$ over $ A$, for $n\ge 1$, the
$n$-fold internal tensor product $X^{\otimes n} :=
X\otimes_ A\dotsm\otimes_ A X$, see e.g. \cite[ch. 4]{lance}, is a
$C^*$-correspondence where $ A$ acts on the left by
$$
\phi^{(n)}(a)(x_1\otimes_ A\dotsm\otimes_ A x_n) :=
(a\cdot x_1)\otimes_ A\dotsm\otimes_ A x_n;
$$
here $a\cdot x_1$ is given by \eqref{left}.
For $n=0$, we take $X^{\otimes 0}$
to be the Hilbert module
$ A$ with left action
$\phi^{(0)}(a) b: = ab$. Then the Hilbert-module direct sum, see \cite[p. 6]{lance},
$$
\mathcal F(X) :=
\widetilde bigoplus_{n=0}^\infty X^{\otimes n}
$$ carries a diagonal left action $\phi_\infty$
of $ A$ in which $\phi_\infty(a)(x):=\phi^{(n)}(a)x$ where $x\in
X^{\otimes n}$. The
$C^*$-correspondence $\mathcal F(X)$ is called
the \emph{Fock space} over the $C^*$-correspondence $X$.
For each $x\in X$, we
define a \emph{creation operator\/}
$T(x)$ on $\mathcal F(X)$ by
$$
T(x)y=\widetilde begin{cases}
x\cdot y
& \text{if $y\in X^{\otimes 0}= A$} \\
x\otimes_ A y
& \text{if $y\in X^{\otimes n}$ for some $n\geq 1$;} \\
\end{cases}
$$
routine calculations show that $T(x)$ is adjointable and its adjoint is the \emph{annihilation operator}
$$
T(x)^*z=\widetilde begin{cases}
0
& \text{if $z\in X^{\otimes 0}= A$} \\
\langle x, x_1\rangle_ A\cdot y
& \text{if $z=x_1\otimes_ A y\in X\otimes_ A X^{\otimes n-1}=
X^{\otimes n}$.}
\\
\end{cases}
$$
One sees that $T:X\to\mathcal L(\mathcal F(X)) $ is an injective linear mapping and since $ A$ is a summand of $\mathcal F(X)$, the map
$\phi_\infty: A \to \mathcal L(\mathcal F(X))$ is injective as well. Actually, $(\phi_\infty,T, \mathcal L(\mathcal F(X)))$ is a
faithul representation of $X$ whose ideal of covariance is $\{0\}$.
\widetilde begin{defn}
The representation $(\phi_\infty, T, \mathcal L(\mathcal F(X)))$ of $X$ in
$\mathcal L(\mathcal F(X))$ defined above is called \emph{Fock representation}
and the \emph{Toeplitz $C^*$-algebra} $\mathcal T(X)$ of $X$ is by
definition the $C^*$-subalgebra of $\mathcal L(\mathcal F(X))$ generated by
$\phi_\infty( A)\cup T(X)$, cf. \cite[Def. 2.4]{ms}, \cite[Def.
1.1]{p}.
\end{defn}
\subsection{Relative Cuntz-Pimsner algebras $\mathcal{O}(X,J)$}\label{Relative Cuntz-Pimsner algebras}
Composing the Fock representation $(\phi_\infty, T, \mathcal L(\mathcal F(X))$
of $X$ with the quotient map one can get a representation of $X$
whose ideal of covariance is an arbitrarily chosen ideal
contained in $J(X)={\phi}^{-1}(\mathcal K(X))$. Namely, let $J$ be an
ideal in $J(X)$ and let $P_0$ be the projection in $\mathcal L(\mathcal F(X))$
that maps $\mathcal F(X)$ onto the first summand $X^{\otimes 0}= A$.
One can show \cite[Lem. 2.17]{ms} that $\phi_\infty(J)P_0$ is contained in $\mathcal T(X)$.
We will write $\mathcal J(J)$ for the ideal in $\mathcal T(X)$ generated by $\phi_\infty(J)P_0$.
\widetilde begin{defn}[\cite{ms}, Def. 2.18]
If $X$ is a $C^*$-correspondence over a $C^*$-algebra $ A$, and
if $J$ is an ideal in $J(X)={\phi}^{-1}(\mathcal K(X))$ we denote by
$\mathcal{O}(J,X)$ the quotient algebra $\mathcal T(X)/\mathcal J(J)$ and call it
\emph{relative Cuntz-Pimsner algebra} determined by $J$.
\end{defn}
The algebras $\mathcal{O}(J,X)$ are characterized by the following universal property.
\widetilde begin{prop}[\cite{fmr}, Prop. 1.3]\label{RCP algebra}
Let $X$ be a $C^*$-correspondence over $ A$, and let $J$ be an
ideal in $J(X)$. Let $q:\mathcal T(X)\to \mathcal{O}(J,X)$ be the quotient map
and put
$$
k_ A= q\circ \phi_\infty \widetilde quad\textrm{ and } \widetilde quad k_X=q \circ T.
$$
Then $(k_ A,k_X, \mathcal{O}(J,X))$ is a representation of $X$ which is covariant on $J$ and satisfies:
\widetilde begin{itemize}
\item[(i)] for every representation $(\pi,t)$ of $X$ which is covariant on $J$, there is a homomorphism $\pi{\times}_J t$ of $\mathcal{O}(J,X)$ such that
$$(\pi{\times}_J t)\circ k_ A=\pi \widetilde quad \textrm{ and } \widetilde quad (\pi{\times}_J t)\circ k_ A=t,$$
\item[(ii)] $\mathcal{O}(J,X)$ is generated as a $C^*$-algebra by $k_X(X)\cup k_ A( A)$.
\end{itemize}
The representation $(k_ A,k_X,\mathcal{O}(J,X))$ is unique in the following sense: if $(k_ A',k_X',B)$ has similar properties, there is an isomorphism $\theta:\mathcal{O}(J,X)\to B$ such that $\theta\circ k_X=k_X'$ and $\theta\circ k_ A=k_ A'$. In particular, there is a strongly continuous gauge action $\gamma:\mathbb T\to Aut\mathcal{O}(J,X)$ where ${\gamma}_z(k_ A(a))=k_ A(a)$ and ${\gamma}_z(k_X(x))=zk_X(x)$ for $a\in A,x\in X$.
\end{prop}
\widetilde begin{rem}\label{ideal of covariance vs universality}
One may show, cf. \cite[Prop. 4.7]{kwa-doplicher}, that the ideal of
coisomtetricity for the universal representation $(k_X,k_ A,
\mathcal{O}(J,X))$ coincides with $J$. Hence $(k_X,k_ A, \mathcal{O}(J,X))$ could
be equivalently defined as a universal triple with respect to
representations whose ideal of covariance not only contains
but actually equals $J$.
\end{rem}
\widetilde begin{table}[hbt]
\widetilde begin{center}
\widetilde begin{tabular}{|c|c|c|c|} \hline
N. & $\phi:A\to \mathcal L(X)$ & $J \triangleleft J(X)$ & $\mathcal{O}(J,X)$
\\ \hline
1. & monomorphism & $J=J(X)$ & Cuntz-Pimsner algebra of $X$
\\ & & & \cite{p}, \cite{fmr}
\\ \hline
2. & arbitrary & $ J=\{0\}$ & Toeplitz algebra of $X$
\\ & & & \cite{fr}, \cite{fmr}
\\ \hline
3. & arbitrary & $J=(\ker\phi)^\widetilde bot\cap J(X)$ & Katsura's algebra of $X$
\\
& & & \cite{katsura1}, \cite{katsura}, \cite{katsura2}
\\ \hline
4. & $\phi(J)=\mathcal K(X)$ & $J=(\ker\phi)^\widetilde bot\cap J(X)$ & crossed product by
\\
& & & the Hilbert bimodule
\\
& & & \cite{aee}
\\ \hline
\end{tabular}
\caption{Different relative Cuntz-Pimsner algebras \label{table 2}}
\end{center}
\end{table}
Table \ref{table 2} presents a juxtaposition of various relative Cuntz-Pimnser algebras studied in the literature obtained from $\mathcal{O}(J,X)$ for different choice of the ideal $J$. To see the coincidence in N.4 of Table \ref{table 2} we refer the reader, for instance, to \cite{katsura1}. In view of the following proposition, the algebra $ A$ embeds into $\mathcal{O}(J,X)$ for all the algebras presented in the Table \ref{table 2}.
\widetilde begin{prop}[\cite{ms}, Prop.~2.21]\label{injectivity of k_A}
Let $X$ be a $C^*$-correspondence over $ A$ and let
$(k_X,k_ A,\mathcal{O}(J,X))$ be the relative Cuntz-Pimsner algebra
associated with $J$. Then $k_ A: A\to\mathcal{O}(J,X)$ is injective if and
only if
\widetilde begin{equation}\label{orthogonal property}
\ker\phi \cap J =\{0\}.
\end{equation}
In particular, cf. Remark \ref{orthogonality remark}, a faithful $J$-covariant representation of $X$ exists if and only if $J$ is orthogonal to $\ker\phi$.
\end{prop}
The analysis presented in the forthcoming sections will show, in
particular, that all the relative Cuntz-Pimsner algebras are in
fact the algebras $\mathcal{O}(J,X)$ where $J\subset (\ker\phi)^{\widetilde bot}$
(see Theorem~\ref{reduction thm}). Moreover, see Theorem
\ref{Katsuras canonical theorem} below, they all can be modeled by
the algebra N.3 in Table 2 or even by the algebra N.4, see
\cite[Thm. 3.1]{aee}, but in the latter case the passage is highly
non-trivial (see also Remark~\ref{6.7}).
\subsection{Crossed products as relative Cuntz-Pimsner algebras}\label{Crossed prod =Relative Cuntz-Pimsner algebras}
Let $( A,\widetilde alphapha)$ be a $C^*$-dynamical system and define the structure of a $C^*$-correspondence over $ A$ on the space
$$
X:=\widetilde alphapha(1) A
$$
by
\widetilde begin{equation}
\label{e-bim-a1}
a \cdot x :=\widetilde alphapha(a)x,
\widetilde qquad
x\cdot a:= xa,
\widetilde qquad
\langle x,y\rangle_ A:=x^*y.
\end{equation}
Then one easily checks that $J(X)= A$ and $\ker\widetilde alphapha =\ker\phi$. We will say that $X$ is the \emph{$C^*$-correspondence of the $C^*$-dynamical system} $( A,\widetilde alphapha)$. The proof of the foregoing proposition in essence follows the argument from \cite[Exm. 1.6]{fmr}.
\widetilde begin{prop}\label{universality proposition}
Let $X$ be the $C^*$-correspondence of a $C^*$-dynamical system
$( A,\widetilde alphapha)$. The relations
\widetilde begin{equation}\
\label{e-bim-a4} U:=t(\widetilde alphapha(1))^*, \widetilde qquad { and}\widetilde qquad t(x):=
U^*\pi(x)
\end{equation}
establish a one-to-one correspondence between representations
$(\pi,U,H)$ of $( A,\widetilde alphapha)$ and representations $(\pi,t,L(H))$
of $X$, under this correspondence the ideal of covariance
for $(\pi,t,L(H))$ coincides with the ideal of covariance for
$(\pi,U,H)$.
\end{prop}
\widetilde begin{Proof} Let $(\pi,t,L(H))$ be a representations of $X$ and put $U:=t(\widetilde alphapha(1))^*$.
Then exploiting \eqref{c*-corr} and \eqref{e-bim-a1} one gets
$$
\pi(\widetilde alphapha(a))=\pi(\langle \widetilde alphapha(1),\widetilde alphapha(a)\rangle_ A)=t(\widetilde alphapha(1))^*t(\widetilde alphapha(a))=U t(a \cdot \widetilde alphapha(1))=U\pi(a)U^*.
$$
Thus, $(\pi,U, H)$ is a representation of $( A,\widetilde alphapha)$.
Moreover, observe that the operator $\phi(a)$ is just $\mathbb Theta_{\widetilde alphapha(a),\widetilde alphapha(1)}$ and since
$$
{(\pi,t,L(H))}^{(1)}(\mathbb Theta_{\widetilde alphapha(a),\widetilde alphapha(1)})=t(\widetilde alphapha(a))t(\widetilde alphapha(1))^*=U^*\pi(\widetilde alphapha(a))U
=U^*U\pi(a)U^*U=U^*U\pi(a)
$$
it follows that the ideal of covariance for $(\pi, t, L(H))$
coincides with the ideal of covariance for $(\pi,U, H)$.
Conversely, for any covariant representation $(\pi,U,H)$ of
$( A,\widetilde alphapha)$ putting $t(x):=U^*\pi(x)$, $x\in X$, we have
$U=t(\widetilde alphapha(1))^*$ and one easily checks conditions
\eqref{c*-corr}.
\end{Proof}
By the universality of $\mathcal{O}(J,X)$ and $C^*( A,\widetilde alphapha,J)$ (recall Definition~\ref{crossed product defn}), see also Remark \ref{ideal of covariance vs universality}, we get the following
\widetilde begin{cor}\label{C-P-cross}
Let $X$ be the $C^*$-correspondence of a $C^*$-dynamical system
$( A,\widetilde alphapha)$ and let $J$ be an ideal in $(\ker\widetilde alphapha)^\widetilde bot$. Then
algebras $\mathcal{O}(J,X)$ and $C^*( A,\widetilde alphapha,J)$ are canonically
isomorphic. In particular,
\widetilde begin{itemize}
\item[(i)] $\mathcal{O}(J,X)$ is generated as a $C^*$-algebra by the partial isometry $u=k_X(\widetilde alphapha(1))^*$ and the $C^*$-algebra $k_ A( A)$.
\item[(ii)] for every $J^\prime$-covariant representation $(\pi,U, H)$ of $( A,\widetilde alphapha)$ with $J^\prime \supset J$, there is a homomorphism $\pi{\times}_J U$ of $\mathcal{O}(J,X)$ uniquely determined by
$$
(\pi{\times}_J U)(u):= U\widetilde qquad \textrm{ and }\widetilde qquad
(\pi{\times}_J U)\circ k_ A :=\pi.
$$
\end{itemize}
\end{cor}
\widetilde begin{cor}\label{corollary stacey's}
Let $X$ be the $C^*$-correspondence of a $C^*$-dynamical system
$( A,\widetilde alphapha)$. Then the algebra $\mathcal{O}( A,X)$ together with the mapping
$k_ A$ and operator $u:=k_X(\widetilde alpha(1))^*$, cf. Proposition \ref{RCP algebra}, forms a Stacey's
crossed product for $( A,\widetilde alphapha)$ (Definition~\ref{Stacey}).
\end{cor}
Note that $\mathcal{O}(J,X)$ is defined for ideals $J$ that are not
necessarily orthogonal to~$\ker\widetilde alphapha$. This along with
Proposition~\ref{universality proposition} may make one guess that
$\mathcal{O}(J,X)$ is a more general object than $C^*( A,\widetilde alphapha,J)$. At
the same time Proposition~\ref{injectivity of k_A} shows that when
$J$ is not orthogonal to $\ker\widetilde alphapha$ the algebra $\mathcal{O}(J,X)$
possesses certain 'degeneracy'. All this stimulates us to take a
closer look and provide a more thorough analysis of the structure
of $\mathcal{O}(J,X)$ and its relation to $C^*( A,\widetilde alphapha,J)$. This is the
theme of the next section, where we present the procedure of
the canonical reduction of $C^*$-correspondences, algebras and
$C^*$-dynamical systems. In particular, as a result of this
reduction we establish the coincidence of $\mathcal{O}(J,X)$ with
appropriate crossed products introduced in the article.
\section{Reductions of $C^*$-correspondences
}\label{Reduction and canonical}
\subsection{Reduction of $C^*$-correspondences}\label{Reduction of Hilbert bimodule}
Let us now fix a $C^*$-correspondence $X$ over $ A$ and an ideal
$J$ in $J(X)$. We will reduce $X$ by taking quotient to
a certain 'smaller' $C^*$-correspondence satisfying
\eqref{orthogonal property} and yielding the same relative
Cuntz-Pimsner algebra as $X$ and $J$.
To this end we note that $C^*$-correspondences behave nice under
quotients. Namely, if $I$ is an ideal in $A$, then
$XI:=\clsp\{xi:x\in X,\, i\in I\}$ is both a right Hilbert
$A$-submodule of $X$ and a right Hilbert $I$-module, as we
have
\widetilde begin{equation}\label{XI equation}
XI=\{xi:x\in X,\, i\in I\}=\{x\in X: \langle x,y\rangle_A\in I \textrm{ for all }y\in X\},
\end{equation}
cf. \cite[Prop. 1.3]{katsura2}. Moreover, we may consider the quotient space $X/XI$ as a right Hilbert $ A/I$-module with an $ A/I$-valued inner product and right action of $ A/I$ given by
\widetilde begin{equation}\label{right action}
\langle q_{XI}(x),q_{XI}(y)\rangle_{ A/I}:=q_{I}(\langle x,y\rangle_ A), \widetilde qquad q_{XI}(x)\cdot q_I(a):=q_{XI}(x\cdot a),
\end{equation}
where $q_I: A\to A/I$ and $q_{XI}: X \to X/ XI$ are the quotient
maps, cf. \cite[Lem. 2.1]{fmr}. However, in order to define a left
action on the quotient $X/XI$ we need to impose the following
condition on an ideal $I$ in $ A$:
\widetilde begin{equation}\label{X-invariance}
\phi(I)X\subset XI.
\end{equation}
An ideal $I$ in $ A$ satisfying \eqref{X-invariance} is called $X$-\emph{invariant}
and for such an ideal the quotient $A/I$-module $X/XI$
is equipped with quotient left action $\phi_{ A/I}: A/I\to
\mathcal L(X/XI)$ given by
\widetilde begin{equation}\label{left action}
\phi_{ A/I}(q_{I}(a))q_{XI}(x):=q_{XI}(\phi(a)x),\widetilde qquad x \in
X,\,\, a\in A.
\end{equation}
and hence $X/XI$ naturally becomes a $C^*$-correspondence over
$ A/I$, cf. \cite[Lem. 2.3]{fmr}, \cite{katsura2}. We refer to it
as to a \emph{quotient $C^*$-correspondence}.
We will apply the following main result of \cite{fmr} to the reduction ideal introduced below.
\widetilde begin{thm}[\cite{fmr}, Thm. 3.1] \label{takie tam aa}
Suppose $X$ is a $C^*$-correspondence over $ A$, $J$ is an ideal
in $J(X)$, and $I$ is an $X$-invariant ideal in $ A$. If we denote
by $\mathcal I(I)$ the ideal in $\mathcal{O}(J,X)$ generated by $k_ A(I)$, then
the quotient $\mathcal{O}(J,X)/\mathcal I(I)$ is canonically isomorphic to
$\mathcal{O}(q_I(J),X/XI)$.
\end{thm}
\widetilde begin{defn}\label{reduction ideal for correspondences}
For any ideal $J$ in $J(X)$ we define recursively an increasing sequence of ideals
\widetilde begin{equation}\label{J-n}
J_0:=\{0\}\,\,\, \textrm{ and }\,\,\, J_{n+1}:=\{a\in J:
\phi(a)X\subset X{J_n}\}\,\,\, \textrm{ for } n\geq 0,
\end{equation}
and call the ideal
$$
J_\infty:=\overline{\widetilde bigcup_{n\in \mathbb N} J_n}
$$
the \emph{reduction ideal} for $X$ and $J$,
\end{defn}
\widetilde begin{rem}\label{universal description of J_infty}
Since $\phi(J_{n+1})X\subset X J_n$ and $J_n\subset J_{n+1}$, the ideals $J_n$, $n\in \mathbb N$, and therefore also $J_\infty$ are $X$-invariant ideals in $ A$. Actually,
let us note that
\widetilde begin{equation}\label{I_infty condition}
a \in J\,\, \textrm{ and } \,\, \phi(a)X\subset X J_\infty \mathcal Longrightarrow a\in J_\infty,
\end{equation}
and this implication characterizes $J_\infty$ in the sense that $J_\infty$ is the smallest $X$-invariant ideal in $ A$ satisfying \eqref{I_infty condition}. In particular, $J_\infty=\{0\}$ if and only if $\ker \phi\cap J=\{0\}$.
Note also that $C^*$-correspondences $X / XJ_{n}$, $n\in \mathbb N$, may
be considered as 'approximations' of $X / XJ_{\infty}$, and if $J_n=J_{n+1}$, for certain $n\in \mathbb N$, then
$J_\infty=J_n$ and $X / XJ_{n}=X / XJ_{\infty}$.
\end{rem}
The next result states, in particular, that the quotient
$C^*$-correspondence $X/XJ_{\infty}$ and the quotient
$C^*$-algebra $ A/J_\infty$ may be identified with the image of
the initial $C^*$-correspondence $X$ and $C^*$-algebra $ A$ in the
relative Cuntz-Pimsner algebra $\mathcal{O}(J,X)$.
\widetilde begin{thm}\label{reduction thm}
Let $X$ be a $C^*$-correspondence over $ A$ and $J$ an ideal in
$J(X)$. Then for $n\in \mathbb N\cup\{\infty\}$, we have a canonical
isomorphism
\widetilde begin{equation}\label{O-n}
\mathcal{O}(J,X) \cong \mathcal{O}(q_{J_n}(J),X/XJ_{n}).
\end{equation}
Moreover, for $n=\infty$ we have
\widetilde begin{equation}\label{ker-infty}
\ker \phi_{ A/J_\infty}\cap q_{J_\infty}(J)=\{0\}
\end{equation}
and thus we have the following (again canonical) isomorphisms
\widetilde begin{equation}\label{iso-infty}
k_ A( A)\cong A/J_\infty, \widetilde qquad k_X(X)\cong X/XJ_\infty.
\end{equation}
\end{thm}
\widetilde begin{Proof} In view of Theorem \ref{takie tam aa} to prove the first part of theorem it is enough to show that for every ideal $J_n$, $n=0,1,..., \infty$, we have $k_ A(J_n)=0$. It is clear that $k_ A(J_0)=0$. Assume that $k_ A(J_n)=0$ and let $a\in J_{n+1}$. Then for every $x\in X$ there exists $y(x)\in X$ and $i(x)\in J_n$ such that $\phi(a)x=y(x)i(x)$ and thus
$$
k_ A(a)k_X(x)=k_X(\phi(a)x)= k_X(y(x)i(x))=k_X(y(x))k_ A(i(x))=0,
$$
that is $k_ A(a)k_X(X)=\{0\}$.
Moreover, since $J_{n+1}\subset J$, and the universal representation is $J$-covariant we have $k_ A(a)={(k_ A,k_X,\mathcal{O}(J,X))}^{(1)}(\phi(a))$, for the mapping given by \eqref{induced representation on K(X)}. By \eqref{induced representation on K(X)}, relation ${(k_ A,k_X,\mathcal{O}(J,X))}^{(1)}(\phi(a))k_X(X)=\{0\}$ imply ${(k_ A,k_X,\mathcal{O}(J,X))}^{(1)}(\phi(a))=0$ and therefore
$k_A(a)=0$. Hence $k_ A(J_{n+1})=0$, and it follows that $k_ A(J_n)=0$ for every
$n=0,1,..., \infty$.
\\
To prove that $ \ker \phi_{ A/J_\infty}\cap
q_{J_\infty}(J)=\{0\} $ take $a\in J$ and suppose that
$\phi_{ A/J_\infty}( q_{J_\infty}(a))=0$. Then by \eqref{left
action} we see that $\phi(a) X \subset XJ_{\infty}$ and by
\eqref{I_infty condition} we have $q_{J_\infty}(a)=0$.
Now it suffices to apply Proposition \ref{injectivity of k_A}.
\end{Proof}
\widetilde begin{rem}
Theorem \ref{reduction thm} shows, in particular, that the ideal $J_\infty$ plays the role of a certain 'measure' of the degree of degeneracy of $\mathcal{O}(J,X)$ -- the bigger $J_\infty$ is
the smaller $\mathcal{O}(J,X)$ is.
In particular, $\mathcal{O}(J,X)=0$ if and only if $J_\infty=
A$. Obviously, $ A$ embeds into $\mathcal{O}(J,X)$ if and only if
$J_\infty=0$ which is equivalent to $X=X/XJ_{\infty}$.
Moreover, it follows that one may always restrict his interest only to the relative Cuntz-Pimsner algebras $\mathcal{O}(J,X)$ determined by ideals such that
$$
J \subset (\ker \phi)^\widetilde bot,
$$
since otherwise one has to pass (either explicitly or implicitly)
to the \emph{reduced $C^*$-correspondence} $X/XJ_\infty$ over the
\emph{reduced $C^*$-algebra} $A/J_{\infty}$ and the \emph{reduced
ideal} $q_{J_\infty}(J)\subset (\ker \phi_{ A/J_\infty})^\widetilde bot$.
\end{rem}
\subsection{Katsura's canonical relations for relative
Cuntz-Pimsner algebras}\label{Katsura's canonical relations}
In \cite{katsura2} T. Katsura associated with a $C^*$-correspondence $X$ a $C^*$-algebra $\mathcal{O}_X$ which is the relative Cuntz-Pimsner algebra $\mathcal{O}(J, X)$ for $J=(\ker \phi)^\widetilde bot\cap J(X)$. By subtle analysis he proved that any relative Cuntz-Pimsner algebra $\mathcal{O}(J, X)$ is isomorphic to $\mathcal{O}_{X_\omega}$ for certain
$X_\omega$.
In this subsection we apply the
reduction procedure presented above to give an alternative
definition of these
$C^*$-cor\-res\-pon\-den\-ces.
The reduction ideal $J_\infty$ defined above
coincides with the ideal $J_{-\infty}$ constructed in \cite[Sect.
11]{katsura2}. It is noted in \cite{katsura2}, that for the pair
$\omega=(J_\infty,J)$ there is a natural (yet a bit
sophisticated)
$C^*$-correspondence structure on the following pair of 'pullbacks'
$$
A_\omega=\{(a,a')\in A/J_\infty \oplus A/J:
q_{J(J_\infty)}(a)=q_{J(J_\infty)}(a')\}
$$
$$
X_\omega=\{(x,x')\in X/X{J_\infty}\oplus X/XJ: q_{XJ(J_\infty)}(x)=q_{XJ(J_\infty)}(x')\}
$$where $J(J_\infty):=(q_{XJ_\infty})^{-1}((\ker \phi_{ A/J_\infty})^\widetilde bot \cap J(X/XJ_\infty))$ (then $J_\infty \subset J\subset J(J_\infty)$) and $q_{J(J_\infty)}$ denotes here the quotient map composed with a corresponding isomorphism $( A/J_\infty)/J(J_\infty)\cong A/J(J_\infty)$ or $( A/J)/J(J_\infty)\cong A/J(J_\infty)$ and similar convention is used for $q_{XJ(J_\infty)}$.
Let us note that the mappings
$$
A/J_\infty \ni a \mapsto a\oplus q_J(a) \in A_\omega, \widetilde qquad X/XJ_\infty \ni x \mapsto x\oplus q_J(x) \in X_\omega
$$
are injective and therefore the pair $( A_\omega,X_\omega)$ is
an extension of the reduced pair $( A/J_\infty, X/XJ_\infty)$ (it
is a nontrivial extension unless $J_\infty=J(J_\infty)$).
Thus
Katsura's construction in fact involves two steps -- reduction (in the sense of Theorem \ref{reduction thm}) and an
extension (which we describe precisely in Definition \ref{Katsura's relations} below); and perhaps exposing them explicitly makes his construction a
bit more transparent.
\widetilde begin{defn}\label{Katsura's relations}
Let $J$ be an ideal in $J(X)$ and $J_X=(\ker \phi)^\widetilde bot \cap J(X)$. If $J$ is orthogonal to $\ker\phi$, then \emph{Katsura's canonical $C^*$-correspondence for} $X$ and $J$ is a $C^*$-correspondence $X_\omega$ over $ A_\omega$
where
$$
A_\omega=\{(a,a')\in A \oplus A/J: q_{J_X}(a)=q_{J_X}(a')\}
$$
$$
X_\omega=\{(x,x')\in X\oplus X/XJ: q_{XJ_X}(x)=q_{XJ_X}(x')\}
$$
and operations are defined simply by coordinates:
$$
(x,x') \cdot (a,a')= (xa,x'a'),\widetilde qquad (a,a')\cdot (x,x') = (ax,a'x')
$$
and
$$
\langle (x,x'), (y,y')\rangle_{ A_\omega} =(\langle x,y \rangle_ A, \langle x',y'\rangle_{ A/J}) \in A_\omega.
$$
\end{defn}
Extending the procedure presented above to the general situation
we naturally arrive at the following
\widetilde begin{defn}\label{Katsura's relations-general}
If $J$ is arbitrary (not necessarily orthogonal to $\ker\phi$), we define a pair $( A_\omega,X_\omega)$ to be
Katsura's canonical $C^*$-correspondence for the reduced
$C^*$-correspondence $X/XJ_\infty$ and the reduced ideal
$q^{J_\infty}(J)$ and also call it \emph{Katsura's canonical
$C^*$-correspondence for} $X$ and $J$.
\end{defn}
\widetilde begin{rem}
By Remark~\ref{universal description of J_infty} the coincidence of notation and terminology in Definitions~\ref{Katsura's relations} and
\ref{Katsura's relations-general} does not cause
confusion: if $J$ is orthogonal to $\ker\phi$, then
$( A_\omega,X_\omega)$ is an extension of $(A,X)$. Moreover
$( A_\omega,X_\omega)\cong(A,X)$ if and only if $J=(\ker
\phi)^\widetilde bot \cap J(X)$.
\end{rem}
One can see that the above Definition~\ref{Katsura's relations-general}
coincides with Katsura's construction and therefore by
\cite[Prop. 11.3]{katsura2} we have
\widetilde begin{thm}\label{Katsuras canonical theorem}
If $J$ is an ideal in $J(X)$ and $( A_\omega,X_\omega)$ is Katsura's canonical $C^*$-correspondence for $X$ and $J$, then we have an isomorphism
$$
\mathcal{O}(J,X)\cong \mathcal{O}((\ker \phi_{\omega})^\widetilde bot\cap J(X_\omega),
X_\omega)
$$
where $\phi_{\omega}$ denotes the left action of $ A_\omega$ on $X_\omega$.
\end{thm}
Additional discussion of Katsura's canonical $C^*$-correspondence
and its relation to the crossed product will be given
in~\ref{canonic-c*-dynam-syst}.
\subsection{Reduction of $C^*$-dynamical systems}\label{Reduction of C*-dynamical systems}
Once we have noted the relation between crossed products and
relative Cuntz-Pimsner algebras in Subsection~\ref{Crossed prod
=Relative Cuntz-Pimsner algebras} and established the reduction
procedure for relative Cuntz-Pimsner algebras in Subection
\ref{Reduction of Hilbert bimodule} it is reasonable to apply
this procedure to crossed products, and this is the theme of
the present subsection. As a by product we also establish the
coincidence of $\mathcal{O}(J,X)$ with appropriate crossed products
introduced in the article.
Let $X$ be the $C^*$-correspondence of a $C^*$-dynamical system $( A,\widetilde alphapha)$ and let $J$ be an ideal in $ A$.
We note that an ideal $I$ satisfies \eqref{X-invariance} if and only if $\widetilde alpha(I)\subset I$ hence $X$-invariance
is equivalent to $\widetilde alpha$-invariance. In particular, the ideals $J_n$, $n=0,1,...,\infty$,
from Definition \ref{reduction ideal for correspondences} are $\widetilde alpha$-invariant. Formulae \eqref{J-n} mean that
\widetilde begin{equation}
\label{e-j-n}
J_n=\underbrace{\widetilde alphapha^{-1}Big(\widetilde alphapha^{-1}\widetilde big(...(\widetilde alphapha^{-1}}_{n\textrm{ times}}(\{0\})\cap J)...\widetilde big)\cap JBig)\cap J , \,\,\,\widetilde qquad n\in \mathbb N,
\end{equation}
that is
$$
J_n=(\ker \widetilde alphapha^{n})\cap \widetilde bigcap_{k=0}^{n-1} \widetilde alphapha^{-k}(J).
$$
\widetilde begin{rem}
\label{remark-inf}
Recalling Remark \ref{universal description of
J_infty} we note that $J_\infty=\overline{\widetilde bigcup_{n\in \mathbb N} J_n}$
is the smallest ideal in $A$ such that
$$
a \in J\,\, \textrm{ and } \,\, \widetilde alpha(a)\in J_\infty \mathcal Longrightarrow
a\in J_\infty,
$$
and $J_\infty=\{0\}$ if and only if $ \ker\widetilde alphapha \cap J=\{0\}$.
\end{rem}
We give an alternative description of $J_\infty$ in the following lemma.
\widetilde begin{lem}\label{lemma about J_infty}
The sets $\overline{\{a \in A: \exists_{n\in\mathbb N}\,\, \widetilde alphapha^n(a)=0\}}$ and $\{a \in A: \lim_{n\to \infty} \widetilde alphapha^n(a)=0\}$ coincide and form the smallest $\widetilde alphapha$-invariant ideal $I_\infty$ in $ A$ such that $\widetilde alphapha$ factors through to a monomorphism on the quotient algebra $ A/I_\infty$. In particular,
\widetilde begin{equation}\label{ideal I_infinity}
J_\infty=\{a \in A: \widetilde alphapha^n(a)\in J \textrm{ for all } n\in \mathbb N \textrm{ and } \lim_{n\to \infty} \widetilde alphapha^n(a)=0\}
\end{equation}
is the largest $\widetilde alphapha$-invariant ideal contained in $J\cap I_\infty$.
\end{lem}
\widetilde begin{Proof}
One sees that both $I_1=\overline{\{a \in A: \exists_{n\in\mathbb N}\,\, \widetilde alphapha^n(a)=0\}}$ and $I_2:=\{a \in A: \lim_{n\to \infty} \widetilde alphapha^n(a)=0\}$ are $\widetilde alphapha$-invariant ideals in $ A$ such that $\widetilde alphapha$ factors through to monomorphisms both on $ A/I_1$ and $ A/I_2$. Moreover, it is clear that $I_1$ is the minimal ideal with the aforementioned properties. In particular, $I_1\subset I_2$ and to see the opposite inclusion note that since $a\mapsto \widetilde alphapha(a)$ factors through to the monomorphism (isometric mapping) on $I_2/I_1$ and $\widetilde alphapha^n(a)\to 0$ for all $a\in I_2$ it follows that $I_2/I_1=\{0\}$.
For the second part of the assertion note that
$ a\in
\widetilde bigcup_{n\in \mathbb N} J_n$ if and only if there is $n=1,2,...,$ such that
$$
\widetilde alpha^n(a)=0, \widetilde qquad \widetilde alpha^{k}(a)\in J, \widetilde qquad k=1,2,...,n-1.
$$
Hence $\widetilde bigcup_{n\in \mathbb N} J_n=(\widetilde bigcup_{n=0}^\infty\ker \widetilde alphapha^{n})\cap \widetilde bigcap_{n=0}^{\infty}\widetilde alpha^{-n}(J)
$ and thus $J_\infty=I_\infty\cap \widetilde bigcap_{n=0}^{\infty}\widetilde alpha^{-n}(J)$.
\end{Proof}
Let $n\in \mathbb N\cup\{\infty\}$. Since the ideal $J_n$ is
$\widetilde alpha$-invariant, we have a quotient $C^*$-dynamical system
$( A/J_n, \widetilde alpha_n)$ where $\widetilde alphapha_n: A/J_{n} \to A/J_n$ is given
by $ \widetilde alphapha_n\circ q_{J_n} =q_{J_n}\circ \widetilde alphapha $ and the quotient
$C^*$-correspondence $X/X J_n$ may be viewed as the
$C^*$-correspondence of $( A/J_n,
\widetilde alphapha_n)$. In particular
\widetilde begin{equation}\label{delta_infinity}
\widetilde alphapha_\infty(a+ J_{\infty}):=\widetilde alphapha(a) + J_\infty
\end{equation}
is an endomorphism of $ A/J_{\infty}$ such that
\widetilde begin{equation}\label{J-infty}
\ker \widetilde alphapha_\infty \cap q_{J_{\infty}}(J)=\{0\}.
\end{equation}
Obviously one may apply Theorem \ref{reduction thm} to each of the systems $( A/J_n,\widetilde alphapha_n)$, $n\in \mathbb N\cup\{\infty\}$, however,
we focus on the case $n=\infty$. Then by virtue of Theorem~\ref{reduction thm} and Corollary~\ref{C-P-cross} along
with \eqref{J-infty} we get
\widetilde begin{prop}\label{reducing C*-Hilbert bimodules}
If $X$ is the $C^*$-correspondence of a $C^*$-dynamical system
$( A,\widetilde alphapha)$ and $J$ is an ideal in $ A$, then
$$\mathcal{O}(J,X)=\mathcal{O}( q_{J_\infty}(J), X/XJ_\infty)= C^* ( A/J_{\infty}, \widetilde alphapha_\infty,q_{J_\infty}(J))$$
is a universal algebra generated by a copy of the algebra $ A/J_{\infty}$ and a partial isometry $u$ subject to relations
$$
ua u^*=\widetilde alphapha_\infty(a),\,\,\, a\in A/J_{\infty}, \widetilde qquad q_{J_\infty}(J)=\{a\in A/J_{\infty}: u^*u a=a\}.
$$
\end{prop}
\widetilde begin{cor}\label{kernel of a covariant representation}
If $(\pi,U, H)$ is a $J$-covariant representation of
$( A,\widetilde alphapha)$, then $ J_\infty \subset \ker \pi. $
\end{cor}
\widetilde begin{ex}\label{reduction of Stacey's crossed product}
Let us apply the above results to Stacey's crossed product
(Definition~\ref{Stacey}), and in particular, obtain a refinement
of \cite[Prop. 2.2]{Stacey}.
Note that if $J=A$, then
$$J_\infty=\{a\in A: \lim_{n\to \infty}\widetilde alphapha^n(a)=0\}=\overline{\widetilde bigcup_{n=0}^\infty \ker\widetilde alpha^n},$$ cf. Lemma \ref{lemma about J_infty}.
Accordingly, the system $({ A_\infty},{\widetilde alphapha_\infty})$, $ A_\infty:= A/J_\infty$, obtained by the quotient of $( A,\widetilde alphapha)$ by $J_\infty$
could be considered as the largest subsystem of $( A,\widetilde alphapha)$ with the property that ${\widetilde alphapha_\infty}:{ A_\infty}\to { A_\infty}$
is a monomorphism. Moreover, by
Corollary \ref{corollary stacey's} and Proposition \ref{reducing C*-Hilbert bimodules}, Stacey's crossed product is a universal $C^*$-algebra generated by a copy of $A_\infty$ and an isometry $u$ such that $ua u^*=\widetilde alphapha_\infty(a)$, for all $a\in A_\infty$. In particular, the Stacey's crossed product is the crossed product
$C^*(A_\infty,\widetilde alphapha_\infty,A_\infty)$ studied in the present paper and it reduces to the zero algebra if and only if $ A=\{a\in A: \lim_{n\to \infty}\widetilde alphapha^n(a)=0\}$.
\end{ex}
\section{Canonical $C^*$-dynamical systems}\label{Canonical C*-dynamical systems}
Looking at Table 1 one can not help feeling that among the ideals
satisfying $\{0\}\subset J \subset (\ker\widetilde alphapha)^\widetilde bot$ the
ideal $J=(\ker\widetilde alphapha)^\widetilde bot$ is somewhat privileged. Taking into
account Cuntz-Pimsner algebras and their established relation to
crossed products it may seem completely natural as
$\mathcal{O}((\ker\widetilde alphapha)^\widetilde bot,X)$ should be considered as 'the smallest'
relative Cuntz-Pimsner algebra containing all the information
about the $C^*$-dynamical system $( A,\widetilde alphapha)$. Now, much in the spirit of Katsura's construction, cf. subsection \ref{Katsura's canonical relations}, yet in a slightly different way we will
show that for an arbitrary choice of $J$ the algebra
$\mathcal{O}(J,X)$ coincides with Cuntz-Pimsner algebra $\mathcal{O}(
(\ker\widetilde alphapha_J)^\widetilde bot, X_J)$ where $X_J$ is the
$C^*$-correspondence of a canonically constructed
$C^*$-dynamical system $( A_J,\widetilde alphapha_J)$ which
will be presented below (see Definition~\ref{definition of the
canon}, Theorem \ref{canon}).
\subsection{Unitization of the kernel of an
endomorphism}\label{unit}
Let us fix a $C^*$-dynamical system $( A,\widetilde alphapha)$ and an ideal $J$ in $ A$.
Above results show us how to reduce the investigation of crossed products to the case where $J$ is orthogonal to $\ker\widetilde alpha$, thereby let us assume for a while that $\ker\widetilde alphapha\cap J=\{0\}$. The first named author described in \cite{kwa4} a procedure of extending
$( A,\widetilde alphapha)$ up to a $C^*$-dynamical system $( A^+,\widetilde alphapha^+)$ with a property that the kernel of $\widetilde alphapha^+$ is unital.
Moreover, the resulting system $( A^+,\widetilde alphapha^+)$ is in a sense the smallest extension of $( A,\widetilde alphapha)$
possessing that property, see \cite{kwa4}.
Let us now slightly generalize this construction, which will be essential for our future purposes.
Our extension construction depends on the choice of an ideal
orthogonal to $I:={\ker}\,\widetilde alphapha$ (recall Definition~\ref{ort}). Thus let us fix a certain ideal $J$ which
satisfies
$$
\{0\} \,\subset \,J\, \subset\, I^\widetilde bot.
$$
By $ A_J$ we denote the direct sum of quotient algebras
$$
A_J=\widetilde big( A/I\widetilde big) \oplus \widetilde big( A/J\widetilde big),
$$
and we set $\widetilde alphapha_J: A_J\to A_J$ by the formula
\widetilde begin{equation}
\label{d_J}
A_J\ni \widetilde big((a +I)\oplus (b
+J)\widetilde big)\stackrel{\widetilde alphapha_J}{\longrightarrow}(\widetilde alphapha(a) +I)\oplus
(\widetilde alphapha(a) +J)\in A_J.
\end{equation}
Since $I={\rm ker}\,\widetilde alphapha$ it follows that an element
$\widetilde alphapha(a)$ does not depend on the choice of a representative
of $a +I$ and so the mapping $\widetilde alphapha_J$ is well defined.
Note that, as $I\cap J = \{0\}$,
$$
\widetilde alphapha_J ([a],[b]) = (0,0)\ \mathcal Leftrightarrow \ \widetilde alphapha(a)\in I \ \text{and} \ \widetilde alphapha(a)\in J \ \mathcal Leftrightarrow \ \widetilde alphapha(a)=0.
$$
And therefore
\widetilde begin{equation}
\label{kerd_J} \ker \widetilde alphapha_J = (0,\, A/J ) \widetilde qquad \text{and}\widetilde qquad (\ker {\widetilde alphapha_J})^\widetilde bot =( A/I,\, 0 ).
\end{equation}
Clearly,
$\widetilde alphapha_J$ is an endomorphism and its kernel is unital with the
unit of the form $(0 + I) \oplus (1 +J)$.
Moreover, the $C^*$-algebra $ A$ embeds into $C^*$-algebra $ A_J$ via
\widetilde begin{equation}
\label{A_in} A \ni a \longmapsto \widetilde big(a +I\widetilde big)\oplus \widetilde big(a +
J\widetilde big)\in A_J.
\end{equation}
Since $I\cap J = \{0\}$ this mapping is injective and we will
treat $ A$ as the corresponding subalgebra of $ A_J$. Under this
identification $\widetilde alphapha_J$ is an extension of $\widetilde alphapha$, and in particular $\widetilde alphapha_J ( A_J)=\widetilde alphapha( A)\subset A$.
\widetilde begin{rem} The larger ideal $J$ is the smaller $ A_J$ is.
Namely, since $ {\rm ker}{\widetilde alphapha_J} = (0,\, A/J ) $
and
$J$ varies from $\{0\}$ to
$I^\widetilde bot$ it follows that the kernel of $\widetilde alphapha_J$ varies from $(0,\, A )$ to
$(0,\, A/I^{\widetilde bot})$. On the other hand,
the image of
$\widetilde alphapha_J$ always concides with $\widetilde alphapha( A)\cong A/I$
and thereby it does not depend on the choice of $J$.
The case when $J=I^\widetilde bot$ was considered in \cite{kwa4}.
\end{rem}
The next statement presents a motivation of the preceding construction.
\widetilde begin{prop}\label{motivation prop}
Let $(\pi,U,H)$ be a faithful $J$-covariant representation of $( A,\widetilde alphapha)$.
Then $J$ is
orthogonal to $I=\ker\widetilde alphapha$ (cf. Corollary \ref{motivation prop1}) and if $( A_J,\widetilde alphapha_J)$ is the extension of $( A,\widetilde alphapha)$ constructed above,
then $\pi$ uniquely extends to the isomorphism $\widetilde{\pi}: A_J\to C^*(\pi( A),U^*U)$ such that
$(\widetilde{\pi},U,H)$ is a faithful covariant representation of $( A_J,\widetilde alphapha_J)$.
Namely $\widetilde{\pi}$ is given by
\widetilde begin{equation}\label{extension of pi to J-algebra}
\widetilde{\pi}(a + I \oplus b + J)= U^*U \pi(a) + (1- U^*U) \pi(b), \widetilde qquad a,b \in A.
\end{equation}
\end{prop}
\widetilde begin{Proof}
If $\widetilde{\pi}: A_J\to C^*(\pi( A),U^*U)$ is onto and
$(\widetilde{\pi},U,H)$ is a faithful covariant representation of
$( A_J,\widetilde alphapha_J)$, then by Proposition \ref{proposition najwazniejsze} we have
$$
U^*U = \widetilde{\pi}(1 + I \oplus 0 + J).
$$
Thus $\widetilde{\pi}$ is of the form \eqref{extension of pi to
J-algebra} where $\pi=\widetilde{\pi}|_{ A}$ and plainly $(\pi,U,H)$ is a faithful $J$-covariant representation of $( A,\widetilde alphapha)$. Conversely, let $(\pi,U,H)$ be a faithful $J$-covariant representation of $( A,\widetilde alphapha)$.
Then by Corollary \ref{ort-id} $$
\pi( I)= [(1-U^*U)\pi( A)] \cap \pi( A), \widetilde qquad \pi(J) = [U^*U\pi( A)]\cap \pi( A),
$$
and consequently, cf. for instance \cite[Lem. 10.1.6]{Kadison}, we have natural isomorphisms
$$
A/ I \cong U^*U \pi( A), \widetilde qquad A/ J \cong [(1-U^*U)\pi( A)].
$$
Thus formula \eqref{extension of pi to
J-algebra} defines an isomorphism $\widetilde{\pi}: A_J\to C^*(\pi( A),U^*U)$. It is readily checked, cf. Proposition \ref{proposition najwazniejsze}, that
$(\widetilde{\pi},U,H)$ is a covariant representation of
$( A_J,\widetilde alphapha_J)$.
\end{Proof}
\widetilde begin{thm}
Let $X$ be the $C^*$-correspondence of a $C^*$-dynamical system
$( A,\widetilde alphapha)$ and let~$J$ be an ideal in $ A$ orthogonal to the
kernel of $\widetilde alphapha$. If $X_J$ is the $C^*$-correspondence of the system
$({ A_J},{\widetilde alphapha_J})$ constructed above, then
$$
\mathcal{O}(J,X)=\mathcal{O}( (\ker{\widetilde alphapha_J})^\widetilde bot, X_J) = C^*( A,\widetilde alphapha, J)=C^*({ A_J},{\widetilde alphapha_J},(\ker{\widetilde alphapha_J})^\widetilde bot)
$$
is a universal algebra generated by a copy of the algebra ${ A_J}$ and a partial isometry $u$ subject to relations
\widetilde begin{equation}\label{reduced realtions1}
u a u^*={\widetilde alphapha_J}(a),\,\,\, a\in { A_J}, \widetilde qquad u^*u\in { A_J}
\end{equation}
(relations \eqref{reduced realtions1} imply that $u^*u$ belongs to the center of ${ A_J}$, cf. Proposition \ref{proposition najwazniejsze}).
\end{thm}
\widetilde begin{proof}
In view of Corollary \ref{C-P-cross} we have natural identifications $\mathcal{O}(J,X)=C^*( A,\widetilde alphapha, J)$ and $\mathcal{O}( (\ker{\widetilde alphapha_J})^\widetilde bot, X_J) =C^*({ A_J},{\widetilde alphapha_J},(\ker{\widetilde alphapha_J})^\widetilde bot)$. In order to prove that $C^*( A,\widetilde alphapha, J)=C^*({ A_J},{\widetilde alphapha_J},(\ker{\widetilde alphapha_J})^\widetilde bot)$ it suffices to show that
we have a one-to-one correspondence between faithful $J$-covariant representations $(\pi,U,H)$ of $( A,\widetilde alphapha)$ and faithful $(\ker{\widetilde alphapha_J})^\widetilde bot$-covariant representations $(\widetilde{\pi},U,H)$ of $({ A_J},{\widetilde alphapha_J})$, but this follows from Proposition \ref{motivation prop} since by Proposition \ref{proposition najwazniejsze} a faithful covariant representation $(\widetilde{\pi},U,H)$ is $(\ker{\widetilde alphapha_J})^\widetilde bot$-covariant if and only if $U^*U\in \widetilde{\pi}( A_J)$.
\end{proof}
\subsection{Canonical $C^*$-dynamical systems}
\label{canonic-c*-dynam-syst}
Proposition \ref{reducing C*-Hilbert
bimodules} describes the natural reduction of relations to the case when $J\subset (\ker\widetilde alphapha)^\widetilde bot$. This proposition along with the argument of Subsection \ref{unit} gives us a tool to achieve the goal of the present section; namely, to reduce the whole construction to the case when $\ker\widetilde alphapha$ is unital and $J=(\ker\widetilde alphapha)^\widetilde bot$.
\widetilde begin{defn}\label{definition of the canon}
Let $( A,\widetilde alphapha)$ be a $C^*$-dynamical system and $J$ an arbitrary ideal in~$ A$. Let $(( A/J_\infty)_{q^{J_\infty}(J)},(\widetilde alphapha_\infty)_{q^{J_\infty}(J)})$ be the above constructed extension of the reduced $C^*$-dynamical system $( A/J_\infty,\widetilde alphapha_\infty)$ given by \eqref{ideal I_infinity}, \eqref{delta_infinity}. We will write
$$
({ A_J},{\widetilde alphapha_J}):=( A/J_\infty)_{q^{J_\infty}(J)},(\widetilde alphapha_\infty)_{q^{J_\infty}(J)})
$$
and say that $({ A_J},{\widetilde alphapha_J})$ is the \emph{canonical $C^*$-dynamical system} associated with $( A,\widetilde alphapha)$ and~$J$.
\end{defn}
\widetilde begin{rem}\label{A_J} By Remark~\ref{remark-inf} the above notation does not cause confusion (in the situation when $
\{0\} \,\subset \,J\, \subset\, I^\widetilde bot $ the pair
$({ A_J},{\widetilde alphapha_J})$ coincides with the corresponding pair
introduced in \ref{unit}) and therefore we keep the notation
$({ A_J},{\widetilde alphapha_J})$ in the general situation.
\end{rem}
Combining Propositions \ref{reducing C*-Hilbert bimodules}, \ref{motivation prop}, see also \cite[Cor. 1.7]{kwa4}, we get
\widetilde begin{thm}\label{canon}
Let $X$ be the $C^*$-correspondence of $( A,\widetilde alphapha)$ and let
$J$ be an ideal in $ A$. If $X_J$ is the $C^*$-correspondence of
the canonical system $({ A_J},{\widetilde alphapha_J})$, then
$$
\mathcal{O}(J,X)=\mathcal{O}( (\ker{\widetilde alphapha_J})^\widetilde bot, X_J) = C^*({ A_J},{\widetilde alphapha_J},(\ker{\widetilde alphapha_J})^\widetilde bot)
$$
is a universal algebra generated by a copy of the algebra ${ A_J}$ and a partial isometry $u$ subject to relations
\widetilde begin{equation}\label{reduced realtions}
u a u^*={\widetilde alphapha_J}(a),\,\,\, a\in { A_J}, \widetilde qquad u^*u\in { A_J}.
\end{equation}
\end{thm}
\widetilde begin{rem}\label{6.7}
The usefulness of canonical $C^*$-dynamical system
$( A_J,\widetilde alphapha_J)$ manifests in reducing
relations \eqref{covariance rel1*},
\eqref{covariance rel3}, that apart from endomorphism involve an
ideal and which may degenerate, to the nondegenerated natural
relations \eqref{reduced realtions}. In fact, one could go even
further and use the construction from \cite{kwa4} to extend, the
canonical system $( A_J,\widetilde alphapha_J)$ up to a $C^*$-dynamical system
$(B,\widetilde \widetilde alphapha)$ possessing a complete transfer operator (cf.
subsection \ref{ABL=Kwa-Leb}). Then $B$ corresponds to the
fixed point subalgebra of $\mathcal{O}(J,X)$ for the gauge action $\gamma$
(Proposition \ref{RCP algebra}), and by \cite[Prop. 1.9]{kwa3} the
$C^*$-correspondence $\widetilde X$ of the $C^*$-dynamical system
$(B,\widetilde \widetilde alphapha)$ is actually a Hilbert bimodule (in the sense of
Remark~\ref{Hilbbert-bim}). Thus $\mathcal{O}(J,X)$ can be modeled not
only by the crossed product of $(B,\widetilde \widetilde alphapha)$, N.3 in Table
\ref{table 1}, cf. Proposition \ref{crossed-ABL1}, but also by
by the $C^*$-correspondence $\widetilde X$, N.4 in
Table \ref{table 2}. Hence the results of \cite{Ant-Bakht-Leb},
\cite{aee} or isomorphism theorem \cite{kwa-ck}
applied to $(B,\widetilde \widetilde alphapha)$ can be exploited in the study of
$\mathcal{O}(J,X)$ in terms of 'Fourier' coefficients.
\end{rem}
We end up by noting that Katsura's 'canonical relations' for
$C^*$-correspondences (see Definitions~\ref{Katsura's
relations},~\ref{Katsura's relations-general}) when applied to the
$C^*$-correspondence $X$ of $( A,\widetilde alphapha)$ also leads to a
certain dynamical system, which however in general is slightly
smaller than $( A_J,\widetilde alphapha_J)$ and is less natural in our context.
Indeed, by passing if necessary to the reduced objects, we need to
consider only the case when $J\subset (\ker\widetilde alpha)^\widetilde bot$, and then
$$
A_\omega=\{(a,q_J(a'))\in A\oplus A/J: q_{(\ker\widetilde alpha)^\widetilde bot}(a)=q_{(\ker\widetilde alpha)^\widetilde bot}(a')\}.
$$
In particular, the mapping
$$
A_\omega\ni (a,q_J(a'))\stackrel{\widetilde alphapha_\omega}{\longmapsto} (\widetilde alpha(a),q_{J}(\widetilde alpha(a))) \in A_\omega
$$
yields a well defined endomorphism $\widetilde alphapha_\omega: A_\omega\to A_\omega$, and one sees that $X_\omega$ coincides with the $C^*$-correspondence of the $C^*$-dynamical systems $( A_\omega,\widetilde alphapha_\omega)$.
Thus we have three $C^*$-dynamical systems $( A,\widetilde alphapha)$,
$( A_\omega,\widetilde alphapha_\omega)$, $( A_J,\widetilde alphapha_J)$, and each of them is
an extension of the proceeding one. Indeed, we have natural
homomorphisms
$$
A \ni a \stackrel{\iota_1}{\longmapsto} a\oplus q_J(a) \in A_\omega, \widetilde qquad A_\omega \ni (a,q_J(a')) \stackrel{\iota_2}{\longmapsto} q_{\ker\widetilde alpha}(a)\oplus q_J(a') \in A_J.
$$
Clearly, $\iota_1$ is injective and to see that $\iota_2$ is injective note that $\iota_2(a,q_J(a'))=0$ means that $a\in\ker\widetilde alphapha$ and $a'\in J\subset (\ker\widetilde alpha)^\widetilde bot $, and then relation $q_{(\ker\widetilde alpha)^\widetilde bot}(a)=q_{(\ker\widetilde alpha)^\widetilde bot}(a')=0$ imply that $a=0$. The monomorphisms $\iota_1, \iota_2 $ make the following diagram commute
$$
\widetilde begin{xy}
\widetilde xymatrix{ A \widetilde ar[d]_{\widetilde alphapha} \widetilde ar[r]^{\iota_1} & A_\omega\widetilde ar[d]_{\widetilde alphapha_\omega} \widetilde ar[r]^{\iota_2} & A_J \widetilde ar[d]^{\widetilde alpha_J}
\\
A \widetilde ar[r]_{\iota_1} & A_\omega \widetilde ar[r]_{\iota_2 }& A_J
}
\end{xy}.
$$
Moreover, $\iota_1$ is an isomorphism iff $J=(\ker\widetilde alpha)^\widetilde bot$ and $\iota_2$ is an isomorphism iff $(\ker\widetilde alpha)^\widetilde bot$ is unital. In particular, in 'Katsura's picture',
the crossed product $C^*( A,\widetilde alphapha, J)$ could be considered as a universal $C^*$-algebra subject to relations
$$
ua u^*=\widetilde alphapha_\omega(a),\widetilde quad a \in A_\omega,\widetilde qquad
\{a\in A_\omega: u^*u a=a\}=(\ker\widetilde alpha_\omega)^\widetilde bot,
$$
which apparently are more complicated than relations \eqref{reduced realtions}.
\widetilde begin{thebibliography}{99}
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No 4, pp. 1133-1141.
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Ergodic Theory Dynam. Systems, Vol 23, (2003), pp. 1733-1750,
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Indiana Univ. Math. J. {\widetilde bf 48} (1999), 155--181.
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\emph{Fundamentals of the theory of operator algebras. Vol.2. Advanced theory}, Academic Press, 1986.
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T. Katsura ''A construction of $C^*$-algebras from $C^*$-correspondences'' Contemp. Math. vol. 335, pp. 173-182, Amer. Math. Soc., Providence (2003)
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Katsura T. ``On $C^*$-algebras associated with $C^*$-correspondences'' J. Funct. Anal. \textbf{217}(2), 366-401 (2004)
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T. Katsura ``Ideal structure of $C^*$-algebras associated with $C^*$-correspondences'' Pacific J. Math. \textbf{230}(1), 107--146 (2007)
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B. K. Kwa\'sniewski: "Covariance algebra of a partial dynamical system", CEJM, 2005, V.3, No 4, pp. 718-765
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to systems with complete transfer operators",
arXiv:\penalty0 math.OA/0703800
\widetilde bibitem[Kwa10]{kwa-ck} B. K. Kwa\'sniewski: "Cuntz-Krieger uniqueness theorem for crossed products by Hilbert bimodules", arXiv:\penalty0 1010.0446
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Rocky J. Math. 42, No 3 (2012), 919-938.
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``Reversible extensions of irreversible dynamical systems: $C^*$-method'',
Math. Sbor., \textbf{199} (2008), No 11, 45-74.
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$C^*$-algebras by partial isometries'', Matemat.\ Sbornik, 2004.
V.~195, No 7, pp.~37--70 (Russian).
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by semigroups of endomorphisms", Journal of Operator Theory, 52 61-87 (2004)
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W.\,L.\ Paschke, "The crossed product of a $C^*$-algebra by an
endomorphism", Proceedings of the AMS, {\widetilde bfseries 80}, No 1, (1980),
p.~113--118.
\widetilde bibitem[Ped79]{Pedersen} G. K. Pedersen: \emph{$C^*$-algebras and their automorphism groups},
Academic Press, London, 1979.
\widetilde bibitem[Pim97]{p} M. V. Pimsner, "A class of $C^*$-algebras generalizing both Cuntz-Krieger algebras and crossed products by $\mathbb Z$", Fields Institute Communications {\widetilde bf 12} (1997), 189--212.
\widetilde bibitem[Sta93]{Stacey}
P.\,J.\ Stacey, "Crossed products of $C^*$-algebras by
$^*$-endomorphisms", J.\ Austral.\ Math.\ Soc. Ser.~A {\widetilde bfseries 54},
(1993), p.~204--212.
\end{thebibliography}
\textbf{B. K. Kwa\'sniewski},
\noindent \textsc{Institute of Mathematics, University of Bialystok},
\textsc{ ul. Akademicka 2, PL-15-267 Bialystok, Poland }\\
\emph{e-mail:} \texttt{[email protected]}, \\
\emph{www:} \texttt{http://math.uwb.edu.pl/$\sim$zaf/kwasniewski}
\widetilde bigskip
\noindent \textbf{A. V. Lebedev},
\noindent \textsc{Department of Mechanics and Mathematics,
Belarus State University,
pr. Nezavisimosti, 4, 220050, Minsk, Belarus}, $\&$ \\
\textsc{Institute of Mathematics, University of Bialystok},
\textsc{ ul. Akademicka 2, PL-15-267 Bialystok, Poland},
\emph{e-mail:} \texttt{[email protected]}
\end{document}
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math
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Curious about the new Caching service running in Windows Azure Platform? In the video, Karandeep Anand, talks about the new service and the different ways you can leverage it within your own apps.
Windows Azure AppFabric Caching service accelerates app performance by providing a distributed, in-memory app cache requiring no installation, configuration, or management.
The CTP release of the Caching service was made available in AppFabric LABS at PDC10. To get started, go to: Windows Azure AppFabric Caching.
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english
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दफनाए हुए बच्चे के शव को निकाला बाहर : राजस्थान
बाय भारत खबर ऑन जून १९, २०१७ ६:१७ प्म कम्मंट ऑफ ऑन दफनाए हुए बच्चे के शव को निकाला बाहर : राजस्थान
नागौर। गैंगरेप के बाद दिया महिला ने बच्चे को जन्म लेकिन पांच दिनों के नवजात की मौत हो गई थी, इस मामले में एक नया मोड़ ले लिया है। अलाय गांव का चर्चित गैंगरेप मामला एक बार फिर से चर्चा में आता नजर आ रहा है। गैंगरेप पीड़िता ने मांग की है उसके मृतक बच्चे का दुबारा डीएनए करने की मांग की है,जिसके बाद पुलिस व प्रशासनिक अधिकारी रविवार दोपहर को अलाय पहुंचे और वहां बालक का दफनाऐ गये शव निकालकर डीएनए टेस्ट के लिए कुछ सैम्पल लिए है।
दफनाऐ गए बच्चे के शव सैम्पल लेने पहुंचे नागौर एसडीएम परसाराम टाक, मकराना सीओ पूनमसिंह व नागौर जेएलएन अस्पताल के चिकित्सकों की टीम मौजूद रही। टीम ने बच्चे के गड़े शव को बाहर निकाला और डीएनए के लिए कुछ सैंपल लिए। पूरे क्षेत्र में ये खबर फैलने के बाद मौके पर बड़ी तादात में ग्रामीण पहुंच गए और बच्चे का शव निकालना चर्चा का विषय बन गया।
आपको बता दे की गत वर्ष ८ अगस्त को श्रीबालाजी थाने में रिपोर्ट देकर अलाय निवासी आरएसी कांस्टेबल की पत्नी ने आरोप लगाया की गांव के एक शिक्षक कैलाश, पुलिसकर्मी सुभाष व एक अन्य युवक सहीराम ने उसके साथ कई बार गैंगरेप किया था।
पीड़िता ने रिपोर्ट में यह भी आरोप लगाया था की उसके पेट में बच्चा है और वह गैंगरेप के चलते हुआ है। मामला दर्ज कराने के कुछ समय बाद यानी २६ अगस्त को पीड़िता ने बीकानेर के पीबीएम अस्पताल में बालक को जन्म दिया, लेकिन जन्म के ५ दिन बाद ही नागौर के जेएलएन अस्पताल में बच्चे की मौत हो गई थी।
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hindi
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// Copyright (c) 2012 The Chromium Authors. All rights reserved.
// Use of this source code is governed by a BSD-style license that can be
// found in the LICENSE file.
#include "content/browser/renderer_host/media/audio_sync_reader.h"
#include <algorithm>
#include "base/process_util.h"
#include "base/shared_memory.h"
#include "base/threading/platform_thread.h"
#include "media/audio/audio_buffers_state.h"
#include "media/audio/audio_util.h"
#if defined(OS_WIN)
const int kMinIntervalBetweenReadCallsInMs = 10;
#endif
AudioSyncReader::AudioSyncReader(base::SharedMemory* shared_memory)
: shared_memory_(shared_memory) {
}
AudioSyncReader::~AudioSyncReader() {
}
bool AudioSyncReader::DataReady() {
return !media::IsUnknownDataSize(
shared_memory_,
media::PacketSizeSizeInBytes(shared_memory_->created_size()));
}
// media::AudioOutputController::SyncReader implementations.
void AudioSyncReader::UpdatePendingBytes(uint32 bytes) {
if (bytes != static_cast<uint32>(media::AudioOutputController::kPauseMark)) {
// Store unknown length of data into buffer, so we later
// can find out if data became available.
media::SetUnknownDataSize(
shared_memory_,
media::PacketSizeSizeInBytes(shared_memory_->created_size()));
}
if (socket_.get()) {
socket_->Send(&bytes, sizeof(bytes));
}
}
uint32 AudioSyncReader::Read(void* data, uint32 size) {
uint32 max_size = media::PacketSizeSizeInBytes(
shared_memory_->created_size());
#if defined(OS_WIN)
// HACK: yield if reader is called too often.
// Problem is lack of synchronization between host and renderer. We cannot be
// sure if renderer already filled the buffer, and due to all the plugins we
// cannot change the API, so we yield if previous call was too recent.
// Optimization: if renderer is "new" one that writes length of data we can
// stop yielding the moment length is written -- not ideal solution,
// but better than nothing.
while (!DataReady() &&
((base::Time::Now() - previous_call_time_).InMilliseconds() <
kMinIntervalBetweenReadCallsInMs)) {
base::PlatformThread::YieldCurrentThread();
}
previous_call_time_ = base::Time::Now();
#endif
uint32 read_size = std::min(media::GetActualDataSizeInBytes(shared_memory_,
max_size),
size);
// Get the data from the buffer.
memcpy(data, shared_memory_->memory(), read_size);
// If amount read was less than requested, then zero out the remainder.
if (read_size < size)
memset(static_cast<char*>(data) + read_size, 0, size - read_size);
// Zero out the entire buffer.
memset(shared_memory_->memory(), 0, max_size);
// Store unknown length of data into buffer, in case renderer does not store
// the length itself. It also helps in decision if we need to yield.
media::SetUnknownDataSize(shared_memory_, max_size);
return read_size;
}
void AudioSyncReader::Close() {
if (socket_.get()) {
socket_->Close();
}
}
bool AudioSyncReader::Init() {
socket_.reset(new base::CancelableSyncSocket());
foreign_socket_.reset(new base::CancelableSyncSocket());
return base::CancelableSyncSocket::CreatePair(socket_.get(),
foreign_socket_.get());
}
#if defined(OS_WIN)
bool AudioSyncReader::PrepareForeignSocketHandle(
base::ProcessHandle process_handle,
base::SyncSocket::Handle* foreign_handle) {
::DuplicateHandle(GetCurrentProcess(), foreign_socket_->handle(),
process_handle, foreign_handle,
0, FALSE, DUPLICATE_SAME_ACCESS);
if (*foreign_handle != 0)
return true;
return false;
}
#else
bool AudioSyncReader::PrepareForeignSocketHandle(
base::ProcessHandle process_handle,
base::FileDescriptor* foreign_handle) {
foreign_handle->fd = foreign_socket_->handle();
foreign_handle->auto_close = false;
if (foreign_handle->fd != -1)
return true;
return false;
}
#endif
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code
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A SpaceX Hyperloop award winning team and globally distributed organisation with over 1300 engineers from more than 59 countries.
rLoop is a decentralised crowdsourced research and development organisation, engineering solutions to some of the world’s greatest challenges. It is designed to enable anyone, anywhere, at any time, to participate in the creation, development, and scrutiny of potentially world-changing technology.
Our mission is to develop and launch innovative technology fueled by a genuine desire to improve the world and humanity.
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english
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Lenses For Ray Diagram Physics. Obd2 Wiring Diagram Gm.
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english
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یہ نفر یس لانڈری چھ کران سہ چھ واریاہ پونٛسہٕ نوان تہٕ زانٛہہ تہ وقتس پؠٹھ واپس یوان
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kashmiri
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Yo Amo el Diseño / I Love Design. May 3rd, 2008.
Open Doors, located in the Miami Modern District will present a weekend-long celebration of distinctive architecture, design and culture. Located in the former International Book Building constructed in 1964 on 7300 Biscayne Boulevard, the 4,000 square foot furniture and accessories boutique invites MiMo fest revelers to join in its Yo Amo el Diseño reception on May 3rd, 2008 from 5:00 p.m. to 8:00 p.m. Open Doors is open throughout the Cinco de MiMo weekend including Friday and Saturday from 11:00 a.m. to 8:00 p.m. and Sunday from noon to 6:00 p.m.
An alternative world of abstract works by ArtCenter/South Florida’s Alekxey Sabido will set the tone at Yo Amo el Diseño, which combines Mexican elements, vibrant colors and Open Doors’ signature Contemporary Tropical style. The urbane space of Open Doors is characterized by a Contemporary Tropical and Transitional style of furniture with clean, natural lines, and is interspersed with vivid furnishings and accessories from abroad.
For more information, please call 305.751.1023.
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english
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जब हम ये सुनते हैं भारतीय रेलवे दुनिया में सबसे ज्यादा लोगों को रोजगार देने वाली संस्था है तो गौरवान्तित महसूस करते हैं लेकिन क्या आपको पता है कि आज भी भारत में ऐसा रेलवे ट्रैक है जो ब्रिटेन के कब्जे...
नई दिल्ली। देश आज अपना ७१वां स्वतंत्रता दिवस मनाएगा। इस मौके पर लाल किले की प्राचीर से प्रधानमंत्री नरेंद्र मोदी तिरंगा फहराया। आज सुबह ७.३० बजे पीएम मोदी लाल किले की प्रचीर पर तिरंगा फहराया। लाल...
लखनऊ। देश की आजादी के हजारो क्रांतिकारियों ने अपनी जान गंवाई लेकिन भारतीय आजादी की लड़ाई के सबसे युवा शहीदों में खुदीराम बोस नाम शामिल था। उन्हें आज ही के दिन ११ अगस्त १९०८ को फांसी दे दी गई थी।...
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hindi
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क्रिकेट | हिम दर्शन समाचार
भारत ने विराट कोहली की गैरमौजूदगी में खेले गए तीन मैचों की सीरीज के आखिरी मैच में मेजबान दक्षिण अफ्रीका को मा...
मुंबई टी-२० को ५ विकेट से जीतकर भारत ने ३-० से किया लंका दहन-
इंडिया ने श्रीलंका को मुंबई टी-२० मैच में ५ विकेट से हराकर ३ मैचों की टी-२० सीरीज ३-० से जीत लिया है। इस मैच में टॉस हारकर पहले बल्लेबाजी करते हुए श्रीलंका की टीम ने भारत को जीत के लिए 1३6 र... रेड मोर
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hindi
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نوٗرِ حق چوں مولوی رومی بدید مثنوی زاں نور آمد در پدید
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kashmiri
|
Guests staying at Zoetry Villa Rolandi Isla Mujeres Cancun quickly find out that indulging in a relaxing state of mind has never been so simple or so close. Located on the small island of Isla Mujeres only miles away from Cancun's coast, this beachfront villa resort is ideally situated between the Caribbean Sea and Lagoon Makax. First class service begins with a private yacht transfer from Cancun to Isla Mujeres and continues throughout your stay. Resort highlights include stunning beaches, gourmet dining and the hotel spa's unique "Thalasso Therapy." Zoetry Villa Rolandi Isla Mujeres Cancun are also blessed once a year with the opportunity to experience whale sharks in their natural habitat. From June through September hundreds of whale sharks gather just north of Isla Mujeres in a seven-mile radius to take advantage of the plankton rich waters created by the joining of the Gulf of Mexico and the Caribbean Sea. The hotel's concierge can offer tours and more detailed information.
Room amenities include private terrace with jacuzzi, mini bar with beer, juice, soft drinks and bottled water, fresh bottle of sparkling wine and fresh fruit daily, satellite television, telephone, wireless internet access, safety deposit box, coffee/tea maker, Bvlgari bath amenities, and hairdryer.
One king bed. Sitting area features a sofa bed and leads to a private terrace with a Whirlpool. Spa-like bathroom with dual therapeutic showers and a steam bath. Stunning ocean views. 498 square feet.
One king bed or two single beds. Sitting area with a sofa-bed. Spa-like bathroom with dual therapeutic showers and a steam bath. Direct pool access terrace with a sea view lounge sitting area. Spectacular ocean views of the Caribbean Sea. 498 sq. ft.
One king bed. Sitting area features a sofa bed and leads to a private terrace with a Whirlpool. Spa-like bathroom with dual therapeutic showers and a steam bath. Stunning ocean views. 818 square feet.
One king bed. The suite features a master bedroom, a separate living & dining area and a private bar and terrace with an ocean view. A private solarium is located on the third level and features an outdoor Jacuzzi. 2,067 square feet.
Casa Rolandi - Casa Rolandi is synonymous with gourmet tradition in Cancun and Isla Mujeres. Its secret lays in the creativity of the Swiss-Northern Italian cuisine and the delicious, exclusive recipes cooked in its firewood oven that seal in the juices and retain their original flavors.
Cozy Bar - Enjoy top-shelf cocktails in the lobby bar.
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english
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Husky gazing with its beautiful blue eyes Animal desktop wallpaper, Husky wallpaper, Dog wallpaper, Animal wallpaper - Animals no. 50568. Download this Husky gazing with its beautiful blue eyes Husky Dog desktop wallpaper in multiple resolutions for free.
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english
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\begin{document}
\title{Entanglement detection with imprecise measurements}
\author{Simon Morelli}\thanks{S.M. and H.Y. contributed equally to this manuscript.}
\author{Hayata Yamasaki}\thanks{S.M. and H.Y. contributed equally to this manuscript.}
\author{Marcus Huber}
\author{Armin Tavakoli}
\affiliation{Institute for Quantum Optics and Quantum Information -- IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria}
\affiliation{Atominstitut, Technische Universit{\"a}t Wien, Stadionallee 2, 1020 Vienna, Austria}
\begin{abstract}
We investigate entanglement detection when the local measurements only nearly correspond to those intended. This corresponds to a scenario in which measurement devices are not perfectly controlled, but nevertheless operate with bounded inaccuracy. We formalise this through an operational notion of inaccuracy that can be estimated directly in the lab. To demonstrate the relevance of this approach, we show that small magnitudes of inaccuracy can significantly compromise several well-known entanglement witnesses. For two arbitrary-dimensional systems, we show how to compute tight corrections to a family of standard entanglement witnesses due to any given level of measurement inaccuracy. We also develop semidefinite programming methods to bound correlations in these scenarios.
\end{abstract}
\maketitle
\textit{Introduction.---} Deciding whether an initially unknown state is entangled is one of the central challenges of quantum information science \cite{Guhne2009, Horodecki2009, Friis2019}. The most common approach is the method of entanglement witnesses, in which one hypothesises that the state is close to a known target and then finds suitable local measurements that can reveal its entanglement \cite{Horodecki1996, Terhal1999, Lewenstein2000}. In principle, this allows for the detection of every entangled state. However, it crucially requires the experimenter to flawlessly perform the stipulated quantum measurements. This is an idealisation to which one may only aspire: even for the simplest system of two qubits, small alignment errors can cause false positives \cite{Seevinck2007, Rosset2012}. In contrast, by adopting a device-independent approach, any concerns about the modelling of the measurement devices can be dispelled. This entails viewing them as quantum black boxes and detecting entanglement through the violation of a Bell inequality \cite{Bancal2011, Moroder2013}. However, Bell experiments are practically demanding \cite{Brunner2014}. Also, many entangled states either cannot, or are not known to, violate any Bell inequality \cite{Werner1989, Augusiak2014}. In addition, for the common purpose of verifying that a non-malicious entanglement source operates as intended, a device-independent approach is to use a sledgehammer to crack a nut. In the interest of a compromise, entanglement detection has also been investigated in steering scenarios, in which some devices are assumed to be perfectly controlled and others are quantum black boxes \cite{Wiseman2007}. Nevertheless, such asymmetry is often not present in non-malicious scenarios, and the approach still suffers from drawbacks similar to both the device-independent case, albeit it milder, and the standard, fully controlled, scenario. A much less explored compromise route is to only assume knowledge of the Hilbert space dimension \cite{Moroder2012, Tavakoli2018}. This essentially adopts the view that the experimenter has no control over the relevant degrees of freedom. Such ideas have also been used to strengthen steering-based entanglement detection \cite{Moroder2016}.
Here, we introduce an approach to entanglement detection that neither assumes flawless control of the measurements nor views them as mostly uncontrolled operations. The main idea is that an experimenter can quantitatively estimate the accuracy of their measurement devices and then base entanglement detection on this benchmark. Such knowledge naturally requires a fixed Hilbert space dimension: the experimenter knows the degrees of freedom on which they operate.
To quantify the inaccuracy between the intended target measurement and the lab measurement, we use a simple fidelity-based notion that can handily be measured experimentally.
In what follows, we first establish the relevance of small inaccuracies by showcasing that the conclusions of well-known entanglement witnesses can be substantially compromised. We show that the magnitude of detrimental influence associated to a small inaccuracy does not have to decrease for higher-dimensional systems. This is important because higher-dimensional entangled systems are increasingly interesting for experiments \cite{Dada2011, Erhard2020, Ecker2019, Herrera2020, Hu2020} but typically cannot be controlled as precisely as qubits. Secondly, we develop entanglement criteria that explicitly take the degree of inaccuracy into account. For two-qubit scenarios, we provide this based on the simplest entanglement witness and the Clauser-Horne-Shimony-Holt (CHSH) quantity. For a pair of systems of any given local dimension, we show that such criteria can be analytically established as corrections to a simple family of standard entanglement witnesses.
Finally, we present semidefinite programming (SDP) relaxations for bounding the set of quantum correlations under measurement inaccuracies. We use this both to estimate the potentially constructive influence of measurement inaccuracy on entanglement-based correlations and to systematically place upper bounds for separable states on linear witnesses.
\textit{Framework.---} We consider sources of bipartite states $\rho=\rho_\text{AB}$ of local dimension $d$. The subsystems are measured individually with settings $x$ and $y$ respectively, producing outcomes $a,b\in\{1,\ldots,o\}$. The experimenter's aim is to measure the first (second) system using a set of projective measurements $\{\tilde{A}_{a|x}\}$ ($\{\tilde{B}_{b|y}\}$). These are called target measurements. However, the measurements actually performed in the lab do not precisely correspond to the targeted measurements, but instead to positive operator-valued measures (POVMs) $\{A_{a|x}\}$ ($\{B_{b|y}\}$). These are called lab measurements and do not need to be projective. The correlations in the experiment are given by the Born-rule
\begin{equation}\label{born}
p(a,b|x,y)=\Tr\left[A_{a|x}\otimes B_{b|y}\rho\right].
\end{equation}
We quantify the correspondence between each of the target measurements and the associated lab measurements through their average fidelity,
\begin{align}\label{fidelity}
& \mathcal{F}^\text{A}_x\equiv \frac{1}{d}\sum_{a=1}^{o}\Tr\left[A_{a|x} \tilde{A}_{a|x} \right],
& \mathcal{F}^{\text{B}}_y\equiv \frac{1}{d}\sum_{b=1}^{o}\Tr\left[B_{b|y} \tilde{B}_{b|y} \right].
\end{align}
The fidelity respects $\mathcal{F}\in[0,1]$ with $\mathcal{F}=1$ if and only if the lab measurement is identical to the target measurement. Importantly, the fidelity admits a simple operational interpretation: it is the average probability of obtaining outcome $a$ ($b$) when the lab measurement is applied to each of the orthonormal states spanning the eigenspace of the $a$-th ($b$-th) target projector. Thus, the fidelities $\{\mathcal{F}^\text{A}_x,\mathcal{F}^\text{B}_y\}$ can be directly determined by probing the lab measurements with single qudits from a well-calibrated, auxiliary, source. This requires no entanglement and can routinely be achieved, see e.g.~Ref.~\cite{Bouchard2018}. It motivates the assumption of a bounded inaccuracy, i.e.~a lower bound on each of the fidelities,
\begin{align}\label{assumption}
& \mathcal{F}^\text{A}_x\geq 1-\varepsilon^\text{A}_x, & \mathcal{F}^\text{B}_y\geq 1-\varepsilon^\text{B}_y,
\end{align}
where the parameter $\varepsilon\in[0,1]$ is the inaccuracy of the considered lab measurement. In the extreme case of $\varepsilon=0$, the lab measurement is identical to the target measurement and our scenario reduces to a standard entanglement witness. In the other extreme, $\varepsilon=1$, only the Hilbert space dimension of the measurement is known. Away from these extremes, one encounters the more realistic scenario, in which the experimenter knows the degrees of freedom, but is only able to control them up to a limited accuracy.
The simplest tests of entanglement use the minimal number of outcomes ($o=2$). In such scenarios the fidelity constrains \eqref{assumption} can be simplified into
\begin{align}\label{observable}
& \Tr\left(A_{x}\tilde{A}_{x}\right)\geq d\left(1-2\varepsilon^\text{A}_x\right), &&\Tr\left(B_{y}\tilde{B}_{y}\right)\geq d\left(1-2\varepsilon^\text{B}_y\right)
\end{align}
where we have defined observables $A_{x}\equiv A_{1|x}-A_{2|x}$ and $B_{y}\equiv B_{1|y}-B_{2|y}$. The observables can be arbitrary Hermitian operators whose extremal eigenvalue is bounded by unity, i.e.~$\norm{A_x}_\infty\leq 1$ and $\norm{B_y}_\infty\leq 1$.
Notice that the proposed framework immediately extends also to multipartite scenarios.
\textit{Impact of inaccuracies in entanglement witnessing.---} A crucial motivating question for our approach is whether, and to what extent, small inaccuracies in the measurement devices ($\varepsilon\ll 1$) impact the analysis of a conventional entanglement witness. We discuss this matter based on several well-known witnesses.
Firstly, consider the simplest entanglement witness for two qubits, involving two pairs local Pauli observables: $\mathcal{W}=\expect{\sigma_X\otimes\sigma_X}+\expect{\sigma_Z\otimes\sigma_Z}$. For separable states we have $\mathcal{W}\leq \mathcal{W}_\text{sep}=1$ and for entangled states $\mathcal{W}\leq \mathcal{W}_\text{ent}=2$. Consider now that the lab observables $\{A_1,A_2\}$ and $\{B_1,B_2\}$ only nearly correspond \eqref{observable} to the target observables $\{\sigma_X,\sigma_Z\}$. Since $\mathcal{W}_\text{ent}=2$ is algebraically maximal, it remains unchanged, but such is not the case for the separable bound $\mathcal{W}_\text{sep}$. Thanks to the simplicity of $\mathcal{W}$, we can precisely evaluate $\mathcal{W}_\text{sep}$ in the prevalent scenario when all measurement devices are equally inaccurate, i.e.~$\varepsilon^\text{A}_x=\varepsilon^\text{B}_y=\varepsilon$. For a product state, we have $\mathcal{W}=\expect{A_1}\expect{B_1}+\expect{A_2}\expect{B_2}\leq
\sqrt{\expect{A_1}^2+\expect{A_2}^2} \sqrt{\expect{B_1}^2+\expect{B_2}^2}$. Since the target measurements are identical on both sites and the factors are independent, they are optimally chosen equal. Then, it is easily shown that the optimal choice of Bloch vectors corresponds to aligning $A_1$ and $A_2$ ($B_1$ and $B_2$) to the extent allowed by $\varepsilon$. This leads to the following tight condition for entanglement detection (see Supplementary Material)
\begin{equation}\label{qubit}
\mathcal{W}_\text{sep}(\varepsilon)= 1+4\left(1-2\varepsilon\right)\sqrt{\varepsilon\left(1-\varepsilon\right)},
\end{equation}
when $\varepsilon\leq \frac{1}{2}-\frac{1}{2\sqrt{2}}$ and $\mathcal{W}_\text{sep}=2$ otherwise. Importantly, the derivative diverges at $\varepsilon\rightarrow 0^+$. Hence, a small $\varepsilon$ induces a large perturbation in the ideal ($\varepsilon=0$) separable bound. In the vicinity of $\varepsilon=0$, it scales as $\mathcal{W}_\text{sep}\sim 1+4\sqrt{\varepsilon}$. For example, $\varepsilon=0.5\%$ leads to $\mathcal{W}_\text{sep}(\varepsilon)\approx 1.28$, which eliminates over a quarter of the range in which standard entanglement detection is possible, indicating the relevance of false positives.
Secondly, consider the CHSH quantity for entanglement detection, namely $\mathcal{W}=\expect{\sigma_X\otimes \left(\sigma_X+\sigma_Z\right)}+\expect{\sigma_Z\otimes \left(\sigma_X-\sigma_Z\right)}$. Here, we have targeted observables optimal for violating the CHSH Bell inequality \cite{CHSH1969}. One has $\mathcal{W}_\text{sep}=\sqrt{2}$ and $\mathcal{W}_\text{ent}=2\sqrt{2}$. In contrast to the previous example, the fact that all correlations from $d$-dimensional separable states constitute a subset of all correlations based on local hidden variables implies that entanglement can be detected for any value of $\varepsilon$. However, as we show in Supplementary Material through an explicit separable model that we conjecture to be optimal, this fact does not qualitatively improve the robustness of idealised ($\varepsilon=0$) entanglement detection to small inaccuracies. We obtain
\begin{align}
\mathcal{W}_\text{sep}=4\left(1-2\varepsilon\right)\sqrt{\varepsilon(1-\varepsilon)}+\sqrt{2-16\varepsilon\left(1-\varepsilon\right)\left(1-2\varepsilon\right)^2},
\end{align}
when $\varepsilon\leq \frac{1}{2}-\frac{1}{2\sqrt{2}}$ and $\mathcal{W}_\text{sep}=2$ otherwise. For small $\varepsilon$, we find $\mathcal{W}_\text{sep}\sim \sqrt{2}+4\sqrt{\varepsilon}$. An inaccuracy of $\varepsilon=0.5\%$ ensures $\mathcal{W}_\text{sep}\gtrsim 1.67 $, which eliminates nearly a fifth of the range in which standard entanglement detection is possible.
\begin{figure}
\caption{Numerically obtained lower bounds on the relative magnitude of the entangled-to-separable gap, $\Delta$, for entanglement witnessing based on two conjugate bases at different degrees of measurement inaccuracy $\varepsilon\in\{0.5\%, 1\%, 2\%, 3\%, 5\%, 10\%\}
\label{FigNumerics}
\end{figure}
Interestingly, it is \textit{a priori} not clear how small $\varepsilon$ should impact standard entanglement witnessing as $d$ increases. On the one hand, the impact ought to increase due to the increasing number of orthogonal directions in Hilbert space. On the other hand, it ought to decrease due to the growing distances in Hilbert space. For instance, the $\varepsilon$ required to transform the computational basis into its Fourier transform scales as $\varepsilon=\frac{\sqrt{d}-1}{\sqrt{d}}$, which rapidly approaches unity. To investigate the trade-off between these two effects, we consider the $d$-dimensional generalisation of the simplest entanglement witness. Both subsystems are subject to the same pair of target measurements, namely the computational basis $\{\ket{e_i}\}_{i=1}^d$ and its Fourier transform $\{\ket{f_i}\}_{i=1}^d$, where $\ket{f_i}=\Omega\ket{e_i}$ with $\Omega_{jk}=\frac{1}{\sqrt{d}}e^{\frac{2\pi i}{d}jk}$. The witness is
$\mathcal{W}^{(d)}=\sum_{i=1}^d \bracket{e_i,e_i}{\rho}{e_i,e_i}+\bracket{f_i,f_i}{\rho}{f_i,f_i}$. Notice that for $d=2$ this only differs from the previous, simplest, witness by a normalisation term. One has $\mathcal{W}^{(d)}_\text{sep}=1+\frac{1}{d}$ and $\mathcal{W}^{(d)}_\text{ent}=2$ \cite{Spengler2012}. Allowing for measurement inaccuracy, we use an alternating convex search algorithm to numerically optimise over the lab measurements and shared separable states to obtain lower bounds on $\mathcal{W}^{(d)}_\text{sep}(\varepsilon)$. See Supplementary Material for details about the method. In order to compare the impact of measurement inaccuracy for different dimensions, we consider the following ratio between the entangled-to-separable gap in the inaccurate and ideal case, $\Delta\equiv \frac{\mathcal{W}^{(d)}_\text{ent}(0)-\mathcal{W}^{(d)}_\text{sep}(\varepsilon)}{\mathcal{W}^{(d)}_\text{ent}(0)-\mathcal{W}^{(d)}_\text{sep}(0)}=\frac{d}{d-1}\left[2-\mathcal{W}^{(d)}_\text{sep}(\varepsilon)\right]$. Notice that the numerator features $\mathcal{W}^{(d)}_\text{ent}(0)$ instead of $\mathcal{W}^{(d)}_\text{ent}(\varepsilon)$ because $\varepsilon$ is not in itself a resource for the experimenter. The results of the numerics are illustrated in Figure~\ref{FigNumerics} for some different choices of $\varepsilon$. We observe that $\Delta$ is not monotonic in $d$, but instead features a maximum, that shifts downwards in $d$ as $\varepsilon$ increases. Beyond this maximum point, the impact of measurement inaccuracies grows as the dimension becomes large.
Finally, for multipartite qubit states, it is natural to expect that the detrimental influence of small $\varepsilon$ grows with the number of qubits under consideration. The reason is that measurement inaccuracies can accumulate separately in the different subsystems. This intuition is confirmed by the models of Ref.~\cite{Rosset2012}, in which small alignment errors are used to spoof, with increasing magnitude, the standard fidelity-based witness of genuine multipartite entanglement for Greenberger-Horne-Zeilinger states \cite{Bourennane2004}.
This further confirms the need of considering measurement inaccuracies.
\textit{High-dimensional entanglement criterion.---} In view of the the relevance of small measurement inaccuracies, it is natural to formulate entanglement criteria that take them explicitly into account beyond the simplest, two-qubit, scenario. Consider a pair of $d$-dimensional systems and $n\in\{1,\ldots,d^2-1\}$ measurements. For system A, the observables ideally correspond to (subsets of) a generalised Bloch basis $\{\lambda_i\}_{i=1}^{n}$ and for system B, the ideal observables are the complex conjugates $\{\bar{\lambda}_i\}_{i=1}^{n}$. Here, $\lambda_i$ is $d$-dimensional, traceless and satisfies $\Tr\left(\lambda_i\lambda_j^\dagger\right)=d\delta_{ij}$ \cite{Bertlmann2008}. Defining $\rho=\frac{1}{d}\left(\openone+\sum_{i=1}^{d^2-1}\mu_i\lambda_i\right)$, one has $\norm{\vec{\mu}}^2\leq d-1$. A simple standard entanglement witness, based on a total of $n$ measurements, is then given by
\begin{equation}\label{oldwitness}
\mathcal{W}^{(d)}=\sum_{i=1}^n \expect{\lambda_i\otimes \bar{\lambda}_i}.
\end{equation}
Using H\"older's inequality, one finds that separable states obey $\mathcal{W}^{(d)}_\text{sep}=d-1$. When the choice of Bloch basis is fixed, entangled states can achieve at most $\mathcal{W}_\text{ent}^{(d)}=\nu_\text{max}\left[\sum_{i=1}^n \lambda_i\otimes \bar{\lambda}_i\right]$, by choosing the state as the eigenvector corresponding to the largest eigenvalue ($\nu_\text{max}$). When the choice of Bloch basis is not fixed, a general upper bound for entanged states is $\mathcal{W}^{(d)}_\text{ent}\leq \min\left\{\sqrt{n\left(d^2-1\right)},n(d-1)\right\}$, as shown in Supplementary Material. Note that $n(d-1)$ only is relevant when $d=2$. Notice also that the maximally entangled state $\ket{\phi^+_d}=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{ii}$ achieves $\mathcal{W}^{(d)}=n$ regardless of the choice of Bloch basis.
Consider now that the lab observables only nearly correspond to $\{\lambda_i\}$ and $\{\bar{\lambda}_i\}$ respectively. We write them as
$A_i=q \lambda_i+\sqrt{1-q^2}\lambda_{i}^{\perp}$ and $B_i=q \bar{\lambda}_i+\sqrt{1-q^2}\bar{\lambda}_{i}^{\perp}$, where $q\in[-1,1]$ is related to the inaccuracy through $q=1-2\varepsilon$ and $\lambda_{i}^{\perp}$ and $\bar{\lambda}_{i}^{\perp}$ are observables orthogonal to $\lambda_i$ and $\bar{\lambda}_i$, respectively, on the generalised Bloch sphere. In Supplementary Material, we prove that the witness $\mathcal{W}^{(d)}=\sum_{i=1}^n\expect{A_i\otimes B_i}$ for separable states obeys
\begin{equation}\label{bound}
\mathcal{W}^{(d)}_\text{sep}(\varepsilon)\leq \left(d-1\right) \left(q+\sqrt{n-1}\sqrt{1-q^2}\right)^2,
\end{equation}
when $q\geq \frac{1}{\sqrt{n}}$ and otherwise $\mathcal{W}^{(d)}_\text{sep}(\varepsilon)\leq n\left(d-1\right)$, which is algebraically maximal. As is intuitive, the window for detecting entanglement shrinks as $\varepsilon$ increases.
We investigate the tightness of the bound. To this end, choose the state as $\ket{\phi^\dagger}\otimes \ket{\phi^T}$, where the local Bloch vector is $\mu_i=\frac{\sqrt{d-1}}{\sqrt{n}}$ and where $\lambda_i\rightarrow \lambda_i^\dagger$ ($\lambda_i\rightarrow \lambda_i^T$) for $\ket{\phi^\dagger}$ ($\ket{\phi^T}$). Choose the observables as $A_i=q\lambda_i+\sum_{j\neq i}\frac{\sqrt{1-q^2}}{\sqrt{n-1}}\lambda_j$ and $B_i=q\bar{\lambda}_i+\sum_{j\neq i}\frac{\sqrt{1-q^2}}{\sqrt{n-1}}\bar{\lambda}_j$. This returns the separable bound \eqref{bound}. However, we need to check that the Bloch vector $\vec{\mu}$ corresponds to a valid state. Curiously, for the most powerful case, namely $n=d^2-1$, tightness would be implied by a positive answer to the long-standing open question of whether there exists a Weyl-Heisenberg covariant symmetric informationally complete (SIC) POVM in dimension $d$. To see the connection, simply choose the Bloch basis as the non-Hermitian Weyl-Heisenberg basis $\{X^uZ^v\}$ for $u,v\in\{0,\ldots,d-1\}$ and $u+v>0$, where $X=\sum_{k=0}^{d-1}\ketbra{k+1}{k}$ and $Z=\sum_{k=0}^{d-1}e^{\frac{2\pi ik}{d}}\ketbra{k}{k}$. It follows immediately that $|\bracket{\phi}{X^uZ^v}{\phi}|=\frac{1}{\sqrt{d+1}}$, which defines a SIC-POVM. Since these SIC-POVMs are conjectured to exist in all dimensions \cite{Zauner2011}, and are known to exist up to well above the first hundred dimensions \cite{Scott2017, Fuchs2017}, our bound is plausibly tight for any $d$.
\textit{SDP methods.---} We develop a hierarchy of SDP relaxations to bound the largest possible value of any linear witness, $\mathcal{W}=\sum_{a,b,x,y} c_{abxy}p(a,b|x,y)$, for some real coefficients $c_{abxy}$. The method applies both for correlations originating from entangled states and from separable states, under any given degree of measurement inaccuracy and arbitrary target measurements. Thus, we systematically establish upper bounds $\mathcal{W}^\uparrow_\text{ent}(\varepsilon)\geq \mathcal{W}_\text{ent}(\varepsilon)$ and $\mathcal{W}^\uparrow_\text{sep}(\varepsilon)\geq \mathcal{W}_\text{sep}(\varepsilon)$. This has a three-fold motivation. Firstly, $\mathcal{W}_\text{ent}$ will generally depend on $\varepsilon$; cases with $\mathcal{W}^{(d)}>\mathcal{W}^{(d)}_\text{ent}(0)$ can be observed when the inaccuracies accumulate in a constructive way (e.g.~a favourable systematic error in the local reference frames). It is relevant to bound such occurances. Secondly, knowledge of $\mathcal{W}^\uparrow_\text{ent}(\varepsilon)$ allows an experimenter to give lower bounds on the inaccuracy of the measurement devices. Thirdly, and most importantly, this enables a general and systematic construction of entanglement witnesses of the form $\mathcal{W}\leq \mathcal{W}^\uparrow_\text{sep}(\varepsilon)$.
We discuss the main features of the method for computing $\mathcal{W}_\text{ent}^\uparrow(\varepsilon)$ and then see how it can be extended to also compute $\mathcal{W}_\text{sep}^\uparrow(\varepsilon)$. To this end, as is standard, the SDP relaxation method is based on the positivity of a moment matrix. This matrix consists of traces of monomials (in the spirit of e.g.~\cite{Burgdorf2012}) which are composed of products of the state, the lab measurements and the target measurements (see Supplementary Material for specifics). Moments corresponding to products of the first two can be used to build a generic linear witness $\mathcal{W}$ via Eq.~\eqref{born}. Moments corresponding to products of the final two can be used to build the constraints on the fidelities $\mathcal{F}^\text{A}_x$ and $\mathcal{F}^\text{B}_y$. Our construction draws inspiration from two established ideas. Firstly, one can capture the constraints of $d$-dimensional Hilbert space, on the level of the moment matrix, by numerically sampling states and measurements \cite{Navascues2015}. Secondly, in scenarios without entanglement, constraints capturing the fidelity of a quantum state with a target can be incorporated into the moment matrix \cite{Tavakoli2021}. We adapt the latter to entanglement-based scenarios and measurement fidelities as needed for Eq.~\eqref{assumption}. Details are given in Supplementary Material. We have applied this method, at low relaxation level, in several different case studies in low dimensions and frequently found that the obtained upper bounds coincide with those obtained from interior point optimisation routines. We note that the computational requirements for this tool can be much reduced since sampling-based symmetrisation methods of Ref.~\cite{Tavakoli2019} can straightforwardly be incorporated.
To extend this method for the computation of $\mathcal{W}_\text{sep}^\uparrow(\varepsilon)$, we must incorporate constraints on the set of quantum states. Since the set of separable states is generally difficult to characterise (see e.g.~\cite{DPS2002}), we instead adopt an approach in which we use the ideal entanglement witness condition, $\mathcal{W}\leq \mathcal{W}_\text{sep}(0)$, which we may realistically assume to possess, in place of the set of separable states. Then, since the probabilities associated to performing the target measurements on the state explicitly appear in our moment matrix, we can introduce it as an additional linear constraint in our SDP. Hence, the optimisation is effectively a relaxation of the subset of entangled states for which the original entanglement witness holds. In fact, since the set of separable states is characterised by infinitely many linear entanglement witnesses, one can in this way continue to introduce linear standard witnesses to constrain the effective state space in the SDP and thus further improve the accuracy of the bound $\mathcal{W}_\text{sep}^\uparrow(\varepsilon)$. In Supplementary Material we exemplify the use of this method, in its basic version, using only a single witness constraint $\mathcal{W}\leq \mathcal{W}_\text{sep}(0)$ on the state space, and show that it returns non-trivial, albeit not tight, bounds for two simple entanglement witnesses for relevant values of $\varepsilon$.
\textit{Discussion.---} We have introduced and investigated entanglement detection when the measurements only nearly correspond to those intended to be performed in the laboratory. We have shown the relevance of the concept, presented explicit entanglement witnesses that take measurement inaccuracy into account, and finally shown how SDP methods can be applied to these types of problems. These results are a step towards a theoretical framework for detecting entanglement based on devices that are quantitatively benchmarked in an operationally meaningful and experimentally accessible manner.
Our work leaves several natural open problems. If given an arbitrary standard entanglement witness, how can we compute corrections due to the introduction of measurement inaccuracies? Our SDP method is a first step towards addressing this problem but better methods are necessary both in terms of computational cost and in terms of the accuracy of the separable bound. Moreover, for a given $d$, what is the smallest number of auxiliary global measurement settings needed to eliminate the diverging derivative for optimal standard entanglement witnesses under small measurement inaccuracy? In addition, can one extend our entanglement witnesses to witnesses of genuine higher-dimensional entanglement, e.g.~by detecting the Schmidt number?
Also, in this first work, we have focused on bipartite entanglement. It would be interesting to identify useful entanglement witnesses for multipartite states at bounded measurement inaccuracy. Finally, the framework proposed here for entanglement detection draws inspiration from ideas proposed in semi-device-independent quantum communications. Given that several frameworks for semi-device-independence recently have been proposed \cite{VanHimbeeck2017, info1, info2, Wang2019, Tavakoli2021}, there may be other similarly inspired avenues for entanglement detection based on quantitative benchmarks.
\appendix
\section{Simplest entanglement witness}\label{AppSimple}
Consider the entanglement witness $\mathcal{W}=\expect{\sigma_X\otimes\sigma_X}+\expect{\sigma_Z\otimes\sigma_Z}$ on a pair of qubits. We allow the lab observables to have an $\varepsilon$-deviation with respect to the target measurements $\{\sigma_X,\sigma_Z\}$ on both sites. This corresponds to the constraints
\begin{align}
& \Tr\left(A_1\sigma_X\right)\geq 2-4\varepsilon, && \Tr\left(A_2\sigma_Z\right)\geq 2-4\varepsilon,\\
& \Tr\left(B_1\sigma_X\right)\geq 2-4\varepsilon, && \Tr\left(B_2\sigma_Z\right)\geq 2-4\varepsilon,
\end{align}
where we have chosen that all measurements are subject to the same magnitude of inaccuracy.
Due to the symmetry of $\mathcal{W}$ under a party swap, we can choose $A_1=B_1$ and $A_2=B_2$. Since the measurements are characterised by a pair of Bloch vectors, we can without loss of generality choose them in the $XZ$-plane of the Bloch sphere. We therefore write $A_k=B_k=\cos\theta_k \sigma_X+\sin\theta_k \sigma_Z$. In the relevant case of equality, the fidelity conditions then become
\begin{align}
&\theta_1=-\arccos\left(1-2\varepsilon\right),\\
&\theta_2=\arcsin\left(1-2\varepsilon\right).
\end{align}
Due to the party symmetry, we can choose a product state on the form $\ket{\phi}\otimes \ket{\phi}$ where $\ket{\phi}=\cos z\ket{0}+\sin z\ket{1}$. Then we obtain
\begin{equation}
\mathcal{W}=1+4\left(1-2\varepsilon\right)\sqrt{\varepsilon(1-\varepsilon)}\sin(4z),
\end{equation}
which is optimal at $z=\frac{\pi}{8}$ when $\varepsilon\leq \frac{1}{2}$. Hence
\begin{equation}
\mathcal{W}_\text{sep}=1+4\left(1-2\varepsilon\right)\sqrt{\varepsilon(1-\varepsilon)}.
\end{equation}
Notice that this is only valid for $\varepsilon\leq \frac{1}{2}-\frac{1}{2\sqrt{2}}$. For larger $\varepsilon$ we have $\mathcal{W}_\text{sep}=2$.
Moreover, we note that the immediate generalisation of this witness, namely $\mathcal{W}=\expect{\sigma_X\otimes\sigma_X}+\expect{\sigma_Y\otimes\sigma_Y}+\expect{\sigma_Z\otimes\sigma_Z}$, in the presence of measurement inaccuracies, can by similar means be shown to admit the separable bound
\begin{equation}
\mathcal{W}_\text{sep}=2+4\sqrt{2}\left(1-2\varepsilon\right)\sqrt{\varepsilon(1-\varepsilon)}-(1-2\varepsilon)^2,
\end{equation}
when $\varepsilon\leq \frac{3-\sqrt{3}}{6}$ and $\mathcal{W}_\text{sep}=3$ otherwise.
\section{Entanglement detection based on the CHSH quantity}\label{AppCHSH}
Consider a pair of qubits, each of which is subject to two measurements. The target observables on both sites are $\sigma_X$ and $\sigma_Z$. The lab observables all have the same inaccuracy bound $\varepsilon$. Thus we have
\begin{align}
& \Tr\left(A_1\sigma_X\right)\geq 2-4\varepsilon, && \Tr\left(A_2\sigma_Z\right)\geq 2-4\varepsilon,\\\label{cons}
& \Tr\left(B_1\sigma_X\right)\geq 2-4\varepsilon, && \Tr\left(B_2\sigma_Z\right)\geq 2-4\varepsilon.
\end{align}
In case of perfect measurements, the CHSH quantity acts as a conventional entanglement witness,
\begin{equation}
\mathcal{W}=\expect{A_1\otimes B_1}+\expect{A_1\otimes B_2}+\expect{A_2\otimes B_1}-\expect{A_2\otimes B_2}\leq \sqrt{2},
\end{equation}
which is respected by all separable states. Evidently, since $\mathcal{W}\leq 2$ for local hidden variable models, which in particular account for the statistics of any measurements performed on a separable state, it follows that entanglement can be detected for arbitrary $\varepsilon$.
We show the potential influence of small measurement inaccuracies through an explicit quantum model. Choose $A_1=B_1$ and associate it to a Bloch vector $\vec{n}_1=\left(\cos \alpha,0,\sin\alpha\right)$ in the XZ-plane. Similarly choose $A_2=B_2$ and associate it to the Bloch vector $\vec{n}_2=\left(\cos \beta,0,\sin\beta\right)$. Our strategy is to align the two Bloch vectors as much as possible under the constraints \eqref{cons}. This implies the choice of
\begin{align}
& \alpha=\arccos\left(1-2\varepsilon\right), && \beta=\arcsin\left(1-2\varepsilon\right).
\end{align}
Then, we choose the product state $\ket{\psi}=\ket{\phi}\otimes\ket{\phi}$ with $\ket{\phi}=\cos z\ket{0}+\sin z\ket{1}$, where
\begin{equation}
z=-\frac{\pi}{4}+\frac{1}{4}\arctan\left(\frac{1}{8\varepsilon-8\varepsilon^2-1}\right).
\end{equation}
The angle has been choosen so as to place the Bloch vector of $\ket{\phi}$ right in the middle of $\vec{n}_1$ and $\vec{n}_2$. This leads to the following value of the CHSH quantity,
\begin{align}
\mathcal{W}=4\left(1-2\varepsilon\right)\sqrt{\varepsilon(1-\varepsilon)}+\sqrt{2-16\varepsilon\left(1-\varepsilon\right)\left(1-2\varepsilon\right)^2},
\end{align}
when $\varepsilon\leq \frac{1}{2}-\frac{1}{2\sqrt{2}}$ and $\mathcal{W}=2$ otherwise. The derivative diverges as $\varepsilon\rightarrow 0^+$, indicating the first-order impact of small measurement inaccuracies. For small $\varepsilon$, the value scales as $\mathcal{W}\sim \sqrt{2}+4\sqrt{\varepsilon}-4\sqrt{2}\varepsilon$. For example, if we choose $\varepsilon=0.5\%$, the separable model achieves $\mathcal{W}=1.67$ which is a perturbation comparable to that obtained in the main text for the simplest two-qubit entanglement witness.
\section{Lower bounds: alternating convex search}\label{AppSeesaw}
Consider that we are given an arbitrary linear functional $\mathcal{W}$, arbitrary target measurements $\{\tilde{A}_{a|x}\}$ and $\{\tilde{B}_{b|y}\}$ and arbitrary measurement inaccuracies $\{\varepsilon^\text{A}_x,\varepsilon^\text{B}_y\}$. Consider a linear functional
\begin{equation}
\mathcal{W}=\sum_{a,b,x,y}c_{abxy} \Tr\left[A_{a|x}\otimes B_{b|y}\rho\right],
\end{equation}
with some real coefficients $c_{abxy}$. We describe a numerical method, based on alternating convex search, to systematically establish lower bounds on both $\mathcal{W}_\text{sep}$ and $\mathcal{W}_\text{ent}$. To this end we consider latter case first.
In order to place a lower bound on $\mathcal{W}_\text{ent}$, we decompose the optimisation problem into three parts: one over the measurements on system A, one over the measurements on system B and one over the global shared state. To this end, we first choose a random set of measurements $\{B_{b|y}\}$ and a random pure state $\rho$. Then, we optimise $\mathcal{W}$ over the measurements $\{A_{a|x}\}$ under the constraint that $\mathcal{F}^\text{A}_x\geq 1-\varepsilon^\text{A}_x$. This optimisation is a semidefinite program and can therefore be efficiently solved. Using the returned measurements $\{A_{a|x}\}$, we optimise $\mathcal{W}$ over the measurements $\{B_{b|y}\}$ under the constraint that $\mathcal{F}^\text{B}_y\geq 1-\varepsilon^\text{B}_y$. This is again a semidefinite program. Finally, using the returned measurements $\{B_{b|y}\}$, we evaluate the Bell operator
\begin{equation}
\mathcal{B}=\sum_{a,b,x,y}c_{abxy}A_{a|x}\otimes B_{b|y}
\end{equation}
and compute its largest eigenvalue. The associated eigenvector is the optimal state, which corresponds to our choice of $\rho$. This routine of two semidefinite programs and one eigenvalue computation can then be iterated in order to find increasingly accurate lower bounds on $\mathcal{W}_\text{ent}$. The procedure depends on the initial starting point and ought therefore to be repeated several times independently.
To place a lower bound on $\mathcal{W}_\text{sep}$, we can proceed analogously to the above when treating the separate optimisations over the measurements $\{A_{a|x}\}$ and $\{B_{b|y}\}$. However, the optimisation over the state is now less straightforward since we require that $\rho=\ketbra{\phi}{\phi}\otimes \ketbra{\psi}{\psi}$. The optimisation over the state can be cast as another alternating convex search, treated as a sub-routine to the main alteranting convex search. In other words, we sample a random $\ket{\phi}$ and evaluate the semidefinite program optimising $\mathcal{W}$ over $\ket{\psi}$. Then, using the returned $\ket{\psi}$, we run a semidefinite program optimising $\mathcal{W}$ over $\ket{\phi}$. This procedure is iterated until desired convergence is obtained.
\section{Bounds on witness}\label{AppWitness}
Let $\{\lambda_i\}_{i=1}^{d^2-1}$ be an orthonormal basis the space of operators acting on $d$-dimensional Hilbert space, with $\Tr\left(\lambda_i\lambda_j^\dagger\right)=d\delta_{ij}$. Then, every qudit state can be written as
\begin{equation}\label{bloch}
\rho=\frac{1}{d}\left(\openone+\vec{\mu}\cdot\vec{\lambda}\right),
\end{equation}
where $\vec{\mu}$ is some complex-valued Bloch vector with entries $\mu_i=\expect{\lambda_i^\dagger}=\Tr\left(\rho \lambda_i^\dagger\right)$. By checking the purity $\Tr\left(\rho^2\right)$, one finds that $\norm{\vec{\mu}}^2=\sum_{i=1}^{d^2-1}\expect{\lambda_i^\dagger}^2\leq d-1$. In general, not every such Bloch vector corresponds to a valid density matrix.
Consider the witness
\begin{equation}
\mathcal{W}^{(d)}=\sum_{i=1}^n \expect{\lambda_i\otimes \bar{\lambda}_i}.
\end{equation}
For separable states, we can evaluate $\mathcal{W}^{(d)}_\text{sep}$ by restricting to product states. Then we have
\begin{align}\nonumber\label{C3}
\mathcal{W}^{(d)}&=\sum_{i=1}^n \expect{\lambda_i}_A \expect{\bar{\lambda}_i}_B\leq \sqrt{\sum_{i=1}^n \expect{\lambda_i}^2_A}\sqrt{\sum_{i=1}^n \expect{\bar{\lambda}_i}_B^2}\\
&\leq d-1 = \mathcal{W}^{(d)}_\text{sep}.
\end{align}
Notice that this is independent of $n$.
For entangled states, we have
\begin{align}
&\mathcal{W}^{(d)}\leq \sum_{i=1}^n \expect{\lambda_i\otimes \bar{\lambda}_i}=\nu_\text{max}\left[\sum_{i=1}^n \lambda_i\otimes \bar{\lambda}_i \right]\\
&\leq n \max_i \nu_\text{max}\left[\lambda_i\otimes \bar{\lambda}_i\right]=n \max_i \nu_\text{max}\left[\lambda_i\right]^2\leq n(d-1),
\end{align}
where we used that $\nu_\text{max}\left[\lambda_i\right]\leq \sqrt{d-1}$. However this, essentially trivial, bound is only tight for $d=2$, in which case it is algebraically maximal. To obtain a bound for $d>2$, we note that the entangled state lives in dimension $d^2$. Hence, its Bloch vector length is at most $\sqrt{d^2-1}$. In other words,
\begin{equation}
\sum_{i=1}^{n} \expect{\lambda_i\otimes \bar{\lambda}_i}^2\leq d^2-1.
\end{equation}
Taking the case of equality, we obtain a bound on the largest value of the witness when all entries in the sum are equal. Thus we require
\begin{equation}
\expect{\lambda_i\otimes \bar{\lambda}_i}=\sqrt{\frac{d^2-1}{n}},
\end{equation}
which gives
\begin{equation}
\mathcal{W}^{(d)}_\text{ent}\leq \sqrt{n}\sqrt{d^2-1}.
\end{equation}
This bound is not necessarily tight.
Consider now the case when we have separable states and inaccurate measurements. Expand $\mathcal{W}^{(d)}$ as follows,
\begin{align}\nonumber
\mathcal{W}^{(d)}=& \sum_{i=1}^n\expect{A_i\otimes B_i}=q^2\sum_{i=1}^n\expect{\lambda_i}_A\expect{\bar{\lambda}_i}_B\\\nonumber
&+q\sqrt{1-q^2}\sum_{i=1}^n\left(\expect{\lambda_i}_A\expect{\bar{\lambda}_{i}^\perp}_B+\expect{\lambda_{i}^\perp}_A\expect{\bar{\lambda}_i}_B\right)\\
&+\left(1-q^2\right)\sum_{i=1}^n\expect{\lambda_{i}^\perp}_A\expect{\bar{\lambda}_{i}^\perp}_B.
\end{align}
We examine these sums one by one. From \eqref{C3}, we see that the first sum is at most $d-1$. Next, we use the Cauchy-Schwarz inequality to write the second sum as
\begin{align}\nonumber
\sum_{i=1}^n& \expect{\lambda_i}_A\expect{\bar{\lambda}_{i}^\perp}_B\leq\sqrt{\sum_{i=1}^n\expect{\lambda_i}_A^2}\sqrt{\sum_{i=1}^n\expect{\bar{\lambda}_{i}^\perp}_B^2}\\
&\leq \sqrt{d-1}\sqrt{\sum_{i=1}^n\expect{\bar{\lambda}_{i}^\perp}_B^2}\leq \left(d-1\right)\sqrt{n-1}.
\end{align}
In the last step, we have used the following lemma. Let $\vec{u}\in\mathbb{R}^n$ and $\vec{v}^i\in\mathbb{R}^n$ be unit vectors such that the $i$'th component of $\vec{v}^i$ is zero, i.e.~$\vec{v}_i^i=0$. Then we have that
\begin{equation}
\sum_{i=1}^n \left(\vec{u}\cdot \vec{v}^i\right)^2\leq \sum_{i=1}^n 1-\vec{u}_i^2=n-1.
\end{equation}
Again using the Cauchy-Schwarz inequality and this lemma also leads to
\begin{align}
&\sum_{i=1}^n\expect{\lambda_{i}^\perp}_A\expect{\bar{\lambda}_{i}}_B\leq \left(d-1\right)\sqrt{n-1},\\
&\sum_{i=1}^n\expect{\lambda_{i}^\perp}_A\expect{\bar{\lambda}_{i}^\perp}_B \leq \left(d-1\right)\left(n-1\right).
\end{align}
Putting it together, we arrive at the bound
\begin{equation}\label{res}
\mathcal{W}_\text{sep}\leq \left(d-1\right)\left(n-1-q^2\left(n-2\right)+2q\sqrt{n-1}\sqrt{1-q^2}\right).
\end{equation}
\section{Semidefinite relaxations}\label{AppSDP}
Consider the task of optimising an arbitrary linear functional over the set of projective quantum strategies with a given inaccuracy to a set of target measurements:
\begin{align}\nonumber\label{optim}
& \qquad \qquad \mathcal{W}_\text{ent}=\max_{\{A_{a|x}\},\{B_{b|y}\},\rho} \mathcal{W}[p] \\\nonumber
& \text{subject to }\quad \Tr\left(\rho\right)=1, \qquad \rho\geq 0, \qquad \rho\in \mathcal{L}(\mathbb{C}^d)\\ \nonumber
& A_{a|x}A_{a'|x}=A_{a|x}\delta_{a,a'}, \qquad B_{b|y}B_{b'|y}=B_{b|y}\delta_{b,b'},\\\nonumber
& \sum_a A_{a|x}=\openone_d, \qquad \sum_b B_{b|y}=\openone_d\\\nonumber
& \mathcal{F}^\text{A}_x\geq 1-\varepsilon^\text{A}_x, \qquad \mathcal{F}^\text{B}_y\geq 1-\varepsilon^\text{B}_y\\
& p(a,b|x,y)=\Tr\left[A_{a|x}\otimes B_{b|y} \rho\right],
\end{align}
where $\mathcal{L}(\mathbb{C}^d)$ is the set of linear operators of dimension $d$. This is generally a difficult optimisation problem. However, it can be relaxed into a hierarchy of increasingly precise criteria, each of which can be evaluated as a semidefinite program.
To this end, define the operator list
\begin{equation}
S=\{\openone_{d^2}, \rho, \{A_{a|x}\}_{a,x},\{B_{b|y}\}_{b,y},\{\tilde{A}_{a|x}\}_{a,x},\{\tilde{B}_{b|y}\}_{b,y}\}.
\end{equation}
Here, the measurement operators are to be understood as spanning the full Hilbert space, e.g.~$A_{a|x}\rightarrow A_{a|x}\otimes \openone_d$. We let $M_k$ denote the set of all monomials, taken from the list $S$, of degree at most $k$. We let $n(k)$ denote the size of the set $M_k$. Then, we define the $n(k)\times n(k)$ tracial moment matrix as
\begin{equation}
\Gamma(u;v)=\Tr\left(uv^\dagger\right),
\end{equation}
for $u,v\in M_k$. A quantum model implies the positivity of $\Gamma$. Moreover, by including enough monomials, we can formulate the objective as a linear function in the moment matrix,
\begin{equation}\label{obj}
\mathcal{W}(\Gamma)=\sum_{a,b,x,y} c_{abxy} \Gamma(\rho A_{a|x}; B_{b|y}).
\end{equation}
Similarly, the inaccuracy constraints can be formulated as the linear constraints
\begin{align}\nonumber\label{sdpcons}
& \frac{1}{d^2}\sum_{a=1}^o \Gamma(A_{a|x}; \tilde{A}_{a|x})\geq 1-\varepsilon^\text{A}_x, \\
& \frac{1}{d^2}\sum_{b=1}^o \Gamma(B_{b|y}; \tilde{B}_{b|y})\geq 1-\varepsilon^\text{B}_y.
\end{align}
In order to capture the constraints of $d$-dimensional Hilbert space and to fix the target measurements in the optimisation, we proceed as follows \cite{Navascues2015, Tavakoli2021}. We randomly sample $\rho$, $\{A_{a|x}\}_{a,x}$ and $\{B_{b|y}\}_{b,y}$ from a $d$-dimensional Hilbert space and construct the list $S$. Note that the target measurements are fixed at all times. Then, we evaluate the moment matrix and label it $\Gamma^{(1)}$. This process is repeated, leading to a list of sampled moment matrices $\{\Gamma^{(1)},\ldots,\Gamma^{(m)}\}$. The sampling is terminated when the next moment matrix is found to be linearly dependent on all the previously sampled moment matrices. Thus, the sampled list constitutes a (non-orthonormal) basis of the space of moment matrices. We then define the total moment matrix as the affine combination
\begin{align}\label{sdpcons2}
& \Gamma=\sum_{i=1}^m s_i \Gamma^{(i)}, && \sum_{i=1}^m s_i=1,
\end{align}
where $\{s_i\}$ serve as optimisation variables.
We can now formulate our relaxation of the optimisation problem \eqref{optim} as $\mathcal{W}_\text{ent}(\varepsilon)\leq \mathcal{W}_\text{ent}^\uparrow(\varepsilon) $ where
\begin{align}
& \mathcal{W}_\text{ent}^\uparrow \equiv \max_{\{s_i\}} \mathcal{W}(\Gamma) \quad \text{ subject to } \quad \Gamma\geq 0
\end{align}
under the constraints \eqref{sdpcons} and \eqref{sdpcons2}. This can be evaluated as a semidefinite program. The relaxation becomes tighter as the list of monomials $M_k$ is extended.
\begin{figure}
\caption{\textbf{Solid lines.}
\label{Fig_SDP}
\end{figure}
In order to instead obtain bounds of the form $\mathcal{W}_\text{sep}(\varepsilon)\leq \mathcal{W}_\text{sep}^\uparrow(\varepsilon) $, we can add the constraint
\begin{equation}\label{ew}
\sum_{a,b,x,y} c_{abxy}\Gamma(\rho \tilde{A}_{a|x}; \tilde{B}_{b|y})\leq \mathcal{W}_\text{sep}(0),
\end{equation}
which corresponds to a standard entanglement witness. Note that we can introduce even more ``target'' measurements in the operator list $S$, thus extending the size $n(k)$ of the moment matrix, and then use them to build additional linear constraint like \eqref{ew} representing standard entanglement witnesses. The introduction of these shrinks the effective state space, thus improving the accuracy of the bound $ \mathcal{W}_\text{sep}^\uparrow(\varepsilon)$, at the price of a larger SDP.
We exemplify a simple version of this method for the case of the two witnesses considered in Appendix~\ref{AppSimple}, namely $\mathcal{W}_2=\expect{\sigma_X\otimes\sigma_X}+\expect{\sigma_Z\otimes\sigma_Z}\leq 1$ and $\mathcal{W}_3=\expect{\sigma_X\otimes\sigma_X}+\expect{\sigma_Y\otimes\sigma_Y}+\expect{\sigma_Z\otimes\sigma_Z}\leq 1$, at inaccuracy $\varepsilon$. These are evaluated with monomial lists of length $46$ and $89$ respectively. The results are illustrated in Figure~\ref{Fig_SDP}. As expected, the returned bounds are not tight, due to the basic relaxation of the separable set to all entangled states obeying $\mathcal{W}\leq 1$. Nevertheless, the bounds are non-trivial for relevant values of $\varepsilon$.
\end{document}
|
math
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सभी प्रिय युजर्स को स्वतंत्रता दिवस की हार्दिक शुभकामनाएँ (इंडिपेंडेंस दए २०१९)। वैसे तो भारत में कई त्योहार मनाए जाते हैं लेकिन स्वतंत्रता दिवस का दिन भारत में सबसे लोकप्रिय तरीके से मनाया जाता है खासकर विद्यार्थियों के जीवन में इसका महत्व बहुत अधिक होता है। इस दिन हम इस दिन को अनेकता में एकता के रूप में भी मनाते हैं ऐसा इसलिए क्योंकि भारत में रहने वाले चाहे किसी भी धर्म का क्यों न हो वे इस मिलकर मनाते हैं। स्कूल, कॉलेजों और सार्वजनिक स्थानों पर इस दिन हर जगह तिरंगा फहराया जाता है।
जानकारी के लिए बता दें कि हमारा देश को ब्रिटिश शासन से १५ अगस्त १९४७ को स्वतंत्रता मिली थी और हमारा देश एक स्वतंत्र देश बन गया था। इसलिए इस दिन (इंडिपेंडेंस दए २०१९) को स्वतंत्रता दिवस के रूपय में मनाया जाता है। हमारे दिश को आजादी दिलाने के लिए कई स्वतंत्रता सेनानियों ने अपनी जान की बाजी लगाई थी और उनके कारण हमें यह आजादी नसीब हुई थी इसलिए उनको भी इस दिन याद किया जाता है।
स्वतंत्रता दिवस से संबंधित कुछ सुविचार:
ना पूछो जमाने से, क्या हमारी कहानी है,
हमारी पहचान तो बस इतनी है कि हम हिंदुस्तानी है....
स्वतंत्रता दिवस की शुभ कामनायें।
सीने में जूनून है, देशभक्ति में सकून है,
चमक दिखता है इस तिरंगे में मुझे्,
इसिलए दुश्मनो की सांसे थम है।
अनेकता में एकता ही हमारी पहचान है इसलिए मेरा भारत महान है...
मुझे तन चाहिए, न धन चाहिए,
जब तक जिंदा हूं इस मार्तभूमि के लिए,
और जब मरू तो इस तिरंगा के लिए।
देशभक्ति का मतलब सिर्फ ध्वज को लहराना नहीं है,
बल्कि अपने देश को मजबूत और सश्कत बनाने में सहायता करना भी है।
तिरंगा हमारा है शान-ए-जिंदगी
वतन परस्ती है वफा-ए-जमी
देश के लिए मर मिटना कुबूल है हमें
अखंड भारत के स्वपन का जूनून है हमें।
|
hindi
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یہ کتھ سا چھکھ ژہ سگ دوان خونہ سیتۍ
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kashmiri
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کیا تہنٛز موج تہ آسہ اتھ منٛز شرکت کران
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kashmiri
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english
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english
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code
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english
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इस पत्थर को जब लोगों ने पीटकर देखा तो निकली ऐसी आवाज जिसे सुनकर लोगों के उड़ गये होश क्लासी न्यूज
होम / उन्कैटेगरिज़्ड / इस पत्थर को जब लोगों ने पीटकर देखा तो निकली ऐसी आवाज जिसे सुनकर लोगों के उड़ गये होश
भारत को ऐसे ही नहीं धार्मिक देश कहा जाता है बल्कि चमत्कारी मंदिर और जगहों की वजह से भी भारत पूरी दुनिया में विख्याद है. भारत में ऐसी बहुत सी जगह हैं जो अपने चमत्कार की वजह से प्रसिद्ध हैं उन्हीं में से सरगुजा जिले के दरिमा एयरपोर्ट के नजदीक छिंदकालो गांव में एक विशाल पत्थर भी काफी प्रसिद्ध है. इस पत्थर का करिश्मा देखकर कोई इसे आसमान का करिश्मा कह रहा हैं तो कोई इसे भगवान का रूप मान रहा हैं. दरअसल इस पत्थर को पीटने पर अजीब आवाज आती है जिसने लोगों को सोचने पर मजबूर कर दिया है.
ये पत्थर कोई आम पत्थर नहीं बल्कि लोगों के लिए आकर्षण का केंद्र बना हुआ है. अभी तक आपने अजीब तरह की वेशभूषा वाले पत्थर देखें होंगे, कुछ पत्थर अलग-अलग तरह के बने होते हैं जो लोगों के लिए चर्चा का विषय बन जाते हैं लेकिन ये पत्थर अपने वेशभूषा की वजह से नहीं प्रसिद्ध है बल्कि अपनी आवाज की वजह से मशहूर है. जी हांइस पत्थर को पीटने पर बर्तनों के खनकने की आवाज आती है.
गांव के लोग इस पत्थर को ठिनठिनी पत्थर के नाम से जानते हैं. इस पत्थर के बर्तनों की आवाज का तो पता नहीं चल पाया है लेकिन गांव वाला का कहना है कि ये पत्थर कई सालों पहले आसमान से गिरा था. ये अनोखा पत्थर ५ फिट ऊंचे एक टिले पर बसा है जिसके आसपास भी कई पत्थर रखे हैं लेकिन इस पत्थर की ही आवाज सबसे अलग पाई गई है.
बताया जाता है कि जिस समय इंदिरा गांधी प्रधानमंत्री थी उस समय से ये पत्थर काफी सुर्खियों में था और इंदिरा गांधी उस वक्त दरिमा हवाई एयरपोर्ट पर आईं थीं, लोगों के शोर से वो भी पत्थर देखने पहुंच गई थीं. उन्होंने भी इस पत्थर को पीटकर देखा था जिसमें किसी धातु की आवाज आ रही थी.
वहीं इस पत्थर के चमत्कार के पीछे वैज्ञानिकों की मानें तो वो भी कोई सटीक उत्तर तो नहीं बता पाये लेकिन उनका मानना है कि दरिमा मैनपाट इलाका खनिज से भरा है जिसकी वजह से पत्थर से धातु की आवाज निकलती है.
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hindi
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اَمہِ پتہٕ ہیوٚت اَمہِ سخت وَدُن تہٕ بڈٕ مُشکلن مٲنۍ رٲو سۄ شہزادن پَنُن بُتھ تھوٚد تُلُن تہٕ رٕژرٕچ وۄمید کَرٕنۍ
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kashmiri
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डिजिटल इंडिया अभियान के तहत डीएवी स्कूली बच्चों हेतु डाक घर द्वारा कार्यशाला आयोजित
(रिपोर्ट एवं. छाया: एस.एस.डोगरा)
राष्ट्रव्यापी डिजिटल इंडिया अभियान के तहत द्वारका सेक्टर छ स्थित डाक घर ने डीएवी स्कूली बच्चों हेतु इन्टरनेट की तर्ज पर नवीनतम प्रक्रिया के बारे में जानकारी देने के उद्देश्य से ओत प्रोत कार्यशाला का आयोजन किया गया.
स्थानीय डाक घर के पोस्ट मास्टर अजीत कुमार ने उक्त स्कुल के लगभग पचास स्कूली छात्र-छात्राओं तथा अध्यापिका सुश्री विनीता एवं सुश्री रिंकी ओझा को भारतीय डाक विभाग द्वारा चालू की गई कोर बैंकिंग, डाक/राजस्व टिकट, अंतरदेशीय लिफाफे, पोस्ट कार्ड, भाइयों को रक्षाबंधन के त्यौहार पर राखी भेजने के लिए वाटर प्रूफ लिफाफा, बचत खाता, रजिस्टर्ड पोस्ट, स्पीड पोस्ट, मनी आर्डर, देश विदेश में डाक व्यवस्था विषय पर आधुनिक इन्टरनेट सुविधा सम्पन्नता पर प्रकाश डालते हुए महत्तवपूर्ण जानकारी साझा की. जबकि अजित जी ने सुझाया कि भारतीय डाक विभाग की उपयोगी ऑफिसियल वेब साईट इंडियपोस्ट.गोव.इन के माध्यम से नवीनतम सुविधाओं के बारे में अनेक योजनाओं की जानकारियां भी प्राप्त की जा सकती है.
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hindi
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Active Family Chiropractic became known as one of the best chiropractic offices in Plymouth, Minnesota for athletes and kids when founded in 2000. Dr. Lori Goodsell, founder and chiropractor, specializes in helping athletes of all ages and levels maximize their performance and utilize their full potential while remaining injury free.
In the field of sports chiropractic, Dr. Goodsell has worked with athletes of all abilities from the young gymnast to the world class triathlete. Her stretching and injury prevention workshops have been utilized throughout Minnesota by many golf courses, run clubs and athletic training organizations. For several years, she was the chiropractic consultant for the Plymouth Life Time Fitness personal training department. She has spoken for many athletic training organizations including Minnesota Distance Runners Association, Minnesota Tri Club, GolfTec, Gear West, Team Ortho, Minnetonka Tennis Club, Runners with Heart, Rush Creek Golf Course, Begin Oaks Golf Course, Wayzata Schools and many others.
Over the years, she has provided Plymouth with programs such as Backpack Safety America for kids and Chiropractic Mothers Morning Out - an educational program for mothers interested in pediatric chiropractic and natural health care. She is a two time peer reviewed author on the subjects of infertility and special needs children. She has also provided corporate wellness services to multiple Fortune 500 companies including General Mills, Carlson Companies and Digital River.
Dr. Lori personally understands the needs of the athlete and is uniquely qualified to help you reach your next level of health. An athlete herself, she has trained and competed in several triathlons, is an avid tennis player and sailor and paddleboarder, former yoga instructor, former competitive soccer player, ski patroller, runner, cyclist and lifelong gym rat. She lives close to the community that she works with her two dogs whose favorite hobbies include barking and napping. She has spent several years volunteering as coach for Orono schools and a tutor for Minneapolis schools mentoring teenagers.
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english
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स्कूलों को जल्द मिलेंगे एलडीसी, भरेंगे रिक्त पद
अब जल्द ही जिले के उच्च माध्यमिक व माध्यमिक स्कूलों में लिपिक ग्रेड-द्वितीय कनिष्ठ सहायक (एलडीसी) के रिक्त पद भरे जाएंगे।
विकल्प परामर्श कैम्प ८ से, मास्क व सैनेटाइजर के साथ आना होगा
लिपिक ग्रेड द्वितीय संयुक्त प्रतियोगिता भर्ती-२०१८
बूंदी. अब जल्द ही जिले के उच्च माध्यमिक व माध्यमिक स्कूलों में लिपिक ग्रेड-द्वितीय कनिष्ठ सहायक (एलडीसी) के रिक्त पद भरे जाएंगे। बूंदी जिले में १९७ कनिष्ठ सहायकों की भर्ती होगी। इस संबंध में माध्यमिक शिक्षा निदेशक बीकानेर ने आदेश जारी कर दिया। लिपिक ग्रेड द्वितीय संयुक्त प्रतियोगिता भर्ती २०१८ के राजस्थान कर्मचारी चयन बोर्ड की ओर से चयनित अभ्यर्थियों की नियुक्ति परामर्श शिविर (काउंसलिंग) के माध्यम से की जाएगी। यह भर्ती २०१८-१९ की है।
राजस्थान कर्मचारी चयन बोर्ड की ओर से चयनित कर्मचारियों की नियुक्ति जिला शिक्षा अधिकारी माध्यमिक ५ चरणों में करेंगे। यहां देवपुरा स्थित प्रवेशिका संस्कृत माध्यमिक विद्यालय में पहले चरण की शुरुआत ८ जुलाई से होगी। जिसके तहत प्रतिदिन ४० अभ्यर्थियों का रजिस्ट्रेशन किया जाएगा। इसके बाद लिपिक ग्रेड- द्वितीय पदों के लिए कर्मचारियों का भौतिक सत्यापन नियुक्त कमेटी से किया जाएगा। इसके बाद वे अपने मनपसंद स्कूलों को चुन सकेंगे। माध्यमिक शिक्षा विभाग की ओर से होने वाली काउंसलिंग को लेकर सोमवार को प्रवेशिका संस्कृत माध्यमिक विद्यालय का जायजा लिया गया यहां जिला शिक्षा अधिकारी माध्यमिक तेजकंवर ने संबंधित अधिकारियों को व्यवस्था स्वस्थ रखने के निर्देश दिए।
कैंप सुबह ८ बजे से शुरू होगा इसके बाद ९ से १० बजे तक रजिस्ट्रेशन किया जाएगा। निर्धारित कार्यक्रम विभागीय वेबसाइट पर भी देख सकेंगे।
जिला शिक्षा अधिकारी (माध्यमिक) तेज कंवर ने बताया कि काउंसलिंग के लिए कर्मचारियों को अपने मूल योग्यता प्रमाण पत्र, पुलिस अधीक्षक की ओर से जारी किया गया चरित्र प्रमाण पत्र, मेडिकल एवं सभी योग्यता प्रमाण पत्रों की छाया प्रति और आवश्यक शपथ पत्र काउंसलिंग के समय अपनी आइडी और दो फोटो आवश्यक रूप से साथ में लाना होगा।
अतिरिक्त जिला शिक्षा अधिकारी (माध्यमिक) ओम गोस्वामी ने बताया कि लिपिक ग्रेड द्वितीय के परामर्श शिविर (काउंसलिंग) पांच चरणों में चलेगी। प्रत्येक चरण में ४० अभ्यार्थियों का रजिस्ट्रेशन किया जाएगा। शुरुआत ८ जुलाई से होगी। जिसका समापन १२ जुलाई को होगा। ऐसे में सभी को अपने जरूरत कागजात साथ में लाने होंगे। ८ को पहला चरण जिसमें क्रमांक १ से ४०, दूसरा चरण ९ को जिसमें 4१ से ८0, तीसरे चरण में ८१ से १२0 तक, चौथे चरण में १२१ से १60 व अंतिम पांचवें चरण में १6१ से १९7 अभ्यार्थियों का रजिस्ट्रेशन किया जाएगा। इसके बाद इसमें चयनित अभ्यर्थियों को स्कूल आवंटित किए जाएंगे। काउंसलिंग के लिए चार कमेटियां गठित की गई।
मेरिट के आधार पर मिलेगी नियुक्ति
लिपिक ग्रेड-द्वितीय में चयनित अभ्यर्थियों को उनकी मेरिट के आधार पर काउंसलिंग के द्वारा नियुक्ति दी जाएगी। ऐसे में सरकार ने दिव्यांग, परित्यक्ता तथा एकल महिला को नियमानुसार रिक्त पदों के आधार पर उनकी नियुक्ति दी जाएगी।
प्रवेश द्वार पर होगी स्क्रीनिंग व सैनेटाइज
लिपिक ग्रेड-द्वितीय संयुक्त प्रतियोगिता भर्ती २०१८ के चयनित आशार्थियों को स्कूल में प्रवेश के साथ ही सभी की स्क्रीनिंग की जाएगी। इसके बाद सभी के हाथों को सैनिटाइज किया जाएगा। काउंसलिंग के दौरान राज्य सरकार की गाइडलाइन की पूर्ण पालना की जाएगी। यहां आने वाले सभी अभ्यार्थियों व कर्मचारियों को मास्क लगाना अनिवार्य होगा।
जिले के लिए आवंटन आशार्थियों का पदस्थापन के लिए स्थान का विकल्प परामर्श कैम्प के माध्यम से किया जाएगा। विकल्प परामर्श शिविर का आयोजन ८ से १२ जुलाई तक किया जाएगा।
तेज कंवर, जिला शिक्षा अधिकारी (मुख्यालय) माध्यमिक, बूंदी
सुरेन्द्र को यातनाएं देने पर फूटा आक्रोश, सौंपा पत्र
बापू के सिद्धांतों के अनुरूप हाइटेक होगी सथूर पंचायत
कारसेवा की यादें ताजा, चहुंओर उल्लास का माहौल
उच्च शिक्षा के लिए रोज लगा रहे ७६ किलोमीटर की दौड़
सप्ताह में दो दिन बन्द रहेंगे बाजार, हाट भी नहीं लगेगा
वैदिक रीति नीति के साथ किया श्रावणीकर्म
प्रवासियों व ग्रामीणों ने रोजगार की मांग को लेकर किया प्रदर्शन
कृषि वैज्ञानिकों ने बांधी काली पट्टी, मांगा अधिकार
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hindi
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I first met the assertion fail in the function minMaxLoc , I am sure that there is nothing wrong with my input because I imwrite and check it successfully.
I finally solve this because I use wrong vc lib. My vs version is 2015 but I use vc15 lib library .After changing to vc14 , it works.
can we see the code, too ?
I have already done. It works in others computers, so I think nothing wrong with this.
I have solve this for a long time ,even change the opencv version and vs version and now everything might be the same.
did you mean "final" ?
please start a new question, if you run into further problems.
I can not answer my owner question. I think I need to tell other new users this point ,what should I do ?
I have finally solved this. vs(Visual Studio) 2015 match to vc(Visual C++)14 lib vs 2017 match to vc15 lib I made a mistake, my vs version is 2015 but the lib library I use vc15 .After changing to vc14, my code finally works.
you can only use minMaxLoc IF your image was loaded correctly. so that call should go into the if block, too !
minMaxLoc only works on single channel images (see docs, again, please). maybe you need to load your image using the IMREAD_GRAYSCALE flag.
you used the 'mask' argument wrong. it should not be the same image, and it would need a single channel, binary image here. i guess, you don't need it at all.
The mask is discreteImg>0 ,sorry to miss this.
however, omitting the IMREAD_GRAYSCALE flag will read a 3 channel image, so please use that flag.
Hi, the outcome is :discreteImg.tpye()=5 discreteImg.channels()=1 I once thought that if the input Mat is umempty ,the function minMaxLoc will work normally.
type() == 5 would mean: float.
so again, there must be code there doing some conversion, which you don't show.
What will happen if the input of minMaxLoc type is 5? I just delete the imwrite and imread part. Now the input is origin image. What conversion should I do?
it should work all fine.
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english
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---
layout: post # page | post
title: 49 Org Mode en Mapas Mentales
date: 2017-09-16
description: Visualización de Org Mode # Argument
keywords: orgmode # Paraules clau
coments: true # Comentaris activats
---
[The Org Mode Manual](http://orgmode.org/manual/index.html)
Org Mode en mapas mentales, lo puedes [descargar](../assets/orgmode/orgmode_mapas.pdf) en PDF. Puede ser útil para tener una visión de conjunto.
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code
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Welcome to www.missourikennels.com. Our intent is to provide a quick and informative means of contacting kennels to board your pet or finding kennels for sale in the geographic location you desire. You may also list your dogs for sale!
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english
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ग्रामीणों की भूमि का मुआवजा देने के निर्देश
गोपेश्वर। मंगलवार सुबह हेलीकाप्टर से देहरादून से गोपेश्वर पहुंचे बीसूका उपाध्यक्ष और बदरीनाथ विधायक राजेंद्र भंडारी ने मंगलवार को दशोली ब्लाक के ग्राम पंचायत दोगड़ी, कांडई, बमियाला और गैर-टंगसा गांवों का भ्रमण कर ग्रामीणों की समस्याएं सुनीं। उन्होंने अधिकारियों को बमियाला सड़क के दौरान पीएमजीएसवाई की ओर से ली गई ग्रामीणों की भूमि का शीघ्र मुआवजा देने के निर्देश दिए। उन्होंने ग्रामीणों को अपनी निधि से महिला और युवक मंगल दलों को धनराशि देने का आश्वासन भी दिया। इस मौके पर ग्राम प्रधान बमियाला माधवी रावत, भरत कनियाल, क्षेत्र पंचायत सदस्य रेखा बिष्ट, नरेंद्र सिंह राणा, महानंद बिष्ट और भरत सिंह रावत आदि मौजूद थे।
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hindi
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url: http://sanskrit.inria.fr/cgi-bin/SKT/sktreader?t=VH;lex=SH;cache=f;st=t;us=f;cp=t;text=rudhirazcaiva;topic=;abs=f;allSol=2;mode=p;cpts=<!DOCTYPE html>
<html><head>
<meta charset="utf-8">
<title>Sanskrit Reader Companion</title>
<meta name="author" content="Gérard Huet">
<meta property="dc:datecopyrighted" content="2016">
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<body class="chamois_back">
<br><h1 class="title">The Sanskrit Reader Companion</h1>
<table class="chamois_back" border="0" cellpadding="0%" cellspacing="15pt" width="100%">
<tr><td>
<p align="center">
<div class="latin16"><a class="green" href="/cgi-bin/SKT/sktgraph?t=VH;lex=SH;cache=f;st=t;us=f;cp=t;text=rudhirazcaiva;topic=;abs=f;cpts=;mode=g">✓</a> Show Summary of Solutions
</p>
Input:
<span class="red16">rudhiraścaiva</span><hr>
<br>
Sentence:
<span class="deva16" lang="sa">रुधिरश्चैव</span><br>
may be analysed as:</div><br>
<hr>
<span class="blue">Solution 1 : <a class="green" href="/cgi-bin/SKT/sktparser?t=VH;lex=SH;cache=f;st=t;us=f;cp=t;text=rudhirazcaiva;topic=;abs=f;cpts=;mode=p;n=1">✓</a></span><br>
[ <span class="blue" title="0"><b>rudhiraḥ</b></span><table class="deep_sky_back">
<tr><th><span class="latin12"><tr><th><span class="latin12">{ nom. sg. m. }[<a class="navy" href="http://sanskrit.inria.fr/DICO/54.html#rudhira"><i>rudhira</i></a>]</span></th></tr></span></th></tr></table>⟨<span class="magenta">ḥ</span><span class="green">|</span><span class="magenta">c</span><span class="blue"> → </span><span class="red">śc</span>⟩]
<br>
[ <span class="blue" title="7"><b>ca</b></span><table class="mauve_back">
<tr><th><span class="latin12"><tr><th><span class="latin12">{ conj. }[<a class="navy" href="http://sanskrit.inria.fr/DICO/26.html#ca"><i>ca</i></a>]</span></th></tr></span></th></tr></table>⟨<span class="magenta">a</span><span class="green">|</span><span class="magenta">ai</span><span class="blue"> → </span><span class="red">ai</span>⟩]
<br>
[ <span class="blue" title="8"><b>aiva</b></span><table class="carmin_back">
<tr><th><span class="latin12"><tr><th><span class="latin12">{ impft. [2] ac. du. 1 }[<a class="navy" href="http://sanskrit.inria.fr/DICO/11.html#i"><i>i</i></a>]</span></th></tr></span></th></tr></table>⟨⟩]
<br>
<br>
<hr>
<span class="blue">Solution 2 : <a class="green" href="/cgi-bin/SKT/sktparser?t=VH;lex=SH;cache=f;st=t;us=f;cp=t;text=rudhirazcaiva;topic=;abs=f;cpts=;mode=p;n=2">✓</a></span><br>
[ <span class="blue" title="0"><b>rudhiraḥ</b></span><table class="deep_sky_back">
<tr><th><span class="latin12"><tr><th><span class="latin12">{ nom. sg. m. }[<a class="navy" href="http://sanskrit.inria.fr/DICO/54.html#rudhira"><i>rudhira</i></a>]</span></th></tr></span></th></tr></table>⟨<span class="magenta">ḥ</span><span class="green">|</span><span class="magenta">c</span><span class="blue"> → </span><span class="red">śc</span>⟩]
<br>
[ <span class="blue" title="7"><b>ca</b></span><table class="mauve_back">
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code
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Oh well, our federal government wasted another $787 billion on a so-called "Stimulus Package." It was touted as a "jobs bill with shovel ready projects" and rushed through unread by the Obama administration including Virginia Sen. Mark Warner and Sen. Jim Webb. They stated, "if we don't pass it, unemployment will go over 8 percent."
Unemployment is currently 9.7 percent and now President Obama has come forward to admit that there really never was any "shovel ready projects." The only thing the "stimulus" seems to have stimulated is unemployment, wasteful spending, and more debt. Maybe Congress should return the unspent stimulus funds to the taxpayers before those funds get wasted as well.
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english
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Viderestilling:
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code
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ASN: AS5588 T-Mobile Czech Republic a.s.
Internet service provider: MediaCenter Hungary Kft.
Page contains advanced information about IP-address 92.43.203.158 that allows you to specify by IP-address the actual location (address and postcode Internet service provider) by IP-address 92.43.203.158 ASN provider of hosting services (hosting) or actual the location of the supporting server, located at IP-address 92.43.203.158 website.
Users of our service, setting IP-address 92.43.203.158, can obtain information about the name of the provider, the ASN number, the ISP, country and region of placement of server equipment, information about the time zone and geographical coordinates (latitude / longitude) of a provider of hosting services. For user convenience, the location of server equipment, in graphical form, shown with a marker on the maps GoogleMaps.
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english
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Dashes are widely used but little understood. Here's everything you need to know.
This entry was posted on Monday, March 25th, 2019 at 3:08 pm and is filed under this week's podcast. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.
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english
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یہِ پتاہ لٔگِتھ کوٚر گلالہٕ شاہَن تٔمۍ سٕنٛزِ گۄڈنچہِ زنانہِ تہٕ تمام گَرٕ بٲژَو ماتم تہٕ ووٚدُکھ
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kashmiri
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یہ چھ مےٚ پؠٹھ
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kashmiri
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گِبن نہِ گرفتٲری کِس نٔتٟجَس منٛز گوٚو سوڈان تہٕ برطانیہَس درمیان بین الاقوٲمی کرٛیکہٕ ناد تہٕ تناؤ پٲدٕ
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kashmiri
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From corporate fleets to private vehicles, we are the panel beating and paintwork specialists! Competitive prices at a fraction of conventional panel beaters and garages.
Most repairs done below excess – keep your no claims bonus, insurance rating and premiums low! We specialise in repairs of scratches, dints, stone chips, scuff marks, rust and paint protection.
At Bakit Panel & Parts we do all major and minor panel beating.
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This page presents the Google satellite map (zoomable and browsable) of Glendale in Canterbury district in New Zealand.
Geographical coordinates are -43.1696 and 172.7572 and its feature type is Hstd. More information below.
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english
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तीन वर्ष से अधिक उम्र वाले बाँस को बूढ़ा बाँस कहा जाता है। बूढ़ा बाँस बड़ा ही सख्त होता है और जल्दी टूट जाता है उसके विपरीत युवा बाँस मुलायम होता है और उसे किसी भी आकार में मोड़ा जा सकता है।
बाँस से बनाई जाने वाली चीज़ों में सबसे आश्चर्यजनक चीज़ मुझे मछली पकड़ने वाला जाल (जकाई) लगी। असम में जकाई नामक विशेष जाल से मछली पकड़ी जाती है और इसे बॉस से बनाया जाता है। इसकी शंकू जैसी विशेष बनावट के कारण ये आश्चर्यजनक लगता है।
कहा जाता है कि इंसान ने जब हाथ से कलात्मक चीज़ें बनानी शुरू कीं, बाँस की चीज़ें तभी से बन रही हैं। जरूरत के अनुसार इसमें बदलाव हुए हैं और अब भी हो रहे हैं। कहते हैं कि बाँस की बुनाई का रिश्ता उस दौर से है, जब इंसान भोजन इकठ्ठा करता था। शायद भोजन इकठ्ठा करने के लिए ही उसने ऐसी डलियानुमा चीज़ें बनाई होंगी। क्या पता बया जैसी किसी चिडि़या के घोंसले से टोकरी के आकार और बुनावट की तरकीब हाथ लगी हो!
बाँस भारत के उत्तर-पूर्वी क्षेत्र के सात राज्यों अरुणाचल प्रदेश, असम, मेघालय, नागालैण्ड, मणिपुर, मिजोरम, व त्रिपुरा में बहुत पैदा होता है।
संगीत - बाँसुरी
मच्छर - बाँस की पत्तियाँ
फर्नीचर - घर का सजावटी सामान
प्रकाशन - कागज बनाना
एक नया संदर्भ - अचार, मकान, औज़ार आदि।
दैनिक उपयोग की वस्तुएँ
जूते, पर्स, वस्त्र, बैग, थैले, बेल्ट आदि।
जमीन पर बिछाने वाले आसन, टोकरियाँ, चटाईयाँ आदि।
पेड़ की छाल
अगरबत्ती, कागज आदि।
उपले, दवाइयाँ, खाद आदि।
बर्तन, मूर्तियाँ आदि।
१. सजावट-सीता के घर की सजावट बढ़िया है।
२. घबराहट-गर्मी के कारण मुझे घबराहट हो रही है।
३. लिखावट-राम की लिखावट बड़ी सुंदर है।
१. सजीला-दूल्हा बड़ा सजीला लग रहा है।
२. चमकीला-इस साड़ी का रंग बड़ा चमकीला है।
३. रसीला-आम बड़ा ही रसीला है।
१. जमाव-यहाँ पर पानी का जमाव हो रहा है।
२. सुझाव-मुझे तुम्हारा सुझाव उत्तम लगा।
३. लगाव-माता-पिता को अपने बच्चों से लगाव होता ही है।
१. पढ़ाई-खेल के साथ हमें पढ़ाई में भी ध्यान देना चाहिए।
२. लड़ाई-तुम्हें इस तरह लड़ाई करना शोभा नहीं देता है।
३. सिलाई-दर्जी आजकल कपड़ों की ठीक से सिलाई नहीं कर रहा है।
(क) वहाँ बाँस की चीजें बनाने की परम्परा रही है।
(ख) हम यहाँ बाँस की एक-दो चीजों के बारे में ही बता पाए हैं।
(ग) उदहारण आसन जैसी छोटी चीज के लिए भी बाँस की प्रत्येक गाँठ से काटा जाता है।
(घ) खपच्चियों से विभिन्न प्रकार की टोपियाँ बनाई जाती हैं।
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hindi
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\begin{document}
\title{The boundary action of a sofic random subgroup of the free group}
\author{Jan Cannizzo}
\address{Stevens Institute of Technology\newline
\indent Department of Mathematical Sciences\newline
\indent Castle Point on Hudson\newline
\indent Hoboken, NJ 07030\newline
}
\email{[email protected]}
\thanks{I am very grateful to my advisor, Vadim Kaimanovich, for his advice and support, and for encouraging me to work on the subject of this paper. I thank the anonymous referee for suggesting several improvements.}
{\mathfrak m}aketitle
\begin{abstract}
We prove that the boundary action of a sofic random subgroup of a finitely generated free group is conservative (there are no wandering sets). This addresses a question asked by Grigorchuk, Kaimanovich, and Nagnibeda, who studied the boundary actions of individual subgroups of the free group. We also investigate the cogrowth and various limit sets associated to sofic random subgroups. We make heavy use of the correspondence between subgroups and their Schreier graphs, and central to our approach is an investigation of the asymptotic density of a given set inside of large neighborhoods of the root of a sofic random Schreier graph.
\end{abstract}
\section{Introduction}
The study of \emph{invariant random subgroups}, meaning subgroups of a given group whose distribution is conjugation-invariant, has recently attracted a lot of attention. Vershik has called for a description of all nonatomic conjugation-invariant measures on the lattice of subgroups of a given countable group \cite{V1} and provided such a description in the case of the infinite symmetric group \cite{V2}. Such measures naturally arise from the boundary actions of self-similar groups, such as the \emph{Basilica group} (see the treatment of D'Angeli, Donno, Matter, and Nagnibeda \cite{ADMN}), or the famous Grigorchuk group (see \cite{Vo}), and progress has recently been made in understanding the spaces of invariant random subgroups of free groups \cite{B} and lamplighter groups \cite{BGK}. Ab\'{e}rt, Glasner, and Vir\'{a}g recently generalized \emph{Kesten's theorem} to invariant random subgroups \cite{AGV}. Moreover, invariant random subgroups are closely connected with the theory of sofic groups (see, for example, the survey \cite{Pe}) and sofic equivalence relations \cite{EL}.
There is a fruitful interplay between groups and graphs, as is evidenced, for instance, in the classic paper of Stallings \cite{S}. Central to our approach is the fact that it is possible to switch back and forth between subgroups and their \emph{Schreier graphs} (objects which generalize Cayley graphs), allowing one to think about subgroups in geometric terms. Accordingly, the study of invariant random subgroups is tantamount to the study of invariant random Schreier graphs (which in turn belongs to the theory of discrete measured equivalence relations established by Feldman and Moore~\cite{FM}).
Intuitively speaking, invariant random Schreier graphs behave rather like Cayley graphs, the analogy being that, whereas a Cayley graph is spatially homogenous, insofar as it is vertex-transitive, i.e.\ invariant upon shifting the root, an invariant random Schreier graph is \emph{stochastically homogenous} (see \cite{K2} for the origin of the term), insofar as its distribution is invariant upon shifting the root. Grigorchuk, Kaimanovich, and Nagnibeda \cite{GKN} recently studied the ergodic properties of the action of a subgroup $H\leqslant{{\mathfrak m}athbb F}_n$ of a finitely generated free group on the boundary $\partial{{\mathfrak m}athbb F}_n$ equipped with the uniform measure (a situation which is analogous to the action of a Fuchsian group on the boundary of the hyperbolic plane equipped with Lebesgue measure). In particular, they used Schreier graphs to describe the \emph{Hopf decomposition} (into conservative and dissipative parts) of this action. Although the boundary action of an arbitrary subgroup may be conservative, dissipative, or such that both its conservative and dissipative parts have positive measure \cite{GKN}, the boundary action of a normal subgroup is necessarily conservative, as follows from \cite{K1}. Our main result is an extension of this result to \emph{sofic random subgroups}. That is, we show that \emph{the boundary action of a sofic random subgroup of a finitely generated free group is conservative} (Theorem~\ref{conservative}), addressing a question asked in \cite{GKN}.
Before proving our main result, we undertake an investigation of the asymptotic density of a given set inside of a random invariant Schreier graph. Our main question of interest (Question~\ref{q}) can be formulated as follows: given a nontrivial subset $A$ of the space of Schreier graphs, must the density of $A$ inside of large neighborhoods of the root of an invariant random Schreier graph be bounded away from zero? If so, then we say the invariant random Schreier graph has \emph{property D}. A positive answer to the question would amount to a new \emph{ergodic theorem} for invariant random graphs (see \cite{BN} for an overview of many ergodic theorems). Unfortunately, we are unable to answer Question~\ref{q}, but by introducing a notion which we call \emph{relative thinness} and assuming that our invariant random Schreier graph $\Gamma$ is sofic, we are able to show that $\Gamma$ fails to satisfy the aforementioned property only if its geometry is quite peculiar (Proposition~\ref{lop}), a fact which allows us to prove Theorem~\ref{conservative}.
The paper is organized as follows: In Section~2, we give an introduction to Schreier graphs and invariant random Schreier graphs, making plain their connection with subgroups. Section~3 is devoted to making precise the question of whether a given set is asymptotically dense inside of large neighborhods of the root of an invariant random Schreier graphs. In Section~4, we introduce sofic invariant subgroups and thereafter, in Section~5, the notion of relative thinness, which allows us to shed some light on Question~\ref{q}. In Section~6, we prove our main result, showing that the boundary action of a sofic random subgroup is conservative (Theorem~\ref{conservative}). Finally, in Section~7, we tease out several consequences of Theorem~\ref{conservative}, namely a bound on the \emph{cogrowth} of a sofic random subgroup (Corollary~\ref{cogrowth}) and a theorem on the size of various \emph{limit sets} associated to sofic random subgroups of ${{\mathfrak m}athbb F}_n$ (Theorem~\ref{lim}). We also give examples showing that the \emph{radial limit set} may have full or zero measure, thus completely characterizing the possible measures of the limit sets of a sofic invariant subgroup.
\section{The space of Schreier graphs of a countable group}
Given a countable group $G$ with generating set ${{\mathfrak m}athcal A}=\{a_i\}_{i\in I}$ and a subgroup $H\leqslant G$, consider the natural action of $G$ on the space of (right) cosets $G/H$. This action is transitive and determines a graph $\Gamma=(\Gamma,H)$ as follows. The vertex set of $\Gamma$ is identified with $G/H$, and two vertices $Hg$ and $Hg'$ are connected with an edge directed from $Hg$ to $Hg'$ and labeled with the generator $a_i$ if and only if $Hga_i=Hg'$. The graph $\Gamma$ (which is \emph{rooted} at $H$, meaning that we distinguish the vertex $H$) is called a (right) \emph{Schreier graph}, and we denote by $\Lambda(G)$ the space of (isomorphism classes) of (right) Schreier graphs of $G$, where two Schreier graphs are said to be isomorphic if there exists a graph isomorphism between them which preserves the edge-labeling and root. Note that Schreier graphs are necessarily $2|{{\mathfrak m}athcal A}|$-regular, meaning that each of their vertices has degree $2|{{\mathfrak m}athcal A}|$ (the degree of a vertex may be defined as the sum of the number of incoming edges and the number of outgoing edges attached to it). Schreier graphs may have both loops (cycles of length one) and multi-edges (multiple edges that join the same pair of vertices). Note also that Schreier graphs naturally generalize Cayley graphs, which arise whenever the subgroup $H$ is normal, i.e.\ when the cosets $Hg$ correspond to the elements of a group.
Let us immediately turn our attention to the space of Schreier graphs of the finitely generated free group of rank $n$ with a fixed set of generators, i.e.\
\[
{{\mathfrak m}athbb F}_n=\langle a_1,\ldots,a_n\rangle.
\]
This is natural since, as we will presently make clear, every Schreier graph is a Schreier graph of a free group. Our first observation is this: Given a Schreier graph $(\Gamma,H)\in\Lambda({{\mathfrak m}athbb F}_n)$, the subgroup $H\leqslant{{\mathfrak m}athbb F}_n$ can be recovered from $\Gamma$ in a very natural way. Namely, $H$ is precisely the fundamental group $\pi_1(\Gamma,H)$, i.e.\ the set of words read upon traversing closed paths that begin and end at the coset $H$. Note that we thereby identify $\pi_1(\Gamma,H)$ with a specific subgroup of ${{\mathfrak m}athbb F}_n$ and are not interested merely in its isomorphism class. By the above discussion, it follows that $\Lambda(G)\subseteq\Lambda({{\mathfrak m}athbb F}_n)$ whenever $G$ is a group with generating set ${{\mathfrak m}athcal A}=\{a_1,\ldots,a_n\}$. It also follows that we could define Schreier graphs ``abstractly,'' without appealing to the coset structure determined by a subgroup of ${{\mathfrak m}athbb F}_n$. That is, we could define a Schreier graph to be a (connected and rooted) $2n$-regular graph whose edges come in $n$ different colors and are colored so that every vertex is attached to precisely one incoming edge of a given color and one outgoing edge of that color.
There is a natural one-to-one correspondence between the lattice of subgroups of ${{\mathfrak m}athbb F}_n$, denoted $L({{\mathfrak m}athbb F}_n)$, and the space of Schreier graphs $\Lambda({{\mathfrak m}athbb F}_n)$. Every subgroup $H\in L({{\mathfrak m}athbb F}_n)$ determines a Schreier graph, and every Schreier graph $\Gamma\in\Lambda({{\mathfrak m}athbb F}_n)$ determines a subgroup of ${{\mathfrak m}athbb F}_n$ (by passing to the fundamental group):
\begin{equation*}
\begin{tikzcd}[column sep=55]
L({{\mathfrak m}athbb F}_n)\arrow[bend left]{r}{(\Gamma,H)}
&\Lambda({{\mathfrak m}athbb F}_n)\arrow[bend left]{l}{\pi_1(\Gamma)}\,.
\end{tikzcd}
\end{equation*}
\noindent The space of Schreier graphs $\Lambda({{\mathfrak m}athbb F}_n)$ has a natural projective structure. Denote by $\Lambda_r({{\mathfrak m}athbb F}_n)$ the set of (isomorphism classes of) $r$-neighborhoods centered at the roots of elements of $\Lambda({{\mathfrak m}athbb F}_n)$, where by an $r$-neighborhood we mean the subgraph of a Schreier graph induced by the set of vertices at distance less than or equal to $r$ from the root. Then $\Lambda({{\mathfrak m}athbb F}_n)$ may be realized as the projective limit
\begin{equation*}\label{proj}
\Lambda({{\mathfrak m}athbb F}_n)=\varprojlim\Lambda_r({{\mathfrak m}athbb F}_n),
\end{equation*}
where the connecting morphisms $\pi_r:\Lambda_{r+1}({{\mathfrak m}athbb F}_n)\to\Lambda_r({{\mathfrak m}athbb F}_n)$ are restriction maps that send an $(r+1)$-neighborhood $V$ to the $r$-neighborhood $U$ of its root. (Looking at things the other way around, $\pi_r(V)=U$ only if there exists an embedding $U\hookrightarrow V$ that sends the root of $U$ to the root of $V$.) By endowing each of the sets $\Lambda_r({{\mathfrak m}athbb F}_n)$ with the discrete topology, we turn $\Lambda({{\mathfrak m}athbb F}_n)$ into a compact Polish space.
Throughout this paper, we will think of an $r$-neighborhood $U\in\Lambda_r({{\mathfrak m}athbb F}_n)$ both as a rooted graph and as the \emph{cylinder set}
\[
U=\{(\Gamma,x)\in\Lambda({{\mathfrak m}athbb F}_n){\mathfrak m}id U_r(x)\cong U\},
\]
where $U_r(x)$ denotes the $r$-neighborhood of the vertex $x$. Note that a finite Borel measure ${\mathfrak m}u$ on $\Lambda({{\mathfrak m}athbb F}_n)$ is the same thing as a family of measures ${\mathfrak m}u_r:\Lambda_r({{\mathfrak m}athbb F}_n)\to{{\mathfrak m}athbb R}$ that satisfies
\[
{\mathfrak m}u_r(U)=\sum_{V\in\pi_r^{-1}(U)}{\mathfrak m}u_{r+1}(V)
\]
for all $U\in\Lambda_r({{\mathfrak m}athbb F}_n)$ and for all $r$. As is customary when working with measure spaces, all statements regarding measurable sets will be understood to be valid \emph{modulo zero}, i.e.\ up to the inclusion or exclusion of null sets (in particular, we will avoid use of qualifying expressions such as ``almost every.'')
By an \emph{invariant random subgroup} of a countable group $G$, we will mean a probability measure on $L(G)$ that is conjugation-invariant, i.e.\ invariant under the action $G\circlearrowright L(G)$ given by $(g,H){\mathfrak m}apsto gHg^{-1}$. Via the correspondence between $L(G)$ and $\Lambda(G)$ (indeed, it is via this correspondence that we endow $L(G)$ with its Borel structure), this determines a continuous action on $\Lambda(G)$ which is easily visualized as follows: Given a Schreier graph $(\Gamma,H)$ and an element $g\in G$, where we assume that $g$ has a fixed presentation in terms of the generators of $G$, it is possible to read the element $g$ starting from the root $H$ (or, indeed, from any other vertex). This is accomplished by following, in the proper order, edges labeled with the generators that comprise $g$ (note that following a generator $a_i^{-1}$ is tantamount to traversing a directed edge labeled with $a_i$ in the direction opposite to which the edge is pointing). Applying the group element $g$ to the graph $(\Gamma,H)$ then amounts simply to ``shifting the root" of $(\Gamma,H)$ in the way just described. That is, one begins at the vertex $H$, then follows the path corresponding to the element $g$, and then declares its endpoint to be the new root. Note that if $G$ has generators of order two, then a path corresponding to an element $g\in G$ may not be unique; nevertheless, the endpoint of any path which represents $g$ is uniquely determined by $g$.
\begin{figure}
\caption{A Schreier graph of the free group ${{\mathfrak m}
\label{fig1}
\end{figure}
The image of a $G$-invariant measure under the identification $H{\mathfrak m}apsto(\Gamma,H)$ is a $G$-invariant measure on $\Lambda(G)$ (and hence, via the inclusion $\Lambda(G)\hookrightarrow\Lambda({{\mathfrak m}athbb F}_n)$, an ${{\mathfrak m}athbb F}_n$-invariant measure on $\Lambda({{\mathfrak m}athbb F}_n)$). We may thus speak of an \emph{invariant random Schreier graph}. In fact, throughout the remainder of the paper we will treat invariant random subgroups and invariant random Schreier graphs as the same objects, using whichever terminology is more appropriate to the context.
The most basic examples of invariant random Schreier graphs are Dirac measures supported on Cayley graphs: indeed, Cayley graphs (equivalently, Schreier graphs of normal subgroups of ${{\mathfrak m}athbb F}_n$) are invariant under conjugation essentially by definition and may be regarded as $1$-periodic points in $\Lambda({{\mathfrak m}athbb F}_n)$. More generally, the uniform measure supported on a finite Schreier graph (of cardinality $k$, say) is an invariant measure, thus giving rise to $k$-periodic points, and it is not difficult to construct examples of infinite periodic Schreier graphs. Of greater interest is the space of \emph{nonatomic} invariant measures, typically supported on aperiodic Schreier graphs. Such measures have recently been the focus of a great deal of research (see \cite{AGV}, \cite{ADMN}, \cite{B}, \cite{BGK}, \cite{V1}, \cite{V2}, and \cite{Vo}), but much remains unknown.
\section{A question regarding the density of sets inside of large neighborhoods}
Our main result is that the boundary action of a sofic random subgroup is conservative. By a theorem of Grigorchuk, Kaimanovich, and Nagnibeda~\cite{GKN}, this assertion is equivalent to the assertion that
\begin{equation}\label{thm}
\lim_{r\to\infty}\frac{|U_r(\Gamma,H)|}{|U_r({{\mathfrak m}athbb F}_n,e)|}=0,
\end{equation}
where the numerator of the above fraction is the size of the $r$-neighborhood of the root of our random Schreier graph and the denominator is the size of the $r$-neighborhood of the identity of the Cayley graph of ${{\mathfrak m}athbb F}_n$. In proving this result, our focus will first be on a considerably more general question regarding the asymptotic density of a given set inside of neighborhoods centered at the root of a random graph. This latter question can be formulated as follows: if $A\subseteq(\Lambda({{\mathfrak m}athbb F}_n),{\mathfrak m}u)$ is a measurable subset of the space of Schreier graphs and ${\mathfrak m}u$ is an invariant measure, then how dense is $A$ inside of ${\mathfrak m}u$-random $r$-neighborhoods $U_r(x)\in\Lambda_r({{\mathfrak m}athbb F}_n)$? An informal---and imprecise---way to say what we mean by the density of $A$ in $U_r(x)$ is in terms the function $\rho_{A,r}:\Lambda({{\mathfrak m}athbb F}_n)\to{{\mathfrak m}athbb Q}$ given by
\[
\rho_{A,r}(\Gamma,x)\colonequals\frac{|A\cap U_r(x)|}{|U_r(x)|}.
\]
To make this rigorous, note that for any $A\subseteq\Lambda({{\mathfrak m}athbb F}_n)$ there is an induced Borel embedding
\[
\Theta_A:\Lambda({{\mathfrak m}athbb F}_n)\to\bigcup_{\Gamma\in\Lambda({{\mathfrak m}athbb F}_n)}\{0,1\}^\Gamma\equalscolon\{0,1\}^{\Lambda({{\mathfrak m}athbb F}_n)}
\]
which sends a Schreier graph $\Gamma$ to the \emph{binary field} ${{\mathfrak m}athbb F}F:\Gamma\to \{0,1\}$ given by
\begin{equation}\label{char}
{{\mathfrak m}athbb F}F(x)=\left\{
\begin{array}{lr}
1,&(\Gamma,x)\in A\\
0,&(\Gamma,x)\notin A
\end{array}
\right.,
\end{equation}
where $(\Gamma,x)$ is the Schreier graph obtained from $\Gamma$ by \emph{rerooting} $\Gamma$ at the vertex $x$. The resulting space of binary configurations over elements of $\Lambda({{\mathfrak m}athbb F}_n)$ (namely, the image of $\Theta_A$) serves to ``highlight'' the set $A$, and the corresponding functions $\rho_{A,r}$ may now be written as
\begin{equation}\label{density}
\rho_r({{\mathfrak m}athbb F}F)=\frac{1}{|U_r(x)|}\sum_{y\in U_r(x)}{{\mathfrak m}athbb F}F(y).
\end{equation}
Note that if ${\mathfrak m}u$ is an invariant measure on $\Lambda({{\mathfrak m}athbb F}_n)$, then $(\Theta_A)_*{\mathfrak m}u$ is an invariant measure on $\{0,1\}^{\Lambda({{\mathfrak m}athbb F}_n)}$. From now on, when talking about the density of a given set $A$ inside of $r$-neighborhoods, we will refer to the functions $\rho_{A,r}$ defined over the binary field constructed as per (\ref{char}), without necessarily making mention of the map $\Theta_A$. We are now ready to formulate our question:
\begin{question}\label{q}
Let ${\mathfrak m}u$ be an invariant random Schreier graph and $A\subseteq\Lambda({{\mathfrak m}athbb F}_n)$ a Borel set, and consider the average densities
\[
{{\mathfrak m}athbb E}(\rho_{A,r})=\int\rho_{A,r}\,d{\mathfrak m}u.
\]
Then supposing ${{\mathfrak m}athbb E}(\rho_{A,0})>0$, what can be said of the averages ${{\mathfrak m}athbb E}(\rho_{A,r})$? Do they converge? Are they bounded away from zero?
\end{question}
More generally, consider an ${{\mathfrak m}athbb F}_n$-invariant measure ${\mathfrak m}u$ on $\{0,1\}^{\Lambda({{\mathfrak m}athbb F}_n)}$ (which needn't necessarily come from a Borel set $A\subseteq\Lambda({{\mathfrak m}athbb F}_n)$ as above). The following example shows that if such an invariant random binary field has a ``fixed geometry,'' meaning that it is supported on a common underlying graph, then it must answer Question~\ref{q} in the positive.
\begin{example}
Let $\Gamma\in\Lambda({{\mathfrak m}athbb F}_n)$ be a Cayley graph, i.e.\ the Schreier graph of a normal subgroup of ${{\mathfrak m}athbb F}_n$, and ${\mathfrak m}u$ an invariant measure on $\{0,1\}^\Gamma$. Then one readily verifies that the average densities ${{\mathfrak m}athbb E}(\rho_r)$ are all the same. Indeed, we have
\begin{align*}
\int\rho_r\,d{\mathfrak m}u&=\int\left(\frac{1}{|U_r(x)|}\sum_{y\in U_r(x)}{{\mathfrak m}athbb F}F(y)\right)d{\mathfrak m}u\\
&=\frac{1}{|U_r(x)|}\sum_{y\in U_r(x)}\left(\int{{\mathfrak m}athbb F}F(y)\,d{\mathfrak m}u\right)\\
&=\frac{1}{|U_r(x)|}\sum_{y\in U_r(x)}{{\mathfrak m}athbb E}(\rho_0)={{\mathfrak m}athbb E}(\rho_0).
\end{align*}
If, however, our random invariant Schreier graph ceases to be so nice (e.g.\ if it ceases to be vertex-transitive), then the averages ${{\mathfrak m}athbb E}(\rho_r)$ can be expected to vary considerably from ${{\mathfrak m}athbb E}(\rho_0)$. Our question is: How much? Can they be arbitrarily close to zero if ${{\mathfrak m}athbb E}(\rho_0)$ is not zero?
\end{example}
Question~\ref{q} asks whether invariance implies that, given a subset of the space of Schreier of positive measure (in other words, a nontrivial property of the root of our random graph), it will be asymptotically dense inside of large $r$-neighborhoods. For the sake of brevity, let us give this property a name.
\begin{definition}(Property $D$)
We say that an invariant random Schreier graph ${\mathfrak m}u$ has \emph{property D} if it answers Question~\ref{q} in the positive, in the sense that, if $A\subseteq\Lambda({{\mathfrak m}athbb F}_n)$ is any subset with ${\mathfrak m}u(A)>0$, then the average densities ${{\mathfrak m}athbb E}(\rho_{A,r})$ of $A$ inside of $r$-neighborhoods are bounded away from zero.
\end{definition}
We will show that, upon placing a mild condition on our random invariant Schreier graph $\Gamma$---namely \emph{soficity}---the averages ${{\mathfrak m}athbb E}(\rho_{A,r})$ can get arbitrarily small only if $\Gamma$ exhibits a rather ``wild'' geometry. To be a little more precise, we will introduce a notion which we call \emph{relative thinness} and show that the average densities ${{\mathfrak m}athbb E}(\rho_{A,r})$ can get arbitrarily small only if $\Gamma$ is arbitrarily relatively thin at different scales. We are then able to deduce the conservativity of the boundary action of a sofic random subgroup via the following argument:
\begin{itemize}
\item[i.] If $\Gamma$ satisfies property $D$ (and is not the Dirac measure concentrated on the Cayley graph of ${{\mathfrak m}athbb F}_n$, a case which is easily dealt with), then there exists a number $k\in{{\mathfrak m}athbb N}$ such that the set of Schreier graphs whose roots belong to a cycle of length $k$ has positive measure, and whose density inside of $r$-neighborhoods is therefore bounded away from zero. The fact that cycles of bounded length are sufficiently dense inside of $\Gamma$ is in turn enough for us to show that $\Gamma$ must satisfy (\ref{thm}).
\item[ii.] If $\Gamma$ does not satisfy property $D$, then its geometry is such that it cannot grow too quickly; in particular, we are again able to show that $\Gamma$ must satisfy the condition (\ref{thm}).
\end{itemize}
It is worth pointing out that, if our property $D$ held for all invariant random Schreier graphs, then the argument of Theorem~\ref{conservative} would imply that the boundary action of any invariant random subgroup (sofic or not) of ${{\mathfrak m}athbb F}_n$ is conservative.
\section{Sofic invariant subgroups}
The class of \emph{sofic groups}, first defined by Gromov~\cite{G} and given their name by Weiss~\cite{W}, is a large class of groups which has recently received a great deal of attention. Roughly speaking, a finitely generated group is sofic if its Cayley graph can be approximated by a sequence of finite Schreier graphs. Amenable groups (for which \emph{F\o lner sequences} determine approximating sequences) and residually finite groups (for which finite quotients serve as approximating sequences) are immediate examples of sofic groups. In fact, so large is the class of sofic groups that it is unknown whether all groups are sofic. For more on sofic groups, we refer the reader to the survey of Pestov~\cite{Pe}.
The notion of soficity, which can be formulated in terms of the weak convergence of measures, naturally generalizes to objects other than groups, such as \emph{unimodular random graphs}---see, for instance, \cite{AL}. In another context, Elek and Lippner~\cite{EL} have recently defined soficity for \emph{discrete measured equivalence relations}, a setting which subsumes invariant random Schreier graphs. To make sense of the definition, observe that the uniform probability measure on a finite Schreier graph $\Gamma$ determines an invariant measure on $\Lambda({{\mathfrak m}athbb F}_n)$, namely the uniform measure supported on the conjugacy class of the associated subgroup $\pi_1(\Gamma)$. The definition now goes as follows:
\begin{definition}\label{sofic}(Sofic random Schreier graph)
An invariant random Schreier graph ${\mathfrak m}u$ is \emph{sofic} if there exists a sequence of finite Schreier graphs $\{\Gamma_i\}_{i\in{{\mathfrak m}athbb N}}$ such that ${\mathfrak m}u_i\to{\mathfrak m}u$ weakly, where ${\mathfrak m}u_i$ is the invariant measure on $\Lambda({{\mathfrak m}athbb F}_n)$ determined by $\Gamma_i$.
\end{definition}
The convergence of which we speak also goes under the name of \emph{Benjamini-Schramm convergence}, after the paper \cite{BS}. We note that (as is also done in \cite{BS}) the weak convergence of measures in Definition~\ref{sofic} can be thought of in more geometric terms as follows: Suppose first that $\Gamma$ is a Cayley graph (which becomes an invariant random Schreier graph when identified with the Dirac measure concentrated on itself). We say that a finite graph $(\Gamma',{\mathfrak m}u)$ equipped with the uniform probability measure is an \emph{$(r,\varepsilon)$-approximation} to $\Gamma$ if there exists a set $A\subseteq\Gamma'$ of measure ${\mathfrak m}u(A)>1-\varepsilon$ such that for all $x\in A$, the $r$-neighborhood of $x$ in $\Gamma'$ is isomorphic (in the category of edge-labeled graphs) to the $r$-neighborhood of the identity (or, indeed, of any other vertex) in $\Gamma$. The graph $\Gamma$ is sofic precisely if it admits an $(r,\varepsilon)$-approximation for any pair $(r,\varepsilon)$, where $r\in{{\mathfrak m}athbb N}$ and $\varepsilon>0$. A group $G$ is thus sofic if, given any Cayley graph $\Gamma$ of $G$, it is possible to construct finite graphs which locally look like $\Gamma$ at almost all of their points. More generally, suppose that $\Gamma$ is a random invariant Schreier graph. The distribution of $\Gamma$ naturally determines a probability measure ${\mathfrak m}u_r$ on $\Lambda_r({{\mathfrak m}athbb F}_n)$, the set of $r$-neighborhoods of Schreier graphs of ${{\mathfrak m}athbb F}_n$, and we again say that a finite graph $(\Gamma,{\mathfrak m}u)$ equipped with the uniform probability measure is an $(r,\varepsilon)$-approximation to $\Gamma$ if for all $U\in\Lambda_r({{\mathfrak m}athbb F}_n)$ we have $|{\mathfrak m}u(U)-{\mathfrak m}u_r(U)|<\varepsilon$. Then, as before, a random invariant Schreier graph is sofic precisely if it admits finite $(r,\varepsilon)$-approximations for any pair $(r,\varepsilon)$.
Our definition does not take exactly the same form as the ones given, for instance, in \cite{EL} or \cite{G}. The main difference is that we require our approximating sequence to consist of bona fide Schreier graphs, and not, as is usually the case, of graphs which need not have the structure of a Schreier graph at all of their points. Let us therefore quickly show that our definition---which we feel is a bit cleaner---is in fact equivalent to the usual one.
\begin{theorem}
If there exist finite graphs $(\Gamma_i,{\mathfrak m}u_i)$ which are a sofic approximation to ${\mathfrak m}u$, then they may be modified to create finite Schreier graphs $(\Gamma_i',{\mathfrak m}u_i')$ which are a sofic approximation to ${\mathfrak m}u$.
\end{theorem}
\begin{remark}
Here the graphs $\Gamma_i$ need not have the structure of a Schreier graph at each of their points, i.e.\ there may exist points whose degree is not $2n$ or are such that the edges attached to them do not have a Schreier labeling. Another caveat that should be pointed out is that a Schreier graph is by definition connected and rooted, although we do not actually impose these conditions in Definition~\ref{sofic} or the above proposition: there is no sense in assigning a root to the graphs of a sofic approximation (as every vertex is effectively treated as a root), and it is often natural for such graphs to have several connected components (e.g.\ if the measure they approximate is supported on a set of several distinct Cayley graphs).
\end{remark}
\begin{proof}
Let $\Gamma_i$ be an $(r,\varepsilon)$-approximation to ${\mathfrak m}u$ and $A\subseteq\Gamma_i$ the set of points at which $\Gamma_i$ does not have the structure of a Schreier graph. Let $\Gamma_i'$ be the subgraph of $\Gamma_i$ induced by the set $\Gamma_i\backslash A$ and $A'\subseteq\Gamma_i'$ the set of points at which $\Gamma_i'$ does not have the structure of a Schreier graph. Note that $A'$ is a subset of the set of neighbors of the removed set $A$, and that therefore ${\mathfrak m}u_i(A\cup A')<\varepsilon$ (since the $r$-neighborhood $U_r(x)\subseteq\Gamma_i$ of any point $x\in A$ does not approximate ${\mathfrak m}u$, neither does the $r$-neighborhood of any neighbor of $x$, provided $r>1$).
Now, the edges attached to points $x\in A'$ are properly labeled with the generators $a_1,\ldots,a_n$ of ${{\mathfrak m}athbb F}_n$---the only problem is that some generators may be missing, i.e.\ it may be that $\deg(x)<2n$. We thus ``stitch up'' the graph $\Gamma_i'$ as follows: for every generator $a_i$ which does not label any of the edges (neither incoming nor outgoing) attached to a given point $x\in A'$, add a loop to $x$ and label it with $a_i$. If, on the other hand, there exists precisely one edge (assume without loss of generality that it is outgoing) attached to $x$ and labeled with a generator $a_i$, then consider the longest path $\gamma$ whose edges are labeled only with $a_i$ and which is attached to $x$. The endpoint of $\gamma$ will be a vertex $y\in A'$ distinct from $x$; to ``complete the cycle,'' we thus need only join $x$ and $y$ with an edge and label this edge with $a_i$ in the obvious way. By repeating this procedure for every vertex in $A'$, we ensure that $\Gamma_i'$ has the structure of a Schreier graph at every point while modifying it only on a set of very small measure. It follows that the sequence of Schreier graphs $(\Gamma_i',{\mathfrak m}u_i')$ is a sofic approximation to ${\mathfrak m}u$.
\end{proof}
Note that Definition~\ref{sofic} readily generalizes to invariant random fields: one must simply define convergence with respect to finite $\{0,1\}$-labeled Schreier graphs. We will make use of the following lemma later.
\begin{lemma}\label{soficfield}
Let ${\mathfrak m}u$ be a sofic random Schreier graph and $A\subseteq\Lambda({{\mathfrak m}athbb F}_n)$ a Borel set. Then the invariant random field $(\Theta_A)_*{\mathfrak m}u$ is also sofic.
\end{lemma}
\begin{proof}
Denote by $A_r$ the collection of cylinder sets $U\in\Lambda_r({{\mathfrak m}athbb F}_n)$ such that ${\mathfrak m}u(A\cap U)>0$. Clearly, $A\subseteq A_r$, and moreover ${\mathfrak m}u(A_r\backslash A)\equalscolon\varepsilon_r\to0$, i.e.\ the sets $A_r$ approximate $A$. Let $\Gamma$ be a finite $(r,\varepsilon)$-approximation to ${\mathfrak m}u$, and construct a binary field ${{\mathfrak m}athbb F}F:\Gamma\to\{0,1\}$ by assigning to a given vertex $x\in\Gamma$ the value $1$ if the cylinder set corresponding to its $r$-neighborhood $U_r(x)$ belongs to $A_r$ and the value $0$ otherwise. Then ${{\mathfrak m}athbb F}F$ is an $(r,\varepsilon)$-approximation to ${\mathfrak m}u_r$ and hence an $(r,\varepsilon+\varepsilon_r)$-approximation to ${\mathfrak m}u$. By constructing fields ${{\mathfrak m}athbb F}F_i$ in this way for a sequence of finite graphs $\Gamma_i$ which are $(r_i,\varepsilon_i)$-approximations to ${\mathfrak m}u$, with $r_i\to\infty$ and $\varepsilon_i\to0$, we obtain a sofic approximation to $(\Theta_A)_*{\mathfrak m}u$.
\end{proof}
Morally speaking, Lemma~\ref{soficfield} allows us to phrase Question~\ref{q} in terms of finite graphs, namely those which come from a sofic approximation. Working with finite graphs in turn has several advantages, as we show in the next section.
\section{Relative thinness}
\noindent In order to investigate Question~\ref{q}, we would like to introduce a notion which we call \emph{relative thinness}. To be more precise, let $\Gamma$ be a Schreier graph, and consider the functions $\tau_r:\Gamma\to{{\mathfrak m}athbb Q}$ defined by
\[
\tau_r(x)\colonequals\sum_{y\in U_r(x)}\frac{1}{|U_r(y)|}.
\]
Note that if, say, all of the $r$-neighborhoods of $\Gamma$ have the same size (as is the case, for instance, when $\Gamma$ is a Cayley graph), then $\tau_r\equiv1$. If, on the other hand, the $r$-neighborhood of a point $x\in\Gamma$ is small compared to the $r$-neighborhoods near it, then one will have $\tau_r(x)<1$ (and if it is large compared to the $r$-neighborhoods near it, then one will have $\tau_r(x)>1$). We thus say that a Schreier graph $\Gamma$ is \emph{relatively thin at scale $r$} at a point $x\in\Gamma$ if $\tau_r(x)<1$ (if a piece of cloth is worn down at a particular spot, then the regions surrounding that spot will have more mass than is to be found at the spot itself).
One feature of relative thinness is that it is ``tempered,'' meaning that if $\Gamma$ is very thin at $x$ and $y$ is a neighbor of $x$, then $\Gamma$ will be thin at $y$ as well. To be more precise, let us say that a function $f:\Gamma\to{{\mathfrak m}athbb R}$ is \emph{$C$-Lipschitz} if whenever $x,y\in\Gamma$ are neighbors,
\[
f(x)\leqslant Cf(y)
\]
for some constant $C\geqslant1$. Likewise, we say that a family of functions $\{f_i:\Gamma_i\to{{\mathfrak m}athbb R}\}_{i\in I}$ is \emph{uniformly $C$-Lipschitz} over the family of graphs $\{\Gamma_i\}_{i\in{{\mathfrak m}athbb N}}$ if each $f_i$ is $C$-Lipschitz for some constant $C\geqslant1$ that does not depend on $i$. We now have the following lemma.
\begin{lemma}\label{lip}
Let $\Gamma\in\Lambda$ be a Schreier graph of ${{\mathfrak m}athbb F}_n$. Then there exists a constant $C\geqslant1$ such that the family of functions $\{\tau_r\}_{r\in{{\mathfrak m}athbb N}}$ is uniformly $C$-Lipschitz over $\Gamma$.
\end{lemma}
\begin{proof}
Note first that if $x$ and $y$ are neighbors in $\Gamma$, then we have the bound
\begin{align}\label{nbrsize}
|U_r(x)|\geqslant\frac{1}{2n-1}|U_r(y)|.
\end{align}
Put $S\colonequals U_r(y)\backslash U_r(x)$, and let $S'$ denote a choice, for each vertex $z\in S$, of a neighbor $z'$ which belongs to $U_r(x)$. Then
\begin{align*}
\tau_r(y)-\tau_r(x)&\leqslant\sum_{z\in S}\frac{1}{|U_r(z)|}\\
&\leqslant(2n-1)^2\sum_{z\in S'}\frac{1}{|U_r(z)|}\\
&\leqslant(2n-1)^2\tau_r(x).
\end{align*}
Here the second line is obtained by applying the inequality (\ref{nbrsize}) and using the fact that points in $S'$ may have at most $2n-1$ neighbors in $S$. It follows that each $\tau_r$ is $C$-Lipschitz with $C=(2n-1)^2+1$.
\end{proof}
Moreover, it turns out that, at least in the model case of a finite Schreier graph (which carries a unique invariant probability measure), thinness and the densities $\rho_{A,r}$ given by (\ref{density}) are directly related to one another.
\begin{proposition}\label{prop1}
Let $(\Gamma,A,{\mathfrak m}u)$ be a finite Schreier graph $\Gamma$ equipped with the uniform probability measure, together with a subset $A\subseteq\Gamma$. Then
\[
\int_\Gamma\rho_{A,r}\,d{\mathfrak m}u=\int_A\tau_r\,d{\mathfrak m}u,
\]
where $\rho_{A,r}$ is the $r$-neighborhood density of the set $A$.
\end{proposition}
\begin{proof}
One must simply observe that, whether summing $\rho_{A,r}$ over $\Gamma$ or $\tau_r$ over $A$, for a given point $x\in\Gamma$ the quantity $1/|U_r(x)|$ is summed exactly once for every point $y\in A$ such that $x\in U_r(y)$.
\end{proof}
\noindent As a corollary, we obtain:
\begin{corollary}\label{one}
Given a finite Schreier graph $(\Gamma,{\mathfrak m}u)$ equipped with the uniform probability measure, $\tau_r$ integrates to one over $\Gamma$.
\end{corollary}
\begin{proof}
Simply choose $A=\Gamma$ in the hypotheses of Proposition~\ref{prop1}. Then $\rho_{A,r}\equiv1$, so that we have
\begin{align*}
\int_\Gamma\tau_r\,d{\mathfrak m}u=\int_\Gamma\rho_{A,r}\,d{\mathfrak m}u=\int_\Gamma1\,d{\mathfrak m}u=1.\tag*{\qedhere}
\end{align*}
\end{proof}
We thus find that the ``average thinness'' of a finite Schreier graph is always one. Proposition~\ref{prop1} can therefore be interpreted as saying that, if the average of $\rho_{A,r}$ over a finite Schreier graph $\Gamma$ is small relative to ${{\mathfrak m}athbb E}(\rho_{A,0})={\mathfrak m}u(A)$, then the set $A$ must be concentrated at points where $\Gamma$ is relatively thin (at scale $r$).
Corollary~\ref{one} tells us that, if $\Gamma$ is a finite Schreier graph, then by integrating the functions $\tau_r$ against the uniform probability measure on $\Gamma$, we obtain a new probability measure $\nu_r$. Suppose now that ${\mathfrak m}u$ is a sofic random Schreier graph, and let $\{\Gamma_i\}_{i\in{{\mathfrak m}athbb N}}$ be a sofic approximation to ${\mathfrak m}u$. Then one readily verifies that the sequence of probability measures $\nu_{r,i}$---those obtained by integrating $\tau_r$ against the uniform measures ${\mathfrak m}u_i$---converges weakly to a probability measure $\nu_r$ on $\Lambda({{\mathfrak m}athbb F}_n)$. That is, soficity implies that $\tau_r$ is a density with respect to ${\mathfrak m}u$.
\begin{proposition}\label{ergseq}
Let ${\mathfrak m}u$ be a sofic random Schreier graph which is ergodic and which does not satisfy property $D$. Then there exist finite Schreier graphs $(\Gamma_i,A_i,{\mathfrak m}u_i)$ together with subsets $A_i\subseteq\Gamma_i$ such that the $\Gamma_i$ are a sofic approximation to ${\mathfrak m}u$, ${\mathfrak m}u_i(A_i)\to1$, and ${{\mathfrak m}athbb E}(\tau_i{\mathfrak m}id A_i)\to0$.
\end{proposition}
\begin{proof}
If ${\mathfrak m}u$ does not satisfy property $D$, then there exists a set $A\subseteq\Lambda({{\mathfrak m}athbb F}_n)$ with ${\mathfrak m}u(A)>0$ such that ${{\mathfrak m}athbb E}(\rho_{A,r})\to0$ along some subsequence of radii $r\in{{\mathfrak m}athbb N}$, and hence such that ${{\mathfrak m}athbb E}(\tau_r{\mathfrak m}id A)\to0$. Let $\{g_i\}_{i\in{{\mathfrak m}athbb N}}$ be an enumeration of ${{\mathfrak m}athbb F}_n$ (e.g.\ the lexicographic order), and put
\[
A_k\colonequals A\cup g_1A\cup\ldots\cup g_kA.
\]
It follows from the fact that the $\tau_r$ are uniformly $C$-Lipschitz (Proposition~\ref{lip}) that ${{\mathfrak m}athbb E}(\tau_r{\mathfrak m}id A_k)\to0$ for any $k$. Indeed, putting
\[
m\colonequals{\mathfrak m}ax_{1\leqslant i\leqslant k}|g_i|,
\]
we have ${{\mathfrak m}athbb E}(\tau_r{\mathfrak m}id A_k)\leqslant C^m{{\mathfrak m}athbb E}(\tau_r{\mathfrak m}id A)\to0$. Moreover, by ergodicity, ${\mathfrak m}u(A_k)\to1$. By Lemma~\ref{soficfield}, there exists a sofic approximation $\{{{\mathfrak m}athbb F}F_{i,k}\}_{i\in{{\mathfrak m}athbb N}}$ for each invariant random field $(\Theta_{A_k})_*$, which is the same thing as a sequence of finite Schreier graphs $(\Gamma_{i,k},A_{i,k},{\mathfrak m}u_{i,k})$ such that the $A_{i,k}$ approximate $A_k$ (just take $A_{i,k}=\{x\in\Gamma_{i,k}{\mathfrak m}id{{\mathfrak m}athbb F}F_{i,k}(x)=1\}$). By choosing an appropriate diagonal sequence, we prove our claim.
\end{proof}
Suppose again that ${\mathfrak m}u$ is a sofic random Schreier graph which is ergodic and does not satisfy property $D$. Our next goal is to show that the geometry of ${\mathfrak m}u$ must be quite peculiar. To do so, we will look at the sofic approximation to ${\mathfrak m}u$ guaranteed by Proposition~\ref{ergseq}, i.e.\ the sequence of finite Schreier graphs $(\Gamma_i,A_i,{\mathfrak m}u_i)$, with ${\mathfrak m}u_i(A_i)\to1$ and ${{\mathfrak m}athbb E}(\tau_i{\mathfrak m}id A_i)\to0$. A trick we will emply is the following: instead of working with the functions $\tau_r$ and letting $r$ vary, we may instead modify the structure of our Schreier graphs and work only with the function $\tau_1$. Thus if $\Gamma_i$ is one of our Schreier graphs (constructed, by default, with respect to the standard generating set ${{\mathfrak m}athcal A}=\{a_1,\ldots,a_n$\}), denote by $\Gamma_i^{(r)}$ what we call the \emph{$r$-contraction of $\Gamma_i$} obtained by regarding it as a Schreier graph of ${{\mathfrak m}athbb F}_n$ constructed with respect to the generating set consisting of all group elements of length less than or equal to $r$. One readily verifies that $\tau_r$ over $\Gamma$ agrees with $\tau_1$ over $\Gamma^{(r)}$, in the sense that the diagram
\begin{equation*}
\begin{tikzcd}
\Gamma_i\arrow[hookrightarrow]{r}\arrow{dr}[swap]{\tau_r}
&\Gamma_i^{(r)}\arrow{d}{\tau_1}\\
&{{\mathfrak m}athbb Q}
\end{tikzcd}
\end{equation*}
commutes (here the upper arrow is the obvious identification between the vertices of $\Gamma_i$ and the vertices of $\Gamma_i^{(r)}$). By modifying the structure of our graphs in this way (for ever larger values of $r$) and choosing an appropriate diagonal sequence, our sofic approximation now takes the form of a sequence of finite Schreier graphs $(\Gamma_i,A_i,{\mathfrak m}u_i)$ such that ${\mathfrak m}u_i(A_i)\to1$ and ${{\mathfrak m}athbb E}(\tau_1{\mathfrak m}id A_i)\to0$.
We do not know of any invariant random Schreier graph which fails to have property $D$. In order to get a sense of what a sequence of graphs satisfying the aforementioned conditions might look like, however, consider the following example.
\begin{example}\label{bip}
Let $X_N$ be a set of $2^N$ points and $Y_N$ a set of $N$ points, and let $\Gamma_N$ denote the complete bipartite graph between $X_N$ and $Y_N$, i.e.\ the graph obtained by adding to the set $X_N\sqcup Y_N$ all possible edges $(x,y)$ such that $x\in X_N$ and $y\in Y_N$. Then the sequence of graphs $(\Gamma_N,X_N,{\mathfrak m}u_N)$ has the property that ${{\mathfrak m}athbb E}(\tau_1{\mathfrak m}id X_N)\to0$. Indeed, it is easy to see that for fixed $N$, $\tau_1$ is constant over each of $X_N$ and $Y_N$, and that $\left.\tau_1\right|_{X_N}\to0$ whereas $\left.\tau_1\right|_{Y_N}\to\infty$. At the same time, we have ${\mathfrak m}u_N(X_N)\to1$.
\end{example}
Note, however, that the graphs constructed in Example~\ref{bip} cannot be realized as a sequence of contracted Schreier graphs. Indeed, suppose that, possibly upon adding loops to the vertices of the bipartite graphs of Example~\ref{bip} and turning some of their edges into multi-edges, we were able to label their edges with generators of ${{\mathfrak m}athbb F}_n$. Then for each vertex $x\in X_N$, it must be the case that one of its ``external edges,'' meaning an edge $(x,y)$ with $y\in Y_N$, is labeled with one of the standard generators $a_1,\ldots,a_n$ (or one of their inverses)---were this not the case, $x$ would be fixed by every $a_i$ and hence by ${{\mathfrak m}athbb F}_n$ itself, a contradiction, since $x$ has ${{\mathfrak m}athbb F}_n$-labeled external edges attached to it. By the pigeonhole principle, there must thus exist a generator $a_i^{\pm1}$ and a subset $X_N'\subseteq X_N$ of measure ${\mathfrak m}u_N(X_N')\geqslant{\mathfrak m}u_N(X_N)/2n$ such that $a_i^{\pm1}X_N'\subseteq Y_N$. But this is again a contradiction, since ${\mathfrak m}u_N(Y_N)\to0$ and ${\mathfrak m}u_N$ is an invariant measure. Alternatively, note that there is an ever widening gap between the values of $\tau_1$ over $X_N$ and $Y_N$, which violates the fact that $\tau_1$ is $C$-Lipschitz (Proposition~\ref{lip}).
The family of graphs constructed in Example~\ref{bip} has what one might call a ``lopsided structure.'' That is to say, graphs in the family split into a set of large measure and a set of small measure in such a way that all of the neighbors of a given vertex in the large set belong to the small set. The next proposition shows that, despite the fact that the bipartite graphs considered above cannot be realized as Schreier graphs, a version of this phenomenon must occur whenever ${\mathfrak m}u$ is a sofic random Schreier graph which is ergodic and does not satisfy property $D$ (see also Figure~2).
\begin{figure}
\caption{Finite (contracted) Schreier graphs $\Gamma$ that approximate invariant random subgroups which do not satisfy property $D$ have a subset $A\subseteq\Gamma$ of large measure such that, for a random point $x\in A$, the large majority of its neighbors belong to the complement $\Gamma\backslash A$.}
\label{fig2}
\end{figure}
\begin{proposition}\label{lop}
Let ${\mathfrak m}u$ be a sofic random Schreier graph which is ergodic and does not satisfy property $D$. Then there exists a sequence of finite (contracted) Schreier graphs $(\Gamma_i,A_i,{\mathfrak m}u_i)$ such that the $\Gamma_i$ are a sofic approximation to ${\mathfrak m}u$, ${\mathfrak m}u_i(A_i)\to1$, and
\[
\lim_{i\to\infty}{{\mathfrak m}athbb E}\left(\left.\frac{\deg_{A_i}(x)}{\deg_{\Gamma_i\backslash A_i}(x)}\,\right| A_i\right)=0,
\]
where $\deg_A(x)$ denotes the number of neighbors of $x$ in the set $A$.
\end{proposition}
\begin{proof}
Let $(\Gamma_i,A_i,{\mathfrak m}u_i)$ be finite contracted Schreier graphs that are a sofic approximation to ${\mathfrak m}u$ and such that ${\mathfrak m}u_i(A_i)\to1$ and ${{\mathfrak m}athbb E}(\tau_1{\mathfrak m}id A_i)\to0$. We have
\begin{align*}
{{\mathfrak m}athbb E}(\tau_1{\mathfrak m}id A_i)&=\frac{1}{|A_i|}\sum_{x\in A_i}\frac{1+\deg_{A_i}(x)}{1+\deg(x)}\\
&=\frac{1}{|A_i|}\sum_{x\in A_i}\frac{1+\deg_{A_i}(x)}{1+\deg_{A_i}(x)+\deg_{\Gamma_i\backslash A_i}(x)}<\varepsilon_i,
\end{align*}
with $\varepsilon_i\to0$. It follows that, for any $K>0$, the subsets $A_{i,K}\subset A_i$ over which $\deg_{\Gamma_i\backslash A_i}(x)\leqslant K\deg_{A_i}(x)$ satisfy ${\mathfrak m}u_i(A_{i,K})\to0$. Indeed, were this not the case, we would have
\begin{align*}
{{\mathfrak m}athbb E}(\tau_1{\mathfrak m}id A_i)&\geqslant{{\mathfrak m}athbb E}(\tau_1{\mathfrak m}id A_{i,K}){\mathfrak m}u_i(A_{i,K})\\
&\geqslant\frac{1}{|A_{i,K}|}\sum_{x\in A_{i,K}}\frac{1+\deg_{A_i}(x)}{1+(K+1)\deg_{A_i}(x)}\,{\mathfrak m}u_i(A_{i,K})\\
&\geqslant\frac{\delta}{K+1}
\end{align*}
for all $i\in{{\mathfrak m}athbb N}$, where $\delta>0$ is a fixed lower bound of the values ${\mathfrak m}u_i(A_{i,K})$. We therefore find that the ratio of the expected number of internal neighbors to external neighbors of points in $A_i$ tends to zero, as desired.
\end{proof}
\section{Conservativity of the boundary action}
There is a natural \emph{boundary}, denoted $\partial{{\mathfrak m}athbb F}_n$, associated to the free group ${{\mathfrak m}athbb F}_n=\langle a_1,\ldots,a_n\rangle$, and it admits a number of interpretations. Viewing elements of ${{\mathfrak m}athbb F}_n$ as finite reduced words in the alphabet ${{\mathfrak m}athcal A}^\pm=\{a_1^{\pm1},\ldots,a_n^{\pm1}\}$, the boundary $\partial{{\mathfrak m}athbb F}_n$ is the space of infinite reduced words in the alphabet ${{\mathfrak m}athcal A}^\pm$ endowed with the topology of pointwise convergence. Equivalently, $\partial{{\mathfrak m}athbb F}_n$ is the projective limit of the spheres $\partial U_r({{\mathfrak m}athbb F}_n,e)$, i.e.\ the sets of words in ${{\mathfrak m}athbb F}_n$ of length $r$, where each such set is given the discrete topology and the connecting maps serve to delete the last symbol of a given word (the space $\partial{{\mathfrak m}athbb F}_n$ is thus a Cantor set provided $n>1$). Taking a more geometric view, $\partial{{\mathfrak m}athbb F}_n$ is naturally homeomorphic to the \emph{space of ends} of the Cayley graph of ${{\mathfrak m}athbb F}_n$. The latter object being a Gromov hyperbolic space, $\partial{{\mathfrak m}athbb F}_n$ may be viewed as the \emph{hyperbolic boundary} of ${{\mathfrak m}athbb F}_n$ (so that ${{\mathfrak m}athbb F}_n\cup\partial{{\mathfrak m}athbb F}_n$ is its \emph{hyperbolic compactification}). And when equipped with the uniform measure ${\mathfrak m}$ (which we will define in a moment), ($\partial{{\mathfrak m}athbb F}_n,{\mathfrak m})$ is naturally isomorphic to the \emph{Poisson boundary} of the simple random walk on ${{\mathfrak m}athbb F}_n$, a fact first established by Dynkin and Malyutov \cite{DM}.
Grigorchuk, Kaimanovich, and Nagnibeda \cite{GKN} recently studied the ergodic properties of the action of a subgroup $H\leqslant{{\mathfrak m}athbb F}_n$ on the boundary of ${{\mathfrak m}athbb F}_n$ equipped with the uniform measure ${\mathfrak m}$. To be explicit, ${\mathfrak m}$ is the probability measure given by
\begin{equation}\label{meas}
{\mathfrak m}(g)=\frac{1}{2n(2n-1)^{|g|-1}},
\end{equation}
where we again allow $g$ to represent both an element of ${{\mathfrak m}athbb F}_n$ (here $|g|$ is the length of $g$) and the cylinder set consisting of those infinite words whose truncations to their first $|g|$ symbols are equal to $g$. Of course, the denominator of (\ref{meas}) is just the cardinality of the sphere $\partial U_{|g|}({{\mathfrak m}athbb F}_n,e)$.
The aforementioned boundary action, which we denote by $H\circlearrowright(\partial{{\mathfrak m}athbb F}_n,{\mathfrak m})$, is analogous to the action of a Fuchsian group on the boundary of the hyperbolic plane $\partial{{\mathfrak m}athbb H}^2\cong{\mathbb S}^1$ equipped with Lebesgue measure: both actions, the latter being a classical object of study, are boundary actions of discrete groups of isometries of a Gromov hyperbolic space. In \cite{GKN}, the combinatorial structure of the space ${{\mathfrak m}athbb F}_n$, and especially the Schreier graphs corresponding to its subgroups, are exploited in order to investigate the action $H\circlearrowright(\partial{{\mathfrak m}athbb F}_n,{\mathfrak m})$. In particular, Theorem~2.12 of \cite{GKN} gives a combinatorial characterization of the Hopf decomposition of this action. Let us review this result.
Let $G\circlearrowright(X,{\mathfrak m}u)$ be a quasi-invariant action of a countable group on a Lebesgue space, i.e.\ a measure space whose nonatomic part is isomorphic to the unit interval equipped with Lebesgue measure. Recall that such an action is \emph{conservative} if every measurable subset $E\subseteq X$ is \emph{recurrent}, meaning that it is contained in the union of its $g$-translates, where $g\in G\backslash\{e\}$. The action is \emph{dissipative} if $(X,{\mathfrak m}u)$ is the union of the translates of a \emph{wandering set}, i.e.\ a subset $E\subseteq X$ whose $G$-translates are pairwise disjoint. Every quasi-invariant action $G\circlearrowright(X,{\mathfrak m}u)$ admits a unique \emph{Hopf decomposition}
\[
X={{\mathfrak m}athcal C}\sqcup{{\mathfrak m}athcal D}
\]
into conservative and dissipative parts (see \cite{A} and the references therein), so that the action of $G$ restricted to ${{\mathfrak m}athcal C}$ is conservative and the action of $G$ restriced to ${{\mathfrak m}athcal D}$ is dissipative.
Turning our attention to the action $H\circlearrowright(\partial{{\mathfrak m}athbb F}_n,{\mathfrak m})$, consider the Schreier graph $(\Gamma,H)$ of $H$, and let $T\subseteq\Gamma$ be a \emph{geodesic spanning tree}, i.e.\ a spanning tree such that $d_T(H,Hg)=d_\Gamma(H,Hg)$ for all vertices (cosets) $Hg$. Such a spanning tree always exists. Let $\Omega_H\subseteq\partial{{\mathfrak m}athbb F}_n$ denote the \emph{Schreier limit set}. It is the set of infinite words (which of course correspond to infinite paths in $\Gamma$) that pass through edges not in $T$ infinitely often. Let ${{\mathfrak m}athcal D}elta_H\subseteq{{\mathfrak m}athbb F}_n$ denote the \emph{Schreier fundamental domain}. It is the set of infinite words that remain in $T$. We then have the following boundary decomposition:
\begin{equation}\label{decomp}
\partial{{\mathfrak m}athbb F}_n=\Omega_H\sqcup\bigsqcup_{h\in H}h{{\mathfrak m}athcal D}elta_H.
\end{equation}
That is, $\partial{{\mathfrak m}athbb F}_n$ is the disjoint union of the Schreier limit set and the $H$-translates of the Schreier fundamental domain. It is shown in \cite{GKN} (see Theorem~2.12) that the decomposition (\ref{decomp}) is in fact the Hopf decomposition of the action $H\circlearrowright(\partial{{\mathfrak m}athbb F}_n,{\mathfrak m})$.
\begin{theorem}\rm(Grigorchuk, Kaimanovich, and Nagnibeda)
\emph{The conservative part of the boundary action $H\circlearrowright(\partial{{\mathfrak m}athbb F}_n,{\mathfrak m})$ coincides with the Schreier limit set $\Omega_H$. The dissipative part coincides with the $H$-translates of the Schreier fundamental domain ${{\mathfrak m}athcal D}elta_H$.}
\end{theorem}
Moreover, Theorem~4.10 of \cite{GKN} shows that the measure of the Schreier fundamental domain is related to the growth of the Schreier graph $(\Gamma,H)$ of $H$.
\begin{theorem}\rm{(Grigorchuk, Kaimanovich, and Nagnibeda)}\label{diss}
\emph{The measure of the Schreier fundamental domain determined by a proper subgroup $H\in L({{\mathfrak m}athbb F}_n)$ is equal to}
\[
{\mathfrak m}({{\mathfrak m}athcal D}elta_H)=\lim_{r\to\infty}\frac{|\partial U_r(\Gamma,H)|}{|\partial U_r({{\mathfrak m}athbb F}_n,e)|},
\]
\emph{and the above sequence of ratios is nonincreasing.}
\end{theorem}
\begin{remark}
Note that Theorem~\ref{diss} remains valid if one replaces the spheres $\partial U_r$ with neighborhoods $U_r$.
\end{remark}
The action $H\circlearrowright(\partial{{\mathfrak m}athbb F}_n,{\mathfrak m})$ may be conservative. This is the case, for example, whenever $H$ is of finite index, or when $H$ is a normal subgroup of ${{\mathfrak m}athbb F}_n$. The action may also be dissipative, which is the case, for instance, whenever $H$ is finitely generated and of infinite index. It may also be the case that both the conservative and dissipative parts of the action have positive measure: see, for instance, Example~4.27 of \cite{GKN}. It is our aim, however, to show that the boundary action of an invariant random subgroup is necessarily conservative. To this end, let us understand a \emph{$k$-cycle} to be a closed path which is isomorphic to a $k$-sided polygon. Our main idea is that an invariant random Schreier graph which satisfies property $D$ must have a certain ``density of $k$-cycles,'' i.e.\ that there exists a $k$ such that a given vertex of an invariant random Schreier graph belongs to a $k$-cycle with positive probability, and that this in turn restricts the growth of our random graph enough to render ${{\mathfrak m}athcal D}elta_H$ a null set.
\begin{theorem}\label{conservative}
The boundary action $H\circlearrowright(\partial{{\mathfrak m}athbb F}_n,{\mathfrak m})$ of a sofic random subgroup of the free group is conservative.
\end{theorem}
\begin{proof}
Suppose first that ${\mathfrak m}u$ is an invariant random Schreier graph that satisfies property $D$. It is not difficult to see that, with the exception of one trivial case, there must always exist a number $k$ such that the Borel set $A$ of Schreier graphs whose roots belong to a $k$-cycle has positive measure. Indeed, if this were not the case, then ${\mathfrak m}u$ would be the Dirac measure concentrated on the Cayley graph of ${{\mathfrak m}athbb F}_n$ (whose boundary action is of course conservative). By assumption, there thus exists an $\varepsilon>0$ such that ${{\mathfrak m}athbb E}(\rho_{A,r})\geqslant\varepsilon$ for all $r$. Put $f(r)\colonequals2n(2n-1)^{r-1}$, let $X_r$ denote the size of the radius-$r$ sphere centered at the root of a ${\mathfrak m}u$-random Schreier graph, let $\ell=\lfloor k/2\rfloor$, and let $r\geqslant1$ be an initial radius. Trivially, ${{\mathfrak m}athbb E}(X_r)\leqslant f(r)$. We are then able to bound ${{\mathfrak m}athbb E}(X_{r+\ell})$ as
\[
{{\mathfrak m}athbb E}(X_{r+\ell})\leqslant f(r+\ell)-\varepsilon f(r)
\]
and, continuing inductively, to obtain the general bound
\begin{align}
{{\mathfrak m}athbb E}(X_{r+(m+1)\ell})&\leqslant(2n-1)^\ell{{\mathfrak m}athbb E}(X_{r+m\ell})-\varepsilon\left({{\mathfrak m}athbb E}(X_{r+m\ell})-\varepsilon{{\mathfrak m}athbb E}(X_{r+(m-1)\ell})\right)\nonumber\\
&=\left((2n-1)^\ell-\varepsilon\right){{\mathfrak m}athbb E}(X_{r+m\ell})+\varepsilon^2{{\mathfrak m}athbb E}(X_{r+(m-1)\ell})\label{recur},
\end{align}
since each $k$-cycle that passes through the boundary of an $(r+m\ell)$-neighborhood allows us to decrease the trivial bound on the size of the boundary of an $(r+(m+1)\ell)$-neighborhood by one. Note that (\ref{recur}) is a linear homogenous recurrence relation with characteristic polynomial
\[
\chi(t)=t^2-\left((2n-1)^\ell-\varepsilon\right)t-\varepsilon^2.
\]
It is easy to see that $\chi$ has distinct real roots. The general solution of the recurrence relation (\ref{recur}) thus yields the bound
\begin{equation}\label{bound}
\begin{split}
{{\mathfrak m}athbb E}(X_{r+m\ell})\leqslant C_0&\left((2n-1)^\ell-\varepsilon+\sqrt{\left((2n-1)^\ell-\varepsilon\right)^2+4\varepsilon^2}\right)^m\\
&+C_1\left((2n-1)^\ell-\varepsilon-\sqrt{\left((2n-1)^\ell-\varepsilon\right)^2+4\varepsilon^2}\right)^m,
\end{split}
\end{equation}
whereupon applying initial conditions readily gives $C_0=C_1=f(r)/2$ (in order to simplify notation, we have doubled the roots of $\chi$). By Theorem~\ref{diss}, we have
\begin{align*}
{{\mathfrak m}athbb E}({\mathfrak m}({{\mathfrak m}athcal D}elta_H))&=\int{\mathfrak m}({{\mathfrak m}athcal D}elta_H)\,d{\mathfrak m}u\\
&=\int\lim_{r\to\infty}\frac{|\partial U_r(\Gamma,H)|}{|\partial U_r({{\mathfrak m}athbb F}_n,e)|}\,d{\mathfrak m}u\\
&=\lim_{r\to\infty}\frac{1}{f(r)}\int|\partial U_r(\Gamma,H)|\,d{\mathfrak m}u\\
&=\lim_{r\to\infty}\frac{1}{f(r)}{{\mathfrak m}athbb E}(X_r).
\end{align*}
Passing to the subsequence $\{r+m\ell\}_{m\in{{\mathfrak m}athbb N}}$ and replacing the second (and clearly smaller) term of (\ref{bound}) with the first, we see that
\begin{align*}
\lim_{r\to\infty}\frac{{{\mathfrak m}athbb E}(X_r)}{f(r)}&\leqslant\lim_{m\to\infty}\frac{f(r)}{f(r+m\ell)}\left((2n-1)^\ell-\varepsilon+\sqrt{\left((2n-1)^\ell-\varepsilon\right)^2+4\varepsilon^2}\right)^m\\
&=\lim_{m\to\infty}\left(1-\frac{\varepsilon}{(2n-1)^\ell}+\sqrt{1-\frac{2\varepsilon}{(2n-1)^\ell}+\frac{5\varepsilon^2}{(2n-1)^{2\ell}}}\right)^m.
\end{align*}
But a simple calculation shows that what is inside the parentheses is less than one, so that the above limit is zero. It follows that ${{\mathfrak m}athbb E}({{\mathfrak m}athcal D}elta_H)=0$ and therefore that the boundary action of our invariant random subgroup is conservative.
Suppose next that ${\mathfrak m}u$ is a sofic random subgroup which is ergodic and does not satisfy property $D$. Then by Proposition~\ref{lop}, it has a lopsided sofic approximation, i.e.\ a sofic approximation consisting of contracted Schreier graphs $(\Gamma_i,A_i,{\mathfrak m}u_i)$ such that ${\mathfrak m}u_i(A_i)\to1$ and the average external degree of vertices in $A_i$ is much smaller than their average external degree, in the sense that their ratio tends to zero. Since ${\mathfrak m}u_i(\Gamma_i\backslash A_i)\to0$, this implies that
\[
{{\mathfrak m}athbb E}(\deg(x){\mathfrak m}id A_i)\ll{{\mathfrak m}athbb E}(\deg(x){\mathfrak m}id \Gamma_i\backslash A_i),
\]
again in the sense that the ratio of these two quantities tends to zero. But the vertex degree of a point in a contracted Schreier graph $\Gamma^{(r)}$ is precisely one less than the size of the $r$-neighborhood of the corresponding uncontracted graph $\Gamma$. We thus find that, over a set of arbitrarily large measure, the ratio of the average size of (arbitrarily large) $r$-neighborhoods in our Schreier graphs to $|U_r({{\mathfrak m}athbb F}_n,e)|$ is arbitraily small, which proves our claim.
\end{proof}
To conclude this section, let us remark that, although Theorem~\ref{conservative} says, in effect, that sofic random subgroups cannot grow as quickly as the free group, it is reasonable to expect that they can still grow very quickly: it is proved in \cite{AGV} (see Theorem~40) that there exists a (nonatomic) regular unimodular random graph whose exponential growth rate is maximal.
\section{Cogrowth and limit sets}
It is interesting to examine other questions considered in \cite{GKN} for sofic random subgroups. Note, for example, that Theorem~\ref{conservative} immediately implies that, unless it is the Dirac measure concentrated on the $2n$-regular tree, a sofic random Schreier graph $\Gamma\in\Lambda({{\mathfrak m}athbb F}_n)$ cannot contain a \emph{branch} of ${{\mathfrak m}athbb F}_n$, i.e.\ a subgraph isomorphic to the unique tree one of whose vertices has degree one and all of whose other vertices have degree $2n$, since the presence of a branch implies the existence of a nontrivial wandering set (another way to say this is that every edge of an invariant random Schreier graph must belong to a cycle). Recall, moreover, that the \emph{cogrowth} of a subgroup $H\leqslant{{\mathfrak m}athbb F}_n$ (i.e.\ the ``growth of $H$ inside of ${{\mathfrak m}athbb F}_n$'') is defined to be
\[
v_H\colonequals\limsup_{r\to\infty}\sqrt[r]{|H\cap U_r({{\mathfrak m}athbb F}_n,e)|}\leqslant2n-1.
\]
By Theorem~4.2 of \cite{GKN}, if $v_H<\sqrt{2n-1}$, then the action $H\circlearrowright(\partial{{\mathfrak m}athbb F}_n,{\mathfrak m})$ is dissipative. We therefore have the following corollary of Theorem~\ref{conservative}.
\begin{corollary}\label{cogrowth}
The cogrowth of a sofic random subgroup $H\in L({{\mathfrak m}athbb F}_n)$ must satisfy $v_H\geqslant\sqrt{2n-1}$.
\end{corollary}
Alternatively, a Schreier graph is \emph{Ramanujan} if and only if its cogrowth does not exceed $\sqrt{2n-1}$, and it is proved in \cite{AGV} (see Theorem~5) that random unimodular $d$-regular graphs are Ramanujan if and only if they are trees, which shows that an invariant random subgroup $H$ satisfies $v_H>\sqrt{2n-1}$.
There are various limit sets associated to a subgroup $H\leqslant{{\mathfrak m}athbb F}_n$ (most of which descend from the general theory of discrete groups of isometries of Gromov hyperbolic spaces). The \emph{radial limit set}, denoted $\Lambda_H^{\rad}$, is the set of limit points (in $\partial{{\mathfrak m}athbb F}_n$) of sequences of elements of $H$ which are contained within a tubular neighborhood of a certain geodesic ray in ${{\mathfrak m}athbb F}_n$. There are the \emph{small horospheric limit set}, denoted $\Lambda_H^{\hor,s}$, which is the set of boundary points $\omega\in\partial{{\mathfrak m}athbb F}_n$ such that any \emph{horosphere} centered at $\omega$ contains infinitely elements of $H$, the Schreier limit set $\Omega_H$, and the \emph{big horospheric limit set}, denoted $\Lambda_H^{\hor,b}$, which is the set of boundary points $\omega\in\partial{{\mathfrak m}athbb F}_n$ such that a certain horosphere centered at $\omega$ contains infinitely elements of $H$. There are also the divergence set of the \emph{Poincar\'{e} series} of $H$, denoted ${\mathbb S}igma_H$, and the \emph{full limit set}, denoted $\Lambda_H$, which is the set of all limit points (in $\partial{{\mathfrak m}athbb F}_n$) of elements of $H$. We refer the reader to \cite{GKN} for the precise definitions of these sets.
As is shown in \cite{GKN}, there is a certain amount of flexibility in the ${\mathfrak m}$-measures of the aforementioned limit sets for arbitrary subgroups $H\leqslant{{\mathfrak m}athbb F}_n$: although several of these sets necessarily have the same measure, the measure of the full limit set $\Lambda_H$ may take on a range of values (and may well be a null set). Once again, however, the situation for sofic random subgroups is more rigid, as the following theorem shows.
\begin{theorem}\label{lim}
Let $H$ be a sofic random subgroup. Then the limit sets $\Lambda_H^{\hor,s}$, $\Omega_H$, $\Lambda_H^{\hor,b}$, ${\mathbb S}igma_H$, and $\Lambda_H$ all have full ${\mathfrak m}$-measure.
\end{theorem}
\begin{proof}
By Theorems~3.20 and 3.21 of \cite{GKN}, the aforementioned limit sets are contained in one another in the order in which we have listed them, i.e.\
\[
\Lambda_H^{\rad}\subseteq\Lambda_H^{\hor,s}\subseteq\Omega_H\subseteq\Lambda_H^{\hor,b}\subseteq{\mathbb S}igma_H\subseteq\Lambda_H,
\]
and the middle four of these have the same ${\mathfrak m}$-measure. By Theorem~\ref{conservative}, ${\mathfrak m}(\Omega_H)=1$. These facts taken together imply the claim.
\end{proof}
By Theorem~3.35 of \cite{GKN} (which is an analogue of the \emph{Hopf-Tsuji-Sullivan theorem}, valid for discrete groups of isometries of $n$-dimensional hyperbolic space), either ${\mathfrak m}(\Lambda_H^{\rad})=1$ or ${\mathfrak m}(\Lambda_H^{\rad})=0$, the former occurring when the simple random walk on $(\Gamma,H)$ is recurrent and the latter when the simple random walk on $(\Gamma,H)$ is transient. The following examples show that the ${\mathfrak m}$-measure of the radial limit set of a nonatomic invariant random subgroup may be either zero or one.
\begin{example}(An invariant random subgroup with the property that ${\mathfrak m}(\Lambda_H^{\rad})=1$) Consider the Cayley graph $\Gamma$ of the group ${{\mathfrak m}athbb Z}^2$ constructed with respect to the standard generators $a=(1,0)$ and $b=(0,1)$. It is a classical result that the simple random walk on ${{\mathfrak m}athbb Z}^2$ is recurrent \cite{Po}, so Theorem~3.35 of \cite{GKN} implies that ${\mathfrak m}(\Lambda_H^{\rad})=1$, where $H$ is the fundamental group of $\Gamma$. The graph $\Gamma$ contains infinitely many ``$a$-chains,'' i.e.\ bi-infinite geodesics labeled with the generator $a$, and by independently reversing the orientations of these $a$-chains or leaving their orientations fixed, we generate a large space of Schreier graphs each of whose underlying unlabeled graphs is isomorphic to the two-dimensional integer lattice (in particular, the simple random walk on these graphs remains recurrent). There is natural uniform measure on this space (the uniform measure on its projective structure), and it is not difficut to see that this measure is invariant.
\end{example}
\begin{example}(An invariant random subgroup with the property that ${\mathfrak m}(\Lambda_H^{\rad})=0$) Consider the Cayley graph $\Gamma$ of the group ${{\mathfrak m}athbb Z}^3$ constructed with respect to the standard generators $a=(1,0,0)$, $b=(0,1,0)$, and $c=(0,0,1)$. It is again a classical result that the simple random walk on ${{\mathfrak m}athbb Z}^3$ (or, indeed, on ${{\mathfrak m}athbb Z}^n$ for $n\geqslant3$) is transient, so that ${\mathfrak m}(\Lambda_H^{\rad})=0$, where $H$ is the fundamental group of $\Gamma$. By employing the same trick as in the previous example, we again generate a large space of Schreier graphs for which the uniform measure is a nonatomic invariant probability measure.
\end{example}
\end{document}
|
math
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/**
* created by Michael Gerlich, Jun 6, 2012 - 11:10:04 AM
*/
package de.ipbhalle.MassBank;
import java.io.BufferedReader;
import java.io.File;
import java.io.FileFilter;
import java.io.FileNotFoundException;
import java.io.FileReader;
import java.io.IOException;
import java.util.ArrayList;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
public class KEGGFileFilter implements FileFilter {
public List<String> ids = new ArrayList<String>();
public Map<String, String> idMap = new HashMap<String, String>();
private boolean unique = false;
public KEGGFileFilter(boolean uniqueOnly) {
this.unique = uniqueOnly;
}
@Override
public boolean accept(File pathname) {
try {
BufferedReader br = new BufferedReader(new FileReader(pathname));
String line = "";
boolean found = false;
while ((line = br.readLine()) != null) {
if (line.startsWith("CH$LINK") && line.contains("KEGG")) {
String id = line.substring(line.indexOf("KEGG") + 5).trim();
idMap.put(pathname.getName().substring(0, pathname.getName().lastIndexOf(".")), id);
if(this.unique) {
if(!ids.contains(id)) { // only add unique identifiers
ids.add(id);
found = true;
break;
}
}
else { // add all identifier, non-unique
ids.add(id);
found = true;
break;
}
}
}
br.close();
if (found)
return true;
else
return false;
} catch (FileNotFoundException e) {
System.err.println("File " + pathname.getAbsolutePath() + " not found!");
return false;
} catch (IOException e) {
System.err.println("IOException for " + pathname.getAbsolutePath());
return false;
}
}
public List<String> getIds() {
return ids;
}
public void setIds(List<String> ids) {
this.ids = ids;
}
public Map<String, String> getIdMap() {
return idMap;
}
public void setIdMap(Map<String, String> idMap) {
this.idMap = idMap;
}
}
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code
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\begin{document}
\title{A class of higher inductive types in Zermelo-Fraenkel set
theory}
\begin{abstract}
We define a class of higher inductive types that can be constructed
in the category of sets under the assumptions of Zermelo-Fraenkel set
theory without the axiom of choice or the existence of uncountable
regular cardinals. This class includes the example of unordered
trees of any arity.
\end{abstract}
\section{Introduction}
\label{sec:introduction}
Higher inductive types are one of the key ideas in homotopy type
theory \cite{hottbook}. We think of an (ordinary) inductive type as
the smallest type closed under certain algebraic operations or
\emph{point constructors}. For instance, we define the type of
countably branching trees $T$ to be the smallest type closed under the
following operations.
\begin{align*}
\mathtt{leaf} &: T \\
\mathtt{node} &: (\omega \to T) \to T
\end{align*}
Within type theory we formalise the idea that $T$ is the smallest type
with the above point constructors using recursion or induction
terms. However, semantically, it is often more convenient to think in
terms of initial algebras. We say an \emph{algebra} for the above
constructors is a type $X$ together with a map $1 + X^\omega \to
X$. $T$ is then the initial object in the category of algebras. This
is a classic example of a \emph{$W$-type}, as defined by Moerdijk and
Palmgren \cite{moerdijkpalmgrenwtypes}.
For higher inductive types, one not only has point constructors, but
also \emph{path constructors}, which add proofs of identities of
terms. Higher inductive types are usually considered within HoTT and
have well understood semantics within models of HoTT
\cite{lumsdaineshulmanhit}, \cite{chmhitsctt},
\cite{cavalloharper}. However, since they are defined within the language of
type theory, one might also consider whether they hold in
interpretations of extensional type theory in locally cartesian closed
categories, and in particular the category of sets, $\mathsf{Set}$.
One of the simplest examples of higher inductive type is pushouts. In
$\mathsf{Set}$ these can be implemented as pushouts in the usual categorical
sense. It follows that $\mathsf{Set}$ contains all of the $n$-dimensional
spheres, although there is not much you can say about them without the
univalence axiom, and indeed they turn out to be trivial in
$\mathsf{Set}$. Quotients and image factorisations are examples of simple
colimits that play a useful role even within models of extensional
type theory \cite{maiettimodular}, \cite{awodeybauerpat}.
There are also more complicated examples of higher inductive types
that are non trivial in extensional type theory, and even $\mathsf{Set}$,
within the framework of \emph{quotient inductive types}
\cite{altenkirchkaposittintt}. In fact our examples of interest fall
within a smaller class with a simpler definition and clearer
semantics. This class was studied by Blass \cite{blassfreealg} under
the name \emph{free algebras subject to identities} and by Fiore,
Pitts and Steenkamp in \cite{fiorepittssteenkamp} under the name
\emph{$QW$-types} (we will refer to them by the latter name). A well
known example of such a type is that of ``unordered countably
branching trees.'' We modify the definition of $T$ above to get the
higher inductive type $T_{\operatorname{Sym}}$ by adding equations as follows,
where we write $\operatorname{Sym}(\omega)$ for the type of permutations
$\omega \to \omega$.
\begin{align*}
\mathtt{leaf} &: T_{\operatorname{Sym}} \\
\mathtt{node} &: (\omega \to T_{\operatorname{Sym}}) \to T_{\operatorname{Sym}} \\
\mathtt{perm} &: \prod_{f \colon \omega \to T_{\operatorname{Sym}}}
\prod_{\pi : \operatorname{Sym}(\omega)} \mathtt{node}(f) =
\mathtt{node}(f \circ \pi)
\end{align*}
Altenkirch, Capriotti, Dijkstra, Kraus and Forsberg include
this in \cite{acdkfqiits}, as a non trivial example of a quotient
inductive(-inductive) type. As they remark, the obvious construction
of $T_{\operatorname{Sym}}$ as a quotient of $T$ requires the axiom of
choice.\footnote{For the special case above, countable choice would be
enough.} Fiore, Pitts and Steenkamp showed that in fact it is an
example of a $QW$-type \cite[Example 2]{fiorepittssteenkamp}.
Blass showed that all $QW$-types can be constructed in $\mathsf{Set}$ under
the assumption that regular cardinals are unbounded in the class of all
ordinals. More generally free algebras can be constructed in
cocomplete categories from the existence of regular cardinals of
sufficiently high cardinality via the general techniques of Kelly
\cite{kellytransfinite}. For example this plays an important role in
the construction of higher inductive types by Lumsdaine and Shulman
\cite{lumsdaineshulmanhit}. The existence of an unbounded class of
regular ordinals is usually a reasonable one. It follows from very
weak versions of choice, such as $\mathbf{WISC}$ \cite{vdbergwisczf}
and a variant is often assumed in constructive set theory \cite[Section
10]{aczelrathjen}. It is also the case that every inaccessible
cardinal is in particular regular.\footnote{Note, however that even in the
presence of inaccessible cardinals it can be useful to have proofs that
are valid in $\mathbf{ZF}$ without further assumption: if $\kappa$ is
inaccessible, then $V_\kappa$ is a transitive model of $\mathbf{ZF}$, and so
any proof valid in $\mathbf{ZF}$ can be carried out inside it (e.g. to
construct HITs that belong to $V_\kappa$), but $V_\kappa$
itself does not contain inaccessible cardinals without further
assumptions on $\kappa$.}
However, Gitik \cite{gitik1980} has constructed a model of $\mathbf{ZF}$ in
which $\omega$ is the only regular cardinal.\footnote{under certain
large cardinal assumptions} Moreover, Blass showed that the
assumption is strictly necessary, by constructing a $QW$-type which is
isomorphic to the collection of ordinals hereditarily of countable
cofinality, if it exists. He deduced by Gitik's result that this gives
an example of a $QW$-type that does not provably exist in $\mathsf{Set}$
under the assumptions of $\mathbf{ZF}$.\footnote{Using Blass' argument and a
later paper by Gitik \cite{gitik85} one can also show the
following. For every successor ordinal $\alpha$ there is a transitive
model of $\mathbf{ZF}$ where Blass' example of a $QW$-type exists and is
precisely the set of all ordinals less than $\omega_\alpha$.}
Fiore, Pitts and Steenkamp in loc. cit. gave an electronically
verified proof that $QW$-types can be constructed in type theory using Agda sized
types and universes closed under inductive-inductive types. We can see
from Blass' counterexample that some combination of these assumptions
for $\mathsf{Set}$ must lead to the existence of uncountable regular
cardinals. In a second paper \cite{fiorepittssteenkamp2} the same
authors showed that the weak choice principle $\mathbf{WISC}$ can be
used in place of Agda sized types, again verifying the result
electronically.
On the other hand, some higher inductive types can be constructed in
$\mathsf{Set}$ without choice or unbounded regular cardinals. In addition to
colimits, as mentioned above, the author showed in \cite{swanwtypered}
that $W$-types with reductions exist in any boolean topos, including
$\mathsf{Set}$. A similar argument shows that Sojakova's notion of
$W$-suspensions \cite{sojakovawsusp} also exist in all boolean
toposes.\footnote{A proof is left as an exercise for the reader.}
In this paper we will see a new class of $QW$-types that can be
constructed in $\mathsf{Set}$ under $\mathbf{ZF}$, without any assumptions of choice
or existence of regular cardinals, that we call \emph{image
preserving} $QW$-types. This will included the example of unordered
countably branching trees above, and more generally unordered trees of
any arity. The proof is based on a construction of a set of all
hereditarily countable sets due to Jech \cite{jechheredctbl}. We will
be able to recover hereditarily countable sets as a special case, and
moreover a later generalisation due to Holmes \cite{holmesheredsmall}.
\section{Image preserving $QW$-types}
We now define our class of higher inductive types that we will
construct in $\mathsf{Set}$. It will be clear by the definition that this is a
special case of $QW$-types \cite{fiorepittssteenkamp}.
\begin{defn}
A $\emph{polynomial}$ is a set $A$ together with a family of sets
$(B_a)_{a \in A}$. We will refer to elements of $A$ as
\emph{constructors} and say $B_a$ is the \emph{arity} of the
constructor $a \in A$.
\end{defn}
\begin{defn}
Given a polynomial $(B_a)_{a \in A}$, an \emph{algebra} is a set $X$
together with a function $s \colon \sum_{a \in A} X^{B_a} \to X$. We
refer to such $s$ as an \emph{algebra structure} on $X$.
If $(X, s)$ and $(Y, t)$ are algebras, we say a \emph{homomorphism}
is a function $h : X \to Y$ such that for all $a \in A$ and $f :
B_a \to X$ we have $h(s(a, f)) = t(a, h \circ f)$.
\end{defn}
\begin{rmk}
Although we assumed $X$ and $Y$ are sets in the definition above, we
can also define algebra structures and homomorphisms for classes by
replacing functions with class functions.
\end{rmk}
\begin{defn}
Given a polynomial $(B_a)_{a \in A}$, an \emph{image preserving
equation} over $(B_a)_{a \in A}$ consists of a set $V$, and
$a, b \in A$ together with $l \colon B_a \to V$ and
$r \colon B_{b} \to V$ such that the image of $l$ is equal to the
image of $r$.
A \emph{family of image preserving equations} consists of a set $E$
together with a family of image preserving equations $(V_e, a_e,
b_e, l_e, r_e)_{e \in E}$.
\end{defn}
\begin{defn}
Given a polynomial $(B_a)_{a \in A}$ and a family of image
preserving equations, $(V_e, a_e, b_e, l_e, r_e)_{e \in E}$, an
\emph{algebra} is an algebra $(X, s)$ for the polynomial
$(B_a)_{a \in A}$ such that for every $e \in E$ and every function
$h \colon V_e \to X$ we have
$s(a_e, h \circ l_e) = s(b_e, h \circ r_e)$.
\end{defn}
\begin{example}
Suppose we are given a set $B$. We consider the polynomial with two
constructors of arity $0$ and $B$. We consider the set of image
preserving equations with set of variables $B$, and
$l, r \colon B \to B$ defined by $l = 1_B$ and $r = \pi$ for each
permutation $\pi \in \operatorname{Sym}(B)$. The initial algebra is then the set
of \emph{unordered trees of arity $B$}. In particular, we can take
$B = \omega$ to get unordered countably branching trees.
\end{example}
\begin{example}
Given any polynomial $(B_a)_{a \in A}$ and set $V$ we can consider the set of
\emph{all} image preserving equations. We will see in the next
section how this allows us to recover hereditarily countable sets
and more generally hereditarily small sets.
\end{example}
\begin{example}
\label{ex:nonexblass}
We will be able to deduce from the main theorem that Blass' example
of a $QW$-type that cannot be constructed in $\mathbf{ZF}$ cannot be viewed
as an image preserving $QW$-type. However, for illumination we will
give a more intuitive direct reason why it does not satisfy the
definition. In Blass' example, the initial algebra is expected to
behave like the collection of all ordinals of countable cofinality. In
particular there is an operation $\sup$ which takes a sequence
$(\alpha_n)_{n < \omega}$ and is expected to behave like the
supremum of the countable sequence of ordinals
$(\alpha_n)_{n < \omega}$. In particular we should identify
$\sup((\alpha_n)_{n < \omega})$ and $\sup((\beta_n)_{n < \omega})$
whenever $\alpha_n$ is cofinal in $\beta_n$ and vice versa. However,
this is much weaker than $\alpha_n$ and $\beta_n$ containing exactly
the same elements (possibly in a different order).
\end{example}
\section{Hereditarily small sets}
\label{sec:hered-small-sets}
As part of \cite{jechheredctbl} Jech showed how to construct the set
of all hereditarily countable sets in $\mathbf{ZF}$. This was later
generalised by Holmes \cite{holmesheredsmall} who showed that for any
set $B$ we can construct a set containing all sets hereditarily with
cardinality less than or equal to $B$. We in fact give a very slight
further generalisation of Holmes' result by instead considering
families of sets $(B_a)_{a \in A}$. In order to derive this from the
main theorem \ref{thm:iptypesexist}, we first need to clarify the
relation between image preserving equations and the class of
hereditarily small sets, which is the topic of this section.
\begin{defn}
Given a family of sets $(B_a)_{a \in A}$ we say a set $X$ is
\emph{small relative to $(B_a)_{a \in A}$} if for some
$a \in A$ there exists a surjection $B_a \twoheadrightarrow X$. We
say $X$ is \emph{hereditarily small relative to $(B_a)_{a \in A}$}
if $X$ is small relative to $(B_a)_{a \in A}$ and all of its
elements are hereditarily small relative to $(B_a)_{a \in A}$.
We write $H((B_a)_{a \in A})$ for the class of hereditarily small
sets.
We view $H((B_a)_{a \in A})$ as an algebra on the polynomial
$(B_a)_{a \in A}$ as follows. Given $a \in A$ and
$f : B_a \to H((B_a)_{a \in A})$, we define $s(a, f)$ to be
$\{ f(b) \;|\; b \in B_a \}$.
\end{defn}
\begin{lemma}
\label{lem:heredsmsatip}
$H((B_a)_{a \in A})$ satisfies all image preserving equations
relative to the polynomial $(B_a)_{a \in A}$.
\end{lemma}
\begin{proof}
Suppose we are given a set $S$ of variables together with $l : B_a
\to S$ and $r : B_c \to S$ and a map $h : S \to
H((B_a)_{a \in A})$. We calculate as follows.
\begin{align*}
s(h \circ l) &= \{ h(l(b)) \;|\; b \in B_a \} \\
&= \{ h(x) \;|\; x \in \operatorname{im}(l) \} \\
&= \{ h(x) \;|\; x \in \operatorname{im}(r) \} \\
&= s(h \circ r)
\end{align*}
\end{proof}
We now fix a set of variables $S := \sum_{a \in A} \mathcal{P}(B_a)$.
\begin{lemma}
\label{lem:outofheredsm}
Suppose $(X, t)$ is an algebra for $(B_a)_{a \in A}$ that satisfies
all image preserving equations for the set of variables $S$
above. Then there is a unique homomorphism
$h : H((B_a)_{a \in A}) \to X$.
\end{lemma}
\begin{proof}
In order for $h$ to be a homomorphism, we need it to satisfy the
following equation whenever $a \in A$ and $f : B_a \to H((B_a)_{a
\in A})$.
\begin{equation*}
h(\{ f(b) \;|\; b \in B_a \}) = t(a, h \circ f)
\end{equation*}
This defines a unique class function by $\in$-induction as long as
the equation above is well defined. That is, given $g : B_c \to H((B_a)_{a
\in A})$ such that $\{ f(b) \;|\; b \in B_a \} = \{ g(b) \;|\; b
\in B_c \}$ we need $t(a, h \circ f) = t(c, h \circ g)$.
Note that the lemma is trivial if $B_a$ is inhabited for each $a$,
since then $H((B_a)_{a \in A})$ is empty. Hence we may assume $B_a$
is empty for some $a$ and so $X$ has a canonical element $x_0$. Now
fix $a, c \in A$ and $f$ and $g$ as above. By induction on rank, we
may assume $h(y)$ is already uniquely defined for
$y \in \{ f(b) \;|\; b \in B_a \}$. We define a map $k : S \to X$ as
follows.
\begin{equation*}
k(a', C) =
\begin{cases}
h(y) & a' = a, C = f^{-1}(y) \text{ for unique } y \in
\{ f(b) \;|\; b \in B_a \} \\
x_0 & \text{otherwise}
\end{cases}
\end{equation*}
We next define $l : B_a \to S$ and $r : B_c \to S$ as follows.
\begin{align*}
l(b) &:= (a, \{ b' \in B_a \;|\; f(b') = f(b) \}) \\
r(b) &:= (a, \{ b' \in B_a \;|\; f(b') = g(b) \})
\end{align*}
Finally, it is straightforward to verify that $l$ and $r$ have the
same image in $S$, and that $k \circ l = h \circ f$
and $k \circ r = h \circ g$. Hence it follows from the image
preserving equation $t(a, k \circ l) = t(c, k \circ r)$ that
$t(a, h \circ f) = t(c, h \circ g)$ as required.
\end{proof}
\begin{thm}
If $H((B_a)_{a \in A})$ is a set, then it is the $QW$-type for the
polynomial $(B_a)_{a \in A}$ with all image preserving equations for
$S$.
\end{thm}
\begin{proof}
This follows directly from lemmas \ref{lem:heredsmsatip} and
\ref{lem:outofheredsm}.
\end{proof}
\begin{thm}
If the $QW$-type for the polynomial $(B_a)_{a \in A}$ with all image
preserving equations for $S$ exists, then $H((B_a)_{a \in A})$ is a
set.
\end{thm}
\begin{proof}
Let $(X, t)$ be the $QW$-type.
We first define a homomorphism from $X$ to $H((B_a)_{a \in A})$. The
essential idea is to use the fact that $X$ is an initial
algebra. However, formally speaking we can only apply initiality of
$X$ once we know that $H((B_a)_{a \in A})$ is an algebra and in
particular a set, which we don't have a priori. Hence we use a trick
of approximating $H((B_a)_{a \in A})$ by a sequence of sets.
For each ordinal $\alpha$ we can define an algebra
structure on the set $(H((B_a)_{a \in A}) \cap V_\alpha) + 1$ that
agrees with the structure $s$ on $H((B_a)_{a \in A})$ when $s(a, f)$
belongs to $H((B_a)_{a \in A}) \cap V_\alpha$ and otherwise sends
$(a, f)$ to the $1$ component. Note furthermore that this
operation satisfies all image preserving equations by lemma
\ref{lem:heredsmsatip}. Hence there is a unique homomorphism
$h_\alpha : X \to (H((B_a)_{a \in A}) \cap V_\alpha) + 1$. Say that
$x \in X$ is \emph{defined at $\alpha$} if
$h(x) \in H((B_a)_{a \in A}) \cap V_\alpha$ and undefined
otherwise. Note that for $\beta \geq \alpha$ the canonical
restriction map
$(H((B_a)_{a \in A}) \cap V_\beta) + 1 \to (H((B_a)_{a \in A}) \cap
V_\alpha) + 1$ is a homomorphism. It follows that if $x$ is defined
at $\alpha$, then it is also defined at $\beta$ and
$h_\beta(x) = h_\alpha(x)$. By induction every element $x$ is
defined at $\alpha$ for some ordinal $\alpha$. Hence this defines a
unique homomorphism $X \to H((B_a)_{a \in A})$. The homomorphism has
an inverse by lemma \ref{lem:outofheredsm}. Hence $H((B_a)_{a \in
A})$ is in bijection with a set, and so a set itself.
\end{proof}
\section{Some useful propositions}
\label{sec:some-usef-prop}
We recall some basic classical set theory that will be useful for the
construction of image preserving $QW$-types.
We fill in some of the details, with the remainder left as an
exercise for the reader.
\begin{prop}
For any ordinals $0 < \alpha < \beta$, there is a canonical
surjection $\beta \twoheadrightarrow \alpha$.
If there is a surjection $X \twoheadrightarrow \beta$ for some set
$X$, then there is also a surjection $X \twoheadrightarrow \alpha$.
\end{prop}
\begin{prop}
For any well ordered set $(X, <)$ (and in particular for sets of
ordinals ordered by $\in$), there is a unique ordinal $\beta$ with a
unique order isomorphism $(X, <) \cong (\beta, \in)$. We refer to
$\beta$ as the \emph{order type} of $(X, <)$.
\end{prop}
\begin{prop}
For any family of sets $(X_i)_{i \in I}$, there is an ordinal
$\aleph((X_i)_{i \in I})$ which is the smallest for which there is
no surjection $X_i \twoheadrightarrow \aleph((X_i)_{i \in I})$ for
any $i \in I$. It is precisely the set of all ordinals $\alpha$ for
which there is a surjection $X_i \twoheadrightarrow \alpha$ for some
$i \in I$.
\end{prop}
\begin{proof}
Note that whenever $X_i \twoheadrightarrow \alpha$, there is an
equivalence relation $\sim$ on $X$, and a well ordering $<$ on
$X/{\sim}$ such that the order type of $(X/{\sim}, <)$ is
$\alpha$. However there is clearly a set of such well orders by
power set, and so there is a set of all such ordinals
$\alpha$. Since this is a downwards closed set of ordinals, it is an
ordinal itself. Since the set cannot contain itself, it is the
least ordinal for which there is no surjection from $X_i$ for any
$i$.
\end{proof}
\begin{prop}
\label{prop:cardinalseqs}
If $\kappa$ is a cardinal number (i.e. an ordinal that is not in
bijection with any lower ordinal), then one can define surjections
\begin{enumerate}
\item \label{sqrt} $\kappa \twoheadrightarrow \kappa \times \kappa$
\item $\kappa \twoheadrightarrow \kappa^n$ for any $n < \omega$
\item \label{timesomega}$\kappa \twoheadrightarrow \omega \times
\kappa$
\item $\kappa \twoheadrightarrow \sum_{n < \omega} \kappa^n$
\end{enumerate}
\end{prop}
\begin{proof}
For \ref{sqrt}, see e.g. \cite[Theorem I.11.30]{kunenfoundations}.
For \ref{timesomega}, suppose we are given a bijective pairing
function $(-,-) \colon \omega \times \omega \to \omega$. Any ordinal
$\alpha$ can be written uniquely as $\alpha = \lambda + (m, n)$
where $\lambda$ is a limit ordinal and $m, n \in \omega$. We then
decode this as the pair $(m, \lambda + n)$, which clearly gives a
bijection.
Deriving the other parts from these two is straightforward.
\end{proof}
\section{The construction of image preserving $QW$-types}
\label{sec:proof}
We now construct image preserving $QW$-types in $\mathsf{Set}$. This is
based on Jech's construction of the set of hereditarily countable sets
\cite{jechheredctbl}.
\begin{defn}
We will define a class function $Q$ from ordinals to sets by
recursion on ordinals.
We define $Q(0)$ to be $\emptyset$ and for limit ordinals $\lambda$,
we define $Q(\lambda)$ to be $\bigcup_{\alpha < \lambda}
Q(\alpha)$.
We define $Q(\alpha + 1)$ as follows. Let $X$ be the set of pairs
$(a, f)$ where $a \in A$ and $f : B_a \to Q(\alpha)$. We
then take $\sim$ to be the equivalence relation on $X$ generated by
identifying $(a_e, t \circ l_e)$ and $(b_e, t \circ r_e)$ whenever
$t \colon V_e \to Q(\alpha)$ for $e \in E$ and we define
$Q(\alpha + 1)$ to be $X/{\sim}$.
\end{defn}
\begin{rmk}
For $\beta \leq \alpha$ we have $Q(\beta) \subseteq Q(\alpha)$, by
exploiting the fact that functions are implemented as relations not
including explicit reference to their codomain, and noting that for
$f : B_a \to Q(\alpha)$ and $g : B_{b} \to Q(\alpha)$ such that
$(a, f) \sim (b, g)$, $f$ factors through the inclusion $Q(\beta)
\subseteq Q(\alpha)$ if and only if $g$ does.
\end{rmk}
We now give a series of definitions and lemmas that apply at any
stage $\alpha \in \mathsf{On}$.
\begin{defn}
Note that we only identify $(a, f)$ and $(b, g)$ when $f$ and $g$
have the same image in $Q(\alpha)$. Hence we have a well defined
image function $\operatorname{im} \colon Q(\alpha) \to \mathcal{P}(Q(\alpha))$, such
that whenever $x = [(a, f)]$, $\operatorname{im}(x)$ is the image of $f$ in
$Q(\alpha)$.
\end{defn}
\begin{defn}
Given an element $x$ of $Q(\alpha)$ of the form $[(a, f)]$, we
defined the \emph{rank} of $x$, $\operatorname{rank}(x)$ to be the smallest ordinal
$\beta$ such that $f$ factors through the inclusion
$Q(\beta) \subseteq Q(\alpha)$. To check this is a well
defined, note that it depends only on the image of $f$.
\end{defn}
Note that $\operatorname{rank}(x) + 1$ is the smallest ordinal $\beta$ such that
$x \in Q(\beta)$.
\begin{defn}
Given a set $X \subseteq Q(\alpha)$, we define the union $\cup X$
by
\begin{equation*}
\cup X := \bigcup_{x \in X} \operatorname{im}(x)
\end{equation*}
We define the \emph{transitive closure} of $x \in Q(\alpha)$,
$\operatorname{TC}(x)$ by
\begin{equation*}
\operatorname{TC}(x) := \bigcup_{1 \leq n < \omega} {\cup}^n \{x\}
\end{equation*}
\end{defn}
\begin{lemma}
\label{lem:rktc}
For all $x \in Q(\alpha)$, we have the following equation.
\begin{equation*}
\operatorname{rank}(x) = \{ \operatorname{rank}(y) \;|\; y \in \operatorname{TC}(x) \}
\end{equation*}
\end{lemma}
\begin{proof}
It is clear that whenever $y \in \operatorname{TC}(x)$ we must have
$\operatorname{rank}(y) < \operatorname{rank}(x)$ since this is the case for any $n < \omega$ and
any $y \in \cup^n \{x\}$ by induction on $n$.
It remains to show that for any $\beta < \operatorname{rank}(x)$, we have
$\beta = \operatorname{rank}(y)$ for some $y \in \operatorname{TC}(x)$. By the definition of rank,
$\operatorname{im}(x)$ cannot be contained in $Q(\beta)$. Hence we must have
$x = [(a, f)]$ and $b \in B_a$ such that $f b \notin Q(\beta)$. For
this $b$ we have $\beta \leq \operatorname{rank}(f b)$. If $\beta = \operatorname{rank}(f b)$, then
$f b \in \cup\{x\} \subseteq \operatorname{TC}(x)$ and so $\beta$ is as
required. Otherwise, $\beta < \operatorname{rank}(f b)$ and so by induction on $\operatorname{rank}(x)$
we may assume $\beta = \operatorname{rank}(y)$ for some $y \in \operatorname{TC}(f b)$. However,
$\operatorname{TC}(f b) \subseteq \operatorname{TC}(x)$, so $y \in \operatorname{TC}(x)$ and $\beta$ is again
as required.
\end{proof}
\begin{defn}
\label{defn:rns}
For $x \in Q(\alpha)$, we write $R_n(x)$ for the set $\{\operatorname{rank}(z) \;|\; z
\in \cup^n\{x\}\}$.
\end{defn}
\begin{lemma}
\label{lem:rkunionrns}
For all $x \in Q(\alpha)$,
\begin{equation*}
\operatorname{rank}(x) = \bigcup_{1 \leq n < \omega} R_n(x)
\end{equation*}
\end{lemma}
\begin{proof}
By lemma \ref{lem:rktc}.
\end{proof}
\begin{defn}
We define $\kappa$ to be $\aleph((B_a)_{a \in A})$. We define
$\kappa^+$ to be the smallest non zero ordinal for which there is no
surjection $\kappa \twoheadrightarrow \kappa^+$.
\end{defn}
\begin{lemma}
\label{lem:rnsurjs}
We define for each $x \in Q(\alpha)$ and each $1 \leq n < \omega$, a
surjection
$F_{x, n} \colon \kappa^n \twoheadrightarrow R_n(x) \cup
\{\emptyset\}$.
\end{lemma}
\begin{proof}
We first consider the case $n = 1$. Suppose that $x = [(a,
f)]$. Note that $R_1(x) := \{\operatorname{rank}(f b) \;|\; b \in B_a \}$ is a set
of ordinals, and so it has an order type $\beta \in \mathsf{On}$, and in
particular we have a unique order isomorphism with $\beta$, say
$\theta \colon \beta \stackrel{\cong}{\to} R_1(x)$. Furthermore, by
definition, there is clearly a surjection from $B_a$ to $R_1(x)$. It
follows that $\beta < \kappa$. Hence we can define a canonical
surjection
$F_{x, 1} \colon \kappa \twoheadrightarrow R_1(x) \cup
\{\emptyset\}$ as follows.
\begin{equation*}
F_{x, 1}(\alpha) :=
\begin{cases}
\theta(\alpha) & \alpha < \beta \\
\emptyset & \text{otherwise}
\end{cases}
\end{equation*}
Now suppose $n = m + 1$. We fix $m$ ordinals less than $\kappa$, say
$\beta_1, \ldots, \beta_m$ and consider the set $Y$ below.
\begin{equation*}
Y := \{ F_{f b, m}(\beta_1, \ldots, \beta_m) \;|\; b
\in B_a \}
\end{equation*}
This is again a set of ordinals with a surjection from $B_a$ for
some $a \in A$, and so as before, we have a canonical surjection
$G \colon \kappa \twoheadrightarrow Y \cup \{ \emptyset \}$. We take
$F_{x, n}( \beta_1, \ldots, \beta_m, \beta_{m + 1} )$ to be
$G(\beta_{m + 1})$. We now simultaneously check that $F_{x, n}$ has
the correct codomain and is surjective.
\begin{align*}
\operatorname{im}(F_{x, n})
&= \bigcup_{\beta_1, \ldots, \beta_m < \kappa} (\{ F_{fb,
m}( \beta_1, \ldots, \beta_m) \;|\; b \in B_a \} \cup \{
\emptyset \}) \\
&= \bigcup_{b \in B_a} \{ F_{fb,
m}( \beta_1, \ldots, \beta_m ) \;|\;
\beta_1, \ldots, \beta_m < \kappa \} \; \cup \; \{\emptyset \} \\
&= \bigcup_{b \in B_a} \{ \operatorname{rank}(z) \;|\; z \in \cup^m\{ f b \} \}
\; \cup \; \{ \emptyset \} \\
&= R_n(x) \; \cup \; \{ \emptyset \}
\end{align*}
\end{proof}
\begin{lemma}
\label{lem:rkbound}
For any $x \in Q(\alpha)$ we have $\operatorname{rank}(x) < \kappa^+$.
\end{lemma}
\begin{proof}
First note that this is clear when $\operatorname{rank}(x) = 0$. Hence we may assume
for the rest of the proof $\operatorname{rank}(x) > 0$.
By the definition of $\kappa^+$, it suffices to define a surjection
$\kappa \twoheadrightarrow \operatorname{rank}(x)$. By proposition
\ref{prop:cardinalseqs} it suffices to define a surjection
$\sum_{1 \leq n < \omega} \kappa^n \twoheadrightarrow
\operatorname{rank}(x)$. However, by lemma \ref{lem:rkunionrns} we can express
$\operatorname{rank}(x)$ as $\bigcup_{1 \leq n < \omega} R_n(x)$. Since
$\operatorname{rank}(x) > 0$, this is the same as
$\bigcup_{1 \leq n < \omega} (R_n(x) \cup \{ \emptyset \})$, and so
we can just combine the surjections defined in lemma
\ref{lem:rnsurjs}.
\end{proof}
\begin{thm}
\label{thm:iptypesexist}
All image preserving $QW$-types exist in $\mathsf{Set}$.
\end{thm}
\begin{proof}
We show that $Q(\kappa^+)$ is an initial algebra. We first need to
show how to define an algebra structure. Suppose we are given $a \in
A$ and a map $f \colon B_a \to Q(\kappa^+)$. Then $[(a, f)]$ is an
element of $Q(\kappa^+ + 1)$. By lemma \ref{lem:rkbound} we have
$\operatorname{rank}([(a, f)]) < \kappa^+$, and so $f$ factors through $Q(\beta)$ for
some $\beta < \kappa^+$. We can then take $\sup(a, f)$ to be
$[(a, f)] \in Q(\beta + 1)$.
We check that this structure respects the equations. Suppose that we
are given $g \colon V_e \to Q(\kappa^+)$. Note that $g \circ l_e$
and $g \circ r_e$ have the same image, and so
$\operatorname{rank}([(a_e, g \circ l_e)]) = \operatorname{rank}([(b_e, g \circ r_e)])$. Hence
$[(a_e, g \circ l_e)]$ and $[(b_e, g \circ r_e)]$ must
have been first added at the same stage, $\alpha + 1$. We can now
see that they were identified in the definition of $Q(\alpha + 1)$.
Finally, it is clear that for any other algebra structure, we can
define a unique homomorphism out of $Q(\kappa^+)$ by
recursion on ordinals.
\end{proof}
\section{The ranks of unordered countably branching trees}
\label{sec:ranks-unord-count}
It follows from Jech's construction that every hereditarily countable
set has rank less than $\omega_2$. In \cite{jechheredctbl} Jech also showed
a converse statement when $\omega_1$ is singular: in this case
there is a hereditarily countable set of rank $\alpha$ for every
$\alpha < \omega_2$. We will show the analogous result for unordered
countably branching trees. We first recall that Jech proved the
following lemma as part of the proof of \cite[Theorem
2]{jechheredctbl}.
\begin{lemma}
\label{lem:choosecofsets}
Suppose that $\omega_1$ is singular and let $\alpha < \omega_2$. For
each fixed surjection $f : \omega_1 \twoheadrightarrow \alpha$ and
choice of countable cofinal sequence in $\omega_1$, we can construct
for each limit ordinal $\lambda \leq \alpha$ a choice of countable
cofinal subset $Y_\lambda \subseteq \lambda$.
\end{lemma}
We can now show the theorem.
\begin{thm}
\label{thm:ucbtrank}
Suppose that $\omega_1$ is singular. Then for every $\alpha <
\omega_2$ there is an unordered countably branching tree of rank
$\alpha$.
\end{thm}
\begin{proof}
Since $\alpha < \omega_2$ and $\omega_1$ is singular there exists a
choice of sets $Y_\lambda$ as in lemma \ref{lem:choosecofsets}
for limit ordinals $\lambda \leq \alpha$.
We construct an unordered countably branching tree $t_\beta$ for
each $\beta \leq \alpha$ by induction. We define
$t_0 := \mathtt{leaf}$. For successor ordinals we define
$t_{\beta + 1} := \mathtt{node}(\lambda n.t_\beta)$. This just
leaves the case of $t_\lambda$ for non zero limit ordinals
$\lambda$. Since $Y_\lambda$ is countable there exists a surjection
$g : \omega \twoheadrightarrow Y_\lambda$. Since $Y_\lambda$ is
cofinal in a limit ordinal it is not finite. Hence there exists a
bijection $h : \omega \stackrel{\cong}{\to} Y_\lambda$, which we
can construct from a choice of surjection $g$ as follows:
\begin{align*}
h(0) &= g(0) \\
h(n + 1) &=
\begin{cases}
g(n + 1) & g(n + 1) \neq h(k) \text{ for all } k \leq
n
\\
\gamma & \text{otherwise},
\gamma \text{ least s.t. } \gamma \neq h(k)\text{ for
all } k \leq n
\end{cases}
\end{align*}
We show how to define $t_\lambda$ from a choice of bijection $h$,
and then check that it is independent of the particular choice and
so well defined.
We define $t_\lambda := \mathtt{node}(\lambda n.t_{h(n)})$. Now
suppose that we are given two such bijections
$h, h' : \omega \stackrel{\cong}{\to} Y_\lambda$. Then
$\pi := h^{-1} \circ h'$ is a permutation of $\omega$ and
$h' = h \circ \pi$. Hence $\mathtt{node}(\lambda n.t_{h'(n)})$ and
$\mathtt{node}(\lambda n.t_{h(n)})$ are identified by one of the
defining equations of unordered countably branching trees, ensuring
that $t_\lambda$ is independent of the choice of $h$, as
required. It is clear that $\operatorname{rank}(t_\beta) = \beta$ for all $\beta
\leq \alpha$.
\end{proof}
\begin{cor}
If $\omega_1$ is singular, then the unique homomorphism
$h : T \to T_{\operatorname{Sym}}$ from countably branching trees to unordered
countably branching trees is not surjective.
\end{cor}
\begin{proof}
Note that $h$ preserves rank. Hence by theorem \ref{thm:ucbtrank} it
suffices to show that the rank of every (ordered) countably
branching tree is less than $\omega_1$.
Fix a surjective function
$s : \omega \twoheadrightarrow \omega + (\omega \times \omega)$. We
construct for each ordered tree $t$ a countable enumeration
$f(t) : \omega \to T$ whose image is the union of the set of proper
subtrees of $t$ and $\{\mathtt{leaf}\}$. We define
$f(\mathtt{leaf})(n) := \mathtt{leaf}$. Now suppose we are given a
tree of the form $t = \mathtt{node}(g)$. By composing with $s$, it suffices
for us to define two functions $f_0 : \omega \to T$ and
$f_1 : \omega \times \omega \to T$ that jointly enumerate all of the
proper subtrees of $t$ (including $\mathtt{leaf}$). We define
$f_0(n) := g(n)$ and $f_1(n, m) := f(g(n))(m)$.
By lemma \ref{lem:rktc} the rank of each $t$ is equal to the set of
ranks of proper subtrees. However, by the above enumeration this is
a countable ordinal and so less than $\omega_1$, as required.
\end{proof}
\section{Conclusion}
\label{sec:conclusion}
We have shown that every image preserving $QW$-type exists in $\mathsf{Set}$
under $\mathbf{ZF}$ without any additional assumptions. Although the proof is
somewhat elaborate, we can see from the results of section
\ref{sec:ranks-unord-count} that some complication is necessary. The
na\"{i}ve construction of defining an equivalence relation on the
$W$-type of the underlying polynomial and quotienting will not work in
general. As we saw, both hereditarily countable sets and unordered
countably branching trees provide counterexamples when $\omega_1$ is
singular.
\subsection{Generalisations and limitations of this method}
\label{sec:gener-limit-this}
The proof in section \ref{sec:proof} made use of the fact that we have
a well defined image operation sending each element of the form
$(a, f)$ to the image of $f$. This may not be strictly necessary for
the proof, since the key lemma \ref{lem:rnsurjs} only used the sets
$R_n$ in definition \ref{defn:rns} which may be definable in
situations without a well defined image operator. So there could be a
more general version of the theorem that uses the same key
idea. However, it is unclear if there are any interesting examples of
$QW$-type that can be constructed by the same general method, but
whose existence isn't already a corollary of theorem
\ref{thm:iptypesexist}. We therefore leave it as an open problem both
to generalise the theorem and to find interesting examples making
essential use of the generalisation.
On the other hand there are some examples where the argument here
cannot possibly apply. As already discussed in example
\ref{ex:nonexblass}, Blass' counterexample cannot be
constructed by this method, since it does not provably exist under
$\mathbf{ZF}$. However, there is also a much simpler example of a $QW$-type
that exists, provably in $\mathbf{ZF}$, but does not seem to be covered by the
argument in this paper, namely free monoids. We consider in particular
the free monoid on a set with four elements $\{a, b, c, d\}$. As a
special case of associativity we have $a(b(cd)) = (ab)(cd)$. However,
it is unclear how to define $R_1$ for this term. Since $\operatorname{rank}(a) = 0$
and $\operatorname{rank}(b(cd)) = 2$, we expect $R_1(a(b(cd))) = \{0, 2\}$. Since
$\operatorname{rank}(ab) = \operatorname{rank}(cd) = 1$, we also expect $R_a((ab)(cd)) = \{1 ,
1\}$. Since $R_n$ play an essential role in lemma \ref{lem:rnsurjs} it
seems that it is impossible to construct free monoids by this method,
even though they do provably exist in $\mathsf{Set}$ under $\mathbf{ZF}$.
Another class of examples of $QW$-types that exist under $\mathbf{ZF}$, but are not
covered directly by the methods of section \ref{sec:proof} are $W$-types with
reductions. By their nature they identify two trees of different rank,
so we don't expect to have a well defined of rank without the
observation in \cite{swanwtypered} that in the presence of the law of
excluded middle they can be viewed as an ordinary $W$-types
of normal forms.
\subsection{Related Open Problems}
\label{sec:relat-open-probl}
The limitations of the result discussed above naturally lead to the
following question.
\begin{openprob}
Is there a general construction of $QW$-types in $\mathsf{Set}$ under $\mathbf{ZF}$
that naturally encompasses all examples known to exist in this
setting?
\end{openprob}
Our proof of lemma \ref{lem:rnsurjs} makes essential use of the fact
that any set of ordinals is order isomorphic to an ordinal, which in
turn uses classical logic. This leaves open the case of intuitionistic
logic, specifically the following problem.
\begin{openprob}
Show all image preserving $QW$-types exist in $\mathsf{Set}$ under the
assumptions of $\mathbf{IZF}$ or find a model of $\mathbf{IZF}$ where they do not.
\end{openprob}
Note that the proof of theorem \ref{thm:ucbtrank} made essential use
of the fact that we have equations for all permutations $\pi$ of
$\omega$. It is unclear what happens when we allow some permutations
but not all, leading to the following question.
\begin{openprob}
We define the $QW$-type of \emph{countably branching weakly
unordered trees} to have the same underlying polynomial as
countably branching trees and equations of the form
$\mathtt{node}(\lambda n.v_n) = \mathtt{node}(\lambda n.v_{\pi(n)})$ where $\pi$ is
a \emph{finitely supported} permutation of $\omega$. Is it provable
in $\mathbf{ZF}$ that the unique homomorphism from countably branching trees
to weakly unordered countably branching trees is a surjection?
\end{openprob}
{}
\end{document}
|
math
|
#ifndef TESTES_H_INCLUDED
#define TESTES_H_INCLUDED
#include "dominios.h"
using namespace std;
// Declaração de classe.
class TUCodigo {
private:
// Definições de constantes para evitar numeros mágicos.
const static int VALOR_VALIDO = 20;
const static int VALOR_INVALIDO = 30;
// Referência para o objeto a ser testado.
Codigo *codigo;
// Estado do teste.
int estado;
// Declarações de métodos.
void setUp();
void tearDown();
void testarCenarioSucesso();
void testarCenarioFalha();
public:
// Definições de constantes para reportar resultado do teste.
const static int SUCESSO = 0;
const static int FALHA = -1;
int run();
};
#endif // TESTES_H_INCLUDED
|
code
|
It Helps When I Post the Correct Link!
Good afternoon and thank you to the Board meeting faithful who let me know that the Board meeting agenda post from the morning contained a link to a login page….not what was intended! Please click the correct link here and we hope to see you there on Tuesday! Many exciting items on the agenda, the most exciting of which may be considering approval of plans and specifications for the 2018 (Measure A funded) Road Improvement Project!
|
english
|
जब दुल्हन के रूप में ईशा का पहला लुक आया तो वहां मौजूद मेहमानों की निगाहें केवल उन्हीं के ब्राइडल लुक पर टिक गई। अपनी शादी में ईशा अंबानी ने आइवरी कलर का हैवी वर्क वाला भारी लहंगा पहना। जिसमें वह किसी अप्सरा से कम नहीं लग रही थीं। इसके साथ उन्होंने मैचिंग हैवी डायमंड जूलरी जिसमें वह बिल्कुल चांद का टुकड़ा लग रही थीं।
नई दिल्ली: देश के सबसे अमीर बिजनेसमैन मुकेश अंबानी की लाड़ली बेटी ईशा अंबानी की शादी बेहद शानदार और गाजे-बाजे के साथ संपन्न हुई। इस शादी में देश-विदेश सहित पूरा बॉलीवुड शामिल हुआ। १२ दिसंबर को विवाह पंचमी के दिन हुई इस शादी में हर किसी को बस ईशा और आंनद की एक झलक पाने का बड़ी बेसब्री से इंतजार था। लेकिन दुल्हन के लिबास में ईशा अपनी की शादी की एक चीज भी पहनें हुए नजर आईं।
जब दुल्हन के रूप में ईशा का पहला लुक आया तो वहां मौजूद मेहमानों की निगाहें केवल उन्हीं के ब्राइडल लुक पर टिक गई। अपनी शादी में ईशा अंबानी ने आइवरी कलर का हैवी वर्क वाला भारी लहंगा पहना। जिसमें वह किसी अप्सरा से कम नहीं लग रही थीं। इसके साथ उन्होंने मैचिंग हैवी डायमंड जूलरी जिसमें वह बिल्कुल चांद का टुकड़ा लग रही थीं।
ईशा जब दुल्हन बनकर सामने आईं तो उनकी सुरमई आंखों के बाद ध्यान खींचा उनकी बड़ी-सी नथ ने। हालांकि दीपिका पादुकोण और प्रियंका चोपड़ा की तरह ये बहुत बड़ी नहीं थी, लेकिन इसकी चमक के आगे तमाम सितारे फीके पड़ रहे थे।
ईशा अंबानी की खूबसूरत ड्रेस को डिजाइनर अबु जानी और संदीप घोसला ने डिजाइन किया है। वहीं स्टाइल एमी पटेल और हेयर स्टाइल पॉम्पी हंस के अलावा मेकअप वरदान नायक ने किया था।
ईशा अंबानी की इस वेडिंग लहंगे में एक चीज बहुत ही खास थी। जी हां ईशा ने अपनी मां नीता अंबानी की शादी का दुपट्टा (पानेतर) भी डाला हुआ था। जो कि सुर्ख लाल रंग का था। जिसे ईशा ने अलग से चुनरी के रुप में लिया हुआ था।
आपको बता दें कि नीता अंबानी को उनके मामा के द्वारा दी गई थी। जो कि रस्म का एक हिस्सा है।
क्या होती है पानेतर साड़ी
यह साड़ी गुजरात में पारंपरिक रूप से शादी पर दुल्हन को उसके मामा द्वारा दी जाती है। यह साडी सफ़ेद रंग में होती और इसका पल्लू चटक लाल रंग का। इसे एक और साड़ी घरचोला (जो दूल्हे के परिवार से आती है) के साथ उपहार स्वरुप दिया जाता है।
परम्परागत रूप से शादी के दौरान शुरू की रस्मों में दुल्हन पानेतर साड़ी पहनती है व बाद की रस्मों में घरचोला साड़ी। यह रस्म चिह्नित करती है कि अब वह दूसरे परिवार का हिस्सा है। खैर आजकल, पुरे शादी समारोह के दौरान दुल्हन सिर पर घरचोला ओढ़नी व पानेतर साड़ी पहनती है।
|
hindi
|
RAPID CITY, S.D. – Regional Health has been named winner of the Silver Star of Excellence Award, presented by the American Technical Education Association (ATEA) and the National Technical Honor Society (NTHS). The award recognizes Regional Health’s support and commitment to postsecondary technical education.
Ann Bolman, President of Western Dakota Tech in Rapid City, nominated Regional Health for the award. She noted that Regional Health is a major employer of Western Dakota Tech graduates. Nearly 10 percent of the class of 2017 joined Regional Health after graduation.
Bolman also cited Regional Health’s generosity in providing meeting space, surgical and laboratory supplies, medical simulation materials, compliance training and other assistance for its programs including practical nursing, surgical technology, medical lab technology, phlebotomy, paramedicine, medical assisting, health information management and pharmacy technology.
Regional Health has also partnered with Western Dakota Tech, South Dakota State University, the University of South Dakota and other community health groups to develop a regional Area Health Education Center, which will provide resources for community health training and medical field development in western South Dakota.
Jill Tice, Director of Quality, Safety and Risk Management at Regional Health, accepted the award on behalf of Regional Health during the ATEA’s National Conference in Indianapolis.
|
english
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विश्व स्वास्थ्य संगठन (डब्लूएचओ) ने कहा है कि सरकारें और उपभोक्ता इलेक्ट्रॉनिक सिगरेट जैसे अन्य उत्पादों के प्रचार पर आसानी से विश्वास न करें।
बयान में कहा गया है कि इलेक्ट्रॉनिक सिगरेट से होने वाला नुकसान कम है, यह तंबाकू कंपनियों के प्रचार की एक रणनीति है। डब्लूएचओ ने अभी जारी २०१९ वैश्विक तंबाकू महामारी रिपोर्ट में बताया कि लंबे समय में तंबाकू उद्योग, तंबाकू नियंत्रण के लिए अपनाए जा रहे कदमों के खिलाफ काम कर रहा है।
कई उद्योगों का कहना है कि पारंपरिक सिगरेट के बदले इलेक्ट्रॉनिक सिगरेट सुरक्षित है और ये सिगरेट पीने की आदत छोड़ने में मदद करता है। प्रमाण के अनुसार अमेरिकी किशोरों में ई-सिगरेट तेजी से लोकप्रिय हो रही है।
डब्लूएचओ ने रिपार्ट जारी की थी कि इस बात के समर्थन में पर्याप्त सबूत नहीं है। जब सिगरेट पीने वाले पूरी तरह से निकोटीन छोड़ देंगे, तभी उन्हें लाभ मिलेगा। अमेरिकी खाद्य और औषधि प्रशासन ने हाल के वर्षो में इलेक्ट्रॉनिक सिगरेट की बिक्री को नियंत्रित करने के लिए कई उपायों की घोषणा की है।
डब्लूएचओ तंबाकू नियंत्रण अधिकारी विनायक प्रसाद ने बताया कि इलेक्ट्रॉनिक सिगरेट और पारंपरिक सिगरेट पीने से होने वाले नुकसान एक जैसे हैं, सबसे बड़ा अंतर यह है कि इलेक्ट्रॉनिक सिगरेट में कोई स्पष्ट धुआं नहीं है। उन्होंने जोर देते हुए कहा कि इलेक्ट्रॉनिक सिगरेट बाजार के पर्यवेक्षण को मजबूत करना चाहिए, जो डब्लूएचओ का एक स्पष्ट लक्ष्य भी है।
|
hindi
|
बैंक ऑफ इंडिया ऑफिसर रिक्रूटमेन्ट बेगिन्स, आप्प्ली ऑनलाइन बेफोर ५ मई - बैंक ऑफ इंडिया में निकली नौकरी, ५ मई तक करें आवेदनऑडत्व्बुसिनेस्शीन्दिमोवीस्क्रिकेथेलफूडतेच्तोप्स्प्रिमआर्टवेड्डींगगूड टाइम्सहोमलाइव टीवीबड़ी ख़बरताज़ातरीनवीडियोदेशविदेशशहरज़रा हटकेराजनीतिबॉलीवुडविचार पेजक्रिकेटअन्य अन्य करियरउत्तर प्रदेशपंजाबगुजरातफोटोस्पोर्ट्ससोशलफूड-फिटनेसलाइफस्टाइलरवीश का पन्नाबिज़नेसऑटोहोम | जॉब्स | बैंक ऑफ इंडिया में निकली नौकरी, ५ मई तक करें आवेदन सुमित राय द्वारा लिखित, अंतिम अपडेट: रविवार अप्रैल २३, २०१७ ११:५0 आम इस्ट शारेंईमेल करेंटिप्पणियांबैंक ऑफ इंडिया (बैंक ऑफ इंडिया) ने ऑफिसर पोस्ट के भर्ती के लिए नोटिफिकेशन जारी किया है. इन पदों के लिए इच्छुक और योग्य अभ्यर्थी ५ मई, २०१७ तक आवेदन कर सकते हैं. बैंक ऑफ इंडिया ने ऑफिसर (क्रेडिट), मैनेजर, सिक्युरिटी ऑफिसर, तकनीकी (मूल्यांकन) और तकनीकी (परिसर) के ७०२ पदों पर भर्ती के लिए आवेदन मंगाए हैं. इन पदों पर आवेदन करने के इच्छुक उम्मीदवार ५ मई २०१७ तक ऑनलाइन आवेदन कर सकते हैं.महत्वपूर्ण तिथियां :
बैंक ऑफ इंडिया के ऑफिसर (क्रेडिट) पोस्ट के लिए आवेदन २० अप्रैल २०17 से शुरू हो गया है और ५ मई २०17 तक आवेदन कर सक हैं. ऑफिसर (क्रेडिट) के अलावा अन्य पदों के लिए आवेदन की प्रक्रिया २६ अप्रैल २०17 से शुरू होगी.इन पदों के लिए कैसे करें आवेदन :
- वेबसाइट पर करियर टैब पर क्लिक करें.
- इसके बाद ऑनलाइन अप्लिकेशन सब्मिशन पेज पर जाएं.
- अप्लिकेशन फॉर्म में डीटेल भरें और डॉक्यूमेंट अपलोड करें.
- आवेदन शुल्क का भुगतान भी ऑनलाइन माध्यम से किया जा सकता है.
यह भी पढ़ें: देना बैंक में प्रोबेशनरी ऑफिसर के पद पर ३०० वैकेंसीशैक्षणिक योग्यता :
इन पदों पर भर्ती के लिए आवेदन करने वाले आवेदक के पास देश के किसी भी बोर्ड, विश्वविद्यालय या संस्थान से ग्रेजुएशन या पोस्ट ग्रेजुएशन की डिग्री होनी चाहिए.चयन प्रक्रिया :
इन पदों के लिए योग्य अभ्यर्थियों का चयन ऑनलाइन लिखित परीक्षा और इंटरव्यू के आधार पर किया जाएगा. १५० नंबर के ऑनलाइन एग्जाम में इंग्लिश लैंग्वेज, जनरल अवेयरनेस, बैंकिंग और फाइनेशियल मैनेजमेंट के सवाल होंगे. इसके लिए कैंडिडेट को १२० मिनट का समय मिलेगा.जॉब्स सेक्शन से जुड़े अन्य ख़बरों को पढ़ने के लिए यहां क्लिक करें.हिन्दी न्यूज
गूगल प्लस पर ज्वॉइन करें, ट्विटर पर फॉलो करे... लोकप्रियगठबंधन में मचे बवाल के बीच लालू यादव ने लगाया नीतीश को फोन...जानें क्या निकला नतीजागस्ट : एक जुलाई से पहले सभी का रजिस्ट्रेशन मुश्किल, आनन-फानन में सरकार ने दी नियमों में छूटजब मौत को टक्कर देने के लिए मछली ने लड़ी सांप से जंग, देखें वीडियो संबंधितप्राइवेट बैंकों के प्रमुखों के मुकाबले भारतीय स्टेट बैंक (सबी) प्रमुख की सैलरी कुछ भी नहींरिजर्व बैंक ऑफ इंडिया ने बैंकों से कहा-ग्राहकों की पासबुक में लेन-देन का पर्याप्त ब्योरा लिखेंज़िला बैंकों को राहत : बदले जाएंगे पुराने नोट, किसान को मदद देने के लिए मिलेगी करेंसीशेयर करें शारें
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hindi
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منز زامُت۔ سُہ اوس ہندوستٲنۍ عٔلمن
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kashmiri
|
विद्या बालन जल्द ही एनटीआर की बायोपिक में नजर आने वाली हैं
विद्या बालन जल्द ही एनटीआर की बायोपिक में नजर आने वाली हैं. इस फिल्म में वो एनटीआर की पत्नी का रोल प्ले कर रही हैं. विद्या को एक बेबाक अभिनेत्री के लिए भी जाना जाता है. हाल ही में एक बातचीत में विद्या से पूछा गया कि वो कौन सा निर्देशक है, जिसके साथ वो कभी भी काम नहीं करेंगी. इस सवाल पर विद्या ने बेहद शॉकिंग जवाब दिया.
विद्या ने दिया शॉकिंग जवाब
हाल ही में विद्या बालन ने एक इंटरव्यू में बेहद शॉकिंग जवाब दिया है. विद्या से पूछा गया कि वो कौन सा निर्देशक है, जिसके साथ वो कभी भी काम नहीं करेंगी. इसके जवाब में विद्या ने तपाक से कहा साजिद खान. विद्या ने इसके साथ ही साजिद के साथ काम न करने की वजह भी बताई. उन्होंने कहा कि, साजिद खान महिलाओं को कभी ठीक तरह से समझ नहीं पाते. बता दें, पिछले दिनों देश-दुनिया की तमाम महिलाओं ने अपने साथ हुए शोषण की कहानी बताई थी.
बुरे दौर से गुजर रहे हैं साजिद
आपको बता दें कि, साजिद खान इन दिनों बुरे दौर से गुजर रहे हैं. उन पर २ से ज्यादा लड़कियों ने यौन शोषण के गंभीर आरोप लगाए थी. इसकी वजह से उनकी खूब आलोचना हुई थी. साथ ही उन्हें हाउसफुल ४ से भी बाहर कर दिया गया था.
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hindi
|
//========================================================================
//
// CompactFontTables.h
//
// Copyright 1999-2003 Glyph & Cog, LLC
//
//========================================================================
#ifndef COMPACTFONTINFO_H
#define COMPACTFONTINFO_H
static char *type1CStdStrings[391] = {
".notdef",
"space",
"exclam",
"quotedbl",
"numbersign",
"dollar",
"percent",
"ampersand",
"quoteright",
"parenleft",
"parenright",
"asterisk",
"plus",
"comma",
"hyphen",
"period",
"slash",
"zero",
"one",
"two",
"three",
"four",
"five",
"six",
"seven",
"eight",
"nine",
"colon",
"semicolon",
"less",
"equal",
"greater",
"question",
"at",
"A",
"B",
"C",
"D",
"E",
"F",
"G",
"H",
"I",
"J",
"K",
"L",
"M",
"N",
"O",
"P",
"Q",
"R",
"S",
"T",
"U",
"V",
"W",
"X",
"Y",
"Z",
"bracketleft",
"backslash",
"bracketright",
"asciicircum",
"underscore",
"quoteleft",
"a",
"b",
"c",
"d",
"e",
"f",
"g",
"h",
"i",
"j",
"k",
"l",
"m",
"n",
"o",
"p",
"q",
"r",
"s",
"t",
"u",
"v",
"w",
"x",
"y",
"z",
"braceleft",
"bar",
"braceright",
"asciitilde",
"exclamdown",
"cent",
"sterling",
"fraction",
"yen",
"florin",
"section",
"currency",
"quotesingle",
"quotedblleft",
"guillemotleft",
"guilsinglleft",
"guilsinglright",
"fi",
"fl",
"endash",
"dagger",
"daggerdbl",
"periodcentered",
"paragraph",
"bullet",
"quotesinglbase",
"quotedblbase",
"quotedblright",
"guillemotright",
"ellipsis",
"perthousand",
"questiondown",
"grave",
"acute",
"circumflex",
"tilde",
"macron",
"breve",
"dotaccent",
"dieresis",
"ring",
"cedilla",
"hungarumlaut",
"ogonek",
"caron",
"emdash",
"AE",
"ordfeminine",
"Lslash",
"Oslash",
"OE",
"ordmasculine",
"ae",
"dotlessi",
"lslash",
"oslash",
"oe",
"germandbls",
"onesuperior",
"logicalnot",
"mu",
"trademark",
"Eth",
"onehalf",
"plusminus",
"Thorn",
"onequarter",
"divide",
"brokenbar",
"degree",
"thorn",
"threequarters",
"twosuperior",
"registered",
"minus",
"eth",
"multiply",
"threesuperior",
"copyright",
"Aacute",
"Acircumflex",
"Adieresis",
"Agrave",
"Aring",
"Atilde",
"Ccedilla",
"Eacute",
"Ecircumflex",
"Edieresis",
"Egrave",
"Iacute",
"Icircumflex",
"Idieresis",
"Igrave",
"Ntilde",
"Oacute",
"Ocircumflex",
"Odieresis",
"Ograve",
"Otilde",
"Scaron",
"Uacute",
"Ucircumflex",
"Udieresis",
"Ugrave",
"Yacute",
"Ydieresis",
"Zcaron",
"aacute",
"acircumflex",
"adieresis",
"agrave",
"aring",
"atilde",
"ccedilla",
"eacute",
"ecircumflex",
"edieresis",
"egrave",
"iacute",
"icircumflex",
"idieresis",
"igrave",
"ntilde",
"oacute",
"ocircumflex",
"odieresis",
"ograve",
"otilde",
"scaron",
"uacute",
"ucircumflex",
"udieresis",
"ugrave",
"yacute",
"ydieresis",
"zcaron",
"exclamsmall",
"Hungarumlautsmall",
"dollaroldstyle",
"dollarsuperior",
"ampersandsmall",
"Acutesmall",
"parenleftsuperior",
"parenrightsuperior",
"twodotenleader",
"onedotenleader",
"zerooldstyle",
"oneoldstyle",
"twooldstyle",
"threeoldstyle",
"fouroldstyle",
"fiveoldstyle",
"sixoldstyle",
"sevenoldstyle",
"eightoldstyle",
"nineoldstyle",
"commasuperior",
"threequartersemdash",
"periodsuperior",
"questionsmall",
"asuperior",
"bsuperior",
"centsuperior",
"dsuperior",
"esuperior",
"isuperior",
"lsuperior",
"msuperior",
"nsuperior",
"osuperior",
"rsuperior",
"ssuperior",
"tsuperior",
"ff",
"ffi",
"ffl",
"parenleftinferior",
"parenrightinferior",
"Circumflexsmall",
"hyphensuperior",
"Gravesmall",
"Asmall",
"Bsmall",
"Csmall",
"Dsmall",
"Esmall",
"Fsmall",
"Gsmall",
"Hsmall",
"Ismall",
"Jsmall",
"Ksmall",
"Lsmall",
"Msmall",
"Nsmall",
"Osmall",
"Psmall",
"Qsmall",
"Rsmall",
"Ssmall",
"Tsmall",
"Usmall",
"Vsmall",
"Wsmall",
"Xsmall",
"Ysmall",
"Zsmall",
"colonmonetary",
"onefitted",
"rupiah",
"Tildesmall",
"exclamdownsmall",
"centoldstyle",
"Lslashsmall",
"Scaronsmall",
"Zcaronsmall",
"Dieresissmall",
"Brevesmall",
"Caronsmall",
"Dotaccentsmall",
"Macronsmall",
"figuredash",
"hypheninferior",
"Ogoneksmall",
"Ringsmall",
"Cedillasmall",
"questiondownsmall",
"oneeighth",
"threeeighths",
"fiveeighths",
"seveneighths",
"onethird",
"twothirds",
"zerosuperior",
"foursuperior",
"fivesuperior",
"sixsuperior",
"sevensuperior",
"eightsuperior",
"ninesuperior",
"zeroinferior",
"oneinferior",
"twoinferior",
"threeinferior",
"fourinferior",
"fiveinferior",
"sixinferior",
"seveninferior",
"eightinferior",
"nineinferior",
"centinferior",
"dollarinferior",
"periodinferior",
"commainferior",
"Agravesmall",
"Aacutesmall",
"Acircumflexsmall",
"Atildesmall",
"Adieresissmall",
"Aringsmall",
"AEsmall",
"Ccedillasmall",
"Egravesmall",
"Eacutesmall",
"Ecircumflexsmall",
"Edieresissmall",
"Igravesmall",
"Iacutesmall",
"Icircumflexsmall",
"Idieresissmall",
"Ethsmall",
"Ntildesmall",
"Ogravesmall",
"Oacutesmall",
"Ocircumflexsmall",
"Otildesmall",
"Odieresissmall",
"OEsmall",
"Oslashsmall",
"Ugravesmall",
"Uacutesmall",
"Ucircumflexsmall",
"Udieresissmall",
"Yacutesmall",
"Thornsmall",
"Ydieresissmall",
"001.000",
"001.001",
"001.002",
"001.003",
"Black",
"Bold",
"Book",
"Light",
"Medium",
"Regular",
"Roman",
"Semibold"
};
static Gushort type1CISOAdobeCharset[229] = {
0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69,
70, 71, 72, 73, 74, 75, 76, 77, 78, 79,
80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
90, 91, 92, 93, 94, 95, 96, 97, 98, 99,
100, 101, 102, 103, 104, 105, 106, 107, 108, 109,
110, 111, 112, 113, 114, 115, 116, 117, 118, 119,
120, 121, 122, 123, 124, 125, 126, 127, 128, 129,
130, 131, 132, 133, 134, 135, 136, 137, 138, 139,
140, 141, 142, 143, 144, 145, 146, 147, 148, 149,
150, 151, 152, 153, 154, 155, 156, 157, 158, 159,
160, 161, 162, 163, 164, 165, 166, 167, 168, 169,
170, 171, 172, 173, 174, 175, 176, 177, 178, 179,
180, 181, 182, 183, 184, 185, 186, 187, 188, 189,
190, 191, 192, 193, 194, 195, 196, 197, 198, 199,
200, 201, 202, 203, 204, 205, 206, 207, 208, 209,
210, 211, 212, 213, 214, 215, 216, 217, 218, 219,
220, 221, 222, 223, 224, 225, 226, 227, 228
};
static Gushort type1CExpertCharset[166] = {
0, 1, 229, 230, 231, 232, 233, 234, 235, 236,
237, 238, 13, 14, 15, 99, 239, 240, 241, 242,
243, 244, 245, 246, 247, 248, 27, 28, 249, 250,
251, 252, 253, 254, 255, 256, 257, 258, 259, 260,
261, 262, 263, 264, 265, 266, 109, 110, 267, 268,
269, 270, 271, 272, 273, 274, 275, 276, 277, 278,
279, 280, 281, 282, 283, 284, 285, 286, 287, 288,
289, 290, 291, 292, 293, 294, 295, 296, 297, 298,
299, 300, 301, 302, 303, 304, 305, 306, 307, 308,
309, 310, 311, 312, 313, 314, 315, 316, 317, 318,
158, 155, 163, 319, 320, 321, 322, 323, 324, 325,
326, 150, 164, 169, 327, 328, 329, 330, 331, 332,
333, 334, 335, 336, 337, 338, 339, 340, 341, 342,
343, 344, 345, 346, 347, 348, 349, 350, 351, 352,
353, 354, 355, 356, 357, 358, 359, 360, 361, 362,
363, 364, 365, 366, 367, 368, 369, 370, 371, 372,
373, 374, 375, 376, 377, 378
};
static Gushort type1CExpertSubsetCharset[87] = {
0, 1, 231, 232, 235, 236, 237, 238, 13, 14,
15, 99, 239, 240, 241, 242, 243, 244, 245, 246,
247, 248, 27, 28, 249, 250, 251, 253, 254, 255,
256, 257, 258, 259, 260, 261, 262, 263, 264, 265,
266, 109, 110, 267, 268, 269, 270, 272, 300, 301,
302, 305, 314, 315, 158, 155, 163, 320, 321, 322,
323, 324, 325, 326, 150, 164, 169, 327, 328, 329,
330, 331, 332, 333, 334, 335, 336, 337, 338, 339,
340, 341, 342, 343, 344, 345, 346
};
#endif
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code
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We offer Cloud Hosting services with guaranteed speeds and uptime underpinned by a Service Level Agreement (SLA). We have a lot of experience hosting and managing complex, database-driven and ecommerce sites.
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english
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बैंकों के खिलाफ सबसे ज्यादा शिकायतें नई दिल्ली में बैंकों के खिलाफ सबसे ज्यादा शिकायतें नई दिल्ली में - छत्तीसगढ़ विकास्पीडिया
होमैन्यूज फीदबैंकों के खिलाफ सबसे ज्यादा शिकायतें नई दिल्ली में न्यूज फीद
ग्राहकों को सेवा देने के मामले में बैंक फिसड्डी साबित हो रहे हैं। इसका अंदाजा सूचना के अधिकार के तहत मिली जानकारी से लगाया जा सकता है। बैंकों के खिलाफ सिर्फ एक जुलाई से २२ दिसंबर, २०१९ के बीच ग्राहकों ने रिजर्व बैंक के पास १.४६ लाख शिकायतें दर्ज कराई हैं। इस दौरान सर्वाधिक शिकायतें राष्ट्रीय राजधानी के नई दिल्ली जिले में दर्ज की गईं।
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hindi
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नेशनल पेमेंट्स कारपोरेशन ऑफ़ इंडिया (नप्सी) भर्ती २०१५-१० रिक्ति प्रबंधक / वरिष्ठ प्रबंधक ब.टेक/ब.ए,म.स्क,म्का
अक्टोबर १, 20१5 प्रिया नेशनल पेमेंट्स कारपोरेशन ऑफ़ इंडिया (एनपीसीआई) भर्ती अधिसूचना के माध्यम से प्रबंधक / वरिष्ठ प्रबंधक की भर्ती के लिए एक भर्ती अधिसूचना जारी की है । पूरा करने वाले उम्मीदवारों को ब.टेक/ब.ए,म.स्क,म्का नेशनल से नई भर्ती के लिए आवेदन कर सकते हैं पेमेंट्स कारपोरेशन ऑफ़ इंडिया (एनपीसीआई)।
योग्य उम्मीदवारों को या उससे पहले प्रबंधक / वरिष्ठ प्रबंधक नौकरी के लिए ऑनलाइन आवेदन कर सकते २८/१०/२०१५ । कैसे आदि लागू करने के लिए नीचे पाया जा सकता आयु सीमा, चयन प्रक्रिया, योग्यता, आवेदन शुल्क, जैसे रिक्ति के बारे में और अधिक जानकारी प्राप्त करें।
कंपनी का नाम : नेशनल पेमेंट्स कारपोरेशन ऑफ़ इंडिया (एनपीसीआई) रिक्ति का नाम : प्रबंधक / वरिष्ठ प्रबंधक शैक्षिक योग्यता : ब.टेक/ब.ए,म.स्क,म्का
रिक्ति की कुल संख्या : १० पद वेतन : एनए नौकरी स्थान : मुंबई की अंतिम तिथि को लागू करने के लिए इस काम के लिए २८/१०/२०१५: नौकरी रिक्ति के लिए पता : नेशनल पेमेंट्स कारपोरेशन ऑफ़ इंडिया, मुंबई
विस्तार नेशनल पेमेंट्स कारपोरेशन ऑफ़ इंडिया (एनपीसीआई) प्रबंधक / वरिष्ठ प्रबंधक भर्ती:
नौकरी विवरण: अंत का विस्तृत ज्ञान एक संदेश / दोहरी संदेश सिस्टम्स, संदेश संहिताओं और ऑनलाइन / ऑफलाइन प्राधिकरण के मापदंडों के ज्ञान सहित कार्ड भुगतान समाशोधन और निपटान, सुलह प्रक्रियाओं को समाप्त करने के लिए। खड़ी होकर प्रसंस्करण परीक्षण और पैरामीटर सेटअप का ज्ञान।
अन्य नेटवर्क के इंटरचेंज शुल्क गणना और इंटरचेंज शुल्क प्रसंस्करण संरचना प्रभावित करने वाले कारकों का विस्तृत ज्ञान। काम कर क्लियरिंग और निपटान, अनुपालन पर अनुभव और संपादन और पूर्वसंपादित प्रसंस्करण सहित पूरे इंटरचेंज प्रणाली की अस्वीकृति हैंडलिंग। चैनलों में नप्सिस कार्ड भुगतान नेटवर्क के सुचारू संचालन के लिए जिम्मेदार है। डेबिट / प्री के लेन-देन का प्रवाह और वापस कार्यालय के संचालन के अनुभव क्रेडिट कार्ड / इकॉम / मोबाइल / एटीएम / आईवीआर / भुगतान किया। निगरानी और उद्योग के उत्पाद के प्रदर्शन ज्ञान, बाजार के रुझान का मूल्यांकन, और प्राधिकरण शुरुआत पूरा लेन-देन चक्र के संबंध में विभिन्न प्रतियोगियों और लेन-देन के निपटारे।
आवश्यक ज्ञान: कार्ड भुगतान प्रोसेस.अबिलिटी का अच्छा ज्ञान विशेष रूप से कार्ड के क्षेत्र में, नियामक नियमों और भारत और प्रमुख बाजारों की अन्य नियामक दिशा-निर्देशों में कार्ड उद्योग से संबंधित नियमों के साथ कार्ड से भुगतान में मार्केट.फेमिलियार वरिष्ठ अधिकारियों के साथ संबंधों का प्रबंधन करने के लिए और व्यापारी सेवा फी.कम्पेटेसी उत्कृष्ट मौखिक और लिखित संचार, बातचीत, परियोजना प्रबंधन और मैट्रिक्स प्रबंधन कौशल बातचीत और आंतरिक और बाह्य व्यापार भागीदारों के साथ प्रभावी ढंग से समन्वय स्थापित करने के लिए महत्वपूर्ण क्षमता है। प्रभावी ढंग से प्रत्यक्ष और कई कार्यों का प्रबंधन करने की क्षमता। स्व-प्रेरित एक टीम के वातावरण में अच्छे लोगों को प्रबंधन कौशल, एक टीम का नेतृत्व करने की क्षमता संचालन करते हुए सफलता देने के एक सिद्ध ट्रैक रिकॉर्ड के साथ
इच्छित प्रोफ़ाइल शिक्षा: स्नातकीय -ब.टेक/बे कंप्यूटर, इलेक्ट्रॉनिक्स / दूरसंचार पीजी म.स्क कंप्यूटर, इलेक्ट्रॉनिक्स, एमसीए कम्प्यूटर डॉक्टरेट अन्य डॉक्टरेट
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hindi
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शिर्डी विच रहन वालेया.....
अक्ख लाई मैं जदों दी तेरे नाल, अक्ख लावां अक्ख न लग्गे
गल मुक्की न सज्जन (साईं) नाल मेरी, रब्बा तेरी रात मुक्क गई
मेरी खुल गई पटक दे के अक्ख नी, गली दे विचों कौन लंगेया
ऐवें खुली नहीं पटक देके अक्ख नी, गली चों मेरा साईं लंगेया
सानू भुल गई ख़ुदाई साईंयां सारी, तेरे नहीं ख्याल भुल्दे
आप यार (ईश्वर) बिना नहीं रहन्दा, लोकां नू कहन्दा चन्गी गल नहीं
वादा कर के सजन (साईं) नहीं आया, उडीकाँ विच रात लंग गई
राँझा (साईं) चौधवीं दे चन्न नालों सोहना, मायें की एहनु झात न लवीं
मायें की कराँ मैं तेरा साडा खैड़ा, राँझा ते मेरा रब्ब वरगा
दोगे दर्शन इस आशा में, जीवन ज्योति जले
अधरों पर है नाम तुम्हारा, हरपल मैंने तुमको पुकारा
दुनिया के इस भवसागर से, तुम बिन कैसे पाऊँ किनारा
देखेंगे हम राह तुम्हारी, जब तक यह साँस चले
मेरा तन-मन मेरा जीवन, कर दिया मैंने तुमको समर्पण
नाथ तुम्हारी शरण में आये, तोड़ के सारे मोह के बन्धन
मिल जाओ तो जन्म- मरण से, मुझको मुक्ति मिले
यमदूत को है देख मेरे प्राण ये घबराये
निकले मेरे मुख से साईं तेरा ही शुभ नाम
ममता का न हो बन्धन माया का नहीं हो दर्पण
आँखों में छवि हो तेरी अधरों पे साईं सुमिरन
तेरे ध्यान में हो मेरे जीवन की यह शाम
मौत का हो डर न पीड़ा कष्ट को हरना
आजाना बन के माझी पार भव से करना
जन्म मरण का कोई फिर रहे न बाबा काम
इक भक्त की अभिलाषा मनोभाव की ये भाषा
कर के कृपा हे साईं कर देना पूरी आशा
तन त्याग मैं जाऊँ अपने ही गुरु के धाम
तेरी शरण में आया हूँ साईं,
मुझको अब तुम दे दो सहारा
छोड़ न देना बीच भवर में,
दिखला दो न मुझको किनारा
राह में तेरी पलकें बिछाए,
बैठा हूँ कब से आस लगाए
रहता है यह दिल बेचैन सा हरपल,
हरपल तुम्हारी याद सताए
आजा ओ तुमको पुकारूँ,
सांई भूल गए क्यूँ रस्ता हमारा
तुमको पुकारे रस्ता निहारूँ,
भूल गए क्यूँ रस्ता हमारा
भटक रहा था तेरे बिना मैं,
अब जो मिले हो तो जाने न दूँगा
तुमको रखूँगा दिल में छुपा के,
तेरी धुन में मैं खोता रहूँगा
साईं सँग मेरे रहना हर पल हर क्षण,
थामे ही रखना हाथ हमारा
भक्ति तुम्हारी तुम मुझे दे दो,
मुक्ति जहाँ से मुझे मिल जायेगी
खिल जायेगी हर कली दिल की,
इच्छा न कोई रह जायेगी
सुन लो न साईं श्रद्धा सुमन और,
सबुरी के गुण से भर दो न यह दामन हमारा
साँई है जीवन, जीवन शिर्डी के साँई
साँई मेरा जीवन सहारा
तेरे बिना साँई सब है अन्धेरा
पार करो मेरी जीवन नईयाँ
चरण लगा लो मुझे साँई कन्हैया
साँई हमारा हम साँई के
ऐसा प्रेम हमारा
साँई राम हमारा
शिर्डी साँई है नाम तुम्हारा
शिर्डी साँई अवतारा
हिन्दू मुस्लिम सिक्ख इसाई
शिर्डी साँई महादेवा
हे साँई महादेवा
निरुपम गुण है साधना साँई
तेरे नीरज नयना
विभुती सुन्दर हे साँई बाबा
महेश्वरा हे साँई बाबा
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hindi
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गुस्ताख़: पढ़ाई लिखाई, हाय रब्बा
पढ़ाई लिखाई, हाय रब्बा
मेरे सुपुत्र का एडमिशन स्कूल में हो गया। यह एडमिशन का मिशन मेरे लिए कितना कष्टकारी रहा, वह सिर्फ मैं जानता हूं। इसलिए नहीं कि बेटे का दाखिला नहीं हो रहा था..बल्कि इसलिए क्योंकि एडमिशन से पहले इंटरव्यू की कवायद में मुझे फिर से उन अंग्रेजी कविताओं-राइम्स-की किताबें पढ़नीं पड़ी, जो हमारे टाइम्स (राइम्स से रिद्म मिलाने के गर्ज से, पढ़ें वक्त) में हमने कभी सुनी भी न थीं।
अंग्रेजी स्कूलों से निकले हमारे साथी जरुर -बा बा ब्लैक शीप, हम्प्टी-डम्प्टी और चब्बी चिक्स जानते होंगे, लेकिन बिहार और बाकी के राज्य सरकारों के स्टेट बोर्ड से निकले मित्र जरुर इस बात से सहमत होंगे कि घरेलू स्तर पर छोड़कर औपचारिक रुप से अंग्रेजी से मुठभेड़ छठी कक्षा में हुआ करती थी।
बहरहाल, शिक्षा के उस वक्त की कमियों की ओर इशारा करना मेरा उद्देश्य नहीं। केजी के मेरे सुपुत्र की किताबों की कीमत ढाई से तीन हजार के आसपास रहने वाली है। मुझे कुछ-कुछ अंदाजा तो था लेकिन सिर्फ किताबों की कीमत इतनी रहने वाली है इस पर मैं श्योर नहीं था।
मुझे अपना वक्त याद आय़ा। जब मैं अपने ज़माने की -हालांकि हमारा ज़माना इतना पीछे नहीं है, सिर्फ अस्सी के दशक के मध्य के बरसों की बात है-की बात करता हूं तो मुझे लगता है कि हम न जाने कितनी दूर चले आए हैं। बहुत सी बातें एकदम से बदल गईं हैं। हालांकि, प्रायः सारे बदलाव सकारात्मक से लगते हैं। लेकिन पढाई के बारे में ऐसा ही कहना, कम से कम पढाई की लागत के बारे में...हम स्वागतयोग्य तो नहीं ही मान सकते।
मेरा पहला स्कूल सरस्वती शिशु मंदिर था। कस्बे के कई स्कूलों को आजमाने के बाद तब के दूसरे सबसे अच्छे स्कूल (संसाधनों के लिहाज से) शिशु मंदिर ही था। हमारे कस्बे का सबसे बेहतर स्कूल कॉर्मेल कॉन्वेंट माना जाता था...अंग्रेजी माध्यम का। पूरे कस्बे में इसमें बच्चे का दाखिला गौरव की बात मानी जाती है, हालांकि जिनके बच्चों का दाखिला इस स्कूल में नहीं हो पाता, वो यह कह कर खुद को दिलासा देते कि स्कूल नहीं पढ़ता बच्चे पढ़ते हैं। और इसकी तैयारी हम घर पर ही करवाएंगे अच्छे से।
तैयारी तो खैर क्या होती होगी। हमारा दाखिला सरस्वती शिशु मंदिर में हुआ, दाखिले की फीस थी ४० रुपये और मासिक शुल्क १५ रुपये। यह सन ८४ की बात होगी। इसमें यूनिफॉर्म था। नीली पैंट सफेद शर्ट...लाल स्वेटर। बस्ता जरुरी था और टिफिन भी ले जाना होता, जो प्रधान जी के मूड के लिहाज से लंबे या छोटे वाले भोजन मंत्र के बाद खाया जाता। भोजन के पहले मंत्रो की इस अनिवार्यता ने ही नास्तिकता के बीज बो दिए थे। चार साल उसमें पढ़ने के बाद जब फीस बढ़कर २५ रुपये हो गया, और तब घरवालों को लगा कि यह शुल्क ज्यादा है।
भैया की नौकरी लग चुकी थी लेकिन उनका वेतन उतना नहीं थी कि हमारे इस मंहगे पढ़ाई का खर्च उठा पाते। सरस्वती शिशु मंदिर से इस नास्तिक को निकाल कर तिलक विद्यालय में भर्ती कराया गया, जिसे राज्य सरकार चलाती थी और यह मशहूर था कि गांधी जी उस स्कूल में आए थे। गांधी जी की वजह से पूरे कस्बे में यह स्कूल गांधी स्कूल भी कहा जाता। एडमिशन फीस ५ रुपये, और सालाना शुल्क १२ रुपये।
गांधी स्कलू की खासियत थी कि हम १० बजे स्कूल में प्रार्थना करने के बाद, जो कि हमारे लिए बदमाशियों का सबसे टीआरपी वक्त होता था...सफाई के लिए मैदान में इकट्ठे होते थे। मैदान में पत्ते और कागज चुनने के बाद, क्लास के फर्श की सफाई का काम होता। लाल रंग के उस ब्रिटिश जमाने के फर्श पर ही बैठना होता था इसलिए सफाई जरुरी थी। यह काम रोल नंबर के लिहाज से बंधा होता।
उस वक्त भी, जो शायद १९८७ का साल था, किताबों की कीमत हमारी ज़द में हुआ करती थी। पांचवी क्लास में ६ रुपये ८० पैसे की विज्ञान की किताब सबसे मंहगी किताब की कीमत थी। बिहार टेक्स्टबुक पब्लिशिंग कॉरपोरेशन किताबें छापा करती थी, सबसे सस्ती थी संस्कृत की किताब और सबसे मंहगी विज्ञान की।
उसमें भी घरवालों की कोशिश रहती कि किसी पुराने छात्र से किताबें सेंकेंडहैंड दिलवा दी जाएँ। आधी कीमत पर। किताब कॉपियां हाथों में ले जाते. सस्ते पेन...बॉल पॉइंट में भी कई स्तर के...३५ पैसे वाले मोटी लिखाई के बॉल पॉइंट रिफिल से लेकर ७५ पैसे में पतले लिखे जाने वाले बॉल पॉइंट पेन तक।
आज वैशाली में ही रहता हूं, बेटे के स्कूल में कंप्यूटरों की भरमार देखकर आया हूं, फीस की रकम देखी...पढाई मंहगी हो गई है या वक्त का तकाजा है...या लोग संपन्न हो गए है या पढाई सुधर गई है...कस्बे और शहर का अंतर....वही सोच रहा हूं। सरकारी स्कूलों में पड़कर और जिंदगी के ढेर सारे साल गरीबी में बिताकर हमने खोया है या पाया है...
लेबल्स: अंतर्मन, नॉस्टेल्जिया, मेरा मधुपुर
आपका यह लेख कल के "चर्चामंच" पे लगाया जा रहा है ,
ओह मंजीत...पुरानी यादें ताजा कर दी। हमारी कहानी भी बिल्कुल यहीं है सिवाय सुपुत्र के। पीछे मुड़कर देखता हूं तो लगता है कि हमने खोया नहीं है पाया ही है। इस पर विवेचना फिर कभी। फिलहाल, पोस्ट के लिए साधुवाद।
आज कल बच्चों को पढ़ना बहुत मंहगा होता जा रहा है ... १९७३-७४ में जब मैंने एम॰ ए ॰ किया था तो अर्थशास्त्र की एक पुस्तक २० रुपये की थी जिसको खरीदने के लिए कई दिन तक विचार किया ... और बाद में पुस्तकालय जा कर नोट्स बनाए ... किताब आखिर तक नहीं ही खरीदी गयी :):)
जमाना बदल गया है सर जी...अब स्कूल, अस्पताल मिशन वाली जगह नहीं रहे.कमर्शियलाइज्ड हो गये हैं..
बढि़या प्रस्तुति, अगले किश्त का इंतजार है.
..भावनाओं के पंख लगा ... तोड़ लाना चाँद नयी पुरानी हलचल में .
मेरी टिप्पणी शायद स्पैम में चली गयी है ...
आह...क्या दिन याद करवा दिए आपने, हमने पच्छीस रुपये महीना दे कर रीजनल इंजीनियरिंग कालेज में पढाई की है...याद है जब हिंद पाकेट बुक्स वाले एक रुपये में अपनी किताब दिया करते थे...उनकी घरेलु लाइब्रेरी योजना में दस रुपये में सात किताबें पोस्ट से आया करती थीं...वो भी क्या दिन थे रे...
बदला कुछ नहीं है बल्कि मैं तो कहूँगा कि शिक्षा का स्तर आज पहले से भी गिरा हुआ है।
आप जैसा बचपन हमारा भी बीता है, बच्चों की एक वर्ष की फीस में हमारी पूरी पढ़ाई हो गयी।
शिक्षा का स्तर अब तो सच में गिर गया है
सरकार ने बच्चो के लिये कई सुविधा निकाल दि है..
ओर शिक्षको के लिये मुसिबत
आज तो शिक्षक केवल सर्टिफिकेट बनाने मात्र रह गये है ..
थी, हूं, रहूंगी... वर्तिका नन्दा का कविता संग्रह
द पोएट चाइल्ड-जिजीविषा की गाथा
बौद्धिकता के बहाने
शहरयार का जाना..
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hindi
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> भाजपा, नीति और नियत
भाजपा, नीति और नियत
आज देश में जितनी भी राजनितिक पार्टिया है वो सभी सत्ता हासिल करने के लिए ही बनी है। भले ही उन्हें इसके लिए वाद-विवाद करना पड़े या सत्ता पर काबिज होने के लिए झगडा। मकसद एक ही है की किसी तरह सत्ता का स्वाद चखना है। फ़िर चाहे वो अपनी बात से मुकर जाए या उन्हें याद ही न हो की अभी कुछ दिनों पहले या चंद समय पहले उन्होंने क्या कहा था। लेकिन अगर कोई पत्रकार उनकी मंशा को भांप जाए और उनकी कथनी और करनी को लेकर कोई सवाल दाग दे तो गिरगिट की तरह ऐसे रंग बदल जाते है जैसे वो यह सब कुछ सत्ता हासिल करने के लिए नही बल्कि जनता के लिए ही कर रहे है। अब भाजपा को ही देख लो। कहने को तो यह पार्टी राष्ट्रीय है और देश की दूसरी सबसे बड़ी पार्टी होने के नाते इस समय विपक्ष में बैठी है। लेकिन इस समय भाजपा के पास जहा नेताओ की भारी कमी खल रही है वही लोकसभा चुनावो के बाद नियत भी साफ़ नजर नही आ रही है। यही कारण है की जहा लोकसभा चुनावो से पहले भाजपा क्षेत्रीय दलों को नकार रही थी वही लोकसभा चुनाव में उसने किस तरह इन दलों का साथ लिया वो सभी ने देखा। भाजपा और कांग्रेस अक्सर कहती आई है की देश में सिर्फ़ राष्ट्रीय पार्टिया ही होनी चाहिए लेकिन जिस तरह सत्ता सुख पाने के लिए ये राजनितिक दल अपनी नीति और नियत दिखाते आए है उससे आम आदमी अक्सर अपने को ठगा सा महसूस करता है। कांग्रेस और भाजपा जहा आज कई प्रदेशो में क्षेत्रीय दलों के साथ मिल कर सरकार चला रही है वो कबीले गौर है और हरियाणा में भाजपा ने विधानसभा चुनाव के लिए क्या-क्या समीकरण बिठाने की कोशिश की वो भी सभी को पता है। हरियाणा में अपने को उभारने के लिए भाजपा ने पहले ओमप्रकाश चौटाला नित इनलो से समझौता किया तो लोकसभा चुनावो में मुहं की खाने के बाद इनलो से दामन छुडाने के पश्चात् भजनलाल नित हरियाणा जनहित कांग्रेस का दामन थामने की कोशिशे की लेकिन दाल नही गली। इस पर अगर भाजपा नेता यह कहे की भाजपा चाहती है की देश में सिर्फ़ क्षेत्रीय दल ही राजनीति में रहे तो ऐसा लगता है की भाजपा मजाक कर रही है। अब हरियाणा भाजपा के सहप्रभारी हरजीत सिंह ग्रेवाल को ही देख लो। कहने को तो उनका नाम व् पद बहुत बड़ा है लेकिन आज वो यह नही समझ पाए की मिडिया के आगे वो क्या बोल गए। अपने संबोधन में उन्होंने कहा की हरियाणा में उनका मुकाबला सीधे कांग्रेस से है क्योंकि अब देश में क्षेत्रीय पार्टियों का कोई वजूद नही रह गया है। साथ ही वो बोल गए की प्रदेश में अब भाजपा को किसी गठबंधन की कोई आवश्यकता नही है। लेकिन जब मैंने इस बात पर उनसे पूछा की भाजपा प्रदेश विधानसभा चुनावो को लेकर किसी क्षेत्रीय दल से गठबंधन को लेकर सबसे ज्यादा उत्साहित थी तो गिरगिट की तरह रंग बदलते हुए वो बोले की वो इस लिए की कांग्रेस को सत्ता से दूर करना आज की जरुरत है। जबकि भाजपा ने जो प्रत्याशी आज मैदान में उतारे है वो सत्ता सुख के लिए नही अपितु जनता की सेवा के लिए चुनाव मैदान में है। जब उनसे पूछा गया की अगर भाजपा क्षेत्रीय पार्टियों को इतना ही अछूत मानती है तो कई प्रदेशो में आज भी भाजपा क्षेत्रीय दलों के साथ मिल कर सरकार चला रही है तो उन्होंने कहा की वो इसलिए की भाजपा ने सरकार में रहते हुए महंगाई को रोक कर रखा था लेकिन कांग्रेस ने जिस तरह महंगाई को बढ़ावा दिया है उससे आम आदमी का जीना दूभर हो गया है। इसलिए भाजपा चाहती है की किसी भी तरह सरकार बना कर कांग्रेस को सत्ता से दूर रखा जाए।तो दिखावा किस बात काअब आप स्वयं ही समझदार है की भाजपा नेता क्या बोलना चाहते है और क्या बोल रहे है। अब यह कौन जाने की भाजपा सत्ता सुख पाना चाहती है की महंगाई को दूर रखने के लिए कांग्रेस को सत्ता से बाहर। लेकिन इनको कौन समझाए की किसको सत्ता में रखना है और किसको नही यह तो मतदान के दिन विचार करके ही जनता मतदान करती है। लेकिन यह राजनीति है यहाँ सब कुछ दिखावे के लिए ही होता है जनता की किसको पड़ी है। क्योंकि सत्ता तो सत्ता ही है।
"इनको कौन समझाए की किसको सत्ता में रखना है और किसको नही यह तो मतदान के दिन विचार करके ही जनता मतदान करती है। लेकिन यह राजनीति है यहाँ सब कुछ दिखावे के लिए ही होता है जनता की किसको पड़ी है। क्योंकि सत्ता तो सत्ता ही है।"क्योंकि इन्होंने कानों में तेल डालकर ऊपर से रूई जो ठूँस रखी है।
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hindi
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इस संसार में उत्पन्न होने वाला प्रत्येक प्राणी अपने जीने के लिए निरंतर सुख-साधनों को जुटाने में लगा रहता है। मनुष्य अपने लिए न केवल आवास का ही निर्माण करता है बल्कि समाज एवं समाज में जीने योग्य नियमों को भी बनाता है जिससे आगे चलकर उसका जीवन सुखमय व्यतीत हो सके। व्यक्ति अपने अनुसार निवास स्थान बनाता है और उसमें निवास करके सुखमय जीवन व्यतीत करने की इच्छा रखता है। जिन निवास स्थानों से मनुष्यों के सुख-दुखों का जुड़ाव होता है उस निवास स्थान पर निर्माण होने के बाद वह मकान पूर्ण रूप से दोषरहित एवं उत्तम होना चाहिए। अतः इन सभी बातों पर ध्यान देते हुए ही हमारे प्राचीन ऋषि-मुनियों ने वास्तुशास्त्र (वस्तु शास्त्र) की रचना की जिसके आधार पर निर्माण कार्य किया जा सके और उसमें रहने वाले लोग सुखी एवं प्रसन्न रह सके।
भूमि की लंबाई को भवन के गर्भगृह के केन्द्रीय हिस्से के चौड़ाई से गुणा करने पर आयाम प्राप्त होता है। आयाम ही किसी भी भवन में रहने वालों के सुख, सौभाग्य, स्वास्थ्य तथा आयु पर अच्छा या बुरा असर डालता है।
आयाम की गणना भवन के सेन्ट्रल एरिया की लंबाई, चौड़ाई के गुणा से ज्ञात किया जा सकता है। लम्बाई व चौड़ाई के गुणन से प्राप्त मान को ९ से गुणा करके ८ से भाग देने पर प्राप्त संख्या १ से ८ का निम्नाकित भविष्य होगा।
एक ध्वजायाम परिवार में सुख, समृद्धि, घर में आर्थिक सम्पन्नता तथा खुशहाली लाने वाला
दो धूमायाम घर के पुरुष मुखिया को बीमारी तथा घर में भयावह गरीबी लाने वाला
तीन सिंहायाम शत्रुओं पर विजय देने वाली तथा भवन के निवासियों के लिए स्वास्थ्य, धन-धान्य तथा सम्पन्नता लाने वाला
चार स्वनायाम भवन में रहने वालों के लिए बीमारियां तथा अशुभ असर लाने वाला
पांच वृषभायाम भवन पर मां लक्ष्मी की साक्षात कृपास्वरूप, धन, सम्पन्नता तथा सौभाग्य देने वाला
छह खरायाम बुरा स्वास्थ्य, जीवन में आकस्मिक दुर्घटनाएं तथा अशुभता लाने वाला
सात गजायाम भवन में रहने वालों के लिए धन, स्वास्थ्य, फेम, बुद्धि तथा भाग्य बढ़ाने वाला
आठ काकायाम भवन के निवासियों के लिए दुखदायी, सभी लोगों की शांति, समृद्धि खत्म कर भयावह गरीबी और बीमारी लाने वाला
शुभ आयाम ध्वज, सिंह, वृषभ, गज
अशुभ आयाम धूम्र, स्वान, खर, काक
आयाम के द्वारा भवन या कमर्शियल स्थान के उपयोग का शुभाशुभ निर्णय :-
गोदाम, वेयरहाउस तथा कोल्ड-स्टोरेज गजायाम
दुकानें तथा कॉमर्शियल कॉम्पलेक्स
गजायाम अथवा सिंहायाम शुभ
ध्वजायाम औसत फल देने वाला
थिएटर, सिनेमा हॉल्स, रिसर्च लैब, सेंटर्स, स्कूल, कॉलेज आदि
जिम, क्लब हाउस, हॉस्टल आदि
गजायाम औसत फल देने वाला
न्यायालय भवन, सार्वजनिक स्थल, पंचायत भवन, विधानसभा आदि सिंहायाम
फैक्टरी तथा औद्योगिक भवन सिंहायाम अथवा ध्वजायाम
मिल्स, चीनी मील, चावल मिल आदि
ध्वजायाम, वृषभायाम शुभ
मैरिज हॉल वृषभायाम अथवा ध्वजायाम
धर्मशाला तथा लॉजिस्टिक सेंटर्स गजायाम सर्वश्रेष्ठ, वृषभायाम व ध्वजायाम औसत फल देने वाला
नोटः पहले फ्लोर (मंजिल) का आयाम पूरी तरह से वही होना चाहिए जोकि ग्राउंड फ्लोर का है। पहले फ्लोर की हाईट (ऊंचाई) भी ग्राउंड फ्लोर की ऊंचाई से ज्यादा नहीं होनी चाहिए।
अयादि की गणना करना
किसी भी भवन को बनाते समय उसमें आयाम के साथ-साथ कई अन्य बातों का भी ध्यान रखा जाता है जो उस भवन की शुभता तथा वास्तु अनुरूपता को बढ़ाकर उसे सब प्रकार से सौभाग्यशाली तथा मंगलदायी बनाती हैं।
प्लिंथ एरिया ( भूमि का या कंस्ट्रक्शन एरिया का क्षेत्रफल) को ८ से गुणा कर उसे १२ का भाग देने पर बचे शेष को धन (वेल्थ) कहते हैं।
प्लिंथ एरिया को ३ से गुणा कर उसमें ८ का भाग देने पर बचे शेष को ऋण (ऋण) कहते हैं।
प्लिंथ एरिया को ९ से गुणा कर उसमें ८ का भाग देने पर बचे शेष को आयाम (आयाम) कहते हैं।
प्लिंथ एरिया को ८ से गुणा कर उसमें ३० का भाग देने पर बचे शेष को तिथि (तिथि) कहते हैं।
प्लिंथ एरिया को ९ से गुणा कर उसमें ७ का भाग देने पर बचे शेष को दिन (दए) कहते हैं।
प्लिंथ एरिया को ८ से गुणा कर उसमें २७ का भाग देने पर बचे शेष को नक्षत्र (स्टार) कहते हैं।
प्लिंथ एरिया को ४ से गुणा कर उसमें २७ से भाग देने पर बचे शेष को योग (योगा) कहते हैं।
प्लिंथ एरिया को ५ से गुणा कर उसमें ११ से भाग देने पर बचे शेष को कर्ण (करना) कहते हैं।
प्लिंथ एरिया को ६ से गुणा कर उसमें ९ का भाग देने पर बचे शेष को (अम्सम) कहते हैं।
प्लिंथ एरिया को ९ से गुणा कर उसमें १२ से भाग देने पर बचे शेष को उस भवन की कुल उम्र (आगे ऑफ थे बिल्डिंग) माना जाता है।
नोट :- धन को सदैव ऋण से अधिक होना चाहिए।
१५ अमावस्या, ३० पूर्णिमा
१, ४, ८, ९, १४ अशुभ
२, ३, ५, ६, ७, १०, ११, 1२, 1३ शुभ
४, ५, ८, १२, १३, 1४, 1५, १७, २१, २३, 2४, २६, २७ सौभाग्यशाली
१५, १३, १, ९, १0, ११, २७, १7, १९ अशुभ
१, २, ३, ४, ५ शुभ फलदायी
१ हानि, २ इम्प्रुवमेंट, ३ धन, ४ चिंता, ५ मृत्यु का भय, ६ चोरी की आशंका, ७ परिवार में बढ़ोतरी, ८ हस्बैंडरी इम्प्रुवमेंट, ९ (अथवा ०) प्रसन्नता तथा सौभाग्यवर्धक
६० वर्ष से अधिक होने पर शुभ तथा इससे कम होने पर अशुभ माना जाता है।
विशेष :- यहां पर आयाम शब्द से तात्पर्य योनि से है। जबकि धन (वेल्थ) का अर्थ आय तथा ऋण का अर्थ व्यय से हैं।
|
hindi
|
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|
english
|
ساروی کھوتہٕ زیادٕ مشہور فلمبوئنٹ عمارتن منٛز چَھ اکھ Sainte-Chapelle de Vincennes (1370s) ،یتھ فرشہٕ پیٹھ چھتس تام شیشِک دیوارٕ چھ۔
|
kashmiri
|
package org.seqcode.data.readdb;
import java.net.*;
import java.util.*;
import java.io.*;
import java.nio.channels.*;
import javax.security.sasl.*;
import javax.security.auth.callback.*;
/**
* <p>API for remote access to the readdb server.
*
* <p>Calls throw IOException on network errors.
* Calls throw ClientException on other errors (authentication, authorization, invalid request ,etc.
*
* <p>Client generally assumes that the hit positions are the 5' end of the hit.
*
* <p>Client IS NOT REENTRANT. Do not overlap calls to a single Client object.
*
* <p>The current version of Client keeps a separate thread that would close the connection if it is idle too long.
*
* <p>Most method parameters that are object types (eg Integer, Boolean) are optional. If a null value
* is passed then no filtering is done based on that parameter.
* <p>Standard parameters shared across methods:
* <ul>
* <li> alignid is the name of the alignment.
* <li> isType2 specifies whether to work on ordinary single-ended reads (false) or the "type 2" reads (true).
* An example of type 2 reads are the R2 reads in paired-end ChIP-exo.
* <li> isPaired specifies whether to work on single-ended reads (false) or paired-end reads (true)
* <li> isLeft if isPaired is true, then isLeft specifies whether to work on the left read (true)
* or right read (false) of the pair
* <li> plusStrand specifies whether to return only reads on the plus strand (true) or minus strand (false). null
* means that reads on both strands should be returned.
* <li> minWeight specifies the minimum weight of reads (or read pairs) to be returned or included in
* the histogram
* <li> start, stop specify the lowest (inclusive) and highest (exclusive) coordinates of reads
* that should be included in the results
* </ul>
*
*
*/
public class Client implements ReadOnlyClient {
public static final String SaslMechanisms[] = {"CRAM-MD5","DIGEST-MD5"};
Socket socket; //the socket to talk to the server. outstream and instream are from the socket
OutputStream outstream;
BufferedInputStream instream;
Thread closeTimerThread=null; //checks time of last activity - closes connection if idle for too long
byte[] buffer; //temporary space for receiving data; contents not persistent between method calls
private static final int BUFFERLEN = 8192*20;
private final int socketLoadDataReadTimeout = 1000*60*8; //socket timeout in ms: set to 8 minutes because we should only be relying on the timeout to detect server shutdowns, and some uses of Client (e.g. loading a lot of reads to ReadDB) can take a long time on the Server.
private final int socketQueryReadTimeout = 60000; //socket timeout in ms for queries
private final int threadSleepTime = 30000; //check time of last activity thread sleep time in ms
private final int connectionIdleTimeLimit = 1000*60*10; //close idle connection after this time (10 minutes)
private long lastActivityTime; //Time of last activity is updated by each query
private boolean connectionOpen=false;
private Request request;
private boolean printErrors;
private String hostname, username, password;
private int portnum;
/** Connects to a ReadDB server on the specified host and port using the specified
* username and password.
* @throws IOException on network errors
* @throws ClientException if the client cannot authenticate to the server
*/
public Client (String hostname,
int portnum,
String username,
String passwd) throws IOException, ClientException {
init(hostname,portnum,username,passwd);
}
/**
* Creates the default connection
* as specified by ~/.readdb_passwd or a readdb_passwd found in the classpath
* Must have keys hostname, port, username, and passwd in a format
* that java.util.PropertyResourceBundle can read
*
* @throws IOException on network errors
* @throws ClientException if the client cannot authenticate to the server
*/
public Client() throws IOException, ClientException {
String homedir = System.getenv("HOME");
String basename = "readdb_passwd";
if (System.getenv("READDBROLE") != null) {
basename = System.getenv("READDBROLE") + basename;
}
String fname = homedir + "/." + basename;
File propfile = new File(fname);
PropertyResourceBundle bundle = null;
if (propfile.exists() && propfile.canRead()) {
bundle = new PropertyResourceBundle(new FileInputStream(propfile));
if (System.getenv("DEBUGPW") != null) {
System.err.println("Opening readdb properties from " + propfile);
}
} else {
ClassLoader cl = ClassLoader.getSystemClassLoader();
URL url = cl.getResource(basename);
if (System.getenv("DEBUGPW") != null) {
System.err.println("Opening readdb properties from " + url);
}
if (url != null) {
bundle = new PropertyResourceBundle(url.openStream());
} else {
throw new IOException("Can't read connection properties from " + url);
}
}
String hostname = bundle.getString("hostname");
String port = bundle.getString("port");
String username = bundle.getString("username");
String password = bundle.getString("passwd");
init(hostname, Integer.parseInt(port), username, password);
}
private void init(String hostname,
int portnum,
String username,
String passwd) throws IOException, ClientException {
this.hostname=hostname;
this.portnum=portnum;
this.username=username;
this.password = passwd;
if(closeTimerThread!=null && closeTimerThread.isAlive())
closeTimerThread.interrupt();
synchronized(this){
socket = new Socket(hostname,portnum);
socket.setTcpNoDelay(true);
socket.setSendBufferSize(BUFFERLEN);
socket.setReceiveBufferSize(BUFFERLEN);
/* linger = true, time = 0 means that the other side gets a reset if the
socket is closed on our end (eg, java exits). We turn this off just before
sending a "bye" to allow for a graceful shutdown. But the RST in
other cases lets the server figure out that we've disappeared
*/
socket.setSoLinger(true,0);
socket.setSoTimeout(socketQueryReadTimeout);
outstream = socket.getOutputStream();
outstream.flush();
instream = new BufferedInputStream(socket.getInputStream());
connectionOpen=true;
lastActivityTime = System.currentTimeMillis();
buffer = new byte[BUFFERLEN];
if (!authenticate(hostname,username,passwd)) {
throw new ClientException("Authentication Exception Failed");
}
request = new Request();
printErrors = false;
//Start a new check alive thread
closeTimerThread = new Thread(new ClientConnectionTimerThread(this));
closeTimerThread.start();
}
}
/**
* Re-connect to the ReadDB sever using the same settings
*/
public void reConnect(){
try {
init(hostname, portnum, username, password);
} catch (IOException e) {
e.printStackTrace();
} catch (ClientException e) {
e.printStackTrace();
}
}
/**
* Pings the ReadDB server,
* @return true if the server pongs
*/
public boolean connectionAlive(){
String response="";
try{
synchronized (this){
if(!connectionOpen)
reConnect();
request.clear();
request.type="ping";
sendString(request.toString());
response = readLine();
}
if (response.equals("pong")) {
return true;
} else {
return false;
}
}catch(IOException e){
//SocketException could be generated by a timeout
return false;
}
}
/**
* Return some basic information about the server
* @return string
*/
public String getServerInfo(){
return("ReadDB\t"+hostname+"\t"+portnum+"\t"+username);
}
/**
* Determines whether the client will print error messages to STDERR. Useful for debugging
* but may produce unwanted screen output.
*/
public void printErrors(boolean b) {printErrors = b;}
/**
* performs the SASL authentication exchange with the server. currently called by the constructor
*/
private boolean authenticate(String hostname, String username, String password) throws IOException {
SaslClient sasl = null;
try {
sendString(username + "\n");
Map<String,String> props = new HashMap<String,String>();
props.put("Sasl.POLICY_NOPLAINTEXT","true");
props.put("Sasl.POLICY_NOANONYMOUS","true");
sasl = Sasl.createSaslClient(SaslMechanisms,
username,
"readdb",
hostname,
props,
new ClientCallbackHandler(username,password));
if (sasl == null) { return false; }
byte[] response = (sasl.hasInitialResponse() ? sasl.evaluateChallenge(new byte[0]) : new byte[0]);
byte continueByte = 1;
while (continueByte != 0) {
outstream.write((response.length + "\n").getBytes());
outstream.write(response);
outstream.flush();
int length = Integer.parseInt(readLine());
byte[] challenge = new byte[length];
int read = 0;
while (read < length) {
read += instream.read(challenge, read, length - read);
}
/* the continueByte tells us whether the server
expects to do another round. Necessary because sometimes
isComplete() returned true here but the server wasn't done
*/
continueByte = (byte)instream.read();
if (!sasl.isComplete()) {
response = sasl.evaluateChallenge(challenge);
} else {
response = new byte[0];
}
}
sasl.dispose();
String status = readLine();
return (status.equals("authenticated as " + username));
} catch (SaslException e) {
e.printStackTrace();
if (sasl != null) {
sasl.dispose();
}
return false;
}
}
/** sends a string to the server and flushes the socket
*/
private void sendString(String s) throws IOException {
outstream.write(s.getBytes());
outstream.flush();
lastActivityTime = System.currentTimeMillis();
}
/** reads one line from the server. blocking.
*/
private String readLine() throws IOException {
int pos = 0;
int i;
while ((i = instream.read()) != -1) {
if (i == '\n') {
break;
} else {
buffer[pos++] = (byte)i;
}
}
String out = new String(buffer,0,pos);
//System.err.println("READ " + out);
lastActivityTime = System.currentTimeMillis();
return out;
}
/**
* Tells the server to shut itself down. Use this to stop the server process.
* @throws IOException on network errors
* @throws ClientException if the user isn't authorized to shut the server down.
*/
public void shutdown() throws IOException, ClientException{
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type = "shutdown";
sendString(request.toString());
}
}
/** this was to fix a bug in the server. You shouldn't need it for general use.
* Regenerate the index for this alignment and chromosome
*/
public void reIndex(String align, int chrom, boolean isType2, boolean paired) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type = "reindex";
request.alignid = align;
request.chromid = chrom;
request.isType2 = isType2;
request.isPaired = paired;
sendString(request.toString());
outstream.flush();
String response = readLine();
if (!response.equals("OK")) {
System.out.println(response);
}
}
}
/** This was to fix a bug in the server. You shouldn't need it for general use.
* Resort the hits for a single-ended alignment and regenerate the index.
*/
public void checksort(String align, int chrom) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type = "checksort";
request.alignid = align;
request.chromid = chrom;
sendString(request.toString());
outstream.flush();
String response = readLine();
if (!response.equals("OK")) {
System.out.println(response);
}
}
}
/**
* Stores a set of SingleHit objects (representing an un-paired or single-ended read
* aligned to a genome) in the specified alignment. The hits are appended
* to any hits that have already been stored in the alignment.
*/
public void storeSingle(String alignid, List<SingleHit> allhits, boolean isType2) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
socket.setSoTimeout(socketLoadDataReadTimeout);
}
int step = 10000000;
for (int pos = 0; pos < allhits.size(); pos += step) {
Map<Integer, List<SingleHit>> map = new HashMap<Integer,List<SingleHit>>();
for (int i = pos; i < pos + step && i < allhits.size(); i++) {
SingleHit h = allhits.get(i);
if (!map.containsKey(h.chrom)) {
map.put(h.chrom, new ArrayList<SingleHit>());
}
map.get(h.chrom).add(h);
}
for (int chromid : map.keySet()) {
synchronized(this){
List<SingleHit> hits = map.get(chromid);
Collections.sort(hits);
int chunk = step;
for (int startindex = 0; startindex < hits.size(); startindex += chunk) {
int count = ((startindex + chunk) < hits.size()) ? chunk : (hits.size() - startindex);
request.clear();
request.type="storesingle";
request.alignid=alignid;
request.chromid = chromid;
request.isType2 = isType2;
request.map.put("numhits",Integer.toString(count));
try{
sendString(request.toString());
String response = readLine();
if (!response.equals("OK")) {
System.err.println("not-OK response to request: " + response);
System.err.println("request was " + request);
throw new ClientException(response);
}
int[] ints = new int[count];
for (int i = startindex; i < startindex + count; i++) {
ints[i - startindex] = hits.get(i).pos;
}
Bits.sendInts(ints, outstream,buffer);
float[] floats = new float[count];
for (int i = startindex; i < startindex + count; i++) {
floats[i - startindex] = hits.get(i).weight;
ints[i - startindex] = Hits.makeLAS(hits.get(i).length, hits.get(i).strand);
}
Bits.sendFloats(floats, outstream,buffer);
Bits.sendInts(ints, outstream,buffer);
System.err.println("Sent " + count + " hits to the server for " + chromid + "," + alignid);
outstream.flush();
response = readLine();
if (!response.equals("OK")) {
throw new ClientException(response);
}
}catch (IOException ioe){
//IOException here is probably a socket time-out.
//I think it's best to kill the process at this point, since we won't know if the sent reads actually got loaded.
System.err.println(ioe);
System.exit(1);
}
}
}
}
}
socket.setSoTimeout(socketQueryReadTimeout);
}
/**
* Stores a set of PairedHit objects (representing an paired-ended read
* aligned to a genome) in the specified alignment. The hits are appended
* to any hits that have already been stored in the alignment
*/
public void storePaired(String alignid, List<PairedHit> allhits) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
socket.setSoTimeout(socketLoadDataReadTimeout);
}
Map<Integer, List<PairedHit>> map = new HashMap<Integer,List<PairedHit>>();
for (PairedHit h : allhits) {
if (!map.containsKey(h.leftChrom)) {
map.put(h.leftChrom, new ArrayList<PairedHit>());
}
map.get(h.leftChrom).add(h);
}
for (int chromid : map.keySet()) {
synchronized(this){
List<PairedHit> hits = map.get(chromid);
System.err.println("SENDING PAIRED HITS n="+hits.size() + " for chrom " + chromid);
int chunk = 1000000;
for (int startindex = 0; startindex < hits.size(); startindex += chunk) {
int count = ((startindex + chunk) < hits.size()) ? chunk : (hits.size() - startindex);
request.clear();
request.type="storepaired";
request.alignid=alignid;
request.chromid=chromid;
request.isLeft=true;
request.map.put("numhits",Integer.toString(count));
try{
sendString(request.toString());
String response = readLine();
if (!response.equals("OK")) {
System.err.println("not-OK response to request: " + response);
System.err.println("request was " + request);
throw new ClientException(response);
}
int[] ints = new int[count];
for (int i = startindex; i < startindex + count; i++) {
ints[i-startindex] = hits.get(i).leftPos;
}
Bits.sendInts(ints, outstream,buffer);
float[] floats = new float[count];
int[] codes = new int[count];
for (int i = startindex; i < startindex + count; i++) {
floats[i-startindex] = hits.get(i).weight;
codes[i-startindex] = hits.get(i).pairCode;
ints[i-startindex] = Hits.makeLAS(hits.get(i).leftLength, hits.get(i).leftStrand,
hits.get(i).rightLength, hits.get(i).rightStrand);
}
Bits.sendFloats(floats, outstream,buffer);
Bits.sendInts(codes, outstream,buffer);
Bits.sendInts(ints, outstream,buffer);
for (int i = startindex; i < startindex + count; i++) {
ints[i-startindex] = hits.get(i).rightChrom;
}
Bits.sendInts(ints, outstream,buffer);
for (int i = startindex; i < startindex + count; i++) {
ints[i-startindex] = hits.get(i).rightPos;
}
Bits.sendInts(ints, outstream,buffer);
System.err.println("Sent " + count + " hits to the server");
outstream.flush();
response = readLine();
if (!response.equals("OK")) {
if (printErrors) {
System.err.println("not-OK response to request: " + response);
System.err.println("request was " + request);
}
throw new ClientException(response);
}
}catch (IOException ioe){
//IOException here is probably a socket time-out.
//I think it's best to kill the process at this point, since we won't know if the sent reads actually got loaded.
System.err.println(ioe);
System.exit(1);
}
}
}
}
socket.setSoTimeout(socketQueryReadTimeout);
}
/** Returns true if the alignment and chromosome exist and are accessible
* to the user. Returns false if they don't exist or if they aren't accessible
*/
public boolean exists(String alignid) throws IOException {
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type="exists";
request.alignid=alignid;
sendString(request.toString());
String response = readLine();
if (response.equals("exists")) {
return true;
} else if (response.equals("unknown")) {
return false;
} else {
return false;
}
}
}
/**
* Deletes an alignment (all chromosomes). isPaired specifies whether to delete
* the paired or single ended reads.
*/
public void deleteAlignment(String alignid, boolean isPaired) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type="deletealign";
request.isPaired = isPaired;
request.alignid=alignid;
sendString(request.toString());
String response = readLine();
if (!response.equals("OK")) {
if (printErrors) {
System.err.println("not-OK response to request: " + response);
System.err.println("request was " + request);
}
throw new ClientException(response);
}
}
}
/**
* Returns the set of chromosomes that exist for this alignment.
*/
public Set<Integer> getChroms(String alignid, boolean isType2, boolean isPaired, Boolean isLeft) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type="getchroms";
request.isType2 = isType2;
request.isLeft = isLeft;
request.isPaired = isPaired;
request.alignid=alignid;
sendString(request.toString());
String response = readLine();
if (!response.equals("OK")) {
if (printErrors) {
System.err.println("not-OK response to request: " + response);
System.err.println("request was " + request);
}
throw new ClientException(response);
}
int numchroms = Integer.parseInt(readLine());
Set<Integer> output = new HashSet<Integer>();
while (numchroms-- > 0) {
output.add(Integer.parseInt(readLine()));
}
return output;
}
}
/**
* Returns the total number of hits in this alignment.
*/
public int getCount(String alignid, boolean isType2, boolean isPaired, Boolean isLeft, Boolean plusStrand) throws IOException, ClientException {
int count = 0;
for (int c : getChroms(alignid, isType2, isPaired, isLeft)) {
count += getCount(alignid, c, isType2, isPaired, null,null,null,isLeft,plusStrand);
}
return count;
}
/**
* Returns the sum of the weights of all hits in this alignment
*/
public double getWeight(String alignid, boolean isType2, boolean isPaired, Boolean isLeft, Boolean plusStrand) throws IOException, ClientException {
double total = 0;
for (int c : getChroms(alignid, isType2, isPaired, isLeft)) {
total += getWeight(alignid, c, isType2, isPaired, null, null, null, isLeft, plusStrand);
}
return total;
}
/**
* Returns the total number of unique positions in this alignment.
*/
public int getNumPositions(String alignid, boolean isType2, boolean isPaired, Boolean isLeft, Boolean plusStrand) throws IOException, ClientException {
int pos = 0;
for (int c : getChroms(alignid, isType2, isPaired, isLeft)) {
pos += getNumPositions(alignid, c, isType2, isPaired, null,null,null,isLeft,plusStrand);
}
return pos;
}
/**
* Returns the total number of unique paired positions in this alignment.
*/
public int getNumPairedPositions(String alignid, boolean isType2, Boolean isLeft) throws IOException, ClientException {
int pos = 0;
for (int c : getChroms(alignid, isType2, true, isLeft)) {
pos += getNumPairedPositions(alignid, c, isType2, null,null,null,isLeft);
}
return pos;
}
/** returns the total number of hits on the specified chromosome in the alignment.
* Any of the object parameters can be set to null to specify "no value"
*/
public int getCount(String alignid, int chromid, boolean isType2, boolean paired, Integer start, Integer stop, Float minWeight, Boolean isLeft, Boolean plusStrand) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type="count";
request.alignid=alignid;
request.chromid=chromid;
request.start = start;
request.end = stop;
request.minWeight = minWeight;
request.isType2 = isType2;
request.isPlusStrand = plusStrand;
request.isPaired = paired;
request.isLeft = isLeft == null ? true : isLeft;
sendString(request.toString());
String response = readLine();
if (!response.equals("OK")) {
if (printErrors) {
System.err.println("not-OK response to request: " + response);
System.err.println("request was " + request);
}
throw new ClientException(response);
}
int numhits = Integer.parseInt(readLine());
return numhits;
}
}
/** returns the total weight on the specified chromosome in this alignment
*/
public double getWeight(String alignid, int chromid, boolean isType2, boolean paired, Integer start, Integer stop, Float minWeight, Boolean isLeft, Boolean plusStrand) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type="weight";
request.alignid=alignid;
request.chromid=chromid;
request.start = start;
request.end = stop;
request.minWeight = minWeight;
request.isType2 = isType2;
request.isPlusStrand = plusStrand;
request.isPaired = paired;
request.isLeft = isLeft == null ? true : isLeft;
sendString(request.toString());
String response = readLine();
if (!response.equals("OK")) {
if (printErrors) {
System.err.println("not-OK response to request: " + response);
System.err.println("request was " + request);
}
throw new ClientException(response);
}
return Double.parseDouble(readLine());
}
}
/** returns the total number of unique positions on the specified chromosome in the alignment.
* Any of the object parameters can be set to null to specify "no value"
*/
public int getNumPositions(String alignid, int chromid, boolean isType2, boolean paired, Integer start, Integer stop, Float minWeight, Boolean isLeft, Boolean plusStrand) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type="numpositions";
request.alignid=alignid;
request.chromid=chromid;
request.start = start;
request.end = stop;
request.minWeight = minWeight;
request.isType2 = isType2;
request.isPlusStrand = plusStrand;
request.isPaired = paired;
request.isLeft = isLeft == null ? true : isLeft;
sendString(request.toString());
String response = readLine();
if (!response.equals("OK")) {
if (printErrors) {
System.err.println("not-OK response to request: " + response);
System.err.println("request was " + request);
}
throw new ClientException(response);
}
int numpos = Integer.parseInt(readLine());
return numpos;
}
}
/** returns the total number of unique paired positions on the specified chromosome in the alignment.
* Note that total number of unique paired positions will depend on the setting of "isLeft":
* e.g. will count pair positions with left read in specified chromosome.
* Any of the object parameters can be set to null to specify "no value"
*/
public int getNumPairedPositions(String alignid, int chromid, boolean isType2, Integer start, Integer stop, Float minWeight, Boolean isLeft) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type="numpairpositions";
request.alignid=alignid;
request.chromid=chromid;
request.start = start;
request.end = stop;
request.minWeight = minWeight;
request.isType2 = isType2;
request.isPlusStrand = null;
request.isPaired = true;
request.isLeft = isLeft == null ? true : isLeft;
sendString(request.toString());
String response = readLine();
if (!response.equals("OK")) {
if (printErrors) {
System.err.println("not-OK response to request: " + response);
System.err.println("request was " + request);
}
throw new ClientException(response);
}
int numpos = Integer.parseInt(readLine());
return numpos;
}
}
/**
* returns the sorted (ascending order) hit positions in the specified range of a chromosome,alignment pair.
*/
public int[] getPositions(String alignid, int chromid, boolean isType2, boolean paired, Integer start, Integer stop, Float minWeight, Boolean isLeft, Boolean plusStrand) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type="gethits";
request.alignid=alignid;
request.chromid=chromid;
request.start = start;
request.end = stop;
request.minWeight = minWeight;
request.isType2 = isType2;
request.isPlusStrand = plusStrand;
request.isPaired = paired;
request.isLeft = isLeft;
request.map.put("wantpositions","1");
sendString(request.toString());
String response = readLine();
if (!response.equals("OK")) {
if (printErrors) {
System.err.println("not-OK response to request: " + response);
System.err.println("request was " + request);
}
throw new ClientException(response);
}
int numhits = Integer.parseInt(readLine());
return Bits.readInts(numhits, instream, buffer);
}
}
/**
* returns the hit weights in the specified range of a chromosome,alignment pair. The weights
* will be in the same order as the sorted positions returned by getPositions()
*/
public float[] getWeightsRange(String alignid, int chromid, boolean isType2, boolean paired, Integer start, Integer stop, Float minWeight, Boolean isLeft, Boolean plusStrand) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type="gethits";
request.alignid=alignid;
request.chromid=chromid;
request.start = start;
request.end = stop;
request.minWeight = minWeight;
request.isType2 = isType2;
request.isPlusStrand = plusStrand;
request.isPaired = paired;
request.isLeft = isLeft;
request.map.put("wantweights","1");
sendString(request.toString());
String response = readLine();
if (!response.equals("OK")) {
if (printErrors) {
System.err.println("not-OK response to request: " + response);
System.err.println("request was " + request);
}
throw new ClientException(response);
}
int numhits = Integer.parseInt(readLine());
return Bits.readFloats(numhits, instream, buffer);
}
}
public List<SingleHit> getSingleHits(String alignid, int chromid, boolean isType2, Integer start, Integer stop, Float minWeight, Boolean plusStrand) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type="gethits";
request.alignid=alignid;
request.chromid=chromid;
request.start = start;
request.end = stop;
request.minWeight = minWeight;
request.isType2 = isType2;
request.isPlusStrand = plusStrand;
request.isPaired = false;
request.map.put("wantpositions","1");
request.map.put("wantweights","1");
request.map.put("wantlengthsandstrands","1");
sendString(request.toString());
String response = readLine();
if (!response.equals("OK")) {
if (printErrors) {
System.err.println("not-OK response to request: " + response);
System.err.println("request was " + request);
}
throw new ClientException(response);
}
List<SingleHit> output = new ArrayList<SingleHit>();
int numhits = Integer.parseInt(readLine());
for (int i = 0; i < numhits; i++) {
output.add(new SingleHit(chromid,0,(float)0.0,false,(short)0));
}
IntBP ints = new IntBP(numhits);
ReadableByteChannel rbc = Channels.newChannel(instream);
Bits.readBytes(ints.bb, rbc);
for (int i = 0; i < numhits; i++) {
output.get(i).pos = ints.get(i);
}
FloatBP floats = new FloatBP(numhits);
Bits.readBytes(floats.bb, rbc);
for (int i = 0; i < numhits; i++) {
output.get(i).weight = floats.get(i);
}
Bits.readBytes(ints.bb, rbc);
for (int i = 0; i < numhits; i++) {
int j = ints.get(i);
SingleHit h = output.get(i);
h.length = Hits.getLengthOne(j);
h.strand = Hits.getStrandOne(j);
}
return output;
}
}
public List<PairedHit> getPairedHits(String alignid, int chromid, boolean isLeft, Integer start, Integer stop, Float minWeight, Boolean plusStrand) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type="gethits";
request.alignid=alignid;
request.chromid=chromid;
request.start = start;
request.end = stop;
request.minWeight = minWeight;
request.isPlusStrand = plusStrand;
request.isLeft = isLeft;
request.isType2 = false;
request.isPaired = true;
request.map.put("wantpositions","1");
request.map.put("wantweights","1");
request.map.put("wantpaircodes","1");
request.map.put("wantlengthsandstrands","1");
request.map.put("wantotherchroms","1");
request.map.put("wantotherpositions","1");
sendString(request.toString());
String response = readLine();
if (!response.equals("OK")) {
if (printErrors) {
System.err.println("not-OK response to request: " + response);
System.err.println("request was " + request);
}
throw new ClientException(String.format("align %s chrom %d: %s", alignid, chromid, response));
}
List<PairedHit> output = new ArrayList<PairedHit>();
int numhits = Integer.parseInt(readLine());
for (int i = 0; i < numhits; i++) {
output.add(new PairedHit(chromid,0,false,(short)0,
chromid,0,false,(short)0,(float)0,0));
}
IntBP ints = new IntBP(numhits);
ReadableByteChannel rbc = Channels.newChannel(instream);
Bits.readBytes(ints.bb, rbc);
if (isLeft) {
for (int i = 0; i < numhits; i++) {
output.get(i).leftPos = ints.get(i);
}
} else {
for (int i = 0; i < numhits; i++) {
output.get(i).rightPos = ints.get(i);
}
}
FloatBP floats = new FloatBP(numhits);
Bits.readBytes(floats.bb, rbc);
for (int i = 0; i < numhits; i++) {
output.get(i).weight = floats.get(i);
}
Bits.readBytes(ints.bb, rbc);
for (int i = 0; i < numhits; i++) {
output.get(i).pairCode = ints.get(i);
}
Bits.readBytes(ints.bb, rbc);
if (isLeft) {
for (int i = 0; i < numhits; i++) {
int j = ints.get(i);
PairedHit h = output.get(i);
h.leftLength = Hits.getLengthOne(j);
h.leftStrand = Hits.getStrandOne(j);
h.rightLength = Hits.getLengthTwo(j);
h.rightStrand = Hits.getStrandTwo(j);
}
} else {
for (int i = 0; i < numhits; i++) {
int j = ints.get(i);
PairedHit h = output.get(i);
h.leftLength = Hits.getLengthTwo(j);
h.leftStrand = Hits.getStrandTwo(j);
h.rightLength = Hits.getLengthOne(j);
h.rightStrand = Hits.getStrandOne(j);
}
}
Bits.readBytes(ints.bb, rbc);
if (isLeft) {
for (int i = 0; i < numhits; i++) {
output.get(i).rightChrom = ints.get(i);
}
} else {
for (int i = 0; i < numhits; i++) {
output.get(i).leftChrom = ints.get(i);
}
}
Bits.readBytes(ints.bb, rbc);
if (isLeft) {
for (int i = 0; i < numhits; i++) {
output.get(i).rightPos = ints.get(i);
}
} else {
for (int i = 0; i < numhits; i++) {
output.get(i).leftPos = ints.get(i);
}
}
return output;
}
}
/**
* returns a TreeMap from positions to counts representing a histogram
* of the hits in a range with the specified binsize. Bins with a count
* of zero are not included in the output.
*
* Ex: getHistgram("foo","1+",1,100,10)
* 6: 5
* 16: 0
* 36: 30
*
* minweight is the minimum weight for reads to be included in the histogram.
*
* dedup is the limit on how many times reads with any given 5' position will be counted.
* A value of zero means no limit. A limit of, eg, 2, means that at most two reads at
* any 5' position will be included in the output. For weighted histograms, the choice of reads
* included is unspecified. For methods that operate on a set of alignments, this many
* reads from each alignment will be included.
*
* Normally, a read is only counted in a single bin as defined by its position (generally
* the 5' end of the read). A non-zero read-Extension counts the read in any
* bin that you cross within that many bases of the read's position. A negative value
* goes backwards (smaller coordinates) and a larger value goes forward. You need
* to get the sign right depending on the strandedness of the chromosome that you're
* working with.
*/
public TreeMap<Integer,Integer> getHistogram(String alignid, int chromid, boolean isType2, boolean paired, int extension, int binsize, Integer start, Integer stop, Float minWeight, Boolean plusStrand) throws IOException, ClientException {
return getHistogram(alignid, chromid, isType2, paired, extension,binsize,0,start,stop,minWeight,plusStrand,true);
}
public TreeMap<Integer,Integer> getHistogram(String alignid, int chromid, boolean isType2, boolean paired, int extension, int binsize, int dedup, Integer start, Integer stop, Float minWeight, Boolean plusStrand, boolean isLeft) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type="histogram";
request.alignid=alignid;
request.chromid=chromid;
request.start = start;
request.end = stop;
request.isLeft = isLeft;
request.minWeight = minWeight;
request.isType2 = isType2;
request.isPlusStrand = plusStrand;
request.isPaired = paired;
request.map.put("binsize",Integer.toString(binsize));
if (dedup > 0) {
request.map.put("dedup",Integer.toString(dedup));
}
if (extension != 0) {
request.map.put("extension",Integer.toString(extension));
}
sendString(request.toString());
String response = readLine();
if (!response.equals("OK")) {
if (printErrors) {
System.err.println("not-OK response to request: " + response);
System.err.println("request was " + request);
}
throw new ClientException(response);
}
int numints = Integer.parseInt(readLine());
int out[] = Bits.readInts(numints, instream, buffer);
TreeMap<Integer,Integer> output = new TreeMap<Integer,Integer>();
for (int i = 0; i < out.length; i += 2) {
output.put(out[i], out[i+1]);
}
return output;
}
}
public TreeMap<Integer,Float> getWeightHistogram(String alignid, int chromid, boolean isType2, boolean paired, int extension, int binsize, Integer start, Integer stop, Float minWeight, Boolean plusStrand) throws IOException, ClientException {
return getWeightHistogram(alignid, chromid, isType2, paired, extension, binsize, 0, start,stop,minWeight,plusStrand, true);
}
public TreeMap<Integer,Float> getWeightHistogram(String alignid, int chromid, boolean isType2, boolean paired, int extension, int binsize, int dedup, Integer start, Integer stop, Float minWeight, Boolean plusStrand, boolean isLeft) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type="weighthistogram";
request.alignid=alignid;
request.chromid=chromid;
request.start = start;
request.end = stop;
request.minWeight = minWeight;
request.isType2 = isType2;
request.isPlusStrand = plusStrand;
request.isLeft = isLeft;
request.isPaired = paired;
request.map.put("binsize",Integer.toString(binsize));
if (dedup > 0)
request.map.put("dedup",Integer.toString(dedup));
if (extension!=0)
request.map.put("extension",Integer.toString(extension));
sendString(request.toString());
String response = readLine();
if (!response.equals("OK")) {
if (printErrors) {
System.err.println("not-OK response to request: " + response);
System.err.println("request was " + request);
}
throw new ClientException(response);
}
int numints = Integer.parseInt(readLine());
int out[] = Bits.readInts(numints, instream, buffer);
float weight[] = Bits.readFloats(numints, instream,buffer);
TreeMap<Integer,Float> output = new TreeMap<Integer,Float>();
for (int i = 0; i < out.length; i++) {
output.put(out[i], weight[i]);
}
return output;
}
}
public TreeMap<Integer,Integer> getHistogram(Collection<String> alignids, int chromid, boolean isType2, boolean paired, int extension, int binsize, Integer start, Integer stop, Float minWeight, Boolean plusStrand) throws IOException, ClientException {
return getHistogram(alignids,chromid,isType2, paired,extension,binsize,0,start,stop,minWeight,plusStrand);
}
public TreeMap<Integer,Integer> getHistogram(Collection<String> alignids, int chromid, boolean isType2, boolean paired, int extension, int binsize, int dedup, Integer start, Integer stop, Float minWeight, Boolean plusStrand) throws IOException, ClientException {
TreeMap<Integer,Integer> output = null;
for (String alignid : alignids) {
TreeMap<Integer,Integer> o = getHistogram(alignid,chromid,isType2, paired,extension,binsize,dedup,start,stop,minWeight,plusStrand,true);
if(paired) //run for isLeft =true & false
o.putAll(getHistogram(alignid,chromid,isType2, paired,extension,binsize,dedup,start,stop,minWeight,plusStrand,false));
for (int k : o.keySet()) {
if ((k - start - binsize / 2) % binsize != 0 ) {
System.err.println(String.format("Bad key %d for binsize %d and start %d in %s,%d",
k, binsize, start, alignid,chromid));
}
}
if (output == null) {
output = o;
} else {
for (int k : o.keySet()) {
if (output.containsKey(k)) {
output.put(k, output.get(k) + o.get(k));
} else {
output.put(k,o.get(k));
}
}
}
}
return output;
}
public TreeMap<Integer,Float> getWeightHistogram(Collection<String> alignids, int chromid, boolean isType2, boolean paired, int extension, int binsize, Integer start, Integer stop, Float minWeight, Boolean plusStrand) throws IOException, ClientException {
return getWeightHistogram(alignids,chromid,isType2, paired,extension,binsize,0,start,stop,minWeight,plusStrand);
}
public TreeMap<Integer,Float> getWeightHistogram(Collection<String> alignids, int chromid, boolean isType2, boolean paired, int extension, int binsize, int dedup, Integer start, Integer stop, Float minWeight, Boolean plusStrand) throws IOException, ClientException {
TreeMap<Integer,Float> output = null;
for (String alignid : alignids) {
TreeMap<Integer,Float> o = getWeightHistogram(alignid,chromid,isType2, paired,extension,binsize,dedup,start,stop,minWeight,plusStrand, true);
if(paired) //run for isLeft =true & false
o.putAll(getWeightHistogram(alignid,chromid,isType2, paired,extension,binsize,dedup,start,stop,minWeight,plusStrand, false));
if (output == null) {
output = o;
} else {
for (int k : o.keySet()) {
if (output.containsKey(k)) {
output.put(k, output.get(k) + o.get(k));
} else {
output.put(k,o.get(k));
}
}
}
}
return output;
}
/**
* Returns a Map from READ, WRITE, and ADMIN to lists of principals that have those privileges on the specified alignment.
*/
public Map<String,Set<String>> getACL(String alignid) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type="getacl";
request.alignid=alignid;
sendString(request.toString());
String response = readLine();
if (!response.equals("OK")) {
if (printErrors) {
System.err.println("not-OK response to request: " + response);
System.err.println("request was " + request);
}
throw new ClientException(response);
}
Map<String,Set<String>> output = new HashMap<String,Set<String>>();
fillPartACL(output);
fillPartACL(output);
fillPartACL(output);
return output;
}
}
/* fills one section of the acl output data structure. A section
is either read, write, or admin.
*/
private void fillPartACL(Map<String,Set<String>> output) throws IOException {
synchronized(this){
if(!connectionOpen)
reConnect();
String type = readLine();
int entries = Integer.parseInt(readLine());
Set<String> out = new HashSet<String>();
while (entries-- > 0) {
out.add(readLine());
}
output.put(type,out);
}
}
/**
* Applies the specified ACLChangeEntry objects to the acl for this experiment/chromosome.
*/
public void setACL(String alignid, Set<ACLChangeEntry> changes) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type="setacl";
request.alignid=alignid;
for (ACLChangeEntry a : changes) {
request.list.add(a.toString());
}
sendString(request.toString());
String response = readLine();
if (!response.equals("OK")) {
if (printErrors) {
System.err.println("not-OK response to request: " + response);
System.err.println("request was " + request);
}
throw new ClientException(response);
}
}
}
/**
* Adds the specified user (princ) to a group.
*/
public void addToGroup(String princ, String group) throws IOException, ClientException {
synchronized(this){
if(!connectionOpen)
reConnect();
request.clear();
request.type="addtogroup";
request.map.put("princ",princ);
request.map.put("group",group);
sendString(request.toString());
String response = readLine();
if (!response.equals("OK")) {
if (printErrors) {
System.err.println("not-OK response to request: " + response);
System.err.println("request was " + request);
}
throw new ClientException(response);
}
}
}
/**
* Closes this connection to the server.
*/
public void close() {
if(closeTimerThread!=null && closeTimerThread.isAlive())
closeTimerThread.interrupt();
if(connectionOpen)
closeConnection();
connectionOpen=false;
}
/**
* Closes this connection to the server.
*/
public void closeConnection() {
if (socket == null) {
return;
}
try {
socket.setSoLinger(false,0);
request.clear();
request.type="bye";
sendString(request.toString());
outstream.close();
outstream = null;
} catch (IOException e) {
e.printStackTrace();
}
try {
instream.close();
instream = null;
} catch (IOException e) {
e.printStackTrace();
}
try {
socket.close();
socket = null;
} catch (IOException e) {
e.printStackTrace();
}
connectionOpen=false;
//System.err.println("Readdb client closed");
}
/**
* ClientConnectionTimerThread closes the connection if it is idle for too long
* @author mahony
*
*/
class ClientConnectionTimerThread implements Runnable{
Client parent; //reference to parent class
public ClientConnectionTimerThread(Client p){
parent = p;
}
public void run() {
while(true){
try {
Thread.sleep(threadSleepTime);
//System.out.println("ClientConnectionTimerThread: checking ("+(System.currentTimeMillis() - lastActivityTime)+")\tconnectionOpen="+connectionOpen);
if(connectionOpen && System.currentTimeMillis() - lastActivityTime >connectionIdleTimeLimit){
parent.closeConnection();
//System.out.println("ClientConnectionTimerThread: connection closed");
}
} catch (InterruptedException e) {
Thread.currentThread().interrupt();
break;
}
}
}
}
}
/**
* SASL callback handler for the authentication exchange
*/
class ClientCallbackHandler implements CallbackHandler {
private String name, pass;
public ClientCallbackHandler(String n, String p) {name = n; pass = p;}
public void handle(Callback[] callbacks) {
for (int i = 0; i < callbacks.length; i++) {
if (callbacks[i] instanceof NameCallback) {
NameCallback nc = (NameCallback)callbacks[i];
nc.setName(name);
}
if (callbacks[i] instanceof PasswordCallback) {
PasswordCallback pc = (PasswordCallback)callbacks[i];
pc.setPassword(pass.toCharArray());
}
}
}
}
|
code
|
۱۹۱۱کس ٲخرس منٛز ہیوت سوُ گُرید غُلامن ہندِ سالار قُطُب دین عیبکن دَلہَ منٛز موٚلہِ تہٕ یتھہٕ کٕنہِ بنووُن غولام سُند غولام
|
kashmiri
|
विरोधी भड़काऊ प्रोकेन एचसीएल के लिए प्रोकेन हाइड्रोक्लोराइड स्थानीय एनेस्थेटिक ड्रग्स
बड़ी छवि : विरोधी भड़काऊ प्रोकेन एचसीएल के लिए प्रोकेन हाइड्रोक्लोराइड स्थानीय एनेस्थेटिक ड्रग्स
उत्पाद का नाम: प्रोकीन हाइड्रोक्लोराइड गर्म उत्पाद: लिडोकीन हाइड्रोक्लोराइड
समारोह: स्थानीय संवेदनाहारी दवाओं, विरोधी भड़काऊ समारोह:: प्रोकीन हाइड्रोक्लोराइड स्थानीय संवेदनाहारी दवाओं
कीवर्ड:: प्रोकीन हाइड्रोक्लोराइड स्थानीय संवेदनाहारी दवाओं व्हेत्सप्प:: ८६-134101217८६
प्रोकेन हाइड्रोक्लोराइड उत्पाद का नाम: प्रोकेन हाइड्रोक्लोराइड
प्रोकेन हाइड्रोक्लोराइड पर्यायवाची: अमीनोकेन; एनाडोलर; एनेस्थेसोल; एनेस्थिल; एटॉक्सिकोकाइन; बेंजोइसीसिड; ४-अमीनो-, २- (डायथाइलीनो) एथिलिस्टर, मोनोसाइड्रोक्लोराइड; बेंजोइसेसिड, पी-एमिनो-, २- (डायथाइलैमिनो) एथिलीनर, मोनोलीनर
प्रोकेन हाइड्रोक्लोराइड कैस: ५१-०५-८
प्रोकेन हाइड्रोक्लोराइड एमएफ: च१३ह२१कन२ओ२
प्रोकेन हाइड्रोक्लोराइड मेगावाट: २७२.७७
प्रोकेन हाइड्रोक्लोराइड ऐनक्स: २००-०७७-२
प्रोकेन हाइड्रोक्लोराइड म्प: १५५-१५६ च (लित.)
प्रोकेन हाइड्रोक्लोराइड बीपी: १९५-१९६ डिग्री सेल्सियस १७ मिमी
प्रोकेन हाइड्रोक्लोराइड एफपी: १९५-१९६ डिग्री सेल्सियस / १७ मिमी
प्रोकेन हाइड्रोक्लोराइड जल घुलनशीलता: घुलनशील
प्रोकीन हाइड्रोक्लोराइड सूरत: ठीक सुई क्रिस्टल या क्रिस्टलीय पाउडर, बिना गंध, थोड़ा कड़वा स्वाद और भांग,
प्रोकेन हाइड्रोक्लोराइड घुलनशीलता: पानी में घुलनशील, शराब में घुलनशील, क्लोरोफॉर्म में थोड़ा घुलनशील।
प्रोकेन हाइड्रोक्लोराइड एस्टर बॉन्ड संरचना को अमीनो एसिड और डायथाइलिनोइथेनॉल के उत्पादन के लिए हाइड्रोलाइज किया जा सकता है, कुछ शर्तों के तहत, अमीनो एसिड आगे डीकार्बाक्सिलेशन विषाक्त एनिलिन हो सकता है।
प्रोकेन हाइड्रोक्लोराइड उपयोग: प्रोकेन हाइड्रोक्लोराइड एक स्थानीय संवेदनाहारी है जो अस्थायी रूप से तंत्रिका फाइबर प्रवाहकत्त्व को अवरुद्ध कर सकता है और इसमें एक मादक प्रभाव होता है, मजबूत, कम विषाक्तता और गैर-नशे की भूमिका, लेकिन त्वचा पर, कमजोर के माध्यम से श्लेष्मा, सामयिक में अस्वस्थ। संज्ञाहरण, मुख्य रूप से नैदानिक घुसपैठ, रीढ़ की हड्डी और चालन संज्ञाहरण में उपयोग किया जाता है।
प्रोकेन हाइड्रोक्लोराइड कोआ
द्वितीयक स्पॉट०.५%, राशि १.०%
०.६यू / मिलीग्राम
प्रोकेन हाइड्रोक्लोराइड उत्पाद टैग: बेंज़ोकेन, लोकेलनेस्टेसिया एपीआईएस, लिडोकेन बेस, लिडोकेन हाइड्रोक्लोराइड, टेट्राकेन हाइड्रोक्लोराइड, प्रमोक्सिन हाइड्रोक्लोराइड, प्रोकेन हाइड्रोक्लोराइड
|
hindi
|
This is the list of every IP Address in our database beginning with 202.157.X.X. If you would like information on a different IP Address, you can search for it at the left. Otherwise, click on the address of interest to you. If you do not see the address you are looking for, that means we have no comments for it. Please add one!
|
english
|
مایکرٛوسافٹ ایٚکسَل چُھ اَکھ بٔڑِس پَیمانس پؠٹھ اِستعمال کرنہٕ یِنہٕ وول تجزِیٲتی ٹوُل زیادٕ تر بُنیٲدی چٟز ہیٚچھنہٕ تہٕ کٲم سَہل بناونہٕ خٲطرٕ
|
kashmiri
|
require 'test_helper'
require 'puppet_proxy/puppet_plugin'
require 'puppet_proxy/customrun'
class CustomRunTest < Test::Unit::TestCase
def setup
@customrun = Proxy::Puppet::CustomRun.new(:nodes => ["host1", "host2"])
end
def test_customrun
::Proxy::Puppet::Plugin.load_test_settings(:customrun_cmd => "/bin/false", :customrun_args => "-ay -f -s")
@customrun.expects(:shell_command).with(["/bin/false", "-ay", "-f", "-s", "host1", "host2"]).returns(true)
@customrun.run
end
def test_customrun_with_array_command_args
::Proxy::Puppet::Plugin.load_test_settings(:customrun_cmd => "/bin/false", :customrun_args => ["-ay", "-f", "-s"])
@customrun.expects(:shell_command).with(["/bin/false", "-ay", "-f", "-s", "host1", "host2"]).returns(true)
@customrun.run
end
def test_customrun_uses_shell_escaped_command
::Proxy::Puppet::Plugin.load_test_settings(:customrun_cmd => "puppet's_run", :customrun_args => "-ay -f -s")
File.stubs(:exist?).with("puppet's_run").returns(true)
@customrun.expects(:shell_command).with(["puppet\\'s_run", "-ay", "-f", "-s", "host1", "host2"]).returns(true)
@customrun.run
end
end
|
code
|
---
title: <image>
type: references
order: 2.4
version: 2.1
---
# <image>
`<image>` 组件用于渲染图片，并且它不能包含任何子组件。新版 Vue 2.0 中**不支持**用 `<img>` 作简写。
需要注意的是，需要明确指定 `width` 和 `height`，否则图片无法显示。
简单例子：
```html
<template>
<div>
<image style="width: 560px;height: 560px;" src="https://img.alicdn.com/tps/TB1z.55OFXXXXcLXXXXXXXXXXXX-560-560.jpg"></image>
</div>
</template>
```
[体验一下](http://dotwe.org/vue/1d6145d98cbdb8c66c69b4d4dcd2744d)
## 子组件
`<image>` 组件不支持任何子组件，因此不要尝试在 `<image>` 组件中添加任何组件。如果需要实现 `background-image` 的效果，可以使用 `<image>` 组件和 `position` 定位来现实，如下面代码。
```html
<template>
<div>
<image style="width:750px; height:750px;" src="https://img.alicdn.com/tps/TB1z.55OFXXXXcLXXXXXXXXXXXX-560-560.jpg"></image>
<div class="title">
<text style="font-size:50px; color: #ff0000">你好，image</text>
</div>
</div>
</template>
<style>
.title{
position:absolute;
top:50;
left:10;
}
</style>
```
[体验一下](http://dotwe.org/vue/0a81d27b5dbc68ea3bf5f9fd56c882e8)
## 特性
`<image>` 组件，包含 `src` 和 `resize` 两个重要特性。
- `src {string}`：定义图片链接，目前图片暂不支持本地图片。
- `resize {string}`：可以控制图片的拉伸状态，值行为和 W3C 标准一致。
可选值为：
- `stretch`：默认值，指定图片按照容器拉伸，有可能使图片产生形变。
- `cover`：指定图片可以被调整到容器，以使图片完全覆盖背景区域，图片有可能被剪裁。
- `contain`：指定可以不用考虑容器的大小，把图像扩展至最大尺寸，以使其宽度和高度完全适应内容区域。
例子：
- `placeholder`: <span class="api-version">v0.9+</span> <string> 当源图片下载中时显示一张占位图。
[体验一下](http://dotwe.org/vue/18e71ab3484bb6751ad77ff7d5195404)
## 样式
- 通用样式：支持所有通用样式
- 盒模型
- `flexbox` 布局
- `position`
- `opacity`
- `background-color`
查看 [组件通用样式](../common-style.html)
## 事件
- `load`: <sup class="api-version">v0.8+</sup>：当图片加载完成时触发。目前在 Android、iOS 上支持，H5 暂不支持。[示例](http://dotwe.org/vue/98ee340348d7cc3e6fbfe68dbaef1eed)
- 事件对象
- `success`: 当图片成功加载时为`true`，否则为`false`
- `size`: 图片的原始尺寸，包含两个参数：`naturalWidth` 代表图片的原始宽度像素值，`naturalHeight` 代表图片的原始高度值。这两个参数的默认值都为`0`
- 通用事件
支持所有通用事件：
- `click`
- `longpress`
- `appear`
- `disappear`
查看 [通用事件](../common-event.html)
## 约束
1. 需要指定宽高；
2. 不支持子组件。
## 示例
```html
<template>
<scroller class="wrapper" >
<div class="page-head" >
<image class="title-bg" resize="cover" src="https://img.alicdn.com/tps/TB1dX5NOFXXXXc6XFXXXXXXXXXX-750-202.png"></image>
<div class="title-box">
<text class="title">Alan Mathison Turing</text>
</div>
</div>
<div class="article">
<text class="paragraph">Alan Mathison Turing ( 23 June 1912 – 7 June 1954) was an English computer scientist, mathematician, logician, cryptanalyst and theoretical biologist. He was highly influential in the development of theoretical computer science, providing a formalisation of the concepts of algorithm and computation with the Turing machine, which can be considered a model of a general purpose computer.Turing is widely considered to be the father of theoretical computer science and artificial intelligence.</text>
<text class="paragraph">During the Second World War, Turing worked for the Government Code and Cypher School (GC&CS) at Bletchley Park, Britain's codebreaking centre. For a time he led Hut 8, the section responsible for German naval cryptanalysis. He devised a number of techniques for speeding the breaking of German ciphers, including improvements to the pre-war Polish bombe method, an electromechanical machine that could find settings for the Enigma machine. Turing played a pivotal role in cracking intercepted coded messages that enabled the Allies to defeat the Nazis in many crucial engagements, including the Battle of the Atlantic; it has been estimated that this work shortened the war in Europe by more than two years and saved over fourteen million lives.</text>
<text class="paragraph">After the war, he worked at the National Physical Laboratory, where he designed the ACE, among the first designs for a stored-program computer. In 1948 Turing joined Max Newman's Computing Machine Laboratory at the Victoria University of Manchester, where he helped develop the Manchester computers and became interested in mathematical biology. He wrote a paper on the chemical basis of morphogenesis, and predicted oscillating chemical reactions such as the Belousov–Zhabotinsky reaction, first observed in the 1960s.</text>
<text class="paragraph">Turing was prosecuted in 1952 for homosexual acts, when by the Labouchere Amendment, "gross indecency" was still criminal in the UK. He accepted chemical castration treatment, with DES, as an alternative to prison. Turing died in 1954, 16 days before his 42nd birthday, from cyanide poisoning. An inquest determined his death as suicide, but it has been noted that the known evidence is also consistent with accidental poisoning. In 2009, following an Internet campaign, British Prime Minister Gordon Brown made an official public apology on behalf of the British government for "the appalling way he was treated." Queen Elizabeth II granted him a posthumous pardon in 2013.</text>
</div>
</scroller>
</template>
<style>
.page-head {
width: 750px;
height: 200px;
}
.title-bg {
width: 750px;
height: 200px;
}
.title-box {
width: 750px;
height: 200px;
justify-content: center;
align-items: center;
position: absolute;
top: 0;
right: 0;
bottom: 0;
left: 0;
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.title {
color: #ffffff;
font-size: 32px;
font-weight: bold;
}
.article {
padding: 20px;
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.paragraph{
margin-bottom: 15px;
}
</style>
```
[try it](http://dotwe.org/vue/e2122bc245beafb0348d79bfd1274904)
|
code
|
The Health and Safety at Work Act turns 40!
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|
english
|
पराग शाह है महाराष्ट्र में सबसे धनी उम्मीदवार, इतने करोड़ है संपत्ति - टारगेट कॉरृप्शन
पराग शाह है महाराष्ट्र में सबसे धनी उम्मीदवार, इतने करोड़ है संपत्ति
बजने के साथ ही नेता अपने चुनावी क्षेत्र से मैदान में उतर चुके हैं व फतह करने की प्रयास में लगे हैं। मुंबई की घाटकोपर ईस्ट की सीट पर पुरे महाराष्ट्र की नजर है। यहां भाजपा के टिकट पर प्रदेश के सबसे धनी उम्मीद्वार पराग शाह चुनाव लड़ रहे हैं। ६ बार के विधायक रहे प्रकाश मेहता को हटाकर शाह को इस सीट से टिकट दिया गया है।
शाह ने अपने शपथ लेटर में भाजपा उम्मीदवार पराग शाह ने ५०० करोड़ की चल-अचल संपत्ति का ऐलान किया है। इसमें ४२२ करोड़ की चल व ७८ करोड़ रुपए की अचल संपत्ति शामिल है। तकरीबन २ करोड़ ६० लाख से ज्यादा के बेशकीमती गहने हैं। पराग शाह की पत्नी के नाम २ करोड़ ४७ लाख की कारें हैं।
मुंबई के घाटकोपर इलाके की दो विधानसभा सीटों पर बीजेपी का अतिक्रमण है। इस बार घाटकोपर पूर्व विधानसभा सीट दो वजहों से चर्चा में है- पहली वजह है इस विधानसभा सीट से करीब ६ बार विधायक रहे प्रकाश मेहता का टिकट कट जाना। दरअसल, मेहता पिछले ३० वर्षों से बीजेपी को इस विधानसभा सीट से रिप्रेजेंट कर रहे थे। साथ ही दो बार वो मंत्री भी रह चुके थे, लेकिन इस बार उनकी टिकट को काट के भाजपा ने इस क्षेत्र के ही नगरसेवक पराग शाह को टिकट दिया है। दूसरी जो सबसे चर्चित बात है वो ये की पराग शाह महारष्ट्र के सबसे धनी उम्मीदवार भी हैं। अपने चुनावी एफिडेविट में पराग शाह ने अपनी ५०० करोड़ से ज्यादा की संपत्ति दिखाई है। देखें- लाइव त्व
बीजेपी उम्मीदवार पराग शाह पेशे से बिल्डर हैं। २०१७ में पहली बार भाजपा में शामिल होकर उन्होंने कॉरपोरेशन का चुनाव लड़ा था व उसमें जीत हासिल की थी। कॉरपोरेशन चुनाव के बाद पराग शाह की लोकप्रियता बहुत ज्यादा बढ़ी जिसकी वजह से भाजपा के इंटरनल सर्वे के बाद घाटकोपर पूर्व विधानसभा सीट से प्रकाश मेहता का टिकट काटकर पराग शाह को मौका दिया गया है।
दरअसल, इसके पीछे एक बड़ी वजह भी है कि प्रकाश मेहता के ऊपर करप्शन के आरोप लगे हैं। इसी आरोप के चलते प्रकाश मेहता को अपना मंत्री पद भी गंवाना पड़ा था। नामांकन के आखिरी दिन पराग शाह के टिकट की घोषणा की थी, जिसके बाद भाजपा उम्मीदवार शाह निवर्तमान विधायक प्रकाश मेहता से मिलने गए थे जहां पर मेहता के समर्थकों ने पराग शाह के ऊपर अटैक कर दिया व उनके विरूद्ध जमकर नारेबाजी की। इसमें शाह के बॉडीगार्ड घायल हुए उनकी गाड़ी भी टूट गई।
इन्हीं कई वजहों के चलते घाटकोपर पूर्व विधानसभा का चुनाव इस बार रोचक हो गया है। पराग शाह के सामने वंचित बहुजन आघाडी से विकास पवार चुनाव लड़ रहे हैं। इनका बोलना है कि विधायक बनने के लिए धनी उम्मद्वार पुरे ढंग से धनबल का उपयोग कर रहे हैं।
इस आरोपों से बेपरवाह पराग का बोलना है कि चुनावी मौसम में इस ढंग के आरोप लगना आम बात है। शाह के मुताबिक उनके सामने विपक्ष है ही नहीं व उनकी जीत पक्की है। इस बार वे अकेले विधायक नहीं बल्कि घाटकोपर की पूरी जनता विधायक बनेगी। साथ ही पराग ने नाराज प्रकाश मेहता को भी मना लिया है।
कुल वोटर- २३४६२१
नार्थ इंडियन- ७%
मुस्लिम- १२ %
अन्य- १४ फीसदी है
पीएम मोदी व जिनपिंग की मुलाकात में होगी इन अहम मुद्दों पर चर्चा, आप भी जाने
लड़ाकू विमान उड़ाने के लिए कोई ऊपरी आयु सीमा नहीं: राजनाथ...
आतंक विरोधी अभियान: सुरक्षाबलों ने ढेर किए इतने आतंकवादी...
भारतीय वायुसेना ने बालाकोट एयर हड़ताल को लेकर बड़ा खुलासा...
हुवी ने अपने माते सीरीज को जर्मनी में आयोजित हुए इवेंट...
यह हो सकती है वनप्लस ७त प्रो के स्पेसिफिकेशन व कीमत, जाने...
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hindi
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class RemoveHandleFromSubgroups < ActiveRecord::Migration[5.2]
def change
Group.where('parent_id is not null').update_all(handle: nil)
end
end
|
code
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// Emacs style mode select -*- C++ -*-
//-----------------------------------------------------------------------------
//
// Copyright (C) 1998-2000 by DooM Legacy Team.
//
// This program is free software; you can redistribute it and/or
// modify it under the terms of the GNU General Public License
// as published by the Free Software Foundation; either version 2
// of the License, or (at your option) any later version.
//
// This program is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
// GNU General Public License for more details.
//-----------------------------------------------------------------------------
/// \file
/// \brief load/unload a DLL at run-time
//#define WIN32_LEAN_AND_MEAN
#define RPC_NO_WINDOWS_H
#include <windows.h>
#ifdef HWRENDER
BOOL Init3DDriver (LPCSTR dllName);
VOID Shutdown3DDriver (VOID);
#endif
#ifdef HW3SOUND
BOOL Init3DSDriver(LPCSTR dllName);
VOID Shutdown3DSDriver(VOID);
#endif
|
code
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NDSU › Publications › Food and Nutrition / Sports Drinks: R They Needed?
Sports drinks, such as Gatorade, Powerade and All Sport, contain carbohydrates and electrolytes, such as sodium, potassium and chloride. They are made for physical activity, to help rehydrate and to keep energy levels high. Are sports drinks really necessary? Not always. You can get these same benefits from other sources. A sports drink is not better for you unless you are active for 60 to 90 minutes or are exercising in very hot conditions. Anything less, and water should be the drink of choice.
Sports drinks, such as Gatorade, Powerade and All Sport, contain carbohydrates and electrolytes, such as sodium, potassium and chloride. They are made for physical activity, to help rehydrate and to keep energy levels high.
Are sports drinks really necessary? Not always. You can get these same benefits from other sources. A sports drink is not better for you unless you are active for 60 to 90 minutes or are exercising in very hot conditions. Anything less, and water should be the drink of choice.
Read the Nutrition Facts labels and compare your choices. A bottle of one popular drink is considered four full servings. The label says 50 calories per serving, so do the multiplication. That’s 200 calories for the bottle! You might not even use this many calories during physical activity. Consider the cost, too. Is a drinking fountain nearby? If so, have a drink of water.
1. Dehydration may cause a person to lose some coordination, concentration, endurance and __________.
2. When is a sports drink considered most beneficial?
3. What nutrients in sports drinks aid in physical performance, hydration and energy?
4. What drink is the best choice when active?
Dehydration results from not drinking enough fluids. This can lead to health problems and decreased physical performance. You may lose your coordination, concentration, endurance and strength. If you ignore your thirst, dehydration can slow you down.
Thirst is not an early warning sign. By the time you feel thirsty, you might already be dehydrated. Other symptoms of dehydration include feeling dizzy and lightheaded, having a dry mouth and/or producing less and darker urine.
Water is the best fluid for keeping hydrated. Cool water may help cool your overheated body.
• Don’t wait until you feel thirsty to have a sip of water because thirst is a sign that your body has needed liquids for a while.
• Drink fluids before, after and during physical activity. Keep a bottle of water with you and take “hydration breaks” every 10 to 15 minutes.
• Following physical activity, drink two cups of fluids for each pound lost.
• Avoid highly caffeinated drinks.
• Try milk or fruit juice. They contain carbohydrates, vitamins and minerals not found in all sports drinks. Milk actually is considered the new sports drink!
If a sports drink is necessary, try making your own. You will save money and have the same benefits.
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english
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\begin{document}
\title*{Is Every Irreducible Shift of Finite Type Flow Equivalent to a Renewal System?}
\author{Rune Johansen}
\institute{Rune Johansen \at Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 K\o benhavn \O, \email{[email protected]}} \maketitle
\abstract*{
Is every irreducible shift of finite type flow equivalent to a renewal system? For the first time, this variation of a classic problem formulated by Adler is investigated, and several partial results are obtained in an attempt to find the range of the Bowen--Franks invariant over the set of renewal systems of finite type. In particular, it is shown that the Bowen--Franks group is cyclic for every member of a class of renewal systems known to attain all entropies realised by shifts of finite type, and several classes of renewal systems with non--trivial values of the invariant are constructed.
}
A renewal system is a shift space consisting of the biinfinite sequences that can be obtained as free concatenations of words from some finite generating list. This simple definition hides a surprisingly rich structure that is in many ways independent of the usual topological and dynamical structure of the shift space.
The present work was motivated by the following problem raised by Adler: Is every irreducible shift of finite type conjugate to a renewal system? Several attempts have been made to answer this question, and the conjugacy of certain special classes of renewal systems is well understood, but there exist only a few results concerning the general problem.
This work is the first to investigate the corresponding question for flow equivalence. The aim has been to find the range of the Bowen--Franks invariant over renewal systems of finite type, and several classes of renewal systems displaying a wide range of values of the invariant are constructed, but it remains unknown whether renewal systems can attain all the values attained by irreducible shifts of finite type.
Section \ref{RJ_sec_introduction} gives an introduction to shift spaces and renewal systems. Section \ref{RJ_sec_rs_lfc} concerns the left Fischer covers of renewal systems and gives conditions under which the Fischer covers of complicated renewal systems can be constructed from simpler building blocks with known presentations. Section \ref{RJ_sec_rs_entropy} gives a flow classification of a class of renewal systems introduced in \cite{RJ_hong_shin}, while Sec.\ \ref{RJ_sec_rs_range} uses the results of the previous two sections to construct classes of renewal systems with interesting values of the Bowen--Franks invariant.
\emph{Acknowledgements}: Supported by \textsc{VILLUM FONDEN} through the experimental mathematics network at the University of Copenhagen. Supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).
\section{Introduction}
\label{RJ_sec_introduction}
\index{shift space}
\index{full shift}
\index{shift map}
\index{shift of finite type}
\index{word}
\index{SFT|see{shift of finite type}}
Here, a short introduction to the basic definitions and properties of shift
spaces is given to make the present paper self--contained.
For a thorough treatment of shift spaces see \cite{RJ_lind_marcus}.
Let $\mathcal{A}$ be a finite set with the discrete topology. The
\emph{full shift} over $\mathcal{A}$ consists of the space $\mathcal{A}^\mathbb{Z}$ endowed
with the product topology and the \emph{shift map} $\sigma \colon
\mathcal{A}^\mathbb{Z} \to \mathcal{A}^\mathbb{Z}$ defined by $\sigma(x)_i = x_{i+1}$ for all $i \in
\mathbb{Z}$. Let $\mathcal{A}^*$ be the collection of finite words (also known as
blocks) over $\mathcal{A}$.
For $w \in \mathcal{A}^*$, $\vert w \vert$ will denote the length of $w$.
A
subset $X \subseteq \mathcal{A}^\mathbb{Z}$ is called a \emph{shift space} if it
is invariant under the shift map and closed.
For each $\mathcal{F} \subseteq \mathcal{A}^*$, define $\mathsf{X}_\mathcal{F}$ to be the set of
bi--infinite sequences in $\mathcal{A}^\mathbb{Z}$ which do not contain any of the
\emph{forbidden words} from $\mathcal{F}$.
A subset $X \subseteq \mathcal{A}^\mathbb{Z}$ is a shift space if and only if there
exists $\mathcal{F} \subseteq \mathcal{A}^*$ such that $X = \mathsf{X}_\mathcal{F}$
(cf. \cite[Proposition 1.3.4]{RJ_lind_marcus}). $X$ is said to be
a \emph{shift of finite type} (SFT) if this is possible for a finite
set $\mathcal{F}$.
\index{language}
The \emph{language} of a shift space $X$ is denoted $\mathcal{B}(X)$ and it is defined to be the set of all words which occur in
at least one $x \in X$. The shift space $X$ is said to be
\emph{irreducible} if there for every $u,w \in \mathcal{B}(X)$ exists $v \in
\mathcal{B}(X)$ such that $uvw \in \mathcal{B}(X)$.
For each $x \in X$ define the \emph{left--ray} of $x$ to be $x^- = \cdots
x_{-2} x_{-1}$ and define the \emph{right--ray} of $x$ to be $x^+ = x_0
x_1 x_2 \cdots$. The sets of all left--rays and all right--rays are,
respectively, denoted $X^-$ and $X^+$. Given a word or ray $x$, $\RJrl(x)$ and $\RJleftl(x)$ will denote respectively the right--most and the left--most letter of $x$.
\index{graph}
\index{path}
\index{edge shift}
\index{irreducible}
A \emph{directed graph} is a quadruple $E = (E^0,E^1,r,s)$ consisting of countable sets $E^0$ and $E^1$, and maps $r,s \colon E^1 \to E^0$.
A \emph{path} $\lambda = e_1 \cdots
e_n$ is a sequence of edges such that $r(e_i) = s(e_{i+1})$ for all $i
\in \{1, \ldots n-1 \}$. The vertices in $E^0$ are considered to be
paths of length $0$.
For each $n \in \mathbb{N}_0$, the set of paths of length $n$ is denoted
$E^n$, and the set of all finite paths is denoted $E^*$.
Extend
the maps $r$ and $s$ to $E^*$ by defining $s(e_1 \cdots e_n) = s(e_1)$
and $r(e_1 \cdots e_n) = r(e_n)$. A directed graph $E$ is said to be
\emph{irreducible} (or transitive) if there for each pair of vertices
$u,v \in E^0$ exists a path $\lambda \in E^*$ with $s(\lambda) = u$
and $r(\lambda) = v$. For a directed graph $E$, the
\emph{edge shift} $(\mathsf{X}_E, \sigma_E)$ is defined by
$
\mathsf{X}_E = \left\{ x \in (E^1)^\mathbb{Z} \mid r(x_i) = s(x_{i+1}) \textrm{ for all }
i \in \mathbb{Z} \right\}$.
\index{conjugacy}
\index{flow equivalence}
\index{Bowen--Franks group}
\index{BF@$\RJBF$|see{Bowen--Franks group}}
\index{Bowen--Franks invariant}
A bijective, continuous and shift commuting map between two shift
spaces is called a \emph{conjugacy}, and when such a map exists,
the two shift spaces are said to be \emph{conjugate}. \emph{Flow
equivalence} is a weaker equivalence relation generated by conjugacy
and \emph{symbol expansion} \cite{RJ_parry_sullivan}.
Let $A$ be the adjacency matrix of a directed graph $E$, then $\RJBF(A) = \mathbb{Z}^n / \mathbb{Z}^n (\RJId - A)$ is called the \emph{Bowen--Franks group} of $A$ and it is an invariant of conjugacy of edge shifts. Let $E$ and $F$ be finite directed graphs for which the edge shifts $\mathsf{X}_E$ and $\mathsf{X}_F$ are irreducible and not flow equivalent to the trivial shift with one element, and let $A_E$ and $A_F$ be the corresponding adjacency matrices. Then $\mathsf{X}_E$ and $\mathsf{X}_F$ are flow equivalent if and only $\RJBF(A_E) \simeq \RJBF(A_F)$ and the signs $\RJsgn \det A_E$ and $\RJsgn \det A_F$ are equal \cite{RJ_franks}. Every SFT is conjugate to an edge shift, so this gives a complete flow equivalence invariant of irreducible SFTs. The pair consisting of the Bowen--Franks group and the sign of the determinant is called the \emph{signed Bowen--Franks group}, and it is denoted $\RJBF_+$. This invariant is easy to compute and easy to compare which makes it appealing to consider flow equivalence rather than conjugacy.
\index{graph!labelled}
\index{shift space!presentation of}
\index{presentation}
\index{presentation!follower separated}
A \emph{labelled graph} $(E, \mathcal{L})$ over an alphabet $\mathcal{A}$ consists
of a directed graph $E$ and a surjective labelling map $\mathcal{L} \colon E^1
\to \mathcal{A}$.
Given a labelled graph $(E, \mathcal{L})$, define the shift space $(\mathsf{X}_{(E, \mathcal{L})}, \sigma)$ by setting
$\mathsf{X}_{(E, \mathcal{L})} = \left\{ \left( \mathcal{L}(x_i) \right)_i \in \mathcal{A}^\mathbb{Z} \mid
x \in \mathsf{X}_E \right\}$,
The
labelled graph $(E, \mathcal{L})$ is said to be a \emph{presentation} of the
shift space $\mathsf{X}_{(E, \mathcal{L})}$, and a \emph{representative} of a word $w \in
\mathcal{B}(\mathsf{X}_{(E, \mathcal{L})}) $ is a path $\lambda \in E^*$ such that
$\mathcal{L}(\lambda) = w$ with the natural extension of $\mathcal{L}$. Representatives of rays are defined analogously.
Let $(E, \mathcal{L})$ be a labelled graph presenting $X$. For
each $v \in E^0$, define the \emph{predecessor set} of $v$ to be the
set of left--rays in $X$ which have a presentation terminating at $v$. This
is denoted $P_\infty^E(v)$, or just $P_\infty(v)$ when $(E, \mathcal{L})$ is understood from the context.
The presentation $(E, \mathcal{L})$ is said to be \emph{predecessor--separated} if $P_\infty^E(u) \neq P_\infty^E(v)$ when $u,v \in E^0$ and $u \neq v$.
A function $\pi \colon X_1 \to X_2$ between shift spaces $X_1$ and
$X_2$ is said to be a \emph{factor map} if it is continuous,
surjective, and shift commuting. A shift space is
called \emph{sofic} \cite{RJ_weiss} if it is the image of an SFT under a factor
map.
Every SFT is sofic, and a sofic shift
which is not an SFT is called \emph{strictly sofic}.
Fischer proved that a shift space is sofic if and only
if it can be presented by a finite labelled graph \cite{RJ_fischer}.
A sofic shift
space is irreducible if and only if it can be presented by an
irreducible labelled graph (see \cite[Sec.\ 3.1]{RJ_lind_marcus}).
Let $(E, \mathcal{L})$ be a finite labelled graph which presents the sofic
shift space $\mathsf{X}_{(E, \mathcal{L})}$, and let $\pi_\mathcal{L} \colon \mathsf{X}_E \to \mathsf{X}_{(E,
\mathcal{L})}$ be the factor map induced by the labelling map $\mathcal{L} \colon
E^1 \to \mathcal{A}$, then the SFT $\mathsf{X}_E$ is called a \emph{cover} of the
sofic shift $\mathsf{X}_{(E, \mathcal{L})}$, and $\pi_\mathcal{L}$ is called the covering map.
Let $X$ be a shift space over an alphabet $\mathcal{A}$. A presentation $(E,\mathcal{L})$ of $X$ is said to be \emph{left--resolving}
if no vertex in $E^0$ receives two edges with the same label.
Fischer proved \cite{RJ_fischer} that up to labelled graph
isomorphism every irreducible sofic shift has a unique left--resolving
presentation with fewer vertices than any other left--resolving
presentation. This is called the \emph{left Fischer cover} of $X$,
and it is denoted $(F, \mathcal{L}_F)$.
\begin{comment}
\begin{definition}
(See \cite[Secs. I and III]{jonoska_marcus} and \cite[Exercise
3.2.8]{lind_marcus})
For $x^+ \in X^+$, define the \emph{predecessor set} of $x^+$ to be
the set of left--rays which may precede $x^+$ in $X$, that is
$P_\infty(x^+) = \{ y^- \in X^- \mid y^- x^+ \in \mathsf{X} \}$.
\end{definition}
\noindent The \emph{follower set} of a left--ray $x^- \in X^-$ is defined
analogously.
\begin{definition}
\label{RJ_def_predecessor_set_of_vertex}
\end{definition}
\end{comment}
\index{predecessor set}
\index{follower set}
\index{intrinsically synchronising}
For $x^+ \in X^+$, define the \emph{predecessor set} of $x^+$ to be
the set of left--rays which may precede $x^+$ in $X$, that is
$P_\infty(x^+) = \{ y^- \in X^- \mid y^- x^+ \in X \}$
(see \cite[Secs. I and III]{RJ_jonoska_marcus} and \cite[Exercise
3.2.8]{RJ_lind_marcus} for details). The \emph{follower set} of a left--ray $x^- \in X^-$ is defined analogously.
The \emph{left Krieger cover} of the sofic shift space $X$ is the labelled graph
$(K, \mathcal{L}_K)$ where $K^0 = \{ P_\infty(x^+) \mid x^+ \in X^+\}$,
and where there is an edge labelled $a \in \mathcal{A}$ from $P \in K^0$ to
$P' \in K^0$ if and only if there exists $x^+ \in X^+$ such that $P
= P_\infty(a x^+)$ and $P' = P_\infty(x^+)$.
A word $v \in \mathcal B(X)$ is said to be \emph{intrinsically synchronising} if $uvw \in \mathcal B(X)$ whenever $u$ and $w$ are words such that $uv, vw \in \mathcal B(X )$.
A ray is said to be \emph{intrinsically synchronising} if it contains an intrinsically synchronising word as a factor.
If a right--ray $x^+$ is intrinsically synchronising, then there is precisely one vertex in the left Fischer cover where a presentation of $x^+$ can start, and this vertex can be identified with the predecessor set $P_\infty(x^+)$ as a vertex in the Krieger cover. In this way, the left Fischer cover can be identified with the irreducible component of the left Krieger cover generated by the vertices that are predecessor sets of intrinsically synchronising right--rays \cite[Lemma 2.7]{RJ_krieger_sofic_I}, \cite[Exercise 3.3.4]{RJ_lind_marcus}. The interplay between the structure of the Fischer and Krieger covers is examined in detail in \cite{RJ_johansen_structure}.
\index{renewal system}
\index{renewal system!generating list of|see{generating list}}
\index{generating list}\index{XL@$\mathsf{X}(L)$}
\index{renewal system!loop graph of}
\index{flower automata}\index{loop system}
Let $\mathcal{A}$ be an alphabet, let $L \subseteq \mathcal{A}^*$ be a finite list of words over $\mathcal{A}$, and define $\mathcal{B}(L)$ to be the set of factors of elements of $L^*$. Then $\mathcal{B}(L)$ is the language of a shift space $\mathsf{X}(L)$ which is said to be the \emph{renewal system}
generated by $L$. $L$ is said to be the \emph{generating list} of $\mathsf{X}(L)$.
A renewal system is an irreducible sofic shift since it can be presented by the labelled graph obtained by writing the generating words on loops starting and ending at a common vertex.
This graph is called the \emph{standard loop graph presentation} of $\mathsf{X}(L)$, and because of this presentation, renewal systems are called \emph{loop systems} or \emph{flower automata} in automata theory (e.g.\ \cite{RJ_berstel_perrin}).
\index{Adler's problem}
\index{Adler's problem!flow equivalence version}
Simple examples show that not every sofic shift---or every SFT---is a renewal system \cite[pp.\ 433]{RJ_lind_marcus}, and these results naturally raise the following question, which was first asked by Adler:
Is every irreducible shift of finite type conjugate to a renewal system?
This question has been the motivation of most of the work done on renewal systems \cite{RJ_goldberger_lind_smorodinsky,RJ_hong_shin_cyclic,RJ_hong_shin,RJ_johnson_madden,RJ_restivo, RJ_restivo_note,RJ_williams_rs}.
The analogous question for sofic shifts has a negative answer \cite{RJ_williams_rs}. The aim of the present work has been to answer another natural variation of Adler's question: Is every irreducible SFT \emph{flow equivalent} to a renewal system?
To answer this question, it is sufficient to find the range of the Bowen--Franks invariant over the set of SFT renewal systems and check whether it is equal to the range over the set of irreducible SFTs.
It is easy to check that a group $G$ is the Bowen--Franks group of an irreducible SFT if and only if it is a finitely generated abelian group and that any combination of sign and Bowen--Franks group can be achieved by the Bowen--Franks invariant.
Hence, the overall strategy of the investigation of the flow equivalence question has been to attempt to construct all these combinations of groups and signs.
However, it is difficult to construct renewal systems attaining many of the values of the invariant. In fact, it is non--trivial to construct an SFT renewal system that is not flow equivalent to a full shift \cite{RJ_johansen_thesis}.
\begin{comment}
\begin{figure}\label{RJ_fig_rs_fe}
\end{figure}
\end{comment}
\section{Fischer covers of renewal systems}
\label{RJ_sec_rs_lfc}
In the attempt to find the range of the Bowen--Franks invariant over the set of SFT renewal systems, it is useful to be able to construct complicated renewal systems from simpler building blocks, but in general, it is non--trivial to study the structure of the renewal system $\mathsf{X}(L_1 \cup L_2)$ even if the renewal systems $\mathsf{X}(L_1)$ and $\mathsf{X}(L_2)$ are well understood.
\begin{comment}
Let $X$ be a shift space over an alphabet $\mathcal{A}$. A presentation $(E,\mathcal{L})$ of $X$ is said to be \emph{left--resolving} if no vertex in $E^0$ receives two edges with the same label. Fischer proved \cite{RJ_fischer} that up to labelled graph
isomorphism every irreducible sofic shift has a unique left--resolving
presentation with fewer vertices than any other left--resolving
presentation. This is called the \emph{left Fischer cover} of $X$,
and it is denoted $(F, \mathcal{L}_F)$.
\end{comment}
The goal of this section is to describe the structure of the left Fischer covers of renewal systems in order to give conditions under which the Fischer cover of $\mathsf{X}(L_1 \cup L_2)$ can be constructed when the Fischer covers of $\mathsf{X}(L_1)$ and $\mathsf{X}(L_2)$ are known.
\index{renewal system!Fischer cover of!$P_0(L)$}\index{P0@$P_0(L)$}
\index{renewal system!Fischer cover of!construction}
\label{RJ_sec_rs_lfc_construction}
Let $L$ be a generating list and define
$ P_0(L) = \left\{ \ldots w_{-2} w_{-1} w_{0} \mid w_i \in L \right \} \subseteq \mathsf{X}(L)^-$.
$P_0(L)$ is the predecessor set of the central vertex in the standard loop graph of $\mathsf{X}(L)$, but it is not necessarily the predecessor set of a right--ray in $\mathsf{X}(L)^+$, so it does not necessarily correspond to a vertex in the left Fischer cover of $\mathsf{X}(L)$. If $p \in \mathcal{B}(\mathsf{X}(L))$ is a prefix of some word in $L$, define
$ P_0(L)p = \left\{ \ldots w_{-2} w_{-1} w_{0} p \mid w_i \in L \right \} \subseteq \mathsf{X}(L)^-$.
\index{partitioning}\index{partitioning!beginning of}\index{partitioning!end of}
Let $L$ be a generating list.
A triple $(n_b,g,l)$ where $n_b,l \in \mathbb{N}$ and $g$ is an ordered list of words $g_1, \ldots , g_k \in L$ with $\sum_{i=1}^k \lvert g_i \rvert \geq n_b+l-1$ is said to be a \emph{partitioning} of the factor $v_{[n_b,n_b+l-1]} \in \mathcal{B}(\mathsf{X}(L))$ of $v = g_1 \cdots g_k$. The \emph{beginning} of the partitioning is the word $v_{[1,n_b-1]}$, and the \emph{end} is the word $v_{[n_b+l,\lvert v \rvert]}$.
A partitioning of a right--ray $x^+ \in \mathsf{X}(L)^+$ is a pair $p = (n_b,(g_i)_{i\in\mathbb{N}})$ where $n_b \in \mathbb{N}$ and $g_i \in L$ such that $wx^+ = g_1 g_2 \cdots$ when $w$ is the \emph{beginning} consisting of the $n_b-1$ first letters of the concatenation $g_1 g_2 \cdots$.
Partitionings of left--rays are defined analogously.
\index{bordering}
\index{bordering!strongly}
\index{word!bordering}
\index{word!bordering!strongly}
\index{generating list!irreducible}
Let $L \subseteq \mathcal{A}^*$ be a finite list, and let $w \in \mathcal{B}(\mathsf{X}(L)) \cup \mathsf{X}(L)^+$ be an allowed word or right--ray. Then $w$ is said to be \emph{left--bordering} if there exists a partitioning of $w$ with empty beginning, and \emph{strongly left--bordering} if every partitioning of $w$ has empty beginning. Right--bordering words and left--rays are defined analogously.
\begin{definition} \label{RJ_def_border_point}
\index{border point}\index{border point!universal}
\index{border point!generator of}
Let $L \subseteq \mathcal{A}^*$ be finite, and let $(F, \mathcal{L}_F)$ be the left Fischer cover of $\mathsf{X}(L)$. A vertex $P \in F^0$ is said to be a \emph{(universal) border point} for $L$ if there exists a (strongly) left--bordering $x^+ \in X^+$ such that $P = P_\infty(x^+)$. An intrinsically synchronizing word $w \in L^*$ is said to be a \emph{generator} of the border point $P_\infty(w) = P_\infty(w^\infty)$, and it is said to be a \emph{minimal generator} of $P$ if no prefix of $w$ is a generator of $P$.
\end{definition}
\index{renewal system!border point of}
The border points add information to the Fischer cover about the structure of the generating lists, and this information will be useful for studying $\mathsf{X}(L_1 \cup L_2)$ when the Fischer covers of $\mathsf{X}(L_1)$ and $\mathsf{X}(L_2)$ are known. If $P$ is a (universal) border point of $L$ and there is no ambiguity about which list is generating $X = \mathsf{X}(L)$, then the terminology will be abused slightly by saying that $P$ is a (universal) border point of $X$ or simply of the left Fischer cover.
\begin{lemma}
\label{RJ_lem_border_point_general_prop}
Let $L$ be a finite list generating a renewal system with left Fischer cover $(F, \mathcal{L}_F)$.
\begin{enumerate}
\item \label{RJ_lem_bp_subset}
If $P \in F^0$ is a border point, then $P_0(L) \subseteq P$, and if $P$ is universal then $P = P_0(L)$.
\item \label{RJ_lem_bp_path_to}
If $P_1,P_2 \in F^0$ are border points and if $w_1 \in L^*$ is a generator of $P_1$, then there exists a path with label $w_1$ from $P_1$ to $P_2$.
\item \label{RJ_lem_bp_path_implies}
If $P_1 \in F^0$ is a border point and $w \in L^*$, then there exists a unique border point $P_2 \in F^0$ with a path labelled $w$ from $P_2$ to $P_1$.
\item
If $\mathsf{X}(L)$ is an SFT, then every border point of $L$ has a generator.
\item \label{RJ_lem_bp_strongly_right_implies}
If $L$ has a strongly right--bordering word $w$, then $x^+ \in \mathsf{X}(L)^+$ is left--bordering if and only if $P_\infty(x^+)$ is a border point.
\end{enumerate}
\end{lemma}
\begin{comment}
\begin{proof}
\begin{enumerate}
\item
Choose a left--bordering $x^+ \in \mathsf{X}(L)^+$ such that $P = P_\infty(x^+)$ and note that $y^-x^+ \in \mathsf{X}(L)$ for each $y^- \in P_0(L)$.
\item
Choose a left--bordering $x^+ \in \mathsf{X}(L)^+$ such that $P_2 = P_\infty(x^+)$. Then $P_\infty(w_1 x^+) = P_1$ since $w_1 x^+ \in \mathsf{X}(L)^+$ and $w_1$ is intrinsically synchronizing, so there is a path labelled $w_1$ from $P_1$ to $P_2$.
\item
Choose a left--bordering $x^+ \in \mathsf{X}(L)^+$ such that $P = P_\infty(x^+)$. Since $w \in L^*$, the right--ray $w x^+$ is also left--bordering.
\item
Let $P = P_\infty(x^+)$ for some left--bordering $x^+ \in \mathsf{X}(L)^+$, and choose an intrinsically synchronizing prefix $w \in L^*$ of $x^+$. Then $P_\infty(x^+) = P_\infty(w)$, so $w$ is a generator of $P$.
\item
If $P_\infty(x^+)$ is a border point, then $w x^+ \in \mathsf{X}(L)^+$, so $x^+$ must be left--bordering. The other implication holds by definition.
\end{enumerate}
\end{proof}
\end{comment}
\begin{proof}
1.
Choose a left--bordering $x^+ \in \mathsf{X}(L)^+$ such that $P = P_\infty(x^+)$ and note that $y^-x^+ \in \mathsf{X}(L)$ for each $y^- \in P_0(L)$.
2.
Choose a left--bordering $x^+ \in \mathsf{X}(L)^+$ such that $P_2 = P_\infty(x^+)$. Then $P_\infty(w_1 x^+) = P_1$ since $w_1 x^+ \in \mathsf{X}(L)^+$ and $w_1$ is intrinsically synchronizing, so there is a path labelled $w_1$ from $P_1$ to $P_2$.
3.
Choose a left--bordering $x^+ \in \mathsf{X}(L)^+$ such that $P = P_\infty(x^+)$. Since $w \in L^*$, the right--ray $w x^+$ is also left--bordering.
4.
Let $P = P_\infty(x^+)$ for some left--bordering $x^+ \in \mathsf{X}(L)^+$, and choose an intrinsically synchronizing prefix $w \in L^*$ of $x^+$. Then $P_\infty(x^+) = P_\infty(w)$, so $w$ is a generator of $P$.
5.
If $P_\infty(x^+)$ is a border point, then $w x^+ \in \mathsf{X}(L)^+$, so $x^+$ must be left--bordering. The other implication holds by definition.
\end{proof}
\noindent
In particular, the universal border point is unique when it exists.
A predecessor set $P_\infty(x^+)$ can be a border point even though $x^+$ is not left--bordering
\begin{figure}\label{RJ_fig_aa_aaa_b}
\end{figure}
\begin{example} \label{RJ_ex_border_points}
Consider the list $L = \{ aa, aaa, b\}$ and the renewal system $\mathsf{X}(L)$. It is straightforward to check that $\mathsf{X}(L) = \mathsf{X}_\mathcal{F}$ for the set of forbidden words $\mathcal{F} = \{ bab \}$, so this is an SFT. For this shift, there are three distinct predecessor sets:
\begin{align*}
P_0 &= P_\infty(b\cdots) = \{ \cdots x_{-1}x_{0} \in \mathsf{X}(L)^- \mid x_0 = b \textnormal{ or } x_{-1}x_{0} = aa \}, \\
P_1 &= P_\infty(a^nb\cdots) = P_\infty(a^\infty) = \mathsf{X}(L)^- , \qquad n \geq 2, \\
P_2 &= P_\infty(ab\cdots) = \{ \cdots x_{-1}x_{0} \in \mathsf{X}(L)^- \mid x_{0} = a \}.
\end{align*}
The information contained in these equations is sufficient to draw the left Krieger cover, and each set is the predecessor set of an intrinsically synchronising right--ray, so the left Fischer cover can be identified with the left Krieger cover. This graph is shown in Fig.\ \ref{RJ_fig_aa_aaa_b}.
Here, $P_0$ is a universal border point because any right--ray starting with a $b$ is strongly left bordering. The generating word $b$ is a minimal generator of $P_0$. The vertex $P_1$ is a border point because $a^nb\cdots$ is left bordering for all $n \geq 2$.
The word $aa$ is a minimal generator of $P_1$, and $aab$ is a non--minimal generator. The vertex $P_2$ is not a border point since there is no infinite concatenation $x^+$ of words from $L$ such that $x^+= ab\cdots$.
Another way to see this is to note that every path terminating at $P_2$ has $a$ as a suffix, so that $P_0$ is not a subset of $P_2$ which together with Lemma \ref{RJ_lem_border_point_general_prop}(\ref{RJ_lem_bp_subset}) implies that $P_2$ is not a border point.
Note also that Lemma \ref{RJ_lem_border_point_general_prop}(\ref{RJ_lem_bp_path_to}) means that there must be paths labelled $b$ from $P_0$ to the two border points, and similarly, paths labelled $aa$ and $aab$ from $P_1$ to the two border points.
\end{example}
\index{renewal system!addition of}
Consider two renewal systems $\mathsf{X}(L_1)$ and $\mathsf{X}(L_2)$. The \emph{sum} $\mathsf{X}(L_1) + \mathsf{X}(L_2)$ is the renewal system $\mathsf{X}(L_1 \cup L_2)$.
Generally, it is non--trivial to construct the Fischer cover of such a sum even if the Fischer covers of the summands are known.
\begin{comment}
\begin{lemma}
If $X_1, X_2$ are renewal systems with disjoint alphabets, then the left Fischer covers of $X_1$ and $X_2$ are subgraphs of the left Fischer cover of $X_1 + X_2$.
\end{lemma}
\begin{proof}
If $x^+,y^+ \in X_1^+$ and $P_\infty(x^+) \neq P_\infty(y^+)$, then $x^+,y^+ \in X_1 + X_2$, and the corresponding predecessor sets are also different in $X_1 + X_2$ since the alphabets are disjoint.
\end{proof}
\noindent
In general, there will also be other vertices in the left Fischer cover of the sum, and it is not trivial to determine how to draw the edges connecting the two subgraphs, however, the results of Sec.\ \ref{RJ_sec_border_point} put some limits on the structure of the Fischer cover of a sum of renewal systems.
\end{comment}
\index{generating list!modular}
\begin{definition}
Let $L$ be a generating list with universal border point $P_0$ and let $(F, \mathcal{L}_F)$ be the left Fischer cover of $\mathsf{X}(L)$.
$L$ is said to be \emph{left--modular} if for all $\lambda \in F^*$ with
$r(\lambda) = P_0$, $\mathcal{L}_F(\lambda) \in L^*$ if and only if $s(\lambda)$ is a border point. \emph{Right--modular} generating lists are defined analogously.
\begin{comment}
Let $L$ be a generating list with universal border point $P_0$, and let $(F, \mathcal{L}_F)$ be the left Fischer cover of $\mathsf{X}(L)$.
$L$ is said to be \emph{left--modular} if $\mathcal{L}_F(\lambda) \in L^*$ whenever $\lambda \in F^*$, $r(\lambda) = P_0$, and $s(\lambda)$ is a border point. \emph{Right--modular} generating lists are defined analogously.
\end{comment}
\end{definition}
\index{renewal system!modular}
\noindent
It is straightforward to check that the list considered in Example \ref{RJ_ex_border_points} is left--modular.
When $L$ is left--modular and there is no doubt about which generating list is used, the renewal system $\mathsf{X}(L)$ will also be said to be \emph{left--modular}.
\begin{lemma} \label{RJ_lem_strongly_bordering_implies_modular}
If $L$ is a generating list with a strongly left--bordering word $w_l$ and a strongly right--bordering word $w_r$, then it is both left-- and right--modular.
\end{lemma}
\begin{proof}
Let $(F, \mathcal{L}_F)$ be the left Fischer cover of $\mathsf{X}(L)$, let $P \in F^0$ be a border point, and choose $x^+ \in \mathsf{X}(L)^+$ such that $w_lx^+ \in \mathsf{X}(L)^+$. Assume that there is a path from $P$ to $P_0(L) = P_\infty(w_lx^+)$ with label $w$. The word $w_r$ has a partitioning with empty end, so there is a path labelled $w_r$ terminating at $P$. It follows that $w_r w w_l x^+ \in \mathsf{X}(L)^+$, so $w \in L^*$. By symmetry, $L$ is also right--modular.
\begin{comment}
Let $(F, \mathcal{L}_F)$ be the left Fischer cover of $\mathsf{X}(L)$, let $P_0(L) = P_\infty(w_l^\infty) \in F^0$ be the universal border point, let $P$ be a border point, and assume that there is a path labelled $w$ from $P$ to $P_0$. Then $P = P_\infty(w w_l^\infty)$, so $w w_l^\infty$ is left--bordering by Lemma \ref{RJ_lem_border_point_general_prop}(\ref{RJ_lem_bp_strongly_right_implies}). It follows that $w_r w w_l^\infty \in \mathsf{X}(L)^+$, so $w \in L^*$. By symmetry, $L$ is also right--modular.
\end{comment}
\end{proof}
For $i \in \{1,2\}$, let $L_i$ be a left--modular generating list and let $X_i = \mathsf{X}(L_i)$ have alphabet $\mathcal{A}_i$ and left Fischer cover $(F_i, \mathcal{L}_i)$. Let $P_i \in F_i^0$ be the universal border point of $L_i$. Assume that $\mathcal{A}_1 \cap \mathcal{A}_2 = \emptyset$.
The left Fischer cover of $X_1 + X_2$ will turn out to be the labelled graph $(F_+, \mathcal{L}_+)$ obtained by taking the union of $(F_1, \mathcal{L}_1)$ and $(F_2, \mathcal{L}_2)$, identifying the two universal border points $P_1$ and $P_2$, and adding certain connecting edges.
To do this formally, introduce a new vertex $P_+$ and define
$F_+^0 = ( F_1^0 \cup F_2^0 \cup \{P_+\} ) \setminus \{ P_1, P_2 \}$.
Define maps $f_i \colon F_i^0 \to F_+^0$ such that for $v \in F_i^0 \setminus \{ P_i \}$, $f_i(v)$ is the vertex in $F_+^0$ corresponding to $v$ and such that $f_i(P_i) = P_+$. For each $e \in F_i^1$, define an edge $e' \in F_+^1$ such that $s(e') = f_i(s(e))$, $r(e') = f_i(r(e))$, and $\mathcal{L}_+(e') = \mathcal{L}_i(e)$. For each $e \in F_1^1$ with $r(e) = P_1$ and each non--universal border point $P \in F_2^0$, draw an additional edge $e' \in F_+^1$ with $s(e') = f_1(s(e))$, $r(e') = f_2(P)$, and $\mathcal{L}_+(e') = \mathcal{L}_1(e)$. Draw analogous edges for each $e \in F_2^1$ with $r(e) = P_2$ and every non--universal border point $P \in F_1^0$.
This construction
is illustrated in Fig.\ \ref{RJ_fig_modular_add}.
\begin{figure}\label{RJ_fig_modular_add}
\end{figure}
\begin{comment}
Define $(F', \mathcal{L}')$ to be the labelled graph obtained by taking the union of $(F_1, \mathcal{L}_1)$ and $(F_2, \mathcal{L}_2)$ and identifying the two universal border points $P_1$ and $P_2$. The vertex obtained by merging $P_1$ and $P_2$ will be denoted $P_+$.
Identify each $P \in F'^0 \setminus \{ P_+ \}$ with the corresponding vertex in $F_1^0$ or $F_2^0$.
Let $(F_+, \mathcal{L}_+)$ be the labelled graph obtained from $(F', \mathcal{L}')$ by adding connecting edges between the two parts of the graph such that for each $e \in F_i^1$ with $r(e) = P_i$ and each non--universal border point $Q$ in the other Fischer cover there is an edge $e'$ with $s(e') = s(e)$, $\mathcal{L}_+(e) = \mathcal{L}_1(e)$, and $r(e') = Q$.
\end{comment}
\begin{proposition}
\label{RJ_prop_addition_modular}
If $L_1$ and $L_2$ are left--modular generating lists with disjoint alphabets, then $L_1 \cup L_2$ is left--modular, the left Fischer cover of $\mathsf{X}(L_1 \cup L_2)$ is the graph $(F_+, \mathcal{L}_+)$ constructed above, and the vertex $P_+ \in F_+^0$ is the universal border point of $L_1 \cup L_2$.
\end{proposition}
\begin{proof}
By construction, the labelled graph $(F_+, \mathcal{L}_+)$ is irreducible, left--resolving, and predecessor--separated, so it is the left Fischer cover of some sofic shift $X_+$ \cite[Cor. 3.3.19]{RJ_lind_marcus}.
Given $w \in L_1^*$, there is a path with label $w$ in the left Fischer cover of $X_1$ from some border point $P \in F_1^0$ to the universal border point $P_1$ by Lemma \ref{RJ_lem_border_point_general_prop}(\ref{RJ_lem_bp_path_implies}). Hence, there is also a path labelled $w$ in $(F_+, \mathcal{L}_+)$ from the vertex corresponding to $P$ to the vertex $P_+$. This means that for every border point $Q \in F_2^0$, $(F_+, \mathcal{L}_+)$ contains a path labelled $w$ from the vertex corresponding to $P$ to the vertex corresponding to $Q$. By symmetry, it follows that every element of $(L_X \cup L_Y)^*$ has a presentation in $(F_+, \mathcal{L}_+)$. Hence, $\mathsf{X}(L_1 \cup L_2) \subset X_+$.
Assume that $awb \in \mathcal{B}(X_+)$ with $a,b \in \mathcal{A}_1$ and $w \in \mathcal{A}_2^*$. Then there must be a path labelled $w$ in $(F_+, \mathcal{L}_+)$ from a vertex corresponding to a border point $P$ of $L_2$ to $P_+$. By construction, this is only possible if there is also a path labelled $w$ from $P$ to $P_2$ in $(F_2, \mathcal{L}_2)$, but $L_2$ is left--modular, so this means that $w \in L_2^*$. By symmetry, $\mathsf{X}(L_1 \cup L_2) = X_+$, and $P_+$ is the universal border point by construction.
\end{proof}
\begin{comment}
\subsection{Constructing renewal systems}
Let $F$ be a finite list of finite words. The goal is to construct an irreducible generating list $L_F$ such that $\mathsf{X}(L_F) = \mathsf{X}_F$. Since not all shifts of finite type are renewal systems, this is not possible for every list $F$. However, it may be possible to do this in a controlled manner whenever $F$ \emph{is} the list of forbidden words for an SFT renewal system.
Let $m = \max_{w \in F} |w|$, and let $L = \{ v \in \mathcal{A}^* \mid |v| \leq m\}$. A \emph{forbidden subword} is a word $v \in L$ for which there exists $w \in F$ such that $v$ is a subword of $w$ or $w$ is a subword of $v$. The set of forbidden subwords is denoted $F_S$.
The \emph{forbidden prefixes} are the words $w_{[1,j]}$ for
which $w \in F$ and $2 \leq j \leq |w|$.
The \emph{forbidden suffixes} are the words $w_{[2,|w|]}$ for which $w \in F$. Define $L_F^+$ to be the subset of $L$ obtained by removing all words which contain a forbidden subword, or have a forbidden prefix or suffix:
\begin{displaymath}
L_F^+ = L \setminus (F_S \cup
\{ vw_{[1,j]} \mid w \in F, 2 \leq j \leq |w| \}
\cup
\{w_{[2,|w|]}v \mid w \in F \}).
\end{displaymath}
In this construction, a direction was chosen, so another list $L_F^-$ can be defined analogously:
\begin{displaymath}
L_F^- = L \setminus (F_S \cup
\{ w_{[j,|w|]}v \mid w \in F, 1 \leq j \leq |w|-1 \}
\cup
\{ v w_{[1,|w|-1]} \mid w \in F \})\;.
\end{displaymath}
Note that the forbidden subwords are common to $L_F^+$ and $L_F^-$ while the forbidden prefixes and forbidden suffixes depend on whether $L_F^+$ or $L_F^-$ is considered. Note also that for each letter $a \in \mathcal{A}$, both $L_F^+$ and $L_F^-$ contain a word which has $a$ as a prefix and a word which has $a$ as a suffix.
\begin{example}
This construction is easy if each word in the list of forbidden words $F$ is a power of one of the letters from $\mathcal{A}$. If, for example, $\mathcal{A} = \{a,b\}$ and $F = \{aaa\}$, then $L_F^+ = L_F^- = \{b, ab, ba, bb, aba, abb, bab, bba, bbb \}$, and $\mathsf{X}_F = \mathsf{X}(L_F^+) = \mathsf{X}(L_F^-)$. Note also, that the same renewal system is generated by the shorter list $\{b, ab,ba,aba \}$.
\end{example}
\begin{lemma} \label{RJ_lem_w_in_BXLF_implies_w_in_BXF}
If $w \in \mathcal{B}(\mathsf{X}(L_F^\pm))$, then $w \in \mathcal{B}(\mathsf{X}_F)$.
\end{lemma}
\begin{proof}
Assume $w \in F \cap \mathcal{B}(\mathsf{X}(L_F^+))$. Since $w$ is not a subword of any word in $L_F^+$ and since no word in $L_F^+$ is a subword of $w$ there must exist non trivial words $u,v \in L_F^+$ such that $w$ is a subword of $uv$. This is in contradiction with the definition of $L_F^+$. An analogous argument works for $L_F^-$.
\end{proof}
Note that $L_F^\pm$ sometimes defines a shift space with a strictly larger set of forbidden words than $F$. Indeed, $\mathsf{X}(L_F^\pm)$ can be a strictly sofic shift even though $F$ is finite.
\begin{proposition}
Let $\mathcal{A} = a_1, \ldots, a_k, a_{k+1}, \ldots ,
a_{k+f}$, let $n_1, \ldots, n_k \in \mathbb{N}$ and $F = \{ a_i^{n_i} | 1 \leq i \leq k \}$. Then $\mathsf{X}_F = \mathsf{X}(L_F^\pm)$.
\end{proposition}
Note that the letters $a_{k+1}, \ldots , a_{k+f}$ are free in the sense that none of the forbidden words contain any of these letters.
\begin{proof}
The proof will be carried out for $L_F^+$; an analogous argument works for $L_F^-$. Assume whitout loss of generality that $n = n_1 \geq n_2 \geq \cdots \geq n_k \geq 2$.
The first goal is to prove that $\mathcal{B}_n(\mathsf{X}_F) = \mathcal{B}_n(\mathsf{X}(L_F^+))$.
One inclusion follows from Lemma
\ref{RJ_lem_w_in_BXLF_implies_w_in_BXF}. Assume that
$w \in \mathcal{B}(\mathsf{X}_F)$ and that $|w| = n$.
Consider the word $w' = w_2 \cdots w_n$. If $w'$ does not have a
forbidden prefix (i.e.\ $a_i^{n_i-1}$for some $i$), then there
exists a word in $L_F^+$ with $w'$ as a prefix, so $w \in
\mathcal{B}(\mathsf{X}(L_F^+))$ since there exists a word in $L_F^+$ with $w_1$ as a
suffix.
On the other hand, if there exists $1 \leq i \leq k$ such that $w =
w_1 a_i^{n_i-1} w_{n_i + 1} \ldots w_n$, then $w_1, w_{n_i + 1} \neq
a_i$. In particular, $w_i a_i$ is an allowed suffix, and $a_i^{n_i-2}
w_{n_i + 1} \ldots w_n$ is an allowed prefix, so $w \in
\mathcal{B}(\mathsf{X}(L_F^+))$.
Now, it is sufficient to use Theorem \ref{RJ_thm_m-step}
to prove that $\mathsf{X}(L_F^+)$ is a $(n-1)$--step shift of finite
type. Consider a word $w \in
\mathcal{B}_{n-1}(\mathsf{X}_F^+) = \mathcal{B}_{n-1}(\mathsf{X}(L_F^+))$ and a letter $a$ such that $aw \in \mathcal{B}_{n}(\mathsf{X}_F^+) = \mathcal{B}_{n}(\mathsf{X}(L_F^+))$. Let $p = p_1 \cdots p_l$ be a minimal partitioning of $w$.\fxfatal{Wrong use of partitioning} Let $w'$ be the shortest word such that $w$ is a prefix of $w'p_2 \cdots p_l$. Now, it is sufficient to prove that there is a word in $L_F^+$ which has $aw'$ as a suffix. Note that $p_1 \in L_F^+$ has $w'$ as a suffix, so this is not an illegal suffix. Choose any $b \neq a$. Then $baw'$ has no illegal prefix either, and hence $baw' \in L_F^+$. By condition 2) from Proposition \ref{RJ_prop_sufficient_for_SFT} this implies that $\mathsf{X}(L_F^+)$ is a $(n-1)$--step shift of finite type.
\end{proof}
\end{comment}
\index{fragmentation}
Let $X$ be a shift space over the alphabet $\mathcal{A}$. Given $a \in \mathcal{A}$, $k \in \mathbb{N}$, and new symbols $a_1, \ldots, a_k \notin \mathcal{A}$ consider the map
$f_{a,k} \colon (\mathcal{A} \setminus \{ a \}) \cup \{ a_1, \ldots , a_k\} \to \mathcal{A}$ defined by
$f_{a,k}(a_i) = a$ for each $1 \leq i \leq k$ and $f_{a,k}(b) = b$ when $b \in \mathcal{A} \setminus \{a \}$. Let $F_{a,k} \colon ((\mathcal{A} \setminus \{ a \}) \cup \{ a_1, \ldots , a_k\})^* \to \mathcal{A}^*$ be the natural extension of $f_{a,k}$.
If $w \in \mathcal{A}^*$ contains $l$ copies of the symbol $a$, then the preimage $F_{a,k}^{-1}(\{w \})$ is the set consisting of the $k^l$ words that can be obtained by replacing the $a$s by the symbols $a_1, \ldots, a_k$.
\begin{definition}
\index{renewal system!fragmentation of}
\index{shift space!fragmentation of}
\index{X*ak@$X_{a,k}$}
\index{F*ak@$F_{a,k}$}
Let $X = \mathsf{X}_\mathcal{F}$ be a shift space over the alphabet $\mathcal{A}$, let $a \in \mathcal{A}$, let $a_1, \ldots, a_k \notin \mathcal{A}$, and let $F_{a,k}$ be defined as above. Then the shift space $X_{a,k} = \mathsf{X}_{F_{a,k}^{-1}(\mathcal{F})}$ is said to be the shift obtained from $X$ by \emph{fragmenting} $a$ into $a_1, \ldots, a_k$.
\end{definition}
\index{L*@$L_{a,k}$}
\noindent
Note that this construction does not depend on the choice of $\mathcal F$ representing $X$, in particular, $\mathcal{B}( X_{a,k} ) = F_{a,k}^{-1}(\mathcal{B}(X))$. Furthermore, $X_{a,k}$ is an SFT if and only if $X$ is an SFT.
If $X$ is an irreducible sofic shift, then the left and right Fischer and Krieger covers of $X_{a,k}$ are obtained by replacing each edge labelled $a$ in the corresponding cover of $X$ by $k$ edges labelled $a_1, \ldots, a_k$.
Note that $X$ and $X_{a,k}$ are not generally conjugate or even flow equivalent.
If $X = \mathsf{X}(L)$ is a renewal system, then $X_{a,k}$ is the renewal system generated by the list $L_{a,k} = F_{a,k}^{-1}(L)$.
\begin{remark}
\label{RJ_rem_fragmentation}
\label{RJ_rem_sum_frag_commute}
Let $A$ be the symbolic adjacency matrix of the left Fischer cover of an SFT renewal system $\mathsf{X}(L)$ with alphabet $\mathcal{A}$. Given $a \in \mathcal{A}$ and $k \in \mathbb{N}$, define $f \colon \mathcal{A} \to \mathbb{N}$ by $f(a) = k$ and $f(b) = 1$ for $b \neq a$. Extend $f$ to the set of finite formal sums over $\mathcal{A}$ in the natural way and consider the integer matrix $f(A)$. Then $f(A)$ is the adjacency matrix of the underlying graph of the left Fischer cover of $\mathsf{X}(L_{a,k})$.
For lists over disjoint alphabets, it follows immediately from the definitions that fragmentation and addition commute.
\end{remark}
\begin{comment}
\begin{lemma}
Let $L_1 \subseteq \mathcal{A}_1^*$, $L_2 \subseteq \mathcal{A}_2^*$ with $\mathcal{A}_1 \cap \mathcal{A}_2 = \emptyset$, let $a \in \mathcal{A}_1$, and choose new symbols $a_1, \ldots , a_k \notin \mathcal{A}_1 \cup \mathcal{A}_2$. Then the renewal system obtained from $\mathsf{X}(L_1 \cup L_2)$ by fragmenting $a$ to $a_1, \ldots , a_k$ is $X_{a,k} + \mathsf{X}(L_2)$ where $X_{a,k}$ is the renewal system obtained from $\mathsf{X}(L_1)$ by fragmenting $a$ to $a_1, \ldots , a_k$.
\end{lemma}
\end{comment}
\section{Entropy and flow equivalence}
\label{RJ_sec_rs_entropy}
Hong and Shin \cite{RJ_hong_shin} have constructed a class $H$ of lists generating SFT renewal systems such that $\log \lambda$ is the entropy of an SFT if and only if there exists $L \in H$ with $h(\mathsf{X}(L)) = \log \lambda$, and this is arguably the most powerful general result known about the invariants of SFT renewal systems. In the following, the renewal systems generated by lists from $H$ will be classified up to flow equivalence.
As demonstrated in \cite{RJ_johansen_thesis}, it is difficult to construct renewal systems with non--cyclic Bowen--Franks groups and/or positive determinants directly, and this classification will yield hitherto unseen values of the invariant.
The construction of the class $H$ of generating lists considered in \cite{RJ_hong_shin} will be modified slightly since some of the details of the original construction are invisible up to flow equivalence. In particular, several words from the generating lists can be replaced by single symbols by using symbol reduction. Additionally, there are extra conditions on some of the variables in \cite{RJ_hong_shin} which will be omitted here since the larger class can be classified without extra work.
\index{generating list!in $B$}
\index{B*@$B$}
Let $r \geq 2$ and let $n_1, \ldots, n_r,c_1, \ldots , c_r,d, N \in \mathbb{N}$, and let $W$ be the set consisting of the following words:
\begin{itemize}
\item $\alpha_i = \alpha_{i,1} \cdots \alpha_{i,n_1}$ for $1 \leq i \leq c_1$
\item $\tilde \alpha_i = \tilde \alpha_{i,1} \cdots \tilde \alpha_{i, n_1}$ for $1 \leq i \leq c_1$
\item $\gamma_{k,i_k} = \gamma_{k,i_k,1} \cdots \gamma_{k,i_k,n_k}$ for $2 \leq k \leq r$ and $1 \leq i_k \leq c_k$
\item $\alpha_{i_1} \gamma_{2,i_2} \cdots \gamma_{r,i_r} \beta_l^N$
for $1 \leq i_j \leq c_j$ and $1 \leq l \leq d$
\item $\beta_l^N \tilde \alpha_{i_1} \gamma_{2,i_2} \cdots \gamma_{r,i_r}$
for $1 \leq i_j \leq c_j$ and $1 \leq l \leq d$.
\end{itemize}
The set of generating lists of this form will be denoted $B$.
\begin{remark}
\label{RJ_rem_R_def}
\index{generating list!in $R$}
\index{R*@$R$}
Symbol reduction can be used to reduce the words $\alpha_i$, $\tilde \alpha_i$, $\gamma_{k,i_k}$, and $\beta_l^N$ to single letters \cite[Lemmas 2.15 and 2.23]{RJ_johansen_thesis}, so up to flow equivalence, the list $W \in B$ considered above can be replaced by the list $W'$ consisting of the one--letter words $\alpha_i$, $\tilde \alpha_i$, and $\gamma_{k,i}$ as well as the words
\begin{itemize}
\item $\alpha_{i_1} \gamma_{2,i_2} \cdots \gamma_{r,i_r} \beta_l$
for $1 \leq i_j \leq c_j$ and $1 \leq l \leq d$
\item $\beta_l \tilde \alpha_{i_1} \gamma_{2,i_2} \cdots \gamma_{r,i_r}$
for $1 \leq i_j \leq c_j$ and $1 \leq l \leq d$.
\end{itemize}
Furthermore, if
\begin{equation} \label{RJ_eq_reduced_list}
L = \{ \alpha, \tilde \alpha,
\alpha \gamma_2 \cdots \gamma_r \beta, \beta \tilde \alpha \gamma_2 \cdots \gamma_r \} \cup \{ \gamma_k \mid 2 \leq k \leq r\} \;,
\end{equation}
then $\mathsf{X}(W')$ can be obtained from $\mathsf{X}(L)$ by fragmenting $\alpha$ to $\alpha_1, \ldots, \alpha_{c_1}$, $\beta$ to $\beta_1, \ldots , \beta_l$ and so on.
Let $R$ be the set of generating lists of the form given in (\ref{RJ_eq_reduced_list}).
\end{remark}
\index{H@$H$}\index{generating list!in $H$}
Next consider generating lists $W_1, \ldots , W_m \in B$ with disjoint alphabets, and let $W = \bigcup_{j=1}^m W_j$.
Let $\tilde W$ be a finite set of words that do not share any letters with each other or with the words from $W$, and consider the generating list $W \cup \tilde W$. Let $\tilde H$ be the set of generating lists that can be constructed in this manner. Let $\mu$ be a Perron number. Then there exists $\tilde L \in \tilde H$ such that $\mathsf{X}(\tilde L)$ is an SFT and $h(\mathsf{X}(\tilde L)) = \log \mu$ \cite{RJ_hong_shin}.
\begin{remark} \label{RJ_rem_H_letter_frag}
If $W \cup \tilde W \in \tilde H$ as above, then symbol reduction can be used to show that $\mathsf{X}(W \cup \tilde W)$ is flow equivalent to the renewal system generated by the union of $W$ and $\lvert \tilde W \rvert$ new letters \cite[Lemma 2.23]{RJ_johansen_thesis}, i.e.\ $\mathsf{X}(W \cup \tilde W)$ is flow equivalent to a fragmentation of $\mathsf{X}(W \cup \{ a \})$ when $a \notin \mathcal{A}(\mathsf{X}(W))$.
\end{remark}
\index{renewal system!entropy of}
Consider a generating list $\tilde L \in \tilde H$ and $p \in \mathbb{N}$. For each letter $a \in \mathcal{A}( \mathsf{X}(\tilde L))$, introduce new letters $a_1, \dots , a_p \notin \mathcal{A}( \mathsf{X}(\tilde L))$, and let $L$ denote the generating list obtained by replacing each occurrence of $a$ in $\tilde L$ by the word $a_1\cdots a_p$.
Let $H$ denote the set of generating lists that can be obtained from $\tilde H$ in this manner. Let $\lambda$ be a weak Perron number. Then there exists $L \in H$ such that $\mathsf{X}(L)$ is an SFT and $h(\mathsf{X}(L)) = \log \lambda$ \cite{RJ_hong_shin}.
\begin{remark}
\label{RJ_rem_H_tilde_fe}
If $L$ is obtained from $\tilde L \in \tilde H$ as above, then $\mathsf{X}(L) \sim_{\textrm{FE}} \mathsf{X}(\tilde L)$ since the modification can be achieved using symbol expansion of each $a \in \mathcal{A}( \mathsf{X}(\tilde L))$.
\end{remark}
\begin{comment}
\begin{proof}
By assumption, there exists $p \in \mathbb{N}$ such that $\mu = \lambda^p$ is a Perron number. Use Lemma \ref{RJ_lem_hs} to find $\tilde L \in \tilde H$ such that $h( \mathsf{X}(\tilde L) ) = \log \mu$. Next replace each letter $a$ in the alphabet by a word $a_1 \cdots a_p$ to obtain the generating list of an SFT renewal system with entropy $\log \lambda$.
\end{proof}
\end{comment}
The next step is to prove that the building blocks in the class $R$ introduced in Remark \ref{RJ_rem_R_def} are left--modular, and to construct the Fischer covers of the corresponding renewal systems. As the following lemmas show, this will allow a classification of the renewal systems generated by lists from $H$ via addition and fragmentation. The first result follows immediately from Remarks \ref{RJ_rem_sum_frag_commute}, \ref{RJ_rem_R_def}, \ref{RJ_rem_H_letter_frag}, and \ref{RJ_rem_H_tilde_fe}.
\begin{lemma}
\label{RJ_lem_entropy_fe}
For each $L \in H$, there exist $L_1, \ldots , L_m \in R$ such that $\mathsf{X}(L)$ is flow equivalent to a fragmentation of $\mathsf{X} ( \bigcup_{j=0}^m L_j )$, where $L_0 = \{ a \}$ for some $a$ that does not occur in $L_1, \ldots , L_m$.
\end{lemma}
\begin{lemma}
\label{RJ_lem_entropy_lfc}
If $L \in R$, then $L$ is left--modular, $\mathsf{X}(L)$ is an SFT, and the left Fischer cover of $\mathsf{X}(L)$ is the labelled graph shown in Fig.\ \ref{RJ_fig_entropy_lfc}.
\end{lemma}
\begin{proof}
Let
\begin{equation}
\label{RJ_eq_R_L}
L = \{ \alpha, \tilde \alpha,
\alpha \gamma_2 \cdots \gamma_r \beta, \beta \tilde \alpha \gamma_2 \cdots \gamma_r \} \cup \{ \gamma_k \mid 2 \leq k \leq r\} \in R \;.
\end{equation}
The word $\alpha \gamma_2 \cdots \gamma_r \beta \beta \tilde \alpha \gamma_2 \cdots \gamma_r$ is strongly left-- and right--bordering, so $L$ is left-- and right--modular by Lemma \ref{RJ_lem_strongly_bordering_implies_modular}. Let $P_0 = P_0(L)$.
If $x^+ \in \mathsf{X}(L)^+$ does not have a suffix of a product of the generating words $\alpha \gamma_2 \cdots \gamma_r \beta$ and $\beta \tilde \alpha \gamma_2 \cdots \gamma_r$ as a prefix, then $x^+$ is strongly left--bordering, so $P_\infty(x^+) = P_0$. Hence, to determine the rest of the predecessor sets and thereby the vertices of the left Fischer cover, it is sufficient to consider right--rays that do have such a prefix.
Consider first $x^+ \in \mathsf{X}(L)^+$ such that $\beta x^+ \in \mathsf{X}(L)^+$. The letter $\beta$ must come from either $\alpha \gamma_2 \cdots \gamma_r \beta$ or $\beta \tilde \alpha \gamma_2 \cdots \gamma_r$, so the beginning of a partitioning of $\beta x^+$ must be either empty or equal to $\alpha \gamma_2 \cdots \gamma_r$.
Assume first that every partitioning of $\beta x^+$ has beginning $\alpha \gamma_2 \cdots \gamma_r$ (i.e.\ that $\tilde \alpha \gamma_2 \cdots \gamma_r$ is not a prefix of $x^+$).
In this case, $\beta x^+$ must be preceded by $\alpha \gamma_2 \cdots \gamma_r$, and the corresponding predecessor sets are:
\begin{align}
\label{RJ_eq_entropy_predecessor_sets}
P_\infty( \alpha \gamma_2 \cdots \gamma_r \beta x^+ ) &= P_0 \nonumber \\
P_\infty( \gamma_2 \cdots \gamma_r \beta x^+ ) &= P_0 \alpha = P_1 \nonumber \\
&\;\, \vdots \\
P_\infty( \gamma_r \beta x^+ ) &= P_0\alpha \gamma_2 \cdots \gamma_{r-1} = P_{r-1} \nonumber \\
P_\infty( \beta x^+ ) &= P_0\alpha \gamma_2 \cdots \gamma_{r-1}\gamma_{r} = P_{r} \;. \nonumber
\end{align}
Assume now that there exists a partitioning of $\beta x^+$ with empty beginning (e.g.\ $x^+ = \beta \tilde \alpha \gamma_2 \cdots \gamma_r^\infty$).
The first word used in such a partitioning must be $\beta \tilde \alpha \gamma_2 \cdots \gamma_r$. Replacing this word by the concatenation of the generating words $\alpha \gamma_2 \cdots \gamma_r \beta$, $\tilde \alpha, \gamma_2, \ldots, \gamma_r$ creates a partitioning of $\beta x^+$ with beginning $\alpha \gamma_2 \cdots \gamma_r $, so in this case:
\begin{align*}
P_\infty( \alpha \gamma_2 \cdots \gamma_r \beta x^+ ) &= P_0 \\
P_\infty( \gamma_2 \cdots \gamma_r \beta x^+ ) &= P_0 \cup P_0\alpha = P_0\\
&\;\, \vdots \\
P_\infty( \gamma_r \beta x^+) &= P_0 \cup P_0\alpha \gamma_2 \cdots \gamma_{r-1} = P_0 \\
P_\infty( \beta x^+) &= P_0 \cup P_0\alpha \gamma_2 \cdots \gamma_{r-1}\gamma_{r} = P_0\;.
\end{align*}
The argument above proves that there are no right--rays such that every partitioning of $\beta x^+$ has empty beginning.
\begin{figure}\label{RJ_fig_entropy_lfc}
\end{figure}
It only remains to investigate right--rays that have a suffix of $\beta \tilde \alpha \gamma_2 \cdots \gamma_r$ as a prefix.
A partitioning of a right--ray $\gamma_r x^+$ may have empty beginning (e.g.\ $x^+ = \gamma_r^\infty$), beginning $\alpha \gamma_2 \cdots \gamma_{r-1}$ (e.g.\ $x^+ = \beta \beta \tilde \alpha \gamma_2 \cdots \gamma_r \cdots$ or $x^+ = \beta \tilde \alpha \gamma_2 \cdots \gamma_r^\infty$), or beginning $\beta \tilde \alpha \gamma_2 \cdots \gamma_{r-1}$ (e.g.\ $x^+ = \gamma_r^\infty)$. Note that there is a partitioning with empty beginning if and only if there is a partitioning with beginning $\beta \tilde \alpha \gamma_2 \cdots \gamma_{r-1}$.
If there exists a partitioning of $\gamma_r x^+$ with beginning $\alpha \gamma_2 \cdots \gamma_{r-1}$, then $\beta$ must be a prefix of $x^+$, so the right--ray $\gamma_r x^+$ has already been considered above.
Hence, it suffices to consider the case where there exists a partitioning of $\gamma_r x^+$ with empty beginning and a partitioning with beginning $\beta \tilde \alpha \gamma_2 \cdots \gamma_{r-1}$ but no partitioning with beginning $\alpha \gamma_2 \cdots \gamma_{r-1}$. In this case, the predecessor sets are
\begin{align*}
P_\infty( \gamma_{r} x^+ ) &= P_0 \cup P_0\beta \tilde \alpha \gamma_2 \cdots \gamma_{r-1} = P_{2r}\\
&\;\,\vdots \\
P_\infty( \gamma_2 \cdots \gamma_{r} x^+ ) &= P_0 \cup P_0\beta \tilde \alpha = P_{r+2}\\
P_\infty( \tilde \alpha \gamma_2 \cdots \gamma_{r} x^+ ) &= P_0 \cup P_0\beta = P_{r+1}\\
P_\infty( \beta \tilde \alpha \gamma_2 \cdots \gamma_{r} x^+ ) &= P_0
\cup P_0\alpha \gamma_2 \cdots \gamma_{r} = P_0 \; .
\end{align*}
\begin{comment}
\begin{align*}
&P_\infty( \beta \tilde \alpha \gamma_2 \cdots \gamma_{r} x^+ ) = P_0
&\textrm{when } \beta \tilde \alpha \gamma_2 \cdots \gamma_{r} x^+ \in \mathsf{X}(L)^+ \\
&P_\infty( \tilde \alpha \gamma_2 \cdots \gamma_{r} x^+ ) = P_0 \cup P_0\beta = P_{r+1}
&\textrm{when } \tilde \alpha \gamma_2 \cdots \gamma_{r} x^+ \in \mathsf{X}(L)^+ \\
&P_\infty( \gamma_2 \cdots \gamma_{r} x^+ ) = P_0 \cup P_0\beta \tilde \alpha = P_{r+2}
&\textrm{when } \gamma_2 \cdots \gamma_{r} x^+ \in \mathsf{X}(L)^+ \\
& \qquad \vdots \\
&P_\infty( \gamma_{r} x^+ ) = P_0 \cup P_0\beta \tilde \alpha \gamma_2 \cdots \gamma_{r-1} = P_{2r} \;.
\end{align*}
\end{comment}
Now all right--rays have been investigated, so there are exactly $2r +1$ vertices in the left Krieger cover of $\mathsf{X}(L)$. The vertex $P_0$ is the universal border point, and the vertices $P_{r+1}, \ldots, P_{2r}$ are border points, while none of the vertices $P_1, \ldots, P_r$ are border points. This gives the information needed to draw the left Fischer cover.
In \cite{RJ_hong_shin} it is proved that all renewal systems in the class $B$ are SFTs. That proof will also work for the related class $R$ considered here, but the result also follows easily from the structure of the left Fischer cover constructed above \cite[Lemma 5.46]{RJ_johansen_thesis}.
\end{proof}
\begin{comment}
\noindent
\begin{lemma}
\label{RJ_lem_entropy_sft}
For each $L \in R$, $\mathsf{X}(L)$ is an SFT.
\end{lemma}
\begin{proof}
Let $L$ be defined as in (\ref{RJ_eq_R_L}). $\mathsf{X}(L)$ is an SFT if and only if the covering map of the left Fischer cover $(F, \mathcal{L})$ is injective. Assume that $\lambda, \mu$ are biinfinite paths in $F$ such that $\mathcal{L}(\lambda) = \mathcal{L}(\mu) = x \in X(L)$.
If there is an upper bound $l$ on the set $\{ i \in \mathbb{Z} \mid x_i = \gamma_r \}$ then $s(\lambda_j) = s(\mu_j) = P_0$ for all $j > l$. Since $(F, \mathcal{L})$ is left--resolving, this implies that $\lambda = \mu$.
Assume now that there is no upper bound on the set.
If $x_i = \gamma_r$, then $s(\lambda_i) = s(\mu_i) = P_{2r}$
unless $x_{i-r}\cdots x_{i-1} = \alpha \gamma_2 \ldots \gamma_{r-1}$, in which case $s(\lambda_{i-r}) = s(\mu_{i-r}) = P_0$. The left Fischer cover is left--resolving, so either way it follows that $s(\lambda_{j}) = s(\mu_{j})$ for all $j \leq i-r$. By assumption, $i$ can be arbitrarily large, so $\lambda = \mu$.
\end{comment}
\begin{comment}
\begin{figure}\label{RJ_fig_entropy_lfc_rc}
\end{figure}
The proof will be finished by showing that the left Fischer cover is right--closing. $P_0$, $P_r$, and $P_{2r}$ are the only vertices that emit multiple edges with the same label, so it is sufficient to consider paths starting at these vertices.
Consider a path $\lambda^+ = \lambda_0 \lambda_1 \ldots \in F^+$ with $\mathcal{L}(\lambda^+) = x^+ \in \mathsf{X}(L)^+$ and assume first that $s(\lambda_0) = P_r$. The vertex $P_r$ emits $r$ edges with label $\beta$: One terminates at $P_0$ and the other terminate $P_{r+2}, \ldots , P_{2r}$.
If $\beta \gamma_i \cdots \gamma_r$ is a prefix of $x^+$, then $r(\lambda_0) = P_{r+i}$, and if not, then $r(\lambda_0) = P_0$. Hence, the left Fischer cover is right--closing at $P_r$ with delay $r+1$.
If $s(\lambda_0) = P_{2r}$, then $x_0 = \gamma_r$ and $r(\lambda_0) \in \{ P_0, P_{r+1}, \ldots, P_{2r} \}$. If $x^+$ has $\gamma_r \tilde \alpha \gamma_1 \cdots \gamma_r$ as a prefix, then $r(\lambda_0) = P_{r+1}$, and if $\gamma_r\gamma_i \cdots \gamma_r$ is a prefix of $x^+$, then $r(\lambda_0) = P_{r+i}$. Otherwise, $r(\lambda_0) = P_0$. Hence, the left Fischer cover is right--closing at $P_{2r}$ with delay $r+1$.
Finally, assume that $s(\lambda_0) = P_0$.
If $x_0 = \tilde \alpha$ or $x_0 = \gamma_i$ for $2 \leq i < r$, then arguments analogous to the ones used above prove that $r(\lambda_0)$ is determined by the first $r+1$ characters of $x^+$, so assume that $x_0 = \alpha$.
If $x_1 x_2 \ldots$ does not have a suffix of $\tilde \alpha \gamma_2 \cdots \gamma_r$ as a prefix, then $r(\lambda_0) = P_0$.
If $\alpha \gamma_i \cdots \gamma_r$ is a prefix of $x^+$ for $2 < i \leq r$, then $r(\lambda_0) = P_{r+i}$, and if $\alpha \tilde \alpha \gamma_2 \cdots \gamma_r$ is a prefix of $x^+$, then $r(\lambda_0) = P_{1}$.
If $\alpha \gamma_2 \cdots \gamma_r$ is a prefix of $x^+$, then $r(\lambda_0)$ is either $P_1$ or $P_{r+2}$. The possible continuations are shown in Fig.\ \ref{RJ_fig_entropy_lfc_rc}:
If $r(\lambda_0) = P_1$, then $\alpha \gamma_2 \cdots \gamma_r \beta$ must be a prefix of $x^+$. On the other hand, if $x^+$ has this prefix and $r(\lambda_0) = P_{r+2}$, then $\alpha \gamma_2 \cdots \gamma_r \beta \tilde \alpha \gamma_2 \cdots \gamma_r$ must also be a prefix of $x^+$, but there are no paths with this label and $r(\lambda_0) = P_1$. Hence, the left Fischer cover is right closing with delay $2r+1$.
\end{comment}
\begin{lemma}
\label{RJ_lem_entropy_cyclic}
Let $L \in R$ and let $X_f$ be a renewal system obtained from $\mathsf{X}(L)$ by fragmentation. Then the Bowen--Franks group of $X_f$ is cyclic, and the determinant is given by (\ref{RJ_eq_det_R}).
\end{lemma}
\begin{proof}
Let $L \in R$ be defined by (\ref{RJ_eq_R_L}). The symbolic adjacency matrix of the left Fischer cover of $\mathsf{X}(L)$ (shown in Fig.\ \ref{RJ_fig_entropy_lfc}) is
\begin{displaymath}
A = \left( \begin{array}{c | c c c c c | c c c c c c}
\gamma & \alpha & 0 & \cdots & 0 & 0 & \gamma+\beta & \tilde \alpha' & \gamma'_2 & \cdots & \gamma'_{r-2} & \gamma'_{r-1} \\
\hline
0 & 0 & \gamma_2 & \cdots & 0 & 0 & & & & & &\\
0 & 0 & 0 & & 0 & 0 & & & & & &\\
\vdots & \vdots & & \ddots & & \vdots & & & & 0 & &\\
0 & 0 & 0 & & 0 & \gamma_r & & & & & &\\
\beta & 0 & 0 & \cdots & 0 & 0 & 0 & \beta & \beta & \cdots & \beta & \beta \\
\hline
0 & & & & & & 0 & \tilde \alpha & 0 & \cdots & 0 & 0 \\
0 & & & & & & 0 & 0 & \gamma_2 & & 0 & 0 \\
0 & & & & & & 0 & 0 & 0 & & 0 & 0 \\
\vdots & & & 0 & & & \vdots & & & \ddots & & \vdots \\
0 & & & & & & 0 & 0 & 0 & & 0 & \gamma_{r-1} \\
\gamma_r & & & & & & \gamma_r & \gamma_r & \gamma_r & \cdots & \gamma_r & \gamma_r \\
\end{array} \right) \; ,
\end{displaymath}
where $ \gamma = \alpha+\tilde \alpha+ \sum_{k=2}^{r-1} \gamma_k$, $\tilde \alpha' = \gamma - \tilde \alpha$, and $\gamma'_k = \gamma - \gamma_k$. Index the rows and columns of $A$ by $0,\ldots, 2r$ in correspondence with the names used for the vertices above, and note that the column sums of the columns $0, r+1, \ldots, 2r$ are all equal to $\alpha+\tilde\alpha +\beta+ \sum_{k=2}^{r} \gamma_k$.
If $X_f$ is a fragmentation of $\mathsf{X}(L)$, then the (non--symbolic) adjacency matrix $A_f$ of the underlying graph of the left Fischer cover of $X_f$ is obtained from $A$ by replacing $\alpha, \tilde \alpha, \beta, \gamma_2, \ldots , \gamma_r$ by positive integers (see Remark \ref{RJ_rem_fragmentation}).
To put $\RJId - A_f$ into Smith normal form, begin by adding each row from number $r+1$ to $2r-1$ to the first row, and subtract the first column from column $r+1, \ldots, 2r$ to obtain
\begin{displaymath}
\RJId - A_f \rightsquigarrow
\left( \begin{array}{c | c c c c c | c c c c c}
1- \gamma & - \alpha & 0 & \cdots & 0 & 0 & -\beta & 0 & \cdots & 0 & -1 \\
\hline
0 & 1 & -\gamma_2 & \cdots & 0 & 0 & & & & &\\
0 & 0 & 1 & & 0 & 0 & & & & &\\
\vdots & \vdots & & \ddots & & \vdots & & & 0 & &\\
0 & 0 & 0 & & 1 & - \gamma_r & & & & &\\
-\beta & 0 & 0 & \cdots & 0 & 1 & \beta & 0 & \cdots & 0 & 0\\
\hline
0 & & & & & & 1 & -\tilde \alpha & \cdots & 0 & 0 \\
0 & & & & & & 0 & 1 & & 0 & 0 \\
\vdots & & & 0 & & & \vdots & & \ddots & & \vdots \\
0 & & & & & & 0 & 0 & & 1 & - \gamma_{r-1} \\
- \gamma_r & & & & & & 0 & 0 & \cdots & 0 & 1 \\
\end{array} \right) \: .
\end{displaymath}
Using row and column addition, this matrix can be further reduced to
\begin{displaymath}
\rightsquigarrow
\left( \begin{array}{c | c c c | c c c}
1- \gamma - b & 0 & \cdots & 0 & 0 & \cdots & t \\
\hline
0 & 1 & \cdots & 0 & & &\\
\vdots & \vdots & \ddots & \vdots & & 0 &\\
0 & 0 & \cdots & 1 & & & \\
\hline
0 & & & & 1 & \cdots & 0 \\
\vdots & & 0 & & \vdots & \ddots & \vdots \\
-\gamma_r & & & & 0 & \cdots & 1 \\
\end{array} \right)
\begin{array}{l}
\\
b = \alpha \beta \gamma_2 \cdots \gamma_r \\
\\
t = \tilde \alpha \gamma_2 \cdots \gamma_{r-1} (b - \beta) -1 \; . \\
\\
\end{array}
\end{displaymath}
Hence, the Bowen--Franks group of $X_f$ is cyclic, and the determinant is
\begin{equation} \label{RJ_eq_det_R}
\det(\RJId - A)
= 1 - \alpha - \tilde \alpha - \sum_{k=2}^r \gamma_k - (\alpha+\tilde \alpha) \beta \gamma_2 \cdots \gamma_r + \alpha \tilde \alpha \beta(\gamma_2 \cdots \gamma_r)^2 \;.
\end{equation}
\end{proof}
\begin{theorem}
For each $L \in H$, the renewal system $\mathsf{X}(L)$ has cyclic Bowen--Franks group and determinant given by (\ref{RJ_eq_entropy_det}). \label{RJ_thm_H_classification}
\end{theorem}
\begin{proof}
By Lemma \ref{RJ_lem_entropy_fe}, there exist $L_1, \ldots , L_m \in R$, $L_0 = \{a \}$ for some letter $a$ that does not appear in any of the lists, and a fragmentation $Y_f$ of $Y = \mathsf{X}( \bigcup_{j=0}^m L_j )$ such that $Y_f \sim_{\textrm{FE}} \mathsf{X}(L)$. For $1 \leq j \leq m$, let
$ L_j = \{ \alpha_j, \tilde \alpha_j, \gamma_{j,k},
\alpha_j \gamma_{j,2} \cdots \gamma_{j,r_j} \beta_j, \beta_j \tilde \alpha_j \gamma_{j,2} \cdots \gamma_{j,r_j} \mid 2 \leq k \leq r_j\} , r_j \in \mathbb{N}$.
Each $L_j$ is left--modular by Lemma \ref{RJ_lem_entropy_lfc},
so $Y$ is an SFT, and the left Fischer cover of $Y$ can be constructed using the technique from Sec.\ \ref{RJ_sec_rs_lfc}: Identify the universal border points in the left Fischer covers of $\mathsf{X}(L_0), \ldots , \mathsf{X}(L_m)$, and draw additional edges to the border points corresponding to the edges terminating at the universal border points in the individual left Fischer covers. Hence, the symbolic adjacency matrix $A$ of the left Fischer cover of $Y$ is
\begin{displaymath}
A =
\left( \!\!\! \begin{array}{c | c | c c c c | c c c c | c | c }
\gamma & & \alpha_j & 0 & \cdots & 0 & \gamma+\beta_ j & \tilde \alpha'_{ j} & \cdots & \gamma'_{ j,{r_ j-1}} & \cdots & \gamma'_{i,k}\\
\hline
&\ddots & & & & & & & & & & \\
\hline
0 & & 0 & \gamma_{ j,2} & \cdots & 0 & & & & & &\\
0 & & 0 & 0 & & 0 & & & & & & \\
\vdots & & \vdots & & \ddots & \vdots & & & 0 & & &\\
0 & & 0 & 0 & & \gamma_{ j,r} & & & & & & \\
\beta_ j & & 0 & 0 & \cdots & 0 & 0 & \beta_ j & \cdots & \beta_ j & & \beta_ j \\
\hline
0 & & & & & & 0 & \tilde \alpha_j & \cdots & 0 & &\\
0 & & & & & & 0 & 0 & & 0 & & \\
\vdots & & & & 0 & & \vdots & & \ddots & \vdots & & \\
0 & & & & & & 0 & 0 & & \gamma_{ j,{r_ j-1}} & &\\
\gamma_{ j,r_ j} & & & & & & \gamma_{ j,r_ j} & \gamma_{ j,r_ j} & \cdots & \gamma_{ j,r_ j} & & \gamma_{ j,r_ j}\\
\hline
& & & & & & & & & & \ddots & \\
\hline
& & & & & & & & & & & \ddots
\end{array} \!\!\! \right) \; .
\end{displaymath}
where $1 \leq j \leq m$, $\gamma = a + \sum_{j=1}^m \left( \alpha_j+\tilde\alpha_j + \sum_{k=2}^{r_j-1} \gamma_{j,k} \right)$, $\tilde \alpha_j' = \gamma - \tilde \alpha_j$, and $\gamma_{j,k}' = \gamma - \gamma_{j,k}$. This matrix has blocks of the same form as in the $m=1$ case considered in Lemma \ref{RJ_lem_entropy_lfc}. The $j$th block is shown together with the first row and column of the matrix---which contain the connections between the $j$th block and the universal border point $P_0$---and together with an extra column representing an arbitrary border point in a different block. Such a border point in another block will receive edges from the $j$th block with the same sources and labels as the edges that start in the $j$th block and terminate at the universal border point $P_0$.
Let $Y_f$ be a fragmentation of $Y$. Then the (non--symbolic) adjacency matrix $A_f$ of the underlying graph of the left Fischer cover of $Y_f$ is obtained by replacing the entries of $A$ by positive integers as described in Remark \ref{RJ_rem_fragmentation}.
In order to put $\RJId - A_f$ into Smith normal form, first add rows $r_j+1$ to $2r_j-1$
in the $j$th block to the first row for each $j$, and then subtract the first column from every column corresponding to a border point in any block.
In this way, $\RJId-A_f$ is transformed into:
\begin{displaymath}
\left(\!\!\! \begin{array}{c | c | c c c c | c c c c | c }
1-\gamma & & -\alpha_j & 0 & \cdots & 0 & -\beta_j & 0 & \cdots & -1 & \\
\hline
&\!\ddots\! & & & & & & & & & \\
\hline
0 & & 1 & -\gamma_{j,2} & \cdots & 0 & & & & & \\
0 & & 0 & 1 & & 0 & & & & & \\
\vdots& & \vdots & & \ddots & \vdots & & & 0 & & \\
0 & & 0 & 0 & & -\gamma_{j,r_j} & & & & & \\
-\beta_j & & 0 & 0 & \cdots & 1 & \beta_j & 0 & \cdots & 0 & \\
\hline
0 & & & & & & 1 & -\tilde \alpha_j & \cdots & 0 & \\
0 & & & & & & 0 & 1 & & 0 & \\
\vdots & & & & 0 & & \vdots & & \ddots & \vdots & \\
0 & & & & & & 0 & 0 & & -\gamma_{j,{r_j-1}} & \\
-\gamma_{j,r_j} & & & & & & 0 & 0 & \cdots & 1 & \\
\hline
& & & & & & & & & & \! \ddots \\
\end{array} \!\!\! \right) \; .
\end{displaymath}
By using row and column addition, and by disregarding rows and columns where the only non--zero entry is a diagonal $1$, $\RJId - A$ can be further reduced to
\begin{displaymath}
\left( \begin{array}{c c c c c c}
S & t_1 & t_2 & \cdots & t_m \\
-\gamma_{1,r_1} & 1 & 0 & & 0 \\
-\gamma_{2,r_2} & 0 & 1 & & 0 \\
\vdots & & & \ddots & \vdots \\
-\gamma_{m,r_m} & 0 & 0 & \cdots & 1 \\
\end{array}\right)
\quad \begin{array}{l}
b_j = \alpha_j \beta_j \gamma_{j,2} \cdots \gamma_{j,r_j} \\ \\
t_j = \tilde \alpha_j \gamma_{j,2} \cdots \gamma_{j,r-1} (b_j-\beta_j) -1 \\ \\
S = 1 - \gamma - \sum_{j=1}^m b_j
\end{array}
\; .
\end{displaymath}
Hence, the Bowen--Franks group is cyclic and the determinant is
\begin{equation}
\label{RJ_eq_entropy_det}
\det( \RJId - A_f ) = 1 - \gamma + \sum_{j=1}^m \left( \gamma_{j,r_j} t_j - b_j \right)\; .
\end{equation}
\end{proof}
\noindent With the results of \cite{RJ_hong_shin}, this gives the following result.
\begin{corollary}
When $\log \lambda$ is the entropy of an SFT, there exists an SFT renewal system $X(L)$ with cyclic Bowen--Franks group such that $h(\mathsf{X}(L)) = \log \lambda$.
\end{corollary}
\section{Towards the range of the Bowen--Franks invariant}
\label{RJ_sec_rs_range}
In the following, it will be proved that the range of the Bowen--Franks invariant over the class of SFT renewal systems contains a large class of pairs of signs and finitely generated abelian groups.
First, the following special case will be used to show that every integer is the determinant of an SFT renewal system.
\index{renewal system!with positive determinant}
\begin{example}
\label{RJ_ex_pos_det}
Consider the generating list \fxnote{Corrected the generating list}
\begin{equation}\label{RJ_eq_pos_det}
L = \{ a, \alpha, \tilde \alpha, \gamma, \alpha \gamma \beta, \beta \tilde \alpha \gamma \} \; .
\end{equation}
By Lemma \ref{RJ_lem_entropy_lfc}, $L$ is left--modular, $\mathsf{X}(L)$ is an SFT, and the symbolic adjacency matrix of the left Fischer cover of $\mathsf{X}(L)$ is
\begin{equation}
\label{RJ_eq_pos_det_adj}
A = \left( \begin{array}{c | c c | c c}
a +\alpha+\tilde \alpha & \alpha & 0 & a+\alpha +\tilde \alpha+\beta & a+\alpha \\
\hline
0 & 0 & \gamma & 0 & 0 \\
\beta & 0 & 0 & 0 & \beta \\
\hline
0 & 0 & 0 & 0 & \tilde \alpha \\
\gamma & 0 & 0 & \gamma & \gamma
\end{array} \right) \; .
\end{equation}
\noindent
By fragmenting $\mathsf{X}(L)$, it is possible to construct an SFT renewal system for which the (non--symbolic) adjacency matrix of the underlying graph of the left Fischer cover has this form with $a, \alpha, \tilde \alpha, \beta, \gamma \in \mathbb{N}$ as described in Remark \ref{RJ_rem_fragmentation}. Let $A_f$ be such a matrix. This is a special case of the shift spaces considered in Theorem \ref{RJ_thm_H_classification}, so the Bowen--Franks group is cyclic and the determinant is
$
\det(\RJId - A_f) = \beta \alpha \tilde \alpha \gamma^2 - \alpha \beta \gamma - \tilde \alpha \beta \gamma -\alpha - \tilde \alpha-\gamma-a +1
$.
\end{example}
\begin{theorem}\label{RJ_thm_rs_det_range}
Any $k \in \mathbb{Z}$ is the determinant of an SFT renewal system with cyclic Bowen--Franks group.
\end{theorem}
\begin{proof}
Consider the renewal system from Example \ref{RJ_ex_pos_det} in the case $\alpha = \tilde \alpha = \beta = 1$, where the determinant is
$ \det(\RJId - A_f) = \gamma^2 - 3\gamma - a -1$,
and note that the range of this polynomial is $\mathbb{Z}$.
\end{proof}
\index{renewal system!with non--cyclic group}
\index{X@$\RJXd{n_1,\dots,n_k}$}
All renewal systems considered until now have had cyclic Bowen--Franks groups, so the next goal is to construct a class of renewal systems exhibiting non--cyclic groups.
Let $k \geq 2$, $\mathcal{A} = \{ a_1, \ldots , a_k \}$, and let $n_1, \ldots , n_k \geq 2$ with $\max_{i}\{n_{i}\} > 2$.
The goal is to define a generating list, $L$, for which $\mathsf X(L) = \mathsf X_\mathcal F$ with $\mathcal{F} = \{ a_i^{n_{i}} \}$. For each $1 \leq i \leq k$, define
\begin{multline}
\label{RJ_eq_Ld}
L_i = \{ a_j a_i^l \mid j \neq i \textrm{ and } 0 < l < n_i -1\} \\ \cup
\{ a_m a_j a_i^l \mid m \neq j \neq i \textrm{ and } 0 < l < n_i -1\} \; .
\end{multline}
Define $L = \bigcup_{i=1}^k L_i \neq \emptyset$, and $\RJXd{n_1,\dots,n_k} = \mathsf{X}(L)$.
\begin{lemma}
\label{RJ_lem_Xd}
Define the renewal system $\RJXd{n_1,\dots,n_k}$ as above. Then $\RJXd{n_1,\dots,n_k} = \mathsf X_\mathcal F$ with $\mathcal F = \{ a_i^{n_{i}} \}$, so $\RJXd{n_1,\dots,n_k}$ is an SFT. The symbolic adjacency matrix of the left Fischer cover of $\RJXd{n_1,\dots,n_k}$ is the matrix in (\ref{RJ_eq_diag_A}).
\end{lemma}
\begin{proof}
Note that for each $i$, $a_i^{n_{i}} \notin \mathcal B (\RJXd{n_1,\dots,n_k})$ by construction. For $1 < l < n_i-1$ and $j \neq i$ the word $a_j a_i^{l}$ has a partitioning in $\RJXd{n_1,\dots,n_k}$ with empty beginning and end. Hence, $a_{i_1} a_{i_2}^{l_2} a_{i_3}^{l_3} \cdots a_{i_m}^{l_m}$ has a partitioning with empty beginning and end whenever $i_j \neq i_{j+1}$, $1 < l_j < n_{i_j}$ for all $1 < j < m$, and $0 < l_m < n_{i_m} -1$. Given $i_1, \ldots , i_m \in \{1, \ldots, k\}$ with $i_j \neq i_{j+1}$ and $m \geq 2$, the word $a_{i_1} a_{i_2} \cdots a_{i_m}$ has a partitioning with empty beginning and end. Hence, every word that does not contain one of the words $ a_i^{n_{i}}$ has a partitioning, so $\RJXd{n_1,\dots,n_k} = \mathsf X_\mathcal F$ for $\mathcal F = \{ a_i^{n_{i}}\}$.
\begin{figure}\label{RJ_fig_Xd_LFC}
\end{figure}
To find the left Fischer cover of $\RJXd{n_1,\dots,n_k}$, it is first necessary to determine the predecessor sets. Given $1 \leq i \leq k$ and $j \neq i$
\begin{align}
P_\infty(a_i a_j \cdots) &= \{ x^- \in \RJXd{n_1,\dots,n_k}^- | x_{-n_i +1} \cdots x_0 \neq a_i^{n_i-1} \} \nonumber \\
P_\infty(a_i^2 a_j \cdots) &= \{ x^- \in \RJXd{n_1,\dots,n_k}^- | x_{-n_i +2} \cdots x_0 \neq a_i^{n_i-2} \} \label{RJ_eq_rs_diag}\\
&\vdots \nonumber \\
P_\infty(a_i^{n_i-1} a_j \cdots) &= \{ x^- \in \RJXd{n_1,\dots,n_k}^- | x_0 \neq a_i \} \nonumber .
\end{align}
Only the first of these predecessor sets is a border point.
Equation \ref{RJ_eq_rs_diag} gives all the information necessary to draw the left Fischer cover of $\RJXd{n_1,\dots,n_k}$. A part of the left Fischer cover is shown in Fig.\ \ref{RJ_fig_Xd_LFC},
and the corresponding symbolic adjacency matrix is:
\begin{align}
\nonumber
&\hspace{14 pt}
\begin{array}{c c c c}
\overbrace{
\phantom{\begin{matrix}
a_1 & \cdots & a_1 & a_1
\end{matrix}}
}^{n_1-1}
&
\overbrace{
\phantom{\begin{matrix}
a_2 & \cdots & a_2 & a_2
\end{matrix}}
}^{n_2-1}
&
\phantom{\cdots}
&
\overbrace{
\phantom{\begin{matrix}
a_k & \cdots & a_k & a_k
\end{matrix}}
}^{n_k-1}
\end{array} \\[-1.5em]
&\left( \begin{array}{ c | c | c | c }
\begin{matrix}
0 & \cdots & 0 & 0 \\
a_1 & \cdots & 0 & 0 \\
& \ddots & & \\
0 & \cdots & a_1 & 0
\end{matrix}
&
\begin{matrix}
a_1 & \cdots & a_1 & a_1\\
0 & \cdots & 0 & 0 \\
& \ddots & & \\
0 & \cdots & 0 & 0
\end{matrix}
&
\cdots
&
\begin{matrix}
a_1 & \cdots & a_1 & a_1\\
0 & \cdots & 0 & 0 \\
& \ddots & & \\
0 & \cdots & 0 & 0
\end{matrix} \\
\hline
\begin{matrix}
a_2 & \cdots & a_2 & a_2\\
0 & \cdots & 0 & 0 \\
& \ddots & & \\
0 & \cdots & 0 & 0
\end{matrix}
&
\begin{matrix}
0 & \cdots & 0 & 0 \\
a_2 & \cdots & 0 & 0 \\
& \ddots & & \\
0 & \cdots & a_2 & 0
\end{matrix}
&
\cdots
&
\begin{matrix}
a_2 & \cdots & a_2 & a_2\\
0 & \cdots & 0 & 0 \\
& \ddots & & \\
0 & \cdots & 0 & 0
\end{matrix} \\
\hline
\vdots & \vdots & \ddots & \vdots \\
\hline
\begin{matrix}
a_k & \cdots & a_k & a_k\\
0 & \cdots & 0 & 0 \\
& \ddots & & \\
0 & \cdots & 0 & 0
\end{matrix}
&
\begin{matrix}
a_k & \cdots & a_k & a_k\\
0 & \cdots & 0 & 0 \\
& \ddots & & \\
0 & \cdots & 0 & 0
\end{matrix}
&
\cdots
&
\begin{matrix}
0 & \cdots & 0 & 0 \\
a_k & \cdots & 0 & 0 \\
& \ddots & & \\
0 & \cdots & a_k & 0
\end{matrix} \\
\end{array} \right) \; . \label{RJ_eq_diag_A}
\end{align} \end{proof}
Let $A$ be the (non--symbolic) adjacency matrix of the underlying graph of the left Fischer cover of $\RJXd{n_1,\dots,n_k}$ constructed above.
Then it is possible to do the following transformation by row and column addition
\begin{align*}
\RJId - A &\rightsquigarrow
\begin{pmatrix}
1 & 1-n_2 & 1-n_3 & \cdots & 1-n_k \\
1-n_1 & 1 & 1-n_3 & \cdots & 1-n_k \\
1-n_1 & 1-n_2 & 1 & \cdots & 1-n_k \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
1-n_1 & 1-n_2 & 1-n_3 & \cdots & 1
\end{pmatrix}
\rightsquigarrow
\begin{pmatrix}
x & 1 & 1 & \cdots & 1 \\
-n_1 & n_2 & 0 & \cdots & 0 \\
-n_1 & 0 & n_3 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
-n_1 & 0 & 0 & \cdots & n_k
\end{pmatrix} \; ,
\end{align*}
where $x = 1-(k-1) n_1$. The determinant of this matrix is
\begin{displaymath}
\det(\RJId - A) = n_2 \cdots n_k \left( x + \sum_{i=2}^k \frac{n_1}{n_i} \right)
= - n_1 n_2 \cdots n_k \left( k-1 - \sum_{i=1}^k \frac{1}{n_i} \right)
< 0 \; .
\end{displaymath}
The inequality is strict since $k-1 - \sum_{i=1}^k \frac{1}{n_i} > \frac{k}{2} -1 \geq 0$.
Given concrete $n_1, \ldots, n_k$, it is straightforward to compute the Bowen--Franks group of $\RJXd{n_1, \ldots, n_k}$, but it has not been possible to derive a general closed form for this group.
\begin{proposition}
\label{RJ_prop_non-cyclic_BF}
Let $n_1, \ldots, n_k \geq 2$ with $n_i | n_{i-1}$ for $2 \leq i \leq k$ and $n_1 > 2$. Let $m = n_1 n_2 ( k-1 - \sum_{i=1}^k \frac{1}{n_i} )$, then $\RJBF_+(\RJXd{n_1, \ldots, n_k}) = - \mathbb{Z} /m\mathbb{Z} \oplus \mathbb{Z} /n_3\mathbb{Z} \oplus \cdots \oplus \mathbb{Z} / n_k\mathbb{Z}$.
\end{proposition}
\begin{proof}
By the arguments above, $\RJXd{n_1, \ldots , n_k}$ is conjugate to an edge shift with adjacency matrix $A$ such that the following transformation can be carried out by row and column addition
\begin{comment}
\begin{align*}
\RJId - A &\rightsquigarrow
\begin{pmatrix}
y & 1 & 1 & \cdots & 1 \\
0 & n_2 & 0 & \cdots & 0 \\
0 & 0 & n_3 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & n_k
\end{pmatrix}
\begin{array}{l}
\\
\\
y = -n_1 \left( k-1- \sum_{i=1}^k 1/n_i \right) \\
\\
\\
\end{array} \\
&\rightsquigarrow
\begin{pmatrix}
0 & 1 & 0 & \cdots & 0 \\
m & 0 & 0 & \cdots & 0 \\
0 & 0 & n_3 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & n_k
\end{pmatrix} \; ,
\end{align*}
\end{comment}
\begin{displaymath}
\RJId - A \rightsquigarrow
\begin{pmatrix}
y & 1 & 1 & \cdots & 1 \\
0 & n_2 & 0 & \cdots & 0 \\
0 & 0 & n_3 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & n_k
\end{pmatrix}
\rightsquigarrow
\begin{pmatrix}
0 & 1 & 0 & \cdots & 0 \\
m & 0 & 0 & \cdots & 0 \\
0 & 0 & n_3 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & n_k
\end{pmatrix} \; ,
\end{displaymath}
where $y = -n_1 \left( k-1- \sum_{i=1}^k 1/n_i \right)$.
It follows that the Smith normal form of $\RJId - A$ is $\RJdiag(m, n_3, \ldots , n_k)$, and $\det(\RJId - A) < 0$.
\end{proof}
Let $G$ be a finite direct sum of finite cyclic groups. Then Proposition \ref{RJ_prop_non-cyclic_BF} shows that $G$ is a subgroup of the Bowen--Franks group of some SFT renewal system, but it is still unclear whether $G$ itself is also the Bowen--Franks group of a renewal system since the term $\mathbb{Z} / m\mathbb{Z}$ in the statement of Proposition \ref{RJ_prop_non-cyclic_BF} is determined by the other terms. Furthermore, the groups constructed in Proposition \ref{RJ_prop_non-cyclic_BF} are all finite. Other techniques can be used to construct renewal systems with groups such as $\mathbb{Z} / (n+1) \oplus \mathbb{Z}$ \cite[Ex. 5.54]{RJ_johansen_thesis}.
\index{renewal system!with positive determinant}
\index{renewal system!with non--cyclic group}
The determinants of all the renewal systems with non--cyclic Bowen--Franks groups considered above were negative or zero, so the next goal is to construct a class of SFT renewal systems with positive determinants and non--cyclic Bowen--Franks groups.
\begin{figure}\label{RJ_fig_addition_Xd}
\end{figure}
\begin{lemma}
\label{RJ_lem_addition_Xd}
Let $L_d$ be the generating list of $\RJXd{n_1, \ldots , n_k}$ as defined in (\ref{RJ_eq_Ld}), and let $(F_d, \mathcal{L}_d)$ be the left Fischer cover of $\RJXd{n_1, \ldots , n_k}$.
Let $L_m$ be a left--modular generating list for which $\mathsf{X}(L_m)$ is an SFT with left Fischer cover $(F_m, \mathcal{L}_m)$.
For $L_{d+m} = L_d \cup L_m \cup_{i=1}^k \{ a_i w \mid w \in L_m \}$, $\mathsf{X}(L_{d+m})$ is an SFT for which the left Fischer cover is obtained by adding the following connecting edges to the disjoint union of $(F_d, \mathcal{L}_d)$ and $(F_m, \mathcal{L}_m)$ (sketched in Fig. \ref{RJ_fig_addition_Xd}):
\begin{itemize}
\item For each $1 \leq i \leq k$ and each $e \in F_m^0$ with $r(e) = P_0(L_m)$ draw an edge $e_i$ with $s(e_i) = s(e)$ and $r(e_i) = P_\infty(a_i a_j \ldots)$ labelled $\mathcal{L}_m(e)$.
\item For each $1 \leq i \leq k$ and each border point $P \in F_m^0$ draw an edge labelled $a_i$ from $P_\infty(a_i a_j \ldots)$ to $P$.
\end{itemize}
\end{lemma}
\begin{proof}
Let $(F_{d+m}, \mathcal{L}_{d+m})$ be the labelled graph defined in the lemma
and sketched in Fig.\ \ref{RJ_fig_addition_Xd}.
The graph is left--resolving, predecessor--separated, and irreducible by construction, so it is the left Fischer cover of some sofic shift $X$ \cite[Cor. 3.3.19]{RJ_lind_marcus}. The first goal is to prove that $X = \mathsf{X}(L_{d+m})$.
By the arguments used in the proof of Lemma \ref{RJ_lem_Xd}, any word of the form $a_{i_0}w_m a_{i_1} a_{i_2}^{l_i} \ldots a_{i_p}^{l_p}$ where
$w_m \in L_m^*$,
$p \in \mathbb{N}$,
$i_j \neq i_{j+1}$ and $l_j < n_{i_j}$ for $1 < j < p$,
and $1 \leq l_p < n_{i_p} -1$
has a partitioning with empty beginning and end in $\mathsf{X}(L_{d+m})$.
Hence, $\mathcal{B}(\mathsf{X}(L_{d+m}))$ is the set of factors of concatenations of words from
$
\left\{ w_m a_i w_d \mid w_m \in L_m^*, 1 \leq i \leq k, w_d \in \mathcal{B}(\RJXd{n_1, \ldots, n_k}), \RJleftl(w_d) \neq a_i \right \}
$.
Since $L_m$ is left--modular, a path $\lambda \in F_m^*$ with $r(\lambda) = P_0(L_m)$ has $\mathcal{L}_m(\lambda) \in L_m^*$ if and only if $s(\lambda)$ is a border point in $F_m$.
Hence, the language recognised by the left Fischer cover $(F_{d+m}, \mathcal{L}_{d+m})$ is precisely the language of $\mathsf{X}(L_{d+m})$.
It remains to show that $(F_{d+m}, \mathcal{L}_{d+m})$ presents an SFT.
Let $1 \leq i \leq k$ and let $\alpha \in \mathcal{B}(\mathsf{X}(L_m))$, then any labelled path in $(F_{d+m}, \mathcal{L}_{d+m})$ with $a_i \alpha$ as a prefix must start at $P_\infty(a_ia_j\cdots)$.
Similarly, if there is a path $\lambda \in F_{d+m}^*$ with $\alpha a_i$ as a prefix of $\mathcal{L}_{d+m}(\lambda)$, then there must be unique vertex $v$ emitting an edge labelled $\alpha$ to $P_0(L)$, and $s(\lambda) = v$.
Let $x \in \mathsf{X}_{(F_{d+m},\mathcal{L}_{d+m})}$. If there is no upper bound on set of $i \in \mathbb{Z}$ such that $x_i \in \{a_1, \ldots, a_k\}$ and $x_{i+1} \in \mathcal{A}(\mathsf{X}(L_m))$ or vice versa, then the arguments above and the fact that the graph is left--resolving prove that there is only one path in $(F_{d+m},\mathcal{L}_{d+m})$ labelled $x$. If there is an upper bound on the set considered above, then a presentation of $x$ is eventually contained in either $F_{d}$ or $F_{m}$. It follows that the covering map of $(F_{d+m}, \mathcal{L}_{d+m})$ is injective, so it presents an SFT.
\begin{comment}
To prove that $X$ is an SFT, note first that the covering maps of $(F_{d}, \mathcal{L}_{d})$ and $(F_{m}, \mathcal{L}_{m})$ are both injective.
Let $\lambda^- = \ldots \lambda_{-1} \lambda_0 \in F_{d+m}^-$ be a left--ray such that $w a_i$ is a suffix of $\mathcal{L}_{d+m}(\lambda^-)$ for some $w \in L_{d+m}^* \setminus \{ \varepsilon \}$
and $1 \leq i \leq k$. Then $s(\lambda_0) = P_i$. The graph is left--resolving, so this means that $\lambda^-$ is the only left--ray with label $\mathcal{L}_{d+m}(\lambda^-)$.
Assume that there exist biinfinite paths $\lambda, \mu \in F_{d+m}^*$ such that $\mathcal{L}_{d+m}(\lambda) = \mathcal{L}_{d+m}(\mu) = x \in X$. If $\lambda \neq \mu$, then the argument above proves that there exist $i \leq j$ such that $s(\lambda_i) = s(\mu_i) \in F_d^0$ and $r(\lambda_l), r(\mu_l) \in F_m^0$ for all $l \geq j$ or vice versa.
Consider the first case and assume without loss of generality that $\mathcal{L}_{d+m}(\lambda_0) = \mathcal{L}_{d+m}(\mu_0) =a_i$ while $r(\lambda_i), r(\mu_i) \in F_{m}$ for all $i \geq 0$.
Then $s(\lambda_0) = s(\mu_0) = P_i$, so $\lambda^- = \mu^-$.
Note that $x^+ = x_1 x_2 \ldots \in \mathsf{X}(L_m)^+$, and $\mathsf{X}(L_m)^+$ is an SFT, so there is only one presentation of $x^+$ in $(F_m, \mathcal{L}_m)$, and hence $\lambda = \mu$. An analogous argument works in the remaining case.
\end{comment}
\end{proof}
\begin{example}
\label{RJ_ex_pos_det+nc_bf}
The next step is to use Lemma \ref{RJ_lem_addition_Xd} to construct renewal systems that share features with both $\RJXd{n_1, \dots, n_k}$ and the renewal systems considered in Example \ref{RJ_ex_pos_det}.
Given $n_1, \ldots , n_k \geq 2$ with $\max_j n_j > 2$, consider the list $L_d$ defined in (\ref{RJ_eq_Ld}) which generates the renewal system $\RJXd{n_1, \ldots, n_k}$,
and the list $L$ from (\ref{RJ_eq_pos_det}).
$L$ is left--modular, and $\mathsf{X}(L)$ is an SFT, so Lemma \ref{RJ_lem_addition_Xd} can be used to find the left Fischer cover of the SFT renewal system $X_+$ generated by
$
L_{+} = L_d \cup L \cup_{i=1}^k \{ a_i w \mid w \in L \}
$,
and the corresponding symbolic adjacency matrix is
\begin{displaymath}
A_+ =
\left( \!\!\! \begin{array}{c | c c c c | c c c c c | c | c c c c c }
b & \alpha & 0 & b +\beta & a+\alpha & b & 0 &\! \cdots \!& 0 & 0 & \!\cdots\! & b & 0 &\! \cdots \!& 0 & 0
\\
\hline
0 & 0 & \gamma & 0 & 0 & 0 & 0 &\! \cdots \!& 0 & 0 & & 0 & 0 &\! \cdots \!& 0 & 0 \\
\beta & 0 & 0 & 0 & \beta & \beta & 0 &\! \cdots \!& 0 & 0 & & \beta & 0 &\! \cdots \!& 0 & 0 \\
0 & 0 & 0 & 0 & \tilde \alpha & 0 & 0 &\! \cdots \!& 0 & 0 & & 0 & 0 &\! \cdots \!& 0 & 0 \\
\gamma & 0 & 0 & \gamma & \gamma & \gamma & 0 &\! \cdots \!& 0 & 0 & & \gamma & 0 &\! \cdots \!& 0 & 0 \\
\hline
a_1 & 0 & 0 & a_1 & a_1 & 0 & 0 & & 0 & 0 & & a_1 & a_1 &\! \cdots \!& a_1 & a_1 \\
0 & 0 & 0 & 0 & 0 & a_1 & 0 & & 0 & 0 & & 0 & 0 & & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & & &\! \ddots \!& & & & & &\! \ddots \!& & \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & & 0 & 0 & & 0 & 0 & & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & & a_1 & 0 & & 0 & 0 & & 0 & 0 \\
\hline
\vdots & & & & & & & & & & \!\ddots\! & & & & & \\
\hline
a_k & 0 & 0 & a_k & a_k & a_k & a_k&\! \cdots \!& a_k & a_k & & 0 & 0 & & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & & 0 & 0 & & a_k & 0 & & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & & &\! \ddots \!& & & & & &\! \ddots \!& & \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & & 0 & 0 & & 0 & 0 & & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & & 0 & 0 & & 0 & 0 & & a_k & 0 \\
\end{array} \!\!\! \right) \; ,
\end{displaymath}
where $b = a + \alpha + \tilde \alpha$. Let $Y_+$ be a renewal system obtained from $X_+$ by a fragmentation of $a$, $\alpha$, $\tilde \alpha$, $\beta$, and $\gamma$. Then the (non--symbolic) adjacency matrix of the left Fischer cover of $Y_+$ is obtained from the matrix $A_+$ above by replacing $a_1, \ldots, a_k$ by $1$, and replacing $a$, $\alpha$, $\tilde \alpha$, $\beta$, and $\gamma$ by positive integers. Let $B_+$ be a matrix obtained in this manner. By doing row and column operations as in the construction that leads to the proof Proposition \ref{RJ_prop_non-cyclic_BF}, and by disregarding rows and columns where the only non--zero entry is a diagonal $1$, it follows that
\begin{displaymath}
\RJId - B_+ \rightsquigarrow
\left( \!\!\! \begin{array}{c | c c c c | c c c c }
1-b & -\alpha & 0 & -b -\beta & -a-\alpha & -b & -b &\! \cdots \! & -b \\
\hline
0 & 1 & -\gamma & 0 & 0 & 0 & 0 &\! \cdots \! & 0 \\
-\beta & 0 & 1 & 0 & -\beta & -\beta & -\beta &\! \cdots \! & -\beta \\
0 & 0 & 0 & 1 & -\tilde \alpha & 0 & 0 &\! \cdots \! & 0 \\
-\gamma & 0 & 0 & - \gamma & 1-\gamma & -\gamma & -\gamma &\! \cdots \! & -\gamma \\
\hline
-1 & 0 & 0 & -1 & -1 & 1 & 1-n_2 & \! \cdots \! & 1-n_k \\
-1 & 0 & 0 & -1 & -1 & 1-n_1 & 1 & & 1-n_k \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & & \! \ddots \! & \vdots \\
-1 & 0 & 0 & -1 & -1 & 1-n_1 & 1-n_2 & \! \cdots \! & 1 \\
\end{array} \!\!\! \right) \; .
\end{displaymath}
Add the third row to the first and subtract the first column from columns $4, \ldots , k+4$ as in the proof of Lemma \ref{RJ_lem_entropy_cyclic}
and choose the variables $a$, $\alpha$, $\tilde \alpha$, $\beta$, and $\gamma$ as in the proof of Theorem \ref{RJ_thm_rs_det_range}. Assuming that $n_i | n_{i-1}$ for $2 \leq i \leq k$, this matrix can be reduced to
\begin{multline*}
\RJId - B_+ \rightsquigarrow \\
\left( \begin{array}{c | c c c c c }
x & -1 & -1 & -1 & \cdots & -1 \\
\hline
2x-1 & 0 & -n_2 & -n_3 & \cdots & -n_k \\
0 & -n_1 & n_2 & 0 & & 0 \\
0 & -n_1 & 0 & n_3 & & 0 \\
\vdots & \vdots & & & \ddots & \vdots \\
0 & -n_1 & 0 & 0 & \cdots & n_k \\
\end{array} \right)
\rightsquigarrow
\left( \begin{array}{c c c c c c }
x & -\sum_{i=1}^k \frac{n_1}{n_i} & -1 & 0 & \cdots & 0 \\
2x-1 & -(k-1)n_1 & 0 & 0 & \cdots & 0 \\
0 & 0 & n_2 & 0 & & 0 \\
0 & 0 & 0 & n_3 & & 0 \\
\vdots & \vdots & & & \ddots & \vdots \\
0 & 0 & 0 & 0 & \cdots & n_k \\
\end{array} \right) \; ,
\end{multline*}
where $x \in \mathbb{Z}$ is arbitrary. Hence, the determinant is
\begin{equation}
\label{RJ_eq_pos_det+nc_bf_det}
\det(\RJId - B_+) = n_2 \cdots n_k \left( (2x-1)\sum_{i=1}^k \frac{n_1}{n_i} - x(k-1)n_1 \right) \; ,
\end{equation}
and there exists an abelian group $G$ with at most two generators such that the Bowen--Franks group of the corresponding SFT is $G \oplus \mathbb{Z} / n_3 \mathbb{Z} \oplus \cdots \oplus \mathbb{Z} / n_k \mathbb{Z}$.
\begin{comment}
If $n_2 | x$\fxnote{Not true?}, then
\begin{equation}
\label{RJ_eq_pos_det+nc_bf_G}
G = \mathbb{Z} / \left( n_2 \left( (2x-1)\sum_{i=1}^k \frac{n_1}{n_i} -
x(k-1)n_1 \right) \right) \mathbb{Z} \; .
\end{equation}
\end{comment}
For $x = 0$, the determinant is negative and the Bowen--Franks group is $\mathbb{Z} \big/ \big(\sum_{i=1}^k \frac{n_2 n_1}{n_i} \big)\mathbb{Z} \oplus \mathbb{Z} /n_3\mathbb{Z} \oplus \cdots \oplus \mathbb{Z} /n_k\mathbb{Z}$.
\end{example}
This gives the first example of SFT renewal systems that simultaneously have positive determinants and non--cyclic Bowen--Franks groups.
\begin{theorem}\label{RJ_thm_nc_bf_det}
Given $n_1, \ldots, n_k \geq 2$ with $n_i | n_{i-1}$ for $2 \leq i \leq k$ there exist abelian groups $G_\pm$ with at most two generators and SFT renewal systems $\mathsf{X}(L_\pm)$ such that $\RJBF_+(\mathsf{X}(L_\pm)) = \pm G_\pm \oplus \mathbb{Z} /n_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z} /n_k \mathbb{Z}$.
\end{theorem}
\begin{proof}
Consider the renewal system from Example \ref{RJ_ex_pos_det+nc_bf}. Given the other variables, (\ref{RJ_eq_pos_det+nc_bf_det}) shows that $x$ can be chosen such that the determinant has either sign.
\end{proof}
The question raised by Adler, and the related question concerning the flow equivalence of renewal systems are still unanswered, and a significant amount of work remains before they can be solved.
However, there is hope that the techniques developed in Sec. \ref{RJ_sec_rs_lfc} and the special classes of renewal systems considered in Sec. \ref{RJ_sec_rs_range} can act as a foundation for the construction of a class of renewal systems attaining all the values of the Bowen--Franks invariant realised by irreducible SFTs.
\end{document}
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math
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अल्पविराम किसे कहते हैं | अलप्वीराम इन हिन्दी | अल्पविराम की परिभाषा क्या है , उदाहरण वाक्य चिन्ह प्रयोग ११त , १२त नोटस इन हिन्दी
अल्पविराम किसे कहते हैं | अलप्वीराम इन हिन्दी | अल्पविराम की परिभाषा क्या है , उदाहरण वाक्य चिन्ह प्रयोग
()अलप्वीराम इन हिन्दी अल्पविराम किसे कहते हैं , अल्पविराम की परिभाषा क्या है , उदाहरण वाक्य चिन्ह प्रयोग कैसे होता है ? हिंदी व्याकरण |
लेखक के भावों और विचारों को स्पष्ट करने के लिए जिन चिह्नों का प्रयोग वाक्य अथवा वाक्यों में किया जाता है, उन्हें विरामचिन कहते हैं।
विराम का शाब्दिक अर्थ होता है-ठहराव । लेखनकार्य में इसी ठहराव के लिए चिह्नों का प्रयोग होता है । लेखनकार्य के अन्तर्गत भावों अथवा विचारों में ठहराव के लिए विराम चिहनों का प्रयोग होता है । इन विराम चिह्नों का प्रयोग न करने से भाव अथवा विचार की स्पष्टता में बाधा पड़ती है तथा वाक्य एक दूसरे से उलझ जाते हैं एवं साथ ही पाठक को भी माथापच्ची करनी पड़ती है । अतः यह कहा जा सकता है कि वाक्य के सुन्दर गठन और भावाभिव्यक्ति की स्पष्टता के लिए इन विरामचिहनों की आवश्यकता एवं उपयोगिता मानी जाती है । प्रत्येक विराम चिहून लेखक की विशेष मनोदशा का एक-एक ठहराव है अथवा विराम का संकेत स्थान है।
विरामचिह्नों का प्रयोग पश्चिमी साहित्य अथवा अंग्रेजी के माध्यम से भारतीय भाषाओं में शुरू हुआ है । इस दृष्टि से हम पश्चिम के ऋणी हैं । १९वीं शती के पूर्वार्द्ध तक भारतीय भाषाओं में विरामचिन्हों का प्रयोग नहीं होता था । संस्कृत भाषा में केवल पूर्णविराम का प्रयोग हुआ है । कारण यह है कि इस भाषा का स्वरूप संश्लिष्ट अथवा सामासिक है एवं गठन अन्वय सापेक्ष । हिन्दी भाषा चूँकि विश्लेषणात्मक है । इसलिए इसमें विरामचिह्नों की आवश्यकता रहती है । यहाँ यह ध्यान में रखना चाहिए कि हमने अंग्रेजी से केवल विराम चिह्न लिये हैं । अंग्रेजी भाषा और व्याकरण को हमने ग्रहण नहीं किया है, क्योंकि हिन्दी की वाक्य रचना अंग्रेजी से भिन्न है।
हिन्दी में प्रयुक्त विरामचिह्न
हिन्दी में मुख्य रूप से निम्नलिखित विरामचिनों का प्रयोग होता है
(१) अर्द्धविराम य
(२) पूर्णविराम- .
(३) अल्पविराम- ,
(५) प्रश्नवाचक चिह्न-?
(६) विस्मयादिबोधक चिह्न-!
(७) उद्धरण चिह्न-
(९) विवरण चिह्न-:
(१०) निर्देशन चिह्न-
( क ) एक वाक्य या वाक्यांश के साथ दूसरे का दूर का सम्बन्ध बतलाने के लिए अर्द्धविराम का प्रयोग होता है । जैसे-यह कलम अधिक दिनों तक नहीं चलेगी; यह बहुत सस्ती है।
(ख) प्रधान वाक्य से सम्बद्ध यदि अन्य सहायक वाक्यांशों का प्रयोग किया जाय तो अर्द्धविराम लगाकर सहायक वाक्यांशों को अलग किया जा सकता है । जैसे-छोटे-छोटे लड़के कम गहरे सरोवर में घुस जाते हैं; पानी उछालते हैं; तरंगों से क्रीड़ा करते हैं।
(ग) सभी तरह की उपाधियों के लेखन में अर्द्धविराम का प्रयोग किया जा सकता है । जैसे-एम० ए०य एल-एल० बी० । प्रायः अर्द्धविराम के प्रयोग में कभी-कभी उलझन की स्थिति भी आ जाती है । कहीं-कहीं तो लोग अल्पविराम के स्थान पर अर्द्धविराम का प्रयोग कर बैठते हैं तथा अर्द्धविराम के स्थान पर अल्पविराम का ।
(२) पूर्णविराम (।)
(क) पूर्णविराम का अर्थ पूर्ण ठहराव । जहाँ विचार की गति एकदम रुक जाय, वहाँ पूर्णविराम का प्रयोग होता है । वस्तुतः वाक्य के अन्त में पूर्ण विराम का प्रयोग होता है । जैसेकृयह लाल घोड़ा है । वह सुन्दर लड़की है।
(ख) कभी-कभी वाक्यांशों के अन्त में भी पूर्णविराम का प्रयोग होता है। जैसे किसी व्यक्ति या वस्तु के सजीव वर्णन में । जैसेश्याम वर्ण । गोल चेहरा । लम्बा कद । बड़ी-बड़ी आँखें । चैड़ा माथा । सफेद पाजामा कुर्ता पहने हुए ।
(३) अल्पविराम (,)
अल्पविराम का अर्थ है थोड़े समय के लिए ठहरना । अपनी मनोदशा के अनुसार लेखक अपने विचारों में अल्प ठहराव ले आता है । ऐसे ठहराव के लिए ही अल्पविराम का प्रयोग किया जाता है ।
(क) जब वाक्य में दो से अधिक समान पदों, पदांशों अथवा वाक्यों में संयोजक अव्यय और की गुंजाइश हो तो उस स्थान पर अल्प विराम का प्रयोग किया जाता है । जैसे-युधिष्ठिर, अर्जुन, भीम, नकुल, सहदेव आ रहे हैं । वाक्यों में मोहन सुबह आता है, झाडू लगाता है, पानी भरता है और चला जाता है।
(ख) जहाँ बार-बार शब्द आ रहे हैं और भावातिरेक में उन पर विशेष बल दिया जाय, वहाँ अल्पविराम का प्रयोग किया जाता है । जैसे-नहीं, नहीं, मुझसे यह काम नहीं होगा।
(ग) वाक्य में यदि कोई वाक्य खंड अथवा अन्तर्वर्ती पद्यांश आ जाय तो वहाँ अल्पविराम का प्रयोग करना चाहिए। जैसे-क्रोध, चाहें जैसा भी हो, मनुष्य को नष्ट कर देता है।
(घ) यदि वाक्य के बीच कुछ अव्यय (जैसे पर, इसी से, इसलिए, किन्तु, परन्तु, अतः क्यों, जिससे, तथापि) प्रयुक्त होते हैं तो उनके पहले अल्पविराम का प्रयोग हो सकता है। जैसे-राम पढ$ने में तेज है, इसीलिए सभी लोग उसकी प्रशंसा करते हैं। मैं कल गोष्ठी में जाता, किन्तु एक आवश्यक कार्य आ पड$ा है।
(च) किसी व्यक्ति को सम्बोधित करते समय अल्पविराम का प्रयोग किया जा सकता है। जैसे-प्रिय महोदय, मैं आपका ध्यान इस घटना की ओर आकर्षित करना चाहता हूँ।
(छ) जिस वाक्य में वह, यह, तब, तो, या, अब आदि लुप्त हों वहाँ अल्पविराम का प्रयोग किया जा सकता है। जैसे-तुम जो कहते हो, ठीक नहीं है (वह लुप्त है)। जब जाना ही है चले जाओ । (तब लुप्त है)।
(ज) किसी आदमी के कथन के पहले अल्पविराम लगाया जा सकता है । जैसेराम ने कहा, मैं तुमको अच्छी तरह जानता हूँ।
(झ) हाँ, नहीं, बस, सचमुच, अच्छा, अतः, वस्तुतः जैसे शब्दों से शुरू होने वाले वाक्यों में इन शब्दों के बाद अल्पविराम का प्रयोग किया जाता है । जैसे-अच्छा, तो चला जाय । नहीं, ऐसा नहीं होगा । बस, इतने से काम चल जायगा ।
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hindi
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The 2014 Genesis. You ready?
It doesn't smell so blue. Not violet, anyway: maybe deeper. The dark edge of it, where the ships fall off. And that pirate bit of it is the first thing requiring addressing: like once you're aboard you gotta look 'em in the eye.
It has that deep mahogany and dried kelp sea captain's cabin reek: more cognac than Royal Navy rum; more Flinders in Baudin's cabin than Baudin visiting the bay-rummed Englishman. Or maybe he dug out his powdered wig especially for it. I can smell that, too, but I reckon it's the Frenchman's.
Wig powder was the beautifully-scented product of the ground roots of irises, just by the way.
Then the currants begin to ooze through the starch and dressed leather, and that prickle of cordite from the powder-keen cannon lads seething belowdecks seems to go quiet when somebody opens the oven with the panforte and dumplings with the blueberries, blackberries and blackcurrants – even some juniper – and it all comes wafting up from the galley.
You're much closer to land than you thought. You can smell Australia in the summer coming over the ocean. And you've gotta get there because it turns out your hull's full of fresh fruit.
And the maritime stuff? The square-riggers? That's what happens when you know too much. A fortnight back, before these Castagnas arrived, I was with another brilliant winemaker, McLaren Vale's Stephen Pannell. As we drove around his vineyards, I remembered the old Jones Block, a legendary patch of Shiraz there on Oliver's Road.
"That's it next door," he said, nodding to the south. That's the vineyard he introduced Julian Castagna to, nearly twenty years before. That's where Julian chose to take most of the Castagna Shiraz cuttings. You can stand in those vines and look west to the Gulf St Vincent, just around the Cape from Encounter Bay, where those rival French and British sea captains just happened to bump into each other in 1802.
I wonder whether the wine would smell like sailing ships if I'd not known that.
Estate-grown Syrah, according to biodynamic principles, with a small proportion of Viognier. It’s a quieter wine than it once was, less flash but more nuance. It’s the vineyard playing its life out, offering different sides to its personality. Or the vineyard combined with the vintage conditions served. We get the meat and the smoke, the spice and the licorice, the belt of black cherry. We get it all with a fresh face too, with a slip of green garden herbs, and with terrific linger to the finish. It has Burgundian Syrah, for want of a better phrase, written all over it. It will age into a complex beauty but it has to be said, it’s drinking like a charm now.
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english
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मुख्यपोषण और स्वास्थ्य: जिम छोड़ने के बाद घंटे के लिए अपने चयापचय को रेव करें
एक पसीना तोड़ने के लिए एक घंटे को अवरुद्ध करना कभी-कभी असंभव महसूस कर सकता है। अच्छी खबर: दक्षिण अफ्रीका में केपटाउन विश्वविद्यालय के एक अध्ययन के मुताबिक, आपको अपने चयापचय को सुधारने के लिए उस समय की आवश्यकता नहीं है-केवल तीव्रता।
किसी भी कसरत के बाद, आपके शरीर को व्यायाम के दौरान उपयोग की जाने वाली ऑक्सीजन को ऑक्सीजन को प्रतिस्थापित करने की आवश्यकता होती है- इसे ऑक्सीजन ऋण कहा जाता है। और आपके ऑक्सीजन जितना अधिक होगा, एंडी स्पीयर के अनुसार, आपके शरीर को कड़ी मेहनत करने के लिए कड़ी मेहनत करने के लिए काम करना होगा, पुरुषों का स्वास्थ्य अगला शीर्ष ट्रेनर और निर्माता अराजकता कसरत.
अध्ययन में बताया गया है कि अध्ययन में एथलीटों ने अपने अधिकतम प्रयासों में से ८० प्रतिशत पर अपने ऑक्सीजन ऋण को दोगुना कर दिया है। दूसरे शब्दों में, वे कम तीव्रता पर व्यायाम करने वाले लोगों की तुलना में कैलोरी जलने वाले लाभों से लगभग दोगुना हो गए।
अपने कैलोरी जला को अधिकतम करने और अपने चयापचय को संशोधित करने के लिए अपने पसीने सत्र के अंत तक एक वसा-फ्राइंग फिनिशर पर टैक करें। उपरोक्त वीडियो में से एक आता है अराजकता कसरत, से एक नई, उच्च तीव्रता फिटनेस योजना पुरुषों का स्वास्थ्य। (कसरत का पीछा करने वाला एक लड़का केवल ६ सप्ताह में १८.६ पाउंड शुद्ध वसा खो गया!)
१२८४५ जवाब दिया
स्क्वाट्ज़िला चुनौती: आपका लेग डे दुःस्वप्न
अगला कौन मिला है?
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hindi
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२६ नवंबर को सुप्रीम कोर्ट ने अपने परिसर के साथ यौन उत्पीड़न की शिकायतों से निपटने के लिए लिंग संवेदीकरण और आंतरिक शिकायत समिति का गठन किया। समिति में शामिल हैं
(आ) ६ सदस्य
(ब) ८ सदस्य
(च) ९ सदस्य
(ड) १० सदस्य
प्रसिद्ध उपन्यासकार जिन्हें ब्रिटिश क्राउन द्वारा हाल ही में `नाइटहुड 'प्रस्तुत किया गया है
(आ) झुम्पा लाहिड़ी
(च) शोभा डे
(ड) सलमान रुश्दी
स्वतंत्र निदेशक के डाटाबैंक में पंजीकरण के लिए सभी मौजूदा स्वतंत्र निदेशकों को क्या समय सीमा दी गई है?
(च) ९ महीने
(ड) १० महीने
विकलांग व्यक्तियों के पुनर्वास के लिए किस राज्य ने उत्कृष्टता का राष्ट्रीय पुरस्कार जीता?
महात्मा गाँधी को 'सत्याग्रह' के दौरान पहली बार कब गिरफ्तार किया गया था ?
किसके द्वारा वायुमंडल के प्रदूषित होने की वजह से अम्ल वर्षा होती है
(आ) कार्बन तथा नाइट्रोजन के ऑक्साइड
(ब) सल्फर तथा नाइट्रोजन के ऑक्साइड
(च) नाइट्रोजन तथा फॉस्फोरस के ऑक्साइड
नोआखाली कहाँ स्थित है ?
निम्नलिखित में से किसे भारत के राष्ट्रपति द्वारा नियुक्त नहीं किया जाता है?
(आ) राज्यों के राज्यपाल
(ब) उच्च न्यायालय के मुख्य न्यायाधीश और न्यायाधीश
(ड) मुख्य न्यायाधीश और उच्चतम न्यायालय के न्यायाधीश
पेंशन सप्ताह अभियान का उद्घाटन निम्नलिखित में से किसने किया?
(आ) संतोष गंगवार
औसत संबंधित प्रश्न उत्तर के साथ
नवीनतम करंट अफेयर प्रश्न २०२० अगस्त ०५
मासिक करंट अफेयर प्रश्नोत्तरी जुलाई २०२०
जीके करंट अफेयर प्रश्न २०२० अगस्त ०४
टुडे करंट अफेयर प्रश्न २०२० - अगस्त ०३
करंट अफेयर प्रश्न २०२० अगस्त ०२
लेटेस्ट एवं नवीनतम करंट अफेयर प्रश्न २०२० अगस्त ०१
प्रतियोगी परीक्षाओं के लिए नाव और धारा प्रश्न
आसान और महत्वपूर्ण करेंट अफेयर्स प्रशन २०२० जुलाई ३१
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hindi
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The Working for the Coast Programme (WftC) of the Department of Environmental Affairs was established to help deal with some of the aforementioned challenges in line with the Integrated Coastal Management Act 28 of 2008. The WftC programme is informed by the broader Expanded Public Works Programme which is using labour intensive methods in its implementation. The programme is also linked to other inland programmes of the EPIP chief directorate aimed at street cleaning and greening, waste management, rehabilitation of degraded areas (catchments), bio remediation of polluted rivers etc. The Integrated Coastal Management Act (ICMA) came into effect on 1 December 2009. The ICMA is a specific management act under the umbrella National Environmental Management Act (NEMA) and is the first legal instrument, in South Africa, dedicated to managing our coastline in an integrated fashion and ensuring the sustainable use of the coast’s natural resources.
The coastal protection zone consists of land falling within an area declared in terms of the Environment Conservation Act, 1989 (Act No. 73 of 1989), as a sensitive coastal area within which activities identified in terms of section 21(1) of that Act may not be undertaken without an authorization. These areas include, among others, any coastal wetland, lake, lagoon or dam which is situated wholly or partially within 100 meters of the high-water mark. The ICM Act provides that initially the coastal protection zone would extend 100 metres inland from the high-water mark in urban areas that have already been zoned for residential, commercial, industrial or multiple-use purposes, and 1 km inland in other areas (rural areas), but these boundaries may be adjusted depending on the sensitivity of the coastline.
For example, in developed areas where the coastal environment has been highly modified, such as Durban’s beach-front and Sea Point in Cape Town, the width of the coastal protection zone could be reduced to less than 100 metres. In other instances such as estuaries where tidal influence extends to further inland to more than 1 km or where dune fields extend further than 1 km inland of the high water mark the width of the coastal protection zone could be extended beyond 1 km.
The coastlands of South Africa are characterized by a coalescence of effects of inland resource practices and those from the marine coastal zone. High population densities and poor techniques of resource development among inland communities create significant impact on coastal resources. Rapid developmental needs have had a negative impact even on un-spoilt yet pristine areas along the coastline.
These include amongst others, coastal construction works, mineral exploration, rapid expansion of urban settlements and most of all serving as a motivation is the economic benefit of the areas without having considered the direct consequences that might come as a result.
Rise in local sea level and extensive coastal erosion etc.
All these, unless averted will lead to a shrinkage in the coastal resource base as even prawns, lobsters, shrimps and fish species will continue to lose their habitats.
All these urban centres receive a growing number of tourists which creates other problems such as overload of facilities during peak periods given the carrying capacity that is mismanaged based on the economic benefit.
Support and expand facilities for education and other social services.
Knowledge about the functioning of marine and coastal ecosystems is very crucial, so are interactions and interdependencies between sea and land, the vulnerability of estuaries and coastal systems, and the impact of human practices on coastal habitats.
The department aims to create access to pristine beaches and a well conserved coastline through these projects. The many other benefits of this project include not only how to contribute to the country being a tourism destination of choice across the world, but also bring about much needed revenue to the coastal towns and communities, whilst creating job opportunities in the tourism sector.
These projects are aimed at generating approximately 2 536 work opportunities and 5 500 full-time equivalent opportunities, over two years. The beneficiaries of this programme will help in achieving the government’s objective of responsible coastal management by contributing to the development and maintenance of coastal infrastructure along the coast. They are also aimed at assisting municipalities in obtaining and maintaining blue flag status for their beaches, regular coastal clean-ups, as well as the removal of invasive alien vegetation.
To create and implement programmes to ensure sustainable and equitable maintenance of the coastal environments.
Coastal environment that is conserved, protected and sustainably enhanced.
Protection and conservation of coastal environment.
NOTE: A checklist of these structures will be developed in collaboration with Oceans & Coastal Management Branch.Workers to be supplied with GPS & Camera for locating and reporting on such developments.
The coastal zone is an area comprising coastal public property, the coastal protection zone, coastal access land and coastal protected areas, the seashore, coastal waters and the exclusive economic zone and includes any aspect of the environment on, in, under and above such area (Figure 1).
The Deputy Minister of Water and Environmental Affairs, Ms Rejoice Mabudafhasi, MP, launched a new cycle of Working for the Coast projects, with a budget of over R292 million, in the Louwville community, Saldanha Bay on 27 September 2013. The new cycle is scheduled to run for the duration of two years covering the entire coastline from Alexander Bay to Kosi Bay. The Working for the Coast projects are part of the Expanded Public Works Programme initiative of the Department of Environmental Affairs (DEA). These projects aim to create job opportunities, training and skills development, particularly in rural communities. Over R19 159 000 of the total budget for the projects is set for allocation on training of beneficiaries by accredited institutions. Primary beneficiaries of this programme are largely local women and the youth.
This initiative is envisioned to provide several thousand temporary work opportunities for people across South Africa, and will have both accredited and non-accredited training initiatives for the successful candidates.
rehabilitation of degraded areas which include dunes, estuaries.
The contribution of the Working for the Coast initiative has been marked in Saldanha Bay over the years, with their presence felt from the developed shores of Saldanha Bay in the North to the pristine shores of the Langebaan Lagoon in the South.
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english
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using System;
using System.Collections.Generic;
using System.Reflection;
using UnityEngine;
using UnityEditor;
using qtools.qhierarchy.pcomponent.pbase;
using qtools.qhierarchy.phierarchy;
using qtools.qhierarchy.phelper;
using qtools.qhierarchy.pdata;
namespace qtools.qhierarchy.pcomponent
{
public class QTagIconComponent: QBaseComponent
{
// PRIVATE
private Color backgroundColor;
// CONSTRUCTOR
public QTagIconComponent()
{
backgroundColor = QResources.getInstance().getColor(QColor.Background);
QSettings.getInstance().addEventListener(QSetting.ShowTagIconComponent , settingsChanged);
QSettings.getInstance().addEventListener(QSetting.ShowTagIconComponentDuringPlayMode, settingsChanged);
settingsChanged();
}
// PRIVATE
private void settingsChanged()
{
enabled = QSettings.getInstance().get<bool>(QSetting.ShowTagIconComponent);
showComponentDuringPlayMode = QSettings.getInstance().get<bool>(QSetting.ShowTagIconComponentDuringPlayMode);
}
// DRAW
public override void layout(GameObject gameObject, QObjectList objectList, ref Rect rect)
{
rect.x -= 18;
rect.width = 18;
}
public override void draw(GameObject gameObject, QObjectList objectList, Rect selectionRect, Rect curRect)
{
string gameObjectTag = "";
try { gameObjectTag = gameObject.tag; }
catch {}
QTagTexture tagTexture = QSettings.getInstance().get<List<QTagTexture>>(QSetting.CustomTagIcon).Find(t => t.tag == gameObjectTag);
EditorGUI.DrawRect(curRect, backgroundColor);
if (tagTexture != null && tagTexture.texture != null)
{
curRect.width = 16;
GUI.DrawTexture(curRect, tagTexture.texture, ScaleMode.ScaleToFit, true);
}
}
}
}
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code
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राधिका आप्टे ने खाई छिपकली, फोटो देख हैरान रह गए फैंस
मुंबई। राधिका आप्टे अपने दमदार अंदाज और शानदार अभिनय के लिए जानी जाती हैं। उन्होंने हाल ही में इंस्टाग्राम पर अपनी एक तस्वीर शेयर की है। जिसे देखकर इनके फैंस शॉक में आ गए है। दरअसल राधिका ने अपने गाल पर छिपकली रख कर एक फोटो शेयर की है, जिसे वो खाने की कोशिश कर रही है।
ना नूवे के लिए टैंगो सीख रहीं तमन्ना भाटिया
फिल्मफेयर अवॉर्ड्स : इन सितारों का रहा जलवा, देखें पूरी लिस्ट
वहीं राधिका की इस फोटो को देखकर घिनौनी फीलिंग भी आती है। लेकिन अगर इस गौर से देखें तो आपको पता चल जाएंगा कि उनके गाल में बैठी वो छिपकली नकली है। राधिका ने इस तस्वीर के साथ कैप्शन में बेहिंद थे सीन लिखा है। फिल्म पैडमैन की शूटिंग के दौरान पूरी टीम मस्ती करती दिखाई दे रही है।
अक्षय कुमार शूटिंग के दौरान सबके साथ मस्ती-मजाक कर रहे है और उनके साथ प्रैंक कर रहे है। सेट पर ही नकली छिपकली बना कर अक्षय पूरी यूनिट के लोगों पर इसे फेंकते दिखाई दे रहे है। अक्षय के इस वीडियो का लिंक राधिका ने अपने इंस्टाग्राम अकाउंट पर भी पोस्ट किया है। एक्ट्रेस की इस तस्वीर पर कई फनी कमेंट्स पढ़े जा सकते हैं।
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hindi
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# Baierophyllites R.K. Jain & T. Delevoryas, 1967 GENUS
#### Status
ACCEPTED
#### According to
Interim Register of Marine and Nonmarine Genera
#### Published in
null
#### Original name
null
### Remarks
null
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code
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गुजरात चुनाव जैसे-जैसे नजदीक आता जा रहा है, बीजेपी और कांग्रेस के बीच चुनावी लड़ाई भी तेज होती जा रही है। इस बीच सूत्रों के मुताबिक एक बार फिर बीजेपी सरकार बहुमत से बनती नज़र आ रही है |
सर्वे में कहा गया है कि बीजेपी गुजरात में छठी बार सरकार बनाएगी। सर्वे के मुताबिक बीजेपी की झोली में १०६ से ११६ सीटें आ सकती हैं जबकि कांग्रेस के खाते में ६३ से ७३ सीटें जाने का अनुमान है। सर्वे के मुताबिक बीजेपी को कुल ४५ फासदी वोट मिलने के आसार हैं जबकि कांग्रेस को ४० फीसदी वोट मिलने की संभावना जताई गई है। अन्य दलों को १५ फीसदी वोट शेयर मिलने का अनुमान जताया गया है। बता दें कि २०१२ के गुजरात विधान सभा चुनाव में बीजेपी को ११६ सीटें मिली थीं जबकि कांग्रेस को ६० और अन्य को ६ सीटें मिली थीं।
ज नेशन ने सर्वे के दौरान जब गुजरात की जनता से ये पूछा कि राज्य में स्टार प्रचारक कौन हैं तो राज्य के सीएम रहे और वर्तमान में पीएम नरेंद्र मोदी लोगों की पहली पसंद बनकर उभरे। ५६ फीसदी लोगों ने माना कि पीएम मोदी सबसे बड़े स्टार प्रचारक हैं जबकि इनके मुकाबले कांग्रेस उपाध्यक्ष राहुल गांधी को सिर्फ ८ फीसदी लोगों ने ही स्टार प्रचारक माना। ४ फीसदी जनता ने पाटीदार नेता हार्दिक पटेल को स्टार प्रचारक के रूप में चुना।
गुजरात के १८२ विधानसभा सीटों पर न्यूज नेशन के १८२ रिपोर्टरों ने जाकर वहां के जनता की राय जानी जिसके आधार पर यह निष्कर्ष निकाला गया है। गुजरात और हिमाचल प्रदेश विधानसभा चुनाव के नतीजे एक ही दिन १८ दिसंबर को आएंगे।
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hindi
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इंटरनेशनल वोमेन'स दए : 'स्तन कर' के विरोध में स्तन काटने वाली महिला को किया गया सम्मानित
यह कहानी केरल की नंगेली की है, जिसने अमानवीय 'ब्रेस्ट टैक्स' के विरोध में अपने स्तन काट दिए थे। अंतर्राष्ट्रीय महिला दिवस के मौके पर प्रतिष्ठित महिलाओं द्वारा राज्य में आयोजित कार्यक्रम में नंगेली को सम्मानित किया गया। केरल के इतिहास के पन्नों में छपी नंगेली की कहानी उस समय की है जब केरल के एक बहुत बड़े भाग में त्रावणकोर के राजा का शासन था।
वहां शामिल प्रतिष्ठित महिलाओं में वकील एवं सक्रियतावादी आभा सिंह, मलिश्का और पर्यावरणविद एल्सी गैबरिल भी मौजूद थीं। सभी महिलाओं ने बहादुर महिला नंगेली की चौंका देने वाली सदियों पुरानी कहानी को याद किया। नंगेली अपने पति के साथ सदियों पहले केरल के चेरथाला क्षेत्र में रहती थीं। उस समय चेरथाला में निचली जाति की महिलाओं पर 'स्तन कर' लगाया गया था। उस समय जातिवाद की जड़ें बहुत गहरी थीं और महिलाओं को स्तन न ढंकने का आदेश था।
बुधवार को आयोजित कार्यक्रम में गैबरिल ने कहा अगर उस समय कोई गरीब और निचली जाति की महिला अगर किसी कपड़े से अपने स्तनों को ढंकती तो उसे त्रावणकोर के राजा को टैक्स देना पड़ता था। गैबरिल ने आगे बताया नंगेली ने यह कर देने से मना कर दिया और अपने स्तन भी ढंके। जिसके बाद राजा के कर संग्राहकों द्वारा उसका शोषण किया गया। इसके बाद नंगेली ने अपने स्तन काट दिए, ज्यादा खून बहने के कारण उसकी मृत्यु हो गई थी।
आभा सिंह ने कहा कि यदि किसी नेता के मरने के बाद मार्ग और सड़कों का नाम उनके नाम पर रख दिया जाता है तो ऐसा उस बहादुर महिला के त्याग के बाद क्यों नहीं किया गया जिसने करीब १०० साल पहले क्रूर प्रणाली के खिलाफ लड़ाई की थी। मलिश्का ने कहा कि मैंने नंगेली की कहानी को मीडिया के जरिये जाना जिसे सुनकर मैं हैरान हो गई थी।
नंगेली सांस्कृतिक फोरम की सचिव देवी के वर्मा ने कहा कि नंगेली ने व्यवस्था के खिलाफ तब लड़ाई छेड़ी थी जब कोई महिला आंदोलन नहीं था। वह इस सवाल के साथ खड़ी हुई कि आखिर उसे राजा को टैक्स क्यों देना चाहिए।
बताई नंगेली के संघर्ष की कहनी
संविधान की प्रारूप समिति के अध्यक्ष के रूप में डॉ. आंबेडकर, संविधान में सभी नागरिकों के लिए सबसे पहले समानता का अधिकार ही सुनिश्चित करते हैं, बावजूद इसके डॉ. आंबेडकर के समानता और जाति उन्मूलन संबंधी प्रश्न आज भी ज्यों के त्यों खड़े हैं।
तकनीक कार्यों के लिए बंगलूरू बना एशिया में सबसे बेहतर शहर
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hindi
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जशपुर जिले में कहाँ बह रहा है विकास आइये जानें
रमन सरकार का विकास देखना है तो आइये जशपुर। विकास का बड़ा ही दिलचस्पग्राम नजारा बुधवार को जशपुर जिले के कुनकुरी ब्लॉक के भेलवांटोली पंचायत में देखने को मिला। जहाँ प्रदेश सरकार की विकास गाथा को जन-जन तक पहुंचाने वाली विकास यात्रा की वाहन गीली सड़क में फँस गई।ग्राम-सुराज,लोक-सुराज में भेलवाटोली मुख्य मार्ग (सुखबासुपारा से तेतरतोली) पर डामरीकरण का मांग हर बार ग्रामीणों द्वारा किया जाता रहा है,मगर इसकी सुध लेने वाला कोई नहीं है। रमन सरकार का विकास दिखाने वाली वाहन का जैसे ही इस पंचायत में प्रवेश हुआ,इसका भी सामना विकास से हो गया जिससे यह गाड़ी दलदल सड़क में फंस गई। जिसे ग्रामीणों की मदद से काफी मशक्कत करने के बाद बाहर निकाला जा सका।
जनता को झूठे विकास दिखाने वाले सरकार आज खुद देख ले कि भेलवाटोली पंचायत पहुँचने में आम जन को कितनी तकलीफें होती है।
प्रेवियस आर्टियलआज का सवाल बने हमारे शो का हिस्सा और रखे अपनी राय बेबाक लिंक ओपन करके कॉमेंट करे आज का सवाल आके शहर के लिए
नेक्स्ट आर्टियलसरकार के नियमो को दरकिनार करते हुए घरेलू गैस सिलेंडर
विकास खण्ड सिसवा आ.ड.ओ.पंचायत द्वारा किया गया जांच
कुलगाम में सेना का ऑपरेशन ऑलआउट, रावण, कुम्भकर्ण और मेघनाद का...
कांग्रेस चुनाव में ३० फीसदी नए चेहरों को मैदान में उतारेगी...
बिहार चम्पारण में वीरप्पन के नाम से कुख्यात सहोदरा थाना के...
डोनाल्ड ट्रंप-किम जोंग ने की ऐतिहासिक मुलाकात, सफल रही बातचीत
देश के सबसे बड़े पोस्ट पेमेंट बैंक की आज होगी शुरुआत,...
बस्ती जिले में आवास घोटाला प्रधानमंत्री आवास योजना में करीब ६८...
केरल के सीएम पी विजयन ने पीएम मोदी पर लगाया अनदेखी...
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hindi
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نَتہٕ یِرِ أمۍ سٕنٛز لاش آبس پؠٹھ تہٕ یِیہِ وَنہِ تہٕ اَسہِ کھالَن پھاسہِ
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kashmiri
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नोकिया ने मोबाइल मार्केट में काफी समय तक राज किया आपका भी पहला फोन शायद नोकिया का ही रहा होगा नोकिया की सक्सेस स्टोरी काफी उतार-चढ़ाव वाली रही है बड़ी इंटरेस्टिंग रही है
२०१४ के अंतराल में नोकिया में १२० देशों से ६१००० कर्मचारी थे नोकिया का बिजनेस १५० देशों से ज्यादा देशों में बिछा हुआ था
नोकिया कंपनी १५० साल पहले १८६५ मैं स्टार्ट हुई थी नोकिया का नाम लंदन में बहने वाली एक नदी के नाम पर रखा गया. इस कंपनी को तीन व्यक्तियों के साझेदारी में खोला गया १८६५ में उनमें से एक व्यक्ति माइनिंग इंजीनियर था इसके बाद १८७१ में इन तीनो ने मिलकर एक कंपनी की स्थापना की जिसका नाम था नोकिया. और अगले ९८ तक उन्होंने मिल में काम किया.१९२२ में नोकिया ने ट्रायो के साथ पार्टनरशिप करके केबल वर्क तथा रबर वर्क स्टार्ट किया. इसके बाद नोकिया रेट टेलीफोन इलेक्ट्रिक केबल और आर्मी के लिए रबर के जूते तथा मास भी बनाती थी १९६७ में ३ बड़ी कंपनियों ने मिलकर नई नोकिया कॉरपोरेशन कंपनी की स्थापना की और नई बनी हुई नोकिया कंपनी टारगेट की गई हुई थी नेटवर्किंग और रेडियो इंडस्ट्री को इसके तहत उन्होंने नए मिलिट्री सामान बनाए इसके बाद 1९८1 से १९९१ तक नोकिया ने मोबाइल रेडियो बनाएं टेलीफोन स्टेटस बनाएं कैपिसिटर और पर्सनल कंप्यूटर पर भी काम किया 1९८1 मोवीरा कंपनी ने जो कि नोकिया की ही एक कंपनी है उन्होंने दुनिया का पहला सेल्यूलर नेटवर्क लॉन्च किया जिसका नाम था गोल्डी मोबाइल टेलीफोन सर्विस. इसके बाद 1९८7 में इस ने अपना पहला मोबाइल फोन लांच किया जिसका नाम था मूवीरा सिटी मेट ९०० था उसके बाद १९९० में नोकिया ने सीमेंस के साथ मिलकर दुनिया का पहला गम नेटवर्क लॉन्च किया दुनिया का सबसे पहला कॉल फिलीपींस प्राइम मिनिस्टर ने नोकिया के ही मोबाइल से किया था वह १९९१ का साल था. जब फिलीपींस के प्राइम मिनिस्टर ने अपना पहला कॉल नोकिया के मोबाइल से किया था
इसके साथ नोकिया का मोबाइल इंडस्ट्री में सफर शुरु हो चुका था
नोकिया का पहला कमर्शियल फोन नोकिया १०११ था जो कि १९९२ में लांच हुआ इसके बाद १९९८ तक दुनिया की सबसे बड़ी इंडस्ट्री मोटोरोला को भी पीछे छोड़ दिया था इसके बाद सबसे ज्यादा बिकने वाला मोबाइल नोकिया ११०० सन २००३ में लॉन्च किया था डिमांड सन २००३ में अधिक होने के कारण तथा सभी द्वारा यही फोन खरीदे जाने के कारण उस समय यह मोबाइल लोगों की पहली पसंद मानी जाती थी या टॉर्च के साथ आता था और रात को भी चलता था
इसके बाद २००३ में गेमिंग इंडस्ट्री के लिए एक पोर्टेबल गेमिंग मोबाइल लांच किया जिसका नाम था नोकिया एंगेज.
इसके बाद नोकिया का पहला कैमरा फोन नोकिया ७६५० लॉन्च हुआ ३६०० और ३६५० भी लॉन्च हुआ यह सबसे पहले नॉर्थ अमेरिका मार्केट में लांच किया गया नोकिया की कैमरा परफॉर्मेंस तो पहले से ही अच्छी रही है
२००६ में एड् ९३ काफी एडवांस कैमरे के साथ लॉन्च हुआ इसके बाद इसके बाद २००७ में एन८ लॉन्च किया गया था इसके बाद सन 200८ में एड ९५ लांच हुआ इसके साथ ही ए७१ की लांच हुआ
इसके बाद मार्केट में सन २०११ के बाद स्क्रीन टच तथा इफोन जैसे मोबाइल आने लगे जिसकी वजह से नोकिया का मार्केट गायब सा होने लगा ,
नोकिया ने सन २०११ में माइक्रोसॉफ्ट के साथ पार्टनरशिप करके उनका सबसे पहला स्मार्टफोन लूमिया ८०० लॉन्च किया इसके बाद नोकिया ने मार्केट में २०१३ तक सामान बिक्री किया मगर फिर भी कंपनी उस लायक नहीं थी कि वह अपने फाइनेंसियल कंडीशन को कम कर सके जिसकी वजह से नोकिया के लिए सप्टेंबर २०१३ साल बहुत बुरा साबित हुआ तथा माइक्रोसॉफ्ट ने नोकिया को टेकओवर कर लिया
इसके बाद भी लोग चाहते थे कि नोकिया एक बार फिर से स्मार्टफोन इंडस्ट्री में इंटर करें तथा सन २०१७ सितंबर में नोकिया ६ पूरे भारत में लॉन्च होने जा रहा है जिसके साथ नोकिया स्मार्टफोन इंडस्ट्री में फिर से अपना जादू दिखाएगा
नोकिया तथा सोनी पिक्चर ने मिलकर ३६० कैमरा लॉन्च किया तथा मुझे लगता है कि यह आज तक का सबसे महंगा कैमरा है जिसका मूल्य है $४४०००.
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hindi
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प्रधानमंत्री के संसदीय क्षेत्र में सरकारी अस्पताल हुआ प्राइवेट अस्पतालों की तर्ज पर हाईटेक - अखंड भारत न्यूज
होम उत्तर प्रदेश वाराणसी प्रधानमंत्री के संसदीय क्षेत्र में सरकारी अस्पताल हुआ प्राइवेट अस्पतालों की तर्ज...
प्रधानमंत्री के संसदीय क्षेत्र में सरकारी अस्पताल हुआ प्राइवेट अस्पतालों की तर्ज पर हाईटेक
मुख्यचिक्तिसा अधिकारी डॉ. एस. के. उपाध्याय बताते है कि हमारे पंडित दीनदयाल अस्पताल में तमाम तरह की सुविधाएं मरीजो को उपलब्ध कराया है| २४ घंटे जांच और मरीजो में दवा वितरण सभी वार्ड में साफ सफाई की व्यवस्था मरीजों को किसी प्रकार की कोई दिक्कत न हो इसके लिए चौबीस घंटे ओपीडी चालू रहती है | अस्पताल में हाईटेक लैब का भी उद्धघाटन कुछ दिन पहले जिलाधिकारी ने किया था जिससे अब मरीजो को बाहर के चक्कर नहीं लगाने पड़ते हैं साथ ही उन्होंने बताया कि शासन की तरफ से सभी अस्पतालों में डेंगू के मरीजों को देखते हुए हाई अलर्ट किया गया है, लेकिन पंडित दीनदयाल अस्पताल में डेंगू के एक भी मरीज नही है ये सबसे बड़ी उपलब्धि है।
सभी उपलब्धियां तो बता दी गई लेकिन अभी तक पंडित दीनदयाल राजकीय चिकित्सालय में सिटी स्कैन मशीन महीनों से बंद मशीन चलाने वाला ऑपरेटर का रिटायरमेंट हो गया है | उसकी जगह अभी तक महीनों से कोई आपरेटर आया नहीं जिससे मरीजों को घोर कष्ट का सामना करना पड़ रहा है | जब सब कुछ हाईटेक हो रहा है तो सीटी स्कैन मशीन क्यों बंद पड़ी है जनहित को ध्यान में रखते हुए सीएमएस साहब इधर भी ध्यान दीजिए
प्रेवियस आर्टियलसांसद भरत सिंह ने बहुप्रतीक्षित ट्रेन संख्या ०९३०७ इंदौर-गोवहाटी एक्सप्रेस को हरी झंडी दिखाकर किया गया रवाना
नेक्स्ट आर्टियलसात वर्ष का बालक लापता परिजन परेशान
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hindi
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مواصلاتک ثقافتی پہلووٕ سمجھنہٕ سۭتۍ چُھ مراد مختلف ثقافتن ہیند علم آسن تاکہٕ کراس کلچرک لوکن سۭتۍ ہکو مؤثر طریقہٕ سۭتۍ کتھ باتھ کٔرتھ۔
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kashmiri
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हेलो दोस्तों केस कैसे हो आप सब. उम्मीद करता हूँ की अच्छे ही होंगे. दोस्तों आज की पोस्ट धमाकेदार है क्योकि आज की पोस्ट व्हेत्सप्प पर है. कई बार क्या होता है कि हम व्हेत्सप्प पर किसी से कोई परसोनल छत करते हो और बाद में आपका मोबाइल आप किसी अपने दोस्त को या किसी परिवार के सदस्य को दे देते हो.और वो इन्सान आपका परसोनल छत भी पढ़ लेता है जो आप नहीं चाहते हो.
ऐसे में अगर आप व्हेत्सप्प पर लॉक लगा देते हो तो किसी को भी शक हो सकता है कि आपके व्हेत्सप्प में कुछ पर्सनल है . अब इसका सॉल्यूशन क्या है. इसके बार एमए आज आपको मैं यहाँ पर बताने वाला हूँ. यहां पर मैं आपको बताने वाला हूँ कि आप व्हेत्सप्प को लॉक करने कि बजाये व्हेत्सप्प के परसोनल छत पर लॉक लगा दीजिये. ऐसा कैसे होगा आपको मैं इस पोस्ट में बताने वाल हूँ.
ये भी जरूर पढ़े :- अगर आपका फ़ोन हैंग होता है तो ये नुस्के अपनाये
दोस्तों इसके लिए सबसे पहले आपको अपने अंड्रॉयड फ़ोन के प्ले स्टोर से एक एप्लीकेशन डाउनलोड करनी होगी. जो मैंने आपको निचे फोटो में दिखा दी है. एप्लीकेशन आपको अपने फ़ोन में इंस्टाल कर लेनी है और फिर इसे ओपन करना है
ओपन करने के बाद आपको अपने फ़ोन में कुछ इस तरह की दिखेगी जैसे मैंने आपको निचे फोटो में दिखाई है . यहाँ पर आपको अपना पासवर्ड त्यार कर लेना है जो भी आप पासवर्ड रखना चाहते हो.
ये भी जरूर पढ़े :- यहाँ पर सवे करे अपनी फोटो और वीडियो को हमेशा के लिए
पासवर्ड त्यार करने के बाद आपको सबसे पहले ये एप्लीकेशन आपके फ़ोन की सेटिंग में ले जाएगी जहां आपको इसकी असिबिलिटी ऑन करनी होगी. जैसा आप अपने निचे फोटो में देख रहे हो.
इसके बाद आपको वापस बऐक आना है. अब ये एप्लीकेशन आपको व्हेत्सप्प पर ले जाएगी. यहां आप जिस जिस छत को लॉक करना चाहते हो उस पर क्लिक करना है . क्लिक करते ही आपका वो चाट लॉक हो जायेगा. अब जब भी आप या कोई भी उस व्हेत्सप्प पर उस छत को ओपन करेगा तो पासवर्ड मांगेगा. पासवर्ड डालने के बाद ही वो छत ओपन हो पायेगा.
अगर आप उस छत पर से लॉक हटाना चाहते हो तो आपको वापस इस एप्लीकेशन पर आना है और इस चाट पर क्लिक कर देना है . आपका लॉक हट जायेगा. उम्मीद करता हूँ की आपो मेरी ये पोस्ट पूरी तरह से समझ आ गयी होगी. अगर आपको अभी भी कोई परेशानी है इस पोस्ट को लेकर तो आप मुझे कमेंट बॉक्स में कमेंट कर सकते है. मुझे आपकी मदद करने में ख़ुशी होगी.
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hindi
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backports again. Thanks Robby for caring about that. (Closes: #396508).
r1203, which contains several bugfixes.
* Have tellico-data depend on kdelibs-data, so we don't have broken symlinks.
(Closes: #387795). Thanks Lars Wirzenius for pointing at it.
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english
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//=============================================================================
// File : dmb_spi.h
//
// Description:
//
// Revision History:
//
// Version Date Author Description of Changes
//-----------------------------------------------------------------------------
// 1.0.0 2011/04/20 yschoi Create
// 1.1.0 2011/09/29 yschoi tdmb_spi.h => dmb_spi.h
//=============================================================================
#ifndef _DMB_SPI_H_
#define _DMB_SPI_H_
#include "dmb_type.h"
//#include <linux/spi/spi.h>
/* ========== Message ID for DMB ========== */
#define DMB_MSG_SPI(fmt, arg...) \
DMB_KERN_MSG_ALERT(fmt, ##arg)
#define DMB_SPI_DEV_NAME "dmb_spi"
void dmb_spi_clear_int(void);
void dmb_spi_init (void);
void* dmb_spi_setup(void);
#if 0
extern int dmb_spi_write_then_read(u8 *txbuf, unsigned n_tx,u8 *rxbuf, unsigned n_rx);
#endif
#endif /* _DMB_SPI_H_ */
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code
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The following is the important information of Mitsubishi DiaMond Plus 120u TFA1105 installation driver.
Running the downloaded file will extract all the driver files and setup program into a directory on your hard drive. The directory these driver are extracted to will have a similar name to the DiaMond Plus 120u TFA1105 model that was downloaded (for example c:\DiaMond Plus 120u TFA1105). The setup program will also automatically begin running after extraction. However, automatically running setup can be unchecked at the time of extracting the driver file.
A list with all driver supported monitors will appear. Select your monitor Mitsubishi DiaMond Plus 120u TFA1105 and double click "NEXT".
Click "FINISH" button and than the "CLOSE" button. Now the driver file for your monitor DiaMond Plus 120u TFA1105 is installed.
Search and consult the Readme file for additional installation drivers instructions for your DiaMond Plus 120u TFA1105.
If during installation in Windows, you will be prompted with a message warning that the driver software for DiaMond Plus 120u TFA1105 has not passed Windows Logo testing. Select Continue Anyway and proceed with the installation. The driver is completely tested and verified by Mitsubishi, and safe to use.
Depending on the environment that you are using, a Windows driver provided by Microsoft may be installed automatically on your computer. However, it is recommended that you use this official driver provided by Mitsubishi for DiaMond Plus 120u TFA1105.
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english
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وُچھِو اَکھ پٔز آشِنۍ تہٕ اَکھ مٲرۍ موٚنٛد لڑکہٕ زاے میٛانہِ خٲطرٕ اکے دۄہہٕ
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kashmiri
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package com.mossle.pim.web;
import java.util.ArrayList;
import java.util.List;
import java.util.Map;
import javax.annotation.Resource;
import javax.servlet.http.HttpServletRequest;
import javax.servlet.http.HttpServletResponse;
import com.mossle.api.scope.ScopeHolder;
import com.mossle.api.user.UserConnector;
import com.mossle.core.hibernate.PropertyFilter;
import com.mossle.core.mapper.BeanMapper;
import com.mossle.core.page.Page;
import com.mossle.core.spring.MessageHelper;
import com.mossle.ext.auth.CurrentUserHolder;
import com.mossle.ext.export.Exportor;
import com.mossle.ext.export.TableModel;
import com.mossle.pim.domain.PimScheduler;
import com.mossle.pim.domain.PimSchedulerParticipant;
import com.mossle.pim.manager.PimSchedulerManager;
import com.mossle.pim.manager.PimSchedulerParticipantManager;
import org.springframework.stereotype.Controller;
import org.springframework.ui.Model;
import org.springframework.web.bind.annotation.ModelAttribute;
import org.springframework.web.bind.annotation.RequestMapping;
import org.springframework.web.bind.annotation.RequestParam;
import org.springframework.web.bind.annotation.ResponseBody;
import org.springframework.web.servlet.mvc.support.RedirectAttributes;
@Controller
@RequestMapping("pim")
public class PimSchedulerController {
private PimSchedulerManager pimSchedulerManager;
private Exportor exportor;
private BeanMapper beanMapper = new BeanMapper();
private UserConnector userConnector;
private MessageHelper messageHelper;
private CurrentUserHolder currentUserHolder;
@RequestMapping("pim-scheduler-list")
public String list(@ModelAttribute Page page,
@RequestParam Map<String, Object> parameterMap, Model model) {
List<PropertyFilter> propertyFilters = PropertyFilter
.buildFromMap(parameterMap);
String userId = currentUserHolder.getUserId();
propertyFilters.add(new PropertyFilter("EQS_userId", userId));
page = pimSchedulerManager.pagedQuery(page, propertyFilters);
model.addAttribute("page", page);
return "pim/pim-scheduler-list";
}
@RequestMapping("pim-scheduler-input")
public String input(@RequestParam(value = "id", required = false) Long id,
Model model) {
if (id != null) {
PimScheduler pimScheduler = pimSchedulerManager.get(id);
model.addAttribute("model", pimScheduler);
}
return "pim/pim-scheduler-input";
}
@RequestMapping("pim-scheduler-save")
public String save(@ModelAttribute PimScheduler pimScheduler,
@RequestParam Map<String, Object> parameterMap,
RedirectAttributes redirectAttributes) {
PimScheduler dest = null;
Long id = pimScheduler.getId();
if (id != null) {
dest = pimSchedulerManager.get(id);
beanMapper.copy(pimScheduler, dest);
} else {
dest = pimScheduler;
String userId = currentUserHolder.getUserId();
dest.setUserId(userId);
}
pimSchedulerManager.save(dest);
messageHelper.addFlashMessage(redirectAttributes, "core.success.save",
"保存成功");
return "redirect:/pim/pim-scheduler-list.do";
}
@RequestMapping("pim-scheduler-remove")
public String remove(@RequestParam("selectedItem") List<Long> selectedItem,
RedirectAttributes redirectAttributes) {
List<PimScheduler> pimSchedulers = pimSchedulerManager
.findByIds(selectedItem);
pimSchedulerManager.removeAll(pimSchedulers);
messageHelper.addFlashMessage(redirectAttributes,
"core.success.delete", "删除成功");
return "redirect:/pim/pim-scheduler-list.do";
}
@RequestMapping("pim-scheduler-export")
public void export(@ModelAttribute Page page,
@RequestParam Map<String, Object> parameterMap,
HttpServletRequest request, HttpServletResponse response)
throws Exception {
List<PropertyFilter> propertyFilters = PropertyFilter
.buildFromMap(parameterMap);
page = pimSchedulerManager.pagedQuery(page, propertyFilters);
List<PimScheduler> pimSchedulers = (List<PimScheduler>) page
.getResult();
TableModel tableModel = new TableModel();
tableModel.setName("pim scheduler");
tableModel.addHeaders("id", "name");
tableModel.setData(pimSchedulers);
exportor.export(request, response, tableModel);
}
@RequestMapping("pim-scheduler-view")
public String view(Model model) {
return "pim/pim-scheduler-view";
}
// ~ ======================================================================
@Resource
public void setPimSchedulerManager(PimSchedulerManager pimSchedulerManager) {
this.pimSchedulerManager = pimSchedulerManager;
}
@Resource
public void setExportor(Exportor exportor) {
this.exportor = exportor;
}
@Resource
public void setUserConnector(UserConnector userConnector) {
this.userConnector = userConnector;
}
@Resource
public void setMessageHelper(MessageHelper messageHelper) {
this.messageHelper = messageHelper;
}
@Resource
public void setCurrentUserHolder(CurrentUserHolder currentUserHolder) {
this.currentUserHolder = currentUserHolder;
}
}
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code
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