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I was like a kid in the proverbial sweet shop during our visit to jeweller Chris Boland's studio at Yorkshire Artspace's Exchange Place Studios. His structural Monolith rings were laid out on a workbench, all beautiful stones and unusual settings, just waiting to be tried on and cooed over. Designed with a flat edge, the rings stand up by themselves so that the stones can be displayed and admired when they're not even being worn.
Chris has picked up a raft of awards in recent years, including the British Jewellers' Association Award for Excellence in Jewellery, and craft&design's Gold Award. It's easy to see why. His designs are incredibly bold and unusual; the oxidised silver rings are particularly striking. Using natural and untreated stones such as topaz, tourmaline, emerald and citrine, Chris's modern designs contrast and compliment the stones beautifully.
This talented young man's studio should be a must-stop on any tour of Yorkshire Artspace's Open Studios in November, and his jewels on every Christmas list.
I make bold, constructed jewellery, handmade to emphasise the qualities of the gemstones inclusions. All work is made from natural stones and precious metals.
I’ve always been inspired by gemstone, crystals and other inorganic structures. My current Monolith collection uses stones with some very visible inclusions. I want to display them in a way that invites the viewer to look into the stone.
I love my studio at Exchange Place! It's probably a little large for a jeweller but it's so light and there's plenty of space for me to spread out, so I can work on lots of projects at once.
Sheffield is a great place to be a jeweller. It's probably got one of the best communities of makers in the country – very open and inviting. The Peak District on your doorstep doesn't hurt either!
I'd like to see the river areas in Sheffield improved. It seems like in every major city the riversides attract huge investments in social and financial developments, but not in Sheffield.
Precious stones and metals glimmering in all directions – Jennie's jewellery workshop is a delight! | english |
تٔمِس گٔیہِ وۄنۍ ؤری جورا بؠکار | kashmiri |
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धनु राशि का स्वामी ग्रह बृहस्पति को माना जाता है। धनु राशि के आराध्य देव दत्तोत्रय होते हैं। इन धार्मिस्था नाम के लड़कों में जांघों और हृदय रोगों का खतरा रहता है। धार्मिस्था नाम के लड़के मोटे होते हैं। इस राशि के धार्मिस्था नाम के लड़के यकृत और रीढ़ की हड्डी और कमज़ोर दृष्टि से परेशान रह सकते हैं। धार्मिस्था नाम के लड़के एडवेंचर्स और नए काम करने और सीखने के लिए तत्पर रहते हैं। धार्मिस्था नाम के लड़के भगवान में आस्था रखने वाले होते हैं इन्हें धार्मिक स्थलों पर घूमना अच्छा लगता है।
अगर आप अपने बच्चे का नाम धार्मिस्था रखने की सोच रहें हैं तो पहले उसका मतलब जान लेना जरूरी है। आपको बता दें कि धार्मिस्था का मतलब धर्म के भगवान, धर्म चाहता है होता है। धर्म के भगवान, धर्म चाहता है मतलब होने के कारण धार्मिस्था नाम बहुत सुंदर बन जाता है। अगर आप अपने बच्चे को धार्मिस्था नाम देते हैं तो जीवनभर के लिए उसका संबंध इस नाम के मतलब यानी धर्म के भगवान, धर्म चाहता है से हो जाएगा। धार्मिस्था नाम रखने से आपका बच्चा भी वो गुण ले लेता है जो इसके अर्थ में समाहित होता है। यह माना जाता है कि यदि आपका नाम धार्मिस्था है और इसका अर्थ धर्म के भगवान, धर्म चाहता है है, तो इसका गहरा प्रभाव व्यक्ति के स्वभाव पर भी पड़ता है। आगे पढ़ें धार्मिस्था नाम की राशि, इसका लकी नंबर क्या है, धार्मिस्था नाम के धर्म के भगवान, धर्म चाहता है मतलब के बारे में विस्तार से जानें।
जिनका नाम धार्मिस्था है, उनका ग्रह स्वामी बृहस्पति व शुभ अंक ३ होता है। धार्मिस्था नाम के लोग क्रिएटिव होते हैं और अपनी बातों से दूसरों को प्रभावित कर देते हैं। इन्हें अपने परिवार के प्रति बहुत प्यार होता है। ये अनुशासित होते हैं और सफलता प्राप्त करने के लिए दृढ़ निश्चय रखते हैं। इस नाम के व्यक्ति किस्मत वाले होते हैं और मुसीबत के समय में कोई ना कोई उनकी मदद कर ही देता है। धार्मिस्था नाम के लोग शारीरिक रूप से हृष्ट-पुष्ट व सेहतमंद रहते हैं। अगर आपका नाम धार्मिस्था है तो आपको बता दें कि आपको अपने जीवन में कभी धन की कमी नहीं होगी।
धार्मिस्था नाम की धनु राशि होती है। आमतौर पर धार्मिस्था नाम के लोग सबके साथ अच्छी तरह से पेश आते हैं लेकिन इन्हें रूढ़िवादी सोच पसंद नहीं है। इसलिए ये इस तरह के लोगों से दूर रहते हैं। इस राशि के लोग अक्सर थोड़े घमंडी होते हैं, जिस कारण लोग इनकी अच्छाइयों को समझ नहीं पाते। धार्मिस्था नाम वाले लोग धार्मिक होते हैं, लेकिन ये अंधविश्वास नहीं करते। इस नाम के लोगों के जीवन में किसी चीज की कमी नहीं रहती है। इस नाम के लोग अपने माता-पिता के काफी नजदीक होते हैं। ये हर रिश्ते का सम्मान करते हैं लेकिन इनके लिए अपना निजी समय भी बहुत महत्व रखता है।
धार्मिस्था की धनु राशि के हिसाब से और नाम | hindi |
Our People — Carr, Gouveia + Associates, CPAs, Inc.
We believe our core business is helping people; which is two-fold: taking care of our clients and taking care of the individuals who do the work. Therefore, we ensure our hiring practice is thorough and thoughtful. Our hiring criteria focuses on (1) the technical knowledge of the applicant, (2) how the applicant communicates their knowledge to clients and colleagues, and (3) whether the applicant will thrive in our work community. We believe a good fit ensures the development of both the firm and the person. Below are the individuals who work at CGA. | english |
सालों तक सत्ता में रहने वाली कांग्रेस को आज तक विपक्ष का रोल अदा करना नहीं आया है, दूसरी तरफ भाजपा आज भी सत्ता में होने के बावजूद विपक्ष की पार्टी लगती है। कांग्रेस के युवराज राहुल गांधी गायब होने के बाद अचानक प्रकट होते हैं और भाषण से लेकर टीवी कैमरों के सामने जुमलेबाजी करते नजर आते हैं। अभी राहुल गांधी ने एक बार फिर प्रधानमंत्री नरेन्द्र मोदी पर निशाना साधते हुए कहा कि वे हवा में बातें करते हैं और उनकी बातों में वजन होना चाहिए। ललित गेट और व्यापमं के मुद्दे पर उन्होंने प्रधानमंत्री की चुप्पी पर भी सवाल उठाए मगर मोदीजी जहां इन मुद्दों पर चुप हैं, लेकिन उनकी पूरी पार्टी पलटवार करने में पीछे नहीं है।
प्रधानमंत्री नरेंद्र मोदी को २.४१ करोड़ रुपए का ड्राफ्ट देता हुआ पंजाब केसरी परिवार
दिल्ली : पिछले दिनों पंजाब केसरी ग्रुप के पाठकों की ओर से नेपाल के भूकंप पीड़ितों को प्राइम मिनिस्टर रिलीफ फंड के लिए भेजे गए २.४१ करोड़ रुपए का ड्राफ्ट प्रधान मंत्री नरेंद्र मोदी को सौंपे गए।
प्रधानमंत्री नरेंद्र मोदी की फोटो गूगल सर्च इंजन में भारत के दस बड़े अपराधियों (टॉप १० क्रिमिनल ऑफ इंडिया) में शो होने पर कंपनी ने एक बयान जारी कर माफी मांगी है। इसके बाद #टॉपलीडर्नमो ट्रेंड के साथ ट्विटर पर टॉप पर आ गया। इस हैशटैग से मोदी समर्थकों ने गूगल द्वारा माफी मांगे जाने के बारे में बताया और प्रधानमंत्री मोदी से जुड़े कामों को पोस्ट किया।
नयी दिल्ली : प्रधानमंत्री नरेंद्र मोदी की अध्यक्षता में हुई एक बैठक में राजग सरकार के एक साल पूरा होने के मौके पर २६ मई को दूरदर्शन के किसान टीवी चैनल की शुरुआत की तैयारियों पर चर्चा हुई.
केंद्र सरकार के पहले वर्ष पर ब्रिटेन से प्रकाशित मशहूर पत्रिका द इकोनॉमिस्ट का क्या कहना है? कि देश में बदलाव का जबरदस्त मौका है, पर मोदी इसके लिए तैयार नहीं। कि मोदी सरकार एक अकेले शख्स का बैंड-बाजा है। मोदी इस तरह काम करते हैं मानो छोटे-छोटे सुधारों से बड़े बदलाव का जलवा हासिल कर लेंगे। मगर यह मुमकिन नहीं है।
जिस वक्त फिक्की मुंबई में अपना सालाना मीडिया सम्मेलन फिक्की फ्रेम्स कर रहा था, उसी दौरान ट्वेंटी फर्स्ट सेंचुरी फॉक्स के सह-सीओओ जेम्स मर्डोक ने गुरुवार, २६ मार्च को प्रधानमंत्री नरेंद्र मोदी से मुलाकात की। इस मुलाकात में मर्डोक के साथ स्टार इंडिया के सीईओ उदय शंकर भी शामिल थे। मर्डोक ने सूचना व प्रसारण मंत्रालय संभाल रहे वित्त व कॉरपोरेट मामलों के मंत्री अरुण जेटली मंत्री से भी मुलाकात की। लेकिन इन बैठकों में क्या हुआ, इसके बारे में ज्यादा कुछ पता नहीं लग सका है।
मोदी को २४ घंटे वालों से डर लगता है!
प्रधानमंत्री नरेन्द्र मोदी ने मीडिया पर टिप्पणी करते हुए कहा कि सांसदों को कुछ बोलते हुए डर लगता है कि २४ घंटे वाले जाने क्या रंग दे डाले। मोदी ने संसद भवन परिसर में बालयोगी सभागार में कांग्रेस के करण सिंह, भाजपा के अरूण जेटली और जदयू के शरद यादव को उत्कृष्ठ सांसद का पुरस्कार दिये जाने के लिए आयोजित समारोह में यह बात कही। | hindi |
Chances are in your work or business you spend time sat at a desk or a computer or both.
Find you slump in your chair or hunch forwards when using the computer or sitting at the desk?
I share how to stop this happening, so you can sit upright, have a better posture and feel pain-free when working at your desk, and computer.
Click on the video above to watch. | english |
मध्यप्रदेश के मुख्यमंत्री कमलनाथ (कमल नाथ) ने प्रधानमंत्री नरेंद्र मोदी (प्म नरेन्द्र मोदी) को पत्र लिखकर गुजरात के गिर राष्ट्रीय उद्यान से एक बब्बर शेर को राज्य के कुनों राष्ट्रीय उद्यान में शीघ्र स्थानांतरित करने का अनुरोध किया है.
प्रधानमंत्री नरेंद्र मोदी के साथ मध्यप्रदेश के मुख्यमंत्री कमलनाथ. (फाइल फोटो)
मध्यप्रदेश के मुख्यमंत्री कमलनाथ (कमल नाथ) ने प्रधानमंत्री नरेंद्र मोदी (प्म नरेन्द्र मोदी) को पत्र लिखकर गुजरात के गिर राष्ट्रीय उद्यान से एक बब्बर शेर (एशियाटिक लॉयन) को राज्य के कुनों राष्ट्रीय उद्यान में शीघ्र स्थानांतरित करने का अनुरोध किया है. कमलनाथ ने प्रधानमंत्री को पत्र लिखकर मामले में हस्तक्षेप करने का अनुरोध किया और कहा कि केंद्रीय वन एवं पर्यावरण मंत्रालय और गुजरात सरकार को इस संबंध में वह शीघ्र कार्रवाई करने के निर्देश दें. मुख्यमंत्री कमलनाथ ने पत्र में लिखा है कि बब्बर शेर (बब्बर शेर) को कुनों राष्ट्रीय उद्यान स्थानांतरित करने के लिए भारतीय वन्यजीव संस्थान और विशेषज्ञों की गठित समिति की अनुशंसाओं को भी प्रदेश सरकार ने लागू कर दिया है.
मध्यप्रदेश सरकार द्वारा २४ गांव (१५४३ परिवारों) का पुनर्वास किया जा चुका है. कुनों राष्ट्रीय उद्यान में गिर के शेर अपना भोजन प्राप्त कर सकें, इसकी भी पूरी व्यवस्था की गई है. राज्य सरकार ने इस पर काफी धन खर्च किया है. अब कुनों राष्ट्रीय उद्यान बब्बर शेर के स्वागत के लिए तैयार है. उन्होंने पत्र में लिखा कि समिति की अनुशंसाओं के अनुरूप ४०४ वर्ग किलोमीटर के अतिरिक्त वन क्षेत्र को भी कुनों राष्ट्रीय उद्यान में जोड़ा जा चुका है.
यह भी पढ़ें: गुजरात : नन्हे शावक के पीछे गाड़ी दौड़ाकर वीडियो बनाने वाले चार लोगों को पुलिस ने पकड़ा
बता दें कि केंद्रीय वन एवं पर्यावरण मंत्रालय ने इस विषय पर चिंता जताते हुए कहा था कि लुप्तप्राय बब्बर शेर के लिए दूसरा घर बनाना अत्यंत आवश्यक है. अगर बब्बर शेर को एक ही जगह रखा गया तो यह प्रजाति विलुप्त हो जाएगी. मध्यप्रदेश का कुनों राष्ट्रीय उद्यान बब्बर शेर के लिए सबसे उपयुक्त स्थान है. सुप्रीम कोर्ट के १५ अप्रैल २०१३ के आदेश के अनुसार छह महीने के भीतर बब्बर शेर को गुजरात से कुनों अभयारण्य में स्थानांतरित किया जाना था.
विडियो: बब्बर शेर के नन्हे बच्चों की अठखेलियां | hindi |
# Maintaining SSC Grid
## 准备开发环境
```
npm install
```
## 开发组件
以TDD开发模式运行测试代码
```
npm run tdd
```
## 开发文档
以开发模式运行文档网站
```
npm run docs
```
```
npm run docs-prod # 运行非调试版文档本地服务器
npm run docs-prod-unoptimized # 运行非调试版文档（未优化）本地服务器
```
## 测试
运行测试代码
```
export CHROME_BIN=chromium-browser ## 如果操作系统为Linux，需要设定该环境变量
export CHROME_BIN=google-chrome ## 如果操作系统为Linux，需要设定该环境变量
npm test
```
## 编译
编译代码以及文档
```
npm run build
```
编译调试版文档到docs-built目录下
```
npm run docs-build
```
## 发版（Releases）
注意：请勿使用`npm publish`命令进行手动发布
注意2：发布之前先`npm run test`确保测试通过
注意3：发布之前先修改`CHANGELOG.md`文件，然后将修改提交到github，然后再进行如下发布流程
```
npm run release patch // 打补丁，默认不添加任何参数，以`dry run`模式运行，防止误操作
npm run release minor // 增加新功能（当major是0的时候，比如0.2.1，那么接口变化也用release minor）
npm run release major // 接口出现变化
npm run release patch -- --run // 进行真实发版
```
### 发布的时候hang住了
如果在发布的时候，hang住了，通常是在git clone文档repo的时候，可以Ctrl+C之后执行（需要首先确认hang在哪里了）：
```
git clone git@github.com:ssc-grid/ssc-grid.github.io.git tmp-docs-repo
cd tmp-docs-repo/
rm -rf assets/ *.html rm -rf assets/ *.html
cp -r ../docs-built/* .
git add -A .
git commit -m 'Release v0.3.0'
git tag -a -m 'v0.3.0' v0.3.0
git push --follow-tags
cd ..
rm -rf tmp-docs-repo/
```
release的流程在[这里](https://github.com/AlexKVal/release-script/blob/master/src/release.js#L198)
### 发布脚本会做如下事情
- 运行lint脚本
- 运行测试脚本
- 修改版本号
- git tag
- 在本地编译源码到`dist`等目录下，并将编译结果发布到npm上
- 在本地编译文档源码到`docs-built`目录，并将`docs-built`目录下的文件push到[https://github.com/ssc-grid/ssc-grid.github.io](https://github.com/ssc-grid/ssc-grid.github.io)项目
### 关于dry run
发版工具默认以`dry run`模式运行，防止误操作导致代码被`git push`到代码仓库，以及
`npm publish`到npmjs.com中。
在运行过`dry run`模式之后，由于`package.json`中的版本号被升级了，所以需要通过`git checkout -- .`来恢复原来的版本。（先确认local没有其他修改）
你可以使用
- 学到如何使用发版工具，以及发版工具是如何运行的。
- 确认发版过程中不会出现其他问题，比如编译失败，或者其他潜在的问题。
## 历史遗留问题
TODO 可能已经废弃了，需要确认
```
grid@0.1.0 /home/chenyang/source/grid
├── UNMET PEER DEPENDENCY history@^1.17.0
└─┬ react-router@1.0.3
└── warning@2.1.0
```
通过`npm install history@2.1.2`解决。 | code |
\begin{document}
\title[Unique continuation]{Uniqueness Properties of Solutions to
Schr\"odinger Equations}
\author{L. Escauriaza}
\address[L. Escauriaza]{UPV/EHU\\Dpto. de Matem\'aticas\\Apto. 644, 48080 Bilbao,
Spain.}
\epsilonmail{luis.escauriaza@ehu.es}
\thanks{The first and fourth authors are supported by MEC grant,
MTM2004-03029, the second and third authors by NSF grants DMS-0968472 and
DMS-0800967 respectively}
\author{C. E. Kenig}
\address[C. E. Kenig]{Department of Mathematics\\University of Chicago\\mathbb Chicago, Il.
60637 \\USA.}
\epsilonmail{cek@math.uchicago.edu}
\author{G. Ponce}
\address[G. Ponce]{Department of Mathematics\\ University of California\\ Santa
Barbara, CA 93106\\ USA.}
\epsilonmail{ponce@math.ucsb.edu}
\author{L. Vega}
\address[L. Vega]{UPV/EHU\\Dpto. de Matem\'aticas\\Apto. 644, 48080 Bilbao, Spain.}
\epsilonmail{luis.vega@ehu.es}
\keywords{Schr\"odinger evolutions}
\sigmaubjclass{Primary: 35Q55}
\maketitle
\sigmaection{Introduction}\lambdabel{S: Introduction}
To place the subject of this paper in perspective, we start out with a brief discussion
of unique continuation. Consider solutions to
\begin{equation}
\lambdabel{aa1}
\Delta u(x)=\sigmaum_{j=1}^n\frac{\partial^2u}{\partial x_j^2}(x)=0,
\epsilonnd{equation}
(harmonic functions) in the unit ball $\,\{x\in \mathbb R^n\,:\,|x|<1\}$. When $n=2$, these functions are real parts
of holomorphic functions, and so, if they vanish of infinite order at $x=0$, they must vanish identically. We call this the strong unique continuation property (s.u.c.p.). The same result holds for $n>2$, since harmonic functions are still real analytic in $\,\{x\in \mathbb R^n\,:\,|x|<1\}$. In fact, it is well-known that if $\,P(x,D)$ is a linear elliptic differential operator with real analytic coefficients, and $\,P(x,D)u=0$ in a open set $\,\Omega\sigmaubset \mathbb R^n$, then $u$ is real analytic in $\,\Omega$. Hence, the (s.u.c.p.) also holds for such solutions. Through the work of Hadamard \cite{Had} on the uniqueness of the Cauchy problem (which is closely related to the strong unique continuation property discussed earlier) it became clear (for applications in nonlinear problems) that it would be desirable to establish the strong unique continuation property for operators whose coefficients are not necessarily real analytic, or even $\, C^{\infty}$. The first results in this direction were found in the pioneering work of Carleman \cite{Car} (when $n=2$) and M\"uller \cite{Mu} (when $n>2$), who proved the (s.u.c.p) for
$$
P(x,D)=\Delta+V(x),\;\;\;\;\;\text{with}\;\;\;\;\;\;V\in L^{\infty}_{loc}(\mathbb R^n).
$$
In order to establish his result, Carleman introduced a method (the method of \lq\lq Carleman estimates") which has permeated the subject ever since. In this context, an example of a Carleman estimate is :
\vskip.05in
\epsilonmph{ For $\,f\in C^{\infty}_0(\{x\in \mathbb R^n\,:\,|x|<1\}-\{0\})$, $\,\alphapha>0$
and
$$
w(r)=r\,\epsilonxp (\,\int_0^r\frac{e^{-s}-1}{s} ds),
$$
one has
\begin{equation}
\lambdabel{1aa}
\alphapha^3\,\int\,w^{-1-2\alphapha}(|x|) f^2(x) dx\leq c\,\int w^{2-2\alphapha}(|x|)\,|\Delta f(x)|^2 dx,
\epsilonnd{equation}
with $\,c\,$ independent of $\,\alphapha$}
\vskip.05in
For a proof of this estimate, see \cite{EsVe}, \cite{BoKe}. The (s.u.c.p.) of Carleman-M\"uller follows easily from \epsilonqref{1aa}
(see \cite{Ke2} for instance).
In the late 1950's and 1960's there was a great deal of activity on the subject of (s.u.c.p.) and the closely related uniqueness
in the Cauchy problem, some highlights being \cite{AKS} and \cite{Cal} respectively, both of which use the method of Carleman estimates. These results and methods
have had a multitude of applications to many areas of analysis, including to non-linear problems. (For a recent example, see \cite{KeMe} for an application to energy critical
non-linear wave equations).
In connection with the Carleman-M\"uller (s.u.c.p.) a natural question is : How fast is a solution $u$ allowed to vanish, before it must vanish identically?
By considering $n=2$, $u(x_1,x_2)=\mathbb Re (x_1+ix_2)^N$, we see that to make sense of the question, a normalization is required, for instance
$$
\sigmaup _{|x|<3/4} |u(x)|\geq 1,\;\;\;\;\;\;\;\|u\|_{L^{\infty}(|x|<1)}<\infty.
$$
We refer to questions of this type as \lq\lq quantitative unique continuation". It is also of interest to consider unique continuation type questions around the point at infinity. For instance, a conjecture of E. M. Landis \cite{KoLa} was : if
$$
\Delta u +V u =0,\;\;\;\;x\in\mathbb R^n,\;\;\;\text{with}\;\;\;\|V\|_{\infty}\leq 1 ,\;\;\;\|u\|_{\infty}<\infty,
$$
and for some $\,\epsilonpsilon>0$ one has
$$
|u(x)|\leq c_{\epsilonpsilon}\,e^{-c_{\epsilonpsilon}|x|^{1+\epsilonpsilon}},
$$
then $\,u\epsilonquiv 0$.
For the case of complex valued potentials $V(x)$,
this conjecture was disproved by Meshkov \cite {Me} who constructed $V,\,u,\,u\not \epsilonquiv 0$ with
$$
|u(x)|\leq c\, e^{-c|x|^{4/3}},\;\;\;\;n\geq 2.
$$
Meshkov also showed that if
$$
|u(x)|\leq c_{\epsilonpsilon}\, e^{-c_{\epsilonpsilon}|x|^{4/3+\epsilonpsilon}},\;\;\;\;\text{for some}\;\;\;\;\epsilonpsilon>0,
$$
then $\,u\epsilonquiv 0$.
It turns out that a \lq\lq quantitative" formulation of this can also be proved, as it was done in \cite{BoKe}, and this was crucial for the resolution in \cite{BoKe} of a long-standing problem in disordered media, namely Anderson localization near the bottom of the spectrum, for the continuous Anderson-Bernoulli model in $\,\mathbb R^n,\,n\geq 1$.
Next, we turn to versions of unique continuation for evolution equations. We start with parabolic equations and consider solutions of
$$
\partial_t u-\Delta u=W\cdot \nabla u+ Vu,\;\;\;\;\;\;\;\text{with}\;\;\;\;\;\;\|W\|_{\infty}+\|V\|_{\infty} <\infty,
$$
(or equivalently $|\partial_tu-\Delta u|\leq M(|\nabla u|+|u|)$).
Using a parabolic analog of the Carleman estimate described earlier, one can show that if
$$
|\partial_tu-\Delta u|\leq M(|\nabla u|+|u|),\;\;\;\;\;\;(x,t)\in \{x\in\mathbb R^n:|x|<4R\}\times [t_0,t_1],\;\;\;R>0,
$$
with $\,|u(x)|\leq A$ and
$$
u\epsilonquiv 0,\;\;\;(x,t)\in \{x\in\mathbb R^n:R<|x|<4R\}\times [t_0,t_1],
$$
then
$$
u\epsilonquiv 0,\;\;\;(x,t)\in \{x\in\mathbb R^n:|x|<R\}\times [t_0,t_1].
$$
We call this type of result \lq\lq unique continuation through spatial boundaries", (see \cite{EsVe}, \cite{Ve} and references therein for this type of result
and strengthenings of it). This result is closely related to the \lq\lq elliptic" (s.u.c.p.) discussed before. On the other hand, for parabolic equations, there is also a
\lq\lq backward uniqueness" principle, which is very useful in applications to control theory (see \cite{lm60} for an early result in this direction) : Consider solutions to
$$
|\partial_tu-\Delta u|\leq M(|\nabla u|+|u|),\;\;\;\;\;(x,t)\in\mathbb R^n\times (0,1],
$$
with $\,\|u\|_{\infty}\leq A$. Then, if $\,u(\cdot,1)\epsilonquiv 0$, we must have $\,u\epsilonquiv 0$.
This result is also proved through Carleman estimates (see \cite{lm60}).
Recently, a strengthening of this result has been obtained in \cite{EsSS}, where one considers solutions only defined in $\,R^n_{+}\times (0,1]$, $\,R^n_{+}=\{(x_1,..,x_n)\in\mathbb R^n\,:\,x_1>0\}$, without any assumptions on $\,u\,$ at $\,x_1=0$, and still obtains the \lq\lq backward uniqueness" result. This strengthening had an important application to non-linear equations, allowing the authors of \cite{EsSS} to establish a long-standing conjecture of J. Leray on regularity and uniqueness of solutions to the Navier-Stokes equations (see also \cite{Se} for a recent extension).
Finally, we turn to dispersive equations. Typical examples of these are the $k$-generalized KdV equation
\begin{equation}
\lambdabel{kdv1}
\partial_t u +\partial_x^3 u+ u^k\partial_xu=0,\;\;\;\;\;\; (x,t)\in \mathbb R\times
\mathbb R,\;\,\;k\in \mathbb Z^+,
\epsilonnd{equation}
and the non-linear Schr\"odinger equation
\begin{equation}
\lambdabel{ae1}
\partial_t u =i( \Delta u \pm |u|^{p-1}u),\;\;\;\;\;\; (x,t)\in
\mathbb{R}^n\times \mathbb R,\;\;\,p>1.
\epsilonnd{equation}
These equations model phenomena of wave propagation and have been extensively studied in the last 30 years or so.
For these equations,\lq\lq unique continuation through spatial boundaries " also holds, as it was shown by Saut-Scheurer \cite{SaSc} for the KdV-type equations and by Izakov \cite{Iza} for Shr\"odinger type equations. (All of these results were established trough Carleman estimates). These equations however are time reversible (no preferred time direction) and so \lq\lq backward uniqueness" is immediate, unlike in parabolic problems.
Once more in connection with control theory, this time for dispersive equations, Zhang \cite{BZ} showed, for solutions of
\begin{equation}
\lambdabel{zhang}
\partial_t u =i( \partial_x^2 u \pm |u|^{2}u),\;\;\;\;\;\; (x,t)\in
\mathbb{R}\times[0,1],
\epsilonnd{equation}
that if $\,u(x,t)=0$ for $(x,t)\in (-\infty,a)\times\{0, 1\}$ (or $(x,t)\in (a,\infty)\times\{0, 1\}$) for some $a\in\mathbb R$, the $\,u\epsilonquiv 0$.
Zhang's proof was based on the inverse scattering method which uses that this is a completely integrable model, and did not apply to
other non-linearities or dimensions. This type of result was extended to the $k$-generalized KdV \epsilonqref{kdv1} and the general non-linear Schr\"odinger equation in \epsilonqref{ae1} in all dimensions
(where inverse scattering is no longer available) using suitable Carleman estimates (see \cite{KPV02}, \cite{IK04}, \cite{IK06}, and references therein).
For recent surveys of the results presented so far, see \cite{Ke1}, \cite{Ke2}.
Returning to \lq\lq backward uniqueness" for parabolic equations, in analogy with Landis' \lq\lq elliptic" conjecture mentioned earlier, Landis-Oleinik \cite{LaOl} conjectured that in the \lq\lq backward uniqueness" result one can replace the hypothesis $\,u(\cdot,1)\epsilonquiv 0$ with the weaker one
$$
|u(x,1)|\leq c_{\epsilonpsilon}\,e^{- c_{\epsilonpsilon}|x|^{2+\epsilonpsilon}},\;\;\;\text{for some }\;\;\;\epsilonpsilon>0.
$$
This is indeed true and was established in \cite{EKPV06a} and \cite{Ng}. Similarly, one can conjecture (as it was done in \cite{EKPV08b}) that for Schr\"odinger equations, if
$$
|u(x,0)|+ |u(x,1)|\leq c_{\epsilonpsilon}\,e^{- c_{\epsilonpsilon}|x|^{2+\epsilonpsilon}},\;\;\;\text{for some }\;\;\;\epsilonpsilon>0,
$$
then $\,u\epsilonquiv 0$. This was established in \cite{EKPV06a}.
In analogy with the improvement of \lq\lq backward uniqueness" in \cite{EsSS}, one can show that it suffices to deal with solutions in $\,\mathbb R^n_{+}\times(0,1]$
(for parabolic problems) and require
$$
|u(x,1)|\leq c_{\epsilonpsilon}\,e^{- c_{\epsilonpsilon}x_1^{2+\epsilonpsilon}},\;\;\;x_1>0,\;\;\;\;\text{for some }\;\;\;\epsilonpsilon>0,
$$
to conclude that $\,u\epsilonquiv 0$ (\cite{Ng}), and that for the Schr\"odinger equations it suffices to have $\,u\,$ a solution in $\,\mathbb R^n_{+}\times[0,1]$, with
$$
|u(x,0)|+ |u(x,1)|\leq c_{\epsilonpsilon}\,e^{- c_{\epsilonpsilon}x_1^{2+\epsilonpsilon}},\;\;\;x_1>0,\;\;\;\;\text{for some }\;\;\;\epsilonpsilon>0,
$$
to conclude that $\,u\epsilonquiv 0$, as we will prove in section 5 of this paper.
In \cite{EKPV06} it was pointed out for the first time (see also \cite{Cha}) that both the results in \cite{EKPV06a} and in \cite{EKPV06}, in the case of the free heat
equation
$$
\partial_tu=\Delta u,
$$
and the free Schr\"odinger equation
$$
\partial_tu=i \Delta u,
$$
respectively, are in fact a corollary of the more precise Hardy uncertainty principle for the Fourier transform, which says :
\vskip.05in
\epsilonmph{If $f(x)=O(e^{-|x|^2/\beta^2})$, $\widehat f(\xi)=O(e^{-4|\xi|^2/\alphapha^2})$ and
$1/\alphapha\beta>1/4$, then $f\epsilonquiv 0$, and if $1/\alphapha\beta=1/4$, $f(x)=ce^{-|x|^2/\beta^2}$} as will be discussed below.
\vskip.05in
Thus, in a series of papers (\cite{EKPV06}-\cite{EKPV10}, \cite{CEKPV}) we took up the task of finding the sharp version of the Hardy uncertainty principle, in the
context of evolution equations. The results obtained have already yielded new results on non-linear equations. For instance in \cite{EKPV08m} and \cite{EKPV10}
we have found applications to the decay of concentration profiles of possible self-similar type blow-up solutions of non-linear Schr\"odnger equations and to the decay of possible solitary wave type solutions of non-linear Schr\"odinger equations.
In the rest of this work we shall review some of our recent results concerning unique continuation
properties of
solutions of Schr\"odinger equations of the form
\begin{equation}
\lambdabel{e1}
\partial_t u =i( \Delta u + F(x,t,u,\bar u)),\;\;\;\;\;\; (x,t)\in
\mathbb{R}^n\times \mathbb R.
\epsilonnd{equation}
We shall be mainly interested in the case where
\begin{equation}
\lambdabel{F1a}
F(x,t,u,\bar u)=V(x,t) u(x,t)
\epsilonnd{equation}
is describing the evolution of the Schr\"odinger flow with a time dependent
potential $V(x,t)$, and
in the semi-linear case
\begin{equation}
\lambdabel{F1b}
F(x,t,u,\bar u)= F(u,\bar u),
\epsilonnd{equation}
with $ F: \mathbb C \times \mathbb C\to \mathbb C$, $F(0,0)=\partial_uF(0,0)=\partial_{\bar u}F(0,0)=0$.
Let us consider a familiar dispersive model,
the $k$-generalized
Korteweg-de Vries equation \epsilonqref{kdv1} and recall a theorem established in \cite{EKPV07} :
\begin{theorem}
\lambdabel{theorem1}
There exists $c_0>0$ such that for any pair
$$
u_1,\,u_2\in C([0,1]:H^4(R)\cap L^2(|x|^2dx))
$$
of solutions of
\epsilonqref{kdv1}
such that if
\begin{equation}
\lambdabel{3:2}
u_1(\cdot,0)-u_2(\cdot,0),\,\;\, u_1(\cdot,1)-u_2(\cdot,1)\in
L^2(e^{c_0x_{+}^{3/2}}dx),
\epsilonnd{equation} then $u_1\epsilonquiv u_2$.
\epsilonnd{theorem}
Above we have used the notation: $ x_{+}=max\{x;\,0\}$.
Notice that taking $u_2\epsilonquiv 0\,$ Theorem \ref{theorem1} gives a restriction on
the possible decay
of a non-trivial solution of
\epsilonqref{kdv1}
at two different times.
The power $3/2$ in the exponent in \epsilonqref{3:2} reflects the asymptotic behavior of
the Airy function.
More precisely, the solution
of the initial value problem (IVP)
\begin{equation}
\begin{aligned}
\begin{cases}
\partial_t v + \partial_x^3 v=0,\\
v(x,0)=v_0(x),
\epsilonnd{cases}
\epsilonnd{aligned}
\epsilonnd{equation}
is given by the group $\{U(t)\,:\,t\in R\}$
$$
U(t)v_0(x)=\frac{1}{\root{3}\of{3t}}\,Ai\left(\frac{\cdot}{\root
{3}\of{3t}}\right)\ast v_0(x),
$$
where
$$
Ai(x)=c\,\int_{-\infty}^{\infty}\,e^{ ix\xi+i \xi^3 }\,d\xi,
$$
is the Airy function which satisfies the estimate
$$
|Ai(x)|\leq c (1+x_{-})^{-1/4}\,e^{-c x_{+}^{3/2}}.
$$
It was also shown in \cite{EKPV07} that Theorem 1 is optimal :
\begin{theorem}
\lambdabel{theorem2}
There exists $ \,u_0\in S(\mathbb R),\;u_0\not \epsilonquiv 0$ and $\Delta T>0$ such that
the IVP associated
to the k-gKdV equation \epsilonqref{kdv1}
with data $u_0$ has solution
$$
u\in C([0,\Delta T] : \mathbb S(\mathbb R)),
$$
satisfying
$$
|u(x,t)|\leq \tilde d \,e^{-x^{3/2}/3}, \;\;\;\;\;\;\;\;x>1,\;\;\;\,t\in [0,\Delta T],
$$
for some constant $\tilde d>0$.
\epsilonnd{theorem}
In the case of the free Schr\"odinger group $\{e^{it\Delta}\,:\,t\in\mathbb R\}$
$$
e^{it\Delta}u_0(x)=(e^{-i|\xi|^2t} \widehat
u_0)^\lor(x)=\frac{e^{i|\cdot|^2/4t}}{(4\pi i t)^{n/2}}*u_0(x),
$$
the fundamental solution does not decay.
However, one has the identity
\begin{equation}
\lambdabel{formula1}
\begin{aligned}
&u(x,t)= e^{it\Delta}u_0(x)= \int_{\mathbb R^n} \frac{e^{i|x-y|^2/4t}}{(4\pi i t)^{n/2}}\,
u_0(y)\,dy\\
\\
&=\frac{e^{i|x|^2/4t}}{(4\pi i t)^{n/2}} \int_{\mathbb R^n}e^{-2ix\cdot y/4t} e^{i|y|^2/4t}
u_0(y)\,dy\\
\\
&= \frac{e^{i|x|^2/4t}}{(2 i t)^{n/2}}\;
\widehat{\;(e^{i|\cdot|^2/4t}u_0)\,}\left(\frac{x}{2 t}\right),
\epsilonnd{aligned}
\epsilonnd{equation}
where
$$
\widehat f(\xi)=(2\pi)^{-n/2} \int_{\mathbb R^n} e^{-i\xi\cdot x} f(x)dx.
$$
Hence,
$$
c_t e^{-i|x|^2/4t} \,u(x,t) = \widehat{(e^{i|\cdot|^2/4t}u_0)}\left(\frac{x}{2
t}\right),\,\,\,\,\,\,\,c_t=(2 i t)^{n/2},
$$
which tells us that
$e^{-i|x|^2/4t} \,u(x,t)$ is a multiple of the rescaled Fourier transform
of $\;e^{i|y|^2/4t}u_0(y)$. Thus, as we pointed out earlier, the behavior of the solution of the free
Schr\"odinger equation is closely
related to uncertainty principles for the Fourier transform.
We shall study
these uncertainty principles
and their relation with the uniqueness properties of the solution of the
Schr\"odinger equation \epsilonqref{e1}.
In the early $1930$'s N. Wiener's remark (see \cite{Hardy}, \cite{In}, and \cite{Mo}):
\vskip.1in
``a pair of transforms $f$ and $g$ ($\widehat f$) cannot
both be very small'',
\vskip.1in
\noindent motivated the works of G. H. Hardy \cite{Hardy}, G. W. Morgan \cite{Mo}, and A. E. Ingham \cite{In} which
will
be considered in detail in this note.
However, before that we shall return to a review of some previous results concerning uniqueness
properties of solutions
of the Schr\"odinger equation which we mentioned earlier and which were not motivated by the formula
\epsilonqref{formula1}.
For solutions $u(x,t)$ of the $1$-D cubic Schr\"odinger equation \epsilonqref{zhang}
B. Y. Zhang \cite{BZ} showed :
\vskip.05in
\epsilonmph{If $u(x,t)=0$ for $(x,t)\in (-\infty, a)\times \{0,1\}\,\, ($or $(x,t)\in
(a,\infty)\times \{0,1\})\,$ for some
$\,a\in \mathbb R$, then $u\epsilonquiv 0$.}
\vskip.03in
As it was mentioned before, his
proof is based on the inverse scattering method, which uses the fact that the
equation in \epsilonqref{zhang} is a completely integrable model.
\vskip.05in
In \cite{KPV02} it was proved under general assumptions on $F$
in \epsilonqref{F1b} that :
\vskip.03in
\epsilonmph{If $u_1,\,u_2\in C([0,1]:H^s(\mathbb R^n))$, with $\,s>\max \{n/2; \,2\}\, $ are
solutions of the equation \epsilonqref{e1} with
$F$ as in \epsilonqref{F1b} such that
$$
u_1(x,t)=u_2(x,t),\;\;\;(x,t)\in \Gamma^c_{x_0}\times \{0,1\},
$$
where $ \Gamma^c_{x_0}$ denotes the complement of a cone $\Gamma_{x_0}$ with vertex
$x_0\in \mathbb R^n$ and opening $<180^0$,
then
$u_1\epsilonquiv u_2$.}
(For further results in this direction see \cite{KPV02}, \cite{IK04}, \cite{IK06},
and
references therein).
\vskip.03in
A key step in the proof in \cite{KPV02} was the following uniform exponential decay
estimate:
\begin{lemma}\lambdabel{ultimo} There exists $\epsilonpsilon_0>0$ such that if
\begin{equation}
\lambdabel{hyp2}
\mathbb V:\mathbb R^n\times [0,1]\to\mathbb C,\;\;\;\;\text{with}\;\;\;\;
\|\mathbb V\|_{L^1_tL^{\infty}_x}\leq \epsilonpsilon_0,
\epsilonnd{equation}
and $u\in C([0,1]:L^2(\mathbb R^n))$ is a strong solution of the IVP
\begin{equation}
\begin{cases}
\begin{aligned}
\lambdabel{eq1}
&\partial_tu=i(\Delta +\mathbb V(x,t))u+\mathbb G(x,t),\\
&u(x,0)=u_0(x),
\epsilonnd{aligned}
\epsilonnd{cases}
\epsilonnd{equation}
with
\begin{equation}
\lambdabel{hyp3} u_0,\,u_1\epsilonquiv u(\,\cdot\,,1)\in
L^2(e^{2\lambdambda\cdot x}dx),\;\mathbb G\in L^1([0,1]:L^2(e^{2\lambdambda\cdot
x}dx)),
\epsilonnd{equation}
for some $\lambdambda\in\mathbb R^n$, then there exists $c_n$ independent of
$\lambdambda$ such that
\begin{equation}
\begin{aligned}
\lambdabel{uno}
&\sigmaup_{0\leq t\leq 1}\| e^{\lambdambda\cdot x} u(\,\cdot\,,t)\|_{L^2(\mathbb \mathbb R^n)} \\
&\leq c_n
\Big(\|e^{\lambdambda\cdot x} u_0\|_{L^2(\mathbb \mathbb R^n)} + \|e^{\lambdambda\cdot x}
u_1\|_{L^2(\mathbb \mathbb R^n)} +\int_0^1
\|e^{\lambdambda\cdot x}\, \mathbb G(\cdot, t)\|_{L^2(\mathbb \mathbb R^n)} dt\Big).
\epsilonnd{aligned}
\epsilonnd{equation}
\epsilonnd{lemma}
Notice that in the above result one assumes the existence of a reference
$L^2$-solution $u$ of the equation \epsilonqref{eq1} and then under the
hypotheses \epsilonqref{hyp2} and \epsilonqref{hyp3} shows that the exponential decay in the
time interval $[0,1]$ is preserved.
The estimate \epsilonqref{uno} can be combined with the subordination formula
\begin{equation}
\lambdabel{est1}
e^{\gamma |x|^p/p}\sigmaimeq \int_{\mathbb R^n} \,e^{\gamma^{1/p}\lambdambda\cdot
x-|\lambdambda|^q/q}\,
|\lambdambda|^{n(q-2)/2}\,d\lambdambda,\,\,\,\forall\, x\in \mathbb R^n\,\,\,\text{and}\,\,\,p>1,
\epsilonnd{equation}
to get that for any $\alphapha>0$ and $ a>1$
\begin{equation}
\lambdabel{dos}
\begin{aligned}
\sigmaup_{0\leq t\leq 1}\| e^{\alphapha|x|^a} u(\,\cdot\,,t)\|_{L^2(\mathbb \mathbb R^n)} &\\
\leq c_n
\Big(\|e^{\alphapha|x|^a} u_0\|_{L^2(\mathbb \mathbb R^n)} +& \|e^{\alphapha|x|^a}
u_1\|_{L^2(\mathbb \mathbb R^n)} +\int_0^1
\|e^{\alphapha|x|^a}\, \mathbb G(\cdot, t)\|_{L^2(\mathbb \mathbb R^n)} dt\Big).
\epsilonnd{aligned}
\epsilonnd{equation}
Under appropriate assumptions on the potential $V(x,t)$ in \epsilonqref{F1a} one writes
$$
V(x,t)u= \chi_{R} V(x,t)u + (1-\chi_{R}) V(x,t)u = \mathbb V(x,t)u + \mathbb G(x,t),
$$
with $\chi_R\in C^{\infty}_0,\,$ $\chi_R(x)=1,\,|x|<R$, supported in $|x|<2R$, and
applies the estimate
\epsilonqref{dos} by fixing $\,R\,$ sufficiently large. Also under appropriate hypothesis
on $F$ and $u$ a similar argument
can be used for
the semi-linear equation in
\epsilonqref{F1b}.
\vskip.03in
The estimate \epsilonqref{dos} gives a control on the decay of the solution in the whole
time interval
in terms of that at the end points and that of the \lq\lq external force''. As we shall see
below a key idea
will be to get improvements of this estimate based on logarithmically convex
versions of it.
We recall that if one considers the equation \epsilonqref{e1} with initial data $u_0\in
\mathbb S(\mathbb R^n)$
and a smooth potential $V(x,t)$ in \epsilonqref{F1a}
or smooth nonlinearity $F$ in \epsilonqref{F1b}, it follows that the corresponding
solution satisfies that
$u\in C([-T,T] :\mathbb S(\mathbb R^n))$. This can be proved using the commutative property
of the operators
$$
L=\partial_t-i\Delta,\;\;\;\;\;\;\text{and}\;\;\;\;\;\;\Gamma_j=x_j+2t\partial_{x_j},\,\,\,j=1,..,n,
$$
see \cite{HKT1}-\cite{HKT2}. From the proof of this fact one also has that the
persistence property of the solution $u=u(x,t)$
(i.e. if the data $u_0\in X$, a function space, then the corresponding solution
$u(\cdot)$ describes
a continuous curve in $X$, $u\in C([-T,T]:X)$, $\,T>0$)
with data $u_0\in L^2(|x|^m)$
can only hold if $u_0\in H^s(\mathbb R^n)$ with $s\geq 2m$.
Roughly speaking, for exponential weights one has a more involved argument where
the time direction plays a role.
Considering the IVP for the one dimensional free
Schr\"odinger equation
\begin{equation}
\lambdabel{*}
\begin{cases}
\begin{aligned}
&\partial_t u=i\partial_x^2u,\;\;\;\;\;\;\;\;\;\;\;\,x,\,t \in \mathbb R,\\
&u(x,0)=u_0(x)\in L^2(\mathbb R),
\epsilonnd{aligned}
\epsilonnd{cases}
\epsilonnd{equation}
and assuming that $ e^{\beta x}u_0 \in L^2(\mathbb R),\,\,\beta>0$, then one formally has
that \
$$
v(x,t)=e^{\beta x}u(x,t)
$$
satisfies the equation
$$
\partial_t v=i(\partial_x-\beta)^2v.
$$
Thus,
$$
v(x,\pm 1)=e^{\beta x}u(x,\pm 1)\in L^2(\mathbb R)\,\;\;\;\text{if}\;\;\;\,e^{\pm 2\beta \xi}\,
\widehat{e^{\beta x}u_0}\in L^2(\mathbb R).
$$
However, if we knew that
$ e^{\beta x}u(x,1),\;\;e^{\beta x}u(x,-1) \in L^2(\mathbb R)$ integrating forward in time
the positive frequencies of $e^{\beta x}u(x,t)$ and backward in time the negative
frequencies of $e^{\beta x}u(x,t)$
one gets an estimate similar to that in \epsilonqref{uno} with $\lambdambda=\beta$ and
$\mathbb G=0$. This argument motivates the
idea behind Lemma \ref{ultimo} and its proof.
\vskip.2in
The rest of this paper is organized as follows: section 2 contains the results
related to Hardy's uncertainty principle including a short discussion on the version of this principle
in terms of the heat flow.
Section 3 those concerned with Morgan's uncertainty principle. In section 4 we
shall consider the limiting case in section 3. Also, section 4 includes the statements
of some related forthcoming results.
Earlier in the introduction we have discussed uniqueness results obtained under the assumption that the
solution vanishes
at two different time in a semi-space
(see \cite{BZ}, \cite{IK04}, \cite{IK06}, \cite{EKPV08b}). In section 2 similar uniqueness results will be
established under a Gaussian decay hypothesis,
in the whole space. In section 5 we shall obtain a unifying result, i.e. a
uniqueness result under Gaussian decay in
a semi-space of $\,\mathbb R^n$ at two different times. The appendix contains an abstract
lemma and a corollary which will be used in
the previous sections.
\sigmaection{Hardy's Uncertainty Principle}\lambdabel{hardy}
In \cite{Hardy} G. H. Hardy's proved the following one dimensional ($n=1$) result:
\vskip.07in
\epsilonmph{If
$f(x)=O(e^{-|x|^2/\beta^2})$, $\widehat f(\xi)=O(e^{-4|\xi|^2/\alphapha^2})$ and
$1/\alphapha\beta>1/4$, then $f\epsilonquiv 0$.
\newline
Also, if $1/\alphapha\beta=1/4$, $f(x)$ is a constant multiple of $e^{-|x|^2/\beta^2}$.}
\vskip.06in
To our knowledge the available proofs of this result and its variants use complex
analysis, mainly appropriate versions
of the
Phragm\'en-Lindel\"of principle.
There has also been considerable interest in a better understanding of this result
and on extensions of it
to other settings:
\cite{bonamie1}, \cite{bonamie2}, \cite{CoPr}, \cite{Ho}, and \cite{SST}. In
particular, the extension of
Hardy's result to higher
dimension $n\geq 2$ (via Radon transform) was given in \cite{SST}.
The formula \epsilonqref{formula1} allows us to re-write this uncertainty principle in terms of the
solution of the IVP
for the free Schr\"odinger equation
$$
\begin{cases}
\begin{aligned}
&\partial_tu=i\triangle u, \;\;\,\,(x,t)\in\mathbb R^n\times (0,+\infty),\\
&u(x,0)=u_0(x),
\epsilonnd{aligned}
\epsilonnd{cases}
$$
in the following manner :
\vskip.05in
\epsilonmph{If $u(x,0)=O(e^{-|x|^2/\beta^2})$, $u(x,T)=O(e^{-|x|^2/\alphapha^2})$ and
$T/\alphapha\beta> 1/4$, then
$u\epsilonquiv 0$. Also, if $T/\alphapha\beta=1/4$, $u$ has as initial data $u_0$ equal to a
constant multiple of
$e^{-\left(1/\beta^2+i/4T\right)|y|^2}$.}
\vskip.05in
The corresponding $L^2$-version of Hardy's uncertainty principle was established in
\cite{CoPr2} :
\vskip.05in
\epsilonmph{If $\,e^{|x|^2/\beta^2}f$, $\,e^{4|\xi |^2/\alphapha^2}\widehat f$ are in
$L^2(\mathbb R^n)$ and
$1/\alphapha\beta\ge 1/ 4$, then $f\epsilonquiv 0$.}
\vskip.05in
In terms of the solution of the Schr\"odinger equation it states :
\vskip.05in
\epsilonmph{If $\,e^{|x|^2/\beta^2}u(x,0)$, $\,e^{|\xi |^2/\alphapha^2}u(x,T)$ are in
$L^2(\mathbb R^n)$ and
$T/\alphapha\beta\ge 1/4$, then $u\epsilonquiv 0$.}
\vskip.05in
More generally, it was shown in \cite{CoPr2} that :
\vskip.05in
\epsilonmph{If $\,e^{|x|^2/\beta^2}f\in L^p(\mathbb R^n)$,
$\,e^{4|\xi |^2/\alphapha^2}\widehat f\in L^q(\mathbb R^n)$,
$p, q\in [1,\infty]\,$ with at least
one of them finite and $1/\alphapha\beta\ge 1/ 4$, then $f\epsilonquiv 0$.}
\vskip.05in
In \cite{EKPV08b} we proved a uniqueness result for solutions of \epsilonqref{e1} with $F$ as
in \epsilonqref{F1a} for bounded potentials
$V$ verifying that either,
$$
V(x,t)=V_1(x)+V_2(x,t),
$$
with $V_1$ real-valued and
$$
\sigmaup_{[0,T]}\|e^{T^2|x|^2/\left(\alphapha
t+\beta\left(T-t\right)\right)^2}V_2(t)\|_{L^\infty(\mathbb R^n)}<+\infty,
$$
or
\begin{equation}
\lambdabel{condition}
\lim_{R\rightarrow +\infty}\int_0^T\|V(t)\|_{L^\infty(\mathbb R^n\sigmaetminus B_R)}\,dt =0.
\epsilonnd{equation}
More precisely, it was shown that the only solution $u\in C([0,T], L^2(\mathbb R^n))$ to
\epsilonqref{e1} with $F=V(x,t)u$,
verifying
\begin{equation}
\lambdabel{E: condicion fundamental}
\|e^{|x|^2/\beta^2}u(0)\|_{L^2(\mathbb R^n)}+\|e^{|x|^2/\alphapha^2}u(T)\|_{L^2(\mathbb R^n)}<+\infty
\epsilonnd{equation}
with $\,T/\alphapha\beta>1/ 2$ and $V$ satisfying one of the above conditions
is the zero solution.
Notice that this result differs by a factor of $1/2$ from that for the solution of
the free
Schr\"odinger equation given by the $L^2$-version of the Hardy uncertainty principle
described above ($T/\alphapha\beta\ge 1/4$).
In \cite{EKPV09} we showed that the
optimal version of Hardy's uncertainty principle in terms of $L^2$-norms, as
established in \cite{CoPr2},
holds for solutions of
\begin{equation}
\lambdabel{E: 1.11}
\partial_tu=i\left(\triangle u+V(x,t)u\right), \,\,\,\, (x,t)\in \mathbb R^n\times [0,T],
\epsilonnd{equation}
such that \epsilonqref{E: condicion fundamental} holds with $T/\alphapha\beta>1/4$ and for
many general bounded potentials $V(x,t)$,
while it fails for some complex-valued potentials in the end-point case,
$T/\alphapha\beta=1/4$.
\vskip.05in
\begin{theorem}\lambdabel{T: hardytimeindepent}
Let $u\in C([0,T]):L^2(\mathbb R^n))$ be a solution of the equation
\epsilonqref{E: 1.11}. If there exist positive constants
$\alphapha$ and $\beta$ such that $T/\alphapha\beta > 1/4$,
and
$$
\|e^{|x|^2/\beta^2}u(0)\|_{L^2(\mathbb R^n)},\,\,\,\,\|e^{|x|^2/\alphapha^2}u(T)\|_{L^2(\mathbb R^n)}<\infty,
$$
and the potential $V$ is bounded and either,
$V(x,t)=V_1(x)+V_2(x,t)$, with $V_1$ real-valued and
$$
\sigmaup_{[0,T]}\|e^{T^2|x|^2/\left(\alphapha t+\beta \left(T-t\right)\right)^2}V_2(t)
\|_{L^\infty(\mathbb R^n)} < +\infty
$$
or
$$
\lim_{R\rightarrow +\infty}\|V\|_{L^1([0,T], L^\infty(\mathbb R^n\sigmaetminus B_R)}=0.
$$
Then, $u\epsilonquiv 0$.
\epsilonnd{theorem}
We remark that there are no assumptions on the size of the potential in the given
class
or on the dimension and that we do not assume any decay of the gradient, neither of
the solutions or
of the time-independent potential or any \it{a priori }\rm regularity on this
potential or the solution.
\vskip.03in
\begin{theorem}\lambdabel{T: hardytimeindepent2}
Assume that $T/\alphapha\beta=1/4$. Then, there is a smooth complex-valued potential
$V$ verifying
$$
|V(x,t)|\lesssim\frac 1{1+|x|^2},\, (x,t)\in \mathbb R^n\times [0,T],
$$
and a nonzero smooth function $u\in C^\infty([0,T],\mathcal S(\mathbb R^n))$ solution of
\epsilonqref{E: 1.11}
such that
\begin{equation}
\lambdabel{007}
\|e^{|x|^2/\beta^2}u(0)\|_{L^2(\mathbb R^n)},\,\,\,\,\|e^{|x|^2/\alphapha^2}u(T)\|_{L^2(\mathbb R^n)}<\infty.
\epsilonnd{equation}
\epsilonnd{theorem}
\vskip.03in
Our proof of Theorem \ref{T: hardytimeindepent} does not use any complex analysis,
giving, in particular, a new proof (up to the end-point) of the $L^2$-version of
Hardy's uncertainty principle
for the Fourier transform. It is based on Carleman estimates for certain evolutions.
More precisely, it is based on the
convexity and log-convexity properties present for the solutions of these evolutions.
Thus, the
convexity and log-convexity of appropriate $L^2$-quantities play the role of the
Phragm\'en-Lindel\"of principle.
We observe that the product of log-convex functions is log-convex which, roughly
speaking, replaces the fact that the
product of analytic functions is analytic.
In \cite{CEKPV} in collaboration with M. Cowling, we gave new proofs, based
only on \it{real variable }\rm techniques, of both the $L^2$-version of the Hardy
uncertainty
principle and the original Hardy's uncertainty
principle $ (L^{\infty}$) $n$-dimensional version for the Fourier transform as stated at the beginning of this
section,
including the end point case $1/\alphapha \,\beta=1/4$.
Returning to Theorem \ref{T: hardytimeindepent} as
a by product of our proof, we obtain the following optimal
interior estimate for
the Gaussian decay of solutions to \epsilonqref{E: 1.11}.
\begin{theorem}\lambdabel{T: lamejora}
Assume that $\,u\,$ and $\,V\,$ verify the hypothesis in Theorem \ref{T:
hardytimeindepent}
and $\,T/\alphapha\beta\le 1/4$. Then,
\begin{equation}
\lambdabel{oda}
\begin{aligned}
&\sigmaup_{[0,T]}\|e^{a(t)|x|^2}u(t)\|_{L^2(\mathbb R^n)} +
\| \sigmaqrt{t(T-t)}\nabla \left(e^{\left(a(t)+\frac{i\dot
a(t)}{8a(t)}\right)|x|^2}u\right)\|_{L^2(\mathbb R^n\times [0,T])}\\
&\le N\left[\|e^{|x|^2/\beta^2}u(0)\|_{L^2(\mathbb R^n)}+
\|e^{|x|^2/\alphapha^2}u(T)\|_{L^2(\mathbb R^n)}\right],
\epsilonnd{aligned}
\epsilonnd{equation}
where
\[a(t)=\frac {\alphapha\beta RT}{2\left(\alphapha t+\beta (T-t)\right)^2+2R^2\left(\alphapha
t - \beta (T-t)\right)^2}\ ,\]
$R$ is the smallest root of the equation
$$
\frac T{\alphapha\beta}=\frac R{2\left(1+R^2\right)}
$$
and $N$ depends on $T$, $\alphapha$, $\beta$ and the conditions on the potential $V$ in
Theorem \ref{T: hardytimeindepent}.
\epsilonnd{theorem}
One has that $1/a(t)$ is convex and attains its minimum value in the interior of
$[0,T]$, when
$$
|\alphapha-\beta|<R^2\left(\alphapha+\beta\right).
$$
To see the optimality of Theorem \ref{T: lamejora}, we write
\begin{equation}
\lambdabel{E: el enemigo}
u_R(x,t)=R^{-\frac n2}\left(t-\frac iR\right)^{-\frac n2}e^{-\frac{|x|^2}{4i(t-\frac
iR)}}=
\left(Rt-i\right)^{-\frac n2}e^{-\frac{(R-iR^2t)}{4(1+R^2t^2)}\,|x|^2},
\epsilonnd{equation}
which is a free wave (i.e. $V\epsilonquiv 0$, in \epsilonqref{E: 1.11}) satisfying in $\mathbb R^n\times
[-1,1]$ the corresponding time
translated conditions in Theorem \ref{T: lamejora} with $T=2$ and
$$
\frac1{\beta^2}=\frac1{\alphapha^2}=\mu=\frac R{4\left(1+R^2\right)}\le\frac 18\, .
$$
Moreover
$$
\frac R{4\left(1+R^2t^2\right)}\, ,
$$
is increasing in the $R$-variable, when $0<R\le 1$ and $-1\le t\le 1$.
Our improvement over the results in \cite{EKPV06} and \cite{EKPV08b} is a consequence
of the possibility of extending
the following argument (for the case of free waves) to prove Theorem \ref{T:
hardytimeindepent}
(a non-free wave case).
We recall the conformal or Appell transformation: If $u(y,s)$ verifies
\begin{equation}
\lambdabel{2.1}
\partial_su=i\left(\triangle
u+V(y,s)u+F(y,s)\right),\;\;\;\;\;\;\;(y,s)\in \mathbb R^n\times [0,1],
\epsilonnd{equation}
and $\alphapha$ and $\beta$ are positive, then
\begin{equation}
\lambdabel{2.2}
\widetilde u(x,t)=\left(\tfrac{\sigmaqrt{\alphapha\beta}}{\alphapha(1-t)+\beta
t}\right)^{\frac n2}u\left(\tfrac{\sigmaqrt{\alphapha\beta}\,
x}{\alphapha(1-t)+\beta t}, \tfrac{\beta t}{\alphapha(1-t)+\beta
t}\right)e^{\frac{\left(\alphapha-\beta\right) |x|^2}{4i(\alphapha(1-t)+\beta
t)}},
\epsilonnd{equation}
verifies
\begin{equation}
\lambdabel{2.3}
\partial_t\widetilde u=i\left(\triangle \widetilde u+\widetilde
V(x,t)\widetilde u+\widetilde F(x,t)\right),\;\; \text{in}\ \mathbb R^n\times
[0,1],
\epsilonnd{equation}
with
\begin{equation}
\lambdabel{potencial}
\widetilde V(x,t)=\tfrac{\alphapha\beta}{\left(\alphapha(1-t)+\beta
t\right)^2}\,V\left(\tfrac{\sigmaqrt{\alphapha\beta}\, x}{\alphapha(1-t)+\beta t},
\tfrac{\beta t}{\alphapha(1-t)+\beta t}\right),
\epsilonnd{equation}
and
\begin{equation}
\lambdabel{externalforce}
\widetilde F(x,t)=\left(\tfrac{\sigmaqrt{\alphapha\beta}}{\alphapha(1-t)+\beta
t}\right)^{\frac n2+2}F\left(\tfrac{\sigmaqrt{\alphapha\beta}\,
x}{\alphapha(1-t)+\beta t}, \tfrac{\beta t}{\alphapha(1-t)+\beta
t}\right)e^{\frac{\left(\alphapha-\beta\right) |x|^2}{4i(\alphapha(1-t)+\beta
t)}}.
\epsilonnd{equation}
Thus,
to prove Theorem \ref{T: hardytimeindepent} for free waves, it suffices to consider $u\in
C([-1,1], L^2(R^n))$
being a solution of
\begin{equation}
\lambdabel{E: free wave}
\partial_tu-=i\triangle u,\,\,\,\,(x,t)\in R\times [-1,1],
\epsilonnd{equation}
and
\begin{equation}
\lambdabel{E: decaimineto}
\|e^{\mu |x|^2}u(-1)\|_{L^2(R^n)}+\|e^{\mu |x|^2}u(1)\|_{L^2(\mathbb R^n)}<+\infty,
\epsilonnd{equation}
for some $\mu >0$.
The main idea consists of showing that either $u\epsilonquiv 0$ or there is a function
$\theta_{R}: [-1,1]\longrightarrow [0,1]$ such that
\begin{equation}
\lambdabel{E: gaussian improvement}
\|e^{\frac{R|x|^2}{4\left(1+R^2t^2\right)}}u(t)\|_{L^2(R^n)}\le
\|e^{\mu |x|^2}u(-1)\|_{L^2(R^n)}^{\theta_{R}(t)}\|e^{\mu
|x|^2}u(1)\|_{L^2(\mathbb R^n)}^{1-\theta_{R}(t)},
\epsilonnd{equation}
where $R$ is the smallest root of the equation
$$
\mu =\frac{R}{4\left(1+R^2\right)}\ .
$$
This gives the optimal improvement of the Gaussian decay of a free wave verifying
\epsilonqref{E: decaimineto}
and we also see that if $\mu > 1/8$, then $u$ is zero.
The proof of these facts relies on new logarithmic convexity properties of free
waves verifying
\epsilonqref{E: decaimineto} and
on those already established in \cite{EKPV08b}. In \cite[Theorem 3]{EKPV08b}, the
positivity of the
space-time commutator
of the symmetric and skew-symmetric parts of the operator,
$$
e^{\mu |x|^2}\left(\partial_t-i\triangle\right)e^{-\mu |x|^2},
$$
is used to prove that $\|e^{\mu |x|^2}u(t)\|_{L^2(\mathbb R^n)}$ is logarithmically convex
in $[-1,1]$. More precisely,
defining
$$
f(x,t) = e^{\mu |x|^2}u(x,t)=e^{it\Delta}u_0(x),
$$
it follows that
$$
e^{\mu |x|^2}\left(\partial_t-i\triangle\right)u=e^{\mu
|x|^2}\left(\partial_t-i\triangle\right)(e^{-\mu |x|^2}f)
=\partial_t f -\mathcal S f-\mathcal A f,
$$
where $\mathcal S$ is symmetric and $\mathcal A$ skew-symmetric with
$$
\mathcal S=- i\mu(4 \,x\cdot \nabla + 2n),\,\;\,\;\;\;\;\mathcal A=i(\Delta+4\mu^2
\,|x|^2),
$$
so that
$$
[\mathcal S;\mathcal A] = - 8 \mu (\nabla\cdot I \nabla) + 16 \mu^2\,|x|^2.
$$
Formally, using the abstract Lemma \ref{L: freq1} (see the appendix) and the
Heisenberg inequality
$$
\|f\|^2_{L^2(\mathbb R^n)}\leq \frac{2}{n} \,\|\,|x|f\|_{L^2(\mathbb R^n)}\,\|\,\nabla f\|_{L^2(\mathbb R^n)},
$$
whose proof follows by integration by parts, one sees that
$$
H(t)=\|f(t)\|^2_{L^2(\mathbb R^n)}=\|e^{\mu |x|^2}u(t)\|_{L^2(\mathbb R^n)}
$$
is logarithmically convex so
$$
\|e^{\mu |x|^2}u(t)\|_{L^2(\mathbb R^n)}\le \|e^{\mu |x|^2} u(-1)\|_{L^2(R^n)}^{\frac{1-t}2}
\|e^{\mu |x|^2} u(1)\|_{L^2(\mathbb R^n)}^{\frac{1+t}2},
$$
when, $-1\le t\le 1$.
Setting $a_1\epsilonquiv \mu$, we begin an iterative process,
where at the $k$-th step, we have $k$ smooth even functions,
$a_j:[-1,1]\longrightarrow (0,+\infty)$, $1\le j\le k$, such that
$$
\mu\epsilonquiv a_1<a_2<\dots<a_k\in (-1,1),
$$
$$
F(a_i)> 0,\ a_j(1)=\mu,\ j=1,\dots,k,
$$
where
$$
F(a)=\frac 1a\left(\ddot a-\frac{3\dot a^2}{2a\,}+32a^3\right)
$$
and functions $\theta_j:[-1,1]\longrightarrow [0,1]$, $1\le j\le k$, such that for
$t\in [.1,1]$
\begin{equation}
\lambdabel{E: algoagradable}
\|e^{a_j(t) |x|^2}u(t)\|_{L^2(R^n)}\le
\|e^{\mu |x|^2}u(-1)\|_{L^2(R^n)}^{\theta_j(t)}\|e^{\mu
|x|^2}u(1)\|_{L^2(\mathbb R^n)}^{1-\theta_j(t)}.
\epsilonnd{equation}
These estimates follow from the construction of the functions $a_i$, while the
method strongly
relies on the following formal
convexity properties of free waves:
\begin{equation}
\lambdabel{E: algo fundamental}
\partial_t\left(\frac 1a\partial_t\log{H_b}\right)\ge -\frac{2\ddot b^2|\xi|^2}{F(a)},
\epsilonnd{equation}
\begin{equation}
\lambdabel{E: el control del gradiente}
\partial_t\left(\frac 1a\partial_tH\right)\ge
\epsilonpsilon_a\int_{R^n}e^{a|x|^2}\left(|\nabla u|^2+|x|^2|u|^2\right)\,dx,
\epsilonnd{equation}
where
$$
H_b(t)=\|e^{a(t)|x+ b(t)\xi|^2}u(t)\|_{L^2(R^n)}^2\ ,\
H(t)=\|e^{a(t)|x|^2}u(t)\|_{L^2(\mathbb R^n)}^2,
$$
$\xi\in \mathbb R^n$ and $a, b: [-1,1]\longrightarrow R$ are smooth functions with
$$
a> 0,\quad \;\;\;\;\;F(a)>0 \;\;\;\;\;\text{in}\;\;\;\;[-1,1].
$$
Once the $k$-th step is completed, we take $a=a_k$ in \epsilonqref{E: algo fundamental}
with a certain choice of
$b=b_k$, verifying $b(-1)=b(1)=0$ and then, a certain test is performed. When the
answer to the test is positive,
it follows that $u\epsilonquiv 0$. Otherwise, the logarithmic convexity associated to
\epsilonqref{E: algo fundamental} allows
us to find a new smooth function $a_{k+1}$ in $[-1,1]$ with
$$
a_1<a_2<\dots<a_k<a_{k+1},\,\, (-1,1),
$$
and verifying the same properties as $\,a_1,\dots,a_k$.
When the process is infinite, we have \epsilonqref{E: algoagradable} for all $k\ge 1$ and
there are two possibilities:
$$
\text{ either }\,\,\,\,\,\,\,
\lim_{k\to +\infty}a_k(0)=+\infty,\,\,\,\,\,\,\, \text{or }\,\,\,\,\,\,\,\lim_{k\to
+\infty}a_k(0)<+\infty.
$$
In the first case and \epsilonqref{E: algoagradable}
one has that $u\epsilonquiv 0$, while in the second, the sequence $a_k$ is shown to
converge to an even function
$a$ verifying
\begin{equation}
\lambdabel{ole}
\begin{cases}
\ddot a-\frac{3\dot a^2}{2a\,\,}+32a^3=0,\,\,\,\,\, [-1,1] \\
a(1)=\mu.
\epsilonnd{cases}
\epsilonnd{equation}
Because
$$
a(t)=\frac{R}{4\left(1+R^2t^2\right)},\,\,\, \,\quad R\in \mathbb R^+,
$$
are all the possible even solutions of this equation, $a$ must be one of them and
$$
\mu =\frac{R}{4\left(1+R^2\right)},
$$
for some $R>0$. In particular, $u\epsilonquiv 0$, when $\mu >1/8$.
\vskip.05in
As it was already mentioned above, our proof of Theorem \ref{T: hardytimeindepent}
(the case of
non-zero potentials $V=V(x,t)$),
is based on the extension of the above convexity properties to the non-free case.
\vskip.05in
Theorem \ref{T: hardytimeindepent2} establishes the sharpness of the result in
Theorem \ref{T: hardytimeindepent}
by giving an example of a complex valued potential $V(x,t)$
verifying \epsilonqref{condition} and a non-trivial solution $u\in C([0,T]:L^2(\mathbb R^n))$
of \epsilonqref{E: 1.11} for which
\epsilonqref{E: condicion fundamental} holds with $T/\alphapha\beta =1/4$.
Thus, one may ask : Is it possible to construct
a real valued potential $V(x,t)$ verifying the same properties, i.e.
satisfying \epsilonqref{condition}
and having a non-trivial solution $u\in C([0,T]:L^2(\mathbb R^n))$ of \epsilonqref{E: 1.11} such
that
\epsilonqref{E: condicion fundamental} holds with $T/\alphapha\beta =1/4\,$?
The same question concerning the sharpness of the above result presents itself in
the case
of time independent potentials $V=V(x)$. In this regard, we consider the stationary
problem
\begin{equation}
\lambdabel{estatic}
\Delta w + V(x) w =0,\,\,\,\, x\in \mathbb R^n,\,\,V\in L^{\infty}(\mathbb R^n),
\epsilonnd{equation}
and recall V. Z. Meshkov's result in \cite{Me} :
\vskip.07in
\epsilonmph{ If $w\in H^2_{loc}(\mathbb R^n)$ is a solution of \epsilonqref{estatic} such that
\begin{equation}
\lambdabel{43}
\int_{\mathbb R^n} e^{a|x|^{4/3}}|w(x)|^2dx<\infty,\,\,\,\,\forall a>0,
\epsilonnd{equation}
then $\,u\epsilonquiv 0$.}
\vskip.04in
Moreover, it was also proved in \cite{Me} that for complex potentials $V$, the
exponent $4/3$ in
\epsilonqref{43} is optimal. However, it has been conjectured that for real valued potentials
the optimal exponent should be 1, (see also \cite{BoKe} for a quantitative form of these results and applications to Anderson localization of
Bernoulli models).
\vskip.05in
More generally, it was established in \cite{EKPV10}, (see also \cite{cruz}) :
\vskip.07in
\epsilonmph{ If $w\in H^2_{loc}(\mathbb R^n)$ is a solution of \epsilonqref{estatic}
with
a complex valued potential $V$ satisfying
$$
V(x)=V_1(x)+V_2(x),
$$
such that
\begin{equation}
\lambdabel{123}
|V_1(x)|\leq \frac{c_1}{(1+|x|^2)^{\alphapha/2}},\,\,\,\,\alphapha\in [0,1/2),
\epsilonnd{equation}
and $V_2$ real valued supported in $\,\{x\,:\,|x|\geq 1\}$
such that
$$
-(\partial_r V_2(x))^- < \frac{c_2}{|x|^{2\alphapha}},\,\,\,\,a^-=\min\{a;0\}.
$$
Then there exists $a=a(\|V\|_{\infty};c_1;c_2;\alphapha)>0$ such that if
\begin{equation}
\lambdabel{43a}
\int_{\mathbb R^n} e^{a|x|^{r}}|w(x)|^2dx<\infty,\,\,\,\,\,r=(4-2\alphapha)/3,
\epsilonnd{equation}
then $\,u\epsilonquiv 0$.}
\vskip.05in
In addition, one can take the value $r=1$ in \epsilonqref{43} by assuming $\alphapha>1/2$ in
\epsilonqref{123}.
\vskip.05in
It was also proved in \cite{cruz} that for complex potentials these results for
$\alphapha\in [0,1/2)$ are sharp.
\vskip.05in
By noticing that given a solution $\phi(x) $ of the eigenvalue problem
\begin{equation}
\lambdabel{eigen}
\Delta \phi + \widetilde V(x) \phi =\lambdambda \phi,\,\,\,\, x\in \mathbb R^n,
\epsilonnd{equation}
with $\lambdambda \in\mathbb R, $ then $V(x)=\widetilde V(x)+\lambdambda$ satisfies the hypothesis
of the previous result
and
$$
u(x,t)= e^{it\lambdambda} \,\phi(x),
$$
solves the evolution equation
\begin{equation}
\lambdabel{evo-notime}
\partial_t u=i(\Delta u + V(x)u),\,\,\,\, x\in \mathbb R^n,\,t\in\mathbb R,
\epsilonnd{equation}
one gets a lower bound for the value of the strongest possible decay rate of non-trivial
solutions
$u(x,t)$ of \epsilonqref{evo-notime} at two different times.
\vskip.05in
As a direct consequence of Theorem \ref{T: hardytimeindepent} we have the
following application concerning
the uniqueness of solutions for semi-linear equations of the form \epsilonqref{e1} with
$F$ as in \epsilonqref{F1b}.
\begin{theorem}
\lambdabel{Theorem NL2}
Let $u_1$ and $u_2$ be strong solutions in $C([0,T],H^k(\mathbb R^n)), \,k>n/2$ of the
equation \epsilonqref{e1} with
$F$ as in \epsilonqref{F1b} such that $\,F\in C^k$ and $F(0)=\partial_uF(0)=\partial_{\bar
u}F(0)=0$.
If there are $\alphapha$ and $\beta$
positive with
$T/\alphapha \beta>1/4$ such that
$$
e^{|x|^2/\beta^2}\left(u_1(0)-u_2(0)\right)\, ,\,\
e^{|x|^2/\alphapha^2}\left(u_1(T)-u_2(T)\right) \in L^2(\mathbb R^n),
$$
then $u_1\epsilonquiv u_2$.
\epsilonnd{theorem}
In Theorem \ref{Theorem NL2} we did not attempt to optimize the regularity
assumption on the solutions $\,u_1,\,u_2$.
By fixing $u_2\epsilonquiv 0$ Theorem \ref{Theorem NL2} provides a restriction on the
possible decay at two different times
of a non-trivial solution $u_1$ of equation \epsilonqref{e1} with
$F$ as in \epsilonqref{F1b}. It is an open question to determine the optimality of this
kind of result. More precisely,
for the standard semi-linear Schr\"odinger equations
\begin{equation}
\lambdabel{NLS}
\partial_t u = i (\Delta u + |u|^{\gamma-1}u),\,\,\,\,\gamma>1,
\epsilonnd{equation}
one has the \it{standing wave }\rm solutions
$$
u(x,t)=e^{\omega \,t} \varphi(x),\,\,\,\omega>0,
$$
where $\varphi$ is the unique (up to translation) positive solution of the elliptic
problem
$$
-\Delta \varphi+\omega \varphi= |\varphi|^{\gamma-1}\varphi,
$$
which has a linear exponential decay, i.e.
$$
\varphi(x)=O(e^{-c|x|}),\,\,\,\text{as}\,\,\,|x|\to\infty,
$$
for an appropriate value of $c>0$ (see \cite{Str}, \cite{BLi}, \cite{BGK}, and
\cite{Kw}). Whether or not
these standing waves are the solutions of \epsilonqref{NLS} having the strongest possible decay at
two
different times is an open question.
\vskip.04in
Hardy's uncertainty principle also admits a formulation in terms of the heat equation
$$
\partial_tu=\Delta u,\;\;\;\;t>0,\;\;x\in\mathbb R^n,
$$
whose solution with data $\,u(x,0)=u_0(x)$ can be written as
$$
u(x,t)= e^{t \Delta}u_0(x)= \int_{\mathbb R^n} \frac{e^{-|x-y|^2/4t}}{(4\pi t)^{n/2}}\,
u_0(y)\,dy.
$$
More precisely, Hardy's uncertainty principle can restated in the following equivalent forms :
\vskip.07in
\epsilonmph{ (i) If $\,u_0\in L^2(\mathbb R^n)$ and there exists $\,T>0$ such that $\,e^{|x|^2/(\delta^2T)}\,e^{T\Delta}u_0\in L^2(\mathbb R^n)\,$ for some $\,\delta \leq 2$, then $\,u_0\epsilonquiv 0$.}
\vskip.04in
\epsilonmph{ (ii) If $\,u_0\in \mathcal S(\mathbb R^n)$ (tempered distribution) and there exists $\,T>0$ such that $\,e^{|x|^2/(\delta^2T)}\,e^{T\Delta}u_0\in L^{\infty}(\mathbb R^n)$ for some $\,\delta< 2$, then $\,u_0\epsilonquiv 0$.
Moreover, if $\,\delta=2$, then $\,u_0$ is a constant multiple of the Dirac delta measure.}
\vskip.03in
In fact, applying Hardy's uncertainty principle to $e^{T \triangle} u_0$ one has that $e^{\frac{|x|^2}{\delta^2 T}}e^{T \triangle} u_0$ and
$e^{ T |\xi|^2}\widehat{e^{T \triangle} u_0}=\widehat u_0$ in $L^2(\mathbb R^n)$ with $\,2\delta\le 4$ implies $e^{\triangle}u_0\epsilonquiv 0$. Then, backward uniqueness arguments
(see for example \cite[Chapter 3, Theorem 11]{lm60}) shows that $u_0\epsilonquiv 0$.
In \cite{EKPV08b} we proved the following weaker extension of this result for parabolic operators with
lower order variable coefficientes :
\vskip.03in
\begin{theorem}\lambdabel{T: toremaparabolicco}
Let $u\in C([0,1] : L^2(\mathbb Rn))\cap L^2([0,T]: H^1(\mathbb Rn))$ be a solution of the IVP
\begin{equation*}
\begin{cases}
\partial_tu=\triangle u+V(x,t)u,\ \text{in}\ \mathbb Rn\times (0,1],\\
u(x,0)=u_0(x),
\epsilonnd{cases}
\epsilonnd{equation*}
where
$$
V\in L^{\infty}(\mathbb Rn\times [0,1]).
$$
If
$$
u_0\;\;\;\;\text{and}\;\;\;\;e^{\frac{|x|^2}{\delta^2}}u(1)\in L^2(\mathbb Rn),
$$
for some $\,\delta <1$, then $u_0\epsilonquiv 0$.
\epsilonnd{theorem}
\vskip.03in
It is natural to expect that Hardy's uncertainty principle holds in this context with
bounded potentials $V$ and with the parameter $\delta$ verifing the condition of the free case, i.e. $\,\delta\leq 2$.
Earlier results in this directions, addressing a question of Landis and Oleinik \cite{LaOl}, were obtained in \cite{EKPV06a} and \cite{Ng}.
\vskip.03in
\sigmaection{Uncertainty Principle of Morgan type}\lambdabel{morgan}
In \cite{Mo} G. W. Morgan proved the following uncertainty principle:
\vskip.07in
\epsilonmph{If $f(x)=O(e^{-\frac{a^p |x|^p}{p}}), \,1<p\leq 2$ and $\widehat
f(\xi)=O(e^{-\frac{(b+\epsilonpsilon)^q |\xi|^q}{q}}),\;1/p+1/q=1,\,\epsilonpsilon>0,$
with
$$
ab>\Big|\cos\left(\frac{p\,\pi}{2}\right)\Big|,
$$
then $f\epsilonquiv 0$.}
\vskip.07in
In \cite{Ho} Beurling-H\"ormander showed :
\vskip.07in
\epsilonmph{ If
$f\in L^1(\mathbb R)$ and
\begin{equation}
\lambdabel{beurling}
\int_{\mathbb R} \int_{\mathbb R} |f(x)| |\widehat f(\xi)| e^{
|x\,\xi|}\,dx\,d\xi<\infty, \;\;\;\text{then}
\;\;\;f\epsilonquiv 0.
\epsilonnd{equation}}
This result was extended to higher dimensions $n\geq 2$ in \cite{bonamie2} and
\cite{ray} :
\vskip.07in
\epsilonmph{If $f\in L^2(\mathbb R^n), n\geq 2$ and
\begin{equation}
\lambdabel{beurlingn}
\int_{\mathbb R^n} \int_{\mathbb R^n} |f(x)| |\widehat f(\xi)| e^{
|x\,\cdot \xi|}\,dx\,d\xi<\infty, \;\;\;\text{then}
\;\;\;f\epsilonquiv 0.
\epsilonnd{equation}}
\vskip.05in
We observe that from \epsilonqref{beurling} and \epsilonqref{beurlingn} it follows that :
\vskip.05in
\epsilonmph{If
$p\in(1,2]$, $\,1/p+1/q=1$, $\,a, \,b>0$, and
\begin{equation}
\lambdabel{primera}
\int_{\mathbb R^n}|f(x)|\, e^{\frac{a^p|x|^p}{p}}dx \,+
\,\int_{\mathbb R^n} |\widehat
f(\xi)| \,e^{\frac{b^q|\xi|^q}{q}}d\xi<\infty,\;\;a b\geq 1\;\mathbb Rightarrow \;f\epsilonquiv 0.
\epsilonnd{equation}}
Notice that in the case $p=q=2$ this gives us an $L^1$-version of Hardy's uncertainty result
discussed above, and
for $p<2$ an $n$-dimensional $L^1$-version of Morgan's uncertainty principle.
In the one-dimensional case ($n=1$), the optimal $L^1$-version of Morgan's result
in \epsilonqref{primera},
\begin{equation}
\lambdabel{primera1}
\int_{\mathbb R}|f(x)|\, e^{\frac{a^p|x|^p}{p}}dx +
\int_{\mathbb R} |\widehat
f(\xi)| \,e^{\frac{b^q|\xi|^q}{q}}d\xi<\infty,\;\;a b>\Big|\cos\left(\frac{p\,\pi}{2}\right)\Big|\;\mathbb Rightarrow \;f\epsilonquiv 0.
\epsilonnd{equation}
was established in \cite{bonamie2} and \cite{monki} (for further results
see \cite{bonamie1} and references therein). A sharp condition for $a,\,b,\,p$ in
\epsilonqref{primera1} in higher dimension
seems to be unknown.
However, in \cite{bonamie2} it was shown :
\vskip.07in
\epsilonmph{ If $f\in
L^2(\mathbb R^n)$, $1<p\leq 2\;$ and $\,1/p+1/q=1\,$ are such that for some $j=1,..,n$,
\begin{equation}
\lambdabel{bonami11}
\int_{\mathbb R^n}
|f(x)|e^{\frac{a^p|x_j|^p}{p}}dx<\infty\;\;+\;\;\int_{\mathbb R^n}
|\widehat f(\xi)|e^{\frac{b^q|\xi_j|^q}{q}}d\xi<\infty.
\epsilonnd{equation}}
\vskip.05in
\epsilonmph{If $a b>\Big|\cos\left(\frac{p\,\pi}{2}\right)\Big|$, then
$\;f\epsilonquiv 0$.}
\vskip.05in
\epsilonmph{If $a b<\Big|\cos\left(\frac{p\,\pi}{2}\right)\Big|$, then there exist
non-trivial functions satisfying \epsilonqref{bonami11}}.
\vskip.05in
Using \epsilonqref{formula1} the above result can be stated in terms of
the solution of the free Schr\"odinger
equation. In particular, \epsilonqref{primera} can be re-written as :
\vskip.05in
\epsilonmph{If $u_0\in L^1(\mathbb R)$ or $u_0\in L^2(\mathbb R^n)$, if $n\geq 2$, and for
some $\,t\neq 0$
\begin{equation}
\lambdabel{pq}
\int_{\mathbb R^n}\;|u_0(x)|\, e^{\frac{a^p|x|^p}{p}}dx \,+
\,\int_{\mathbb R^n}\,
|\,e^{it\Delta}u_0(x)|
\,e^{\frac{b^q|x|^q}{q(2t)^q}}dx<\infty,
\epsilonnd{equation}
with
$$
ab>\Big|\cos\left(\frac{p\,\pi}{2}\right)\Big|\;\;\;\;\text{if}\;\;\;n=1,\;\;\;\;\text{and}\;\;\;\;ab>1\;\;\;\;\text{if}\;\;\;\;\;n\geq 2,
$$
then $u_0\epsilonquiv 0$. }
\vskip.07in
Related with Morgan's uncertainty principle one has the following result due to Gel'fand and Shilov.
In \cite{GeShi} they considered the class $Z^p_p,\,p> 1$,
defined as the space of all functions $\varphi(z_1,..,z_n)$ which are
analytic for all values of $z_1,..,z_n\in \mathbb C$ and such that
$$
|\varphi(z_1,..,z_n)|\leq C_0\, e^{\sigmaum_{j=1}^n\,\epsilonpsilon_j\,C_j\,|z_j|^p},
$$
where the $C_j,\,j=0,1,..,n$ are positive constants and $\epsilonpsilon_j=1$ for
$z_j$ non-real and $\epsilonpsilon_j=-1$ for $z_j$ real, $j=1,..,n$, and showed
that the Fourier transform of the function space $Z_p^p$ is the space
$Z_q^q$,
with $\,1/p+1/q=1$.
\vskip.05in
Notice that the class $Z_p^p$ with $p\geq 2$ is closed with respect to
multiplication by $\,e^{i c |x|^2}$. Thus, if $u_0\in Z^p_p,\,p\geq 2$, then by
\epsilonqref{formula1}
one has that
$$
|e^{it\Delta}u_0(x)|\leq d(t)\,e^{-a(t)|x|^q},
$$
for some functions
$\,d,\,a\,:\,\mathbb R\to(0,\infty)$.
\vskip.03in
In \cite{EKPV08m} the following results were established:
\begin{theorem}\lambdabel{Theorem 22}
Given $\,p\in(1,2)$ there exists $\,M_p>0$ such that for any solution $u
\in C([0,1] :L^2(\mathbb Rn))$ of
\begin{equation*}
\lambdabel{E: 1.111}\partial_tu=i\left(\triangle
u+V(x,t)u\right),\;\;\;\text{in}\;\;\;\;\;\mathbb Rn\times [0,1],
\epsilonnd{equation*}
with $V=V(x,t)$ complex valued, bounded (i.e.
$\|V\|_{L^{\infty}(\mathbb R^n\times[0,1])}\leq C$)
and
\begin{equation}
\lambdabel{14}
\lim_{R\rightarrow +\infty}\|V\|_{L^1([0,1] : L^\infty(\mathbb Rn\sigmaetminus B_R))}=0,
\epsilonnd{equation}
satisfying that for some constants $\,a_0,\,a_1,\,a_2>0$
\begin{equation}
\lambdabel{12}
\int_{\mathbb R^n} |u(x,0)|^2\,e^{2a_0 |x|^p}dx < \infty,
\epsilonnd{equation}
and for any $k\in\mathbb Z^+$
\begin{equation}
\lambdabel{13}
\int_{\mathbb R^n} |u(x,1)|^2\,e^{2k |x|^p}dx < a_2 e^{2 a_1 k^{q/(q-p)}},
\epsilonnd{equation}
$1/p+1/q=1$, if
\begin{equation}
\lambdabel{conditionp}
\,a_0\,a_1^{(p-2)} > M_p,
\epsilonnd{equation}
then $\,u\epsilonquiv 0$.
\epsilonnd{theorem}
\begin{corollary}\lambdabel{Corollary 22}
Given $\,p\in(1,2)$ there exists $N_p>0$ such that if
\newline $u\in C([0,1]:L^2(\mathbb R^n))$ is a solution of
$$
\partial_t u=i (\Delta u +V(x,t)u),
$$
with $V=V(x,t)$ complex valued, bounded (i.e.
$\|V\|_{L^{\infty}(\mathbb R^n\times[0,1])}\leq C$) and
$$
\lim_{R\to\infty} \,\int_0^1\,\sigmaup_{|x|>R} |V(x,t)| dt=0,
$$
and there exist $\,\alphapha,\,\beta>0$ such that
\begin{equation}
\lambdabel{con1}
\int_{\mathbb R^n}
|u(x,0)|^2e^{2\,\alphapha^p\,|x|^p/p}dx\;\,\,+\,\,\;\int_{\mathbb
R^n}
|u(x,1)|^2e^{2\,\beta^q\,|x|^q/q}dx<\infty,
\epsilonnd{equation}
$\,1/p+1/q=1$, with
\begin{equation}
\lambdabel{conditionp2}
\;\alphapha\,\beta > N_p,
\epsilonnd{equation}
then $\;u\epsilonquiv 0$.
\epsilonnd{corollary}
As a consequence of Corollary \ref{Corollary 22} one obtains the
following result concerning the uniqueness of solutions for the semi-linear
equations \epsilonqref{e1} with $F$ as in \epsilonqref{F1b}
\begin{equation}
\lambdabel{E: NLS}
i \partial_t u + \triangle u = F(u,\overline u).
\epsilonnd{equation}
\begin{theorem}
\lambdabel{Theorem 23}
Given $\,p\in(1,2)$ there exists $\,N_p>0$ such that if
$$
u_1,\,u_2 \in C([0,1] : H^k(\mathbb R^n)),
$$
are strong solutions of \epsilonqref{E: NLS} with $k\in \mathbb Z^+$, $k>n/2$,
$F:\mathbb C^2\to \mathbb C$, $F\in C^{k}$ and $F(0)=\partial_uF(0)=\partial_{\bar
u}F(0)=0$, and there exist $\,\alphapha,\,\beta>0$
such that
\begin{equation}
\lambdabel{con2}
e^{\alphapha^p\,|x|^p/p}\left(u_1(0)-u_2(0)\right),\;\;\;\
e^{\beta^q\,|x|^q/q}\left(u_1(1)-u_2(1)\right) \in L^2(\mathbb R^n),
\epsilonnd{equation}
$1/p+1/q=1$, with
\begin{equation}
\lambdabel{conditionp2b}
\,\alphapha\,\beta > N_p,
\epsilonnd{equation}
then $u_1\epsilonquiv u_2$.
\epsilonnd{theorem}
Notice that the conditions \epsilonqref{conditionp} and \epsilonqref{conditionp2} are
independent of the size of
the potential and there is not any \it{a priori }\rm regularity assumption on the
potential $V(x,t)$.
The result in \cite{bonamie2}, see \epsilonqref{bonami11}, can be extended to our setting
with an non-optimal constant. More precisely,
\begin{corollary}\lambdabel{Corollary 24}
The conclusions in Corollary \ref{Corollary 22} still hold with a different
constant $N_p>0$ if one replaces the hypothesis \epsilonqref{con1} by the following one
dimensional version
\begin{equation}
\lambdabel{conn=1}
\int_{\mathbb R^n}
|u(x,0)|^2e^{2\,\alphapha^p\,|x_j|^p/p}dx<\infty\,\;\,\,+\,\,\;\int_{\mathbb
R^n}
|u(x,1)|^2e^{2\,\beta^q\,|x_j|^q/q}dx<\infty,
\epsilonnd{equation}
for some $j=1,..,n$.
\epsilonnd{corollary}
Similarly, the non-linear version of Theorem \ref{Theorem 23}
still holds, with different constant $N_p>0$, if one replaces the hypothesis
\epsilonqref{con2} by
$$
e^{\alphapha^p\,|x_j|^p/p}\left(u_1(0)-u_2(0)\right),\;\;\;\
e^{\beta^q\,|x_j|^q/q}\left(u_1(1)-u_2(1)\right) \in L^2(\mathbb R^n),
$$
for $j=1,..,n$.
In \cite{EKPV08m} we did not attempt to give an estimate
of the universal constant $N_p$.
The limiting case $\,p=1$ will be considered in the next section.
The main idea in the proof of these results is to combine an
upper estimate with a lower one to obtain the desired result. The upper estimate is based on the decay hypothesis on the solution at two different
times
(see Lemma \ref{ultimo}).
In previous works we had been able to establish these estimates from assumptions
that at time $t=0$
and $t=1$ involving
the
same weight. However, in our case (Corollary \ref{Corollary 22}) we have
different weights at time $t=0$ and $t=1$. To
overcome this difficulty, we carry out the details with the weight
$e^{a_j|x|^p},\,1<p<2$, $j=0$ at $t=0$ and $j=1$ at $t=1$, with $a_0$ fixed and
$a_1=k\in\mathbb Z^+$
as in \epsilonqref{13}. Although the powers $\,|x|^p\,$ in the exponential are
equal at time $t=0$ and $t=1$ to apply our estimate (Lemma \ref{ultimo})
we also need to have the same constant in front of them. To achieve this we
apply the conformal or Appell transformation described above, to get solutions and
potentials,
whose bounds depend on $k\in\mathbb Z^+$.
Thus we have to consider a family of solutions and obtain estimates on their
asymptotic value as
$k\uparrow \infty$.
The proof of the lower estimate is based on the
positivity of the commutator operator obtained by conjugating the equation
with the appropriate exponential weight, (see Lemma \ref{L: freq1} in the appendix)
\sigmaection{Paley-Wiener Theorem and Uncertainty Principle of Ingham type}\lambdabel{ingham}
This section is concerned with the limiting case $p=1$ in the previous section.
It is easy to see that if $f\in L^1(\mathbb R^n)$ is non-zero and has compact support, then
$\,\widehat f $
cannot satisfy a condition of the type
$\widehat f(y)=O(e^{-\epsilonpsilon |y|})$ for any $\epsilonpsilon>0$.
However, it may be possible to have $ f\in L^1(\mathbb R^n)$ a non-zero function with
compact support, such that
$\widehat f(\xi)=O(e^{-\epsilonpsilon(y) |y|})$, $\epsilonpsilon(y)$ being a positive function
tending to zero as
$|y|\to \infty$.
In the one-dimensional case ($n=1$) soon after Hardy's result described above, A. E.
Ingham \cite{In}
proved the following :
\vskip.05in
\epsilonmph{There exists $f\in L^1(\mathbb R)$ non-zero, even, vanishing outside an interval such
that
$\widehat f(y)=O(e^{-\epsilonpsilon(y) |y|})$ with $\epsilonpsilon(y)$ being a positive
function tending to zero at infinity
if and only if
$$
\int^{\infty} \frac{\epsilonpsilon(y)}{y}\,dy<\infty.
$$}
In a similar direction the Paley-Wiener Theorem \cite{PW} gives a characterization of
a function
or distribution with compact support in term of analyticity properties of its
Fourier transform.
Regarding our results discussed above it would be interesting to identify a class
of potentials $V(x,t)$ for which
a result of the following kind holds:
\vskip.05in
If $u\in C([0,1]:L^2(\mathbb R^n))$ is a non-trivial solution of the IVP
\begin{equation}
\lambdabel{0007}
\begin{cases}
\begin{aligned}
&\partial_tu=i(\triangle u+V(x,t)u), \;\;\,\,(x,t)\in\mathbb R^n\times [0,1],\\
&u(x,0)=u_0(x),
\epsilonnd{aligned}
\epsilonnd{cases}
\epsilonnd{equation}
with $u_0\in L^2(\mathbb R^n)$ having compact support, then $ e^{\epsilonpsilon
|x|}\,u(\cdot,t)\notin L^2(\mathbb R^n)$
for any $\epsilonpsilon>0$ and any $t\in(0,1]$.
\vskip.03in
In this direction we have the following result which will appear in \cite{EKPV12}:
\begin{theorem}\lambdabel{2012} Assume that
$u\in C([0,1]:L^2(\mathbb R^n))$ is a strong solution of the IVP \epsilonqref{007}
with
\begin{equation}
\lambdabel{hyp1-2012}
supp\,u_0\sigmaubset B_R(0)=\{x\in\mathbb R^n\,:\,|x|\leq R\},
\epsilonnd{equation}
\begin{equation}
\lambdabel{hyp2-2012}
\int_{\mathbb R^n}\,e^{2a_1|x|}\,|u(x,1)|^2\,dx<\infty,\;\;\;\;\;\;a_1>0,
\epsilonnd{equation}
and
\begin{equation}
\lambdabel{hyp3-2012}
\|V\|_{L^{\infty}(\mathbb R^n\times [0,1])}=M_0,
\epsilonnd{equation}
with
\begin{equation}
\lambdabel{hyp4-2012}
\lim_{R\rightarrow +\infty}\|V\|_{L^1([0,1] : L^\infty(\mathbb Rn\sigmaetminus B_R))}=0.
\epsilonnd{equation}
Then, there exists $b=b(n)>0$ (depending only on the dimension $n$)
such that if
$$
\frac{a_1}{R\,(1+M_0)}\geq b,
$$
then $\,u\epsilonquiv 0$.
\epsilonnd{theorem}
\vskip.05in
A similar question can be raised for results of the type described above due to A.
E. Ingham in \cite{In}
and possible extensions to higher dimensions $n\geq 2$.
\vskip.05in
It would be interesting to obtain extensions of the above results
characterizing the
decay of the solution $u(x,t)$ to the equation \epsilonqref{e1} with $F$ as in \epsilonqref{F1b}
associated
to data $u_0\in L^2(\mathbb R^n)$ with compact support or with $u_0\in C_0^{\infty}(\mathbb R^n)$.
In this direction, some results can be deduced as a consequence of Theorem \ref{2012},
see \cite{EKPV12}.
\sigmaection{Hardy's Uncertainty Principle in a half-space}\lambdabel{half}
In the introduction we have briefly reviewed some uniqueness results established for
solutions
of the Schr\"odinger equation vanishing at two different times in a semi-space of
$\,\mathbb R^n$,
(see \cite{BZ}, \cite{ds}, \cite{IK04}, \cite{IK06}, \cite{EKPV08b}). In section 2,
we have studied
uniqueness results gotten under the hypothesis that the solution
of the Schr\"odinger equation at two different times has an appropriate Gaussian
decay, in the whole space $\mathbb R^n$.
In this section, we shall deduce a unified result, i.e. a uniqueness result
under the hypothesis that at two different times the solution of the Schr\"odinger
equation has Gaussian decay in
just a semi-space of $\,\mathbb R^n$.
\begin{theorem}\lambdabel{hardyhalf} Assume that
$u\in C([0,1]:L^2((0,\infty)\times \mathbb R^{n-1}))$ is a strong solution of the IVP
\begin{equation}
\begin{cases}
\begin{aligned}
\lambdabel{eq441}
&\partial_tu=i(\Delta + V(x,t))u,\\
&u(x,0)=u_0(x),
\epsilonnd{aligned}
\epsilonnd{cases}
\epsilonnd{equation}
with
\begin{equation}
\lambdabel{extrahyp}
\int_0^1\,\int_{1/2}^{3/2}\,|\partial_{x_1}u(x,t)|^2\,dx\,dt<\infty,
\epsilonnd{equation}
\begin{equation}
\lambdabel{hyp442}
V:\mathbb R^n\times [0,1]\to\mathbb C,\,\,\,\,\,\,\,V\in L^{\infty}(\mathbb
R^n\times [0,1]),
\epsilonnd{equation}
and \begin{equation}
\lambdabel{condition44}
\lim_{R\rightarrow +\infty}\int_0^1\|V(t)\|_{L^\infty(\{x_1>R\})}\,dt =0.
\epsilonnd{equation}
Assume that
\begin{equation}
\begin{aligned}
\lambdabel{443}
&\int_{x_1>0} \,e^{c_0\,|x_1|^2}\,|u(x,0)|^2\, dx <\infty,\\
\\
&\int_{x_1>0} \,e^{c_1\,|x_1|^2}\,|u(x,1)|^2\, dx <\infty,
\epsilonnd{aligned}
\epsilonnd{equation}
with $c_0,\,c_1>\,0$ sufficiently large.
Then $\,u\epsilonquiv 0$.
\epsilonnd{theorem}
\underline{Remarks} : (a) Note that in Theorem \ref{hardyhalf}, the solution does not need to be defined for $\,x_1\leq 0$.
In this sense, this is a stronger result that the uniqueness results in \cite{BZ}, \cite{KPV02}, \cite{IK04}, \cite{IK06},
and \cite{ds}, which required that the solution be defined in $\,\mathbb R^n\times [0,1]$ and be $C([0,1]:L^2(\mathbb R^n))$.
On the other hand, we need to assume the condition \epsilonqref{extrahyp}. Note that \cite{KPV02} also needs an extra assumption on $\,\nabla u$,
stronger that \epsilonqref{extrahyp}, but that in \cite{IK04}, which among other things removed any extra assumption on $\,\nabla u$, but still required
the solution to be defined in $\,\mathbb R^n\times [0,1]$ and be in $C([0,1]:L^2(\mathbb R^n))$. If in the setting of Theorem \ref{hardyhalf}
we know that $\,u\,$ is a solution in $\,\mathbb R^n\times [0,1]$ and is in $C([0,1]:L^2(\mathbb R^n))$, then we can dispose the hypothesis
\epsilonqref{extrahyp} as follows:
First as in the first step of the proof of Theorem \ref{hardyhalf}, we can use the Appell transformation to reduce to the case
$c_1=c_2=2\gamma$. Then, using $\,\varphi(x_1)$ a \lq\lq regularized" convex function which agrees with $\,x_1^+$ for $x_1>1$ , $x_1<-1$,
an application of Lemma \ref{L: freq1} and Corollary \ref{last} in the appendix yields the estimate
$$
\sigmaup_{0\leq t\leq 1}\,\int\,e^{2\gamma(x_1^+)^2}|u(x,t)|^2dx+\int_0^1\int_{x_1>2}\,t(1-t)|\nabla u(x,t)|^2e^{2\gamma(x_1^+)^2}dxdt<\infty.
$$
Once this is obtained, by restricting our attention to
$$
(2,\infty)\times \mathbb R^{n-1}\times [\delta,1-\delta],
$$
for each $\,\delta>0$, we are in the situation of Theorem \ref{hardyhalf}, and hence $\,u\epsilonquiv 0$ on $\{x_1>2\}\times[0,1]$. Finally, Izakov's result
in \cite{Iza} concludes that $\,u\epsilonquiv 0$ (more precisely, the version of Izakov's result proved in \cite{IK04}, which does not require $\,\nabla u$ to exist
for $-1<x_1<1).$
\vskip.05in
(b) We have seen that Theorem \ref{hardyhalf} includes many of the uniqueness results for solutions vanishing at two different times
in a semi-space. In comparison with the results in section 2, since the extra assumption \epsilonqref{extrahyp} can be recovered as in remark (a) when
the solution is defined in $\,\mathbb R^n\times [0,1]$ and is in $C([0,1]:L^2(\mathbb R^n))$, the only weakness is that the provide an optimal
estimate for the constants $\,c_1,\,c_2$, but on the other hand deals with solutions only defined in $(0,\infty)\times \mathbb R^{n-1}\times [0,1]$.
\vskip.05in
(c) In Theorem \ref{hardyhalf} the direction $\,\vec e_1$ can be replaced by any
other $\,\omega\in\mathcal S^{n-1}$.
\vskip.1in
\underline{Proof of Theorem \ref{hardyhalf}}: The strategy of the proof follows closely the one in \cite{EKPV06}. We divide the proof into three steps.
\vskip.05in
\underline{First Step } : Reduction to the case $c_0=c_1=2 \gamma$.
\vskip.05in
This follows by using the conformal or Appell transformation introduced in section 2
(see \epsilonqref{2.1}-\epsilonqref{externalforce}), combined with the observation that the set $\{x_1>0\}$ remains invariant.
\vskip.05in
\underline{Second Step } : Upper Bounds.
\vskip.05in
We define
$$
v(x,t)=\theta(x_1)\,u(x,t),
$$
with $\,\theta \in C^{\infty}(\mathbb R)$, non-decreasing with $\,\theta(x_1)\epsilonquiv 1\,$ if
$\,x_1>3/2$, and $\theta(x_1)\epsilonquiv 0\,$ if $\,x_1<1/2$.
Therefore,
\begin{equation}
\lambdabel{FFF}
\partial_t v=i\,\Delta v + i\,V(x,t) v +
i\,F(x,t),\;\;\;\,\,\,\;\;\;F(x,t)=2\,\partial_{x_1}u\,\theta'(x_1)+
u\,\theta''(x_1).
\epsilonnd{equation}
Using \epsilonqref{extrahyp} we can apply Lemma \ref{ultimo} to get that
\begin{equation}
\begin{aligned}
\lambdabel{uno44}
&\sigmaup_{0\leq t\leq 1}\| e^{\lambdambda\cdot x_1} v(\,\cdot\,,t)\|_{L^2(\mathbb \mathbb R^n)} \\
&\leq c_n
\Big(\|e^{\lambdambda\cdot x_1} v(0)\|_{L^2(\mathbb \mathbb R^n)} + \|e^{\lambdambda\cdot x_1}
v(1)\|_{L^2(\mathbb \mathbb R^n)}\\
& +\int_0^1
\|e^{\lambdambda\cdot x_1}\, F(\cdot, t)\|_{L^2(\mathbb \mathbb R^n)} dt
+ \int_0^1
\|e^{\lambdambda\cdot x_1}\, V\,\chi_{\{x_1<R\}}v(\cdot, t)\|_{L^2(\mathbb \mathbb R^n)} dt\Big),
\epsilonnd{aligned}
\epsilonnd{equation}
for some fixed $R$ sufficiently large. Thus, using \epsilonqref{extrahyp}
\begin{equation}
\begin{aligned}
\lambdabel{uno444}
&\sigmaup_{0\leq t\leq 1}\| e^{\lambdambda\cdot x_1} v(\,\cdot\,,t)\|_{L^2(\mathbb \mathbb R^n)} \\
&\leq c_n
\Big(\|e^{\lambdambda\cdot x_1} v(0)\|_{L^2(\mathbb \mathbb R^n)} + \|e^{\lambdambda\cdot x_1}
v(1)\|_{L^2(\mathbb \mathbb R^n)}\\
& + c\,e^{c\,|\lambdambda|}
+ c\,\|V\|_{\infty} \,e^{c\,|\lambdambda|\,R}\Big).
\epsilonnd{aligned}
\epsilonnd{equation}
Thus, from the formula \epsilonqref{est1} (with $p=2$ and $n=1$) and \epsilonqref{uno444}
we obtain that
$$
\alphaigned
&\sigmaup_{0\leq t\leq 1}\| e^{\gamma\,|x_1|^2} v(\,\cdot\,,t)\|_{L^2(\mathbb \mathbb R^n)}
\\
&
\,\,\,\leq
\Big(\| e^{\gamma\,|x_1|^2} v(0)\|_{L^2(\mathbb \mathbb R^n)} + \| e^{\gamma\,|x_1|^2}
v(1)\|_{L^2(\mathbb \mathbb R^n)}
+c +\,\|V\|_{\infty} \,e^{c\,\gamma\,R^2}\Big).
\epsilonndaligned
$$
Thus,
\begin{equation}
\lambdabel{step2a}
\sigmaup_{0\leq t\leq 1}\| e^{\gamma\,|x_1|^2} v(\,\cdot\,,t)\|_{L^2(\mathbb \mathbb R^n)}\leq
c_{\gamma}.
\epsilonnd{equation}
Combining this and the equation for $\,v\,$ we shall get a smoothing estimate.
Using the notation
$$
H(t)=\|f\|^2_{L^2(\mathbb R^n)}=\|f\|^2,
$$
with
$$
f(x,t)= e^{\gamma|x_1|^2}\,v(x,t)
$$
and the abstract Lemma \ref{L: freq1} (see the appendix) one formally has that
\begin{equation}
\lambdabel{upper-smooth}
\begin{aligned}
\partial_t^2H &\leq 2\partial_t\text{\it Re}\left(\partial_tf-\mathcal Sf-\mathcal
Af,f\right)\\
&+ 2\left(\mathcal S_tf+\left[\mathcal S,\mathcal A\right]f,f\right) + \|\,e^{\gamma
|x_1|^2}(F + V\,v)\|^2,
\epsilonnd{aligned}
\epsilonnd{equation}
with
$$
e^{\gamma|x_1|^2}(\partial_t-i\,\Delta) (e^{-\gamma|x_1|^2}f) = \partial_t f
-\mathcal Sf-\mathcal Af= e^{\gamma |x_1|^2}(F + V v),
$$
where $\, \mathcal S = - i \gamma (4x_1\,\partial_{x_1}+2)$ is symmetric,
$\mathcal A=i(\Delta + 4\gamma x_1^2)$ is skew-symmetric, and $\,F\,$ as in
\epsilonqref{FFF}.
Since,
$$
[\mathcal S ; \mathcal A] = -8 \gamma \partial_{x_1}^2+ 16 \gamma^2\,x_1^2.
$$
using the inequality
$$
\alphaigned
&\int_{\mathbb R^n}\,(|\partial_{x_1}f|^2+4\gamma^2|x_1|^2|f|^2)\,dx =
\int_{\mathbb R^n}\,e^{2\,\gamma|x_1|^2}\,(|\partial_{x_1}u|^2-2\gamma\,|u|^2)dx\\
&\geq 2\,\gamma\,\int_{\mathbb R^n}\,|f|^2\,dx.
\epsilonndaligned
$$
together with Corollary \ref{last} we conclude that
\begin{equation}
\lambdabel{009}
\int_0^1\,\int \,t(1-t) \,|\partial_{x_1}
v(x,t)|^2\,e^{2\,\gamma|x_1|^2}\,e^{2\,\gamma |x_1|^2}\,dx\,dt \leq c_{\gamma}.
\epsilonnd{equation}
Combining and \epsilonqref{step2a} and \epsilonqref{009} one gets that
\begin{equation}
\lambdabel{00step2}
\begin{aligned}
&\sigmaup_{0\leq t\leq 1}\| e^{\gamma\,|x_1|^2} v(\,\cdot\,,t)\|_{L^2(\mathbb \mathbb R^n)}\\
&
+ \int_0^1 \int t(1-t) |\partial_{x_1} v(x,t)|^2\,e^{2\,\gamma|x_1|^2} e^{2\,\gamma
|x_1|^2}| dx dt\leq c_{\gamma}.
\epsilonnd{aligned}\epsilonnd{equation}
\underline{Step3}
We recall the following result which is a slight variation of that proven in detail
in \cite{EKPV06} (Lemma 3.1, page 1818) :
\begin{lemma}\lambdabel{CPDE}
Assume that $ R>0$ and $\,\varphi : [0,1] \to \mathbb R$ is a smooth function. Then, there
exists
$\,c=c(n;\|\varphi'\|_{\infty}+\|\varphi''\|_{\infty})>0$ such that the inequality
\begin{equation}
\lambdabel{cpde1}
\frac{\alphapha^{3/2}}{R^2}\,\Big\|\,e^{\alphapha |\frac{x_1-x_{0_1}}{R}+\varphi(t)|^2}g
\Big\|_{L^2(dxdt)}
\leq c\, \Big\|\,e^{\alphapha |\frac{x_1-x_{0_1}}{R}+\varphi(t)|^2}(i
\partial_t+\Delta) g \Big\|_{L^2(dxdt)}
\epsilonnd{equation}
holds when $\,\alphapha > c R^2 \,$ and $\,g\in C^{\infty}_0(\mathbb R^{n+1})\,$ is supported in
the set
$$
\{(x,t)=(x_1,..,x_n,t)\,\in\mathbb R^{n+1}\,:\, |\frac{x_1-x_{0_1}}{R}+\varphi(t)|\geq 1\}.
$$
\epsilonnd{lemma}
Now, we will chose $\,x_{0_1}=R/2$, $\;0\leq \varphi(t)\leq a,$ with $\,a=3/2-1/R$,
$\,\varphi(t)=a,$ on $\,3/8\leq t\leq 5/8$,
$\,\varphi(t)=0, $ for $\,t\in [0,1/4]\cup [3/4,1]$, and $\,\theta_R\in
C^{\infty}(\mathbb R)$ with $\,\theta_R(x_1)=1\,$ on $\,1<x_1<R-1$,
and $\,\theta_R(x_1)=0\,$ for $\,x_1<1/2$ or $\,x_1>R$.
Also we chose $\,\epsilonta \in C^{\infty}(\mathbb R)$ with $\,\epsilonta(x_1)=0,\;x_1\leq 1$ and
$\,\epsilonta(x_1)=1,\;x_1\geq 1+1/2R$.
We notice that up to translation we can assume that
\begin{equation}
\lambdabel{b}
\int_{3/8}^{5/8} \,\int_{2<x_1<3} \,|u(x,t)|^2dx dt=b\neq 0,
\epsilonnd{equation}
otherwise we would have
$$
u(x,t)=0\;\;\,\;\;\,\;\text{on}\,\;\;\,\;(x,t)\;\,\,s.t.\,\,\,(x_1,t)\in
(0,\infty)\times (3/8,5/8),
$$
and thus by Izakov's result \cite{Iza} we would get that $u\epsilonquiv 0$.
We let
\begin{equation}
\lambdabel{defg}
g(x,t)=\theta_R(x_1)\,\epsilonta\Big(\frac{x_1-R/2}{R}+\varphi(t)\Big)\,u(x,t).
\epsilonnd{equation}
It is easy to see that $\,g$ is supported on the set
\begin{equation}
\lambdabel{domain}
\{(x,t)\in\mathbb R^{n+1}\,:\, 1/2<x_1<R,\,\, 1/32<t<31/32,\;\;|\frac{x_1-R/2}{R}+\varphi(t)|\geq 1\}.
\epsilonnd{equation}
so satisfies the hypothesis of Lemma \ref{CPDE}. Also if $(x_1,t)\in (2,3)\times
(3/8,5/8)$ one has
$\varphi=a$, $\,\epsilonta\Big(\frac{x_1-R/2}{R}+a\Big)=1$ and $\theta_R=1$, hence in this domain
$$
g(x,t)=u(x,t).
$$
Thus, from \epsilonqref{domain}
it follows that
$$
|\frac{x_1-R/2}{R}+\varphi(t)|\geq 1 + 1/R,
$$
so we have the lower bound of \epsilonqref{cpde1}
$$
\frac{\alphapha^{3/2}}{R^2} \,b\,e^{\alphapha(1+1/R)^2},
$$
with $\,b\,$ as in \epsilonqref{b}.
Now we shall estimate the right hand side of \epsilonqref{cpde1}. Thus,
\begin{equation}
\lambdabel{07}
\begin{aligned}
&(i\partial_t-\Delta)g= - \theta_R(x_1)\epsilonta\Big(\frac{x_1-R/2}{R}+\varphi(t)\Big)
V(x,t) u(x,t)\\
&\;\;\;+\epsilonta\Big(\frac{x_1-R/2}{R}+\varphi(t)\Big)(2\theta'(x_1)\,\partial_{x_1}u+u\,\theta_R''(x_1))\\
&\;\;\;+(i\epsilonta'(\cdot)\,\varphi'(t)+\epsilonta''(\cdot)\,\frac{1}{R^2}) \theta_R(x_1)
u(x,t)\epsilonquiv E_1+E_2+E_3.
\epsilonnd{aligned}
\epsilonnd{equation}
Choosing $R>>\|V\|_{\infty}$, and recalling the fact that $\,\alphapha>c R^2\,$ we see
that the contribution of the term
$E_1$ involving the potential $V$ can be absorbed by the term in the left hand side
of \epsilonqref{cpde1}.
Next, we notice that the terms in $E_2$ involve derivatives of $\,\theta_R$
($\theta_R'$ or $\theta_R''$) so they are supported in
the $(x,t)\in \mathbb R^n\times [0,1]$ such that
$$
1/2<x_1<1,\,\,\,\,\,\text{or}\,\,\,\,\,R-1<x_1<R.
$$
But, if $1/2<x_1<1$, it follows that
$$
\frac{x_1-R/2}{R}+\varphi(t)\leq
1/R-1/2+3/2-1/R=1,\,\,\,\,\text{so}\,\,\,\,\epsilonta\Big(\frac{x_1-R/2}{R}+\varphi(t)\Big)=0.
$$
Thus, we only get contribution from the $(x,t)\in \mathbb R^n\times [0,1]$ such that
$\,R-1<x_1<R$, which can be bounded by
$$
c\,\int_{1/32}^{31/32}\,\int_{R-1<x_1<R}\,(|u|^2+|\partial_{x_1}u|^2)(x,t)\,e^{\alphapha(2-1/R)^2}\,dx\,dt.
$$
Finally, we look at the contribution of the term in $E_3$ in \epsilonqref{07}. In those
the derivatives fall on $\,\epsilonta$, thus they
are supported in the region
$$
1\leq \frac{x_1-R/2}{R}+\varphi(t)\leq 1 +
\frac{1}{2R},\;\;\,\,\,\,\,\frac{1}{2}<x_1<R,\,\,\,\,\,\;\;\frac{1}{32}<t<\frac{31}{32}.
$$
Hence, their contribution in \epsilonqref{cpde1} is bounded by
$$
c\,\int_{1/32}^{31/32}\,\int_{1/2<x_1<R}\,|u(x,t)|^2\,e^{\alphapha(1+1/(2R))^2}\,dx\,dt
\leq c_{\gamma}\,e^{\alphapha(1+1/(2R))^2}.
$$
Defining
\begin{equation}
\lambdabel{defdelta}
\delta(R)=\int_{1/32}^{31/32}\,\int_{R-1<x_1<R}\,(|u|^2+|\partial_{x_1}u|^2)(x,t)\,dx\,dt,
\epsilonnd{equation}
and collecting the above information using that $\,\alphapha=c_n\,R^2$ we get
$$
c\,R\,b\,e^{\alphapha(1+1/R)^2}\leq c\,\delta(R)\,e^{\alphapha(2-1/R)^2}+
\,c_{\gamma}\,e^{\alphapha(1+1/(2R))^2}.
$$
Therefore, for $\,R\,$ sufficiently large it follows that (since $\,b\neq 0$)
$$
c\,R\,b\,e^{\alphapha(1+1/R)^2}\leq c\,\delta(R)\,e^{\alphapha(2-1/R)^2},
$$
and since $\,\alphapha=c_n\,R^2$ one has that
$$
\delta(R)\geq b\,e^{-c_n R^2}.
$$
To conclude we recall that the upper bounds in \epsilonqref{00step2} gave us
$$
\delta(R)\leq c\,e^{-\gamma R^2},
$$
hence if $\gamma>c_n/2$ we conclude that $\,b=0$, which yields the desired result
$\,u\epsilonquiv 0$.
\sigmaection{Appendix}\lambdabel{aaa}
Above we have used the following abstract results established in \cite{EKPV08b}:
\begin{lemma}\lambdabel{L: freq1}
Let $\mathcal S$ be a symmetric operator, $\mathcal A$ be a skew-symmetric one, both
allowed to depend on the time variable. Let $G$ be
a positive function, $f(x,t)$ a reasonable function,
\begin{equation*}
\alphaigned
&\;H(t)=\left( f, f\right)=\|f\|^2_{L^2(\mathbb R^n)}=\|f\|^2\ ,\,\,\,\,\,\ D(t)=\left(
\mathcal Sf, f\right),\\
&\; \partial_t\mathcal S=\mathcal S_t
\quad \,\,\,\,\,\text{and}\,\,\,\,\,\quad N(t)=\frac{D(t)}{H(t)}\ .
\epsilonndaligned
\epsilonnd{equation*}
Then,
\begin{multline}
\lambdabel{E: derivadasegunda}
\begin{aligned}
\partial_t^2H &= 2\partial_t\text{\it Re}\left(\partial_tf-\mathcal Sf-\mathcal
Af,f\right)+
2\left(\mathcal S_tf+\left[\mathcal S,\mathcal A\right]f,f\right)\\
&+\|\partial_tf-\mathcal Af+\mathcal Sf\|^2-
\|\partial_tf-\mathcal Af-\mathcal Sf\|^2
\epsilonnd{aligned}
\epsilonnd{multline}
and
\begin{equation*}
\dot N(t)\ge \left(\mathcal S_tf +\left[\mathcal S,\mathcal A\right]f, f\right)/H-
\|\partial_tf-\mathcal Af-\mathcal Sf\|^2/\left(2H\right).
\epsilonnd{equation*}
Moreover, if
\begin{equation}\lambdabel{E: condicionesbase}
|\partial_tf-\mathcal Af-\mathcal Sf|\le M_1|f| +G,\ \text{in}\ \mathbb Rn\times
[0,1],\quad \mathcal S_t+\left[\mathcal S,\mathcal A\right]\ge -M_0,
\epsilonnd{equation}
and
\[M_2=\sigmaup_{[0,1]}{\|G(t)\|/\|f(t)\|}\]
is finite, then
$\log H(t)$ is \lq\lq logarithmically convex\rq\rq\ in $[0,1]$ and there is a
universal constant $N$ such that
\begin{equation}\lambdabel{E: convexidadlogaritmica}
H(t)\le e^{N\left(M_0+M_1+M_2+M_1^2+M_2^2\right)}H(0)^{1-t}H(1)^t,\ \text{when}\
0\le t\le 1.
\epsilonnd{equation}
\epsilonnd{lemma}
By multiplying the formula \epsilonqref{E: derivadasegunda} by $\,t(1-t)$, integrating the
result over $[0,1]$ and using integration by parts,
one gets the following \lq\lq smoothing" inequality
\begin{corollary}\lambdabel{last}
With the same hypotheses and notation as in Lemma \ref{L: freq1}
\begin{equation}
\lambdabel{lastformula}
\begin{aligned}
&2\int_0^1\,t(1-t)\left(\mathcal S_tf+\left[\mathcal S,\mathcal
A\right]f,f\right)\,dt+\int_0^1 \,H(t)\,dt\leq H(0)+H(1)\\
& + 2\int_0^1\,(1-2t)\text{\it Re}\left(\partial_tf-\mathcal Sf-\mathcal
Af,f\right)\,dt\\
&+
\int_0^1\,t(1-t)\|\partial_tf-\mathcal Af-\mathcal Sf\|^2_2\,dt.
\epsilonnd{aligned}
\epsilonnd{equation}
\epsilonnd{corollary}
\begin{thebibliography}{99}
\bibitem{AKS} N. Aronszajn, A. Krzywicki, and J. Szarski,
\epsilonmph{Unique continuation theorem for exterior differential forms on Riemannian manifolds},
Ark. Math. {\bf 4} (1962) 417--453.
\bibitem{monki} S. Ben Farah, and K. Mokni,
\epsilonmph{Uncertainty principles and the $L^p-L^q$-version of Morgan's theorem on some
groups},
Russian J. Math. Physics {\bf 10} (2003) 245--260.
\bibitem{BLi} H. Berestycki, and P.-L. Lions,
\epsilonmph{Nonlinear scalar field equations},
Arch. Rational Mech. Anal. {\bf 82} (1983) 313--375.
\bibitem{BGK} H. Berestycki, T. Gallou\"et, and O. Kavian,
\epsilonmph{\'Equations de champs scalaires Euclidiens non lin\'eaires dans de plan},
C. R. Acad. Sci. Paris, ser I Math {\bf 297} (1983) 307--310.
\bibitem{bonamie1} A. Bonami, and B. Demange, \epsilonmph{A survey on uncertainty
principles related to quadratic forms}, Collect.
Math. Vol. Extra (2006) 1--36.
\bibitem{bonamie2} A. Bonami, B. Demange, and P. Jaming,
\epsilonmph{Hermite functions and uncertainty principles for the Fourier and
the windowed Fourier transforms}, Rev. Mat. Iberoamericana {\bf 19} (2006) 23--55.
\bibitem{BoKe} J. Bourgain, and C. E. Kenig, \epsilonmph{On localization in the
continuous Anderson-Bernoulli model in higher dimensions,} Invent. Math.
{\bf 161}, 2 (2005) 1432--1297.
\bibitem{Cal} A. P. Calder\'on, \epsilonmph{Uniqueness in the Cauchy problem for partial differential equations}, Amer. J. Math., {\bf 80} (1958),
16--36.
\bibitem{Car} T. Carleman, \epsilonmph{Sur un probl\'eme d'unicit\'e pour les syst\'emes d' equations aux deriv\'ees partielles \'a deux variables
ind\'ependantes}, Ark. Math.
{\bf 26B}, (1939) 1--9.
\bibitem{Cha} S. Chanillo, \epsilonmph{Uniqueness of solutions to Schr\"odinger equations on complex semi-sin=mple Lie groups}, Proc. Indian Acad. Sci.
Math. Sci. {\bf 117} (2007),
325--331.
\bibitem{CEKPV} M. Cowling, L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega,
\epsilonmph{The Hardy Uncertainty Principle Revisited}, to appear in Indiana
U. Math. J.
\bibitem{CoPr} M. Cowling, and J. F. Price,
\epsilonmph{Generalizations of Heisenberg's inequality}, Harmonic Analysis (Cortona, 1982)
Lecture Notes in Math.,{\bf 992} (1983), 443-449, Springer, Berlin.
\bibitem{CoPr2} M. Cowling, and J. F. Price,
\epsilonmph{Bandwidth versus time concentration: the Heisenberg--Pauli--Weyl inequality},
SIAM J. Math. Anal. {\bf 15} (1984) 151--165.
\bibitem{cruz} J. Cruz-Sanpedro,
\epsilonmph{Unique continuation at infinity of solutions to Schr\"odinger equations with
complex potentials},
Proc. Roy. Soc. Edinburgh. {\bf 42} (1999) 143--153.
\bibitem{ds} H. Dong, and W. Staubach.
\epsilonmph{Unique continuation for the Schr\"odinger equation with gradient vector
potentials},
Proc. AMS
{\bf 135}, 7 (2007) 2141--2149.
\bibitem{EKPV06} L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega,
\epsilonmph{On Uniqueness Properties of Solutions of Schr\"odinger Equations,}
Comm. PDE. {\bf 31}, 12 (2006) 1811--1823.
\bibitem{EKPV07} L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega,
\epsilonmph{On Uniqueness Properties of Solutions of the k-generalized KdV,}
J. of Funct. Anal. {\bf 244}, 2 (2007) 504--535.
\bibitem{EKPV06a} L. Escauriaza, C. E. Kenig, G. Ponce, L. Vega, \epsilonmph{Decay at
Infinity of Caloric Functions within Characteristic Hyperplanes}, Math. Res. Letters
{\bf 13}, 3 (2006) 441--453.
\bibitem{EKPV08a} L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega,
\epsilonmph{Convexity of Free Solutions of Schr\"odinger Equations with Gaussian Decay},
Math. Res. Lett. {\bf 15}, 5 (2008) 957--971.
\bibitem{EKPV08b} L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega,
\epsilonmph{Hardy's Uncertainty Principle, Convexity and Schr\"odinger Evolutions},
J. European Math. Soc. {\bf 10}, 4 (2008) 883--907.
\bibitem{EKPV08m} L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega,
\epsilonmph{Uncertainty principle of Morgan type and Schr\"odinger evolution}, J. London
Math. Soc.
{\bf 81}, (2011) 187--207.
\bibitem{EKPV09} L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega,
\epsilonmph{The sharp Hardy Uncertainty Principle for Schr\"odinger evolutions}, Duke Math. J.
{\bf 155}, (2010) 163--187.
\bibitem{EKPV10} L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega,
\epsilonmph{Unique continuatuion for Schr\"odinger evolutions, with
applications to profiles of concentration and traveling waves},
Comm. Math. Phys. {\bf 305}, (2011) 487--512.
\bibitem{EKPV12} L. Escauriaza, C. E. Kenig, G. Ponce, L. Vega, \epsilonmph{A Theorem of Paley-Wiener Type for Schr\"odinger Evolutions}, to appear.
\bibitem{EsSS} L. Escauriaza, G. Seregin, and V. \v Sver\'ak \epsilonmph{$L^{3,\infty}$ solutions to the Navier-Stokes equations and backward uniqueness}, Russ. Math. Surv.
{\bf 58}, (2003) 211-250.
\bibitem{EsVe} L. Escauriaza, and S. Vessella \epsilonmph{Optimal three cylinder inequalities for solutions to parabolic equations with Lipschitz leading coefficients}, Contemp. Math.
{\bf 333}, Inverse Problems theory and applications (2002) 79--87.
\bibitem{GeShi} I. M. Gel'fand, and G. E. Shilov, \epsilonmph{Fourier transforms of
rapidly increasing functions and questions of uniqueness of the solution
of Cauchy's problem}, Uspehi Matem. Nauk {\bf 8}, (1953), 3--54.
\bibitem{Had} J. Hadamard, \epsilonmph{Le probl\'ome de Cauchy et les \'equations aux deriv\'ees partielles lin\'eaires hyperboliques},
Hermann, Paris (1932)
\bibitem{Hardy} G. H. Hardy, \epsilonmph{A Theorem Concerning Fourier Transforms}, J.
London Math. Soc. s1-8 (1933) 227--231.
\bibitem{HKT1} N. Hayashi, K. Nakamitsu, and M. Tsutsumi,
\epsilonmph{On solutions of the initial value problem for the nonlinear Schr\"odinger
equations in one space dimension},
Math. Z. {\bf 192} (1986) 637--650.
\bibitem{HKT2} N. Hayashi, K. Nakamitsu, and M. Tsutsumi, \epsilonmph{On solutions of the
initial
value problem for the nonlinear Schr\"odinger equations}, J. Funct. Anal. {\bf 71}
(1987) 218--245.
\bibitem{Ho} L. H\"ormander, \epsilonmph{A uniqueness theorem of Beurling for Fourier
transform pairs}, Ark. Mat. {\bf 29}, 2 (1991) 237--240.
\bibitem{In} A. E. Ingham,
\epsilonmph{A note on Fourier transforms} J. London Math. Soc. s1-9 (1934) 29--32.
\bibitem{IK04} A. D. Ionescu, and C. E. Kenig,
\epsilonmph{$L^p$-Carleman inequalities and uniqueness of solutions of nonlinear
Schr\"odinger equations,}
Acta Math. {\bf 193}, 2 (2004) 193--239.
\bibitem{IK06} A. D. Ionescu, and C. E. Kenig,
\epsilonmph{Uniqueness properties of solutions of Schr\"odinger equations,} J. Funct. Anal.
{\bf 232} (2006) 90--136.
\bibitem{Iza} V. Izakov
\epsilonmph{Carleman type estimates in an anisotropic case and applications,} J. Diff . Eqs.
{\bf 105} (1993) 217--238.
\bibitem{Ke1} C. E. Kenig,
\epsilonmph{Some recent quantitative unique continuation theorem}, Rendiconti Accademia Nazionale dell Scienze, vol. {\bf XXIX} (2005) 231--242.
\bibitem{Ke2} C. E. Kenig,
\epsilonmph{Some recent applications of unique continuations}, Contemp. Math.
{\bf 439} (2007) 25--56.
\bibitem{KeMe} C. E. Kenig, and F. Merle
\epsilonmph{Global well-posedness scattering and blow-up for the energy-critical focusing non-linear wave equation}, Acta Math.
{\bf 201} (2008) 147--212.
\bibitem{KPV02} C. E. Kenig, G. Ponce, and L. Vega,
\epsilonmph{On unique continuation for nonlinear Schr\"odinger equations}, Comm. Pure
Appl. Math.
{\bf 60} (2002) 1247--1262.
\bibitem{KoLa} V. A. Kondratiev, and E. M. Landis
\epsilonmph{Qualitative properties of the solutions of a second-order nonlinear equation}, Encyclopedia of Math. Sci. 32
(Partial Differential equations III) Springer-Verlag, Berlin
{\bf } (1988).
\bibitem{Kw} K. M. Kwong,
\epsilonmph{Uniqueness of positive solutions of $\Delta u-u+u=0$ in $\mathbb R^n$},
Arch. Rational Mech. Anal. {\bf 105} (1989) 243-266.
\bibitem{LaOl} E. M. Landis, and O. A. Oleinik, \epsilonmph{Generalized analyticity and certain properties, of solutions of elliptic and parabolic equations, that are connected with it},
Uspehi Mat. Nauk {\bf 29} (1974), 190-206.
\bibitem{lm60} J. L. Lions, and B. Malgrange, \epsilonmph{Sur l'unicit\'e r\'etrograde dans les probl\`emes mixtes paraboliques,} Math. Scan. {\bf8} (1960) 277--286.
\bibitem{Me} V. Z. Meshkov, \epsilonmph{On the possible rate of decay at infinity
of solutions of second-order partial differential equations}, Math. USSR Sbornik
{\bf 72} (1992), 343--361.
\bibitem{Mo} G. W. Morgan, \epsilonmph{A note on Fourier transforms}, J. London Math. Soc.
{\bf 9} (1934), 187-192.
\bibitem{Mu} C. M\"uller, \epsilonmph{On the behavior of the solution of the differential equation $\Delta u=f(x,u)$ in the neighborhood of a point},
Comm. Math. Pure Appl.
{\bf 1} (1954), 505--515.
\bibitem{ray} E. K. Naranayan, and S. K. Ray, \epsilonmph{Beurling's theorem in $\mathbb
R^n$}, preprint.
\bibitem{Ng} T. Nguyen, \epsilonmph{On a question of Landis and Oleinik}, Trans. Amer. Math. Soc. {\bf 362} (2010) 2875--2899
\bibitem{PW} R. Paley, and N. Wiener, \epsilonmph{Fourier transform in the complex domain},
Amer. Math. Soc Providence RI (1934).
\bibitem{SaSc} J.-C. Saut, and B. Scheurer,
\epsilonmph {Unique continuation for some evolution equations},
J. Diff. Eqs. {\bf 66}
(1987), 118--139.
\bibitem{Se} G. Seregin, \epsilonmph{A certain necessary condition of potential blow-up for Navier-Stokes equations}, preprint, arXiv: 1104.3615
\bibitem{SST} A. Sitaram, M. Sundari, and S. Thangavelu, \epsilonmph{Uncertainty
principles on certain Lie groups},
Proc. Indian Acad. Sci. Math. Sci.
{\bf 105} (1995), 135-151.
\bibitem{Str} W. A. Strauss, \epsilonmph{Existence of solitary waves in higher dimensions},
Comm. Math. Phys.
{\bf 55} (1977), 149--162.
\bibitem{Ve} S. Vessella \epsilonmph{Three cylinder inequalities and unique continuation properties for parabolic equations}, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. {\bf 13} (2002), no. 2, 107Ð120
\bibitem{BZ} B. Y. Zhang, \epsilonmph{Unique continuation properties of the
nonlinear Schr\"odinger equations}, Proc. Roy. Soc. Edinburgh. {\bf 127} (1997)
191--205.
\epsilonnd{thebibliography}
\epsilonnd{document} | math |
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\begin{document}
\markright{
}
\markboth{
{\footnotesize\rm S. Chiou, J. Kim AND J. Yan}
}
{
{\footnotesize\rm AFT Models with GEE}
}
\renewcommand{\thefootnote}{}
$\ $\par
\fontsize{10.95}{14pt plus.8pt minus .6pt}\selectfont
\centerline{\large\bf Semiparametric Multivariate Accelerated Failure Time Model }
\centerline{\large\bf with Generalized Estimating Equations}
\centerline{Sy Han Chiou, Junghi Kim, and Jun Yan}
\centerline{\it University of Conecticut, University of Minnesota }
\centerline{\it and University of Connecticut Health Center}
\fontsize{9}{11.5pt plus.8pt minus .6pt}\selectfont
\begin{quotation}
\noindent {\it Abstract:}
The semiparametric accelerated failure time model is not as widely used
as the Cox relative risk model mainly due to computational difficulties.
Recent developments in least squares estimation and induced
smoothing estimating equations provide promising tools to make
the accelerate failure time models more attractive in practice.
For semiparametric multivariate accelerated failure time models,
we propose a generalized estimating equation approach to account for
the multivariate dependence through working correlation structures.
The marginal error distributions can be either identical as in
sequential event settings or different as in parallel event settings.
Some regression coefficients can be shared across margins as needed.
The initial estimator is a rank-based estimator with Gehan's weight,
but obtained from an induced smoothing approach with computation ease.
The resulting estimator is consistent and asymptotically normal,
with a variance estimated through a multiplier resampling method.
In a simulation study, our estimator was up to three
times as efficient as the initial estimator, especially with
stronger multivariate dependence and heavier censoring percentage.
Two real examples demonstrate the utility of the proposed method.
\noindent {\it Key words and phrases:}
efficiency; induced smoothing; least squares; multivariate survival.
\end{quotation}\par
\fontsize{10.95}{14pt plus.8pt minus .6pt}\selectfont
\section{Introduction}
\label{sect:intr}
Multivariate failure times are frequently encountered in
biomedical research where failure times are clustered.
For example, a diabetic retinopathy study assessed the
efficacy of a laser treatment on decelerating vision loss,
measured by time to blindness in the left eye and in the
right eye from the same patient with diabetes \citep{DRS:1976};
a colon cancer study evaluated the treatment effects on prolonging
the time to tumor recurrence and time to death \citep{Lin:cox:1994}.
The failure times within the same cluster are associated.
Even though the primary interest most often lies in the
marginal effects of covariates on the failure times,
accounting for the within-cluster dependence may lead to
more efficient regression coefficient estimators.
For non-censored multivariate data, the generalized estimating
equations (GEE) approach \citep{Lian:Zege:long:1986} has become an
important piece in statisticians' toolbox for marginal regression.
For censored multivariate failure times, the marginal accelerated
failure time (AFT) model is a counterpart of the marginal model.
This paper aims to develop a GEE approach to make inferences
for multivariate AFT models, taking advantage of recent
developments on AFT models with least squares and induced smoothing.
A semiparametric AFT model is a linear model for the logarithm
of the failure times with error distribution unspecified.
A nice interpretation is that the effect of a covariate
is to multiply the predicted failure time by some constant.
It provides an attractive alternative to the
popular relative risk model \citep{Cox:regr:1972}.
Three main classes of estimator exist for univariate AFT models.
The Buckley--James (BJ) estimator extends the least squares principle
to accommodate censoring through an expectation--maximization (EM)
algorithm which iterates between imputing the censored failure times
and least squares estimation \citep{Buck:Jame:line:1979}.
Despite the nice asymptotic properties
\citep{Rito:esti:1990, Lai:Ying:larg:1991}, the BJ estimator may
be hard to get as the EM algorithm may not converge.
Further, the limiting covariance matrix is difficult to estimate
because it involves the unknown hazard function of the error term.
The second class is the rank-based estimator motivated by
inverting the weighted log-rank test \citep{Pren:line:1978}.
Its asymptotic properties has been rigorously studied by
\citet{Tsia:esti:1990} and \citet{Ying:larg:1993}.
Due to lack of efficient and reliable computing algorithm, the
rank-based estimator has not been widely used in practice until
recently, with numerical strategies for drawing inference developed
by \citet{Huan:cali:2002} and \citet{Stra:acce:2005}.
The third class is obtained by minimizing an inverse probability
of censoring weighed (IPCW) loss function \citep{Robi:Rotn:reco:1992}.
The IPCW estimator is easy to compute, consistent and asymptotically normal
\citep{Zhou:$m$-:1992, Stut:cons:1993, Stut:dist:1996}, but it requires
correct specification of the conditional censoring distribution and
overlapping of the supports of the censoring time and the failure time.
More recent works have led to a promising perspective on
bringing AFT models into routine data analysis practice.
For rank-based inference, \citet{Jin:Lin:Wei:Ying:rank:2003}
proposed a linear programming approach, exploiting the fact that the
weighted rank estimating equation is the gradient of an objective
function which can be readily solved by linear programming.
Variances of the estimators are obtained from a resampling method.
A computationally more efficient approach for rank-based
inference with Gehan's weight \citep{Geha:gene:1965} is the
induced smoothing procedure of \citet{Brow:Wang:indu:2007}.
This approach is an application of the general induced smoothing
method of \citet{Brow:Wang:stan:2005}, where the discontinuous
estimating equations are replaced with a smoothed version, whose
solutions are asymptotically equivalent to those of the former.
The smoothed estimating equations are differentiable, which
facilitates rapid numerical solution and sandwich variance estimator.
\citet{Jin:Lin:Ying:on:2006} suggested an iterative least-squared
procedure that starts from a consistent and asymptotically
normal initial estimator such as the one obtained from the
rank-based method of \citet{Jin:Lin:Wei:Ying:rank:2003}.
The resulting estimator is consistent and asymptotically normal, with
variance estimated from a multiplier resampling approach.
For multivariate AFT models, \citet{Jin:Lin:Ying:rank:2006}
developed rank-based estimating equations that are solved
via linear programming for marginal regression parameters.
\citet{John:Stra:indu:2009} extended the induced smoothing
approach for a rank-based estimator with Gehan's weight
to the case of clustered failure times and showed that the
smoothed estimates perform as well as those from the best
competing methods at a fraction of the computational cost.
\citet{Jin:Lin:Ying:on:2006} considered their least squares
method with marginal models for multivariate failure times.
All these approaches used independent working model and
left the within-cluster dependence structure unspecified.
\citet{Li:Yin:gene:2009} developed a generalized method of
moments approach for rank-based estimator using the quadratic
inference function approach \citep{Qu:Lind:Li:impr:2000}
to incorporate within-cluster dependence.
\citet{Wang:Fu:rank:2011} incorporated within-cluster ranks
for the Gehan type estimator with the aid of induced smoothing.
To the best of our knowledge, little work has been done to
extend the GEE approach to the setting of multivariate AFT models
except a technical report \citep{Horn:Hame:comb:1996},
where the BJ estimator was combined with GEE.
Nevertheless, having no access to recent advances on AFT models,
they did not solve the convergence problems, and their asymptotic
variance estimator formula could not be easily computed because it
depends on the derivatives of imputed failure times with
respect to regression parameters, which might explain
their overestimation of the variance.
We propose an iterative GEE procedure to account for multivariate
dependence through a working covariance or weight matrix.
This method has the same spirit as GEE in that misspecification
of the working covariance matrix does not affect the consistency
of the parameter estimator in the marginal AFT models; when
the working covariance is close to the unknown truth, the
estimator has higher efficiency than that from working
independence as used in \citet{Jin:Lin:Ying:on:2006}.
Our initial estimator is the computationally efficient, rank-based
estimator from \citet{John:Stra:indu:2009}, whose consistency and
asymptotic normality is inherited by the resulting GEE estimator.
We develop methods for cases where all marginal distributions are
identical and for cases where at least two margins are different.
Regression coefficients can be the same or partially the same
across margins as needed.
The rest of the article is organized as follows.
The semiparametric multivariate accelerated failure time model
and the notation are introduced in Section~\ref{sect:maft}.
In Section~\ref{sect:gee}, we propose an iterative GEE procedure
to update a consistent and asymptotically normal initial estimator
and present asymptotic properties of our estimator.
A large scale simulation study is reported in Section~\ref{sect:simu}
to assess the properties of the proposed estimator.
The proposed methods are illustrated with the two
aforementioned real applications in Section~\ref{sect:appl}.
In particular, some new findings are
reported in analyzing the diabetic retinopathy study.
A discussion concludes in Section~\ref{sect:disc}.
The sketch of proofs are relegated to the appendix.
\section{Multivariate Accelerated Failure Time Model}
\label{sect:maft}
There are two types of multivariate failure times depending
on whether the multiple events are parallel or sequential.
The difference between the two types is that the dimension is
fixed for parallel data while random for sequential data.
In a regression model, we generally have different covariates
and different coefficients at each margin for parallel data.
For sequential data, however, some or all covariates and
covariate coefficients may be the same across margins.
In general, it is desirable to allow some of the regression
coefficients to be shared across margins as needed.
We develop the methodology for parallel data for notational simplicity
but comment when appropriate on how to adapt to sequential data.
Consider a random sample formed by $n$ clusters.
For parallel data, all clusters are of size $K$
while for sequential data, cluster $i$ may have size $K_i$.
For ease of notation, assume at the moment that
the cluster sizes are all equal to $K$.
For $i = 1, \cdots,n$ and $k = 1, \cdots, K$, let
$T_{ik}$ and $C_{ik}$ be, respectively, the log-transformed
failure time and censoring time for margin $k$ in cluster $i$.
Let $Y_{ik} = \min(T_{ik}, C_{ik})$ and $\Delta_{ik} = I(T_{ik} < C_{ik})$.
We stack $Y_{ik}$, $T_{ik}$, $C_{ik}$, and $\Delta_{ik}$,
$k = 1, \ldots, K$, to form $K\times 1$ vector $Y_{i}$, $T_{i}$,
$C_{i}$, and $\Delta_i$, respectively.
Let $X_{i} = (X_{i1}, \ldots, X_{iK})^{\top}$ be a $K \times p$
covariate matrix, with the $k$th row denoted by $X_{ik}$.
The observed data are independent and identically
distributed copies of $\{Y, \Delta, X\}$:
$\{(Y_{i}, \Delta_{i}, X_{i}): i = 1, \ldots, n\}$.
We assume that $T_{i}$ and $C_{i}$ are conditionally independent
given $X_i$.
Our multivariate accelerated failure time model is
\begin{equation}
\label{equ:maft}
T_{i} = X_{i} \beta + \epsilon_{i},
\end{equation}
where $\beta$ is a $p \times 1$ vector of regression coefficients,
and $\epsilon_{i} = (\epsilon_{i1}, \ldots, \epsilon_{iK})^{\top}$ is a
random error vector with an unspecified multivariate distribution.
This formulation accommodates margin-specific regression
coefficients, in which case, $\beta$ is a stack of all
marginal coefficients, and $X_{i}$ is a block diagonal matrix.
The error vectors $\epsilon_{i}$'s, $i = 1, \ldots, n$,
are independent and identically distributed.
For parallel data, the $K$ marginal distributions can be all different,
while for sequential data, the number of unique marginal distributions
may be smaller or even one as in a recurrent event setting.
With right censoring, \citet{Buck:Jame:line:1979}
replaced each response $T_{ik}$ with its conditional expectation
$\hat{Y}_{ik}(\beta) = E_{\beta}(T_{ik}| Y_{ik}, \Delta_{ik}, X_{ik})$,
where the expectation is evaluated at regression coefficients $\beta$.
Let $\hat Y_i(\beta) =
\big(\hat Y_{i1}(\beta), \ldots, \hat Y_{iK}(\beta)\big)^{\top}$.
\citet{Jin:Lin:Ying:on:2006} defined
\begin{equation}
\label{equ:U}
U_n(\beta, b)= \sum_{i=1}^n\left(X_{i} - \bar{X}\right)^\top\left(\hat{Y}_{i}(b)-X_{i} \beta \right)=0,
\end{equation}
where
$\bar X = \sum_{i=1}^n X_{i} / n$, and $b$ is an initial estimator of $\beta$.
The solution for $U_n(\beta, \beta)$ is the Buckley-James estimator.
The advantage for fixing the initial value $b$ is to avoid solving
for $U_n(\beta, \beta)$ which is neither continuous nor monotone in $\beta$.
Let the $L_n(b)$ be the solution for $U_n(\beta, b)=0$ given $b$.
Then $L_n(b)$ has a closed-form,
\begin{equation}
\label{equ:Lk}
L_n(b) = \left[ \sum_{i=1}^{n}(X_{i}-\bar{X})^\top (X_{i}-\bar{X}) \right]^{-1}\left[ \sum_{i=1}^{n}(X_{i}-\bar{X})^\top \left(\hat{Y}_{i}(b)-\bar{Y}(b)\right)\right],
\end{equation}
where
$\bar{Y}(b) = \sum_{i=1}^n \hat Y_{i}(b) / n$.
Equation~\eqref{equ:Lk} leads to an iterative algorithm:
$\hat{\beta}^{(m)}_n=L_n(\hat{\beta}^{(m-1)}_n)$, $m \geq 1$.
If the initial estimator $b$ is consistent and asymptotically normal,
$\hat{\beta}^{(m)}_n$ is consistent and asymptotically normal for every $m$.
Although this estimator is consistent, its efficiency might be
low because it completely ignores the within-cluster dependence.
We next propose to accommodate dependence using the GEE approach,
which covers the estimator of \citet{Jin:Lin:Ying:on:2006}
as a special case with working independence.
\section{Inference with GEE}
\label{sect:gee}
For a given initial estimator $b$ of $\beta$, we propose an updated
estimator by solving the GEE
\begin{equation}
\label{equ:gee}
U_n(\beta, b, \alpha) = \sum_{i = 1}^{n} (X_i - \bar X)^{\top} \Omega_{i}^{-1}\big(\alpha(b)\big) \left(\hat{Y}_{i}(b)-X_{i} \beta \right)=0,
\end{equation}
where $\bar X = \sum_{i=1}^n X_i / n$, and
$\Omega_{i}^{-1}\big(\alpha(b)\big)$ is a $K \times K$ nonsingular working weight matrix
which may involve additional working parameters $\alpha$, which
may depend on $b$.
For given $\alpha$ and $b$, the solution of the GEEs~\eqref{equ:gee}
has a closed-form
\begin{equation}
\label{equ:Lb}
L_n(b, \alpha) = \left [ \sum_{i=1}^{n}(X_{i}-\bar{X})^{\top} \Omega_{i}^{-1}\big(\alpha(b)\big) (X_{i}-\bar{X}) \right ]^{-1}\left [ \sum_{i=1}^{n}(X_{i}-\bar{X})^\top \Omega_{i}^{-1}\big(\alpha(b)\big) \left(\hat{Y}_{i}(b)-\bar{Y}(b)\right)\right ].
\end{equation}
This process can be carried out iteratively, summarized as follows.
\begin{enumerate}
\item[1.]
Obtain an initial estimate $\hat{\beta}^{(0)}_n = b_n$ of $\beta$ and
initialize with $m = 1$.
\item[2.]
Obtain an estimate $\hat\alpha_n$ of $\alpha$ given $\hat\beta^{(m-1)}_n$,
$\hat\alpha_n(\hat{\beta}_n^{m-1})$.
\item[3.]
Update with $\hat\beta^{(m)}_n = L_n(\hat\beta^{(m-1)}_n, \hat\alpha_n)$.
\item[4.]
Increase $m$ by one and repeat 2 and 3 until convergence.
\end{enumerate}
As in \citet{Jin:Lin:Ying:on:2006}, a consistent and asymptotically
normal estimator is important for avoiding convergence problems.
We propose to use the rank-based estimator with Gehan's weight
from the induced smoothing approach of \citet{John:Stra:indu:2009}.
This estimator has the same asymptotic property as the non-smoothed
version in \citet{Jin:Lin:Wei:Ying:rank:2003}, but can be obtained
with computation ease; its finite sample performance was also reported
to be as well as the best competing methods \citep{John:Stra:indu:2009}.
The GEEs are most efficient when $\Omega_{i}$ is
chosen to be the covariance matrix of $\hat{Y}_{i}(b)$.
When $\Omega_{i}$'s are the identity matrix (working independence
with all marginal variances the same), our estimator reduces to
the least squares estimator of \citet{Jin:Lin:Ying:on:2006}.
The working covariance matrix $\Omega_{i}$'s are the same when
all clusters have the same size $K$; they only vary with
$i$ when the cluster sizes are not equal.
For convenience, we assume from now on that
$E(\epsilon_{ik}) = 0$, $i = 1, \ldots, n$, $k = 1, \ldots, K$.
This can be achieved by incorporating appropriate columns of ones
in $X_i$, and, hence, adding intercepts in $\beta$.
Our construction of working covariance involves filling element
$\Omega_{kl}$, for $k, l\in\{1, \cdots, K\}$,
of the working covariance matrix $\Omega$.
To allow arbitrary number of unique marginal distributions,
let $m_k \in \{1, \ldots, \kappa\}$ be the index of the $k$th
margin among the $\kappa$ unique marginal distributions.
The conditional expectation $\hat{Y}_{ik}(b)$
is computed as
\begin{equation*}
\hat{Y}_{ik}(b) = \Delta_{ik}Y_{ik}+(1-\Delta_{ik})\left [ \frac{\int_{e_{ik}(b)}^{\infty}u \mathrm{d} \hat{F}_{k, b}(u)}{1 - \hat{F}_{k,b}\left \{ e_{ik}(b) \right \}} + X_{{ik}}^{\top}b \right ],
\end{equation*}
where $e_{ik}(b) = Y_{ik}-X_{{ik}}^{\top}b$ is the right-censored
error evaluated at $b$, and $\hat{F}_{k, b}$ is the pooled
Kaplan--Meier estimator of the distribution function $F_{k, b}$
from the transformed data $\{ e_{ir}(b), \Delta_{ir}: m_r = m_k \}$,
which share the same margin $m_k$.
Specifically, $\hat{F}_{k, b}$ is
\begin{equation*}
\hat{F}_{k, b}(t) = 1 - \prod_{1\leq i \leq n, 1 \le r \le K : m_r = m_k, e_{ir} < t}\left( 1-\frac{\Delta_{ir}}{\sum_{j=1}^n\sum_{1 \le l \le K: m_l = m_k} I\left(e_{jl}(b) \geq e_{ir}(b)\right)}\right).
\end{equation*}
To fill the diagonal elements $\Omega_{kk}$, $1 \le k \le K$, evaluate
the conditional second moment of $\epsilon_{ik}(b)$ given the observed data:
\begin{equation}
\label{equ:Vik}
\hat{V}_{ik}(b)= \Delta_{ik} e_{ik}^2(b) + (1 - \Delta_{ik})
\frac{\int_{e_{ik}(b)}^{\infty}u^2 \mathrm{d} \hat{F}_{k, b}(u)}{1-\hat{F}_{k, b}\left \{ e_{ik}(b) \right \}},
\qquad i = 1, \ldots, n, \quad k = 1, \ldots, K.
\end{equation}
For a given $b$, we fill $\Omega_{kk}$ by an unbiased estimator
of $\mathrm{Var}\big(\epsilon_{ik}(b)\big)$
\begin{equation}
\label{equ:vhat}
\hat \Omega_{kk}(b) = \frac{\sum_{1 \le i \le n, 1 \le r \le K: m_r = m_k} \hat V_{ik}(b)} {n \sum_{1 \le r \le K} I\{m_r = m_k\}}.
\end{equation}
To fill the off-diagonal elements $\Omega_{kl}$, $k \ne l$, define
\begin{equation}
\label{equ:ehat}
\hat e_{ik}(b) = \hat Y_{ik}(b) - X_{ik}^{\top} b,
\qquad i = 1, \ldots, n, \quad k = 1, \ldots, K,
\end{equation}
the conditional expectation of $\epsilon_{ik}(b)$ given the observed data.
Only when $\Delta_{ik} = 1$ is $\hat e_{ik}(b)$ equal to $e_{ik}(b)$.
For a given $b$, we fill $\Omega_{kl}$, $k \ne l$, by
\begin{equation}
\label{equ:chat}
\hat{\Omega}_{kl}(b)= \frac{1}{n} \sum_{i=1}^{n} \hat e_{ik}(b) \hat e_{il}(b).
\end{equation}
Because the construction of $\hat e_{ik}(b)$ does not
involve the dependence between pair $(k,l)$ in cluster $i$,
$\hat e_{ik}(b) \hat e_{il}(b)$ does not have expectation
$\mathrm{Cov}\big(\epsilon_{ik}(b), \epsilon_{il}(b)\big)$ unless
$\Delta_{ik} = \Delta_{il} = 1$.
Nevertheless, $\hat\Omega_{kl}(b)$ is still usable for
its simplicity in constructing working covariance.
Parsimonious working covariance structures such as exchangeable
(EX) or autoregressive with order 1 (AR1) can be imposed.
Parameters $\alpha$ in the working covariance can be estimated with
method of moment estimator $\hat\alpha_n$ based on $\hat\Omega$
as in the non-censored case \citep{Lian:Zege:long:1986}.
When there is no censoring, the working covariance matrix
$\hat{\Omega}$ converges to the true covariance matrix.
This is no longer true when censoring is present.
Nevertheless, $\hat{\Omega}$, and consequently, $\hat{\alpha}_n$,
still converges to some limit which helps to
improve the efficiency of the GEE estimation.
Extension to unequal cluster sizes as in a
recurrent event setting is straightforward.
In this case, it is reasonable to assume identical marginal error
distributions, hence, identical marginal variances.
The working covariance matrix $\Omega_i$ with dimension $K_i\times K_i$
can be constructed with an given estimator $\hat\alpha_n$ for $\alpha$
for a specified working covariance structure.
Under certain regularity conditions, the proposed estimator is consistent
to the true regression coefficients $\beta_0$ and asymptotically normal.
The asymptotic results are summarized in the following theorems,
whose proofs are sketched in the Appendix.
\begin{thm}
\label{thm:cons}
Under conditions A1--A9 in the Appendix, $\hat{\beta}^{(m)}_n$
is a consistent estimator of the true parameter $\beta_0$ for each $m \ge 1$.
\end{thm}
\begin{thm}
\label{thm:norm}
Under conditions A1--A9 in the Appendix,
$n^{1/2}(\hat{\beta}^{(m)}_n - \beta_0)$
converges in distribution to multivariate normal with mean zero
for each $m \ge 1$.
\end{thm}
The resampling approach developed by \citet{Jin:Lin:Ying:on:2006}
is adapted to estimate the covariance matrix of $\hat{\beta}^{(m)}_n$.
Let $Z_i$, $i = 1, \cdots, n$, be independent and identically
distributed positive random variables, independent of
the observed data, with $E(Z_{i}) = \mathrm{Var}(Z_{i})=1$.
Define
\begin{equation*}
\hat{Y}_{ik}^*(b) = \Delta_{ik} Y_{ik}+(1-\Delta_{ik})\left [ \frac{\int_{e_{ik}(b)}^{\infty} u \mathrm{d} \hat{F}_{k, b}^*(u)}{1-\hat{F}_{k, b}^*\left \{ e_{ik}(b) \right \}}+X_{{ik}}^{\top}b \right ],
\end{equation*}
where
\begin{equation*}
\hat{F}^*_{k, b}(t) = 1-\prod_{1 \le i \le n, 1 \le r \le K: m_r=m_k, e_{ir}<t}\left( 1-\frac{Z_i\Delta_{ir}}{\sum_{j=1}^n\sum_{1 \le l \le K : m_l = m_k} Z_iI\left(e_{jl}(b) \geq e_{ir}(b)\right)}\right).
\end{equation*}
Then the multiplier resampling version of equation~\eqref{equ:Lb} has the following form,
\begin{equation*}
L^{*}_n(b, \alpha) = \left [ \sum_{i=1}^{n}Z_{i}(X_{i} - \bar{X}) \Omega^{-1}_i\big(\alpha(b)\big) (X_{i}-\bar{X}) \right]^{-1}\left [ \sum_{i=1}^{n}Z_{i}(X_{i}-\bar{X})\Omega^{-1}_i\big(\alpha(b)\big)\left \{ \hat{Y}_{i}^*(b)- \bar{Y}^*(b)\right \}\right ],
\end{equation*}
where
$\alpha(b)$ is an estimator of working correlation parameter given
regression coefficients evaluated at $b$ and
$\bar{Y}^*(b) = \sum_{i=1}^{n} \hat{Y}^*_{i}(b) / n$.
For a realization of $(Z_1, \ldots, Z_n)$ and an initial estimator
$\hat\beta_n^{(0)}$, a bootstrap estimator of $\beta$ is obtained from
iteration $\hat{\beta}^{(m)*}_n = L_n^*(\hat{\beta}^{(m-1)*}_n)$.
The covariance matrix of $\hat\beta^{(m)}_n$ can be estimated from the
sample covariance matrix of a bootstrap sample of $\hat\beta^{(m)*}_n$.
The consistency of this variance estimator can be proved following
arguments similar to those in \citet{Jin:Lin:Ying:on:2006}.
\section{Simulation Study}
\label{sect:simu}
We conducted two simulation studies to assess the performance
of proposed estimators and compared its efficiency with
the initial estimators from \cite{John:Stra:indu:2009}.
The first study had a clustered failure time setting with
identical regression coefficients across margins
and identical marginal error distributions.
The cluster sizes were fixed at three.
For cluster $i$, the multivariate failure time
$T_i = (T_{i1}, T_{i2}, T_{i3})$ was generated from
\begin{equation*}
\log T_{ik} = 2 + X_{1ik} + X_{2ik} + \epsilon_{ik},
\end{equation*}
where $X_{1ik}$ was Bernoulli with rate 0.5,
$X_{2ik}$ was $N(0, 0.5^2)$, and
$\epsilon_i = (\epsilon_{i1}, \epsilon_{i2}, \epsilon_{i3})$
was a trivariate random vector specified by identical marginal
error distributions and a copula for the dependence structure.
Three marginal error distributions were considered:
standard normal, standard logistic, and standard Gumbel,
abbreviated by N, L, and G, respectively; the tail of
the three distributions gets heavier from N to L to G.
The dependence structure was specified by a Clayton copula with three
levels of dependence measured by Kendall's tau: 0, 0.3, and 0.6.
Censoring times were independently generated from uniform
distributions over $(0, c)$, where $c$ was selected for each margin to achieve
three levels of censoring percentage: 0\%, 25\%, and 50\%.
We considered random samples of size $n = 200$ clusters.
Rank-based estimator with Gehan's weight from the induced
smoothing approach of \citet{John:Stra:indu:2009}, denoted by JS,
was used as the initial estimator for GEE estimators.
Two working covariance structures, EX and AR1,
were used for the proposed iterative GEE procedure.
The covariance matrix of the estimator was obtained from the
resampling approach with 200 bootstrap size in Section~\ref{sect:gee}.
For each configuration, we did 1000 replicates.
The results are summarized in Table~\ref{tab:seq}.
To save space, only results for nonzero Kendall's tau were reported.
All estimators appear to be virtually unbiased.
The empirical variation of the estimates and the estimated variation
based on the resampling procedure agree closely for all estimators.
For a given censoring percentage, as the dependence level increases,
the variance of the JS estimator changes little, but the variance
of the GEE estimators with both working covariance structures decreases.
Further, the variance from the EX structure is in general smaller
than that from the AR1 structure, which is expected because the true
covariance structure is exchangeable in this simulation setting.
For a fixed dependence level, the effect of censoring percentage on the
variances of the estimator depends on the marginal error distributions.
The variance increases clearly as the censoring gets heavier when
the errors are normally distributed, but this pattern is not
observed with Gumbel or logistic marginal error distributions.
The relative efficiency of the proposed GEE estimator in relative
to the rank-based JS estimator is up to 3.5 in the table
(with logistic margin and Kendall's tau 0.6 for $\beta_2$).
\begin{table}[tbp]
\begin{center}
\caption{
Summary of simulation results with identical regression coefficients
and identical marginal error distributions based on 1000 replications.
Empirical SE is the standard deviation of the parameter estimates;
Estimated SE is the mean of the standard error of the estimator;
RE is the empirical relative efficiencies in relative to the JS estimator.
}
\label{tab:seq}
\renewcommand\tabcolsep{3pt}
\begin{tabular}{cccc rrr r rrr r rrr r rr r}
\toprule
Marg & $\tau$ & Cens & $\beta$ & \multicolumn{3}{c}{Bias} && \multicolumn{3}{c}{Empirical SE} && \multicolumn{3}{c}{Estimated SE} && \multicolumn{2}{c}{RE}&\\
\cmidrule(lr){5-7} \cmidrule(lr){9-11} \cmidrule(lr){13-15} \cmidrule(lr){17-19}
&&&& JS & EX & AR1 && JS &EX&AR1&&JS&EX&AR1&&EX&AR1\\
\midrule
N & 0.3 & 0\% & $\beta_1$ & $-$0.002 & $-$0.003 & $-$0.004 & & 0.087 & 0.072 & 0.075 & & 0.084 & 0.068 & 0.072 & & 1.492 & 1.376 \\
& & & $\beta_2$ & 0.001 & 0.002 & 0.002 & & 0.083 & 0.072 & 0.074 & & 0.084 & 0.068 & 0.071 & & 1.349 & 1.264 \\
& & 25\% & $\beta_1$ & $-$0.008 & $-$0.012 & $-$0.013 & & 0.091 & 0.073 & 0.076 & & 0.089 & 0.073 & 0.077 & & 1.543 & 1.415 \\
& & & $\beta_2$ & $-$0.003 & $-$0.005 & $-$0.003 & & 0.093 & 0.075 & 0.079 & & 0.090 & 0.075 & 0.078 & & 1.550 & 1.384 \\
& & 50\% & $\beta_1$ & $-$0.006 & $-$0.011 & $-$0.011 & & 0.101 & 0.084 & 0.088 & & 0.099 & 0.086 & 0.090 & & 1.467 & 1.316 \\
& & & $\beta_2$ & $-$0.004 & $-$0.009 & $-$0.010 & & 0.102 & 0.084 & 0.090 & & 0.102 & 0.089 & 0.093 & & 1.484 & 1.281 \\
& 0.6 & 0\% & $\beta_1$ & 0.002 & 0.001 & 0.001 & & 0.082 & 0.047 & 0.050 & & 0.083 & 0.046 & 0.050 & & 3.130 & 2.691 \\
& & & $\beta_2$ & 0.005 & 0.001 & 0.001 & & 0.082 & 0.045 & 0.050 & & 0.084 & 0.046 & 0.050 & & 3.316 & 2.697 \\
& & 25\% & $\beta_1$ & $-$0.007 & $-$0.009 & $-$0.009 & & 0.092 & 0.050 & 0.055 & & 0.088 & 0.052 & 0.057 & & 3.322 & 2.826 \\
& & & $\beta_2$ & $-$0.003 & $-$0.008 & $-$0.007 & & 0.090 & 0.053 & 0.058 & & 0.090 & 0.054 & 0.058 & & 2.931 & 2.432 \\
& & 50\% & $\beta_1$ & $-$0.003 & $-$0.008 & $-$0.008 & & 0.101 & 0.063 & 0.069 & & 0.100 & 0.069 & 0.074 & & 2.567 & 2.144 \\
& & & $\beta_2$ & 0.000 & $-$0.005 & $-$0.004 & & 0.103 & 0.070 & 0.077 & & 0.102 & 0.071 & 0.077 & & 2.142 & 1.815 \\
L & 0.3 & 0\% & $\beta_1$ & $-$0.001 & 0.002 & 0.004 & & 0.138 & 0.123 & 0.130 & & 0.142 & 0.124 & 0.130 & & 1.258 & 1.128 \\
& & & $\beta_2$ & $-$0.006 & $-$0.004 & $-$0.004 & & 0.145 & 0.125 & 0.130 & & 0.142 & 0.123 & 0.128 & & 1.352 & 1.250 \\
& & 25\% & $\beta_1$ & $-$0.020 & $-$0.022 & $-$0.021 & & 0.140 & 0.117 & 0.121 & & 0.145 & 0.121 & 0.128 & & 1.442 & 1.341 \\
& & & $\beta_2$ & $-$0.013 & $-$0.017 & $-$0.018 & & 0.153 & 0.124 & 0.131 & & 0.147 & 0.121 & 0.128 & & 1.512 & 1.369 \\
& & 50\% & $\beta_1$ & $-$0.011 & $-$0.012 & $-$0.012 & & 0.164 & 0.133 & 0.140 & & 0.162 & 0.135 & 0.143 & & 1.524 & 1.363 \\
& & & $\beta_2$ & $-$0.008 & $-$0.013 & $-$0.014 & & 0.164 & 0.137 & 0.148 & & 0.166 & 0.137 & 0.145 & & 1.428 & 1.231 \\
& 0.6 & 0\% & $\beta_1$ & 0.006 & 0.001 & 0.000 & & 0.145 & 0.084 & 0.093 & & 0.141 & 0.085 & 0.093 & & 2.966 & 2.419 \\
& & & $\beta_2$ & 0.001 & 0.002 & 0.001 & & 0.142 & 0.082 & 0.090 & & 0.142 & 0.085 & 0.092 & & 3.020 & 2.505 \\
& & 25\% & $\beta_1$ & $-$0.011 & $-$0.014 & $-$0.015 & & 0.145 & 0.080 & 0.088 & & 0.145 & 0.080 & 0.087 & & 3.245 & 2.679 \\
& & & $\beta_2$ & $-$0.014 & $-$0.013 & $-$0.013 & & 0.149 & 0.080 & 0.088 & & 0.146 & 0.081 & 0.088 & & 3.494 & 2.868 \\
& & 50\% & $\beta_1$ & $-$0.009 & $-$0.011 & $-$0.012 & & 0.164 & 0.089 & 0.099 & & 0.162 & 0.094 & 0.102 & & 3.439 & 2.778 \\
& & & $\beta_2$ & $-$0.006 & $-$0.011 & $-$0.012 & & 0.161 & 0.092 & 0.102 & & 0.165 & 0.095 & 0.104 & & 3.036 & 2.479 \\
G & 0.3 & 0\% & $\beta_1$ & $-$0.001 & 0.004 & 0.005 & & 0.092 & 0.092 & 0.096 & & 0.094 & 0.093 & 0.096 & & 0.982 & 0.911\\
& & & $\beta_2$ & 0.000 & $-$0.004 & $-$0.005 & & 0.093 & 0.094 & 0.096 & & 0.094 & 0.093 & 0.096 & & 0.973 & 0.942\\
& & 25\% & $\beta_1$ & $-$0.007 & $-$0.015 & $-$0.017 & & 0.095 & 0.086 & 0.089 & & 0.093 & 0.085 & 0.088 & & 1.221 & 1.155 \\
& & & $\beta_2$ & $-$0.007 & $-$0.012 & $-$0.014 & & 0.094 & 0.088 & 0.092 & & 0.094 & 0.086 & 0.089 & & 1.140 & 1.048 \\
& & 50\% & $\beta_1$ & $-$0.008 & $-$0.012 & $-$0.012 & & 0.099 & 0.089 & 0.091 & & 0.095 & 0.090 & 0.093 & & 1.255 & 1.187 \\
& & & $\beta_2$ & $-$0.009 & $-$0.013 & $-$0.014 & & 0.100 & 0.090 & 0.094 & & 0.097 & 0.092 & 0.095 & & 1.235 & 1.128 \\
& 0.6 & 0\% & $\beta_1$ & 0.000 & $-$0.004 & $-$0.005 & & 0.095 & 0.075 & 0.081 & & 0.094 & 0.072 & 0.077 & & 1.614 & 1.374 \\
& & & $\beta_2$ & 0.001 & $-$0.002 & $-$0.002 & & 0.094 & 0.074 & 0.079 & & 0.094 & 0.071 & 0.077 & & 1.592 & 1.426 \\
& & 25\% & $\beta_1$ & $-$0.013 & $-$0.015 & $-$0.016 & & 0.090 & 0.065 & 0.070 & & 0.093 & 0.065 & 0.070 & & 1.911 & 1.644 \\
& & & $\beta_2$ & $-$0.013 & $-$0.016 & $-$0.015 & & 0.099 & 0.066 & 0.071 & & 0.093 & 0.066 & 0.071 & & 2.231 & 1.918 \\
& & 50\% & $\beta_1$ & $-$0.012 & $-$0.011 & $-$0.011 & & 0.093 & 0.069 & 0.074 & & 0.095 & 0.074 & 0.079 & & 1.835 & 1.561 \\
& & & $\beta_2$ & $-$0.008 & $-$0.013 & $-$0.013 & & 0.096 & 0.073 & 0.079 & & 0.097 & 0.077 & 0.083 & & 1.729 & 1.448 \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
The second simulation setting had multiple event data with different
regression coefficients and different marginal error distributions.
The cluster sizes were still fixed at three.
For cluster $i$, the multivariate failure times were generated from
\begin{equation*}
\log T_{ik} = \beta_{0k} + \beta_{1k} X_{1ik} + \beta_{2k}X_{2ik} + \epsilon_{ik},
\end{equation*}
where $(\beta_{0k}, \beta_{1k}, \beta_{2k})$, $k = 1, 2, 3$,
was the regression coefficient vector for margin $k$,
and $\epsilon_i = (\epsilon_{i1}, \epsilon_{i2}, \epsilon_{i3})$
was a trivariate random vector specified by three
marginal distributions and a copula for dependence.
The marginal distributions of $\epsilon_i$ were standard
normal, standard logistic, and standard Gumbel,
respectively, for the first, second and third margin;
their copula was Clayton with three dependence
levels measured by Kendall's tau: 0, 0.3, and 0.6.
The regression coefficients $(\beta_{0k}, \beta_{1k}, \beta_{2k})$
were set to be $(-1, 1, -1)$, $(1, -1, 1)$, and $(1, 1, 1)$,
respectively for $k = 1$, 2, and 3.
Other settings such as the covariates, censoring time, sample size,
initial estimator, bootstrap sample size for variance estimation,
replication size were all the same as in the first simulation setting.
In addition to the JS estimator, GEE estimators with two working
covariance structures were considered: EX and unstructured (UN).
\begin{table}[tbp]
\begin{center}
\caption{
Summary of simulation results with different regression coefficients
and different marginal error distributions based on 1000 replications.
Empirical SE is the standard deviation of the parameter estimates;
Estimated SE is the mean of the standard error of the estimator;
RE is the empirical relative efficiencies in relative to the JS estimator.}
\label{tab:par}
\renewcommand\tabcolsep{3pt}
\begin{tabular}{ccc rrr r rrr r rrr r rr}
\hline
&&&\multicolumn{3}{c}{EST}&&\multicolumn{3}{c}{Empirical SE}&&\multicolumn{3}{c}{Estimated SE}&&\multicolumn{2}{c}{RE}\\
\cmidrule(lr){4-6} \cmidrule(lr){8-10} \cmidrule(lr){12-14} \cmidrule(lr){16-17}
$\tau$ & Cen &$\beta$ & JS & EX & UN && JS & EX & UN && JS & EX & UN && EX&UN\\
\hline
0.3 & 0\% & $\beta_{11}$ & 0.008 & 0.003 & 0.003 & & 0.143 & 0.122 & 0.123 & & 0.146 & 0.120 & 0.119 & & 1.370 & 1.351 \\
& & $\beta_{21}$ & 0.000 & $-$0.003 & $-$0.004 & & 0.151 & 0.130 & 0.130 & & 0.146 & 0.120 & 0.119 & & 1.340 & 1.346 \\
& & $\beta_{12}$ & $-$0.000 & $-$0.003 & $-$0.002 & & 0.164 & 0.163 & 0.164 & & 0.166 & 0.160 & 0.159 & & 1.014 & 1.006 \\
& & $\beta_{22}$ & $-$0.001 & $-$0.005 & $-$0.005 & & 0.162 & 0.160 & 0.161 & & 0.166 & 0.158 & 0.157 & & 1.023 & 1.012 \\
& & $\beta_{13}$ & 0.002 & $-$0.004 & $-$0.003 & & 0.242 & 0.219 & 0.219 & & 0.247 & 0.217 & 0.217 & & 1.221 & 1.220 \\
& & $\beta_{23}$ & 0.007 & $-$0.001 & $-$0.003 & & 0.254 & 0.227 & 0.228 & & 0.249 & 0.217 & 0.217 & & 1.257 & 1.248 \\
& 25\% & $\beta_{11}$ & 0.008 & 0.004 & 0.003 & & 0.154 & 0.131 & 0.132 & & 0.156 & 0.127 & 0.127 & & 1.374 & 1.368 \\
& & $\beta_{21}$ & $-$0.005 & $-$0.007 & $-$0.006 & & 0.160 & 0.132 & 0.132 & & 0.158 & 0.129 & 0.128 & & 1.476 & 1.478 \\
& & $\beta_{12}$ & $-$0.006 & $-$0.001 & $-$0.000 & & 0.161 & 0.151 & 0.151 & & 0.165 & 0.148 & 0.147 & & 1.147 & 1.150 \\
& & $\beta_{22}$ & $-$0.003 & $-$0.010 & $-$0.010 & & 0.170 & 0.154 & 0.154 & & 0.167 & 0.149 & 0.149 & & 1.217 & 1.209 \\
& & $\beta_{13}$ & 0.002 & $-$0.006 & $-$0.006 & & 0.262 & 0.228 & 0.230 & & 0.260 & 0.220 & 0.219 & & 1.315 & 1.295 \\
& & $\beta_{23}$ & $-$0.000 & $-$0.011 & $-$0.012 & & 0.262 & 0.229 & 0.228 & & 0.264 & 0.221 & 0.221 & & 1.310 & 1.321 \\
& 50\% & $\beta_{11}$ & 0.010 & 0.001 & $-$0.000 & & 0.170 & 0.144 & 0.145 & & 0.177 & 0.146 & 0.145 & & 1.381 & 1.376 \\
& & $\beta_{21}$ & $-$0.018 & $-$0.008 & $-$0.007 & & 0.180 & 0.150 & 0.150 & & 0.181 & 0.148 & 0.147 & & 1.443 & 1.434 \\
& & $\beta_{12}$ & $-$0.006 & $-$0.005 & $-$0.004 & & 0.176 & 0.153 & 0.152 & & 0.169 & 0.149 & 0.148 & & 1.319 & 1.342 \\
& & $\beta_{22}$ & 0.014 & 0.004 & 0.002 & & 0.185 & 0.165 & 0.166 & & 0.172 & 0.153 & 0.152 & & 1.261 & 1.241 \\
& & $\beta_{13}$ & 0.018 & 0.001 & 0.000 & & 0.315 & 0.270 & 0.271 & & 0.309 & 0.262 & 0.260 & & 1.364 & 1.352 \\
& & $\beta_{23}$ & 0.029 & 0.006 & 0.007 & & 0.327 & 0.283 & 0.283 & & 0.314 & 0.264 & 0.262 & & 1.339 & 1.339 \\
0.6 & 0\% & $\beta_{11}$ & 0.004 & $-$0.000 & $-$0.001 & & 0.149 & 0.089 & 0.087 & & 0.146 & 0.084 & 0.092 & & 2.813 & 2.919 \\
& & $\beta_{21}$ & $-$0.015 & $-$0.003 & $-$0.002 & & 0.140 & 0.085 & 0.085 & & 0.146 & 0.082 & 0.090 & & 2.700 & 2.722 \\
& & $\beta_{12}$ & $-$0.010 & 0.000 & $-$0.001 & & 0.167 & 0.126 & 0.126 & & 0.165 & 0.120 & 0.142 & & 1.754 & 1.744 \\
& & $\beta_{22}$ & $-$0.001 & $-$0.000 & $-$0.000 & & 0.169 & 0.124 & 0.124 & & 0.165 & 0.119 & 0.166 & & 1.873 & 1.853 \\
& & $\beta_{13}$ & 0.003 & $-$0.004 & $-$0.005 & & 0.245 & 0.159 & 0.156 & & 0.248 & 0.156 & 0.192 & & 2.370 & 2.451 \\
& & $\beta_{23}$ & $-$0.003 & $-$0.001 & $-$0.000 & & 0.238 & 0.158 & 0.156 & & 0.248 & 0.154 & 0.189 & & 2.279 & 2.326 \\
& 25\% & $\beta_{11}$ & 0.009 & 0.003 & 0.002 & & 0.155 & 0.093 & 0.092 & & 0.157 & 0.091 & 0.113 & & 2.783 & 2.858 \\
& & $\beta_{21}$ & $-$0.007 & $-$0.004 & $-$0.005 & & 0.155 & 0.093 & 0.092 & & 0.159 & 0.093 & 0.112 & & 2.763 & 2.798 \\
& & $\beta_{12}$ & 0.000 & $-$0.003 & $-$0.002 & & 0.166 & 0.113 & 0.113 & & 0.166 & 0.111 & 0.114 & & 2.145 & 2.168 \\
& & $\beta_{22}$ & $-$0.003 & $-$0.006 & $-$0.006 & & 0.168 & 0.118 & 0.118 & & 0.167 & 0.112 & 0.114 & & 2.036 & 2.033 \\
& & $\beta_{13}$ & 0.001 & 0.000 & 0.000 & & 0.266 & 0.160 & 0.160 & & 0.260 & 0.155 & 0.175 & & 2.769 & 2.771 \\
& & $\beta_{23}$ & 0.011 & $-$0.000 & 0.000 & & 0.264 & 0.153 & 0.152 & & 0.262 & 0.155 & 0.174 & & 2.991 & 3.028 \\
& 50\% & $\beta_{11}$ & 0.007 & 0.004 & 0.004 & & 0.174 & 0.112 & 0.111 & & 0.176 & 0.112 & 0.112 & & 2.404 & 2.471 \\
& & $\beta_{21}$ & $-$0.015 & $-$0.005 & $-$0.005 & & 0.192 & 0.120 & 0.119 & & 0.179 & 0.118 & 0.117 & & 2.567 & 2.587 \\
& & $\beta_{12}$ & $-$0.009 & 0.002 & 0.003 & & 0.180 & 0.120 & 0.120 & & 0.169 & 0.119 & 0.120 & & 2.235 & 2.229 \\
& & $\beta_{22}$ & 0.017 & 0.005 & 0.003 & & 0.176 & 0.127 & 0.127 & & 0.172 & 0.125 & 0.126 & & 1.911 & 1.923 \\
& & $\beta_{13}$ & $-$0.000 & $-$0.006 & $-$0.003 & & 0.307 & 0.199 & 0.196 & & 0.312 & 0.200 & 0.203 & & 2.387 & 2.444 \\
& & $\beta_{23}$ & 0.036 & 0.004 & 0.004 & & 0.322 & 0.207 & 0.205 & & 0.315 & 0.204 & 0.203 & & 2.423 & 2.471 \\
\hline
\end{tabular}
\end{center}
\end{table}
The results are summarized in Tables~\ref{tab:par}.
Similar to the first simulation study, all estimators are
virtually unbiased, and their variance estimators are
generally close to the empirical variances of the replicates.
The variance of the GEE estimators decreases as the
dependence gets stronger at any level of censoring percentage.
Holding the dependence level, as the censoring percentage
increases, the variance increases at the normal margin,
but the pattern is different for the other two margins.
The variance has little changes at the logistic margin.
At the Gumbel margin, it remains its level as the censoring
percentage increases from 0 to 25\%, but increases notably
as the censoring percentage increases from 25\% to 50\%.
There is almost no difference between the two working
covariance structures, both leading to about the same relative
efficiency compared to the rank-based JS estimator.
The relative efficiency of both GEE estimators almost
double as Kendall's tau is increased from 0.3 to 0.6.
\section{Application}
\label{sect:appl}
The diabetic retinopathy study (DRS) was started in 1971
\citep{DRS:1976} with the aim to investigate the efficacy of laser
photocoagulation in delaying the onset of severe vision loss.
Diabetic retinopathy is the most common and serious eye complication
of diabetes, which may lead to poor vision or even blindness.
A subset of the DRS data for patients with ``high-risk'' diabetic
retinopathy, categorized by risk group 6 or higher, has been
analyzed by many authors \citep[e.g.,][]{Hust:Broo:Self:mode:1989,
Lian:Self:Chan:mode:1993, Lee:Wei:Ying:line:1993, Spie:Lin:chec:1996}.
Each of the 197 patients in this subset had one eye randomized
to laser treatment and the other eye received no treatment.
The outcomes of interest were the actual times from
initiation of treatment to the time when visual acuity dropped
below 5/200 at two visits in a row (defined as ``blindness'').
The scientific interest was the effectiveness of the
laser treatment and the influence of other risk factors.
In addition to the treatment indicator,
three covariates are available:
age at diagnosis of diabetes,
type of diabetes (1 = adult, 0 = juvenile),
and risk group (6 to 12, rescaled to 0.5 to 1.0).
Since the interaction between treatment and diabetes
type was found to be significant in \citet{Spie:Lin:chec:1996},
we also include this interaction in the model.
\begin{table}[tbp]
\caption{Results of analyzing Diabetic Retinopathy Study.}
\label{tab:dia}
\begin{center}
\begin{tabular}{cc rrr rrr rr}
\toprule
&&\multicolumn{2}{c}{JS}& &\multicolumn{2}{c}{IND}& &\multicolumn{2}{c}{EX}\\
\cmidrule(lr){3-4} \cmidrule(lr){6-7} \cmidrule(lr){9-10}
Margin & Effects & EST & SE && EST & SE && EST &SE\\
\midrule
\multicolumn{10}{l}{Identical error margins and identical regression coefficients:}\\
pooled& risk group & $-$2.659 & 0.739 && $-$2.408 & 0.859 && $-$2.306 & 0.775 \\
&age& $-$0.010 & 0.012 && $-$0.010 & 0.013 && $-$0.010 & 0.014 \\
&diabetes& $-$0.140 & 0.349 && $-$0.065 & 0.440 && $-$0.065 & 0.369 \\
&treatment&0.520 & 0.197 && 0.545 & 0.330 && 0.542 & 0.263 \\
&interaction& 1.116 & 0.301 && 0.961 & 0.466 && 0.964 & 0.410 \\
[1ex]
\multicolumn{10}{l}{Different error margins and different regression coefficients:}\\
left &risk group& $-$2.819 & 1.114 && $-$2.832 & 1.195 && $-$2.654 & 1.242 \\
&age &$-$0.042 & 0.016 && $-$0.037 & 0.019 && $-$0.036 & 0.020 \\
&diabetes & 0.825 & 0.463 && 0.706 & 0.554 && 0.702 & 0.544 \\
&treatment& 0.925 & 0.422 && 0.645 & 0.549 && 0.652 & 0.489 \\
&interaction&1.719 & 0.650 && 1.742 & 0.855 && 1.739 & 0.820 \\
right &risk group& $-$2.087 & 1.013 && $-$1.944 & 1.316 && $-$1.805 & 1.283 \\
& age & 0.011 & 0.014 && 0.009 & 0.016 && 0.009 & 0.018 \\
& diabetes & $-$0.770 & 0.432 && $-$0.640 & 0.528 && $-$0.639 & 0.656 \\
& treatment & 0.383 & 0.326 && 0.481 & 0.381 && 0.477 & 0.446 \\
&interaction&0.752 & 0.476 && 0.600 & 0.639 && 0.603 & 0.646 \\
[1ex]
\multicolumn{10}{l}{Identical error margins with partial common regression coefficients:}\\
left&age&$-$0.039 & 0.015 && $-$0.036 & 0.021 && $-$0.036 & 0.022 \\
&diabetes & 0.892 & 0.406 && 0.848 & 0.607 && 0.846 & 0.621 \\
right &age & 0.011 & 0.015 && 0.009 & 0.019 && 0.009 & 0.017 \\
& diabetes & $-$0.870 & 0.435 && $-$0.837 & 0.499 && $-$0.835 & 0.574 \\
common & treatment & 0.630 & 0.227 && 0.606 & 0.250 && 0.607 & 0.267 \\
& risk group & $-$2.588 & 0.747 && $-$2.409 & 1.034 && $-$2.264 & 0.938\\
& interaction & 1.067 & 0.318 && 1.014 & 0.344 && 1.014 & 0.409 \\
\hline
\end{tabular}
\end{center}
\end{table}
We first fit a bivariate AFT model with identical error margins and
identical regression coefficients for both left and right eyes.
The second AFT model we fit was the opposite, with different error
margins and different regression coefficients for left and right eyes.
For each model, we report GEE estimators with working
independence and working exchangeable covariance structures,
in addition to the rank-based JS estimator in Table~\ref{tab:dia}.
The GEE estimator with exchangeable working structure from the
first model suggests that the treatment was significant in
delaying the onset of vision loss; it had a significant
higher effect for adult than for juvenile, and patients
in higher risk groups tended to lose vision sooner.
Note that the treatment effect was not significant if
working independence were used in the GEE estimator.
The second model offered a possibility to check whether the
marginal error distributions and regression coefficients
should indeed be identical as assumed in the first model.
Figure~\ref{fig:surv} shows the the Kaplan--Meier survival curves
of the censored residuals for the left margin and right margin
respectively, overlaid with the pooled estimate from the first model.
All three curves appear to be mingled together tightly.
A naive log-rank test to compare the two margins, ignoring that
the regression coefficients were not known but estimated, yielded
a p-value of 0.907, confirming the visual observation.
Our joint model also allows hypothesis testing of equal coefficients
for each covariate across the two margins with Wald-type tests.
The coefficients of treatment, risk group, and treatment-diabetes
interaction were found to be not significantly different across
the two margins, with p-values 0.400, 0.278, and 0.147, respectively.
The coefficients of age and diabetes were found to be significantly
different across the two margins, with p-values 0.036 and 0.042, respectively.
We then fit an bivariate AFT model with identical error margins, same
coefficients for treatment, risk group and treatment-diabetes
interaction, and different coefficients for age and diabetes.
This is one of the many models with intermediate
complexity between the first model and the second model.
Results are summarized in the last section of Table~\ref{tab:dia}.
This time, the shared coefficients of treatment, risk group,
and treatment-diabetes interaction remained significant as before.
An interesting finding is that the difference between the coefficient
of diabetes ($0.846$ versus $-0.835$) is significantly nonzero
with a p-value 0.002, suggesting that the adult diabetes have
sooner onset of vision loss in right eye than in left eye.
This finding has not been reported in existing analyses.
\begin{figure}
\caption{Kaplan--Meier survival curves for censored residuals of the
two applications. Left: the DRS Study. Right: the colon cancer study.}
\label{fig:surv}
\end{figure}
The second application is a colon cancer study \citep{Lin:cox:1994}.
Through randomization, 315, 310 and 304 patients with stage~C
colon cancer received observation, levamisole alone (Lev), and
levamisole combined with fluorouracil (Lev + 5FU), respectively.
\citet{Lin:cox:1994} considered bivariate models for
the time to first recurrence and the time to death.
The research interest was the effectiveness of the treatment
in prolonging the time to recurrence and time to death.
Gender and age are available as covariates besides treatment.
\begin{table}[tbp]
\caption{Result of analyzing Colon Cancer Study}
\label{tab:colo}
\begin{center}
\begin{tabular}{cc rrr rrr rr}
\toprule
&&\multicolumn{2}{c}{JS}& &\multicolumn{2}{c}{EX}\\
\cmidrule(lr){3-4} \cmidrule(lr){6-7}
Margin & Effects & EST & SE & & EST & SE\\
\midrule
recurrence& Lev & 0.010 & 0.124 && 0.012 & 0.173 \\
& Lev + 5FU & 0.940 & 0.138 && 0.931 & 0.185 \\
& gender & 0.310 & 0.111 && 0.274 & 0.161 \\
& age & 0.011 & 0.004 & &0.012 & 0.006 \\
death &Lev & $-$0.009 & 0.104 && $-$0.038 & 0.131 \\
& Lev + 5FU & 0.458 & 0.108 && 0.307 & 0.136 \\
& gender & 0.064 & 0.090 && 0.066 & 0.111 \\
& age & $-$0.003 & 0.004 && $-$0.004 & 0.004 \\
\hline
\end{tabular}
\end{center}
\end{table}
In this application, the error distributions and regression
coefficients have no reason to be identical across margins.
We report results with different error margin and different
regression coefficients in Table~\ref{tab:colo}.
Since all covariates are at the cluster level, the exchangeable
and independent working covariance structure give the
same results \citep[e.g.,][]{Hin:Care:Wang:crit:2007}.
The Kaplan--Meier survival curves for the two error margins
are shown in Figure~\ref{fig:surv}, which clearly exhibits
no similarity; a naive log-rank test gives p-value $0.0008$.
The treatment of levamisole combined with fluorouracil appears
to have a significant positive effect on both event times.
The gender and age are found not to be significant for either time.
The estimated difference between the combined treatment effect
on recurrence and on death ($0.931$ versus $0.307$) has a
standard error $0.103$, suggesting that the combined treatment
has a higher effect on recurrence than on death.
\section{Discussion}
\label{sect:disc}
The working covariance structure of the proposed GEE approach
is different from that in a generalized linear model setting,
where the variance is assumed to be a function of the mean.
The errors at each margin are assumed to be independent and
identically distributed, and hence have the same variance.
This assumption may be relaxed by imposing a structure
on the variance of the errors.
For instance, in model~\eqref{equ:maft}, we replace
$\epsilon_{ik}$ with $\sigma_{ik} \nu_{ik}$, where
$\nu_{ik}$'s are independent and identically distributed for
$i = 1, \ldots, n$ with mean zero and variance one, and the scale
$\sigma_{ik}$ may be described by a regression model with covariates.
Such specification leads to heteroskedasticity in errors
and merits further investigation.
For applications like the DRS study, where there are reasons
to impose identical distribution across margins, a rigorous test
to compare the survival curves of the residuals would be desirable.
We used naive tests ignoring the fact that the residuals
were calculated based on estimated regression coefficients.
A rigorous test procedure should take into account of the variation
caused by the estimation procedure.
\par
\appendix
\setcounter{section}{0}
\setcounter{subsection}{0}
\renewcommand\thesection {\Alph{section}}
\section{Sketch of the Proofs}
We impose the following regularity conditions:
\begin{enumerate}[{A}1:]
\item $\| X_i\| \leq B$ for all $i = 1, \cdots, n$ and some nonrandom constant $B$, where $\|\cdot\|$ is matrix norm.
\item The density function of $F_{k, \beta}$ exists such that $\int_{-\infty}^\infty t^2\mathrm{d} F_{k, \beta}(t) < \infty$, for $k=1, \cdots, K$.
\item The distribution function $F_{k, \beta}$ is twice differentiable with density $f_{k, \beta}$ such that $$\int_{-\infty}^\infty \left( \frac{f_{k, \beta}^\prime(t)}{f_{k, \beta}(t)}\right)^2 \mathrm{d} F_{k, \beta}(t) < \infty$$ where $1 \leq k \leq K$, and both $f_{k, \beta}(t)$ and $f^\prime_{k, \beta}(t)$ are bounded functions.
\item $E[\exp(\theta\epsilon_{ik}^-)]+ \sup_{k\in\{1, \cdots, K\}} E[\exp(\theta C_{ik}^- )] < \infty$ for some $\theta > 0$, where $a^-=|a|I_{\{a\leq 0\}}$.
\item $\sup_{| b | < \infty; -\infty < t < \infty}\sum_{i=1}^n\sum_{k=1}^K \Pr(t \leq C_{ik} - X_{ik}^\top b \leq t +h) = O(nh)$ as $h \to 0$ and $nh \to \infty$.
\item As $n\to\infty$, $\hat{\alpha}_n$ is bounded and is $n^{1/2}$ consistent to $\alpha_0$ given $\beta$.
\item As $n\to\infty$, initial estimator $b_n$ is $n^{1/2}$ consistent to $\beta_0$ and $\sqrt{n}( b_n - \beta_0)$ is asymptoticly normal with zero mean.
\item The slope matrices $n^{-1} \partial U_n / \partial \beta$ and $n^{-1} \partial U_n / \partial b$ evaluated at $(\beta_0, \beta_0, \alpha_0)$ converge to nondegenerate, finite limit $A$ and $B$, respectively.
\item The derivative $\partial \Omega_i^{-1}(\alpha) / \partial\alpha$ is finite for all $i = 1, 2, \ldots n$.
\end{enumerate}
Conditions A1--A5 are standard and ensure the existence of
the solution of equation~\eqref{equ:U} \citep{Lai:Ying:larg:1991}.
It is natural to assume that the working covariance matrix $\Omega$ in
equation~\eqref{equ:gee} is a symmetric positive definite matrix.
Then there exist a $K\times K$ nonsingular matrix, $\Gamma$, such
that $\Omega(\alpha_0) = \Gamma^{1/2} \Gamma^{1/2}$.
Let $\mathbb X_i = \Gamma^{-1/2} X_i$, $\mathbb T_i = \Gamma^{-1/2} Y_i$,
$\mathbb C_i = \Gamma^{-1/2} C_i$, and $\omega_i = \Gamma^{-1/2} \epsilon_i$.
Then equation~\eqref{equ:gee} evaluated at $\alpha = \alpha_0$
can be viewed as equation~\eqref{equ:U} with the transformed
data $\mathbb X_i$ and $\mathbb Y_i = \min(\mathbb Y_i, \mathbb C_i)$,
with error $\omega_i$, $i = 1, \ldots, n$.
The existence of the solution to equation~\eqref{equ:gee} can be
verified by the same arguments as in \citet{Lai:Ying:larg:1991},
with assumptions similar to A1 to A5 on the transformed data.
The consistency and asymptotic normality of the estimator
given $\alpha = \alpha_0$ follow from the same arguments
as in \citet{Jin:Lin:Ying:on:2006}.
The extra complexity here comes from the fact that equation~\eqref{equ:gee}
is solved at $\alpha = \hat\alpha_n$, an estimator of $\alpha_0$.
Under condition A9, the $i$th term in the summation of
$\partial U_n / \partial \alpha$ evaluated at
$(\beta_0, \beta_0, \alpha_0)$ is a linear function
of $\hat{Y}_i(\beta_0)-X_i^\top\beta_0$,
$i = 1, \ldots, n$, with expectation zero.
By the law of large number, $n^{-1}\partial U_n/\partial \alpha$
evaluated at $(\beta_0, \beta_0, \alpha_0)$
converges to zero in probability.
\subsection{Proof of Theorem ~\ref{thm:cons}}
At the solution $\hat\beta_n^{(1)}$ given $b_n$ and $\hat\alpha_n$,
we have $n^{-1} U_n(\hat{\beta}_n^{(1)}, b_n, \hat{\alpha}_n) = 0$.
Taylor expansion at $(\beta_0, \beta_0, \alpha_0)$ gives
\begin{align}
\nonumber
0 =\,& \frac{1}{n}U_n(\beta_0, \beta_0, \alpha_0) + \frac{1}{n} \frac{\partial}{\partial \beta}\left[ U_n(\beta_0, \beta_0, \alpha_0) \right](\hat{\beta}_n^{(1)}-\beta_0) \\
\nonumber
\,& + \frac{1}{n}\frac{\partial}{\partial b}\left[ U_n(\beta_0, \beta_0, \alpha_0) \right](b_n-\beta_0) + \frac{1}{n} \frac{\partial}{\partial \alpha}\left[ U_n(\beta_0, \beta_0, \alpha_0) \right](\hat{\alpha}_n-\alpha_0) + o_p(n^{-1/2}) \\
=\,& \frac{1}{n}U_n(\beta_0, \beta_0, \alpha_0) + A_n (\hat{\beta}_n^{(1)}-\beta_0) +B_n (b_n-\beta_0)+C_n(\hat{\alpha}_n-\alpha_0) + o_p(n^{-1/2}).
\label{equ:taylor}
\end{align}
With regularity conditions A1--A5, the first term converges
in probability to zero by the law of large number.
The convergence of $b_n$ and $\alpha_n$ in A6 and A7, combined
with the limit condition in A8 and A9, then gives consistency of
$\hat\beta_n^{(1)}$ to $\beta_0$.
By induction, $\hat{\beta}^{(m)}_n$ is consistent for $\beta_0$ at every $m$.
\subsection{Proof of Theorem ~\ref{thm:norm}}
Under regularity conditions $\sqrt{n}(\hat{\beta}_n^{(1)}-\beta_0)$
can be expressed as
\begin{equation}
\sqrt{n}(\hat{\beta}_n^{(1)}-\beta_0) = \left[A_n\right] ^{-1}\left[ \frac{1}{\sqrt{n}} U_n(\beta_0, \beta_0, \alpha_0)+B_n\sqrt{n}(b_n-\beta_0)+C_n\sqrt{n}(\hat{\alpha}_n - \alpha_0)\right] + o_p(1).
\label{equ:app}
\end{equation}
With condition A9, $C_n$ converges to zero in probability, and, hence, with
$\sqrt{n}$ consistency of $\hat\alpha_n$,
$C_n \sqrt{n} (\hat \alpha_n - \alpha_0) = o_p(1)$.
Equation~\eqref{equ:app} is then asymptotically equivalent to
\begin{equation*}
\left[A_n\right] ^{-1}\left[ \frac{1}{\sqrt{n}} U_n(\beta_0, \beta_0, \alpha_0)+B_n\sqrt{n}(b_n-\beta_0)\right].
\end{equation*}
With the assumption that $b_n-\beta_0$ is asymptoticly normal,
there exist some nonrandom functions $\eta_i$ with zero mean such that,
\begin{equation*}
\sqrt{n}(b_n - \beta_0) = n^{-1/2}\sum_{i=1}^n\eta_i + o_p(\|b_n-\beta_0\|).
\end{equation*}
On the other hand, $U_n(\beta_0, \beta_0, \alpha_0)$ is a
sum of independent and identically distributed quantities
with zero mean, denoted by $\phi_i$'s, $i = 1, \ldots, n$.
Equation~\eqref{equ:app} reduces to
\begin{equation*}
\sqrt{n}(\hat{\beta}_n^{(1)}-\beta_0) = \left[A_n\right] ^{-1}\left[ n^{-1/2}\sum_{i=1}^n \left(\phi_i+B_n\eta_i\right)\right] + o_p(\|b_n-\beta_0\|).
\end{equation*}
By multivariate central limit theorem for sums of independent
random vectors, the asymptotic distribution for $\hat{\beta}_n^{(1)}$
is zero mean multivariate normal as $n\to\infty$.
The limit covariance matrix $\Sigma$ have the form
$A^{-1}\Phi A^{-1}$, where
$\Phi = \lim_{n\to\infty}n^{-1}\sum_{i=1}^n \imath_i \imath_i^{\top}$
with $\imath_i = \phi_i + B\eta_i$.
Induction then implies that $\hat{\beta}_n^{(m)}$ is
multivariate normal for every $m$.
\vskip .65cm \noindent Department of Statistics, University of Connecticut,
215 Glenbrook Rd. U-4120, Storrs, CT 06269, U.S.A.
\vskip 2pt \noindent E-mail: (steven.chiou@uconn.edu and jun.yan@uconn.edu)
\vskip .65cm \noindent Division of Biostatistics, School of Public Health, University of Minnesota,
A460 Mayo Building, MMC 303, 420 Delaware St., S.E. Minneapolis, MN 55455
\vskip 2pt \noindent E-mail: (junghikim0@gmail.com)
\vskip .65cm \noindent Institute for Public Health Research, University of Connecticut Health Center,
99 Ash Street, 2nd Floor, MC 7160, East Hartford, CT 06108
\vskip 2pt \noindent E-mail: (jun.yan@uconn.edu) \vskip .3cm
\end{document} | math |
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\begin{document}
\title{Producing symmetrical facts for lists induced by the list reversal mapping in Isabelle/HOL}
\begin{abstract}
Many facts possess symmetrical counterparts that often require a separate formal proof, depending on the nature of the involved symmetry.
We introduce a method in Isabelle/HOL which produces such a symmetrical fact for the list datatype and the symmetry induced by the list reversal mapping.
The method is implemented as an attribute and its result is based on user-declared symmetry rules.
Besides general rules, we provide rules that are aimed to be applied in the domain of Combinatorics on Words.
\end{abstract}
\vskip 32pt
\section{Introduction}
While formalizing a piece of mathematical knowledge, one probably hopes that some part of the tedious work will be done by the machine.
One such mechanical tasks are proofs that follow ``by symmetry'' which can be seen as a variation of ``without loss of generality'' \cite{wlog}.
One such ``by symmetry'' usually stands for a proper description of the symmetry involved and the procedure of how lemmas involving the symmetry should be used to obtain the symmetrical claim.
In this article, we exhibit a partial, yet quite useful, solution to ``by symmetry'' in the case of lists and the reversal mapping in the proof assistant Isabelle/HOL \cite{IsabelleHOLBook}.
The reversal, or mirror mapping, is the mapping reversing the order of elements in a list.
This mapping interconnects many pairs of definitions over lists in the spirit of the following duality: the list $p$ is a prefix of the list $w$ if and only if the reversal of $p$ is a suffix of the reversal of $w$.
We situate this solution in the context of Combinatorics on words, a mathematical domain which studies words, i.e., lists and their various properties including equations on words.
First, we give a short overview of mathematical context along with examples of the symmetry in question.
In \Cref{sec:solution}, we shortly describe possible approaches to the solution and then we describe our solution which is part of the ongoing project of formalization of Combinatorics on Words \cite{CoW_gitlab}.
We conclude by describing the limits of the current solution in \Cref{sec:limits} and conclude by final remarks in \Cref{sec:conclusion}
\section{Mathematical context and examples of the symmetry} \label{sec:context_and_examples}
We work with \emph{words}, which are finite sequences $(a_i)_{i=0}^{n}$ with $a_i \in A$ with $A$ usually being a finite set.
The set of all words over $A$ is denoted $A^*$ (where $^*$ is the Kleene star).
The \emph{reversal mapping}, denoted $\rev{}$, is a mapping $A^* \to A^*$ which maps the word $w = (a_i)_{i=0}^{n}$ to the word $\rev{w} = (a_{n-i})_{i=0}^{n}$, or simply put, it reads the letters of the word in the reverse order.
The reversal mapping is an \emph{involutive antimorphism} with respect to the operation of concatenation of two words, that is, $\rev{} \circ \rev{} = \id$ and $\rev{v \cdot w} = \rev{w} \cdot \rev{v}$ where $\cdot$ is the binary operation of concatenation.
It follows that \rev{} is also a bijection.
In our ongoing project \cite{CoW_gitlab} of formalization of Combinatorics on Words, we formalize many elementary preparatory lemmas dealing with a handful of notions.
Many of these notions have a symmetrical counterpart, and many facts are symmetrical, and their proof is just copy and paste of the proof of the original lemma.
We continue with examples that exhibit this symmetry.
\subsection{Example 1} \label{sec:ex1}
If a word $w$ can be written as a concatenation of two words $p$ and $s$, i.e., $w = p \cdot s$, we say that $p$ is a \emph{prefix} of $w$ and $s$ is a \emph{suffix}.
An elementary example of a symmetrical pair of claims involving prefix and suffix is the following.
\begin{lemma} \label{pair1}
If $p$ is a prefix of $v$, then $p$ a prefix of $v \cdot w$.
\end{lemma}
\begin{proof}
If $p$ is a prefix of $v$, there exists a word $s$ such that $v = p \cdot s$.
Hence, $v\cdot w = p\cdot s\cdot w$, and $p$ is a prefix of $v \cdot w$.
\end{proof}
By the symmetrical counterpart of \Cref{pair1} we mean the following claim.
\begin{lemma}\label{pair2}
If $s$ is suffix of $v$, then $s$ is a suffix of $w \cdot v$.
\end{lemma}
Its proof can be done as the presented proof of \Cref{pair1}, however, stating in follows ``by symmetry'' from \Cref{pair1} would be no exception in literature.
In order to formally exploit the symmetry in a full proof, we have to make the intended symmetry between a prefix and a suffix explicit:
\begin{lemma}\label{p_to_s}
The word $p$ is a prefix of $w$ if and only if $\rev{p}$ is a suffix of $\rev{w}$.
\end{lemma}
A full proof of \Cref{pair2}, by symmetry, is as follows.
\begin{proof}[Proof of \Cref{pair2}]
Fix $w$ and assume that $s$ is a suffix of $v$.
Let $s'$, $v'$, and $w'$ be the words such that
\[
s' = \rev{s}, \quad v' = \rev{v} \quad \text { and } \quad w' = \rev{w}.
\]
As the reversal is an involution, it follows that
\[
s = \rev{s'}, \quad v = \rev{v'} \quad \text { and } \quad w = \rev{w'}.
\]
As $s$ is a suffix of $v$, the word $\rev{s'}$ is a suffix of $\rev{v'}$.
By \Cref{p_to_s}, $s'$ is a prefix of $v'$.
Using \Cref{pair1}, $s'$ is a prefix of $v' \cdot w'$.
Again, by \Cref{p_to_s}, $\rev{s'}$ is a suffix of $\rev{v' \cdot w'}$.
Since
\[
\rev{v' \cdot w'} = \rev{w'}\cdot \rev{v'} = w\cdot v,
\]
we conclude that $s$ is a suffix of $w\cdot v$.
\end{proof}
\subsection{Example 2} \label{sec:ex2}
Since the next examples are in the framework of Isabelle/HOL, we first recall our setting.
A word is represented by the datatype of list, which is is specified via 2 constructors: \isaterm{Nil}{} (denoted \isaterm{{\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}}), the empty list/word, and \isaterm{Cons}{} (denoted \isaterm{{\isacharhash}}), the recursive constructor allowing to add an element to the list at its beginning.
The reversal mapping is represented by the function \isaterm{rev}:
\begin{isaframe}
\isacommand{primrec}\isamarkupfalse
\ rev\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ list\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ list{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
{\isachardoublequoteopen}rev\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}\ {\isacharbar}{\kern0pt}\isanewline
{\isachardoublequoteopen}rev\ {\isacharparenleft}{\kern0pt}x\ {\isacharhash}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ rev\ xs\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}x{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}
\end{isaframe}
with \isaterm{{\isacharat}} being the notation for list \isaterm{append}, i.e., concatenation of two words.
The predicates for prefix and suffix are already part of the Isabelle distribution in the theory HOL-Library.Sublist:
\begin{isaframe}
\isacommand{definition}\isamarkupfalse
\ prefix\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ list\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ list\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}
\ \ \isakeyword{where}\ {\isachardoublequoteopen}prefix\ xs\ ys\ {\isasymlongleftrightarrow}\ {\isacharparenleft}{\kern0pt}{\isasymexists}zs{\isachardot}{\kern0pt}\ ys\ {\isacharequal}{\kern0pt}\ xs\ {\isacharat}{\kern0pt}\ zs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{definition}\isamarkupfalse
\ suffix\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ list\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a\ list\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}
\ \ \isakeyword{where}\ {\isachardoublequoteopen}suffix\ xs\ ys\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymexists}zs{\isachardot}{\kern0pt}\ ys\ {\isacharequal}{\kern0pt}\ zs\ {\isacharat}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}
\end{isaframe}
The second example is constituted by the pair of symmetric definitions of the first and the last letter of a word, i.e., element of a list.
In Isabelle/HOL, the first element of a list is its head, realized as one of two selectors, named \isaterm{hd}, of the list constructor \isaterm{Cons}{}.
The last letter is the recursive function \isaterm{last}:
\begin{isaframe}
\isacommand{primrec}\isamarkupfalse
\ {\isacharparenleft}{\kern0pt}nonexhaustive{\isacharparenright}{\kern0pt}\ last\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharprime}{\kern0pt}a\ list\ {\isasymRightarrow}\ {\isacharprime}{\kern0pt}a{\isachardoublequoteclose}\ \isakeyword{where}\isanewline
\ \ {\isachardoublequoteopen}last\ {\isacharparenleft}{\kern0pt}x\ {\isacharhash}{\kern0pt}\ xs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}if\ xs\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ then\ x\ else\ last\ xs{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}
\end{isaframe}
\noindent given in the main theory List.
To obtain a simple enough symmetry rule for {\tt hd} and {\tt last}, it suffices to notice
that they behave the same way on the empty list.
\begin{isaframe}
\isacommand{lemma}
\ hd{\isacharunderscore}{\kern0pt}last{\isacharunderscore}{\kern0pt}Nil{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}hd\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ last\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isachardoublequoteclose}
\isacommand{unfolding}\isamarkupfalse
\ hd{\isacharunderscore}{\kern0pt}def\ last{\isacharunderscore}{\kern0pt}def\ \isacommand{by}\isamarkupfalse
\ simp
\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse
\ hd{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}last{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}hd{\isacharparenleft}{\kern0pt}rev\ xs{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ last\ xs{\isachardoublequoteclose}
\isacommand{by}\isamarkupfalse
\ {\isacharparenleft}{\kern0pt}induct\ xs{\isacharcomma}{\kern0pt}\ simp\ add{\isacharcolon}{\kern0pt}\ hd{\isacharunderscore}{\kern0pt}last{\isacharunderscore}{\kern0pt}Nil{\isacharcomma}{\kern0pt}\ simp{\isacharparenright}{\kern0pt}
\end{isaframe}
The pair of symmetrical claims is the following.
\begin{isaframe}
\isacommand{lemma}\isamarkupfalse
\ example{\isadigit{2}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}u\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymLongrightarrow}\ \ prefix\ u\ v\ {\isasymLongrightarrow}\ hd\ u\ {\isacharequal}{\kern0pt}\ hd\ v{\isachardoublequoteclose}
\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse
\ example2\_sym{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}u\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymLongrightarrow}\ \ suffix\ u\ v\ {\isasymLongrightarrow}\ last\ u\ {\isacharequal}{\kern0pt}\ last\ v{\isachardoublequoteclose}
\end{isaframe}
The goal is to obtain example2\_sym from example{\isadigit{2}} by symmetry.
We proceed analogously to the proof of \Cref{pair2} above using standard methods in Isabelle/HOL:
\begin{isaframe}
example{\isadigit{2}}{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}rev\ u{\isachardoublequoteclose}\ {\isachardoublequoteopen}rev\ v{\isachardoublequoteclose}{\isacharcomma}{\kern0pt}\ unfolded\ rev{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}Nil{\isacharunderscore}{\kern0pt}conv\ suffix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}prefix{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}\ hd{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}last{\isacharbrackright}{\kern0pt}
\end{isaframe}
where \isaterm{rev{\isacharunderscore}{\kern0pt}is{\isacharunderscore}{\kern0pt}Nil{\isacharunderscore}{\kern0pt}conv}
is
\isaterm{{\isacharparenleft}{\kern0pt}rev\ xs\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}xs\ {\isacharequal}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}{\isacharparenright}{\kern0pt}}
and \isaterm{suffix{\isacharunderscore}{\kern0pt}to{\isacharunderscore}{\kern0pt}prefix{\isacharbrackleft}{\kern0pt}symmetric{\isacharbrackright}{\kern0pt}} is \isaterm{prefix\ {\isacharparenleft}{\kern0pt}rev\ xs{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rev\ ys{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ suffix\ xs\ ys}.
That is, we instantiate every variable of example{\isadigit{2}} by its reversal to obtain
\begin{isaframe}
{\isachardoublequoteopen}rev\ u\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymLongrightarrow}\ prefix\ {\isacharparenleft}{\kern0pt}rev\ u{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rev\ v{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ hd\ {\isacharparenleft}{\kern0pt}rev\ u{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ hd\ {\isacharparenleft}{\kern0pt}rev\ v{\isacharparenright}{\kern0pt}{\isachardoublequoteclose},
\end{isaframe}
and then rewrite the terms using appropriate symmetry rules,
via the unfolded attribute (which is analogous to what was done in the proof of \Cref{pair2} above).
We end up with example2\_sym and the proof by symmetry is done.
\subsection{Example 3} \label{sec:ex3}
The next example is the following pair of symmetric facts.
\begin{isaframe}
\isacommand{lemma}\isamarkupfalse
\ example{\isadigit{3}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefix\ u\ {\isacharparenleft}{\kern0pt}p\ {\isacharat}{\kern0pt}\ w\ {\isacharat}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow} \ length\ p\ {\isasymle}\ length\ u\ \ {\isasymLongrightarrow}\ length\ u\ {\isasymle}\ length\ {\isacharparenleft}{\kern0pt}p\ {\isacharat}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow} \isanewline
\ {\isasymexists}r{\isachardot}{\kern0pt}\ u\ {\isacharequal}{\kern0pt}\ p\ {\isacharat}{\kern0pt}\ r\ {\isasymand}\ prefix\ r\ w{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse
\ example3\_sym{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}suffix\ u\ {\isacharparenleft}{\kern0pt}p\ {\isacharat}{\kern0pt}\ w\ {\isacharat}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow} \ length\ q\ {\isasymle}\ length\ u\ {\isasymLongrightarrow}\ length\ u\ {\isasymle}\ length\ {\isacharparenleft}{\kern0pt}w\ {\isacharat}{\kern0pt}\ q{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
\ {\isasymexists}r{\isachardot}{\kern0pt}\ u\ {\isacharequal}{\kern0pt}\ r\ {\isacharat}{\kern0pt}\ q\ {\isasymand}\ suffix\ r\ w{\isachardoublequoteclose}\isanewline
\end{isaframe}
Applying the same strategy as for example2 fails, since trying to obtain example3\_sym from
\begin{isaframe}
example{\isadigit{3}}{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}rev\ u{\isachardoublequoteclose}\ {\isachardoublequoteopen}rev\ p{\isachardoublequoteclose}\ {\isachardoublequoteopen}rev\ w{\isachardoublequoteclose}\ {\isachardoublequoteopen}rev\ q{\isachardoublequoteclose}{\isacharcomma}{\kern0pt}\ unfolded\ symmetry{\isacharunderscore}{\kern0pt}rules{\isacharbrackright}{\kern0pt},
\end{isaframe}
\noindent where \isaterm{symmetry{\isacharunderscore}{\kern0pt}rules} is a list of appropriate symmetry rules, leaves us with
\begin{isaframe}
\ {\isachardoublequoteopen}suffix\ u\ {\isacharparenleft}{\kern0pt}{\isacharparenleft}{\kern0pt}q\ {\isacharat}{\kern0pt}\ w{\isacharparenright}{\kern0pt}\ {\isacharat}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}
\ \ length\ p\ {\isasymle}\ length\ u\ {\isasymLongrightarrow} \isanewline\ length\ u\ {\isasymle}\ length\ {\isacharparenleft}{\kern0pt}w\ {\isacharat}{\kern0pt}\ p{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ {\isasymexists}r{\isachardot}{\kern0pt}\ rev\ u\ {\isacharequal}{\kern0pt}\ rev\ p\ {\isacharat}{\kern0pt}\ r\ {\isasymand}\ prefix\ r\ {\isacharparenleft}{\kern0pt}rev\ w{\isacharparenright}{\kern0pt}{\isachardoublequoteclose},
\end{isaframe}
which is not yet in the form of example3\_sym.
This is not unexpected, the claim contains a bound variable \isaterm{r}.
To finish the conversion, it suffices to realize that $(\exists x. \ P(x)) \leftrightarrow (\exists x. \ P(\rev{} x))$ holds.
Thus, we may replace the last two occurrences of \isaterm{r} with \isaterm{rev r}, and apply appropriate symmetric rules.
The next section addresses our realization of automatic production of symmetric rules, preceded by a discussion on the use of existing tools.
\section{Automated production of symmetrical claims} \label{sec:solution}
Before describing our solution to the automation of producing symmetrical claims, we discuss if and how might our task be achieved using tools for theorem reuse available in Isabelle/HOL.
We have not found any ready made tool in Isabelle/HOL that could achieve our objectives.
We shall briefly discuss two existing tools that achieve a similar task, namely reusing of a theory in a homomorphic setting.
The first tool are locales.
Locale is a mechanism for abstraction via interpretation and locale expressions \cite{Ballarin2010,Ballarin2014}.
We could see the ``by symmetry'' argument as two instantiations of the same claim: first in lists, and second in reversed lists.
It would require to prove all claims about lists in an abstract setting, and then apply it to lists and reversed lists.
The abstract setting would mean some kind of ``axiomatic theory of lists'', that is, of free monoids, as in \cite{HolubVeroff}.
While this may be the correct idea mathematically, we do not see how to naturally recreate it in Isabelle/HOL using locales.
The second tool is the infrastructure of transfer \cite{lifting_and_transfer,phd_kuncar}.
Its main purpose is to transfer facts between two datatypes, e.g., from natural integers to integers, via user specified transfer rules.
Although, in principle, it should be possible to use this powerful tool, we encountered several problems using it and we did not find a way how to employ it for our purposes without producing undesired limitations.
For example, it is not clear how to specify whether in the case of transferring a fact containing $\isaterm{{\isacharprime}a\ list}list$ the transfer rule should be applied to $\isaterm{{\isacharprime}a\ list}$ or $\isaterm{{\isacharprime}a\ list}list = \isaterm{{\isacharprime}b\ list}$.
Since it seems from the above discussion that there is no direct way how to achieve the desired automation of the symmetry, we propose a ``lightweight'' solution which closely mimics the simple reproving of each individual claim ``on the fly'' as indicated by the examples in~\Cref{sec:context_and_examples}.
Our solution is very simple but at the same time it proves to be very practical and sufficiently versatile.
It is created as a single attribute called ``reversed''.
The symmetry rules are collected as a list of theorems called ``reversal\_rule'', i.e., a user can add and remove them any time.
By default, rules are required to eliminate reversal images, thus the reversal images are supposed to be on the left side of the equalities serving as rules.
For instance, the symmetry rule \Cref{p_to_s} is stored in this form
\begin{isaframe}
\ {\isachardoublequoteopen}suffix\ {\isacharparenleft}{\kern0pt}rev\ p{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rev\ w{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ prefix\ p\ w{\isachardoublequoteclose}.
\end{isaframe}
The execution follows examples of \Cref{sec:ex2,sec:ex3}:
first, all schematic variables of type list of the fact being reversed are instantiated by their reversals.
Before the application of the symmetry rules, bound variables need to be treated.
Let us indicate this procedure on example3 of \Cref{sec:ex3} which contains one bound variable.
As indicated above, the idea is to use the equivalence $(\exists x. \ P(x)) \leftrightarrow (\exists x. \ P(\rev{} x))$.
We introduce a helper (private) definition and 2 claims as follows:
\begin{isaframe}
\isacommand{definition}\isamarkupfalse
\ Ex{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}wrap\ {\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}{\isacharprime}{\kern0pt}a\ list\ {\isasymRightarrow}\ bool{\isacharparenright}{\kern0pt}\ {\isasymRightarrow}\ bool{\isachardoublequoteclose}\isanewline
\ \ \isakeyword{where}\ {\isachardoublequoteopen}Ex{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}wrap\ P\ {\isacharequal}{\kern0pt}\ {\isacharparenleft}{\kern0pt}{\isasymexists}x{\isachardot}{\kern0pt}\ P\ {\isacharparenleft}{\kern0pt}rev\ x{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse
\ Ex{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}wrapI{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}{\isasymexists}x{\isachardot}{\kern0pt}\ P\ x\ {\isasymequiv}\ Ex{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}wrap\ P{\isachardoublequoteclose}\isanewline
\isanewline
\isacommand{lemma}\isamarkupfalse
\ Ex{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}wrapE{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}Ex{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}wrap\ {\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ P\ x{\isacharparenright}{\kern0pt}\ {\isasymequiv}\ {\isasymexists}x{\isachardot}{\kern0pt}\ P\ {\isacharparenleft}{\kern0pt}rev\ x{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\isanewline
\end{isaframe}
The application of these claims can be seen as
\begin{isaframe}
\ example{\isadigit{3}}{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}rev\ u{\isachardoublequoteclose}\ {\isachardoublequoteopen}rev\ p{\isachardoublequoteclose}\ {\isachardoublequoteopen}rev\ w{\isachardoublequoteclose}\ {\isachardoublequoteopen}rev\ q{\isachardoublequoteclose}{\isacharcomma}{\kern0pt}unfolded\ Ex{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}wrapI{\isacharbrackright}{\kern0pt},
\end{isaframe}
which yields
\begin{isaframe}
\ {\isachardoublequoteopen}prefix\ {\isacharparenleft}{\kern0pt}rev\ u{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rev\ p\ {\isacharat}{\kern0pt}\ rev\ w\ {\isacharat}{\kern0pt}\ rev\ q{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}
length\ {\isacharparenleft}{\kern0pt}rev\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\ length\ {\isacharparenleft}{\kern0pt}rev\ u{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
length\ {\isacharparenleft}{\kern0pt}rev\ u{\isacharparenright}{\kern0pt}\ {\isasymle}\ length\ {\isacharparenleft}{\kern0pt}rev\ p\ {\isacharat}{\kern0pt}\ rev\ w{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}
Ex{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}wrap\ {\isacharparenleft}{\kern0pt}{\isasymlambda}r{\isachardot}{\kern0pt}\ rev\ u\ {\isacharequal}{\kern0pt}\ rev\ p\ {\isacharat}{\kern0pt}\ r\ {\isasymand}\ prefix\ r\ {\isacharparenleft}{\kern0pt}rev\ w{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}.
\end{isaframe}
The next step is
\begin{isaframe}
example{\isadigit{3}}{\isacharbrackleft}{\kern0pt}of\ {\isachardoublequoteopen}rev\ u{\isachardoublequoteclose}\ {\isachardoublequoteopen}rev\ p{\isachardoublequoteclose}\ {\isachardoublequoteopen}rev\ w{\isachardoublequoteclose}\ {\isachardoublequoteopen}rev\ q{\isachardoublequoteclose}{\isacharcomma}{\kern0pt}unfolded\ Ex{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}wrapI{\isacharcomma}{\kern0pt}\ unfolded\ Ex{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}wrapE{\isacharbrackright}{\kern0pt}
\end{isaframe}
resulting in
\begin{isaframe}
{\isachardoublequoteopen}prefix\ {\isacharparenleft}{\kern0pt}rev\ u{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rev\ p\ {\isacharat}{\kern0pt}\ rev\ w\ {\isacharat}{\kern0pt}\ rev\ q{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}
length\ {\isacharparenleft}{\kern0pt}rev\ p{\isacharparenright}{\kern0pt}\ {\isasymle}\ length\ {\isacharparenleft}{\kern0pt}rev\ u{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\isanewline
length\ {\isacharparenleft}{\kern0pt}rev\ u{\isacharparenright}{\kern0pt}\ {\isasymle}\ length\ {\isacharparenleft}{\kern0pt}rev\ p\ {\isacharat}{\kern0pt}\ rev\ w{\isacharparenright}{\kern0pt}\ {\isasymLongrightarrow}\ {\isasymexists}r{\isachardot}{\kern0pt}\ rev\ u\ {\isacharequal}{\kern0pt}\ rev\ p\ {\isacharat}{\kern0pt}\ rev\ r\ {\isasymand}\ prefix\ {\isacharparenleft}{\kern0pt}rev\ r{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}rev\ w{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}.
\end{isaframe}
Note that the name of the bound variable is preserved in this step.
It is due to \isaterm{{\isacharparenleft}{\kern0pt}{\isasymlambda}x{\isachardot}{\kern0pt}\ P\ x{\isacharparenright}{\kern0pt}} being present in
\isaterm{Ex{\isacharunderscore}{\kern0pt}rev{\isacharunderscore}{\kern0pt}wrapE} rather than just \isaterm{P}.
This two step rewriting using the definition \isaterm{Ex\_rev\_wrap} in the intermediate step is to prevent an infinite loop of rewriting while trying to go directly from
\isaterm{{\isasymexists}x{\isachardot}{\kern0pt}\ P\ x}
to
\isaterm{{\isasymexists}x{\isachardot}{\kern0pt}\ P\ {\isacharparenleft}{\kern0pt}rev\ x{\isacharparenright}}.
The last form is ready for the application of symmetry rules, and we almost obtain our goal, example3\_sym.
The remaining difference is the order of application of \isaterm{\isacharat}, i.e., the arrangement of parentheses.
The operation \isaterm{\isacharat} is associative and this final adjustment is left to be done manually, if desirable.
The implementation deals with other types of bound variables in a similar manner using a definition analogous to \isaterm{Ex\_rev\_wrap} and its two associated wrapping and unwrapping rules.
In a similar spirit, a special care for the constructors \isaterm{Nil}{} and \isaterm{Cons}{} is also part of the reversing process.
The described implementation is available at \cite{RevSymArch_gitlab}.
\section{Limits of the approach} \label{sec:limits}
To show the current limits, consider the following pair of symmetric claims:
\begin{isaframe}
\isacommand{lemma}\isamarkupfalse
\ example{\isadigit{4}}{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}prefix\ ps\ ws\ {\isasymLongrightarrow}\ prefix\ {\isacharparenleft}{\kern0pt}concat\ ps{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}concat\ ws{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}
\isanewline \isanewline
\isacommand{lemma}\isamarkupfalse
\ example{\isadigit{4}}{\isacharunderscore}{\kern0pt}sym{\isacharcolon}{\kern0pt}\ {\isachardoublequoteopen}suffix\ ps\ ws\ {\isasymLongrightarrow}\ suffix\ {\isacharparenleft}{\kern0pt}concat\ ps{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}concat\ ws{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}\
\end{isaframe}
Applying the attribute reversed on example{\isadigit{4}} produces:
\begin{isaframe}
{\isachardoublequoteopen}suffix\ ps\ ws\ {\isasymLongrightarrow}\ prefix\ {\isacharparenleft}{\kern0pt}concat\ {\isacharparenleft}{\kern0pt}rev\ ps{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}concat\ {\isacharparenleft}{\kern0pt}rev\ ws{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}{\isachardoublequoteclose}
\end{isaframe}
The problem here is that we are dealing with variables of type \isaterm{{\isacharprime}a\ list}list{}, representing factorizations or decomposition of words.
As they are of type \isaterm{{\isacharprime}b\ list}{}, the reversing happens only on this level, whereas to produce example4\_sym one would need the reversing to act on \isaterm{{\isacharprime}a\ list}{} =
\isaterm{\isacharprime{}b}.
Namely, the additional required action of the symmetry on \isaterm{ps} and \isaterm{ws} is the application of \isaterm{map rev}.
The reason that this represents a current limit is that the choice of correct reversal rules for variables of type \isaterm{{\isacharprime}a\ list}list{} becomes crucial and it is no more clear what are the correct reversal rules.
\section{Concluding remarks} \label{sec:conclusion}
Although the implemented attribute seems to be very simple, together with many delicately selected reversal rules it is very useful in our current project of formalization of Combinatorics on Words \cite{CoW_gitlab}.
As the attribute is a part of a living project, and the time period between the acceptance and publication of this article was noticeable, the obstacle exhibited in the previous section has been already surmounted in a way to suit the needs of the project.
However, the goal to properly deal with variables of any type remains.
In order to do that, our tentative model of the symmetry in question needs to be generalized and validated.
\end{document} | math |
Cartagena, COLOMBIA–Colombian gold stands out for its fiery hues and authentic lustre — today as it did about four thousand years ago. That’s right, Colombians have been master goldsmith’s since 1500 BC and before any European had ever stepped food on its lush soil.
American metallurgy has its origins in Peru, but eventually made its way throughout South America.
What we know today as “Pre Colombino” jewellery is rooted in the native Zenú culture, whose people inhabited the grasslands and developed a talent, not only for working metals like copper and gold, but also for weaving (a technique that eventually gave birth to the filigree technique).
Colombia’s Cano Family, which founded the Cano chain of boutiques, has been in large part responsible for the preservation of traditional Colombian jewellery. In addition to their stores, the Cano family was responsible for furnishing much of Cartagena’s gold museum, the Museo del Oro Zenú.
It all started in 1890, when the family’s patriarch, Nemesio Cano uncovered important archeological sites filled with gold treasures. His discoveries ignited a passion for collecting indigenous artefacts that was passed on to his sons Jose and Felix and later on to his grandson Guillermo.
Though still expanding, CANO can be found not only in Colombia, but also Costa Rica, Spain and the United States. By visiting their website you can also choose from their collection and have it shipped to you internationally for 35 USD more.
As starting point of the process the artisans choose the desired mold.
The wax outcome is then examined by the artisan to detect any flaws in the mold.
The second step is casting, the liquid metal is poured into the mold and then spun on centrifuge to ensure perfection.
The hot pieces are then chilled under water and inspected.
The opaque pieces are polished in the third phase.
After brushing each piece is carefully polished with special tools to enhance each detail of the design.
Careful hands take part in every step of creation.
Many pieces are then soldered to compose the final masterpiece.
The final masterpiece is a synthesis of hours of work, many master hands and years of history. | english |
package it.stasbranger.rotarylive.service;
import java.io.IOException;
import java.util.List;
import org.bson.types.ObjectId;
import org.springframework.beans.factory.annotation.Autowired;
import org.springframework.dao.DuplicateKeyException;
import org.springframework.data.domain.Page;
import org.springframework.data.domain.Pageable;
import org.springframework.stereotype.Service;
import org.springframework.web.multipart.MultipartFile;
import it.stasbranger.rotarylive.dao.ClubRepository;
import it.stasbranger.rotarylive.domain.Attach;
import it.stasbranger.rotarylive.domain.Club;
@Service("ClubService")
public class ClubServiceImpl implements ClubService {
@Autowired private ClubRepository clubRepository;
@Autowired private AttachService attachService;
public void create(Club club){
if(clubRepository.findByName(club.getName())!=null) throw new DuplicateKeyException("this club already exists");
clubRepository.save(club);
}
public Club update(Club club){
return clubRepository.save(club);
}
public void delete(Club club){
clubRepository.delete(club);
}
public void delete(ObjectId id){
clubRepository.delete(id);
}
public Club findOne(ObjectId id){
return clubRepository.findOne(id);
}
public List<Club> findAll(){
return clubRepository.findAll();
}
public Page<Club> findAll(Pageable pageable){
return clubRepository.findAll(pageable);
}
public Page<Club> findByNameContainingIgnoreCase(String name, Pageable pageable){
return clubRepository.findByNameContainingIgnoreCase(name, pageable);
}
public Club addImage(Club club) throws IOException {
MultipartFile image = club.getFile();
if(image != null){
if(club.getId() == null) club = update(club);
String type = "LOGO_CLUB";
Attach attach = attachService.createAttach(image, type);
club.setLogoId(attach.getId());
club = clubRepository.save(club);
}
return club;
}
}
| code |
اَکھ گٲنٛٹا ہیوٗ گٔژھِتھ ییٚلہِ یہِ جماعت امہِ لۄکچہِ میٚژِ پہرِ کِنۍ پکان ٲس | kashmiri |
यूनिस्टो: अइसन कोनो सिरप नइखे जो चार घंटे में पथरी के ठीक करदी - थीप मीडिया
यूनिस्टो: अइसन कोनो सिरप नइखे जो चार घंटे में पथरी के ठीक करदी
कोनो यूनिहर्ब्स हेल्थकेयर नाम के कंपनी बाटे?
का ई हो सकेला की यूनिस्टो नाम से सिरप कोनो और कंपनी बनावत होई?
एगो दावा जउन के सोशल मीडिया में करल जात रहे की यूनिहर्ब्स हेल्थकेयर जो यूनी-स्टो नाम से सिरप बनैले रहे ऊ ४ घण्टे में आदमी के शरीर से पथरी रोग के ठीक करदी। हमनी के जब जांच करनी तो पता चलल की अइसन कोनो सिरप हइये नइखे। दावा झूट निकलल।
एगो फेसबुक उपयोग करेवाला एगो पैकेट के फोटो डालके हर्बल सिरप के दवा करत रहलन। पैकेट पे लिखल जानकारी से पता चलल की बनाये वाले कंपनी के नाम यूनिहर्ब्स हेल्थकेयर बा और सिरप के नाम यूनी-स्टो बा। पैकेट में पढ़े से पता चलल की ओमे गुर्दा के दिक्कत के लेके रस्याण सूत्रीकरण लिखल बाटे। फोटो के साथ इहो लेखल रहे की ई सिरप ४ घण्टे में आदमी के शरीर से पथरी रोग के ठीक करेला।
पोस्ट का संग्रहीत संस्करण यहां देखल जा सकेला।
पोस्ट के आशुचित्र निचे देल गइल बा।
दावा: यूनिहर्ब्स हेल्थकेयर के जो यूनी-स्टो सिरप बा ऊ ४ घंटा में गुर्दा के पथरी ठीक कर दे बा।
हमनी के पड़ताल करनी कॉर्पोरेट कार्य मंत्रालय, भारत सरकार से। पता चलल युनीहर्ब्स हेल्थ केयर नाम से कोनो कंपनी दर्ज नइखे। दुगो कंपनी मिलल अइसन मिलता जुलता नाम से:
युनिहरबल लाइफ साइंस प्राइवेट लिमिटेड कंपनी जो आयुर्वेदिक दवा बनावेला
युनिहर्ब्स ४ लाइफ प्रोडक्ट्स प्राइवेट लिमिटेड जो आयुर्वेदिक दवाई बनावेला मगर ज्यादातर थर्ड पार्टी उत्पादक बाटे
हमनी के पड़ताल कइनी युनिहर्बल लाइफ साइंस प्राइवेट लिमिटेड से। कंपनी हमनी के पक्का कईलस की युनि-स्टो उनकर उत्पाद हईये नइखे।
हमनी के युनिहर्ब्स ४ लाइफ प्रोडक्ट्स प्राइवेट लिमिटेड में भी गइनी ई पता करें की खातिर की कही ई कंपनी कउनो दूसर कंपनी ला उत्पाद तां ना करेला। पर अभी तक हमनी के कुच्छो जवाब उँहा से ना मिलल। हमनी के जैसे कउनो जानकारी मिली रउवा लोग के बता देईल जाई।
हालाँकि युनिहर्ब्स हेल्थ केयर के लोगो एकदम मेल ना खाइल जौनो कंपनी के नाम ऊपर लिखल बाटे। और जो नाम कंपनी के पैकेट पे लिखल बाटे ऊ नाम ऊपर देल गइल कंपनी से नइखे मिलत नाही भारत में ऐइसन कोनो कंपनी बा।
ता अइसन शायदे हो सकेला की अइसन टाइप के दवाई या कोनो पोषण वाला दवा मिल सकी।
यूनिस्टो कंपनी व्यापार चिन्ह के साथ एक पंजीकृत स्वीटसेरलैंड कंपनी बाटे जे सुरक्षा उत्पाद में कार्य करेला। तो अइसन ना हो सकेला की यूनिस्टो नाम से वैध उतपाद कउनो समान्नित कंपनी के होई।
हालाँकि भारत में बहुते आयुर्वेदा सूत्रीकरण से सामान बनावल और बेचल जाइला, कोई कुछ सिमित समय ला बनाके पहर रोकदेला, ता इ कहल बारे मुश्किल बा की कभी इ सिरप के प्रचारकरण भारत के कोनो हिस्सा में भइल रहे या ना।
हमनी के एगो आयुर्वेदिक डॉक्टर से दिल्ली में भेट भएल जिनकर नाम रहे डॉक्टर राजीब भटिआ, बीए एम्स, जे पुष्टि कइलन की आयुर्वेदा सूत्रीकरण पथरी रोग पर कुछ हद तक कारगर सिद्ध भइल बा। हम युनि-स्टो नाम से कोनो सिरप के नाम नइखे सुनली।
अगला लेखफाबिफ्लू बनाम लीवफावीर: कीमत के अंतर के मैसेज झूठ बा | hindi |
چُھ جسم واراہ بڑان تۂ نظام چُھ ہیوان کام کرنی | kashmiri |
Book Description: Africa: Volume 1 begins a series of books which adopt a new perspective on African history and culture, surveying the wide array of societies and states that have existed on the African continent and introducing readers to the diversity of African experiences and cultural expressions. Toyin Falola has brought together African studies professors from a variety of schools and settings. Writing from their individual areas of expertise, these authors work together to break general stereotypes about Africa, focusing instead on the substantive issues of the African past from an African perspective. The texts are richly illustrated and include maps and timelines to make cultural and historical movements clearer, and suggestions for further reading will help readers broaden their own particular interests. Africa provides new perspectives that challenge the accepted ways of studying Africa, flexibility for instructors to structure courses, and encouragement for readers who are eager to learn about the diversity of the African experience.
Volume 2, African Cultures and Societies Before 1885 provides a broad view of precolonial experiences and expressions in Africa. The book focuses on culture as a means of understanding both the traditions that thrived throughout Africa and the efforts of modern Africans to reclaim their cultural past in lands that have been divided and exploited by Western imperial powers.
by Max Essex et al.
Book Description In this stunning memoir, veteran Washington Post correspondent Lynne Duke takes readers on a wrenching but riveting journey through Africa during the pivotal 1990s and brilliantly illuminates a continent where hope and humanity thrive amid unimaginable depredation and horrors. | english |
/* umask.c, created from umask.def. */
#line 23 "d:/jicamasdk/progs/bash-2.05/builtins/umask.def"
#line 35 "d:/jicamasdk/progs/bash-2.05/builtins/umask.def"
#include <config.h>
#include "../bashtypes.h"
#include "filecntl.h"
#ifndef _MINIX
# include <sys/file.h>
#endif
#if defined (HAVE_UNISTD_H)
#include <unistd.h>
#endif
#include <stdio.h>
#include <chartypes.h>
#include "../shell.h"
#include "posixstat.h"
#include "common.h"
#include "bashgetopt.h"
#ifdef __LCC__
#define mode_t int
#endif
/* **************************************************************** */
/* */
/* UMASK Builtin and Helpers */
/* */
/* **************************************************************** */
static void print_symbolic_umask __P((mode_t));
static int symbolic_umask __P((WORD_LIST *));
/* Set or display the mask used by the system when creating files. Flag
of -S means display the umask in a symbolic mode. */
int
umask_builtin (list)
WORD_LIST *list;
{
int print_symbolically, opt, umask_value, pflag;
mode_t umask_arg;
print_symbolically = pflag = 0;
reset_internal_getopt ();
while ((opt = internal_getopt (list, "Sp")) != -1)
{
switch (opt)
{
case 'S':
print_symbolically++;
break;
case 'p':
pflag++;
break;
default:
builtin_usage ();
return (EX_USAGE);
}
}
list = loptend;
if (list)
{
if (DIGIT (*list->word->word))
{
umask_value = read_octal (list->word->word);
/* Note that other shells just let you set the umask to zero
by specifying a number out of range. This is a problem
with those shells. We don't change the umask if the input
is lousy. */
if (umask_value == -1)
{
builtin_error ("`%s' is not an octal number from 000 to 777",
list->word->word);
return (EXECUTION_FAILURE);
}
}
else
{
umask_value = symbolic_umask (list);
if (umask_value == -1)
return (EXECUTION_FAILURE);
}
umask_arg = (mode_t)umask_value;
umask (umask_arg);
if (print_symbolically)
print_symbolic_umask (umask_arg);
}
else /* Display the UMASK for this user. */
{
umask_arg = umask (022);
umask (umask_arg);
if (pflag)
printf ("umask%s ", (print_symbolically ? " -S" : ""));
if (print_symbolically)
print_symbolic_umask (umask_arg);
else
printf ("%04lo\n", (unsigned long)umask_arg);
}
fflush (stdout);
return (EXECUTION_SUCCESS);
}
/* Print the umask in a symbolic form. In the output, a letter is
printed if the corresponding bit is clear in the umask. */
static void
print_symbolic_umask (um)
mode_t um;
{
char ubits[4], gbits[4], obits[4]; /* u=rwx,g=rwx,o=rwx */
int i;
i = 0;
if ((um & S_IRUSR) == 0)
ubits[i++] = 'r';
if ((um & S_IWUSR) == 0)
ubits[i++] = 'w';
if ((um & S_IXUSR) == 0)
ubits[i++] = 'x';
ubits[i] = '\0';
i = 0;
if ((um & S_IRGRP) == 0)
gbits[i++] = 'r';
if ((um & S_IWGRP) == 0)
gbits[i++] = 'w';
if ((um & S_IXGRP) == 0)
gbits[i++] = 'x';
gbits[i] = '\0';
i = 0;
if ((um & S_IROTH) == 0)
obits[i++] = 'r';
if ((um & S_IWOTH) == 0)
obits[i++] = 'w';
if ((um & S_IXOTH) == 0)
obits[i++] = 'x';
obits[i] = '\0';
printf ("u=%s,g=%s,o=%s\n", ubits, gbits, obits);
}
int
parse_symbolic_mode (mode, initial_bits)
char *mode;
int initial_bits;
{
int who, op, perm, bits, c;
char *s;
for (s = mode, bits = initial_bits;;)
{
who = op = perm = 0;
/* Parse the `who' portion of the symbolic mode clause. */
while (member (*s, "agou"))
{
switch (c = *s++)
{
case 'u':
who |= S_IRWXU;
continue;
case 'g':
who |= S_IRWXG;
continue;
case 'o':
who |= S_IRWXO;
continue;
case 'a':
who |= S_IRWXU | S_IRWXG | S_IRWXO;
continue;
default:
break;
}
}
/* The operation is now sitting in *s. */
op = *s++;
switch (op)
{
case '+':
case '-':
case '=':
break;
default:
builtin_error ("bad symbolic mode operator: %c", op);
return (-1);
}
/* Parse out the `perm' section of the symbolic mode clause. */
while (member (*s, "rwx"))
{
c = *s++;
switch (c)
{
case 'r':
perm |= S_IRUGO;
break;
case 'w':
perm |= S_IWUGO;
break;
case 'x':
perm |= S_IXUGO;
break;
}
}
/* Now perform the operation or return an error for a
bad permission string. */
if (!*s || *s == ',')
{
if (who)
perm &= who;
switch (op)
{
case '+':
bits |= perm;
break;
case '-':
bits &= ~perm;
break;
case '=':
bits &= ~who;
bits |= perm;
break;
/* No other values are possible. */
}
if (*s == '\0')
break;
else
s++; /* skip past ',' */
}
else
{
builtin_error ("bad character in symbolic mode: %c", *s);
return (-1);
}
}
return (bits);
}
/* Set the umask from a symbolic mode string similar to that accepted
by chmod. If the -S argument is given, then print the umask in a
symbolic form. */
static int
symbolic_umask (list)
WORD_LIST *list;
{
int um, bits;
/* Get the initial umask. Don't change it yet. */
um = umask (022);
umask (um);
/* All work is done with the complement of the umask -- it's
more intuitive and easier to deal with. It is complemented
again before being returned. */
bits = parse_symbolic_mode (list->word->word, ~um);
if (bits == -1)
return (-1);
um = ~bits & 0777;
return (um);
}
| code |
\begin{document}
\title{Observables of Angular Momentum as Observables on the Fedosov
Quantized Sphere}
\author{Philip Tillman$^{1}$, George Sparling$^{2}$ \\
$^{1}$Department of Physics and Astronomy, University of Pittsburgh,
Pittsburgh, PA, USA\\
$^{2}$Department of Mathematics, University of Pittsburgh, Pittsburgh, PA,
USA\\
email:$^{1}$phil.tillman@gmail.com $^{2}$sparling@twistor.org}
\date{\today }
\maketitle
\begin{abstract}
In this paper we construct quantum mechanical observables of a single free
particle that lives on the surface of the two-sphere $\mathbb{S}^{2}$ by
implementing the Fedosov $\ast $-formalism. The Fedosov $\ast $ is a
generalization of the Moyal star product on an arbitrary symplectic
manifold. After their construction we show that they obey the standard
angular momentum commutation relations in ordinary nonrelativistic quantum
mechanics. The purpose of this paper is three-fold. One is to find an exact,
non-perturbative solution of these observables. The other is to verify that
the commutation relations of these observables correspond to angular
momentum commutation relations. The last is to show a more general
computation of the observables in Fedosov $\ast $-formalism; essentially an
undeformation of Fedosov's algorithm.
\end{abstract}
\section{Introduction}
The Moyal star product formalism is an equivalent way to do quantum
mechanics.$\left[ \text{\hyperlink{3}{3}}\right] $ \ The idea is that
instead of using abstract linear operators on a Hilbert space such as
position $\hat{x}$ and momentum $\hat{p}$, we may use classical variables $x$
and $p$ however we change the product so that the commutation relations are
the same as in the Hilbert space formalism. \ Namely:
\begin{equation*}
\left[ \hat{x}^{a},\hat{p}_{b}\right] =i\hbar \delta _{b}^{a}\text{ \ \ },
\text{ \ \ }\left[ \hat{x}^{a},\hat{x}^{b}\right] =0=\left[ \hat{p}_{a},\hat{
p}_{b}\right]
\end{equation*}
become:
\begin{equation*}
\left[ x^{a},p_{b}\right] _{\ast }=i\hbar \delta _{b}^{a}\text{ \ \ },\text{
\ \ }\left[ x^{a},x^{b}\right] _{\ast }=0=\left[ p_{a},p_{b}\right] _{\ast }
\end{equation*}
we use the convention that the lower case indices run from $1,\ldots ,n$ and
capital ones run from $1,\ldots 2n$ and:
\begin{equation*}
\left[ f,g\right] _{\ast }=f\ast g-g\ast f
\end{equation*}
where $f$ and $g$ are any 2 functions of $x$ and $p$.
We note that the limit $\hbar \rightarrow 0^{+}$ gives the ordinary product
of functions.
The definition of the Moyal star for $\mathbb{R}^{2n}$\ explicitly is:
\begin{equation*}
f\ast g=fe^{\frac{i\hbar }{2}\omega ^{AB}\overleftarrow{\partial }_{A}
\overrightarrow{\partial }_{B}}g=fg+\frac{i\hbar }{2}\omega ^{AB}\left(
\partial _{A}f\right) \left( \partial _{B}g\right) -\frac{\hbar ^{2}}{8}
\omega ^{CE}\omega ^{AB}\left( \partial _{C}\partial _{A}f\right) \left(
\partial _{E}\partial _{B}g\right) +\cdots
\end{equation*}
where $\partial _{A}=\left( \frac{\partial }{\partial x^{a}},\frac{\partial
}{\partial p_{a}}\right) $ and the arrow determines the direction that the
derivative acts and the operator $\omega ^{AB}\overleftarrow{\partial }_{A}
\overrightarrow{\partial }_{B}$\ is called the Poisson bracket.
There is an invertible map called the Weyl transform $\mathcal{W}$ that
translates from the Hilbert space formalism to the Moyal formalism.\ The
main property of this transform is that an arbitrary Taylor series operator
on the Hilbert space:\footnote{
Note that this is effectively an arbitrary operator since we can use the
commutators to rearrange each term so that the $x$'s are to the left and the
$p$'s are to the right.}
\begin{equation*}
\hat{A}=\sum_{m,n}A_{a_{1}\cdots a_{m}}^{~~~~~~~~b_{1}\cdots b_{n}}\hat{x}
^{a_{1}}\cdots \hat{x}^{a_{m}}\hat{p}_{b_{1}}\cdots \hat{p}_{b_{n}}
\end{equation*}
becomes by applying the Weyl transform:
\begin{equation*}
\mathcal{W}\left( \hat{A}\right) =A=\sum_{m,n}A_{a_{1}\cdots
a_{m}}^{~~~~~~~~b_{1}\cdots b_{n}}x^{a_{1}}\ast \cdots \ast x^{a_{m}}\ast
p_{b_{1}}\ast \cdots \ast p_{b_{n}}
\end{equation*}
in a mechanical way by simply replacing each $\hat{x}$ with $x$, $\hat{p}$
with $p$ and placing stars between each of them as is done above.$\left[
\text{\hyperlink{3}{3}}\right] $
The trace over an operator of compact support goes to:
\begin{equation*}
Tr\left( \hat{A}\right) \overset{\mathcal{W}}{\leftrightarrow }Tr_{\ast
}\left( A\right) :=\frac{1}{\left( 2\pi \hbar \right) ^{n}}\int \frac{\omega
^{n}}{n!}A
\end{equation*}
So if we are given the Hamiltonian $\hat{H}$ and the density matrix $\hat{
\rho}$ we may map:
\begin{equation*}
\hat{H}\overset{\mathcal{W}}{\leftrightarrow }H\text{ \ , \ \ }\hat{\rho}
\overset{\mathcal{W}}{\leftrightarrow }\rho
\end{equation*}
We thus can get the time-independent Schr\"{o}dinger equation by mapping:
\begin{equation*}
\hat{H}\hat{\rho}_{n}=E_{n}\hat{\rho}_{n}\text{ \ \ , \ \ }\left[ \hat{H},
\hat{\rho}_{n}\right] =0
\end{equation*}
to:
\begin{equation*}
H\ast \rho _{n}=E_{n}\rho _{n}\text{ \ \ , \ \ }\left[ H,\rho _{n}\right]
_{\ast }=0
\end{equation*}
where $\rho _{n}$\ are called the Wigner functions.\ This also works with
the time-dependent Schr\"{o}dinger equation.\footnote{
See Fedosov for clarification.$\left[ 1\right] $}
Also expectation values become:
\begin{equation*}
Tr\left( \hat{\rho}\hat{A}\right) \leftrightarrow Tr_{\ast }\left( \rho \ast
A\right)
\end{equation*}
The Moyal $\ast $ has been generalized to an arbitrary smooth symplectic
manifold $\left( \mathcal{N},\omega ,D\right) $ endowed with a preserved
two-form $\omega $ (called the symplectic form) and a phase-space connection
$D$ by Fedosov.$\left[ \text{\hyperlink{1}{1}}\right] $(an excellent summary
is $\left[ \text{\hyperlink{2}{2}}\right] $) For any such manifold $\left(
\mathcal{N},\omega ,D\right) $ he gives a perturbative expansion for his $
\ast $-product. However, the convergence issues of the Fedosov $\ast $, in
general, remain unknown.
The properties of the Fedosov $\ast $ are:
\begin{itemize}
\item It is an associative (but not commutative) map $\ast :C^{\infty
}\left( \mathcal{N}\right) \times C^{\infty }\left( \mathcal{N}\right)
\rightarrow C^{\infty }\left( \mathcal{N}\right) $.
\item Invariant under \underline{all} smooth coordinate transformations of
the phase-space variables $x$ and $p$.
\item No assumed Hamiltonian.
\item The Fedosov $\ast $ is given perturbatively given any symplectic
manifold $\left( \mathcal{N},\omega ,D\right) $.
\item In the limit $\hbar \rightarrow 0^{+},$ $\ast $ becomes the ordinary
pointwise multiplication of functions on $\mathcal{N}$.
\item To first order in $\hbar $ the commutator is the Poisson bracket: $
\left[ f,g\right] _{\ast }=i\hbar \left\{ f,g\right\} +\mathcal{O}\left(
\hbar ^{2}\right) $.
\item When $\mathcal{N}=T^{\ast }\mathbb{E}^{n}$ (i.e. the phase space or\
the cotangent bundle of $\mathbb{E}^{n}$)\footnote{
Here $\mathbb{E}^{n}$ stands for Euclidean $n$-dimensional space.} we get
the Moyal $\ast $.
\end{itemize}
In this paper we restrict $\mathcal{N}$ to be the cotangent bundle of a
manifold with metric $g$ $\left( \mathcal{M},g\right) $ denoted $T^{\ast }
\mathcal{M}$.\footnote{
The cotangent bundle of any manifold is known to be a symplectic manifold.}\
The reason to do this is that the cotangent bundle of a manifold is the
phase-space of that manifold (i.e. the space of all coordinates $x$ and
momentum $p$). In quantum mechanics using the Moyal $\ast $ the phase-space
is the arena for quantization by giving proper $\ast $-commutation relations
between the $x$'s and $p$'s. The importance of the Fedosov $\ast $-formalism
is that it is a coordinate invariant way of constructing these commutation
relations on general $T^{\ast }\mathcal{M}$ in such a way that they patch
consistently to any coordinate map of the cotangent bundle. Also another
important point is that it can be constructed at least perturbatively for
any cotangent bundle.
However unlike Fedosov who defines a formulation based on the deformation of
covectors (i.e. covectors equipped with a Moyal-like product between them)
we will not. We will introduce a Heisenberg algebra generated by $\tilde{s}$
and $\tilde{k}$ ($\left[ \tilde{s}^{i},\tilde{s}^{j}\right] =\left[ \tilde{k}
_{i},\tilde{k}_{j}\right] =0,~\left[ \tilde{s}^{i},\tilde{k}_{j}\right]
=i\hbar \delta _{j}^{i}$ where $i$ and $j$ run from $1$ through $2n$) at
every point of our phase-space $T^{\ast }\mathcal{M}$. The motivation to do
this instead of Fedosov's way is to make a more direct connection between
ordinary quantum mechanics involving Heisenberg algebras and the state
spaces that the algebra acts on called Hilbert spaces. We then define this
algebra to be linear operators on a Hilbert space which, of course, will
eventually contain our states. This new construction will still preserve all
of the essential properties of the original Fedosov $\ast $ albeit
reformulated so as to apply to different objects. It will \ be a
quantization procedure i.e. a map of the variables on the phase-space $x$
and $p$ to the observables $\hat{x}$ and $\hat{p}$ which are linear
operators on the Hilbert space.
The properties of the Fedosov $\ast $-quantization in our construction are:
\begin{itemize}
\item $\hat{x}$ and $\hat{p}$ form an associative but noncommutative algebra.
\item The map from $\left( x,p\right) \rightarrow \left( \hat{x},\hat{p}
\right) $ is invariant under \underline{all} smooth canonical coordinate
transformations of the phase-space variables $x$ and $p$.
\item No assumed Hamiltonian.
\item We can construct the $\hat{x}$ and $\hat{p}$ perturbatively given any $
\left( T^{\ast }\mathcal{M},\omega ,D\right) $.
\item In the limit $\hbar \rightarrow 0^{+},$ $\hat{x}$ and $\hat{p}$ become
$x$ and $p$ respectively i.e. the ordinary variables on $T^{\ast }\mathcal{M}
$.
\item To first order in $\hbar $ the commutator is the Poisson bracket: $
\left[ \hat{f},\hat{g}\right] =i\hbar \left\{ \hat{f},\hat{g}\right\} +
\mathcal{O}\left( \hbar ^{2}\right) $.
\item When $\mathcal{M}=\mathbb{
\mathbb{R}
}^{n}$ we get the ordinary quantum mechanics.
\end{itemize}
In the present work we take as our symplectic manifold$\mathcal{\ }T^{\ast }
\mathbb{S}^{2}$, the phase space of a single particle on the 2-sphere, $
\mathbb{S}^{2}$. For this space we construct the Fedosov observables
non-perturbatively. The advantage of choosing $\mathbb{S}^{2}$ is that we
had suspected previous to the calculation that the commutators are the same
as the usual angular momentum commutators in nonrelativistic quantum
mechanics. Saying in fact that the theory of angular momentum is the
quantization of the two-sphere without the need for it to be embedded in $
\mathbb{
\mathbb{R}
}^{3}$.
\subsection{Outline}
We will follow the basic scheme of keeping derivations sufficiently general
so as to apply to a completely general manifold with metric $\left( \mathcal{
M},g\right) $ and then state results from our specific case of the sphere.
In section 2 we introduce the phase-space connection. We introduce the basis
of covectors of matrices/operators $\hat{y}^{A}$ on the cotangent bundle in
section 3. In section 4 we attempt to motivate and solve for a new
derivation $\hat{D}$. Also we talk a bit about $\hat{D}$'s\ ambiguities.
Moving into section 5 we explicitly compute the quantities $\hat{x}$ and $
\hat{p}$. In section 6 we compute the commutators $\left[ \hat{x}^{a},\hat{x}
^{b}\right] ,~\left[ \hat{x}^{a},\hat{p}_{b}\right] $ and $\left[ \hat{p}
_{a},\hat{p}_{b}\right] $ using the explicit forms of the operators. Section
7 explains how one would construct states of angular momentum on $T^{\ast }
\mathbb{S}^{2}$ by finally introducing the standard Hamiltonian in ordinary
nonrelativistic quantum mechanics. Up until this point no Hamiltonian was
assumed.
\section{The Phase-Space Connection for $T^{\ast }\mathbb{S}^{2}$}
Before we begin, we note the use of the convention that the lower case are
the indices of $\mathcal{M}$ (these run from $1,\ldots ,n$) and capital ones
are the indices of the phase-space $T^{\ast }\mathcal{M}$ (these run from $
1,\ldots ,2n$).
We start with the phase space of a single classical particle confined to a
general manifold $\left( \mathcal{M},g\right) $. \ The objects needed are
the phase space, $T^{\ast }\mathcal{M}$ which is the cotangent bundle of $
\mathcal{M}$, an affine connection on the phase space $D$ and the symplectic
form $\omega $ of $T^{\ast }\mathcal{M}$.
A phase-space connection's action on all functions $f\left( x,p\right) \in
T^{\ast }\mathcal{M}$ and a basis of covectors $\Theta ^{A}\in T^{\ast
}T^{\ast }\mathcal{M}$ are:
\begin{equation*}
Df=df=\frac{\partial f}{\partial x^{a}}dx^{a}+\frac{\partial f}{\partial
p_{a}}dp_{a}
\end{equation*}
\begin{equation*}
D\otimes \Theta ^{A}=\Gamma _{~B}^{A}\otimes \Theta ^{B}=\Gamma
_{~BC}^{A}\Theta ^{C}\otimes \Theta ^{B}
\end{equation*}
in such a way as to preserve the symplectic form $\omega =dp_{a}\wedge
dx^{a} $ on $T^{\ast }\mathbb{S}^{2}$ ($D\otimes \omega =0$) where $D=\Theta
^{C}D_{C}$, $D_{C}\Theta ^{A}=\Gamma _{~BC}^{A}\Theta ^{B}$ and $\Gamma
_{~BC}^{A}$ is the Christoffel symbol in this basis.
Additionally we impose that $D$ be torsion-free ($D^{2}f=0$) and that it
corresponds to the Levi-Civita connection on $\mathcal{M}$ when it acts on
functions of $x$ and $dx$. Of course we extend to vectors and higher tensors
by the Leibnitz rule.
In the specific case of $\mathbb{S}^{2}$ ($T^{\ast }\mathbb{S}^{2}$) we
employ the convention that the lower/upper-case indices be of the embedding
space $\mathbb{E}^{3}$ ($T^{\ast }\mathbb{E}^{3}$) running from $1,2,3$ ($
1,\ldots ,6$) instead of $1,2$ ($1,\ldots ,4$). We note before continuing
that the calculation of the Fedosov observables is \underline{inherently}
two space-time dimensional. The third coordinate is merely for convenience.
We see this fact manifest itself by the two conditions (e.g. $\underline{x}
\cdot \underline{x}=1$ and $\underline{x}\cdot \underline{p}=0$) on the
three coordinates every step of the way.
The natural objects and quantities on $T^{\ast }\mathbb{S}^{2}$ are:
\begin{itemize}
\item The induced $\mathbb{S}^{2}$ metric $g$ by the $\mathbb{E}^{3}$
embedding metric $\delta $.
\item The induced $T^{\ast }\mathbb{S}^{2}$ symplectic form $\omega $\ by
the $T^{\ast }\mathbb{E}^{3}$ embedding symplectic form.
\item Also the equations defining $T^{\ast }\mathbb{S}^{2}$ inside of $
T^{\ast }\mathbb{E}^{3},$ $\underline{x}\cdot \underline{x}=\delta
_{ab}x^{a}x^{b}=1$ and $\underline{x}\cdot \underline{p}=x^{a}p_{a}=0$.
\item A torsion-free phase-space connection $D=\Theta ^{A}D_{A}$ on $T^{\ast
}\mathbb{S}^{2}$ that preserves all of the above conditions along with the
symplectic form $\omega $ and there subsequent derivatives. In other words:
\begin{equation*}
D^{l}\otimes g=D^{l}\otimes \omega =D^{l}\left( \delta
_{ab}x^{a}x^{b}\right) =D^{l}\left( x^{a}p_{a}\right) =0
\end{equation*}
\end{itemize}
for all positive integers $l$\ where $g=g_{ab}dx^{a}\vee dx^{b},~\omega
=\omega _{AB}\Theta ^{A}\wedge \Theta ^{B}$ , where $\Theta ^{A}$ is basis
of forms and $\vee ,\wedge $ are the symmetric, antisymmetric tensor
products respectively that we will omit because it will be clear when we
mean the one or the other.
We define a basis of covectors or forms by:
\begin{equation*}
\Theta ^{A}=\left( \theta ^{a},\alpha _{a}\right)
\end{equation*}
where the $\theta $'s are the first three $\Theta $'s and the $\alpha $'s
are the last three $\Theta $'s. $\theta $ and $\alpha $ are defined to be:
\begin{equation*}
\underline{\alpha }:=\underline{x}\times d\underline{p}
\end{equation*}
\begin{equation*}
\underline{\theta }:=\underline{x}\times d\underline{x}
\end{equation*}
The metric on $\mathbb{S}^{2}$ is:
\begin{equation*}
g=\underline{\theta }\cdot \underline{\theta }
\end{equation*}
The phase-space connection we use for $T^{\ast }\mathbb{S}^{2}$ is:
\begin{equation*}
D\underline{x}:=d\underline{x}=\underline{\theta }\times \underline{x}
\end{equation*}
\begin{equation*}
D\underline{p}:=d\underline{p}=\underline{\alpha }\times \underline{x}-
\underline{p}\times \underline{\theta }
\end{equation*}
\begin{equation}
D\otimes \underline{\theta }=\underline{\theta }\otimes _{\times }\underline{
\theta } \tag{$D\theta $}
\end{equation}
\begin{equation}
D\otimes \underline{\alpha }=\underline{\theta }\otimes _{\times }\underline{
\alpha }-\frac{2}{3}\left( \underline{\theta }\times \underline{x}\right)
\otimes \left( \underline{p}\cdot \underline{\theta }\right) +\frac{1}{3}
\left( \underline{p}\cdot \underline{\theta }\right) \otimes \left(
\underline{\theta }\times \underline{x}\right) \tag{$D\alpha $}
\end{equation}
And its corresponding curvature:
\begin{equation*}
D^{2}\underline{x}:=0
\end{equation*}
\begin{equation*}
D^{2}\underline{p}:=0
\end{equation*}
\begin{equation}
D^{2}\otimes \underline{\theta }=\tilde{\omega}\otimes \left( \underline{x}
\times \underline{\theta }\right) \tag{$D^{2}\theta $}
\end{equation}
\begin{equation}
D^{2}\otimes \underline{\alpha }=\tilde{\omega}\otimes \left( \underline{x}
\times \underline{\alpha }\right) +\frac{1}{3}\left( \underline{\alpha }
\left( \underline{\theta }\otimes _{\cdot }\underline{\theta }\right) -
\underline{\theta }\left( \underline{\alpha }\otimes _{\cdot }\underline{
\theta }\right) -2\omega \otimes \underline{\theta }\right)
\tag{$D^{2}\alpha $}
\end{equation}
\section{Introducing the $\hat{y}$'s}
Following Fedosov, we are going to introduce some machinery namely the
operators $\hat{y}$'s to calculate the observables on general manifold $
\mathcal{M}$. However, unlike Fedosov who defines these $\hat{y}$'s as
covectors equipped with a Moyal-like product between them we choose a
different starting point. We define the $\hat{y}$'s at fixed point to be a
Heisenberg algebra $\left[ \hat{y}^{A},\hat{y}^{B}\right] =i\hbar \omega
^{AB}$ where $\omega ^{AB}$ is the inverse of $\omega _{AB}$\ with $\omega
^{AB}\omega _{BC}=\delta _{C}^{A}$. More explicitly $\hat{y}$'s are huge
(infinite dimensional) matrices that act on a Hilbert space:
\begin{equation*}
\hat{y}^{A}=\left(
\begin{array}{ccc}
y_{11}^{A}\left( x,p\right) & y_{12}^{A}\left( x,p\right) & \cdots \\
y_{21}^{A}\left( x,p\right) & y_{22}^{A}\left( x,p\right) & \cdots \\
\vdots & \vdots & \ddots
\end{array}
\right)
\end{equation*}
where for each $A$, $j$, and $k$ $y_{jk}^{A}\in C^{\infty }\left( T^{\ast }
\mathcal{M}\right) $.
To make a connection with a more familiar form of the Heisenberg algebra we
use Darboux's theorem. Darboux's theorem says that in the neighborhood of
each point of $q\in T^{\ast }\mathcal{M}$ there exist $2n$ local coordinates
$\left( \tilde{x}^{1},\ldots ,\tilde{x}^{n},\tilde{p}_{1},\ldots ,\tilde{p}
_{n}\right) $\footnote{
Note that these $2n$ coordinates and are different from the $2n+2$ embedding
coordinates $\left( x^{\mu },p_{\mu }\right) $.}, called canonical or
Darboux coordinates, such that the symplectic form $\omega $ may be written
by means of these coordinates as $\omega =d\tilde{p}_{1}d\tilde{x}
^{1}+\cdots +d\tilde{p}_{n}d\tilde{x}^{n}$. Thus in this coordinate system
at $q$ the $\hat{y}$'s are expressed as $2n$\ operators $\left( \tilde{s}
^{1},\ldots ,\tilde{s}^{n},\tilde{k}_{1},\ldots ,\tilde{k}_{n}\right) $
which have the commutators $\left[ \tilde{s}^{i},\tilde{s}^{j}\right] =\left[
\tilde{k}_{i},\tilde{k}_{j}\right] =0,~\left[ \tilde{s}^{i},\tilde{k}_{j}
\right] =i\hbar \delta _{j}^{i}$ where $i$ and $j$ run from $1$ through $2n$
. And so at each point the $\hat{y}$'s establish a Heisenberg algebra which
acts on a Hilbert space.
\textbf{Important Note:} Fedosov actually begins with the $\hat{y}$'s as
being an arbitrary basis of ordinary covectors with a Moyal-like product
between themselves.$\left[ \text{\hyperlink{1}{1}}\right] $ We take the
point of view that the specific form of the product is irrelevant. All that
matters is that we have an algebra with same commutation relations and the
action of the connection is same on the $\hat{y}$'s.
\underline{Defining Properties of $\hat{y}$:}
\begin{equation*}
\left[ \hat{y}^{A},\hat{y}^{B}\right] =i\hbar \omega ^{AB}
\end{equation*}
\begin{equation*}
D\hat{y}^{A}=\Gamma _{~B}^{A}\hat{y}^{B}=\Gamma _{~BC}^{A}\Theta ^{C}\hat{y}
^{B}~~~,~~~\Theta ^{A}=\left( \theta ^{a},\alpha _{a}\right)
\end{equation*}
The $\hat{y}$'s commute with the set of quantities $\left\{
x,p,dx,dp,g,\omega ,\hbar ,i\right\} $ where $i$ is the complex unit.
\textbf{Note:} The action of the phase-space connection on $\hat{y}$ is the
same as the one on $\Theta $ ($D\otimes \Theta ^{A}=\Gamma _{~BC}^{A}\Theta
^{C}\otimes \Theta ^{B}$) and so we regard it as a basis of operator or
matrix-valued covectors.\footnote{
One may be tempted to quantize the manifold by mapping $\left(
x^{1},x^{2},x^{3},p_{1},p_{2},p_{3}\right) $ to the matrices $\left( \hat{y}
^{1},\hat{y}^{2},\hat{y}^{3},\hat{y}^{4},\hat{y}^{5},\hat{y}^{6}\right) $,
but we want a coordinate independent formalism and, in general, this is not
coordinate independent.} This tells us how to parallel transport the
Heisenberg algebra (the $\hat{y}$'s) at one point to the Heisenberg algebra
of every other point in a consistent way.
\underline{Introducing terminology:}
In this paper when we say $f$ is a function/form we define it to be a
complex Taylor series in its variables\footnote{
The set of all of these type of functions is sometimes called the enveloping
algebra of its arguments.}. Explicitly:
\begin{equation}
f\left( u,\ldots ,v\right) =\sum_{l,j\text{'s}}f_{j_{1}\cdots
j_{l}}u^{j_{1}}\cdots v^{j_{l}}\text{ \ \ (}j\text{'s are powers not indices)
} \notag
\end{equation}
where $f_{j_{1}\cdots j_{l}}$\ are constants while $u$ and $v$ could be any
of the set $\left\{ x,p,dx,dp,\omega ,\hbar ,i\right\} $.
So if $f$ is a function/form of some subset or all of the quantities $
x,p,dx,dp,\omega ,\hbar $ and $i$ it then commutes with the $\hat{y}$'s and
will be called a complex-valued function/form. On the contrary a
matrix-valued function/form is a complex Taylor series in $\hat{y}$ and
possibly some subset or all of the quantities $x,p,dx,dp,\omega ,\hbar $ and
$i$.
So if $f\left( x,p,dx,dp,\omega ,\hbar ,i\right) $ is a complex-valued
function/form it then commutes with the $\hat{y}$'s. More explicitly with
the matrix indices written:
\begin{equation*}
\left( \hat{y}^{A}\hat{y}^{B}\right) _{jk}=\Sigma _{l}\hat{y}_{jl}^{A}\hat{y}
_{lk}^{B}
\end{equation*}
\begin{equation*}
\left( \left[ \hat{y}^{A},f\right] \right) _{jk}:=\hat{y}_{jk}^{A}f-f\hat{y}
_{jk}^{A}=0
\end{equation*}
On the contrary a matrix-valued function/form does not. From now on we will
not write the matrix indices explicitly.
\underline{\textbf{The End Goal:}}
The idea for Fedosov's introduction of the $\hat{y}$'s is to associate to
each $f\left( x,p\right) \in C^{\infty }\left( T^{\ast }\mathcal{M}\right) $
a unique observable $\hat{f}\left( x,p,\hat{y}\right) $:
\begin{equation}
\hat{f}\left( x,p,\hat{y}\right) =\sum_{l}f_{A_{1}\cdots A_{l}}\hat{y}
^{A_{1}}\cdots \hat{y}^{A_{l}} \tag{$\hat{f}$} \label{fhat}
\end{equation}
where $f_{A_{1}\cdots A_{l}}$ are some unknown functions of $x$ and $p$\ to
be determined.
\textbf{Important Note:} Most of the rest of the sections will be dedicated
to finding a solution for $\hat{f}$ (i.e. the coefficients functions $
f_{A_{1}\cdots A_{l}}$) for each $f\left( x,p\right) \in C^{\infty }\left(
T^{\ast }\mathbb{S}^{2}\right) $ up to some "reasonable" ambiguity
(discussed in sections 4.1 and 5).
\subsection{$T^{\ast }\mathbb{S}^{2}$ Explicitly}
Specifically for $T^{\ast }\mathbb{S}^{2}$ we have the induced symplectic
form $\omega $ of $T^{\ast }
\mathbb{R}
^{3}$ onto $T^{\ast }\mathbb{S}^{2}$ being:
\begin{equation*}
\omega =\underline{\alpha }\cdot \underline{\theta }=\left( \delta
_{b}^{a}-x^{a}x_{b}\right) \alpha _{a}\theta ^{b}
\end{equation*}
We make the convention\footnote{
Note that the indices go from $1$ to $2n+2$ and are different from the $2n$
operators defined above by $\left( \tilde{s}^{1},\ldots ,\tilde{s}^{n},
\tilde{k}_{1},\ldots ,\tilde{k}_{n}\right) $. The difference between them is
the same as the difference between the embedding coordinates $\left(
x^{1},\ldots ,x^{n+1},p_{1},\ldots ,p_{n+1}\right) $ and $\left( \tilde{x}
^{1},\ldots ,\tilde{x}^{n},\tilde{p}_{1},\ldots ,\tilde{p}_{n}\right) $.}:
\begin{equation*}
\hat{y}^{A}=\left( s^{a},k_{a}\right)
\end{equation*}
where the $s$'s are the first three $\hat{y}$'s and the $k$'s are the last
three $\hat{y}$'s. Using the above formulas we then write the commutation
relations:
\begin{equation*}
\left[ s^{a},s^{b}\right] =0=\left[ k_{a},k_{b}\right] \ ,\ \left[
s^{a},k_{b}\right] =i\hbar \left( \delta _{b}^{a}-x^{a}x_{b}\right)
\end{equation*}
We may assume w.l.o.g. that $\underline{x}\cdot \underline{s}=\underline{x}
\cdot \underline{k}=0$ because we observe that the only part of $s$ and $k$
that affect the commutators are the parts that are perpendicular to $x$. The
irrelevance of the part of $s$ and $k$ parallel to $x$ stems from the above
relations because $\left[ x_{a}s^{a},k_{b}\right] =0$ and $\left[
s^{a},k_{b}x^{b}\right] =0$ and so we could always subtract off the part of $
s$ and $k$ parallel to $x$ and get the same commutators. Since $\underline{x}
\cdot \underline{s}=\underline{x}\cdot \underline{k}=0$ we have four
independent operators which is required since (one for each direction on $
T^{\ast }\mathbb{S}^{2}$).
The action of the connection and curvature acting on $\underline{s}$ \& $
\underline{k}$ is written down directly from the equations $\left( D\theta
\right) ,~\left( D\alpha \right) ,~\left( D^{2}\theta \right) ,$ and $\left(
D^{2}\alpha \right) $:
\begin{equation*}
D\underline{s}=\underline{\theta }\times \underline{s}
\end{equation*}
\begin{equation*}
D\underline{k}=\underline{\theta }\times \underline{k}-\frac{2}{3}\underline{
\theta }\times \underline{x}\left( \underline{p}\cdot \underline{s}\right) +
\frac{1}{3}\left( \underline{p}\cdot \underline{\theta }\right) \left(
\underline{s}\times \underline{x}\right)
\end{equation*}
\begin{equation*}
D^{2}\underline{s}=\tilde{\omega}\left( \underline{x}\times \underline{s}
\right)
\end{equation*}
\begin{equation*}
D^{2}\underline{k}=\tilde{\omega}\left( \underline{x}\times \underline{k}
\right) +\frac{1}{3}\left( \underline{\alpha }\left( \underline{s}\cdot
\underline{\theta }\right) +\left( \underline{s}\cdot \underline{\alpha }
\right) \underline{\theta }-2\omega \underline{s}\right)
\end{equation*}
\section{Constructing the global derivation $\hat{D}$}
Following Fedosov, we now introduce a global derivation as a matrix
commutator $\hat{D}=\left[ \hat{Q},\cdot \right] $ which is central to
constructing the coefficients $f_{A_{1}\cdots A_{l}}$ in equation $\left(
\text{\hyperref[fhat]{$\hat{f}$}}\right) $\ for each $f\left( x,p\right) \in
C^{\infty }\left( T^{\ast }\mathcal{M}\right) $. One possible physical
motivation for $\hat{D}$ is that in the next section we will require that
all observables $\hat{f}$ must satisfy the equation $\left( D-\hat{D}\right)
\hat{f}\left( x,p,\hat{y}\right) =0$. We see that on $\hat{f}$ $\hat{D}$ is
an infinitesimal translation matrix operator equivalent to $D$. We then
reason that matrix operators corresponding to infinitesimal translations on
the cotangent bundle should exist i.e. $\hat{D}$. The reason that we require
that they must exist is because we are constructing the set of \textit{all}
physical matrix operators on states and certainly infinitesimal translations
are in this set. If this reasoning is correct then the equation $\left( D-
\hat{D}\right) \hat{f}=0$ must be satisfied for all observables $\hat{f}$.
Also the case of $T^{\ast }
\mathbb{R}
^{n}$ may provide some insight since it is the overlap of this formalism and
quantum mechanics using the Moyal $\ast $ (see in \hyperlink{D}{Appendix D}
for the example of $T^{\ast }
\mathbb{R}
^{n}$).
Define the derivation $\hat{D}$ by the graded commutator\footnote{
Graded commutators have the property that $\left[ \hat{Q}_{A}\Theta ^{A},w
\right] =\left[ \hat{Q}_{A},w\right] \Theta ^{A}=\left( \hat{Q}_{A}w-w\hat{Q}
_{A}\right) \Theta ^{A}$ where $w$ is an $l$-form with coefficients $
w_{A_{1}\cdots A_{l}}$ which are complex-valued functions of the variables $
x,p$ and $\hat{y}$.}:
\begin{equation}
\hat{D}=\left[ \hat{Q},\cdot \right] =\left[ \hat{Q}_{A}\Theta ^{A},\cdot
\right] \tag{$\hat{D}$}
\end{equation}
\begin{equation*}
\hat{Q}_{A}=\sum_{l}Q_{AA_{1}\cdots A_{l}}\hat{y}^{A_{1}}\cdots \hat{y}
^{A_{l}}
\end{equation*}
where $\Theta ^{A}=\left( \theta ^{a},\alpha _{a}\right) $ and $
Q_{AA_{1}\cdots A_{l}}$\ are complex-valued functions of $x$ and $p$ that
need to be determined. We reiterate that complex-valued functions are\ not
matrices hence they commute with the $\hat{y}$'s.
Again following Fedosov, we can partially determine the functions $
Q_{AA_{1}\cdots A_{l}}$ by the mysterious equation\footnote{
Fedosov adds an additional condition that makes his $\hat{D}$ unique from a
fixed $D$ being $\hat{d}^{-1}r_{0}=0$ where $\hat{d}^{-1}$ is what he calls $
\delta ^{-1}$ (an operator used in a de Rham decomposition) and $r_{0}$ is
the first term in the recursive solution. We regard this choice as being
artificial and thus omit it from the paper.}:
\begin{equation}
\left( D-\hat{D}\right) ^{2}\hat{y}^{A}=0 \tag{cond $\hat{D}$}
\label{cond Dhat}
\end{equation}
The physical motivation for this equation is still unclear and may lurk in
the work of Fedosov. One reason for the above requirement is that in the
next section we want to solve the equation $\left( D-\hat{D}\right) \hat{f}
=0 $ for $\hat{f}$ and the above is an integrability condition for the
solvability of this equation.
We now let $\hat{Q}$ be the sum of 2 parts the first being the solution in
the case of $T^{\ast }
\mathbb{R}
^{n}$ (Christoffels$=\Gamma =0$):
\begin{equation}
\hat{Q}_{A}\Theta ^{A}=\omega _{AB}\hat{y}^{A}\Theta ^{B}+r \tag{$\hat{Q}$}
\label{Qhat}
\end{equation}
where:
\begin{equation*}
r=\sum_{l}r_{AA_{1}\cdots A_{l}}\Theta ^{A}\hat{y}^{A_{1}}\cdots \hat{y}
^{A_{l}}
\end{equation*}
and $r_{AA_{1}\cdots A_{l}}$ are complex-valued functions of $x$ and $p$
that need to be determined. In general, we assume that $r$ has terms that
are cubic or higher powers in the $\hat{y}$'s (see \hyperlink{B}{Appendix B}
and Fedosov $\left[ \text{\hyperlink{1}{1}}\right] $ for clarification).
We rewrite the condition $\left( \text{\hyperref[cond Dhat]{cond $\hat{D}$}}
\right) $ as:
\begin{equation*}
\left( D-\hat{D}\right) ^{2}\hat{y}^{A}=\left[ \Omega -Dr+\hat{d}r+r^{2},
\hat{y}^{A}\right] =0
\end{equation*}
where $\Omega :=\frac{1}{2i\hbar }\omega _{FN}R_{~BCE}^{F}\Theta ^{C}\Theta
^{E}\hat{y}^{N}\hat{y}^{B}$ is the phase-space curvature ($D^{2}\otimes
\Theta ^{A}=R_{~BCE}^{A}\Theta ^{C}\Theta ^{E}\otimes \Theta ^{B}$) as a
commutator and $\hat{d}h=\frac{1}{i\hbar }\left[ \omega _{AB}\hat{y}
^{A}\Theta ^{B},h\right] $ where $h$ is a matrix-valued function of $
x,~p,~dx,~dp$ and $\hat{y}$ (see \hyperlink{A}{Appendix A} for the proof).
From now on we let:
\begin{equation}
\Omega -Dr+\hat{d}r+r^{2}=0 \tag{$r$} \label{r}
\end{equation}
and keep it in the back of our minds that we could add something that
commutes with all $\hat{y}$'s to $\Omega -Dr+\hat{d}r+r^{2}$.
\textbf{Important:} To emphasize the importance of this equation the reader
should note that the whole Fedosov $\ast $-formalism hinges on this $r$
existing. We know solutions exists perturbatively in general (Fedosov has
the recursive solution for it $\left[ \text{\hyperlink{1}{1}}\right] \left[
p.144\right] $), however convergence issues still remain unresolved. On a
technical note we have found that solving for $r$ to be the hardest point of
the computation of the Fedosov observables because of the need for the right
ansatz and the nonlinear equation $\left( \hyperref[r]{r}\right) $ above
that it must solve.
Specifically for the case of $T^{\ast }\mathbb{S}^{2}$\ the solution for the
curvature as a commutator $\Omega $ is:
\begin{equation*}
\Omega :=\frac{1}{3}\left( \left( \underline{s}\cdot \underline{\alpha }
\right) \left( \underline{s}\cdot \underline{\theta }\right) -s^{2}\omega
\right) +\left( \underline{x}\times \underline{k}\right) \cdot \underline{s}
\tilde{\omega}
\end{equation*}
We then verify that it gives the curvature as commutators:
\begin{equation*}
\left[ \Omega ,\underline{s}\right] =\left[ -\underline{k}\cdot \left(
\underline{x}\times \underline{s}\right) \tilde{\omega},\underline{s}\right]
=\tilde{\omega}\left( \underline{x}\times \underline{s}\right)
\end{equation*}
\begin{equation*}
\left[ \Omega ,\underline{k}\right] =\frac{1}{3}\left( \underline{\alpha }
\left( \underline{s}\cdot \underline{\theta }\right) +\left( \underline{s}
\cdot \underline{\alpha }\right) \underline{\theta }-2\omega \underline{s}
\right) +\left( \underline{x}\times \underline{k}\right) \tilde{\omega}
\end{equation*}
To simplify the calculations we set $i\hbar =1$ which we will eventually put
back in the end.
Fedosov at this point would implement an algorithm to construct $r$
perturbatively, however rather than do this we will make an ansatz for $r$
by exploiting the rotational symmetry of the sphere. This will give us an
exact solution for $r$.\footnote{
On a technical note: we ran the Fedosov algorithm a few times to help us see
what form the ansatz should take. Also remember that when we require $\Omega
-Dr+\hat{d}r+r^{2}=0$ modulo terms that commute with the $\hat{y}$'s.}
Our ansatz for $r$ is:
\begin{equation}
r=r_{0}+f\left( s^{2}\right) \underline{z}\cdot \underline{s}\left(
\underline{x}\times \underline{s}\right) \cdot \underline{\theta }+g\left(
s^{2}\right) \underline{z}\cdot \left( \underline{x}\times \underline{s}
\right) \underline{s}\cdot \underline{\theta }+h\left( s^{2}\right)
\underline{s}\cdot \underline{\theta } \tag{r ansatz} \label{r ansatz}
\end{equation}
where $\underline{z}=\underline{p}-\underline{x}\times \underline{k}$ and $
r_{0}=\frac{1}{3}\left( \left( \underline{k}\cdot \underline{\theta }\right)
s^{2}-\underline{k}\cdot \underline{s}\left( \underline{s}\cdot \underline{
\theta }\right) \right) $.
We will now state the results of our calculations because the calculations
are just too space consuming and yet at the same time straight forward.\
Given the formulas for $r$ and $\Omega $ and performing lengthy calculations
eventually we get:
\begin{equation*}
Dr=\left( \frac{1}{9}-\frac{2g}{3}+\frac{f}{3}\right) s^{2}\underline{p}
\cdot \underline{s}\tilde{\omega}+f\underline{\alpha }\cdot \left(
\underline{x}\times \underline{s}\right) \left( \underline{x}\times
\underline{s}\right) \cdot \underline{\theta }-g\left( \underline{s}\cdot
\underline{\alpha }\right) \underline{s}\cdot \underline{\theta }
\end{equation*}
\begin{equation*}
\hat{d}r=-\Omega +\left( 2f^{\prime }s^{2}+3f+g\right) \underline{z}\cdot
\underline{s}\tilde{\omega}-g\left( \underline{s}\cdot \underline{\alpha }
\right) \underline{s}\cdot \underline{\theta }+f\underline{\alpha }\cdot
\left( \underline{x}\times \underline{s}\right) \left( \underline{x}\times
\underline{s}\right) \cdot \underline{\theta }
\end{equation*}
\begin{equation*}
r^{2}=\left( \frac{1}{9}-\frac{2g}{3}+\frac{f}{3}\right) s^{2}\underline{p}
\cdot \underline{s}\tilde{\omega}+\left( 2gf^{\prime }s^{2}+gf-f^{2}-\frac{2f
}{3}+\frac{g}{3}-\frac{1}{9}\right) s^{2}\underline{z}\cdot \underline{s}
\tilde{\omega}
\end{equation*}
$\allowbreak $where $f^{\prime }=\frac{\partial f}{\partial \left(
s^{2}\right) }$ for all functions.
Putting these into the equation $\left( \hyperref[r]{r}\right) $ we obtain a
condition for $g$:
\begin{equation*}
g=\frac{s^{2}\left( \left( f+\frac{1}{3}\right) ^{2}-2f^{\prime }\right)
\allowbreak -3f}{s^{2}\left( \left( f+\frac{1}{3}\right) +2s^{2}f^{\prime
}\right) +1}
\end{equation*}
while $f$ and $h$ are left arbitrary as long as $g$ is well-defined. This is
a necessary and sufficient condition for the equation $\left( \hyperref[r]{r}
\right) $\ to hold.
We note that $f=-\frac{1}{3},g=1$ and $f=-\frac{1}{12},g=\frac{1}{4}$ are
the only solutions where $f$ and $g$ are constant. We will choose to work
with the $f=-\frac{1}{3},~g=1,~h=0$ solution from now on. We choose this
solution for the sake of clarity because it turns out to be the easiest to
use in the next few sections. However the reader should note that we
calculated the commutators for the general solutions for $g,~f$ and $h$ and
obtained the same result for all of them. See section 6 for the exact result
of the commutators for the particular solution $f=-\frac{1}{3},~g=1,~h=0$
(and hence the solution for the general solutions for $g,~f$ and $h$).
The solution for $r$ for $f=-\frac{1}{3},~g=1,~h=0$ is:
\begin{equation}
r=-\frac{1}{3}\left( \underline{p}\cdot \underline{s}\right) \left( \left(
\underline{x}\times \underline{s}\right) \cdot \underline{\theta }\right) +
\underline{z}\cdot \left( \underline{x}\times \underline{s}\right)
\underline{s}\cdot \underline{\theta } \tag{r soln} \label{r soln}
\end{equation}
\subsection{Ambiguities in $r$}
It is worthwhile to note that the condition $\left( \text{\text{\hyperref[cond Dhat
]{cond $\hat{D}$}}}\right) $ does not uniquely define $\hat{D}$ given a
fixed $D$.\footnote{
Fedosov adds an additional condition that makes his $\hat{D}$ unique from a
fixed $D$ being $\hat{d}^{-1}r_{0}=0$ where $\hat{d}^{-1}$ is what he calls $
\delta ^{-1}$ (an operator used in a de Rham decomposition) and $r_{0}$ is
the first term in the recursive solution. We regard this choice as being
artificial and thus omit it from the paper.} It appears however that the
most of the ambiguities in constructing $\hat{D}$ when given a fixed phase
space connection $D$ can be absorbed by a basis change (in other words a
gauge transformation). It is easy to see this in a Darboux chart because the
connection may be expressed as a commutator:
\begin{equation*}
\tilde{D}\hat{y}^{A}=\left[ \tilde{Q},\hat{y}^{A}\right]
\end{equation*}
where $\tilde{D}=D-\hat{D}$, $\tilde{Q}=Q-\hat{Q}$ and $D=\left[ Q,\cdot
\right] $. The gauge transformation takes the form:
\begin{equation*}
\hat{y}^{A}\rightarrow \hat{y}_{new}^{A}:=U\hat{y}^{A}U^{-1}\text{ \ \ , \ \
}\tilde{D}\hat{y}^{A}\rightarrow \tilde{D}_{new}\hat{y}_{new}^{A}:=\left[ U
\tilde{Q}U^{-1},U\hat{y}^{A}U^{-1}\right] =U\left( \tilde{D}\hat{y}
^{A}\right) U^{-1}
\end{equation*}
where $U$ is some invertible function of the $x$'s, $p$'s and $\hat{y}$'s.
Thus the physical content of this theory is independent of $U$ because the
commutators remain unchanged.
This can be seen as follows:
\begin{equation*}
r\rightarrow r+r^{\prime }
\end{equation*}
where $r$ is a solution to the equation $\left( \hyperref[r]{r}\right) $ and
$r^{\prime }$ is some unknown series:
\begin{equation*}
r^{\prime }=\sum_{l}r_{AA_{1}\cdots A_{l}}^{\prime }\Theta ^{A}\hat{y}
^{A_{1}}\cdots \hat{y}^{A_{l}}
\end{equation*}
Putting $r\rightarrow r+r^{\prime }$ into $\left( \hyperref[r]{r}\right) $
we obtain:
\begin{equation*}
\Omega -D\left( r+r^{\prime }\right) +\left[ \underline{s}\cdot \underline{
\alpha }-\underline{k}\cdot \underline{\theta },\left( r+r^{\prime }\right)
\right] +\left( r+r^{\prime }\right) ^{2}=0
\end{equation*}
modulo the equation $\left( \hyperref[r]{r}\right) $\ to get:
\begin{equation*}
-Dr^{\prime }+\left[ \underline{s}\cdot \underline{\alpha }-\underline{k}
\cdot \underline{\theta },r^{\prime }\right] +\left( r^{\prime }\right) ^{2}+
\left[ r,r^{\prime }\right] =0
\end{equation*}
\begin{equation*}
\implies \tilde{D}r^{\prime }-\left( r^{\prime }\right) ^{2}=0
\end{equation*}
This tells us that if $r^{\prime }$ is of the form:
\begin{equation*}
r^{\prime }=\left( \tilde{D}U\right) U^{-1}
\end{equation*}
for any $U$ which corresponds to a gauge transformation in the enveloping
algebra then the resulting $r_{new}=r+r^{\prime }$ will solve equation $
\left( \hyperref[r]{r}\right) $. \ In other words once we have one solution
we have actually have huge class of equivalent solutions. We suspect this
class of equivalent solutions are all of the solutions for a
simply-connected manifold.
\textbf{Note:} There is another source of ambiguity namely the ambiguity in
the phase-space connection $D$.\ Given a connection $D$ we may add to it a
tensor $\Delta _{\text{ }BC}^{A}$ where if we lower by $\Delta _{ABC}=\omega
_{AE}\Delta _{\text{ \ }BC}^{E}$ it is symmetric in all three indices. The
new connection still preserves the symplectic form $\omega $. Our curvature
becomes:
\begin{equation*}
\left( D+\Delta \right) ^{2}=D^{2}+D\left( \Delta \right) +\Delta ^{2}
\end{equation*}
It is unclear what this ambiguity means so we will leave it for a future
discussion.
\section{Computing $\hat{x}$ and $\hat{p}$}
At this point in Fedosov's algorithm we have all the tools in place to
associate an observable $\hat{f}$ to every $f\in C^{\infty }\left( T^{\ast }
\mathcal{M}\right) $. Following Fedosov we require that every observable $
\hat{f}\left( x,p,\hat{y}\right) $ must satisfy the equation:
\begin{equation}
\left( D-\hat{D}\right) \hat{f}\left( x,p,\hat{y}\right) =0 \notag
\end{equation}
where $f_{A_{1}\cdots A_{l}}$ are some unknown functions of $x$ and $p$\
such that:
\begin{equation*}
\ell o\left( \hat{f}\left( x,p,\hat{y}\right) \right) =f\left( x,p\right)
\end{equation*}
$\ell o$ (short for leading order in $\hat{y}$ and $\hbar $) picks out the
term which has no $\hat{y}$'s and no $\hbar $'s in it. Explicitly:
\begin{equation}
\hat{f}\left( x,p,\hat{y}\right) =f\left( x,p\right) +\mathcal{O}\left( \hat{
y},\hbar \right) \notag
\end{equation}
where $f$ has no $\hbar $'s in it.
And so the condition to solve (we believe unique up to unitary
transformations) for an observable $\hat{f}$ for every $f\in C^{\infty
}\left( T^{\ast }\mathcal{M}\right) $ is:
\begin{equation}
\left( D-\hat{D}\right) \hat{f}\left( x,p,\hat{y}\right) =0~~~,~~~\ell
o\left( \hat{f}\left( x,p,\hat{y}\right) \right) =f\left( x,p\right)
\tag{cond $\hat{f}$} \label{cond fhat}
\end{equation}
If we have determined our $D$ and $\hat{D}$ we can find solutions for the
operators $\hat{x}^{a}$ and $\hat{p}_{a}$ (i.e. their coefficients $
b_{A_{1}\cdots A_{l}}^{a}\,$\ and $c_{aA_{1}\cdots A_{l}}$):
\begin{equation}
\hat{x}^{a}=\sum_{l}b_{A_{1}\cdots A_{l}}^{a}\hat{y}^{A_{1}}\cdots \hat{y}
^{A_{l}} \tag{$\hat{x}$} \label{xhat}
\end{equation}
\begin{equation}
\hat{p}_{a}=\sum_{l}c_{aA_{1}\cdots A_{l}}\hat{y}^{A_{1}}\cdots \hat{y}
^{A_{l}} \tag{$\hat{p}$} \label{phat}
\end{equation}
where $b_{A_{1}\cdots A_{l}}^{a}\,$\ and $c_{aA_{1}\cdots A_{l}}$ are
complex-valued functions of $x$ and $p$ (which are the coefficients $
f_{A_{1}\cdots A_{l}}$ in equation $\left( \text{\hyperref[fhat]{$\hat{f}$}}
\right) $ where the first terms in the series is $f=b^{a}=x^{a}$ or $
f=c_{a}=p_{a}\,$\ respectively) and will be determined by the equations:
\begin{equation}
\left( D-\hat{D}\right) \hat{x}^{a}=0~~~~,~~\ ~\ell o\left( \hat{x}
^{a}\right) =x^{a} \tag{cond $\hat{x}$} \label{cond xhat}
\end{equation}
\begin{equation}
\left( D-\hat{D}\right) \hat{p}_{a}=0~~~,~~~\ell o\left( \hat{p}_{a}\right)
=p_{a} \tag{cond $\hat{p}$} \label{cond phat}
\end{equation}
Again see the example in \hyperlink{D}{Appendix D} for solutions to $\hat{x}$
and $\hat{p}$ in the case of $T^{\ast }
\mathbb{R}
^{n}$ where $D=d$.
If we invert the equations $\left( \text{\hyperref[xhat]{$\hat{x}$}}\right) $
and $\left( \text{\hyperref[phat]{$\hat{p}$}}\right) $ once we have solved
for the coefficients $b_{A_{1}\cdots A_{l}}^{a}$ and $c_{A_{1}\cdots
A_{l}}^{a}$ to get $\hat{y}$ as matrix-valued function of $x,p,\hat{x}$ and $
\hat{p}$ (i.e. $\hat{y}^{A}=\hat{y}^{A}\left( x,p,\hat{x},\hat{p}\right) $)
and then substitute it into the equation for an arbitrary observable $\left(
\text{\hyperref[fhat]{$\hat{f}$}}\right) $ and get:
\begin{equation}
\hat{f}\left( \hat{x},\hat{p}\right) =\sum_{lm}f_{a_{1}\cdots
a_{l}}^{b_{1}\cdots b_{m}}\hat{x}^{a_{1}}\cdots \hat{x}^{a_{l}}\hat{p}
_{b_{1}}\cdots \hat{p}_{b_{m}} \tag{$\hat{f}$ soln} \label{fhat soln}
\end{equation}
where $f_{a_{1}\cdots a_{l}}^{b_{1}\cdots b_{m}}$ are constant coefficients.
\footnote{
To prove this act $D-\hat{D}$ on this equation.}
However, once have our $\hat{x}$ and $\hat{p}$ there is the ambiguity of how
to order each variable when you map a function $f\left( x,p\right) $ to $
\hat{f}\left( \hat{x},\hat{p}\right) $. For example does the function $
f\left( x,p\right) =x^{1}p_{1}$ go to $\hat{x}^{1}\hat{p}_{1}$, $\hat{p}_{1}
\hat{x}^{1}$ or some linear combination of the two? We should expect this in
any well defined quantization procedure because such ordering ambiguities
arise in quantum mechanics. We will, for now, regard the ordering of each $
\hat{f}$ to be undetermined.\footnote{
Fedosov chooses Weyl ordering.}
\subsection{$T^{\ast }\mathbb{S}^{2}$ Explicitly}
Fedosov at this point would implement an algorithm to construct $\hat{x}$
and $\hat{p}$ perturbatively$\left[ \text{\hyperlink{1}{1}}\right] \left[
p.146\right] $ for our specific case of $T^{\ast }\mathbb{S}^{2}$. We
instead try to find exact solutions to them.\footnote{
We, again, ran the Fedosov algorithm a few times to help us see what for the
ansatz should take.} Specifically for the case of $T^{\ast }\mathbb{S}^{2}$
we have the ansatz for both $\hat{x}$ and $\hat{p}$ as:
\begin{equation*}
\underline{\hat{x}}=v\left( s^{2}\right) \underline{x}+w\left( s^{2}\right)
\underline{x}\times \underline{s}+y\left( s^{2}\right) \underline{s}
\end{equation*}
\begin{equation*}
\underline{\hat{p}}=\left( \underline{z}\cdot \underline{s}t\left(
s^{2}\right) +\underline{z}\cdot \left( \underline{x}\times \underline{s}
\right) q\left( s^{2}\right) \right) \underline{x}+\underline{z}n\left(
s^{2}\right) +\underline{z}\times \underline{x}u\left( s^{2}\right)
\end{equation*}
with some functions $v,~w,~y,~t,~q,~n$ and $u$ to be determined and the
requirements that $\ell o\left( \underline{\hat{x}}\right) =\underline{x}$
and $\ell o\left( \underline{\hat{p}}\right) =\underline{p}$.
The conditions $\left( \text{\hyperref[cond xhat]{cond $\hat{x}$}}\right) $
and $\left( \text{\hyperref[cond phat]{cond $\hat{p}$}}\right) $\ become the
following equations:
\begin{eqnarray*}
0 &=&\left( D-\hat{D}\right) \underline{\hat{x}}=\left( \left( -2v^{\prime
}\left( s^{2}+1\right) +w\right) \left( \underline{s}\cdot \underline{\theta
}\right) -y\left( \underline{x}\times \underline{s}\right) \cdot \underline{
\theta }\right) \underline{x} \\
&&+\left( \left( -\frac{v}{s^{2}}-2w^{\prime }\left( s^{2}+1\right) -w\left(
1+\frac{1}{s^{2}}\right) \right) \left( \underline{s}\cdot \underline{\theta
}\right) -y\frac{1}{s^{2}}\left( \underline{x}\times \underline{s}\right)
\cdot \underline{\theta }\right) \underline{x}\times \underline{s} \\
&&+\left( \left( \frac{v}{s^{2}}+w\frac{1}{s^{2}}\right) \left( \underline{x}
\times \underline{s}\right) \cdot \underline{\theta }+\left( -2y^{\prime
}\left( s^{2}+1\right) -y\left( 1+\frac{1}{s^{2}}\right) \right) \left(
\underline{s}\cdot \underline{\theta }\right) \right) \underline{s}
\end{eqnarray*}
and:
\begin{eqnarray*}
0 &=&\left( D-\hat{D}\right) \underline{\hat{p}}=\left(
\begin{array}{c}
\left(
\begin{array}{c}
-2\underline{z}\cdot \underline{s}t^{\prime }\left( s^{2}+1\right) -\left(
\underline{z}\cdot \underline{s}\right) \frac{1}{s^{2}}t-2\underline{z}\cdot
\left( \underline{x}\times \underline{s}\right) q^{\prime }\left(
s^{2}+1\right) \\
+\underline{z}\cdot \left( \underline{x}\times \underline{s}\right) \left( 1-
\frac{1}{s^{2}}\right) q-\left( \underline{z}\cdot \underline{s}\right)
\frac{1}{s^{2}}u+\underline{z}\cdot \left( \underline{x}\times \underline{s}
\right) \frac{1}{s^{2}}n
\end{array}
\right) \left( \underline{s}\cdot \underline{\theta }\right) \\
\left(
\begin{array}{c}
-\left( \underline{z}\cdot \left( \underline{x}\times \underline{s}\right)
\left( 1+\frac{1}{s^{2}}\right) \right) t+\left( \underline{z}\cdot
\underline{s}\right) \frac{1}{s^{2}}q \\
+\underline{x}\cdot \left( \underline{z}\times \underline{s}\right) \frac{1}{
s^{2}}u-\left( \underline{z}\cdot \underline{s}\right) \frac{1}{s^{2}}n
\end{array}
\right) \left( \underline{x}\times \underline{s}\right) \cdot \underline{
\theta }
\end{array}
\right) \underline{x} \\
&&+\left(
\begin{array}{c}
\left(
\begin{array}{c}
-\underline{z}\cdot \underline{s}t-\underline{z}\cdot \left( \underline{x}
\times \underline{s}\right) q+2\underline{z}\cdot \left( \underline{x}\times
\underline{s}\right) n \\
-\left( \underline{z}\cdot \underline{s}\right) u+2\left( \underline{z}\cdot
\underline{s}\right) \left( s^{2}+1\right) u^{\prime }-2\left( \underline{z}
\times \underline{x}\right) \cdot \underline{s}\left( s^{2}+1\right)
n^{\prime }
\end{array}
\right) \left( \underline{s}\cdot \underline{\theta }\right) \\
+\underline{z}\cdot \left( \underline{x}\times \underline{s}\right) \left(
\underline{x}\times \underline{s}\right) \cdot \underline{\theta }u
\end{array}
\right) \frac{1}{s^{2}}\underline{x}\times \underline{s} \\
&&+\left(
\begin{array}{c}
\left(
\begin{array}{c}
2\underline{z}\cdot \left( \underline{x}\times \underline{s}\right) u+\left(
\underline{z}\cdot \underline{s}\right) n \\
-2\left( \underline{z}\cdot \underline{s}\right) \left( s^{2}+1\right)
n^{\prime }-2\left( \left( \underline{z}\times \underline{x}\right) \cdot
\underline{s}\right) \left( s^{2}+1\right) u^{\prime }
\end{array}
\right) \underline{s}\cdot \underline{\theta } \\
+\left( \underline{z}\cdot \underline{s}t+\underline{z}\cdot \left(
\underline{x}\times \underline{s}\right) q-\underline{z}\cdot \left(
\underline{x}\times \underline{s}\right) n\right) \left( \underline{x}\times
\underline{s}\right) \cdot \underline{\theta }
\end{array}
\right) \frac{1}{s^{2}}\underline{s}
\end{eqnarray*}
So the conditions that $\tilde{D}\underline{\hat{x}}=0$ and $\tilde{D}
\underline{\hat{p}}=0$ becomes 6+6 equations because $\left( \underline{s}
\cdot \underline{\theta }\right) ^{2}=0=\left( \left( \underline{x}\times
\underline{s}\right) \cdot \underline{\theta }\right) ^{2}$ and $\left(
\underline{s}\cdot \underline{\theta }\right) \left( \underline{x}\times
\underline{s}\right) \cdot \underline{\theta }=\tilde{\omega}$ where $\tilde{
\omega}_{ab}$ is invertible. We then solve the subsequent differential
equations for the functions $v,~w,~y,~t,~q,~n$ and $u$ along with requiring
that they have the correct term with no $\hat{y}$'s ($\ell o\left(
\underline{\hat{x}}\right) =\underline{x}$ and $\ell o\left( \underline{\hat{
p}}\right) =\underline{p}$) in the Taylor expansion to obtain the solutions:
\begin{equation}
\underline{\hat{x}}=\left( \underline{x}-\underline{x}\times \underline{s}
\right) \left( s^{2}+1\right) ^{-\frac{1}{2}} \tag{$\hat{x}$ soln}
\end{equation}
\begin{equation}
\underline{\hat{p}}=\left( \underline{z}\cdot \left( \underline{x}\times
\underline{s}\right) \underline{x}+\underline{z}\right) \left(
s^{2}+1\right) ^{\frac{1}{2}} \tag{$\hat{p}$ soln} \label{phat soln}
\end{equation}
where $\underline{z}=\underline{p}-\underline{x}\times \underline{k}$ with
the following conditions holding:
\begin{equation*}
\ell o\left( \underline{\hat{x}}\right) =\underline{x}~~~,~~~\ell o\left(
\underline{\hat{p}}\right) =\underline{p}
\end{equation*}
\begin{equation}
\underline{\hat{p}}\cdot \underline{\hat{x}}=\underline{\hat{x}}\cdot
\underline{\hat{p}}-2i\hbar =0 \tag{$\hat{x}\hat{p}$ conds}
\end{equation}
We note at this point that there is not much insight looking at these
formulas except for what we get for the commutators in the next
section.\pagebreak
\section{The Commutators $\left[ \hat{x}^{a},\hat{x}^{b}\right] ,\left[ \hat{
x}^{a},\hat{p}_{b}\right] $ and $\left[ \hat{p}_{a},\hat{p}_{b}\right] $}
Once we have $\hat{x}^{a}$ and $\hat{p}_{a}$ i.e. the coefficients $
b_{A_{1}\cdots A_{l}}^{a}$ and $c_{A_{1}\cdots A_{l}}^{a}$\ we work out the
commutation relations $\left[ \hat{x}^{a},\hat{x}^{b}\right] ,\left[ \hat{x}
^{a},\hat{p}_{b}\right] $ and $\left[ \hat{p}_{a},\hat{p}_{b}\right] $ using
the formulas $\left( \text{\hyperref[xhat]{$\hat{x}$}}\right) $ and $\left(
\text{\hyperref[phat]{$\hat{p}$}}\right) $ in the previous section in a
brute force calculation. Remember that the $\ast $-commutators is the
\newline
Poisson bracket on $T^{\ast }\mathcal{M}$ to first order in $\hbar $:
\begin{equation*}
\left[ \hat{f}\left( \hat{x},\hat{p}\right) ,\hat{g}\left( \hat{x},\hat{p}
\right) \right] =\hat{h}\left( \hat{x},\hat{p}\right)
\end{equation*}
\begin{equation}
\left[ f_{\ast }\left( x,p\right) ,g_{\ast }\left( x,p\right) \right] _{\ast
}=h_{\ast }\left( x,p\right) =i\hbar \left\{ f,g\right\} _{\mathcal{M}}+
\mathcal{O}\left( \hbar ^{2}\right) \tag{$\ast $-comm} \label{*-comm}
\end{equation}
where $\hat{f}$, $\hat{g}$, $\hat{h}$ and $f_{\ast }$, $g_{\ast }$, $h_{\ast
}$ are fuctions defined by:
\begin{equation*}
\hat{f}\left( \hat{x},\hat{p}\right) =\sum_{lm}f_{ja_{1}\cdots
a_{l}}^{b_{1}\cdots b_{m}}\hbar ^{j}\hat{x}^{a_{1}}\cdots \hat{x}^{a_{l}}
\hat{p}_{b_{1}}\cdots \hat{p}_{b_{m}}
\end{equation*}
\begin{equation*}
f_{\ast }\left( x,p\right) =\sum_{lm}f_{ja_{1}\cdots a_{l}}^{b_{1}\cdots
b_{m}}\hbar ^{j}x^{a_{1}}\ast \cdots \ast x^{a_{l}}\ast p_{b_{1}}\ast \cdots
\ast p_{b_{m}}
\end{equation*}
where $f_{ja_{1}\cdots a_{l}}^{b_{1}\cdots b_{m}}$ are constants.
These two sets, one of all $f_{\ast }$'s $\left\{ f_{\ast }\right\} $ and
one of all $\hat{f}$'s $\left\{ \hat{f}\right\} $ defined above are
isomorphic.\pagebreak
\subsection{$T^{\ast }\mathbb{S}^{2}$ Explicitly}
In our case of $T^{\ast }\mathbb{S}^{2}$ we find:
\begin{equation*}
\left[ \hat{x}^{a},\hat{x}^{b}\right] =0
\end{equation*}
\begin{equation*}
\left[ \hat{x}^{a},\hat{p}_{b}\right] =i\hbar \left( \delta _{b}^{a}-\hat{x}
^{a}\hat{x}_{b}\right)
\end{equation*}
\begin{equation*}
\left[ \hat{p}_{a},\hat{p}_{b}\right] =2i\hbar \hat{x}_{[b}\hat{p}_{a]}
\end{equation*}
\begin{equation*}
\underline{\hat{x}}\cdot \underline{\hat{x}}=1,\text{ }\underline{\hat{p}}
\cdot \underline{\hat{x}}=\underline{\hat{x}}\cdot \underline{\hat{p}}
-2i\hbar =0
\end{equation*}
We now define $\underline{\hat{L}}$ because we argue below that it is a more
"natural" momentum:
\begin{equation*}
\underline{\hat{L}}:=-\underline{\hat{p}}\times \underline{\hat{x}}=
\underline{\hat{x}}\times \underline{\hat{p}}=\underline{x}\times \underline{
z}+\left( \underline{z}\cdot \underline{s}\right) \underline{x}-\underline{z}
\cdot \left( \underline{x}\times \underline{s}\right) \underline{s}
\end{equation*}
again with the computed conditions:
\begin{equation*}
\underline{\hat{L}}\cdot \underline{\hat{x}}=\underline{\hat{x}}\cdot
\underline{\hat{L}}=0~~~,~~~\underline{\hat{x}}\cdot \underline{\hat{x}}=1
\end{equation*}
\begin{equation*}
\ell o\left( \underline{\hat{L}}\right) =\underline{L}=\underline{x}\times
\underline{p}
\end{equation*}
We easily recognize that $\underline{\hat{L}}$ is the more "natural"
variable compared to $\underline{\hat{p}}$. This is because $\underline{\hat{
p}}\cdot \underline{\hat{x}}=0$ and $\underline{\hat{x}}\cdot \underline{
\hat{p}}=2i\hbar $ are very "unnatural" conditions since there is no
physical reason why it shouldn't be $\underline{\hat{x}}\cdot \underline{
\hat{p}}=0$ and $\underline{\hat{p}}\cdot \underline{\hat{x}}=-2i\hbar $. We
could define $\underline{\hat{p}}_{new}=\underline{\hat{p}}+A\underline{\hat{
x}}$ where $A$ is an arbitrary constant and obtain the same commutators. On
the other hand the symmetry between $\underline{\hat{L}}\cdot \underline{
\hat{x}}=\underline{\hat{x}}\cdot \underline{\hat{L}}=0$ seems to suggest
that $\underline{\hat{L}}$ should\ be the preferred quantity over $
\underline{\hat{p}}$. \ In other words the relevant component of $\underline{
\hat{p}}$ is the one perpendicular to $\underline{\hat{x}}$ which is
precisely what $\underline{\hat{L}}$\ is.
Therefore the part of $\hat{p}$ parallel to $\hat{x}$ is irrelevant:
\begin{equation}
\underline{\hat{x}}=\left( \underline{x}-\underline{x}\times \underline{s}
\right) \left( s^{2}+1\right) ^{-\frac{1}{2}} \tag{$\hat{x}$ soln}
\label{xhat soln}
\end{equation}
\begin{equation}
\underline{\hat{L}}=\underline{x}\times \underline{z}+\left( \underline{z}
\cdot \underline{s}\right) \underline{x}-\underline{z}\cdot \left(
\underline{x}\times \underline{s}\right) \underline{s} \tag{$\hat{L}$ soln}
\label{Lhat soln}
\end{equation}
where $\underline{z}=\underline{p}-\underline{x}\times \underline{k}$ with
conditions:
\begin{equation}
\underline{\hat{L}}\cdot \underline{\hat{x}}=\underline{\hat{x}}\cdot
\underline{\hat{L}}=0~~~,~~~\underline{\hat{x}}\cdot \underline{\hat{x}}=1
\tag{$\hat{x}\hat{L}$ conds} \label{xhatLhat conds}
\end{equation}
Again we note at this point that there is not much insight looking at these
formulas except for what we get for the commutators in the remainder of this
section.
We compute the commutators:
\begin{equation}
\left[ \hat{x}^{a},\hat{x}^{b}\right] =0 \tag{$xx$} \label{xx}
\end{equation}
\begin{equation}
\left[ \hat{x}^{a},\hat{L}_{b}\right] =i\hbar \varepsilon _{~bc}^{a}\hat{x}
^{c} \tag{$xL$} \label{xL}
\end{equation}
\begin{equation}
\left[ \hat{L}_{a},\hat{L}_{b}\right] =i\hbar \varepsilon _{~ab}^{c}\hat{L}
_{c} \tag{$LL$} \label{LL}
\end{equation}
along with:
\begin{equation}
\underline{\hat{x}}\cdot \underline{\hat{x}}=1,\text{ }\underline{\hat{L}}
\cdot \underline{\hat{x}}=\underline{\hat{x}}\cdot \underline{\hat{L}}=0
\tag{cond $xL$} \label{cond xL}
\end{equation}
Once we know these relations we know the whole algebra of functions since
the algebra is associative. And thus we are done!
And so in the case of $T^{\ast }\mathbb{S}^{2}$\ a general element $\hat{f}$
(the function $\left( \text{\hyperref[fhat]{$\hat{f}$}}\right) $ we were
looking for and the specific form of the solution $\left( \text{\hyperref[fhat soln
]{$\hat{f}$ soln}}\right) $)\ in the space of all observables of $\hat{x}$
and $\hat{L}$ is
\begin{equation*}
\hat{f}\left( \hat{x},\hat{L}\right) =\sum_{lm}f_{a_{1}\cdots
a_{l}}^{b_{1}\cdots b_{m}}\hat{x}^{a_{1}}\cdots \hat{x}^{a_{l}}\hat{L}
_{b_{1}}\cdots \hat{L}_{b_{m}}
\end{equation*}
where $f_{a_{1}\cdots a_{l}}^{b_{1}\cdots b_{m}}$ are constants. This is the
enveloping algebra of the operators of angular momentum and position on a
Hilbert space.
Clearly we see that the $\hat{L}$'s generate the standard angular momentum
algebra and the $\hat{x}$'s transform properly under rotations. However both
the $\hat{x}$'s and the $\hat{L}$'s\ form a constrained version of the
standard $
\mathbb{R}
^{3}$ Euclidean algebra with invariant constraints given by the last
equations.
\section{Angular Momentum States}
Since we now have the algebra of observables we can ask about Hamiltonians
and states. \ The free single quantum particle Hamiltonian in ordinary
quantum mechanics is $\hat{H}=\frac{\hat{p}^{2}}{2m}=\frac{\hat{p}_{r}^{2}}{
2m}+\frac{\underline{\hat{L}}\cdot \underline{\hat{L}}}{mr^{2}}$ where $\hat{
p}_{r}$ is the radial component of momentum and $\underline{\hat{L}}$ is the
angular momentum. \ In other words the natural choice for the Hamiltonian on
our $\mathbb{S}^{2}$ (which we are free to choose) is $\hat{H}=\underline{
\hat{L}}\cdot \underline{\hat{L}}$, $r=1,m=1\,$\ because it is just the
restricted version of the $\mathbb{E}^{3}$ free particle Hamiltonian onto $
\mathbb{S}^{2}$. We then construct our angular momentum states in the usual
way by solving the eigenvalue equation:
\begin{equation}
\hat{H}\left\vert \phi \right\rangle =E\left\vert \phi \right\rangle
\tag{Schroedinger} \label{Schroedinger}
\end{equation}
where $E\in
\mathbb{R}
$.
We won't do it because it is standard physics that one is able to do as an
undergraduate physics student.
\section{Conclusions}
We have explicitly constructed an exact non-perturbative solutions to the
observables in the Fedosov $\ast $-formalism on $T^{\ast }\mathbb{S}^{2}$
and showed that they obeyed the angular momentum commutation relations. In
other words we took the phase space of a single classical particle confined
to a sphere, quantized it and got the quantum angular momentum algebra
(which we expected). This is done by starting with a chosen phase-space
connection $D$ and constructing an explicit formula for $\hat{D}$. Via the
equation $\left( D-\hat{D}\right) \hat{f}=0$ that defines the algebra i.e.
the algebra of all $\hat{f}$'s we then explicitly constructed $\underline{
\hat{x}}$ and $\underline{\hat{p}}$ (the operator analogues of $\underline{x}
$ and $\underline{p}$) and computed their commutators. We realized (by
defining $\underline{\hat{L}}=\underline{\hat{x}}$ $\times \underline{\hat{p}
}$) that the enveloping algebra of all $\underline{\hat{x}}$'s and $
\underline{\hat{p}}$'s gives the angular momentum algebra.
Subsequently we defined a Hamiltonian $\underline{\hat{L}}\cdot \underline{
\hat{L}}$ that would have eigenstates of angular momentum, however we did
not explicitly construct it because it is standard physics.
Another main point was that most of the ambiguity given a fixed phase space
connection $D$ of the construction of $\hat{D}$, it seemed, stemmed from the
freedom of a change of basis ($\hat{f}\rightarrow U\hat{f}U^{-1}$) given by
the argument in section 4.1. And finally the matrix form of the $\hat{y}$'s
did not change anything from a Moyal-like object as is done in deformation
quantization.
We conclude that we would arrive at the same answer given any algebraic
object $\hat{y}$ that had the same commutators along with the same action of
the connection on them. We then view the Fedosov $\ast $-formalism as a
general algebraic construction and less tied to the deformation aspect of
its original formulation. Thus our formulation using Heisenberg algebras and
their subsequent representation spaces (Hilbert spaces) makes a more direct
connection to the standard formulation of ordinary quantum mechanics.
\section{Appendix A}
\hypertarget{A}{}We now show that the equation $\left( D-\hat{D}\right) ^{2}
\hat{y}^{A}=0$ is equivalent to $\left[ \Omega -Dr+\hat{d}r+r^{2},\hat{y}^{A}
\right] =0$:
Proof:
\begin{equation*}
\left( D-\hat{D}\right) ^{2}\hat{y}^{A}=\left( D^{2}-D\hat{D}-\hat{D}D+\hat{D
}^{2}\right) \hat{y}^{A}
\end{equation*}
\begin{equation*}
\left( D\hat{D}+\hat{D}D\right) \hat{y}^{A}=\left[ D\left( \omega _{AB}\hat{y
}^{A}\Theta ^{B}+r\right) ,\hat{y}^{A}\right] =\left[ Dr,\hat{y}^{A}\right]
\end{equation*}
\begin{eqnarray*}
\hat{D}^{2}\hat{y}^{A} &=&\left[ \hat{Q},\left[ \hat{Q},\hat{y}^{A}\right]
\right] =\hat{Q}\left( \hat{Q}\hat{y}^{A}-\hat{y}^{A}\hat{Q}\right) +\left(
\hat{Q}\hat{y}^{A}-\hat{y}^{A}\hat{Q}\right) \hat{Q} \\
&=&\left[ \hat{Q}^{2},\hat{y}^{A}\right] _{-}=\left[ \left( \omega _{AB}\hat{
y}^{A}\Theta ^{B}+r\right) ^{2},\hat{y}^{A}\right] _{-}=\left[ \left( \omega
_{AB}\hat{y}^{A}\Theta ^{B}\right) ^{2}+\left[ \omega _{AB}\hat{y}^{A}\Theta
^{B},r\right] +r^{2},\hat{y}^{A}\right] _{-}
\end{eqnarray*}
\begin{equation*}
2\left( \omega _{AB}\hat{y}^{A}\Theta ^{B}\right) ^{2}=\left[ \omega _{AB}
\hat{y}^{A}\Theta ^{B},\omega _{CE}\hat{y}^{C}\Theta ^{E}\right] =\left[
\hat{y}^{A},\hat{y}^{C}\right] \omega _{AB}\Theta ^{B}\omega _{CE}\Theta
^{E}=\omega _{AB}\Theta ^{A}\Theta ^{B}
\end{equation*}
\begin{equation*}
\implies \hat{D}^{2}\hat{y}^{A}=\left[ \left[ \omega _{AB}\hat{y}^{A}\Theta
^{B},r\right] +r^{2},\hat{y}^{A}\right] _{-}
\end{equation*}
where $\left[ A,B\right] _{-}=AB-BA$ for any $A$ and $B$.
The curvature $D^{2}$ acting on $\Theta ^{A}$ is:
\begin{equation*}
D^{2}\otimes \Theta ^{A}=R_{B}^{\text{ \ }A}\otimes \Theta ^{B}
\end{equation*}
Thus the curvature $D^{2}$ acting on $\hat{y}^{A}$ is:
\begin{equation*}
D^{2}\hat{y}^{A}=R_{B}^{\text{ \ }A}\hat{y}^{B}
\end{equation*}
Knowing this we define $\Omega $ as the curvature $D^{2}$ acting on $\hat{y}
^{A}$ as a commutator, namely:
\begin{equation*}
\frac{1}{i\hbar }\left[ \Omega ,\hat{y}^{A}\right] =R_{B}^{\text{ \ }A}\hat{y
}^{B}
\end{equation*}
we can immediately write a solution for $\Omega $ knowing $\left[ \hat{y}
^{A},\hat{y}^{B}\right] =i\hbar \omega ^{AB}$, $\omega ^{AB}\omega
_{BC}=\delta _{C}^{A}$ and using the symmetries of the curvature tensor:
\begin{equation*}
\Omega :=-\frac{1}{2}\omega _{AC}R_{B}^{\text{ \ }A}\hat{y}^{B}\hat{y}^{C}
\end{equation*}
Thus we may rewrite the condition $\left( D-\hat{D}\right) ^{2}\hat{y}^{A}=0$
as:
\begin{equation*}
\left( D-\hat{D}\right) ^{2}\hat{y}^{A}=\left[ \Omega -Dr+\hat{d}r+r^{2},
\hat{y}^{A}\right] =0
\end{equation*}
\section{Appendix B}
\hypertarget{B}{}Here we present an argument as to why $r$ only has terms
that are cubic or higher powers in the $\hat{y}$'s.
Given:
\begin{equation*}
\hat{D}=\left[ \hat{Q},\cdot \right] =\left[ \hat{Q}_{A}\Theta ^{A},\cdot
\right]
\end{equation*}
\begin{equation*}
\hat{Q}_{A}=\sum_{l}Q_{AA_{1}\cdots A_{l}}\hat{y}^{A_{1}}\cdots \hat{y}
^{A_{l}}
\end{equation*}
we require:
\begin{equation*}
\left( D-\hat{D}\right) ^{2}\hat{y}^{A}=0
\end{equation*}
If we let:
\begin{equation*}
\hat{Q}_{A}\Theta ^{A}=\omega _{AB}\hat{y}^{A}\Theta ^{B}+r
\end{equation*}
\begin{equation*}
r=\sum_{l}r_{AA_{1}\cdots A_{l}}\Theta ^{A}\hat{y}^{A_{1}}\cdots \hat{y}
^{A_{l}}
\end{equation*}
If we want $r$ to be globally defined for \underline{all} manifolds we must
define it out of non-degenerate tensors namely the metric, the symplectic
form and the curvature. This is because $\Omega $ is degree 2 in the $\hat{y}
$'s (i.e. $\Omega :=-\frac{1}{2}\omega _{AC}R_{B}^{\text{ \ }A}\hat{y}^{B}
\hat{y}^{C}$ has 2 $\hat{y}$'s). The degree is defined by:
\begin{equation*}
\deg \left( a\right) =\left( \text{number of }\hat{y}\text{'s}\right)
+2\left( \text{number of }\hbar \text{'s}\right)
\end{equation*}
A linear $r$ would yield:
\begin{equation*}
\underset{2}{\underbrace{\Omega }}-\underset{1}{\underbrace{Dr}}+\underset{0}
{\underbrace{\hat{d}r}}+\underset{1}{\underbrace{r^{2}}}
\end{equation*}
and this cannot be zero for $\Omega \neq 0$. This means that $r$ must have a
quadratic term in it.
If $r$ is quadratic ($r=\sum_{l=0}^{2}r_{AA_{1}\cdots A_{l}}\Theta ^{A}\hat{y
}^{A_{1}}\cdots \hat{y}^{A_{l}}$), in general, there is no way to construct
the degree 2 coefficient $r_{AA_{1}A_{2}}$ out of invariant tensors. Thus we
require that $r$ has terms that are cubic or higher powers in the $\hat{y}$
's. Fedosov mentions this fact also.$\left[ \text{\hyperlink{1}{1}}\right] $
For a specific manifold there might be an $r$ that is quadratic. The
argument above is meant for an $r$ in a \underline{general} construction for
a \underline{general} manifold and so we give a counterexample in the case
when the manifold $\mathcal{M}$ is $\mathbb{E}^{n}$.
There is always the trivial solution to $r$:
\begin{equation*}
r=-\frac{1}{2}\omega _{CB}\Gamma _{~A}^{C}\hat{y}^{A}\hat{y}^{B}
\end{equation*}
where $\Gamma _{~A}^{C}=\Gamma _{~BA}^{C}\Theta ^{B}$ are the Christoffel
symbols associated to $D$. One can easily observe that this is a solution
knowing $\left[ \hat{y}^{A},\hat{y}^{B}\right] =i\hbar \omega ^{AB}$, $
\omega ^{AB}\omega _{BC}=\delta _{C}^{A}$ and using the symmetries of the
Christoffel symbols. However the $\Gamma $'s are not necessarily globally
defined and if we find an $r$ in one coordinate patch on $T^{\ast }\mathcal{M
}$ there is no guarantee that it will be well-defined in another. However if
$\mathcal{M}=\mathbb{E}^{n}$ then this is a global $r$.
\section{Appendix C}
\hypertarget{C}{}Useful identities:
\begin{equation*}
d\underline{p}=\underline{\alpha }\times \underline{x}-\underline{p}\times
\underline{\theta }
\end{equation*}
\begin{equation*}
\theta ^{a}\theta ^{b}=\tilde{\omega}\varepsilon ^{abc}x_{c}
\end{equation*}
\begin{equation*}
\underline{z}\times \underline{x}=\underline{p}\times \underline{x}-
\underline{k}
\end{equation*}
\begin{equation*}
\underline{z}=\underline{p}-\underline{x}\times \underline{k}
\end{equation*}
\begin{equation*}
\theta ^{a}\theta ^{b}=\theta ^{\lbrack a}\theta ^{b]}=\frac{1}{2}
\varepsilon ^{abc}\left( \underline{\theta }\times \underline{\theta }
\right) _{c}=\tilde{\omega}\varepsilon ^{abc}x_{c}
\end{equation*}
\begin{equation*}
\left( \underline{v}\times \underline{w}\right) \times \underline{u}=\delta
_{ab}v^{a}\underline{w}u^{b}-\underline{v}\left( \underline{w}\cdot
\underline{u}\right)
\end{equation*}
\begin{equation*}
\underline{v}\times \left( \underline{w}\times \underline{u}\right) =\delta
_{ab}v^{a}\underline{w}u^{b}-\left( \underline{v}\cdot \underline{w}\right)
\underline{u}
\end{equation*}
for all 3-D vectors assuming nothing about $\left[ v_{a},w_{b}\right] ,\left[
v_{a},u_{b}\right] $ or $\left[ w_{a},u_{b}\right] $.
\begin{equation*}
\left( \underline{v}\cdot \underline{\theta }\right) \left( \underline{x}
\times \underline{w}\right) \cdot \underline{\theta }=\tilde{\omega}\left(
\underline{v}\cdot \underline{w}\right)
\end{equation*}
for all 3-D vectors assuming $\left[ \theta ^{a},v_{b}\right] =\left[ \theta
^{a},w_{b}\right] =0$ and assuming nothing about $\left[ v_{a},w_{b}\right] $
.
For two vectors such that $\underline{v}\cdot \underline{x}=\underline{w}
\cdot \underline{x}=0$ we have the identities:
\begin{equation*}
\underline{v}\times \underline{w}=\left( \left( \underline{v}\times
\underline{w}\right) \cdot \underline{x}\right) \underline{x}\sim \underline{
x}
\end{equation*}
\begin{equation*}
\underline{z}\cdot \left( \underline{x}\times \underline{s}\right) =
\underline{p}\cdot \left( \underline{x}\times \underline{s}\right) -t
\end{equation*}
\begin{equation*}
\left[ s^{2},\left( \underline{x}\times \underline{k}\right) \cdot
\underline{s}\right] =0
\end{equation*}
\begin{equation*}
s_{a}f\left( \underline{k}\cdot \underline{s}\right) =f\left( \underline{k}
\cdot \underline{s}+1\right) s_{a}
\end{equation*}
\begin{equation*}
\left[ r_{0},\underline{s}\right] =\frac{1}{3}\left( \left( \underline{s}
\cdot \underline{\theta }\right) \underline{s}-s^{2}\underline{\theta }
\right)
\end{equation*}
\begin{equation*}
\left[ r_{0},\left( \underline{s}\cdot \underline{\theta }\right) \right] =0
\end{equation*}
\begin{equation*}
\left[ r_{0},s^{2}\right] =0=\left[ \underline{z}\cdot \underline{s},s^{2}
\right]
\end{equation*}
\begin{equation*}
\left[ r_{0},\underline{k}\right] =\frac{1}{3}\left( 2\underline{s}\left(
\underline{k}\cdot \underline{\theta }\right) -\underline{\theta }t-\left(
\underline{s}\cdot \underline{\theta }\right) \underline{k}\right)
\end{equation*}
\begin{equation*}
\left[ r_{0},\underline{z}\right] =\frac{1}{3}\left( \left( \underline{s}
\cdot \underline{\theta }\right) \underline{x}\times \underline{k}-
\underline{\theta }\times \underline{x}t-2\underline{x}\times \underline{s}
\left( \underline{k}\cdot \underline{\theta }\right) \right)
\end{equation*}
\begin{equation*}
\tilde{D}\underline{s}=\underline{\theta }\times \underline{s}-\left( 1+
\frac{1}{s^{2}}\right) \left( \underline{s}\cdot \underline{\theta }\right)
\underline{s}-\frac{1}{s^{2}}\left( \left( \underline{x}\times \underline{s}
\right) \cdot \underline{\theta }\right) \underline{x}\times \underline{s}
\end{equation*}
\begin{equation*}
\tilde{D}\underline{x}=D\underline{x}=\underline{\theta }\times \underline{x}
=\frac{1}{s^{2}}\left( \left( \underline{x}\times \underline{s}\right) \cdot
\underline{\theta }\right) \underline{s}-\left( \underline{s}\cdot
\underline{\theta }\right) \underline{x}\times \underline{s}
\end{equation*}
\begin{eqnarray*}
\tilde{D}\underline{z} &=&\underline{\theta }\times \underline{z}+\left(
\left( \underline{z}\cdot \underline{s}\right) \left( \underline{s}\cdot
\underline{\theta }\right) -\underline{z}\cdot \left( \underline{x}\times
\underline{s}\right) \left( \left( \underline{x}\times \underline{s}\right)
\cdot \underline{\theta }\right) \right) \frac{1}{s^{2}}\underline{s} \\
&&+2\underline{z}\cdot \left( \underline{x}\times \underline{s}\right)
\left( \underline{s}\cdot \underline{\theta }\right) \frac{1}{s^{2}}
\underline{x}\times \underline{s}
\end{eqnarray*}
\section{Appendix D: $T^{\ast }
\mathbb{R}
^{n}$}
\begin{itemize}
\item \hypertarget{D}{}In the case of $T^{\ast }
\mathbb{R}
^{n}$ we solve equation $\left( \hyperref[r]{r}\right) $ above for $r$ when $
D\otimes \Theta ^{A}=0$ therefore $D\hat{y}^{A}=0$ and hence $\Omega =0$ and
get the solution $r=0$. This gives us $\hat{D}$ by the formulas $\left( \hat{
D}\right) $ and $\left( \hat{Q}\right) $:
\begin{equation*}
\hat{D}=\frac{1}{i\hbar }\left[ \omega _{AB}\hat{y}^{A}\Theta ^{B},\cdot
\right] =\frac{1}{i\hbar }\left[ \underline{s}\cdot d\underline{p}-
\underline{k}\cdot d\underline{x},\cdot \right] =\frac{1}{i\hbar }\left[
\left( \underline{x}+\underline{s}\right) \cdot d\underline{p}-\left(
\underline{p}+\underline{k}\right) \cdot d\underline{x},\cdot \right]
\end{equation*}
where $s$ and $k$ are the first $n$ $\hat{y}$'s and the last $n$ $\hat{y}$'s
respectively (i.e. $\hat{y}^{A}=\left( s^{a},k_{a}\right) $) also we have $
\left[ s^{a},s^{b}\right] =0=\left[ k_{a},k_{b}\right] ,~\left[ s^{a},k_{b}
\right] =i\hbar \delta _{b}^{a}$ and $Ds^{a}=0=Dk_{a}$.
All operators are required to satisfy:
\begin{equation*}
\frac{\partial \hat{f}}{\partial x^{a}}dx^{a}+\frac{\partial \hat{f}}{
\partial p_{a}}dp_{a}-\hat{D}\hat{f}=0
\end{equation*}
\begin{equation*}
\implies \frac{\partial \hat{f}}{\partial x^{a}}dx^{a}+\frac{\partial \hat{f}
}{\partial p_{a}}dp_{a}=\frac{1}{i\hbar }\left[ \left( \underline{x}+
\underline{s}\right) \cdot d\underline{p}-\left( \underline{p}+\underline{k}
\right) \cdot d\underline{x},\hat{f}\right]
\end{equation*}
This equation is the specific case of the equation $\left( \text{\hyperref[cond fhat
]{cond $\hat{f}$}}\right) $ for $T^{\ast }
\mathbb{R}
^{n}$ introduced in section 2.4. The above equation tells us that $\hat{f}$
is a function of $\hat{x}^{a}=x^{a}+s^{a}$ and $\hat{p}_{a}=p_{a}+k_{a}$ ($
\hat{f}=\hat{f}\left( \hat{x},\hat{p}\right) $) which are solutions to the
equation $\left( \text{\hyperref[cond fhat]{cond $\hat{f}$}}\right) $ i.e.
the coefficients $b_{A_{1}\cdots A_{l}}^{a}\,$\ and $c_{aA_{1}\cdots A_{l}}$
in the case of $T^{\ast }
\mathbb{R}
^{n}$ introduced in the section 2.4 when $\ell o\left( \hat{f}\right) =x^{a}$
and $\ell o\left( \hat{f}\right) =p_{a}$ respectively. The equation above
implies that $\frac{1}{i\hbar }\left[ \cdot ,\hat{p}_{a}\right] $ generates
the translation on the cotangent bundle in the $x^{a}-$direction and $\frac{1
}{i\hbar }\left[ \hat{x}^{a},\cdot \right] $ generates the translation on
the cotangent bundle in the $p_{a}-$direction on all observables $\hat{f}$.
See Fedosov for more details on motivating the need for $\hat{D}$.$\left[
\text{\hyperlink{1}{1}}\right] \qquad {\tiny \square }$
\end{itemize}
$\left[ \text{\hypertarget{1}{1}}\right] $\ Boris Fedosov,\textit{\
Deformation Quantization and Index Theory,} Akademie, Berlin 1996.
$\left[ \text{\hypertarget{2}{2}}\right] $\ M. Gadella, M. A. del Olmo and
J. Tosiek,\textit{\ Geometrical Origin of the }$\ast -$\textit{product in
the Fedosov Formalism,} \href{http://xxx.lanl.gov/abs/hep-th/0405157}{
hep-th/0405157v1}.
$\left[ \text{\hypertarget{3}{3}}\right] $\ J. Hancock, M. Walton, B.
Wynder, \textit{Quantum Mechanics Another Way,} \href{http://xxx.lanl.gov/abs/physics/0405029v1
}{physics/0405029v1}.
$\left[ \text{\hypertarget{4}{4}}\right] $ H. Omori, Y. Maeda and A.
Yoshioka, \textit{Lett. Math. Phys.} \textbf{26}, 285 (1992).
$\left[ \text{\hypertarget{5}{5}}\right] $ A. Connes, M. Flato and D.
Sternheimer, \textit{Lett. Math. Phys.} \textbf{24}, 1 (1992).
$\left[ \text{\hypertarget{6}{6}}\right] $ V. I. Arnold, \textit{
Mathematical Methods of Classical Mechanics}, Springer, New York 1978.
$\left[ \text{\hypertarget{7}{7}}\right] $ N. Woodhouse, \textit{Geometric
Quantization}, Oxford Univ. Press, New York 1980.
$\left[ \text{\hypertarget{8}{8}}\right] $ J. E. Moyal, Proc. Camb. Phil.
Soc. \textbf{45}, 99 (1949).
$\left[ \text{\hypertarget{9}{9}}\right] $ H. Weyl, \textit{The Theory of
Groups and Quantum Mechanics}, Dover, New York 1931.
$\left[ \text{\hypertarget{10}{10}}\right] $ E. P. Wigner, Phys. Rev.
\textbf{40}, 749 (1932).
$\left[ \text{\hypertarget{11}{11}}\right] $ H. J. Groenewold, Physica
\textbf{12}, 405 (1946).
$\left[ \text{\hypertarget{12}{12}}\right] $~P. A. M. Dirac, \textit{The
Principles of Quantum Mechanics}, Oxford Univ. Press, Oxford
1958.
\end{document} | math |
<?php
namespace Core\Listeners;
use Zend\EventManager\ListenerAggregateInterface;
use Zend\EventManager\EventManagerInterface;
use Zend\Mvc\MvcEvent;
use \Core\Acl\Service\Acl as Acl;
use \Core\Acl\Model\Resource as AclResource;
use \Core\Acl\Model\Role as AclRole;
use \Core\Acl\Model\Rule as AclRule;
class AclListener implements ListenerAggregateInterface
{
protected $_listeners = array();
/**
* Attach one or more listeners
*
* Implementors may add an optional $priority argument; the EventManager
* implementation will pass this to the aggregate.
*
* @param EventManagerInterface $events
*/
public function attach(EventManagerInterface $events)
{
$this->_listeners[] = $events->attach(MvcEvent::EVENT_DISPATCH, array(
$this,
'initAcl'
), 100);
}
/**
* Detach all previously attached listeners
*
* @param EventManagerInterface $events
*/
public function detach(EventManagerInterface $events)
{
foreach ($this->_listeners as $index => $listener) {
if ($events->detach($listener)) {
unset($this->_listeners[$index]);
}
}
}
public function initAcl(MvcEvent $e)
{
/* @var \Zend\Mvc\Application $app */
$app = $e->getApplication();
// Get SM
$sm = $app->getServiceManager();
/* @var Acl $acl */
$acl = $sm->get('Core/Acl/Service/Acl');
// Get params 'controller', 'action' and 'privilege' from route match
$matches = $e->getRouteMatch();
// Resource based on request params
$resource = new AclResource();
$resource->setController($matches->getParam('controller'));
$resource->setAction($matches->getParam('action', 'index'));
// 404 response if resource does not exist
if (! $acl->hasResource($resource, true)) {
$e->getResponse()->setStatusCode(404);
return;
}
// Get config
$config = $sm->get('config');
$configAcl = $config['Acl'];
// Role
$auth = $sm->get('Core/Acl/Service');
$role = new AclRole();
$role->setName($auth->hasIdentity() ? $auth->getIdentity()->$configAcl['field_role'] : $configAcl['default_role']);
// Query ACL
$result = $acl->isAllowed($role, $resource, $e->getRequest()->getMethod());
// 403 Unauthorized
if ($result === false) {
// Create ViewModel
$model = new \Zend\View\Model\JsonModel();
$model->setVariable('exception', new \Exception(Acl::ERROR_UNAUTHORIZED));
$e->getResponse()->setReasonPhrase(Acl::ERROR_UNAUTHORIZED);
// Add $model as a child and set 403 status code
$e->getResponse()->setStatusCode(403);
// Stop propagation
$e->stopPropagation();
return;
}
}
}
| code |
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دیوارن خلاف ظاہری زورک مقابلہ اوس تپون ہنٛد وزن تہٕ پتہٕ وڑون تنہٕ سۭتۍ کرنہٕ آمُت۔ | kashmiri |
ये सब मैं नहीं कह रही क्बी रिपोर्ट कहता है, सुप्रीम कोर्ट कहता है, जमीनी हकीकत कहता है वर्षा डोंगरे ,पुलिस अधिकारी छत्तीसगढ़ बॉस्केट
वर्षा डोंगरे की रिपोर्ट भी कह रही संविधान को मानने ,नागरिक अधिकार बहाली और आदिवासियों का विश्वास जीतकर ही बस्तर में हिंसा को ख़त्म किया जा सकता है
छत्तीसगढ़ में तैनात सुरक्षा बल जंगलों में जाते हैं, आदिवासी महिलाओं के साथ बलात्कार करते हैं,
जंगल में लड़कियों के स्तनों को दबा कर दूध निकाल कर सिपाही जांच करते हैं कि लडकियां शादी शुदा हैं कि नहीं ?
थानों में लड़कियों को नंगा रखा जाता है बिजली से उनके स्तन जलाए जाते हैं ?
राष्ट्रीय मानवाधिकार जांच करता है और इसे सच घोषित करता है,
लेकिन आप इसे मानने के लिए तैयार नहीं हैं,
क्योंकि आप शहर में बैठ कर हराम की खा रहे हैं,
अगर सिपाही गाँव और जंगल से लूट कर नहीं लायेंगे तो आप शहर में बैठ कर क्या खायेंगे ?
आप एक हिंसक और लुटेरी अर्थव्यवस्था और राज्य व्यवस्था का समर्थन कर रहे हैं,
हिंसा का समर्थन मत कीजिये, अपनी हिंसा का भी समर्थन मत कीजिये,
जिसमें दम हो आये हमारे साथ हम एक अहिंसक और शांतिप्रिय समाज बनाने का रास्ता जानते हैं,
इसके साथ रायपुर जेल की डिप्टी जेलर वर्षा डोंगरे का स्टेट्स शेयर कर रहा हूँ जिन्होंने जेल में ऐसी जलाई गयी आदिवासी लड़कियों को देखा है जिनके स्तनों को थानों में बिजली से जलाया गया है
#पूंजीवादी #व्यवस्था #के #शिकार #जवान #शहीदों #को #सत #सत #नमन्
मगर मुझे लगता है कि एक बार हम सभी को अपना गिरेबान झांकना चाहिए, सच्चाई खुदबखुद सामने आ जाऐगी घटना में दोनों तरफ मरने वाले अपने देशवासी हैंभारतीय हैं । इसलिए कोई भी मरे तकलिफ हम सबको होती है । लेकिन पूँजीवादी व्यवस्था को आदिवासी क्षेत्रों में जबरदस्ती लागू करवाना उनकी जल जंगल जमीन से बेदखल करने के लिए गांव का गांव जलवा देना, आदिवासी महिलाओं के साथ बलात्कार, आदिवासी महिलाऐं नक्सली है या नहीं इसका प्रमाण पत्र देने के लिए उनका स्तन निचोड़कर दुध निकालकर देखा जाता है । टाईगर प्रोजेक्ट के नाम पर आदिवासियों के जल जंगल जमीन से बेदखल करने की रणनीति बनती है जबकि संविधान अनुसार ५ वी अनुसूची में शामिल होने के कारण सैनिक सरकार को कोई हक नहीं बनता आदिवासियों के जल जंगल और जमींन को हड़पने का.
प्रेवियस सेना के जवान हमारे दुश्मन नहीं: माओवादी- बीबीसी
नेक्स्ट यह युद्ध थमना चाहिए क्योंकि इसमें जीत किसी की नहीं: बेला भाटिया | hindi |
مےٚ چھُ ٹرینہِ ذٔریعہٕ آتھٕوارٕ دۄہ صبحٲے پانٛژھِ بجہِ میسور گژھُن | kashmiri |
\begin{document}
\title{Bounded-Regret MPC via Perturbation Analysis: Prediction Error, Constraints, and Nonlinearity}
\renewcommand{\arabic{footnote}}{\fnsymbol{footnote}}
\renewcommand{\arabic{footnote}}{\arabic{footnote}}
\begin{abstract}
We study Model Predictive Control (MPC) and propose a general analysis pipeline to bound its dynamic regret. The pipeline first requires deriving a perturbation bound for a finite-time optimal control problem. Then, the perturbation bound is used to bound the per-step error of MPC, which leads to a bound on the dynamic regret. Thus, our pipeline reduces the study of MPC to the well-studied problem of perturbation analysis, enabling the derivation of regret bounds of MPC under a variety of settings. To demonstrate the power of our pipeline, we use it to generalize existing regret bounds on MPC in linear time-varying (LTV) systems to incorporate prediction errors on costs, dynamics, and disturbances. Further, our pipeline leads to regret bounds on MPC in systems with nonlinear dynamics and constraints.
\end{abstract}
\section{Introduction}\label{sec:intro}
\begin{comment}
Main contributions:
\begin{enumerate}
\item A refined pipeline (Sec 2) that takes in perturbation bounds and outputs the performance guarantee (dynamic regret or competitive ratio). Compared with \cite{lin2021perturbation}, the new pipeline is more general because: 1) It works for non-linear dynamics; 2) It works when the exp-decay perturbation bound does not hold globally; 3) It can even work for polynomial perturbation bounds.
\item The first competitive ratio bound for MPC in unconstrained LTV systems when there are prediction errors on disturbances.
\item The first dynamic regret bound for MPC in unconstrained LTV systems when there are prediction errors on dynamical matrices.
\item The first dynamic regret bound for MPC in constrained LTV systems and non-linear LTV systems. A good initial guess of $\mathsf{OPT}$ trajectory is needed. Compared with \cite{xu2019exponentially} and \cite{na2020superconvergence}, our method does not require the counter-intuitive replan window $L$.
\item We provide a simple example where exp-decay perturbation bound does not hold. We show that no online algorithm (include MPC) cannot perform well in this case.
\end{enumerate}
\end{comment}
Model Predictive Control (MPC) is an optimal control approach that solves a Finite-Time Optimal Control Problem (FTOCP) using future predictions in a receding horizon manner \cite{garcia1989model}. It is a flexible approach that is able to accommodate nonlinear and time-varying dynamics, state and actuation constraints, and general cost functions \cite{rosolia2017learning, korda2018linear, allgower2012nonlinear, falcone2007linear}. As a result, it is broadly applied in a wide spectrum of control problems, including robotics \cite{wieber2006trajectory, gu2006receding, shim2003decentralized, diedam2008online, neunert2016fast}, autonomous vehicles \cite{morgan2014model, richards2006robust, cairano2008MPC, stewart2008model, amari2008unified, Hatanaka2009explicit, cairano2013vehicle}, power systems \cite{gonzalez2019powergrid, santhosh2020windforecast, lin2012online, li2018model, shetaya2017model, parisio2014model, yaramasu2014predictive}, process control \cite{wang2016combined, clarke1988application, ellis2014tutorial}, etc.
Despite the popularity of MPC, its theoretic analysis has been quite challenging. Early works along this line focused on the stability and recursive feasibility of MPC \cite{diehl2010lyapunov, angeli2011average, angeli2016theoretical, grune2020economic}. More recently, there has been tremendous interest in providing \text{finite-time learning-theoretic} performance guarantees for MPC, such as regret and/or competitive ratio bounds \cite{yu2020competitive, yu2020power}. For example, progress has recently been made toward (i) regret analysis of MPC in linear time-invariant (LTI) systems with prediction errors on the trajectory to track \cite{zhang2021regret}, (ii) the dynamic regret and competitive ratio bounds of MPC under linear time-varying (LTV) dynamics with exact predictions \cite{lin2021perturbation}, and (iii) exponentially decaying perturbation bounds of the finite-time optimal control problem in time-varying, constrained, and non-linear systems \cite{shin2020decentralized, shin2021controllability}. Beyond MPC, providing regret and/or competitive ratio guarantees for a variety of (predictive) control policies has been a focus in recent years. Examples include RHGC \cite{li2020online, li2019online} and AFHC \cite{lin2012online, chen2015online} for online control/optimization with prediction horizons, OCO-based controllers \cite{agarwal2019online, agarwal2019logarithmic} for no-regret online control, and variations of ROBD for competitive online control without predictions \cite{goel2019beyond, shi2020online} or with delayed observations \cite{pan2021online}. In addition, regret lower bounds have been studied in known LTI systems \cite{goel2020power} and unknown LTV systems \cite{minasyan2021online}.
A promising analysis approach that has emerged from the literature studying MPC and, more generally, predictive control, is the use of perturbation analysis techniques, or more particularly, the use of so-called exponential decaying perturbation bounds. Such techniques underlie the results in \cite{lin2021perturbation, shin2020decentralized, shin2021controllability, zhang2021regret}.
This research direction is particularly promising since perturbation bounds exist for FTOCP in many dynamical systems, e.g., \cite{xu2019exponentially, na2020superconvergence, shin2021exponential, lee2006continuity, fiacco1990sensitivity}, and thus it potentially allows the derivation of regret and/or competitive ratio bounds in a variety of settings.
However, to this point the approach has only yielded results in unconstrained linear systems with no prediction errors (e.g., \cite{lin2021perturbation}), and often requires adjusting MPC to include a counter-intuitively large re-planning window due to technical challenges in the analysis (e.g., \cite{xu2019exponentially, na2020superconvergence}).
Thus, though perturbation analysis techniques might seem promising, many important questions about applying them for the study of predictive control remain open. Firstly, one of the major reasons for the extensive application of MPC is its flexibility in incorporating constraints and nonlinear dynamics \cite{borrelli2017predictive}. However, none of the existing results and approaches can analyze the performance of MPC under constraints and/or nonlinear dynamics. In fact, the anlyasis of MPC under constraints or nonlinearity has long been known to be challenging because of the intractable form of cost-to-go functions and optimal solutions.
Secondly, prediction error is inevitable for real-world implementations of MPC due to unpredictable noise and model mismatch, yet the analysis of MPC subject to prediction errors is limited.
Thirdly, existing approaches analyze MPC in a case-by-case manner and, in most cases, the analysis framework is specific to the assumptions of the particular case (e.g. quadratic costs, perfect predictions, etc) in a way that does not generalize to other settings \cite{yu2020power, zhang2021regret, lin2021perturbation, xu2019exponentially, na2020superconvergence}.
\textbf{Contributions.} In this paper, we propose a general analysis pipeline (Section \ref{sec:pipeline}) that converts perturbation bounds for an FTOCP into dynamic regret bounds for MPC across a variety of settings.
More specifically, the pipeline consists of three steps (see \Cref{fig:flowchart}). In Step 1, we obtain the required perturbation bounds for the specific setting. In Step 2, as shown in \Cref{lemma:pipeline-step2}, the perturbation bounds are used to bound the \textit{per-step error}, which is defined to be the error of the MPC action against the clairvoyant optimal action (see \Cref{def:per-step-error}). In Step 3, the per-step error bound is converted to a dynamic regret bound for MPC, as shown in \Cref{thm:per-step-error-to-performance-guarantee}. The full pipeline is summarized into a \emph{Pipeline Theorm} (\Cref{thm:the-pipeline-theorem}), which directly converts perturbation bounds into bounds on the dynamic regret of MPC in general settings, including those with time-variation, prediction error, constraints, and nonlinearities. The key technical insight that enables the pipeline is the following recursive relationship between Step 2 and Step 3 (\Cref{lemma:pipeline-step2} and \Cref{thm:per-step-error-to-performance-guarantee}): Step 2 guarantees a ``small'' per-step error $e_t$ once the current state $x_t$ of MPC is ``near'' the offline optimal trajectory ($\mathsf{OPT}$), while Step 3 guarantees the next state $x_{t+1}$ of MPC will be near $\mathsf{OPT}$ if all previous per-step errors ($\{e_\tau\}_{\tau \leq t}$) are small. Thus Step 2 and Step 3 work together to guarantee MPC states are always near $\mathsf{OPT}$ and thus MPC per-step errors are always small (\Cref{thm:the-pipeline-theorem}).
To demonstrate the power of the proposed pipeline, we apply it to a range of settings, as summarized in \Cref{table:settings}. Our first applications are to two settings with linear time-varying (LTV) dynamics and prediction errors on (i) disturbances, \Cref{sec:unconstrained:disturbances}, and (ii) the dynamical matrices and cost functions, \Cref{sec:unconstrained:dynamics}. The state-of-the-art results in the LTV setting are \cite{lin2021perturbation}, which requires exact knowledge of the disturbances and of the dynamics. To the best of our knowledge, our work provides the first regret result for MPC with prediction error on the dynamics (see \Cref{thm:perturbation:unconstrained-LTV-pred-err-dynamics}), a result that enables the bounds in settings where MPC is applied to learned dynamics \cite{papadimitriou2020control}.
Our second application is to a setting with nonlinear dynamics and constraints (\Cref{sec:general}). We show the first dynamic regret bound for MPC under state and actuation constraints in nonlinear systems with general costs (\Cref{thm:perturbation:general-system}). Very few prior results exist for MPC in this setting, even with nonlinear dynamics or constraints individually. The most related works are \cite{xu2019exponentially}, which studies constrained MPC, and \cite{na2020superconvergence}, which studies nonlinear MPC. In both cases, a counter-intuitive re-planning window is added to MPC to facilitate the analysis, a downside that our pipeline could avoid. Besides, \cite{xu2019exponentially} and \cite{na2020superconvergence} require exact predictions of the cost functions, dynamics, and constraints for the exponential convergence property of MPC to hold, while our result can apply to more general noisy predictions.
\section{Preliminaries}\label{sec:preliminaries}
In this section, we first introduce the general predictive online control problem including the settings, the objective, available information, and the predictive controller class. Then, we introduce the MPC algorithm, which is a widely-used predictive controller that we focus on in this work. Specifically, we consider a general, finite-horizon, discrete-time optimal control problem with \emph{time-varying costs, dynamics and constraints}, namely
\begin{align}\label{equ:online_control_problem}
\min_{x_{0:T}, u_{0:T-1}} &\sum_{t = 0}^{T-1} f_t(x_t, u_t; \xi_t^*) + F_T(x_T; \xi_T^*) \nonumber\\*
\text{ s.t. }&x_{t+1} = g_{t}(x_{t}, u_{t}; \xi_{t}^*), &\forall 0 \leq t < T,\nonumber\\*
&s_t(x_t, u_t; \xi^*_t) \leq 0, &\forall 0 \leq t < T,\\*
&x_0 = x(0).\nonumber
\end{align}
Here, $x_t \in \mathbb{R}^n$ is the \textit{state}, $u_t \in \mathbb{R}^m$ is the \textit{control input} or \textit{action}; $f_t$ is a time-varying \textit{stage cost} function, $g_t$ is a time-varying \textit{dynamical} function, and $s_t$ is a time-varying \textit{constraint} function, all parameterized by a ground-truth parameter $\xi_t^*$ (unknown to an online controller); and $F_T$ is a terminal cost function parameterized by $\xi_T^*$ that regularizes the terminal state.
The offline optimal trajectory $\mathsf{OPT}$ is obtained by solving \eqref{equ:online_control_problem} with the full knowledge of the true parameters $\xi_{0:T}^*$. In contrast, an online controller can only observe noisy estimations of the parameters in a fixed prediction horizon to decide its current action $u_t$ at each time step $t$. For example, MPC picks $u_t$ by calculating the optimal sub-trajectory confined to the prediction horizon. The objective is to design an online controller that can compete against the offline optimal trajectory $\mathsf{OPT}$.
We use \textit{dynamic regret} as the performance metric, which is widely used to evaluate the performance of online controllers/algorithms in the literature of online control \cite{lin2021perturbation, yu2020competitive, zhang2021regret} and online optimization \cite{li2020online, goel2019beyond, lin2020online}.
Specifically, for a concrete problem instance $(x(0), \xi_{0:T}^*)$, let $\mathrm{cost}(\mathsf{OPT})$ denote the total cost incurred by $\mathsf{OPT}$, and $\mathrm{cost}(\mathsf{ALG})$ denote the total cost incurred by an online controller $\mathsf{ALG}$. The \textit{dynamic regret} is defined as the worst-case additional cost incurred by $\mathsf{ALG}$ against $\mathsf{OPT}$, i.e., $\sup_{x(0), \xi_{0:T}^*} \left(\mathrm{cost}(\mathsf{ALG}) - \mathrm{cost}(\mathsf{OPT})\right)$.
The formulation in \eqref{equ:online_control_problem} is general enough to include a variety of challenging settings. In this paper, we consider three important settings to illustrate how to apply our analysis pipeline. The settings differ in (a) the form of costs, dynamics, and constraints, and (b) the quantities in the system to be predicted (i.e., parameterized by $\xi_t^*$), and the prediction error allowed. An overview of the settings is presented in Table \ref{table:settings} below.
\vspace*{-4pt}
\begin{table}[H]
\caption{Overview of the settings considered in this paper}\label{table:settings}
\footnotesize\centering\vspace*{-6pt}
\begin{tabular}{c|ccc|cc}
\specialrule{1.0pt}{0pt}{0pt}
\textbf{Section} & \textbf{Costs} & \textbf{Dynamics} & \textbf{Constraints} & \textbf{Prediction $\bm{\xi_t}$} & \textbf{Prediction error} \\\hline
\ref{sec:unconstrained:disturbances} & decomposable & LTV & none & disturbance: $w_t$ & arbitrary \\\hline
\ref{sec:unconstrained:dynamics} & quadratic & LTV & none & \tabincell{c}{cost: $Q_t, R_t, \bar{x}_t$ \\ dynamics: $A_t, B_t$}& sufficiently small \\\hline
\ref{sec:general} & general & \tabincell{c}{non-linear\\time-varying} & \tabincell{c}{non-linear\\stage constraint} & \tabincell{c}{cost: $f_t$\\dynamics: $g_t$\\constraints: $s_t$} & sufficiently small \\
\specialrule{1.0pt}{0pt}{0pt}
\end{tabular}
\end{table}
\vspace*{-10pt}
In each setting, we impose different assumptions on cost functions, dynamical systems, constraints, and properties of the predicted quantities as functions of parameter $\xi_t$. In general, we require well-defined costs, Lipschitz and uniformly controllable dynamics, and Lipschitzness of the predicted quantities with regard to $\xi_t$. For constraints, additional assumptions characterizing the active constraints along and near the optimal trajectory are imposed. Detailed definitions and statements are deferred to Appendix \ref{appendix:assumptions} and Sections \ref{sec:pipeline}, \ref{sec:unconstrained}, and \ref{sec:general}. To facilitate the statement of the pipeline, we assume the following \textit{universal properties} hold throughout the paper:
\begin{itemize}
\item \textit{Stability of $\mathsf{OPT}$:} there exists a constant $D_{x^*}$ such that $\norm{x_t^*} \leq D_{x^*}$ for every state $x_t^*$ on the offline optimal trajectory $\mathsf{OPT}$.
\item \textit{Lipschitz dynamics:} the ground-truth dynamical function $g_t(\cdot, \cdot; \xi_t^*)$ is Lipschitz in action; i.e., for any feasible $x_t, u_t, u'_t$, $g_t$ satisfies
$\norm{g_t(x_t, u_t; \xi_t^*) - g_t(x_t, u_t'; \xi_t^*)} \leq L_g \norm{u_t - u_t'}.$
\item \textit{Well-conditioned costs:} every stage cost $f_t(\cdot, \cdot; \xi_t^*)$ and the terminal cost $F_T(\cdot ; \xi_T^*)$ are nonnegative, convex, and $\ell$-smooth in $(x_t, u_t)$ and $x_T$, respectively.
\end{itemize}
\subsection{Predictive Online Control}\label{sec:setting:MPC}
While Step 3 (\Cref{thm:per-step-error-to-performance-guarantee}) in our pipeline can be generally applied to all online controllers, in the subsequent applications we focus on \textit{Model Predictive Control (MPC)}, a popular classical controller. In this subsection, we first define the available information (predictions) as well as its quality (prediction power), and how general predictive online controllers make decisions. Then, we define a useful optimization problem called FTOCP, and introduce MPC as a predictive online controller.
We represent the uncertainties in cost functions, dynamics, constraints, and terminal costs as function families parameterized by $\xi_t$:
$\mathcal{F}_t \coloneqq \{ f_t(x_t, u_t; \xi_t) \mid \xi_t \in \varXi_t \}, \mathcal{G}_t \coloneqq \{ g_t(x_t, u_t; \xi_t) \mid \xi_t \in \varXi_t \},$ $\mathcal{S}_t \coloneqq \{ s_t(x_t, u_t; \xi_t) \mid \xi_t \in \varXi_t \},$ and $\mathcal{F}_T \coloneqq \{ F_T(x_T; \xi_T) \mid \xi_T \in \varXi_T \}$. The online controller knows the function families $\mathcal{F}_{0:T}$, $\mathcal{G}_{0:T-1}$, and $\mathcal{S}_{0:T-1}$ as prior knowledge, but it does not know the true parameters $\xi_{0:T}^* \in \prod_{\tau = 0}^{T} \varXi_{\tau}$. Instead, at time step $t$, the online controller has access to noisy predictions of these parameters for the future $k$ time steps (where $k$ is called the \textit{prediction horizon}), represented by $\xi_{t:t+k\mid t} \in \prod_{\tau = t}^{t+k} \varXi_{\tau}$. The parameter space $\varXi_t$ at each time step $t$ may have different dimensions.
We formally define the quality of predictions by introducing the following notion of prediction error.
\begin{definition}\label{def:pred-oracle}
The prediction error is defined as $\rho_{t, \tau} \coloneqq \norm{\xi_{t + \tau \mid t} - \xi_{t+\tau}^*}$ for an integer $\tau \geq 0$. The power of $\tau$-step-away predictions (for parameter $\xi$) is defined as $P(\tau) \coloneqq \sum_{t = 0}^{T-\tau} \rho_{t, \tau}^2$.
\end{definition}
Under this noisy prediction model, a general predictive online controller $\mathsf{ALG}$ decides the control action based on the current state and the latest available predictions of future parameters. We formally define the class of predictive online controllers considered in this paper in \Cref{def:online-controller}, which includes MPC as a special case.
\begin{definition}\label{def:online-controller}
A predictive online controller $\mathsf{ALG}$ is a function that takes the current state $x_t$ and the available predictions $\xi_{t:t+k\mid t}$ as inputs at time $t$ and outputs the current control action $u_t$, i.e.,
$u_t = \mathsf{ALG}(x_t, \xi_{t:t+k\mid t}).$
We use $x_0 \xrightarrow{u_0} x_1 \xrightarrow{u_1} \cdots \xrightarrow{u_{T-1}} u_T$ to denote the trajectory achieved by $\mathsf{ALG}$, and use $x_0 \xrightarrow{u_0^*} x_1^* \xrightarrow{u_1^*} \cdots \xrightarrow{u_{T-1}^*} u_T^*$ to denote the offline optimal trajectory $\mathsf{OPT}$.
\end{definition}
A core component of both the design of online controllers and our analysis is the following \textit{finite-time optimal control problem} (FTOCP). Given a time interval $[t_1, t_2]$, the FTOCP solves the optimal sub-trajectory subjected to the given initial state $z$, terminal cost $F$, and a sequence of (potentially noisy) parameters $\xi_{t_1:t_2-1}, \zeta_{t_2}$, as formalized in the following definition.
\begin{definition}\label{def:FTOCP}
The finite-time optimal control problem (FTOCP) over the horizon $[t_1, t_2]$, with initial state $z$, parameters $\xi_{t_1: t_2-1}$ and $\zeta_{t_2}$, and terminal cost $F(\cdot; \cdot)$, is defined as
\begin{align}\label{equ:auxiliary_control_problem}
\iota_{t_1}^{t_2}(z, \xi_{t_1:t_2-1}, \zeta_{t_2}; F) \coloneqq \min_{y_{t_1:t_2}, v_{t_1:t_2-1}} &\sum_{t = t_1}^{t_2-1} f_t(y_t, v_t; \xi_t) + F(y_{t_2}; \zeta_{t_2})\nonumber\\*
\text{ s.t. }&y_{t+1} = g_{t}(y_{t}, v_{t}; \xi_{t}), &\forall t_1 \leq t < t_2,\nonumber\\*
&s_t(y_t, v_t; \xi_t) \leq 0, &\forall t_1 \leq t < t_2,\\*
&y_{t_1} = z,\nonumber
\end{align}
and a corresponding optimal solution as $\psi_{t_1}^{t_2}(z, \xi_{t_1:t_2-1}, \zeta_{t_2}; F)$. We shall use the shorthand notation $\psi_{t_1}^{t_2}(z, \xi_{t_1:t_2}; F) \coloneqq \psi_{t_1}^{t_2}(z, \xi_{t_1:t_2-1}, \xi_{t_2}; F)$ when the context is clear.
\end{definition}
Note that the formulation of the FTOCP in \Cref{def:FTOCP} does not include a terminal constraint set. To compensate for this, we allow the terminal cost $F(\cdot; \zeta_{t_2})$ to take value $+\infty$ in some subset of $\mathbb{R}^n$, and $\zeta_{t_2}$ is not necessarily an element in $\varXi_{t_2}$. For example, a terminal cost function that we frequently use later is the indicator function of the terminal parameter $\zeta_{t_2}$, where $\zeta_{t_2} \in \mathbb{R}^n$. We use $\mathbb{I}$ to denote such indicator terminal cost (i.e., $\mathbb{I}(y_{t_2}; \zeta_{t_2}) = 0$ if $y_{t_2} = \zeta_{t_2}$ and $\mathbb{I}(y_{t_2}; \zeta_{t_2}) = +\infty$ otherwise).
Finally, given the definition of the FTOCP, we are ready to formally introduce MPC. The pseudocode of this online controller is given in Algorithm \ref{alg:mpc}. Basically, at time step $t$, $\mathsf{MPC}_k$ solves a $k$-step predictive FTOCP using the latest available parameter predictions, and commits the first control action in the solution. When there are only fewer than $k$ steps left, $\mathsf{MPC}_k$ directly solves a $(T-t)$-step FTOCP at time $t$ until the end of the horizon, using the predicted real terminal cost $F_T(\cdot; \xi_{T \mid t})$. This MPC controller (and its variants) has a wide range of real-world applications.
\vspace*{-8pt}
\begin{algorithm}[H]
\caption{Model Predictive Control ($\mathsf{MPC}_k$)}\label{alg:mpc}
\begin{algorithmic}[1]
\Require Specify the terminal costs $F_t$ for $k \leq t < T$.
\For{$t = 0, 1, \ldots, T-1$}
\State $t' \gets \min\{ t+k, T \}$
\State Observe current state $x_t$ and obtain predictions $\xi_{t:t' \mid t}$.
\State Solve and commit control action $u_t := \psi_t^{t'}(x_t, \xi_{t:t'\mid t}; F_{t'})_{v_t}$.
\EndFor
\end{algorithmic}
\end{algorithm}
\begin{comment}
\begin{example}
Consider a trajectory tracking problem in a linear time-varying (LTV) system $x_{t+1} = A_t^* x_t + B_t^* u_t + w_t^*$. Suppose the trajectory to track is $\bar{x}_{0:T}$ and the cost functions $f_t(x_t, u_t) = (x_t - \bar{x}_t^*)^\top Q_t^* (x_t^* - \bar{x}_t^*) + u_t^\top R_t^* u_t$ are known exactly by the controller from the start of the game, while the dynamical system is subject to prediction errors. Thus, at time $t$, the online controller receives the noisy predictions of the LTV dynamics for the future $k$ steps, given by $A_{t:t+k\mid t}, B_{t:t+k\mid t}, w_{t:t+k\mid t}$. We also know the true dynamical system at any time step must be in some bounded sets given by $\mathcal{A}, \mathcal{B}, \mathcal{W}$ with diameters $\Delta_{\mathcal{A}}, \Delta_{\mathcal{B}},$ and $\Delta_{\mathcal{W}}$ respectively. In this case, we can set $\xi_t^* \coloneqq \frac{1}{\sqrt{3}}\left(A_t^*/\Delta_{\mathcal{A}}, B_t^*/\Delta_{\mathcal{B}}, w_t^*/\Delta_{\mathcal{W}}\right)$ and $\xi_{\tau\mid t} \coloneqq \frac{1}{\sqrt{3}}\left(A_{\tau\mid t}/\Delta_{\mathcal{A}}, B_{\tau \mid t}/\Delta_{\mathcal{B}}, w_{\tau \mid t}/\Delta_{\mathcal{W}}\right)$. Here, $\varXi_t = \frac{\mathcal{A}}{\sqrt{3}\Delta_\mathcal{A}} \times \frac{\mathcal{B}}{\sqrt{3}\Delta_\mathcal{B}} \times \frac{\mathcal{W}}{\sqrt{3}\Delta_\mathcal{W}}$ satisfies that $diam(\varXi_t) \leq 1$.
\end{example}
\end{comment}
\begin{comment}
\item $\mathcal{G}_t := \{ g_t(x_t, u_t; \xi_t) \mid \xi_t \in \varXi_t \}$ is a family of dynamics functions parameterized by $\xi_t$, working as valid predictions of $g_t$. The true parameter $\xi_t^* \in \varXi_t$ is unknown to the online controller, and we stick to the convention that $g_t(x_t, u_t; \xi_t^*) = g_t(x_t, u_t)$.
\begin{itemize}
\item The online controller knows $g_{0:T-1}$ from time step $0$, but it does not know the true parameters $\xi_{0:T-1}^*$. At time step $t$, the online controller has access to a noisy prediction of these parameters for future $k$ time steps, given by $\xi_{t:t+k-1\mid t}$.
\item In linear systems we have decomposition $g_t(x_t, u_t; \xi_t) = A_t(\xi_t) x_t + B_t(\xi_t) u_t + w_t(\xi_t)$.
\end{itemize}
\end{comment}
\begin{comment}
We make the following standard assumptions on the system:
\begin{assumption}\label{assump:system}
For time steps $t = 0, 1, \ldots, T$, we assume $f_t(\cdot, \cdot; \delta_t)$ is $\mu_f$-strongly convex and $\ell_f$-smooth in $(x_t, u_t)$ for any fixed $\delta_t \in \varDelta$, and $g_t(\cdot, \cdot; \xi_t)$ is $L_g$-Lipschitz in $(x_t, u_t)$ for any fixed $\xi_t \in \varXi$. Further, we assume that the terminal cost function $F_T$ is convex and $\ell_F$-smooth, and the constraint sets $\mathcal{X}_t$ and $\mathcal{U}_t$ are both convex.
\end{assumption}
\end{comment}
\section{The Pipeline: Bounded Regret via Perturbation Analysis}\label{sec:pipeline}
The goal of this section is to give an overview of a novel analysis pipeline that converts a perturbation bound into a bound on the dynamic regret. We begin by highlighting the form of perturbation bounds required in the pipeline, and then describe the 3-step process of applying the pipeline. In subsequent sections, we apply this pipeline to obtain new regret bounds for MPC in different settings.
\subsection{Per-Step Error and Perturbation Bounds}
A key challenge when comparing the performance of an online controller against the offline optimal trajectory is that the online controller's state $x_t$ is different from the offline optimal state $x_t^*$ at time step $t$. Due to such discrepancy in states, we cannot simply evaluate the online controller's action $u_t$ via comparison against the offline optimal action $u_t^*$. To address this challenge, our pipeline uses the notion of per-step error (\Cref{def:per-step-error}) inspired by the performance difference lemma and its proofs in reinforcement learning (RL) \cite{lin2021perturbation}. Specifically, we compare $u_t$ to the clairvoyant optimal action one may adopt at the same state $x_t$ if all true future parameters $\xi_{t:T}^*$ are known, which leads to the definition of \textit{per-step error} as follows.
\begin{definition}\label{def:per-step-error}
The per-step error $e_t$ incurred by a predictive online controller $\mathsf{ALG}$ at time step $t$ is defined as the distance between its actual action $u_t$ and the clairvoyant optimal action, i.e.,
\[e_t \coloneqq \norm{u_t - \psi_t^T(x_t, \xi_{t:T}^*; F_T)_{v_t}}, \text{ where }u_t = \mathsf{ALG}(x_t, \xi_{t:t+k\mid t}).\]
The clairvoyant optimal trajectory starting from $x_t$ is defined as $x_{t:T\mid t}^* \coloneqq \psi_t^T(x_t, \xi_{t:T}^*; F_T)_{y_{t:T}}$.
\end{definition}
Note that the clairvoyant optimal trajectory can be viewed as being generated by an MPC controller with long enough prediction horizon and exact predictions. This notion highlights the reason why MPC can compete against the clairvoyant optimal trajectory, since the per-step error in a system controlled by $\mathsf{MPC}_k$ becomes
$e_t = \norm{\psi_t^{t+k}(x_t, \xi_{t:t+k\mid t}; F_{t+k})_{v_t} - \psi_t^T(x_t, \xi_{t:T}^*; F_T)_{v_t}}.$
Intuitively, the per-step error converges to zero as the prediction horizon $k$ increases and the quality of predictions improves (i.e. $\norm{\xi_{t:t+k\mid t} - \xi_{t:t+k}^*} \to 0$).
This intuition highlights the important role of perturbation bounds in comparing online controllers against (offline) clairvoyant optimal trajectories.
As we have discussed in Section \ref{sec:intro}, many previous works \cite{xu2019exponentially, na2020superconvergence, shin2020decentralized, shin2021controllability} have established (local) decaying sensitivity/perturbation bounds for different instances of the FTOCP \eqref{equ:auxiliary_control_problem}. These bounds may take different forms, but for the application of our pipeline we require two types of perturbation bounds that are both common in the literature:
\begin{enumerate}[nosep,leftmargin=.2in,label=(\alph*)]
\item \textit{Perturbations of the parameters $\xi_{t_1:t_2}$ given a fixed initial state $z$}:
\begin{equation}\label{equ:perturbation-bound-fix-initial}
\norm{\psi_{t_1}^{t_2}\left(z, \xi_{t_1:t_2}; F\right)_{v_{t_1}} - \psi_{t_1}^{t_2}\left(z, \xi_{t_1:t_2}'; F\right)_{v_{t_1}}} \leq \left(\sum_{t=t_1}^{t_2} q_1(t - t_1) \delta_t\right) \norm{z} + \sum_{t=t_1}^{t_2} q_2(t - t_1) \delta_t,
\end{equation}
where $\delta_t \coloneqq \norm{\xi_t - \xi_t'}$ for $t \in [t_1, t_2]$, and scalar functions $q_1$ and $q_2$ satisfy
$\lim_{t\to\infty}q_i(t) = 0$, $\sum_{t=0}^\infty q_i(t) \leq C_i$ for constants $C_i \geq 1, i=1, 2$.
This perturbation bound is useful in bounding the per-step error $e_t$, as we will discuss in \Cref{lemma:pipeline-step2}.
\item \textit{Perturbation of the initial state $z$ given fixed parameters $\xi_{t_1:t_2}$}:
\begin{equation}\label{equ:perturbation-bound-fix-parameters}
\norm{\psi_{t_1}^{t_2}\left(z, \xi_{t_1:t_2}; F\right)_{y_t/v_t} - \psi_{t_1}^{t_2}\left(z', \xi_{t_1:t_2}; F\right)_{y_t/v_t}} \leq q_3(t - t_1) \norm{z - z'}, \text{ for }t \in [t_1, t_2],
\end{equation}
where the scalar function $q_3$ satisfies $\sum_{t=0}^\infty q_3(t) \leq C_3$ for some constant $C_3 \geq 1$. This bound is useful in preventing the accumulation of per-step errors $e_t$ throughout the horizon (see \Cref{thm:per-step-error-to-performance-guarantee}). Compared with \eqref{equ:perturbation-bound-fix-initial}, the right hand side of \eqref{equ:perturbation-bound-fix-parameters} has a simpler form.
\end{enumerate}
Existing perturbation bounds usually combine the above two types (\eqref{equ:perturbation-bound-fix-initial} and \eqref{equ:perturbation-bound-fix-parameters}) into a single equation that characterizes perturbations on $z$ and $\xi_{t_1:t_2}$ simultaneously, e.g., \cite{lin2021perturbation, shin2021controllability}. Here, we decompose them into two separate types because they are used in different parts of our pipeline.
\begin{comment}
\begin{equation}\label{equ:general-perturbation-bound}
\norm{\psi_{t_1}^{t_2}\left(z, \xi_{t_1:t_2}; F\right)_{y_t} - \psi_{t_1}^{t_2}\left(z', \xi_{t_1:t_2}'; F\right)_{y_t}} \leq q(t - t_1) \norm{z - z'} + \sum_{\tau = t_1}^{t_2} q(\abs{t-\tau}) \norm{\xi_\tau - \xi_\tau'},
\end{equation}
where $q$ is a decaying scalar function that satisfies $\lim_{t \to \infty} q(t) = 0$. Intuitively, this inequality implies the impact of a perturbation on the initial state $z$ or a parameter $\xi_\tau$ on a decision variable $y_t$ decays with respect to their distance ($(t-t_1)$ or $\abs{t - \tau}$) on the time axis. It is worth noticing that the right hand side of a perturbation bound can be more complicated than \eqref{equ:general-perturbation-bound} in some cases. For example, when there are prediction errors in the dynamics, the right hand side will include additional terms like the product of $\norm{\xi_\tau - \xi_\tau'}$ and $\norm{z}$ (see Section \ref{sec:unconstrained:dynamics}).
\end{comment}
\subsection{A 3-Step Pipeline from Perturbation Bounds to Regret}\label{sec:pipeline:flowchart}
\begin{wrapfigure}{r}{140pt}
\vspace*{-35pt}
\centering
\input{Figures/pipeline-simple}
\vspace*{-15pt}
\caption{Illustrative diagram of the 3-step pipeline from perturbation analysis to bounded regret.}\label{fig:flowchart}
\vspace*{-20pt}
\end{wrapfigure}
An overview of the pipeline is given in Figure \ref{fig:flowchart}, which illustrates the high-level ideas of the pipeline that starts by obtaining perturbation bounds, proceeds to bound the per-step error using perturbation bounds, and finally combines the per-step error and perturbation bounds to bound the dynamic regret. In the following we describe each step in detail.
\textbf{Step 1: Obtain the perturbation bounds given in (\ref{equ:perturbation-bound-fix-initial}) and (\ref{equ:perturbation-bound-fix-parameters}).} The form of the perturbation bounds depends heavily on the specific form of the FTOCP, and thus the derivation requires case-by-case study (e.g., see Section \ref{sec:unconstrained} and Section \ref{sec:general}).
However, off-the-shelf bounds are available in most cases, as there has been a rich literature on perturbation analysis of control systems (e.g., \cite{xu2019exponentially, na2020superconvergence, shin2020decentralized, shin2021controllability, lin2021perturbation} and the references therein). The following property summarizes precisely what is expected to be derived for bounds \eqref{equ:perturbation-bound-fix-initial} and \eqref{equ:perturbation-bound-fix-parameters} in Steps 2 and 3.
\begin{property}\label{assump:pipeline-perturbation-bounds}
Suppose there exists a positive constant $R$ such that the perturbation bound \eqref{equ:perturbation-bound-fix-initial} holds for the following specifications: with $t_1 = t$ and $t_2 = t+k$ for $t < T-k$, \eqref{equ:perturbation-bound-fix-initial} holds for $F: \mathbb{R}^n \to \mathbb{R}^n$ be the identity function $\mathbb{I}$, and
\[z \in \mathcal{B}(x_t^*, R);~ \xi_{t:t+k-1} \in \varXi_{t:t+k-1}, \xi_{t:t+k-1}' = \xi_{t:t+k-1}^*;~ \xi_{t+k}, \xi_{t+k}' \in \mathcal{B}(x_{t+k}^*, R) \subseteq \mathbb{R}^n;\]
with $t_1 = t$ and $t_2 = T$ for $t \geq T-k$, \eqref{equ:perturbation-bound-fix-initial} holds for
$z \in \mathcal{B}(x_t^*, R);~ \xi_{t:T} \in \varXi_{t:T}, \xi_{t:T} = \xi_{t:T}^*;~ F = F_T.$
Further, perturbation bound \eqref{equ:perturbation-bound-fix-parameters} holds for any $z, z' \in \mathcal{B}(x_t^*, R)$ and $\xi_{t_1:t_2} = \xi_{t_1:t_2}^*$.
\end{property}
As a remark, note that for the first specification of Property \ref{assump:pipeline-perturbation-bounds} with $t_1 = t$ and $t_2 = t + k$, $\xi_{t+k}$ and $\xi_{t+k}'$ live in the state space $\mathbb{R}^n$ rather than $\varXi_{t+k}$ because they represent the target terminal state of the FTOCP solved by $\mathsf{MPC}_k$. Intuitively, Property \ref{assump:pipeline-perturbation-bounds} states that perturbation bounds \eqref{equ:perturbation-bound-fix-initial} and \eqref{equ:perturbation-bound-fix-parameters} hold in a small neighborhood (specifically, a ball with radius $R$) around the offline optimal trajectory $\mathsf{OPT}$, which is much weaker than the global exponentially decaying perturbation bounds required by previous work (e.g., \cite{lin2021perturbation}) in the following sense:
(i) in the general settings where the dynamical function $g_t$ is non-linear, or where there are constraints on states and actions, one cannot hope the perturbation bound to hold globally for all possible parameters \cite{shin2021exponential,shin2021controllability,na2020superconvergence};
(ii) the decay functions $\{q_i\}_{i=1, 2, 3}$ are only required to converge to zero and satisfy $\sum_{\tau = 0}^\infty q_i(\tau) \leq C_i$, which means the exponential decay rate as in \cite{lin2021perturbation} is not necessary --- in fact, polynomial decay rates can also satisfy these properties, which greatly broadens the applicability of our pipeline.
\textbf{Step 2: Bound the per-step error $\bm{e_t}$.} The core of the analysis is to apply the perturbation bounds to bound the per-step error. For $\mathsf{MPC}_k$, under Property \ref{assump:pipeline-perturbation-bounds}, this step can be done in a universal way, as summarized in \Cref{lemma:pipeline-step2} below. A complete proof of \Cref{lemma:pipeline-step2} can be found in \Cref{appendix:lemma-pipeline-step2}.
\begin{lemma}\label{lemma:pipeline-step2}
Let Property \ref{assump:pipeline-perturbation-bounds} hold. Suppose the current state $x_t$ satisfies $x_t \in \mathcal{B}(x_t^*, {R}/{C_3})$ and the terminal cost $F_{t+k}$ of $\mathsf{MPC}_k$ is set to be the indicator function of some state $\bar{y}(\xi_{t+k\mid t})$ that satisfies $\bar{y}(\xi_{t+k\mid t}) \in \mathcal{B}(x_{t+k}^*, R)$ for $t < T - k$. Then, the per-step error of $\mathsf{MPC}_k$ is bounded by
\begin{equation}\label{lemma:pipeline-step2:conclusion}
e_t \leq \sum_{\tau = 0}^{k}\left(\left(\frac{R}{C_3} + D_{x^*}\right) \cdot q_1(\tau) + q_2(\tau)\right)\rho_{t, \tau} + 2R\left(\left(\frac{R}{C_3} + D_{x^*}\right) \cdot q_1(k) + q_2(k)\right).
\end{equation}
\end{lemma}
\Cref{lemma:pipeline-step2} is a straight-forward implication of perturbation bound \eqref{equ:perturbation-bound-fix-initial} specified in Property \ref{assump:pipeline-perturbation-bounds}. To see this, for $t < T - k$, note that the per-step error $e_t$ can be bounded by
\begin{subequations}\label{lemma:pipeline-step2:e0}
\begin{align}
e_t &= \norm{\psi_t^{t+k}(x_t, \xi_{t:t+k-1\mid t}, \bar{y}(\xi_{t+k\mid t}); \mathbb{I})_{v_t} - \psi_t^T(x_t, \xi_{t:T}^*; F_T)_{v_t}} \label{lemma:pipeline-step2:e0:s0}\\
&= \norm{\psi_t^{t+k}(x_t, \xi_{t:t+k-1\mid t}, \bar{y}(\xi_{t+k\mid t}); \mathbb{I})_{v_t} - \psi_t^{t+k}(x_t, \xi_{t:t+k-1}^*, x_{t+k\mid t}^*; \mathbb{I})_{v_t}} \label{lemma:pipeline-step2:e0:s1}\\
&\leq \sum_{\tau = 0}^{k-1} \big( \norm{x_t} \cdot q_1(\tau) + q_2(\tau) \big) \rho_{t, \tau} + \big( \norm{x_t} \cdot q_1(k) + q_2(k)\big)\norm{\bar{y}(\xi_{t+k\mid t}) - x_{t+k\mid t}^*}.\label{lemma:pipeline-step2:e0:s2}
\end{align}
\end{subequations}
Here, we apply the principle of optimality to conclude that the optimal trajectory from $x_t$ to $x_{t+k\mid t}^*$ (i.e., $\psi_t^{t+k}(x_t, \xi_{t:t+k-1}^*, x_{t+k\mid t}^*; \mathbb{I})$ in \eqref{lemma:pipeline-step2:e0:s1}) is a sub-trajectory of the clairvoyant optimal trajectory from $x_t$ (i.e., $\psi_t^T(x_t, \xi_{t:T}^*; F_T)$ in \eqref{lemma:pipeline-step2:e0:s0}), and \eqref{lemma:pipeline-step2:e0:s2} is obtained by directly applying perturbation bound \eqref{equ:perturbation-bound-fix-initial}. Note that $\norm{x_t} \leq \frac{R}{C_3} + D_{x^*}$, and that both $\bar{y}(\xi_{t+k\mid t})$ and $x_{t+k\mid t}^*$ are in $\mathcal{B}(x_{t+k}^*; R)$ by assumption and by perturbation bound \eqref{equ:perturbation-bound-fix-parameters} specified in Property \ref{assump:pipeline-perturbation-bounds}, we conclude that \eqref{lemma:pipeline-step2:conclusion} hold for $t < T - k$. The case $t \geq T - k$ can be shown similarly. We defer the detailed proof to \Cref{appendix:lemma-pipeline-step2}.
\textbf{Step 3: Bound the dynamic regret by $\bm{\sum_{t=0}^{T-1} e_t^2}$.} This final step builds upon perturbation bound \eqref{equ:perturbation-bound-fix-parameters}, and aims at deriving dynamic regret bounds in a universal way, as stated in \Cref{thm:per-step-error-to-performance-guarantee} below. Specifically, under the assumption that a local decaying perturbation bound in the form of \eqref{equ:perturbation-bound-fix-parameters} holds around the offline optimal trajectory $\mathsf{OPT}$, and the property that per-step errors $e_t$ are sufficiently small, we can show that the online controller will not leave the ``safe region'' near the offline optimal trajectory as specified in Property \ref{assump:pipeline-perturbation-bounds}, and thus the dynamic regret of $\mathsf{ALG}$ is bounded as in \eqref{thm:per-step-error-to-performance-guarantee:dynamic-regret} (note that $\mathsf{ALG}$ is not confined to MPC, but is allowed to be any algorithm with bounded per-step errors). A complete proof of \Cref{thm:per-step-error-to-performance-guarantee} can be found in Appendix \ref{appendix:per-step-error-to-performance-guarantee}.
\begin{lemma}\label{thm:per-step-error-to-performance-guarantee}
Let Property \ref{assump:pipeline-perturbation-bounds} hold. If the per-step errors of $\mathsf{ALG}$ satisfy $e_\tau \leq {R}/{(C_3^2 L_g)}$ for all time steps $\tau < t$, the trajectory of $\mathsf{ALG}$ will remain close to $\mathsf{OPT}$ at time $t$, i.e. $x_t \in \mathcal{B}(x_t^*, {R}/{C_3})$.
Further, if $e_t \leq {R}/{(C_3^2 L_g)}$ for all $t < T$, the dynamic regret of $\mathsf{ALG}$ is upper bounded by
\begin{equation}\label{thm:per-step-error-to-performance-guarantee:dynamic-regret}
\mathrm{cost}(\mathsf{ALG}) - \mathrm{cost}(\mathsf{OPT}) = O\left(\sqrt{\mathrm{cost}(\mathsf{OPT})\cdot \sum_{t=0}^{T-1} e_t^2} + \sum_{t=0}^{T-1} e_t^2\right).
\end{equation}
\end{lemma}
\textbf{Summary.} Combining Steps 2 and 3 of the pipeline yields the following \textit{Pipeline Theorem} for $\mathsf{MPC}_k$ (see \Cref{thm:the-pipeline-theorem}). Basically it states that, when the prediction horizon $k$ is sufficiently large and the prediction errors $\rho_{t, \tau}$ are sufficiently small, \Cref{lemma:pipeline-step2} and \Cref{thm:per-step-error-to-performance-guarantee} can work together to make sure that $\mathsf{MPC}_k$ never leaves a $(R/C_3)$-ball around the offline optimal trajectory $\mathsf{OPT}$; thus we obtain a dynamic regret bound.
\begin{theorem}[The Pipeline Theorem]\label{thm:the-pipeline-theorem}
Let Property \ref{assump:pipeline-perturbation-bounds} hold. Suppose the terminal cost $F_{t+k}$ of $\mathsf{MPC}_k$ is set to be the indicator function of some state $\bar{y}(\xi_{t+k\mid t})$ that satisfies $\bar{y}(\xi_{t+k\mid t}) \in \mathcal{B}(x_{t+k}^*, R)$ for all time steps $t < T - k$. Further, suppose the prediction errors $\rho_{t, \tau}$ are sufficiently small and the prediction horizon $k$ is sufficiently large, such that
\[\sum_{\tau = 0}^{k}\left(\left(\frac{R}{C_3} + D_{x^*}\right) \cdot q_1(\tau) + q_2(\tau)\right)\rho_{t, \tau} + 2R\left(\left(\frac{R}{C_3} + D_{x^*}\right) \cdot q_1(k) + q_2(k)\right) \leq \frac{R}{C_3^2 L_g}.\]
Then, the trajectory of $\mathsf{MPC}_k$ will remain close to $\mathsf{OPT}$, i.e. $x_t \in \mathcal{B}(x_t^*, {R}/{C_3})$ for all time steps $t$, and the dynamic regret of $\mathsf{MPC}_k$ is upper bounded by
\begin{equation}\label{thm:the-pipeline-theorem:statement}
\mathrm{cost}(\mathsf{MPC}_k) - \mathrm{cost}(\mathsf{OPT}) = O\left(\sqrt{\mathrm{cost}(\mathsf{OPT}) \cdot E} + E\right),
\end{equation}
where $E \coloneqq \sum_{\tau = 0}^{k-1}\left(q_1(\tau) + q_2(\tau)\right)P(\tau) + \left(q_1(k)^2 + q_2(k)^2\right) T$.
\end{theorem}
The proof of \Cref{thm:the-pipeline-theorem} can be found in \Cref{appendix:the-pipeline-theorem}. To interpret the dynamic regret bound in \eqref{thm:the-pipeline-theorem:statement}, note that we have $\mathrm{cost}(\mathsf{OPT}) = O(T)$ as a result of our model assumptions. Thus, the dynamic regret of $\mathsf{ALG}$ is in the order of $\sqrt{T E} + E$. When there is no prediction error, the regret bound $O((q_1(k) + q_2(k))\cdot T)$ reproduces the result in \cite{lin2021perturbation}, and the bound will degrade as the prediction error increases. It is also worth noticing that, when the prediction power improves over time as the online controller learns the system better and $k = \Omega(\ln{T})$, the dynamic regret can be $o(T)$.
\begin{comment}
The core requirement of \Cref{thm:per-step-error-to-performance-guarantee} is the perturbation bound in \eqref{equ:perturbation-bound-fix-parameters}. While previous work has derived dynamic regret bounds via perturbation bounds in restricted settings (unconstrained LTV), e.g., \cite{lin2021perturbation}, the assumption \eqref{equ:local-sensitivity-wrt-initial-state} is much weaker than the ones in \cite{lin2021perturbation} in multiple ways. First, we only require \eqref{equ:local-sensitivity-wrt-initial-state} to hold for perturbations on the initial states. If all parameters of $\psi_t^T$ are perturbed as in \cite{lin2021perturbation}, one cannot expect to retain the decay coefficients $q$ in general (see Section \ref{sec:unconstrained:dynamics} for an example). Second, we only require \eqref{equ:local-sensitivity-wrt-initial-state} to hold in a small neighborhood around the offline optimal trajectory $\mathsf{OPT}$, while \cite{lin2021perturbation} requires the perturbation bound to hold globally. In general settings where the dynamics $g_t$ is non-linear or there are state/action constraints, one cannot hope the perturbation bound to hold globally for every possible parameters (see Section \ref{sec:general}). Finally, third, the decay function $q$ in \eqref{equ:local-sensitivity-wrt-initial-state} is only required to satisfy that $\sum_{\tau = 0}^\infty q(\tau) \leq C_3$, which means the exponential decay rate on the coefficients as in \cite{lin2021perturbation} is not necessary. Polynomial decay rates can also satisfies our assumptions, which broadens the applicability of the framework dramatically.
1) Use perturbation bound \eqref{equ:perturbation-bound-fix-parameters} to show
\begin{equation}\label{equ:pipeline-initial-bound-clairvoyant}
\norm{x_{t+k\mid t}^*} = q(k) \norm{x} + O(1).
\end{equation}
2) Use perturbation bound \eqref{equ:perturbation-bound-fix-initial} to show
\begin{equation}\label{equ:pipeline-initial-bound-e-t}
e_t \leq \left(p_1 + q_1(k)\right)\norm{x_t} + p_2 + q_2(k).
\end{equation}
3) Use perturbation bound \eqref{equ:perturbation-bound-fix-parameters} to show
\begin{equation}\label{equ:pipeline-initial-bound-x-t}
\norm{x_t} \leq O\left(\sum_{\tau=0}^{t-1} q(\tau) e_{t-1-\tau}\right) + O(1).
\end{equation}
\begin{enumerate}[leftmargin=.2in]
\item \textit{Obtain the two types of perturbation bounds given \eqref{equ:perturbation-bound-fix-initial} and \eqref{equ:perturbation-bound-fix-parameters}.} The perturbation bounds depend heavily on the specific instance of FTOCP, and so they require case-by-case study (e.g., see Section \ref{sec:unconstrained} and Section \ref{sec:general}) and can often be obtained from the existing literature (e.g., \cite{xu2019exponentially, na2020superconvergence, shin2020decentralized, shin2021controllability, lin2021perturbation}).
\item \textit{Bound the per-step error $e_t$.} The core of the analysis is to use the perturbation bounds to bound the per-step error. This can be done as follows:
\begin{enumerate}[i)]
\item Use perturbation bound \eqref{equ:perturbation-bound-fix-initial} to obtain an initial bound on $e_t$ that may involve $\norm{x_t}$, i.e.,
\begin{equation}\label{equ:pipeline-initial-bound-e-t}
e_t \leq \left(p_1 + q_1(k)\right)\norm{x_t} + p_2 + q_2(k),
\end{equation}
where $p_1, p_2$ depend on the prediction errors $\rho_{t, \tau}$, and $q_1(k), q_2(k)$ decays to $0$ as prediction horizon $k$ increases. Specific example are given in \Cref{thm:perturbation:unconstrained-LTV-pred-err-disturbance}, \Cref{thm:perturbation:unconstrained-LTV-pred-err-dynamics}, etc. \guannan{make this sound like an direct consequence of the perturbation bound (i think it is a trivial step if the perturbation bound is given) }\yiheng{Sure, Guannan, I will think about this.}
\item Use perturbation bound \eqref{equ:perturbation-bound-fix-parameters} to obtain an initial bound on $\norm{x_t}$ that involves $\{e_\tau\}_{\tau < t}$, i.e.,
\begin{equation}\label{equ:pipeline-initial-bound-x-t}
\norm{x_t} \leq O\left(\sum_{\tau=0}^{t-1} q(\tau) e_{t-1-\tau}\right) + (\text{constant}),
\end{equation}
where $q(\tau)$ decays to $0$ as $\tau$ increases. This bound can be derived by an intermediate result of \Cref{thm:per-step-error-to-performance-guarantee} when $\norm{x_t^*}$ is uniformly bounded (see Section \ref{sec:pipeline:universal-thm}).
\item Use alternating bound refinement to obtain bounds for $e_t$ and $\norm{x_t}$ that are independent with each other, i.e., $e_t \leq p_3 + q_3(k)$ and $\norm{x_t} \leq \text{(constant)}$, where $p_3$ depends on prediction errors $\rho_{t, \tau}$, and $q_3(k)$ decays to $0$ as prediction horizon $k$ increases (see Section \ref{sec:unconstrained} and \ref{sec:general}). The refinement is done by substituting \eqref{equ:pipeline-initial-bound-e-t} into \eqref{equ:pipeline-initial-bound-x-t} and do an induction. \guannan{make this sound like direct consequence. Ideally, each step in the pipeline should be something that either there is a universal result (like thm 3.1), or it is a trivial/direct step but needs to do it in a case by case manner. }
\end{enumerate}
\item \textit{Bound the dynamic regret of \mathsf{ALG} by $\sum_{t=0}^{T-1} e_t^2$.} This final step relies on the perturbation bound \eqref{equ:perturbation-bound-fix-parameters} and can be done in an universal way as we will show in \Cref{thm:per-step-error-to-performance-guarantee}. Specifically,
\begin{equation}\label{equ:pipeline-regret bound}
\mathrm{cost}(\mathsf{ALG}) - \mathrm{cost}(\mathsf{OPT}) = O\left(\sqrt{\mathrm{cost}(\mathsf{OPT}) \cdot \sum_{t=0}^{T-1}e_t^2} + \sum_{t=0}^{T-1} e_t^2\right).
\end{equation}
\end{enumerate}
\end{comment}
\begin{comment}
The bound may involve $\norm{x_t}, \norm{x_t^*}$, the prediction error $\rho_{t, \tau}$, and some additive term that decays with respect to $k$.
\item \textit{Bound the states on $\mathsf{ALG}$'s trajectory $\norm{x_t}$.} Under the assumption that decaying perturbation bound holds (Assumption \ref{assump:local-sensitivity-wrt-initial-state}), $\norm{x_t}$ can be bounded by the accumulation of previous per-step errors $e_\tau, \tau < t$ (\Cref{thm:per-step-error-to-performance-guarantee}).
\item \textit{Alternating bound refinement.} The goal is to obtain bounds for $e_t$ and $\norm{x_t}$ that are independent with each other.
\item \textit{Bound the dynamic regret of $\mathsf{ALG}$.} Under the assumption that decaying perturbation bound holds (Assumption \ref{assump:local-sensitivity-wrt-initial-state}), the total squared distance between $\mathsf{ALG}$ and $\mathsf{OPT}$ can be bounded by the sum of squared per-step errors (\Cref{thm:per-step-error-to-performance-guarantee}).
\end{comment}
\begin{comment}
For a vector $x \in \mathbb{R}^n$ and a constant $r > 0$, let $\mathcal{B}(x, r)$ denote the closed Euclidean ball with radius $r$ centered at $x$, i.e., $\mathcal{B}(x, r) = \{y \in \mathbb{R}^n \mid \norm{x - y} \leq r\}$.
\begin{assumption}\label{assump:local-sensitivity-wrt-initial-state}
There exists a constant $R$ and a function $q: \mathbb{N} \to \mathbb{R}_{\geq 0}$ such that the following conditions hold for any time step $t$: For arbitrary $z \in \mathcal{B}(x_t^*, R)$, there exists a unique optimal solution to $\iota_t^T(z, \xi_{t:T}^*; F_T)$, denoted as $\psi_t^T(z, \xi_{t:T}^*; F_T)$, that satisfies for any $h \in [t, T]$,
\begin{subequations}\label{equ:local-sensitivity-wrt-initial-state}
\begin{align}
\norm{\psi_t^T(z, \xi_{t:T}^*; F_T)_{y_h} - \psi_t^T(z', \xi_{t:T}^*; F_T)_{y_h}} &\leq q(h - t)\cdot \norm{z - z'},\\
\norm{\psi_t^T(z, \xi_{t:T}^*; F_T)_{v_h} - \psi_t^T(z', \xi_{t:T}^*; F_T)_{v_h}} &\leq q(h - t)\cdot \norm{z - z'}
\end{align}
\end{subequations}
for all $z, z' \in \mathcal{B}(x_t^*, R)$. Further, suppose the decay function $q$ satisfies that $\sum_{\tau = 0}^\infty q(\tau) \leq C_3$ for some constant $C_3 \geq 1$.
\end{assumption}
As has been utilized in \cite{lin2021perturbation, xu2019exponentially, na2020superconvergence}, we want the decay function $q(\cdot)$ to display some kind of decay property. Intuitively, any strong enough decay property would guarantee that the error injected by the per-step error will not propagate along the trajectory at a non-negligible magnitude. \guannan{I think the connection between Assump 2.1 and the per step error is a bit weak.. feels there is gap between Assump 2.1 and Thm 2.1. } \tongxin{Consider giving this assumption a name, e.g., stability/Lipschitz continuity of optimal solutions (https://epubs.siam.org/doi/epdf/10.1137/S1052623498348274) and emphasize why it is necessary. }
\yang{per-step error $e_t$ generally has the form $e_t = O((\sum_{\rho_{t,\tau}} + \lambda^k) \norm{x_t}) + O(\sum_{\rho_{t,\tau}}) + \text{decaying term in}~k$. If $\norm{x_t}$ has a constant upper bound (stability), we would have a constant upper bound on $e_t$.\\
$e_t$ bound related to $\norm{x_t}$ -> bound on $\norm{x_t}$ -> $e_t$ bound without $\norm{x_t}$}
\textbf{The fundamental theorem of perturbation-based analysis.} Now we are ready to present our main result, which contains the stability and the cost difference bound. Intuitively, if an algorithm $\mathsf{ALG}$ incurs a small enough per-step error, we shall first guarantee that the trajectory never goes out of the ball $\mathcal{B}(x^*_t, R)$ (so the trajectory is stable); then, we shall apply the general perturbation bound to establish an upper bound on the ``deviation'' of $\mathsf{ALG}$.
\end{comment}
\begin{comment}
\begin{theorem}\label{thm:per-step-error-to-performance-guarantee}
Suppose there exists a constant $R$ and a function $q: \mathbb{N} \to \mathbb{R}_{\geq 0}$ such that the following conditions hold for any time step $t$: For arbitrary $z \in \mathcal{B}(x_t^*, R)$ \footnote{For a vector $x \in \mathbb{R}^n$ and a constant $r > 0$, let $\mathcal{B}(x, r)$ denote the closed Euclidean ball with radius $r$ centered at $x$, i.e., $\mathcal{B}(x, r) = \{y \in \mathbb{R}^n \mid \norm{x - y} \leq r\}$.}, there exists a unique optimal solution to $\iota_t^T(z, \xi_{t:T}^*; F_T)$, denoted as $\psi_t^T(z, \xi_{t:T}^*; F_T)$, that satisfies for any $h \in [t, T]$,
\begin{subequations}\label{equ:local-sensitivity-wrt-initial-state}
\begin{align}
\norm{\psi_t^T(z, \xi_{t:T}^*; F_T)_{y_h} - \psi_t^T(z', \xi_{t:T}^*; F_T)_{y_h}} &\leq q(h - t)\cdot \norm{z - z'},\\
\norm{\psi_t^T(z, \xi_{t:T}^*; F_T)_{v_h} - \psi_t^T(z', \xi_{t:T}^*; F_T)_{v_h}} &\leq q(h - t)\cdot \norm{z - z'}
\end{align}
\end{subequations}
for all $z, z' \in \mathcal{B}(x_t^*, R)$, where the decay function $q$ satisfies that $\sum_{\tau = 0}^\infty q(\tau) \leq C_3$ for some constant $C_3 \geq 1$. Assume the dynamical function $g_t$ satisfies that
\[\norm{g_t(x_t, u_t; \xi_t^*) - g_t(x_t, u_t'; \xi_t^*)} \leq L_g \norm{u_t - u_t'}, \text{for any feasible } x_t, u_t, u_t'.\]
If the per-step error of a predictive online control algorithm $\mathsf{ALG}$ is sufficiently small, such that $e_t \leq \frac{R}{C_3^2 L_g}$ for all time step $t$, the trajectory of $\mathsf{ALG}$ satisfies that
\begin{equation}\label{thm:per-step-error-to-performance-guarantee:stable}
x_t \in \mathcal{B}\left(x_t^*, \frac{R}{C_3}\right) \subseteq \mathcal{B}(x_t^*, R), \text{ for } t = 1, \ldots, T.
\end{equation}
Further, the distance between $\mathsf{ALG}$'s trajectory and the offline optimal trajectory satisfies that $\norm{x_t - x_t^*} \leq L_g \sum_{\tau=0}^{t-1} q(\tau) e_{t-1-\tau}$.
If we additionally assume that every stage cost $f_t(\cdot, \cdot; \xi_t^*)$ and the terminal cost $F_T(\cdot ; \xi_T^*)$ are nonnegative, convex, and $\ell$-smooth in $(x_t, u_t)$ and $x_T$ respectively, the dynamic regret of $\mathsf{ALG}$ can be upper bounded by
\begin{equation}\label{thm:per-step-error-to-performance-guarantee:dynamic-regret}
\mathrm{cost}(\mathsf{ALG}) - \mathrm{cost}(\mathsf{OPT}) = O\left(\sqrt{\mathrm{cost}(\mathsf{OPT})\cdot \sum_{t=0}^{T-1} e_t^2} + \sum_{t=0}^{T-1} e_t^2\right).
\end{equation}
\end{theorem}
\yang{Perhaps we can reorganize the sections to make our framework more modulized? For example, shall we first prove the form of $e_t$ in this subsection under some general assumptions, and then just plug them in for the following sections? I'm thinking of this because, at least, Section 4.1 and 4.2 will use very similar results on $e_t$.} \yang{And, maybe, describe how we can get regret and competitive ratio from this fundamental theorem (in the form of corollaries, or in plain English)?}
\yiheng{I add a paragraph below about our general framework below:}
We provide a high-level intuition on how we will derive the dynamic regret bound using \Cref{thm:per-step-error-to-performance-guarantee} in the following sections: As shown in the second inequality, the goal is to show the RHS, $\sum_{t=0}^{T-1} e_t^2$, is given by the some form of $O\left(q(k) + (\text{prediction error})\right)\cdot T$. However, the challenge is that the per-step error $e_t$ usually also depends on the current state $x_t$, which creates a relationship like the chicken or the egg causality dilemma: The accumulation of per-step errors results in the deviation of $x_t$ from the offline optimal $x_t^*$, while $x_t$ with a large norm or deviation from the offline optimal can result in a large per-step error $e_t$. To break this dilemma, we will first derive a bound on the per-step $e_t$ that may depend on $\norm{x_t}$ and/or $\norm{x_t - x_t^*}$. Then, we substitute the bound for $e_t$ into \eqref{thm:per-step-error-to-performance-guarantee:e0-1} to obtain a recursive relationship on $x_t$ like
\[\norm{x_t} \leq \sum_{\tau = 0}^{t-1} \eta_\tau \norm{x_\tau} + C_3,\]
where $\sum_{\tau = 0}^{t-1} \eta_\tau \leq 1 - \delta$ for some $\delta \in (0, 1)$. This allows us to show that $\norm{x_t}$ is uniformly bounded by some constant that is independent of $\{e_\tau\}$ and does not grow w.r.t. $t$. Thus, we can use the bound on $\norm{x_t}$ to derive a bound on $e_t$. Note that we can always improve the prediction quality or increase the prediction horizon $k$ to meet the requirement on $e_t$. Finally, we use \eqref{thm:per-step-error-to-performance-guarantee:e0-2} to show the dynamic regret bound.
\end{comment}
\section{Unconstrained LTV Systems}\label{sec:unconstrained}
We now illustrate the use of the Pipeline Theorem by applying it in the context of (unconstrained) LTV systems with prediction errors, either on disturbances or the dynamical matrices.
\subsection{Prediction Errors on Disturbances}\label{sec:unconstrained:disturbances}
In this section, we consider the following special case of problem \eqref{equ:online_control_problem}, where the dynamics is LTV and the prediction error can only occur on the disturbances $w_t$:
\begin{align}\label{equ:online_control_problem:unconstrained-LTV-disturbance}
\min_{x_{0:T}, u_{0:T-1}} &\sum_{t = 0}^{T-1} \left(f_t^x(x_t) + f_t^u(u_t)\right) + F_T(x_T)\nonumber\\*
\text{ s.t. }&x_{t+1} = A_{t} x_{t} + B_{t} u_{t} + w_{t}(\xi_t^*), &\forall 0 \leq t < T,\\*
&x_0 = x(0).\nonumber
\end{align}
All necessary assumptions on the system are summarized below in Assumption \ref{assump:unconstrained:disturbances}.
\begin{assumption}\label{assump:unconstrained:disturbances}
Assume the following holds for the online control problem instance \eqref{equ:online_control_problem:unconstrained-LTV-disturbance}:
\begin{itemize}[nosep,leftmargin=.2in]
\item \textit{Cost functions:} $\{f_t^x\}_{t=0}^{T-1}, \{f_t^u\}_{t=0}^{T-1}, F_T$ are nonnegative $\mu$-strongly convex and $\ell$-smooth. And we assume $f_t^x(0) = f_t^u(0) = F_T(0) = 0$ without the loss of generality.
\item \textit{Dynamical systems:} the LTV system $\{A_t, B_t\}$ is $\sigma$-uniform controllable with controllability index $d$, and $\norm{A_t} \leq a,~ \norm{B_t} \leq b,~ \text{and}~\Vert B_t^\dagger\Vert \leq b'$ hold for all $t$, where $B_t^\dagger$ denotes the Moore–Penrose inverse of matrix $B_t$.. The detailed definitions can be found in Assumption \ref{assump:unconstrained-LTV-pred-err-disturbance} in Appendix \ref{appendix:unconstrained-LTV-disturbances}.
\item \textit{Predicted quantities:} $\norm{w_t(\xi_t)} \leq D_w$ holds for all $\xi_t \in \varXi_t$ and all $t$. For every time step $t$, $w_t(\xi_t)$ is a $L_w$-Lipschitz function in $\xi_t$, i.e.,
$\norm{w_t(\xi_t) - w_t(\xi_t')} \leq L_w \norm{\xi_t - \xi_t'}, \forall \xi_t, \xi_t' \in \varXi_t.$
\end{itemize}
\end{assumption}
Under Assumption \ref{assump:unconstrained:disturbances}, we can again apply the perturbation bounds shown in \cite{lin2021perturbation} to show Property \ref{assump:pipeline-perturbation-bounds}. In particular, we already know that for some constants $H_1 \geq 1$ and $\lambda_1 \in (0, 1)$, perturbation bounds \eqref{equ:perturbation-bound-fix-initial} and \eqref{equ:perturbation-bound-fix-parameters} hold globally for $q_1(t) = 0$, $q_2(t) = H_1 \lambda_1^t$, and $q_3(t) = H_1 \lambda_1^t$. Since both of these perturbation bounds hold globally, radius $R$ in Property \ref{assump:pipeline-perturbation-bounds} can be set arbitrarily, and we shall take $R \coloneqq \max\left\{ D_{x^*}, \frac{2L_g H_1^3}{(1 - \lambda_1)^3} \right\}$ so that \Cref{thm:the-pipeline-theorem} can be applied to $\mathsf{MPC}_k$ with terminal cost $F_{t+k}(\cdot; \xi_{t\mid t+k}) \equiv \mathbb{I}(\cdot;0)$. This leads to the following dynamic regret bound:
\begin{theorem}\label{thm:perturbation:unconstrained-LTV-pred-err-disturbance}
In the unconstrained LTV setting \eqref{equ:online_control_problem:unconstrained-LTV-disturbance}, under Assumption \ref{assump:unconstrained:disturbances}, when the prediction horizon $k$ is sufficiently large such that $k \geq \ln\left(\frac{4 H_1^3 L_g}{(1 - \lambda_1)^2}\right)/\ln(1/\lambda_1)$, the dynamic regret of $\mathsf{MPC}_k$ (Algorithm \ref{alg:mpc}) with terminal cost $F_{t+k}(\cdot; \xi_{t\mid t+k}) \equiv \mathbb{I}(\cdot; 0)$ is bounded by
$\mathrm{cost}(\mathsf{MPC}_k) - \mathrm{cost}(\mathsf{OPT}) \leq O\left(\sqrt{T \cdot \sum_{\tau = 0}^{k-1} \lambda_1^\tau P(\tau) + \lambda_1^{2k} T^2} + \sum_{\tau = 0}^{k-1} \lambda_1^\tau P(\tau)\right).$
\end{theorem}
A complete proof of \Cref{thm:perturbation:unconstrained-LTV-pred-err-disturbance} can be found in Appendix \ref{appendix:unconstrained-LTV-disturbances}. When there are no prediction errors, the bound in \Cref{thm:perturbation:unconstrained-LTV-pred-err-disturbance} reduces to $O(\lambda_1^k T)$, which reproduces the result of \cite{lin2021perturbation}. Further, it is also worth noticing that due to the form of discounted sum $\sum_{\tau=0}^{k-1} \lambda_1^{\tau} P(\tau)$, prediction errors for the near future matter more than those for the far future.
\begin{comment}
While the dynamic regret bound in \Cref{thm:perturbation:unconstrained-LTV-pred-err-disturbance} will deteriorate to $\Omega(T)$ if we do not have any prediction about the parameters $\xi_t^*$, the following result shows $\mathsf{MPC}_k$ can still achieve a constant competitive ratio in this case.
\begin{theorem}\label{thm:perturbation:unconstrained-LTV-pred-err-disturbance:CR}
Under Assumption \ref{assump:unconstrained:disturbances}, suppose $P(\tau) \leq \eta_\tau \cdot \mathrm{cost}(\mathsf{OPT})$ for $0\leq \tau <k$, then the competitive ratio of $\mathsf{MPC}_k$ is bounded by
$1 + O\left(\sqrt{\sum_{\tau = 0}^{k-1} \lambda_1^\tau \eta_\tau + \lambda_1^{2k}} + \sum_{\tau = 0}^{k-1} \lambda_1^\tau \eta_\tau\right).$
\end{theorem}
\yiheng{To do: Add discussion for \Cref{thm:perturbation:unconstrained-LTV-pred-err-disturbance:CR}.}
For this setting, we use $\mathsf{MPC}_k$ (Algorithm \ref{alg:mpc}) with the terminal cost $F_t = \mathbb{I}_0$ for $t \in [k, T-1]$. We implement the pipeline in Section \ref{sec:pipeline:flowchart} to obtain the dynamic regret bound in \Cref{thm:perturbation:unconstrained-LTV-pred-err-disturbance}:
\begin{enumerate}[leftmargin=.2in]
\item \textit{Obtain the two types of perturbation bounds given \eqref{equ:perturbation-bound-fix-initial} and \eqref{equ:perturbation-bound-fix-parameters}.} For constants $C_1 > 0, \lambda_1 \in (0, 1)$:
\begin{align*}
&\text{Form \eqref{equ:perturbation-bound-fix-initial}: } \norm{\psi_{t_1}^{t_2}\left(z, \xi_{t_1:t_2}; F\right)_{v_{t_1}} - \psi_{t_1}^{t_2}\left(z, \xi_{t_1:t_2}'; F\right)_{v_{t_1}}} \leq C_1 L_w \sum_{\tau = t_1}^{t_2} \lambda_1^{\tau - t_1} \norm{\xi_\tau - \xi_\tau'},\\
&\text{Form \eqref{equ:perturbation-bound-fix-parameters}: } \norm{\psi_{t_1}^{t_2}\left(z, \xi_{t_1:t_2}; F\right)_{y_t} - \psi_{t_1}^{t_2}\left(z', \xi_{t_1:t_2}; F\right)_{y_t}} \leq C_1 \lambda_1^{t - t_1} \norm{z - z'}, \text{ for }t\in [t_1, t_2].
\end{align*}
For the detailed form of $F$ and other constants, see Appendix \ref{appendix:unconstrained-LTV-disturbances}.
\item \textit{Bound the per-step error $e_t$.}
\begin{enumerate}[i)]
\item \textit{Initial bound on $e_t$}: $e_t \leq C_1^2 \lambda_1^{2k} \norm{x_t} + C_1 L_w \sum_{\tau=0}^{k-1}\lambda_1^{\tau} \rho_{t, \tau} + O(\lambda_1^k)$. Recall that $\rho_{t, \tau}$ is the prediction error defined in \Cref{def:pred-oracle}.
\item \textit{Initial bound on $\norm{x_t}$}: $\norm{x_t} \leq C_1 b \sum_{\tau = 0}^{t-1} \lambda_1^{\tau} e_{t-1-\tau} + O(1)$.
\item When the prediction horizon $k$ is sufficiently large, alternatively refining i) and ii) gives $e_t = O\left(\sum_{\tau=0}^{k-1}\lambda_1^{\tau} \rho_{t, \tau} + \lambda_1^k\right)$ and $\norm{x_t} = O(1)$.
\end{enumerate}
\item \textit{Bound the dynamic regret of ALG by $\sum_{t=0}^{T-1} e_t^2$.} Recall that we define $\rho(\tau) \coloneqq \sum_{t=0}^{T-\tau} \rho_{t, \tau}^2$, thus we see $\sum_{t=1}^{T-1} e_t^2 = O\left(\sum_{\tau = 0}^{k-1} \lambda_1^\tau P(\tau) + \lambda_1^{2k} T\right)$. This gives the result in \Cref{thm:perturbation:unconstrained-LTV-pred-err-disturbance}.
\end{enumerate}
We adopt the same assumptions as \cite{lin2021perturbation} (see Assumption \ref{assump:unconstrained-LTV-pred-err-disturbances} in Appendix \ref{appendix:unconstrained-LTV-disturbances} for details). \yiheng{Explain what the assumptions mean.} To handle the prediction error that was not considered in \cite{lin2021perturbation}, we need the following additional assumption:
\begin{assumption}\label{assump:Lipschitz-parameterization}
For every time step $t$, $w_t(\xi_t)$ is a $L_w$-Lipschitz function in $\xi_t$, i.e.,
\[\norm{w_t(\xi_t) - w_t(\xi_t')} \leq L_w \norm{\xi_t - \xi_t'}, \forall \xi_t, \xi_t' \in \varXi_t.\]
\end{assumption}
\end{comment}
\begin{comment}
\begin{theorem}
Under Assumption \ref{assump:Lipschitz-parameterization} and Assumption \ref{assump:unconstrained-LTV-pred-err-disturbance}, the FTOCP \eqref{equ:auxiliary_control_problem} always has a unique primal-dual optimal solution. Assumption \ref{assump:local-sensitivity-wrt-initial-state} holds for $R = +\infty$ and $q(h) = C_1 \lambda_1^h$ for some constants $C_1 > 0, \lambda_1 \in (0, 1)$. Further, if the terminal function $F_{t+k}$ of $PC_k$ is set to be $\mathbb{I}_0$, the per-step error $e_t$ of $\mathsf{MPC}_k$ is upper bounded by
\begin{equation*}
e_t \leq \begin{cases}
C_1 \left(C_1 \lambda_1^{2k} \norm{x_t - x_t^*} + L_w \sum_{\tau=0}^{k-1} \lambda_1^\tau \rho_{t, \tau} + \lambda_1^k \norm{x_{t+k}^*}\right) & \text{ if } t < T - k,\\
C_1 L_w\sum_{\tau = 0}^{T - t - 1} \lambda_1^\tau \rho_{t, \tau} & \text{ otherwise.}
\end{cases}
\end{equation*}
\end{theorem}
The proof of \Cref{thm:perturbation:unconstrained-LTV-pred-err-disturbance} can be found in Appendix \ref{appendix:unconstrained-LTV-disturbances}.
\yiheng{Make the dynamic regret bound first.}
\begin{definition}\label{def:power-of-pred-err}
Recall that $\rho_{t, \tau} \coloneqq \norm{\xi_{t+\tau\mid t} - \xi_{t+\tau}^*}$ denotes the error of predicting the true parameter after $\tau$ steps at time $t$. The power of the prediction error for predicting parameters after $\tau$ time steps is defined as
$P(\tau) = \sum_{t=0}^{T - \tau} \rho_{t, \tau}^2.$
\end{definition}
\begin{corollary}\label{coro:perturbation:unconstrained-LTV-pred-err-disturbance}
Suppose the prediction horizon $k$ is sufficiently large such that
\[k \geq \frac{1}{2} \ln\left(C_1^3 \left(1 + \frac{2 C_1 L_g^2}{1 - \lambda_1}\right)\left(1 + \frac{C_1}{1 - \lambda_1}\right) \left(1 + C_1 + \frac{L_w^2}{1 - \lambda_1}\right)\right)/\ln(1/\lambda).\]
Under the assumption that $\norm{w_t(\xi_t)} \leq D$ holds for all $\xi_t \in \varXi_t$ for all time step $t$,
the dynamic regret of $\mathsf{MPC}_k$ is upper bounded by
\[\mathrm{cost}(\mathsf{MPC}_k) - \mathrm{cost}(\mathsf{OPT}) = O\left(\sqrt{\frac{1}{T}\sum_{\tau = 0}^{k-1} \lambda_1^\tau P(\tau) + \lambda_1^{2k}}\cdot T\right).\]
If the magnitude of prediction errors satisfies $P(\tau) \leq \delta_\tau \cdot \mathrm{cost}(\mathsf{OPT})$ for $0 \leq \tau < k$, the competitive ratio of $\mathsf{MPC}_k$ is upper bounded by
\[1 + O \left(\sum_{\tau = 0}^{k-1} \lambda_1^\tau \delta_\tau + \lambda_1^{2k} + \sqrt{\sum_{\tau = 0}^{k-1} \lambda_1^\tau \delta_\tau + \lambda_1^{2k}}\right).\]
\end{corollary}
\textbf{Remark}: \yiheng{Discuss the robust MPC which predicts $\xi_{t:t+k \mid t} = 0$.} \guannan{Also discuss the exact prediction case?}
\end{comment}
\subsection{Prediction Error on Costs and Dynamical Matrices}\label{sec:unconstrained:dynamics}
We now consider prediction errors on cost functions and dynamics, rather than disturbances. Specifically, we consider the following instance of problem \eqref{equ:online_control_problem}:
\begin{align}\label{equ:online_control_problem:unconstrained-LTV-dynamics}
\min_{x_{0:T}, u_{0:T-1}} &\sum_{t = 0}^{T-1} \left( (x_t - \bar{x}_t(\xi_t^*))^\top Q_t(\xi_t^*) (x_t - \bar{x}_t(\xi_t^*)) + u_t^{\top} R_t(\xi_t^*) u_t \right) + F_T(x_T; \xi_t^*)\nonumber\\*
\text{ s.t. }&x_{t+1} = A_t(\xi_{t}^*)\cdot x_{t} + B_t(\xi_{t}^*)\cdot u_{t} + w_t(\xi_{t}^*), \hspace{6em}\forall 0 \leq t < T,\\*
&x_0 = x(0), \nonumber
\end{align}
where the terminal cost is given by $F_T(x_T; \xi_T^*) \coloneqq (x_T - \bar{x}_T(\xi_T^*))^\top P_T(\xi_T^*) (x_T - \bar{x}(\xi_T^*))$.
All necessary assumptions on the system are summarized below in Assumption \ref{assump:unconstrained:dynamics}.
\begin{assumption}\label{assump:unconstrained:dynamics}
Assume the following holds for the online control problem instance \eqref{equ:online_control_problem:unconstrained-LTV-dynamics}:
\begin{itemize}[nosep,leftmargin=.2in]
\item \textit{Cost:} $\mu I \preceq Q_t(\xi_t) \preceq \ell I, \mu I \preceq R_t(\xi_t) \preceq \ell I,$ and $\mu I \preceq P_T(\xi_T) \preceq \ell I$, $\forall \xi_t \in \varXi_t,$ $
\forall t$.
\item \textit{Dynamical systems:} both the ground-truth LTV system $\{A_t(\xi_t^*), B_t(\xi_t^*)\}_{t=0}^{T-1}$ and any predicted LTV system $\{A_t(\xi_{t+\tau\mid t}), B_t(\xi_{t+\tau\mid t})\}_{\tau = 0}^{k-1}$ (for all $\xi_t \in \varXi_t$ and all $t$) satisfy the controllability assumptions in Assumption \ref{assump:unconstrained-LTV-pred-err-dynamics} in Appendix \ref{appendix:unconstrained-LTV-dynamics}.
\item \textit{Predicted quantities:} bounds $\norm{w_t(\xi_t)} \leq D_w, \norm{\bar{x}_t(\xi_t)} \leq D_{\bar{x}}, \norm{A_t(\xi_t)} \leq a, \norm{B_t(\xi_t)} \leq b$ hold for all $\xi_t \in \varXi_t$ and all $t$. $L_A$ is a uniform Lipschitz constant such that
$\norm{A_t(\xi_t) - A_t(\xi_t')} \leq L_A \norm{\xi_t - \xi_t'}, \forall \xi_t, \xi_t' \in \varXi_t$ holds for all $t$, and $L_B, L_Q, L_R, L_{\bar{x}}, L_w$ are defined similarly.
\end{itemize}
\end{assumption}
Under Assumption \ref{assump:unconstrained:dynamics}, we can show that for some constants $H_2 \geq 1$ and $\lambda_2 \in (0, 1)$, perturbation bounds \eqref{equ:perturbation-bound-fix-initial} and \eqref{equ:perturbation-bound-fix-parameters} hold globally for $q_1(t) = H_2 \lambda_2^{2t}$, $q_2(t) = H_2 \lambda_2^t$, and $q_3(t) = H_2 \lambda_2^t$ under the specifications of Property \ref{assump:pipeline-perturbation-bounds}. Thus, Property \ref{assump:pipeline-perturbation-bounds} holds for arbitrary $R$, and we can set $R = D_x^* + D_{\bar{x}}$ so that \Cref{thm:the-pipeline-theorem} can be applied to $\mathsf{MPC}_k$ with terminal cost $F_{t+k}(\cdot; \xi_{t\mid t+k}) = \mathbb{I}(\cdot ;\bar{x}(\xi_{t\mid t+k}))$, which leads to the following dynamic regret bound:
\begin{theorem}\label{thm:perturbation:unconstrained-LTV-pred-err-dynamics}
In the unconstrained LTV setting \eqref{equ:online_control_problem:unconstrained-LTV-dynamics}, under Assumption \ref{assump:unconstrained:dynamics}, when the prediction horizon $k \geq O(1)$ \footnote{When we say $z \geq O(1)$, we mean there exists $c = O(1)$ such that $z \geq c$ holds.} and the prediction errors satisfy $\sum_{\tau=0}^k \lambda_2^{2\tau} \rho_{t, \tau} \leq \Omega(1)$ for all $t$, the dynamic regret of $\mathsf{MPC}_k$ (Algorithm \ref{alg:mpc}) with terminal cost $F_{t+k}(\cdot; \xi_{t\mid t+k}) = \mathbb{I}(\cdot ;\bar{x}(\xi_{t\mid t+k}))$ is bounded by
$\mathrm{cost}(\mathsf{MPC}_k) - \mathrm{cost}(\mathsf{OPT}) \leq O\left(\sqrt{T \cdot \sum_{\tau = 0}^{k-1} \lambda_2^\tau P(\tau) + \lambda_2^{2k} T^2} + \sum_{\tau = 0}^{k-1} \lambda_2^\tau P(\tau)\right).$
\end{theorem}
The exact constants and a complete proof of \Cref{thm:perturbation:unconstrained-LTV-pred-err-dynamics} can be found in \Cref{appendix:unconstrained-LTV-dynamics}. Compared with \Cref{thm:perturbation:unconstrained-LTV-pred-err-disturbance}, \Cref{thm:perturbation:unconstrained-LTV-pred-err-dynamics} additionally requires the discounted total prediction errors $\sum_{\tau=0}^k \lambda_2^{2\tau} \rho_{t, \tau}$ to be less than or equal to some constant. This is actually expected, and emphasizes the critical difference between the prediction errors on dynamical matrices $(A_t, B_t)$ and the prediction errors on $w_t$, since an online controller cannot even stabilize the system when the predictions on $(A_t, B_t)$ can be arbitrarily bad. It is worth noting that Assumption \ref{assump:unconstrained:dynamics} requires the uniform controllability to hold for the unknown ground-truth LTV dynamics and any predicted dynamics. The goal is to ensure the perturbation bounds for KKT matrix inverse hold in \Cref{thm:diff-inverse-sophisticated}. Intuitively, this assumption is necessary because otherwise the solution of MPC (by solving FTOCP induced by the predicted dynamics) can be unbounded. We provided two examples (Example \ref{example:inverted-pendulum} and \ref{example:frequency-regulation}) that satisfy Assumption \ref{assump:unconstrained:dynamics} while the true dynamics are unknown.
\begin{comment}
For this setting, we use $\mathsf{MPC}_k$ (Algorithm \ref{alg:mpc}) with the terminal cost $F_t = \mathbb{I}_0$ for $t \in [k, T-1]$. We implement the pipeline in Section \ref{sec:pipeline:flowchart} to obtain the dynamic regret bound in \Cref{thm:perturbation:unconstrained-LTV-pred-err-dynamics}:
\begin{enumerate}[leftmargin=.2in]
\item \textit{Obtain the two types of perturbation bounds given \eqref{equ:perturbation-bound-fix-initial} and \eqref{equ:perturbation-bound-fix-parameters}.} For constants $C_2 > 0, \lambda_2 \in (0, 1)$:
\begin{align*}
&\text{Form \eqref{equ:perturbation-bound-fix-initial}: } \norm{\psi_{t_1}^{t_2}\left(z, \xi_{t_1:t_2}; F\right)_{v_{t_1}} - \psi_{t_1}^{t_2}\left(z, \xi_{t_1:t_2}'; F\right)_{v_{t_1}}}\\
&\leq O\left(\norm{z} \cdot \sum_{t = t_1}^{t_2} \lambda_2^{2(t - t_1)} \norm{\xi_t - \xi_t'} + \sum_{t=t_1}^{t_2} \lambda_2^{t - t_1} \norm{\xi_t - \xi_t'} + \lambda_2^{t_2 - t_1} \cdot \norm{\xi_{t_2}} \cdot \sum_{t=t_1}^{t_2}\norm{\xi_t - \xi_t'}\right),\\
&\text{Form \eqref{equ:perturbation-bound-fix-parameters}: } \norm{\psi_{t_1}^{t_2}\left(z, \xi_{t_1:t_2}; F\right)_{y_t} - \psi_{t_1}^{t_2}\left(z', \xi_{t_1:t_2}; F\right)_{y_t}} \leq C_2 \lambda_2^{t - t_1} \norm{z - z'}, \text{ for }t\in [t_1, t_2].
\end{align*}
For the detailed form of $F$ and other constants, see Appendix \ref{appendix:unconstrained-LTV-disturbances}.
\item \textit{Bound the per-step error $e_t$.}
\begin{enumerate}[i)]
\item \textit{Initial bound on $e_t$}: $e_t \leq O\left(\sum_{\tau = 0}^k \lambda_2^{2\tau}\rho_{t, \tau} + k \lambda_2^{2k}\right) \norm{x_t} + O\left(\sum_{\tau = 0}^k \lambda_2^{\tau} \rho_{t, \tau} + k \lambda_2^k\right)$.
\item \textit{Initial bound on $\norm{x_t}$}: $\norm{x_t} \leq C_2 b \sum_{\tau = 0}^{t-1} \lambda_2^{\tau} e_{t-1-\tau} + O(1)$.
\item When the prediction horizon $k$ is sufficiently large and the prediction errors $\rho_{t, \tau}$ are sufficiently small, alternatively refining i) and ii) gives $e_t = O\left(\sum_{\tau=0}^{k-1}\lambda_2^{\tau} \rho_{t, \tau} + k \lambda_2^k\right)$ and $\norm{x_t} = O(1)$.
\end{enumerate}
\item \textit{Bound the dynamic regret of ALG by $\sum_{t=0}^{T-1} e_t^2$.} Recall that we define $\rho(\tau) \coloneqq \sum_{t=0}^{T-\tau} \rho_{t, \tau}^2$, thus we see $\sum_{t=1}^{T-1} e_t^2 = O\left(\sum_{\tau = 0}^{k-1} \lambda_2^\tau P(\tau) + k^2 \lambda_2^{2k} T\right)$. This gives the result in \Cref{thm:perturbation:unconstrained-LTV-pred-err-dynamics}.
\end{enumerate}
\begin{theorem}
Under Assumption \ref{assump:unconstrained-LTV-pred-err-dynamics}, the FTOCP \eqref{equ:auxiliary_control_problem} always has a unique primal-dual optimal solution. Assumption \ref{assump:local-sensitivity-wrt-initial-state} holds for $R = +\infty$ and $q(h) = C_2 \lambda_2^h$ for some constants $C_2 > 0, \lambda_2 \in (0, 1)$. Further, if the terminal function $F_{t+k}$ of $PC_k$ is set to be $\mathbb{I}_{\bar{x}_{t+k}(\xi_{t+k\mid t})}$, the per-step error $e_t$ is upper bounded by
\begin{align*}
e_t \leq \begin{cases}
O\left(L_2\sum_{\tau = 0}^k \lambda_2^{2\tau}\rho_{t, \tau} + k \lambda_2^{2k}\right) \norm{x_t} + O\left(\sum_{\tau = 0}^k \lambda_2^{\tau} \rho_{t, \tau} + k \lambda_2^k\right) & \text{ if }t< T-k,\\
O\left((L_2 + L_P)\sum_{\tau = 0}^{T-t} \lambda_2^{2\tau}\rho_{t, \tau}\right) \norm{x_t} + O\left(\sum_{\tau = 0}^{T-t} \lambda_2^{\tau} \rho_{t, \tau}\right) & \text{ otherwise,}
\end{cases}
\end{align*}
where $L_2 \coloneqq L_A + L_B + L_Q + L_R$.
\end{theorem}
The proof of \Cref{thm:perturbation:unconstrained-LTV-pred-err-dynamics} can be found in Appendix \ref{appendix:unconstrained-LTV-dynamics}.
\begin{corollary}\label{coro:perturbation:unconstrained-LTV-pred-err-dynamics}
Under the assumption that
$L_2\sum_{\tau = 0}^k \lambda_2^{2\tau}\rho_{t, \tau} + k \lambda_2^{2k} \leq \delta_2$ for all time steps $t$,
where $\delta_2 = \delta_2(C_2, \lambda_2, L_g)$ is a constant in $(0, 1)$, we have $\norm{x_t} \leq D_2$ holds for all time steps $t$. The dynamic regret of $\mathsf{MPC}_k$ is upper bounded by
\[\mathrm{cost}(MPC) - \mathrm{cost}(\mathsf{OPT}) = O\left(\sqrt{\mathrm{cost}(\mathsf{OPT}) \cdot \left(\sum_{\tau = 0}^k \lambda_2^{\tau} P(\tau) + k^2 \lambda_2^{2k} T\right)}\right).\]
\end{corollary}
\textbf{Remark:} The assumption of Corollary \ref{coro:perturbation:unconstrained-LTV-pred-err-dynamics} require that $A_t, B_t, Q_t, R_t$ must be `insensitive' to any change on $\xi_t$. While this may seem unnatural at first, we can think about the case when $A_t, B_t, Q_t, R_t, \bar{x}_t, w_t$ are parameterized by $\xi_t^A, \xi_t^B, \xi_t^Q, \xi_t^R, $
\end{comment}
\begin{comment}
C_2' \sum_{\tau = 0}^k \lambda_2^{2\tau} \rho_{t, \tau} \cdot \norm{x_t} + C_2 \lambda_2^{2k} \left(C_2' \sum_{\tau=0}^k \rho_{t, \tau} + C_2\right) \norm{x_t - \bar{x}_{t}(\xi_t^*)}\nonumber\\
&+ \left(C_2 L_w + \frac{2 C_2' (q D_{\bar{x}} + D_w)}{1 - \lambda_2}\right) \sum_{\tau = 0}^{k} \lambda_2^{\tau} \rho_{t, \tau} + \frac{2 C_2 C_2' D_{traj}}{1 - \lambda_2} \cdot \lambda_2^k \sum_{\tau=0}^k \rho_{t, \tau}\nonumber\\
&+ C_2 \lambda_2^k L_{\bar{x}} \rho_{t, k} + \frac{2 C_2^2 D_{traj}}{1 - \lambda_2} \cdot \lambda_2^k.
\end{comment}
\section{General Dynamical Systems}\label{sec:general}
We now move beyond unconstrained linear systems to constrained nonlinear systems given by the general online control problem \eqref{equ:online_control_problem} in Section \ref{sec:preliminaries}. All necessary assumptions are summarized in \Cref{assump:general-dynamical-system} in \Cref{appendix:general}. Perhaps surprisingly, decaying perturbation bounds can hold even in this case. In particular, using Theorem 4.5 in \cite{shin2021exponential}, we can show that there exists a small constant $R$ such that, for some constants $H_3 \geq 1$ and $\lambda_3 \in (0, 1)$, perturbation bounds \eqref{equ:perturbation-bound-fix-initial} and \eqref{equ:perturbation-bound-fix-parameters} hold for $q_1(t) = 0$, $q_2(t) = H_3 \lambda_3^t$, and $q_3(t) = H_3 \lambda_3^t$. Thus, Property \ref{assump:pipeline-perturbation-bounds} holds (see \Cref{appendix:general} for formal statements) and we can apply \Cref{thm:the-pipeline-theorem} to obtain the following dynamic regret bound:
\begin{theorem}\label{thm:perturbation:general-system}
In the general system \eqref{equ:online_control_problem}, under \Cref{assump:general-dynamical-system} in \Cref{appendix:general}, Property \ref{assump:pipeline-perturbation-bounds} holds for some positive constant $R$ and $q_1(t) = 0$, $q_2(t) = H_3 \lambda_3^t$, and $q_3(t) = H_3 \lambda_3^t$. Suppose the terminal cost $F_{t+k}$ of $\mathsf{MPC}_k$ is set to be the indicator function of some state $\bar{y}(\xi_{t+k\mid t})$ that satisfies $\bar{y}(\xi_{t+k\mid t}) \in \mathcal{B}(x_{t+k}^*, R)$ for $t < T - k$. Suppose the prediction errors $\rho_{t, \tau}$ are sufficiently small and the prediction horizon $k$ is sufficiently large such that
$H_3\sum_{\tau = 0}^{k-1}\lambda_3^\tau \rho_{t, \tau} + 2R H_3 \lambda_3^k \leq \frac{(1 - \lambda_3)^2 R}{H_3^2 L_g}.$
Then, the dynamic regret of $\mathsf{MPC}_k$ is upper bounded by
$\mathrm{cost}(\mathsf{MPC}_k) - \mathrm{cost}(\mathsf{OPT}) \leq O\left(\sqrt{T \cdot \sum_{\tau = 0}^{k-1} \lambda_3^\tau P(\tau) + \lambda_3^{2k} T^2} + \sum_{\tau = 0}^{k-1} \lambda_3^\tau P(\tau)\right).$
\end{theorem}
A complete proof of \Cref{thm:perturbation:general-system} can be found in \Cref{appendix:general}. An assumption in \Cref{thm:perturbation:general-system} that is difficult to satisfy in general is that the reference terminal states $\bar{y}(\xi_{t+k\mid t})$ of $\mathsf{MPC}_k$ must be close enough to the offline optimal state $x_{t+k}^*$, i.e., $\bar{y}(\xi_{t+k\mid t}) \in \mathcal{B}(x_{t+k}^*, R)$, while the offline optimal state $x_{t+k}^*$ is generally unknown. This can be achieved in some special cases, for example, when we know $\norm{\xi_t^*}$ is sufficiently small. In this case, one can first solve FTOCP $\psi_0^T\left(x_0, \mathbf{0}; F_T\right)$ and use it as a reference to set the terminal states of $\mathsf{MPC}_k$. This intuition is formally shown in \Cref{appendix:general}. Another limitation is that \Cref{thm:perturbation:general-system} is only a bound on the cost of $\mathsf{MPC}$, not its feasibility. There are many ways to guarantee recursive feasibility of $\mathsf{MPC}$ \cite{borrelli2017predictive}, which we leave as future work. We also discuss how to verify Assumption \ref{assump:general-dynamical-system} in two simple examples that arise from a simple inventory dynamics in Appendix \ref{appendix:inventory-control}. The first positive example shows that Assumption \ref{assump:general-dynamical-system} is not vacuous, and the second negative example shows exponentially decaying perturbation bounds may not hold when Assumption \ref{assump:general-dynamical-system} is not satisfied.
\begin{comment}
\subsection{Constrained LTV Systems}\label{sec:general:constrained}
In this section, we consider a general trajectory tracking problem given by \eqref{equ:online_control_problem:constrained-LQR}:
\begin{align}\label{equ:online_control_problem:constrained-LQR}
\min_{x_{0:T}, u_{0:T-1}} &\sum_{t = 0}^{T-1} \left( (x_t - \bar{x}_t)^\top Q_t (x_t - \bar{x}_t) + u_t^{\top} R_t u_t \right) + (x_T - \bar{x}_T)^\top Q_T (x_T - \bar{x}_T) \nonumber\\*
\text{ s.t. }& x_{t+1} = A_t(\xi_{t})\cdot x_{t} + B_t(\xi_{t})\cdot u_{t} + w_t(\xi_{t}),~ \forall 0 \leq t < T,\nonumber\\*
& \tilde{\varGamma}^x_{t+1} x_{t+1} + \tilde{\varGamma}^u_{t} u_{t} \geq \tilde{\gamma},~ \forall 0 \leq t < T,\nonumber\\*
&x_0 = x(0).
\end{align}
To simplify the problem, we assume that all predictions are exact
\begin{theorem}\label{thm:perturbation:general-constrained-lqr}
Under Assumption \ref{assump:general-constrained}, the FTOCP \eqref{equ:auxiliary_control_problem} always has a unique primal-dual optimal solution. Assumption \ref{assump:local-sensitivity-wrt-initial-state} holds for $R = ???$ and $q(h) = ???$ for some constants $C_2 > 0, \lambda_2 \in (0, 1)$. Further, if the terminal function $F_{t+k}$ of $PC_k$ is set to be ????, the per-step error $e_t$ is upper bounded by
\begin{align*}
e_t \leq ???
\end{align*}
\end{theorem}
The proof of \Cref{thm:perturbation:general-constrained-lqr} can be found in Appendix \ref{appendix:general-constrained-proofs}.
\begin{corollary}\label{coro:perturbation:general-constrained-lqr}
Under the assumption ????, we have $\norm{x_t} \leq D_2$ holds for all time steps $t$. The dynamic regret of $\mathsf{MPC}_k$ is upper bounded by
\[\mathrm{cost}(MPC) - \mathrm{cost}(\mathsf{OPT}) = O \left(\sqrt{\mathrm{cost}(\mathsf{OPT}) \cdot ???} \right).\]
\end{corollary}
\textbf{Remark:} Discussion on the results.
\subsection{Nonlinear Systems}\label{sec:general:nonlinear}
In this section, we consider the general unconstrained form of problem \eqref{equ:online_control_problem}:
\begin{align}\label{equ:online_control_problem:general-nonlinear}
\min_{x_{0:T}, u_{0:T-1}} &\sum_{t = 0}^{T-1} f_t(x_t, u_t; \xi_t^*) + F_T(x_T; \xi_T^*) \nonumber\\*
\text{ s.t. }&x_{t+1} = g_{t}(x_{t}, u_{t}; \xi_{t}^*), &\forall 0 \leq t < T,\nonumber\\*
&x_0 = x(0),
\end{align}
\begin{theorem}\label{thm:perturbation:general-nonlinear}
Under Assumption \ref{assump:general-nonlinear}, the FTOCP \eqref{equ:auxiliary_control_problem} always has a unique primal-dual optimal solution. Assumption \ref{assump:local-sensitivity-wrt-initial-state} holds for $R = ???$ and $q(h) = ???$ for some constants $C_2 > 0, \lambda_2 \in (0, 1)$. Further, if the terminal function $F_{t+k}$ of $PC_k$ is set to be ????, the per-step error $e_t$ is upper bounded by
\begin{align*}
e_t \leq ???
\end{align*}
\end{theorem}
The proof of \Cref{thm:perturbation:general-nonlinear} can be found in Appendix \ref{appendix:general-nonlinear-proofs}.
\begin{corollary}\label{coro:perturbation:general-nonlinear}
Under the assumption ????, we have $\norm{x_t} \leq D_2$ holds for all time steps $t$. The dynamic regret of $\mathsf{MPC}_k$ is upper bounded by
\[\mathrm{cost}(MPC) - \mathrm{cost}(\mathsf{OPT}) = O \left(\sqrt{\mathrm{cost}(\mathsf{OPT}) \cdot ???} \right).\]
\end{corollary}
\textbf{Remark:} Discussion on the results.
\end{comment}
\begin{theorem}\label{thm:exp-decay-impossible}
Consider the specific instance of \eqref{equ:auxiliary_control_problem} where $n = m = 1$. Suppose
\[A_t = B_t = 1, \mathcal{X}_t = \{x\mid -1 \leq x \leq 1\}, \mathcal{U}_t = \{u\mid -1 \leq u \leq 1\}, Q_t = 1, R_t = 0, \delta_t = 0\]
holds for all $t$. Let $w_t = 1$ if $t$ is odd, and $w_t = -1$ if $t$ is even. For any $H$ is even, we have
\[\abs{\psi_{0}^H\left(0, -\frac{1}{2}, w_{1:H-1}, 0\right)_{x_h} - \psi_{0}^H\left(0, -\frac{1}{2} + \frac{1}{2(H - 1)}, w_{1:H-1}, 0\right)_{x_h}} = \frac{1}{2(H-1)}\]
holds for any $h \in \{1, \ldots, H\}$. For any $H$ is odd, we have
\[\abs{\psi_{0}^H\left(0, \frac{1}{2}, w_{1:H-1}, 0\right)_{x_h} - \psi_{0}^H\left(0, \frac{1}{2} + \frac{1}{2H}, w_{1:H-1}, 0\right)_{x_h}} = \frac{1}{2H}\]
holds for any $h \in \{1, \ldots, H\}$.
\end{theorem}
\textbf{Remark:} \Cref{thm:exp-decay-impossible} implies that exponentially decaying perturbation bound cannot hold even for a simple class of constrained control problems. Further, the decay rate in the perturbation bound cannot be better than $\Omega(1)$.
\begin{figure}
\caption{Illustration of the perturbation lower bound in \Cref{thm:exp-decay-impossible}
\label{fig:perturbation_lower_bound}
\end{figure}
\begin{proof}[Proof of \Cref{thm:exp-decay-impossible}]
It is straightforward to verify that when $H$ is even,
\begin{align*}
\psi_{0}^H\left(0, -\frac{1}{2}, w_{1:H-1}, 0\right)_{x_{h}} &= \begin{cases}
\frac{1}{2}, \text{ if }h\equiv 1 (\text{mod }2),\\
-\frac{1}{2}, \text{ otherwise,}
\end{cases}\\
\psi_{0}^H\left(0, -\frac{1}{2}+\frac{1}{2(H-1)}, w_{1:H-1}, 0\right)_{x_h} &= \begin{cases}
\frac{1}{2} + \frac{1}{2(H-1)}, \text{ if }h\equiv 1 (\text{mod }2),\\
-\frac{1}{2} + \frac{1}{2(H-1)}, \text{ otherwise,}
\end{cases}
\end{align*}
for all $h \in \{1, \ldots, H\}$. We also see that when $H$ is odd,
\begin{align*}
\psi_{0}^H\left(0, \frac{1}{2}, w_{1:H-1}, 0\right)_{x_{h}} &= \begin{cases}
\frac{1}{2}, \text{ if }h\equiv 1 (\text{mod }2),\\
-\frac{1}{2}, \text{ otherwise,}
\end{cases}\\
\psi_{0}^H\left(0, \frac{1}{2}+\frac{1}{2H}, w_{1:H-1}, 0\right)_{x_h} &= \begin{cases}
\frac{1}{2} + \frac{1}{2H}, \text{ if }h\equiv 1 (\text{mod }2),\\
-\frac{1}{2} + \frac{1}{2H}, \text{ otherwise,}
\end{cases}
\end{align*}
for all $h \in \{1, \ldots, H\}$.
\end{proof}
Recall that in \cite{lin2021perturbation}, the authors showed that MPC can achieve a competitive ratio of $1 + O(\rho^k)$ when $\mathcal{X}_t = \mathbb{R}^n, \mathcal{U}_t = \mathbb{R}^m$. Their result heavily relies on a exponentially decaying perturbation bound on the optimal solution. Indeed, the perturbation bound is critical for the online controller's performance because \yiheng{explain the reason...}
Given that the perturbation bound cannot achieve better than $1/H$ decay rate, a natural question to ask is whether it implies a corresponding lower bound on the competitive ratio for any online controllers. The following corollary answers this question:
\begin{theorem}\label{thm:lower-bound-online-controller}
Consider a special class of the online control problem \eqref{equ:online_control_problem} where
\[A_t = B_t = 1, \mathcal{X}_t = \{x\mid -1 \leq x \leq 1\}, \mathcal{U}_t = \{u\mid -1 \leq u \leq 1\}, Q_t = 1, R_t = 0\]
for all time $t$. Suppose an online controller $ALG$ has access to $k$ steps of exact predictions of $(w_t, \delta_t)$. For arbitrary prediction horizon $k$, there exists a problem instance that satisfies Assumption \ref{assump:controllability} and Assumption \ref{assump:uniform-controllability} where
\[\frac{\mathrm{cost}(ALG)}{\mathrm{cost}(OPT)} \geq 1 + \Omega\left(\frac{1}{k^2}\right).\]
\end{theorem}
\begin{proof}[Proof of \Cref{thm:lower-bound-online-controller}]
Consider the same problem instance as constructed in \Cref{thm:exp-decay-impossible} where the initial state is $0$ and
\begin{equation*}
w_t = \begin{cases}
1 & \text{ if } t \equiv 1(\text{mod }2),\\
0 & \text{ otherwise.}
\end{cases}
\end{equation*}
After the online controller picks the $u_0$ and the system evolves to $x_1$, the adversary decides $w_{k+1}$ and $w_{k+2}$ based on $x_1$:
\begin{enumerate}
\item If $k$ is odd and $x_1 \geq \frac{1}{2} + \frac{1}{4(H - 1)}$, the adversary picks $w_{k+1} = -1, w_{k+2} = 0$;
\item If $k$ is odd and
\end{enumerate}
If $k$ is odd, the adversary picks
\begin{align*}
w_{k+1} = \begin{cases}
-1 & \text{ if } x_1 \geq \frac{1}{2} + \frac{1}{4(H - 1)},\\
0 & \text{ otherwise.}
\end{cases}
\end{align*}
By picking the terminal state in this way, we see that: If $x_1 \geq \frac{1}{2} + \frac{1}{4(H - 1)}$,
\end{proof}
\begin{comment}
\section{Exponentially Decaying Perturbation Bounds}
\subsection{Constrained SOCO}
\begin{theorem}\label{thm:SOCO-sensitivity}
Given a tuple $\left(\hat{y}_0, \hat{y}_H, \zeta_{1:H-1}^f, \zeta_{0:H-1}^c\right)$ that contains the initial state, the terminal state, and the parameters for the hitting/switching costs in this order, we consider the optimal solution of the constrained SOCO problem
\begin{align}\label{equ:large-scale-SOCO}
\hat{\psi}\left(\hat{y}_0, \hat{y}_H, \zeta_{1:H-1}^f, \zeta_{0:H-1}^c\right) \coloneqq \argmin_{\hat{x}_{1:H-1}}& \sum_{\tau=1}^{H-1} \hat{f}_\tau(\hat{x}_\tau, \zeta_\tau^f) + \sum_{\tau=1}^{H} \hat{c}_\tau(\hat{x}_\tau, \hat{x}_{\tau - 1}, \zeta_{\tau-1}^c)\nonumber\\*
\text{ s.t. }& \hat{x}_\tau \in \hat{\mathcal{X}}_\tau \subseteq \mathbb{R}^n, &\forall 0 < \tau < H,\nonumber\\*
&\hat{x}_0 = \hat{y}_0, \hat{x}_H = \hat{y}_H,
\end{align}
indexed by $1, \ldots, H-1$. Assume the hitting cost $\hat{f}_\tau: \hat{\mathcal{X}}_\tau \times \mathbb{R}^{r_f} \to \mathbb{R}$ is convex and $\hat{\ell}_f$-smooth, and $\hat{f}_\tau\left(\cdot, \zeta_\tau^f\right)$ is $\hat{\mu}_f$-strongly convex for any fixed parameter $\zeta_\tau^f$. The switching cost $\hat{c}_\tau: \hat{\mathcal{X}}_\tau \times \hat{\mathcal{X}}_{\tau-1} \times \mathbb{R}^{r_c} \to \mathbb{R}$ is convex and $\hat{\ell}_c$-smooth. Here $\hat{\mathcal{X}}_\tau$ is a convex subset of $\mathbb{R}^n$ for $\tau = 0, \ldots, H$. Then, for any $\left(\hat{y}_0, \hat{y}_H, \zeta_{1:H-1}^f, \zeta_{0:H-1}^c\right), \left(\hat{y}_0', \hat{y}_H', (\zeta_{1:H-1}^f)', (\zeta_{0:H-1}^c)'\right) \in \hat{\mathcal{X}}_0 \times \hat{\mathcal{X}}_H \times \mathbb{R}^{H \times r} \times \mathbb{R}^{H \times r}$,
\begin{align*}
&\norm{\hat{\psi}\left(\hat{y}_0, \hat{y}_H, \zeta_{1:H-1}^f, \zeta_{0:H-1}^c\right)_h - \hat{\psi}\left(\hat{y}_0', \hat{y}_H', (\zeta_{1:H-1}^f)', (\zeta_{0:H-1}^c)'\right)_h}\\
\leq{}& C_0\bigg(\lambda_0^{h}\norm{\hat{y}_0 - \hat{y}_0'} + \sum_{\tau=1}^{H-1} \lambda_0^{\abs{h - \tau}}\norm{\zeta_\tau^f - (\zeta_\tau^f)'} + \sum_{\tau = 0}^{H-1}\lambda_0^{\abs{h - \tau}}\norm{\zeta_\tau^c - (\zeta_\tau^c)'} + \lambda_0^{H-h}\norm{\hat{y}_H - \hat{y}_H'} \bigg)
\end{align*}
for all $1 \leq h \leq H-1$, where
\begin{align*}
\lambda_0 &= \lambda_0(\hat{\mu}_f, \hat{\ell}_f, \hat{\ell}_c) = \left(1 - \frac{\hat{\mu}_c}{\hat{\ell}_f + 2\hat{\ell}_c}\right)^{\frac{1}{2}},\\
C_0 &= C_0(\hat{\mu}_f, \hat{\ell}_f, \hat{\ell}_c) = \frac{\hat{\ell}_f + 2\hat{\ell}_c + \sqrt{(\hat{\ell}_f + 2\hat{\ell}_c)^2 + 4(\hat{\ell}_f + 2\hat{\ell}_c + \max(\hat{\ell}_f, \hat{\ell}_c))}}{2\hat{\mu}_f \lambda_0^{\frac{1}{2}}}.
\end{align*}
\end{theorem}
\yiheng{Compared with existing SOCO perturbation bounds, \Cref{thm:SOCO-sensitivity} are novel in the following ways:
\begin{enumerate}
\item It does not require the cost functions to have second order derivatives (Most important);
\item The constraint set $\hat{\mathcal{X}}_\tau$ can be arbitrary convex set. In our ICML submission, it has to be a polytope with non-empty interior;
\item The parameter in hitting cost function is also considered, so we can study the prediction error on hitting cost functions.
\end{enumerate}
}
A key lemma we need to show \Cref{thm:SOCO-sensitivity} is \Cref{lemma:Lipschitzness-of-global-optimal-solution} about the Lipschitzness of the global optimal solution of \eqref{equ:large-scale-SOCO}.
\begin{lemma}\label{lemma:Lipschitzness-of-global-optimal-solution}
Under the same assumption as \Cref{thm:SOCO-sensitivity}, the optimal solution $\hat{\psi}(\hat{y}_0, \zeta, \hat{y}_H)$ is $L_1$-Lipschitz in its parameters, i.e.,
\begin{align*}
&\norm{\hat{\psi}\left(\hat{y}_0, \hat{y}_H, \zeta_{1:H-1}^f, \zeta_{0:H-1}^c\right) - \hat{\psi}\left(\hat{y}_0', \hat{y}_H', (\zeta_{1:H-1}^f)', (\zeta_{0:H-1}^c)'\right)}\\
\leq{}& L_1 \sqrt{\norm{\hat{y}_0 - \hat{y}_0'}^2 + \sum_{\tau = 1}^{H - 1} \norm{\zeta_\tau^f - (\zeta_\tau^f)'}^2 + \sum_{\tau = 0}^{H - 1} \norm{\zeta_\tau^c - (\zeta_\tau^c)'}^2 + \norm{\hat{y}_H - \hat{y}_H'}^2},
\end{align*}
where $L_1 = L_1(\hat{\mu}_f, \hat{\ell}_f, \hat{\ell}_c) = \frac{\hat{\ell}_f + 2\hat{\ell}_c + \sqrt{(\hat{\ell}_f + 2\hat{\ell}_c)^2 + 4(\hat{\ell}_f + 2\hat{\ell}_c + \max(\hat{\ell}_f, \hat{\ell}_c))}}{2\hat{\mu}_f}$.
\end{lemma}
\begin{proof}[Proof of \Cref{lemma:Lipschitzness-of-global-optimal-solution}]
\input{Proofs/Lipschitz-global-opt-sol}
\end{proof}
Before showing \Cref{thm:SOCO-sensitivity}, we first recall a result about the convergence rate of gradient descent for a well-conditioned function $\hat{F}$ on a constraint set $\hat{\mathcal{Q}}$.
\begin{theorem}\label{thm:well-cond-grad-descent-with-constraints}
Let $\hat{F}$ be a $\hat{\mu}_F$-strongly convex and $\hat{\ell}_F$-smooth function on $\mathbb{R}^{n_0}$. Assume the constraint set $\hat{\mathcal{Q}} \subseteq \mathbb{R}^{n_0}$ is a closed convex set. Consider the iterative gradient-based update rule
\begin{align}\label{thm:well-cond-grad-descent-with-constraints:update-rule}
z(s+1) = z(s) - \frac{1}{\hat{\ell}_F} g_{\hat{\mathcal{Q}}}\left(z(s); \hat{\ell}_F\right), \text{ for }s = 0, 1, \ldots, \text{ with } z(0) \in \hat{\mathcal{Q}},
\end{align}
where $g_{\hat{\mathcal{Q}}}\left(z(s); \hat{\ell}_F\right)$ denotes the gradient mapping of $\hat{F}$ on $\hat{\mathcal{Q}}$, which is defined as
\begin{subequations}\label{thm:well-cond-grad-descent-with-constraints:gradient-map}
\begin{align}
z_{\hat{\mathcal{Q}}}\left(\bar{z}; \gamma\right) ={}& \argmin_{z \in \hat{\mathcal{Q}}}\left(\hat{F}(\bar{z}) + \langle \nabla\hat{F}(\bar{z}), z - \bar{z}\rangle + \frac{\gamma}{2}\norm{z - \bar{z}}^2\right),\label{thm:well-cond-grad-descent-with-constraints:gradient-map:s1}\\
g_{\hat{\mathcal{Q}}}(\bar{z}; \gamma) ={}& \gamma\left(\bar{z} - z_{\hat{\mathcal{Q}}}(\bar{z}; \gamma)\right)\label{thm:well-cond-grad-descent-with-constraints:gradient-map:s2}
\end{align}
for any fixed $\gamma > 0$. Define $z^* \coloneqq \argmin_{z \in \hat{\mathcal{Q}}}\hat{F}(z)$. Then, we have that
\[\norm{z(s) - z^*}^2 \leq \left(1 - \frac{\hat{\mu}_F}{\hat{\ell}_F}\right)^s \norm{z(0) - z^*}^2.\]
\end{subequations}
\end{theorem}
Now we come back to the proof of \Cref{thm:SOCO-sensitivity}.
\begin{proof}[Proof of \Cref{thm:SOCO-sensitivity}]
\input{Proofs/SOCO-sensitivity}
\end{proof}
\subsection{Constrained LTV System}
We will follow the two-time-scale scheme illustrated in Figure \ref{fig:LTV-reduction-to-SOCO} to reduce the constrained LTV optimization problem to a constrained SOCO problem on a larger time-scale. The new switching cost will be defined as
\[\hat{c}_\tau\left(\hat{x}_\tau, \hat{x}_{\tau-1}, \zeta_{\tau-1}^c\right) \coloneqq \iota_{(\tau-1) d}^{\tau d}\left(y_{(\tau - 1) d}, y_{\tau d}, w_{(\tau - 1)d+1:\tau d-1}, \delta_{(\tau - 1)d:\tau d-1}\right),\]
where $\hat{x}_\tau = y_{\tau d}, \hat{x}_{\tau - 1} = y_{(\tau - 1)d}$, and $\zeta_{\tau-1}^c = \left(w_{(\tau - 1)d+1:\tau d-1}, \delta_{(\tau - 1)d:\tau d-1}\right)$. We will show the new switching cost is convex and smooth by studying the property of $\psi_{t_1}^{t_2}$. To simplify the notation, we will use $\upsilon_{t_1}^{t_2}$ to denote the parameter tuple $\left(y_{t_1}, y_{t_2}, w_{t_1+1:t_2-1}, \delta_{t_1:t_2-1}\right)$.
\begin{figure}
\caption{Illustration of the reduction from constrained LTV system to constrained SOCO.}
\label{fig:LTV-reduction-to-SOCO}
\end{figure}
\begin{lemma}\label{lemma:switching-cost-opt-sol}
When $t_1$ and $t_2$ are fixed and satisfies $t_2 \in [t_1 + d, t_1 + 2d)$, $\psi_{t_1}^{t_2}$ is a well-defined function from $\mathcal{X}_{t_1} \times \mathcal{X}_{t_2} \times (\mathbb{R}^{n})^{t_2 - t_1 - 1} \times \left(\mathbb{R}^{m}\right)^{t_2 - t_1}$ to $(\mathbb{R}^{n})^{t_2 - t_1 - 1} \times \left(\mathbb{R}^{m}\right)^{t_2 - t_1}$. Further, $\psi_{t_1}^{t_2}$ is continuous and piece-wise affine in its parameter $\upsilon_{t_1}^{t_2}$, and satisfies
\[\norm{\psi_{t_1}^{t_2}\left(\upsilon_{t_1}^{t_2}\right) - \psi_{t_1}^{t_2}\left((\upsilon_{t_1}^{t_2})'\right)} \leq L_2 \norm{\upsilon_{t_1}^{t_2} - (\upsilon_{t_1}^{t_2})'}\]
where
\[L_2 = L_2()\]
\end{lemma}
\begin{proof}[Proof of \Cref{lemma:switching-cost-opt-sol}]
\input{Proofs/switching-cost-opt-sol}
\end{proof}
\end{comment}
\begin{comment}
The checklist follows the references. Please
read the checklist guidelines carefully for information on how to answer these
questions. For each question, change the default \answerTODO{} to \answerYes{},
\answerNo{}, or \answerNA{}. You are strongly encouraged to include a {\bf
justification to your answer}, either by referencing the appropriate section of
your paper or providing a brief inline description. For example:
\begin{itemize}
\item Did you include the license to the code and datasets? \answerYes{See Section~\ref{gen_inst}.}
\item Did you include the license to the code and datasets? \answerNo{The code and the data are proprietary.}
\item Did you include the license to the code and datasets? \answerNA{}
\end{itemize}
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limit. In your paper, please delete this instructions block and only keep the
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\end{comment}
\appendix
\section{Notation Summary}\label{appendix:notations}
\input{A1-notation}
\section{Assumptions Overview}\label{appendix:assumptions}
\input{A2-assumption}
\section{Proof of Lemma \ref{lemma:pipeline-step2}}\label{appendix:lemma-pipeline-step2}
\input{Proofs/lemma-step-2}
\section{Proof of Lemma \ref{thm:per-step-error-to-performance-guarantee}}\label{appendix:per-step-error-to-performance-guarantee}
\input{Proofs/per-step-error-to-performance-guarantee}
\section{Proof of Theorem \ref{thm:the-pipeline-theorem}}\label{appendix:the-pipeline-theorem}
\input{Proofs/the-pipeline-theorem}
\section{Assumptions and Proofs of Section \ref{sec:unconstrained:disturbances}}\label{appendix:unconstrained-LTV-disturbances}
\input{Proofs/unconstrained-LTV-disturbances}
\section{Assumptions and Proofs of Section \ref{sec:unconstrained:dynamics}}\label{appendix:unconstrained-LTV-dynamics}
\input{Proofs/unconstrained-LTV-dynamics}
\section{Assumptions and Proofs of Section \ref{sec:general}}\label{appendix:general}
\input{Proofs/general-dynamical-system}
\section{Inventory control with constraints}\label{appendix:inventory-control}
\input{Proofs/inventory_control}
\end{document} | math |
On April 2, 2018, Joel Ericson, Director of the Fulbright Program in Russia, Institute of International Education, Olga Petrova, Fulbright Program Officer and administrators from a number of community colleges from the USA met at KNRTU.
The visit was within a Fulbright Grant Program aimed at developing academic cooperation between Russia and the USA in the sphere of higher education. American delegation was represented by Dr. Barbara S. Abromitis (Director of Grants, College of DuPage), Dr. Warren Brown (President of North Seattle College), Dr. Julie Lavender (Vice President for Academic Affairs), Dr. Daniel Rodkin (Associate Vice President, Santa Fe College), Dr. Dean Roughton (Academic Dean, College of the Albemarle), Nadine Russell (Global Leading Director, Central Piedmont Community College).
The meeting addressed a number of topics: the history of higher education in the USA, community colleges and their role in the institutional system. The delegates emphasized that community colleges offer education on a wide range of disciplines and provide a set of competences for the students that let them work for the certain position after graduation. Another important trait of community colleges is the possibility of enrollment for the third year of the Bachelor’s Degree Programs at Higher Education Institutions. The delegation representatives discussed the cooperation issues between schools, colleges and universities regarding dual education.
The most important and interesting part of the meeting was designated to communication with the second year students from Russia and Nigeria (KNRTU Faculty of Petroleum and Petrochemistry), that are studying at bilingual program. The students introduced themselves, spoke on their studies and interests with the foreign guests. When the official part of the meeting ended, the students surrounded American professors and discussed all the questions that they were interested in.
KNRTU has had partnership with the Fulbright Program in Russia for a long time. Since 2011, KNRTU has hosted 4 Fulbright summer schools in nanomaterials and nanotechnologies and much work has been done in the sector of faculty academic exchange. | english |
विराट कोहली से कम है इस इंग्लिश खिलाड़ी की उम्र, और इसने पहले टेस्ट में बनाया नया रिकॉर्ड अनी न्यूज इंडिया
इंग्लैंड क्रिकेट टीम के कप्तान जो रूट ने बर्मिंघम में भारत के खिलाफ खेले गए पहले टेस्ट के पहले दिन ही बल्ले से नया क्रीर्तिमान रच दिया। आइए आपको इस खिलाड़ी की पर्सनल लाइफ के बारे में बताते है।
सबसे कम उम्र में रूट का करिश्मा:
इंग्लैंड के कप्तान जो रूट सबसे कम उम्र में ६ हजार टेस्ट रन पूरे करने वाले दुनिया के तीसरे बल्लेबाज बन गए। रूट ने भारत के खिलाफ पहले टेस्ट मैच में इंग्लैंड की पहली पारी के दौरान ४०वां रन पूरा करते हुए ये कामयाबी हासिल की, रूट ६ हजार टेस्ट रन पूरे करने वाले दुनिया के ६5वें और इंग्लैंड के १५वें बल्लेबाज हैं। रूट ने टीम इंडिया के खिलाफ पहली पारी में ८० रन बनाए।
७०वें मैच में बल्ले से दिखाया पराक्रम:
अपना ७०वां टेस्ट मैच खेल रहे रूट अभी २७ साल के हैं, और वो ६ हजार रन के मुकाम पर पहुंचने वाले तीसरे युवा बल्लेबाज हैं। सचिन तेंदुलकर ने 2६ साल ३३१ दिन और रूट के साथी खिलाड़ी एलिस्टेयर कुक ने २७ साल ३३ दिन में ये मुकाम हासिल किया था।
२०१२ में हुई टेस्ट क्रिकेट में एंट्री:
जोसेफ एडवर्ड रूट का जन्म ३० दिसंबर १९९० को शेफील्ड योर्कशायर में हुआ था, रूट ने अपने टेस्ट क्रिकेट की शुरुआत १३ दिसंबर २०१२ में की थी, रूट इंग्लैंड की तरफ से सीधे हाथ से बल्लेबाजी करते है।
साल २०१४ में गर्लफ्रेंड से रचाई सगाई:
रूट के पिता का नाम मैट रूट है जबकि मां का नाम हेलेन रूट है, इनके बड़े भाई का नाम बिली रूट है, जिनका नाता भी क्रिकेट से है, इसके अलावा जो रूट के २ बहनें भी है। जो रूट की गर्लफ्रेंड का नाम कैरी केटरिल है, साल २014 में इन दोनों ने सगाई कर ली थी।
विजडन क्रिकेटर का मिला अवॉर्ड:
साल २०१४ में ही जो रूट को विजडन क्रिकेट ऑफ द ईयर का खिताब भी मिला था, रूट को गिटार बजाने का शौक भी है, इनके पसंदीदा बल्लेबाज माइकल वॉन है जबकि पसंदीदा गेंदबाज जेम्स एंडरसन है। साल २०१३ में जो रूट का झगड़ा ऑस्ट्रेलियन खिलाड़ी डेविड वॉर्नर से हो गया था।
इंग्लैंड के दिग्गज तेज गेंदबाज जेम्स एंडरसन आईपीएल
डेविड वॉर्नर ने की धमाकेदार वापसी, १८ छक्कों के साथ
इंडव्सेंग: कोहली का एक और विराट
उज्जैन के लिए प्रस्थान कावड़ियों का नगर में फूलो की वर्षा करते नगरवासियो ने किया स्वागत
प्रियंका के बाद कैटरीना को मिली भारत, कहा- फिल्म की स्क्रिप्ट कमाल की है | hindi |
आम आदमी की जेब पर फिर पड़ी महंगाई की मार, आज से महंगा हुआ गैस सिलेंडर राष्ट्र चांदिका
होम / व्यवसाय / आम आदमी की जेब पर फिर पड़ी महंगाई की मार, आज से महंगा हुआ गैस सिलेंडर
दिल्ली विधानसभा चुनाव के नतीजे आने के अगले दिन ही रसोई गैस सिलेंडर महंगा हो गया है। देश की सबसे बड़ी ऑयल मार्केटिंग कंपनी इंडेन (ल्प्ग गैस सिलिन्दर प्राइस) ने इसके दाम में इफाजा किया है, जिससे आम आदमी को झटका लगा है ।सब्सिडी वाले रसोई गैस सिलेंडर में आज बुधवार १२ फरवरी से करीब १५० रुपये का इजाफा हुआ है।
दिल्ली ८५८.५० रुपये
कोलकाता ८९६.०० रुपये
मुंबई ८२९.५० रुपये
चेन्नई ८८१ रुपये
देश के बड़े महानगरों की बात करें तो बिना सब्सिडी वाले १४ किलो के रसोई गैस सिलिंडर के दाम में १४4.५० रुपये से १४9 रुपये तक की बढोतरी कर दी गई है, जो आज से लागू हो गई हैं। आपको बता दें कि इससे पहले १ जनवरी २०२० को रसोई गैस के दाम बढ़ाए गए थे। हर महीने सब्सिडी और मार्केट रेट में बदलाव होता है, लेकिन फरवरी की शुरुआत में कोई बदलाव नहीं किया गया था।
गौरतलब है कि इंडियन ऑयल देश में प्रति दिन ३० लाख इंडेन गैस सिलेंडर की सप्लाई करता है। भारत में एलपीजी सिलेंडर की कीमत दो चीजों पर निर्भर करती है। इसमें पहला है एलपीजी का इंटरनेशनल बेंचमार्क रेट और दूसरा है यूएस डॉलर और रुपये का एक्सचेंज रेट। फ्यूल रिटेलर्स एलपीजी सिलेंडर को बाजार कीमत पर बेचते हैं, लेकिन सरकार प्रत्येक परिवार को हर साल १२ सिलेंडर में सीधे सब्सिडी प्रदान करती है।
प्रेवियस कम योगी के पिता की फिर बिगड़ी तबियत, एम्स में भर्ती
नेक्स्ट टेरर फंडिंग के दो मामलों में हाफिज सईद के खिलाफ सुनवाई १८ फरवरी को
देश के सबसे बड़े सरकारी बैंक भारतीय स्टेट बैंक (सबी) के ग्राहकों के लिए बड़ी | hindi |
/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.ignite.internal.processors.cache;
import java.util.Collections;
import java.util.HashSet;
import java.util.Map;
import java.util.Set;
import javax.cache.configuration.Factory;
import javax.cache.processor.EntryProcessor;
import javax.cache.processor.EntryProcessorResult;
import javax.cache.processor.MutableEntry;
import org.apache.ignite.Ignite;
import org.apache.ignite.IgniteCache;
import org.apache.ignite.cache.CacheAtomicityMode;
import org.apache.ignite.cache.CacheMode;
import org.apache.ignite.cache.affinity.rendezvous.RendezvousAffinityFunction;
import org.apache.ignite.cache.store.CacheStore;
import org.apache.ignite.configuration.CacheConfiguration;
import org.apache.ignite.configuration.IgniteConfiguration;
import org.apache.ignite.configuration.NearCacheConfiguration;
import org.apache.ignite.internal.util.typedef.G;
import org.apache.ignite.internal.util.typedef.internal.U;
import org.apache.ignite.spi.discovery.tcp.TcpDiscoverySpi;
import org.apache.ignite.spi.discovery.tcp.ipfinder.TcpDiscoveryIpFinder;
import org.apache.ignite.spi.discovery.tcp.ipfinder.vm.TcpDiscoveryVmIpFinder;
import org.apache.ignite.testframework.junits.common.GridCommonAbstractTest;
import org.apache.ignite.transactions.Transaction;
import org.apache.ignite.transactions.TransactionConcurrency;
import org.apache.ignite.transactions.TransactionIsolation;
import org.jetbrains.annotations.Nullable;
import static org.apache.ignite.cache.CacheAtomicityMode.TRANSACTIONAL;
import static org.apache.ignite.cache.CacheMode.PARTITIONED;
import static org.apache.ignite.cache.CacheWriteSynchronizationMode.FULL_SYNC;
/**
*
*/
public abstract class IgniteCacheInvokeReadThroughAbstractTest extends GridCommonAbstractTest {
/** */
private static final TcpDiscoveryIpFinder IP_FINDER = new TcpDiscoveryVmIpFinder(true);
/** */
private static volatile boolean failed;
/** */
protected boolean client;
/** {@inheritDoc} */
@Override protected IgniteConfiguration getConfiguration(String igniteInstanceName) throws Exception {
IgniteConfiguration cfg = super.getConfiguration(igniteInstanceName);
((TcpDiscoverySpi)cfg.getDiscoverySpi()).setIpFinder(IP_FINDER);
cfg.setClientMode(client);
return cfg;
}
/** {@inheritDoc} */
@Override protected void beforeTestsStarted() throws Exception {
super.beforeTestsStarted();
failed = false;
startNodes();
}
/** {@inheritDoc} */
@Override protected void beforeTest() throws Exception {
super.beforeTest();
IgniteCacheAbstractTest.storeMap.clear();
}
/** {@inheritDoc} */
@Override protected void afterTestsStopped() throws Exception {
stopAllGrids();
super.afterTestsStopped();
}
/**
* @return Store factory.
*/
protected Factory<CacheStore> cacheStoreFactory() {
return new IgniteCacheAbstractTest.TestStoreFactory();
}
/**
* @param data Data.
* @param cacheName Cache name.
* @throws Exception If failed.
*/
protected void putDataInStore(Map<Object, Object> data, String cacheName) throws Exception {
IgniteCacheAbstractTest.storeMap.putAll(data);
}
/**
* @throws Exception If failed.
*/
protected abstract void startNodes() throws Exception;
/**
* @param ccfg Cache configuration.
* @throws Exception If failed.
*/
@SuppressWarnings("unchecked")
protected void invokeReadThrough(CacheConfiguration ccfg) throws Exception {
Ignite ignite0 = ignite(0);
ignite0.createCache(ccfg);
try {
int key = 0;
for (Ignite node : G.allGrids()) {
if (node.configuration().isClientMode() && ccfg.getNearConfiguration() != null)
node.createNearCache(ccfg.getName(), ccfg.getNearConfiguration());
}
for (Ignite node : G.allGrids()) {
log.info("Test for node: " + node.name());
IgniteCache<Object, Object> cache = node.cache(ccfg.getName());
for (int i = 0; i < 50; i++)
checkReadThrough(cache, key++, null, null);
Set<Object> keys = new HashSet<>();
for (int i = 0; i < 5; i++)
keys.add(key++);
checkReadThroughInvokeAll(cache, keys, null, null);
keys = new HashSet<>();
for (int i = 0; i < 100; i++)
keys.add(key++);
checkReadThroughInvokeAll(cache, keys, null, null);
if (ccfg.getAtomicityMode() == TRANSACTIONAL) {
for (TransactionConcurrency concurrency : TransactionConcurrency.values()) {
for (TransactionIsolation isolation : TransactionIsolation.values()) {
log.info("Test tx [concurrency=" + concurrency + ", isolation=" + isolation + ']');
for (int i = 0; i < 50; i++)
checkReadThrough(cache, key++, concurrency, isolation);
keys = new HashSet<>();
for (int i = 0; i < 5; i++)
keys.add(key++);
checkReadThroughInvokeAll(cache, keys, concurrency, isolation);
keys = new HashSet<>();
for (int i = 0; i < 100; i++)
keys.add(key++);
checkReadThroughInvokeAll(cache, keys, concurrency, isolation);
}
}
for (TransactionConcurrency concurrency : TransactionConcurrency.values()) {
for (TransactionIsolation isolation : TransactionIsolation.values()) {
log.info("Test tx2 [concurrency=" + concurrency + ", isolation=" + isolation + ']');
for (int i = 0; i < 50; i++)
checkReadThroughGetAndInvoke(cache, key++, concurrency, isolation);
}
}
}
}
ignite0.cache(ccfg.getName()).removeAll();
}
finally {
ignite0.destroyCache(ccfg.getName());
}
}
/**
* @param cache Cache.
* @param key Key.
* @param concurrency Transaction concurrency.
* @param isolation Transaction isolation.
* @throws Exception If failed.
*/
private void checkReadThrough(IgniteCache<Object, Object> cache,
Object key,
@Nullable TransactionConcurrency concurrency,
@Nullable TransactionIsolation isolation) throws Exception {
putDataInStore(Collections.singletonMap(key, key), cache.getName());
Transaction tx = isolation != null ? cache.unwrap(Ignite.class).transactions().txStart(concurrency, isolation)
: null;
try {
Object ret = cache.invoke(key, new TestEntryProcessor());
assertEquals(key, ret);
if (tx != null)
tx.commit();
}
finally {
if (tx != null)
tx.close();
}
checkValue(cache.getName(), key, (Integer)key + 1);
}
/**
* @param cache Cache.
* @param key Key.
* @param concurrency Transaction concurrency.
* @param isolation Transaction isolation.
* @throws Exception If failed.
*/
private void checkReadThroughGetAndInvoke(IgniteCache<Object, Object> cache,
Object key,
TransactionConcurrency concurrency,
TransactionIsolation isolation) throws Exception {
putDataInStore(Collections.singletonMap(key, key), cache.getName());
try (Transaction tx = cache.unwrap(Ignite.class).transactions().txStart(concurrency, isolation)) {
cache.get(key);
Object ret = cache.invoke(key, new TestEntryProcessor());
assertEquals(key, ret);
tx.commit();
}
checkValue(cache.getName(), key, (Integer)key + 1);
}
/**
* @param cache Cache.
* @param keys Key.
* @param concurrency Transaction concurrency.
* @param isolation Transaction isolation.
* @throws Exception If failed.
*/
private void checkReadThroughInvokeAll(IgniteCache<Object, Object> cache,
Set<Object> keys,
@Nullable TransactionConcurrency concurrency,
@Nullable TransactionIsolation isolation) throws Exception {
Map<Object, Object> data = U.newHashMap(keys.size());
for (Object key : keys)
data.put(key, key);
putDataInStore(data, cache.getName());
Transaction tx = isolation != null ? cache.unwrap(Ignite.class).transactions().txStart(concurrency, isolation)
: null;
try {
Map<Object, EntryProcessorResult<Object>> ret = cache.invokeAll(keys, new TestEntryProcessor());
assertEquals(ret.size(), keys.size());
for (Object key : keys) {
EntryProcessorResult<Object> res = ret.get(key);
assertNotNull(res);
assertEquals(key, res.get());
}
if (tx != null)
tx.commit();
}
finally {
if (tx != null)
tx.close();
}
for (Object key : keys)
checkValue(cache.getName(), key, (Integer)key + 1);
}
/**
* @param cacheName Cache name.
* @param key Key.
* @param val Expected value.
*/
private void checkValue(String cacheName, Object key, Object val) {
for (Ignite ignite : G.allGrids()) {
assertEquals("Unexpected value for node: " + ignite.name(),
val,
ignite.cache(cacheName).get(key));
}
assertFalse(failed);
}
/**
* @param cacheMode Cache mode.
* @param atomicityMode Atomicity mode.
* @param backups Number of backups.
* @param nearCache Near cache flag.
* @return Cache configuration.
*/
@SuppressWarnings("unchecked")
protected CacheConfiguration cacheConfiguration(CacheMode cacheMode,
CacheAtomicityMode atomicityMode,
int backups,
boolean nearCache) {
CacheConfiguration ccfg = new CacheConfiguration(DEFAULT_CACHE_NAME);
ccfg.setReadThrough(true);
ccfg.setWriteThrough(true);
ccfg.setCacheStoreFactory(cacheStoreFactory());
ccfg.setWriteSynchronizationMode(FULL_SYNC);
ccfg.setAtomicityMode(atomicityMode);
ccfg.setCacheMode(cacheMode);
ccfg.setAffinity(new RendezvousAffinityFunction(false, 32));
if (nearCache)
ccfg.setNearConfiguration(new NearCacheConfiguration());
if (cacheMode == PARTITIONED)
ccfg.setBackups(backups);
return ccfg;
}
/**
*
*/
static class TestEntryProcessor implements EntryProcessor<Object, Object, Object> {
/** {@inheritDoc} */
@Override public Object process(MutableEntry<Object, Object> entry, Object... args) {
if (!entry.exists()) {
failed = true;
fail();
}
Integer val = (Integer)entry.getValue();
if (!val.equals(entry.getKey())) {
failed = true;
assertEquals(val, entry.getKey());
}
entry.setValue(val + 1);
return val;
}
}
}
| code |
दिल्ली के साबिक वज़ीर सोमनाथ भारती को एक और नोटिस भेजी जाएगी - थे शियासत डेली
होम / क्राइम / दिल्ली के साबिक वज़ीर सोमनाथ भारती को एक और नोटिस भेजी जाएगी
दिल्ली के साबिक वज़ीर सोमनाथ भारती को एक और नोटिस भेजी जाएगी
नयी दिल्ली: दिल्ली पुलिस ने आज कहा कि वह दिल्ली के साबिक वज़ीर ( कानून) सोमनाथ भारती को दूसरी नोटिस भेजेगा. भारती पर उनकी बीवी की शिकायत की बुनियाद पर क़त्ल करने की कोशिश , घरेलू तशद्दुद और दिगर मुजरिमाना इल्ज़ामात में मामला दर्ज किया गया है. यह पहल भारती के पूछताछ के लिए मौजूद न होने के सबब की जायेगी.
पुलिस के ज्वाइंट कमिशन (जुनूबी मगरिबी) देवेन्द्र पाठक ने कहा कि, भारती जांच से बच रहे हैं और हम उन्हें दूसरा नोटिस भेजेंगे. वह कल पूछताछ के लिए नहीं आए हालांकि उन्हें एक रात पहले ही नोटिस भेजा गया था. उन्होंने दावा किया कि पहली नोटिस भेजने के बाद भारती कल ११ बजे सुबह द्वारका जुनूबी पुलिस थाने में आने के रिये राज़ी हो गए थे लेकिन उन्होंने अपना वादा पूरा नहीं किया.
पाठक ने कहा, बाद में जब पुलिस टीम मालवीय नगर वाके उनके रिहायशगाह पर गई वहां ताला लगा हुआ था. पुलिस ने उनका पता लगाने की कोशिश की लेकिन पता नहीं लग सका. उन्होंने कहा कि दूसरा नोटिस आज उनके रिहायशगाह पर चस्पा कर दिया जायेगा जिसमें उनसे फौरन जांच के लिए आने को कहा जायेगा.
बुध के रोज़ दिल्ली पुलिस ने आप लीडर के खिलाफ एफआईआर दर्ज किया था. तीन महीने पहले उनकी बीवी लिपिका मित्रा ने उनके खिलाफ घरेलू तशद्दुद का मामला दर्ज कराया था और उनके खिलाफ संगीन इल्ज़ाम लगाये थे जिसमें उनकी कलाई काटने की कोशिश भी शामिल है.
प्रेवियस गैस सिलिंडर में ब्लास्ट, ८२ लोगों की मौत
नेक्स्ट कम-सिन लड़के को बोरवेल से बाहर निकाल लिया गया | hindi |
Christmas has always been a time for sharing and so it was that I opted to call on Santa a little early this year and with a rather specific request – three teams to back for an away day treble from the teams that make up League One and League Two.
And lo and behold, the big red man has duly delivered, with the treble of Leyton Orient, Bournemouth and Southend, offering returns of £320 for anyone seeking to utilise Bwin’s free £20 bet for registering.
It could make for the perfect gift this year or at least help cover the costs of the seasonal period.
Eddie Howe has certainly spread the Christmas cheer around the south coast of late, with Bournemouth currently enjoying a run of 11 games unbeaten.
Fresh from a 3-1 away win over Carlisle in the FA Cup, the Cherries will be confident of continuing their fine string of away results, which have seen the team go unbeaten in five on the road, winning three of their last four.
By contrast, Scunthorpe have endured a few weeks to forget, claiming just two wins in their last 13 matches with Brian Laws displaying all the tact of Basil Fawlty on crack following his team’s recent capitulation at Doncaster.
With Bournemouth heading into this match unbeaten in their last two meetings with Scunny, an away win at 1/1 could be a tasty punt.
It may only be a coincidence that Leyton Orient boss Russell Slade shares his surname with that of a band known for a festive favourite, but it is certainly shaping up to be a December to remember for the O’s.
Orient are currently on an incredible run of seven straight victories, and head to Gigg Lane in confident mood, with their hosts Bury claiming just one win in their last four encounters.
Home form has hardly been much better for the Greater Manchester club, with just one win in their last three home matches.
By contrast, Leyton Orient have the best away record in the division, accruing 15 points from a possible 18 and winning each of their last five competitive matches on the road.
Expect the visitors to add to that number with an away win at 11/5.
It’s fair to say that Fleetwood Town are in a state of crisis. On a run of three consecutive defeats, the team suffered at the hands of Aldershot Town last time out in the FA Cup.
The club is also managerless after boss Micky Mellon was sacked this week and could well be in for a drubbing against Southend come Saturday.
The Shrimpers are on a nine-game unbeaten run and have won four of their last five.
Available at 3/2 for an away win, they are certainly one to watch. | english |
मुंबई भगदड़: पीयूष गोयल ने दिये उच्च स्तरीय जांच के आदेश
होम > देश > मुंबई भगदड़: पीयूष गोयल ने दिये उच्च स्तरीय जांच के आदेश
रेल मंत्री पीयूष गोयल।
नई दिल्ली (भाषा)। रेल मंत्री पीयूष गोयल ने मुंबई में एल्फिंस्टन रोड और परेल उपनगरीय रेलवे स्टेशनों को जोड़ने वाले फुटओवर ब्रिज पर मची भगदड़ में कम से कम २२ लोगों के मारे जाने की घटना की उच्च स्तरीय जांच के आदेश दिये हैं।
केंद्रीय मंत्री पीयूष गोयल ने ट्वीट किया है, अब मुंबई पहुंचा हूं। एल्फिंस्टन रोड फुटओवर ब्रिज पर मची दुर्भाग्यपूर्ण भगदड़ में मासूम जिंदगियों के नुकसान से शोक संतप्त हूं। उन्होंने लिखा है, शोक संतप्त परिवारों के प्रति मेरी गहरी संवेदनाएं। मैं घायलों के जल्दी स्वस्थ्य होने की प्रार्थना करता हूं।
ये भी पढ़ें:मुंबई के परेल रेलवे स्टेशन के फुटओवर ब्रिज पर भगदड़, २२ लोगों की मौत, ३० घायल
गोयल ने ट्वीट किया है, मैंने पश्चिम रेलवे के मुख्य सुरक्षा अधिकारी के नेतृत्व में उच्चस्तरीय जांच के आदेश दिये हैं। इससे पहले प्रधानमंत्री नरेन्द्र मोदी ने घटना पर शोक जताते हुए कहा था कि रेल मंत्री पीयूष गोयल वहां मौजूद हैं और हरसंभव सहायता सुनिश्चित कर रहे हैं।
प्रधानमंत्री मोदी ने ट्विटर पर लिखा है, मुंबई में मची भगदड़ में जिन लोगों की जान गयी है उनके प्रति मेरी गहरी संवेदनाएं। मेरी प्रार्थनाएं घायलों के साथ हैं। मुंबई में हालात पर लगातार नजर रखी जा रही है। एपीयूषगोयल मुंबई में हैं और हालात का जायजा लेते हुए हरसंभव सहायता सुनिश्चित कर रहे हैं। गौरतलब है कि आज सुबह करीब पौने ग्यारह बजे मुंबई में एल्फिंस्टन रोड और परेल उपनगरीय रेलवे स्टेशनों को जोड़ने वाले फुटओवर ब्रिज पर मची भगदड़ में कम से कम २२ लोग मारे गये हैं जबकि कई अन्य घायल हुए हैं। | hindi |
#ifndef SELECT_CFLAGS
#define SELECT_CFLAGS ""
#endif
| code |
With the release of the all new UFC uniforms by Reebok, UFC asked us to develop a :30 UFCstore.com launch spot that featured the new gear and reflected the brand’s rising status as a lifestyle brand. They needed a spot that could be used as a template for sustain spots that could shift from featuring official athlete apparel to general lifestyle gear. It had to be modular enough that their internal teams could takeover and expand upon. We developed a flexible script framework and produced a modular shot library of both lifetime and fight preparation scenarios. For this initiative I worked with UFC E-Commerce group as both agency CD and as Director for the spot. Produced at Troika. | english |
क्वेटा। पाकिस्तान के बलूचिस्तान प्रांत में एक बस पर फायरिंग की घटना हुई है। बताया जा रहा है कि आतंकी हमले में १४ लोगों की मौत हो गई है। गुरुवार को यह घटना बलूचिस्तान के दक्षिणी पश्चिमी हिस्से में हुई है। हमलावर सेना की यूनिफॉर्म में थे और यूनिर्फॉ फ्रंटियर कोर की थी। प्रांत के गृह मंत्री हैदर अली ने न्यूज एजेंसी एएफपी को यह जानकारी दी है। हैदर अली के मुताबिक हमलावरों ने मकरान कोस्टल हाइवे पर बसों को रोका और १४ लोगों की हत्या कर दी। यह बसें ओरमारा से कराची की तरफ जा रही थीं।
१५ से २० हमलावरों ने बोला हमला
अभी तक किसी भी संगठन ने इस हमले की जिम्मेदारी नहीं ली है। पिछले दिनों क्वेटा में एक हमला हुआ था जिसमें २० लोगों की मौत हो गई थी। उस हमले के एक हफ्ते के अंदर ही यह हमला होने से सनसनी मच गई है। हमला बुजी टॉप इलाके के मकरान कोस्टल हाईवे पर हुआ है। । सेना जैसी वर्दी पहने करीब १५ से २० अज्ञात बंदूकधारियों ने कराची और ग्वादर के बीच चलने वाली पांच से छह बसों को रोका। उन्होंने ब्लूचिस्तान के ओरमारा इलाके में मकरान तटीय राजमार्ग पर एक बस को रोका, यात्रियों के पहचान पत्रों की जांच की और उनमें से करीब १६ को नीचे उतार लिया।
बलूचिस्तान का बॉर्डर अफगानिस्तान और ईरान से सटा हुआ है। यह पाकिस्तान का सबसे बड़ा और गरीब प्रांत है। यहां पर कई इस्लामिक आतंकी संगठनों को बोलबाला है। हमले में एक नेवी ऑफिसर और एक कोस्ट गार्ड मेंबर की मौत हो गई है। बलूचिस्तान के मस्तुंग इलाके में २०१५ में इसी तरह की घटना हुई थी। तब हमलावरों ने कराची जाने वाली बस से २४ लोगों को उतारकर अगवा कर लिया था। उनमें से १९ लोगों की गोली मारकर हत्या कर दी गई थी।
यह भी पढ़ें-लोकसभा चुनाव की हर बड़ी खबर के बारे में जानें यहां | hindi |
It's time to get wet and wild in statement cut-out swimsuits. Rule the pool in the fierce Evita costume, featuring a square neckline, high rise leg and cutout with tie up detailing to the front. Perfect for those beach days, or you could it as a bodysuit with denim shorts. | english |
मिलावट का बढ़ता कारोबार
आज बाज़ार से सामान खरीदने जाते वक्त यह ध्यान नहीं आता कि उसकी गुणवत्ता कैसी होगी? हकीकत तो यह है कि जब हम और आप खरीददारी करने जाते हैं तो हमें प्रोडक्ट का नाम याद हो न हो, पर इसे लगाकर दिखाने वाले अदाकार का नाम बाखूबी याद रहता है, इसलिए हम दुकानदार को बड़ी खुशी से कहते हैं कि वह क्रीम देना जिसकी ऐड फलां हीरोइन ने की है। मुझे तो इसके अलावा कोई क्रीम सूट ही नहीं करती। हद तो तब हो जाती है जब यह बड़ी-बड़ी कम्पनियां पैसे के बलबूते पर अपनी हानिकारक वस्तुएं हर पांच-सात मिनट बाद टीवी पर विज्ञापनों के जरिये हमारे समक्ष पेश करती रहती हैं और चाहते न चाहते हुए हम इसके मुरीद बन जाते हैं। खाने की वस्तुएं भी विज्ञापनों की महामारी का शिकार हो रही हैं। ये बड़ी कम्पनियां मुनाफे के नाम पर सिर्फ यहां तक सीमित नहीं रहतीं, बल्कि हमारे ही द्वारा घरेलू पैदा किया दूध हमीं से खरीदकर, उसकी क्रीम निकालकर, पैकेट बनाकर हमें ही महंगे दामों पर बेचने की डगर पर चल पड़ी हैं, जिसे पीकर बच्चे बीमार हो रहे हैं। इतना ही नहीं, बाज़ार में खुले बड़े-बड़े शो रूमों, दुकानों में सजी रंग-बिरंगी मिठाइयों को हम बड़े चाव से खरीदते व खाते हैं, मगर क्या दुकानदार उनमें डाले गये हानिकारक रंग, चीनी व कैमिकल की जानकारी आपको देता है, जिसकी वजह से आप पेट की बीमारियों से ग्रस्त हो सकते हैं? मिलावट व पैसे की भूख में अंधे इन व्यापारियों ने हमारी रोज़मर्रा की चीजों को भी नहीं छोड़ा है। आज कोई भी दाल बाज़ार से खरीद लाओ तो उसमें उसी रंग व आकार के पत्थर डाल दिए जाते हैं, जिससे अक्सर लोगों में पत्थरी की शिकायत सुनने को मिलती है। पिसा हुआ गेहूं का आटा भी मिलावट से अछूता नहीं। इसमें अखरोट, चावलों का आटा व मैदा मिलाया जाता है। कभी नकल को एक कला समझ कर सराहा जाता था पर आज तो नकल व नकली के व्यापारियों ने हमसे असल चीजों के रंग, रूप व स्वाद तक को छीन लिया है। आज पैसा फैंकों व नकली दूध, नकली लस्सी, दही, शक्कर, चावल, आटा, शहद, मक्खन व सब्ज़ियां व नकली फल भी आपको मिल जायेंगे। जिन्हें देखकर आप असल को भूल कर इन्हीं के स्वाद के अनुरूप ढल जायेंगे। मिलावट के इन चोरों ने हमारी दवाओं को भी अपने मोह पाश में फंसा लिया है। इसलिए आज रोग छोटा हो या बड़ा उसमें न ही आराम आता है और न ही चैन।
मिलावट का कारोबार आज हमारे जीवन के हर पहलू से इस कद्र जुड़ चुका है कि उससे बाहर आना मुश्किल ही नहीं, असम्भव भी लगता है। पर हम कोशिश भी न करें तो यह गलत बात है। हमारी सेहत के साथ हर तरफ से खिलवाड़ करते इन कारिन्दों पर नकेल कसने का वक्त आ चुका है। आपको जहां भी गलत काम नज़र आए उसके विरुद्ध आवाज़ उठाइए। याद रखिये कि कोशिशें ही कामयाब होती हैं। इसलिए कोशिश कीजिये, धीरे-धीरे ही सही पर वक्त बदलेगा ज़रूर।
विडियो : पेट्रोल में पानी की मिलावट को लेकर लोगों ने पेट्रोल पंप पर किया हंगामा
१५ जुल, २०१९ १३:४५
विडियो : स्वास्थ्य विभाग ने ३५० किलो मिलावटी खोया किया बरामद
०६ नोव, २०१८ ११:२५
मिलावटी रिफाइंड पाम आयल से भरे दो टैंकर ज़ब्त
०८ सेप, २०१८ ०१:२६
विडियो : मिलावटी मिठाई के खिलाफ लोगों द्वारा प्रदर्शन
३१ आग, २०१८ १०:५४
८० ग्राम हेरोइन समेत लड़का-लड़की काबू | hindi |
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| code |
تِم ٲسۍ دَوان تہٕ سعید تُلۍ تَن تھال سۄرمہٕ دٲنۍ تہٕ پرٛون ژھیوٚنمُت خرقہٕ تہٕ بیوٗٹھ بٔدرٕ پٟٹھس پؠٹھ | kashmiri |
LADY MARGARET sat in her bower-door, Sewing at her silken seem, When by it came Prince Heathen then, An gae to her a gay gold ring. ‘O bonny may, what do you now?’‘Ye heathenish dog, dying for you. ‘O bonny may, ye do greet now:’‘Ye heathenish dog, but nae for you.
He turnd about, an gied a bow; She said, Begone, I love na you; When he sware by his yellow hair That he woud gar her greet fu sair.
But she sware by her milk-white skin Prince Heathen shoud gar her greet nane: But she sware by her milk-white skin Prince Heathen shoud gar her greet nane: ‘O bonny may, winna ye greet now?’‘Ye heathenish dog, nae yet for you.
‘He’s put her in a vault o stone, Where five an thirty locks hing on; Naebody there coud eer her see, Prince Heathen kept the keys him wi. But ae she cried, What shall I do!The heathenish dog has gart me rue.
He’s taen her out upon the green, Where she saw women never ane, But only him and ‘s merry young men, Till she brought hame a bonny young son.
‘I will lend you my horse’s sheet, That will row him baith head and feet.’ As soon’s she took it in her han, Tears oer her cheeks down rapping ran.
‘Ye’ll row my young son in the silk, An ye will wash him wi the milk, An lay my lady very saft, That I may see her very aft.’ When hearts are broken, bands will bow; Sae well’s he loved his lady now! | english |
package net.martenscs.client.stock.trader;
import org.eclipse.swt.graphics.Point;
import org.eclipse.ui.application.ActionBarAdvisor;
import org.eclipse.ui.application.IActionBarConfigurer;
import org.eclipse.ui.application.IWorkbenchWindowConfigurer;
import org.eclipse.ui.application.WorkbenchWindowAdvisor;
public class ApplicationWorkbenchWindowAdvisor extends WorkbenchWindowAdvisor {
public ApplicationWorkbenchWindowAdvisor(IWorkbenchWindowConfigurer configurer) {
super(configurer);
}
public ActionBarAdvisor createActionBarAdvisor(
IActionBarConfigurer configurer) {
return new ApplicationActionBarAdvisor(configurer);
}
public void preWindowOpen() {
IWorkbenchWindowConfigurer configurer = getWindowConfigurer();
configurer.setInitialSize(new Point(400, 300));
configurer.setShowCoolBar(false);
configurer.setShowStatusLine(false);
configurer.setShowPerspectiveBar(true);
configurer.setTitle("Stock Trade Client");
}
}
| code |
وارِیاہ زیاد مُشکِلات پیمٔتِہ وؠتراؤنِہ | kashmiri |
بس ابي ملابسهم تجننننننننننننننننننن .... *-*.. | kashmiri |
Since the bus and coach operation started in 1989, our fleet has grown from strength to strength . . .
We now operate 37 vehicles, including 11 coaches and 26 services buses.
All the front line coaches are LEZ (Low Emission Zone) making them compliant for entry into London and other major European cities. Coaches also have air conditioning for your comfort. Our executive coaches have a host of other luxury features for your added journeying pleasure.
At Marshalls everyone works together as a team, from the fitters to the drivers through to the Management.
“Please let me introduce our experienced staff”.
Simon Read, Transport Manager: Simon joined Marshalls in October 2014. Simon has been in the industry all of his life. He started as a mechanic and has worked his way up to the present role that he is in now as Transport Manager. Simon's knowledge and background will be very important to us as a company.
Kenneth Tagg, Operations Manager: Kenneth joined Marshalls in 1998 as the Office Manager and succeeded to the post of Operations Manager in 2001. He mainly looks after the day to day logistics of ensuring that vehicles and drivers are in the right place at the right time and ensuring that all passengers get the quality of service we aim to provide.
Tracey Cull, Office Administrator: Tracey is the newest member of the team, joining Marshalls in January 2018. Tracey's responsibilities include organising the day excursions programme, marketing and promotion, correspondences and general office duties.
John Marshall, Managing Director: John was the founder of the business in 1989 and has guided the business into becoming the successful operation it is today. For many years he was the main driver, but he now he then a more managerial role, getting involved with CPT and other bus and coach organisations. John still has strong links with the bus and coach industry and spends his time restoring older vehicles.
Sally Sloan, Financial Director: Sally joined Marshalls in 2001 on a part time basis to look after the accounts of the business, overseeing every penny earned and spent!
Paul Marshall, Engineering Director: Paul is in charge of looking after our fleet of vehicles and making sure that we always have enough vehicles to fulfil our requirements on a daily basis. He and his team undertake all types of engineering work from routine checking to full mechanical overhauls.
......”and now to introduce our engineers”.
Paul, Bob, Jacob, Robert & Tony along with Aaron our apprentice all maintain our fleet to DVSA standards. Our service buses are inspected every 4 weeks and coaches every 6 weeks. Our OCRS records show we are rated at the highest level of compliance. | english |
#ifndef OPENCL_NODE_PLATFORM_CONTAINER_H
#define OPENCL_NODE_PLATFORM_CONTAINER_H
#include <iostream>
#include <string>
#include <list>
#include "support.h"
#include "OpenCLPlatform.h"
#include "NodePlatformContainer.h"
using namespace std;
// ****************************************************************************
// Class: OpenCLNodePlatformContainer
//
// Purpose:
// A container for all OpenCL platforms on a node.
//
// Notes: Extends the generic node platform container class
//
// Programmer: Gabriel Marin
// Creation: September 22, 2009
//
// Modifications:
//
// ****************************************************************************
namespace SHOC {
class OpenCLNodePlatformContainer : public NodePlatformContainer<OpenCLPlatform>
{
private:
static const int MAGIC_KEY_OPENCL_NODE_CONTAINER;
public:
// constructor collects information about all platforms on this node
OpenCLNodePlatformContainer (bool do_initialize = true);
OpenCLNodePlatformContainer (const OpenCLNodePlatformContainer &ondc);
OpenCLNodePlatformContainer& operator= (const OpenCLNodePlatformContainer &ondc);
~OpenCLNodePlatformContainer () { }
void Print (ostream &os) const;
void initialize();
virtual void writeObject (ostringstream &oss) const;
virtual void readObject (istringstream &iss);
bool operator< (const OpenCLNodePlatformContainer &ndc) const;
bool operator> (const OpenCLNodePlatformContainer &ndc) const;
bool operator== (const OpenCLNodePlatformContainer &ndc) const;
};
};
#endif
| code |
We have all heard of Vitamin C, right? We've also been told to take our vitamins and I'm sure by our mothers to take good care of our skin. Obviously we all age, but it is something that tends to take us quite a few decades to really realize how much of an effect we can have on it ourselves. I was recently asked to review an Even Glow Serum - Antioxidant Skin Treatment for the company Valentia Skin Care.
I agreed quickly after reading this contained 98% natural ingredients which can help with things like an uneven skin tone, boosting collagen production which of course reduces wrinkles and fine lines. Ingredients include Organic Rosehip Seed Oil & Organic Sea Buckthorn Oil, Resistem, Hyaluronic Acid (a personal favorite ingredient), and Green Tea Extact.
Now although I haven't noticed any wrinkles just yet, I am seeing my age quickly catch up with me. I have a pretty even skin tone as well on my face but it just doesn't have that bright glow it used to when I was younger. With the sensitive skin I have, I am very weary about what i put on myself but this sounded like something I could trust putting on my face. I also decided to use this on my arms because of the uneven dark spots I have from scarring.
I love the fact that this absorbed quickly, leaving no greasy or oily film on my face and arms. I felt very refreshed especially after adding this to my face and under my eyes! I am hoping that with using this for a prolonged amount of time, I will see even more drastic changes. I even found that I had ZERO breakouts after using this like I normally would from trying something new!
Recommendation: I think the regular price is beyond expensive for this product BUT at the sale price, I would recommend buying yourself a bottle or two! This isn't a remedy or miracle product, its a natural product which is going to help keep your skin healthy. | english |
<?php
/*
* This file is part of PHPExifTool.
*
* (c) 2012 Romain Neutron <imprec@gmail.com>
*
* For the full copyright and license information, please view the LICENSE
* file that was distributed with this source code.
*/
namespace PHPExiftool\Driver\Tag\Theora;
use JMS\Serializer\Annotation\ExclusionPolicy;
use PHPExiftool\Driver\AbstractTag;
/**
* @ExclusionPolicy("all")
*/
class ImageWidth extends AbstractTag
{
protected $Id = 7;
protected $Name = 'ImageWidth';
protected $FullName = 'Theora::Identification';
protected $GroupName = 'Theora';
protected $g0 = 'Theora';
protected $g1 = 'Theora';
protected $g2 = 'Video';
protected $Type = 'int32u';
protected $Writable = false;
protected $Description = 'Image Width';
}
| code |
/**
* Created by rockyren on 14-10-25.
*/
requirejs.config({
//相对于public目录
baseUrl: '/',
paths:{
'home': 'js/home',
'jquery': 'packages/bower/jquery/dist/jquery',
'bootstrap': 'packages/bower/bootstrap/dist/js/bootstrap.min',
'common_lib': 'js/common/common_lib'
},
shim: {
'bootstrap': {
deps: ['jquery'],
exports: 'bootstrap'
}
}
});
requirejs(['jquery','bootstrap','home/home-common'],function($,bootstrap,common){
common.run();
}); | code |
سکوٹر چھِ اَکھ مشین یَتھ زٕ ٹٲر چھِ آسان - یہِ چھِ اؠکہِ جاہے پؠٹھ بیٚیِس جاے تام واتٕنَس مَنٛز مَدَتھ کران- اَمہِ سٟتؠ چھِ اس کنٹن ہُنٛد سفر منٹنن منر کران | kashmiri |
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
namespace PokemonBattleOnline.Game.Host
{
/// <summary>
/// thread unsafe, do not access properties or methods concurrently
/// </summary>
public class GameContext : IDisposable
{
public readonly int Id;
private readonly Controller Controller;
private bool gaming;
internal GameContext(int id, IGameSettings settings, IPokemonData[,][] pokemons)
{
Id = id;
Controller = new Controller(settings, pokemons);
}
public event Action<GameEvent[], InputRequest[,]> GameUpdated
{
add { Controller.GameUpdated += value; }
remove { Controller.GameUpdated -= value; }
}
public event Action GameEnd
{
add { Controller.GameEnd += value; }
remove { Controller.GameEnd -= value; }
}
public event Action<int[,]> TimeUp
{
add { Controller.Timer.TimeUp += value; }
remove { Controller.Timer.TimeUp -= value; }
}
public event Action<bool[,]> WaitingNotify
{
add { Controller.Timer.WaitingNotify += value; }
remove { Controller.Timer.WaitingNotify -= value; }
}
public event Action Error;
public IGameSettings Settings
{ get { return Controller.GameSettings; } }
public void Start()
{
try
{
gaming = true;
Controller.StartGameLoop(); //想用异步...
}
catch
{
Error();
}
}
public void TryContinue()
{
try
{
Controller.TryContinueGameLoop();
}
catch
{
Error();
}
}
private bool Input(IInput input, Controller controller, Tile tile)
{
bool r = false;
if (input.SendOutIndex > 0) r = controller.InputSendOut(tile, input.SendOutIndex);
else
{
var pm = tile.Pokemon;
if (input.Move > 0)
{
foreach (MoveProxy m in pm.Moves)
{
if (m.MoveE.Id == input.Move)
{
Tile target = input.TargetTeam > 0 ? controller.Board[input.TargetTeam - 1][input.TargetX - 1] : null;
r = controller.InputSelectMove(m, target, input.Mega, input.Zmove);
break;
}
}
}
else r = controller.InputStruggle(pm);
}
return r;
}
public bool InputAction(int teamId, int teamIndex, ActionInput input)
{
if (gaming)
{
if (input.GiveUp) Controller.InputGiveUp(teamId, teamIndex);
else
{
try
{
for (int i = 0; i < Controller.GameSettings.Mode.OnboardPokemonsPerPlayer(); ++i)
{
var iai = input.Get(i);
if (iai != null)
{
if (!Input(iai, Controller, Controller.Board[teamId][teamIndex + i])) return false;
}
}
}
catch
{
Error();
}
}
return Controller.CheckInputSucceed(teamId, teamIndex);
}
return false;
}
public ReportFragment GetFragment()
{
return Controller.ReportBuilder.GetFragment(); //is null possible?
}
private bool _isDisposed;
public void Dispose()
{
if (!_isDisposed)
{
_isDisposed = true;
Controller.Dispose();
}
}
}
}
| code |
موتی لال ساقی تہِ چھہ یہ بٲتھ دوان تہ بدخواہ بدلہ چھہ بدکار لیکھان۔ تسند بند چھہ یتھہ پاٹھۍ: | kashmiri |
मध्य प्रदेश असेंबली इलेक्शन्स २०१८ हिन्दी न्यूज पांच साल में भी पश्चिमी रिंग रोड का असमंजस नहीं हुआ दूर, कॉलोनियां भी नहीं हो सकीं वैध
विधानसभा क्षेत्र क्रमांक एक के दो एेसे बूथों का जायजा लिया तो कई जमीनी हकीकत सामने आई। यहां विकास तो खूब हुआ, लेकिन कुछ स्थाई मुद्दों पर पांच साल के बाद भी असमंजस की स्थिति बरकरार है।
इंदौर. विधानसभा क्षेत्र क्रमांक एक में पांच साल पहले की स्थिति को देखें तो अवैध कॉलोनियों का विकास और इनको वैध करोन का मुद्दा प्रमुख रहा। वह आज भी वहीं का वहीं है। हालांकि, विधायक और प्रतिद्वंद्वी उम्मीदवारों ने इनमें विकास के रास्ते जरूर निकाले और वोटरों के विश्वास पर खरे उतरने के प्रयास भी किए। पत्रिका ने विधानसभा क्षेत्र क्रमांक एक के दो एेसे बूथों का जायजा लिया तो कई जमीनी हकीकत सामने आई। यहां विकास तो खूब हुआ, लेकिन कुछ स्थाई मुद्दों पर पांच साल के बाद भी असमंजस की स्थिति बरकरार है। भाजपा विधायक सुदर्शन गुप्ता ने पांच साल तक क्षेत्र की जनता के साथ अपने संपर्क को जीवंत बनाए रखा। वहीं, कांग्रेस उम्मीदवार प्रदीप यादव दीपू अपनी पार्षद पत्नी के साथ पूरी तरह सक्रिय रहे। दोनों क्षेत्रों की कई समस्याओं का स्थाई हल तो निकला, लेकिन राज नगर बूथ के आसपास रहने वाले मतदाताओं के मन में पश्चिम रिंग रोड को लेकर असमंजस अभी भी बरकरार है। लोगों का कहना है, विधायक व पार्षद के बीच संवाद की कमी से कुछ मामले अटके हुए हैं। बाणगंगा मेन रोड के रहवासियों के लिए मेन रोड का चौड़ीकरण प्रमुख मुद्दा रहा।
विधायक गुप्ता को क्षेत्र के वार्ड क्रमांक ५ के बूथ क्रमांक २१८ में सर्वाधिक मत मिले थे। यह बूथ राजनगर के ई व एफ सेक्टर को मिला कर बनाया गया था। यहां 1५00 से ज्यादा मतदाता हैं। २०१३ में ७० प्रतिशत मतदान हुआ था, इसमें गुप्ता को ७९७ मत मिले थे। क्षेत्र के व्यापारी रवि प्रजापति कहते हैं, पांच साल पहले यह क्षेत्र काफी पिछड़ा था। यहां असामाजिक तत्व सक्रिय थे। चंदे व अवैध वसूली से लोग परेशान रहते थे। इस स्थिति में अब काफी सुधार आया है। विधायक के प्रयास व जन सहयोग से यहां पर पुलिस चौकी की स्थापना हुई। अपराधिक तत्वों का घूमना-फिरना बंद हुआ। रहवासी निलेश जैन का कहना है, काम तो हुआ। मूलभूत समस्याएं भी हल हुईं, लेकिन कॉलोनियों को वैध करने का मामला अभी भी अटका है। रहवासी किरण व्यास का कहना है, विधायक और पार्षद में समन्वयय नहीं होने से हमारी गली की सड़क का काम एक माह से अटका पड़ा है।
२०१३ में कांग्रेस उम्मीदवार रहे प्रदीप यादव दीपू को बाणगंगा मेन रोड के मतदाताओं से बने बूथ क्रमांक ६९ से सर्वाधिक मत मिले थे। १३८२ मतदाता वाले इस बूथ पर ६९ प्रतिशत मतदान हुआ था। दीपू को ६३३ मत मिले थे। यह बूथ वार्ड क्र. १० का हिस्सा है। यहां से पार्षद विनितिका यादव हैं। क्षेत्र के व्यापारी नरेश यादव का कहना है, पांच साल में स्थिति काफी सुधरी है। गलियां बनी हैं। गर्मी में पानी की समस्या काफी रहती है। टैंकर से पानी उपलब्ध करवाया जाता है। रहवासी राजेश कुमावत कहते हैं, हम क्या बोले। सारी स्थिति आपके सामने है। यहां कोई एेसा बड़ा मुद्दा नहीं है। सड़क-पानी-बिजली ही हमारी समस्या है। जहां तक अपराध का सवाल है, कोई विशेष फर्क नहीं पड़ा।
इस बूथ पर भाजपा विधायक सुदर्शन गुप्ता को मिले सर्वाधिक मत
बूथ नंबर - २१८
बूथ के क्षेत्र - राजनगर ई व एफ सेक्टर
कुल मतदाता - १५६३
पुरुष - ८३६
कुल वोट पड़े -११०३
भाजपा - ७९७
कांग्रेस - ७८
इस बूथ पर मिले कांग्रेस उम्मीदवार प्रदीप यादव को सर्वाधिक मत
बूथ नंबर - ६९
बूथ क्षेत्र - बाणगंगा मेन रोड
कुल मतदाता - १३८२
पुरुष - ७४१
कुल मतदान - ९५४
कांग्रेस - ६३३
भाजपा - २५७
- सुदर्शन गुप्ता, विधायक
- प्रदीप यादव, कांग्रेस प्रत्याशी २०१३
विधायक के साथ मिलकर पूरे वार्ड के विकास की रूप रेखा बनाई गई। उनके और महापौर के सहयोग से ही काम होता है। रिंग रोड आइडीए बनाएंगा। अभी इस योजना पर काम आगे नहीं बढ़ा है।
- राजेश चौहान, पार्षद वार्ड ५
लगातार प्रयास के बाद भी जयहिंद नगर की टंकी बन कर तैयार नहीं हुई है। स्ट्रीट लाइट लगाने के लिए प्रयास कर रहे हैं। जिससे गलियों में उजाला रहे और घटनाएं नहीं हो।
- विनितिका यादव, पार्षद, वार्ड १० | hindi |
एलईडी पट्टी के लिए डीसी बिजली की आपूर्ति
होम > उत्पादों > एलईडी पट्टी के लिए डीसी बिजली की आपूर्ति
(एलईडी पट्टी के लिए डीसी बिजली की आपूर्ति के लिए कुल २४ उत्पादों)
एलईडी पट्टी के लिए डीसी बिजली की आपूर्ति - निर्माता, कारखाने, आपूर्तिकर्ता चीन से
हम विशेष हैं एलईडी पट्टी के लिए डीसी बिजली की आपूर्ति निर्माताओं और आपूर्तिकर्ताओं / कारखाने चीन से। कम कीमत / सस्ते के रूप में उच्च गुणवत्ता के साथ थोक एलईडी पट्टी के लिए डीसी बिजली की आपूर्ति, चीन से अग्रणी ब्रांडों में से एक एलईडी पट्टी के लिए डीसी बिजली की आपूर्ति में से एक, डोंगउअन चेंग्लियांग इलेक्ट्रॉनिक टेक्नोलॉजी को.,लैड.।
थोक चीन से एलईडी पट्टी के लिए डीसी बिजली की आपूर्ति , लेकिन कम कीमत के अग्रणी निर्माताओं के रूप में सस्ते एलईडी पट्टी के लिए डीसी बिजली की आपूर्ति खोजने की आवश्यकता है। बस एलईडी पट्टी के लिए डीसी बिजली की आपूर्ति पर उच्च गुणवत्ता वाले ब्रांडों पा कारखाना उत्पादन, आप आप क्या चाहते हैं, बचत शुरू करते हैं और हमारे एलईडी पट्टी के लिए डीसी बिजली की आपूर्ति का पता लगाने के बारे में भी राय, आप में सबसे तेजी से उत्तर हम करूँगा कर सकते हैं। | hindi |
The air comes out from the top and then returns to the bottom. It takes off the dust from the body with gravity together. It will not produce the whirlpool and ensure the purifying effect.
The traditional air shower at market – simple structure, double blow system. It is more easy to cause whirlpool by irregular air flow. And the dust isn’t easy to take out immediately.
Electronic, biological manufacturing industry, medical industry, food processing, space, air, automobile manufacturing industry, dust-free painting industry and other industries. | english |
This month, Adobe’s Design and Sustainability & Social Impact teams joined forces to host two design thinking events benefitting nonprofit partners. The goals of these events, called Pro Bono Design Challenge, were threefold: to quickly generate design solutions for nonprofit partners, to spread the core tenets of design thinking internally, and to integrate the aptitudes and perspective of a diverse group of Adobe talent for social impact. And the results? Inspiring.
Adobe brought the power of design thinking to pro bono in a big way. Let’s dig in on the details to see how they did it.
Mission: To build democracy through citizen service, civic leadership, and social entrepreneurship.
Challenge: In this time of federal spending cuts, how do we create and reinforce a culture of national service?
Mission: To mentor young people in the digital media arts to help them affect positive change in their communities.
Challenge: How do we help young people bridge the gap between high school, college, and career when they have no networks in place?
Adobe’s not new to channeling its business assets to drive social good. The company has a broad philanthropic portfolio and, in particular, has been committed to engaging its people in pro bono service since 2012.
This year, Adobe took their lessons from the Oxfam pilot and teamed up with Taproot to bring this innovative model of corporate pro bono to new nonprofit beneficiaries through the “Pro Bono Design Challenge” events held in San Francisco and Lehi, Utah. These events leveraged the talent of 100+ Adobe employees from across the business, connecting Adobe talent in cross-functional teams led by an Adobe who specialized in design thinking.
Over the course of one very full day, teams designed and iterated on creative solutions that address key organizational and sector-wide challenges. Adobe participants delivered impressive and varied solutions at both events. For example, one team suggested City Year address their challenge around reinforcing a culture of service through a visual analytics dashboard to share the impact each Corps Member contributes during their year and how that lives on long after their service. Another team suggested City Year focus on corporate partnerships in order to remove barriers to service like free housing, transportation, and general cost of living expenses.
Expert insights: A cross-sector panel of leaders that helped prepare Adobe talent for the day by providing crucial context about the challenge and its broader social implications.
Design thinking focus: A day designed around bringing the best of design thinking to the social sector – to make sure that Adobe participants were aligned on their design thinking approach, a design thinking training for all participating Adobe talent was conducted before the pro bono sessions began.
Cross-department collaboration: An event open to all employees, Adobe strategically built teams that brought colleagues from diverse departments to encourage creative thinking.
Opportunities to shine: Every team presented on their solution to the challenge at the end of the day.
For the nonprofit participants, the event was an opportunity to get fresh and innovative insights on trenchant issues. And for Adobe participants, the experience highlighted the complexity of real-world challenges that nonprofits face and demonstrated how their professional expertise can make a positive difference on society.
All in all, both nonprofits and Adobe pro bono volunteers found this summer’s Design Events an inspiring, exciting, and exhausting (in the best possible way!) day of creativity and problem solving. Cheers to Adobe for finding new and impactful ways to apply their talent to drive positive social change! | english |
\begin{document}
\title{Rich Words in the Block Reversal of a Word}
\author{Kalpana Mahalingam, Anuran Maity, Palak Pandoh}
\address
{Department of Mathematics,\\
Indian Institute of Technology Madras,
Chennai, 600036, India}
\email{kmahalingam@iitm.ac.in, anuran.maity@gmail.com, palakpandohiitmadras@gmail.com}
\keywords{Combinatorics on words, rich words, run-length encoding, block reversal}
\ensuremath{\mathfrak m}aketitle
\begin{abstract}
The block reversal of a word $w$, denoted by $\ensuremath{\mathfrak m}athtt{BR}(w)$, is a generalization of the concept of the reversal of a word, obtained by concatenating the blocks of the word in the reverse order. We characterize non-binary and binary words whose block reversal contains only rich words. We prove that for a binary word $w$, richness of all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ depends on $l(w)$, the length of the run sequence of $w$. We show that if all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich, then $2\leq l(w)\leq 8$. We also provide the structure of such words.
\end{abstract}
\section{Introduction}
Inversions, insertions, deletions, duplications, substitutions and translocations are some of the operations that transform a DNA sequence from a primitive sequence (see \cite{Cantone2013,Cantone2010,Zhong2004,mahalingam2020}).
A rearrangement of chromosomes can happen when a single sequence undergoes breakage and one or more segments of the chromosome are shifted by some form of dislocation (\cite{Zhong2004}).
Mahalingam et al. (\cite{blore}) defined the block reversal of a word which is a rearrangement of strings when dislocations happen through inversions. The authors generalized the concept of the reversal of a word where in place of reversing individual letters, they decomposed the word into factors or blocks and considered the new word such that the blocks appear in the reverse order.
The block reversal operation of a word $w$, denoted by $\ensuremath{\mathfrak m}athtt{BR}(w)$, is represented in Figure \ref{f2}.
\begin{figure}
\caption{For $w \in \Sigma^*$, $w'\in \ensuremath{\mathfrak m}
\label{f2}
\end{figure}
If the word $w$ can be expressed as a concatenation of its factors or blocks $B_i$ such that $w=B_1B_2\cdots B_k$, then $w'=B_kB_{k-1}\cdots B_1$ is an element of $\ensuremath{\mathfrak m}athtt{BR}(w)$. Since there are multiple ways to divide a word into blocks, the block reversal of a word forms a set.
Mahalingam et al. (\cite{blore}) proved that there is a strong connection between the block reversal and the non-overlapping inversion of a word. A non-overlapping inversion of a word is a set of inversions that do not overlap with each other.
In 1992, Sch\"oniger et al. (\cite{Schon1992}) presented a heuristic for computing the edit distance when non-overlapping inversions are allowed.
They presented an $\ensuremath{\mathfrak m}athcal{O}$($n^
6$) exact solution for the
alignment with the non-overlapping inversion problem and showed the non-overlapping inversion operation ensures that all inversions occur in one mutation step. Instances of problems considering the non-overlapping inversions include the string alignment problem, the edit distance problem, the approximate matching problem, etc. (\cite{Cantone2010,Cantone20,Kece93,Augusto2006}). Kim et al. (\cite{Kim2015}) studied the non-overlapping inversion on strings from a formal language theoretic approach.
A word is a palindrome if it is equal to its reverse. Let $|w|$ be the length of the word $w$. It was proved by Droubay et al. (\cite{epi}) that a word $w$ has at most $|w|$ non-empty distinct palindromic factors. The words that achieve the bound were referred to as rich words by Glen et al. (\cite{palrich}). Several properties of rich words were studied in the literature (see \cite{pal3,epi,palrich,guo}). Droubay et al. (\cite{epi}) proved that a word $w$ contains exactly $|w|$ non-empty distinct palindromic factors iff the longest
palindromic suffix of any prefix $p$ of $w$ occurs exactly once in $p$. Guo et al. (\cite{guo}) provided necessary and sufficient conditions for richness in terms of the run-length encoding of binary words. It is known that on a binary alphabet, the set of rich words contain factors of the period-doubling words, factors of Sturmian words, factors of complementary symmetric Rote words, etc. (see \cite{palrot1, epi, schaclo}). In a non-binary alphabet, the set of rich words contain, for example factors of Arnoux–Rauzy words and factors of words coding symmetric interval exchange.
There are many results in the literature regarding the occurrence of rich words in infinite and finite words, but there are significantly fewer results about the occurrence of rich words in a language. The occurrence of rich words in the conjugacy class of a word $w$, denoted by $C(w)$, is a well-studied concept in literature (see \cite{careymusic, palrich, restivobwt, oeis1}).
Shallit et al. (\cite{oeis1}) calculated the number of binary words $w$ of a particular length such that every conjugate of $w$ is rich.
A word $w$ is said to be circularly rich if all of the conjugates of $w$ (including itself) are rich, and $w$ is a product of two palindromes.
Glen et al. (\cite{palrich}) studied circularly rich words and proved the equivalence conditions for circularly rich words. They proved that a word $w$ is circularly rich iff the infinite word $w^\omega$ is rich iff $ww$ is rich where $w^\omega$ is a word formed by concatenating infinite copies of $w$.
Restivo et al. (\cite{restivobwt, bwtRESTIVO}) outlined relationships between circularly rich words and the Burrows–Wheeler transform, a highly efficient data compression algorithm.
In many musical contexts, scale and rhythmic patterns are extended beyond a single iteration of the interval of periodicity. From the equivalent conditions for circularly rich words, proved by Glen et al. (\cite{palrich}), if $w$ is the step pattern of an octave-based scale and is rich, and if $ww$ is also rich, then the property can be extended without limit $(w^\omega)$. Lopez et al. (\cite{lopezmusic}) examined that circular palindromic richness is inherent in numerous musical contexts, including all well-formed and maximally even sets and also in non well-formed scales which display three different step sizes.
Carey (\cite{careymusic}) also deeply studied circularly rich words from a music theory perspective. He proposed that perfectly balanced scales that display circular palindromic richness and also exhibit relatively few step differences may prove to be advantageous from a cognitive and musical perspective. Since block reversal operation is a generalization of conjugate operation, the study of rich words in the block reversal of the word has possible applications in music theory and data compression techniques.
In this paper, we characterize words whose block reversal contains only rich words.
We find a necessary and sufficient condition for a non-binary word such that all elements in its block reversal are rich.
For a binary word $w'$, we prove that the richness of elements of $\ensuremath{\mathfrak m}athtt{BR}(w')$
depends on $l(w')$ which is the length of the run sequence of $w'$. We show that for a binary word $w$, if all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich, then $2\leq l(w) \leq 8$. We also find the structure of binary words whose block reversal consists of only rich words.
The paper is organized as follows.
In Section \ref{sec3}, we prove that for a non-binary word $w$, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich iff $w$ is either of the form $a_1a_2a_3 \cdots a_k$ or $a_j^{|w|}$ where each $a_i\in \Sigma$ is distinct.
In Section \ref{sec4}, we show that for a binary word $w$, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich if $l(w) =2$. We also show that if all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich, then $2\leq l(w) \leq 8$. We discuss the case when $3\leq l(w)\leq 8$ separately in detail and provide the structure of words such that all elements in their block reversal are rich. We end the paper with a few concluding remarks.
\section{Basic definitions and notations}\label{sec2}
Let $\Sigma$ be a non-empty set of letters. A word $w=[a_{i}]$ over $\Sigma$ is a finite sequence of letters from $\Sigma$ where $a_i$ is the $i$-$th$ letter of $w$. We denote the empty word by $\lambda$. By $\Sigma^*$, we denote the set of all words over $\Sigma$ and $\Sigma^+=\Sigma^*\setminus \{\lambda\}$. The length of a word $w$, denoted by $|w|$, is the number of letters in $w$. $\Sigma^n$ and $\Sigma^{\geq n}$ denote the set of all words of length $n$ and the set of all words of length greater than or equal to $n$, respectively. For $a \in \Sigma$, $|w|_a$ denotes the number of occurrences of $a$ in $w$. A word $u$ is a factor or block of the word $w$ if $w=puq$ for some $p, q\in \Sigma^*$. If $p= \lambda$, then $u$ is a prefix of $w$ and if $q = \lambda$, then $u$ is a suffix of $w$. Let $Fac(w)$ denote the set of all factors of the word $w$. $\ALPH(w)$ denotes the set of all letters in $w$. Two words $u$ and $v$ are called conjugates of each other if there exist $x,y\in \Sigma^*$ such that $u=xy$ and $v=yx$.
For a word $w= w_1w_2\cdots w_n$ such that $w_i\in \Sigma$, the reversal of $w$, denoted by $w^R$, is the word $w_n\cdots w_2w_1$. A word $w$ is a palindrome if $w=w^R$. By $P(w)$, we denote the number of all non-empty palindromic factors of $w$. A word $w$ has at most $|w|$ distinct non-empty palindromic factors. The words that achieve the bound are called rich words.
Every non-empty word $w$ over $\Sigma$ has a unique encoding of the form $w = a_1^{n_1}a_2^{n_2}\ldots a_k^{n_k},$ where $n_i \geq 1$, $a_i \neq a_{i+1}$ and $a_i\in \Sigma$ for all $i$. This encoding is called run-length encoding of $w$ (\cite{guo}). The word $a_1 a_2 \ldots a_k$ is called the trace of $w$. The sequence $(n_1, n_2, \ldots, n_k)$ is called the run sequence of $w$ and the length of the run sequence of $w$ is $k$. For any binary word $w$ over $\Sigma = \{a,b\}$, the complement of $w$, denoted by $w^c$, is the word $\phi(w)$ where, $\phi$ is a morphism such that $\phi(a)=b$ and $\phi(b)=a$. For example, if $w= ababb$, then $w^c = babaa$.
We recall the definition of the block reversal of a word from Mahalingam et al. (\cite{blore}).
\begin{definition}\cite{blore} \label{br}
Let $w,\;B_i \in \Sigma^+$ for all $i$. The block reversal of $w$, denoted by $\ensuremath{\mathfrak m}athtt{BR}(w)$, is the set $$\ensuremath{\mathfrak m}athtt{BR}(w) =\{ B_tB_{t-1}\cdots B_1 \;:\; \;w=B_1B_2 \ldots B_t, \; ~t\ge 1\}.$$
\end{definition}
Note that a word can be divided into a maximum of $|w|$ blocks. We illustrate Definition \ref{br} with the help of an example.
\begin{example}\label{e2}
Let $\Sigma=\{a, b,c\}$. Consider $u=abbc$ over $\Sigma$. Then, $$\ensuremath{\mathfrak m}athtt{BR}(u) = \{ cbab, cbba, cabb, bbca, bcab, abbc, bcba \}.$$
\end{example}
For more information on words, the reader is referred to Lothaire (\cite{Lothaire1997}) and Shyr (\cite{Shyr2001}).
\section{Block Reversal of Non-binary Words}\label{sec3}
It is well known that a rich word $w$ contains exactly $|w|$ distinct palindromic factors.
In this section, we find a necessary and sufficient condition for a non-binary word such that all elements in its block reversal are rich.
We recall the following from Glen et al. (\cite{palrich}).
\begin{theorem}\cite{palrich}\label{tglen}
For any word $w$, the following properties are equivalent:\\
(i) $w$ is rich;\\
(ii) for any factor $u$ of $w$, if $u$ contains exactly two occurrences of a palindrome $p$ as a prefix and as a suffix only, then $u$ is itself a palindrome.
\end{theorem}
\begin{lemma}\cite{palrich}\label{rich}
If $w$ is rich, then
\begin{itemize}
\item all factors of $w$ are rich.
\item $w^R$ is rich.
\end{itemize}
\end{lemma}
We first give a necessary condition under which $\ensuremath{\mathfrak m}athtt{BR}(w)$ contains at least one rich word.
\begin{lemma}\label{45tn}
Let $w \in \Sigma^n$. If $\ensuremath{\mathfrak m}athtt{BR}(w)$ has no rich element, then $|\ALPH(w)|< n-1$.
\end{lemma}
\begin{proof}
Let $w \in \Sigma^n$ such that $\ensuremath{\mathfrak m}athtt{BR}(w)$ contains no rich element. We prove that if $|\ALPH(w)| \geq n-1$, then there exists at least one rich word in $\ensuremath{\mathfrak m}athtt{BR}(w)$.
If $|\ALPH(w)|=n $, then all elements in $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich.
If $|\ALPH(w)|=n-1 $, then for $u_1, u_2, u_3 \in \Sigma^*$, $w = u_1 a u_2 a u_3$ such that $a \notin \ALPH(u_i)$ for all $i$ and
$\ALPH(u_i) \cap \ALPH(u_j) = \emptyset$ for $i\neq j$. Now, $u_3u_2a^2u_1 \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is a rich word.
\end{proof}
We now give an example of a word $w \in \Sigma^n$ with $|\ALPH(w)|=n-2$ such that $\ensuremath{\mathfrak m}athtt{BR}(w)$ contains no rich word.
\begin{example}\label{45tnr}
For $a, b \in \Sigma$, consider $w= u_1 \textbf{a} u_2 \textbf{b} u_3 \textbf{b} u_4 \textbf{a} u_5$ such that $a, b \notin \ALPH(u_i)$, $|u_i|\geq 3$ for each $i$, $\ALPH(u_i) \cap \ALPH(u_j)=\emptyset$ for $i\neq j$ and $\sum_{i=1}^{i=5}|\ALPH(u_i)|=|w|-4$. Then, $|\ALPH(w)|=|w|-2$.
We denote by $\pi(w)$, the set of all permutations of the word $w$, i.e., $\pi(w)=\{u\in \Sigma^*|\; |u|_a=|w|_a \text{ for all } a\in \Sigma \}$. One can easily observe that $\ensuremath{\mathfrak m}athtt{BR}(w)$ is a subset of $\pi(w)$.
If $\pi(w)$ has no rich words, then $\ensuremath{\mathfrak m}athtt{BR}(w)$ also has no rich words. Suppose there is a $ v \in \pi(w)$ such that $v$ is rich then, as $|\ALPH(v)|=|v|-2$, $|v|_a=|v|_b=2$,
by Theorem \ref{tglen}, we have, $\{ a \alpha a, b \alpha' b~ |~ \alpha, \alpha' \in \Sigma^{\geq 2} \text{ such that } \alpha \neq bzb, \alpha' \neq az'a \text{ where } z, z' \in \Sigma \cup \{\lambda\} \}$ $\cap \; Fac(v) = \emptyset$. Otherwise, if $a \alpha a$ or $b \alpha' b$ lies in $Fac(v)$, then as $\alpha \neq bzb$, $\alpha' \neq az'a$ where $z, z' \in \Sigma \cup \{\lambda\}$, $\alpha, \alpha' \in \Sigma^{\geq 2}$, $|\alpha|_a = 0$ and $|\alpha'|_b = 0$, we have, $a \alpha a$ and $b \alpha' b$ are not palindromes, which contradicts Theorem \ref{tglen}.
This implies that $v$ is of one of the following forms:
\begin{align}\label{algrt1}
v_1 x^2 v_2 y^2 v_3,\; v_1 x x_1 x v_2 y x_2 y v_3, \; v_1 xy^2x v_2,\; v_1 xx_1x v_2 y^2 v_3, \; v_1 y^2 v_2 xx_1x v_3, \; v_1 xyxy v_2, \; v_1 x y x_1 y x v_2
\end{align}
where $x \neq y \in \{a, b\}$, $ x_1, x_2 \in \Sigma \setminus \{a, b\}$ and $v_i \in \Sigma^*$ for all $i$.
Now, from the structure of $w$, we observe that $\ensuremath{\mathfrak m}athtt{BR}(w)$ does not contain any element of forms in (\ref{algrt1}).
Thus, $v \notin \ensuremath{\mathfrak m}athtt{BR}(w)$. Now, $\ensuremath{\mathfrak m}athtt{BR}(w) \subseteq \pi(w)$ and $v\in\pi(w) $ is rich implies no element of $\ensuremath{\mathfrak m}athtt{BR}(w)$ is rich.
\end{example}
Note that some elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ may not be rich even when $w$ is rich.
For example, the word $w=abbc$ is rich, but $ bcab \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich.
We now give a necessary and sufficient condition on a non-binary word $w$ such that all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich. We need the following results.
\begin{lemma} \label{nori}
Let $w = a_1^{n_1}a_2uv$ where $a_1\neq a_2$, $u \in \{a_1,a_2\}^+$, $v \in (\Sigma \setminus \{ a_1, a_2\})^+$ and $n_1 \geq 1$. Then, there exists an element in $\ensuremath{\mathfrak m}athtt{BR}(w)$ which is not rich.
\end{lemma}
\begin{proof}
Let $w = a_1^{n_1} a_2 u v$ such that $a_1\neq a_2$, $u \in \{a_1,a_2\}^+$, $v \in (\Sigma \setminus \{ a_1, a_2\})^+$ and $n_1 \geq 1$. Let $u = u' a_i$
where $i=1$ or $2$. Then, $w = a_1^{n_1} a_2 u'a_i v $.
Note that $w'=u' a_i v a_2 a_1^{n_1} \in \ensuremath{\mathfrak m}athtt{BR}(w)$ and $w''=a_i v a_1^{n_1} a_2 u' \in \ensuremath{\mathfrak m}athtt{BR}(w)$. The factor $ a_i v a_2 a_1$ of $w'$ is not a palindrome for $i=1$ as $a_i\notin \ALPH(v)$ and similarly the factor $ a_iv a_1^{n_1} a_2$ of $w''$ is not a palindrome for $i=2$. Hence, by Theorem \ref{tglen} and Lemma \ref{rich}, $w'\in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich when $i=1$ and $w''\in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich when $i=2$.
\end{proof}
We also have the following:
\begin{lemma}\label{notrich}
For $u_1, u_2, u_3, u_4, u_5 \in \Sigma^*$, consider $w= u_1 a_i u_2 a_j u_3 a_{k} u_4 a_i u_5$ such that $a_j \neq a_k$, $a_j \neq a_i \neq a_k$ and $a_k$ is not a suffix of $u_3$. Then, there exists an element in $\ensuremath{\mathfrak m}athtt{BR}(w)$ which is not rich.
\end{lemma}
\begin{proof}
For $u_1, u_2, u_3, u_4, u_5 \in \Sigma^*$, consider $w= u_1 a_i u_2 a_j u_3 a_{k} u_4 a_i u_5$ where $a_j \neq a_k$, $a_j \neq a_i \neq a_k$ and $a_k$ is not a suffix of $u_3$. We have the following cases:
\begin{itemize}
\item $a_i\notin \ALPH(u_3) : $
Then, $w'= u_5 u_4 a_i a_k u_3 a_j a_i u_2 u_1 \in \ensuremath{\mathfrak m}athtt{BR}(w)$ and $a_j \neq a_k$ implies $a_i a_k u_3 a_j a_i$ is not a palindromic factor of $w'$. Then, by Theorem \ref{tglen}, $w'$ is not rich.
\item $a_i\in \ALPH(u_3) : $ Then, let $u_3 = u_3' a_i u_3''$ such that $u_3', u_3'' \in \Sigma^*$ and $|u_3''|_{a_i}=0$.
Now, $w''= u_5 u_4 a_i a_{k} u_3'' a_i u_1 a_i u_2 a_j u_3' \in \ensuremath{\mathfrak m}athtt{BR}(w)$. If $w''$ is not rich, then we are done. If $w''$ is rich, then $a_i a_{k} u_3'' a_i\in Fac(w'')$ and since, $|u_3''|_{a_i}=0$, by Theorem \ref{tglen}, $a_i a_{k} u_3'' a_i$ is a palindrome. This implies $u_3''=\lambda$ as $a_k$ is not a suffix of $u_3$. Then, $w=u_1 a_i u_2 a_j u_3' a_i a_{k} u_4 a_i u_5$. Now, $w'''= u_4 a_i u_5 u_3' a_i a_{k} a_j a_i u_2 u_1 \in \ensuremath{\mathfrak m}athtt{BR}(w) $. Then, $a_i a_{k} a_j a_i \in Fac(w''')$ is not a palindrome as $a_j\neq a_k$. Therefore, by Theorem \ref{tglen}, $w'''$ is not rich.
\end{itemize}
\end{proof}
Now, we find a necessary and sufficient condition for a non-binary word such that all elements in its block reversal are rich.
\begin{theorem}
Let $w$ be a non-binary word. Then, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich iff $w$ is either of the form $a_1a_2a_3 \cdots a_k$ or $a_i^{|w|}$ where $a_i\in \Sigma$ are distinct.
\end{theorem}
\begin{proof}
Let $w \in \Sigma^*$. If $|\ALPH(w)|=1$, we are done. Assume $|\ALPH(w)|\geq 3$ and consider the run-length encoding of $w$ to be $ a_1^{n_1}a_2^{n_2}a_3^{n_3} \cdots a_k^{n_k}$ where $a_i \neq a_{i+1}\in \Sigma$, $k \geq 3$ and $n_i \geq 1$. Let all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ be rich. We have the following cases:
\begin{itemize}
\item All $a_t$'s are distinct for $1\leq t\leq k$ : We prove that $n_t=1$ for $1\leq t\leq k$.
Assume if possible that there exists at least one $n_j \geq 2$ for some $j$, i.e., $n_j = 2m +s$ for $m\geq 1$ and $ s\in \{0,1\}$. Let $\gamma = a_1^{n_1} a_2^{n_2} \cdots a_{j-1}^{n_{j-1}}$ and $\gamma'=a_{j+1}^{n_{j+1}} a_{j+2}^{n_{j+2}} \cdots a_k^{n_k}$.
Then, $w = \gamma a_j^{2m+s}\gamma'$.
Since, $a_j^m \gamma' \gamma a_j^m a_j^s \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is rich and all $a_t$'s are distinct, by Theorem \ref{tglen}, $u = a_j^m \gamma' \gamma a_j^m$ is a palindrome. Now, as $|\ALPH(w)|\geq 3$ and all $a_t$'s are distinct, $u$ is not a palindrome which is a contradiction. Therefore, $n_t = 1$ for $1\leq t\leq k$ and $w=a_1a_2a_3 \cdots a_k$.
\item Otherwise, suppose $i$ is the least index such that $|a_1a_2\cdots a_k|_{a_i}\geq 2$ and $a_j=a_i$ where $a_l\neq a_i$ for $i+1\leq l\leq j-1$ i.e., $j$ is the first position at which $a_i$ repeats for $i<j$. We have the following cases :
\begin{itemize}
\item $i\geq 3$ : Note that $a_i \neq a_1$ and $a_i \neq a_2$. Let $n_i \geq n_j$ such that $n_i = n_j + s' $ where $s' \geq 0$. Now, for $\delta = a_3^{n_3} a_4^{n_4} \cdots a_{i-2}^{n_{i-2}}$, $\alpha = a_{i+1}^{n_{i+1}} a_{i+2}^{n_{i+2}} \cdots a_{j-1}^{n_{j-1}}$ and $\beta = a_{j+1}^{n_{j+1}} a_{j+2}^{n_{j+2}}\cdots a_{k}^{n_{k}}$, we have,
$$w = a_1^{n_1}a_2^{n_2} \delta a_{i-1}^{n_{i-1}} a_i^{n_j + s'} \alpha a_j^{n_j} \beta. $$
Since, $\beta a_j^{n_j} a_1^{n_1} a_2^{n_2} \delta a_{i-1}^{n_{i-1}} a_i^{n_j + s'} \alpha \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is rich, by Theorem \ref{tglen}, we get, $a_j^{n_j} a_1^{n_1} a_2^{n_2} \delta a_{i-1}^{n_{i-1}} a_i^{n_j}$ is a palindrome, and hence, $a_1 = a_{i-1}$. Similarly, as $\beta a_j^{n_j} a_2^{n_2} \delta a_{i-1}^{n_{i-1}} a_i^{n_j} a_i^{s'} \alpha a_1^{n_1} \in \ensuremath{\mathfrak m}athtt{BR}(w)$, by Theorem \ref{tglen}, we have, $a_j^{n_j} a_2^{n_2} \delta a_{i-1}^{n_{i-1}} a_i^{n_j}$ is a palindrome, which gives $a_2 = a_{i-1}$. Thus, $a_1 = a_2$ which is a contradiction. A symmetrical argument holds for the case $n_i< n_j$.
\item $i\leq 2$ : Let $i=1$, i.e., $a_1$ has a repetition and there exists an index $l>1$ such that $a_1 = a_l$. If all elements in $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich, then by Lemma \ref{notrich}, there exists at most one distinct letter between $a_1$ and $a_l$. Similarly, between any two occurrences of $a_2$, there exists at most one distinct letter. Then, $w$ can only be of forms $a_1^{n_1} a_2^{n_2} a_3^{n_3} a_2 z'$ or $ a_1^{n_1} a_2 u v$ where $a_1 \neq a_3$, $z' \in \{\Sigma \setminus \{a_1\}\}^*$, $u \in \{ a_1, a_2 \}^+$ and $v \in (\Sigma \setminus \{ a_1, a_2 \} )^+$. If $w$ is in form $a_1^{n_1} a_2^{n_2} a_3^{n_3} a_2 z'$, then $z'a_2 a_3^{n_3} a_1^{n_1} a_2^{n_2} \in \ensuremath{\mathfrak m}athtt{BR}(w)$. Since, $a_2 a_3^{n_3} a_1^{n_1} a_2$ is not a palindrome, by Theorem \ref{tglen}, $z'a_2 a_3^{n_3} a_1^{n_1} a_2^{n_2} $ is not rich, a contradiction. Now, consider $w$ is in form $ a_1^{n_1} a_2 u v$.
Note that as $|\ALPH(w)|\geq 3,$ $v\neq \lambda$.
By Lemma \ref{nori}, there exists an element in $\ensuremath{\mathfrak m}athtt{BR}(w)$ which is not rich, a contradiction.
\end{itemize}
\end{itemize}
Thus, if $|\ALPH(w)|\geq 3$ and all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich, then $w=a_1a_2a_3 \cdots a_k$ where each $a_i\in \Sigma$ are distinct.\\
The converse is straightforward.
\end{proof}
\section{Block Reversal of Binary Words}\label{sec4}
Anisiu et al. (\cite{pcofw}) showed that any binary word of length greater than $8$, has at least $8$ non-empty palindromic factors. A set of words that achieve the bound of having exactly $8$ palindromic factors was given by Fici et al. (\cite{lepin}).
We recall the definition of $k$-$th$ power of $u \in \Sigma^*$ from Brandenburg (\cite{FUPH}) as the prefix of least length $u'$ of $u^n$ where $n\geq k$ such that $|u'|\geq k|u|$. For example, given a word $aba$, the $\frac{5}{3}$-$th$ power of $aba$ is $ aba^{(\frac{5}{3})} = abaab$. Fici et al. (\cite{lepin}) showed that for all $u\in C(v)$ where $v= abbaba$, $P(u^{(\frac{n}{6})})=8$, $n\geq 9$. Mahalingam et al. (\cite{lep}) characterized words $w$ such that $P(w) =8$. They proved that a binary word $w$ has $8$ palindromic factors iff $w$ is of the form $u^{(\frac{n}{6})}$ where $u\in C(v) \cup C(v^R)$ and $v=abbaba$.
In this section, we discuss the case of binary words. Let $l(w)$ be the length of the run sequence of a binary word $w$. We prove that
if $l(w)=2$, then all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich and if $l(w)\geq 9$, then there exists an element in $\ensuremath{\mathfrak m}athtt{BR}(w)$ that is not rich.
Then, we study the block reversal of binary words with $3\leq l(w)\leq 9$. The results in this section also hold for complement words as we have considered unordered alphabet $\Sigma=\{a,b\}$.
\subsection{\textbf{Block reversal of binary words $\bf{w}$ with} $\bf{l(w)= 2}$ $\bf{\&}$ $\bf{l(w)\geq 9}$}
Now, we discuss the block reversal of binary words $w$ with $l(w)=2$ $\&$ $l(w)\geq 9$.
We first recall the following from Guo et al. (\cite{guo}).
\begin{proposition} \cite{guo}\label{runlen}
Every binary word having a run sequence of length less than or equal to $4$ is rich.
\end{proposition}
It was verified by Anisiu et al. (\cite{pcofw}) that for all short binary words (up to $|w|=7$), $P(w)=|w|$.
We observe that for words $w$ with $|w|> 7$, some elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ may not be rich even when $w$ is rich.
For example, $w = a^2b^3a^3$ is rich but $a^2bab^2a^2 \in \ensuremath{\mathfrak m}athtt{BR}(w)$, is not rich. We discuss the case when $l(w)=2$ for a word $w$ in the following.
\begin{proposition}\label{u1}
If $w = a^{n_1}b^{n_2}$ where $n_1, n_2 \geq 1$, then all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich.
\end{proposition}
\begin{proof}
Let $w=a^{n_1}b^{n_2}$ where $n_1, n_2 \geq 1$. Then,
$$\ensuremath{\mathfrak m}athtt{BR}(w) = \{b^{n_2'}a^{n_1'}b^{n_2-n_2'}a^{n_1-n_1'}\; : \;0 \leq n_1' \leq n_1, \; 0 \leq n_2' \leq n_2\}.$$
Since the length of the run sequence of each element of $\ensuremath{\mathfrak m}athtt{BR}(w)$ is less than or equal to $4$, by Proposition \ref{runlen}, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich.
\end{proof}
We now prove that for a binary word $w$ with length of the run sequence greater than $8$, there exists an element in $\ensuremath{\mathfrak m}athtt{BR}(w)$ that is not rich.
We recall the following from Mahalingam et al. (\cite{blore}).
\begin{lemma}\label{hhh}\cite{blore}
$\ensuremath{\mathfrak m}athtt{BR}(v) \ensuremath{\mathfrak m}athtt{BR}(u) \subseteq \ensuremath{\mathfrak m}athtt{BR}(uv)$ for $u,\; v\in \Sigma^*$.
\end{lemma}
We now have the following:
\begin{proposition}\label{u3}
Let $w \in \{a, b\}^*$ such that $l(w)\geq 9$. Then, there exists an element in $\ensuremath{\mathfrak m}athtt{BR}(w)$ that is not rich.
\end{proposition}
\begin{proof}
Let $w \in \{a, b\}^*$ such that $l(w) \geq 9$. Since, $l(w) \geq 9$, then for $n_i\geq 1$, consider $w' = a^{n_1} b^{n_2} a^{n_3} b^{n_4} a^{n_5} b^{n_6}\\ a^{n_7} b^{n_8} a^{n_9}$ to be a prefix of $w$. If $w$ is not rich, then we are done. Otherwise, $w$ is rich, then by Lemma \ref{rich}, all factors of $w$ are rich. Also, from Lemma \ref{hhh}, we have, $\ensuremath{\mathfrak m}athtt{BR}(v) \ensuremath{\mathfrak m}athtt{BR}(u) \subseteq {\ensuremath{\mathfrak m}athtt{BR}(uv)}$ for $u,\; v\in \Sigma^*$. We show that there exists an element in $\ensuremath{\mathfrak m}athtt{BR}(w')$ that is not rich to complete the proof. Suppose to the contrary that all elements of $\ensuremath{\mathfrak m}athtt{BR}(w')$ are rich. Let $w_1, w_2, w_3 \in \ensuremath{\mathfrak m}athtt{BR}(w')$ where
$$w_1 = a^{n_9+n_7} b^{n_8+n_6} a b a^{n_5-1+n_3}b^{n_4-1+n_2}a^{n_1},$$
$$w_2 = b^{n_8} a^{n_9+n_7} b a b^{n_6-1+n_4} a^{n_5-1+n_3+n_1}b^{n_2} \text{ \;\;and}$$
$$ w_3 = a^{n_9} b^{n_8+n_6} a^{n_7+n_5} b a b^{n_4-1+n_2} a^{n_3-1+n_1}.$$ We have the following:
\begin{itemize}
\item If $n_3+n_5\geq 3$, then $a^2 b^{n_8+n_6} a b a^2$ is a factor of $w_1$ that contains exactly two occurrences of a palindrome $a^2$ as a prefix and as a suffix. By Theorem \ref{tglen}, if $w_1$ is rich, then $a^2 b^{n_8+n_6} a b a^2$ is a palindrome which is a contradiction. Hence, $n_3 = n_5 =1$.
\item If $n_4+n_6\geq 3$, then $a^2 b a b^{n_6-1+n_4} a^2$ is a factor of $w_2$ that contains exactly two occurrences of a palindrome $a^2$ as a prefix and as a suffix. By Theorem \ref{tglen}, if $w_2$ is rich, then $ a^2 bab^{n_6-1+n_4} a^2$ is a palindrome which is a contradiction. Hence, $n_4 =n_6=1$.
\item If $n_2 \geq 2$, then
$b^2 a^{n_7+n_5} b a b^2$
is a factor of $w_3$ that contains exactly two occurrences of a palindrome $b^2$ as a prefix and as a suffix. By Theorem \ref{tglen}, if $w_3$ is rich, then $ b^2 a^{n_5+n_7} b a b^2$ is a palindrome which is a contradiction. Hence, $n_2=1$.
\end{itemize}
Hence, we have $n_2=n_3=n_4=n_5=n_6=1$ and $w' = a^{n_1} b a b a b a^{n_7} b^{n_8} a^{n_9}$.
Let $w_4,\; w_5\in \ensuremath{\mathfrak m}athtt{BR}(w')$ where $w_4= a^{n_9+n_7} b^{n_8} a b^2 a^{n_1+1} b$ and $w_5 = a^{n_9+n_7} b^{n_8+1} a b^2 a^{1+n_1}$. Now,
$a^2 b^{n_8}a b^2 a^2$
is a factor of $w_4$ that contains exactly two occurrences of a palindrome $a^2$ as a prefix and as a suffix. By Theorem \ref{tglen}, since $w_4$ is rich, $n_8=2$. Also, $a^2 b^{n_8+1}a b^2 a^2$
is a factor of $w_5$ that contains exactly two occurrences of a palindrome $a^2$ as a prefix and as a suffix. By Theorem \ref{tglen}, since $w_5$ is rich, $n_8=1$, which is a contradiction. Thus, there always exists an element in $\ensuremath{\mathfrak m}athtt{BR}(w')$ that is not rich.
\end{proof}
\subsection{\bf{Block reversal of binary words $\bf{w}$ with} $\bf{3\leq l(w)\leq 8}$}
We now consider the case for a binary word $w$ such that $3\leq l(w)\leq 8$. We observe that the result varies with the structure of the word. We compile all results towards the end of this section. We first recall the following from Anisiu et al. (\cite{pcofw}).
\begin{theorem}\cite{pcofw}\label{7ric} If $w$ is a binary word of length less than $8$, then $P(w)=|w|$. If $w$ is a binary word of length $8$, then $7\leq P(w)\leq 8$ and $P(w)= 7$ iff $w$ is of the form $aabbabaa$ or $aababbaa$.
\end{theorem}
Now, with the help of examples, we illustrate that all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ may be rich for a binary word $w$ such that $3\leq l(w)\leq 8$.
\begin{example}\label{u2}
For $3\leq l(w)\leq 8$, consider \[w=\left \{\begin{array}{cc}
a(ba)^i & \text{for $i=\frac{l(w)-1}{2}$ and $l(w)$ odd,}\\
(ab)^i & \text{for $i=\frac{l(w)}{2}$ and $l(w)$ even.}\end{array} \right.\]
It can be observed that all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich.
\end{example}
Thus, from Propositions \ref{u1} and \ref{u3} and Example \ref{u2}, we conclude the following.
\begin{theorem}
Let $w$ be a binary word and $l(w)$ be the length of the run sequence of $w$. If all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich, then $2\leq l(w)\leq 8$.
\end{theorem}
We now consider the following example of a binary word $v$ with $3\leq l(v)\leq 8$ such that
there exists an element in $\ensuremath{\mathfrak m}athtt{BR}(v)$ that is not rich.
\begin{example}\label{u4}
For $3\leq l(v)\leq 8$, consider \[v=\left \{\begin{array}{cc}
a^2b^3a^3(ba)^i & \text{for $i=\frac{l(v)-3}{2}$ and $l(v)$ odd,}\\
a^2b^3a^3(ba)^ib & \text{for $i=\frac{l(v)-4}{2}$ and $l(v)$ even.}\end{array} \right.\]
and
\[v'=\left \{\begin{array}{cc}
(ba)^i a^2 b^2 a b a^2 & \text{for $i=\frac{l(v)-3}{2}$ and $l(v)$ odd,}\\
(ba)^i b a^2 b^2 a b a^2 & \text{for $i=\frac{l(v)-4}{2}$ and $l(v)$ even.}
\end{array} \right.\]
It can observed that
$v' \in \ensuremath{\mathfrak m}athtt{BR}(v)$. Note that by Lemma \ref{rich}, $v'$ is not rich as $a^2 b^2 a b a^2$ is not rich.
\end{example}
We conclude from Examples \ref{u2} and \ref{u4} that all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ may or may not be rich for a binary word $w$ such that $3\leq l(w)\leq 8$. Now, we find the structure of binary words $w$ with $3\leq l(w)\leq 8$ such that the block reversal of $w$ contains only rich words.
We recall the following from Mahalingam et al. (\cite{blore}).
\begin{lemma}\label{f1}\cite{blore}
Let $w\in \Sigma^+$, $(\ensuremath{\mathfrak m}athtt{BR}(w))^R = \ensuremath{\mathfrak m}athtt{BR}(w^R)$.
\end{lemma}
We conclude the following from Lemmas \ref{rich} and \ref{f1}.
\begin{remark}\label{o1}
All elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich iff all elements of $\ensuremath{\mathfrak m}athtt{BR}(w^R)$ are rich.
\end{remark}
We first study the case when the length of the run sequence of the word is equal to $8$.
\begin{proposition}\label{g8}
Let $w\in \{a, b\}^*$ and $l(w)=8$. Then, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich iff $w=abababab$.
\end{proposition}
\begin{proof}
Let $w$ be a binary word with $l(w)=8$ such that all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich. Consider $w=a^{n_1}b^{n_2}a^{n_3}b^{n_4}a^{n_5}b^{n_6}a^{n_7}b^{n_8}$ to be the run-length encoding of $w$ where $n_i\geq 1$ for all $i$. Let $$w'=b^{n_8+n_6} a^{n_7+n_5} b a b^{n_4-1+n_2} a^{n_3-1+n_1} \in \ensuremath{\mathfrak m}athtt{BR}(w).$$
Then, $w'$ is rich. If $n_4 \geq 2$ or $n_2\geq 2$, then since $v=b^{2} a^{n_7+n_5} b a b^{2}\in Fac(w')$ and $v$ contains exactly two occurrences of $b^2$,
by Theorem \ref{tglen}, $b^{2} a^{n_7+n_5} b a b^{2}$ is a palindrome which is a contradiction. Thus, $n_2=n_4=1$. Now, by Remark \ref{o1}, we get, $n_7=n_5=1$. Thus, $w=a^{n_1}ba^{n_3}bab^{n_6}ab^{n_8}$.
Now, consider $$w''= b^{n_8} a^{2} b^{n_6+1} a b a^{n_3-1+n_1} \in \ensuremath{\mathfrak m}athtt{BR}(w).$$
Then, $w''$ is rich. If $n_1\geq 2$ or $n_3\geq 2$, then since $v'=a^{2} b^{n_6+1} a b a^{2} \in Fac(w'')$ and $v'$ contains exactly two occurrences of $a^2$, by Theorem \ref{tglen}, $ a^{2} b^{n_6+1} a b a^{2}$ is a palindrome which is a contradiction. Thus, $n_1=n_3=1$. Now, by Remark \ref{o1}, we get, $n_8=n_6=1$. Thus, $w=abababab$.
The converse follows from Theorem \ref{7ric}.
\end{proof}
We conclude the following from Proposition \ref{g8}.
\begin{remark}
Let $w$ be a binary word such that $l(w)=8$ and $|w|>8$. Then, there exists an element in $\ensuremath{\mathfrak m}athtt{BR}(w)$ that is not rich.
\end{remark}
We now consider the case when the length of the run sequence of the word is $7$. For a binary word $w$, if $l(w)=7$, then $|w|\geq 7$. If $|w|=7$, it is well known (\cite{pcofw}) that all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich. We consider the case when $|w|>7$ in the following.
\begin{proposition}\label{g7}
Let $w$ be a binary word with $|w|>7$ and $l(w)=7$. Then, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich iff $w$ is $ababab^2a$, $abab^2aba$ or $ab^2ababa$.
\end{proposition}
\begin{proof}
Let $w$ be a binary word with $l(w)=7$ such that all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich. Consider $w=a^{n_1}b^{n_2}a^{n_3}b^{n_4}a^{n_5}b^{n_6}a^{n_7}$ to be the run-length encoding of $w$ where $n_i\geq 1$ for all $i$. Let $$\alpha = a^{n_7-1+n_5} b^{n_6} a b^{n_4+n_2} a^{n_3+n_1} \in \ensuremath{\mathfrak m}athtt{BR}(w)$$ and $$\beta =a^{n_7-1+n_5} b^{n_6} a b^{n_4} a^{n_3+n_1} b^{n_2} \in \ensuremath{\mathfrak m}athtt{BR}(w) .$$ Then, $\alpha,\; \beta$ are rich.
If $n_7 \geq 2$ or $n_5 \geq 2$, then by Theorem \ref{tglen}, $a^{2} b^{n_6} a b^{n_4+n_2} a^{2}$ and $a^{2} b^{n_6} a b^{n_4} a^{2}$ are palindromic factors of $\alpha$ and $\beta $, respectively. This implies $n_6=n_4+n_2$ and $n_6=n_4$, which is a contradiction to the fact that $n_2 \geq 1$. Thus, $n_7 = n_5 =1$. \\
Since, $n_7=n_5=1$, by Remark \ref{o1}, we get, $n_1=n_3=1$. Thus, $w=a b^{n_2}a b^{n_4}a b^{n_6} a$.
We now show that $n_6\leq2$. Consider $\gamma = b^{n_6-1} aa b a b^{n_4+n_2} a \in \ensuremath{\mathfrak m}athtt{BR}(w)$. Then, $\gamma$ is rich. If $n_6\geq 3$, then by Theorem \ref{tglen}, $b^{2} a^2 b a b^{2}$ is a palindromic factor of $\gamma$ which is a contradiction. Thus, $n_6\leq 2$. We have the following cases:
\begin{enumerate}
\item $n_6=2: $ If $n_4\geq 2$ or $n_2\geq 2$, then $b^2 a^2 ba b^{n_4-1+n_2} a \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction. Thus, in this case, $n_4=n_2=1$. We have, $w=a b a b a b^{2} a$.
\item $n_6=1 :$ Here, $w=a b^{n_2}a b^{n_4}a b a$. If $n_4\geq 2$ and $n_2\geq 2$, $b^{n_4} a b a^2 b^{n_2} a \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction. So, either $n_2=1$ or $n_4=1$. Note that if $n_2=n_4=1$, then $|w|=7$, which is a contradiction. We are left with the following cases:
\begin{itemize}
\item $n_2=1$ and $n_4 \geq 2$ : Here, $w=a b a b^{n_4}a b a$. If $n_4\geq 3$, $b^{n_4-1} a b a^2 b^2 a \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich, a contradiction. Thus, $n_4 =2$ and $w=a b a b^{2}a b a$.
\item $n_4=1$ and $n_2\geq 2$ : Here, $w=a b^{n_2} a ba b a$. If $n_2\geq 3$, $ a b^2 a b a^2 b^{n_2-1} \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich, a contradiction. Thus, $n_2=2$ and $w=a b^2 a ba b a$.
\end{itemize}
\end{enumerate}
The converse follows from Theorem \ref{7ric}.
\end{proof}
We conclude the following from Proposition \ref{g7}.
\begin{remark}
Let $w$ be a binary word such that $l(w)=7$ and $|w|>8$. Then, there exists an element in $\ensuremath{\mathfrak m}athtt{BR}(w)$ that is not rich.
\end{remark}
We now consider the case when the length of the run sequence of the word is $6$.
We need the following:
\begin{remark}\label{rem2}
We consider the block reversal of the following words:
\begin{enumerate}
\item Let $w=a^2 b a b a b^{n_6}$ for $n_6<4$.
\begin{itemize}
\item If $n_6=1$ or $2$, then as $|w|\leq 8$, by Theorem \ref{7ric}, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich.
\item If $n_6=3$, then $b a b^2 a^2 bab \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which implies that
not all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich.
\end{itemize}
\item Let $w=a^3 b a b a b^{n_6}$ for $n_6<4$.
\begin{itemize}
\item If $n_6=1$, then by Theorem \ref{7ric}, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich.
\item If $n_6=2$, then $a b a b^2 a^2 b a \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which implies that
not all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich.
\item If $n_6=3$, then $ba b^2 a^3 bab \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which implies that
not all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich.
\end{itemize}
\item Let $w=a^{n_1} b a b a b$ for $n_1\geq 1$.
We show that all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich.
Let $w=D_1 D_2 D_3 \cdots D_k$ where $k \geq 2$ and each $D_i \in \Sigma^+$. Then, either $D_1 = a^j$, $D_2=a^{n_1-j}x$ and $D_3 \cdots D_k = y$ where $x, y \in \Sigma^*$, $x y =babab$ and $1 \leq j <n_1$
or
$D_1 = a^{n_1} x'$ and $D_2 D_3 \cdots D_k=y'$ where $x' \in \Sigma^*, y' \in \Sigma^+$ and $x' y' = babab$. Then, for distinct elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$, we can divide $w$ in at most seven non-empty blocks.
\begin{itemize}
\item When we divide $w$ in two non-empty blocks, then $\ensuremath{\mathfrak m}athtt{BR}(w)$ contains the following:\\
Let, $A_2 = \{ b a^{n_1} baba, ab a^{n_1} bab, bab a^{n_1} ba, abab a^{n_1} b, babab a^{n_1}, a^{n_1-i}babab a^i~ |~ 1 \leq i \leq n_1-1 \}.$ We can observe that each element of $A_2$ is rich.
\item When we divide $w$ in three non-empty blocks, then $\ensuremath{\mathfrak m}athtt{BR}(w)$ contains the following:\\
Let, $A_3 = \{ ba a^{n_1} bab, bba a^{n_1}ba, baba a^{n_1}b, bbaba a^{n_1}, b a^{n_1-i} baba a^{i}, abba^{n_1}ba, ababa^{n_1}b, ab bab a^{n_1},\\ ab a^{n_1-i} bab a^{i}, babba a^{n_1}, bab a^{n_1-i} ba a^{i}, abab b a^{n_1}, abab a^{n_1-i} b a^{i}, babab a^{n_1} ~|~ 1 \leq i \leq n_1-1 \}$. We can observe that each element of $A_3$ is rich.
\item When we divide $w$ in four non-empty blocks, then $\ensuremath{\mathfrak m}athtt{BR}(w)$ contains the following:\\
Let, $A_4 = \{ bab a^{n_1} ba, baab a^{n_1} b, babab a^{n_1}, ba a^{n_1-i} bab a^{i}, bbaa a^{n_1} b, bb aba a^{n_1}, bba a^{n_1-i} ba a^{i}, baba a^{n_1-i} b a^{i},\\ b baba a^{n_1}, abba a^{n_1} b, abbba a^{n_1}, abb a^{n_1-i} ba a^{i}, ababb a^{n_1}, abab a^{n_1-i} b a^{i}, abb ab a^{n_1}, babba a^{n_1}, ababb a^{n_1} ~|~ 1 \leq i \leq n_1-1 \}$. We can observe that each element of $A_4$ is rich.
\item When we divide $w$ in five non-empty blocks, then $\ensuremath{\mathfrak m}athtt{BR}(w)$ contains the following:\\
Let, $A_5 =\{ baba a^{n_1} b, babba a^{n_1}, bab a^{n_1-i} ba a^{i}, baab b a^{n_1}, baab a^{n_1-i} b a^{i}, ba bab a^{n_1}, bbaa b a^{n_1}, bb aa a^{n_1-i} b a^{i},\\ bbaba a^{n_1}, abbab a^{n_1}, abba a^{n_1-i} b a^{i}, ababb a^{n_1}, a bbb a a^{n_1} ~|~ 1 \leq i \leq n_1-1 \}$. We can observe that each element of $A_5$ is rich.
\item When we divide $w$ in six non-empty blocks, then $\ensuremath{\mathfrak m}athtt{BR}(w)$ contains the following:\\
Let, $A_6 = \{ babab a^{n_1}, baba a^{n_1-i} b a^{i}, babba a^{n_1}, baabb a^{n_1}, bbaab a^{n_1}, abb ab a^{n_1} ~|~ 1 \leq i \leq n_1-1\}$. We can observe that each element of $A_6$ is rich.
\item When we divide $w$ in seven non-empty blocks, then $\ensuremath{\mathfrak m}athtt{BR}(w)$ contains the following:\\
Let, $A_7=\{ babab a^{n_1} \}$. Clearly, $babab a^{n_1}$ is rich.
Thus, each element of $A_7$ is rich. Also, as $babab a^{n_1}$ is rich, $w=a^{n_1} babab $ is rich.
\end{itemize}
Therefore, as $\ensuremath{\mathfrak m}athtt{BR}(w)= \{w\} \cup A_2 \cup A_3 \cup A_4 \cup A_5 \cup A_6 \cup A_7$, each element of $\ensuremath{\mathfrak m}athtt{BR}(w)$ is rich.
In a similar fashion one can also show that each element of $\ensuremath{\mathfrak m}athtt{BR}(ababab^{n_1})$ is also rich.
\end{enumerate}
\end{remark}
We now have the following:
\begin{proposition}\label{g6}
Let $w$ be a binary word with $|w|>7$ and $l(w)=6$. Then, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich iff $w\in \{u, (u^c)^R\;|\; u\in T\}$ where $$T=\{a b^2 a b a^2 b, a b a b^2 a^2 b, a b a b a^2 b^2, a^2 b a b^2 a b, a^2 b a b a b^2, a b a^2 b^2 a b, a^{n_1} b a b a b\; |\;n_1\geq 3\}.$$
\end{proposition}
\begin{proof}
Let $w$ be a binary word with $|w|>7$ and $l(w)=6$ such that all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich. Consider $w=a^{n_1}b^{n_2}a^{n_3}b^{n_4}a^{n_5}b^{n_6}$ to be the run-length encoding of $w$ where $n_i\geq 1$ for all $i$. If $n_5\geq 3$, then by Theorem \ref{tglen}, $ b^{n_6-1} a^{n_5-1} b a b^{n_4+n_2} a^{n_3+n_1} \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich. This implies, $n_5\leq 2$.
By Remark \ref{o1}, we get, $n_2\leq 2$. We have the following cases:
\begin{itemize}
\item $n_5 = 2 : $ If $n_1 \geq 2$ or $n_3\geq 2$, then by Theorem \ref{tglen}, $ b^{n_6-1} a^{2} b a b^{n_4+n_2} a^{n_3-1+n_1} \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction. Thus, $n_1 =n_3= 1$. Thus, in this case, we get,
$w = a b^{n_2} a b^{n_4} a^2 b^{n_6}$ where $n_2\leq 2$.
We have the following cases:
\begin{itemize}
\item $n_2=2 : $ Then, $w^R = b^{n_6} a^{2} b^{n_4} a b^{2} a$. By Remark \ref{o1}, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w^R)$ are rich. Here, $n_4=n_6=1$, otherwise $a b^2 a b a^2 b^{n_4-1+n_6} \in \ensuremath{\mathfrak m}athtt{BR}(w^R)$ is not rich. Thus, $w = a b^{2} a b a^2 b$.
\item $n_2 = 1 : $ Then, $w = a b a b^{n_4} a^2 b^{n_6}$. We have the following cases:
\begin{itemize}
\item $n_4 \geq 2$ : If $n_6 \geq 2$, then by Theorem \ref{tglen}, $b^{n_6} a^2 b a b^{n_4} a \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction. Otherwise, $n_6 =1$ and $w = a b a b^{n_4} a^2 b$. If $n_4 \geq 3$, then by Theorem \ref{tglen}, $b^{n_4-1} a^2 b a b^2 a \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction. Thus, $n_4 =2$ and $w = a b a b^{2} a^2 b$.
\item $n_4=1$ : If $n_6 \geq 3$, then by Theorem \ref{tglen}, $ b^{n_6-1} a^2 b a b^2 a \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction. Thus, $n_6\leq 2$. If $n_6=1$, then $|w|=7$, thus, $n_6=2$ and $w = a b a b a^2 b^{2}$.
\end{itemize}
\end{itemize}
\item $n_5 = 1 : $ Here, $w=a^{n_1}b^{n_2}a^{n_3}b^{n_4} a b^{n_6}$. We have the following cases:
\begin{itemize}
\item $n_1 \geq 2$ : If $n_3\geq 2$, then by Theorem \ref{tglen}, $ a^{n_3} b^{n_4} a b^{n_6} a^{n_1} b^{n_2} \in \ensuremath{\mathfrak m}athtt{BR}(w)$ for $n_4 \neq n_6$ ($ a^{n_3} b^{n_4} a b^{n_6+n_2} a^{n_1} \in \ensuremath{\mathfrak m}athtt{BR}(w)$ for $n_4 = n_6$, respectively) is not rich which is a contradiction. So, $n_3=1$ and $w= a^{n_1}b^{n_2} a b^{n_4}a b^{n_6}$ where $n_2\leq 2$. We have the following cases:
\begin{itemize}
\item $n_2 = 2 : $ Now, $w^R = b^{n_6} a b^{n_4} a b^2 a^{n_1}$. By Remark \ref{o1}, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w^R)$ are rich. If $n_4\geq 2$ or $n_6\geq 2$, then $a^{n_1-1} b^2 a b a^2 b^{n_4-1+n_6} \in \ensuremath{\mathfrak m}athtt{BR}(w^R)$
is not rich, a contradiction. Thus, $n_4 = n_6=1$. We get, $w= a^{n_1}b^{2} a b a b $. If $n_1\geq 3$, then by Theorem \ref{tglen}, $b a^{n_1-1} b^2 a b a^2 \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction. Thus, $n_1=2$ and $w= a^2 b^{2} a b a b$.
\item $n_2 = 1 : $ Then, $w= a^{n_1} b a b^{n_4}a b^{n_6}$. We have the following cases: \begin{itemize}
\item $n_4\geq 2$ : If $n_6 \geq 2$, then by Theorem \ref{tglen}, $b^{n_6} a^{n_1} b a b^{n_4}a \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction. Otherwise, $n_6=1$. Here, $n_1=2$, otherwise, by Theorem \ref{tglen}, $b a^{n_1-1} b a b^{n_4} a^2 \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich. Thus, $w= a^{2} b a b^{n_4} a b$. Also, $n_4=2$, otherwise, by Theorem \ref{tglen}, $ab b^{n_4-2} a^2 b a b^2 \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich. Thus, $w= a^{2} b a b^{2} a b$.
\item $n_4 = 1 : $ If $n_6\geq 4$, then by Theorem \ref{tglen}, $b^{n_6-2} a^{n_1} b a b a b^2\in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction. Thus, $w=a^{n_1} b a b a b^{n_6}$
for $1\leq n_6\leq 3$. In this case, if $n_1\geq 4$ and $2\leq n_6 \leq 3$, then by Theorem \ref{tglen}, $a^{n_1-2} b a b a b^{n_6} a^2 \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich. We are left with the words
$a^2 b a b a b^{n_6}$, $a^3 b a b a b^{n_6}$ for $n_6<4$ and $a^{n_1} b a b a b$ for $n_1\geq 1$. By Remark \ref{rem2}, one can conclude that if all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich, then $w$ is $a^{n_1} b a b a b$ or $a^{2} b a b a b^{2}$ where $n_1 \geq 3$.
\end{itemize}
\end{itemize}
\item $n_1=1$
: Similar to the case $n_1=2$, one can prove that if all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich, then $w$ is of one of the following forms:
$a b^2 a^2 b a b,\; a b a^2 b^2 a b, \; a b a^2 b a b^2,\; a b a b a b^{n_6} $, where $n_6 \geq 3$.
\end{itemize}
\end{itemize}
The converse follows from Theorem \ref{7ric} and Remark \ref{rem2}.
\end{proof}
We now consider the case when the length of the run sequence of the word is $5$.
\begin{proposition}\label{g5}
Let $w$ be a binary word with $|w|>7$ and $l(w)=5$. Then, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich iff $w$ is $a^{n_1}ba^{n_3}ba^{n_5}$, $ab^{n_2}ab^{n_4}a^2$, $a^2b^{n_2}ab^{n_4}a$,
$ab^{n_2}a^2b^{n_4}a$ where $n_2+n_4 =4$ for $n_i\geq 1$.
\end{proposition}
\begin{proof}
Let $w$ be a binary word with $|w|>7$ and $l(w)=5$ such that all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich. Consider $w=a^{n_1}b^{n_2}a^{n_3}b^{n_4}a^{n_5}$ to be the run-length encoding of $w$ where $n_i\geq 1$ for all $i$.
If $n_2 = n_4 = 1$, then by Theorem \ref{tglen}, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich. Now, consider $n_2 + n_4 \geq 3$. If $n_5 \geq 3$, then by Theorem \ref{tglen}, $a^{n_5-1} b a b^{n_4-1+n_2} a^{n_3+n_1} \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction. Thus, $n_5\leq 2$. By Remark \ref{o1}, we get, $n_1\leq 2$. We are left with the following cases:
\begin{itemize}
\item $n_5=2 : $ Then, $w=a^{n_1}b^{n_2}a^{n_3}b^{n_4}a^{2}$ where $n_2 + n_4 \geq 3$ and $n_1\leq 2$. If $n_3\geq 2$ or $n_1= 2$, then by Theorem \ref{tglen}, $a^{2} b a b^{n_4-1+n_2} a^{n_3-1+n_1} \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction. Thus, $n_3=n_1=1$ and in this case, $w=a b^{n_2} a b^{n_4} a^{2}$ where $n_2 + n_4 \geq 3$. As $|w|>7$, we get, $n_2+n_4\geq 4$. If $n_2+n_4= 4$, then by Theorem \ref{7ric}, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich. We now consider the case when $n_2 + n_4 \geq 5$. We have the following cases:
\begin{itemize}
\item $n_2=i$ for $1\leq i\leq 3$: Then, $n_4 \geq 5-i$ and $w=a b^i a b^{n_4} a^{2}$. By Theorem \ref{tglen}, we get, $b^{n_4-3+i} a^2 b a b^2 a \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction.
\item $n_2 \geq 4 : $ Then, $n_4 \geq 1$ and $w=a b^{n_2} a b^{n_4} a^{2}$. If $n_4 = 1$, then $b^{n_2-2} a b a^2 b^2 a \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction. If $n_4 \geq 2$, then $b^{n_4} a^2 b a b^{n_2-1} a \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction.
\end{itemize}
Hence, if $n_5=2$ and all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich, then $w=a b^{n_2} a b^{n_4} a^{2}$ where $n_2+n_4=4$.
\item
$n_5= 1 $: If $n_1=2$, then by Remark \ref{o1}, we get from the case $n_5=2$, if all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich, then $w= a^2 b^{n_2} a b^{n_4} a$
where $n_2+n_4=4$. Otherwise, $n_1=1$. Then, $w=ab^{n_2}a^{n_3}b^{n_4}a$ where $n_2+n_4 \geq 3$. We have the following cases:
\begin{itemize}
\item $n_3 \geq 3$ : If $n_2=n_4$, then since $n_2+n_4 \geq 3$, we get both $n_2,\;n_4 \geq 2$. By Theorem \ref{tglen}, $a^{n_3-1} b^{n_4} a b a^{2} b^{n_2-1} \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction. Thus, $n_2 \neq n_4$. By Theorem \ref{tglen}, $a^{n_3-1} b^{n_4} a b^{n_2} a^{2} \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction.
\item $n_3\leq 2$ : Then, $w = a b^{n_2} a^{n_3} b^{n_4} a$ where $n_2+n_4\geq 3$. As $|w|>7$, we get, $n_2+n_4\geq 4$. If $n_2+n_4 = 4$, then by Theorem \ref{7ric}, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich. Thus, $w = a b^{n_2} a^{n_3} b^{n_4} a$ where $n_2+n_4 = 4$.
Now, we consider the case when $n_2+n_4\geq 5$. We have the following cases:
\begin{itemize}
\item $n_2=i$ for $1\leq i\leq 3$: Then, $n_4 \geq 5-i$ and $w=a b^i a^{n_3} b^{n_4} a$. By Theorem \ref{tglen}, we get, $b^{n_4-3+i} a b a^2 b^2 a \in \ensuremath{\mathfrak m}athtt{BR}(w)$ for $n_3=2$ ( $b^{n_4-3+i} a^2 b a b^2 \in \ensuremath{\mathfrak m}athtt{BR}(w)$ for $n_3=1$, respectively) is not rich which is a contradiction.
\item $n_2 \geq 4 : $ Then, $n_4 \geq 1$ and $w = a b^{n_2} a^{n_3} b^{n_4} a$. If $n_4=1$, then by Theorem \ref{tglen}, $b^{n_2-2} a^2 b a b^2 a \in \ensuremath{\mathfrak m}athtt{BR}(w)$ for $n_3 =2$ ( $b^{n_2-2} a b a^2 b^2 \in \ensuremath{\mathfrak m}athtt{BR}(w)$ for $n_3 =1$, respectively ) is not rich which is a contradiction. Otherwise, $n_4 \geq 2$. By Theorem \ref{tglen}, $b^{n_4} a b a^2 b^{n_2-1} a \in \ensuremath{\mathfrak m}athtt{BR}(w)$ for $n_3=2$ ( $b^{n_4} a^2 b a b^{n_2-1} \in \ensuremath{\mathfrak m}athtt{BR}(w)$ for $n_3=1$, respectively) is not rich which is a contradiction.
\end{itemize}
\end{itemize}
\end{itemize}
The converse follows from Theorems \ref{tglen} and \ref{7ric}.
\end{proof}
We consider the case when the length of the run sequence of the word is $4$ and the length of the word is greater than $7$. We have the following result.
\begin{proposition}\label{g4}
Let $w$ be a binary word with $|w|>7$ and $l(w)=4$. Then, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich iff $w$ is of the form $ab^{n_2}ab^{n_4}$ or $a^{n_1}b a^{n_3}b$ or $w\in S$ where
\[S=\big\{
a^{n_1}b^{n_2}a^{n_3}b^{n_4}| (n_2,n_4), (n_1,n_3) \in \{(3,1),(2,2),(1,3)\}\big\} \]
and $n_i\geq 1$.
\end{proposition}
\begin{proof}
Let $w$ be a binary word with $l(w)=4$ such that all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich. Consider $w=a^{n_1}b^{n_2}a^{n_3}b^{n_4}$ to be the run-length encoding of $w$ where $n_i\geq 1$ for all $i$.
If $n_1=n_3=1$ or $n_2= n_4 = 1$, then all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich. Now, we consider the case $n_1+n_3 \geq 3$ and $n_2 + n_4 \geq 3$. If $n_3 \geq 4$, then for $j \neq r$ and $i+j=n_4$, and $r+s = n_2$, consider $\alpha = b^i a^{n_3-2} b^j a b^r a^{1+n_1} b^s \in \ensuremath{\mathfrak m}athtt{BR}(w)$. By Theorem \ref{tglen}, $\alpha$ is not rich which is a contradiction. Thus, $n_3 \leq 3$. We have the following cases:
\begin{itemize}
\item $n_3=3$ : Then, for $j' \neq r'$ and $i'+j'=n_4$, and $r'+s' = n_2$, consider $\beta = b^{i'} a^{2} b^{j'} a b^{r'} a^{n_1} b^{s'} \in \ensuremath{\mathfrak m}athtt{BR}(w)$. If $n_1 \geq 2$, then by Theorem \ref{tglen}, $\beta$ is not rich which is a contradiction. Thus, if $n_3=3$, then $n_1=1$.
\item $n_3=2: $ Then, for $j'' \neq n_2$ and $i''+j''=n_4$, consider $\gamma = b^{i''} a^{2} b^{j''} a b^{n_2} a^{n_1-1} \in \ensuremath{\mathfrak m}athtt{BR}(w) $. If $n_1 \geq 3$, $\gamma $ is not rich which is a contradiction. Thus, if $n_3=2$, then $n_1\leq 2$.
\item $n_3=1 :$ Then, by Theorem \ref{tglen}, for $n_1 \geq 4$, $a^{n_1-2}b^{n_2}a b^{n_4} a^2 \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich when $n_2 \neq n_4$ and $ b a^{n_1-2}b^{n_2}a b^{n_4-1} a^2 \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich when $n_2 = n_4$. Thus, if $n_3=1$, then $n_1\leq 3$.
\end{itemize}
Now, $w^R = b^{n_4} a^{n_3} b^{n_2} a^{n_1}$. Then from Remark \ref{o1},
one can similarly deduce that $n_2\leq 3$ and we also conclude the following:
\begin{itemize}
\item If $n_2=3$, then $n_4=1$.
\item If $n_2=2$, then $n_4\leq 2$.
\item If $n_2=1$, then $n_4\leq 3$.
\end{itemize}
Hence, as $|w|>7$, we get, $w\in S$ where
\[S=\big\{
a^{n_1}b^{n_2}a^{n_3}b^{n_4}| (n_2,n_4), (n_1,n_3) \in \{(3,1),(2,2),(1,3)\}\big\} \]
for $n_i\geq 1$ and $n_1+n_2+n_3+n_4>7$.\\
The converse follows from Theorems \ref{tglen} and \ref{7ric}.
\end{proof}
We consider the case when the length of the run sequence of the word is $3$ and the length of the word is greater than $7$. We need the following result.
\begin{lemma}\label{l1}
Let $w=a^{n_1} b^{n_2} a$ or $a b^{n_2} a^{n_1}$ where $n_1\geq 1$ and $n_2\in \{3, 4\}$. Then, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich.
\end{lemma}
\begin{proof}
We prove the result for $w=a^{n_1} b^{n_2} a$. The proof for $w=a b^{n_2} a^{n_1}$ is similar. Let $w=a^{n_1} b^{n_2} a$ where $n_1\geq 1$ and $n_2\in \{3, 4\}$. Consider $w' \in \ensuremath{\mathfrak m}athtt{BR}(w)$. Then, one can observe that $2\leq l(w')\leq 6$. If $l(w')\leq 4$, then from Proposition \ref{runlen}, $w'$ is rich. We are left with the following cases:
\begin{itemize}
\item $l(w')=5 :$ Then, $w'$ is either $a b^{i_1} a^{i_2} b^{i_3} a^{i_4}$ or $b^{j_1} a b^{j_2} a^{n_1} b^{j_3}$ where $i_1+i_3=n_2$, $i_2+i_4 = n_1$ and $j_1+j_2+j_3=n_2$. By Theorem \ref{tglen}, $w'$ is rich.
\item $l(w')=6 :$ Then, $w'= b^{k_1} a b^{k_2} a^{k_3} b^{k_4} a^{k_5} $ where $k_1+k_2+k_4=n_2$ and $k_3+k_5=n_1$. By Theorem \ref{tglen}, $w'$ is rich.
\end{itemize}
\end{proof}
We have the following:
\begin{proposition}\label{g3}
Let $w$ be a binary word with $|w|>7$ and $l(w)=3$. Then, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich iff $w$ is of one of the following forms:
\begin{itemize}
\item $a^2 b^4 a^2$, $ab^{m_2}a$, $a^{m_1}ba^{m_3}$, $a^{m_1}b^2a^{m_3}$.
\item $a b^{n_2} a^{n_3}, a^{n_1} b^{n_2} a $ where $n_2 \in \{ 3, 4\}$ and $ n_1, n_3 \geq 3$.
\end{itemize}
where $m_i\geq 1$.
\end{proposition}
\begin{proof}
Let $w$ be a binary word with $l(w)=3$ such that all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich. Consider $w=a^{n_1}b^{n_2}a^{n_3}$ to be the run-length encoding of $w$ where $n_i\geq 1$ for all $i$.
If $n_1=n_3=1$ or $n_2 \in \{1, 2\}$, then all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich. Here, $w=ab^{n_2}a$ or $a^{n_1}ba^{n_3}$ or $a^{n_1}b^2a^{n_3}$.
Now, we consider $n_1+n_3 \geq 3$ and $n_2 \geq 3$.
Let $n_2\geq 5$ and for $j \neq r$, $i+j=n_3$ and $r+s=n_1$, consider $\alpha = a^i b^2 a^j b a^r b^{n_2-3} a^s\in \ensuremath{\mathfrak m}athtt{BR}(w)$. Since, $b^2 a^j b a^r b^{2}$ is not a palindrome, by Theorem \ref{tglen}, $\alpha$ is not rich which is a contradiction. Thus, $n_2\leq 4$. So, $n_2 \in \{ 3, 4\}$. We have the following cases:
\begin{itemize}
\item $n_3\geq 3$ : Then, consider $\beta = a^{n_3-1} b a b^2 a^{n_1} b^{n_2-3} \in \ensuremath{\mathfrak m}athtt{BR}(w)$. If $n_1\geq 2$, then by Theorem \ref{tglen}, $a^{2} b a b^2 a^{2}$ is a palindrome, which is a contradiction. Thus, if $n_3 \geq 3$, then $n_1=1$. Here, $w= a b^{n_2} a^{n_3}$ where $n_2 \in \{ 3, 4\}$ and $ n_3 \geq 3$.
\item $n_3=2$ : Then, consider $\beta' = a^2 b a b^{n_2-1} a^{n_1-1}\in \ensuremath{\mathfrak m}athtt{BR}(w)$. If $n_1\geq 3$, then $\beta'$ is not rich. Thus, if $n_3=2$, then $n_1 \leq 2$. Thus, as $|w|\geq 8$, $w= a^2 b^4 a^2$.
\item $n_3=1$ : Then by Lemma \ref{l1}, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich.
\end{itemize}
The converse follows from Theorems \ref{tglen} and \ref{7ric} and Lemma \ref{l1}.
\end{proof}
We conclude the following from Propositions \ref{g8}, \ref{g7}, \ref{g6}, \ref{g5}, \ref{g4} and \ref{g3} for the binary words with $3\leq l(w) \leq 8.$
\begin{theorem}
Let $w$ be a binary word with $|w|>7$. Then, all elements of $\ensuremath{\mathfrak m}athtt{BR}(w)$ are rich for
\begin{enumerate}
\item $l(w)=8$ iff $w=abababab$
\item $l(w)=7$ iff $w$ is $ababab^2a$, $abab^2aba$ or $ab^2ababa$
\item $l(w)=6$ iff $w\in \{u, (u^c)^R\;|\; u\in T\}$ where $$T=\{a b^2 a b a^2 b, a b a b^2 a^2 b, a b a b a^2 b^2, a^2 b a b^2 a b, a^2 b a b a b^2, a b a^2 b^2 a b, a^{n_1} b a b a b\; |\;n_1\geq 3\}$$
\item $l(w)=5$ iff $w$ is $a^{n_1}ba^{n_3}ba^{n_5}$, $ab^{n_2}ab^{n_4}a^2$, $a^2b^{n_2}ab^{n_4}a$,
$ab^{n_2}a^2b^{n_4}a$ where $n_2+n_4 =4$
\item $l(w)=4$ iff $w$ is $ab^{n_2}ab^{n_4}$ or $a^{n_1}b a^{n_3}b$ or $w\in S$ where
\[S=\big\{
a^{n_1}b^{n_2}a^{n_3}b^{n_4}| (n_2,n_4), (n_1,n_3) \in \{(3,1),(2,2),(1,3)\}\big\} \]
\item $l(w)=3$ iff $w$ is of one of the following forms:
\begin{itemize}
\item $a^2 b^4 a^2$, $ab^{n_2}a$, $a^{n_1}ba^{n_3}$, $a^{n_1}b^2a^{n_3}$
\item $a b^{n_2} a^{n_3}, a^{n_1} b^{n_2} a $ where $n_2 \in \{ 3, 4\}$ and $ n_1, n_3 \geq 3$
\end{itemize}
\end{enumerate}
where $n_i\geq 1$.
\end{theorem}
\section{Conclusions}
In this paper, we have characterized words whose block reversal contains only rich words. We have found necessary and sufficient conditions for a non-binary word such that all elements in its block reversal are rich.
For a binary word, we have showed that the result varies with the length of the run sequence of the word and the structure of the word.
In future, we would like to find an upper bound on the number of elements in the block reversal of the binary word. It would also be interesting to study other combinatorial properties such as counting primitive words, bordered and unbordered words in the block reversal set of a word.
\section{Appendix}
Proof of the subcase $n_5=1$ and $n_1=1$ in Proposition \ref{g6}:
\begin{proof}
\item $n_1 = 1 : $ Then, $w= a b^{n_2} a^{n_3} b^{n_4} a b^{n_6}$ where $n_2\leq 2$. We have the following cases:
\begin{itemize}
\item $n_2 = 2 : $ Then, $w^R = b^{n_6} a b^{n_4} a^{n_3} b^{2} a$. If $n_4 \geq 2$ or $n_6 \geq 2$, then by Theorem \ref{tglen}, $b^2 a b a^{n_3+1} b^{n_4-1+n_6} \in \ensuremath{\mathfrak m}athtt{BR}(w^R)$ is not rich which is a contradiction. Thus, $n_4 = n_6=1$ and $w= a b^{2} a^{n_3} b a b $. If $n_3\geq 3$, then by Theorem \ref{tglen}, $a^{n_3-1} b a b^3 a^2 \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction. Thus, as $|w|>7$, $n_3=2$ and $w= a b^{2} a^{2} b a b $.
\item $n_2 = 1 : $ Then, $w= a b a^{n_3} b^{n_4} a b^{n_6}$.
If $n_4 \geq 2$ and $n_6 \geq 2$, then by Theorem \ref{tglen}, for $n_3 \geq 2$, $b^{n_6} a b a^{n_3} b^{n_4}a \in \ensuremath{\mathfrak m}athtt{BR}(w)$ and for $n_3=1$, $b^{n_6} a^2 b a b^{n_4} \in \ensuremath{\mathfrak m}athtt{BR}(w)$ are not rich which is a contradiction. Then, we have the following cases:
\begin{itemize}
\item $n_4 \geq 2$ and $n_6 = 1 : $ Then by Theorem \ref{tglen}, for $n_3 \geq 3$, $b a^{n_3-1} b^{n_4} a b a^2 \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction. For $n_3=2$, $w= a b a^2 b^{n_4} a b$. If $n_4\geq 3$, then by Theorem \ref{tglen}, $ab b^{n_4-2} a b a^2 b^2 \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction. Thus, $n_4=2$ and $w= a b a^2 b^{2} a b$. For $n_3=1$, $w= a b a b^{n_4} a b$. Since, $|w|>7$, $n_4 \geq 3$. Then, by Theorem \ref{tglen}, $b^2 a^2 b a b^{n_4-1} \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction.
\item $n_6 \geq 2$ and $n_4 = 1 : $ Then, $w= a b a^{n_3} b a b^{n_6}$. Now, by Theorem \ref{tglen}, for $n_3 \geq 3$, $a^{n_3-1} b a b^{n_6} a^2 b\in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction. For $n_3=2$, $w= a b a^{2} b a b^{n_6}$. Then by Theorem \ref{tglen}, for $n_6 \geq 3$, $b^{n_6-1}a^2 b a b^2 a \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction.
Thus, $n_6=2$ and $w= a b a^{2} b a b^{2}$. For $n_3=1$, $w= a b a b a b^{n_6}$.
\item $n_4 = 1 $ and $n_6 = 1 : $ Then, $w= a b a^{n_3} b a b$. Now, by Theorem \ref{tglen}, for $n_3 \geq 3$, $a^{n_3-1} b a b^2 a^2 \in \ensuremath{\mathfrak m}athtt{BR}(w)$ is not rich which is a contradiction. For $n_3\leq 2$, $|w|<8$. Hence, we omit this.
\end{itemize}
\end{itemize}
\end{proof}
\end{document} | math |
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/* -*- Mode: C; tab-width: 8; indent-tabs-mode: t; c-basic-offset: 8 -*-
*
* Copyright (C) 2010 Richard Hughes <richard@hughsie.com>
*
* Licensed under the GNU General Public License Version 2
*
* 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.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
*/
#include "config.h"
#include <sys/types.h>
#include <unistd.h>
#include <glib-object.h>
#include <glib/gi18n.h>
#include <gio/gio.h>
#include <locale.h>
#include <gtk/gtk.h>
#include <unique/unique.h>
#include <lcms2.h>
#include "egg-debug.h"
#include "mcm-calibrate-argyll.h"
#include "mcm-colorimeter.h"
#include "mcm-profile-store.h"
#include "mcm-utils.h"
#include "mcm-xyz.h"
static GtkBuilder *builder = NULL;
static GtkWidget *info_bar_hardware = NULL;
static GtkWidget *info_bar_hardware_label = NULL;
static McmCalibrate *calibrate = NULL;
static McmProfileStore *profile_store = NULL;
static const gchar *profile_filename = NULL;
static gboolean done_measure = FALSE;
enum {
MCM_PREFS_COMBO_COLUMN_TEXT,
MCM_PREFS_COMBO_COLUMN_PROFILE,
MCM_PREFS_COMBO_COLUMN_LAST
};
/**
* mcm_picker_set_pixbuf_color:
**/
static void
mcm_picker_set_pixbuf_color (GdkPixbuf *pixbuf, gchar red, gchar green, gchar blue)
{
gint x, y;
gint width, height, rowstride, n_channels;
guchar *pixels, *p;
n_channels = gdk_pixbuf_get_n_channels (pixbuf);
width = gdk_pixbuf_get_width (pixbuf);
height = gdk_pixbuf_get_height (pixbuf);
rowstride = gdk_pixbuf_get_rowstride (pixbuf);
pixels = gdk_pixbuf_get_pixels (pixbuf);
/* set to all the correct colors */
for (y=0; y<height; y++) {
for (x=0; x<width; x++) {
p = pixels + y * rowstride + x * n_channels;
p[0] = red;
p[1] = green;
p[2] = blue;
}
}
}
/**
* mcm_picker_measure_cb:
**/
static void
mcm_picker_measure_cb (GtkWidget *widget, gpointer data)
{
GtkWindow *window;
gboolean ret;
GError *error = NULL;
/* reset the image */
widget = GTK_WIDGET (gtk_builder_get_object (builder, "image_preview"));
gtk_image_set_from_file (GTK_IMAGE (widget), DATADIR "/icons/hicolor/64x64/apps/mate-color-manager.png");
/* get value */
window = GTK_WINDOW (gtk_builder_get_object (builder, "dialog_picker"));
ret = mcm_calibrate_spotread (calibrate, window, &error);
if (!ret) {
egg_warning ("failed to get spot color: %s", error->message);
g_error_free (error);
}
}
/**
* mcm_picker_refresh_results:
**/
static void
mcm_picker_refresh_results (void)
{
McmXyz *xyz = NULL;
GtkImage *image;
GtkLabel *label;
GdkPixbuf *pixbuf = NULL;
gdouble color_xyz[3];
guint8 color_rgb[3];
gdouble color_lab[3];
gdouble color_error[3];
gchar *text_xyz = NULL;
gchar *text_lab = NULL;
gchar *text_rgb = NULL;
gchar *text_error = NULL;
cmsHPROFILE profile_xyz;
cmsHPROFILE profile_rgb;
cmsHPROFILE profile_lab;
cmsHTRANSFORM transform_rgb;
cmsHTRANSFORM transform_lab;
cmsHTRANSFORM transform_error;
/* nothing set yet */
if (profile_filename == NULL)
goto out;
/* get new value */
g_object_get (calibrate, "xyz", &xyz, NULL);
/* create new pixbuf of the right size */
pixbuf = gdk_pixbuf_new (GDK_COLORSPACE_RGB, FALSE, 8, 200, 200);
/* get values */
g_object_get (xyz,
"cie-x", &color_xyz[0],
"cie-y", &color_xyz[1],
"cie-z", &color_xyz[2],
NULL);
/* lcms scales these for some reason */
color_xyz[0] /= 100.0f;
color_xyz[1] /= 100.0f;
color_xyz[2] /= 100.0f;
/* get profiles */
profile_xyz = cmsCreateXYZProfile ();
profile_rgb = cmsOpenProfileFromFile (profile_filename, "r");
profile_lab = cmsCreateLab4Profile (cmsD50_xyY ());
/* create transforms */
transform_rgb = cmsCreateTransform (profile_xyz, TYPE_XYZ_DBL, profile_rgb, TYPE_RGB_8, INTENT_PERCEPTUAL, 0);
if (transform_rgb == NULL)
goto out;
transform_lab = cmsCreateTransform (profile_xyz, TYPE_XYZ_DBL, profile_lab, TYPE_Lab_DBL, INTENT_PERCEPTUAL, 0);
if (transform_lab == NULL)
goto out;
transform_error = cmsCreateTransform (profile_rgb, TYPE_RGB_8, profile_xyz, TYPE_XYZ_DBL, INTENT_PERCEPTUAL, 0);
if (transform_error == NULL)
goto out;
cmsDoTransform (transform_rgb, color_xyz, color_rgb, 1);
cmsDoTransform (transform_lab, color_xyz, color_lab, 1);
cmsDoTransform (transform_error, color_rgb, color_error, 1);
/* destroy lcms state */
cmsDeleteTransform (transform_rgb);
cmsDeleteTransform (transform_lab);
cmsDeleteTransform (transform_error);
cmsCloseProfile (profile_xyz);
cmsCloseProfile (profile_rgb);
cmsCloseProfile (profile_lab);
/* set XYZ */
label = GTK_LABEL (gtk_builder_get_object (builder, "label_xyz"));
text_xyz = g_strdup_printf ("%.3f, %.3f, %.3f", color_xyz[0], color_xyz[1], color_xyz[2]);
gtk_label_set_label (label, text_xyz);
/* set LAB */
label = GTK_LABEL (gtk_builder_get_object (builder, "label_lab"));
text_lab = g_strdup_printf ("%.3f, %.3f, %.3f", color_lab[0], color_lab[1], color_lab[2]);
gtk_label_set_label (label, text_lab);
/* set RGB */
label = GTK_LABEL (gtk_builder_get_object (builder, "label_rgb"));
text_rgb = g_strdup_printf ("%i, %i, %i (#%02X%02X%02X)",
color_rgb[0], color_rgb[1], color_rgb[2],
color_rgb[0], color_rgb[1], color_rgb[2]);
gtk_label_set_label (label, text_rgb);
mcm_picker_set_pixbuf_color (pixbuf, color_rgb[0], color_rgb[1], color_rgb[2]);
/* set error */
label = GTK_LABEL (gtk_builder_get_object (builder, "label_error"));
text_error = g_strdup_printf ("%.1f%%, %.1f%%, %.1f%%",
ABS ((color_error[0] - color_xyz[0]) / color_xyz[0] * 100),
ABS ((color_error[1] - color_xyz[1]) / color_xyz[1] * 100),
ABS ((color_error[2] - color_xyz[2]) / color_xyz[2] * 100));
gtk_label_set_label (label, text_error);
/* set image */
image = GTK_IMAGE (gtk_builder_get_object (builder, "image_preview"));
gtk_image_set_from_pixbuf (image, pixbuf);
out:
g_free (text_xyz);
g_free (text_lab);
g_free (text_rgb);
g_free (text_error);
if (xyz != NULL)
g_object_unref (xyz);
if (pixbuf != NULL)
g_object_unref (pixbuf);
}
/**
* mcm_picker_xyz_notify_cb:
**/
static void
mcm_picker_xyz_notify_cb (McmCalibrate *calibrate_, GParamSpec *pspec, gpointer user_data)
{
GtkWidget *widget;
/* set expanded */
widget = GTK_WIDGET (gtk_builder_get_object (builder, "expander_results"));
gtk_expander_set_expanded (GTK_EXPANDER (widget), TRUE);
gtk_widget_set_sensitive (widget, TRUE);
/* we've got results so make sure it's sensitive */
done_measure = TRUE;
mcm_picker_refresh_results ();
}
/**
* mcm_picker_close_cb:
**/
static void
mcm_picker_close_cb (GtkWidget *widget, gpointer data)
{
GMainLoop *loop = (GMainLoop *) data;
g_main_loop_quit (loop);
}
/**
* mcm_picker_help_cb:
**/
static void
mcm_picker_help_cb (GtkWidget *widget, gpointer data)
{
mcm_mate_help ("picker");
}
/**
* mcm_picker_delete_event_cb:
**/
static gboolean
mcm_picker_delete_event_cb (GtkWidget *widget, GdkEvent *event, gpointer data)
{
mcm_picker_close_cb (widget, data);
return FALSE;
}
/**
* mcm_picker_colorimeter_setup_ui:
**/
static void
mcm_picker_colorimeter_setup_ui (McmColorimeter *colorimeter)
{
gboolean present;
gboolean supports_spot;
gboolean ret;
GtkWidget *widget;
present = mcm_colorimeter_get_present (colorimeter);
supports_spot = mcm_colorimeter_supports_spot (colorimeter);
ret = (present && supports_spot);
/* change the label */
if (present && !supports_spot) {
/* TRANSLATORS: this is displayed the user has not got suitable hardware */
gtk_label_set_label (GTK_LABEL (info_bar_hardware_label), _("The attached colorimeter is not capable of reading a spot color."));
} else if (!present) {
/* TRANSLATORS: this is displayed the user has not got suitable hardware */
gtk_label_set_label (GTK_LABEL (info_bar_hardware_label), _("No colorimeter is attached."));
}
/* hide some stuff */
widget = GTK_WIDGET (gtk_builder_get_object (builder, "button_measure"));
gtk_widget_set_sensitive (widget, ret);
widget = GTK_WIDGET (gtk_builder_get_object (builder, "expander_results"));
gtk_widget_set_sensitive (widget, ret && done_measure);
gtk_widget_set_visible (info_bar_hardware, !ret);
}
/**
* mcm_picker_colorimeter_changed_cb:
**/
static void
mcm_picker_colorimeter_changed_cb (McmColorimeter *colorimeter, gpointer user_data)
{
mcm_picker_colorimeter_setup_ui (colorimeter);
}
/**
* mcm_picker_message_received_cb
**/
static UniqueResponse
mcm_picker_message_received_cb (UniqueApp *app, UniqueCommand command, UniqueMessageData *message_data, guint time_ms, gpointer data)
{
GtkWindow *window;
if (command == UNIQUE_ACTIVATE) {
window = GTK_WINDOW (gtk_builder_get_object (builder, "dialog_picker"));
gtk_window_present (window);
}
return UNIQUE_RESPONSE_OK;
}
/**
* mcm_window_set_parent_xid:
**/
static void
mcm_window_set_parent_xid (GtkWindow *window, guint32 xid)
{
GdkDisplay *display;
GdkWindow *parent_window;
GdkWindow *our_window;
display = gdk_display_get_default ();
parent_window = gdk_window_foreign_new_for_display (display, xid);
our_window = gtk_widget_get_window (GTK_WIDGET (window));
/* set this above our parent */
gtk_window_set_modal (window, TRUE);
gdk_window_set_transient_for (our_window, parent_window);
}
/**
* mcm_picker_error_cb:
**/
static void
mcm_picker_error_cb (cmsContext ContextID, cmsUInt32Number errorcode, const char *text)
{
egg_warning ("LCMS error %i: %s", errorcode, text);
}
/**
* mcm_prefs_space_combo_changed_cb:
**/
static void
mcm_prefs_space_combo_changed_cb (GtkWidget *widget, gpointer data)
{
gboolean ret;
GtkTreeIter iter;
GtkTreeModel *model;
McmProfile *profile = NULL;
/* no selection */
ret = gtk_combo_box_get_active_iter (GTK_COMBO_BOX(widget), &iter);
if (!ret)
goto out;
/* get profile */
model = gtk_combo_box_get_model (GTK_COMBO_BOX(widget));
gtk_tree_model_get (model, &iter,
MCM_PREFS_COMBO_COLUMN_PROFILE, &profile,
-1);
if (profile == NULL)
goto out;
profile_filename = mcm_profile_get_filename (profile);
egg_debug ("changed picker space %s", profile_filename);
mcm_picker_refresh_results ();
out:
if (profile != NULL)
g_object_unref (profile);
}
/**
* mcm_prefs_set_combo_simple_text:
**/
static void
mcm_prefs_set_combo_simple_text (GtkWidget *combo_box)
{
GtkCellRenderer *renderer;
GtkListStore *store;
store = gtk_list_store_new (2, G_TYPE_STRING, MCM_TYPE_PROFILE);
gtk_tree_sortable_set_sort_column_id (GTK_TREE_SORTABLE (store), MCM_PREFS_COMBO_COLUMN_TEXT, GTK_SORT_ASCENDING);
gtk_combo_box_set_model (GTK_COMBO_BOX (combo_box), GTK_TREE_MODEL (store));
g_object_unref (store);
renderer = gtk_cell_renderer_text_new ();
g_object_set (renderer,
"ellipsize", PANGO_ELLIPSIZE_END,
"wrap-mode", PANGO_WRAP_WORD_CHAR,
"width-chars", 60,
NULL);
gtk_cell_layout_pack_start (GTK_CELL_LAYOUT (combo_box), renderer, TRUE);
gtk_cell_layout_set_attributes (GTK_CELL_LAYOUT (combo_box), renderer,
"text", MCM_PREFS_COMBO_COLUMN_TEXT,
NULL);
}
/**
* mcm_prefs_combobox_add_profile:
**/
static void
mcm_prefs_combobox_add_profile (GtkWidget *widget, McmProfile *profile, GtkTreeIter *iter)
{
GtkTreeModel *model;
GtkTreeIter iter_tmp;
const gchar *description;
/* iter is optional */
if (iter == NULL)
iter = &iter_tmp;
/* also add profile */
model = gtk_combo_box_get_model (GTK_COMBO_BOX(widget));
description = mcm_profile_get_description (profile);
gtk_list_store_append (GTK_LIST_STORE(model), iter);
gtk_list_store_set (GTK_LIST_STORE(model), iter,
MCM_PREFS_COMBO_COLUMN_TEXT, description,
MCM_PREFS_COMBO_COLUMN_PROFILE, profile,
-1);
}
/**
* mcm_prefs_setup_space_combobox:
**/
static void
mcm_prefs_setup_space_combobox (GtkWidget *widget)
{
McmProfile *profile;
guint i;
const gchar *filename;
McmColorspace colorspace;
gboolean has_profile = FALSE;
gboolean has_vcgt;
gboolean has_colorspace_description;
gchar *text = NULL;
GPtrArray *profile_array = NULL;
GtkTreeIter iter;
/* get new list */
profile_array = mcm_profile_store_get_array (profile_store);
/* update each list */
for (i=0; i<profile_array->len; i++) {
profile = g_ptr_array_index (profile_array, i);
/* only for correct kind */
has_vcgt = mcm_profile_get_has_vcgt (profile);
has_colorspace_description = mcm_profile_has_colorspace_description (profile);
colorspace = mcm_profile_get_colorspace (profile);
if (!has_vcgt && has_colorspace_description &&
colorspace == MCM_COLORSPACE_RGB) {
mcm_prefs_combobox_add_profile (widget, profile, &iter);
/* set active option */
filename = mcm_profile_get_filename (profile);
if (g_strcmp0 (filename, profile_filename) == 0)
gtk_combo_box_set_active_iter (GTK_COMBO_BOX (widget), &iter);
has_profile = TRUE;
}
}
if (!has_profile) {
/* TRANSLATORS: this is when there are no profiles that can be used; the search term is either "RGB" or "CMYK" */
text = g_strdup_printf (_("No %s color spaces available"),
mcm_colorspace_to_localised_string (MCM_COLORSPACE_RGB));
gtk_combo_box_append_text (GTK_COMBO_BOX(widget), text);
gtk_combo_box_set_active (GTK_COMBO_BOX (widget), 0);
gtk_widget_set_sensitive (widget, FALSE);
}
if (profile_array != NULL)
g_ptr_array_unref (profile_array);
g_free (text);
}
/**
* main:
**/
int
main (int argc, char *argv[])
{
GOptionContext *context;
guint retval = 0;
GError *error = NULL;
GMainLoop *loop;
GtkWidget *main_window;
GtkWidget *widget;
UniqueApp *unique_app;
guint xid = 0;
McmColorimeter *colorimeter = NULL;
const GOptionEntry options[] = {
{ "parent-window", 'p', 0, G_OPTION_ARG_INT, &xid,
/* TRANSLATORS: we can make this modal (stay on top of) another window */
_("Set the parent window to make this modal"), NULL },
{ NULL}
};
/* setup translations */
setlocale (LC_ALL, "");
bindtextdomain (GETTEXT_PACKAGE, LOCALEDIR);
bind_textdomain_codeset (GETTEXT_PACKAGE, "UTF-8");
textdomain (GETTEXT_PACKAGE);
/* setup LCMS */
cmsSetLogErrorHandler (mcm_picker_error_cb);
context = g_option_context_new (NULL);
/* TRANSLATORS: tool that is used to pick colors */
g_option_context_set_summary (context, _("MATE Color Manager Color Picker"));
g_option_context_add_group (context, egg_debug_get_option_group ());
g_option_context_add_group (context, gtk_get_option_group (TRUE));
g_option_context_add_main_entries (context, options, NULL);
g_option_context_parse (context, &argc, &argv, NULL);
g_option_context_free (context);
/* block in a loop */
loop = g_main_loop_new (NULL, FALSE);
/* are we already activated? */
unique_app = unique_app_new ("org.mate.ColorManager.Picker", NULL);
if (unique_app_is_running (unique_app)) {
egg_debug ("You have another instance running. This program will now close");
unique_app_send_message (unique_app, UNIQUE_ACTIVATE, NULL);
goto out;
}
g_signal_connect (unique_app, "message-received",
G_CALLBACK (mcm_picker_message_received_cb), NULL);
/* get UI */
builder = gtk_builder_new ();
retval = gtk_builder_add_from_file (builder, MCM_DATA "/mcm-picker.ui", &error);
if (retval == 0) {
egg_warning ("failed to load ui: %s", error->message);
g_error_free (error);
goto out;
}
main_window = GTK_WIDGET (gtk_builder_get_object (builder, "dialog_picker"));
gtk_window_set_icon_name (GTK_WINDOW (main_window), MCM_STOCK_ICON);
g_signal_connect (main_window, "delete_event",
G_CALLBACK (mcm_picker_delete_event_cb), loop);
widget = GTK_WIDGET (gtk_builder_get_object (builder, "button_close"));
g_signal_connect (widget, "clicked",
G_CALLBACK (mcm_picker_close_cb), loop);
widget = GTK_WIDGET (gtk_builder_get_object (builder, "button_help"));
g_signal_connect (widget, "clicked",
G_CALLBACK (mcm_picker_help_cb), NULL);
widget = GTK_WIDGET (gtk_builder_get_object (builder, "button_measure"));
g_signal_connect (widget, "clicked",
G_CALLBACK (mcm_picker_measure_cb), NULL);
widget = GTK_WIDGET (gtk_builder_get_object (builder, "image_preview"));
gtk_widget_set_size_request (widget, 200, 200);
/* add application specific icons to search path */
gtk_icon_theme_append_search_path (gtk_icon_theme_get_default (),
MCM_DATA G_DIR_SEPARATOR_S "icons");
/* use the color device */
colorimeter = mcm_colorimeter_new ();
g_signal_connect (colorimeter, "changed", G_CALLBACK (mcm_picker_colorimeter_changed_cb), NULL);
/* set the parent window if it is specified */
if (xid != 0) {
egg_debug ("Setting xid %i", xid);
mcm_window_set_parent_xid (GTK_WINDOW (main_window), xid);
}
/* use argyll */
calibrate = MCM_CALIBRATE (mcm_calibrate_argyll_new ());
g_signal_connect (calibrate, "notify::xyz",
G_CALLBACK (mcm_picker_xyz_notify_cb), NULL);
/* use an info bar if there is no device, or the wrong device */
info_bar_hardware = gtk_info_bar_new ();
info_bar_hardware_label = gtk_label_new (NULL);
gtk_info_bar_set_message_type (GTK_INFO_BAR(info_bar_hardware), GTK_MESSAGE_INFO);
widget = gtk_info_bar_get_content_area (GTK_INFO_BAR(info_bar_hardware));
gtk_container_add (GTK_CONTAINER(widget), info_bar_hardware_label);
gtk_widget_show (info_bar_hardware_label);
/* add infobar to devices pane */
widget = GTK_WIDGET (gtk_builder_get_object (builder, "vbox1"));
gtk_box_pack_start (GTK_BOX(widget), info_bar_hardware, FALSE, FALSE, 0);
/* disable some ui if no hardware */
mcm_picker_colorimeter_setup_ui (colorimeter);
/* maintain a list of profiles */
profile_store = mcm_profile_store_new ();
/* default to AdobeRGB */
profile_filename = "/usr/share/color/icc/Argyll/ClayRGB1998.icm";
/* setup RGB combobox */
widget = GTK_WIDGET (gtk_builder_get_object (builder, "combobox_colorspace"));
mcm_prefs_set_combo_simple_text (widget);
mcm_prefs_setup_space_combobox (widget);
g_signal_connect (G_OBJECT (widget), "changed",
G_CALLBACK (mcm_prefs_space_combo_changed_cb), NULL);
/* setup results expander */
mcm_picker_refresh_results ();
/* setup initial preview window */
widget = GTK_WIDGET (gtk_builder_get_object (builder, "image_preview"));
gtk_image_set_from_file (GTK_IMAGE (widget), DATADIR "/icons/hicolor/64x64/apps/mate-color-manager.png");
/* wait */
gtk_widget_show (main_window);
g_main_loop_run (loop);
out:
g_object_unref (unique_app);
if (profile_store != NULL)
g_object_unref (profile_store);
if (colorimeter != NULL)
g_object_unref (colorimeter);
if (calibrate != NULL)
g_object_unref (calibrate);
if (builder != NULL)
g_object_unref (builder);
g_main_loop_unref (loop);
return retval;
}
| code |
کٲملن لہجہ کٲشِرس بدلوو راے ظٲہر کران چھِ کم تامَتھ کٲمل چٲنۍ طرح کھٔتۍ عرشس غزلُک فن وُپھٕ ناوُن چھُے سا کٲمل یہ ونان چھُ بٔلۍ تہ باضے تٔتھۍ حض چھِ ونان غزل جنابو غزلہٕ بنجٲرۍ یہ کٲمل چھُ دِوان نارٕ طرح بختہٕ شٮ۪ہجٲرۍ لُکن پیمٕژ چھے نیہہ آبس منز کٲملن بوٗزِتھ طرح دار کلام سونچ پیوو بدلُن غزل خوانن تاں دٮ۪و کانہہ کٲمل بخشکھ رنگ اَسہ تُل خٲلِص لہجن دَس | kashmiri |
Award-winning music sensation, Tiwa Savage, has entered the second quarter of the year in grand style, as she got for herself a customized exotic car.
Savage, a mother of one, who got separated from her hubby Tee Bilz, in 2018, came to limelight with the signing of the recording contract with Mavin Records in 2012.
Her debut studio album, `Once Upon a Time,’ was released on July 3, 2013. It was supported by the singles “Kele Kele Love’’, “Love Me (3x)”, “Without My Heart”, “Ife Wa Gbona”, “Folarin”, “Olorun Mi” and “Eminado’’.
In 2018, ace music superstar, Wizkid featured her in the video for his acclaimed song ‘Fever’, in ‘intimate scenes’ which further fueled fans’ speculations of a relationship between them. | english |
\begin{document}
\title[Fusion procedure]{Fusion procedure for cyclotomic BMW algebras}
\author[Weideng Cui]{Weideng Cui}
\address{School of Mathematics, Shandong University, Jinan, Shandong 250100, P.R. China.}
\email{cwdeng@amss.ac.cn}
\begin{abstract}
Inspired by the work [IMOg2], in this note, we prove that the pairwise orthogonal primitive idempotents of generic cyclotomic Birman-Murakami-Wenzl algebras can be constructed by consecutive evaluations of a certain rational function. In the appendix, we prove a similar result for generic cyclotomic Nazarov-Wenzl algebras.
\end{abstract}
\maketitle
\section{Introduction}
\subsection{}
The primitive idempotents of a symmetric group $\mathfrak{S}_n,$ showed by Jucys [Juc], can be obtained by taking a certain limiting process on a rational function. The process is now commonly known as the fusion procedure, which has been further developed in the situation of Hecke algebras [Ch]; see also [Na2-4]. In [Mo], Molev has presented another approach of the fusion procedure for $\mathfrak{S}_n,$ which depends on the existence of a maximal commutative subalgebra generated by the Jucys-Murphy elements. In his approach, the primitive idempotents are obtained by consecutive evaluations of a certain rational function. The new version of the fusion procedure has been generalized to the Hecke algebras of type $A$ [IMO], to the Brauer algebras [IM, IMOg1], to the Birman-Murakami-Wenzl algebras [IMOg2], to the complex reflection groups of type $G(d,1,n)$ [OgPA1], to the Ariki-Koike algebras [OgPA2], to the wreath products of finite groups by the symmetric group [PA], to the degenerate cyclotomic Hecke algebras [ZL], to the Yokonuma-Hecke algebras [C1], to the cyclotomic Yokonuma-Hecke algebras [C2, Appendix] and to the degenerate cyclotomic Yokonuma-Hecke algebras [C3].
\subsection{}
The Birman-Murakami-Wenzl (for brevity, BMW) algebra was algebraically defined by Birman and Wenzl [BW], and independently Murakami [Mu], which is an algebra generated by some elements satisfying certain particular relations. These relations are in fact implicitly modeled on the ones of certain algebra of tangles studied by Kauffman [Ka] and Morton and Traczyk [MT], which is known as a Kauffman tangle algebra. BMW algebra are closely related to Artin braid groups of type $A,$ Iwahori-Hecke algebras of type $A,$ quantum groups, Brauer algebras and other diagram algebras; see [Eny1-2, HuXi, Hu, LeRa, MW, RuSi4-6, RuSo, Xi] and the references therein.
Motivated by studying link invariants, H\"{a}ring-Oldenburg [HO] introduced a class of finite dimensional associative algebras called cyclotomic Birman-Murakami-Wenzl (for brevity, BMW) algebras, generalizing the notions of BMW algebras. Such algebras are closely related to Artin braid groups of type $B,$ cyclotomic Hecke algebras and other research objects, and have been studied by a lot of authors from different perspectives; see [Go1-4, GoHM1-2, HO, OrRa, RuSi2-3, RuXu, Si, WiYu1-3, Xu, Yu] and so on.
\subsection{}
Inspired by the work [IMOg2] on the fusion procedures of BMW algebras, in this note we prove that a complete set of pairwise orthogonal primitive idempotents of cyclotomic BMW algebras can be derived by consecutive evaluations of a certain rational function in several variables. In the appendix, we prove a similar result for generic cyclotomic Nazarov-Wenzl algebras.
This paper is organized as follows. In Section 2, we recall some preliminaries and introduce the the primitive idempotents $E_{\mathcal{T}}$ of cyclotomic BMW algebras. In Section 3, we establish the fusion formula for the primitive idempotent $E_{\mathcal{T}}.$ In Section 4 (Appendix), we develop the fusion formulas for the primitive idempotents of cyclotomic Nazarov-Wenzl algebras.
\section{Preliminaries}
\subsection{Cyclotomic Birman-Murakami-Wenzl algebras}
\begin{definition}
Assume that $\mathbb{K}$ is an algebraically closed field containing $\delta_{j},$ $0\leq j\leq d-1,$ and some nonzero elements $\rho,$ $q$, $q-q^{-1}$ and $v_i,$ $1\leq i\leq d$, and that they satisfy the relation $\rho-\rho^{-1}=(q-q^{-1})(\delta_{0}-1).$\vskip2mm
Fix $n\geq 1.$ The cyclotomic Birman-Murakami-Wenzl algebra $\mathscr{B}_{d, n}$ is the $\mathbb{K}$-algebra generated by the elements $X_{1}^{\pm 1}, T_{i}^{\pm 1}$ and $E_{i}$ ($1\leq i\leq n-1$) with the following relations:\vskip2mm
(1) (Inverses) $T_{i}T_{i}^{-1}=T_{i}^{-1}T_{i}=1$ and $X_{1}X_{1}^{-1}=X_{1}^{-1}X_{1}=1.$
(2) (Idempotent relations) $E_{i}^{2}=\delta_{0} E_{i}$ for $1\leq i\leq n-1.$
(3) (Affine braid relations)
\hspace{0.7cm}(a) $T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1}$ and $T_{i}T_{j}=T_{j}T_{i}$ if $|i-j|\geq 2.$
\hspace{0.7cm}(b) $X_{1}T_{1}X_{1}T_{1}=T_{1}X_{1}T_{1}X_{1}$ and $X_{1}T_{j}=T_{j}X_{1}$ if $j\geq 2.$
(4) (Tangle relations)
\hspace{0.7cm}(a) $E_{i}E_{i\pm 1}E_{i}=E_{i}.$
\hspace{0.7cm}(b) $T_{i}T_{i\pm 1}E_{i}=E_{i\pm 1}E_{i}$ and $E_{i}T_{i\pm 1}T_{i}=E_{i}E_{i\pm 1}.$
\hspace{0.7cm}(c) For $1\leq j\leq d-1,$ $E_{1}X_{1}^{j}E_{1}=\delta_{j}E_{1}.$
(5) (Kauffman skein relations) $T_{i}-T_{i}^{-1}=(q-q^{-1})(1-E_{i})$ for $1\leq i\leq n-1.$
(6) (Untwisting relations) $T_{i}E_{i}=E_{i}T_{i}=\rho^{-1}E_{i}$ for $1\leq i\leq n-1.$
(7) (Unwrapping relations) $E_{1}X_{1}T_{1}X_{1}=\rho E_{1}=X_{1}T_{1}X_{1}E_{1}.$
(8) (Cyclotomic relation) $(X_1-v_1)(X_1-v_2)\cdots (X_1-v_d)=0.$
\end{definition}
In $\mathscr{B}_{d, n}$, We define inductively the following elements:
\begin{equation}\label{JMur-elements}
X_{i+1} :=T_{i}X_iT_i\quad\mbox{for}~i=1,\ldots,n-1.
\end{equation}
It can be easily checked that the elements $X_1,\ldots,X_n$ commute with each other, and moreover, we have
\begin{equation}\label{JMur-elements1}
E_{i}X_{i}X_{i+1}=X_{i}X_{i+1}E_{i}=E_{i}\quad\mbox{for}~i=1,\ldots,n-1.
\end{equation}
We now define the following elements (see [IMOg2, (2.15)]):
\begin{equation}\label{Baxterized-elements11}
T_{i}(u,v)=T_{i}+\frac{(q-q^{-1})u}{v-u}+\frac{(q-q^{-1})u}{u+\rho qv}E_{i}\quad\mbox{for}~i=1,\ldots,n-1.
\end{equation}
Note that $E_{i}^{2}=\delta_{0}E_{i}$, where $\delta_{0}=\frac{(q^{-1}+\rho^{-1})(\rho q-1)}{q-q^{-1}}.$ By using this, it can be easily checked that (see [IMOg2, (2.17-18)])
\begin{equation}\label{Baxterized-elements111}
T_{i}(u,v)T_{i}(v,u)=f(u,v)\quad\mbox{for}~i=1,\ldots,n-1,
\end{equation}
where
\begin{equation}\label{Baxterized-elements1111}
f(u,v)=f(v,u)=\frac{(u-q^{2}v)(u-q^{-2}v)}{(u-v)^{2}}.
\end{equation}
\subsection{Combinatorics}
$\lambda=(\lambda_{1},\ldots,\lambda_{k})$ is called a partition of $n$ if it is a finite sequence of weakly decreasing nonnegative integers whose sum is $n.$ We set $|\lambda| :=n.$ We shall identify a partition $\lambda$ with a Young diagram, which is the set $$[\lambda] :=\{(i,j)\:|\:i\geq 1~\mathrm{and}~1\leq j\leq \lambda_{i}\}.$$ We shall regard $\lambda$ as a left-justified array of boxes such that there exist $\lambda_{j}$ boxes in the $j$-th row for $j=1,\ldots,k.$ We write $\theta=(a,b)$ if the box $\theta$ lies in row $a$ and column $b.$
Similarly, a $d$-partition of $n$ is an ordered $d$-tuple $\bm{\lambda}=(\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(d)})$ of partitions $\lambda^{(k)}$ such that $\sum_{k=1}^{d}|\lambda^{(k)}|=n.$ We denote by $\mathcal{P}_{d}(n)$ the set of $d$-partitions of $n.$ We shall identify a $d$-partition $\bm{\lambda}$ with its Young diagram, which is the ordered $d$-tuple of the Young diagrams of its components. We write $\bm{\theta}=(\theta, s)$ if the box $\theta$ lies in the component $\lambda^{(s)}.$
Assume that $\bm{\lambda}$ and $\bm{\mu}$ are two $d$-partitions. We say that $\bm{\lambda}$ is obtained from $\bm{\mu}$ by adding a box if there exists a pair $(j,t)$ such that $\lambda_{j}^{(t)}=\mu_{j}^{(t)}+1$ and $\lambda_{i}^{(s)}=\mu_{i}^{(s)}$ for $(i,s)\neq (j,t).$ In this case, we will also say that $\bm{\mu}$ is obtained from $\bm{\lambda}$ by removing a box.
Set \[\Lambda_{d,n}^{+} :=\{(l,\bm{\lambda})\:|\:0\leq l\leq \lfloor n/2\rfloor, \bm{\lambda}\in \mathcal{P}_{d}(n-2l)\}.\]
The combinatorial objects appearing in the representation theory of $\mathscr{B}_{d, n}$ will be updown tableaux. For $(f, \bm{\lambda})\in \Lambda_{d,n}^{+},$ an $n$-updown $\bm{\lambda}$-tableau, or more simply an updown $\bm{\lambda}$-tableau, is a sequence $\mathcal{T}=(\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{n})$ of $d$-partitions such that $\mathcal{T}_{n}=\bm{\lambda}$ and $\mathcal{T}_{i}$ is obtained from $\mathcal{T}_{i-1}$ by either adding or removing a box, for $i=1,\ldots,n$, where we set $\mathcal{T}_{0}=\emptyset.$ Let $\mathscr{T}_{n}^{ud}(\bm{\lambda})$ be the set of updown $\bm{\lambda}$-tableaux of $n.$
Suppose that $(f, \bm{\lambda})\in \Lambda_{d,n}^{+}$ and $\mathcal{U}=(\mathcal{U}_{1},\ldots,\mathcal{U}_{n})\in \mathscr{T}_{n}^{ud}(\bm{\lambda}).$ Let
\begin{align}\label{symme-forms}
\mathrm{c}(\mathcal{U}|k)=
\begin{cases}
v_{s}q^{2(j-i)} & \text{if } \mathcal{U}_{k}=\mathcal{U}_{k-1}\cup ((i,j),s),
\\
v_{s}^{-1}q^{2(i-j)} & \text{if } \mathcal{U}_{k-1}=\mathcal{U}_{k}\cup ((i,j),s).
\end{cases}
\end{align}
Given a box $\bm{\alpha}=((i,j),s),$ we define the content of it by
\begin{align}\label{symme-forms11113344}
\mathrm{c}(\mathcal{U}|\bm{\alpha})=
\begin{cases}
v_{s}q^{2(j-i)} & \text{if }\bm{\alpha}\text{ is an addable box of }\mathcal{U},
\\
v_{s}^{-1}q^{2(i-j)} & \text{if }\bm{\alpha}\text{ is a removable box of }\mathcal{U}.
\end{cases}
\end{align}
We give the generalizations of some constructions in [IM, Section 3]. Suppose that $(f, \bm{\lambda})\in \Lambda_{d,n}^{+}$ and $\mathcal{T}=(\mathcal{T}_{1},\ldots,\mathcal{T}_{n})\in \mathscr{T}_{n}^{ud}(\bm{\lambda}).$ Set $\bm{\mu}=\mathcal{T}_{n-1}$ and consider the updown $\bm{\mu}$-tableau $\mathcal{U}=(\mathcal{T}_{1},\ldots,\mathcal{T}_{n-1}).$ We now define two $d$-tuples of infinite matrices
\[M(\mathcal{U})=(m_{1}(\mathcal{U}),\ldots,m_{d}(\mathcal{U}))\quad \mathrm{ and }\quad \overline{M}(\mathcal{U})=(\overline{m}_{1}(\mathcal{U}),\ldots,\overline{m}_{d}(\mathcal{U})),\]
here the rows and columns of each $m_{s}(\mathcal{U})$ or $\overline{m}_{s}(\mathcal{U})$ are labelled by positive integers and only a finite number of entries in each of the matrices are nonzero. The entry $m_{ij}^{s}$ of the matrix $m_{s}(\mathcal{U})$ (respectively, the entry $\overline{m}_{ij}^{s}$ of the matrix $\overline{m}_{s}(\mathcal{U})$) equals the number of times that the box $((i,j),s)$ is added (respectively, removed) in the sequence $(\emptyset, \mathcal{T}_{1},\ldots, \mathcal{T}_{n-1}).$
For each $k\in \mathbb{Z}$ and $1\leq s\leq d,$ we define two nonnegative integers $d_{k}^{s}=d_{k}(m_{s}(\mathcal{U}))$ and $\overline{d}_{k}^{s}=d_{k}(\overline{m}_{s}(\mathcal{U}))$ as the sums of the entries of the matrices $m_{s}(\mathcal{U})$ and $\overline{m}_{s}(\mathcal{U})$ on the $k$-th diagonal, that is,
\begin{equation}\label{dsk-dskbar}
d_{k}^{s}=\sum_{j-i=k}m_{ij}^{s}\quad\mathrm{ and }\quad \overline{d}_{k}^{s}=\sum_{j-i=k}\overline{m}_{ij}^{s}.
\end{equation}
Furthermore, we define the indexes $g_{k}^{s}=g_{k}(m_{s}(\mathcal{U}))$ and $\overline{g}_{k}^{s}=g_{k}(\overline{m}_{s}(\mathcal{U}))$ as follows:
\begin{equation}\label{index-indexbar}
g_{k}^{s}=\delta_{k0}+d_{k-1}^{s}+d_{k+1}^{s}-2d_{k}^{s}\quad\mathrm{ and }\quad \overline{g}_{k}^{s}=\overline{d}_{k-1}^{s}+\overline{d}_{k+1}^{s}-2\overline{d}_{k}^{s}.
\end{equation}
Finally, we define some integer $p_{1},\ldots,p_{n}$ associated to $\mathcal{T}$ inductively such that $p_{k}$ depends only on the first $k$ $d$-partitions $(\mathcal{T}_{1},\ldots,\mathcal{T}_{k})$ of $\mathcal{T}.$ Therefore, it is enough to define $p_{n}.$ We set
\begin{equation}\label{integer-indexbar}
p_{n}=1-g_{k_{n}}(m_{s_{n}}(\mathcal{U}))
\end{equation}
if $\mathcal{T}_{n}$ is obtained from $\mathcal{T}_{n-1}$ by adding a box $((i_n,j_n),s_n)$, where $k_n=j_n-i_n;$
\begin{equation}\label{integer-indexbar11}
p_{n}=1-g_{k_{n}'}(\overline{m}_{s_{n}'}(\mathcal{U}))
\end{equation}
if $\mathcal{T}_{n}$ is obtained from $\mathcal{T}_{n-1}$ by removing a box $((i_{n}',j_{n}'),s_{n}')$, where $k_{n}'=j_{n}'-i_{n}'.$
Assume that $(f, \bm{\lambda})\in \Lambda_{d,n}^{+},$ $\mathcal{T}=(\mathcal{T}_{1},\ldots,\mathcal{T}_{n})$ is an $n$-updown $\bm{\lambda}$-tableau and that
$\mathcal{U}=(\mathcal{T}_{1},\ldots,\mathcal{T}_{n-1}).$ We then define the element $f(\mathcal{T})$ inductively by
\begin{equation}\label{hooklength-indexbar11}
f(\mathcal{T})=f(\mathcal{U})\varphi(\mathcal{U}, \mathcal{T}),
\end{equation}
where
\begin{equation*}
\varphi(\mathcal{U}, \mathcal{T})=\prod_{\substack{k\neq k_{n}\\k\in \mathbb{Z}}}(q^{2k_{n}}-q^{2k})^{g_{k}^{s_{n}}}\prod_{\substack{1\leq t\leq d; t\neq s_{n}\\k\in \mathbb{Z}}}\hspace{-2mm}(v_{s_{n}}q^{2k_{n}}-v_{t}q^{2k})^{g_{k}^{t}} \prod_{\substack{1\leq r\leq d\\k\in \mathbb{Z}}}(v_{s_{n}}q^{2k_{n}}-v_{r}^{-1}q^{-2k})^{\overline{g}_{k}^{r}}
\end{equation*}
if $\mathcal{T}_{n}$ is obtained from $\mathcal{T}_{n-1}$ by adding a box $((i_n,j_n),s_n)$, where $k_n=j_n-i_n;$
\begin{equation*}
\varphi(\mathcal{U}, \mathcal{T})=\prod_{\substack{k\neq k_{n}'\\k\in \mathbb{Z}}}(q^{-2k_{n}'}-q^{-2k})^{\overline{g}_{k}^{s_{n}'}}\hspace{-1.5mm}
\prod_{\substack{1\leq t\leq d; t\neq s_{n}'\\k\in \mathbb{Z}}}\hspace{-2mm}(v_{s_{n}'}^{-1}q^{-2k_{n}'}-v_{t}^{-1}q^{-2k})^{\overline{g}_{k}^{t}} \prod_{\substack{1\leq r\leq d\\k\in \mathbb{Z}}}(v_{s_{n}'}^{-1}q^{-2k_{n}'}-v_{r}q^{2k})^{g_{k}^{r}}
\end{equation*}
if $\mathcal{T}_{n}$ is obtained from $\mathcal{T}_{n-1}$ by removing a box $((i_{n}',j_{n}'),s_{n}')$, where $k_{n}'=j_{n}'-i_{n}'.$
In the special situation when $f=0,$ that is, $\bm{\lambda}$ is a $d$-partition of $n,$ there is a natural bijection between the set of $n$-updown $\bm{\lambda}$-tableaux and the set of standard $\bm{\lambda}$-tableaux defined in [DJM, Definition (3.10)]. The following proposition is inspired by [IM, Proposition 3.3] and can be proved similarly.
\begin{proposition}\label{special-propo}
If $\bm{\lambda}$ is a $d$-partition of $n$ and $\mathcal{T}=(\mathcal{T}_{1},\ldots,\mathcal{T}_{n})$ is an $n$-updown $\bm{\lambda}$-tableau, then $p_1,\ldots,p_{n}$ are all equal to zero, and $f(\mathcal{T})$ is exactly equal to $\emph{F}_{\bm{\lambda}}^{-1}$ defined in $[\emph{OgPA}2, \emph{Section } 2.2(12)]$ when $d=m.$
\end{proposition}
\subsection{Idempotents of $\mathscr{B}_{d, n}$}
Following [RuXu, Definition 3.4], we say that $\mathscr{B}_{d, n}$ is generic if the parameters $v_i$, $1\leq i\leq d$, and $q$ satisfy the conditions (1) the order $o(q^{2})$ of $q^{2}$ satisfies $o(q^{2})> 2n$; (2) $|r|\geq 2n$ whenever there exists $r\in \mathbb{Z}$ such that either $v_{i}v_{j}^{\pm 1}=q^{2r}$ for $i\neq j,$ or $v_{i}=\pm q^{r}.$ Following [WiYu1, Corollary 4.5], we say that $\mathscr{B}_{d, n}$ is admissible if the set $\{E_{1}, E_{1}X_{1},\ldots,E_{1}X_{1}^{d-1}\}$ is linearly independent in $\mathscr{B}_{d, 2}.$ It has been proved by Goodman [Go2, Theorem 4.4] that this admissible condition coincides with the $\bm{\mathrm{u}}$-admissible condition defined in [RuXu, Definition 2.27].
From now on, we always assume that $\mathscr{B}_{d, n}$ is generic and admissible. Thus, by [RuXu, Lemma 3.5], we have $\mathcal{S}=\mathcal{T}$ if and only if $\mathrm{c}(\mathcal{S}|k)=\mathrm{c}(\mathcal{T}|k)$ for all $1\leq k\leq n.$ Therefore, the set $\{X_1,\ldots,X_n\},$ as a family of JM-elements for $\mathscr{B}_{d, n}$ in the abstract sense defined in [Ma, Definition 2.4], satisfies the separation condition associated to the weakly cellular basis of $\mathscr{B}_{d, n}$ constructed in [RuXu, Theorem 4.19]. Note that the results in [Ma] also hold for $\mathscr{B}_{d, n}$ with respect to the weakly cellular basis. In particular, we can construct the primitive idempotents of $\mathscr{B}_{d, n}$ following the arguments in [Ma, Section 3].
For each $1\leq k\leq n,$ we define the following set:
\[\mathcal{R}(k) :=\{\mathrm{c}(\mathcal{S}|k)\:|\:\mathcal{S}\in \mathscr{T}_{n}^{ud}(\bm{\lambda})
\text{ for some }(f, \bm{\lambda})\in \Lambda_{d,n}^{+}\}.\]
Suppose that $(f, \bm{\lambda})\in \Lambda_{d,n}^{+}$ and $\mathcal{T}\in \mathscr{T}_{n}^{ud}(\bm{\lambda}).$ We set
\begin{equation}\label{hooklength-idempotentelement11}
E_{\mathcal{T}}=\prod_{k=1}^{n}\bigg(\prod_{\substack{c\in \mathcal{R}(k)\\c\neq \mathrm{c}(\mathcal{T}|k)}}\frac{X_{k}-c}{\mathrm{c}(\mathcal{T}|k)-c}
\bigg).
\end{equation}
By standard arguments in [Ma, Section 3], the elements $\{E_{\mathcal{T}}\:|\:\mathcal{T}\in \mathscr{T}_{n}^{ud}(\bm{\lambda})
\text{ for some }(f, \bm{\lambda})\in \Lambda_{d,n}^{+}\}$ form a complete set of pairwise orthogonal primitive idempotents of $\mathscr{B}_{d, n}.$ Moreover, the elements $X_1,\ldots,X_n$ generate a maximal commutative subalgebra of $\mathscr{B}_{d, n}.$ We also have
\begin{equation}\label{hooklength-idempotentelement1111}
X_{k}E_{\mathcal{T}}=E_{\mathcal{T}}X_{k}=\mathrm{c}(\mathcal{T}|k)E_{\mathcal{T}}.
\end{equation}
\section{Fusion procedure for cyclotomic BMW algebras}
Assume that $(f, \bm{\lambda})\in \Lambda_{d,n}^{+}$ and that $\mathcal{T}=(\mathcal{T}_{1},\ldots,\mathcal{T}_{n})$ is an $n$-updown $\bm{\lambda}$-tableau. Set $\bm{\mu}=\mathcal{T}_{n-1}$ and $\mathcal{U}=(\mathcal{T}_{1},\ldots,\mathcal{T}_{n-1})$ as an updown $\bm{\mu}$-tableau. Let $\bm{\theta}$ be the box that is addable to or removable from $\bm{\mu}$ to get $\bm{\lambda}.$ For simplicity, we set $\mathrm{c}_{k} :=\mathrm{c}(\mathcal{T}|k).$ By \eqref{hooklength-idempotentelement11}, we can rewrite $E_{\mathcal{T}}$ inductively as follows:
\begin{equation}\label{idempotentele-induc}
E_{\mathcal{T}}=E_{\mathcal{U}}\frac{(X_{n}-b_1)\cdots (X_{n}-b_k)}{(\mathrm{c}_{n}-b_1)\cdots (\mathrm{c}_{n}-b_k)},
\end{equation}
where $b_1,\ldots,b_k$ are the contents of all boxes except $\bm{\theta},$ which can be addable to or removable from $\bm{\mu}$ to get a $d$-partition.
We denote by $\{\Lambda_{1},\ldots,\Lambda_{h}\}$ the set of all $d$-partitions obtained from $\bm{\mu}$ by adding a box or removing one. Set $\mathscr{T}_{j} :=(\mathcal{T}_{1},\ldots,\mathcal{T}_{n-1},\Lambda_{j})$ for $1\leq j\leq h.$ Note that $\mathcal{T}\in \{\mathscr{T}_{1},\ldots,\mathscr{T}_{h}\}.$ Since $\mathscr{B}_{d, n}$ is generic, hence it is semisimple. By [RuSi3, (4.16)] we have
\begin{equation}\label{sum-formula11}
E_{\mathcal{U}}=\sum_{j=1}^{h}E_{\mathscr{T}_{j}}.
\end{equation}
The property \eqref{hooklength-idempotentelement1111} implies that the following rational function
\begin{equation}\label{rational-function11}
E_{\mathcal{U}}\frac{u-\text{c}_n}{u-X_{n}}
\end{equation}
is regular at $u=\text{c}_n,$ and by \eqref{sum-formula11}, we get
\begin{equation}\label{sum-function1111}
E_{\mathcal{U}}\frac{u-\text{c}_n}{u-X_{n}}\Big|_{u=\text{c}_n}=E_{\mathcal{T}}.
\end{equation}
For $1\leq i\leq n-1,$ we set
\begin{align}\label{Q-function41}
Q_{i}(u,v;c) :=T_{i}+\frac{q-q^{-1}}{\rho^{-1}cuv-1}+\frac{q-q^{-1}}{1+qcuv}E_{i}.
\end{align}
Let $\phi_{1}(u) :=\frac{cuX_{1}-\rho}{u-X_1}.$ For $k=2,\ldots,n$, we set
\begin{align}\label{phi-function42}
\phi_k(u_1,\ldots,u_{k-1},u)& :=Q_{k-1}(u_{k-1},u;c)\phi_{k-1}(u_1,\ldots,u_{k-2},u)T_{k-1}(u_{k-1}, u)\notag\\
=Q_{k-1}&(u_{k-1},u;c)\cdots Q_{1}(u_{1},u;c)\phi_{1}(u)T_{1}(u_{1}, u)\cdots T_{k-1}(u_{k-1}, u).
\end{align}
From now on, we always set $c :=-q^{-1}.$ The following lemma is inspired by [IMOg2, Lemma 1] and can be proved similarly.
\begin{lemma}\label{phi-phi-phi111}
Assume that $n\geq 1.$ We have
\begin{align}\label{F-PhiEu43}
E_{\mathcal{U}}\phi_n(\mathrm{c}_1,\ldots,\mathrm{c}_{n-1},u)\prod_{r=1}^{n-1}f(u, \mathrm{c}_{r})^{-1}=E_{\mathcal{U}}\frac{cuX_{n}-\rho}{u-X_n}.
\end{align}
\end{lemma}
\begin{proof}
We shall prove \eqref{F-PhiEu43} by induction on $n.$ For $n=1,$ the situation is trivial.
We set
\begin{align}\label{phi-function421}
\phi'_n(\mathrm{c}_1,&\ldots,\mathrm{c}_{n-1},u)\notag\\
&=Q_{n-1}(\mathrm{c}_{n-1},u;c)\cdots Q_{1}(\mathrm{c}_{1},u;c)\phi_{1}(u)T_{1}(u, \mathrm{c}_{1})^{-1}\cdots T_{n-1}(u, \mathrm{c}_{n-1})^{-1}.
\end{align}
By \eqref{Baxterized-elements111} and \eqref{phi-function421}, in order to show \eqref{F-PhiEu43}, it suffices to prove that
\begin{align}\label{F-PhiEu4321}
E_{\mathcal{U}}\phi'_n(\mathrm{c}_1,\ldots,\mathrm{c}_{n-1},u)=E_{\mathcal{U}}\frac{cuX_{n}-\rho}{u-X_n}.
\end{align}
By the induction hypothesis, it boils down to proving the following equality:
\begin{align}\label{EUEU-PhiEu5}
E_{\mathcal{U}}Q_{n-1}(\mathrm{c}_{n-1},u;c)\frac{cuX_{n-1}-\rho}{u-X_{n-1}}T_{n-1}(u, \mathrm{c}_{n-1})^{-1}=E_{\mathcal{U}}\frac{cuX_{n}-\rho}{u-X_n}.
\end{align}
Since $X_{n}$ commutes with $E_{\mathcal{U}},$ we can rewrite \eqref{EUEU-PhiEu5} as follows:
\begin{align}\label{EUEU-PhiEu6}
E_{\mathcal{U}}(u-X_n)Q_{n-1}(&\mathrm{c}_{n-1},u;c)(cuX_{n-1}-\rho)\notag\\
&=E_{\mathcal{U}}(cuX_{n}-\rho)T_{n-1}(u, \mathrm{c}_{n-1})(u-X_{n-1}).
\end{align}
By \eqref{Baxterized-elements11} and \eqref{Q-function41}, the equality \eqref{EUEU-PhiEu6} becomes
\begin{align}\label{EUEU-PhiEu7}
E_{\mathcal{U}}(u&-X_n)\Big(T_{n-1}+\frac{q-q^{-1}}{\rho^{-1}cu\mathrm{c}_{n-1}-1}+\frac{q-q^{-1}}{1+qcu\mathrm{c}_{n-1}}E_{n-1}\Big)(cuX_{n-1}-\rho)\notag\\
&=E_{\mathcal{U}}(cuX_{n}-\rho)\Big(T_{n-1}+\frac{(q-q^{-1})u}{\mathrm{c}_{n-1}-u}+\frac{(q-q^{-1})u}{u+\rho q\mathrm{c}_{n-1}}E_{n-1}\Big)(u-X_{n-1}).
\end{align}
By definition, we have $T_{n-1}X_{n-1}=X_{n}T_{n-1}-(q-q^{-1})X_{n}+(q-q^{-1})X_{n}E_{n-1}.$ Thus, we get that \eqref{EUEU-PhiEu7} is equivalent to
\begin{align}\label{EUEU-PhiEu8}
E_{\mathcal{U}}(u&-X_n)\Big(cu(X_{n}T_{n-1}-(q-q^{-1})X_{n}+(q-q^{-1})X_{n}E_{n-1})-\rho T_{n-1}\notag\\
&\hspace{2cm}+(q-q^{-1})\rho+\frac{q-q^{-1}}{1+qcu\mathrm{c}_{n-1}}E_{n-1}(cuX_{n-1}-\rho)\Big)\notag\\
&=E_{\mathcal{U}}(cuX_{n}-\rho)\Big(-X_{n}T_{n-1}+(q-q^{-1})X_{n}-(q-q^{-1})X_{n}E_{n-1}+uT_{n-1}\notag\\
&\hspace{2cm}-(q-q^{-1})u+\frac{(q-q^{-1})u}{u+\rho q\mathrm{c}_{n-1}}E_{n-1}(u-X_{n-1})\Big).
\end{align}
Since we have
\begin{align*}
cu^{2}&X_{n}T_{n-1}-cuX_{n}^{2}T_{n-1}-(q-q^{-1})cuX_{n}(u-X_n)-\rho(u-X_n)(T_{n-1}-(q-q^{-1}))\notag\\
=&-cuX_{n}^{2}T_{n-1}+\rho X_nT_{n-1}+(q-q^{-1})(cuX_{n}-\rho)X_{n}+u(cuX_{n}-\rho)(T_{n-1}-(q-q^{-1})),
\end{align*}
it is easy to see that the equality \eqref{EUEU-PhiEu8} comes down to the following equality:
\begin{align}\label{EUEU-PhiEu9}
E_{\mathcal{U}}&(u-X_n)\Big(cuX_{n}E_{n-1}+\frac{1}{1+qcu\mathrm{c}_{n-1}}E_{n-1}(cuX_{n-1}-\rho)\Big)\notag\\
&=E_{\mathcal{U}}(cuX_{n}-\rho)\Big(-X_{n}E_{n-1}+\frac{u}{u+\rho q\mathrm{c}_{n-1}}E_{n-1}(u-X_{n-1})\Big).
\end{align}
By definition, we have $E_{\mathcal{U}}X_{n-1}=\mathrm{c}_{n-1}E_{\mathcal{U}}.$ Hence, we get $E_{\mathcal{U}}X_{n}E_{n-1}=\frac{1}{\mathrm{c}_{n-1}}E_{\mathcal{U}}E_{n-1}$ by \eqref{JMur-elements1}.
According to this, by comparing the coefficients of the terms involving $E_{\mathcal{U}}E_{n-1}X_{n-1}$, we see that it suffices to show that
\begin{align}\label{EUEU-PhiEu11}
\frac{1}{1+qcu\mathrm{c}_{n-1}}\cdot \frac{cu^{2}\mathrm{c}_{n-1}-cu}{\mathrm{c}_{n-1}}=\frac{u}{u+\rho q\mathrm{c}_{n-1}}\cdot \frac{\rho\mathrm{c}_{n-1}-cu}{\mathrm{c}_{n-1}}.
\end{align}
By comparing the coefficients of the terms involving $E_{\mathcal{U}}E_{n-1}$, it suffices to show that
\begin{align}\label{EUEU-PhiEu12}
\frac{cu^{2}-\rho}{\mathrm{c}_{n-1}}+\frac{1}{1+qcu\mathrm{c}_{n-1}}\cdot \frac{\rho-\rho u\mathrm{c}_{n-1}}{\mathrm{c}_{n-1}}=
\frac{u}{u+\rho q\mathrm{c}_{n-1}}\cdot \frac{cu^{2}-\rho u\mathrm{c}_{n-1}}{\mathrm{c}_{n-1}}.
\end{align}
Noting that $c=-q^{-1},$ it is easy to verify that \eqref{EUEU-PhiEu11} and \eqref{EUEU-PhiEu12} are true. Thus, \eqref{EUEU-PhiEu9} holds. The lemma is proved.
\end{proof}
Let $\overline{\phi}_{1}(u) :=(u-v_{1})\cdots (u-v_{d})\frac{cuX_{1}-\rho}{u-X_1}.$ For $k=2,\ldots,n$, we set
\begin{align}\label{phi-function424242}
\overline{\phi}_k(u_1,\ldots,u_{k-1},u)& :=Q_{k-1}(u_{k-1},u;c)\overline{\phi}_{k-1}(u_1,\ldots,u_{k-2},u)T_{k-1}(u_{k-1}, u)\notag\\
=Q_{k-1}&(u_{k-1},u;c)\cdots Q_{1}(u_{1},u;c)\overline{\phi}_{1}(u)T_{1}(u_{1}, u)\cdots T_{k-1}(u_{k-1}, u).
\end{align}
We also define the following rational function:
\begin{align}\label{Phi-function111}
\Phi(u_1,\ldots,u_n) :=\overline{\phi}_1(u_1)\cdots \overline{\phi}_{n-1}(u_1,\ldots,u_{n-1})\overline{\phi}_n(u_1,\ldots,u_{n}).
\end{align}
Recall that the integers $p_{1},\ldots,p_{n}$ associated to $\mathcal{T}$ have been defined as in \eqref{integer-indexbar} or \eqref{integer-indexbar11}.
Now we can state the main result of this paper.
\begin{theorem}\label{main-theorem11112}
The idempotent $E_{\mathcal{T}}$ of $\mathscr{B}_{d, n}$ corresponding to an $n$-updown $\bm{\lambda}$-tableau $\mathcal{T}$ can be derived by the following consecutive evaluations$:$
\begin{equation}\label{idempotents111}
E_{\mathcal{T}}=\frac{1}{f(\mathcal{T})}\Big(\prod_{k=1}^{n}\frac{(u_{k}-\mathrm{c}_{k})^{p_{k}}}{cu_{k}\mathrm{c}_{k}-\rho}\Big)
\Phi(u_1,\ldots,u_n)\Big|_{u_{1}=\emph{c}_1}\cdots\Big|_{u_{n}=\emph{c}_{n}}.
\end{equation}
\end{theorem}
\begin{proof}
We shall prove the theorem by induction on $n.$ For $n=1,$ we have $p_{1}=0$ by Proposition \ref{special-propo}. Thus, we get that the right-hand side of \eqref{idempotents111} is equal to
\begin{align}\label{n-1-istrue}
\frac{1}{f(\mathcal{T})}&\frac{(u_{1}-v_{1})\cdots (u_{1}-v_{d})}{cu_{1}\mathrm{c}_{1}-\rho}
\frac{cu_{1}X_{1}-\rho}{u_{1}-X_1}\Big|_{u_{1}=\mathrm{c}_1}\notag\\
&=\frac{1}{f(\mathcal{T})}\frac{(u_{1}-v_{1})\cdots (u_{1}-v_{d})}{u_{1}-\mathrm{c}_{1}}\frac{u_{1}-\mathrm{c}_{1}}{cu_{1}\mathrm{c}_{1}-\rho}\frac{cu_{1}X_{1}-\rho}{u_{1}-X_1}\Big|_{u_{1}=\mathrm{c}_1}.
\end{align}
Moreover, by \eqref{hooklength-indexbar11}, we have \[f(\mathcal{T})=\prod_{1\leq k\leq d;v_{k}\neq \mathrm{c}_{1}}(\mathrm{c}_{1}-v_{k}).\]
Therefore, it is easy to see that \eqref{n-1-istrue} is equal to $E_{\mathcal{T}}$ by \eqref{hooklength-idempotentelement1111} and \eqref{sum-function1111}.
For $n\geq 2,$ by the induction hypothesis we can write the right-hand side of \eqref{idempotents111} as follows:
\begin{align}\label{n-1-istrue2}
\frac{f(\mathcal{U})}{f(\mathcal{T})}\frac{(u_{n}-\mathrm{c}_{n})^{p_{n}}}{cu_{n}\mathrm{c}_{n}-\rho}E_{\mathcal{U}}
\overline{\phi}_n(\mathrm{c}_{1},\ldots,\mathrm{c}_{n-1},u_n)\Big|_{u_{n}=\mathrm{c}_{n}}.
\end{align}
Note that $\overline{\phi}_n(\mathrm{c}_{1},\ldots,\mathrm{c}_{n-1},u_n)=(u_{n}-v_{1})\cdots (u_{n}-v_{d})\phi_n(\mathrm{c}_{1},\ldots,\mathrm{c}_{n-1},u_n).$ By \eqref{F-PhiEu43}, we can rewrite the expression \eqref{n-1-istrue2} as
\begin{align}\label{n-1-istrue3}
\frac{f(\mathcal{U})}{f(\mathcal{T})}\frac{(u_{n}-\mathrm{c}_{n})^{p_{n}}}{cu_{n}\mathrm{c}_{n}-\rho}(u_{n}-v_{1})\cdots (u_{n}-v_{d})\prod_{r=1}^{n-1}f(u_{n}, \mathrm{c}_{r})E_{\mathcal{U}}\frac{cu_{n}X_{n}-\rho}{u_{n}-X_n}\Big|_{u_{n}=\mathrm{c}_{n}}.
\end{align}
By \eqref{hooklength-indexbar11}, we see that
\begin{align*}
\frac{f(\mathcal{U})}{f(\mathcal{T})}(u_{n}&-v_{1})\cdots (u_{n}-v_{d})\prod_{r=1}^{n-1}f(u_{n}, \mathrm{c}_{r})(u_{n}-\mathrm{c}_{n})^{p_{n}-1}\notag\\
&=\frac{f(\mathcal{U})}{f(\mathcal{T})}(u_{n}-v_{1})\cdots (u_{n}-v_{d})\prod_{r=1}^{n-1}\frac{(u_{n}-q^{2}\mathrm{c}_{r})(u_{n}-q^{-2}\mathrm{c}_{r})}{(u_{n}-\mathrm{c}_{r})^{2}}(u_{n}-\mathrm{c}_{n})^{p_{n}-1}
\end{align*}
is regular at $u_n=\mathrm{c}_{n}$ and is equal to $1.$ Thus, the expression \eqref{n-1-istrue3} equals
\begin{align}\label{n-1-istrue4}
E_{\mathcal{U}}\frac{u_{n}-\mathrm{c}_{n}}{u_{n}-X_n}\frac{cu_{n}X_{n}-\rho}{cu_{n}\mathrm{c}_{n}-\rho}\Big|_{u_{n}=\mathrm{c}_{n}}.
\end{align}
By \eqref{sum-function1111}, we see that \eqref{n-1-istrue4} is equal to
\begin{align}\label{n-1-istrue5}
E_{\mathcal{T}}\frac{cu_{n}X_{n}-\rho}{cu_{n}\mathrm{c}_{n}-\rho}\Big|_{u_{n}=\mathrm{c}_{n}}.
\end{align}
By \eqref{hooklength-idempotentelement1111}, we have $E_{\mathcal{T}}X_{n}=\mathrm{c}_{n}E_{\mathcal{T}}.$ Thus, we get that the expression \eqref{n-1-istrue5}, that is, the right-hand side of \eqref{idempotents111} equals $E_{\mathcal{T}}.$
\end{proof}
\begin{remark}\label{remark111}
Let $\mathscr{H}_{d, n}$ be the cyclotomic Hecke algebra defined in [AK]. It has been proved in [RuXu, Proposition 4.1] that $\mathscr{H}_{d, n}$ is isomorphic to the quotient of $\mathscr{B}_{d, n}$ by the two-sided ideal generated by all $E_{i}.$ In the process of taking quotient, the parameter $\rho$ disappears; however, the parameter $c$ is reserved and can be arbitrary. If we replace the $T_{i}(u, v),$ $Q_{i}(u,v;c),$ $\phi_{1}(u)$ in \eqref{phi-function42} with
\begin{align*}
\overline{T}_{i}(u,v)=T_{i}+\frac{(q-q^{-1})u}{v-u},\quad \overline{Q}_{i}(u,v;c) :=T_{i}+\frac{q-q^{-1}}{cuv-1},\quad \psi_{1}(u) :=\frac{cuX_{1}-1}{u-X_1},
\end{align*}
it is easy to see that the analogue of Lemma \ref{phi-phi-phi111} holds.
Let $\overline{\psi}_{1}(u) :=(u-v_{1})\cdots (u-v_{d})\frac{cuX_{1}-1}{u-X_1},$ and for $k=2,\ldots,n$, set
\begin{align*}
\overline{\psi}_k(u_1,\ldots,u_{k-1},u)& :=\overline{Q}_{k-1}(u_{k-1},u;c)\overline{\psi}_{k-1}(u_1,\ldots,u_{k-2},u)\overline{T}_{k-1}(u_{k-1}, u).
\end{align*}
We also define a rational function by
\begin{align*}
\Upsilon(u_1,\ldots,u_n) :=\overline{\psi}_1(u_1)\cdots \overline{\psi}_{n-1}(u_1,\ldots,u_{n-1})\overline{\psi}_n(u_1,\ldots,u_{n}).
\end{align*}
Then it is easy to see that the analogue of Theorem \ref{main-theorem11112} is true. Thus, we get a one-parameter family of the fusion procedures for cyclotomic Hecke algebras, generalizing the results obtained in [OgPA2].
\end{remark}
\section{Appendix. Fusion procedure for cyclotomic Nazarov-Wenzl algebras}
When studying the representations of Brauer algebras, Nazarov [Na1] introduced a class of infinite dimensional algebras under the name affine Wenzl algebras. In order to study finite dimensional irreducible representations of affine Wenzl algebras, Ariki, Mathas and Rui [AMR] defined the finite dimensional quotients of them, known as the cyclotomic Nazarov-Wenzl algebras. Cyclotomic Nazarov-Wenzl algebras are related to degenerate cyclotomic Hecke algebras just in the same way that cyclotomic BMW algebras are connected with cyclotomic Hecke algebras. Cyclotomic Nazarov-Wenzl algebras have been studied by many authors; see [Go3-4, RuSi1-2, Xu] and so on.
\subsection{Cyclotomic Nazarov-Wenzl algebras}
\begin{definition}
Suppose that $\mathbb{K}$ is an algebraically closed field containing $\omega_{j}$ ($0\leq j\leq d-1$), $v_i$ ($1\leq i\leq d$), and the invertible element $2.$\vskip2mm
Fix $n\geq 1.$ The cyclotomic Nazarov-Wenzl algebra $\mathscr{W}_{d, n}$ is the $\mathbb{K}$-algebra generated by the elements $S_{i}, E_{i}$ ($1\leq i\leq n-1$) and $X_{j}$ ($1\leq j\leq n$) satisfying the following relations:\vskip2mm
(1) (Involutions) $S_{i}^{2}=1$ for $1\leq i\leq n-1.$
(2) (Idempotent relations) $E_{i}^{2}=\omega_{0} E_{i}$ for $1\leq i\leq n-1.$
(3) (Affine braid relations)
\hspace{0.7cm}(a) $S_{i}S_{i+1}S_{i}=S_{i+1}S_{i}S_{i+1}$ and $S_{i}S_{j}=S_{j}S_{i}$ if $|i-j|\geq 2.$
\hspace{0.7cm}(b) $S_{i}X_{j}=X_{j}S_{i}$ if $j\neq i, i+1.$
(4) (Tangle relations)
\hspace{0.7cm}(a) $E_{i}E_{i\pm 1}E_{i}=E_{i}.$
\hspace{0.7cm}(b) $S_{i}S_{i\pm 1}E_{i}=E_{i\pm 1}E_{i}$ and $E_{i}S_{i\pm 1}S_{i}=E_{i}E_{i\pm 1}.$
\hspace{0.7cm}(c) For $1\leq k\leq d-1,$ $E_{1}X_{1}^{k}E_{1}=\omega_{k}E_{1}.$
(5) (Untwisting relations) $S_{i}E_{i}=E_{i}S_{i}=E_{i}$ for $1\leq i\leq n-1.$
(6) (Skein relations) $S_{i}X_{i}-X_{i+1}S_{i}=E_{i}-1$ for $1\leq i\leq n-1.$
(7) (Anti-symmetry relations) $E_{i}(X_{i}+X_{i+1})=(X_{i}+X_{i+1})E_{i}=0$ for $1\leq i\leq n-1.$
(8) (Commutative relations)
\hspace{0.7cm}(a) $S_{i}E_{j}=E_{j}S_{i}$ and $E_{i}E_{j}=E_{j}E_{i}$ if $|i-j|\geq 2.$
\hspace{0.7cm}(b) $E_{i}X_{j}=X_{j}E_{i}$ if $j\neq i, i+1.$
\hspace{0.7cm}(c) $X_{i}X_{j}=X_{j}X_{i}$ for $1\leq i,j \leq n.$
(9) (Cyclotomic relation) $(X_1-v_1)(X_1-v_2)\cdots (X_1-v_d)=0.$
\end{definition}
We define the following elements:
\begin{equation}\label{Baxterized-elements11cde}
S_{i}(u,v)=S_{i}+\frac{1}{v-u}-\frac{1}{v-u+\frac{\omega_{0}}{2}-1}E_{i}\quad\mbox{for}~1\leq i\leq n-1.
\end{equation}
By using the fact that $E_{i}^{2}=\omega_{0}E_{i}$, we can easily get
\begin{equation}\label{Baxterized-elements111cde}
S_{i}(u,v)S_{i}(v,u)=g(u,v)\quad\mbox{for}~1\leq i\leq n-1,
\end{equation}
where
\begin{equation}\label{Baxterized-elements1111cde}
g(u,v)=g(v,u)=\frac{(u-v+1)(u-v-1)}{(u-v)^{2}}.
\end{equation}
\subsection{Combinatorics}
Suppose that $(f, \bm{\lambda})\in \Lambda_{d,n}^{+}$ and $\mathfrak{s}=(\mathfrak{s}_{1},\ldots,\mathfrak{s}_{n})\in \mathscr{T}_{n}^{ud}(\bm{\lambda}).$ We can define the integers $d_{k}^{s},$ $\overline{d}_{k}^{s},$ $g_{k}^{s},$ $\overline{g}_{k}^{s}$ and some integers $p_{1},\ldots,p_{n}$ associated to $\mathfrak{s}$ in exactly the same way as those related to some $\mathcal{T}$ defined in Subsection 2.2. We shall follow the notations and only emphasize the differences.
Set
\begin{align}\label{symme-formscde}
\mathrm{c}(\mathfrak{s}|k)=
\begin{cases}
v_{s}+j-i & \text{if } \mathfrak{s}_{k}=\mathfrak{s}_{k-1}\cup ((i,j),s),
\\
-v_{s}+i-j & \text{if } \mathfrak{s}_{k-1}=\mathfrak{s}_{k}\cup ((i,j),s).
\end{cases}
\end{align}
Given a box $\bm{\beta}=((i,j),s),$ we define the content of it by
\begin{align}\label{symme-forms11113344cde}
\mathrm{c}(\mathcal{U}|\bm{\beta})=
\begin{cases}
v_{s}+j-i & \text{if }\bm{\beta}\text{ is an addable box of }\mathfrak{s},
\\
-v_{s}+i-j & \text{if }\bm{\beta}\text{ is a removable box of }\mathfrak{s}.
\end{cases}
\end{align}
Assume that $(f, \bm{\lambda})\in \Lambda_{d,n}^{+},$ $\mathfrak{t}=(\mathfrak{t}_{1},\ldots,\mathfrak{t}_{n})$ is an $n$-updown $\bm{\lambda}$-tableau and that
$\mathfrak{u}=(\mathfrak{t}_{1},\ldots,\mathfrak{t}_{n-1}).$ We then define the element $g(\mathfrak{t})$ inductively by
\begin{equation}\label{hooklength-indexbar11cde}
g(\mathfrak{t})=g(\mathfrak{u})\psi(\mathfrak{u}, \mathfrak{t}),
\end{equation}
where
\begin{equation*}
\psi(\mathfrak{u}, \mathfrak{t})=\prod_{\substack{k\neq k_{n}\\k\in \mathbb{Z}}}(k_{n}-k)^{g_{k}^{s_{n}}}\prod_{\substack{1\leq t\leq d; t\neq s_{n}\\k\in \mathbb{Z}}}\hspace{-2mm}(v_{s_{n}}-v_{t}+k_{n}-k)^{g_{k}^{t}}\prod_{\substack{1\leq r\leq d\\k\in \mathbb{Z}}}(v_{s_{n}}+v_{r}+k_{n}+k)^{\overline{g}_{k}^{r}}
\end{equation*}
if $\mathfrak{t}_{n}$ is obtained from $ \mathfrak{t}_{n-1}$ by adding a box $((i_n,j_n),s_n)$, where $k_n=j_n-i_n;$
\begin{equation*}
\psi(\mathfrak{u}, \mathfrak{t})=\prod_{\substack{k\neq k_{n}'\\k\in \mathbb{Z}}}(-k_{n}'+k)^{\overline{g}_{k}^{s_{n}'}}\prod_{\substack{1\leq t\leq d; t\neq s_{n}'\\k\in \mathbb{Z}}}(-v_{s_{n}'}+v_{t}-k_{n}'+k)^{\overline{g}_{k}^{t}}\prod_{\substack{1\leq r\leq d\\k\in \mathbb{Z}}}(-v_{s_{n}'}-v_{r}-k_{n}'-k)^{g_{k}^{r}}
\end{equation*}
if $\mathfrak{t}_{n}$ is obtained from $\mathfrak{t}_{n-1}$ by removing a box $((i_{n}',j_{n}'),s_{n}')$, where $k_{n}'=j_{n}'-i_{n}'.$
The following proposition is inspired by [IM, Proposition 3.3] and can be proved similarly.
\begin{proposition}\label{special-propocde}
If $\bm{\lambda}$ is a $d$-partition of $n$ and $\mathfrak{t}=(\mathfrak{t}_{1},\ldots,\mathfrak{t}_{n})$ is an $n$-updown $\bm{\lambda}$-tableau, then $p_1,\ldots,p_{n}$ are all equal to zero, and $g(\mathfrak{t})$ is exactly equal to $\Theta_{\bm{\lambda}}(Q)^{-1}$ defined in $[\emph{ZL}, (3.2)]$ when $d=m$ and $v_{s}=q_{s}$ for $1\leq s\leq m.$
\end{proposition}
\subsection{Idempotents of $\mathscr{W}_{d, n}$}
Following [AMR, Definition 4.3], we say that $\mathscr{W}_{d, n}$ is generic if the parameters $v_i$, $1\leq i\leq d$, satisfy the conditions (1) the characteristic $p$ of $\mathbb{K}$ satisfies $p=0$ or $p> 2n;$ (2) $|r|\geq 2n$ whenever there exists $r\in \mathbb{Z}$ such that either $v_{i}\pm v_{j}=r$ and $i\neq j,$ or $2v_{i}=r.$ Following [Go3, Definition 4.2], we say that $\mathscr{W}_{d, n}$ is admissible if the set $\{E_{1}, E_{1}X_{1},\ldots,E_{1}X_{1}^{d-1}\}$ is linearly independent in $\mathscr{B}_{d, 2}.$ It has been proved by Goodman [Go3, Theorem 5.2] that this admissible condition coincides with the $\bm{\mathrm{u}}$-admissible condition defined in [AMR, Definition 3.6].
From now on, we always assume that $\mathscr{W}_{d, n}$ is generic and admissible. Thus, by [AMR, Lemma 4.4], we have $\mathfrak{s}=\mathfrak{t}$ if and only if $\mathrm{c}(\mathfrak{s}|k)=\mathrm{c}(\mathfrak{t}|k)$ for all $1\leq k\leq n.$ Therefore, the set $\{X_1,\ldots,X_n\},$ as a family of JM-elements for $\mathscr{W}_{d, n}$ in the abstract sense defined in [Ma, Definition 2.4], satisfies the separation condition associated to the cellular basis of $\mathscr{W}_{d, n}$ constructed in [AMR, Theorem 7.17]. In particular, we can construct the primitive idempotents of $\mathscr{W}_{d, n}$ following the arguments in [Ma, Section 3].
For each $1\leq k\leq n,$ we define the following set:
\[\mathscr{R}(k) :=\{\mathrm{c}(\mathfrak{s}|k)\:|\:\mathfrak{s}\in \mathscr{T}_{n}^{ud}(\bm{\lambda})
\text{ for some }(f, \bm{\lambda})\in \Lambda_{d,n}^{+}\}.\]
Suppose that $(f, \bm{\lambda})\in \Lambda_{d,n}^{+}$ and $\mathfrak{t}\in \mathscr{T}_{n}^{ud}(\bm{\lambda}).$ We set
\begin{equation}\label{hooklength-idempotentelement11cde}
E_{\mathfrak{t}}=\prod_{k=1}^{n}\bigg(\prod_{\substack{a\in \mathscr{R}(k)\\a\neq \mathrm{c}(\mathfrak{t}|k)}}\frac{X_{k}-a}{\mathrm{c}(\mathfrak{t}|k)-a}
\bigg).
\end{equation}
By standard arguments in [Ma, Section 3], the elements $\{E_{\mathfrak{t}}\:|\:\mathfrak{t}\in \mathscr{T}_{n}^{ud}(\bm{\lambda})
\text{ for some }(f, \bm{\lambda})\in \Lambda_{d,n}^{+}\}$ form a complete set of pairwise orthogonal primitive idempotents of $\mathscr{W}_{d, n}.$ Moreover, the elements $X_1,\ldots,X_n$ generate a maximal commutative subalgebra of $\mathscr{W}_{d, n}.$ We also have
\begin{equation}\label{hooklength-idempotentelement1111cde}
X_{k}E_{\mathfrak{t}}=E_{\mathfrak{t}}X_{k}=\mathrm{c}(\mathfrak{t}|k)E_{\mathfrak{t}}.
\end{equation}
\subsection{Fusion procedure for cyclotomic Nazarov-Wenzl algebras}
Assume that $(f, \bm{\lambda})\in \Lambda_{d,n}^{+}$ and that $\mathfrak{t}=(\mathfrak{t}_{1},\ldots,\mathfrak{t}_{n})$ is an $n$-updown $\bm{\lambda}$-tableau. Set $\bm{\mu}=\mathfrak{t}_{n-1}$ and $\mathfrak{u}=(\mathfrak{t}_{1},\ldots,\mathfrak{t}_{n-1})$ as an updown $\bm{\mu}$-tableau. Let $\bm{\theta}$ be the box that is addable to or removable from $\bm{\mu}$ to get $\bm{\lambda}.$ For simplicity, we set $\mathrm{c}_{k} :=\mathrm{c}(\mathfrak{t}|k).$ By \eqref{hooklength-idempotentelement11cde}, we can rewrite $E_{\mathfrak{t}}$ inductively as follows:
\begin{equation}\label{idempotentele-induccde}
E_{\mathfrak{t}}=E_{\mathfrak{t}}\frac{(X_{n}-a_1)\cdots (X_{n}-a_k)}{(\mathrm{c}_{n}-a_1)\cdots (\mathrm{c}_{n}-a_k)},
\end{equation}
where $a_1,\ldots,a_k$ are the contents of all boxes except $\bm{\theta},$ which can be addable to or removable from $\bm{\mu}$ to get a $d$-partition.
We denote by $\{\Delta_{1},\ldots,\Delta_{e}\}$ the set of all $d$-partitions obtained from $\bm{\mu}$ by adding a box or removing one. Set $\mathscr{S}_{j} :=(\mathfrak{t}_{1},\ldots,\mathfrak{t}_{n-1},\Delta_{j})$ for $1\leq j\leq e.$ Note that $\mathfrak{t}\in \{\mathscr{S}_{1},\ldots,\mathscr{S}_{e}\}.$ Since $\mathscr{W}_{d, n}$ is generic, hence it is semisimple. By [AMR, Theorem 5.3 a)] we have
\begin{equation}\label{sum-formula11cde}
E_{\mathfrak{u}}=\sum_{j=1}^{e}E_{\mathscr{S}_{j}}.
\end{equation}
The equality \eqref{hooklength-idempotentelement1111cde} implies that the following rational function
\begin{equation}\label{rational-function11cde}
E_{\mathfrak{u}}\frac{u-\text{c}_n}{u-X_{n}}
\end{equation}
is regular at $u=\text{c}_n,$ and by \eqref{sum-formula11cde}, we get
\begin{equation}\label{sum-function1111cde}
E_{\mathfrak{u}}\frac{u-\text{c}_n}{u-X_{n}}\Big|_{u=\text{c}_n}=E_{\mathfrak{t}}.
\end{equation}
For $1\leq i\leq n-1,$ we set
\begin{align}\label{Q-function41cde}
R_{i}(u,v;c) :=S_{i}+\frac{1}{u+v+c}-\frac{1}{u+v}E_{i}.
\end{align}
Let $\varphi_{1}(u) :=\frac{u+X_{1}+c}{u-X_1}.$ For $k=2,\ldots,n$, we set
\begin{align}\label{phi-function42cde}
\varphi_k(u_1,\ldots,u_{k-1},u)& :=R_{k-1}(u_{k-1},u;c)\varphi_{k-1}(u_1,\ldots,u_{k-2},u)S_{k-1}(u_{k-1}, u)\notag\\
=R_{k-1}&(u_{k-1},u;c)\cdots R_{1}(u_{1},u;c)\varphi_{1}(u)S_{1}(u_{1}, u)\cdots S_{k-1}(u_{k-1}, u).
\end{align}
From now on, we always set $c :=1-\frac{\omega_{0}}{2}.$ The following lemma is inspired by [IMOg2, Lemma 1] and can be proved similarly.
\begin{lemma}\label{phi-phi-phi111cde}
Assume that $n\geq 1.$ We have
\begin{align}\label{F-PhiEu43cde}
E_{\mathfrak{u}}\varphi_n(\mathrm{c}_1,\ldots,\mathrm{c}_{n-1},u)\prod_{r=1}^{n-1}g(u, \mathrm{c}_{r})^{-1}=E_{\mathfrak{u}}\frac{u+X_{n}+c}{u-X_n}.
\end{align}
\end{lemma}
\begin{proof}
We shall prove \eqref{F-PhiEu43cde} by induction on $n.$ For $n=1,$ the situation is trivial.
We set
\begin{align}\label{phi-function421cde}
\varphi'_n(\mathrm{c}_1,&\ldots,\mathrm{c}_{n-1},u)\notag\\
&=R_{n-1}(\mathrm{c}_{n-1},u;c)\cdots R_{1}(\mathrm{c}_{1},u;c)\varphi_{1}(u)S_{1}(u, \mathrm{c}_{1})^{-1}\cdots S_{n-1}(u, \mathrm{c}_{n-1})^{-1}.
\end{align}
By \eqref{Baxterized-elements111cde} and \eqref{phi-function421cde}, in order to show \eqref{F-PhiEu43cde}, it suffices to prove that
\begin{align}\label{F-PhiEu4321cde}
E_{\mathfrak{u}}\varphi'_n(\mathrm{c}_1,\ldots,\mathrm{c}_{n-1},u)=E_{\mathfrak{u}}\frac{u+X_{n}+c}{u-X_n}.
\end{align}
By the induction hypothesis, it boils down to proving the following equality:
\begin{align}\label{EUEU-PhiEu5cde}
E_{\mathfrak{u}}R_{n-1}(\mathrm{c}_{n-1},u;c)\frac{u+X_{n-1}+c}{u-X_{n-1}}S_{n-1}(u, \mathrm{c}_{n-1})^{-1}=E_{\mathfrak{u}}\frac{u+X_{n}+c}{u-X_n}.
\end{align}
Since $X_{n}$ commutes with $E_{\mathfrak{u}},$ we can rewrite \eqref{EUEU-PhiEu5cde} as follows:
\begin{align}\label{EUEU-PhiEu6cde}
E_{\mathfrak{u}}(u-X_n)R_{n-1}(&\mathrm{c}_{n-1},u;c)(u+X_{n-1}+c)\notag\\
&=E_{\mathfrak{u}}(u+X_{n}+c)S_{n-1}(u, \mathrm{c}_{n-1})(u-X_{n-1}).
\end{align}
By \eqref{Baxterized-elements11cde} and \eqref{Q-function41cde}, the equality \eqref{EUEU-PhiEu6cde} becomes
\begin{align}\label{EUEU-PhiEu7cde}
E_{\mathfrak{u}}&(u-X_n)\Big(S_{n-1}+\frac{1}{\mathrm{c}_{n-1}+u+c}-\frac{1}{\mathrm{c}_{n-1}+u}E_{n-1}\Big)(u+X_{n-1}+c)\notag\\
&=E_{\mathfrak{u}}(u+X_{n}+c)\Big(S_{n-1}+\frac{1}{\mathrm{c}_{n-1}-u}-\frac{1}{\mathrm{c}_{n-1}-u+\frac{\omega_{0}}{2}-1}E_{n-1}\Big)(u-X_{n-1}).
\end{align}
By definition, we have $S_{n-1}X_{n-1}=X_{n}S_{n-1}+E_{n-1}-1.$ Thus, we get that \eqref{EUEU-PhiEu7cde} is equivalent to
\begin{align}\label{EUEU-PhiEu8cde}
E_{\mathfrak{u}}(u&-X_n)\Big(uS_{n-1}+(X_{n}S_{n-1}+E_{n-1}-1)+cS_{n-1}+1\notag\\
&\hspace{2cm}-\frac{1}{\mathrm{c}_{n-1}+u}E_{n-1}(u+X_{n-1}+c)\Big)\notag\\
&=E_{\mathfrak{u}}(u+X_{n}+c)\Big(uS_{n-1}-(X_{n}S_{n-1}+E_{n-1}-1)-1\notag\\
&\hspace{2cm}-\frac{1}{\mathrm{c}_{n-1}-u+\frac{\omega_{0}}{2}-1}E_{n-1}(u-X_{n-1})\Big).
\end{align}
It is easy to see that the equality \eqref{EUEU-PhiEu8cde} comes down to the following equality:
\begin{align}\label{EUEU-PhiEu9cde}
(c+2u)&E_{\mathfrak{u}}-E_{\mathfrak{u}}(u-X_n)\frac{1}{\mathrm{c}_{n-1}+u}E_{n-1}(u+X_{n-1}+c)\notag\\
&=-E_{\mathfrak{u}}(u+X_{n}+c)\frac{1}{\mathrm{c}_{n-1}-u+\frac{\omega_{0}}{2}-1}E_{n-1}(u-X_{n-1}).
\end{align}
By definition, we have $E_{\mathfrak{u}}X_{n-1}=\mathrm{c}_{n-1}E_{\mathfrak{u}}.$ Hence, we get $E_{\mathfrak{u}}X_{n}E_{n-1}=-\mathrm{c}_{n-1}E_{\mathfrak{u}}E_{n-1}$ by definition.
According to this, by comparing the coefficients of the terms involving $E_{\mathfrak{u}}E_{n-1}X_{n-1}$, we see that it suffices to show that
\begin{align}\label{EUEU-PhiEu11cde}
\frac{-u-\mathrm{c}_{n-1}}{\mathrm{c}_{n-1}+u}=\frac{u-\mathrm{c}_{n-1}+c}{\mathrm{c}_{n-1}-u+\frac{\omega_{0}}{2}-1}.
\end{align}
By comparing the coefficients of the terms involving $E_{\mathfrak{u}}E_{n-1}$, it suffices to show that
\begin{align}\label{EUEU-PhiEu12cde}
(c+2u)+\frac{-(c+u)(\mathrm{c}_{n-1}+u)}{\mathrm{c}_{n-1}+u}=\frac{u(-u+\mathrm{c}_{n-1}-c)}{\mathrm{c}_{n-1}-u+\frac{\omega_{0}}{2}-1}.
\end{align}
Noting that $c=1-\frac{\omega_{0}}{2},$ it is easy to verify that \eqref{EUEU-PhiEu11cde} and \eqref{EUEU-PhiEu12cde} are true. Thus, \eqref{EUEU-PhiEu9cde} holds. The lemma is proved.
\end{proof}
Let $\overline{\varphi}_{1}(u) :=(u-v_{1})\cdots (u-v_{d})\frac{u+X_{1}+c}{u-X_1}.$ For $k=2,\ldots,n$, we set
\begin{align}\label{phi-function424242cde}
\overline{\varphi}_k(u_1,\ldots,u_{k-1},u)& :=R_{k-1}(u_{k-1},u;c)\overline{\varphi}_{k-1}(u_1,\ldots,u_{k-2},u)S_{k-1}(u_{k-1}, u)\notag\\
=R_{k-1}&(u_{k-1},u;c)\cdots R_{1}(u_{1},u;c)\overline{\varphi}_{1}(u)S_{1}(u_{1}, u)\cdots S_{k-1}(u_{k-1}, u).
\end{align}
We also define the following rational function:
\begin{align}\label{Phi-function111cde}
\Psi(u_1,\ldots,u_n) :=\overline{\varphi}_1(u_1)\cdots \overline{\varphi}_{n-1}(u_1,\ldots,u_{n-1})\overline{\varphi}_n(u_1,\ldots,u_{n}).
\end{align}
Recall that the integers $p_{1},\ldots,p_{n}$ associated to $\mathfrak{t}$ have been defined as in \eqref{integer-indexbar} or \eqref{integer-indexbar11}.
Now we can state the main result of this paper.
\begin{theorem}\label{main-theorem11112cde}
The idempotent $E_{\mathfrak{t}}$ of $\mathscr{W}_{d, n}$ corresponding to an $n$-updown $\bm{\lambda}$-tableau $\mathfrak{t}$ can be derived by the following consecutive evaluations$:$
\begin{equation}\label{idempotents111cde}
E_{\mathfrak{t}}=\frac{1}{g(\mathfrak{t})}\Big(\prod_{k=1}^{n}\frac{(u_{k}-\mathrm{c}_{k})^{p_{k}}}{u_{k}+\mathrm{c}_{k}+c}\Big)
\Psi(u_1,\ldots,u_n)\Big|_{u_{1}=\emph{c}_1}\cdots\Big|_{u_{n}=\emph{c}_{n}}.
\end{equation}
\end{theorem}
\begin{proof}
We shall prove the theorem by induction on $n.$ For $n=1,$ we have $p_{1}=0$ by Proposition \ref{special-propocde}. Thus, we get that the right-hand side of \eqref{idempotents111cde} is equal to
\begin{align}\label{n-1-istruecde}
\frac{1}{g(\mathfrak{t})}&\frac{(u_{1}-v_{1})\cdots (u_{1}-v_{d})}{u_{1}+\mathrm{c}_{1}+c}
\frac{u_{1}+X_{1}+c}{u_{1}-X_1}\Big|_{u_{1}=\mathrm{c}_1}\notag\\
&=\frac{1}{g(\mathfrak{t})}\frac{(u_{1}-v_{1})\cdots (u_{1}-v_{d})}{u_{1}-\mathrm{c}_{1}}\frac{u_{1}-\mathrm{c}_{1}}{u_{1}+\mathrm{c}_{1}+c}\frac{u_{1}+X_{1}+c}{u_{1}-X_1}\Big|_{u_{1}=\mathrm{c}_1}.
\end{align}
Moreover, by \eqref{hooklength-indexbar11cde}, we have \[g(\mathfrak{t})=\prod_{1\leq k\leq d;v_{k}\neq \mathrm{c}_{1}}(\mathrm{c}_{1}-v_{k}).\]
Therefore, it is easy to see that \eqref{n-1-istruecde} is equal to $E_{\mathfrak{t}}$ by \eqref{hooklength-idempotentelement1111cde} and \eqref{sum-function1111cde}.
For $n\geq 2,$ by the induction hypothesis we can write the right-hand side of \eqref{idempotents111cde} as follows:
\begin{align}\label{n-1-istrue2cde}
\frac{g(\mathfrak{u})}{g(\mathfrak{t})}\frac{(u_{n}-\mathrm{c}_{n})^{p_{n}}}{u_{n}+\mathrm{c}_{n}+c}E_{\mathfrak{u}}
\overline{\varphi}_n(\mathrm{c}_{1},\ldots,\mathrm{c}_{n-1},u_n)\Big|_{u_{n}=\mathrm{c}_{n}}.
\end{align}
Note that $\overline{\varphi}_n(\mathrm{c}_{1},\ldots,\mathrm{c}_{n-1},u_n)=(u_{n}-v_{1})\cdots (u_{n}-v_{d})\varphi_n(\mathrm{c}_{1},\ldots,\mathrm{c}_{n-1},u_n).$ By \eqref{F-PhiEu43cde}, we can rewrite the expression \eqref{n-1-istrue2cde} as
\begin{align}\label{n-1-istrue3cde}
\frac{g(\mathfrak{u})}{g(\mathfrak{t})}\frac{(u_{n}-\mathrm{c}_{n})^{p_{n}}}{u_{n}+\mathrm{c}_{n}+c}(u_{n}-v_{1})\cdots (u_{n}-v_{d})\prod_{r=1}^{n-1}g(u_{n}, \mathrm{c}_{r})E_{\mathfrak{u}}\frac{u_{n}+X_{n}+c}{u_{n}-X_n}\Big|_{u_{n}=\mathrm{c}_{n}}.
\end{align}
By \eqref{hooklength-indexbar11cde}, we see that
\begin{align*}
\frac{g(\mathfrak{u})}{g(\mathfrak{t})}(u_{n}&-v_{1})\cdots (u_{n}-v_{d})\prod_{r=1}^{n-1}g(u_{n}, \mathrm{c}_{r})(u_{n}-\mathrm{c}_{n})^{p_{n}-1}\notag\\
&=\frac{g(\mathfrak{u})}{g(\mathfrak{t})}(u_{n}-v_{1})\cdots (u_{n}-v_{d})\prod_{r=1}^{n-1}\frac{(u_{n}-\mathrm{c}_{r}+1)(u_{n}-\mathrm{c}_{r}-1)}{(u_{n}-\mathrm{c}_{r})^{2}}(u_{n}-\mathrm{c}_{n})^{p_{n}-1}
\end{align*}
is regular at $u_n=\mathrm{c}_{n}$ and is equal to $1.$ Thus, the expression \eqref{n-1-istrue3cde} equals
\begin{align}\label{n-1-istrue4cde}
E_{\mathfrak{u}}\frac{u_{n}-\mathrm{c}_{n}}{u_{n}-X_n}\frac{u_{n}+X_{n}+c}{u_{n}+\mathrm{c}_{n}+c}\Big|_{u_{n}=\mathrm{c}_{n}}.
\end{align}
By \eqref{sum-function1111cde}, we see that \eqref{n-1-istrue4cde} is equal to
\begin{align}\label{n-1-istrue5cde}
E_{\mathfrak{t}}\frac{u_{n}+X_{n}+c}{u_{n}+\mathrm{c}_{n}+c}\Big|_{u_{n}=\mathrm{c}_{n}}.
\end{align}
By \eqref{hooklength-idempotentelement1111cde}, we have $E_{\mathfrak{t}}X_{n}=\mathrm{c}_{n}E_{\mathfrak{t}}.$ Thus, we get that the expression \eqref{n-1-istrue5cde}, that is, the right-hand side of \eqref{idempotents111cde} equals $E_{\mathfrak{t}}.$
\end{proof}
\begin{remark}\label{remark111cde}
Let $\mathscr{D}_{d, n}$ be the degenerate cyclotomic Hecke algebra. It has been proved in [AMR, Proposition 7.2] that $\mathscr{D}_{d, n}$ is isomorphic to the quotient of $\mathscr{W}_{d, n}$ by the two-sided ideal generated by all $E_{i}.$ In the process of taking quotient, the parameter $\omega_{0}$ disappears; however, the parameter $c$ is reserved and can be arbitrary. If we replace the $S_{i}(u, v),$ $R_{i}(u,v;c),$ $\varphi_{1}(u)$ in \eqref{phi-function42cde} with
\begin{align*}
\overline{S}_{i}(u,v)=S_{i}+\frac{1}{v-u},\quad \overline{R}_{i}(u,v;c) :=S_{i}+\frac{1}{u+v+c},\quad \chi_{1}(u) :=\frac{u+X_{1}+c}{u-X_1},
\end{align*}
it is easy to see that the analogue of Lemma \ref{phi-phi-phi111cde} holds.
Let $\overline{\chi}_{1}(u) :=(u-v_{1})\cdots (u-v_{d})\frac{u+X_{1}+c}{u-X_1},$ and for $k=2,\ldots,n$, set
\begin{align*}
\overline{\chi}_k(u_1,\ldots,u_{k-1},u)& :=\overline{R}_{k-1}(u_{k-1},u;c)\overline{\chi}_{k-1}(u_1,\ldots,u_{k-2},u)\overline{S}_{k-1}(u_{k-1}, u).
\end{align*}
We also define a rational function by
\begin{align*}
\Omega(u_1,\ldots,u_n) :=\overline{\chi}_1(u_1)\cdots \overline{\chi}_{n-1}(u_1,\ldots,u_{n-1})\overline{\chi}_n(u_1,\ldots,u_{n}).
\end{align*}
Then it is easy to see that the analogue of Theorem \ref{main-theorem11112cde} is true. Thus, we get a one-parameter family of the fusion procedures for degenerate cyclotomic Hecke algebras, generalizing the results obtained in [ZL].
\end{remark}
\noindent{\bf Acknowledgements.}
The author is deeply indebted to Dr. Shoumin Liu for posing the question about fusion procedures for cyclotomic Nazarov-Wenzl algebras to him.
\end{document} | math |
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\begin{document}
\title{Oriented Birkhoff sections of Anosov flows}
\begin{abstract}
This paper gives $3$ different proofs (independently obtained by the $3$ authors) of the following fact:
given an Anosov flow on an oriented $3$ manifold, the existence of a positive Birkhoff section is equivalent to the fact that the flow is $\mathbb{R}$-covered positively twisted.
\end{abstract}
\tableofcontents
\section*{Introduction}
\addcontentsline{toc}{section}{Introduction}
\subfile{Section/Introduction}
\paragraph{Acknowledgments.} MA
was supported by the JSPS Kakenki Grants 18K03276.
TM is grateful to P.Dehornoy and to the Max Plank Institute in Bonn.
\subfile{Section/Preliminar}
\section{Oriented partial sections}
\label{Section:Marty}
\subfile{SectionMarty/SectionMarty}
\section{Drift along Birkhoff sections}
\label{Section:Asaoka}
\subfile{SectionAsaoka/SectionAsaoka}
\section{Holonomy in the bi-foliation of pseudo-Anosov map.}
\label{Section:Bonatti}
\subfile{SectionBonatti/SectionBonatti}
\begin{thebibliography}{MM}
\bibitem[Barb95a]{Ba1} Barbot, Thierry. \emph{Caract\'erisation des flots d'Anosov en dimension 3 par leurs feuilletages faibles},
Ergodic Theory Dynamical Systems 15 (1995) 247-270.
\bibitem[Barb95b]{Ba2} Barbot, Thierry. \emph{Mise en position optimale de tores par rapport \`a un flot d'Anosov}. (French) [Optimal positioning of tori with respect to an Anosov flow] Comment. Math. Helv. 70 (1995), no. 1, 113-160.
\bibitem[BBY]{BeBoYu} B\'eguin, François; Bonatti, Christian; Yu, Bin \emph{Building Anosov flows on 3-manifolds} Geometry and Topology 21 (2017), no. 3, 1837–1930
\bibitem[BoIa]{BoIa} Bonatti, Christian; Iakovoglou, Ioannis \emph{Anosov flows on 3-manifolds: the surgeries and the foliations }
preprint arXiv:2007.11518
\bibitem[BoGu]{BG10}
C.~Bonatti and N.~Guelman,
Axiom A diffeomorphisms derived from Anosov flows.
J. Mod. Dyn. 4 (2010), no. 1, 1--63.
\bibitem[Fe94]{Fe1} Fenley, Sergio. \emph{ Anosov flows in 3-manifolds}, Ann. of Math. (2) 139 (1) (1994) 79-115.
\bibitem[Fe95]{Fe95a}
S.~Fenley, Quasigeodesic Anosov flows and homotopic properties of flow lines.
J. Differential Geom. 41 (1995), no. 2, 479--514.
\bibitem[Fe98]{Fe2}Fenley, Sergio. \emph{The structure of branching in Anosov flows of 3-manifolds}, Comment. Math. Helv. 73 (2) (1998) 259-297.
\bibitem[FiHa]{FiHa}
T.~Fisher and B.~Hasselblatt,
Hyperbolic flows.
Zurich Lectures in Advanced Mathematics, Eurpean Math. Soc.,
Berlin, 2019. xiv+723 pp.
\bibitem[FraWi]{FrWi} Franks, John; Williams, Bob. \emph{Anomalous Anosov flows.}
Global theory of dynamical systems
, pp. 158-174,
Lecture Notes in Math., 819, Springer, Berlin, 1980.
\bibitem[Fri]{Fri} Fried, David. \emph{Transitive Anosov flows and pseudo-Anosov maps.} Topology 22 (1983), no. 3, 299-C303.
\bibitem[Goo]{Go} Goodman,S. \emph{Dehn surgery on Anosov flows}, from: "Geometric dynamics", (J Palis, Jr, editor), Lecture Notes in Math. 1007, Springer, Berlin (1983) 300–307
\bibitem[GomSt]{GS}
R.~E.~Gompf and A.~I.~Stipsicz,
$4$-manifold and Kirby calculus.
Graduate Studies in Mathematics, 20.
American Mathematical Society, Providence, RI, 1999. xvi+558 pp.
\bibitem[HaTh]{HT} Handel, M. ; Thurston, W. \emph{Anosov flows on new three manifolds.} Invent. Math. 59 (1980), no. 2, 95–103.
\bibitem[Sh]{Sh} Shannon, M. \emph{} Phd Thesis, Universit\'e de Bourgogne 2020.
\bibitem[Ve]{Verjovsky}
Alberto Verjovsky. \newblock Codimension one Anosov flows.
\newblock {\em Bol. Soc. Mat. Mexicana (2) 19 (1974), no. 2, 49–77.}
\end{thebibliography}
\addcontentsline{toc}{section}{References}
\end{document} | math |
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अपना नाम लिखे (जो पान कार्ड पर है )
आप कया सेल करना चाहते है सेलेक्ट करे
अगर आप मेरा भी प्रोडक्ट बाय करना चाहते है तो यहाँ क्लिक करे:
एवन, अगर हमारे पास कोई एक गुड्स है तो हम उसे डाइरेक्ट प्रोडक्ट में अड कर सकते है और उसे सले कर सकते है।
अगर आप एक ब्लॉग्जर है या किसी काम में एक्सपर्ट है तो उसकी ईबुक बना कर भी आप यहाँ सले कर सकते है।
मै आशा करता हू, अब आप एकाउन्ट बनाने के बाद इसे अच्चे से उसे कर सकते हो। क्युकि ये बहुत ही पॉपुलर और सफ़े है ओर ये एक एह्सा ऑनलाइन स्टोर है जिसे हम आसानी से प्रोमोटे कर सकते है अपनी सोशियल साइट्स पर। अंत में, अगर आपका इसके रिलेटेड कोई सवाल है तो आप मुझे कमेंट में पूछ सकते हो। | hindi |
If you’re looking for activities to keep toddlers and young children entertained this summer then we’ve created a guide just for you.
From local events and days out to science experiments and getting outside to explore, there’s plenty of ideas to try.
It’s 2018 Gala day at Beaumont Park , fun fair, traditional games, craft and charity stalls, real donkeys, dog show, LB Brass playing at the Bandstand, burgers/hot dogs, curry stall, cake stall, plant stall, cheerleaders and lots more.
Cartoon Capers: family storywalks for under 8s.
Thursday July 26 - 2pm, Birstall Library and Information Centre.
Monday August 6 - 10.30am, Crow Nest Park cafe, Crow Nest Park.
Tuesday August 7 - 2.30pm, Kirkburton Library, Turnshaw Road, Kirkburton.
Wednesday August 8 - 2pm,, Marsden Mechanics Hall, Peel Street, Marsden.
Thursday August 9 - 10.30am, Greenhead Park Conservatory, Trinity Street, Huddersfield.
Age 1 – 2 years £5.25; 3+ years £12.95. Come rain or shine, the Eureka! Big Summer festival will take over the museum’s grounds, offering a fun mix of science busking, interactive storytelling and craft activities, plus kids (and grown-ups) can try their hand at coding, animation and more with a changing programme of digital activities.
Lawrence Batley Theatre: Lost and Found, Saturday August 4, 55 minutes, £8.
Based on the award-winning book by children’s author Oliver Jeffers. This simple story of true friendship comes to life with puppetry, songs and music.
It tells the story of a little boy who found a penguin at his door. He didn’t know where it had come from or who it belonged to. The penguin looked sad and the boy thought it must be lost.
But no one seems to be missing a penguin. So the boy decides to take the penguin home himself, and they set out in his rowing boat on a journey to the South Pole. But when they get there, the boy discovers that maybe home wasn’t what the penguin was looking for after all.
Paddle in the waters edge at Burnsall, there is nearby parking (fee charging), an ice-cream van, toilets and shops selling outdoor toys.
Alternatively, the river’s edge beyond Standegde Tunnel in Marsden, is a great place for little ones to paddle, plus you can explore the Standedge visitor centre and take a trip on the boat.
Millennium Square will transform into an urban beach once again this summer, promising an oasis of fun from Saturday 12 August until Sunday 3 September. Entry is free but around the beach will be a selection of pay on admission children’s rides and activities including the popular Jungle Funhouse, Bamboo Bungees, Miama Surf and Waikiki cars. It’s open from 10.30am to 6pm.
In addition to the usual pit activities and outdoor play area, there’s a giant sandpit with deckchairs; family crafts and beach huts.
It opens on July 21 for the summer season.
As well as the maze there’s pedal go-karts, hoppers and bales alongside traditional and themed games/puzzles and water games.
You don’t need to splash the cash as there’s loads of activities you can do for free.
Home-grown ideas include making a den; creating a treasure hunt for children; camping out in the garden; make a ‘splash zone’ in the garden with buckets of water; get your hands dirty and make a mud kitchen with old pots and pans.
The Woodland Trust’s Nature Detectives has some great ideas for children aged up to two.
Online activity sheets include encouraging tots to listen to the sounds of summer, a floral sensory activity, going on a mini-beat hunt and colouring pages to print out.
Visit the Woodland Trust activity pages by clicking here for more details.
Cannon Hall Farm is a great day out for all the family, with toddler-friendly outdoor play too.
Little Owl Farm in Saddleworth is perfect for toddlers (and older kids too).
The family-run farms is small, but you can get up close to the animals and feed the animals, which includes rabbits, goats, owls and donkeys.
Visit www.facebook.com/LittleOwlFarm for more details.
Stirley Farm, Huddersfield, the monthly Nature Tots for under 5s will run on Friday August 3.
Stirley Farm Community Farm,at Hall Bower.
The farm also hosts Stirley Summer Fun days for children. For dates visit www.ywt.org.uk to explore events for children at Stirley Farm and other Yorkshire Wildlife Trust sites.
Ideas to do at home (or in the garden) can be an introduction to science.
Fizzy magic: Mixing baking soda and vinegar is a great way to introduce ‘chemical reactions’ and science to young children.
Baking Soda is alkaline and reacts with acids such as vinegar. The fizz produced is brilliant fun for children to get stuck into - and make it more fun by adding food colouring or ice. Parents can try this first to see how it reacts.
You can also make fizzy paint by adding food colouring, cornflour/plain flour, water, baking soda and vinegar.
Create a colour lab: set up a colour laboratory with water, bubbles and different containers. Use food colouring or paint and let children add them to water to exploring colour mixing.
Introduce them to gravity: get balloons and fill them with varying amounts of air, play throw and catch with balls and balloons and talk about how gravity affects how they fall.
Paint rocks: collect rocks, give them a make-over and then leave them in village centres as a surprise for people to discover.
Start saving boxes, loo rolls, plastic bottles and so on: use toilet rolls and paint them different colours and use them as giant threading beads on string; make a ‘hungry caterpillar’ a ‘snake’ or more with them.
Manipulate loo rolls into different shapes (a circle, square and triangle are easy) and use them for mark making. | english |
भारत में बढ़ी है धूम्रपान करने वालों की संख्या
यह बहुत ही चिंता जनक है की पिछले १७ साल में भारत में धूम्रपान करने वालों की संख्या में १०८ मिलियन की वृद्धि हुई है। एक भारतीय अनुसन्धान में यह पाया गया की सिगरेट ने आम तम्बाकू के पदार्थों को पीछे छोड़ते हुए पहला स्थान बना लिया है।
इस अनुसन्धान में यह पाया गया की तम्बाकू का सबसे अधिक उपयोग करने वालों में १५-६९ वर्ष के लोग हैं जिनमें से १.७ मिलियन पुरुष हैं। अगर 20१0 के आकड़ों पर नज़र डाली जाये तो यह पाया गया की करीब १0% मौतें सिगरेट का सेवन करने से हुई।
इस संख्या में बढ़ोतरी का कारण बढ़ती जनसंख्या और युवा वर्ग का नशे के प्रति आकर्षण भी है। सिगरेट ने जहाँ भारत में सस्ते नशों में सबसे अहम स्थान ले लिया है वहीं दूसरी तरफ इसकी लत युवा वर्ग को बर्बाद भी कर रही है।
सिगरेट से कई जानलेवा बीमारियां हो जाती है सिगरेट न ही सिर्फ पीने वाले को अथवा साथ में खड़े लोगों को भी प्रभावित करती है। अब शायद वक़्त आ गया है जब हम सभी को इस विषय पर गंभीरता से सोचना चाहिए क्योँकि यह एक धीमा ज़हर है जो हमारी युवा पीढ़ी को अंदर ही अंदर खाए जा रहा है।
तो आइये हम सब मिलकर यह प्रण लें की ना तो हम सिगरेट पिएंगे और न ही किसी को इसका सेवन करने देंगे। | hindi |
package com.indeed.proctor.consumer;
import com.google.common.base.Strings;
import com.google.common.collect.Maps;
import com.indeed.proctor.common.Identifiers;
import com.indeed.proctor.common.Proctor;
import com.indeed.proctor.common.ProctorResult;
import com.indeed.proctor.common.model.TestType;
import org.apache.commons.lang3.StringUtils;
import org.apache.logging.log4j.LogManager;
import org.apache.logging.log4j.Logger;
import javax.annotation.Nonnull;
import javax.annotation.Nullable;
import javax.servlet.http.Cookie;
import javax.servlet.http.HttpServletRequest;
import javax.servlet.http.HttpServletResponse;
import java.util.Collections;
import java.util.Iterator;
import java.util.Map;
import java.util.Map.Entry;
import java.util.Optional;
public class ProctorConsumerUtils {
private static final Logger LOGGER = LogManager.getLogger(ProctorConsumerUtils.class);
/**
* plain old "forceGroups" is already in use by JASX for SERP groups
* @param allowForcedGroups if true, parses force group parameters from request / cookie, sets new cookie
*/
public static final String FORCE_GROUPS_PARAMETER = "prforceGroups";
public static final String FORCE_GROUPS_COOKIE_NAME = "prforceGroups";
public static final String FORCE_GROUPS_HEADER = "X-PRFORCEGROUPS";
public static ProctorResult determineBuckets(final HttpServletRequest request, final HttpServletResponse response, final Proctor proctor,
final String identifier, final TestType testType, final Map<String, Object> context, final boolean allowForcedGroups) {
final Identifiers identifiers = new Identifiers(testType, identifier);
return determineBuckets(request, response, proctor, identifiers, context, allowForcedGroups);
}
/**
* calculates ProctorResult (determined groups) and also handles forcedGroups, setting cookie if necessary
* @param allowForcedGroups if true, parses force group parameters from request / cookie, sets new cookie
*/
public static ProctorResult determineBuckets(final HttpServletRequest request, final HttpServletResponse response, final Proctor proctor,
final Identifiers identifiers, final Map<String, Object> context, final boolean allowForcedGroups) {
final Map<String, Integer> forcedGroups;
if (allowForcedGroups) {
forcedGroups = parseForcedGroups(request);
createForcedGroupsCookieUnlessEmpty(request.getContextPath(), forcedGroups)
.ifPresent(response::addCookie);
} else {
forcedGroups = Collections.emptyMap();
}
return proctor.determineTestGroups(identifiers, context, forcedGroups);
}
/**
* Consumer is required to do any privilege checks before getting here
*
* @param request a {@link HttpServletRequest} which may contain forced groups parameters from URL, Header or Cookie.
* @return a map of test names to bucket values specified by the request. Returns an empty {@link Map} if nothing was specified
*/
@Nonnull
public static Map<String, Integer> parseForcedGroups(@Nonnull final HttpServletRequest request) {
final String forceGroupsList = getForceGroupsStringFromRequest(request);
return parseForceGroupsList(forceGroupsList);
}
/**
* @return proctor force groups if set in request, returns first found of: parameter, header, cookie
*/
@Nonnull
public static String getForceGroupsStringFromRequest(@Nonnull final HttpServletRequest request) {
final String param = request.getParameter(FORCE_GROUPS_PARAMETER);
if (param != null) {
return param;
}
final String header = request.getHeader(FORCE_GROUPS_HEADER);
if (header != null) {
return header;
}
final Cookie[] cookies = request.getCookies();
if (cookies == null) {
return "";
}
for (int i = 0; i < cookies.length; i++) {
if (FORCE_GROUPS_COOKIE_NAME.equals(cookies[i].getName())) {
final String cookieValue = cookies[i].getValue();
return Strings.nullToEmpty(cookieValue);
}
}
return "";
}
@Nonnull
public static Map<String, Integer> parseForceGroupsList(@Nullable final String payload) {
if (payload == null) {
return Collections.emptyMap();
}
// using single char in split regex avoids Pattern creation since java8
final String[] pieces = payload.split(",");
final Map<String, Integer> forcedGroups = Maps.newHashMapWithExpectedSize(pieces.length);
// detect integer number from end of string
for (final String rawPiece : pieces) {
final String piece = rawPiece.trim();
if (piece.isEmpty()) {
continue;
}
int bucketValueStart = piece.length() - 1;
for (; bucketValueStart >= 0; bucketValueStart--) {
if (!Character.isDigit(piece.charAt(bucketValueStart))) {
break;
}
}
// if no name or no value was found, it's not a valid proctor test bucket name
if ((bucketValueStart == piece.length() - 1) || (bucketValueStart < 1)) {
continue;
}
// minus sign can only be at the beginning of a run
if (piece.charAt(bucketValueStart) != '-') {
bucketValueStart++;
}
// bucketValueStart should now be the index of the minus sign or the first digit in a run of digits going to the end of the word
final String testName = piece.substring(0, bucketValueStart).trim();
final String bucketValueStr = piece.substring(bucketValueStart);
try {
final Integer bucketValue = Integer.valueOf(bucketValueStr);
forcedGroups.put(testName, bucketValue);
} catch (final NumberFormatException e) {
LOGGER.error("Unable to parse bucket value " + bucketValueStr + " as integer", e);
}
}
return forcedGroups;
}
/**
* Unless forceGroups is empty, set a cookie that will be parsed by {@link #parseForcedGroups(HttpServletRequest)}.
* Cookie expires at end of browser session
*
* @param forceGroups parsed force groups
* @deprecated use {@link ProctorConsumerUtils#createForcedGroupsCookieUnlessEmpty}, modify as needed, then add to the response
*/
@Deprecated
public static void setForcedGroupsCookie(final HttpServletRequest request, final HttpServletResponse response, final Map<String, Integer> forceGroups) {
createForcedGroupsCookieUnlessEmpty(request.getContextPath(), forceGroups)
.ifPresent(response::addCookie);
}
/**
* Create a cookie that will be parsed by {@link #parseForcedGroups(HttpServletRequest)}. Cookie expires at end of browser session
* @param contextPath request.contextPath
* @param forceGroups parsed force groups
*/
public static Optional<Cookie> createForcedGroupsCookieUnlessEmpty(final String contextPath, final Map<String, Integer> forceGroups) {
// don't overwrite other cookie with empty; this would be relevant in a race condition where
// there is a forceGroups request simultaneous with a non-forceGroups request
if (forceGroups.isEmpty()) {
return Optional.empty();
}
return Optional.of(createForcedGroupsCookie(contextPath, forceGroups));
}
/**
* @Deprecated use {@link ProctorConsumerUtils#createForcedGroupsCookieUnlessEmpty}
*/
@Deprecated // not safe, see comment in createForcedGroupsCookieUnlessEmpty
public static Cookie createForcedGroupsCookie(final String contextPath, final Map<String, Integer> forceGroups) {
return doCreateForcedGroupsCookie(contextPath, forceGroups);
}
// TODO: can be merged into createForcedGroupsCookieUnlessEmpty once createForcedGroupsCookie() was deleted
private static Cookie doCreateForcedGroupsCookie(final String contextPath, final Map<String, Integer> forceGroups) {
// be sure to quote cookies because they have characters that are not allowed raw
final StringBuilder sb = new StringBuilder(10 * forceGroups.size());
sb.append('"');
for (final Iterator<Entry<String, Integer>> iterator = forceGroups.entrySet().iterator(); iterator.hasNext(); ) {
final Entry<String, Integer> next = iterator.next();
sb.append(next.getKey()).append(next.getValue());
if (iterator.hasNext()) {
sb.append(',');
}
}
sb.append('"');
final String cookiePath;
if (StringUtils.isBlank(contextPath)) {
cookiePath = "/";
} else {
cookiePath = contextPath;
}
final Cookie cookie = new Cookie(FORCE_GROUPS_COOKIE_NAME, sb.toString());
cookie.setPath(cookiePath);
return cookie;
}
}
| code |
Just beyond the sprawling outskirts of Phoenix, the heat is 111 degrees and the rush-hour traffic shimmers like a mirage. A billboard filled with verdant pine trees beckons: ”Come to the real Arizona.” To the northeast, the landscape climbs from 1,500 feet in the Valley of the Sun to 7,000 feet along the forested Mogollon Rim, a magnet for vacation homes and retirement communities drawn by the promise of trout in cool streams and elk grazing beneath the ponderosa pines.
This day in early summer, the magnet is on fire. A stream of evacuees in pickups and RVs is heading the other way. Soon, flurries of ashes hiss against the windshield, air tankers lumber overhead toward the mountains and a thunderhead of smoke boils up from flames raging through the forest canopy 10 miles away. Arizona is suffering the largest fire in its history.
The blaze, known as the Rodeo-Chediski fire, tops the Mogollon Rim and is racing eastward. Most of the town of Overgaard and parts of Heber are quickly overrun. In the development of Pinecrest Lakes, 166 of 200 double-wide mobile homes are literally vaporized. In Bison Ranch, dozens of faux-log cabins are left twisted and smoking. Undeterred by an army of 4,500 firefighters and a squadron of air tankers, the fire surges outward along a 400-mile perimeter. To the north, it has jumped the highway in several places, making a fast run toward the high desert.
Up ahead, at 8,500 feet in the hamlet of Alpine, potentially in the path of the advancing flames, Arizona State University professor Stephen J. Pyne and his wife, Sonja, are packing up their cabin. The inferno is coming, just as he had long predicted. One of the world’s leading authorities on fire, Pyne has for years advanced the paradoxical prophecy that the greatest threat to America’s forests is not too much fire, but too little. In Pyne’s paradox, without regular, low-intensity wildfires to clear out undergrowth, the landscape will one day explode in devastating firestorms like the one nearby, fires so intense that they obliterate the forests and the creatures within them.
The Pynes’ cabin is a house with rounded wooden siding meant to look like logs, set in a sparse subdivision of similar houses carved into dense forest. Pyne greets me at the door. He is tall and lanky, with an open, bright face and a nearly constant smile. He’s younger looking than his 53 years and more relaxed than might be expected in light of his prodigious scholarly output. So far he has written 14 books, 11 of which compose a panoramic history of the world as seen through the lens of fire–an extraordinary ouevre for which he was awarded a MacArthur ”genius” fellowship in the late 1980s.
Inside, Sonja is carefully packing important items, leaving what they don’t mind losing–which she cheerfully admits includes the house itself if forced to choose between it and the ancient ponderosa pines outside. Pyne is packing too, in between telephone calls from the media. The national press this year is reporting a growing consensus that the ”fire deluge,” as Pyne and others have argued, is the unintended consequence of a century of government policy of putting out fires in the woods. What’s more, Pyne says, our attitudes toward fire ignore the historical reality that man’s role in nature has been to start fires, usually unintentionally, but almost always to healthy effect.
A landscape some distance from the cabin illustrates his point. Under a low, leaden ceiling, a forest of black poles stands in a rolling black landscape punctuated by still-smoking stumps. White flakes of ash as big as autumn snowflakes blow lazily through the air and across the charred crust the soil has become, building low dunes against stumps and filling holes where trees had been. There is a pleasant, sweet smell of burning pine and juniper, exactly like a campfire. Every living thing has been killed, transformed into charcoal. Hiroshima in the pines–luckily with no human victims, though nationally 21 firefighters have already died this season.
Ultimately, the White Mountain Apache reservation, where the Rodeo-Chediski fire started, is the hardest hit. Of the acreage burned, two-thirds is reservation land. Timber is the among the top three sources of income on the reservation, which has 60% unemployment. The fire claimed 12 years of their harvest, worth perhaps $241 million and 400 jobs at the two mills. According to the Forest Service, the forest will take at least 100 years to return to what it was.
The news reports of this fire bring a parade of government officials repeating Pyne’s scenario like a mantra: There is too much fuel in the woods. But beyond that, the discussion quickly dissolves into a cacophony of old arguments, with the same antagonists drawing the same battle lines that have frustrated Pyne for years–environmentalists versus loggers arguing about U.S. forest policy, neither of them seeing the bigger picture.
In this fire season, Arizona Republican Sen. Jon Kyl blames ”radical environmentalists. They would rather the forests burn than to see sensible forest management,” which to him means thinning out the forests by logging and burning the debris left behind.
Pyne has heard it all before. ”I see the same sort of distressing political polarization” as in past seasons, he says. ”Nobody is willing to deal with the fire problem as a fire problem. They want to use it for something else.” That ”something else” is a series of agendas that beg a central question: How can we keep our forests and wildlife healthy?
For the logging lobby, fire is a disaster, eating up a valuable resource, ”standing board feet,” as trees are called. The industry’s approach is to ”go in and cut it out–which doesn’t solve the problem at all,” Pyne says, because it takes the largest trees and leaves behind smaller trees and other highly flammable material.
For the angry politicians, fire is the enemy, threatening the homeland–or at least its second homes. It’s the same view that guided the era of vigorous fire suppression and it leads back to the same precipice, Pyne says.
The flaw in that reasoning is a point that underlies much of Pyne’s research. Pyne insists that fire is not, in any useful sense, simply natural. The most common natural source of fire is lightning, but many landscapes in North America see little of it. Instead, for more than 10,000 years, when human hunters first came into North America, many fires, perhaps most of them, were set by people. Even in places with abundant lightning, like Arizona, the forests are also unmistakably shaped by ”anthropogenic” fire, which in Pyne’s lexicon, means fire started by people.
While the squabbling continues, the danger rises. By government estimates, the U.S. needs to reduce fuel on 70 million to 80 million acres; right now it is managing to deal with 2 million acres per year. The record-breaking fire season of 2002 is just one more hot summer in a ratcheting national bonfire. By mid-August, Oregon and Colorado had joined Arizona in suffering the worst fires in their history.
The week after his high school graduation in 1967, Pyne went to Grand Canyon National Park to take a job as a summer laborer. Instead he was offered a place on a five-man North Rim fire crew. Eighteen years old, living away from home for the first time, he fell in love in the classic coming-of-age sense: ”We were living on the rim of the Grand Canyon and getting paid for it: clearing the fire roads, cutting the limbs off trees, digging out of the mud,” and hiking all over the backcountry to find the source of smoke reported by lookouts. ”You become very sensitized to the landscape around you, to the things that matter: how the seasons come and where the winds are; to soil; to duff, because that’s where fire persists. You’re going to have to mop it up, you’re going to spend hours, days, at the wrong end of a shovel spading over smoking pine needles.
With his PhD in hand but without an academic post in the lean job market of the late 1970s, still supporting himself as a seasonal firefighter, Pyne decided as a last gambit to combine his love of fire with history. The resulting book, ”Fire in America,” published in 1982, became an instant classic in the new field of environmental history and won him the MacArthur in 1988. The award funded a five-year peregrination to all five continents, where he did the research that underpins his five-volume ”Cycle of Fire,” the fire histories of America, Australia, the World, Antarctica (a place distinguished by the absence of fire), and what is acknowledged to be his masterpiece, ”Vestal Fire,” a fire history of Europe and European colonialism, published in 1997. His most recent book, “Year of the Fires,” while not part of the MacArthur series, touches on familiar territory in addressing government policy on firefighting before and after the year 1910, when a wave of wildfires threatened to consume the Western United States.
By ”treating fire as a larger cultural force,” putting it at the center of the narrative, Pyne has uncovered an analytical tool of surprising scope and power. In his work, fire is paradoxically both natural and cultural, acting like a Promethean ur-species, sometimes competing with, sometimes domesticated by, people. Nature gave us fire in the form of lightning, and we took it and altered it, making it the fire in the hearth of civilization and the engine that, by clearing land, helped create agriculture and allow its expansion into inhospitable zones. It is embedded in every society and landscape–as the biology of every continent except Antarctica has been, to some extent, fashioned by human fire use. In Australia, for example, Pyne argues convincingly that fire-stick-wielding aborigines literally created the landscape by favoring the fire-loving eucalyptus over other species.
Arizona is no exception. It has the highest incidence of fire caused by lightning in America and a long history of human occupation. Nowhere is this combination more keenly felt than in the ponderosa pinelands, where periodic fires fed by thick grasses left larger trees standing but cleaned out smaller trees and brush. Burned often enough, ponderosa forests can be largely fireproof. But when Euro-American settlement began 150 years ago, overgrazing soon stripped the grasses and bouts of logging cut out the old growth trees. Firefighting, believed to protect the resource, instead eliminated the janitor. Smaller trees grew into dense thickets, transpiring scarce water through their leaves so that springs and creeks dried up.
At the peak of the Rodeo-Chediski fire, I leave the Pynes to their packing and drive to Show Low, a town of 7,700 named for the Wild West card game by which a ranchhand won the land, now evacuated and serving as the base for the firefighting effort. I’m looking for Jim Paxon, a district ranger in the Gila National Forest in New Mexico and Forest Service spokesman on the fire.
It is eerily calm. Police cruisers and military units in Humvees patrol empty streets under a milky orange sun filtered through a light fall of ash. On the playing fields at the high school, swarms of blue crew tents are massed like jellyfish in a green sea. Parking lots stretching two blocks are jammed with TV satellite trucks. Five times daily, Paxon appears before TV cameras to announce the numbers of acres burned, of homes ”saved” and of those afforded no salvation. A Texan with a thick mustache and a laconic drawl, Paxon is the very definition of a grizzled veteran: since 1969, he has worked on some of the biggest wildfires in American history, and has seen the bodies of scores of firefighters taken out of the mountains.
”I’m a prescribed burner,” he tells me. ”Some of my own peers call me a pyromaniac. I burned 86,000 acres in the last four years in my little bitty district”–the Black Range, 560,000 acres of ponderosa and pinyon-juniper pine woodlands within the national forest. He can quantify the results: ”Where I’ve burned, I’ve got creeks that have been dry for 15, 20 years that are flowing water year round now.” Some areas of pinyon-juniper forest that have been thinned are yielding about 700 pounds per acre of grass, he says. “I can show you a biological desert, where there’s 400 juniper trees an acre, and you’re getting 200 pounds of forage [grass] an acre. I can show you where we burned and there’s 1,200 pounds.
Paxon is the first to admit that his is a rare situation. His district is mostly wilderness in a near-empty quarter of a sparsely populated state, 50 miles from the closest thing to a city–Truth or Consequences, N.M., named after the game show. Successes like this do not constitute a nationwide model.
Paxon was also lead information officer on the May 2000 ”Los Alamos” fire in New Mexico, a prescribed fire at Bandelier National Monument that turned out of its intended track and blew toward the town of Los Alamos and the nearby national laboratory, where nuclear materials are kept. Conditions in Upper Frijoles Canyon, the target of the fuel-reduction burn, were probably too moist for burning, according to Pyne’s post mortem, while conditions in the ponderosas around Los Alamos were superb for a firestorm.
In spite of intensive firefighting efforts, the blaze was unstoppable, burning 42,849 acres, 235 homes and part of the lab itself. In its wake, prescribed burning fell under a cloud of smoke. Its record is, on the whole, good: In the last few decades, less than 3% of controlled burns have become uncontrollable–exactly the same percentage as the number of unintentional wildfires that defeat suppression efforts. But the exceptions make the perception. Some of the worst fires of the past 20 years have been escapes, Pyne says. Seven have killed firefighters; one burned a small town in Michigan to the ground.
Burning remains an art, not a science–one with innumerable variables that must line up for success, and a steep price for failure. Of those prescribed burns that are attempted, many end in failure–fizzling out, going the wrong way when the wind shifts or just not burning hot enough, singeing trees enough to kill them but not fell them, leaving standing deadwood that is more flammable than live trees. The burning also creates smoke, which few people associate with a wholesome weekend in the mountains.
”Urban people,” Pyne says, ”the only fire they see is a disaster–the car is burning, there was an accident, the house is burning, they see it on TV, it’s a crisis.” In their cabins in the pines, they can’t see the ”large fuels,” in firefighter parlance, for the trees. It’s tough to get them to cut down the trees that brush against their wood-shake roofs; it’s even harder to persuade them to welcome prescribed fire into the nearby woods.
Burning programs nationwide are routinely stifled by complaints about drifting smoke. The White Mountain Apache tribe, historically among the most aggressive controlled burners in the country, has seen its program severely curtailed when smoke sinks down into smog-choked Phoenix, pushing it into violation of clean-air statutes. In 1996, a fire closed Phoenix’s Sky Harbor airport for three days to all but instrument landings.
Against that backdrop, advocates of prescribed burning have had to give ground to those who see a greater role for chainsaws in bringing the forests back into balance.
The nation unquestionably needs to find a way to put fire back into the landscape, on a huge scale. But how? It’s a riddle Pyne has struggled with for most of his career.
There is consensus that forest ”treatment” must involve a combination of cutting and burning. Smaller trees would be cut down and, along with other debris from the forest floor, either ground up or burned after the snow falls. In a given forest, the process might have to be repeated, and it might take several years.
Instead, the recipe must be tailored to the habitat. What works in a ponderosa pine forest won’t apply to a coastal Douglas fir forest, a lodgepole pine forest, a pinyon-juniper forest or chaparral. It will differ on north-facing slopes and south-facing slopes. And, to top Pyne’s paradox with another, he points out that, in some high-altitude forests, periodic ”stand-replacing” fires that strip away every living thing are ecologically normal and necessary. Thinning would only deprive them of the fuel they require.
The greatest barrier to free-burning fire is the growing phalanx of homes being built in fire-prone environments. Houses in the woods form the most bitterly contended front in the fire wars: the ”wildland/urban interface.” Now resources have to be committed to saving structures, at great cost and risk to firefighters trained and equipped to fight for territory, not lives or property. ”Suddenly the firefighter is even more compromised. Do you save the houses or the trees?” Pyne asks.
In California, an explosive hillside intermix is a traditional form of urbanism, and it has shown a discouraging pattern. When a neighborhood burns in Malibu or Berkeley, the owners rebuild, generally with insurance and disaster relief money, building bigger and more expensive houses. Property values rise, the area becomes more desirable, and more houses are built even higher on the slopes. Eventually, another fire torches the neighborhood, and the insurance and relief payments, being larger, cover the cost of rebuilding even bigger structures, and so on. So far, no limit to this perverse cycle has been reached.
The federal land management agencies spent their entire fire-suppression budgets this year by mid-July. The Forest Service drained half of its reserve on just five fires. To continue the effort through the fire season, the agencies will have to ask for at least $1 billion in emergency appropriations–money the White House is loath to grant during a budget crunch and an election year, but which influential members of Congress are loath to deny constituents.
After the disastrous 2000 season, the Clinton administration secured passage of the National Fire Plan, with $2 billion in funding for mitigation efforts, including fuel reduction. To date, fewer than half of the national forest fire plans required by the legislation have been prepared–hence, little progress has been made. The money is regarded as just a down payment on a fuel-reduction bill that is expected to reach $12 billion over a decade.
By the end of August, the national burn tally exceeded 6.2 million acres–double the mark last summer, surpassing the year-to-date burn tally for the devastating 2000 fire season, the worst ever recorded. Various governments spent $1.6 billion fighting those fires, calling out 30,000 firefighters.
Pointing to these numbers, the Bush administration is pushing Congress to exempt 10 million acres from environmental laws to clear the way for “treatment”–mostly by for-profit logging, not burning. Environmentalists counter that the White House is using the fires as a smoke screen for letting loggers back into the wilderness, setting back decades of environmental gains.
Pyne is not optimistic. ”You’re going to see some change around the houses. Congress is tired of having houses burning on TV,” he says. Some national parks and monuments will have a form of preventive treatment, including prescribed fires. But the vast expanses of generic public lands, the 50% of the West owned by federal agencies like the Forest Service and the Bureau of Land Management, remain in danger of catastrophic fires and, in their wake, cascading biological shifts that no one can predict.
Arizona’s Rodeo-Chediski fire eventually destroyed 467 structures and blackened 468,638 acres. Its eastward advance, toward the Pynes’ cabin, was stopped by an Apache fire crew working day and night in a last-ditch effort along a fire line that had been cleared two years before in a controlled burn. The earlier fire had successfully removed enough undergrowth and small trees that the new fire was deprived of fuel. When it hit the line, the inferno ”just lay down and died,” says Rick Lupe, a 24-year firefighting veteran and head of the Apache crew. The Pynes’ cabin was spared.
The same thing happened to Colorado’s Hayman fire. After joining with the Arizona fire, it, too, ran into a previous prescribed burn and died.
As I left the cabin, Sonja gave me two gifts: a bag of cookies and a sack of ponderosa pine cones she had soaked in paraffin to make foolproof firestarters for the fireplace.
THE PROPOSED sale of federal land in Washington County, Utah, is spectacular, in the scale of both its greed and its shamelessness. Legislation has been drafted to allow county officials to steal 25,000 acres of public land near Zion National Park to benefit themselves and well-connected private developers.
I stay “steal” because the draft bill — the Washington County Growth and Conservation Act, to be sponsored by Utah Republican Sen. Robert F. Bennett and Democratic Rep. Jim Matheson — would require the federal government to sell land to private developers, then use the proceeds to buy other, less valuable land from the same developers at inflated prices. All without paying the taxpayers for their property.
As land grabs go, it is impressive, but it isn’t original. The scheme’s promoters are following to the letter the tired plot line of the oldest script in the West. It’s Chinatown, Jake, all over again.
The movie “Chinatown,” you may recall, was a fictionalized version of how, 100 years ago, Los Angeles stole water from the Owens Valley to make land development more profitable. Now, as then, politicians are trying to scare the public into believing that the future of their community depends on taxpayers footing the tab. Now, as then, the real goal is to make more profit for developers by subdividing land that the community, if it had been consulted, probably wouldn’t want developed. Now, as then, the boogeyman is water scarcity. That’s why the other goal is to open up protected public lands for a 130-mile-long pipeline to bring water from Lake Powell on the Colorado River.
The $1-billion price tag for the pipeline would be conveniently covered by the taxpayers of Utah, not just Washington County, through a sales-tax increase that would last for 15 years.
To put this brazen plan into perspective, consider Washington County’s recent history. County officials turned a blind eye as illegal roads were bulldozed across protected federal lands in an effort to claim them as county roads. One city, La Verkin, has declared itself a “United Nations-free zone.” The town of Hildale is a polygamist stronghold whose leaders are reportedly under investigation for alleged child sexual abuse.
But Washington County’s attempted water heist isn’t just a local crackpot scheme: 25 million people in seven states (California, Arizona, Nevada, Utah, New Mexico, Colorado and Wyoming) and Mexico depend on the Colorado River. A new tap puts everyone’s supplies in jeopardy. The county seat, St. George, already has the highest water consumption of any desert city in America: 335 gallons a person a day, twice the national average. (By comparison, Phoenix uses just 170 gallons a person a day by using basic conservation measures). The plans to expand St. George sevenfold indicate a total disconnect from reality because Washington is the driest county in the second-driest state in the country.
Despite the scare campaign, the county doesn’t need new water. By simply wasting less, it would have enough for growth for 50 years, a study commissioned by the Grand Canyon Trust has shown. Even Las Vegas added 250,000 new residents between 2002 and 2005 while cutting its water use by 20 billion gallons — in part by paying homeowners $1 per square foot of lawn changed to Xeriscape. Even without the pipeline, Washington County has already grown 73% from 1990 to 2000. It is sure to keep expanding because it has few if any land-use controls. Bringing it more cheap water would be like handing dope to an addict.
The irony is that there probably won’t be enough water to fill the pipe even if it is built, because the Colorado River is already too depleted to satisfy the demands of those who hold its water rights. In the future, as the current rights holders put more straws in the Colorado, Lake Powell will be less than one-quarter full most of the time. With global warming expected to cut the Colorado River’s flow by 14% to 18% in coming decades, the West has an even bigger water problem.
If Washington County succeeds in hoodwinking the taxpayers, it will be a sad commentary on how little we have learned from our history of private pillage of public resources. But it is the county that will pay the highest price, in ill-conceived sprawl disfiguring one of the most magnificent regions in America and in a dangerous dependence on water supplies that are drying up.
Utah’s leaders ought to see that their state grows through responsible planning and stewardship, not through delusion, distortion and duplicity.
Congressional leaders of both parties and the public should stop this bill before it is introduced. And federal and state authorities should do a better job of safeguarding the public lands of Utah. | english |
\begin{document}
\title{Effects of decoherence on the radiative and squeezing properties in a coherently driven trapped
two-level atom}
\author{Sintayehu Tesfa}
\affiliation{Physics Department, Addis Ababa University, P. O. Box 1176, Addis Ababa, Ethiopia}
\email{sint_tesfa@yahoo.com}
\date{\today}
\begin{abstract}Analysis of the effects of decoherence on the radiative and
squeezing properties of a coherently driven two-level atom trapped
in a resonant cavity applying the corresponding master equation is
presented. The atomic dynamics as well as the squeezing and
statistical properties of the emitted radiation are investigated.
It is found that the atom stays in the lower energy level more
often at steady state irrespective of the strength of the coherent
radiation and thermal fluctuations entering the cavity. Moreover, a
strong external coherent radiation results the splitting of the
line of the emission spectrum, whereas the decoherence broadens
the width and significantly decreases the height. It is also found
that the emitted radiation exhibits photon anti-bunching,
super-Poissonian photon statistics and squeezing, despite the
presence of the decoherence which is expected to destroy the
quantum features.
\end{abstract}
\keywords{two-level atom, atomic dynamics, quadrature squeezing,
emission spectrum, decoherence}
\maketitle
\section{Introduction}
Interaction of a single two-level atom with a radiation
has received a great deal of interest in recent years
\cite{jmo,jmo451859,prl561917,jmo34821,el10237,jobqso4142,jmo48347,jmo46379,pr1881969,pra414083,pra426873,pra373867,prl582539,oc118143,jmo431555,pra534439,pra533633,pra532846,pra74063817}.
In the absence of the external light, spontaneous emission of an
excited two-level atom results due to the fluctuations in a
continuum of vacuum modes that play a role of a reservoir.
However, when the vacuum modes are replaced with, let us say, a
squeezed vacuum reservoir, the radiative properties of the atom
are significantly modified
\cite{jmo,jmo451859,prl561917,jmo34821,el10237,jobqso4142,jmo48347,jmo46379}.
In this respect, the resonance fluorescence and absorption spectra
of a driven two-level atom coupled to a squeezed vacuum reservoir
have been analyzed by various authors \cite{jmo,jmo451859}. It is,
in general, found that the rate of emission is greater than the
rate of absorption, in which the squeezed input inhibits
absorption somehow and broadens the spectrum with decreasing
height. Moreover, whenever the atom is driven on resonance by a strong
external monochromatic laser beam, the structure of the atomic
energy level changes dramatically, that is, the atomic dynamics and
properties of the emitted radiation would be appreciably altered.
It is known for long that the resonance fluorescence spectrum
splits into Mollow triplets in the strong driving regime
\cite{pr1881969}. It is also a well-established fact that the radiative
properties of the atom and squeezing properties of the emitted
radiation considerably depend on the amplitude of the driving
radiation.
It can be learned from earlier works that the
successively emitted photons from the two-level atom in the cavity
are correlated due to the atomic coherence induced by the driving
mechanism, whereby the emitted radiation is found to exhibit
nonclassical features. For instance, D'Souza {\it{et al.}}
\cite{pra414083} analyzed the quantum nature of the light emitted
by the two-level atom coupled to a squeezed vacuum reservoir in
the strong driving limit with the aid of atomic-dressed state
earlier. Based on the phase sensitivity of the Mandel's response
function, they claimed that the emitted light exhibits squeezing.
Most recently, the squeezing properties of the radiation emitted by
a coherently driven two-level atom coupled to a broadband squeezed
vacuum reservoir is studied using the variance of the
atomic-dipole operator in the normal ordering. Successively
emitted radiation turns out to be in squeezed state even in the
absence of the squeezed input for certain values of the
amplitude of the external radiation. In addition, although it was predicted earlier that the emitted
radiation exhibits sub and super-Poissonian photon statistics,
from the curve of the Mandel's response function \cite{pra414083}, recent analysis
based on the two-time second-order correlation function shows that it exhibits super-Poissonian photon statistics for
larger delayed time \cite{jmo}. It is also found that the two-time
second-order correlation function oscillates with the delayed time,
where the frequency of its oscillation increases with the
amplitude of the driving radiation, whereas its height decays fast
with the squeeze parameter.
In the actual experimental setting, the two-level atom in the cavity is
unavoidably coupled to the fluctuations in the surrounding
environment via the walls of the cavity. In general, the phenomenon in which
the quantum system losses its nonclassical features due to its
interaction with the environment is defined as decoherence. It is
not difficult to realize, therefore, that decoherence is basically
related to unbiased noise fluctuations in the modes of the
environment that able to interact with the system. Though various ways of including the effects of
decoherence are possible \cite{pra445401}, its contribution can be readily modeled as
thermal fluctuations of the walls of the cavity that can be taken
usually as thermal reservoir. It is a well-known fact that a squeezed
vacuum reservoir introduces a biased noise fluctuations to the
system, as a result it induces additional coherence, whereas
the thermal reservoir, on the other hand, adds decoherence into the system. In view of the contribution of the squeezed vacuum reservoir towards
the nonclassical features of the emitted radiation that have been
reported, it appears natural to ask how the radiative properties
of the atom as well as the squeezing and statistical properties of
the emitted radiation could possibly be modified by decoherence due to the
presumed thermal heating entering the cavity via the vibration of
the walls of the container? On the basis of the properties of unbiased noise fluctuations associated with the thermal heating, it seams reasonable to expect that the quantum features of the radiation would be degraded by the decoherence. The main task of this work is,
thereupon, devoted to investigate this basic issue. Earlier, effects of the thermal light as incoherent relaxation on the
collapse and revival as well as the photon anti-bunching have been
considered by Puri and Agarwal \cite{pra353433}. They
found that the oscillations of the collapse and revival become
more irregular with the intensity of the thermal radiation and the
thermal light characteristically destroys the photon anti-bunching
phenomenon.
In this communication, the effects of decoherence on a radiative,
squeezing and statistical properties of a coherently driven
two-level atom would be analyzed. It is a common knowledge that the effects of squeezed input are related to the amplitude and phase fluctuations of the reservoir modes. Nonetheless, which of these two would be predominant one of the issues in this work. To achieve this goal, the squeezed input is replaced with an unbiased thermal fluctuations whose mean phase fluctuations are readily averaged out to be zero. In accordance to this, throughout, the results previously obtained for squeezed input elsewhere \cite{jmo} are compared with the effects of the thermal fluctuations so that which of the fluctuations in the reservoir modes would actually be essential in bringing about a significant modification in a radiative and squeezing properties ie evident. Though methods from the
stochastic simulation of the Bloch equations in secular
approximation \cite{pra426873,pra373867} to diagonalizing the
coefficient matrix \cite{jobqso4142,jmo48347,pra414083} have been
used in previous contributions, the differential equations
associated with the expectation values of the atomic and energy
operators following from the master equation would be
simultaneously solved in view of the procedure recently applied.
It is believed that this approach helps in overcoming the
inevitable limitations corresponding to the approximations and
computer simulation frequently employed. Usually the effects of
the external coherent radiation either in a weak or strong driving
limit have been studied, but in here an arbitrary amplitude of the
driving radiation is taken. In particular, the population
inversion, probability for the atom to be in the upper energy
level, emission spectrum, two-time second-order correlation
function and quadrature variance for the cavity radiation in terms
of the atomic polarization would be calculated.
\section{Atomic Dynamics}
It is a common knowledge that driving a two-level atom on
resonance by a coherent light amounts to pumping the two-level
atom continuously by an external laser beam whose frequency
matches with the atomic transition frequency. Treating the driving
radiation classically, the Hamiltonian describing the interaction
of two-level atom with a radiation in the rotating-wave and
electric-dipole approximations in the interaction picture can be
expressed as
\begin{align}\label{tla01}\hat{H}=i{\Omega\over2}\big(\hat{\sigma}_{+}-\hat{\sigma}_{-}\big),\end{align}
where $\Omega$ is the positive-real constant proportional to the
amplitude of the external coherent radiation,
$\hat{\sigma}_{+}\;(\hat{\sigma}_{-})$ is the creation
(annihilation) atomic operator defined as
$\hat{\sigma}_{+}=|a\rangle\langle b|$ and
$\hat{\sigma}_{-}=|b\rangle\langle a|$ in which $|a\rangle$ and
$|b\rangle$ represent the upper and lower atomic energy levels. It
is a well known fact that the master equation of a two-level atom
coupled to a thermal reservoir can be derived applying the
Born-Markov approximation. Hence following the
standard procedure \cite{lou}, it is possible to verify for a two-level atom driven on resonance
by a
coherent light and coupled to a
thermal reservoir that
\begin{align}\label{tla02}\frac{d\hat{\rho}}{dt}& =
\frac{\Omega}{2}\big(\hat{\sigma}_{+}\hat{\rho} -
\hat{\rho}\hat{\sigma}_{+}-\hat{\sigma}_{-}\hat{\rho}
-\hat{\rho}\hat{\sigma}_{-}\big) \notag\\&+
\frac{\gamma\big(\bar{n}+1\big)}{2}\big[2\hat{\sigma}_{-}\hat{\rho}\hat{\sigma}_{+}
- \hat{\sigma}_{+}\hat{\sigma}_{-}\hat{\rho} -
\hat{\rho}\hat{\sigma}_{+}\hat{\sigma}_{-}\big] \notag\\&+
\frac{\gamma\bar{n}}{2}\big[2\hat{\sigma}_{+}\hat{\rho}\hat{\sigma}_{-}
- \hat{\sigma}_{-}\hat{\sigma}_{+}\hat{\rho} -
\hat{\rho}\hat{\sigma}_{-}\hat{\sigma}_{+}\big],\end{align} where
$\gamma$ is the atomic damping constant and $\bar{n}$ is the mean
photon number corresponding to the reservoir modes, which is the
measure of the intensity of the unbiased noise fluctuations of the
broadband environment modes.
Making use of the master equation \eqref{tla02}, the time
evolution of the expectation values of the atomic creation,
annihilation and energy operators can be obtained,
\begin{align}\label{tla03}\frac{d}{dt}\langle\hat{\sigma}_{-}(t)\rangle & =
-
\frac{\gamma}{2}\big(2\bar{n} + 1\big)
\langle\hat{\sigma}_{-}(t)\rangle-\frac{\Omega}{2}\langle\hat{\sigma}_{z}(t)\rangle,
\end{align}
\begin{align}\label{tla04}\frac{d}{dt}\langle\hat{\sigma}_{+}(t)\rangle & =
-
\frac{\gamma}{2}\big(2\bar{n} + 1\big)
\langle\hat{\sigma}_{+}(t)\rangle-\frac{\Omega}{2}\langle\hat{\sigma}_{z}(t)\rangle,
\end{align}
\begin{align}\label{tla05}\frac{d}{dt}\langle\hat{\sigma}_{z}(t)\rangle & =
- \gamma\big(2\bar{n} +
1\big)\langle\hat{\sigma}_{z}(t)\rangle
\notag\\&+\Omega\big(\langle\hat{\sigma}_{-}(t)\rangle +
\langle\hat{\sigma}_{+}(t)\rangle\big)- \gamma.\end{align}
Following the procedure outlined in Ref. \cite{jmo}, it is
possible to show that
\begin{align}\label{tla06}\langle\hat{\sigma}_{z}(t)\rangle & =
\left(\langle\hat{\sigma}_{z}(0)\rangle +
\frac{\gamma^{2}(1+2\bar{n})}{2\alpha\beta}\right)e^{-\beta t} -
\frac{\gamma^{2}(1+2\bar{n})}{2\alpha\beta} \notag\\&+
\left[\frac{\beta - \gamma\big(2\bar{n} + 1\big)}{\beta -
\alpha}\langle\hat{\sigma}_{z}(0)\rangle +
\frac{\gamma^{2}(1+2\bar{n})}{2\alpha\big(\beta -
\alpha\big)}\right.\notag\\&\left.+ \frac{\Omega}{\beta -
\alpha}\big(\langle\hat{\sigma}_{-}(0)\rangle +
\langle\hat{\sigma}_{+}(0)\rangle\big)- \frac{\gamma}{\beta -
\alpha}\right]\notag\\&\times\big(e^{-\alpha t} - e^{-\beta
t}\big),\end{align} where
\begin{align}\label{tla07}\alpha = \frac{\gamma}{4}(6\bar{n} +
3) - \xi,\end{align}
\begin{align}\label{tla08}\beta &= \frac{\gamma}{4}(6\bar{n} + 3) +
\xi,\end{align}
in which
\begin{align}\label{tla09}\xi & =
\left[\frac{\gamma^{2}}{16}\big(2\bar{n} + 1\big)^{2} - \Omega^{2}
\right]^{1/2}.\end{align}
It may worth mentioning that in the
forthcoming discussions various quantities of interest can be
determined using Eq. \eqref{tla06}.
Applying Eqs. \eqref{tla06}, \eqref{tla07}, \eqref{tla08},
\eqref{tla09}
and the fact that the population inversion,
$W(t)=\langle\hat{\sigma}_{z}(t)\rangle$, it is found at steady
state that
\begin{align}\label{tla10}W=-{1
\over(1+2\bar{n})\left({2\Omega^{2}\over\gamma^{2}}+1\right)}.\end{align}
\begin{center}
\begin{figure}
\caption{Plot of the atomic inversion at steady state. }
\end{figure}
\end{center}
The population inversion is defined as the difference of the
population in the lower and upper energy levels,
$\rho_{aa}-\rho_{bb}$. From the result shown in Fig. 1, it is not
difficult to observe that the population inversion increases with
the amplitude of the external coherent radiation and intensity of the thermal
fluctuations. Moreover, further scrutiny reveals that the
inversion decreases with the intensity of the thermal fluctuations
for larger values of the amplitude of the coherent radiation. As clearly shown in our previous work \cite{jmo}, the
population inversion decreases with the increasing degree of
squeeze parameter for larger values of $\Omega/\gamma$. However, comparison of the dependence of the population inversion on the intensity of the noise fluctuations in the two systems indicates that the squeezed input slightly enhances the decrement of the population inversion. Since the unbiased noise fluctuations in the thermal vibrations are presumed to be washed out in the process of calculating the mean values and hence $\bar{n}$ accounts for the intensity of the fluctuations alone. This is one of the essential differences in atomic
dynamics in cases of biased or unbiased noise fluctuations are
allowed to enter the cavity. It can also be inferred from this
result that the atom stays more often in the lower energy level at
steady state, since the population inversion is found to be
negative for all values of the parameters under consideration.
Furthermore, on the basis of the fact that the probability for
the atom to be in the upper energy level is given by
$\rho_{aa}(t)= {\langle\hat{\sigma}_{z}(t)\rangle+1\over2}$
and
making use of Eq. \eqref{tla06} one gets at steady state
\begin{align}\label{tla11}\rho_{aa} =
\frac{{\Omega^{2}\over\gamma^{2}}+\bar{n}(1+2\bar{n})}{{2\Omega^{2}\over\gamma^{2}}+(2\bar{n}+1)^{2}}.\end{align}
\begin{center}
\end{center}
It can readily be seen from Fig. 2 that the probability for the
atom to be in the upper energy level increases with the mean
photon number of the reservoir modes for smaller values of
$\Omega/\gamma$, but decreases for larger values. In relation to
this, Tanas {\it{et al.}} \cite{jmo451859} recently found that
the absorption spectrum of the driving field or the stationary
line shape, the quantity which is twice of the value in Eq.
\eqref{tla11}, is less than 1 at resonance when a strongly driven
two-level atom is coupled to a finite band squeezed vacuum
reservoir. The dependence of the
probability for the atom to be in the upper energy level on the
squeezed parameter has also the same form as indicated in Fig. 2. It,
hence, can be inferred that when there is an external radiation
the atom may absorb a photon from the cavity and then excited to
the upper energy level. Even then irrespective of the strength
of the external radiations (coherent driving and thermal
fluctuations) the rate of emission is relatively stronger than the
rate of absorption. As can readily be seen, $\rho_{aa}$ takes
values between 0 and 0.5, which implies that this mechanism
perhaps be employed in preparing the atom in an arbitrary
coherent superposition of the upper and lower energy levels by
adjusting the required amplitude of the external coherent
radiation. Despite previous claim that inhibition of absorption is
related to the phase difference between the coherent and squeezed
radiations, the result shown in this Section, rather, indicates
that the inhibition of absorption is predominantly depends on the
intensity of the fluctuations. It is good to note that there is a slight variation due to the phase sensitivity of the noise of
course.
\section{Emission Spectrum}
Emission spectrum that corresponds to the power spectrum of a
radiation emitted by a two-level atom can be conveniently
expressed in terms of the atomic creation and annihilation
operators as
\begin{align}\label{tla12}S(\omega)=2Re\int_{0}^{\infty}\langle\hat{\sigma}_{+}(t)\hat{\sigma}_{-}
(t+\tau)\rangle_{ss}e^{i\omega\tau}d\tau.\end{align}
Following the approach in Ref. \cite{jmo} along with the aid of
the properties of the atomic operators that
$\langle\hat{\sigma}^{2}_{-}\rangle=0$ and
$\langle\hat{\sigma}_{+}\hat{\sigma}_{z}\rangle=-\langle\hat{\sigma}_{+}\rangle$,
one can obtain
\begin{align}\label{tla13}&\langle\hat{\sigma}_{+}(t)\hat{\sigma}_{-}(t +
\tau)\rangle =
\langle\sigma_{+}(t)\sigma_{-}(t)\rangle\left\{2e^{-\frac{\gamma}{2}(2\bar{n}+1)\tau}
\right.\notag\\&\left.-
\frac{\Omega^{2}}{2\left(\frac{\gamma(1+2\bar{n})}{2} -
\beta\right)(\beta-\alpha)}(e^{-\beta\tau} - e^{-\alpha\tau})\right\}
\notag\\&+\langle\hat{\sigma}_{+}(t)
\rangle\left\{\frac{-\Omega}{2\left(\frac{\gamma(1+2\bar{n})}{2} - \beta\right)}\left[
\frac{\gamma}{\beta - \alpha} - \frac{\gamma^{2}(1+2\bar{n})}{2\beta(\beta - \alpha)} \right.\right.\notag\\&\left.\left.+ {\alpha -
\gamma(2\bar{n}+1)\over\beta-\alpha}\right]e^{-\beta\tau} - \frac{\Omega}{2\left(\frac{\gamma(1+2\bar{n})}
{2} - \alpha\right)}\left[\frac{\gamma^{2}(1+2\bar{n})}
{2\alpha(\beta - \alpha)} \right.\right.\notag\\&\left.\left.-\frac{\gamma}{\beta -
\alpha} + \frac{\gamma(2\bar{n}+1)-\beta}{\beta -
\alpha}\right]e^{-\alpha\tau} \right.\notag\\&\left.+
\frac{\Omega\gamma^{2}}{2\beta\alpha}\left(1 -
e^{-\frac{\gamma(1+2\bar{n})}{2}\tau}\right)\right\},\end{align}
\begin{align}\label{tla14}\langle\hat{\sigma}_{+}\rangle_{ss}={\Omega\gamma^{2}\over2\alpha\beta}.\end{align}
In order to
study the dependence of the emission spectrum on the amplitude of
the coherent radiation and intensity of the thermal fluctuations
more closely, two cases of interest are considered. For a strong
driving field, $\Omega \gg \gamma$, it is possible to easily see
from Eq. \eqref{tla09} that $\xi = i\Omega$, as a result, $\beta -
\alpha = i2\Omega$, $\frac{c\gamma}{2} - \beta = -i\Omega$,
$\frac{c\gamma}{2} - \alpha = i\Omega$, $\alpha\beta =
\Omega^{2}$, and $\langle\sigma_{+}(t)\rangle_{ss} = 0$. Moreover,
since
$\rho_{aa}(t)=\langle\hat{\sigma}_{+}(t)\hat{\sigma}_{-}(t)\rangle$
Eq. \eqref{tla11} reduces for $\Omega\gg\gamma$ and modest values
of $\bar{n}$ to
\begin{align}\label{tla15}\langle\sigma_{+}(t)\sigma_{-}(t)\rangle_{ss} =
\frac{1}{2}.\end{align} It can be realized that, at steady state,
the population is independent of the strength of the decoherence
which is consistent with the result shown in Fig. 2. This would
strengthen the already established fact that to prepare a
two-level atom in a possible maximum coherent superposition of the
two energy levels, driving it with a strong external coherent
radiation is sufficient. Furthermore, it is not difficult to see
that
\begin{align}\label{tla16}\langle\hat{\sigma}_{+}(t)\hat{\sigma}_{-}(t +
\tau)\rangle_{ss} &=
\frac{1}{4}e^{-\frac{\gamma}{2}(2\bar{n}+1)\tau} \notag\\&+
\frac{1}{8}e^{(i\Omega -\frac{\gamma}{4}(6\bar{n} + 3))\tau}
\notag\\&+ \frac{1}{8}e^{-(\frac{\gamma}{4}(6\bar{n}+3) +
i\Omega)\tau},\end{align}
from which follows
\begin{align}\label{tla17}S(\omega) &=
\frac{\frac{\gamma}{16}(6\bar{n}+3)}{(\Omega + \omega)^{2} +
[\frac{\gamma}{4}(6\bar{n}+3)]^{2}} \notag\\&+
\frac{\frac{\gamma}{16}(6\bar{n}+3)}{(\Omega - \omega)^{2} +
[\frac{\gamma}{4}(6\bar{n}+3)]^{2}} \notag\\&+
\frac{\frac{\gamma}{4}(1+2\bar{n})}{\omega^{2} +
[\frac{\gamma}{2}(1+2\bar{n})]^{2}}.\end{align}
\begin{center}
\end{center}
\begin{center}
\end{center}
\begin{center}
\end{center}
It is not difficult to observe that the emission spectrum has
three well defined peaks at $\omega = 0$ and $\omega = \pm\Omega$
with line width of $\frac{\gamma(1+2\bar{n})}{2}$ and
$\frac{\gamma(6\bar{n} +3)}{4}$. For $\bar{n}=0$, this result goes
over to the usual Mollow type resonant fluorescent spectrum
\cite{pr1881969}. As can easily be seen from Fig. 3, the stronger
the intensity of the thermal fluctuations of the noise, the wider the splitting
and the shorter the height of the spectrum would be. In addition,
comparison of the results given in Figs. 4 and 5 shows that the
width of the central line and sidebands broadened with the
intensity of the decoherence, whereas the heights decreased. In
connection to this, Parkins \cite{pra426873} has simulated the
resonance fluorescence of a two-level atom coupled to a two-mode
squeezed vacuum reservoir and found that all the three peaks
exhibit subnatural line widths for particular choice of the phase
in a strong driving limit for a moderate squeezed input. On the
other hand, Tanas {\it{et al.}} \cite{jmo451859} recently
found that the spectral lines of the resonance fluorescence of the
two-level atom coupled to finite
band squeezed vacuum reservoir are narrower than for the ordinary
vacuum and the side bands are slightly shifted. It is now evident that the profile of the spectra are the same as what is obtained here even when the biased noise is replaced by unbiased noise fluctuations. Nonetheless, comparison with
previous results shows that the biased fluctuations in the
squeezed vacuum modes suppress the height of the central peak
prominently than the unbiased noise fluctuations in the
decoherence phenomenon, which is basically related to the phase
sensitivity in the squeezed input. On the basis of this understanding, one can then come to conclude that except for such minor
differences, the essential mechanism in emission-absorption process depends
on the intensity of the fluctuations of the noise associated with
the environment rather than the phase.
In a weak driving limit, $\Omega\approx0$, it follows from Eq.
\eqref{tla13} that
\begin{align}\label{tla18}\langle\hat{\sigma}_{+}(t)\hat{\sigma}_{-}(t +
\tau)\rangle_{ss} & =
2\langle\hat{\sigma}_{+}(t)\hat{\sigma}_{-}(t)\rangle_{ss}
\left[e^{-\frac{\gamma}{2}(2\bar{n}+1)\tau}\right],\end{align}
which leads, making use of Eq. \eqref{tla11} at steady state,
$\langle\hat{\sigma}_{+}(t)\hat{\sigma}_{-}(t)\rangle_{ss} =
\frac{\bar{n}}{2\bar{n}+1}$, to
\begin{align}\label{tla19}\langle\hat{\sigma}_{+}(t)\hat{\sigma}_{-}(t+\tau)\rangle_{ss} =
\frac{\bar{2n}}{2\bar{n}+1}\left[e^{-\frac{\gamma}{2}(2\bar{n}+1)\tau}\right].\end{align}
It is, hence, observed that in the weak driving limit the noise
associated to the thermal fluctuations can excite the atom to the
upper energy level, namely, for strongly intense thermal light
there is nearly 50\% probability for the atom to be found in the
upper atomic energy level at steady state. Just like the coherent
driving radiation, the thermal fluctuations entering the cavity
through the walls of the mirror can also be employed in preparing
the atom in arbitrary coherent superposition of the two atomic
energy levels. One can easily see that the atom would be
completely in the ground state at steady state for vacuum
reservoir.
Moreover, it can be deduced from Eq. \eqref{tla19} that for a weak driving limit, $\Omega=0$, the emission spectrum
generally does not split. Therefore, it is possible to infer that the spectral splitting is associated with the strength
of the external coherent radiation, whereas broadening of the
width with the intensity of the fluctuations entering the cavity.
\section{Photon statistics of the cavity radiation}
Currently available literatures indicate that the photon
statistics of the cavity radiation can be investigated using the
normalized two-time second-order correlation function that can be
expressed for the two-level atom in terms of the creation and
annihilation atomic operators in the form
\begin{align}\label{tla21}g^{(2)}(\tau)={\langle\hat{\sigma}_{+}(t)
\hat{\sigma}_{+}(t+\tau)\hat{\sigma}_{-}(t+\tau)\hat{\sigma}_{-}(t)\rangle\over
\langle\hat{\sigma}_{+}(t)\hat{\sigma}_{-}(t)\rangle^{2}}.\end{align}
Therefore, in view of the property of the atomic operators,
\begin{align}\label{tla22}\langle\hat{\sigma}_{+}(t+\tau)\hat{\sigma}_{-}(t+\tau)\rangle={
\langle\hat{\sigma}_{z}(t+\tau)\rangle+1\over2},\end{align} one
gets
\begin{align}\label{tla23}g^{(2)}(\tau)={\langle\hat{\sigma}_{+}(t)\hat{\sigma}_{-}(t)\rangle
+\langle
\hat{\sigma}_{+}(t)\hat{\sigma}_{z}(t+\tau)\hat{\sigma}_{-}(t)\rangle\over2
\langle\hat{\sigma}_{+}(t)\hat{\sigma}_{-}(t)\rangle^{2}},\end{align}
from which follows
\begin{align}\label{tla24}g^{(2)}(\tau) & =
{1\over2\langle\hat{\sigma}_{+}(t)\hat{\sigma}_{-}(t)\rangle}
\left[{2\alpha\beta-\gamma^{2}(1+2\bar{n})\over2\alpha\beta}\right.\notag\\&\left.
+{2\beta\alpha-\gamma^{2}(1+2\bar{n})-4\beta\gamma
\bar{n}\over2\beta(\beta-\alpha)}e^{-\beta\tau}\right.\notag\\&\left.+{\gamma^{2}(1+2\bar{n})
-2\beta\alpha+4\alpha\gamma
\bar{n}\over2\alpha(\beta-\alpha)}e^{-\alpha\tau}\right].\end{align}
For a strong driving limit one finds at steady state
\begin{align}\label{tla25}g^{(2)}(\tau) & =
1- \cos(\Omega\tau)e^{-{\gamma\over4}(6\bar{n}+3)\tau}.\end{align}
\begin{center}
\end{center}
\begin{center}
\end{center}
\begin{center}
\end{center}
It is known for long that the two-time second-order correlation
function describes the delayed coincidence between the
successively emitted light. It is not difficult to see from Figs.
6, 7 and 8 that $g^{(2)}(\tau)>g^{(2)}(0)$ for all cases under
consideration, which indicates that the emitted light exhibits
photon anti-bunching, despite the fact that the unbiased noise
fluctuations entering the cavity destroys the quantum features of
the radiation. The anti-bunching phenomenon can be interpreted as
the atom goes over to the lower energy level after emitting a
photon needs time before it absorbs a photon and excited to the
upper energy level to emit the next photon. It can be deduced that
this is one of the fundamental properties of the absorption and
emission processes of the two-level atom, which is independent of
the external coherent radiation and reservoir to which the cavity
is coupled. It is also possible to identify the photon statistics
of the cavity radiation employing the normalized two-time
second-order correlation function. As can readily be seen from
Figs. 6, 7 and 8, the two-time second-order correlation function
oscillates between $g^{(2)}(\tau)>1$ and $g^{(2)}(\tau)<1$ for
smaller values of the delayed time. This can be interpreted as the
photon statistics oscillates between sub and supper-Poissonian in
this case. However, for larger delayed time there is a possibility
that $g^{(2)}(\tau)>1$ or $g^{(2)}(\tau)<1$ depending on the
amplitude of the coherent radiation and the strength of the
decoherence. The super-Poissonian photon statistics becomes more
prominent for stronger intensity of decoherence. As can be
observed from Eq. \eqref{tla25} the emitted photon exhibits
Poissonian photon statistics for modest values of the amplitude of
the external radiation and for larger delayed time. Similar
oscillatory nature of the two-time second-order correlation
function with delayed time has been discussed by various authors
\cite{jmo,pra414083}. In particular, D'Souza {\it{et al.}}
\cite{pra414083} earlier predicted that the emitted radiation
exhibits both sub and super-Poissonian photon statistics based on
the phase between the coherent and squeezed lights from the curve
of the Mandel's response function, whereas recent study shows that
for larger delayed time the cavity radiation exhibits a
super-Poissonian photon statistics when the cavity is coupled to a
broadband squeezed vacuum reservoir \cite{jmo}. It is evident from
these works that the photon statistics at larger delayed time is
dominated by the properties of the light entering the cavity.
\section{Squeezing of the cavity radiation}
The squeezing properties of the cavity radiation can be studied
applying the variances of the atomic-dipole operators in the
normal order. In order to determine the variances of the
atomic-dipole operators in the normal order, it is possible to
begin with a well established fact that the emitted radiation can
be described in terms of the electric field. If the two quadrature
components of the electric field satisfy the commutation relation,
\begin{align}\label{tla26}\big[\hat{E}_{\theta},\;\hat{E}_{\theta-\pi/2}\big]=i2C,\end{align}
then the usual uncertainty relation,
\begin{align}\label{tla27}\langle(\Delta\hat{E}_{\theta})^{2}\rangle\langle(\Delta
\hat{E}_{\theta-\pi/2})^{2}\rangle\ge C^{2},\end{align} holds.
The radiation represented by this electric field is in squeezed
state, provided that either
$\langle(\Delta\hat{E}_{\theta})^{2}\rangle$ or $\langle(\Delta
\hat{E}_{\theta-\pi/2})^{2}\rangle$ is below the vacuum limit $C$.
In general, one of the variances of the electric field can be put
in the normal order as
\begin{align}\label{tla28}\langle:(\Delta\hat{E}_{\theta})^{2}:\rangle
=\langle(\Delta \hat{E}_{\theta})^{2}\rangle- C,\end{align} where
the symbol :: stands for the operator put in the normal order.
Therefore, the squeezing can be related to the requirement that
either $\langle:(\Delta\hat{E}_{\theta})^{2}:\rangle$ or
$\langle:(\Delta \hat{E}_{\theta-\pi/2})^{2}:\rangle$ is less than
zero. On the other hand, making use of the relation between the
electric field and atomic operators, the variance in the field
operator can be defined in terms of the atomic-dipole operators.
In this regard, Ficek and Tanas \cite{pr372369} have expressed the
variance of the atomic-dipole operator in the normal order in the
form
\begin{align}\label{tla29}\langle:(\Delta\hat{\sigma}_{i})^{2}:\rangle
=\langle(\Delta
\hat{\sigma}_{i})^{2}\rangle+{\langle\hat{\sigma}_{z}\rangle\over2},\end{align}
where $i=x,y$ and
\begin{align}\label{tla30}\hat{\sigma}_{x}={1\over\sqrt{2}}\big(\hat{\sigma}_{+}+\hat{\sigma}_{-}\big),\end{align}
\begin{align}\label{tla31}\hat{\sigma}_{y}={i\over\sqrt{2}}\big(\hat{\sigma}_{-}-\hat{\sigma}_{+}\big).\end{align}
Then with the aid of the fact that at steady state
$\langle\hat{\sigma}_{-}(t)\rangle_{ss}=\langle\hat{\sigma}_{+}(t)\rangle_{ss}$,
one finds
\begin{align}\label{tla32}\langle:(\Delta\hat{\sigma}_{x})^{2}:\rangle
={1\over2}\big(1+\langle\hat{\sigma}_{z}\rangle\big)-2\langle\hat{\sigma}_{+}\rangle^{2}_{ss},\end{align}
\begin{align}\label{tla33}\langle:(\Delta\hat{\sigma}_{y})^{2}:\rangle
={1\over2}\big(1+\langle\hat{\sigma}_{z}\rangle\big).\end{align}
\begin{center}
\end{center}
It is not difficult to realize based on the definition of the
population inversion along with the result shown in Fig. 1 that
$\langle\Delta\hat{\sigma}_{z}\rangle>-1$, which implies that
$\langle:(\Delta\hat{\sigma}_{y})^{2}:\rangle$ never be negative
at steady state. As can also be seen from Eq. \eqref{tla14},
$\langle\hat{\sigma}_{+}\rangle_{ss}=\gamma^{2}/2\Omega$, which
approaches zero for strong driving radiation. In this case as
well, it can readily be seen that
$\langle:(\Delta\hat{\sigma}_{x})^{2}:\rangle$ never be negative.
This indicates that the emitted radiation does not exhibit
squeezing when the atom is pumped externally with a strong
coherent radiation at steady state. As opposed to this,
$\langle:(\Delta\hat{\sigma}_{x})^{2}:\rangle$ can be less than
zero for certain values of the amplitude of the external radiation
as clearly shown in Fig. 9 for smaller values of $\Omega/\gamma$.
On the basis of the criterion set for squeezing, the light
emitted by the two-level atom exhibits squeezing even in the
presence of a significant amount of decoherence that is believed
to destroy the quantum features of the light. Unfortunately, the
squeezing is found to exist only for narrow pockets of the values
of the amplitude of the coherent radiation which, of course,
depend on the strength of the decoherence. This result
demonstrates that the atomic coherence induced between the upper
and lower energy levels by the coherent radiation, which is
responsible for the squeezing, is too strong to be destroyed by
the unbiased noise fluctuations. It is believed that this must be
the reason for observing a considerable entanglement in a
correlated emission laser even in the presence of a strong
decoherence \cite{jpbamop402373}. Moreover, critical survey of
Fig. 9 reveals that the squeezing exists for values of
$\Omega/\gamma$ for which the squeezing disappears in the absence
of the decoherence. This may be related to a recent claim that a
decoherence due to environment enhances entanglement in a
two-level atomic system by providing an indirect correlation
between totally uncorrelated quantum states \cite{pra75012101}.
\section{Conclusion}
In this contribution, a thorough study of the effects of the
external coherent radiation and thermal fluctuations corresponding
to the vibration of the walls of the mirrors due to their coupling
with the external environment on the atomic dynamics, squeezing
properties and photon statistics of the radiation produced by a
coherently driven two-level atom trapped in a resonant cavity is
presented. It is found that though the atom absorbs the radiation
from the available cavity modes, including the driving, emitted
and thermal light entering the cavity, and makes a transition to
the upper energy level, it prefers to stay in the ground state
more often irrespective of the amplitude of the coherent radiation
and the strength of the intensity of the decoherence. In this
regard, in comparison to previous works, the fundamental
phenomenon in absorption and emission processes, in which the rate
of emission is greater than the rate of absorption, is basically found to
depend on the strength of the fluctuations associated to the
environment rather the phase difference. This, on the other hand,
indicates that except for the minor differences in its degree the
inhibition of absorption is resulted when the cavity is coupled to
both biased and unbiased noise fluctuations. Therefore, it is
possible to deduce from this study that predominantly the atomic
dynamics is affected by the mean photon number of the reservoir
modes rather than whether the reservoir is squeezed vacuum or
thermal. In addition to this, it is not difficult to realize that
the two-level atom can be prepared in arbitrary coherent
superposition of the upper and lower energy levels by varying the
intensity of the thermal fluctuations in the environment. It is
believed that this approach perhaps would be practically
attractive in the preparation of the injected atomic coherence required in
multi-level atomic laser \cite{jpbamop402373}.
It was previously discussed that the emission spectrum is
broadened and the height is reduced by the squeezed input. In the
same way, the thermal fluctuation is found to broaden the spectrum
and reduce the height significantly, but it does not contribute to
the splitting of the central line into triplet. Comparison with
the previous works indicates that the biased noise fluctuations in
the squeezed vacuum modes decrease the height of the central peak
more than the unbiased noise fluctuations in decoherence.
Moreover, the emitted radiation is found to exhibit anti-bunching
photon statistics independent of the type of the reservoir. It is,
rather, a fundamental property of a driven two-level atom related
to the time required for absorbing a radiation to make a
transition to the upper energy level after the atom emits a
photon. As opposed to this, the super-Poissonian statistics is
found to be enhanced by biased noise fluctuations. In addition to
this, the cavity radiation exhibits appreciable squeezing for some
pockets of the amplitude of the driving radiation that depends on
the strength of the intensity of decoherence. In conclusion, even
though the successively emitted photons are separated in time,
they are strongly correlated which leads to the appearance of the nonclassical
features even in the presence of decoherence which presumably
destroys the quantum properties.
\end{document} | math |
module ProfessionalContributionsHelper
end
| code |
\betaegin{document}
\muaketitle
\setcounter{page}{1}
\partialgestyle{myheadings}
\muarkboth{SHANJIAN TANG}{DYNAMIC PROGRAMMING FOR STOCHASTIC LQ}
\betaegin{abstract}
We are concerned with the linear-quadratic optimal stochastic control problem where all the coefficients of the control system and the running weighting matrices in the cost functional are allowed to be predictable (but essentially bounded) processes and the terminal state-weighting matrix in the cost functional is allowed to be random. Under suitable conditions,
we prove that the value field $V(t,x,\omegamega), (t,x,\omegamega)\in [0,T]\tauimes R^n\tauimes \Omegamega$, is quadratic in $x$, and has the following form: $V(t,x)=\lambdaangle K_tx, x\mathop{\rangle}gle$ where $K$ is an essentially bounded nonnegative symmetric matrix-valued adapted processes. Using the dynamic programming principle (DPP), we prove that $K$ is a continuous semi-martingale of the form $$K_t=K_0+\int_0^t \, dk_s+\sum_{i=1}^d\int_0^tL_s^i\, dW_s^i, \quaduad t\in [0,T]$$
with $k$ being a continuous process of bounded variation and
$$E\lambdaeft[\lambdaeft(\int_0^T|L_s|^2\, ds\right)^p\right] <\infty, \quaduad \varphiorall p\gammae 2; $$
and that $(K, L)$ with $L:=(L^1, \cdotots, L^d)$ is a solution to the associated backward stochastic Riccati equation (BSRE), whose generator is highly nonlinear in the unknown pair of processes. The uniqueness is also proved via a localized completion of squares in a self-contained manner for a general BSRE. The existence and uniqueness of adapted solution to a general BSRE was initially proposed by the French mathematician J. M. Bismut
[in \tauextit{SIAM J. Control \& Optim.}, 14(1976), pp.\ 419--444, and in \tauextit{S\'eminaire de Probabilit\'es} XII, Lecture Notes in Math. 649,
C. Dellacherie, P. A. Meyer, and M. Weil, eds., Springer-Verlag, Berlin, 1978, pp.\ 180--264], and subsequently listed by Peng [in \tauextit{Control of Distributed Parameter and Stochastic Systems (Hangzhou, 1998)},
S. Chen, et al., eds., Kluwer Academic Publishers, Boston, 1999, pp.\ 265--273] as the first open problem for backward stochastic differential equations. It had remained to be open until a general solution by the author [in \tauextit{SIAM J. Control \& Optim.}, 42(2003), pp.\ 53--75] via the stochastic maximum principle with a viewpoint of stochastic flow for the associated stochastic Hamiltonian system. The present paper is its companion, and gives the {\it second but more comprehensive} (seemingly much simpler, but appealing to the advanced tool of Doob-Meyer decomposition theorem, in addition to the DDP) adapted solution to a general BSRE via the DDP. Further extensions to the jump-diffusion control system and to the general nonlinear control system are possible.
\varepsilonnd{abstract}
\betaegin{keywords}
linear quadratic optimal stochastic control, random coefficients,
Riccati equation, backward stochastic differential equations, dynamic programming, semi-martingale
\varepsilonnd{keywords}
\betaegin{AMS}
93E20, 49K45, 49N10, 60H10
\varepsilonnd{AMS}
\section{Formulation of the problem and basic assumptions}\lambdaabel{sec1}
Consider the following linear quadratic optimal stochastic control (SLQ in short
form) problem: minimize over $u\in
\muathscr{L}^2_\muathscr{F}(0,T; \widehatboxox{\rm l\nuegthinspace R}^m)$ the following quadratic cost functional:
\betaegin{equation}\lambdaabel{eq1.1}
J(u;0,x):=E^{0,x;u}\lambdaeft[\lambdaangle MX_T,X_T\mathop{\rangle}gle +\int_0^T(\lambdaangle
Q_sX_s,X_s\mathop{\rangle}gle +\lambdaangle N_su_s,u_s\mathop{\rangle}gle )\, ds\right],
\varepsilonnd{equation}
where $X$ is the solution of the following linear stochastic
control system:
\betaegin{equation}\lambdaabel{eq1.2}
\lambdaeft\{\varepsilonq{dX_t=&\;(A_tX_t+B_tu_t)\, dt
+\sum_{i=1}^d(C_t^iX_t+D_t^iu_t)\, dW_t^i,\cr X_0=&\;x\in \widehatboxox{\rm l\nuegthinspace R}^n.
\cr}\right.
\varepsilonnd{equation}
Here, $\{W_t:=(W_t^1, \lambdadots, W_t^d)', 0\lambdae t\lambdae T\}$ is a
$d$-dimensional standard Brownian motion defined on some
probability space $(\Omega, \muathscr{F}, P)$. Denote by $\{\muathscr{F}_t,0\lambdae t\lambdae T\}$ the augmented natural filtration of the
standard Brownian motion $W$. The control $u$ belongs to the
Banach space $\muathscr{L}^2_\muathscr{F}(0,T;
\widehatboxox{\rm l\nuegthinspace R}^m)$, which consists of all $\widehatboxox{\rm l\nuegthinspace R}^m$-valued square integrable
$\{\muathscr{F}_t, 0\lambdae t\lambdae T\}$-adapted processes. Denote by $\muathbb{S}^n$ the totality of $n\tauimes n$ symmetric matrices, and by $\muathbb{S}^n_+$ the totality of $n\tauimes n$ nonnegative matrices.
Throughout this paper, we make the following two assumptions on the coefficients of
the above problem.
(A1) Assume that the matrix processes $A: [0,T]\tauimes \Omega\tauo
\widehatboxox{\rm l\nuegthinspace R}^{n\tauimes n}$, $B:[0,T]\tauimes \Omega\tauo \widehatboxox{\rm l\nuegthinspace R}^{n\tauimes m}$;
$C^i:[0,T]\tauimes \Omega\tauo \widehatboxox{\rm l\nuegthinspace R}^{n\tauimes n}$, $D^i:[0,T]\tauimes \Omega\tauo
\widehatboxox{\rm l\nuegthinspace R}^{n\tauimes m}$, $i=1,\lambdads,d$;
$Q:[0,T]\tauimes \Omega\tauo \muathbb{S}^n_+$, $N:[0,T]\tauimes \Omega\tauo \muathbb{S}^m_+ $
and the random matrix $M:\Omega\tauo \muathbb{S}^n_+$ are uniformly bounded and
$\{{\cal F }_t, 0\lambdae t\lambdae T\}$-adapted or ${\cal F}_T$-measurable.
(A2) Assume that the control weighting matrix process $N$ is uniformly
positive.
\vskip0.5cm
Define for $(t,K, L)\in [0,T]\tauimes \muathbb{S}^n\tauimes (\muathbb{S}^n)^d$,
\betaegin{equation}
\varepsilonq{\muathscr{N}_t(K):=& N_t+\sum_{i=1}^d(D^i_t)'KD^i_t, \cr
\muathscr{M}_t(K,L):=& KB_t+\sum_{i=1}^d(C^i_t)'KD^i_t+\sum_{i=1}^dL^iD^i_t. \cr}
\varepsilonnd{equation}
For $(t, K)\in [0,T]\tauimes \muathbb{S}^n_+$ and $L=(L^1,\lambdadots, L^d)\in (\muathbb{S}^n)^d$, define
\betaegin{equation}\lambdaabel{G}
\varepsilonq{G(t, K,L):=&\ A'_tK+KA_t+Q_t+\sum_{i=1}^d(C^i_t)'KC^i_t+\sum_{i=1}^d[(C^i_t)'L^i+L^iC^i_t]\cr
&\quaduad\quaduad -\muathscr{M}_t(K,L)\muathscr{N}_t^{-1}(K)\muathscr{M}_t'(K,L).\cr}
\varepsilonnd{equation}
Here, we use the prime to denote the transpose of a vector or a matrix.
Associated to the above SLQ problem is the following backward stochastic Riccati equation (BSRE):
\betaegin{equation}\lambdaabel{bsre}
\lambdaeft\{\varepsilonq{dK_t=&\;-G(t, K_t,L_t)\, dt+\sum_{i=1}^dL_t^i\,
dW_t^i, \quaduad t\in [0,T);\cr K_T=&\;M,\quadquad L_t:=(L_t^1, \lambdadots, L_t^d).\cr}\right.
\varepsilonnd{equation}
The generator is {\it highly nonlinear} in the unknown pair of variables $(K, L)$.
\betaegin{mydefinition}\lambdaabel{solution bsre}
A solution of BSRDE (\ref{bsre}) is defined as a pair $(K,L)$
of matrix-valued adapted processes such that
{\rm (i)} $\int_0^T|L_t|^2\, dt + \int_0^T|G(t,K_t,L_t)|\, dt<\infty,\ a.s.;$
{\rm (ii)} The $m\tauimes m$ matrix-valued process $\{\muathscr{N}_t(K_t), t\in [0,T]\}$ is $a.s.a.e.$
positive; and
{\rm (iii)} $K_t=M+\int_t^TG(s, K_s, L_s)\, ds-\int_t^T\sum_{i=1}^dL^i_s\, dW^i_s $ a.s. for all
$t\in [0,T].$
\varepsilonnd{mydefinition}
\mus
The adapted solution to a general BSRE~(\ref{bsre}) was initially proposed by the French mathematician J. M. Bismut~\cite{Bismut1976,Bismut1978}, and subsequently listed by Peng~\cite{Peng1999} as the first open problem for backward stochastic differential equations. It had remained to be open until a general solution by the author~\cite{Tang2003} via the stochastic maximum principle and using a viewpoint of stochastic flow for the associated stochastic Hamiltonian system. For more details on the historical studies on BSRE~(\ref{bsre}) and the progress, see the author's previous paper~\cite[Section 4, pages 60--61]{Tang2003} and the plenary lecture by Peng~\cite{Peng2010} at the International Congress of Mathematicians in 2010. In the paper, we shall give a novel proof to the existence for BSRE~(\ref{bsre}) via dynamic programming principle. A crucial point is that we can show the value field is a semi-martingale of both ``sufficiently good" parts of bounded variation and martingale.
The rest of our paper is organized as follows. Section 2 gives preliminaries. In Section 3, we prove that the value field $V(t,x, \omegamega)$ is quadratic in $x$.
In Section 4, we prove that the value field is a semi-martingale and that BSRE~(\ref{bsre}) has an adapted solution. Section 5 is concerned with a verification theorem for the SLQ problem, and the uniqueness of solution to BSRE~(\ref{bsre}). Finally, in Section 6, we give some comments and possible extensions.
\section{Preliminaries}\lambdaabel{sec2}
For each $u\in \muathscr{L}^2_\muathscr{F}(0,T; \widehatboxox{\rm l\nuegthinspace R}^m)$, the following linear stochastic differential equation
\betaegin{equation}\lambdaabel{eq2.2}
\lambdaeft\{\varepsilonq{dX_t=&(A_tX_t+B_tu_t)\, dt
+\sum_{i=1}^d(C_t^iX_t+D_t^iu_t)\, dW_t^i,\quadquad \tauau \lambdae t\lambdae T,
\cr X_s=& x\in \muathbb{R}^n,
\cr}\right.
\varepsilonnd{equation}
has a unique strong solution (see Bismut~\cite{Bismut1978}), denoted by $X^{s,x;u}$ with the superscripts indicating the dependence on the initial data $(s,x)$ and the control action.
We have the following well-known quantitative dependence of the solution $X^{s,x;u}$ on the initial data $(s,x)$ and the control action $u$.
\betaegin{lemma}\lambdaabel{sde estimate} Let assumption (A1) be satisfied. For any $p\gammae 1$, there is a positive constant $C_p$ such that for any initial state $\xi\in L^p(\Omegamega, \muathscr{F}_s, P;\muathbb{R}^n)$ and predictable control $u$ with
$$
E\lambdaeft[\lambdaeft(\int_s^T|u_r|^2\, dr\right)^{p/2}\right]<\infty,
$$
we have
\betaegin{equation}\lambdaabel{state-estimate}
E\lambdaeft[\muax_{t\in [s,T]}|X^{s,\xi;u}|^p\betaiggm | \muathscr{F}_s\right]\lambdae C_p\lambdaeft(|\xi|^p+E\lambdaeft[\lambdaeft(\int_s^T|u_r|^2\, dr\right)^{p/2}\betaiggm| \muathscr{F}_s\right]\right).
\varepsilonnd{equation}
\varepsilonnd{lemma}
Consider the initial-data-parameterized
SLQ problem: minimize over $u\in \muathscr{L}^2_\muathscr{F}(0,T; \widehatboxox{\rm l\nuegthinspace R}^m)$ the quadratic cost functional
\betaegin{equation}\lambdaabel{eq2.1}
\widehatspace*{12pt}J(u; s,x):=E^{s,x;u}\lambdaeft[\lambdaangle MX_T, X_T\mathop{\rangle}gle
+\int_s^T(\lambdaangle Q_rX_r, X_r\mathop{\rangle}gle
+\lambdaangle N_ru_r,u_r\mathop{\rangle}gle )\, dr\betaiggm | \muathscr{F}_s\right].
\varepsilonnd{equation}
Define the value field
\betaegin{equation}\lambdaabel{value}
V(s,x):=\varepsilonssinf_{u\in \muathscr{L}^2_{\muathscr{F}_t}(s,T; \muathbb{R}^m)}J(u;s,x), \quaduad (s,x)\in [0,T]\tauimes \muathbb{R}^n.
\varepsilonnd{equation}
Assumptions (A1) and (A2) imply that the above
SLQ problem has a unique optimal control for any $\xi\in L^2(\Omegamega, \muathscr{F}_s, P;\muathbb{R}^n)$, that is, there is unique ${\omegaverline u}\in \muathscr{U}_s$ such that
$$
V(s, \xi)=J({\omegaverline u}; s, \xi).
$$
See Bismut~\cite{Bismut1978} for the
proof of such a result. A further step is to characterize the
optimal control.
We easily prove the following
\betaegin{lemma}\lambdaabel{bound} Let Assumptions (A1) and (A2) be satisfied. There is a positive constant $\lambdaambda$ such that
$$
0\lambdae V(s,\xi)\lambdae J(0;s,\xi)\lambdae \lambdaambda |\xi|^2, \quaduad \varphiorall (s,\xi) \in [0,T]\tauimes L^2(\Omegamega, \muathscr{F}_s, P;\muathbb{R}^n).
$$
\varepsilonnd{lemma}
\betaegin{proof} In view of assumption (A1) and the definition of the value field $V$, it is sufficient to show $J(0;s,\xi)\lambdae \lambdaambda |\xi|^2$, which is an immediate consequence of Lemma~\ref{state-estimate} and the following estimate:
$$\varepsilonq{J(0;s,\xi)\lambdae\ & \lambdaambda E\lambdaeft[|X_T^{0,\xi;0}|^2+\int_0^T|X_t^{0,\xi;0}|^2\, dt\ \betaiggm |\muathscr{F}_s\right]\cr
\lambdae\ & \lambdaambda (1+T) E\lambdaeft[\muax_{t\in [0,T]}|X_t^{0,\xi;0}|^2\betaiggm |\muathscr{F}_s\right].\cr}
$$
\varepsilonnd{proof}
\section{The value field $V$ is quadratic in the space variable}
This section is an adaptation of Faurre~\cite{Faurre1968} to our SLQ problem with random coefficients.
We have
\betaegin{theorem} \lambdaabel{quadratic} Let Assumptions (A1) and (A2) be satisfied. The value field $V(s,x)$ is quadratic in $x$. Moreover, there is an essentially bounded continuous nonnegative matrix-valued process $K$ such that
\betaegin{equation}
V(s,x)=\lambdaangle K_s x,x \mathop{\rangle}gle, \quaduad \varphiorall (s,x)\in [0,T]\tauimes \muathbb{R}^n.
\varepsilonnd{equation}
\varepsilonnd{theorem}
The state-quadratic property follows from the following lemma.
\betaegin{lemma} Let Assumptions (A1) and (A2) be satisfied. The value field has the following two laws in the state variable $x$ of (i) square homogeneity
$$
V(s, \xi x)=\xi^2 V(s,x), \quaduad \varphiorall (s, x, \xi) \in [0,T]\tauimes \muathbb{R}^n\tauimes L^\infty(\Omegamega, \muathscr{F}_s, P)
$$
and (ii) parallelogram
$$
V(s, x+y)+V(s,x-y)=2 V(s,x)+2V(s,y), \quaduad \varphiorall (s, x, y) \in [0,T]\tauimes \muathbb{R}^n\tauimes \muathbb{R}^n.
$$
\varepsilonnd{lemma}
\betaegin{proof} It is easy to derive from the linearity of the control system and the quadratic structure of the cost functional the following two identities for any $u\in \muathscr{U}_s$,
$$
\xi X^{s,x;u}=X^{s,\xi x; \xi u} , \quaduad \xi^2 J(u;s, x)= J(\xi u; s, \xi x).
$$
Therefore, we have
$$
\xi^2 V(s,x)=\xi^2 \varepsilonssinf_{u\in \muathscr{U}_s} J(u;s,x)=\varepsilonssinf_{u\in \muathscr{U}_s} \xi^2 J(u;s,x)=\varepsilonssinf_{u\in \muathscr{U}_s} J(\xi u;s,\xi x),
$$
which is equal to $V(s, \xi x)$ by definition, immediately giving assertion (i).
Let us show assertion (ii). It is easy to see (see Bismut~\cite{Bismut1978})
that there are $\alpha, \beta \in \muathscr{U}_s$ such that
$$
V(s,x+y)=J(\alpha; s,x+y), \quaduad V(s,x-y)=J(\beta;s,x-y).
$$
Then, we easily see that
$$
V(s, (x+y)\pm (x-y))\lambdae J(\alpha\pm \beta; s, (x+y)\pm (x-y))
$$
and therefore,
$$
V(s, 2x)+V(s, 2y)\lambdae J(\alpha + \beta; s, 2x)+J(\alpha- \beta; s, 2y).
$$
Since $J(u;s,x)$ is quadratic in the pair $(u,x)$ and satisfies the parallelogram
$$
2J(\alpha + \beta; s, 2x)+2J(\alpha- \beta; s, 2y)= J(2\alpha; s,2(x+y))+J(2\beta;s,2(x-y)),
$$
we have
$$
V(s, 2x)+V(s, 2y)\lambdae {1\omegaver2}[ J(2\alpha; s,2(x+y))+J(2\beta;s,2(x-y))],
$$
and therefore by the square homogeneity of $J(u;s,x)$ in the pair $(u,x)$
$$
V(s, x+y)+V(s,x-y)\lambdae 2J(\alpha; s, x+y)+2J(\beta;s,x-y)=2 V(s,x)+2V(s,y).
$$
By symmetry, it holds for $x':=x+y$ and $y':=x-y$:
$$
V(s,(x+y)+(x-y))+V(s,(x+y)-(x-y))\lambdae 2 V(s, x+y)+2V(s, x-y)
$$
which leads by assertion (i) to the following desired reverse inequality
$$
4V(s,x)+4V(s,y)=V(s,2x)+V(s,2y)\lambdae 2V(s,x+y)+2V(s,x-y).
$$
The proof is then complete.
\varepsilonnd{proof}
The nonnegativity and the essential bound of the process $K$ are immediate consequences of Lemma~\ref{bound}.
\section{Dynamic programming principle and the semi-martingale property of the value field}\lambdaabel{sec3} For simplicity, define the function
\betaegin{equation}
l(t,x,u):=\lambdaangle Q_tx, x\mathop{\rangle}gle
+\lambdaangle N_tu,u\mathop{\rangle}gle, \quaduad (t,x,u)\in [0,T]\tauimes \muathbb{R}^n\tauimes \muathbb{R}^m
\varepsilonnd{equation}
and the set
\betaegin{equation}
\muathscr{U}_s:=\muathscr{L}^2_{\muathscr{F}}(s,T; \muathbb{R}^m).
\varepsilonnd{equation}
We denote by $\muathbb{V}(t, \cdotot)$ the restriction of $V(t, \cdotot)$ to $\muathbb{R}^n$. By definition, we have almost surely
$$
V(t,x)=\muathbb{V}(t,x), \quaduad \varphiorall \ x \in \muathbb{R}^n.
$$
For any $\xi\in L^2(\Omegamega, \muathscr{F}_t, P; \muathbb{R}^n)$, in an analogous way to the proof of Peng~\cite[Lemma 6.5, page 122]{Peng1997}, we also have almost surely
$$
V(t,\xi)=\muathbb{V}(t,\xi).
$$
We have
\betaegin{theorem} \lambdaabel{BP}(Bellman's Principle). Let Assumptions (A1) and (A2) be satisfied. We have
(i) For $s\lambdae t\lambdae T$ and $\xi\in L^2(\Omegamega, \muathscr{F}_s, P; \muathbb{R}^n)$,
$$
\muathbb{V}(s,\xi)=\mubox{\rm ess.}\inf_{u\in {\muathscr U}_s} E^{s,\xi; u} \lambdaeft \{\int_s^tl(r, X_r, u_r)\, dr+ \muathbb{V}(t, X_t) \betaiggm | {\muathscr F}_s\right\}.
$$
For the optimal control $\omegaverline{u}\in \muathscr{U}_s$, we have
$$
\muathbb{V}(s,\xi)= E^{s,\xi; \omegaverline{u}} \lambdaeft \{\int_s^tl(r, X_r, \omegaverline{u}_r)\, dr+ \muathbb{V}(t, X_t) \betaiggm | {\muathscr F}_s\right\}.
$$
(ii) For $(s,x,u)\in [0,T]\tauimes \muathbb{R}^n\tauimes {\muathscr U}_s$, the process
$$\betaegin{array}{rcl}
\kappa_t^{s,x;u}&:=&\deltaisplaystyle \muathbb{V}(t,X_t^{s,x;u}) +\int_s^tl(r, X_r^{s,x;u}, u_r)\, dr
\varepsilonnd{array}
$$
defined for $t\in [s,T]$, is a submartingale w.r.t. $\{{\muathscr F}_t\}$; and for the optimal control $\omegaverline{u}\in \muathscr{U}_s$, the process $\kappa_t^{s,x;\omegaverline{u}}, t\in [s,T]$, is a martingale w.r.t. $\{{\muathscr F}_t\}$.
\varepsilonnd{theorem}
\betaegin{proof} It is easy to check that Assertion (ii) is an immediate consequence of Assertion (i). Assertion (i) is more or less standard, and the proof is similar to that of Krylov~\cite[Theorem 6, Section 3, Chapter 3, page 150]{Krylov1977} or Peng~\cite[Theorem 6.6, page 123]{Peng1997}.
\varepsilonnd{proof}
From assertion (i), we have
\betaegin{corollary}\lambdaabel{con mean cont} We have the following time continuity of $\muathbb{V}$ and $K$: for any $(s,x)\in [0,T]\tauimes \muathbb{R}^n$,
$$
\lambdaim_{t\tauo s}E[\muathbb{V}(t,x)-\muathbb{V}(s,x)\, |\muathscr{F}_s]=\ 0, \quaduad
\lambdaim_{t\tauo s} E[K_t-K_s\, |\muathscr{F}_s]=\ 0, \quaduad a.s..
$$
\varepsilonnd{corollary}
\betaegin{proof} In view of Theorem~\ref{quadratic}, the second limit easily follows from the first one. It remains to prove the first limit.
Assume without loss of generality that $s\lambdae t$. We have
$$
\muathbb{V}(s,x)= E^{s,x; \omegaverline{u}} \lambdaeft \{\int_s^tl(r, X_r, \omegaverline{u}_r)\, dr+ \muathbb{V}(t, X_t) \betaiggm | {\muathscr F}_s\right\}
$$
where $\omegaverline{u}\in \muathscr{U}_s$ is the optimal control. Therefore,
$$
|E[\muathbb{V}(t,x)-\muathbb{V}(s,x)\, |\muathscr{F}_s]|\lambdae E^{s,x; \omegaverline{u}} \lambdaeft \{\int_s^tl(r, X_r, \omegaverline{u}_r)\, dr+ |\muathbb{V}(t, X_t)-\muathbb{V}(t,x)| \betaiggm | {\muathscr F}_s\right\}.
$$
Since
$$
|\muathbb{V}(t, X_t^{s,x; \omegaverline{u}})-\muathbb{V}(t,x)|\lambdae \lambda (|x|+|X_t^{s,x; \omegaverline{u}}|) |X_t^{s,x; \omegaverline{u}}-x|,
$$
using estimate~(\ref{state-estimate}), we have
$$
\varepsilonq{&|E[\muathbb{V}(t,x)-\muathbb{V}(s,x)\, |\muathscr{F}_s]|\lambdae \ \lambda E^{s,x; \omegaverline{u}} \lambdaeft \{\int_s^t ( |X_r|^2+|\omegaverline{u}_r|^2)\, dr \betaiggm | {\muathscr F}_s\right\}\cr
&\ +\lambda \lambdaeft\{|x|+E^{s,x; \omegaverline{u}}\lambdaeft[ \lambdaeft(\int_s^t|\omegaverline{u}_r|^2\, dr\right)^{1/2} \betaiggm | {\muathscr F}_s\right]\right\} E^{s,x; \omegaverline{u}}\lambdaeft[ \lambdaeft(\int_s^t|\omegaverline{u}_r|^2\, dr\right)^{1/2} \betaiggm | {\muathscr F}_s\right], \cr}
$$
which implies the desired limit.
\varepsilonnd{proof}
Using Theorems~\ref{quadratic} and ~\ref{BP} , we can prove the following
\betaegin{theorem} \lambdaabel{rep}
The value field $V$ is a semi-martingale of the following representation:
\betaegin{equation}\lambdaabel{quadratic form}
\muathbb{V}(t,x)=\lambdaangle K_tx, x\mathop{\rangle}gle
\varepsilonnd{equation} where $K$ is an essentially bounded nonnegative symmetric matrix-valued continuous semi-martingale of the form
\betaegin{equation}\lambdaabel{representation} K_t=K_0-\int_0^t d k_s +\sum_{i=1}^d\int_0^tL_s^i\, dW_s^i, \quaduad t\in [0,T]; \quaduad K_T=M\varepsilonnd{equation}
with $k$ being an $n\tauimes n$ atrix-valued continuous process of bounded variation such that
\betaegin{equation}\lambdaabel{formula}
\varepsilonq{ dk_s= &\ G(s,K_s,L_s)\, ds, \quaduad \widehatboxox{ \rm almost everywhere } (s,\omegamega) \in [0,T]\tauimes \Omegamega. \cr}
\varepsilonnd{equation}
and
\betaegin{equation}
\lambdaabel{p-estimate}
E\lambdaeft[\lambdaeft(\int_0^T|L_s|^2\, ds\right)^p\right] <\infty, \quaduad \varphiorall p\gammae 2. \varepsilonnd{equation}
\varepsilonnd{theorem}
\betaegin{proof} Theorem~\ref{quadratic} states that there is an essentially bounded nonnegative symmetric matrix-valued process $K$ such that (\ref{quadratic form}) holds true. The rest of the proof is divided into the following three steps.
{\betaf Step 1. $K$ is a semi-martingale of form~(\ref{representation}) in the Doob-Meyer decomposition. }
Let $e_i$ be the unit column vector of $\muathbb{R}^n$ whose $i$-th component is the number $1$ for $i=1,\lambdadots,n$. In view of Assertion (ii) of Theorem~\ref{BP}, we see that for $x=e_i, e_i+e_j, e_i-e_j, i,j=1,\lambdadots,n$,
$\{\kappa_t^{0,x;0}, t\in [0,T]\}$ is a sub-martingale, and since
$$|\kappa_t^{0,x;0}|\lambdae \lambda |X_t^{0,x;0}|^2+\int_0^t |X_s^{0,x;0}|^2\, ds\lambdae \lambda \muax_{t\in [0,T]}|X_t^{0,x;0}|^2 \in L^1(\Omegamega, \muathscr{F}_T, P),$$
it is of class $D$. Since $V(t,x)$ is continuous in the sense of conditional mean in $t$ (see corollary~\ref{con mean cont}), $\{\kappa_t^{0,x;0}, t\in [0,T]\}$ is continuous in the sense of conditional mean in $s$. In view of Doob-Meyer decomposition (see Protter~\cite[Theorem 11, page 112]{Protter2005}), its bounded variational process is continuous and increasing in time, and $\{\kappa_t^{0,x;0}, t\in [0,T]\}$ is sample continuous.
Define the $n\tauimes n$ symmetric matrix-valued process
\betaegin{equation}\lambdaabel{submartingales}
\Gammaamma_t:=(\kappa_t(i,j))_{1\lambdae i, j\lambdae n}
\varepsilonnd{equation}
where
\betaegin{equation}\lambdaabel{kappa}
\kappa_t(i,i):=\kappa_t^{0,e_i;0}, \quaduad \kappa_t(i,j):={1\omegaver 4} [\kappa_t^{0,e_i+e_j;0}-\kappa_t^{0,e_i-e_j;0}], \quaduad 1\lambdae i\nuot=j\lambdae n.
\varepsilonnd{equation}
It is a $n\tauimes n$ matrix-valued semi-martingale and the bounded variational process in the Doob-Meyer decomposition is continuous in time.
Define
$$
\Phi_t:=(X_t^{0,e_1;0}, \cdotots, X_t^{0,e_n;0}), \quaduad t\in [0,T].
$$
Then, we have
\betaegin{equation}\lambdaabel{submartingales1}
\Gammaamma_t=\Phi_t'K_t\Phi_t+\int_0^t\Phi_r'Q_r\Phi_r\, dr, \quaduad t\in [0,T];
\varepsilonnd{equation}
and $\Phi$ satisfies the following matrix-valued stochastic differential equation (SDE):
\betaegin{equation}\lambdaabel{SDE}
d\Phi_t= A_t \Phi_t\, dt+C_t^i\Phi_t\, dW_t^i, \quaduad t\in (0,T]; \quaduad \Phi_0= I_n.
\varepsilonnd{equation}
It is well-known that $\Phi_t$ has an inverse $\Psi_t:=\Phi_t^{-1}$, satisfying the following SDE:
\betaegin{equation}\lambdaabel{SDE1}
d\Psi_t= \Psi_t (-A_t+C_t^iC_t^i)\, dt-\Psi_t C_t^i\, dW_t^i, \quaduad t\in (0,T]; \quaduad \Psi_0= I_n.
\varepsilonnd{equation}
Therefore, we have
\betaegin{equation}\lambdaabel{semimartingales1}
K_t= \Psi_t'\lambdaeft( \Gammaamma_t -\int_0^t\Phi_r'Q_r\Phi_r\, dr \right)\Psi_t, \quaduad t\in [0,T].
\varepsilonnd{equation}
Since $\Gammaamma$ is a semi-martingale, using It\^o-Wentzell formula, we see that $K$ is a semi-martingale of form~(\ref{representation}) from the Doob-Meyer decomposition, with the bounded variational process $k$ being continuous in time. It remains to derive the formula~(\ref{formula}) for $k$ and the estimate~(\ref{p-estimate}) for $L$.
{\betaf Step 2. Formula for the bounded variational process $k$. } Define the function:
\betaegin{equation}\lambdaabel{Hamiltonian}
\varepsilonq{ F(t, x,v; K, L)=& 2\lambdaangle Kx, A_t x+B_tv\mathop{\rangle}gle + 2\lambdaangle L^ix, C_t^ix+D_t^iv\mathop{\rangle}gle\cr
& +\lambdaangle L^i (C_t^ix+D_t^iv), C_t^ix+D_t^iv \mathop{\rangle}gle, \cr}
\varepsilonnd{equation}
for $(t,x,v, K,L)\in [0,T]\tauimes \muathbb{R}^n\tauimes \muathbb{R}^m\tauimes \muathbb{S}^n\tauimes (\muathbb{S}^n)^m$.
Using It\^o-Wentzell formula, we have
\betaegin{equation}\lambdaabel{submartingales2}\lambdaeft\{
\varepsilonq{ dV(t, X_t^{0,x;v})=&\betaiggl [-\lambdaangle d k_t X_t^{0,x;v}, X_t^{0,x;v}\mathop{\rangle}gle+F(t,X_t^{0,x;v}, v; K_t, L_t)\, dt\betaiggr]\cr
&+\betaiggl[\lambdaangle K_t (C_t^iX_t^{0,x;v}+D_t^iv), X_t^{0,x;v}\mathop{\rangle}gle \cr
&+ \lambdaangle K_t X_t^{0,x;v}, (C_t^iX_t^{0,x;v}+D_t^iv)\mathop{\rangle}gle\cr
&\quaduad\quaduad +\lambdaangle L_t^iX_t^{0,x;v}, X_t^{0,x;v}\mathop{\rangle}gle\betaiggr]\, dW_t^i,\quaduad t\in [0,T); \cr
V(T,X_T^{0,x;v})=&\lambdaangle M X_T^{0,x;v}, X_T^{0,x;v}\mathop{\rangle}gle. \cr}\right.
\varepsilonnd{equation}
and
\betaegin{equation}\lambdaabel{submartingales3}
\varepsilonq{ \kappa_t^{0,x;v}=&\lambdaangle K_0 x, x\mathop{\rangle}gle+ \int_0^t\betaiggl [-\lambdaangle d k_s X_s^{0,x;v}, X_s^{0,x;v}\mathop{\rangle}gle+F(s,X_s^{0,x;v}, v; K_s, L_s)\, ds\cr
&+l(s, X_s^{0,x;v}, v)\, ds\betaiggr]+\int_0^t\betaiggl[\lambdaangle K_s (C_s^iX_s^{0,x;v}+D_s^iv), X_s^{0,x;v}\mathop{\rangle}gle\cr
& + \lambdaangle K_s X_s^{0,x;v}, (C_s^iX_s^{0,x;v}+D_s^iv)\mathop{\rangle}gle +\lambdaangle L_s^iX_s^{0,x;v}, X_s^{0,x;v}\mathop{\rangle}gle\betaiggr]\, dW_s^i, \quaduad t\in [0,T].\cr}
\varepsilonnd{equation}
Assertion (ii) of Theorem~\ref{BP} states that $ \{\kappa_t^{0,x;v}, t\in [0,T]\}$ is a sub-martingale for any $(v,x)\in \muathbb{R}^m\tauimes \muathbb{R}^n$, yielding the following fact: for any $(x,v)\in \muathbb{R}^n\tauimes \muathbb{R}^m$, we have $E\int_0^T\varepsilonta (s,x) \gammaamma(ds,x; v)\lambdae 0$ for any essentially bounded nonnegative predictable process $\varepsilonta$ on $[0,T]\tauimes \Omegamega$, where
\betaegin{equation}\lambdaabel{submartingales4}
\varepsilonq{ \gammaamma(ds,x;v):=& -\lambdaangle d k_s X_s^{0,x;v}, X_s^{0,x;v}\mathop{\rangle}gle+F(s,X_s^{0,x;v}, v; K_s, L_s)\, ds\cr
& \quaduad \quaduad +l(s, X_s^{0,x;v}, v)\, ds; \cr}
\varepsilonnd{equation}
and for the optimal control $\omegaverline{u}\in \muathscr{U}_0$, the process $\kappa_t^{s,x;\omegaverline{u}}, t\in [s,T]$, is a martingale w.r.t. $\{{\muathscr F}_t\}$, yielding the following fact:
for any $x\in \muathbb{R}^n$, we have $E\int_0^T\varepsilonta (s,x) \gammaamma(ds,x;\omegaverline{u})= 0$ for any essentially bounded nonnegative predictable process $\varepsilonta$ on $[0,T]\tauimes \Omegamega$, where
\betaegin{equation}\lambdaabel{martingales4}
\varepsilonq{ \gammaamma(ds,x;\omegaverline{u}):=& -\lambdaangle d k_s X_s^{0,x;\omegaverline{u}}, X_s^{0,x;\omegaverline{u}}\mathop{\rangle}gle+F(s,X_s^{0,x;\omegaverline{u}}, \omegaverline{u}_s; K_s, L_s)\, ds\cr
& \quaduad \quaduad +l(s, X_s^{0,x;\omegaverline{u}}, \omegaverline{u}_s)\, ds. \cr}
\varepsilonnd{equation}
It is well-known that the stochastic flow $X_s^{0,x;v}, x\in \muathbb{R}^n$ has an inverse $Y_s^{0,x;v}, x\in \muathbb{R}^n$. Since (see Yong and Zhou~\cite[Theorem 6.14, page 47]{YongZhou})
\betaegin{equation}\lambdaabel{sde flow}
X_s^{0,x;v}=\Phi_t x+\Phi_t\int_0^t\Psi_s(B_sv-C_s^iD_s^iv)\, ds+\Phi_t\int_0^t\Psi_s D_s^i v\, dW_s^i
\varepsilonnd{equation}
for $t\in [0,T]$, we have
\betaegin{equation}\lambdaabel{inverse flow}
Y_s^{0,x;v}=\Psi_t x-\int_0^t\Psi_s(B_sv-C_s^iD_s^iv)\, ds-\int_0^t\Psi_s D_s^i v\, dW_s^i, \quaduad t\in [0,T].
\varepsilonnd{equation}
More generally, we define for any $u\in \muathscr{U}_0$ and $t\in [0,T]$,
\betaegin{equation}\lambdaabel{g-inverse flow}
Y_s^{0,x;u}=\Psi_t x-\int_0^t\Psi_s(B_su_s-C_s^iD_s^iu_s)\, ds-\int_0^t\Psi_s D_s^i u_s\, dW_s^i.
\varepsilonnd{equation}
We have
\betaegin{equation}
X_s^{0,y;u} \betaiggm|_{y=Y_s^{0,x;u}}=x, \quaduad \varphiorall x\in \muathbb{R}^n.
\varepsilonnd{equation}
Incorporating the composition of $\gammaamma(s,\cdotot;v)$ with the inverse flow $Y_s^{0,x;v}, x\in \muathbb{R}^n$, we have
\betaegin{equation}\lambdaabel{submartingales5}
\varepsilonq{ 0\lambdae &\ \gammaamma(ds,Y_s^{0,x;v};v)\cr
=& -\lambdaangle d k_s x, x\mathop{\rangle}gle+\lambdaeft[F(s,x, v; K_s, L_s) +l(s, x, v)\right]\, ds \cr}
\varepsilonnd{equation}
and in a similar way, we have for almost everywhere $(s,\omegamega) \in [0,T]\tauimes \Omegamega$,
\betaegin{equation}\lambdaabel{martingales5}
\varepsilonq{ 0=&\ \gammaamma(ds,Y_s^{0,x;\omegaverline{u}};\omegaverline{u})\cr
=& -\lambdaangle d k_s x, x\mathop{\rangle}gle+\lambdaeft[F(s,x, \omegaverline{u}_s; K_s, L_s) +l(s, x, \omegaverline{u}_s)\right]\, ds.\cr}
\varepsilonnd{equation}
Therefore, we have
\betaegin{equation}\lambdaabel{drift}
\lambdaangle d k_s x, x\mathop{\rangle}gle= \muin_{v\in \muathbb{R}^m} \lambdaeft[F(s,x,v; K_s, L_s) +l(s, x, v)\right]\, ds, \quaduad \varphiorall x\in \muathbb{R}^n,
\varepsilonnd{equation}
which implies formula~(\ref{formula}).
{\betaf Step 3. Estimate for $L$. }
From the theory of BSDEs, we have from BSDE~(\ref{submartingales2})
\betaegin{equation}\lambdaabel{L-estimate0}
\varepsilonq{ &\int_0^T\lambdaeft|\lambdaangle K_t X_t^{0,x;v}, (C_t^iX_t^{0,x;v}+D_t^iv)\mathop{\rangle}gle +\lambdaangle L_t^iX_t^{0,x;v}, X_t^{0,x;v}\mathop{\rangle}gle\right|^2\, dt\cr
= &\ |\lambdaangle M X_T^{0,x;v}, X_T^{0,x;v}\mathop{\rangle}gle|^2-|V(t, X_t^{0,x;v})|^2\cr
&+2\int_0^TV(t, X_t^{0,x;v})\betaiggl [\lambdaangle k_t X_t^{0,x;v}, X_t^{0,x;v}\mathop{\rangle}gle-F(t,X_t^{0,x;v}, v; K_t, L_t)\betaiggr]\, dt\cr
&-\int_0^TV(t, X_t^{0,x;v})\betaiggl[2\lambdaangle K_t (C_t^iX_t^{0,x;v}+D_t^iv), X_t^{0,x;v}\mathop{\rangle}gle-\lambdaangle L_t^iX_t^{0,x;v}, X_t^{0,x;v}\mathop{\rangle}gle\betaiggr] \, dW_t^i. \cr}
\varepsilonnd{equation}
Since $V(t, X_t^{0,x;v})\gammae 0$, taking $v=0$ and using the inequality~(\ref{submartingales4}), we have
\betaegin{equation}\lambdaabel{L-estimate2}
\varepsilonq{ &\int_0^T\lambdaeft|\lambdaangle K_t X_t^{0,x;0}, C_t^iX_t^{0,x;0}\mathop{\rangle}gle +\lambdaangle L_t^iX_t^{0,x;0}, X_t^{0,x;0}\mathop{\rangle}gle\right|^2\, dt\cr
\lambdae &\ |M| |X_T^{0,x;0}|^4+2\int_0^TV(t, X_t^{0,x;0})l(t,X_t^{0,x;0}, 0)\, dt\cr
&-\int_0^TV(t, X_t^{0,x;0})\lambdaeft[2\lambdaangle K_t C_t^iX_t^{0,x;0}, X_t^{0,x;0}\mathop{\rangle}gle-\lambdaangle L_t^iX_t^{0,x;0}, X_t^{0,x;0}\mathop{\rangle}gle\right] \, dW_t^i. \cr}
\varepsilonnd{equation}
Since $V(t, X_t^{0,x;0})=\lambdaangle K_t X_t^{0,x;0}, X_t^{0,x;0}\mathop{\rangle}gle$ and $K$ is uniformly bounded, there is a positive constant $\lambdaambda$ such that
\betaegin{equation}\lambdaabel{L-estimate3}
\varepsilonq{ &\int_0^T\lambdaeft|\lambdaangle L_t^iX_t^{0,x;0}, X_t^{0,x;0}\mathop{\rangle}gle\right|^2\, dt\cr
\lambdae &\ 2\int_0^T\lambdaeft|\lambdaangle K_t X_t^{0,x;0}, C_t^iX_t^{0,x;0}\mathop{\rangle}gle\right|^2\, dt+2|M| |X_T^{0,x;0}|^4\cr
& +4\int_0^TV(t, X_t^{0,x;0})l(t,X_t^{0,x;0}, 0)\, dt\cr
&-2\int_0^TV(t, X_t^{0,x;0})\lambdaeft[2\lambdaangle K_t C_t^iX_t^{0,x;0}, X_t^{0,x;0}\mathop{\rangle}gle-\lambdaangle L_t^iX_t^{0,x;0}, X_t^{0,x;0}\mathop{\rangle}gle\right] \, dW_t^i\cr
\lambdae & \ \lambdaambda \muax_{t\in [0,T]}| X_t^{0,x;0}|^4-4\int_0^TV(t, X_t^{0,x;0})\lambdaangle K_t C_t^iX_t^{0,x;0}, X_t^{0,x;0}\mathop{\rangle}gle\, dW_t^i \cr
&+2\int_0^TV(t, X_t^{0,x;0})\lambdaangle L_t^iX_t^{0,x;0}, X_t^{0,x;0}\mathop{\rangle}gle \, dW_t^i. \cr}
\varepsilonnd{equation}
Therefore, for $p\gammae 1$, we have
\betaegin{equation}\lambdaabel{L-estimate4}
\varepsilonq{ &E\lambdaeft(\int_0^T\lambdaeft|\lambdaangle L_t^iX_t^{0,x;0}, X_t^{0,x;0}\mathop{\rangle}gle\right|^2\, dt\right)^p\cr
\lambdae & \ \lambdaambda_p E\lambdaeft[\muax_{t\in [0,T]}| X_t^{0,x;0}|^{4p}\right]\cr
&+\lambdaambda_p E \lambdaeft|\int_0^TV(t, X_t^{0,x;0})\lambdaangle K_t C_t^iX_t^{0,x;0}, X_t^{0,x;0}\mathop{\rangle}gle\, dW_t^i\right|^p\cr
&+\lambdaambda_p E\lambdaeft|\int_0^TV(t, X_t^{0,x;0})\lambdaangle L_t^iX_t^{0,x;0}, X_t^{0,x;0}\mathop{\rangle}gle \, dW_t^i\right|^p \cr
\lambdae & \ \lambdaambda_p E\lambdaeft[\muax_{t\in [0,T]}| X_t^{0,x;0}|^{4p}\right]\cr
&+\lambdaambda_p E \lambdaeft[\int_0^T\lambdaeft|V(t, X_t^{0,x;0})\lambdaangle K_t C_t^iX_t^{0,x;0}, X_t^{0,x;0}\mathop{\rangle}gle\right|^2\, dt\right]^{p/2}\cr
&+\lambdaambda_p E\lambdaeft[\int_0^T\lambdaeft|V(t, X_t^{0,x;0})\lambdaangle L_t^iX_t^{0,x;0}, X_t^{0,x;0}\mathop{\rangle}gle \right|^2\, dt\right]^{p/2} \cr
\lambdae & \ \lambdaambda_p E\lambdaeft[\muax_{t\in [0,T]}| X_t^{0,x;0}|^{4p}\right]+\lambdaambda_p E\lambdaeft[\int_0^T|\lambdaangle L_t^iX_t^{0,x;0}, X_t^{0,x;0}\mathop{\rangle}gle|^2|X_t^{0,x;0}|^4\, dt\right]^{p/2} \cr
\lambdae & \ \lambdaambda_p E\lambdaeft[\muax_{t\in [0,T]}| X_t^{0,x;0}|^{4p}\right]\cr
&\quaduad\quaduad+\lambdaambda_p E\lambdaeft[\lambdaeft(\int_0^T|\lambdaangle L_t^iX_t^{0,x;0}, X_t^{0,x;0}\mathop{\rangle}gle|^2\, dt\right)^{p/2}\muax_{t\in [0,T]} |X_t^{0,x;0}|^{2p}\right]\cr
\lambdae & \ \lambdaambda_p E\lambdaeft[\muax_{t\in [0,T]}| X_t^{0,x;0}|^{4p}\right]+{1\omegaver2} E\lambdaeft[\lambdaeft(\int_0^T|\lambdaangle L_t^iX_t^{0,x;0}, X_t^{0,x;0}\mathop{\rangle}gle|^2\, dt\right)^p\right]. \cr}
\varepsilonnd{equation}
Consequently, we have for any $x\in \muathbb{R}^n$,
\betaegin{equation}\lambdaabel{L-estimate5}
E\lambdaeft(\int_0^T\lambdaeft|\lambdaangle L_t^iX_t^{0,x;0}, X_t^{0,x;0}\mathop{\rangle}gle\right|^2\, dt\right)^p \lambdae \ 2\lambdaambda_p E\lambdaeft[\muax_{t\in [0,T]}| X_t^{0,x;0}|^{4p}\right]\lambdae \lambdaambda_p' |x|^{4p},
\varepsilonnd{equation}
which implies the following inequality
\betaegin{equation}\lambdaabel{L-estimate6}
E\lambdaeft(\int_0^T\lambdaeft|\Phi_t' L_t^i\Phi_t\right|^2\, dt\right)^p \lambdae \lambdaambda_p.
\varepsilonnd{equation}
Hence,
\betaegin{equation}\lambdaabel{L-estimate7}
\varepsilonq{ & E\lambdaeft(\int_0^T\lambdaeft|L_t^i\right|^2\, dt\right)^p
\lambdae \ E\lambdaeft(\int_0^T\lambdaeft|\Psi_t'\Phi_t' L_t^i\Phi_t\Psi_t\right|^2\, dt\right)^p \cr
\lambdae & \ E\lambdaeft(\int_0^T|\Psi_t'|^2|\Psi_t|^2 \lambdaeft|\Phi_t' L_t^i\Phi_t\right|^2\, dt\right)^p \cr
\lambdae & \ E\lambdaeft[\lambdaeft(\int_0^T \lambdaeft|\Phi_t' L_t^i\Phi_t\right|^2\, dt\right)^p \muax_{t\in [0,T]}|\Psi_t|^{4p}\right]\cr
\lambdae & \ \lambdaeft\{E\lambdaeft[\lambdaeft(\int_0^T \lambdaeft|\Phi_t' L_t^i\Phi_t\right|^2\, dt\right)^{2p}\right]E\lambdaeft[\muax_{t\in [0,T]}|\Psi_t|^{8p}\right]\right\}^{1/2}\lambdae \
\lambdaambda_p. \cr}
\varepsilonnd{equation}
The proof is complete.
\varepsilonnd{proof}
\betaegin{remark} We have shown in Steps 1 and 2 that $(K,L)$ solves BSRE~(\ref{bsre}) with $K$ being nonnegative and uniformly bounded. Then from Tang~\cite[Theorem 5.1, page 62]{Tang2003}, we have the desired estimate. Here we have given a different proof to the estimate~(\ref{p-estimate}).
\varepsilonnd{remark}
Immediately, we have the following existence of adapted solution to BSRE~(\ref{bsre}).
\betaegin{corollary} (Existence result for BSRE). Let assumptions (A1) and (A2) be satisfied. Then $(K,L)$ is an adapted solution to BSDE~(\ref{bsre}).
\varepsilonnd{corollary}
\section{Verification theorem and uniqueness result for BSRE}
In the theory of linear quadratic optimal stochastic control, the Riccati equation as a nonlinear system of backward (stochastic) differential equations is an equivalent form of the underlying Bellman equation as a nonlinear backward (stochastic) partial differential equations, and both the optimal control and the value function are expected to be given in terms of the solution to the Riccati equation. The following verification theorem illustrates such a philosophy, which, however, has more or less been addressed in the author's work~\cite[Theorem 3.2, page 60]{Tang2003}.
\betaegin{theorem} (Verification Theorem). Let assumptions (A1) and (A2) be satisfied. Let $(K,L)$ be an adapted solution to BSDE~(\ref{bsre}) such that $K$ is essentially bounded and nonnegative (and consequently $L$ satisfies estimate~(\ref{p-estimate}) in view of Tang~\cite[Theorem 5.1, page 62]{Tang2003}). Then, (i) the following linear SDE
\betaegin{equation}\lambdaabel{closed system}
\lambdaeft\{\varepsilonq{ d\omegaverline{X}_t=&\lambdaeft[A_t-B_t\muathscr{N}_t^{-1}(K_t)\muathscr{M}_t'(K_t, L_t)\right]\omegaverline{X}_t\, dt\cr
&+ \sum_{i=1}^d \lambdaeft[C_t^i-D_t^i\muathscr{N}_t^{-1}(K_t)\muathscr{M}_t'(K_t, L_t)\right]\omegaverline{X}_t\, dW_t^i, \quaduad t\in [0,T]; \cr
\omegaverline{X}_0=&x \cr}\right.
\varepsilonnd{equation}
has a unique strong solution $\omegaverline{X}$ such that
\betaegin{equation}\lambdaabel{square integrable}
E\lambdaeft[\muax_{t\in [0,T]}|\omegaverline{X}_t|^2\right]< \infty;
\varepsilonnd{equation}
(ii) the following given process
\betaegin{equation}\lambdaabel{control}
\omegaverline{u}_t=-\muathscr{N}_t^{-1}(K_t)\muathscr{M}_t'(K_t, L_t)\omegaverline{X}_t, \quaduad t\in [0,T],
\varepsilonnd{equation}
belongs to $\muathscr{L}^2_{\muathscr{F}}(0,T;\muathbb{R}^m)$, and is the optimal control for the SLQ;
and (iii) the value field $V$ is given by
\betaegin{equation}\lambdaabel{value vield}
V(t,x)=\lambdaangle K_tx, x\mathop{\rangle}gle, \quaduad (t,x) \in [0,T]\tauimes \muathbb{R}^n.
\varepsilonnd{equation}
\varepsilonnd{theorem}
\betaegin{remark} A proof using the stochastic maximum principle (the so-called stochastic Hamilton system) is given in Tang~\cite[Section 3, pages 58--60]{Tang2003}. The main difficulty of the proof comes from the appearance of $L$ in the optimal feedback law~(\ref{control}) since $L$ is in general not expected to be essentially bounded. Since the coefficients of the optimal closed system~(\ref{closed system}) contain $L$, we could directly have neither the integrability~(\ref{square integrable}) nor the square integrability of $\omegaverline{u}$, which prevent us from going through the conventional method of ``completion of squares" in a straightforward way. In what follows, we get around the difficulty via the technique of localization by stopping times, and develop a localized version of the conventional method of ``completion of squares", which give a different self-contained proof.
\varepsilonnd{remark}
\betaegin{proof} Since the coefficients of the optimal closed system~(\ref{closed system}) is square integrable on $[0,T]$ almost surely, SDE~(\ref{closed system}) has a unique strong solution ${\omegaverline X}$ (see Gal'chuk~\cite{Galchuk1978}). Define for sufficiently large integer $j$, the stopping time $\tauau_j$ as follows:
\betaegin{equation}\lambdaabel{localization}
\tauau_j:=T\wedge \muin \{t\gammae 0: \ |{\omegaverline X}_t|\gammae j\},
\varepsilonnd{equation}
with the convention that $\muin \varepsilonmptyset=\infty$. It is obvious that $\tauau_j\uparrow T$ almost surely as $j\uparrow \infty$.
Then, we have
\betaegin{equation}\lambdaabel{identity}
\lambdaangle K_0x,x \mathop{\rangle}gle= E\lambdaangle K_{\tauau_j}{\omegaverline X}_{\tauau_j}, {\omegaverline X}_{\tauau_j}\mathop{\rangle}gle +E\int_0^{\tauau_j}l(t,{\omegaverline X}_t, \omegaverline{u}_t )\, dt,
\varepsilonnd{equation}
which together with assumption (A2) implies the following (with the constant $\delta>0$)
\betaegin{equation}
E\int_0^{\tauau_j}|{\omegaverline u}_t|^2 \, dt \lambdae\ \delta^{-1} E\int_0^{\tauau_j}\lambdaangle N_t {\omegaverline u}_t, \omegaverline{u}_t\mathop{\rangle}gle \, dt\lambdae \ \delta^{-1}\lambdaangle K_0x,x \mathop{\rangle}gle.
\varepsilonnd{equation}
Using Fatou's lemma, we have $\omegaverline{u}\in \muathscr{L}^2_{\muathscr{F}}(0,T;\muathbb{R}^m)$. Since ${\omegaverline X}=X^{0,x;\omegaverline{u}}$, we have from estimate~(\ref{state-estimate}) the integrability~(\ref{square integrable}). Assertion (i) has been proved.
From Assertion (i), we see that
$$
0\lambdae \lambdaangle K_{\tauau_j}{\omegaverline X}_{\tauau_j}, {\omegaverline X}_{\tauau_j}\mathop{\rangle}gle \lambdae \lambdaambda\muax_{t\in [0,T]} |{\omegaverline X}_t|^2 \in L^1(\Omegamega, \muathscr{F}_T, P)
$$
and
$$
0\lambdae \int_0^{\tauau_j}l(t,{\omegaverline X}_t, \omegaverline{u}_t )\, dt \ \lambdae \ (\widehatboxox{ \rm and } \betaigm\uparrow ) \ \int_0^T l(t,{\omegaverline X}_t, \omegaverline{u}_t )\, dt \in L^1(\Omegamega, \muathscr{F}_T, P).
$$
Using Lebesgue's dominant convergence theorem, we have
\betaegin{equation}\lambdaabel{dominant convergence}
\varepsilonq{\lambdaim_{j\tauo \infty}E\lambdaangle K_{\tauau_j}{\omegaverline X}_{\tauau_j}, {\omegaverline X}_{\tauau_j}\mathop{\rangle}gle=& E\lambdaangle K_T{\omegaverline X}_T, {\omegaverline X}_T\mathop{\rangle}gle, \cr
\lambdaim_{j\tauo \infty} E\int_0^{\tauau_j}L(t,{\omegaverline X}_t, \omegaverline{u}_t )\, dt=& E\int_0^T l(t,{\omegaverline X}_t, \omegaverline{u}_t )\, dt.\cr}
\varepsilonnd{equation}
In view of the equality~(\ref{identity}), we have
\betaegin{equation}\lambdaabel{identity at limit}
\lambdaangle K_0x,x \mathop{\rangle}gle=E\lambdaangle K_T{\omegaverline X}_T, {\omegaverline X}_T\mathop{\rangle}gle+E\int_0^T l(t,{\omegaverline X}_t, \omegaverline{u}_t )\, dt=J(\omegaverline{u};0,x).
\varepsilonnd{equation}
It remains to prove that for any $u\in \muathscr{L}^2_{\muathscr{F}}(0,T;\muathbb{R}^m)$, we have $J(u;0,x)\gammae \lambdaangle K_0x,x \mathop{\rangle}gle$.
For given $u\in \muathscr{L}^2_{\muathscr{F}}(0,T;\muathbb{R}^m)$ and sufficiently large integer $j$, define the stopping time $\tauau_j^u$ as follows:
\betaegin{equation}\lambdaabel{localization*}
\tauau_j^u:=T\wedge \muin \{t\gammae 0: \ |X_t^u|\gammae j\},
\varepsilonnd{equation}
with the notation $X^u:=X^{0,x;u}$. It is obvious that $\tauau_j^u\uparrow T$ almost surely as $j\uparrow \infty$.
Define
\betaegin{equation}\lambdaabel{control*}
\widetilde{u}_t:=-\muathscr{N}_t^{-1}(K_t)\muathscr{M}_t'(K_t, L_t)X^u_t, \quaduad t\in [0,T].
\varepsilonnd{equation}
Then, the restriction of $\widetilde{u}$ to the random time interval $[0, \tauau_j^u]$ lies in $\muathscr{L}^2_{\muathscr{F}}(0, \tauau_j^u;\muathbb{R}^m)$ for any $j$. Using BSRE~(\ref{bsre}) to complete the square in a straightforward manner, we have
\betaegin{equation}\lambdaabel{identity*}
\varepsilonq{ & E\lambdaangle K_{\tauau_j^u}X^u_{\tauau_j^u}, X^u_{\tauau_j^u}\mathop{\rangle}gle +E\int_0^{\tauau_j^u}l(t, X^u_t, u_t )\, dt\cr
&\quaduad\quaduad\quaduad\quaduad= \ \lambdaangle K_0x,x \mathop{\rangle}gle+E\int_0^{\tauau_j^u}\lambdaangle \muathscr{N}_t^{-1}(K_t) (u_t- \widetilde{u}_t), u_t- \widetilde{u}_t\mathop{\rangle}gle\, dt. \cr}
\varepsilonnd{equation}
Therefore, we have
\betaegin{equation}\lambdaabel{inequality*}
E\lambdaangle K_{\tauau_j^u}X^u_{\tauau_j^u}, X^u_{\tauau_j^u}\mathop{\rangle}gle +E\int_0^{\tauau_j^u}l(t, X^u_t, u_t )\, dt\gammae \lambdaangle K_0x,x \mathop{\rangle}gle.
\varepsilonnd{equation}
In view of estimate~(\ref{state-estimate}) in Lemma~\ref{sde estimate}, we see that
$$
0\lambdae \lambdaangle K_{\tauau_j^u}X^u_{\tauau_j^u}, X^u_{\tauau_j^u}\mathop{\rangle}gle\lambdae \lambdaambda\muax_{t\in [0,T]} |X_t^u|^2\in L^1(\Omegamega, \muathscr{F}_T, P)
$$
and
$$0\lambdae \int_0^{\tauau_j^u}l(t, X^u_t, u_t )\, dt \ \lambdae \ (\widehatboxox{ \rm and } \betaigm\uparrow ) \ \int_0^T l(t, X^u_t, u_t )\, dt\in L^1(\Omegamega, \muathscr{F}_T, P).
$$
Passage to the limit in inequality~(\ref{inequality*}), again using Lebesgue's dominant convergence theorem, we have
\betaegin{equation}
J(u; s,x)= E\lambdaangle K_TX^u_T, X^u_T\mathop{\rangle}gle +E\int_0^Tl(t, X^u_t, u_t )\, dt\gammae \lambdaangle K_0x,x \mathop{\rangle}gle.
\varepsilonnd{equation}
The proof is then complete.
\varepsilonnd{proof}
Immediately, we have the following uniqueness of adapted solution to BSRE~(\ref{bsre}).
\betaegin{corollary} (Uniqueness result for BSRE). Let assumptions (A1) and (A2) be satisfied. Let $(\widetilde{K}, \widetilde{L})$ be an adapted solution to BSDE~(\ref{bsre}) such that ${\widetilde K}$ is essentially bounded and nonnegative and $\widetilde{L}$ satisfies estimate~(\ref{p-estimate}). Then, ${\widetilde K}=K$ and $\widetilde{L}=L$.
\varepsilonnd{corollary}
The corollary and its proof can be found in Tang~\cite[the beginning paragraph of Section 8, page 70]{Tang2003}.
\section{Comments and possible extensions}
The results of this paper can be adapted to the singular case ($N$ is allowed to be
only nonnegative) but with suitable additional conditions such as the following:
(A3) Assume that the matrix process $\sum_{i=1}^d(D^i)'D^i$
and the terminal state weighting random matrix $M$ are uniformly positive.
This subject will be detailed elsewhere.
The singular case has received much recent interests because of its appearance in
financial mean-variance problems. More generally, $N$ can also be possibly
negative---this is the so-called indefinite case. On these
features, the interested reader is referred to Chen and Yong
\cite{new5}, Hu and Zhou~\cite{HuZhou2003},
Kohlmann and Tang \cite{8, 11}, Yong and Zhou \cite{YongZhou}, and the references therein.
Finally, the main results of the paper can also be adapted to the quadratic optimal control problem for linear
stochastic differential system driven by jump-diffusion processes under suitable assumptions. The details will be presented elsewhere.
Consider a general non-Markovian nonlinear optimal stochastic control problem. Let $A$ be a separable metric space,
and ${\muathscr U}_s$ be the set of $A$-valued predictable processes on $[s,T]$.
For any triplet $(u, s, \xi)\in {\muathscr U}_s\tauimes [0,T]\tauimes L^2(\Omegamega, \muathscr{F}_s, P; \muathbb{R}^n)$, consider the following SDE:
$$
X_t=\xi+\int_s^t\sigmagma(r,X_r, u_r)\, dW_r+\int_s^tb(r,X_r,u_r)\, dr, \quaduad t\in [s,T].
$$
Assume that the following functions
$$\betaegin{array}{c}
\sigmagma(t,x,\alphalpha)\in \muathbb{R}^{n\tauimes d},\quaduad b(t,x,\alphalpha)\in \muathbb{R}^n,\\
l(t,x, \alpha)\in \muathbb{R}, \quaduad g(x)\in \muathbb{R}; \quadquad (t,x,\alphalpha)\in [0,T]\tauimes \muathbb{R}^n\tauimes A
\varepsilonnd{array}$$ are continuous in $(x, \alphalpha)$ and continuous in $x$ uniformly over $\alphalpha$ for each $(t, \omegamega)$. Also, assume thatthere is positive constant $\lambda$ such that
$$\betaegin{array}{rcl}
\|\sigmagma(t,x,\alphalpha)-\sigmagma(t,y,\alphalpha)\|+|b(t,x,\alphalpha)-(t,y,\alphalpha)|&\lambdae& \lambda |x-y|,\\
\|\sigmagma(t,x,\alphalpha)\|+|b(t,x,\alphalpha)|&\lambdae& \lambda (1+|x|),\\
|l(t,x, \alpha)|+|g(x)|&\lambdae& \lambda (1+|x|)^m.
\varepsilonnd{array}$$
For $(s, \xi, u) \in [s,T]\tauimes L^2(\Omegamega, \muathscr{F}_s, P; \muathbb{R}^n)\tauimes {\muathscr U}_s,$ define
$$\betaegin{array}{rcl}
J(u;s,\xi)&=&\deltaisplaystyle E^{s,\xi;u}\lambdaeft[\int_s^Tl(t,X_t, u_t)\, dt+g(X_T)\betaiggm | {\muathscr F}_s \right],\\[4mm]
V(s,\xi)&:=&\deltaisplaystyle\widehatboxox{\rm ess.}
\inf_{u\in {\muathscr U}_s} J(u; s, \xi).
\varepsilonnd{array}$$
Denote by $\muathbb{V}(s,\cdotot)$ the restriction of $V(s,\cdotot)$ to $\muathbb{R}^n$. In the nonlinear context, the restricted value field $\muathbb{V}$ can be proved to satisfy the stochastic dynamic programming principle:
(i) For $s\lambdae t\lambdae T$ and $\xi\in L^2(\Omegamega, \muathscr{F}_s, P; \muathbb{R}^n)$,
$$
\muathbb{V}(s,\xi)=\mubox{\rm ess.}\inf_{u\in {\muathscr U}_s} E^{s,\xi; u} \lambdaeft \{\int_s^t l(r, X_r, u_r)\, dr+ \muathbb{V}(t, X_t) \betaiggm | {\muathscr F}_s\right\}.
$$
(ii) For $(s,x,u)\in [0,T]\tauimes R^n\tauimes {\muathscr U}_s$, the process
$$\betaegin{array}{rcl}
\kappa_t^{s,x;u}&:=&\deltaisplaystyle \muathbb{V}(t,X_t^{s,x;u}) +\int_s^tl(r, X_r^{s,x;u}, u_r)\, dr
\varepsilonnd{array}
$$
defined for $t\in [s,T]$, is a submartingale w.r.t. $\{{\muathscr F}_t\}$.
Using the above dynamic programming principle and Kunita's stochastic calculus~\cite{Kunita1994}, we can still show that $\muathbb{V}$ is a Sobolev space valued semi-martingale and satisfy the associated backward Bellman equation in the strong sense. All the details shall be given in our forthcoming paper to extend Krylov~\cite{Krylov1972} to the non-Markovian framework for optimal stochastic control problem.
\section*{Acknowledgment} The main results and the methodology of the paper has been announced in my plenary talk at the 7th international symposium on backward stochastic differential equations (June 22-27, 2014), Weihai, Shandong Provence, China. The author would thank the organizers for kind hospitality.
\rm
\betaegin{thebibliography}{99}
\betaibitem{Bismut1976}
{\sc J.-M. Bismut}, {\it Linear quadratic optimal stochastic control
with random coefficients}, SIAM J. Control Optim., 14 (1976), pp.\ 419--444.
\betaibitem{Bismut1978}
{\sc J. M. Bismut}, {\it Controle des systems lineares quadratiques:
applications de l'integrale stochastique}, in S\'eminaire de Probabilit\'es XII,
Lecture Notes in Math.\ 649,
C. Dellacherie, P. A. Meyer, and M. Weil, eds., Springer-Verlag, Berlin, 1978,
pp.\ 180--264.
\betaibitem{new5}
{\sc S. Chen and J. Yong}, \tauextit{Stochastic linear quadratic optimal
control problems}, Appl.\ Math.\ Optim., 43 (2001), pp.\ 21--45.
\betaibitem{Faurre1968} {\sc P. Faurre},
\tauextit{Sur les points conjugu¨¦s en commande optimale (French)},
C. R. Acad. Sci. Paris S¨¦r. A-B, 266(1968), pp. A1294--A1296.
\betaibitem{Galchuk1978}
{\sc L. I. Gal'chuk}, {\it Existence and uniqueness of a solution for stochastic
equations with respect to semimartingales}, Theory Probab.\ Appl.,
23 (1978), pp.\ 751--763.
\betaibitem{HuZhou2003} {\sc Y. Hu and X. Zhou}, {\it Indefinite stochastic Riccati equations},
SIAM J. Control Optim., 42 (2003), pp. 123--137.
\betaibitem{8}
{\sc M. Kohlmann and S. Tang},
{\it Minimization of risk and LQ theory},
SIAM J. Control Optim., 42 (2003), pp. 1118--1142.
\betaibitem{9}
{\sc M. Kohlmann and S. Tang}, {\it Global adapted solution of one-dimensional
backward stochastic Riccati equations, with application to the mean-variance hedging},
Stochastic Process.\ Appl., 97 (2002), pp.\ 255--288.
\betaibitem{10}
{\sc M. Kohlmann and S. Tang}, {\it Multidimensional backward stochastic
Riccati equations and applications}, SIAM J. Control Optim., 41 (2003), pp. 1696--1721.
\betaibitem{11}
{\sc M. Kohlmann and S. Tang}, {\it New developments in backward stochastic
Riccati equations and their applications}, in
Mathematical Finance (Konstanz, 2000), M. Kohlmann and S. Tang, eds., Birkh\"auser, Basel, 2001,
pp.\ 194--214.
\betaibitem{Krylov1972} {\sc N. V. Krylov}, {\it Control of a solution of a stochastic integral equation},
Theory of Probability and Its Applications, 17 (1972), pp. 114--131.
\betaibitem{Krylov1977} {\sc N. V. Krylov}, {\sl Controlled Diffusion Processes}, Springer, New York, 1980.
\betaibitem{Kunita1994} {\sc H. Kunita},
{\it Stochastic flows acting on Schwartz distributions}, J. Theor. Probab., 7(1994), pp. 247--278.
\betaibitem{15}
{\sc E. Pardoux and S. Peng}, {\it Adapted solution of backward
stochastic equation}, Systems Control Lett., 14 (1990), pp.\ 55--61.
\betaibitem{Peng1992}
{\sc S. Peng}, {\it Stochastic Hamilton--Jacobi--Bellman equations},
SIAM J. Control Optim., 30 (1992), pp.\ 284--304.
\betaibitem{Peng1997}
{\sc S. Peng}, {\it Backward stochastic differential equations---theory of stochastic optimazation and viscosity solution of HJB equations}, in: Jia-an Yan, Shige Peng, Shizan Fang, Liming Wu, Selected Lectures on Stochastic Analysis (Chinese), Science press, Beijing, 1997, pp. 85--138.
\betaibitem{Peng1999}
{\sc S. Peng}, {\it Open problems on backward stochastic differential equations},
in Control of Distributed Parameter and Stochastic Systems (Hangzhou, 1998),
S. Chen, et al., eds.,
Kluwer Academic Publishers, Boston, 1999, pp.\ 265--273.
\betaibitem{Peng2010}
{\sc S. Peng}, {\it Backward stochastic differential equation, nonlinear expectation and their applications}, in: Proceedings of the International Congress of Mathematicians, Volume I, pp. 393--432, Hindustan Book Agency, New Delhi, 2010.
\betaibitem{Protter2005}
{\sc Philip E. Protter}, {\it Stochastic Integration and Differential Equations} (second edition, Version 2.1), Springer-Verlag, Berlin, Heidelberg, 2005
\betaibitem{Tang2003}
{\sc S. Tang}, {\it General linear quadratic optimal stochastic control problems with random coefficients: linear stochastic Hamilton systems and backward stochastic Riccati equations},
SIAM J. Control Optim., 42 (2003), pp.\ 53--75.
\betaibitem{YongZhou}
{\sc J. Yong and X. Zhou}, {\it Stochastic Controls: Hamiltonian Systems and
HJB Equations}, Springer-Verlag, Berlin, New York, 1999.
\varepsilonnd{thebibliography}
\varepsilonnd{document} | math |
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शाओमी ने बनाया रीकोर्ड, लोन्च से लेकर ४५ दिनो में बिके रेडमी नोट ४ के १ मिलियन युनिट उपकोमिंग.को.इन
होम मोबाइल अंड्रॉयड शाओमी ने बनाया रीकोर्ड, लोन्च से लेकर ४५ दिनो में बिके रेडमी...
शाओमी ने बनाया रीकोर्ड, लोन्च से लेकर ४५ दिनो में बिके रेडमी नोट ४ के १ मिलियन युनिट
चीन की जानी मानी स्मार्टफोन निर्माता कंपनी शाओमी आज कल भारत में एक उभरती हुई कंपनी बन चुकी है। कंपनी के कुछ स्मार्टफोन भारत में काफी धोंस जमां चुके है। बात करें इस साल कंपनी ने लोन्च कीए अपने स्मार्टफोन के बारे में, तो कंपनी ने इस साल अपने रेडमी नोट ४ स्मार्टफोन को लोन्च किया है जो की भारत में काफी लोकप्रिय बनता जा रहा है। हाल ही में इस स्मार्टफोन ने एक रीकोर्ड तक कायम कर लिया है।
खबरों के मुताबिक कंपनी ने जनवरी महीने में अपने स्मार्टफोन रेडमी नोट ४ को लोन्च किया था। जो की एक बजट स्मार्टफोन था। लेकीन अपने लूक्स और फिचर की वजह से यह स्मार्टफोन काफी चर्चा का विषय बना रहा। इसके चलते यह स्मार्टफोन अपनी पहली बिक्री में सिर्फ कुछ ही मिनिट में सोल्ड आउट हो गया।
हाल ही में कंपनी के एक प्रवक्ता ने इस बारे में बताते हुए कहां की रेडमी नोट ४ स्मार्टफोन भारत में काफी ज्यादा मात्रा में बिक चुका है। लोन्च से लेकर सिर्फ ४5 दिनो में इस स्मार्टफोन के १ मिलियन युनिट बिक चुके है। जिसके मुताबिक मानो के हर चार सेकंड में स्मार्टफोन की एक युनिट बिकती है। भारत में इतने कम समय में इतनी ज्यादा युनिट बेचने वाला यह आज तक का पहला स्मार्टफोन बन चुका है। साथ ही कंपनी ने इस स्मार्टफोन का मैटे ब्लैक कलर वेरीएंट भी लोन्च किया था जो की काफी क्लासिक और बोल्ड लग रहा था।
आपको बता दे की कंपनी ने इस स्मार्टफोन को तीन वेरीएंट में लोन्च किया था जो की रैम और स्टोरज के मामले में अलग अलग है। इस स्मार्टफोन की शरुआती कीमत ९९९९ रुपये रखी है। जो की एज बजट रेंज का स्मार्टफोन है। कुछ स्पेसिफिकेशन की बात करें तो इस स्मार्टफोन में ५.५ इंच की फुल एचडी डिस्प्ले दि गई है। वहीं स्नैपड्रैगन 62५ प्रोसेसर के साथ ही इस स्मार्टफोन में १३ मेगापिक्सल का रीयर कैमरा और ५ मेगापिक्सल का फ्रंट कैमरा दिया गया है।
पावर बैकअप के लिए इस स्मार्टफोन में ४१०० एमएएच की बैटरी दि गई है। साथ ही इस स्मार्टफो में कनेक्टिविटी के लिए ड्युल सिम सपोर्ट, वाईफाई, ब्लूटूथ, जीपीएस जैसे फीचर भी शामिल किए गए। इस स्मार्टफोन को ग्रे, गोल्ड और मैटे ब्लैक कलर वेरीएंट में लोन्च किया गया है, और इस स्मार्टफोन को एक्सक्लुजिन साईट इ-कोमर्स पर बिक्री के लिए उपलब्ध किया गया है।
प्रेवियस आर्टियलसोनी के प्रीमियम स्मार्टफोन एक्सपीरिया एक्सजेड की कीमत में हुई १०,००० की भारी कटोती
नेक्स्ट आर्टियलस्मार्टफोन में पहले से मोजुद रहता है यह मालवेयर, क्या आपके स्मार्टफोन में तो नही है? | hindi |
घुसपैठ के बाद नया दांव चल रहा है चीन, सीमाओं पर कर रहा है अब ऐसी हरकतें - न्यूज़ट्रेंड
होम/ब्रेकिंग न्यूज़/घुसपैठ के बाद नया दांव चल रहा है चीन, सीमाओं पर कर रहा है अब ऐसी हरकतें
घुसपैठ के बाद नया दांव चल रहा है चीन, सीमाओं पर कर रहा है अब ऐसी हरकतें
चीन अब सीमा पर अपने देश से जुड़े प्रतीक बना रहा है। चीन ने फिंगर ४ और फिंगर ५ के बीच एक बहुत बड़ा मैंडेरिन प्रतीक (मंदरीन सिमबोल) और चीन का मानचित्र बनाया है। ये निशान इतना बड़ा है कि इसे सैटेलाइल इमेज में आसानी से देखा जा सकता है। इस प्रतीक की लंबाई लगभग ८१ मीटर और चौड़ाई 2५ मीटर के आसपास है।
आपको बता दें कि लद्दाख में पैंगॉन्ग लेक के जिन इलाकों को लेकर भारत और चीन के बीच विवाद चल रहा है। चीन ने उसी जगह पर ये प्रतीक और मानचित्र बनाया है। इतना ही नहीं इसी हफ्ते तिब्बत में मौजूद चीनी सेना के ओवरऑल कमांडर वांग हाईजांग की एक तस्वीर सामने आई थी। जिसमें ये भारत-चीन सीमा पर लिखे हुए चीन को पेंट कर रहे थे। यानी साफ है कि चीन अब सीमाओं पर अपने निशान बनाकर तनाव पैदा कर रहा है। क्योंकि फिंगर ४ और फिंगर ५ को लेकर ही इस वक्त भारत और चीन के विवाद चल रहा है और चीन फिंगर ४ और फिंगर ५ को अपना बताने के लिए यहां पर चीनी प्रतीक बना रहा है।
फिंगर ४ को लेकर है दोनों देशों में विवाद
दरअसल भारत फिंगर १ से फिंगर ८ तक अपना अधिकार मानता है। वहीं चीन फिंगर ८ से फिंगर ४ पर अपना अधिकार जताता है। इस समय फिंगर ४ दोनों देशों के बीच सीमा बनी हुई हैं और इसी जगह पर चीन ने हाल ही में चौंकी बना ली थी। जिसके बाद जवानों के बीच झड़पें हुई थीं। फिंगर ४ इलाके में काफी बड़ी संख्या में चीनी सेना मौजूद है और ये सैनिक भारतीय सेना को फिंगर ८ तक पेट्रोलिंग करने से रोकती है।
फिंगर ४ पर चल रही हैं गतिविधियां
चीन की और से फिंगर ४ पर कई सारी गतिविधियां चल रही है और चीन अपने सैनिकों की तादाद भी बढ़ाने में लगा हुआ है। वहीं तनाव को कम करने के लिए मंगलवार को लद्दाख के चुशुल में लेफ्टिनेंट जनरल स्तर पर दोनों देशों के बीच बातचीत हुई थी। ये बातचीत करीब १२ घंटे तक चली थी और रात को ११:०० बजे खत्म हुई थी। इस बैठक में भारत ने चीन से २२ जून को हुए समझौते का पालन करने और अप्रैल २०२० की यथास्थिति को लाक पर कायम रखने की बात कही है।
हालांकि ये बैठक सफल रही है कि नहीं इसपर अभी कुछ कहा नहीं जा सकता है। लेकिन जिस तरह से चीन अब अपने देश के प्रतीक को विवाद वाली जगह पर बना रहा है। उससे साफ जाहिर है कि चीन विवाद को कम करने की जगह उसे बढ़ाने पर जोर दो रहा है।
गौरतलब है कि पूर्वी लद्दाख की गलवान घाटी में पिछले महीने १५ जून को चीनी सैनिकों के साथ हिंसक झड़प हुई थी। इस हिंसक झड़प में भारतीय सेना के एक कर्नल सहित २० सैनिक शहीद हो गए थे। ये झड़प चीन की और से शुरू की गई थी और इस झड़प में चीन ने रॉड और छड़ों से हमला किया था। हालांकि भारतीय सैनिकों ने इस झड़प में चीन को उसी की भाषा में जवाब दिया था और चीन के सैनिकों को बुरी तरह से पीटा था। इस हिंसक झड़प में चीन के ४० के करीब सैनिक मारे गए थे। लेकिन चीन इस झड़प में मारे गए अपने सैनिकों की संख्या छुपा रहा है।
वहीं इस हिंसक झड़प के बाद से कई बार दोनों देशों के बीच लेफ्टिनेंट जनरल स्तर पर बातचीत हुई है। लेकिन अभी तक सीमा पर स्थिति पहले जैसे ही बनीं हुई है।
रजनीकांत ने फिर किया का का समर्थन, बोले- केंद्र सरकार का के बारे में पूरी तरह से स्पष्ट है | hindi |
\begin{document}
\author{Abhishek \and
Marko~Boon \and
Rudesindo~N\'u\~nez-Queija
}
\institute{Abhishek $\&$ Rudesindo~N\'u\~nez~Queija \at
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Netherlands\\
\email{\{Abhishek, nunezqueija\}@uva.nl}
\and
Marko~Boon \at
Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands \\
\email{m.a.a.boon@tue.nl}
}
\date{\today}
\title{Heavy-traffic analysis of the $M^X/ ext{semi-Markov}
\begin{abstract}
In this paper we analyze a single server queue with batch arrivals and semi-Markovian service times. We also include the feature that the first service of each busy period might have a different distribution than subsequent service times. Our generating function based approach allows us to determine the heavy traffic limit of the scaled queue-length distribution.
It turns out that this distribution converges to an exponential distribution. Nonsurprisingly, the exceptional first service does not influence this limiting distribution. We identify a sufficient and necessary condition under which the dependence between successive service times disappears in the limit, which we illustrate in a numerical example.
\end{abstract}
\keywords{batch arrivals, $M^X/SM/1$ queue, correlated service times, queue length, heavy-traffic analysis.}
\section{Introduction}
In many systems, successive service times of customers are not independent. The service type of a customer may depend on the type and the service duration of the preceding customer. Queueing systems with correlated service durations arise in many applications: logistics, production/inventory systems, computer and telecommunication networks. The model considered in this paper, is specifically motivated by its application to a road traffic setting, in which a stream of vehicles on a minor road merges with, or crosses a stream on a main road at an unsignalized intersection. The queueing model of vehicles on the minor road is known to be a single server queue (possibly with batch arrivals) with semi-Markovian service times and a different service time distribution for vehicles that arrive when no queue is present (see \cite{AbhishekMMOR,AbhishekWaitingTimes}). The dependence between successive service times (i.e. the time to wait for a sufficiently large gap and crossing the road) is caused by either platoon forming on the major road, or by the fact that the remaining part of a gap between successive vehicles on the major road may be used by the following vehicle.
In this paper, we consider a single-server queue with batch arrivals and correlated service times. The correlations are modeled with different service types, which form a Markov chain that itself depends on the sequence of service lengths. In addition, the first customer in a busy period has a different service time distribution than regular customers served in the busy period, which was firstly introduced in the framework of the $M/G/1$ queueing model by Welch \cite{welch} and by Yeo \cite{yeo}.
Queues with correlated service times have been studied for many years \cite{QUESTA2017,cin_s,gaver,neuts66,neuts77a,neuts77b}. One of the first studies for Markov-modulated single-server queueing systems in heavy traffic (HT) was by Burman and Smith \cite{burman}, who study the mean delay and the mean number in queue in a single-server system in both light-traffic and heavy-traffic regimes, where customers arrive according to a nonhomogeneous Poisson process with rate equal to a function of the state of an independent Markov process. In their model, service times are independent and identically distributed. Later, G. Falin and A. Falin \cite{falin} suggest another approach to analyze the same queueing model, which is based on certain ‘semi-explicit’ formulas for the stationary distribution of the virtual waiting time and its mean value under heavy traffic. Dimitrov \cite{dimitrov} applies the same approach to a single-server queueing system with arrival rate and service time depending on the state of Markov chain at an arrival epoch, and shows that the distribution of the scaled stationary virtual waiting time is exponential under a HT scaling. Several other authors \cite{asmussen,thorsdottir} also study Markov-modulated $M/G/1$-type queueing systems in heavy traffic. However, we are not aware of any prior work analyzing the $M^X/\text{SM}/1$ queue with exceptional first service under a HT scaling.
The current model is a slight extension of that in \cite{QUESTA2017}. We allow the service duration of a customer arriving into an empty system to have a distribution that differs from the service-time distributions of other customers. For the stationary analysis of the model this requires minor adaptations of that in \cite{QUESTA2017}. In addition, we investigate the stationary distribution in the heavy-traffic regime.
The remainder of this paper is organized as follows. In Section \ref{description}, we present the description of the queueing model. In Section \ref{stationary queue length}, we first determine the stationary probability generating function of the queue length of the system at the departure time of a customer. Subsequently, we use that result to derive the generating functions of the stationary queue length at an arbitrary time, at batch arrival instants, and at customer arrival instants. Using these results, we obtain the heavy-traffic distribution of the scaled stationary queue length in Section \ref{heavy-traffc}. In Section \ref{numerical_results}, a numerical example is presented to demonstrate the impact of the correlated service times on the queue length distribution in the heavy-traffic regime.
\section{Model description}\label{description}
We consider a single-server queuing system. Customers arrive in batches at the system according to a Poisson process with rate $\lambda$. The arriving batch size is denoted by the random variable $B$, with generating function $B(z)$, for $|z|\leq 1$ (zero-sized batches are not allowed, i.e. $B\geq 1$). Customers are served individually, and the first customer in a busy period has a different service time distribution than regular customers served in the busy period. There are $N$ types of customers, which we number $1,2,\dots,N$. Denote by $J_n$ the type of the $n$th customer and $G^{(n)}$ its service time, $n=1,2,\dots$. The type of a customer is only determined at the moment its service begins. More specifically, the type of the $n$th customer depends on the type, and on the service duration of the $(n-1)$th customer, as well as on whether the queue is empty at the departure time of the $(n-1)$th customer. We introduce, for $i=1,2,\dots,N$,
\begin{align}
\tilde{G}_{ij}(s)&=\E[e^{-sG^{(n)}}1_{\{J_{n+1}=j\}}|J_n=i, X_{n-1}\geq 1],\label{G_{ij}(s)}\\
\tilde{G}^*_{ij}(s)&=\E[e^{-sG^{(n)}}1_{\{J_{n+1}=j\}}|J_n=i, X_{n-1}=0]\label{G*_{ij}(s)},
\end{align} where $X_{n-1}$ is the number of customers in the system immediately after the departure of the $(n-1)$th customer. \\
In particular, for $i,j=1,2,\dots,N$, we define
\begin{align}
P_{ij}=\tilde{G}_{ij}(0)=\P(J_{n+1}=j|J_n=i,X_{n-1}\geq 1),\label{p_ij}\\
P^*_{ij}=\tilde{G}^*_{ij}(0)=\P(J_{n+1}=j|J_n=i,X_{n-1}= 0).\label{p^*_ij}
\end{align}
In the literature, the service process considered in this paper is referred to as a semi-Markov (SM) process, and thus the queuing system is referred to as the $M^X/SM/1$. In fact, in the gap acceptance literature, the single server queue with exceptional first service is commonly referred to as the $M/G2/1$ queue, a term seemingly introduced by Daganzo \cite{daganzo1977}, which motivates us to denote this model (with batch arrivals and exceptional first service) as the $M^X/SM2/1$ queue.\\
We assume that $P=[P_{ij}]_{i,j\in\{1,2,\dots,N\}}$ is the transition probability matrix of an irreducible discrete time Markov chain, with stationary distribution $\pi=(\pi_1,\pi_2,\dots,\pi_N)$ such that
\begin{align}\label{rel: piandP}
\pi P=\pi.
\end{align}
For intuition we may think of $\pi$ as the conditional equilibrium distribution of $J_n$ in case the queue would never empty.
Using Cramer's rule with the normalizing equation $\sum_{i=1}^{N}\pi_i=1$, the solutions of the system of equations \eqref{rel: piandP} are given by
\begin{align}\label{pi_i}
\pi_i=\frac{d_i}{d},
\end{align}
where $d=\sum_{i=1}^Nd_i,$ and $d_i$ is the cofactor of the entry in the $i$-th row and the first column of the matrix $(I-P)$, which is given by
\begin{align}
d_1=&\begin{vmatrix}
1-P_{22} & -P_{23} & \dots & -P_{2N}\\
-P_{32} & 1-P_{33} & \dots & -P_{3N}\\
\vdots & \vdots & \ddots & \vdots \\
-P_{N2} & -P_{N3} & \dots & 1-P_{NN}
\end{vmatrix}, \label{d_1}\\
d_i=&(-1)^{i+1}\begin{vmatrix}
-P_{12} & -P_{13} & \dots & -P_{1N}\\
\vdots & \vdots & \ddots & \vdots \\
-P_{i-12} & -P_{i-13} & \dots & -P_{i-1N}\\
-P_{i+12} & -P_{i+13} & \dots & -P_{i+1N}\\
\vdots & \vdots & \ddots & \vdots \\
-P_{N2} & -P_{N3} & \dots & 1-P_{NN}
\end{vmatrix},\quad i=2,3,\dots,N-1, \label{d_i}\\
d_N=&(-1)^{N+1}\begin{vmatrix}
-P_{12} & -P_{13} & \dots & -P_{1N}\\
1-P_{22} & -P_{23} & \dots & -P_{2N}\\
\vdots & \vdots & \ddots & \vdots \\
-P_{N-12} & -P_{N-13} & \dots & -P_{N-1N}
\end{vmatrix}. \label{d_N}
\end{align}
In the next section, to study the queue length distribution at departure times of customers, we denote by $A_{n}$ the number of arrivals during the service time of the $n$th customer (counting the individual customers inside the batches). We introduce, for $i=1,2,\dots,N$, \begin{align}
A_i(z)&=\sum_{j=1}^{N}A_{ij}(z),\label{A_i(z)}\\
A^{*}_i(z)&=\sum_{j=1}^{N}A^{*}_{ij}(z),\label{A^*_i(z)}
\end{align} with
\begin{align}
A_{ij}(z)=\E[z^{A_{n}}1_{\{J_{n+1}=j\}}|J_n=i,X_{n-1}\geq 1], \label{A_ij(z)}\\
A^{*}_{ij}(z)=\E[z^{A_{n}}1_{\{J_{n+1}=j\}}|J_n=i,X_{n-1}=0]. \label{A_ijstar(z)}
\end{align}
Let us define
\begin{align}\label{rho}
\rho&=\sum_{i=1}^{N}\pi_i\alpha_i,
\end{align}
where
\begin{align}
\alpha_i&=\sum_{j=1}^N\alpha_{ij}, \label{alph_i}
\end{align} with
\begin{align}\label{alpha_ij}
\alpha_{ij}=\E[A_n 1_{\{J_{n+1}=j\}}|J_n=i,X_{n-1}\geq 1].
\end{align}
Intuitively, we can think of $\rho$ as being the expected number of arrivals during a service time if the process $(J_n,X_{n-1})$ would never hit the level $X_{n-1}=0$. Introducing some further notations:
\begin{align}
\alpha^{*}_i=\sum_{j=1}^N\alpha^{*}_{ij}, \label{alph_i*}
\end{align} with
\begin{align}\label{alpha_ij*}
\alpha^*_{ij}=\E[A^*_n 1_{\{J_{n+1}=j\}}|J_n=i,X_{n-1}=0].
\end{align}
Note that the number of arrivals during the service time of a customer is a batch Poisson process. Therefore, we can write the following relations:
\begin{align}
A_{ij}(z)=&\tilde{G}_{ij}(\lambda(1-B(z))),\label{rel_A_G} \\
A^{*}_{ij}(z)=&\tilde{G}^{*}_{ij}(\lambda(1-B(z))), \quad \text{ for } i,j=1,2,\dots,N. \label{rel_A*_G*}
\end{align}
To derive the stability condition for our model we use the results from \cite{QUESTA2017}. Note that the dynamics in the current model only differs from that in \cite{QUESTA2017} when the queue length is zero. More specifically, the two processes have identical transition rates, except in a finite number of states. This implies that the two processes are either both positive recurrent, both null recurrent or both transient. The condition for stability reads $\rho<1$, in accordance with \cite{QUESTA2017}, and similarly, both processes are null recurrent if $\rho=1$. Hence, if we modify the parameters such that $\rho\uparrow1$, the processes move from positive recurrence to null recurrence. In particular, $\P[X=0]>0$ if $\rho<1$ and $\P[X=0]\rightarrow0$ as $\rho\uparrow1$.
\section{ Stationary queue length analysis}\label{stationary queue length}
In this section, we shall first determine the steady-state joint distribution of the number of customers in the system immediately after a departure, and the type of the next customer to be served. Subsequently, we will use this result to derive the generating functions of the stationary number of customers at an arbitrary time, at batch arrival instants, and at customer arrival instants.
\subsection{ Stationary queue length analysis: departure epochs}
Starting-point of the analysis is the following recurrence relation:
\begin{align}
X_n = \left\{
\begin{array}{l l}
\ X_{n-1}-1+A_n & \quad \text{if $X_{n-1} \geq 1$ }\\
A_n +B_n-1& \quad \text{if $X_{n-1} =0$}
\end{array} \right., ~~~ n=1,2,3,\dots,
\label{recurA}
\end{align} where $X_n$ is the number of customers at the departure times of the $n$th customer and $B_n$ is the size of the batch in which $n$th customer arrived, with generating function $B(z)$, for $|z|\leq 1$. Due to dependent successive service times, $X_n$ here is not a Markov chain. In order to obtain a Markovian model, it is required to keep track of the type of a departing customer together with the number of customers in the system immediately after the departure of that customer. As a consequence, $(X_n,J_{n+1})$ forms a Markov chain. \\
Taking generating functions and exploiting the fact that $X_{n-1}$ and ($A_n,J_{n+1}$) are conditionally independent, given $J_n$ and $X_{n-1}\geq 1$, we find:
\begin{align}
&\E[z^{X_{n}}1_{\{J_{n+1}=j\}}]
=\sum_{i=1}^{N}\E[z^{X_{n-1}-1}|J_{n}=i,X_{n-1} \geq 1]\E[z^{A_{n}}1_{\{J_{n+1}=j\}}|J_{n}=i,X_{n-1} \geq 1]\P(X_{n-1} \geq 1,J_{n}=i)\nonumber\\
& +\frac{B(z)}{z}\sum_{i=1}^{N}\E[z^{A_{n}}1_{\{J_{n+1}=j\}}|J_{n}=i,X_{n-1}=0]\P(X_{n-1}=0,J_{n}=i) \nonumber\\
=&\frac{1}{z}\sum_{i=1}^{N} \E[z^{X_{n-1}}1_{\{J_{n}=i\}}]\E[z^{A_{n}}1_{\{J_{n+1}=j\}}|J_{n}=i,X_{n-1} \geq 1]\nonumber\\
\ \ \ & +\frac{1}{z}\sum_{i=1}^{N}\Big(B(z)\E[z^{A_{n}}1_{\{J_{n+1}=j\}}|J_{n}=i,X_{n-1}=0]-\E[z^{A_{n}}1_{\{J_{n+1}=j\}}|J_{n}=i,X_{n-1}\geq 1]\Big)\P(X_{n-1}=0,J_{n}=i),\nonumber\\
\ \ \ & \text{for} ~~ n=1,2,3,\dots, ~~~ j=1,2,\dots,N. \nonumber\\
\label{twentysixAA}
\end{align}
Now, we restrict ourselves to the stationary situation, assuming that the stability condition holds.\\
Introduce, for $i,j=1,2,\dots,N$ and $|z| \leq 1$:
\begin{equation}
f_i(z)= {\rm lim}_{n \rightarrow \infty} \E[z^{X_n}1_{\{J_{n+1}=i\}}],
\label{fizA}
\end{equation}
with, for $i=1,2,\dots,N$,
\begin{equation}
f_i(0)= {\rm lim}_{n \rightarrow \infty} \P(X_n=0,J_{n+1}=i)
\label{fi0A}
\end{equation} such that
\begin{equation}
F(z)=\sum_{i=1}^{N}f_i(z). \label{F(z)}
\end{equation}
In stationarity, Equation \eqref{twentysixAA} leads to the following $N$ equations:
\begin{align}
& (z-A_{jj}(z))f_j(z)-\sum_{i=1,i\neq j}^{N}A_{ij}(z)f_i(z)
=\sum_{i=1}^{N}(B(z)A^{*}_{ij}(z)-A_{ij}(z))
f_i(0),\quad \quad ~~~~~~ j=1,2,\dots,N. \label{twentysevenA}
\end{align}
We can also write these $N$ linear equations in matrix form as
\begin{align*}
M(z)^{T}f(z)=b(z),
\end{align*}
where \begin{align}
M(z)=&
\begin{bmatrix}
z-A_{11}(z) & -A_{12}(z) & \dots & -A_{1N}(z)\\
-A_{21}(z) & z-A_{22}(z) & \dots & -A_{2N}(z)\\
\vdots & \vdots & \ddots & \vdots \\
-A_{N1}(z) & -A_{N2}(z) & \dots & z-A_{NN}(z)
\end{bmatrix}, \label{M(z)}\\
f(z)=&
\begin{bmatrix}
f_1(z)\\
f_2(z)\\
\vdots\\
f_N(z)
\end{bmatrix},
b(z)=
\begin{bmatrix}
b_1(z)\\
b_2(z)\\
\vdots \\
b_N(z)
\end{bmatrix}, \text{ with } b_j(z)=\sum_{i=1}^{N}(B(z)A^{*}_{ij}(z)-A_{ij}(z))f_i(0).\label{b(z)}
\end{align}
Therefore, by Cramer's rule, solutions of the non-homogeneous linear system $M(z)^{T}f(z)=b(z)$ are in the form
\begin{align}
f_i(z)=\frac{\det L_i(z)}{\det M(z)^{T}}, \quad \det M(z)^{T}\neq 0, \quad i=1,2,\dots,N, \label{f(z)_N}
\end{align} where $L_i(z)$ is the matrix formed by replacing the $i$-th column of $M(z)^T$ by the column vector $b(z)$:
\begin{align}
\det L_1(z)=&\begin{vmatrix}
b_1(z) & -A_{21}(z)&\dots& -A_{N1}(z) \\
b_2(z) & z-A_{22}(z)&\dots& -A_{N2}(z) \\
\vdots & \vdots & \ddots &\vdots \\
b_N(z) & -A_{2N}(z)&\dots& z-A_{NN}(z) \\
\end{vmatrix}\label{L_1(z)_D},\\
\det L_i(z)=&\begin{vmatrix}
z-A_{11}(z)&\dots& -A_{i-11}(z)& b_1(z) & -A_{i+11}(z)&\dots& -A_{N1}(z) \\
-A_{12}(z)&\dots& -A_{i-12}(z)& b_2(z) & -A_{i+12}(z)&\dots& -A_{N2}(z) \\
\vdots & \ddots &\vdots & \vdots & \vdots & \ddots &\vdots \\
-A_{1N}(z)&\dots& -A_{i-1N}(z)& b_N(z) & -A_{i+1N}(z)&\dots& z-A_{NN}(z) \\
\end{vmatrix}, \quad i=2,3,\dots,N.\label{L_i(z)_D}
\end{align}
It remains to find the values of $f_1(0),f_2(0),\dots,f_N(0)$.
We shall derive $N$ linear equations for $f_1(0),f_2(0),\dots,f_N(0)$.
\\
\paragraph{First equation:}
Note that $\det M(z)^T=\det M(z)$.
After replacing the first column by sum of all $N$ columns in \eqref{M(z)}, and using \eqref{A_i(z)}, we get,
\begin{align}
\det M(z)^T=&
\begin{vmatrix}
z-A_{1}(z) & -A_{12}(z) & \dots & -A_{1N}(z)\\
z-A_{2}(z) & z-A_{22}(z) & \dots & -A_{2N}(z)\\
\vdots & \vdots & \ddots & \vdots \\
z-A_{N}(z) & -A_{N2}(z) & \dots & z-A_{NN}(z)
\end{vmatrix}
.\label{Add: detM(z)}
\end{align}
This implies that
\begin{align}\label{sumform: detM(z)}
\det M(z)^T= \sum_{i=1}^{N}(z-A_i(z))u_{i1}(z),
\end{align}
where $u_{i1}(z)$ is the cofactor of the entry in the $i$-th row and the first column of the matrix in Equation~\eqref{Add: detM(z)}.
Note that $\{z-A_i(z)\}|_{z=1}=0, \frac{d}{dz}\{z-A_i(z)\}|_{z=1}=1-\alpha_i$, and $u_{i1}(1)=d_i$, where $d_i$ are given by Equations \eqref{d_1},\eqref{d_i},\eqref{d_N}, and $\alpha_i$ are defined in \eqref{alph_i*}, for $i=1,2,\dots,N$. Therefore, we obtain
\begin{align}\label{D: detM(z)^T}
\frac{d}{dz}\{\det M(z)^T\}\Big|_{z=1}&=\sum_{i=1}^{N}(1-\alpha_i)d_i
=d-\sum_{i=1}^{N}\alpha_id_i
=d(1-\rho).
\end{align}
This implies that
\begin{align}\label{L_i(z)}
\det L_i(z)=\sum_{j=1}^Nb_j(z)r_{ji}(z), \quad i=1,2,\dots,N,
\end{align} where $b_j(z)$ is given by \eqref{b(z)}, and $r_{ji}(z)$ is the cofactor of the entry in the $j$th row and $i$th column of the matrix $L_i(z)$, which is given by
\begin{align}
r_{11}(z)=&\begin{vmatrix}
z-A_{22}(z)&-A_{32}(z)&\dots& -A_{N2}(z) \\
-A_{23}(z)&z-A_{33}(z)&\dots& -A_{N3}(z) \\
\vdots & \vdots &\ddots &\vdots \\
-A_{2N}(z)& -A_{3N}(z)& \dots& z-A_{NN}(z) \\
\end{vmatrix}\label{r(11)_D},\\
r_{1i}(z)=&(-1)^{i+1}\begin{vmatrix}
-A_{12}(z)&\dots& -A_{i-12}(z) & -A_{i+12}(z)&\dots& -A_{N2}(z) \\
-A_{13}(z)&\dots& -A_{i-13}(z)& -A_{i+13}(z)&\dots& -A_{N3}(z) \\
\vdots & \ddots &\vdots & \vdots & \ddots &\vdots \\
-A_{1N}(z)&\dots& -A_{i-1N}(z) & -A_{i+1N}(z)&\dots& z-A_{NN}(z) \\
\end{vmatrix}, \quad i=2,3,\dots,N,\label{r(i1)_D}\\
r_{j1}(z)=&(-1)^{j+1}\begin{vmatrix}
-A_{21}(z)& -A_{31}(z) & -\dots& -A_{N1}(z) \\
\vdots & \vdots & \ddots &\vdots \\
-A_{2j-1}(z)& -A_{3j-1}(z)& \dots& -A_{Nj-1}(z) \\
-A_{2j+1}(z)& -A_{3j+1}(z)& \dots& -A_{Nj+1}(z) \\
\vdots & \vdots & \ddots &\vdots \\
-A_{2N}(z)& -A_{3N}(z) &\dots& z-A_{NN}(z) \\
\end{vmatrix}, \quad j=2,3,\dots,N\label{r(1j)_D}\\
r_{ji}(z)=&(-1)^{i+j}\begin{vmatrix}
z-A_{11}(z)&\dots& -A_{i-11}(z) & -A_{i+11}(z)&\dots& -A_{N1}(z) \\
\vdots & \ddots &\vdots & \vdots & \ddots &\vdots \\
-A_{1j-1}(z)&\dots& -A_{i-1j-1}(z)& -A_{i+1j-1}(z)&\dots& -A_{Nj-1}(z) \\
-A_{1j+1}(z)&\dots& -A_{i-1j+1}(z)& -A_{i+1j+1}(z)&\dots& -A_{Nj+1}(z) \\
\vdots & \ddots &\vdots & \vdots & \ddots &\vdots \\
-A_{1N}(z)&\dots& -A_{i-1N}(z) & -A_{i+1N}(z)&\dots& z-A_{NN}(z) \\
\end{vmatrix}, \quad i,j=2,3,\dots,N.\label{r(ij)_D}
\end{align}
Subsequently,
\begin{align}\label{L_i_prime(z)}
\frac{d}{dz}\{\det L_i(z)\}|_{z=1}&=\sum_{j=1}^N(b_j(1)r_{ji}^\prime(1)+b^\prime_j(1)r_{ji}(1))\nonumber \\
&=\sum_{j=1}^N\sum_{k=1}^{N}\Big(r_{ji}^\prime(1)(P^*_{kj}-P_{kl})+r_{ji}(1)(\E[B]P^*_{kj}+\alpha^*_{kj}-\alpha_{kj})\Big)f_k(0).
\end{align}
After replacing the first row by the sum of all $N$ rows of $\det L_i(z)$ in \eqref{L_i(z)_D}, we obtain $\det L_i(z)$, $i=2,3,\dots,N$, as
\begin{align}
\det L_i(z)=&\begin{vmatrix}
z-A_{1}(z)&\dots& z-A_{i-1}(z)& \sum_{j=1}^N b_j(z) & z-A_{i+1}(z)&\dots& z-A_{N}(z) \\
-A_{12}(z)&\dots& -A_{i-12}(z)& b_2(z) & -A_{i+12}(z)&\dots& -A_{N2}(z) \\
\vdots & \ddots &\vdots & \vdots & \vdots & \ddots &\vdots \\
-A_{1N}(z)&\dots& -A_{i-1N}(z)& b_N(z) & -A_{i+1N}(z)&\dots& z-A_{NN}(z) \\
\end{vmatrix}.\label{L_i_add(z)}
\end{align}
In particular,
\begin{align*}
\det L_i(1)=&\begin{vmatrix}
0 &\dots & 0 & 0 & 0 & \dots & 0 \\
-P_{12}& \dots & -P_{i-12}& b_2(1) & -P_{i+12}&\dots& -P_{N2} \\
\vdots & \ddots &\vdots & \vdots & \vdots & \ddots &\vdots \\
-P_{1N}&\dots& -P_{i-1N}& b_N(1) & -P_{i+1N}&\dots& 1-P_{NN} \\
\end{vmatrix},\quad i=2,3,\dots,N,\\
=&\ 0.
\end{align*}
Following the same steps, one can show that $\det L_1(1)=0$ and $\det M(1)^T=0$. Therefore, for $i=1,2,\dots,N$, we obtain,
\begin{align}
f_i(1)&=\lim_{z\to 1}\frac{\det L_i(z)}{\det M(z)^T}\nonumber\\
&=\frac{\frac{d}{dz}\{\det L_i(z)\}|_{z=1}}{\frac{d}{dz}\{\det M(z)^T\}|_{z=1}}.\label{f_i(1)}
\end{align}
Note that $F(1)=1$, which implies that $\sum_{i=1}^Nf_i(1)=1$. And, as a consequence, we obtain,
\begin{align}
\frac{\sum_{i=1}^N\frac{d}{dz}\{\det L_i(z)\}|_{z=1}}{\frac{d}{dz}\{\det M(z)^T\}|_{z=1}}&=1. \nonumber
\end{align}
This implies that
\begin{align}
\sum_{k=1}^{N}\Bigg(\sum_{i=1}^{N}\sum_{j=1}^N\Big(r_{ji}^\prime(1)(P^*_{kj}-P_{kl})+r_{ji}(1)(\E[B]P^*_{kj}+\alpha^*_{kj}-\alpha_{kj})\Big)\Bigg)f_k(0)&=d(1-\rho).\label{first: normalizingEq}
\end{align}
\paragraph{$(N-1)$ equations:}
Under the stability condition, $\det M(z)^T$ has exactly $N-1$ zeros in $|z|<1$, denoted by $\hat{z_l}$, $l=1,2,\dots,N-1$ (see in \cite{QUESTA2017}), and $F(z)$ is an analytical function in $|z|<1$. Therefore, the numerator of $F(z)$ also has $(N-1)$ zeros in $|z|<1$. As a consequence, these $(N-1)$ zeros provide $(N-1)$ linear equations for $f_1(0),f_2(0),\dots,f_N(0)$:
\begin{align}\label{N-1: equations}
\sum_{i=1}^{N}\det L_i(\hat{z_l})&=0, \quad |\hat{z_l}|<1\nonumber\\
\implies \sum_{i=1}^{N}\sum_{j=1}^Nb_j(\hat{z_l})r_{ji}(\hat{z_l})&=0 \nonumber\\
\implies \sum_{k=1}^N\Bigg(\sum_{i=1}^{N}\sum_{j=1}^Nr_{ji}(\hat{z_l})\Big( B(\hat{z_l})A^*_{kj}(\hat{z_l})-A_{kj}(\hat{z_l})\Big)\Bigg)f_k(0)&=0,\quad l=1,2,\dots,N-1.
\end{align}
\subsection{Special case: $N=2$}
For $N=2$, we can solve \eqref{twentysevenA} and find an explicit expression for the steady-state probability generating function of the number of customers.
\begin{align}
f_1(z)=\frac{\sum_{i=1}^{2}\Big((z-A_{22}(z))(B(z)A^{*}_{i1}(z)-A_{i1}(z))+A_{21}(z)(B(z)A^{*}_{i2}(z)-A_{i2}(z)) \Big)f_i(0)}{(z-A_{11}(z))(z-A_{22}(z))-A_{12}(z)A_{21}(z)}, \label{twentynine}
\end{align}
\begin{align}
f_2(z)=\frac{\sum_{i=1}^{2}\Big((z-A_{11}(z))(B(z)A^{*}_{i2}(z)-A_{i2}(z))+A_{12}(z)(B(z)A^{*}_{i1}(z)-A_{i1}(z)) \Big)f_i(0)}{(z-A_{11}(z))(z-A_{22}(z))-A_{12}(z)A_{21}(z)}. \label{thirty}
\end{align}
In particular,
\begin{align}
&f_1(1)=\lim_{z\to 1}\frac{\sum_{i=1}^{2}\Big((z-A_{22}(z))(B(z)A^{*}_{i1}(z)-A_{i1}(z))+A_{21}(z)(B(z)A^{*}_{i2}(z)-A_{i2}(z)) \Big)f_i(0)}{(z-A_{11}(z))(z-A_{22}(z))-A_{12}(z)A_{21}(z)}\nonumber\\
=&\frac{\sum_{i=1}^{2}\Big(P_{21}(P^{*}_{i1}\E[B]+\alpha^{*}_{i1}-\alpha_{i1}) +(1-\alpha_{22})(P^*_{i1}-P_{i1}) +P_{21}(P^{*}_{i2}\E[B]+\alpha^{*}_{i2}-\alpha_{i2}) +\alpha_{21}(P^*_{i2}-P_{i2}) \Big)f_i(0)}{(1-P_{11})(1-\alpha_{22})+(1-P_{22})(1-\alpha_{11})-P_{12}\alpha_{21}-P_{21}\alpha_{12}}\nonumber\\
=&\frac{\sum_{i=1}^{2}\Big(P_{21}(\E[B]+\alpha^*_i-\alpha_i)+(1-\alpha_2)(P^*_{i1}-P_{i1})\Big)f_i(0)}{(P_{12}+P_{21})\left(1-\frac{P_{21}}{P_{12}+P_{21}}\alpha_1-\frac{P_{12}}{P_{12}+P_{21}}\alpha_2\right)}\nonumber\\
=&\frac{\sum_{i=1}^{2}\Big(P_{21}(\E[B]+\alpha^*_i-\alpha_i)+(1-\alpha_2)(P^*_{i1}-P_{i1})\Big)f_i(0)}{(P_{12}+P_{21})\left(1-\rho \right)}\label{f_1(1)}.
\end{align}
Similarly,
\begin{align}
f_2(1)=\frac{\sum_{i=1}^{2}\Big(P_{12}(\E[B]+\alpha^*_i-\alpha_i)+(1-\alpha_1)(P^*_{i2}-P_{i2})\Big)f_i(0)}{(P_{12}+P_{21})\left(1-\rho \right)}\label{f_2(1)}.
\end{align}
As a consequence of $f_1(1)+f_2(1)=1$, we obtain
\begin{align}\label{F(1)}
\sum_{i=1}^{2}\Big((P_{12}+P_{21})(\E[B]+\alpha^*_i-\alpha_i)+(\alpha_1-\alpha_2)(P^*_{i1}-P_{i1})\Big)f_i(0)=(P_{12}+P_{21})\left(1-\rho \right).
\end{align}
After substituting the values of $f_1(z)$ and $f_2(z)$ from Equations \eqref{twentynine} and \eqref{thirty}, respectively, in \eqref{F(z)}, we obtain
\begin{align}
F(z)=\frac{\sum_{i=1}^{2}\Big((z+A_{12}(z)-A_{22}(z))(B(z)A^{*}_{i1}(z)-A_{i1}(z))+(z+A_{21}(z)-A_{11}(z))(B(z)A^{*}_{i2}(z)-A_{i2}(z)) \Big)f_i(0)}{(z-A_{11}(z))(z-A_{22}(z))-A_{12}(z)A_{21}(z)}.\label{F_2(z)}
\end{align}
Let $z=\hat{z}$ be the zero of the denominator of $F(z)$ such that $|\hat{z}|<1$. Since $z=\hat{z}$ must also be the zero of the numerator of $F(z)$, we obtain the following equation in terms of $f_1(0)$ and $f_2(0)$:
\begin{align}\label{second: eq}
\sum_{i=1}^{2}\Big((\hat{z}+A_{12}(\hat{z})-A_{22}(\hat{z}))(B(\hat{z})A^{*}_{i1}(\hat{z})-A_{i1}(\hat{z}))+(\hat{z}+A_{21}(\hat{z})-A_{11}(\hat{z}))(B(\hat{z})A^{*}_{i2}(\hat{z})-A_{i2}(\hat{z})) \Big)f_i(0)=0.
\end{align}
Solving Equations \eqref{F(1)}and \eqref{second: eq} yields
\begin{align}
f_1(0)=&\frac{-(P_{12}+P_{21})\left(1-\rho \right)R_{12}}{\det R}, \label{N: f1(0)}\\
f_2(0)=&\frac{(P_{12}+P_{21})\left(1-\rho \right)R_{11}}{\det R}, \label{N: f2(0)}
\end{align} where $\det R$ is the determinant of the matrix $R=[R_{ij}]$, whose elements are given by
\begin{align*}
R_{1j}&=(\hat{z}+A_{12}(\hat{z})-A_{22}(\hat{z}))(B(\hat{z})A^{*}_{j1}(\hat{z})-A_{j1}(\hat{z}))+(\hat{z}+A_{21}(\hat{z})-A_{11}(\hat{z}))(B(\hat{z})A^{*}_{j2}(\hat{z})-A_{j2}(\hat{z})),\\
R_{2j}&=(P_{12}+P_{21})(\E[B]+\alpha^*_j-\alpha_j)+(\alpha_1-\alpha_2)(P^*_{j1}-P_{j1}), \quad j=1,2.
\end{align*}
\subsection{Stationary queue length analysis: arrival and arbitrary epochs}
In the previous subsection, we determined the probability generating function of the stationary queue length distribution at the departure epoch of an arbitrary customer for general batch arrivals. As customers arrive at the system according to a batch Poisson process with rate $\lambda$, from the PASTA property, the distribution of the number of customers in the system at the arrival time of a batch is the same as the distribution of the number of customers at an arbitrary time. After using PASTA and level-crossing arguments (see \cite{QUESTA2017} for more details), we obtain the following relations:
\begin{align}
\E[z^{X}]=\E[z^{X^{\textit{ca}}}] = \E[z^{X^{\textit{ba}}}]\frac{1-B(z)}{\E[B](1-z)}, \label{rel:queue}
\end {align}
with,
\begin{align}
\E[z^{X^{\textit{arb}}}]=\E[z^{X^{\textit{ba}}}] ,
\label{F(z):arbitrary}
\end{align}
where $X$ and $X^{\textit{ca}}$ are the number of customers at the departure and the arrival epoch of the customer respectively; $X^{\textit{arb}}$ and $X^{\textit{ba}}$ are the number of customers at an arbitrary time and the arrival time of a batch respectively.
From these relations, we can obtain all the required distributions.
\section{Heavy-traffic analysis}\label{heavy-traffc}
In this section, we shall determine the HT limit of the scaled queue length at departure epochs. In particular, we will show that under some conditions the distribution of the scaled stationary queue length in heavy traffic is exponential. This will be formally stated in Theorem \ref{thm:ch6}.
Let us define the HT limit for the LST of the scaled queue length at departure epochs, $(1-\rho)X$, for $i=1,2,\dots,N$:
\begin{align*}
\bar{F}(s)=\lim_{\rho \uparrow 1}\E[e^{-s(1-\rho)X}]=\sum_{i=1}^N\bar{f}_i(s),
\end{align*}
with
\begin{align*}
\bar{f}_i(s)=\lim_{\rho \uparrow 1}\E[e^{-s(1-\rho)X_n}1_{\{J_{n+1}=i\}}].
\end{align*}
Firstly, we introduce the following notation. For $i=1,2,\dots,N$, $|z|\leq 1$,
\begin{align}\label{mathcal: A_i}
\mathcal{A}_{i}(z)=\E[z^{A_n}|J_{n+1}=i,X_{n-1}\geq 1],
\end{align}
and
\begin{align}\label{gamma_i}
\gamma_i=\E[A_n|J_{n+1}=i,X_{n-1}\geq 1].
\end{align}
To find the limiting HT distribution, we substitute
\begin{align}
z=e^{-s(1-\rho)}=1-s(1-\rho)+\frac{1}{2}s^2(1-\rho)^2+O((1-\rho)^3), ~~~~~\text{as } \rho \uparrow 1.\label{HT_z}
\end{align}
With this substitution, we can write the generating function of the batch size, $B(z)$, as
\begin{align}
B(e^{-s(1-\rho)})=1-s(1-\rho)\E[B]+\frac{1}{2}s^2(1-\rho)^2\E[B^2]+O((1-\rho)^3). \label{HT_B(z)}
\end{align}
Similarly, from Equations \eqref{A_ij(z)}, \eqref{A_ijstar(z)}, and \eqref{mathcal: A_i}, we obtain for $\rho \uparrow 1$,
\begin{align}
A_{ij}(e^{-s(1-\rho)})=&P_{ij}-s(1-\rho)\alpha_{ij}+\frac{1}{2}s^2(1-\rho)^2\hat{\alpha}_{ij}+O((1-\rho)^3),\label{HT_A_ij}\\
A^*_{ij}(e^{-s(1-\rho)})=&P^*_{ij}-s(1-\rho)\alpha^*_{ij}+\frac{1}{2}s^2(1-\rho)^2\hat{\alpha}^*_{ij}+O((1-\rho)^3), \label{HT_A^*_ij}\\
\mathcal{A}_{i}(e^{-s(1-\rho)})=&1-s(1-\rho)\gamma_{i}+\frac{1}{2}s^2(1-\rho)^2\hat{\gamma}_{i}+O((1-\rho)^3),~\text{as } \rho \uparrow 1,\label{HT_mathcal_A_i}
\end{align} where $\alpha_{ij},\alpha^*_{ij}$, and $\gamma_i$ are respectively defined in Equations \eqref{alpha_ij}, \eqref{alpha_ij*} and \eqref{gamma_i}, and
\begin{align}
\hat{\alpha}_{ij}=&\E[(A_n)^2 1_{\{J_{n+1}=j\}}|J_n=i,X_{n-1}\geq 1],\label{hat: alpha_ij}\\
\hat{\alpha}^*_{ij}=&\E[(A^*_n)^2 1_{\{J_{n+1}=j\}}|J_n=i,X_{n-1}=0],\label{hat: alpha^*_ij}\\
\hat{\gamma}_i=&\E[(A_n)^2|J_{n+1}=i,X_{n-1}\geq 1].\label{hat: gamma_i}
\end{align}
Using $A_i(z)=\sum_{j=1}^{N}A_{ij}(z)$ and $A^*_i(z)=\sum_{j=1}^{N}A^*_{ij}(z)$, we obtain,
\begin{align}
A_{i}(e^{-s(1-\rho)})=&1-s(1-\rho)\alpha_{i}+\frac{1}{2}s^2(1-\rho)^2\hat{\alpha}_{i}+O((1-\rho)^3),\label{HT_A_i}\\
A^*_{i}(e^{-s(1-\rho)})=&1-s(1-\rho)\alpha^*_{i}+\frac{1}{2}s^2(1-\rho)^2\hat{\alpha}^*_{i}+O((1-\rho)^3),~\text{as } \rho \uparrow 1, \label{HT_A^*_i}
\end{align}
where $\alpha_{i}$ and $\alpha^*_{i}$ are respectively defined in Equations \eqref{alph_i} and \eqref{alph_i*}, and
\begin{align}
\hat{\alpha}_i=&\sum_{j=1}^N\hat{\alpha}_{ij},\label{hat: alpha_i}\\
\hat{\alpha}^{*}_i=&\sum_{j=1}^N\hat{\alpha}^{*}_{ij}.\label{hat: alpha^*_i}
\end{align}
After substituting the values of $B(z),A_{ij}(z)$ and $A^*_{ij}(z)$ from Equations \eqref{HT_B(z)}, \eqref{HT_A_ij} and \eqref{HT_A^*_ij} respectively, we obtain $b_j(z)$, with $z=e^{-s(1-\rho)}$, from \eqref{b(z)} as
\begin{align}
&b_j(e^{-s(1-\rho)})=\sum_{i=1}^N\Big(\ (P^*_{ij}-P_{ij})-s(1-\rho)(P^*_{ij}\E[B]+\alpha^*_{ij}-\alpha_{ij})\nonumber\\
&+\frac{s^2(1-\rho)^2}{2}(\E[B^2]P^*_{ij}+2\E[B]\alpha^*_{ij}+\hat{\alpha}^*_{ij}-\hat{\alpha}_{ij})+O((1-\rho)^3)\Big)f_i(0),~\text{as } \rho \uparrow 1.\label{HT_b_j}
\end{align}
Summing over $j$ and using $\sum_{j=1}^NP_{ij}=\sum_{j=1}^N P^*_{ij}=1$ gives
\begin{align}
\sum_{j=1}^N b_j(e^{-s(1-\rho)})&=-s(1-\rho)\sum_{i=1}^N\Big((\E[B]+\alpha^*_{i}-\alpha_{i})-\frac{s(1-\rho)}{2}(\E[B^2]\nonumber\\
&+2\E[B]\alpha^*_{i}+\hat{\alpha}^*_{i}-\hat{\alpha}_{i})+O((1-\rho)^2)\Big)f_i(0), ~\text{as } \rho \uparrow 1.\label{HT_sum_b_j}
\end{align}
Substituting the values of $z,A_{ij}(z),A_{i}(z),b_j(z)$, and $\sum_{j=1}^{N}b_j(z)$ from Equations \eqref{HT_z}, \eqref{HT_A_ij}, \eqref{HT_A_i}, \eqref{HT_b_j}, and \eqref{HT_sum_b_j}, respectively, in Equation \eqref{L_i_add(z)}, and after simplification, we obtain $\det L_i(z)$, with $z=e^{-s(1-\rho)}$, as
\begin{align}
&\det L_i(e^{-s(1-\rho)})=-s(1-\rho)\nonumber\\
&\times\begin{vmatrix}
1-\alpha_{1}&\dots& 1-\alpha_{i-1}& \sum_{k=1}^N(\E[B]+\alpha^*_{k}-\alpha_{k})f_k(0) & 1-\alpha_{i+1}&\dots&1-\alpha_{N}\\
-P_{12}&\dots& -P_{i-12}& \sum_{k=1}^N(P^*_{k2}-P_{k2})f_k(0) & -P_{i+12}&\dots& -P_{N2} \\
\vdots & \ddots &\vdots & \vdots & \vdots & \ddots &\vdots \\
-P_{1N}&\dots& -P_{i-1N}&\sum_{k=1}^N(P^*_{kN}-P_{kN})f_k(0) & -P_{i+1N}&\dots& 1-P_{NN} \\
\end{vmatrix}\nonumber\\
&+c_is^2(1-\rho)^2+O((1-\rho)^3),\quad i=2,3,\dots,N,\label{HT_Li1}
\end{align} where $c_i$ is the coefficient of $s^2(1-\rho)^2$ term such that
\begin{align}\label{lim_ci}
\lim_{\rho \uparrow 1} c_i=0.
\end{align}
This coefficient exists, because
\[\lim_{\rho \uparrow 1}\P(X=0)= \lim_{\rho \uparrow 1} \sum_{k=1}^{N} f_k(0) =0,\]
implying that $\lim_{\rho \uparrow 1} f_k(0)=0$ for all $1\leq k \leq N$.
Now, differentiating Equation \eqref{L_i_add(z)} w.r.t. $z$, and substituting $z=1$, we get,
\begin{align}\label{HT_Li2}
&\frac{d}{dz}\{\det L_i(z)\}|_{z=1}=\begin{vmatrix}
1-\alpha_{1}&\dots& 1-\alpha_{i-1}& \sum_{k=1}^N(\E[B]+\alpha^*_{k}-\alpha_{k})f_k(0) & 1-\alpha_{i+1}&\dots&1-\alpha_{N}\\
-P_{12}&\dots& -P_{i-12}& \sum_{k=1}^N(P^*_{k2}-P_{k2})f_k(0) & -P_{i+12}&\dots& -P_{N2} \\
\vdots & \ddots &\vdots & \vdots & \vdots & \ddots &\vdots \\
-P_{1N}&\dots& -P_{i-1N}&\sum_{k=1}^N(P^*_{kN}-P_{kN})f_k(0) & -P_{i+1N}&\dots& 1-P_{NN} \\
\end{vmatrix}.
\end{align}
After using Equations \eqref{D: detM(z)^T}, \eqref{f_i(1)} and \eqref{HT_Li2} in Equation \eqref{HT_Li1}, we can write
\begin{align}
&\det L_i(e^{-s(1-\rho)})\nonumber\\
&=-s(1-\rho)\frac{d}{dz}\{\det L_i(z)\}|_{z=1}+c_is^2(1-\rho)^2+O((1-\rho)^3)\nonumber\\
&=-s(1-\rho)\frac{d}{dz}\{\det M(z)^T\}|_{z=1}f_i(1)+c_is^2(1-\rho)^2+O((1-\rho)^3)\nonumber\\
&=-sd(1-\rho)^2(f_i(1)-\frac{c_is}{d})+O((1-\rho)^3),\quad i=2,3,\dots,N.\nonumber
\end{align}
Similarly,
\begin{align*}
\det L_1(e^{-s(1-\rho)})&=-sd(1-\rho)^2(f_1(1)-\frac{c_1s}{d})+O((1-\rho)^3).
\end{align*}
Hence, we can write, for $i=1,2,\dots,N$,
\begin{align}
\det L_i(e^{-s(1-\rho)})&=-sd(1-\rho)^2(f_i(1)-\frac{c_is}{d})+O((1-\rho)^3).\label{HT_Li(s)}
\end{align}
From Equation \eqref{Add: detM(z)}, $\det M(z)^T$ is given by
\begin{align}
&\det M(z)^T=
\begin{vmatrix}
z-A_{1}(z) & -A_{12}(z) & \dots & -A_{1N}(z)\\
z-A_{2}(z) & z-A_{22}(z) & \dots & -A_{2N}(z)\\
\vdots & \vdots & \ddots & \vdots \\
z-A_{N}(z) & -A_{N2}(z) & \dots & z-A_{NN}(z)
\end{vmatrix}\nonumber\\
&=\frac{1}{\prod_{i=1}^{N}\pi_i}
\begin{vmatrix}
\pi_1(z-A_{1}(z)) & -\pi_1 A_{12}(z) & \dots & -\pi_1 A_{1N}(z)\\
\pi_2(z-A_{2}(z)) & \pi_2(z-A_{22}(z)) & \dots & -\pi_2A_{2N}(z)\\
\vdots & \vdots & \ddots & \vdots \\
\pi_N(z-A_{N}(z)) & -\pi_NA_{N2}(z) & \dots & \pi_N(z-A_{NN}(z))
\end{vmatrix},\nonumber\\
&\qquad \qquad \qquad \qquad \qquad\text{ since } \pi_i\neq 0, i=1,2,\dots,N.\label{HT: M(z)0}
\end{align}
Using $\lim_{\rho \uparrow 1}f_k(0)=0$, we will first show that $\lim_{\rho \uparrow 1}f_j(1)=\pi_j$ for all $1\leq j\leq N$. To do so, we first write $\lim_{\rho \uparrow 1}f_j(1)$ as
\begin{align*}
\lim_{\rho \uparrow 1}f_j(1)&=\lim_{\rho \uparrow 1}\P(J_{n+1}=j)\\
&=\lim_{\rho \uparrow 1}(\P(J_{n+1}=j,X_{n-1}=0)+\P(J_{n+1}=j,X_{n-1}\geq 1))\\
&=\lim_{\rho \uparrow 1}\P(J_{n+1}=j|X_{n-1}\geq 1)\P(X_{n-1}\geq 1)\\
&=\lim_{\rho \uparrow 1}\sum_{i=1}^{N}\P(J_{n+1}=j|J_n=i,X_{n-1}\geq 1)\P(J_{n}=i|X_{n-1}\geq 1)\P(X_{n-1}\geq 1)\\
&=\lim_{\rho \uparrow 1}\sum_{i=1}^{N}P_{ij}P(J_{n}=i,X_{n-1}\geq 1)\\
&=\lim_{\rho \uparrow 1}\sum_{i=1}^{N}P_{ij}(\P(J_n=i)-\P(J_{n}=i,X_{n-1}=0))\\
&=\lim_{\rho \uparrow 1}\sum_{i=1}^{N}P_{ij}(f_i(1)-f_i(0))\\
&=\sum_{i=1}^{N}P_{ij}\lim_{\rho \uparrow 1}f_i(1), \quad \text{for } j=1,2,\dots,N.
\end{align*}
As $P=[P_{ij}]_{i,j\in\{1,2,\dots,N\}}$ is the transition probability matrix of an irreducible discrete time Markov chain, with stationary distribution $\pi=(\pi_1,\pi_2,\dots,\pi_N)$, $\pi$ is the unique solution of the system of equations $\pi (I-P)=0$, and, hence, $\lim_{\rho \uparrow 1}f_j(1)=\pi_j$ for all $1\leq j\leq N$. As a consequence, we obtain $\lim_{\rho \uparrow 1} \P(J_{n}=j|X_{n-1}\geq 1)=\pi_j$.
Furthermore,
\begin{align}
\lim_{\rho \uparrow 1}\P(J_{n+1}=j|X_{n-1}\geq 1)&=\lim_{\rho \uparrow 1}\sum_{i=1}^{N}\P(J_{n+1}=j|J_n=i,X_{n-1}\geq 1)\P(J_{n}=i|X_{n-1}\geq 1)\nonumber\\
&=\sum_{i=1}^{N}P_{ij}\pi_i \nonumber\\
&=\pi_j.\label{P(J_{n+1}|X>0)}
\end{align}
As a consequence, $\lim_{\rho \uparrow 1} \mathcal{A}_j(z)$ is given by
\begin{align}
\lim_{\rho \uparrow 1} \mathcal{A}_j(z)&=\lim_{\rho \uparrow 1}\frac{\E[z^{A_n}1_{\{J_{n+1}=j\}}|X_{n-1}\geq 1]}{\P(J_{n+1}=j|X_{n-1}\geq 1)}\nonumber\\
&=\lim_{\rho \uparrow 1}\frac{\sum_{i=1}^{N} \P(J_{n}=i|X_{n-1}\geq 1) \E[z^{A_n}1_{\{J_{n+1}=j\}}|J_n=i,X_{n-1}\geq 1]}{\P(J_{n+1}=j|X_{n-1}\geq 1)} \nonumber\\
&=\frac{\sum_{i=1}^{N} \pi_i A_{ij}(z)}{\pi_j}. \label{Rel: mathcalA_jA_ij}
\end{align}
Subsequently, we obtain,
\begin{align}
\gamma_j=\frac{\sum_{i=1}^{N} \pi_i \alpha_{ij}}{\pi_j},\quad \text{as } \rho \uparrow 1.\label{Rel: gammaj_alphaij}
\end{align}
Replacing the first row by the sum of all $N$ rows in Equation \eqref{HT: M(z)0}, and using $\mathcal{A}_j(z)=\frac{1}{\pi_j}\sum_{i=1}^{N}\pi_iA_{ij}(z)$ as $\rho \uparrow 1$ and $\sum_{i=1}^{N}\pi_i=1$ , we obtain $\det M(z)^T$ as, for $\rho \uparrow 1$,
\begin{align}
\det M(z)^T&=\frac{1}{\prod_{i=1}^{N}\pi_i}
\begin{vmatrix}
z-\sum_{i=1}^{N}\pi_iA_i(z) & \pi_2(z- \mathcal{A}_{2}(z)) & \dots & \pi_N(z-\mathcal{A}_{N}(z))\\
\pi_2(z-A_{2}(z)) & \pi_2(z-A_{22}(z)) & \dots & -\pi_2A_{2N}(z)\\
\vdots & \vdots & \ddots & \vdots \\
\pi_N(z-A_{N}(z)) & -\pi_NA_{N2}(z) & \dots & \pi_N(z-A_{NN}(z))
\end{vmatrix}\nonumber\\
&=\frac{1}{\pi_1}
\begin{vmatrix}
z-\sum_{i=1}^{N}\pi_iA_i(z) & \pi_2(z-\mathcal{A}_{2}(z)) & \dots & \pi_N(z-\mathcal{A}_{N}(z))\\
z-A_{2}(z) & z-A_{22}(z) & \dots & -A_{2N}(z)\\
\vdots & \vdots & \ddots & \vdots \\
z-A_{N}(z) & -A_{N2}(z) & \dots & z-A_{NN}(z)
\end{vmatrix}.\label{HT_M(z)1}
\end{align}
Substituting the values of $z,A_{ij}(z),A_{i}(z)$, and $\mathcal{A}_{i}(z)$ from Equations \eqref{HT_z}, \eqref{HT_A_ij}, \eqref{HT_A_i}, and \eqref{HT_mathcal_A_i}, respectively, in Equation \eqref{HT_M(z)1}, and after simplification, with $z=e^{-s(1-\rho)},\rho=\sum_{i=1}^{N}\pi_i\alpha_i,\hat{\alpha}=\sum_{i=1}^{N}\pi_i\hat{\alpha}_i,\pi_1=\frac{d_1}{d}$, we obtain
\begin{align}
&\det M(e^{-s(1-\rho)})^T\nonumber \\
&=\frac{d}{d_1}
\begin{vmatrix}
-s(1-\rho)^2(1-\frac{s}{2}(1-\hat{\alpha})) & -\pi_2s(1-\rho)(1-\gamma_2)& \dots & -\pi_Ns(1-\rho)(1-\gamma_N)\\
-s(1-\rho)(1-\alpha_2) & 1-P_{22} & \ddots & -P_{2N}\\
\vdots & \vdots & \ddots & \vdots \\
-s(1-\rho)(1-\alpha_N) & -P_{N2} & \ddots & 1-P_{NN}
\end{vmatrix}+O((1-\rho)^3)\nonumber\\
&=\frac{-sd(1-\rho)^2}{d_1} \begin{vmatrix}
1-\frac{s}{2}(1-\hat{\alpha}) & -\pi_2s(1-\gamma_2)& \dots & -\pi_Ns(1-\gamma_N)\\
1-\alpha_2 & 1-P_{22} & \ddots & -P_{2N}\\
\vdots & \vdots & \ddots & \vdots \\
1-\alpha_N & -P_{N2} & \ddots & 1-P_{NN}
\end{vmatrix}
+O((1-\rho)^3)\nonumber\\
&=\frac{-sd(1-\rho)^2}{d_1}\Big((1-\frac{s}{2}(1-\hat{\alpha}))d_1-s\sum_{k=2}^{N}\pi_k(1-\gamma_k)q_k\Big)+O((1-\rho)^3)\nonumber\\
&=-sd(1-\rho)^2\left(1+s\left(\frac{\hat{\alpha}-1}{2}-\frac{1}{d_1}\sum_{k=2}^{N}\pi_k(1-\gamma_k)q_k\right)\right)+O((1-\rho)^3),\label{MT_M(s)^T}
\end{align} where $d_1$ is defined in Equation \eqref{d_1}, and $q_k$, $k=2,3,\dots,N$, is the cofactor of the entry in the first row and the $k$-th column of the matrix
\[\begin{bmatrix}
1-\frac{s}{2}(1-\hat{\alpha}) & -\pi_2s(1-\gamma_2)& \dots & -\pi_Ns(1-\gamma_N)\\
1-\alpha_2 & 1-P_{22} & \ddots & -P_{2N}\\
\vdots & \vdots & \ddots & \vdots \\
1-\alpha_N & -P_{N2} & \ddots & 1-P_{NN}
\end{bmatrix},\]
which is given by
\begin{align}
q_2=&-\begin{vmatrix}
1-\alpha_2 & -P_{23} & -P_{24} & \dots &-P_{2N}\\
1-\alpha_3 & 1-P_{33} & -P_{34} & \dots &1-P_{3N}\\
\vdots & \vdots & \vdots &\ddots & \vdots\\
1-\alpha_N & -P_{N3} & -P_{N4} & \dots &1-P_{NN}
\end{vmatrix},\label{q1}\\
q_k=&(-1)^{k+1}\begin{vmatrix}
1-\alpha_2 & 1-P_{22} & \dots & -P_{2k-1} & -P_{2K+1}& \dots &-P_{2N}\\
1-\alpha_3 & -P_{32} & \dots & -P_{3k-1} & -P_{3K+1}& \dots &1-P_{3N}\\
\vdots & \vdots & \ddots & \vdots & \vdots &\ddots & \vdots\\
1-\alpha_N & -P_{N2} & \dots & -P_{Nk-1} & -P_{NK+1}& \dots &1-P_{NN}
\end{vmatrix},\label{qk}
\end{align}
for $k=3,4,\dots,N$.
Therefore,
\begin{align}
\bar{f}_i(s)&=\lim_{\rho \uparrow 1} \frac{\det L_i(e^{-s(1-\rho)})}{\det M(e^{-s(1-\rho)})^T}\nonumber\\
&=\lim_{\rho \uparrow 1} \frac{-sd(1-\rho)^2(f_i(1)-\frac{c_is}{d})+O((1-\rho)^3)}{-sd(1-\rho)^2\left(1+s\left(\frac{\hat{\alpha}-1}{2}-\frac{1}{d_1}\sum_{k=2}^{N}\pi_k(1-\gamma_k)q_k\right)\right)+O((1-\rho)^3)}\nonumber\\
&=\frac{\pi_i}{1+s\left(\frac{\bar{\hat{\alpha}}-1}{2}-\frac{1}{d_1}\sum_{k=2}^{N}\pi_k(1-\bar{\gamma}_k)\bar{q}_k\right)},\label{HT_fi(s)}
\end{align} where we define $\lim_{\rho \uparrow 1}\hat{\alpha}=\bar{\hat{\alpha}},\lim_{\rho \uparrow 1}\gamma_k=\bar{\gamma}_k$ and $\lim_{\rho \uparrow 1}q_k=\bar{q}_k$.\\
This finally gives us the HT limit of the scaled queue length, which we formulate in the following theorem.
\begin{theorem}\label{thm:ch6}
If $\E[B^2]$ and $\hat{\alpha}_{ij}$ are finite for $i,j=1,2,\dots,N$, then
\begin{align}
\bar{F}(s)= \lim_{\rho \uparrow 1} \E[e^{-s(1-\rho)X}]= \frac{1}{1+s\left(\frac{\bar{\hat{\alpha}}-1}{2}-\frac{1}{d_1}\sum_{k=2}^{N}\pi_k(1-\bar{\gamma}_k)\bar{q}_k\right)},
\label{HT_F(s)}
\end{align}
provided $\left(\frac{\bar{\hat{\alpha}}-1}{2}-\frac{1}{d_1}\sum_{k=2}^{N}\pi_k(1-\bar{\gamma}_k)\bar{q}_k\right)>0$, which is the LST of an exponentially distributed random variable with rate parameter $$\eta=\frac{1}{\frac{\bar{\hat{\alpha}}-1}{2}-\frac{1}{d_1}\sum_{k=2}^{N}\pi_k(1-\bar{\gamma}_k)\bar{q}_k}.$$
\end{theorem}
\begin{remark}\label{remark: equal_alpha_i}
If $\alpha_i=\alpha$ for all $i=1,2,\dots,N$, then Equation \eqref{rho} implies that $\rho=\alpha$. In that case, the system is in heavy traffic when $\alpha\uparrow 1$ and, as a consequence, when $\alpha_i\uparrow 1$ for all $i=1,2,\dots,N$.
Note that each element of the first column of $q_k$, $k=2,3,\dots,N$, tends to zero, as $\alpha_i\uparrow 1$ for all $i=1,2,\dots,N$. It follows that $q_k=0$, which implies that $\bar{q}_k=0$ for all $k=2,3,\dots,N$, and
\begin{align*}
\bar{F}(s) \rightarrow \frac{1}{1+s\left(\frac{\bar{\hat{\alpha}}-1}{2}\right)},\quad \text{as }\alpha=\rho \uparrow 1,
\end{align*} which is the HT limit of the scaled queue length of the standard $M^X/G/1$ without dependencies at the departure epochs. Furthermore, we can conclude that the term $-\frac{s}{d_1}\sum_{k=2}^{N}\pi_k(1-\bar{\gamma}_k)\bar{q}_k$ in Equation \eqref{HT_F(s)} appears due to the dependent service times.
\end{remark}
\begin{remark}\label{remark: N=2}
For $N=2$, Equation~\eqref{HT_F(s)} reduces to
\begin{align*}
\bar{F}(s)=\frac{1}{1+s\left(\frac{\bar{\hat{\alpha}}-1}{2}+\frac{\left(1-\bar{\alpha}_2\right)}{P_{12}+P_{21}}\left(\frac{P_{12}}{P_{21}}(1-\bar{\alpha}_{22})-\bar{\alpha}_{12} \right)\right)}.
\end{align*}
Additionally, when $\frac{\left(1-\bar{\alpha}_2\right)}{P_{12}+P_{21}}\left(\frac{P_{12}}{P_{21}}(1-\bar{\alpha}_{22})-\bar{\alpha}_{12} \right)=0$,
then $\bar{F}(s)$ becomes $(1+s((\bar{\hat{\alpha}}-1)/{2}))^{-1}$, which is the
HT limit of the scaled queue length at departure epochs of the standard $M^X/G/1$ queue without dependencies.
\end{remark}
\begin{remark}
After using Equation \eqref{HT_F(s)} in \eqref{rel:queue} and \eqref{F(z):arbitrary}, it can be shown by substituting $z=e^{-s(1-\rho)}$ and taking $\rho\uparrow 1$ that the HT distribution of the scaled stationary queue length at an \emph{arbitrary epoch} is the same as the HT distribution of the scaled stationary queue length at a departure epoch.
\end{remark}
\section{Numerical example}\label{numerical_results}
In this section we would like to given an example of the interesting situation described in Remark~\ref{remark: N=2}, where we carefully construct the dependencies between successive service times in such a way that they disappear as $\rho$ tends to one.
For simplicity, we take $N=2$, $B(z)=z$ and $\tilde{G}^*_{ij}(s)=\tilde{G}_{ij}(s)$ for all $i,j=1,2$, i.e., there are two customer types, the batch size is one, and customers arriving in an empty system have the same service-time distributions as regular customers. The conditional service times are Erlang distributed random variables, with
\begin{align*}
G_{ij}(x) &= \Big(1-\sum_{m=0}^{k_{ij}-1} \frac{(\mu_{ij} x)^m}{m!}e^{-\mu_{ij}x}\Big)P_{ij},
\end{align*}
where $k_{ij}=i+j,\mu_{ij} > 0,$ $i,j=1, 2$.
We can use Equation \eqref{rel_A_G} to obtain
\begin{align*}
A_{ij}(z)=P_{ij}\left(\frac{\mu_{ij}}{\lambda(1-B(z))+\mu_{ij}}\right)^{k_{ij}}, \quad \text{for } i,j=1,2.
\end{align*}
We choose model parameters $P_{11}=0.9$, $\alpha_{11}=\lambda, \alpha_{12}=3\lambda,\alpha_{21}=10 \lambda$, and $\alpha_{22}=20 \lambda$. To ensure that $\frac{P_{12}}{P_{21}}(1-\bar{\alpha}_{22})-\bar{\alpha}_{12}=0$,
we take $P_{22}=0.951138$.
\begin{figure}
\caption{The mean scaled queue length versus the number of arrivals per time unit.}
\label{fig:EXscaledexample}
\end{figure}
Indeed, it can be observed in Figure \ref{fig:EXscaledexample} that the HT limits of the mean queue lengths in both models, with and without correlated service times, are the same. Note that the \emph{light-traffic} limits, when $\rho\downarrow0$, are also the same. This, however, is caused by the fact that we chose an example with \emph{single} arrivals. In the batch arrival case, the queue-length distributions would also be different in light traffic, due to the correlation between service times of customers inside one batch. It is interesting to see, however, that when $\rho$ tends to $1$, the dependence between subsequent service times no longer influences the mean scaled queue length, and thus the system can be analyzed as an $M/G/1$ queueing system, in this particular example. Furthermore, in Figure \ref{fig:DistXscaledexample}, it can be seen that the density of the scaled queue length converges to the limiting density of an exponential distribution when the traffic intensity $\rho$ approaches $1$.
\begin{figure}
\caption{The density of the scaled queue length.}
\label{fig:DistXscaledexample}
\end{figure}
\section*{Acknowledgments} The research of Abhishek and Rudesindo~N\'u\~nez-Queija is partly funded by NWO Gravitation project {\sc Networks}, grant number 024.002.003. The authors thank Onno Boxma (Eindhoven University of Technology) and Michel Mandjes (University of Amsterdam) for helpful discussions.
\end{document} | math |
لسہٕ خان فِدا صٲبن چھُ " | kashmiri |
Some companies may require a 10% deposit, some may not require any. For larger projects, you'll almost always need to give a deposit, and it's good to ask how much the deposit will be before going forward with a project. The Berthoud contractor will know how much they need for a deposit when they've given you the estimate, so ask early.
Always ask for references, and always check them. Keep in mind though that the only references you'll get are people who the contractor is certain will give him a good review. You'd also be well advised to check online review sources in Berthoud as well, which will give you a broader view of customer satisfaction for a given company. | english |
مے گوو یٔژ شوق زین گُفتارِ خوشبو بہ کشمیری زبان زیباکرن بو | kashmiri |
یا تٔتِہ چُھ تَمیٚوک اِنفراسٹرکچا چُھ آسن کوچِنگ سنٹَرَن سُہ چُھ | kashmiri |
میڈیکل کالجَن منٛز چھِ سِکمٕز تہٕ سرینگرَس منٛز گورنمنٹ میڈیکل کالج تہٕ ایمٕز وجے پوُر شٲمِل | kashmiri |
Simple and addictive puzzle gameplay, stunning graphics!
Well-loved match-3 puzzle genre, a lot of secrets and mysteries.
Launched Mar 25, 2019 (28 days ago). | english |
The text examined here, Sūr ī Saxwan, is a banquet speech which I thought may be of interest. The text is a blessing of a banquet, and of the hosts and guests, by a eulogist. It should be noticed, however, that there is a religious / sacrificial aspect to the speech. The text is also of interest for it provides information on Sasanian court culture, including administrative structure and courtier hierarchy. One can imagine a banquet at the court at the time of Xusrō I or Xusrō II, where the ranking of guests is apparent and their functions are emphasized. Eulogists were taught how to go about blessing the deities, the king and the courtiers. The order of dignitaries mentioned is also interesting in that it may provide a glimpse into the now lost Sasanian Gāh-nāmag (Notitia Dignitatum), similar to the extant Armenian text (Gāhnāmak) regarding the Naxarars and the Armenian court. This list may also be compared with the Notitia Dignitatum of the Roman world, albeit a much shorter version.
But the Sūr ī Saxwan is more similar in content to another text which is found in Late Antiquity in the Mediterranean world. This is the κλητορολόγιον of Philotheos, completed in 899 CE. The word κλητορολόγιον is very much linked with κλεσισ “invitation” and κλητοριον “banquet,” which is very much matched with our text, usually known as a dinner speech. The second chapter of the κλητορολόγιον is important in that it lists the highest dignitaries who join the emperor’s table: the Patriarch of Constantinople, Caesar, and other dignitaries. One sees a similar list in the Sur ī Saxwan, but the progression of the list of deities, heavens and offices mentioned is Zoroastrian in nature. The text begins with an order that is both spiritual and corporeal. First, Ohrmazd is mentioned, followed by the Amaharspandān (Holy Immortals) who are said to be in paradise, then Ohrmazd’s name is repeated. Following this, the seven heavens are mentioned, from the lowest station to the highest where Ohrmazd resides. This is followed by a list of the seven Kišwars (climes or continents), finishing with the central clime of Xwanirah. Then, the three sacred fires are praised, followed by Mihr, Srōš, Rašn, Wahrām, Wāy, Aštād and Frawahr.
After the mention of the deities, the corporeal order of things begins. Naturally, the Šāhān Šāh (King of Kings) is mentioned first. Then are listed princes of the blood, the Grand Minister, the Generals of the four quarters of the empire, Judge of the Empire, the Chief Councilor of the Mages, and the performer of the Drōn ceremony. One may be able to make several connections between the spiritual and corporeal worlds represented, since the realm of Ohrmazd and his cohorts is mirrored by the Šāhān Šāh and his court. Enumerating the order of the courtiers also gives us a general view of the gāh or place of particular dignitaries relative to the King of Kings. The text itself may be divided into two sections: The first part (passages 1-17) which is the before-banquet speech, and the second part (passages 18-22) which is the after-banquet speech, where the eulogist is full of food and wine, and gives thanks to the deities and the host.
1) It is befitting to say and consider gratitude for the Gods and the Good Ones at every moment and time, especially at such a day in such a manner.
2) Listen you good ones who have here so that I speak to praise this banquet, of the Gods and gratitude towards this host.
3) May it be worthy of all offerings: Worthy of all the offering (is) the Lord Ohrmazd, who among the spiritual and material world is the greatest, who created all of the creatures and creations, (and) it is its guardian and preserver.
4) Worthy of all the offerings (are) these seven Holy Immortals who are in Paradise, Ohrmazd, Wahman, and Ardwahišt and Šahrewar and Spandarmad and Hordād and Amurdād.
5) Worthy of all the offerings (are) these seven heavens which through arrangement are above (one another): one at cloud-station, two at star-station, three at moon-station, four at sun-station, five at Harborz-station, six at [Endless Light], seven at Rōšn Garōdmān, full of light, beautiful radiance, full of goodness, full of beneficence, which is before the Lord Ohrmazd himself, ruling over the spiritual realm which are these fifteen (and) the seven (climes) which are these seven: Arzah and Sawah and Fradadafš and Wīdadfš and Wōrūbarist and Wōrūjariš, which in the middle is the glorious Xwanīrah, is the store of many people (who are) full of goodness.
6) Worthy of all the offerings (are) Ādur-farrōybāy and Ādur-Gušnasp and Ādur-Burzēnmihr and other sacred fires and fires seated at their place of creation (i.e., designated place), may they always be burning, always worshipped, and always (receiving) offerings first.
7) Worthy of all the offerings (is) Mihr, possessor of the wide pastures, and Srōš the strong, and Rašn the truthful, and Wahrām the powerful, and Wāy the good and the Good Religion of the Mazda-worshipping religion, and Aštād the prominent creator of the corporeal world, and Frawahr of the righteous ones.
8) Worthy of all offerings (are) the great and good spirits who at the time of Sīh Rōzag (each of their names) are revealed.
9) Worthy of all offerings (is) the King of Kings, foremost of men.
10) Worthy of all offerings (are) the principal sons of the king, most fortunate of the foremost creatures, most necessary in the corporeal world.
11) Worthy of all offerings (is) the Wuzurg Framādār, who in greatness is great and in sovereignty is the sovereign and among the created (i.e., men) is greater and better.
12) Worthy of all offerings (is) the Spāhbed of Xwarāsān, worthy of all offerings (is) the Spāhbed of Xwarwarān, worthy of all offerings (is) the Spāhbed of Nēmrōz.
13) Worthy of all offerings (is) the Chief Judge of the Empire (Šahr Dādwarān).
14) Worthy of all offerings (is) the Chief Councilor of the Mages (Mowān Handarzbed) and worthy of all offerings (is) the Leader of a Thousand (Hazārbed), worthy of all offerings (is) the performer of the Drōn ceremony.
15) Worthy of all offerings is the great and good (things) which the Gods have provided in this meal, may he quickly give sovereignty to Ērānšahr, and splendor amidst it, as it was during the sovereignty of Jamšēd of good herds, (may) the day of blessed good ones continue with pleasure, (may) the Gods accept it a thousand times, and also bless the man who is the host.
16) Especially may he bless this that for his own people, (provide) health, long life and increase in wealth, may it be in this way as is it manifest from the Avesta.
17) When they praise us, it is as if all of the material world will become more pleasant and continuously bless this house that it increase, in this house many swift horse and glory, manly man, able in a gathering to speak with reason (and) memory, and have much gold with silver, much barley with wheat, much storage of goodness and blissful and delightful, good time and good year and good month and good day and goodness for this host (and) for being better.
18) Thanks to Ohrmazd, thanks to Holly Immortals, and thanks priests (asrōnān) and thanks warriors (artēšdārān) and thanks husbandmen (wāstaryōšān) and thanks artisans (hutūxšān) and thanks the fires of the material world, thanks cooks and thanks entertainers and thanks the guardians of the palace, thanks this host who planned and prepared and arranged this day, good is our food and grand is our banquet, and excellent is our gathering and praiseworthy, and there is no other thing greater than thoughtful speech and action.
19) But I must say more before you good ones, that I am satiated from food and full of wine and blissful from pleasure; but it is not possible to praise the Gods and bless the good ones completely, you good ones who have come here, whoever knows to say it better say it.
20) Because I am evermore joyous, because I (am) buzzed on the account of much wine I have drank, (I) will sleep pleasantly and I will dream of the Gods and will rise well and will be diligent in doing work and in deed, because from the beginning of creation to the end, his work is evermore joyous, whom the Gods value his diligence through righteousness. May it be so, it will be so.
21) May there come blessing in the manner that I have said, the width of earth and the length of river and the height of sun.
22) Finished with salutations, happiness and pleasure unto every righteous doer.
* The complete version of this article can be found in T. Daryaee, “The Middle Persian Text Sur i Saxwan and the Late Sasanian Court,” Des Indo-Grecs aux Sassanides: Donnees pour l’historie et la geographie historique , Res Orientales XVII, 2007, pp. 65-72.
Notitia dignitatum omnium tam civilium quam militarium in partibus Occidentis (Latin version), for the East.
“Philotheos, Kletorologion of,” Dictionary of Byzantium, vol. 3, p. 1662.
For a discussion of the titles which appear here see R. Gyselen, La Géographie administrative de l’empire Sassanide, Les témoignages sigillographiques, Peeters, Leuven, 1989.
The wuzurg framādār was certainly the most important personage in the court after the King of Kings and the princes from the fourth century CE onwards. The holders of this title include Xusrō Yazdgerd (from the Syriac sources as harmadārā rabbā, see J.B. Chabot, Synodicon Orientale ou Recueil de synods nestoriens, Paris, 1902, p. 260; Mihr-Narseh in the fifth century CE, see W.B. Henning, “The Inscription of Firuzabad,” Asia Major, vol. 4, 1954, pp. 99-100. For other figures who may have been a wuzurg framādār see M.L. Chaumont, “Framadār,” Encyclopaedia Iranica, ed. E. Yarshater, Bibliotehca Persica Press, 2001, pp. 125-126. While wuzurg framadār appears in the early Sasanian inscription, wuzurg framādār (with the long ā) is matched by the orthography of the seals from sixth and seventh centuries CE which also appears in the text under study (I would like to thank R. Gyselen for bringing this fact to my attention).
The omission of the spāhbed of Abāxtar / Ādurbādagān in this text certainly suggests the religious / ritualistic nature of the text where hamāg zōhr can not be directed towards it. For example in the Bundahišn (XIV.27-28) because Mašānē poured milk as libation towards the north / abāxtar, the demons became stronger, see F. Pakzad, Bundahišn: Zoroastrische Kosmogonie und Kosmologie, Kritische Edition, Ancient Iranian Studies Series, Centre for the Great Islamic Encyclopaedia, Tehran, 2005, pp. 187-188. I would suggest that is the reason for the omission of this spāhbed in this text as compared with others, see T. Daryaee, Šahrestānīhā ī Ērānšahr, A Middle Persian Text on Late Antique Geography, Epic, and History, Mazda Publishers, Costa Mesa, 2002, pp. 7-11. For some other suggestions see Gnoli, op. cit., p. 269. For the latest and comprehensive evidence for the spāhbeds see R. Gyselen, The Four Generals of the Sasanian Empire: Some Sigillographic Evidence, Roma, 2001.
This title echoes the early hāmšahr dādwar “Judge of the whole empire,” in the third century CE which was held first by Kerdīr. We come across the title again for Mār Qardag who held the title of šahr dādwar, P. Bedjan, Histoire de Mar-Jabalaha, de trios autres patriarches, d’un prêtre et de deux laïques nestoriens, Paris, 1895, p. 228. On Mār Qardag see now J. Walker, The Legend of Mar Qardagh: Narrative and Christian Heroism in Late Antique Iraq, California University Press, 2006. In the Mādīyān ī Hazār Dādestān we have the title of šahr dādwarān dādwar, which according to M. Shaki was introduced during the reign of Yazdgerd II (439-457 CE), “Dādwar, Dādwarīh,” Encyclopaedia Iranica, ed. E. Yarshater, Mazda Publishers, 1993, p. 558.
It appears that the holder of the office had both administrative, but more importantly legal skills. We also come across this title in the Armenian sources as mogac‘ anderjapet and movan anderjapet (Ełīšē, 8, p. 315; Łazar P‘arpec‘i, 2.55, 57, p. 326, 345, 349; and P‘awstos Buzand, 4.47, all quoted by M.L. Chaumont, “Andarzbed,” Encyclopaedia Iranica, ed. E. Yarshater, Routledge & Kegan Paul, p. 23.
It appears that the office of hazārbed came into being in the Sasanian period and that in the late third, early fourth centuries CE at the court of king of kings, Narseh a Affarban held this title and was one of the two officers (along with the hargbed) that remained with him when the Roman representatives came to the court, see R.M. Shayegan, “Hazārbed,” Encyclopaedia Iranica, ed. E. Yarshater, vol. XII, 2005, p. 94.
As any important dinner a drōn-yaz, someone who is in charge of making ritual offering of a portion of the food to the deities must have been present at the court. The office is rarely mentioned and so the Sūr ī Saxwan is important for this title. For the Drōn ceremony see J.K. Choksy, “Drōn,” Encyclopaedia Iranica, ed. E. Yarshater, vol. VII, Mazda Publishers, 1996, pp. 554-555.
Tavadia wylwk “man” and has added the number 1000 after it; Mazdapour has nylng “spell.” Orian follows Tavadia and inserts the 1000 as well. I would suggest wyl’d “arrange, prepare,” in the sense of arranging the heavens.
There is a lacuna here, but from Zoroastrian cosmology it is certain that we should have ’sl-lwšnyh. See A. Panaino, “Uranographia Iranica I: The Three Heavens in the Zoroastrian Tradition and the Mesopotamian Background,” Au Carrefour des religions, mélanges offert à Philippe Gignoux, Res Orientales, vol. VII, Paris, 1995, pp. 205-221. Orian has left the lacuna.
Tavadia emends to pwl-GDE and Orian has accepted his reading. The manuscript, however, clearly shows pwl-hwyh which Mazdapour suggests as well.
DP provides the numeral 15 which makes sense here as Ohrmazd along with the Amaharspandān and the seven Wahišts would come to fifteen (Ohrmazd counted twice).
Tavadia has also inserted kyšwr correctly as the list of the climes follows it. Mazdapour does not insert the word, nor does Orian.
Mazdāpour and Orian do not omit.
Tavadia inserts d’tbl after štrٰٰ which is not necessary; Mazdapour emends the word to d’t and Orian reads it as d’tbl. The MK manuscript which is now in the process of being published clearly shows that the word is štr. I would like to thank Professor A. Hintze who will publish the manuscript for providing me with the pages of MK containing this text.
Tavadia makes the only logical suggestion as the text has pšnwst plm’nٰ to read as pyh’n and connect the last three letters with the next word as W stpl. This is followed by Mazdapour and Orian who read the first word as pyh.
If the preceding reading is accepted the next word can be emended as wsn’d.
Decolonizing Persian History – Review of British Museum's "Forgotten Empire, the World of Ancient Persia"
Nuts About Fruits – What Fruits and Nuts to Eat in Ancient Persia? | english |
कांग्रेस ने इस मामले के पीछे प्रधानमंत्री नरेन्द्र मोदी और भाजपा का हाथ बताया नई दिल्ली। नेशनल हेराल्ड मामले में कांग्रेस अध्यक्ष सोनिया गांधी व उपाध्यक्ष राहुल गांधी को जमानत मिल गई है। दोनों को ५०-५० हजार रुपये के निजी मुचलके पर जमानत दे दी गई। कपिल सिब्बल ने सोनिया व राहुल गांधी की ओर से दलील दी और जमानत बॉन्ड भरने की बात मान ली। याचिकाकर्ता सुब्रमण्यम स्वामी ने जमानत देने का विरोध किया और कहाकि दोनों देश छोड़कर भाग सकते हैं।इससे पहले कांग्रेस ने इस मामले के पीछे प्रधानमंत्री नरेन्द्र मोदी और भाजपा का हाथ बताया। कांग्रेस नेताओं ने प्रेस कांफ्रेंस कर आरोप लगाया कि सुब्रमण्यम स्वामी को सोनिया गांधी व राहुल गांधी पर केस करने का इनाम सरकारी मकान व जेड सिक्योरिटी के रूप में दिया गया- सोनिया गांधी और राहुल गांधी कोर्ट पहुंचे- प्रियंका गांधी लेंगी सोनिया व राहुल गांधी की जमानत- गुलाम नबी आजाद के घर कांग्रेस नेताओं की बैठक- विपक्ष की राज्य सरकारों को गिराने का षडय़ंत्र रचा जा रहा है- पीएम के इशारे पर सबकुछ हो रहा है- विपक्ष मुक्त भारत केन्द्र सरकार का एजेंडा- सुब्रमण्यम स्वामी को नियमों के परे जाकर जेड सिक्योरिटी और सरकारी मकान दिया- नेशनल हेराल्ड मामले में पीएम नरेन्द्र मोदी, उनके मंत्री और भाजपा का हाथ- कांग्रेस की प्रेस कांफ्रेंस- दोपहर डेढ़ बजे गुलाम नबी आजाद के घर मिलेंगे कांग्रेस नेता, राहुल-सोनिया भी होंगे शामिल- राहुल गांधी घर से निकले- रायबरेली से आए कांग्रेसी कार्यकर्ताओं का कोर्ट के बाहर प्रदर्शन।- भाजपा का सोनिया की पेशी से लेना देना नहीं: केन्द्रीय मंत्री रविशंकर- सुब्रहमण्यम स्वामी के पीछे पीएम मोदी का हाथ: कांग्रेसजमानत लेंगे सोनिया-राहुलसोनिया ने शुक्रवार को कहा कि वह और राहुल कोर्ट में पेश होंगे। हालांकि उन्होंने बेल बॉन्ड के सवाल पर कोई जवाब नहीं दिया। ऐसी अटकलें हैं कि शायद दोनों नेता जमानत के लिए आवेदन न करें। हालांकि सूत्रों के हवाले से यह भी खबर है कि सोनिया-राहुल कोर्ट की कार्यवाही में पूरा साथ देेंगे और अगर जमानत लेने को कहा जाएगा, तो वे आवेदन भी करेंगे। अभी तक सोनिया-राहुल ने बेल बॉन्ड नहीं भरा है। अगर आरोपी जमानत के लिए आवेदन नहीं करता है तो अदालत उसे न्यायिक हिरासत में ले सकती है। इस मामले के अन्य आरोपियों मोतीलाल वोरा, ऑस्कर फर्नांडीस के भी कोर्ट में पेश होने की संभावना है।कोर्ट पर सुरक्षाबलों का पहरावहीं कोर्ट के आसपास सुरक्षाबलों को तैनात कर दिया गया है। इसके तहत एसपीजी, सीआरपीएफ,दिल्ली पुलिस के ५०0 जवान तैनात किए गए है और सीसीटीवी से चप्पे चप्पे पर नजर रखी जा रही है। इसी बीच कांग्रेस समर्थकों ने सोनिया गांधी के समर्थन में और सुब्रमण्यम स्वामी के विरोध में जगह-जगह पोस्टर लगाए हैं। इस मामले को लेकर कांग्रेस ने पीएम मोदी पर राजनीतिक प्रतिशोध का आरोप लगाया। प्रवक्ता रणदीप सिंह सुरजेवाला ने कहाकि सुब्रमण्यम स्वामी पीएम मोदी के मोहरे हैं।
यह भी पढ़े : नेशनल हेराल्ड केस: सोनिया-राहुल को क्या मिला, जानिए पूरा मामला
जेठमलानी का सोनिया गांधी को फ्री में केस लडऩे का ऑफर नेशनल हेराल्ड केस में कोर्ट में पेश होंगी सोनिया, बेल बॉन्ड पर चुप्पी
लालू यादव व राबड़ी देवी का व्विप दर्जा खत्म, अब एयरपोर्ट पर होगी जांच | hindi |
مُتعلقہٕ مظاہرن منٛز سپرنووا دھماکہ، گاما شعاعین ہنٛد پھٹنہٕ، کواسار، بلزار، پلسر، تہٕ کائنأتی مائکروویو پس منظرچ تابکاری چِھ شأمل۔ | kashmiri |
नवादा : नगर थाना क्षेत्र के भदौनी हाट पर निवासी बाबूलाल चौधरी की हत्या के बाद उनके छोटे भाई मुनेश्वर चौधरी ने ट्रेन से कटकर अपनी जान दे दी. मंगलवार की सुबह केजी रेलखंड पर सनोखरा गांव के समीप अज्ञात शव बरामद किया गया था.
बुधवार को सदर अस्पताल में परिजनों ने मृतक की पहचान की. मृतक मुनेश्वर चौधरी के पुत्र ने बताया कि १५ सितंबर की रात बड़े पापा बाबूलाल चौधरी की हत्या की गई थी. तब से मेरे पापा यह कह रहे थे कि बड़े भैया जिंदा नहीं रहे तो मैं भी नहीं बचूंगा. १६ सितंबर को बड़े पापा के दाह संस्कार के बाद वे गायब हो गए. काफी खोजबीन के बाद भी पता नहीं चला.
आज सुबह सदर अस्पताल में अज्ञात शव की जानकारी मिली तो यहां आकर पहचान की. बता दें कि १५ सितंबर को बाबूलाल चौधरी की हत्या पीट पीटकर की गई थी. उस मामले में पड़ोस के ही पवन चौधरी को गिरफ्तार किया गया था. इधर परिवार के दो सदस्यों की मौत के बाद परिजनों पर दुखों का पहाड़ टूट पड़ा है.
ट्रेन से कटकर दे दी जान | hindi |
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| code |
\begin{document}
\mathfrak{t}itle{On a Diophantine problem with two primes and $s$ powers of two}
\author{A.~LANGUASCO and A.~ZACCAGNINI}
\date{}
\mathfrak{m}aketitle
\begin{abstract}
We refine a recent result of Parsell \cite{Parsell2003}
on the values of the form
$
\lambda_1p_1
+
\lambda_2p_2
+
\mathfrak{m}u_1 2^{m_1}
+
\dotsm
+
\mathfrak{m}u_s 2^{m_s},
$
where $p_1,p_2$ are prime numbers, $m_1,\dotsc, m_s$ are positive
integers, $\lambda_1 / \lambda_2$ is negative and irrational and
$\lambda_1 / \mathfrak{m}u_1$, $\lambda_2/\mathfrak{m}u_2 \in \mathbb{Q}$. \par
\mathfrak{m}edskip
\noindent
2000 AMS Classification: 11D75, 11J25, 11P32, 11P55. \par
\noindent
Keywords: Goldbach-type theorems, Hardy-Littlewood method, diophantine inequalities.
\end{abstract}
\section{Introduction}
\allowdisplaybreaks
In this paper we are interested to study the values of the form
\begin{equation}
\label{linear-form}
\lambda_1p_1
+
\lambda_2p_2
+
\mathfrak{m}u_1 2^{m_1}
+
\dotsm
+
\mathfrak{m}u_s 2^{m_s},
\end{equation}
where $p_1,p_2$ are prime numbers, $m_1,\dotsc, m_s$
are positive integers, and the coefficients $\lambda_1$, $\lambda_2$ and
$\mathfrak{m}u_1$, \dots, $\mathfrak{m}u_s$ are real numbers satisfying suitable relations.
This is clearly a variation of the so-called Goldbach-Linnik problem,
\emph{i.e.} to prove that every sufficiently large even integer is
a sum of two primes and $s$ powers of two,
where $s$ is a fixed integer.
Concerning this problem the first result was proved by Linnik himself
\cite{Linnik51,Linnik53} who remarked that a suitable $s$ exists
but he gave no explicitly estimate of its size. Other results
were proved by Gallagher \cite{Gallagher1975},
Liu, Liu \& Wang \cite{LiuLW98a,LiuLW98b, LiuLW99},
Wang \cite{Wang99} and Li \cite{Li2000, Li2001}.
Now the best conditional result is due to Pintz \& Ruzsa
\cite{PintzR2003} and Heath-Brown \& Puchta \cite{Heath-BrownP2002}
($s=7$ suffices under the assumption of the Generalized Riemann Hypothesis),
while, unconditionally, it is due to Heath-Brown \& Puchta
\cite{Heath-BrownP2002} ($s = 13$ suffices).
Elsholtz, in unpublished work, improved it to $s=12$.
We should also remark that Pintz \& Ruzsa announced a proof for the
case $s=8$ in their paper \cite{PintzR2006} which is as yet
unpublished.
Looking for the size of the exceptional set of the Goldbach problem
we recall the fundamental paper
by Montgomery-Vaughan \cite{MontgomeryVaughan1975}
in which they showed that the number of even integers up to $X$ that
are not the sum of two primes is $\ll X^{1-\delta}$.
Pintz recently announced that $\delta=1/3$ is admissible
in the previous estimate. Concerning the
exceptional set for the Goldbach-Linnik problem, the authors
of this paper in a joint work with Pintz \cite{LanguascoPZ2007}
proved that for every $s \ge 1$, there
are $\ll X^{3/5} (\log X)^{10}$ even integers in $[1,X]$ that are not
the sum of two primes and $s$ powers of two.
This obviously corresponds to the case
$\lambda_1 = \lambda_2 = \mathfrak{m}u_1 = \dots = \mathfrak{m}u_s = 1$.
In diophantine approximation several results were proved
concerning the linear forms with primes that, in some
sense, can be considered as the real analogous
of the binary and ternary Goldbach problems.
On this topic we recall the papers by
Vaughan \cite{Vaughan1974a,Vaughan1974b,Vaughan1976},
Harman \cite{Harman1991},
Br\"udern-Cook-Perelli \cite{BrudernCP1997},
and Cook-Harman \cite{CookH2006}.
Concerning the problem in \eqref{linear-form}, we can consider it
as a real analogous of the Goldbach-Linnik problem.
We have the following
\begin{Nonumthm}
Suppose that $\lambda_1,\lambda_2$ are real numbers
such that $\lambda_1/\lambda_2$ is negative and irrational
with $\lambda_1>1$, $\lambda_2<-1$ and
$\vert \lambda_1/\lambda_2 \vert \geq1$.
Further suppose that $\mathfrak{m}u_1, \dotsc, \mathfrak{m}u_s$ are nonzero
real numbers such that
$\lambda_i/\mathfrak{m}u_i \in \mathbb{Q}$, for $i\in\{1,2\}$, and denote by
$a_i/q_i$ their reduced representations as rational numbers.
Let moreover $\eta$ be a sufficiently small positive constant such that
$\eta<\mathfrak{m}in(\lambda_1/a_1;\vert \lambda_2/a_2 \vert)$.
Finally,
for $\lambda_1/\lambda_2$ transcendental, let
\begin{equation}
\label{s0-def-transc}
s_0
=
2
+
\mathfrak{m}athcal{B}igl\lceil
\frac
{
\log (2C(q_1,q_2)\vert \lambda_1 \lambda_2\vert)
-
\log \eta
}
{-\log (0.91237810306)}
\mathfrak{m}athcal{B}igr\rceil,
\end{equation}
while, for $\lambda_1/\lambda_2$ algebraic, let
\begin{equation}
\label{s0-def-alg}
s_0
=
2+
\mathfrak{m}athcal{B}igl\lceil
\frac
{
\log (2C(q_1,q_2)\vert \lambda_1 \lambda_2\vert)
-
\log \eta
}
{-\log (0.83372131685)}
\mathfrak{m}athcal{B}igr\rceil,
\end{equation}
where $C(q_1,q_2)$
verifies
\begin{equation}
\label{Cq1q2-def}
C(q_1,q_2)
=
\mathfrak{m}athcal{B}igl(
\log{2}
+
C \cdot
\mathfrak{S}^{\prime}(q_1)
\mathfrak{m}athcal{B}igr)^{1/2}
\mathfrak{m}athcal{B}igl(
\log{2}
+
C \cdot
\mathfrak{S}^{\prime}(q_2)
\mathfrak{m}athcal{B}igr)^{1/2},
\end{equation}
with
\begin{equation}
\label{singseries-def}
\mathfrak{S}^{\prime}(n)
=
\prod_{\substack{p \mathfrak{m}id n \\ p > 2}} \frac{p - 1}{p - 2}
\end{equation}
and
$C = 10.0219168340$.
\noindent
Then for every real number $\gamma$ and every integer
$s\geq s_0$ the inequality
\begin{equation}
\label{main-inequality}
\vert
\
\lambda_1p_1
+
\lambda_2p_2
+
\mathfrak{m}u_1 2^{m_1}
+
\dotsm
+
\mathfrak{m}u_s 2^{m_s}
+
\gamma
\
\vert
<
\eta
\end{equation}
has infinitely many solutions in primes
$p_1,p_2$ and positive integers $m_1,\dotsc, m_s$.
\end{Nonumthm}
The only result on this problem we know is by Parsell \cite{Parsell2003};
our values in \eqref{s0-def-transc}-\eqref{s0-def-alg}
improve Parsell's
one given by
\begin{equation}
\label{Parsell-1}
s_0
=
2+
\mathfrak{m}athcal{B}igl\lceil
\frac
{
\log (2C_1(q_1,q_2)\vert \lambda_1 \lambda_2\vert)
-
\log \eta
}
{-\log (0.954)}
\mathfrak{m}athcal{B}igr\rceil,
\end{equation}
where
\begin{equation}
\label{Parsell-2}
C_1(q_1,q_2)
=
25 (\log 2q_1)^{1/2}(\log 2q_2)^{1/2}.
\end{equation}
Checking the proof in \cite{Parsell2003} one can see that \eqref{Parsell-2}
is in fact
\begin{equation}
\label{Parsell-3}
C_1(q_1,q_2,\epsilon)
=
\mathfrak{m}athcal{B}igl(
1+ C_1 \cdot
\mathfrak{S}^{\prime}(q_1)
\mathfrak{m}athcal{B}igr)^{1/2}
\mathfrak{m}athcal{B}igl(
1+ C_1 \cdot
\mathfrak{S}^{\prime}(q_2)
\mathfrak{m}athcal{B}igr)^{1/2}
+ \epsilon,
\end{equation}
and $C_1 = 11.4525218267$.
Comparing the numerical values involved in
\eqref{s0-def-transc}-\eqref{Cq1q2-def} with \eqref{Parsell-1} and \eqref{Parsell-3},
without considering the contribution of the $\log 2$
which in \eqref{Parsell-3} is replaced by $1$,
we see that the our gain is
about $50$\%
in the transcendental case and about $75$\%
in the algebraic case.
For instance, taking $\lambda_1= \sqrt{3}= \mathfrak{m}u_1^{-1}$,
$\lambda_2= -\sqrt{2}= \mathfrak{m}u_2^{-1}$ and $\eta=1$, we get $s_0= 61$
while for $\lambda_1= \pi = \mathfrak{m}u_1^{-1}$,
$\lambda_2= -\sqrt{2}= \mathfrak{m}u_2^{-1}$ and $\eta=1$, we get $s_0= 119$.
In both cases, \eqref{Parsell-1} gives $s_0=267$.
Moreover we remark that the work of Rosser-Schoenfeld
\cite{RosserS1962} on $n/\varphi(n)$, see Lemma \ref{sing-series-estim} below,
gives for $\mathfrak{S}'(q)$ a sharper estimate than $2\log (2q)$,
used in \eqref{Parsell-2}, for large values of $q$.
With respect to \cite{Parsell2003},
our main gain comes
from enlarging the size of the major arc since this lets us
use sharper estimates on the minor arc.
In particular, on the major arc we replaced the technique used in
\cite{Parsell2003}
with a well-known argument involving the
Selberg integral; this also simplified the actual
work to get a ``good'' major arc contribution.
On the minor arc we used Br\"udern-Cook-Perelli's \cite{BrudernCP1997}
and Cook-Harman's \cite{CookH2006} technique to deal with the exponential sum
on primes ($S(\alpha)$) while, in order to work with the exponential sum over powers of two
($G(\alpha)$), we inserted Pintz-Ruzsa's \cite{PintzR2003} algorithm to estimate
the measure of the subset of the minor arc on which $\vert G(\alpha)\vert$ is ``large''.
These two ingredients lead to
a sharper estimate on the minor arc
and let us improve the size of the denominators in
\eqref{s0-def-transc}-\eqref{s0-def-alg}.
It is in this step that we have to distinguish whether $\lambda_1/\lambda_2$
is an algebraic or a transcendental number; this fact leads to
two different estimates for the minor arc and, \emph{a fortiori},
using Pintz-Ruzsa's algorithm (see Lemma \ref{minor-arc-power-of-two-estim}),
to two different constants in \eqref{G-algebraic}-\eqref{G-transcendental}
and \eqref{s0-def-transc}-\eqref{s0-def-alg}.
A second, less important, gain arises from our Lemma \ref{Dioph-equation}
which improves the values
in \eqref{Cq1q2-def} comparing with
the ones in \eqref{Parsell-3} (obtained in \cite{Parsell2003}, Lemma 3).
Such an improvement
comes from using the Prime Number Theorem
(to get $\log 2$ instead of $1$)
and Khalfalah-Pintz's \cite{KhalfalahP2006}
computational estimates for the number of representations
of an integer as a difference of powers of two, see Lemma \ref{KP-Lemma}.
Finally we remark that assuming a suitable form of the twin-prime conjecture,
\emph{i.e.} $B=1$ in Lemma \ref{BD-Thm2}, we get that \eqref{Cq1q2-def}
holds with $C= 2.5585042082$.
Using the notation
$\bm{\lambda}=(\lambda_1,\lambda_2)$,
$\bm{\mathfrak{m}u}=(\mathfrak{m}u_1,\mathfrak{m}u_2)$, as a consequence of the Theorem we have the
\begin{Nonumcor}
Suppose that $\lambda_1,\lambda_2$ are real numbers
such that $\lambda_1/\lambda_2$ is negative and irrational.
Further suppose $\mathfrak{m}u_1, \dotsc, \mathfrak{m}u_s$ are nonzero
real numbers such that $\lambda_i/\mathfrak{m}u_i \in \mathbb{Q}$, for $i\in\{1,2\}$,
and denote by $a_i/q_i$ their reduced representations as rational numbers.
Let moreover $\eta$ be a sufficiently small positive constant such that
$\eta<\mathfrak{m}in(\vert \lambda_1/a_1\vert ;\vert \lambda_2/a_2 \vert)$ and
$\mathfrak{t}au\geq\eta>0$.
Finally let $s_0=s_0(\bm{\lambda},\bm{\mathfrak{m}u},\eta)$ as defined
in \eqref{s0-def-transc}-\eqref{s0-def-alg}.
Then
for every real number $\gamma$ and every integer
$s\geq s_0$ the inequality
\begin{equation}
\label{general-inequality}
\vert
\
\lambda_1p_1
+
\lambda_2p_2
+
\mathfrak{m}u_1 2^{m_1}
+
\dotsm
+
\mathfrak{m}u_s 2^{m_s}
+
\gamma
\
\vert
<
\mathfrak{t}au
\end{equation}
has infinitely many solutions in primes
$p_1,p_2$ and positive integers $m_1,\dotsc, m_s$.
\end{Nonumcor}
This Corollary immediately follows from the Theorem
since, multiplying by a suitable constant both sides
of \eqref{general-inequality}, we can always reduce ourselves
to study the case $\lambda_1>1$, $\lambda_2<-1$ and
$\vert \lambda_1/\lambda_2 \vert \geq1$.
Hence the Theorem assures us that \eqref{main-inequality}
has infinitely many solutions and the Corollary immediately
follows from the condition $\mathfrak{t}au\geq\eta$.
We finally remark that the condition
about about the rationality of the two ratios $\lambda_i/\mathfrak{m}u_i$,
$i=1,2$,
which, at first sight, could appear a ``weird'' one,
is in fact quite natural in the sense that otherwise the numbers $\lambda x+
\mathfrak{m}u y$, $x,y\in \mathbb{Z}$, are dense in $\mathbb{R}$ by Kronecker's Theorem, see
also the remark after Lemma \ref{Dioph-equation}.
\paragraph{Acknowledgements.}
We would like to thank J\'{a}nos Pintz, Umberto Zannier and Carlo Viola
for a discussion, Imre Ruzsa
for sending us his original U-Basic code for
Lemma \ref{minor-arc-power-of-two-estim}
and Karim Belabas for helping us in improving the performance
of our PARI/GP code for the Pintz-Ruzsa algorithm.
\section{Definition}
Let $\epsilon$ be a sufficiently small positive constant, $X$ be a large parameter,
$M=\vert \mathfrak{m}u_1 \vert + \dotsm +\vert \mathfrak{m}u_s \vert$
and $L=\log_2 (\epsilon X/(2M))$, where $\log_2 v$ is
the base $2$ logarithm of $v$.
We will use the Davenport-Heilbronn variation of the Hardy-Littlewood
method to count the number of solutions
$\mathfrak{m}athfrak{N}(X)$ of the inequality \eqref{main-inequality} with
$\epsilon X \leq p_1, p_2 \leq X$
and $1\leq m_1, \dotsc, m_s \leq L$.
Let now $e(u) = \exp(2\pi i u)$ and
\[
S(\alpha) = \sum_{\epsilon X\leq p \leq X} \log p\ e(p\alpha)
\quad
\mathfrak{t}extrm{and}
\quad
G(\alpha)= \sum_{1 \leq m \leq L} e(2^m\alpha).
\]
For $\alpha\neq 0$, we also define
\begin{equation}
\notag
K(\alpha,\eta)=
\mathfrak{m}athcal{B}igl(\frac{\sin\pi \eta \alpha}{\pi \alpha}\mathfrak{m}athcal{B}igr)^2
\end{equation}
and hence both
\begin{equation}
\label{K-transform}
\widehat{K}(t,\eta)
=
\int_\mathbb{R} K(\alpha,\eta) e(t\alpha ) \mathfrak{m}athrm{d}\alpha
=
\mathfrak{m}ax(0; \eta -\vert t\vert)
\end{equation}
and
\begin{equation}
\label{K-inequality}
K(\alpha,\eta)
\ll
\mathfrak{m}in(\eta^2; \alpha^{-2})
\end{equation}
are well-known facts.
Letting
\begin{equation}
\notag
I(X ; \mathbb{R})
=
\int_{\mathbb{R}}
S(\lambda_1 \alpha) S(\lambda_2 \alpha)
G(\mathfrak{m}u_1 \alpha)\dotsm G(\mathfrak{m}u_s \alpha)
e(\gamma \alpha)
K(\alpha,\eta)
\mathfrak{m}athrm{d} \alpha,
\end{equation}
it follows from \eqref{K-transform} that
\[
I(X ; \mathbb{R})
\ll
\eta \log^2 X \cdot \mathfrak{m}athfrak{N}(X).
\]
We will prove that
\begin{equation}
\label{I-lower-bound}
I(X ; \mathbb{R})
\gg_{s,\bm{\lambda},\epsilon}
\eta^2 X (\log X)^s
\end{equation}
thus obtaining
\[
\mathfrak{m}athfrak{N}(X) \gg_{s,\bm{\lambda},\epsilon} \eta X (\log X)^{s-2}
\]
and hence the Theorem follows.
To prove \eqref{I-lower-bound}
we first dissect the real line in the major, minor and trivial arcs,
by choosing $P=X^{1/3}$ and letting
\begin{equation}
\label{dissect-def}
\mathfrak{m}athfrak{M} = \{\alpha\in \mathbb{R}: \vert \alpha \vert \leq P/X \},
\quad
\mathfrak{m} = \{\alpha\in \mathbb{R}: P/X<\vert \alpha \vert \leq L^2\},
\end{equation}
and $\mathfrak{t} = \mathbb{R} \setminus(\mathfrak{m}athfrak{M}\cup\mathfrak{m})$.
Accordingly, we write
\begin{equation}
\label{integral-dissect}
I(X ; \mathbb{R})
=
I(X ; \mathfrak{m}athfrak{M}) + I(X ; \mathfrak{m}) + I(X ; \mathfrak{t}).
\end{equation}
We will prove that the inequalities
\begin{equation}
\label{major-goal}
I(X ; \mathfrak{m}athfrak{M})
\geq
c_1 \eta^2 X L^s,
\end{equation}
\begin{equation}
\label{trivial-goal}
\vert
I(X ; \mathfrak{t})
\vert
=
\odi{XL^s},
\end{equation}
hold for all sufficiently large $X$, and
\begin{equation}
\label{minor-goal}
\vert
I(X ; \mathfrak{m})
\vert
\leq
c_2(s) \eta X L^s,
\end{equation}
where $c_2(s)>0$ depends on $s$,
$c_2(s)\mathfrak{t}o 0$ as $s\mathfrak{t}o+\infty$, and
$c_1=c_1(\epsilon,\bm{\lambda})>0$ is a constant such that
\begin{equation}
\label{constant-condition}
c_1 \eta -c_2(s) \geq c_3\eta
\end{equation}
for some absolute positive constant $c_3$ and $s\geq s_0$.
Inserting \eqref{major-goal}-\eqref{constant-condition} into \eqref{integral-dissect},
we finally obtain that \eqref{I-lower-bound} holds thus proving the Theorem.
\section{Lemmas}
Let $1\leq n \leq (1-\epsilon)X/2$ be an integer and $p,p^{\prime}$ two prime numbers.
We define the twin prime counting function as follows
\begin{equation}
\label{Z-def}
Z(X; 2n)
=
\sum_{\epsilon X \leq p \leq X}
\sum_{\substack{p^{\prime}\leq X \\ p^{\prime} -p =2n}} \log p \log p^{\prime}.
\end{equation}
Moreover we denote by $\mathfrak{S}(n)$ the singular series and set
$\mathfrak{S}(n) = 2 c_0 \mathfrak{S}^{\prime}(n)$ where $\mathfrak{S}^{\prime}(n)$
is defined in \eqref{singseries-def} and
\begin{equation}
\label{c0-def}
c_0
=
\prod_{p > 2} \mathfrak{m}athcal{B}ig( 1 - \frac{1}{(p-1)^2} \mathfrak{m}athcal{B}ig).
\end{equation}
Notice that $\mathfrak{S}^{\prime}(n)$ is a multiplicative function.
According to Gourdon-Sebah \cite{GourdonS2001},
we can also write that
$0.66016181584<c_0<0.66016181585$.
Let further $k\geq 1$ be an integer and $r_{k,k}(m)$ be the
number of representations of an integer $m$ as
$\sum_{i=1}^{k} 2^{u_i} - \sum_{i=1}^{k} 2^{v_i}$, where
$1\leq u_i, v_i \leq L$ are integers, so that
$r_{k,k}(m)=0$ for sufficiently large $\vert m\vert$.
Define
\[
S(k,L) =
\sum_{m\in \mathbb{Z}\setminus \{0\}}
r_{k,k}(m)
\mathfrak{S}(m).
\]
The first Lemma is about the behaviour of $S(k,L)$
for sufficiently large $X$.
\begin{Lemma}[Khalfalah-Pintz \cite{KhalfalahP2006}, Theorem 2]
\label{KP-Lemma}
For any given $k\geq 1$,
there exists $A(k)\in \mathbb{R}$ such that
\[
\lim_{L\mathfrak{t}o +\infty}
\mathfrak{m}athcal{B}igl(
\frac{S(k,L)}{2L^{2k}}
-1
\mathfrak{m}athcal{B}igr)
=
A(k).
\]
\end{Lemma}
Moreover they also proved numerical estimates
for $A(k)$ when $1\leq k\leq 7$. We will just need
\begin{equation}
\label{A(1)-estim}
A(1) < 0.2792521041.
\end{equation}
The second lemma is an upper bound for the multiplicative part of
the singular series.
\begin{Lemma}
\label{sing-series-estim}
For $n\in \mathbb{N}$, $n\geq 3$, we have that
\[
\mathfrak{S}^{\prime}(n)
<
\frac{n}{c_0\varphi(n)}
<
\frac{e^{\gamma} \log \log n}{c_0}
+
\frac{2.50637}{c_0 \cdot \log \log n},
\]
where $\gamma=0.5772156649\dotsc$ is the Euler constant.
\end{Lemma}
\begin{Proof}
Let $n\geq 3$.
The first estimate follows immediately remarking
that
\[
\mathfrak{S}^{\prime}(n)
=
\prod_{\substack{p \mathfrak{m}id n \\ p > 2}}
\frac{(p - 1)^2}{p(p - 2)}
\prod_{\substack{p \mathfrak{m}id n \\ p > 2}}
\frac{p}{p - 1}
<
\prod_{\substack{p > 2}}
\frac{(p - 1)^2}{p(p - 2)}
\prod_{\substack{p \mathfrak{m}id n}}
\frac{p}{p - 1}
=
\frac{1}{c_0}
\frac{n}{\varphi(n)}.
\]
The second estimate is a direct application of Theorem 15 of
Rosser and Schoenfeld \cite{RosserS1962}.
\end{Proof}
Letting $f(1)=f(2)=1$ and
$ f(n) = n/(c_0\varphi(n)) $
for $n \geq 3$,
we can say that the inequality $\mathfrak{S}^{\prime}(n) \leq f(n)$
is sharper than Parsell's estimate $\mathfrak{S}^{\prime}(n) \leq 2 \log(2 n)$,
see page 7 of \cite{Parsell2003}, for every $n\geq 1$.
Since it is clear that computing the exact value of $f(n)$ for
large values of $n$ it is not easy (it requires the knowledge of
every prime factor of $n$), we also remark that
the second estimate in Lemma \ref{sing-series-estim} leads to
a sharper bound than $\mathfrak{S}^{\prime}(n) \leq 2 \log(2 n)$
for every $n\geq 14$.
The next lemma is a famous result
of Bombieri and Davenport.
\begin{Lemma}[Theorem 2 of Bombieri-Davenport \cite{BombieriD1966}]
\label{BD-Thm2}
There exists a positive constant $B$ such that,
for every positive integer $n$, we have
\[
Z(X; 2n)
<
B\ \mathfrak{S}(n) X,
\]
where $Z(X; 2n)$ and $\mathfrak{S}(n)$ are defined in
\eqref{singseries-def} and \eqref{Z-def}-\eqref{c0-def},
provided that $X$ is sufficiently large.
\end{Lemma}
Chen \cite{Chen1978} proved that $B =3.9171$
can be used in Lemma \ref{BD-Thm2}.
The assumption of a suitable form of the twin prime conjecture,
\emph{i.e.} $Z(X; 2n) \sim \mathfrak{S}(n) X$
for $X\mathfrak{t}o +\infty$, implies that in this case we can take $B=1$.
Now we state some lemmas we need to estimate $I(X ;\mathfrak{m})$.
The first one is
\begin{Lemma}
\label{Dioph-equation}
Let $X$ be a sufficiently large parameter and let $\lambda, \mathfrak{m}u \neq
0$ be two real numbers such that $\lambda / \mathfrak{m}u\in \mathbb{Q}$.
Let $a,q\in \mathbb{Z}\setminus\{0\}$ with $q>0$, $(a,q)=1$ be such
that $\lambda/\mathfrak{m}u= a/q$.
Let further $0<\eta < \vert \lambda/a \vert $.
We have
\[
\int_{\mathbb{R}}
\vert
S(\lambda \alpha) G(\mathfrak{m}u \alpha)
\vert^2
K(\alpha, \eta)
\mathfrak{m}athrm{d}\alpha
<
\eta X L^2
\mathfrak{m}athcal{B}igl(
(1-\epsilon)\log{2}
+
C \cdot
\mathfrak{S}'(q)
\mathfrak{m}athcal{B}igr)
+
\Odip{M,\epsilon}{\eta X L},
\]
where $C = 10.0219168340$.
\end{Lemma}
\begin{Proof}
First of all we remark that the constant
$C $ is in fact $2 B (1+A(1))$,
where $B = 3.9171$ is the constant
in Lemma \ref{BD-Thm2} and $A(1)$ is
estimated in \eqref{A(1)-estim}.
This
should be compared with the value
$C_1 =11.4525218267$
obtained in \cite{Parsell2003}.
Assuming $B=1$ in Lemma \ref{BD-Thm2}, we get
$C=2.5585042082$.
Letting now
\[
I
=
\int_{\mathbb{R}}
\vert
S(\lambda \alpha) G(\mathfrak{m}u \alpha)
\vert^2
K(\alpha, \eta)
\mathfrak{m}athrm{d}\alpha,
\]
by \eqref{K-transform} we immediately have
\begin{equation}
\label{expanded}
I
=
\sum_{\epsilon X \leq p_1, p_2\leq X}
\sum_{1 \leq m_1, m_2 \leq L}
\log p_1 \log p_2
\mathfrak{m}ax
\mathfrak{m}athcal{B}igl(
0;
\eta - \vert
\lambda(p_1-p_2) +\mathfrak{m}u(2^{m_1}-2^{m_2})
\vert
\mathfrak{m}athcal{B}igr).
\end{equation}
Let $\delta= \lambda(p_1-p_2) + \mathfrak{m}u(2^{m_1}-2^{m_2})$.
For a sufficiently small $\eta>0$, we claim that
\begin{equation}
\label{delta0}
\vert
\delta
\vert
< \eta
\quad
\mathfrak{t}extrm{is equivalent to}
\quad
\delta = 0.
\end{equation}
Recall our hypothesis on $a$ and $q$, and assume that
$\delta\neq 0$ in \eqref{delta0}.
For $\eta<\vert \lambda/a\vert$ this leads to a contradiction. In fact we have
\[
\frac{1}{\vert a \vert}
>
\frac{\eta}{\vert \lambda\vert}
>
\mathfrak{m}athcal{B}igl\vert
(p_1-p_2) + \frac{q}{a}(2^{m_1}-2^{m_2})
\mathfrak{m}athcal{B}igr\vert
=
\mathfrak{m}athcal{B}igl\vert
\frac{a(p_1-p_2) + q(2^{m_1}-2^{m_2})}{a}
\mathfrak{m}athcal{B}igr\vert
\geq
\frac{1}{\vert a \vert},
\]
since $a(p_1-p_2) + q(2^{m_1}-2^{m_2})\neq 0$ is a linear integral
combination.
Inserting \eqref{delta0} in \eqref{expanded},
for $\eta<\vert \lambda/a\vert$ we can write that
\begin{equation}
\label{expanded1}
I
=
\eta
\sum_{\epsilon X \leq p_1, p_2\leq X}
\sum _{\substack{
1\leq m_1, m_2 \leq L \\ \hskip-1.7cm
\lambda(p_1-p_2) +\mathfrak{m}u(2^{m_1}-2^{m_2}) =0
}}
\log p_1 \log p_2.
\end{equation}
The diagonal contribution in \eqref{expanded1}
is equal to
\begin{equation}
\label{diagonal-contrib}
\eta
\sum_{\epsilon X \leq p \leq X}
\log^2 p
\sum_{1\leq m \leq L}
1
=
\eta
XL^2
(1-\epsilon)\log{2}
+\Odip{M,\epsilon}{\eta X L}
\end{equation}
where we used the Prime Number Theorem instead
of trivially estimate the contribution of $\log p_i$
as in \cite{Parsell2003}.
Now we have to estimate
the contribution $I'$ of the non-diagonal solutions
of $\delta=0$ and we will achieve this by connecting $I'$ with
the singular series of the twin prime problem.
Recalling that $\lambda/\mathfrak{m}u = a/q \neq 0$, $(a,q)=1$,
by Lemma \ref{BD-Thm2} and the fact that
$Z(X;(q/a)(2^{m_2}-2^{m_1}))\neq0$
if and only if
$a \mathfrak{m}id (2^{m_2}-2^{m_1})$, we have,
since $\mathfrak{S}(v)=\mathfrak{S}(2^{u}v)$
for every $u,v\in \mathbb{N}$, $u\geq1$, that
\begin{equation}
\label{Sol-estim1}
\begin{split}
I'
\leq
2 \eta
\sum_{1\leq m_1 < m_2 \leq L}
Z
\mathfrak{m}athcal{B}igl(
X;\frac{q}{a}(2^{m_2}-2^{m_1})
\mathfrak{m}athcal{B}igr)
<
2
B
X \eta
\sum_{1\leq m_1 < m_2 \leq L}
\mathfrak{S}
\mathfrak{m}athcal{B}igl(
\frac{q}{a}(2^{m_2}-2^{m_1})
\mathfrak{m}athcal{B}igr).
\end{split}
\end{equation}
Using the multiplicativity of
$\mathfrak{S}^{\prime} (n)$ (defined in \eqref{singseries-def}),
we get
\[
\mathfrak{S}^{\prime}
\mathfrak{m}athcal{B}igl(
\frac{q}{a}(2^{m_2}-2^{m_1})
\mathfrak{m}athcal{B}igr)
\leq
\mathfrak{S}^{\prime}(q)
\mathfrak{S}^{\prime}
\mathfrak{m}athcal{B}igl(
\frac{2^{m_2}-2^{m_1}}{a}
\mathfrak{m}athcal{B}igr)
\leq
\mathfrak{S}^{\prime}(q)
\mathfrak{S}^{\prime}(2^{m_2}-2^{m_1})
\]
and so, by Lemma \ref{KP-Lemma},
\eqref{A(1)-estim} and \eqref{Sol-estim1},
we can write, for every sufficiently large $X$, that
\begin{equation}
\label{I-estim2}
\begin{split}
I'
&
\leq
2
B
X \eta \mathfrak{S}^{\prime}(q)
\sum_{1\leq m_1 < m_2 \leq L}
\mathfrak{S}(2^{m_2}-2^{m_1})
=
B
X \eta
\mathfrak{S}^{\prime}(q)
S(1,L)
\\
&
<
2B (1+A(1))
\mathfrak{S}^{\prime}(q)
X \eta L^2.
\end{split}
\end{equation}
Hence, by \eqref{expanded1}-\eqref{diagonal-contrib}
and \eqref{I-estim2}, we finally get
\[
I
<
\eta X L^2
\mathfrak{m}athcal{B}igl(
(1-\epsilon)\log{2}
+
2B (1+A(1))
\mathfrak{S}'(q)
\mathfrak{m}athcal{B}igr)
+
\Odip{M,\epsilon}{\eta X L},
\]
this way proving
Lemma \ref{Dioph-equation}.
\end{Proof}
We remark that if in Lemma \ref{Dioph-equation} we consider also the case
$\lambda/\mathfrak{m}u \in \mathbb{R}\setminus\mathbb{Q}$,
we can just find $\eta=\eta(X)\mathfrak{t}o 0$ as $X\mathfrak{t}o+\infty$ and this
implies that $s_0\approx |\log \eta| \mathfrak{t}o +\infty$,
see equations \eqref{s0-def-transc}-\eqref{s0-def-alg}
for the precise definition of $s_0$. This essentially depends on the fact
that, for $\lambda/\mathfrak{m}u\in\mathbb{R}\setminus\mathbb{Q}$ and $m,n\in \mathbb{Z}$, it
is not possible to find a function $f(X)$ such that
$\vert \lambda m + \mathfrak{m}u n \vert
\geq f(X)$ and $f(X)\mathfrak{t}o c>0$ as $X \mathfrak{t}o +\infty$ since
the set of values of $\lambda m + \mathfrak{m}u n$
is dense in $\mathbb{R}$.
A different, but related, way to see this phenomenon is to remark
that the inequality $|\alpha n + m| < \eta$ is equivalent to the
pair of inequalities $\Vert n \alpha \Vert < \eta$ or
$\Vert n \alpha \Vert > 1 - \eta$, where $\Vert u \Vert$
is the distance of $u$ from the nearest integer.
When $\alpha$ is irrational, it has $\sim 2 \eta X$ solutions with
$n \le X$, since the sequence $\Vert n \alpha \Vert$ is uniformly
distributed modulo 1.
To estimate the contribution of $G(\alpha)$ on the minor arc
we use Pintz-Ruzsa's method as developed in
\cite{PintzR2003}, \S 3-7.
\begin{Lemma}[Pintz-Ruzsa \cite{PintzR2003}, \S~7]
\label{minor-arc-power-of-two-estim}
Let $0< c <1$. Then there exists $\nu=\nu(c)\in (0,1)$
such that
\[
\vert
E(\nu)
\vert
:=
\vert
\{
\alpha \in (0,1) \
\mathfrak{t}extrm{such that} \
\vert
G(\alpha)
\vert
> \nu L
\}
\vert
\ll_{M,\epsilon}
X^{-c}.
\]
\end{Lemma}
To obtain explicit values for $\nu$ we had to write
our own version of Pintz-Ruzsa algorithm since
in this application the estimates has to be performed
for a different choice of parameters than the ones
they used in \cite{PintzR2003}.
We used the PARI/GP \cite{PARI2} scripting language and the
gp2c compiling tool to be able to compute fifty decimal digits
(but we write here just ten) of the constant
involved in the following Lemma.
We will write two different estimates that we will use in the case
$\lambda_1/\lambda_2$ is a transcendental or an algebraic
number.
Running the program in our cases, Lemma \ref{minor-arc-power-of-two-estim}
gives the following results:
\begin{equation}
\label{G-algebraic}
\vert
G(\alpha)
\vert
\leq
0.83372131685 \cdot L
\end{equation}
if
$\alpha \in [0,1] \setminus E$ where
$\vert E \vert \ll_{M,\epsilon} X^{-2/3-10^{-20}}$, to be
used when $\lambda_1 / \lambda_2$ is algebraic,
and
\begin{equation}
\label{G-transcendental}
\vert
G(\alpha)
\vert
\leq
0.91237810306 \cdot L
\end{equation}
if
$\alpha \in [0,1] \setminus E$ where
$\vert E \vert \ll_{M,\epsilon} X^{-4/5-10^{-20}}$, to be
used when $\lambda_1 / \lambda_2$ is transcendental.
The computing time to get \eqref{G-algebraic}-\eqref{G-transcendental}
on a double quad-core PC of the NumLab laboratory of the
Department of Pure and Applied Mathematics of the
University of Padova was equal in the first case to
24 minutes and 40 seconds (but to get 30 correct digits just
3 minutes and 24 seconds suffice) and to 29 minutes
(but to get 30 correct digits just 3 minutes and 50 seconds suffice)
in the second case.
You can download the PARI/GP source code of our program
together with the cited numerical values at the following link:
\url{www.math.unipd.it/~languasc/PintzRuzsaMethod.html}.
Now we state some lemmas we will use to work on the major arc.
Let $\mathfrak{t}heta(x)=\sum_{p\leq x} \log p$,
\begin{equation}
\label{Selberg-int-def}
J(X,h)
=
\int_{\epsilon X}^X
(\mathfrak{t}heta(x+h)- \mathfrak{t}heta(x) -h)^2
\mathfrak{m}athrm{d} x
\end{equation}
be the Selberg integral and
\[
U(\alpha)
=
\sum_{\epsilon X\leq n \leq X} e(\alpha n).
\]
Applying a famous Gallagher's lemma (\cite{Gallagher1970}, Lemma 1)
on the truncated $L^2$-norm
of exponential sums to $S(\alpha) - U(\alpha)$, one gets the following
well-known statement which we cite from
Br\"udern-Cook-Perelli \cite{BrudernCP1997}, Lemma 1.
\begin{Lemma}
\label{BCP-Gallagher}
For $1/X \leq Y \leq 1/2$ we have
\[
\int_{-Y}^Y
\vert
S(\alpha) - U(\alpha)
\vert^2
\mathfrak{m}athrm{d} \alpha
\ll_{\epsilon}
\frac{\log X}{Y}
+
Y^2X
+
Y^2 J \mathfrak{m}athcal{B}igl( X,\frac{1}{Y} \mathfrak{m}athcal{B}igr),
\]
where $J(X,h)$ is defined in \eqref{Selberg-int-def}.
\end{Lemma}
To estimate the Selberg integral, we use the next result.
\begin{Lemma}[Saffari-Vaughan \cite{SaffariV1977a}, \S6]
\label{Saffari-Vaughan}
For any $A>0$ there exists $B=B(A)>0$ such that
\[
J(X,h)
\ll_{\epsilon}
\frac{h^2X}
{(\log X)^A}
\]
uniformly for $h\geq X^{1/6} (\log X)^B$.
\end{Lemma}
\section{The major arc}
Letting
\begin{equation}
\label{T-def-estim}
T(\alpha)
=
\int_{\epsilon X}^{X}e(t\alpha)\mathfrak{m}athrm{d} t
\ll_{\epsilon}
\mathfrak{m}in \mathfrak{m}athcal{B}igl(X, \frac{1}{\vert\alpha\vert} \mathfrak{m}athcal{B}igr),
\end{equation}
we first write
\begin{equation}
\label{I-splitting}
\begin{split}
I(X ; \mathfrak{m}athfrak{M})
&=
\int_\mathfrak{m}athfrak{M}
T(\lambda_1 \alpha)
T(\lambda_2 \alpha)
G(\mathfrak{m}u_1\alpha)
\dotsm
G(\mathfrak{m}u_s\alpha)
e(\gamma \alpha)
K(\alpha, \eta) \mathfrak{m}athrm{d} \alpha
\\
& +
\int_\mathfrak{m}athfrak{M}
\mathfrak{m}athcal{B}igl(S(\lambda_1 \alpha) - T(\lambda_1 \alpha)\mathfrak{m}athcal{B}igr)
T(\lambda_2 \alpha)
G(\mathfrak{m}u_1\alpha)
\dotsm
G(\mathfrak{m}u_s\alpha)
e(\gamma \alpha)
K(\alpha, \eta) \mathfrak{m}athrm{d} \alpha \\
& +
\int_\mathfrak{m}athfrak{M}
S(\lambda_1 \alpha)
\mathfrak{m}athcal{B}igl(S(\lambda_2 \alpha) - T(\lambda_2 \alpha)\mathfrak{m}athcal{B}igr)
G(\mathfrak{m}u_1\alpha)
\dotsm
G(\mathfrak{m}u_s\alpha)
e(\gamma \alpha)
K(\alpha, \eta) \mathfrak{m}athrm{d} \alpha \\
&=
J_1+ J_2+J_3,
\end{split}
\end{equation}
say.
In what follows we will prove that
\begin{equation}
\label{J1-lower-bound}
J_1
\geq
\frac{1-(7/2)\lambda_1\epsilon}
{2 \vert\lambda_1\lambda_2 \vert}
\eta^2 XL^s
\end{equation}
and
\begin{equation}
\label{J2-estim}
J_2 + J_3
=
\odi{\eta^2XL^s},
\end{equation}
thus obtaining by \eqref{I-splitting}-\eqref{J2-estim} that
\[
I(X ; \mathfrak{m}athfrak{M})
\geq
\frac{1-4\lambda_1\epsilon}
{2 \vert\lambda_1\lambda_2 \vert}
\eta^2 XL^s.
\]
Thus we will prove that \eqref{major-goal} holds with
$c_1= (1-4\lambda_1\epsilon)/ (2 \vert\lambda_1\lambda_2 \vert)$.
\paragraph{Estimation of $J_2$ and $J_3$.}
We first estimate $J_3$.
We remark that, by the partial summation formula, we have
$T(\alpha) - U(\alpha) \ll (1 + X\vert \alpha \vert)$.
So, recalling $P=X^{1/3}$, \eqref{dissect-def}
and $\vert S(\lambda_1\alpha) \vert \ll X \log X$,
we get
\[
\int_{\mathfrak{m}athfrak{M}}
\vert
T(\lambda_2\alpha)
-
U(\lambda_2\alpha)
\vert
\vert S(\lambda_1\alpha) \vert
\mathfrak{m}athrm{d} \alpha
\ll
X \log X
\int_{\mathfrak{m}athfrak{M}}
(1 + X\vert \lambda_2 \alpha \vert)
\mathfrak{m}athrm{d} \alpha
\ll_{\bm{\lambda}}
X^{2/3} \log X.
\]
Hence,
using the trivial estimates
$\vert G(\mathfrak{m}u_i \alpha)\vert \leq L$, $K(\alpha,\eta)\ll \eta^2$, we can write
\[
J_3
=
\int_\mathfrak{m}athfrak{M}
S(\lambda_1 \alpha)
\mathfrak{m}athcal{B}igl(S(\lambda_2 \alpha) - U(\lambda_2 \alpha)\mathfrak{m}athcal{B}igr)
G(\mathfrak{m}u_1\alpha)
\dotsm
G(\mathfrak{m}u_s\alpha)
e(\gamma \alpha)
K(\alpha, \eta) \mathfrak{m}athrm{d} \alpha
+
\Odip{\bm{\lambda},M}
{\eta^2X^{2/3} L^{s+1}}.
\]
Now using \eqref{dissect-def}, the Cauchy-Schwarz inequality, the Prime Number Theorem,
Lemmas \ref{BCP-Gallagher}-\ref{Saffari-Vaughan}
with $A=3$, $Y=P/X$, $P=X^{1/3}$, and again
the trivial estimates
$\vert G(\mathfrak{m}u_i \alpha)\vert \leq L$, $K(\alpha,\eta)\ll \eta^2$,
we have that
\[
\begin{split}
J_3
&
\ll
\eta^2 L^s
\mathfrak{m}athcal{B}igl(
\int_{\mathfrak{m}athfrak{M}}
\vert
S(\lambda_2\alpha)
-
U(\lambda_2\alpha)
\vert^2
\mathfrak{m}athrm{d} \alpha
\mathfrak{m}athcal{B}igr)^{1/2}
\mathfrak{m}athcal{B}igl(
\int_{\mathfrak{m}athfrak{M}}
\vert
S(\lambda_1\alpha)
\vert^2
\mathfrak{m}athrm{d} \alpha
\mathfrak{m}athcal{B}igr)^{1/2}
+
\Odip{\bm{\lambda}, M}
{\eta^2X^{2/3} L^{s+1}}
\\
&
\ll_{\bm{\lambda}, M, \epsilon}
\eta^2 L^s
\frac{X^{1/2}}
{(\log X)^{3/2}}
\mathfrak{m}athcal{B}igl(
\int_{0}^{1}
\vert
S(\alpha)
\vert^2
\mathfrak{m}athrm{d} \alpha
\mathfrak{m}athcal{B}igr)^{1/2}
+
\eta^2X^{2/3} L^{s+1}
\ll_{\bm{\lambda},M,\epsilon}
\eta^2 X L^{s-1}
=
\odi{\eta^2XL^s}.
\end{split}
\]
The integral $J_2$ can be estimated analogously using \eqref{T-def-estim}
instead of the Prime Number Theorem.
Hence \eqref{J2-estim} holds.
\paragraph{Estimation of $J_1$.}
Recalling that $P= X^{1/3}$
and using \eqref{dissect-def}, \eqref{T-def-estim} and \eqref{I-splitting}
we obtain
\begin{equation}
\label{J1J-relation}
J_1
=
\sum_{1\leq m_1 \leq L} \dotsm \sum_{1\leq m_s \leq L}
J
\mathfrak{m}athcal{B}igl(
\mathfrak{m}u_1 2^{m_1} + \dotsm + \mathfrak{m}u_s 2^{m_s}
+ \gamma
\mathfrak{m}athcal{B}igr)
+
\Odip{\epsilon}{\eta^2 X^{2/3} L^s},
\end{equation}
where $J(u)$ is defined by
\[
J(u)
:=
\int_\mathbb{R}
T(\lambda_1 \alpha)
T(\lambda_2 \alpha)
e(u \alpha)
K(\alpha, \eta) \mathfrak{m}athrm{d} \alpha
=
\int_{\epsilon X}^{X}
\int_{\epsilon X}^{X}
\widehat{K}(\lambda_1u_1+\lambda_2u_2 + u)
\mathfrak{m}athrm{d} u_1 \mathfrak{m}athrm{d} u_2
\]
and the second relation follows by interchanging the order of integration.
Assume
now that $\vert u \vert \leq \epsilon X$ and that
$2\epsilon\lambda_1 X
\leq
\vert \lambda_2\vert u_2
\leq
(1-\epsilon\lambda_1)X$.
For
$\eta<2\epsilon(\lambda_1-1)X$
and
$X$ sufficiently large, we have, by \eqref{K-transform},
that there exists an interval
for $u_1$, of length $\geq \eta/\lambda_1$ and contained
in $[\epsilon X,X]$, on which
$\widehat{K}(\lambda_1u_1+\lambda_2u_2 + u) \geq \eta/2$.
Thus we have
\begin{equation}
\label{J-lower-bound}
J(u)
\geq
\frac{1-3\lambda_1\epsilon}
{2 \vert\lambda_1\lambda_2 \vert}
\eta^2 X.
\end{equation}
For a sufficiently large $X$, it is clear that
$\vert \mathfrak{m}u_1 2^{m_1} + \dotsm + \mathfrak{m}u_s 2^{m_s}
+ \gamma
\vert \leq \epsilon X$
while the other condition on the size of
$\vert \lambda_2\vert u_2$ follows from the hypothesis
$\vert \lambda_1/\lambda_2\vert \geq 1$
and
$\lambda_2<-1$.
Hence,
from \eqref{J1J-relation}-\eqref{J-lower-bound},
we obtain that \eqref{J1-lower-bound} holds.
\section{The trivial arc}
Recalling \eqref{dissect-def}, the trivial estimate
$\vert G(\mathfrak{m}u_i \alpha)\vert \leq L$
and using the Cauchy-Schwarz inequality, we get
\[
\vert I(X ; \mathfrak{t}) \vert
\ll
L^s
\mathfrak{m}athcal{B}igl(
\int_{L^2}^{+\infty}
\vert
S(\lambda_1\alpha)
\vert^2
K(\alpha,\eta)
\mathfrak{m}athrm{d} \alpha
\mathfrak{m}athcal{B}igr)^{1/2}
\mathfrak{m}athcal{B}igl(
\int_{L^2}^{+\infty}
\vert
S(\lambda_2\alpha)
\vert^2
K(\alpha,\eta)
\mathfrak{m}athrm{d} \alpha
\mathfrak{m}athcal{B}igr)^{1/2}
\]
By \eqref{K-inequality}
and making a change of variable, we have,
for $i=1,2$, that
\[
\begin{split}
\int_{L^2}^{+\infty}
\vert
S(\lambda_i\alpha)
\vert^2
K(\alpha,\eta)
\mathfrak{m}athrm{d} \alpha
&
\ll_{\bm{\lambda}}
\int_{\lambda_i L^2}^{+\infty}
\frac
{\vert
S(\alpha)
\vert^2
}
{\alpha^2}
\mathfrak{m}athrm{d} \alpha
\ll
\sum_{n\geq \lambda_i L^2}
\frac{1}{(n-1)^2}
\int_{n-1}^{n}
\vert
S(\alpha)
\vert^2
\mathfrak{m}athrm{d} \alpha
\\
&
\ll_{\bm{\lambda}}
L^{-2}
\int_0^{1}
\vert
S(\alpha)
\vert^2
\mathfrak{m}athrm{d} \alpha
\ll_{\bm{\lambda},M,\epsilon}
\frac{X}{\log X},
\end{split}
\]
by the Prime Number Theorem,
and hence \eqref{trivial-goal} holds.
\section{The minor arc: $\lambda_1/\lambda_2$ algebraic}
Recalling first
\[
I(X ; \mathfrak{m})
=
\int_{\mathfrak{m}}
S(\lambda_1 \alpha) S(\lambda_2 \alpha)
G(\mathfrak{m}u_1 \alpha)\dotsm G(\mathfrak{m}u_s \alpha)
e(\gamma \alpha)
K(\alpha,\eta)
\mathfrak{m}athrm{d} \alpha,
\]
and letting $c\in (0,1)$ to be chosen later,
we first split $\mathfrak{m}$ as $\mathfrak{m}_1 \cup \mathfrak{m}_2$,
$\mathfrak{m}_1 \cap \mathfrak{m}_2=\emptyset$,
where $\mathfrak{m}_2$ is the set of $\beta\in \mathfrak{m}$
such that $\vert G(\beta)\vert > \nu(c) L$
and $\nu(c)$ is defined in Lemma \ref{minor-arc-power-of-two-estim}.
We will choose $c$ to get
$\vert I(X ; \mathfrak{m}_2) \vert = \odi{\eta X}$,
since, again by Lemma \ref{minor-arc-power-of-two-estim},
we know that $\vert \mathfrak{m}_2 \vert \ll_{M,\epsilon} s L^2 X^{-c}$.
To this end, we first use the trivial estimates
$\vert G(\mathfrak{m}u_i \alpha)\vert \leq L$ and $K(\alpha,\eta)\ll \eta^2$,
and the Cauchy-Schwarz inequality thus obtaining
\begin{equation}
\label{minor-arc-2}
\begin{split}
\vert I(X ; \mathfrak{m}_2) \vert
& \leq
L^{s}
\mathfrak{m}athcal{B}igl(
\int_{\mathfrak{m}_2}
\vert S(\lambda_1 \alpha) S(\lambda_2 \alpha) \vert ^2
K(\alpha,\eta)
\mathfrak{m}athrm{d} \alpha
\mathfrak{m}athcal{B}igr)^{1/2}
\mathfrak{m}athcal{B}igl(
\int_{\mathfrak{m}_2}
K(\alpha,\eta)
\mathfrak{m}athrm{d} \alpha
\mathfrak{m}athcal{B}igr)^{1/2}
\\
&
\ll
\eta L^s
\vert \mathfrak{m}_2 \vert ^{1/2}
\mathfrak{m}athcal{B}igl(
\int_{\mathfrak{m}_2}
\vert S(\lambda_1 \alpha) S(\lambda_2 \alpha) \vert ^2
K(\alpha,\eta)
\mathfrak{m}athrm{d} \alpha
\mathfrak{m}athcal{B}igr)^{1/2}.
\end{split}
\end{equation}
We can now argue as in section 4 of
Br\"udern-Cook-Perelli \cite{BrudernCP1997}
thus getting
\begin{equation}
\label{minor-arc-2-low}
\int_{\mathfrak{m}_2}
\vert S(\lambda_1 \alpha) S(\lambda_2 \alpha) \vert ^2
K(\alpha,\eta)
\mathfrak{m}athrm{d} \alpha
\ll_{\epsilon}
\eta X^{8/3+\epsilon'}.
\end{equation}
Hence, by \eqref{minor-arc-2-low},
\eqref{minor-arc-2} becomes
\[
\vert I(X ; \mathfrak{m}_2) \vert
\ll_{M,\epsilon}
s^{1/2} \
\eta^{3/2}
X^{4/3+2\epsilon'-c/2}.
\]
Taking $c=2/3+10^{-20}$ and using \eqref{G-algebraic}, we get,
for $\nu= 0.83372131685$ and a
sufficiently small $\epsilon'>0$, that
\begin{equation}
\label{minor-arc-2-final}
\vert I(X ; \mathfrak{m}_2) \vert
=
\odi{\eta X}.
\end{equation}
Now we evaluate the contribution of $\mathfrak{m}_1$.
Using Lemma \ref{Dioph-equation}
and the Cauchy-Schwarz inequality,
we have
\begin{align}
\vert I(X ; \mathfrak{m}_1) \vert
&
\leq
(\nu L)^{s-2}
\notag
\\
& \mathfrak{t}imes
\mathfrak{m}athcal{B}igl(
\int_{\mathfrak{m}}
\vert S(\lambda_1 \alpha) G(\mathfrak{m}u_1 \alpha) \vert ^2
K(\alpha,\eta)
\mathfrak{m}athrm{d} \alpha
\mathfrak{m}athcal{B}igr)^{1/2}
\mathfrak{m}athcal{B}igl(
\int_{\mathfrak{m}}
\vert S(\lambda_2 \alpha) G(\mathfrak{m}u_2 \alpha) \vert ^2
K(\alpha,\eta)
\mathfrak{m}athrm{d} \alpha
\mathfrak{m}athcal{B}igr)^{1/2}
\notag
\\
&
<
\nu^{s-2}
C(q_1,q_2)
\eta X L^{s},
\label{minor-arc-1}
\end{align}
where, recalling Lemmas \ref{sing-series-estim} and
\ref{Dioph-equation},
$C(q_1,q_2)$ is defined as we did in \eqref{Cq1q2-def}.
Hence, by \eqref{minor-arc-2-final} and \eqref{minor-arc-1},
for $X$ sufficiently large
we finally get
\begin{equation}
\notag
\vert I(X ; \mathfrak{m}) \vert
<
(0.83372131685)^{s-2}
C(q_1,q_2)
\eta X L^{s}
\end{equation}
whenever $\lambda_1/\lambda_2$ is an algebraic number.
This means that \eqref{minor-goal} holds, in this case, with
$c_2(s)=(0.83372131685)^{s-2}
C(q_1,q_2)$.
\section{The minor arc: $\lambda_1/\lambda_2$ transcendental}
We will act on $\mathfrak{m}_1$ as in \eqref{minor-arc-1}
of the previous section thus obtaining
\begin{equation}
\label{minor-arc-1-transc}
\vert I(X ; \mathfrak{m}_1) \vert
<
\nu^{s-2}
C(q_1,q_2)
\eta X
L^{s},
\end{equation}
where $C(q_1,q_2)$ is defined in
\eqref{Cq1q2-def}.
Now we proceed to estimate $I(X ; \mathfrak{m}_2)$.
First we argue as in the previous section until \eqref{minor-arc-2}
and then we work as in section 8 of Cook-Harman \cite{CookH2006}
and pp.~221-223 of Harman \cite{Harman1991} thus obtaining
\[
\int_{\mathfrak{m}_2}
\vert S(\lambda_1 \alpha) S(\lambda_2 \alpha) \vert ^2
K(\alpha,\eta)
\mathfrak{m}athrm{d} \alpha
\ll
\eta^2
X^{14/5+\epsilon'}
+
\eta
X^{13/5+\epsilon'}.
\]
This, using \eqref{minor-arc-2}, leads to
\[
\vert I(X ; \mathfrak{m}_2) \vert
\ll_{M,\epsilon}
s^{1/2}
X^{-c/2}
(
\eta^{2}
X^{7/5+\epsilon'}
+
\eta^{3/2}
X^{13/10+\epsilon'}
).
\]
Taking $c=4/5+10^{-20}$ and using \eqref{G-transcendental}, we get,
for $\nu= 0.91237810306$ and a
sufficiently small $\epsilon'>0$,
that
\begin{equation}
\label{minor-arc-2-transc}
\vert I(X ; \mathfrak{m}_2) \vert
=
\odi{\eta X}.
\end{equation}
Hence, by \eqref{minor-arc-1-transc} and \eqref{minor-arc-2-transc},
for $X$ sufficiently large
we finally get
\begin{equation}
\notag
\vert I(X ; \mathfrak{m}) \vert
<
(0.91237810306)^{s-2}
C(q_1,q_2)
\eta X L^{s}
\end{equation}
whenever $\lambda_1/\lambda_2$ is a transcendental number.
This means that \eqref{minor-goal} holds, in this case, with
$c_2(s)=(0.91237810306)^{s-2}
C(q_1,q_2)$.
\section{Proof of the Theorem}
We have to verify if there exists an $s_0\in\mathbb{N}$ such that
\eqref{constant-condition} holds for $X$ sufficiently large.
Combining the inequalities \eqref{major-goal}-\eqref{minor-goal},
where
$c_2(s)= (0.83372131685)^{s-2}C(q_1,q_2)$
if $\lambda_1/\lambda_2$ is algebraic and,
if $\lambda_1/\lambda_2$ is transcendental,
$c_2(s)= (0.91237810306)^{s-2}C(q_1,q_2)$,
we obtain for $s\geq s_0$, $s_0$ defined
in \eqref{s0-def-transc}-\eqref{s0-def-alg},
that \eqref{constant-condition} holds in both cases.
This completes the proof of the Theorem.
\normalsize
\begin{tabular}{l@{\hskip 14mm}l}
A.~Languasco & A.~Zaccagnini \\
Universit\`a di Padova & Universit\`a di Parma \\
Dipartimento di Matematica & Dipartimento di Matematica \\
Pura e Applicata & Parco Area delle Scienze, 53/a \\
Via Trieste 63 & Campus Universitario \\
35121 Padova, Italy & 43100 Parma, Italy \\
{\it e-mail}: languasco@math.unipd.it & {\it e-mail}:
alessandro.zaccagnini@unipr.it
\end{tabular}
\end{document} | math |
تہنٛد مٲل کمہ قٕسمٕک نفر چھ | kashmiri |
/*L
* Copyright Moxie Informatics.
*
* Distributed under the OSI-approved BSD 3-Clause License.
* See http://ncip.github.com/calims/LICENSE.txt for details.
*/
package gov.nih.nci.calims2.domain.workflow.l10n;
import java.util.ListResourceBundle;
import gov.nih.nci.calims2.domain.workflow.enumeration.MethodStatus;
/**
* @author connollym@moxieinformatics.com
*
*/
public class MethodStatusBundle extends ListResourceBundle {
private static final Object[][] CONTENTS = {
{MethodStatus.APPROVALPENDING.name(), "Approval Pending"},
{MethodStatus.APPROVED.name(), "Approved"},
{MethodStatus.NOTAPPROVED.name(), "Not Approved"},
{MethodStatus.INPROGRESS.name(), "In Progress"},
{MethodStatus.COMPLETED.name(), "Completed"},
{MethodStatus.INREVIEW.name(), "In Review"},
{MethodStatus.REVIEWED.name(), "Reviewed"},
{MethodStatus.PUBLISHED.name(), "Published"}};
/**
* {@inheritDoc}
*/
protected Object[][] getContents() {
return CONTENTS;
}
} | code |
// Copyright 2012, Google Inc. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package vtgate
import (
mproto "github.com/youtube/vitess/go/mysql/proto"
"github.com/youtube/vitess/go/vt/proto/vtrpc"
"github.com/youtube/vitess/go/vt/vterrors"
"github.com/youtube/vitess/go/vt/vtgate/proto"
)
// A list of all vtrpc.ErrorCodes, ordered by priority. These priorities are
// used when aggregating multiple errors in VtGate.
// Higher priority error codes are more urgent for users to see. They are
// prioritized based on the following question: assuming a scatter query produced multiple
// errors, which of the errors is the most likely to give the user useful information
// about why the query failed and how they should proceed?
const (
PrioritySuccess = iota
PriorityTransientError
PriorityQueryNotServed
PriorityDeadlineExceeded
PriorityCancelled
PriorityIntegrityError
PriorityNotInTx
PriorityUnknownError
PriorityInternalError
PriorityResourceExhausted
PriorityUnauthenticated
PriorityPermissionDenied
PriorityBadInput
)
var errorPriorities = map[vtrpc.ErrorCode]int{
vtrpc.ErrorCode_SUCCESS: PrioritySuccess,
vtrpc.ErrorCode_CANCELLED: PriorityCancelled,
vtrpc.ErrorCode_UNKNOWN_ERROR: PriorityUnknownError,
vtrpc.ErrorCode_BAD_INPUT: PriorityBadInput,
vtrpc.ErrorCode_DEADLINE_EXCEEDED: PriorityDeadlineExceeded,
vtrpc.ErrorCode_INTEGRITY_ERROR: PriorityIntegrityError,
vtrpc.ErrorCode_PERMISSION_DENIED: PriorityPermissionDenied,
vtrpc.ErrorCode_RESOURCE_EXHAUSTED: PriorityResourceExhausted,
vtrpc.ErrorCode_QUERY_NOT_SERVED: PriorityQueryNotServed,
vtrpc.ErrorCode_NOT_IN_TX: PriorityNotInTx,
vtrpc.ErrorCode_INTERNAL_ERROR: PriorityInternalError,
vtrpc.ErrorCode_TRANSIENT_ERROR: PriorityTransientError,
vtrpc.ErrorCode_UNAUTHENTICATED: PriorityUnauthenticated,
}
// rpcErrFromTabletError translate an error from VTGate to an *mproto.RPCError
func rpcErrFromVtGateError(err error) *mproto.RPCError {
if err == nil {
return nil
}
return &mproto.RPCError{
Code: int64(vterrors.RecoverVtErrorCode(err)),
Message: err.Error(),
}
}
// aggregateVtGateErrorCodes aggregates a list of errors into a single error code.
// It does so by finding the highest priority error code in the list.
func aggregateVtGateErrorCodes(errors []error) vtrpc.ErrorCode {
highCode := vtrpc.ErrorCode_SUCCESS
for _, e := range errors {
code := vterrors.RecoverVtErrorCode(e)
if errorPriorities[code] > errorPriorities[highCode] {
highCode = code
}
}
return highCode
}
func aggregateVtGateErrors(errors []error) error {
if len(errors) == 0 {
return nil
}
return vterrors.FromError(
aggregateVtGateErrorCodes(errors),
vterrors.ConcatenateErrors(errors),
)
}
// AddVtGateErrorToQueryResult will mutate a QueryResult struct to fill in the Err
// field with details from the VTGate error.
func AddVtGateErrorToQueryResult(err error, reply *proto.QueryResult) {
if err == nil {
return
}
reply.Err = rpcErrFromVtGateError(err)
}
// AddVtGateErrorToQueryResultList will mutate a QueryResultList struct to fill in the Err
// field with details from the VTGate error.
func AddVtGateErrorToQueryResultList(err error, reply *proto.QueryResultList) {
if err == nil {
return
}
reply.Err = rpcErrFromVtGateError(err)
}
// AddVtGateErrorToSplitQueryResult will mutate a SplitQueryResult struct to fill in the Err
// field with details from the VTGate error.
func AddVtGateErrorToSplitQueryResult(err error, reply *proto.SplitQueryResult) {
if err == nil {
return
}
reply.Err = rpcErrFromVtGateError(err)
}
// AddVtGateErrorToBeginResponse will mutate a BeginResponse struct to fill in the Err
// field with details from the VTGate error.
func AddVtGateErrorToBeginResponse(err error, reply *proto.BeginResponse) {
if err == nil {
return
}
reply.Err = rpcErrFromVtGateError(err)
}
// AddVtGateErrorToCommitResponse will mutate a CommitResponse struct to fill in the Err
// field with details from the VTGate error.
func AddVtGateErrorToCommitResponse(err error, reply *proto.CommitResponse) {
if err == nil {
return
}
reply.Err = rpcErrFromVtGateError(err)
}
// AddVtGateErrorToRollbackResponse will mutate a RollbackResponse struct to fill in the Err
// field with details from the VTGate error.
func AddVtGateErrorToRollbackResponse(err error, reply *proto.RollbackResponse) {
if err == nil {
return
}
reply.Err = rpcErrFromVtGateError(err)
}
// RPCErrorToVtRPCError converts a VTGate error into a vtrpc error.
func RPCErrorToVtRPCError(rpcErr *mproto.RPCError) *vtrpc.RPCError {
if rpcErr == nil {
return nil
}
return &vtrpc.RPCError{
Code: vtrpc.ErrorCode(rpcErr.Code),
Message: rpcErr.Message,
}
}
// VtGateErrorToVtRPCError converts a vtgate error into a vtrpc error.
// TODO(aaijazi): rename this guy, and correct the usage of it everywhere. As it's currently used,
// it will almost never return the correct error code, as it's only getting executeErr and reply.Error.
// It should actually just use reply.Err.
func VtGateErrorToVtRPCError(err error, errString string) *vtrpc.RPCError {
if err == nil && errString == "" {
return nil
}
message := ""
if err != nil {
message = err.Error()
} else {
message = errString
}
return &vtrpc.RPCError{
Code: vterrors.RecoverVtErrorCode(err),
Message: message,
}
}
| code |
\begin{document}
\title{Periodic self maps and thick ideals in the stable motivic homotopy category over \C\ at odd primes}
\begin{abstract}In this article we study thick ideals defined by periodic self maps in the stable motivic homotopy category over \ensuremath{\mathbb{C}}. In addition, we extend some results of Ruth Joachimi about the relation between thick ideals defined by motivic Morava K-theories and the preimages of the thick ideals in the stable homotopy category under Betti realization.
\end{abstract}
\setcounter{tocdepth}{2}
\tableofcontents
\section{Introduction}
There are two famous results by Hopkins and Smith in \cite{HS} that provide a complete description of the thick subcategories in the stable homotopy category of finite topological spectra.
\begin{defin}
A thick subcategory of a tensor triangulated category is a nonempty, full, triangulated subcategory that is closed under retracts. A thick ideal is a thick subcategory that is closed under tensoring with arbitrary objects.
\end{defin}
The thick subcategory theorem states that if we localize at a prime \(l\) the thick subcategories (in fact thick ideals) of the category \(\mathcal{SH}_{(l)}^{fin}\) are given by a chain
\[\mathcal{SH}_{(l)}^{fin}=\mathcal{C}_0 \supsetneq \mathcal{C}_1 \supsetneq \mathcal{C}_2 \supsetneq ... \supsetneq \mathcal{C}_\infty=\{0\}\]
and each thick ideal \(\mathcal{C}_{i+1}, 0\leq i < \infty\), is characterized by the vanishing of the \(i\)-th Morava K-theory \(K(i)\), where \(K(0)=H\ensuremath{\mathbb{Q}}\) by convention. The periodicity theorem states that these thick subcategories can also be described by the property of admitting a special kind of periodic self map; a so called \(v_n\)-self map that induces an isomorphism in \(K(n)\) and nilpotent maps in \(K(m),\ m\neq n\). Using the older Nilpotence theorem of Devinatz, Hopkins and Smith in \cite{DHS}, Hopkins and Smith showed that the full subcategory \(\mathcal{C}_{v_n}\) of finite spectra admitting such self maps is in fact thick, and thus equal to one of the categories \(\mathcal{C}_i\). For algebraic reasons (see \cite[3.3.11]{RAV2}) the category \(\mathcal{C}_{v_n}\) must be nested in the following way: \[\mathcal{C}_{n+1}\subset\mathcal{C}_{v_n}\subset\mathcal{C}_{n}\]
Therefore, by the thick subcategory theorem, the existence of at least one spectrum \(X_n\) in \(\mathcal{C}_{n}\) admitting such a self map proves the equality \(\mathcal{C}_{v_n}=\mathcal{C}_{n}\). Using an earlier construction of Smith, they prove that there is indeed such a spectrum \(X_n\) that admits a \(v_n\)-self map.\\
The fact that the motivic Hopf map \(\eta\) is not nilpotent suggests that the picture looks very different in the motivic context, even over the complex numbers. Ruth Joachimi showed in her dissertation that algebraic Morava K-theories, originally defined by Borghesi, define a similar chain of thick subcategories \(\mathcal{C}_{AK(n)}\) for odd primes over the base field \ensuremath{\mathbb{C}} (\cite[9.6.4]{JOA}), but also that there are a more thick ideals in the motivic homotopy category (\cite[Chapter 7]{JOA}). In addition, she relates the thick ideals \(\mathcal{C}_{AK(n)}\) to the thick ideals \(\text{thickid}(c\mathcal{C}_n)\) and \(R^{-1}(\mathcal{C}_n)\) provided by the classical thick ideals via the constant simplicial presheaf and Betti realization functors, respectively.\\
The purpose of this article is to explore the motivic equivalents of the constructions by Hopkins and Smith.
In Theorem \ref{ThickSubcat}, we prove that periodic motivic self maps defined by algebraic Morava K-theory define a thick subcategory, but we need to make use of a conjectural weakened version of a motivic nilpotence lemma. In Theorem \ref{SelfMapExample} we lift a construction by Hopkins and Smith in \cite{HS} to the motivic world to show that examples of these self maps exist. Finally, in the last two sections, we use some of our computations in the preceding sections to settle \cite[Conjecture 7.1.7.3]{JOA}. We furthermore provide a counterexample to the asserted inclusion \(\text{thickid}(c\mathcal{C}_2) \subset \mathcal{C}_{AK(1)}\) in \cite[Chapter 9, last section]{JOA} and we identify an error in \cite[Proposition 8.7.3]{JOA}, on which the assertion is based. The counterexample also proves that the inclusion \(\mathcal{C}_{AK(1)}\subset R^{-1}(\mathcal{C}_2)\) is actually proper and hence that the thick subcategories defined by algebraic Morava K-theories are distinct from the preimages of the topological thick ideals under Betti realization.\\
This research was originally part of my dissertation under supervision by Jens Hornbostel, to who I am grateful for his support. It was conducted in the framework of the research training group
\emph{GRK 2240: Algebro-Geometric Methods in Algebra, Arithmetic and Topology},
which is funded by the DFG.
\section{Background}
We work in the motivic stable homotopy category \(\mathcal{SH}_\ensuremath{\mathbb{C}}\), whose objects are \(\mathbb{P}^1\)-spectra of motivic spaces over the base field \(\ensuremath{\mathbb{C}}\). The construction of this category is due to Voevodsky and Morel (see \cite{VOE} and \cite{MV}) and mimicks the construction of the topological stable homotopy category, where smooth schemes take the place of topological spaces. There are two kinds of spheres in the motivic world, a simplicial and a geometric one; therefore suspensions, homotopy, homology and cohomology are all not singly graded but bigraded; and there are two common conventions for how to grade them. We index them according to the following convention:
\begin{defin}
Define \(S^{1,0}\) as the \(\mathbb{P}^1\)-suspension spectrum of the simplicial sphere \((S^1,1)\) and \(S^{1,1}\) as the \(\mathbb{P}^1\)-suspension spectrum of \((\mathbb{A}^1-0,1)\). The suspension spectrum of \(\mathbb{P}^1\) is then equivalent to \(S^{2,1}\). Define
\[S^{p,q}\defeq(S^{1,0})^{\wedge(p-q)}\wedge (S^{1,1})^q.\]
This relates to the other common notation of \(S^{\alpha} = S^{1,1}\) by \(S^{p,q}=S^{p-q+q\alpha}\).\\
The motivic homotopy groups of a motivic spectrum \(X\in\mathcal{SH}_k\) are then defined as:
\[
\pi_{p,q}(X)\defeq[S^{p,q},X]_{\mathcal{SH}_k}
\]
\end{defin}
\ \\
There is a topological realization functor \(R: \mathcal{SH}_\ensuremath{\mathbb{C}} \rightarrow \mathcal{SH}_{top}\) called Betti realization. There are many reviews of the construction and basic properties of this functor. We rely on the account in \cite[4.3]{JOA}.
Betti realization maps the suspension spectrum of a smooth scheme over \ensuremath{\mathbb{C}}\ to the suspension spectrum of the topological space of its complex points, endowed with the analytic topology. In particular the image of the motivic sphere \(S^{p,q}\) under Betti realization is the topological sphere \(S^p\). This functor is a strict symmetric monoidal left Quillen functor.
Because Betti realization maps the motivic spheres to the topological ones, it induces maps on homotopy groups
\[R: \pi_{pq}(X)\rightarrow \pi_p(R(X))\]
for every motivic spectrum \(X\in\mathcal{SH}_\ensuremath{\mathbb{C}}\). Therefore for every motivic spectrum \(E\in\mathcal{SH}_\ensuremath{\mathbb{C}}\) it also induces maps \[R: E_{pq}(X)\rightarrow R(E)_p(R(X))\]
and
\[R: E^{pq}(X)\rightarrow R(E)^p(R(X))\] on homology and cohomology associated to that spectrum. Betti realization has a strict symmetric monoidal right inverse
\[
c: \mathcal{SH}_{top} \rightarrow \mathcal{SH}_\ensuremath{\mathbb{C}}
\]
called the constant simplicial presheaf functor. It is a result of Levine(see \cite[Theorem 1]{LEV}) that \(c\) is not only faithful but also full.
\subsection{Cellular motivic spectra}
We intent to construct an example \(v_n\)-self map \(v: \ensuremath{\mathbb{X}}_n\rightarrow \ensuremath{\mathbb{X}}_n\) on the motivic equivalent of the space \(X_n\) used by Hopkins and Smith. This space is constructed as a retract of a finite cell spectrum. In classical topology, a retract of a finite cell spectrum is a finite cell spectrum again, but this does not necessarily need to be the case motivically. Therefore, we want to consider the slightly larger thick envelope \(\mathcal{SH}_{\ensuremath{\mathbb{C}}}^{qfin}\) (defined in \ref{finiteType}) of the subcategory of finite spectra in \(\mathcal{SH}_\ensuremath{\mathbb{C}}\) in the definition of motivic \(v_n\)-self maps and for the study of thick subcategories characterized by \(v_n\)-self maps.
In contrast to classical algebraic topology, not all motivic spectra are cellular in the following sense:
\begin{defin}
\label{finiteType}
\begin{enumerate}
\item The category of cellular spectra \(\mathcal{SH}_k^{cell}\) in \(\mathcal{SH}_k\) is defined (c.f. \cite[Definition 2.1]{DI2}) as the smallest full subcategory that satisfies
\begin{itemize}
\item The spheres \(S^{p,q}\) are contained in the subcategory \(\mathcal{SH}_k^{cell}\).
\item If a spectrum \(X\) is contained in the subcategory \(\mathcal{SH}_k^{cell}\), then so are all spectra which are weakly equivalent to \(X\).
\item If \(X\rightarrow Y \rightarrow Z\) is a cofiber sequence and two of the three spectra are contained in the subcategory \(\mathcal{SH}_k^{cell}\), then so is the third.
\item The subcategory \(\mathcal{SH}_k^{cell}\) is closed under arbitrary colimits.
\end{itemize}
\item The subcategory of \emph{finite cellular spectra} \(\mathcal{SH}_k^{fin}\) in \(\mathcal{SH}_k\) is defined similarly as the smallest full subcategory that satisfies the first three conditions (see \cite[Definition 8.1]{DI2}).\\
\item We define the category of \emph{quasifinite cellular spectra} \(\mathcal{SH}_k^{qfin}\) as the smallest full triangulated subcategory of \(\mathcal{SH}_k\) that contains \(\mathcal{SH}_k^{fin}\) and is closed under retracts. The spectra in \(\mathcal{SH}_k^{qfin}\) are exactly finite cell spectra and their retracts, since the cofiber of two retracts of finite cell spectra is a retract of a finite cell spectrum by the octahedral axiom.
\item By \cite[Lemma 2.2]{ROEN} a motivic spectrum is cellular if and only if it admits a cell presentation, i.e. it can be built by successively attaching cells \(S^{s,t}\). A motivic cell spectrum \(X\) is called \emph{of finite type} if it admits a cell presentation with the following property: there exists a \(k\in \ensuremath{\mathbb{N}} \) such that there are no cells in dimensions satisfying \(s-t<k\) and such that there exist only finitely many cells in dimensions \((s+t,t)\) for each \(s\).
\end{enumerate}
\end{defin}
\ \\
\subsection{Completions}
For the sake of studying periodic self maps it is useful to consider one prime at a time, because these maps are detected by a collection of cohomology theories called Morava K-theory, which are defined with regard to a specific prime. In our case this prime will usually be odd, i.e. different from two. Topologically one can implement this by studying the localized or completed homotopy category via the tool of Bousfield localization at an appropiate Moore spectrum. Motivically this works as well (a discussion of this in the motivic setting can be found in \cite[Section 3]{OR}): We define the \(l\)-completed motivic homotopy category \(\mathcal{SH}_{k,l}^\wedge\) as the Bousfield localization of the category \(\mathcal{SH}_k\) at the mod-l Moore spectrum \(S/l\).\\
\begin{defin}
Let \(l\) be any prime number, and let $X$ be a motivic spectrum in \(\mathcal{SH}_k\). The \emph{$l$-completion \(X^{\wedge}_l\) of $X$} is the Bousfield localization of $X$ at the mod-\(l\) Moore spectrum \(S/l\). One can also describe this completion as:
\[X^{\wedge}_l\defeq L_{S/l}X \simeq \underset{\leftarrow}{\text{holim}}~ X/{l^n}\]
\end{defin}
\begin{defin}
We define the subcategory \(\mathcal{SH}_{k,l}^{\wedge,cell}\) of \emph{\(l\)-complete cellular spectra} in \(\mathcal{SH}_{k,l}^\wedge\) as the full subcategory of \(l\)-completions of cellular spectra. Similarly, we define the subcategories \(\mathcal{SH}_{k,l}^{\wedge,fin}\) of \(l\)-complete finite cellular spectra and \(\mathcal{SH}_{k,l}^{\wedge,qfin}\) of \(l\)-complete quasifinite cellular spectra as the full subcategories of \(l\)-completions of spectra in \(\mathcal{SH}_{k}^{fin}\) and \(\mathcal{SH}_{k}^{qfin}\).
\end{defin}
\subsection{Motivic Spanier-Whitehead duality}
We are going to make use of Spanier-Whitehead duality when we study periodic self maps. The sources we want to quote use different, but equivalent definitions of dualizability, so we collect a number of basic definitions and facts about Spanier-Whitehead duality that we are going to use in one place. Our primary source is \cite[III.1]{LMS} where categorical duality is explained with great detail.\\
Consider a spectrum \(X\) in \(\mathcal{SH}_k\) or \(\mathcal{SH}_{k,l}^{\wedge}\). Both categories are closed symmetric monoidal categories (see \cite{JAR}), and therefore for an arbitrary motivic spectrum \(Y\) there exists a function spectrum \(F(X,Y)\). The unit and counit of the canonical tensor-hom adjunction are given by maps
\[\eta_{X,Y}: X\rightarrow F(Y,X\wedge Y)\]
and by the evaluation
\[\epsilon_{X,Y}: F(X,Y)\wedge X\rightarrow Y\]
and furthermore there is a natural pairing
\[
F(X,Y)\wedge F(X',Y')\rightarrow F(X\wedge X',Y\wedge Y')
\]
which provides a natural map
\[\nu_{X,Y}:F(X,S)\wedge Y \rightarrow F(X,Y)\]
by specializing to the case \(X'=Y=S\) and using the fact \(F(S,Y')\cong Y'\).
\begin{prop}
\label{SWEQUI}
Let \(X\) be a spectrum in \(\mathcal{SH}_k\) or \(\mathcal{SH}_{k,l}^{\wedge}\). Then the following three conditions are equivalent:
\begin{enumerate}
\item The canonical map \[\nu_{X,Y}:F(X,S)\wedge Y \rightarrow F(X,Y)\] is an isomorphism for all spectra \(Y\).
\item The canonical map \[\nu_{X,X}:F(X,S)\wedge X \rightarrow F(X,X)\] is an isomorphism.
\item There is a coevaluation map \(coev: S\rightarrow X \wedge F(X,S)\) such that the diagram
\[
\xymatrix@C=1.5cm@R=1.5cm
{
S \ar[r]^{coev}\ar[d]^{\eta_{S,X}}&
X \wedge F(X,S)\ar[d]^T\\
F(X,X)&
F(X,S)\wedge X\ar[l]_{\nu_{X,X}}\\
}
\]
commutes, where \(T\) denotes the transposition map.
\end{enumerate}
\end{prop}
\begin{proof}
Clearly the first point implies the second. The second point implies the third, because one can define \(coev\) as the composite \(T\circ\nu_{X,X}^{-1}\circ\eta_{S,X}\). Finally, the third point implies the first (c.f. \cite[Proposition III.1.3(ii)]{LMS}) because one can define an inverse to \[\nu_{X,Y}:F(X,S)\wedge Y \rightarrow F(X,Y)\]
as the following composite:
\begin{align*}
\nu_{X,Y}^{-1}: F(X,Y)&\cong F(X,Y)\wedge S \overset{id\wedge coev}\longrightarrow F(X,Y)\wedge X \wedge F(X,S) \overset{\epsilon_{X,Y}\wedge id}\longrightarrow\\
&\longrightarrow Y\wedge F(X,S)\overset{T}\longrightarrow F(X,S)\wedge Y
\end{align*}
\end{proof}
\begin{defin}
If \(X\) satisfies any of the preceding conditions, it is called \emph{strongly dualizable}.\\
The spectrum \(DX=F(X,S)\) is called the \emph{(motivic) Spanier-Whitehead dual} of \(X\). By definition, \(D\defeq F(-,S)\) is a contravariant functor \[D: \mathcal{SH}_k\rightarrow \mathcal{SH}_k\] and similarly \(D\defeq F(-,S^\wedge_{l})\) is a contravariant functor \[D:\mathcal{SH}_{k,l}^{\wedge}\rightarrow \mathcal{SH}_{k,l}^{\wedge}\]
on the category of \(l\)-complete spectra. In fact, the obvious map \[F(-,S)\rightarrow F(-,S^\wedge_{l})\]
is a completion at \(l\), but we will neither need nor prove it.
\end{defin}
We will need the following general facts about strongly dualizable spectra, which are proven in \cite[Proposition III.1.3 (i, iii)]{LMS}:
\begin{lemma}
\begin{enumerate}
\item If \(X\) is strongly dualizable, then \(DDX\cong X\).
\item If \(X\) and \(Y\) are strongly dualizable, then the natural map \[F(X,S) \wedge F(Y,S) \rightarrow F(X \wedge Y, S)\]
is an isomorphism. In particular, \(X \wedge Y\) is strongly dualizable.
\end{enumerate}
\end{lemma}
The spectrum \(DX\wedge X\) has the structure of a homotopy ring spectrum by the same arguments as in\cite[Proof of Corollary 5.1.5]{RAV2}:
\begin{rem}
\label{SWRingSpectrum}
If \(X\) is strongly dualizable, then the unit map \[e: S \overset{\eta_{S,X}}{\longrightarrow} F(X,X) \cong F(X,S) \wedge X=DX\wedge X\]
and the multiplication map
\[ \mu: DX \wedge X \wedge DX \wedge X \overset{D(e)}\longrightarrow DX\wedge S \wedge X \cong DX \wedge X\]
endow \(DX\wedge X\) with the structure of motivic homotopy ring spectrum (in fact an \(A_\infty\)-structure, but we are not going to use or prove it), where we use
\[X\wedge DX\cong DDX\wedge DX=D(DX\wedge X)\]
in the definition of \(D(e)\).
\end{rem}
\begin{lemma}
\label{DCofib} The functor \(D\) maps cofiber sequences to cofiber sequences, and the full subcategory of strongly dualizable spectra in \(\mathcal{SH}_k\) is thick.
\end{lemma}
\begin{proof}
For the first statement, let \(X\rightarrow Y \rightarrow Z\) be a cofiber sequence. Because \(\mathcal{SH}_k\) is the homotopy category of a pointed monoidal model category, the functor \(F(-,A)\) maps cofiber sequences to fiber sequences for any \(A\) in \(\mathcal{SH}_k\) (c.f. \cite[6.6]{HOV}). In particular this is true for \(D(-)=F(-,S)\). Because \(\mathcal{SH}_k\) is stable, fiber and cofiber sequences agree, and \(DZ \rightarrow DY \rightarrow DX\) is a cofiber sequence again.\\
For the second statement we only need to show that a retract of a strongly dualizable spectrum is again strongly dualizable, so let \(A\) be a retract of a strongly dualizable spectrum \(X\). Note that by the first point of \ref{SWEQUI} we have to show that the canonical map
\[
F(A,S)\wedge Y \rightarrow F(A,Y)
\]
is an isomorphism for all motivic spectra \(Y\), and we already now this statement is true if we replace \(A\) with \(X\). But this follows immediately from the following diagram:
\[
\xymatrix{
F(X,S)\wedge Y \ar @`{(-10,10),(10,10)}^{id} \ar[r]^\cong\ar[d]&F(X,Y)\ar[d]\ar @`{(15,10),(35,10)}^{id}\\
F(A,S)\wedge Y \ar[r]\ar@/^/[u]& F(A,Y)\ar@/_/[u]\\
}
\]
\end{proof}
\begin{lemma}
\label{CellDualizable}
All spectra in \(\mathcal{SH}_\ensuremath{\mathbb{C}}^{qfin}\) are strongly dualizable in \(\mathcal{SH}_\ensuremath{\mathbb{C}}\), and \(\mathcal{SH}_\ensuremath{\mathbb{C}}^{qfin}\) is closed under taking duals.\\
As a consequence, all spectra in \(\mathcal{SH}_{k,l}^{\wedge,qfin}\) are strongly dualizable in \(\mathcal{SH}_{\ensuremath{\mathbb{C}}(l)}\), and \(\mathcal{SH}_{k,l}^{\wedge,qfin}\) is closed under taking duals.
\end{lemma}
\begin{proof}
Finite cell spectra are contained in the thick subcategory of compact spectra, and compact spectra are dualizable(See \cite[Remark 4.1]{NOS} or \cite[5.2.7]{JOA}). Therefore the thick subcategory generated by finite cell spectra is dualizable.\\
To show that \(\mathcal{SH}_\ensuremath{\mathbb{C}}^{qfin}\) is closed under taking duals, we only have to check that the duals of finite cell spectra and their retracts are in \(\mathcal{SH}_\ensuremath{\mathbb{C}}^{qfin}\) again by \ref{finiteType}. This is true for finite cell spectra by cellular induction, because the duals of suspensions of the sphere spectrum are suspensions of the sphere spectrum.
If \(X\) is a retract of a spectrum \(F \in \mathcal{SH}_\ensuremath{\mathbb{C}}^{qfin}\) such that \(DF\in \mathcal{SH}_\ensuremath{\mathbb{C}}^{qfin}\), with maps \(r:F\rightarrow X\) and \(s:X\rightarrow F\) such that \(r\circ s=id_X\), then \(DX\) is a retract of \(DF\in \mathcal{SH}_\ensuremath{\mathbb{C}}^{qfin}\) with maps \(Ds:DF\rightarrow DX\) and \(Dr:DX\rightarrow DF\) because \(Ds\circ Dr=id_{DX}\).
\end{proof}
\subsection{The motivic Steenrod algebra and the dual motivic Steenrod algebra}
One key ingredient for the Adams spectral sequence is knowlegde of the Steenrod algebra or of the dual Steenrod algebra. Motivically, the Steenrod Algebra was described by Voevodsky for fields of characteristic zero and later by Hoyois, Kelly and {\O}stv{\ae}r in positive characteristic. While some interesting phenomenas happen at the prime two, the motivic Steenrod algebra is more closely related to the classical topological Steenrod algebra at odd primes. To describe the motivic Steenrod algebra it is sufficient to know the coefficients of motivic coholomogy with \(\ensuremath{\mathbb{Z}}/l\ensuremath{\mathbb{Z}}\)-coefficients:
\begin{prop}For \(l\neq 2\) a prime and \(k=\ensuremath{\mathbb{C}}\) the coefficients \(H\ensuremath{\mathbb{Z}}/l^{**}\) of motivic cohomology are given as a ring by \[H\ensuremath{\mathbb{Z}}/l^{**}\cong \ensuremath{\mathbb{Z}}/l [\tau] \] with \(|\tau|=(0,1)\), and the image of \(\tau\) under Betti realization is nonzero.
\end{prop}
\begin{proof}
We know that \(H\ensuremath{\mathbb{Z}}/l^{**}=0\) for $q<p$ ((cf. \cite[Theorem 3.6]{MVW})).\\ Let \(q\geq p\). Then there is an isomorphism from motivic to \'etale cohomology: \[H^{p,q}(Spec(k), \mathbb{Z}/l) \cong H_{\acute{e}t}^p(k,\mu_l^{\otimes q})\]
This isomorphism respects the product structure(\cite[1.2,4.7]{GL}).\\
The \'etale cohomology groups \(H_{\acute{e}t}^p(k,\mu_l^{\otimes q})\) can be computed as the Galois cohomology of the separable closure of the base field (in both cases the complex numbers) with coefficients in the $l$-th roots of unity. The action of the absolute Galois group $G$ is given by the trivial action if $k=\mathbb{C}$ and by complex conjugation if $k=\mathbb{R}$:
\[H_{\acute{e}t}^p(k,\mu_l^{\otimes q})\cong H(G, \mu_l^{\otimes q}(\ensuremath{\mathbb{C}}))\]
For $k=\mathbb{C}$, these groups all vanish for $p\neq 0$ by triviality of the Galois action, and they are $\mathbb{Z}/l$ in the degree $p=0$ for all $q\geq 0$. The multiplicative structure is given by the tensor product of the modules.\\
\end{proof}
\begin{rem}
We denote the image of \(\tau\) under \(H\ensuremath{\mathbb{Z}}/l_{**}=H\ensuremath{\mathbb{Z}}/l^{-*,-*}\) with the same name. This image has bidegree \(|\tau|=(0,-1)\).
\end{rem}
The motivic mod-$l$ Steenrod algebra over basefields of characteristic 0 has been computed by Voevodsky in \cite{VOE2}. The implications for the dual motivic Steenrod algebra are for example written down in the introduction of \cite{HKO}. In our special case it has the following shape:
\begin{prop}
Let $k=\ensuremath{\mathbb{C}}$ as above, and let $l$ be an odd prime. The dual motivic Steenrod algebra \(A_{**}\) and its Hopf algebroid structure can be described as follows:
\[A_{**}=H\ensuremath{\mathbb{Z}}/l_{**}[\tau_0,\tau_1,\tau_2,...,\xi_1,\xi_2,...]/(\tau_i^2=0)\]
Here \(|\tau_i|=(2l^i-1,l^i-1)\) and \(|\xi_i|=(2l^i-2,l^i-1)\).\\ The comultiplication is given by
\[\Delta(\xi_n) = \sum_{i=0}^n \xi_{n-i}^{l^i} \otimes \xi_i\] where \(\xi_0:=1\), and
\[\Delta(\tau_n) = \tau_n\otimes 1 + \sum_{i=0}^n \xi_{n-i}^{l^i} \otimes \tau_i\]
\end{prop}
\subsection{Generalized motivic Adams spectral sequences}
We will use the homological motivic Adams spectral sequence to compute the coefficients of the \(l\)-completed motivic Brown-Peterson spectrum \(ABP^\wedge_l\).
The motivic Adams spectral sequence was inspired by Morels computation of the zeroth motivic stable stem(c.f. \cite{Mor}) and was used by Dugger and Isaksen for extensive computations over \(\ensuremath{\mathbb{C}}\) at the prime 2 (c.f. \cite{DI}). They also use additional information available in the MASS to deduce new information about the classical Adams spectral sequence. In other work Isaksen has extended these computations to the base field \(\mathbb{R}\). Generalized motivic Adams spectral sequences can be constructed for \(E\) an arbitrary motivic ring spectrum and \(X\) a motivic spectrum.
Define \(\bar{E}\) as the cofiber of the unit map \(S\rightarrow E\). Smashing the cofiber sequence \(\bar{E} \rightarrow S \rightarrow E\) with \(\bar{E}^s\wedge X\) yields cofiber sequences
\[\bar{E}^{\wedge(s+1)}\wedge X \rightarrow \bar{E}^{\wedge s}\wedge X \rightarrow E\wedge \bar{E}^{\wedge s}\wedge X\]
giving rise to the following tower, called the canonical \(E_{**}\)-Adams resolution:
\[
\xymatrix
{
...\ar[r]\ar[d]&
\bar{E}^{\wedge(s+1)} \wedge X \ar[r]\ar[d] &
\bar{E}^{\wedge s} \wedge X \ar[r]\ar[d]&
...\ar[r]\ar[d]&
\bar{E}\wedge X \ar[r]\ar[d]&
X\ar[d]\\
...&
E\wedge \bar{E}^{\wedge(s+1)} \wedge X &
E\wedge \bar{E}^{\wedge s} \wedge X &
...&
E\wedge \bar{E} \wedge X &
E \wedge X\\
}
\]
The long exact sequences of homotopy groups associated to these cofiber sequences forms a trigraded exact couple
\[
\xymatrix{
\pi_{**}(\bar{E}^{\wedge*}\wedge X) \ar[rr]&
&
\pi_{**}(\bar{E}^{\wedge*}\wedge X)\ar[dl]\\
&
\pi_{**}(E\wedge\bar{E}^{\wedge*}\wedge X)\ar[lu]
&\\
}
\]
and thus give rise to a trigraded spectral sequence \(E_r^{s,t,u}(E,X)\) with differentials \(d_r: E_r^{s,t,u} \longrightarrow E_r^{s+r,t+r-1,u}\).\\
If one furthermore assumes that $E_{**}E$ is flat as a (left) module over the coefficients \(E_{**}\) it is possible to identify the \(E_2\)-term via homological algebra. In this case one can associate a flat Hopf algebroid to \(E\) (See \cite[Lemma 5.1]{NOS} for the statement and \cite[Appendix 1]{RAV} for the definition and basic properties of Hopf algebroids), and the category of comodules over this Hopf Algebroid is abelian and thus permits homological algebra. Because \(E_{**}E\) is flat over \(E_{**}\) there is also an isomorphism (see \cite[Lemma 5.1(i)]{NOS})
\[\pi_{**}(E\wedge E \wedge X)\cong E_{**}(E)\underset{E_{**}}{\otimes}E_{**}(X)\]
allowing us to identify the long exact sequences of homotopy groups of the canonical \(E_{**}\)-Adams resolution with the (reduced) cobar complex \(C^*(E_{**}(X))\). For this reason the resolution is also referred to as the geometric cobar complex. The \(E_2\)- page of the $E$-Adams spectral sequence can then be described as:
\[E_2^{s,t,u}(E,X)=\text{Cotor}^{s,t,u}_{E_{**}(E)}(E_{**},E_{**}(X))\]
Here \(\text{Cotor}\) denotes the derived functors of the cotensor product in the category of \(E_{**}(E)\)-comodules and can be computed as the homology of the cobar complex \(C^*(E_{**}(X))\).
\begin{rem}
Assume now that \(k=\ensuremath{\mathbb{C}}\). Then Betti realization induces a map of spectral sequences \(R_{E,X}: E_r^{s,t,u}(E,X)\rightarrow E_r^{s,t}(R(E),R(X))\)
\end{rem}
This can be checked by going through the definitions: Because Betti realization preserves cofiber sequences and smash products, we have \(R( \bar{E})=\overline{R(E)}\), and the realization of the canonical \(E_{**}\)-Adams resolution for \(X\) is the canonical \(R(E)_{*}\)-Adams resolution for the topological spectrum \(R(X)\). If we consider the induced maps on the long exact sequences of homotopy groups defining the exact couple, we get the following commutative diagram:
\[
\xymatrix@C-=0.3cm{
...\ar[r]\ar[d]^R&
\pi_{p,*}(\bar{E}^{\wedge(s+1)}\wedge X)\ar[r]\ar[d]^R&
\pi_{p,*}(\bar{E}^{\wedge s}\wedge X)\ar[r]\ar[d]^R&
\pi_{p,*}(E\wedge\bar{E}^{\wedge s}\wedge X)\ar[r]\ar[d]^R&
...\ar[d]^R\\
...\ar[r]&
\pi_{p}(\overline{R(E)}^{\wedge(s+1)}\wedge R(X))\ar[r]&
\pi_{p}(\overline{R(E)}^{\wedge s}\wedge R(X))\ar[r]&
\pi_{p}(R(E)\wedge \overline{R(E)}^{\wedge s}\wedge R(X))\ar[r]&
...
}\]
In particular, Betti realization induces a map of exact couples and hence a map of spectral sequences.\\
Convergence of the spectral sequence has been studied for the case \(E=H\ensuremath{\mathbb{Z}}/l\) by Hu, Kriz and Ormsby in \cite[Theorem 1]{HKO}. It turns out that over the complex numbers, the spectral sequence will just converge to the $l$-completion \(X^{\wedge}_l\) of $X$, which one can either describe as the Bousfield localization of $X$ at the mod-\(l\) Moore spectrum \(S/l\) or explicitely as:
\[X^{\wedge}_l\defeq L_{S/l}X \simeq \underset{\leftarrow}{\text{holim}}~ X/{l^n}\]
The homotopy groups of $X$ and its $l$-completion are related by the following short exact sequence(\cite[End of section 3]{OR}):
\[0 \rightarrow \text{Ext}^1(\ensuremath{\mathbb{Z}}/{l^\infty},\pi_{**}X)\rightarrow \pi_{**}X^\wedge_l \rightarrow \text{Hom}(\ensuremath{\mathbb{Z}}/{l^\infty},\pi_{*-1,*}X) \rightarrow 0\]
We will see later that for our case of interest, where \(X=ABP^\wedge_l\), the spectral sequence actually converges strongly because of a vanishing line.
\subsection{The algebraic Morava-K-theories \(AK(n)\)}
As before we work over the complex numbers, and the prime \(l\) will be odd. In particular this prime is implicit in the definition of the motivic Brown-Peterson-spectrum \(ABP\) and of the algebraic Morava-K-theory spectrum \(AK(n)\). In this section we show that the algebraic Morava-K-theory spectra \(AK(n)\) admit the structure of a commutative homotopy ring spectrum similar to their classical counterparts. These spectra were originally defined by Borghesi in \cite{Bor}. In addition we rely on the description provided in \cite[Def. 6.3.1]{JOA}:
\begin{defin}
The connective n-th motivic Morava K-theory is defined as
\[Ak(n)=ABP/(v_0,...,v_{n-1},v_{n+1},v_{n+2},...)\]
and the n-th motivic Morava K-theory spectrum \(AK(n)\) is defined as: \[AK(n)=v_n^{-1}ABP/(v_0,...,v_{n-1},v_{n+1},v_{n+2},...)\]
In particular, both spectra are \(MGL_{(l)}\)-modules.
\end{defin}
\(AK(n)\) and \(Ak(n)\) are genuinely motivic in the sense that they are derived from the spectrum representing algebraic cobordism. We will need some of the properties of \(AK(n)\) proven in \cite{JOA}, namely:
\begin{rem}
\begin{enumerate}
\item The Betti realization of the (connective) motivic Morava K-theory is the classical (connective) Morava K-theory (\cite[Lemma 6.3.2]{JOA}):
\[R_{\ensuremath{\mathbb{C}}}(AK(n))=K(n)\]
and
\[R_{\ensuremath{\mathbb{C}}}(Ak(n))=k(n)\]
\item By \cite[Lemma 6.3.7]{JOA} the coefficients of algebraic Morava K-theory are given by: \[AK(n)_{**}=H\ensuremath{\mathbb{Z}}/(l)_{**}\underset{\ensuremath{\mathbb{Z}}/(l)}{\otimes}K_{*}\]
\item If \(X\) is a finite motivic cell spectrum such that \(H\ensuremath{\mathbb{Z}}/(l)^{**}(X)\) is free over the coefficients, then the motivic Adams spectral sequence for \(Y=Ak(n)\wedge X\) will converge strongly to \(Ak(n)_{**}(X)\). (See \cite[8.3.3]{JOA})
\end{enumerate}
\end{rem}
At least for odd primes, the topological spectra \(K(n)\) can be shown to be homotopy ring spectra. As remarked in \cite[Remark 6.3.3(6)]{JOA}, it is not known in general if the motivic Morava K-theory spectrum \(AK(n)\) can be endowed with the structure of a motivic homotopy ring spectrum. In the special case \(k=\ensuremath{\mathbb{C}}\), \(l\neq 2\) however Joachimi proved that the spectrum
\[AP(n)\defeq ABP/(v_0=l,v_1,...,v_{n-1}),\]
another quotient of \(MGL\), admits a unital homotopy associative product \cite[9.3]{JOA}, and with the work done by her it is no longer difficult to do the same for \(AK(n)\).\\
We want to use and extend the results in \cite[9.3]{JOA} and follow the notation used there to make comparison easier. In particular \(\eta\) will not denote the motivic Hopf map in this chapter, but a different map to be defined later. The only exception is the name of the prime \(l\), which is referred to as \(p\) in \cite{JOA}.\\
Let \(R \in \mathcal{SH}_k\) be a strictly commutative ring spectrum with multiplication map \(m: R\wedge R \rightarrow R\) and unit map \(i:S\rightarrow R\). The example that we have in mind is \(MGL_{(l)}\), which is a strictly commutative motivic ring spectrum by the reasoning given in the beginning of \cite[9.3]{JOA}.\\
Classically one can study the products on \(R\)-modules of the form \(R/x\) and use them to gain information about products on quotients of the form \(R/X\) where \(X\) is a countable regular sequence of homogeneous elements.
In contrast to the classical situation, the coefficients of \(MGL_{(l)}\) are not known, but the coefficients \(MGL_{(l)}/l\) are. Therefore motivically one has to consider \(R\)-modules of the form \(R/(x,y)\).\\
In the section immediately preceding \cite[9.3.7]{JOA} and in the proof of \cite[Lemma 9.3.8]{JOA} Joachimi constructs a product on quotients of this form and proves the following statement:
\begin{lemma}
\label{ProductLemma}
Let \(y\in\pi_{k',l'}(R)\) and let \(x\in \pi_{k,l}(R)\). Define the \(R\)-modules \(M\defeq R/y\) and \(N\defeq M/x\) and denote the structure map of \(M\) as \(\nu_M: R\wedge M\rightarrow M\). Write \(\eta'\) for the canonical map \[\eta': R\rightarrow M=R/y\] and \(\eta\) for the canonical map \[\eta: M\rightarrow N=R/(x,y)\].\\
If \(\pi_{2k'+1,2l'}(M)=0\) and \(\pi_{2k+1,2l}(N)=0\), there are maps of \(R\)-modules
\[\mu_M: M\wedge M\rightarrow M\]
\[\nu_{M,N}: M\wedge N \rightarrow M \]
\[\mu_N: N\wedge N \rightarrow N\]
making the following diagrams commute up to homotopy:
\begin{equation}
\label{ProductLemmaDiagram1}
\xymatrix{
R \wedge R \ar[rr]^{\eta'\wedge \eta'} \ar[dd]^{m} \ar[dr]^{1\wedge \eta'} & & M\wedge M \ar[dd]^{\mu_M}\\
& R \wedge M \ar[dr]^{\nu_M} \ar[ur]^{\eta'\wedge 1}& \\
R \ar[rr]^{\eta'} & & M\\
}
\end{equation}
\begin{equation}
\label{ProductLemmaDiagram2}
\xymatrix{
M \wedge M \ar[rr]^{\eta\wedge \eta} \ar[dd]^{\mu_M} \ar[dr]^{1\wedge \eta} & & N \wedge N \ar[dd]^{\mu_N} \\
& M \wedge N \ar[dr]^{\nu_{M,N}} \ar[ur]^{\eta\wedge 1}& \\
M \ar[rr]^{\eta} & & N\\
}
\end{equation}
In particular, if we choose the maps \(\eta'\circ i\) and \(\eta\circ\eta'\circ i\) as unit maps, \(\mu_M\) and \(\mu_N\) are unital products on \(M\) and \(N\) respectively.
\end{lemma}
Furthermore the following result of Joachimi \cite[Lemma 9.3.8]{JOA} proves associativity, and we wish to extend it to include commutativity:
\begin{lemma}
\label{HomotopyAssociativityLemma}
If \(\pi_{k'+1,l'}(M)=\pi_{2k'+2,2l'}(M)=\pi_{3k'+3,3l'}(M)=0\), then \(\mu_M\) is homotopy associative.\\
If furthermore \(\pi_{k+1,l}(N)=\pi_{2k+2,2l}(N)=\pi_{3k+3,3l}(N)=0\), then \(\mu_N\) is also homotopy associative.
\end{lemma}
We need the following lemma of Joachimi \cite[Lemma 9.3.3]{JOA} in the proof of commutativity:
\begin{lemma}
\label{Joachimi933}
Let \(R'\) be a (homotopy) ring spectrum, \(M'\) a left \(R'\)-module, and \(\pi_{k,l}(M')=0\). Then any \(R'\)-module map \(\psi:S^{k,l}\wedge R' \rightarrow M'\) is homotopically trivial.
\end{lemma}
\begin{prop}
\label{CommutativityLemma}
Let R be a homotopy ring spectrum and commutative up to homotopy. Let M and N be quotient modules defined as in \ref{ProductLemma}.
If \(\pi_{k'+1,l'}(M)=\pi_{2k'+2,2l'}(M)=0\), then \(\mu_M\) is homotopy commutative.\\
If furthermore the homotopy groups of \(N\) satisfy \(\pi_{k+1,l}(N)=\pi_{2k+2,2l}(N)=0\), then \(\mu_N\) is also homotopy commutative.
\end{prop}
\begin{proof}
The \(R\) module \(M=R/y\) is defined by the following cofiber sequence:
\[
\Sigma^{k',l'}R\overset{\phi}\rightarrow R \overset{\eta'}\rightarrow M \overset{\delta}{\rightarrow} \Sigma^{k'+1,l'}R
\]
Recall that \(m:R\wedge R\rightarrow R\) is the product on the ring spectrum \(R\). To show that the product \(\mu_M: M\wedge M\rightarrow M\) is commutative, it suffices to show \[\theta\defeq\mu_M\circ(1-T):M\wedge M\rightarrow M\]
is homotopic to the zero map, where \(T\) is the transposition map. The map \[\theta'\defeq (\eta' \wedge id_M)\circ \theta\]fits into the following diagram of R modules
\[
\xymatrix{
R\wedge R \ar[dd]_{id_R\wedge \eta'}\ar[rr]^{m \circ (1-T) = 0} \ar[ddrr]^{0} && R \ar[dd]^{\eta'}\\\\
R\wedge M \ar[dd]\ar[rr]^{\theta'}&&M\\\\
\Sigma^{k'+1,l'}R\wedge R \ar@{-->}[uurr]^{\bar{\theta'}}
}
\]
which commutes by \ref{ProductLemmaDiagram1}. The top horizontal map is zero up to homotopy because \(m\) is homotopy commutative by assumption, and the first column is the cofiber sequence defining \(M\), smashed with \(R\). Together, this implies the existence of the dashed map \(\bar{\theta'}\).
Now \(R\) is a \(R \wedge R\) module via the product map \(m\) and we can consider this diagram as a diagram of \(R \wedge R\) modules. Then proposition \ref{Joachimi933}, applied to the ring spectrum \(R \wedge R\), implies that \(\bar{\theta'}\) is null homotopic by our assumptions on the homotopy groups of \(M\). Therefore \(\theta'\) is null homotopic as well. We then get the following commutative diagram for \(\theta\): \[
\xymatrix{
&&R\wedge M \ar[dd]_{\eta'\wedge id_M}\ar[rr]^{\theta' = 0} \ar[ddrr]^{0} && M \ar@{=}[dd]\\\\
&&M\wedge M \ar[dd]\ar[rr]^{\theta}&&M\\\\
\Sigma^{k'+1,l'}R\wedge R \ar[rr]^{id_R \wedge \eta'} &&\Sigma^{k'+1,l'}R\wedge M \ar@{-->}[uurr]^{\bar{\theta}}\ar[rr] && \Sigma^{2k'+2,2l'}R\wedge R\ar@{.>}^{\tilde{\theta}}[uu]
}
\]
Once again the first column is a cofiber sequence, which implies the existence of the dashed map.
The composite \(\bar{\theta}\circ(id_R \wedge \eta')\) is null homotopic because this diagram is a diagram of \(R\wedge R\) modules again, so we can use the same argument as before. This implies the existence of the dotted map \(\tilde{\theta}\). This map also vanishes by the second condition on the homotopy groups of M, which in turn implies that \(\bar{\theta}\) is zero up to homotopy. Therefore \(\theta\) also vanishes, so \(\mu_M\) is homotopy commutative.
Because we used only the fact that \(R\) is a homotopy ring spectrum and not strict commutativity, and because diagram \ref{ProductLemmaDiagram2} in \ref{ProductLemma} commutes, we can then repeat the same proof with \(M\) replacing \(R\) and \(N\) replacing \(M\). Note that this would not have been possible if we worked over \(R\)-modules, because it is not clear that \(R/x\) is a strictly commutative ring spectrum again.
\end{proof}
\begin{lemma}
\label{CommutativityLemma2}
Let \(k=\ensuremath{\mathbb{C}}\) and \(l\neq 2\). The spectrum \(AP(n)\) admits a unital, homotopy associative and homotopy commutative product
\[\mu_{AP(n)}:AP(n)\wedge AP(n) \rightarrow AP(n)\]
and so do the spectra \(A_i=AP(n)/(v_{n+1},...,v_{n+i})\).
\end{lemma}
\begin{proof}
Except for the statement about commutativity, the first part of this lemma is the content of \cite[9.3.9]{JOA}. The essential argument in the proof of the cited lemma is as follows: if one has a sequence of elements \(J\subset R_{**}\) and one knows that \(A\defeq R/(J-\{x,y\})\) is a homotopy associative and commutative ring spectrum, then one can describe the product on \(R/J\cong A/(x,y) \cong A \underset{R}\wedge R/(x,y)\) by
\[(N\underset{R}\wedge A) \wedge (N\underset{R}\wedge A) \overset{\tau}\longrightarrow (N\wedge N)\underset{R}\wedge (A\wedge A)\overset{id_N\wedge id_N \wedge \mu_A}\longrightarrow N\wedge N\underset{R}\wedge A \overset{\mu_N \wedge id_A}\longrightarrow N\wedge A\]
and thus has to prove the vanishing of the obstruction groups to associativity only after application of \((-)\underset{R}\wedge A\) to the associativity diagram.\\
Now choose \(R=MGL_{(l)}\) and \(A=ABP\) and \(J\) such that \(MGL_{(l)}/J=AP(n)\). Then the relevant obstruction groups are trivial because for odd primes \(l\neq 2\), \(ABP_{**}\) is concentrated in bidegrees where the first degree is divisible by 4. We can then show that there is a homotopy associative product on \(AP(n)\) by induction; because \(AP(n)/(v_0,...,v_n)\), we only have to do finitely many steps, and we can use the fact (see \cite[Lemma 9.3.7]{JOA}) that for any sequence \((l)\subset J'\):
\[
ABP/(J'\cup \{y\})\cong MGL_{(l)}/(l,y)\underset{MGL_{(l)}}\wedge ABP/J'\]
We can use the same argument to prove commutativity: if we apply \((-)\underset{R}\wedge A\) to all the relevant diagrams in \ref{CommutativityLemma}, we see that the obstructions to commutativity lie in groups \(\pi_{i,j}(M\underset{R}\wedge A)\) and \(\pi_{i,j}(N\underset{R}\wedge A)\) which are trivial because 4 does not divide \(i\) in the relevant bidegrees. Therefore the product on \(AP(n)\) is in fact homotopy commutative.\\
Now consider the spectra \(A_i\). To define them, we add finitely many elements, namely \(v_{n+1},...,v_{n+i}\), to the sequence \(J\). The proof of \cite[Lemma 9.3.7]{JOA} carries through verbatim and we can conclude that there is a product map \(A_i\wedge A_i \rightarrow A_i\). Similarly, because we had to add only finitely many elements to \(J\), we can repeat the induction argument above for the spectra \(A_i\). This shows that the multiplication on \(A_i\) is in fact homotopy associative and homotopy commutative.
\end{proof}
By essentially classical arguments, this allows us to conclude that \(Ak(n)\) has the desired ring structure:
\begin{prop}
Let \(k=\ensuremath{\mathbb{C}}\) and let \(l\) be an odd prime. Then the connective algebraic Morava K-theory spectrum
\[Ak(n)=\underset\longrightarrow{\textnormal{hocolim }} A_i=ABP/(v_0,v_1,...v_{n-1},v_{n+1},v_{n+2},...)\]
admits the structure of a homotopy associative and homotopy commutative motivic ring spectrum.
\end{prop}
\begin{proof}
By \cite[Corollary 9.3.5]{JOA} the elements \(v_i, i\neq n\) act trivially on \(Ak(n)\). This is in particular the case for \(v_0=l\). Therefore \cite[Lemma 6.7]{STR} holds for \(A=M=Ak(n)\) (although Strickland considers rings in R-modules, the only necessary modification is replacing the map \(\rho^*\) by the map \(\rho^*: [R/(l,x_i)\underset{R}\wedge B, M]\rightarrow[R/x_i\underset{R}\wedge B, M]\rightarrow[B,M]\)), and we can use the arguments of \cite[Proposition 6.8]{STR} to conclude that the constructed products on \(A_i\) induce a unital, homotopy associative product on \(Ak(n)\). As noted in the proof of Stricklands proposition, this product is commutative if and only if the maps \(A_i\rightarrow Ak(n)\) commute with themself (see \cite[Definition 6.1]{STR} for a definition of this notion). Because the product on \(A_i\) is commutative, this is the case for every map out of \(A_i\).
\end{proof}
\begin{corol}
Let \(k=\ensuremath{\mathbb{C}}\) and let \(l\) be an odd prime. The algebraic Morava K-theory spectrum \(AK(n)=v_n^{-1}Ak(n)\) admits the structure of a commutative and associative motivic homotopy ring spectrum.
\end{corol}
\begin{proof}
We have an isomorphism \(AK(n)\cong v_n^{-1}MGL_{(l)}\underset{MGL_{(l)}}\wedge Ak(n)\) and both smash factors admit a homotopy commutative and associative product(\cite[Proposition 6.6]{STR}). Therefore we can endow \(AK(n)\) with the desired structure as in the proof of \ref{CommutativityLemma2}.
\end{proof}
It remains to show that this product induces the same product structure on \(AK(n)_{**}\) as one would expect from the computation of these coefficients:
\begin{lemma}
The multiplication map \[\mu_{AK(n)}: AK(n)\wedge AK(n)\rightarrow AK(n)\]
induces the multiplication on \(AK(n)_{**}\) given by the multiplication on \(K(n)_*\) and the isomorphism \(AK(n)_{**}\cong HZ/l_{**}\underset{\ensuremath{\mathbb{Z}}/l}\otimes K(n)_*\) of \cite[Lemma 6.3.7]{JOA}.
\end{lemma}
\begin{proof}
The proof is similar to the proof of \cite[Lemma 9.3.10]{JOA}
\end{proof}
\section{Thick subcategories characterized by motivic \(v_n\)-self maps}
Let \(l\) be an odd prime and let \(k=\ensuremath{\mathbb{C}}\). The aim of this section is to show that the existence of \(v_n\)-self maps characterizes thick subcategories in \(\mathcal{SH}_{\ensuremath{\mathbb{C}}}^{qfin}\) and hence also in the motivic homotopy category. We consider only the case \(n>0\).\\
\begin{defin}
Let \(X\) be a motivic spectrum in \(\mathcal{SH}_{\ensuremath{\mathbb{C}}}^{qfin}\) or \(\mathcal{SH}_{\ensuremath{\mathbb{C}},l}^{\wedge,qfin}\). A map \(f: \Sigma^{t,u} X \rightarrow X\) is a motivic \(v_n\) self-map if it satisfies the following conditions:
\begin{enumerate}
\item \(AK(m)_{**} f\) is nilpotent if \(m \neq n\)
\item \(AK(m)_{**} f\) is given by multiplication with an invertible element of \(H\ensuremath{\mathbb{Q}}_{**}\) if \(m=n=0\).
\item \(AK(m)_{**} f\) is an isomorphism if \(m=n\neq 0\).
\end{enumerate}
\end{defin}
\ \\
As mentioned before, the topological nilpotence theorem is a key ingredient in the proof that topological finite cell spectra spectra admitting a \(v_n\)-self map form a thick subcategory: A map of finite spectra is nilpotent if and only if it induces zero in all Morava K-theories. The motivic equivalent of this theorem does not hold: For example, the motivic Hopf map \(\eta\) is a non-nilpotent map in \(\mathcal{SH}_{\ensuremath{\mathbb{C}}}\), but induces the zero map in motivic Morava K-theory for degree reasons. It seems likely however that a weaker version of the theorem applies, where we only consider maps of a certain bidegree. For the remainder of this subsection we assume that the following motivic nilpotence conjecture holds:
\begin{conj}
\label{NilpotenceConjecture}
Let \(k=\ensuremath{\mathbb{C}}\), let \(l\) be an odd prime and \(n>0\) be an integer. If \(X\) is a motivic spectrum in \(\mathcal{SH}_{\ensuremath{\mathbb{C}}}^{qfin}\) or \(\mathcal{SH}_{\ensuremath{\mathbb{C}},l}^{\wedge,qfin}\) and \(f:\Sigma^{p,q}X\rightarrow X\) is a motivic map such that \((p,q)\) is a multiple of \((2l^n-2,l^n-1)\), then: \[\forall m\in \ensuremath{\mathbb{N}}: AK(m)_{**}(f)=0 \implies \exists k\in \ensuremath{\mathbb{N}}: f^k\simeq 0\]
\end{conj}
\ \\
The known examples of non-nilpotent motivic self maps that induce the zero map in motivic Morava-K-theory (A variety of examples can be found in \cite{HOR}, and Boghdan George has constructed a whole family of such maps detected by exotic motivic Morava K-theories) do not contradict this conjecture.\\
\ \\
To prove the motivic equivalent of asymptotic uniqueness, we want to use Betti realization to compare the motivic situation to the classical one. To do this, we need to study the effect of Betti Realization on homology groups of \(AK(n)\). We will show that the kernel of the map induced by Betti realization is precisely the \(\tau\)-primary torsion elements. To do this, we need to compare \(K(n)_*\) and \(AK(n)_{**}\)-modules, which is only possible after inverting \(\tau\). We also need the fact that the \(AK(n)\)-homology of a (quasi)-finite motivic cell spectrum is finitely generated over the coefficients:
\begin{lemma}
\label{FinGen}
Let \(X\) be a motivic spectrum in \(\mathcal{SH}_{\ensuremath{\mathbb{C}}}^{qfin}\) or \(\mathcal{SH}_{\ensuremath{\mathbb{C}},l}^{\wedge,qfin}\). Then
\begin{enumerate}
\item \(AK(n)_{**}(X)\) is finitely presented as an \(AK(n)_{**}\)-module.
\item \(\text{Hom}_{AK(n)_{**}}(AK(n)_{**}(X),M)\) is finitely presented as an \(AK(n)_{**}\)-module for every finitely presented \(AK(n)_{**}\)-module \(M\). In particular, \[\text{End}_{AK(n)_{**}}(AK(n)_{**}(X))\]
is finitely presented.
\end{enumerate}
\end{lemma}
\begin{proof}
Note that \(AK(n)_{**}\) is a quotient of a polynomial ring in the three variables \(v_n, v_n^{-1}, \tau\) over the field \(\mathbb{F}_l\) and hence Noetherian. Therefore a \(AK_{**}\)-module is finitely presented if and only if it is finitely generated.
\begin{enumerate}
\item We will show the statement for finite cell spectra by cellular induction and then show that it also holds for retracts of finite cell spectra.
The claim is trivially true for the sphere spectrum. If the statement holds for a spectrum, it also holds for retracts of this spectrum because \(AK(n)_{**}\) is Noetherian and submodules of finitely generated modules are again finitely generated.
It remains to show that if the spectra \(X\) and \(Y\) in a cofiber sequence \(X\overset{f}\longrightarrow Y \overset{g}\longrightarrow Z \) satisfy the statement, then so does \(Z\). Consider the long exact sequence in \(AK(n)\)-homology
\[...\rightarrow AK(n)_{**}(Y) \overset{g}\longrightarrow AK(n)_{**}(Z)\overset{\delta}\longrightarrow AK(n)_{*-1,*}(X)\overset{f}\longrightarrow AK(n)_{*-1,*}(Y)\rightarrow ...\]
associated to this cofiber sequence. We can break it up into short exact sequences in the canonical way:
\[0\rightarrow \textnormal{coker}(f)\overset{\bar{g}}\longrightarrow AK(n)_{**}(Z)\overset{\bar{\delta}_p}\longrightarrow \textnormal{ker}(f)[-1]\rightarrow 0\]
The two outer terms in the short exact sequence are finitely generated: \(\textnormal{ker}(f)[-1]\) as a submodule of a finitely generated module over a Noetherian ring, and \(\textnormal{coker}(f)\) as a quotient of a finitely generated module. Therefore the middle term is also finitely generated.
\item By the first part of this lemma, \(AK(n)_{**}(X)\) is finitely generated as an \(AK(n)_{**}\)-module. Therefore there is a surjection \(R^k\rightarrow AK(n)_{**}\) from a free and finitely generated \(AK(n)_{**}\)-module \(R^k\) onto \(AK(n)_{**}(X)\). Then
\[\text{Hom}_{AK(n)_{**}}(R^k,M)\cong M^k\]
is a free and finitely generated \(AK(n)_{**}\)-module. Because \(AK(n)_{**}\) is a Noetherian ring,
\[\text{Hom}_{AK(n)_{**}}(AK(n)_{**}(X),M)\]
is finitely generated as a submodule of this finitely generated module.
\end{enumerate}
\end{proof}
\begin{rem}
\label{ExtScalarFlat}
One can regard \(K(n)_{*}\) and its modules as a bigraded ring and bigraded modules concentrated in degree 0 with respect to the second bidegree. Then every \(AK(n)_{**}[\tau^{-1}]\)-module has the structure of a bigraded \(K(n)_{*}\)-module where \(v_n^{top}\) acts via \(\tau^{l^n-1}v_n\). (This of course implies that \((v_n^{top})^{-1}\) acts via \(\tau^{-l^n+1}v^{-1}_n\), so it only makes sense after inverting \(\tau\).) With this module structure, \(AK(n)_{**}[\tau^{-1}]\) is free (with basis \(\tau^{k}, k\in \mathbb{Z},- l^n+1< k < l^n-1\)) and in particular flat as a \(K(n)_{*}\)-module. Likewise it is flat as an \(AK(n)_{**}\)-module, because it is a localization. We will implicitly use this in the following statements and sometimes write \(-[\tau^{-1}]\) for \(-\underset{AK(n)_{**}}{\otimes}AK(n)_{**}[\tau^{-1}]\), and \(-[\tau,\tau^{-1}]\) for \(-\underset{K(n)_*}{\otimes}AK(n)_{**}[\tau^{-1}]\) as an abbreviation.
\end{rem}
\begin{lemma}
(Compare \cite[2.7 + 2.8]{DI})\\
\label{taurealization}
\begin{enumerate}
\item Let \(X\) be a motivic spectrum in \(\mathcal{SH}_{\ensuremath{\mathbb{C}}}^{qfin}\) or \(\mathcal{SH}_{\ensuremath{\mathbb{C}},l}^{\wedge,qfin}\).\\
We can define a map of bigraded \(AK(n)_{**}[\tau^{-1}]\)-modules (even a map of bigraded algebras if \(X\) is a ring spectrum) natural in \(X\)
\[R: AK(n)_{**}(X)\underset{AK(n)_{**}}{\otimes}AK(n)_{**}[\tau^{-1}]\rightarrow K(n)_{*}(R_\ensuremath{\mathbb{C}}(X))\underset{K(n)_*}{\otimes}AK(n)_{**}[\tau^{-1}]\]
via the assignment
\[x\otimes \tau^k\mapsto R_\ensuremath{\mathbb{C}}(x)\otimes \tau^{-q+k} \]
where \(q\) is the motivic weight of \(x\in AK(n)_{p,q}(X)\).\\
This map is an isomorphism.
\item The induced map
\[\bar{R}_{End,X}: \text{End}_{AK(n)_{**}}(AK(n)_{**}(X))[\tau^{-1}]\rightarrow \text{End}_{K(n)_{*}}(K(n)_{*}(R_\ensuremath{\mathbb{C}}(X)))[\tau,\tau^{-1}]\]
is an isomorphism of bigraded \(AK(n)_{**}[\tau^{-1}]\)-algebras.
\item A homogeneous element \(f\in\text{End}_{AK(n)_{**}}(AK(n)_{**}(X))\) maps to zero under the map
\[
R_{End,X}: \text{End}_{AK(n)_{**}}(AK(n)_{**}(X))\rightarrow \text{End}_{K(n)_{*}}(K(n)_{*}(R_\ensuremath{\mathbb{C}}(X)))
\]
induced by motivic realization if and only if it is \(\tau\)-primary torsion.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item The statement about naturality and the module/algebra structure follow from the properties of motivic realization. It remains to show that the map is an isomorphism for spectra \(X\) in \(\mathcal{SH}_{\ensuremath{\mathbb{C}}}^{qfin}\) or \(\mathcal{SH}_{\ensuremath{\mathbb{C}},l}^{\wedge,qfin}\). We will prove this using cellular induction, and then show that it remains an isomorphism under taking retracts.\\
Consider the case of the sphere spectrum \(X=S\): The map \[R: AK(n)_{**}[\tau^{-1}]\rightarrow K(n)_{*}\underset{K(n)_*}{\otimes}AK(n)_{**}[\tau^{-1}]\] sends \(\tau\) to \(\tau\) and \(v_n\in AK(n)_{2(l^n-1),l(n-1)}\) to \(v_n^{top}\otimes \tau^{-l^n+1}=1\otimes \tau^{-l^n+1}\tau^{l^n-1}v_n=v_n\), so it is an isomorphism.\\
If \(X\) is a retract of a spectrum \(F\) for which the statement holds, then \(AK(n)_{**}(X)\) is a direct summand of \(AK(n)_{**}(F)\) and all squares in the following diagram commute:
\[
\xymatrix{
AK(n)_{p,q}(X)[\tau^{-1}]\ar[r]^{AK(n)_{**}(s)}\ar[d]^{R_X} \ar@/^2.0pc/[rr]_{id}&
AK(n)_{p,q}(F)[\tau^{-1}]\ar[d]_{R_F}^\cong \ar[r]^{AK(n)_{**}(r)} &
AK(n)_{p,q}(X)[\tau^{-1}]\ar[d]^{R_X}
\\
K(n)_{p}(R_\ensuremath{\mathbb{C}}(X))[\tau,\tau^{-1}]\ar[r]_{K(n)_{*}(R_\ensuremath{\mathbb{C}}(s))}\ar@/_2.0pc/[rr]_{id} &
K(n)_{p}(R_\ensuremath{\mathbb{C}}(F))[\tau,\tau^{-1}]\ar[r]_{K(n)_{*}(R_\ensuremath{\mathbb{C}}(r))}&
K(n)_{p}(R_\ensuremath{\mathbb{C}}(X))[\tau,\tau^{-1}]
\\
}
\]
Therefore \(R_X\) is surjective and injective via a simple diagram chase.\\
Finally, suppose \(X\rightarrow Y \rightarrow Z\) is a cofiber sequence and the statement holds for \(X\) and \(Y\). Then the long exact sequence for \(AK(n)\)-homology maps to the long exact sequence for \(K(n)\)-homology associated to the cofiber sequence \(R_\ensuremath{\mathbb{C}}(X)\rightarrow R_\ensuremath{\mathbb{C}}(Y) \rightarrow R_\ensuremath{\mathbb{C}}(Z)\), and the five lemma tells us that the statement also holds for \(Z\):
\[
\xymatrix@C=7pt{
... \ar[r]\ar[d]_{\cong}^{R_X}&
AK(n)_{pq}(Y)[\tau^{-1}] \ar[r]\ar[d]_{\cong}^{R_Y}&
AK(n)_{pq}(Z)[\tau^{-1}] \ar[r]\ar[d]^{R_Z}&
AK(n)_{p-1,q}(X)[\tau^{-1}]\ar[r] \ar[d]_{\cong}^{R_X}&
... \ar[d]_{\cong}^{R_Y}\\
... \ar[r]&
K(n)_{p}(R_\ensuremath{\mathbb{C}}(Y))[\tau][\tau^{-1}] \ar[r]&
K(n)_{p}(R_\ensuremath{\mathbb{C}}(Z))[\tau][\tau^{-1}] \ar[r]&
K(n)_{p-1}(R_\ensuremath{\mathbb{C}}(X))[\tau][\tau^{-1}]\ar[r]&
...\\
}
\]
\item Let \(M\) be a finitely presented \(K(n)_{*}\)-module and \(N\) be an arbitrary \(K(n)_{*}\)-module. As noted in \ref{ExtScalarFlat}, \(AK(n)_{**}[\tau^ -1]\) is a flat \(K(n)_{*}\)-module. By \cite[§2.10, Proposition 11]{BOUR} there is a canonical isomorphism:
\[\text{Hom}_{K(n)_{*}}(M,N)[\tau,\tau^{-1}]\overset\cong\longrightarrow \text{Hom}_{K(n)_{*}[\tau,\tau^{-1}]}(M[\tau,\tau^{-1}],N[\tau,\tau^{-1}])\]
Likewise, let \(M\) be a finitely presented \(AK(n)_{**}\)-module and \(N\) be an arbitrary \(AK(n)_{**}\)-module. Because \(AK(n)_{**}[\tau^{-1}]\) is a flat \(AK(n)_{**}\)-module, there is also a canonical isomorphism:
\[\text{Hom}_{AK(n)_{**}}(M,N)[\tau^{-1}]\overset\cong\longrightarrow \text{Hom}_{AK(n)_{**}[\tau^{-1}]}(M[\tau^{-1}],N[\tau^{-1}])\]
The module \(AK(n)_{**}(X)\) is finitely presented by \ref{FinGen}. Specializing to the case \(M=N=AK(n)_{**}(X)\), these two isomorphisms fit in the following commutative diagram:
\[
\xymatrix{
\text{End}_{AK(n)_{**}}(AK(n)_{**}(X))[\tau^{-1}] \ar[r]^\cong\ar[d]&
\text{End}_{AK(n)_{**}[\tau^{-1}]}(AK(n)_{**}(X)[\tau^{-1}]) \ar[d] \\
\text{End}_{K(n)_{*}}(K(n)_{*}(R_\ensuremath{\mathbb{C}}(X)))[\tau,\tau^{-1}] \ar[r]^\cong &
\text{End}_{K(n)_{*}[\tau,\tau^{-1}]}(K(n)_{*}(R_\ensuremath{\mathbb{C}}(X)))[\tau,\tau^{-1}])\\
}
\]
The first statement of the lemma tells us that \(K(n)_{*}[\tau,\tau^{-1}]\cong AK(n)_{**}[\tau^{-1}]\) and \(K(n)_{*}(R_\ensuremath{\mathbb{C}}(X)))[\tau,\tau^{-1}]\cong AK(n)_{**}(X)[\tau^{-1}]\), so the right vertical map is an isomorphism. It follows that the left vertical map is also an isomorphism.
\item Let \[P: \text{End}_{K(n)_{*}}(K(n)_{*}(R_\ensuremath{\mathbb{C}}(X)))[\tau,\tau^{-1}]\rightarrow \text{End}_{K(n)_{*}}(K(n)_{*}(R_\ensuremath{\mathbb{C}}(X)))\] be the map of \(K(n)_{**}\)-algebras defined by sending \(\tau\) to 1 and elements of \(\text{End}_{K(n)_{*}}(K(n)_{*}(R_\ensuremath{\mathbb{C}}(X)))\) to themselves. Then we have a commutative diagram of \(K(n)_{*}\)-algebras:
\[
\xymatrix{
\text{End}_{AK(n)_{**}}(AK(n)_{**}(X))[\tau^{-1}]\ar[d]^{\bar{R}_{End,X}}&
\text{End}_{AK(n)_{**}}(AK(n)_{**}(X))\ar[l]\ar[d]^{R_{End,X}}\\
\text{End}_{K(n)_{*}}(K(n)_{*}(R_\ensuremath{\mathbb{C}}(X)))[\tau,\tau^{-1}]\ar[r]^{\quad P}& \text{End}_{K(n)_{*}}(K(n)_{*}(R_\ensuremath{\mathbb{C}}(X)))\\
}
\]
A homogeneous element maps to zero under the top horizontal map if and only if it is \(\tau\)-primary torsion. By the second statement of this lemma, the left vertical map is an isomorphism, and there are no homogeneous elements in the kernel of \(P\). All this together implies the desired result.
\end{enumerate}
\end{proof}
\begin{rem}
If \(X\) is strongly dualizable, the map \(DX\wedge X= F(X,S)\wedge X \rightarrow F(X,X)\) is a weak equivalence, and we have a corresponding isomorphism on homotopy groups \(\pi_{pq}(X\wedge DX)\cong \textnormal{End}(X)_{pq}\). With regard to motivic Morava K-theory the situation is more complicated. Using Spanier-Whitehead duality we have:
\begin{align*}AK(n)_{pq}(X\wedge DX)&=[S, AK(n) \wedge X \wedge DX]_{pq}\\
&=[X, AK\wedge X]_{pq}\\
&=[AK \wedge X, AK\wedge X]_{AK,pq}
\end{align*}
The last term is related to \(\textnormal{End}_{AK(n)_{**}}(AK(n)_{**}(X))_{pq}\) via the Universal coefficient spectral sequence(c.f \cite[Prop. 7.7]{DI2}]. The \(E_2\)-term of this spectral sequence is given by \[\textnormal{Ext}_{AK(n)_{**}}(AK(n)_{**}(X),AK(n)_{**}(X))\]
and it converges conditionally to \([AK \wedge X, AK\wedge X]_{AK,pq}\). In particular, if \(AK(n)_{**}(X)\) is free or just projective as an \(AK(n)_{**}\)-module, this spectral sequence collapses at the \(E_2\)-page because it is concentrated in the 0-line, and we get an isomorphism:
\[AK(n)_{pq}(X\wedge DX)\cong \textnormal{End}_{AK(n)_{**}}(AK(n)_{**}(X))_{pq}\]
However, there is no general reason why \(AK(n)_{**}(X)\) should be free or projective. In contrast to this, all graded modules over the graded field \(K(n)_{*}\) are free, and therefore we always have an isomorphism \[K(n)_{**}(X\wedge DX)\cong \textnormal{End}_{K(n)_{*}}(K(n)_{*}(X))\] for all finite topological cell spectra \(X\). As a consequence, instead of working with \(AK(n)_{**}(X\wedge DX)\), we will work directly with \(\textnormal{End}_{AK(n)_{**}}(AK(n)_{**}(X))\) motivically.\\
\end{rem}
\ \\
Every element in \(AK(n)_{**}\) induces a map in \(\text{End}_{AK(n)_{**}}(AK(n)_{**}(X))\) given by multiplication with that element. We will denote this map by the same name as the element. We can now prove the motivic equivalent of asymptotic uniqueness:
\begin{lemma}
\label{Lemma611}
Let \(X\) be a motivic spectrum in \(\mathcal{SH}_{\ensuremath{\mathbb{C}}}^{qfin}\) or \(\mathcal{SH}_{\ensuremath{\mathbb{C}},l}^{\wedge,qfin}\) and \[f: X\rightarrow X\] a motivic \(v_n\)-self map. Then there exist integers \(i\) and \(j\) such that: \[AK(n)_{**}(f^i)=v^j_n\]
\end{lemma}
\begin{proof}
We will use the classical statement for \(v_n^{top}\)-self maps in the topological stable homotopy category. In addition, it is known that for any unit \(u\) in a \(K(n)_*\)-algebra that is finitely generated as a \(K(n)_*\)-module (c.f. \cite[Lemma 3.2]{HS} or \cite[Proof of Lemma 6.1.1]{RAV2}) there is a power of that element such that \(u^i=(v_n^{top})^j\). We will deduce the motivic statement by applying these classical lemmas twice. On the one hand, one can divide out the ideal generated by \(\tau\), which yields a finitely generated \(K(n)_*\)-algebra; on the other hand, one can apply Betti realization.\\
\ \\
Our first step is to show that the map
\[\tau: AK(n)_{**}(X)\rightarrow AK(n)_{**}(X)\]
can not be a unit in \(\text{End}_{AK(n)_{**}}(AK(n)_{**}(X))\):\\
The element \(\tau \in AK(n)_{**}\) is not a unit; if we fix the first degree \(p\) in \(AK(n)_{pq}\) and vary the height \(q\), then there is a maximum height such that \(AK(n)_{pq}=0\) for all larger heights \(q\). If \(\tau\) were a unit, all its powers \(\tau^k\in AK(n)_{0,-k}\) would need to have an inverse \(\tau^{-k}\in AK(n)_{0,k}\) in arbitrarily high weights, which is a contradiction to the previous statement. By the same argument the image of \(\tau\) cannot be a unit in any finitely generated \(AK(n)_{**}\)-module. But \(\text{End}_{AK(n)_{**}}(AK(n)_{**}(X))\) was finitely generated by \ref{FinGen}, so the multiplication-by-\(\tau\)-map cannot be a unit.\\
\ \\
In the second step, we show that the statement is true modulo \(\tau\):\\
The motivic \(v_n\)-self map \(f\) induces an isomorphism in \(AK(n)_{**}\)-homology, i.e. a unit in \(\text{End}_{AK(n)_{**}}(AK(n)_{**}(X))\). In the previous step we showed that \(\tau\) cannot be a unit; this implies that it cannot divide \(AK(n)_{**}(f)\), for if it did, \(\tau\) would also be a unit.\\
Therefore \(AK(n)_{**}(f)\) does not map to zero under the quotient map
\[\text{End}_{AK(n)_{**}}(AK(n)_{**}(X))\rightarrow \text{End}_{AK(n)_{**}/(\tau)}(AK(n)_{**}(X)/(\tau)))\]
and its image \(\overline{AK(n)_{**}(f)}\) is thus a unit in the second ring.\\
If we forget the second bidegree, \(AK(n)_{**}/(\tau)\) is isomorphic to \(K(n)_{*}\), and \(\text{End}_{AK(n)_{**}/(\tau)}(AK(n)_{**}(X)/(\tau))\) is a finitely generated \(K(n)_{*}\)-algebra. In this case we know that there are integers \(i\) and \(j\) such that \(\overline{AK(n)_{**}(f)}^i=(v^{top}_n)^j\). Hence \[AK(n)_{**}(f)^i=v_n^j+\tau \tilde{x}\]
for some element \(\tilde{x}\in AK(n)_{**}(X)\).\\
\ \\
For the last step, suppose now that \(\tilde{x}\) is \(\tau\)-primary torsion. For the fixed prime \(l\) and any \(k\in \ensuremath{\mathbb{N}}\) we can consider powers \(AK(n)_{**}(f)^{ikl}=v_n^{jkl}+(\tau \tilde{x})^{kl}\). If \(k\) is sufficiently large, the second term vanishes and we are done.\\
Suppose then that \(\tilde{x}\) is not \(\tau\)-primary torsion. Motivic realization induces a map \[\text{End}_{AK(n)_{**}}(AK(n)_{**}(X))\rightarrow \text{End}_{K(n)_{*}}(K(n)_{*}(X))\]
By the classical statement we know that there are integers \(i'\) and \(j'\) such that \(R_{End,X}(f)^{i'}=(v_n^{top})^{j'}\).
Replace \(i,i'\) and \(j,j'\) with their products \(i\cdot i'\) and \(j\cdot j'\) and call the result \(i\) and \(j\) again. Then \(AK(n)_{**}(f)^i=v_n^j+\tau \tilde{x}\) realizes to \(v_n^j\), so \(\tau \tilde{x}\) realizes to \(0\). Because \(\tau\) realizes to 1, \(\tilde{x}\) realizes to \(0\) and by \ref{taurealization} is therefore \(0\) itself.
\end{proof}
\begin{lemma}
\label{Centrality}
Assume that the motivic nilpotence conjecture holds. Let \(X \in \mathcal{SH}_{\ensuremath{\mathbb{C}}}^{qfin}\), which implies \(DX \in \mathcal{SH}_{\ensuremath{\mathbb{C}}}^{qfin}\) by \ref{CellDualizable}.
If \[f: \Sigma^{p,q}X\rightarrow X\] is a motivic \(v_n\)-self map and \[x\in \pi_{p,q}(DX\wedge X)\] is the element corresponding to \(f\) under motivic Spanier Whitehead duality, then there exists an integer \(i \in \ensuremath{\mathbb{N}}\) such that \(x^i\) is in the center of \(\pi_{p,q}(DX\wedge X)\).
\end{lemma}
\begin{proof}
The proof is essentially similar to \cite[Lemma 3.5]{HS} and \cite[Lemma 6.1.2]{RAV2}, but we will have to use the motivic Nilpotence conjecture at one point.\\
For all \(a\in \pi_{**}(DX\wedge X)\) there is an abstract map of rings
\[ad(a):\pi_{**}(DX\wedge X)\rightarrow\pi_{**}(DX\wedge X)\]
defined by \(ad(a)(b)=ab-ba\), and the element \(a\) is central if and only if \(ad(a)\) is the zero map. This map is realized in homotopy by the composite (here we write \(R\) for \(DX\wedge X\) and \(T\) for the transposition map):
\[S^{p,q}\wedge R \overset{a\wedge id_R}{\rightarrow} R\wedge R \overset{1-T}{\rightarrow} R\wedge R \overset{\mu}{\rightarrow} R\]
We also denote this composite by \(ad(a)\).\\
It now suffices to show that \(ad(x)\) is nilpotent because of the following classical formula (proved in \cite[Lemma 6.1.2]{RAV2}):
\[ad(x^i)(b)=\sum_{j=1}^{i}{{i}\choose{j}}ad^j(x)(b)x^{i-j}\]
If we choose \(i=l^N\) for a sufficiently large \(N\), all summands in this formula vanish either because of the nilpotence of \(ad(x)\) or because the binomial coefficient annihilates \(ad(x)\).\\
Note that \(AK(n)_{**}(DX\wedge X)\) is a finitely generated \(AK(n)_{**}\)-algebra that maps to \(K(n)_{*}(DR(X)\wedge R(X))\) under Betti realization. It follows by the same reasoning as in the proof of Lemma \ref{Lemma611} that a suitable power of \(AK(n)_{**}(x)\) is given by \(v_n^i\) for some \(i\in \ensuremath{\mathbb{N}}\), which is in the image of \(AK(n)_{**}\) in \(AK(n)_{**}(DX\wedge X)\) and hence central. Replace \(x\) with that power and name it \(x\) again. Then \(AK(n)_{**}(ad(x))\) is zero, so \(ad(x)\) is nilpotent by the nilpotence conjecture.
\end{proof}
\begin{lemma}
Let \(X\) be a motivic spectrum in \(\mathcal{SH}_{\ensuremath{\mathbb{C}},(l)}^{qfin}\) or \(\mathcal{SH}_{\ensuremath{\mathbb{C}},l}^{\wedge,qfin}\). Assume that the motivic nilpotence conjecture \ref{NilpotenceConjecture} holds. If \(f,g: X\rightarrow X\) are two motivic \(v_n\)-self maps, then there exist integers \(i,j \in \ensuremath{\mathbb{N}}\) such that \(f^{i}=g^{j}\).
\end{lemma}
\begin{proof}
This lemma corresponds to \cite[Lemma 3.6]{HS} and \cite[Lemma 6.1.3]{RAV2}. By the previous two lemmas, we can assume that \(f\) and \(g\), after replacing them with appropiate powers of themselves, commute with each other in regard to composition, and furthermore that \[AK(n)_{**}(f^{i'}-g^{j'})=0.\]
Using the nilpotence conjecture, we can conclude that \(f^{i'}-g^{j'}\) is nilpotent. Then \cite[Lemma 3.4]{HS} gives us the desired statement.
\end{proof}
\begin{lemma}
\label{ExtendedUniqueness}
Assume that the motivic nilpotence conjecture \ref{NilpotenceConjecture} holds. If \(f: X\rightarrow X\) and \(g: Y\rightarrow Y\) are two \(v_n\) self maps of \(X\) and \(Y\) and \(h:X\rightarrow Y\) is any map, then there exist integers \(i,j \in \ensuremath{\mathbb{N}}\) such that \(h\circ f^{i}=g^{l^m}\circ h\).
\end{lemma}
\begin{proof}
The proof is entirely similar to \cite[6.1.4]{RAV2}
\end{proof}
\begin{theorem}
\label{ThickSubcat}
Let \(k=\ensuremath{\mathbb{C}}\) and \(l\) be an odd prime. Assume that the motivic nilpotence conjecture \ref{NilpotenceConjecture} holds. Then the full subcategories of \(\mathcal{SH}_{\ensuremath{\mathbb{C}},(l)}^{qfin}\) and \(\mathcal{SH}_{\ensuremath{\mathbb{C}},l}^{\wedge,qfin}\) consisting of spectra admitting motivic \(v_n\)-self maps are thick.
\end{theorem}
\begin{proof}
First we prove that the category of spectra admitting motivic \(v_n\)-self maps is closed under retracts:\\
Let \(e: X\rightarrow Y\) be a retract with right inverse \(s: Y \rightarrow X\) and assume that there is a \(v_n\)-self map \(f:X\rightarrow X\).
By \ref{Centrality} a power of \(f\) commutes with \(s\circ e\), so \(e\circ f \circ s\) is a \(v_n\)-self map.\\
Furthermore the category of spectra admitting motivic \(v_n\)-self maps is closed under cofiber sequences:\\
Let \(X\) and \(Y\) be two spectra with motivic \(v_n\)-self maps \(f: \Sigma^{a ,b}X\rightarrow X\) and \(g: \Sigma^{c,d} Y\rightarrow Y\) and let \(h:X\rightarrow Y\) be any map. By \ref{ExtendedUniqueness} we can, after replacing the self maps with suitable powers, assume that \((a,b)=(c,d)\) and \(h\circ f = g\circ h\). Therefore there exists a map \(k:\Sigma^{a,b}C_h \rightarrow C_h\) making the following diagram commute:
\[
\xymatrix{
X \ar[r]^{h}& Y \ar[r] & C_h\\
\Sigma{a,b}X\ar[u]^f \ar[r]^h & \Sigma^{a,b}Y \ar[r] \ar[u]^g & \Sigma^{a,b}C_h \ar@{.>}[u]^k\\
}
\]
It follows by the five lemma and basic facts about triangulated categories that \(k^2\) is a \(v_n\)-self map on \(C_h\) as desired.
\end{proof}
\section{Existence of a self map on \(\ensuremath{\mathbb{X}}_n\)}
In \cite{HS} Hopkins and Smith used the Adams spectral sequence to prove the existence of a self map on a spectrum \(X_n\) constructed by Smith. In this section we use their proof together with a suitable motivic spectrum \(\ensuremath{\mathbb{X}}_n\) constructed by Joachimi to show that at least one spectrum in \(\mathcal{SH}_{\ensuremath{\mathbb{C}}}^{qfin}\) or \(\mathcal{SH}_{\ensuremath{\mathbb{C}},l}^{\wedge,qfin}\) actually has a motivic \(v_n\)-self map. The classical proof relies on computing \[K(n)_{p}(X_n\wedge DX_n)\cong \textnormal{End}_{K(n)_{*}}(K(n)_{**}(X_n))_{p}\] via the Adams spectral sequence, so we run into the same kind of problem as in the previous chapter: Because motivically not all graded modules over \(AK(n)_{**}\) are free, we first have to show that \(AK(n)_{**}(\ensuremath{\mathbb{X}}_n)\) is in fact free. This also provides us with a Künneth isomorphism for products involving \(AK(n)_{**}(\ensuremath{\mathbb{X}}_n)\).\\
\ \\
The proof of the existence of a \(v_n\)-self map also relies on the approximation lemma, which relates the cohomology of the Steenrod algebra in certain degrees to the cohomology of certain subalgebras. We need the motivic analogue of this lemma. To this end we need to make two definitions:
\begin{defin}
\begin{enumerate}
\item Let \(X\) be a motivic spectrum. Call \(X\) \emph{\(k\)-bounded below} if \(\pi_{m,n}=0\) for \(m\leq k\). Similarly, call a bigraded module \(M_{m,n}\) over the motivic Steenrod algebra \emph{\(k\)-bounded below} if \(M_{m,n}=0\) for \(m\leq k\).\\
\item A module over the motivic Steenrod algebra has a vanishing line of slope \(m\) and intercept \(b\) if \(\text{Ext}_A^{s,t,u}(M,H\ensuremath{\mathbb{Z}}/l^{**})=0\) for \(s>m(t-s)+b\).
\end{enumerate}
\end{defin}
Note that the preceding definition is exactly like the classic one and the weight is not involved.
\begin{defin}
\begin{enumerate}
\item Let \(\beta\) denote the motivic Bockstein homomorphism, and \(Sq^i\) resp. \(P^i\) denote the motivic Square- and Power operations as constructed by Voevodsky in \cite{VOE2}.
If \(l=2\), define \(A_n\) as the subalgebra of the motivic Steenrod algebra generated by \(Sq^1,Sq^2,...,Sq^{2^n}\) over \(H\ensuremath{\mathbb{Z}}_l^{**}\).\\
If \(l \neq 2\), define \(A_n\) as the subalgebra of the motivic Steenrod algebra generated by \(\beta, P^1,...P^{n-1}\) for \(n\neq 0\) and by \(\beta\) for \(n=0\).
\item Fix the monomial \(\ensuremath{\mathbb{Z}}/l\)-basis for the dual motivic Steenrod algebra defined by the elements \(\tau\), \(\xi_i\) and \(\tau_i\) (if \(l\neq 2\)). The elements \(P^s_t\) in the motivic Steenrod algebra are defined as the dual elements to \(\xi^{p^s}_t\), and the elements \(Q_i\) are defined as the dual elements to \(\tau_i\) if \(l\neq2\) and as \(Q_i=P^0_{i+1}\) in the case \(l=2\).
\item Write \(\Lambda(Q_n)\) for the exterior algebra over the ground ring \(H\ensuremath{\mathbb{Z}}/l^{**}\) in the generator \(Q_n\). This is a subalgebra of the motivic Steenrod algebra.
\end{enumerate}
\end{defin}
\ \\
We can now prove the motivic analogon to the approximation lemma (c.f. \cite[6.3.2]{RAV2}):
\begin{prop}
Let \(M\) be a bounded below module over the motivic Steenrod algebra such that
\(\text{Ext}_A^{s,t}(M,H\ensuremath{\mathbb{Z}}/l^{**})\) has a vanishing line of slope \(m\) and intercept \(b\). \\
For sufficiently large \(N\) the restriction map \[\text{Ext}_A^{s,t}(M,H\ensuremath{\mathbb{Z}}/l^{**})\rightarrow \text{Ext}_{A_N}^{s,t}(M,H\ensuremath{\mathbb{Z}}/l^{**})\]
is an isomorphism in degrees \(s\geq m(t-s)+b'\), where \(b'\) can be chosen arbitrarily low for sufficiently large \(N\).
\end{prop}
\begin{proof}
Define \(C\) as the kernel of the surjective map of \(A\)-modules\\ \(A\underset{A_N}{\otimes}M \rightarrow M\). As an \(A_N\)-module \(C\) is given by \(M\otimes \overline{A//A_N}\), where \(A//A_N=A\underset{A_N}{\otimes}\ensuremath{\mathbb{Z}}/(l)\) and the bar denotes the augmentation ideal.
The motivic Steenrod squares \(Sq^i\) live in bidegrees \((2i,i)\) if \(i\) is even and \((2i+1,i)\) if it is odd and the motivic Power operations \(P^i\) live in bidegrees \((2i(l-1),i(l-1))\). Hence \(\overline{A//A_N}\) will be \(k\)-bounded below, and \(k\) can be chosen arbitrarily high if \(N\) is sufficiently large. Therefore \(C\) has a vanishing line of the same slope as \(M\) and arbitrarily low intercept for sufficiently large \(N\) , cf. \cite{HS}[4.4]. The short exact sequence defining \(C\) and the change-of-rings isomorphism for \(A_N\) and \(A\) provide the following diagram:
\[
\xymatrix{
\text{Ext}^{s-1}_{A}(C,H\ensuremath{\mathbb{Z}}/l^{**}) \ar[d]\ & \\
\text{Ext}^{s}_{A}(M,H\ensuremath{\mathbb{Z}}/l^{**})\ar[d]\ar[dr]^{\phi} & \\
\text{Ext}^{s}_{A}(A\underset{A_N}{\otimes}M,H\ensuremath{\mathbb{Z}}/l^{**})
\ar[d]\ar[r]^{\cong} &
\text{Ext}^{s}_{A_{N}}(M,H\ensuremath{\mathbb{Z}}/l^{**}) \\
\text{Ext}^{s}_{A}(C,H\ensuremath{\mathbb{Z}}/l^{**}) &
\\
}
\]
If the upper and lower term in the diagram vanish - which is the case above the vanishing line of \(C\) - the map \(\phi\) is the composite of two isomorphisms and hence an isomorphism itself.
\end{proof}
\ \\
In \cite[Theorem 8.5.12]{JOA} Joachimi defined a motivic cell spectrum \(\ensuremath{\mathbb{X}}_n\) analogous to the Smith-construction spectrum \(X_n\) in \cite{HS}(see also \cite{RAV2}) by splitting off a wedge summand of a finite cell spectrum via an idempotent. We need some of the details of the construction of \(\ensuremath{\mathbb{X}}_n\) and its properties for the construction of the \(v_n\)-self map, so we recall and collect all those that are relevant in one place:
\begin{defin}
The spectrum \(\ensuremath{\mathbb{X}}_n\) is defined as \[\ensuremath{\mathbb{X}}_n=e_V(\mathbb{B}_{(l)}^{\wedge k_V})=\underset{\rightarrow}{\text{hocolim }}\mathbb{B}_{(l)}^{\wedge k_V}\underset{e_V}{\rightarrow}\mathbb{B}_{(l)}^{\wedge k_V}\underset{e_V}{\rightarrow}...\]
where
\begin{itemize}
\item \(\mathbb{B}_{(l)}\) is a motivic \(l\)-local finite cellular spectrum defined in \cite[8.5]{JOA}, implicitly depending on \(n\).
\item \(V=H\ensuremath{\mathbb{Z}}/l^{**}(\mathbb{B}_{(l)})=H\ensuremath{\mathbb{Z}}/l^{**}(a,b)/(a^2,b^{l^n})\), where \(|a|=(1,1)\) and \(|b|=(2,1)\) (\cite[8.5.10]{JOA})
\item \(k_V\) is an integer dependent on \(V\).
\item \(e_V\) is an idempotent of the groupring \(\ensuremath{\mathbb{Z}}_{(l)}[\Sigma_{k_V}]\), which acts on \(\mathbb{B}_{(l)}^{\wedge k_V}\) by permuting the smashfactors and adding maps.
\item On the level of cohomology, the effect of this idempotent is to split of a free, nonzero \(H\ensuremath{\mathbb{Z}}/l^{**}\)-submodule of \(V^{\otimes k_V}\). In particular, the motivic cohomology of \(\ensuremath{\mathbb{X}}_n\) is bounded below as a module over the Steenrod algebra.
\end{itemize}
\end{defin}
Furthermore Joachimi proves the following statements about \(\ensuremath{\mathbb{X}}_n\):
\begin{theorem}
\label{XnFacts}
\begin{enumerate}
\item \(AK(s)_{**}(\ensuremath{\mathbb{X}}_n)=0\) for \(s<n\) and \(AK(n)(\ensuremath{\mathbb{X}}_n)\neq 0\) (\cite[Theorem 8.5.12]{JOA})
\item The operation \(Q_n\) acts trivially on \(H\ensuremath{\mathbb{Z}}/l^{**}(\mathbb{B}_{(l)})\). This follows for degree reasons from the description of \(H\ensuremath{\mathbb{Z}}/l^{**}(\mathbb{B}_{(l)}^{\wedge k_V})\) in the previous remark. Since \(H\ensuremath{\mathbb{Z}}/l^{**}(\ensuremath{\mathbb{X}}_n)\) is a \(H\ensuremath{\mathbb{Z}}/l^{**}\)-submodule of this module, \(Q_n\) acts trivially on \(H\ensuremath{\mathbb{Z}}/l^{**}(\ensuremath{\mathbb{X}}_n)\).
\item \(R(\ensuremath{\mathbb{X}}_n)=X_n\). (\cite[8.6]{JOA})
\end{enumerate}
\end{theorem}
\ \\
By \ref{CellDualizable} \(\ensuremath{\mathbb{X}}_n\) is dualizable, and its dual is the retract of a finite cell spectrum.
\ \\
Because the spectrum \(\ensuremath{\mathbb{X}}_n\) is dualizable, it satisfies the expected relation between homology and cohomology once we show that its cohomology is free:
\begin{lemma}
\begin{enumerate}
\item Let \(E\) be a cellular motivic ring spectrum and \(X\) be a dualizable cellular motivic spectrum. If \(E^{**}(X)\) is a free module over the coefficients \(E^{**}\), then \(\text{Hom}_{E^{**}}(E^{**}(X),E^{**})\cong E_{**}(X)\).
\item Let \(X\) be a dualizable cellular motivic spectrum such that
\begin{itemize}
\item \(H\ensuremath{\mathbb{Z}}/l^{**}(X)\) is free over \(H\ensuremath{\mathbb{Z}}/l^{**}\)
\item \(Q_n\) acts trivially on \(H\ensuremath{\mathbb{Z}}/l^{**}(X)\).
\end{itemize}
Then we have an additive bigraded isomorphism \[\text{Ext}^{s,t,u}_{\Lambda(Q_n)}(H\ensuremath{\mathbb{Z}}/l^{**}(X),H\ensuremath{\mathbb{Z}}/l^{**})\cong H\ensuremath{\mathbb{Z}}/l_{**}(X)[v_n]\]
where \(|v_n|=(1,2(l^n-1),l^n-1)\). Here \(s\) is the homological degree and \(t,u\) correspond to the internal bidegree. (The result also holds multiplicatively, but we are not going to need this.)
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item This is the content of \cite[8.1.2]{JOA}, using the universal coefficient spectral sequence of \cite[7.7]{DI2} and the fact that this spectral sequence collapses if \(E^{**}(X)\) is free over \(E^{**}\). Note that the cited corollary is stated only for finite cell spectra and the case \(E=H\ensuremath{\mathbb{Z}}/l\), but the only properties of \(X\) actually used are cellularity and dualizability, and that the proof also works for any cellular motivic ring spectrum \(E\).
\item This is a classical result that can be proven in the following way:\\
Consider the following resolution of free \(\Lambda(Q_n)\)-modules
\[...\overset{\cdot Q_n}\rightarrow \Lambda(Q_n)\overset{\cdot Q_n}\rightarrow \Lambda(Q_n)\overset{\cdot Q_n}\rightarrow \Lambda(Q_n)\overset{\epsilon}\rightarrow H\ensuremath{\mathbb{Z}}/l^{**}\]
where the last map is the projection \(\epsilon: \Lambda(Q_n)\rightarrow H\ensuremath{\mathbb{Z}}/l^{**}\) and apply \((-)\underset{H\ensuremath{\mathbb{Z}}/l^{**}}{\otimes}H\ensuremath{\mathbb{Z}}/l^{**}(X)\).\\
The resulting long exact sequence
\[...\overset{\cdot Q_n}\rightarrow \Lambda(Q_n)\underset{H\ensuremath{\mathbb{Z}}/l^{**}}{\otimes}H\ensuremath{\mathbb{Z}}/l^{**}(X)\overset{\cdot Q_n}\rightarrow \Lambda(Q_n)\underset{H\ensuremath{\mathbb{Z}}/l^{**}}{\otimes}H\ensuremath{\mathbb{Z}}/l^{**}(X)\overset{\epsilon}\rightarrow H\ensuremath{\mathbb{Z}}/l^{**}(X)\]
is a resolution of the \(\Lambda(Q_n)\)-module \(H\ensuremath{\mathbb{Z}}/l^{**}(X)\). Here we use the assumption that \(Q_n\) acts trivially on this module in the claim that the last map is a map of \(\Lambda(Q_n)\)-modules.\\
Now apply \(\text{Hom}_{\Lambda(Q_n)}((-),H\ensuremath{\mathbb{Z}}/l^{**})\) and take cohomology. All maps are zero because the target has the trivial \(\Lambda(Q_n)\)-module structure. Using the isomorphism from the previous part, we can rewrite degreewise:
\begin{align*}
&\text{Hom}_{\Lambda(Q_n)}(\Lambda(Q_n)\underset{H\ensuremath{\mathbb{Z}}/l^{**}}{\otimes}H\ensuremath{\mathbb{Z}}/l^{**}(X),H\ensuremath{\mathbb{Z}}/l^{**})\\
&\cong \text{Hom}_{H\ensuremath{\mathbb{Z}}/l^{**}}(H\ensuremath{\mathbb{Z}}/l^{**}(X),H\ensuremath{\mathbb{Z}}/l^{**})\\
&\cong H\ensuremath{\mathbb{Z}}/l_{**}(X)
\end{align*}
\end{enumerate}
\end{proof}
\ \\
Recall that the coefficient rings of the classical Morava K-theories are graded fields in the sense that all graded modules over it are free. This is not true of the motivic Morava K-theories in general. The algebraic Morava K-theory of the spectrum \(\ensuremath{\mathbb{X}}_n\) however is free and finitely generated. To see this, we need to go through the steps of its construction.
\begin{prop}
Let \(k=\ensuremath{\mathbb{C}}\) and \(l\) be an odd prime. Then \(AK(n)_{**}(\ensuremath{\mathbb{X}}_n)\) is a free, finitely generated \(AK(n)_{**}\)-module.
\end{prop}
\begin{proof}
To prove the statement it suffices to show that
\begin{itemize}
\item \(AK(n)_{**}(\ensuremath{\mathbb{X}}_n)\) is a finitely generated \(AK(n)_{**}\)-module
\item \(AK(n)_{**}(\ensuremath{\mathbb{X}}_n)\) has no \(\tau\)-torsion.
\end{itemize}
We are going to show both claims in three steps: First we compute \(Ak(n)(\mathbb{B}_{(l)})\) using the motivic Adams spectral sequence. We show that it is finitely generated and does not have \(\tau\)-torsion, which implies that \(AK(n)(\mathbb{B}_{(l)})\) is finitely generated and torsionfree. Then we use the Künneth theorem to show the same statement for \(AK(n)(\mathbb{B}_{(l)}^{\wedge k_V})\). Finally we use the definition of the idempotent defining \(\ensuremath{\mathbb{X}}_n\) to show that \(AK(n)_{**}(\ensuremath{\mathbb{X}}_n)\) satisfies both claims.\\
We begin with the first step: The motivic Adams spectral sequence for
\(Ak(n)\wedge \mathbb{B}_{(l)}\) converges strongly to \(Ak_{**}(\mathbb{B}_{(l)})\) (\cite[8.3.3]{JOA}). We claim that there are no nontrivial differentials in this spectral sequence.
The \(E_2\)-term of this motivic Adams spectral sequence can be written as \[\text{Ext}_{\Lambda(Q_n)}(H\ensuremath{\mathbb{Z}}/l^{**}(\mathbb{B}_{(l)}),H\ensuremath{\mathbb{Z}}/l^{**})\] by change of rings (\cite[8.2.3]{JOA}).
Recall that \(H\ensuremath{\mathbb{Z}}/l^{**}(\mathbb{B}_{(l)})=H\ensuremath{\mathbb{Z}}/l^{**}(a,b)/(a^2,b^{l^n})\).
The element \(Q_n\) acts trivially on this free and finitely generated \(H\ensuremath{\mathbb{Z}}/l^{**}\)-module, which implies by the previous lemma \[\text{Ext}^{***}_{\Lambda(Q_n)}(H\ensuremath{\mathbb{Z}}/(l)^{**}(\mathbb{B}_{(l)})), H\ensuremath{\mathbb{Z}}/l^{**})\cong H\ensuremath{\mathbb{Z}}/l_{**}[v_n]\underset{H\ensuremath{\mathbb{Z}}/(l)_{**}}{\otimes} H\ensuremath{\mathbb{Z}}/(l)_{**}(\mathbb{B}_{(l)})\]
The right hand side is a tensor product of polynomial algebras, and the position of the polynomial generators and of \(v_n\) in the spectral sequence imply that they cannot support a nontrivial differential at any stage. In the following sketch of the spectral sequence in an abuse of notation \(a\) and \(b\) denote the dual of the cohomology classes with the same name. Note that the spectral sequence to the right of the depicted area looks very similar to the displayed area - the same elements appear in the same configuration, just multiplied by some power of \(v_n\). In the standard Adams grading the differential \(d_r\) maps one entry to the left and \(r\) entries up. Thus it is clear that no potentially nontrivial differential can have a target different from zero.
\[
\xy
(-7,19)*{\mathrm{s}};(23,-8)*{\mathrm{t-s}};
(-3,3)*{0};(-3,18)*{1};(-3,30)*{...};
(-3,3)*{0};(3,-3)*{0};(9,-3)*{1};
(15,-3)*{2};(21,-3)*{...};(36,-3)*{l^n-1};(54,-3)*{2(l^n-1)};
(3,3)*{a};
(9,3)*{b};
(9,9)*{ab};
(15,3)*{b^2};
(15,9)*{ab^2};
(21,3)*{...};
(36,3)*{b^{l^n-1}};
(36,9)*{ab^{l^n-1}};
(54,18)*{v_n};
(54,24)*{av_n};
\ar@{-}(0,0);(0,40);
\ar@{-}(0,0);(60,0);
\endxy
\]
Therefore \(Ak(n)(\mathbb{B}_{(l)})\) is finitely generated over \(Ak(n)_{**}\) and does not have \(\tau\)-primary torsion. For all cellular spectra \(X\) we have \(AK(n)_{**}(X)\cong v_n^{-1}Ak(n)_{**}(X)\). Therefore \(AK(n)(\mathbb{B}_{(l)})\) is free and finitely generated over \(Ak(n)_{**}\).
\\
The second step is now easy: Since \(AK\) is a cellular spectrum and since we just proved that the cellular spectrum \(\mathbb{B}_{(l)}\) has free \(AK\)-homology over the coefficients, we can apply the Künneth theorem ((\cite{DI2}[Remark 8.7])) and obtain \[AK(n)_{**}(\mathbb{B}_{(l)}^{\wedge k_V})\cong AK(n)_{**}(\mathbb{B}_{(l)})^{\otimes k_V}\]
Therefore also \(AK(n)_{**}(\mathbb{B}_{(l)}^{\wedge k_V})\) is free and finitely generated over the coefficients.\\
For the last step, note that \(AK(n)_{**}(\ensuremath{\mathbb{X}}_n)\) is a finitely generated \(AK(n)_{**}\)-module as well, since it is a submodule of the finitely generated module \(AK(n)_{**}(\mathbb{B}_{(l)}^{\wedge k_V})\) over the noetherian ring \(AK(n)_{**}\).\\
It remains to show that no torsion occurs. The idempotent \(e_V\in Z_{(l)}[\Sigma_{k_V}]\) acts on \(AK(n)_{**}(\mathbb{B}_{(l)}^{\wedge k_V})\) by permutation of the tensor factors and multiplication by integers. No \(\tau\)-Torsion can occur in \(e_V(AK(n)_{**}(\mathbb{B}_{(l)}^{\wedge k_V}))=AK(n)_{**}(\ensuremath{\mathbb{X}}_n)\) because the order of an element in a fixed bidegree in \(AK(n)_{**}(\mathbb{B}_{(l)}^{\wedge k_V})\) is the same as that of the \(\tau\)-multiples of that element. Consequently, \(AK(n)_{**}(\ensuremath{\mathbb{X}}_n)\) is a free \(AK(n)_{**}\)-module.
\end{proof}
\begin{defin}
Define \(R=D\ensuremath{\mathbb{X}}_n \wedge \ensuremath{\mathbb{X}}_n\). It is a quasifinite cell spectrum by definition and by \ref{SWRingSpectrum} it can be endowed with the structure of a motivic homotopy ring spectrum, with unit map \(e: S\rightarrow D\ensuremath{\mathbb{X}}_n\wedge X_n\) and multiplication map \(\mu: R\wedge R \rightarrow R\).
\end{defin}
\ \\
As a corollary of the preceding proposition we get the following:
\begin{corol}
\label{Kunneth}
Let \(k=\ensuremath{\mathbb{C}}\) and \(R=D\ensuremath{\mathbb{X}}_n \wedge \ensuremath{\mathbb{X}}_n\). There are Künneth isomorphisms
\begin{enumerate}
\item \[AK(n)^{**}(R)\overset{\cong}{\rightarrow}AK(n)^{**}(D\ensuremath{\mathbb{X}}_n)\underset{AK(n)^{**}}{\otimes}AK(n)^{**}(\ensuremath{\mathbb{X}}_n)\]
\item \[AK(n)_{**}(R)\overset{\cong}{\rightarrow}AK(n)_{**}(D\ensuremath{\mathbb{X}}_n)\underset{AK(n)_{**}}{\otimes}AK(n)_{**}(\ensuremath{\mathbb{X}}_n)\]
\end{enumerate}
\end{corol}
\begin{proof}
The Milnor short exact sequence for \(\ensuremath{\mathbb{X}}_n\) and \(AK(n)\)-cohomology is
\[0\rightarrow \underset{\leftarrow}{\text{lim}^1}AK(n)^{*-1,*}(\mathbb{B}_{(l)}^{\wedge k_V})\rightarrow AK(n)^{**}(\ensuremath{\mathbb{X}}_n) \rightarrow \underset{\leftarrow}{\text{lim }}AK(n)^{**}(\mathbb{B}_{(l)}^{\wedge k_V})\rightarrow 0\]
Because the map \(e_V\) over which the homotopy colimit defining \(\ensuremath{\mathbb{X}}_n\) is taken is an idempotent, the system \(AK(n)^{**}(\mathbb{B}_{(l)}^{\wedge k_V})\) is Mittag-Leffler, which implies that the \(\underset{\leftarrow}{\text{lim}^1}\)-term vanishes. By the same argument, we have:
\[\underset{\leftarrow}{\text{lim }} AK(n)^{**}(D\ensuremath{\mathbb{X}}_n \wedge \mathbb{B}_{(l)}^{\wedge k_V})\cong AK(n)^{**}(D\ensuremath{\mathbb{X}}_n \wedge \ensuremath{\mathbb{X}}_n)\]
Here the limit is taken over the maps \(id\wedge e_V\).\\
Since \(\mathbb{B}_{(l)}^{\wedge k_V} \) is the \(l\)-localization of a finite cell spectrum and \(AK(n)^{**}(\mathbb{B}_{(l)}^{\wedge k_V})\) is a free module over \(AK(n)^{**}\), we can use the Künneth-isomorphism of Dugger and Isaksen \cite[Remark 8.7]{DI2} to see that \[AK(n)^{**}(D\ensuremath{\mathbb{X}}_n \wedge \mathbb{B}_{(l)}^{\wedge k_V})\overset{\cong}{\rightarrow}AK(n)^{**}(D\ensuremath{\mathbb{X}}_n)\underset{AK(n)^{**}}{\otimes}AK(n)^{**}(\mathbb{B}_{(l)}^{\wedge k_V})\]
It remains to rewrite the inverse limit over the right hand side: \(AK(n)^{**}(D\ensuremath{\mathbb{X}}_n)\) is a free \(AK(n)^{**}\)-module because \(AK(n)_{**}(\ensuremath{\mathbb{X}}_n)\) is a free \(AK(n)_{**}\)-module, so using the earlier isomorphism we get:
\[ \underset{\leftarrow}{\text{lim }}\big(AK(n)^{**}(D\ensuremath{\mathbb{X}}_n)\underset{AK(n)^{**}}{\otimes}AK(n)^{**}(\mathbb{B}_{(l)}^{\wedge k_V})\big)\cong AK(n)^{**}(D\ensuremath{\mathbb{X}}_n)\underset{AK(n)^{**}}{\otimes}AK(n)^{**}(\ensuremath{\mathbb{X}}_n)\]\\
The Künneth-isomorphism in \(AK(n)\)-homology can either be derived from the one in cohomology or from the Künneth-isomorphism of the \(l\)-local finite cell spectra \(\mathbb{B}_{(l)}^{\wedge k_V}\) and the fact that homology commutes with direct limits.
\end{proof}
\ \\
We also need the following vanishing line:
\begin{lemma}
Let \(l\) be odd and \(R=D\ensuremath{\mathbb{X}}_n \wedge \ensuremath{\mathbb{X}}_n\) as before. The \(A^{**}\)-module \[\textnormal{Ext}_{A^{**}}(H\ensuremath{\mathbb{Z}}/l^{**}(R),H\ensuremath{\mathbb{Z}}/l^{**})\] has a vanishing line of slope \(1/2(l^n-1)\).
\end{lemma}
\begin{proof}
Over odd primes, the motivic Steenrod-algebra is just the classical Steenrod algebra (where the generators are understood to live in the appropiate motivic bidegrees) base changed to \(H\ensuremath{\mathbb{Z}}/l^{**}\). Similarly, \(H\ensuremath{\mathbb{Z}}/l^{**}(D\ensuremath{\mathbb{X}}_n \wedge \ensuremath{\mathbb{X}}_n)\) corresponds to \(H\ensuremath{\mathbb{Z}}/l_{*}(DX_n\wedge X_n)\) basechanged to \(H\ensuremath{\mathbb{Z}}/l^{**}\), where the generators are once again understood to live in the appropiate bidegree.\\
Consequently \(\text{Ext}_{A^{**}}(H\ensuremath{\mathbb{Z}}/l^{**}(D\ensuremath{\mathbb{X}}_n \wedge \ensuremath{\mathbb{X}}_n),H\ensuremath{\mathbb{Z}}/l^{**})\), which maps to the classical Ext-term \(\text{Ext}_{A^{*}_{\text{top}}}(H\ensuremath{\mathbb{Z}}/l^{*}(DX_n\wedge X_n),H\ensuremath{\mathbb{Z}}/l^{*})\), is just that classical Ext-term base changed to \(H\ensuremath{\mathbb{Z}}/l^{**}\) and in particular does not contain \(\tau\)-torsion. The existence of the vanishing line then follows from the existence of a vanishing line with the same slope in the classical case for the spectrum \(X_n\)(see \cite[6.3.1]{RAV2}).
\end{proof}
\ \\
Furthermore, we need the following duality isomorphisms:
\begin{prop}
Let \(R=D\ensuremath{\mathbb{X}}_n \wedge \ensuremath{\mathbb{X}}_n\) as before:
\leavevmode
\begin{enumerate}
\item \(\textnormal{Hom}_{H\ensuremath{\mathbb{Z}}/l^{**}}(H\ensuremath{\mathbb{Z}}/l^{**}(R),H\ensuremath{\mathbb{Z}}/l^{**})\cong H\ensuremath{\mathbb{Z}}/l_{**}(R)\)
\item \(AK(n)_{**}(D\ensuremath{\mathbb{X}}_n)\cong AK(n)^{**}(\ensuremath{\mathbb{X}}_n)\cong \textnormal{Hom}_{AK(n)_{**}}(AK(n)_{**}(\ensuremath{\mathbb{X}}_n), AK(n)_{**})\)
\item \(AK(n)^{**}(D\ensuremath{\mathbb{X}}_n)\cong AK(n)_{**}(\ensuremath{\mathbb{X}}_n)\cong \textnormal{Hom}_{AK(n)^{**}}(AK(n)^{**}(\ensuremath{\mathbb{X}}_n), AK(n)^{**})\)
\end{enumerate}
\end{prop}
\begin{proof}
\begin{enumerate}
\item \(R=D\ensuremath{\mathbb{X}}_n\wedge\ensuremath{\mathbb{X}}_n\) is a dualizable cell spectrum since \(\ensuremath{\mathbb{X}}_n\) and \(D\ensuremath{\mathbb{X}}_n\) are. Therefore we can consider the universal coefficient spectral sequence of \cite[7.7]{DI2}. As explained in \cite[8.1.2]{JOA}, this spectral sequence collapses if \(H\ensuremath{\mathbb{Z}}/l^{**}(R)\) is free over \(H\ensuremath{\mathbb{Z}}/l^{**}\). (Note that the cited corollary is stated for finite cell spectra, but the only properties actually used are cellularity and dualizability.) To show the freeness of \(H\ensuremath{\mathbb{Z}}/l^{**}(R)\) as a \(H\ensuremath{\mathbb{Z}}/l^{**}\)-module, observe that \(H\ensuremath{\mathbb{Z}}/l^{**}(\ensuremath{\mathbb{X}}_n)\) is free by construction (\cite[8.5.3]{JOA}). This implies the existence of a Künneth isomorphism for \(\ensuremath{\mathbb{X}}_n\), and thus \[H\ensuremath{\mathbb{Z}}/l^{**}(R)=H\ensuremath{\mathbb{Z}}/l^{**}(D\ensuremath{\mathbb{X}}_n)\underset{H\ensuremath{\mathbb{Z}}/l^{**}}\otimes H\ensuremath{\mathbb{Z}}/l^{**}(\ensuremath{\mathbb{X}}_n)\] is free.
\item The first isomorphism follows directly from the canonical bijection. The second isomorphism is proven by the same argument as in the proof of part 1, using the universal coefficient spectral sequence \cite[7.7]{DI2} together with the facts that \(AK\) is a cellular spectrum and that \(AK(n)_{**}(\ensuremath{\mathbb{X}}_n)\) is free over the coefficients.
\item This is proven just as in part 1 or part 2.
\end{enumerate}
\end{proof}
\begin{corol}
\begin{enumerate}
\item There exists a well defined coevaluation map \[coev: AK_{**}\rightarrow AK_{**}(\ensuremath{\mathbb{X}}_n)^\vee \underset{AK_{**}}{\otimes} AK_{**}(\ensuremath{\mathbb{X}}_n)\]
Here \((-)^\vee\) denotes the linear dual \(\textnormal{Hom}_{AK_{**}}(-, AK_{**})\).
It is induced by the map \(T\circ e: S\rightarrow \ensuremath{\mathbb{X}}_n \wedge D\ensuremath{\mathbb{X}}_n\), where \(e:S\rightarrow D\ensuremath{\mathbb{X}}_n \wedge \ensuremath{\mathbb{X}}_n\) is the unit map of \(R=D\ensuremath{\mathbb{X}}_n\wedge \ensuremath{\mathbb{X}}_n\) and \(T\) is the map that transposes the two factors.
\item Under the composition \[AK_{**}\rightarrow AK_{**}(R)\rightarrow \textnormal{Hom}_{AK_{**}}(AK_{**}(\ensuremath{\mathbb{X}}_n),AK_{**}(\ensuremath{\mathbb{X}}_n))\] an element \(v\in AK_{**}\) maps to multiplication by that element.
\end{enumerate}
\end{corol}
\begin{proof}
\begin{enumerate}
\item The coevalution map of \ref{SWEQUI}, which is the same as \(T\circ e\), induces the claimed map in \(AK(n)\)-homology, together with the identification \[AK(n)_{**}(D\ensuremath{\mathbb{X}}_n)\cong AK^{**}(\ensuremath{\mathbb{X}}_n)\cong \textnormal{Hom}_{AK_{**}}(AK_{**}(\ensuremath{\mathbb{X}}_n), AK_{**})\] of the preceding proposition. Because \(AK(n)_{**}(\ensuremath{\mathbb{X}}_n)\) is a free and finitely generated \(AK(n)_{**}\)-module, there is also an algebraic coevalution defined via choosing a basis as for a vector space, and the two maps coincide since they both satisfy the equivalent of the condition of the first point of \ref{SWEQUI} for projective and finitely generated modules.
\item The element \(1\in AK_{**}\) maps to the coevaluation of \(AK_{**}(\ensuremath{\mathbb{X}}_n)\) under the first map, using the identification \(AK_{**}(R)\cong AK_{**}(\ensuremath{\mathbb{X}}_n)^\vee \underset{AK_{**}}{\otimes} AK_{**}(\ensuremath{\mathbb{X}}_n)\) implied by the Künneth and duality isomorphisms. Hence an element of \(AK_{**}\) maps to that element times the coevaluation. The coevaluation maps to the identity under the second map. Consequently an element in \(AK_{**}\) times the coevaluation maps to multiplication by that element.
\end{enumerate}
\end{proof}
\ \\
We now have all the ingredients to use the classical proof in the motivic setting (\cite[Theorem 4.12]{HS}, see also \cite[6.3]{RAV2}):
\begin{theorem}
\label{SelfMapExample}
Let \(k=\ensuremath{\mathbb{C}}\) and \(l\) be an odd prime. The spectrum \(\mathbb{X}_n\) has a motivic \(v_n\) self-map \(f\) satisfying \[AK(m)_*f=\delta_{mn}v_n^{p^{N_m}}\]
for a sufficiently large integer \(N_m\).
\end{theorem}
\begin{proof}
The aim is to construct a permanent cycle
\[v\in \text{Ext}_{A_{**}}(H\ensuremath{\mathbb{Z}}/l_{*}(R),H\ensuremath{\mathbb{Z}}/l_{*})\]
that maps to a power of \(v_n\) in \(Ak(n)_{**}(R)\) and to a nilpotent element in \(Ak(m)_{**}(R)\) if \(m\neq n\). The diagram below will specify the meaning of "maps". Under motivic Spanier Whitehead duality such a class corresponds to a self-map of the described form on \(\ensuremath{\mathbb{X}}_n\).
\\
The cohomology of the point, \(H\ensuremath{\mathbb{Z}}/l^{**}\), is concentrated in simplicial degree 0. Therefore the operations \(Q_n\) act trivially on this module over the motivic Steenrod algebra. They act trivially on \(H\ensuremath{\mathbb{Z}}/l^{**}(R)\) since they act trivially on \(H^{**}(\ensuremath{\mathbb{X}}_n)\). If we write \(P(v_n)\) for the polynomial algebra in one generator with respect to the base ring \(H\ensuremath{\mathbb{Z}}/(l)_{**}\), this provides us with the following isomorphisms of trigraded algebras:
\begin{align}
\text{Ext}^{***}_{\Lambda(Q_n)}(H\ensuremath{\mathbb{Z}}/l^{**},H\ensuremath{\mathbb{Z}}/l^{**})&\xrightarrow{\cong} P(v_n)\otimes H\ensuremath{\mathbb{Z}}/l_{**}\\
\text{Ext}^{***}_{\Lambda(Q_n)}(H\ensuremath{\mathbb{Z}}/l^{**}(R),H\ensuremath{\mathbb{Z}}/l^{**})&\xrightarrow{\cong} P(v_n)\otimes H\ensuremath{\mathbb{Z}}/l_{**}(R)
\end{align}
Here \(v_n\) has homological degree 1 and internal bidegree \((2(l^n-1,l^n-1)\).\\
\ \\
Together with the change-of-rings morphisms related to the subalgebras \(\Lambda(Q_n)\) and \(A_N\) these fit into the following diagram:
\[
\xymatrix{
\text{Ext}_{A^{**}}(H\ensuremath{\mathbb{Z}}/l^{**},H\ensuremath{\mathbb{Z}}/l^{**}) \ar[r]^{i}\ar[d]^{\phi} & \text{Ext}_{A^{**}}(H\ensuremath{\mathbb{Z}}/l^{**}(R),H\ensuremath{\mathbb{Z}}/l^{**}) \ar[d]^{\phi}
\\
\text{Ext}_{A_N^{**}}(H\ensuremath{\mathbb{Z}}/l^{**},H\ensuremath{\mathbb{Z}}/l^{**}) \ar[r]^{i} \ar[d]^{\lambda} &
\text{Ext}_{A_N^{**}}(H\ensuremath{\mathbb{Z}}/l^{**}(R),H\ensuremath{\mathbb{Z}}/l^{**}) \ar[d]^{\lambda} &
\\
\text{Ext}_{\Lambda(Q_n)}(H\ensuremath{\mathbb{Z}}/l^{**},H\ensuremath{\mathbb{Z}}/l^{**}) \ar[r]^{i} \ar[d]^{\cong (1)} &
\text{Ext}_{\Lambda(Q_n)}(H\ensuremath{\mathbb{Z}}/l^{**}(R),H\ensuremath{\mathbb{Z}}/l^{**}) \ar[d]^{\cong (2)} &
\\
P(v_n)\underset{H\ensuremath{\mathbb{Z}}/(l)_{**}}{\otimes} H\ensuremath{\mathbb{Z}}/l_{**} \ar[r]^{i} \ar[d]&
P(v_n)\underset{H\ensuremath{\mathbb{Z}}/(l)_{**}}{\otimes} H\ensuremath{\mathbb{Z}}/l_{**}(R) \ar[d]&
\\
Ak(n)_{**} \ar[r]^{i} &
Ak(n)_{**}(R)&
\\
}
\]
\ \\
Step 1: Consider the element \(\widetilde{v_n}\in \text{Ext}_{\Lambda(Q_n)}(H\ensuremath{\mathbb{Z}}/l_{*}(R),H\ensuremath{\mathbb{Z}}/l_{*})\) that corresponds to \(v_n\otimes 1\in P(v_n)\otimes H_{**}(R)\) under the isomorphism (2).
\begin{prop} \(\forall N\geq n\) there is an integer \(t>0\) and an element \(x \in \text{Ext}_{A_{N,**}}(H\ensuremath{\mathbb{Z}}/l_{*},H\ensuremath{\mathbb{Z}}/l_{*})\) such that \(\lambda(x)=v_n^t\). The image of \(x\) under \(i\) is central in \(\text{Ext}_{A_{N,**}}(H\ensuremath{\mathbb{Z}}/l_{*}(R),H\ensuremath{\mathbb{Z}}/l_{*})\), where central is meant in respect to graded commutativity in the first, but not in the second bidegree.
\end{prop}
\begin{proof}
This statement is a corollary of \cite[Theorem 4.12]{HS}. Since the motivic cohomology of the point \(H\ensuremath{\mathbb{Z}}/(l)_{**}=\ensuremath{\mathbb{Z}}/(l)[\tau]\) is concentrated in simplicial degree 0, the action of the motivic Steenrod algebra is trivial on this module. Hence we can basechange the statement of the cited theorem to \(\ensuremath{\mathbb{Z}}/(l)[\tau]\).
\end{proof}
\ \\
Step 2: The module \(\text{Ext}_{A^{**}}(H\ensuremath{\mathbb{Z}}/l^{**}(R),H\ensuremath{\mathbb{Z}}/l^{**})\) has a vanishing line of slope \(1/(2^l-2)\) and a fixed intercept \(b\). By the motivic approximation lemma, the morphism \[\phi:\text{Ext}_{A^{**}}(H\ensuremath{\mathbb{Z}}/l^{*}(R),H\ensuremath{\mathbb{Z}}/l^{**})\rightarrow \text{Ext}_{A_N^{**}}(H\ensuremath{\mathbb{Z}}/l^{**}(R),H\ensuremath{\mathbb{Z}}/l^{**})\]
is an isomorphism above a line with slope \(1/2(l^n-1)\) and arbitrarily low intercept for sufficiently large \(N\). Since the element \(x\)(and therefore also \(i(x)\)) has tridegree (\(t,2(l^n-1),(l^n-1)\)), it lies above that line for a sufficiently large choice of \(N\). Define \(y\in \text{Ext}_{A^{**}}(H\ensuremath{\mathbb{Z}}/l^{**}(R),H\ensuremath{\mathbb{Z}}/l^{**})\) as the preimage of \(i(x)\) under \(\phi\). Since \(i(x)\) is central (in the graded sense with respect to the first bidegree but not with respect to the second) in \(\text{Ext}_{A_N^{**}}(H\ensuremath{\mathbb{Z}}/l^{**}(R),H\ensuremath{\mathbb{Z}}/l^{**})\), it commutes with all elements in the image of \(\phi\), in particular with all elements above the line defined by the approximation lemma.\\
\ \\
Step 3: The element \(y\) and its powers, as well as the images of \(y\) and its powers under the differentials of the motivic Adams spectral sequence all satisfy the requirement of the last statement, so they commute with each other. By induction we can assume that a power \(\tilde{y}\) of \(y\) survives up to the \(r\)th page. We wish to show that \(\tilde{y}^l\) is a \(r\)-cycle, i.e. \(d_r(\tilde{y}^l)=0\). This is true since \(d_r(\tilde{y}^l)=l\cdot \tilde{y}^{l-1}d_r(\tilde{y})=0\). After a finite number of pages, the differential will point in the area of the spectral sequence above the vanishing line, and we can stop the process. We end with a power \(\tilde{y}\) of \(y\) that is a permanent cycle in the motivic Adams spectral sequence and hence represents an element of \(Ak(n)_{**}(R)\).\\
\ \\
Step 4: The permanent cycle \(\tilde{y}\) represents an element \(\bar{y}\in \pi_{**}(R)\). Choose \(m\) such that \(v_n^m\) has the same degree as \(\bar{y}\). By the exact same arguments as in \cite{HS} we can choose a power of \(\bar{y}\) such that \(Ak(n)_{**}(\bar{y}^g)=v_n^{gm}\) and define \(f\) as the map corresponding to that power of \(\bar{y}\) under motivic Spanier Whitehead duality.\\
\ \\
Step 5: For \(m \neq n\) it follows just as in the topological case that the image of \(v\) in \(AK(m)_{**}\) is nilpotent either for trivial reasons (\(m<n\)) or because of a vanishing line with tighter slope in the Adams spectral sequence computing \(Ak(m)_{**}(R)\) (\(m>n\)).
\end{proof}
\section{The relation of \(\mathcal{C}_\eta\) and \(\mathcal{C}_{AK(n)}\)}
As a corollary of the Künneth isomorphism, we can settle one of the open conjectures in Ruth Joachimis dissertation \cite[Conjecture 7.1.7.3]{JOA} which concerns the relation of the thick ideal \(\text{thickid}(C_\eta)\) generated by the cone of the motivic Hopf map \(C_\eta\) and the thick ideals \(\mathcal{C}_{AK(n)}\) characterized by the vanishing of motivic Morava K-theory.
\begin{lemma}
Let \(m\in \ensuremath{\mathbb{N}}\) be any integer. Then the coefficients of the cone \(C_\eta\) of \(\eta: \Sigma^{1,1}S\rightarrow S\) in \(AK(m)_{**}\)-homology are given by: \[AK(m)_{**}(C_\eta)\cong AK(m)_{**}\oplus AK(m)_{*-2,*-1}\]
In particular, they are free over \(AK(m)_{**}\).
\end{lemma}
\begin{proof}
The long exact sequence induced by the cofiber sequence \[S^{1,1}\rightarrow S^{0,0}\rightarrow C_\eta \rightarrow S^{2,1}\] defining \(C_\eta\) splits into short exact sequences \[0 \rightarrow AK(m)_{**}\rightarrow AK(m)_{**}(C_\eta)\rightarrow AK(m)_{*-2,*-1}\rightarrow 0\]
because \(\eta\) induces the zero map in \(AK(m)_{**}\)-homology. The sequence splits because the outer terms are free \(AK(m)_{**}\)-modules, yielding the result.
\end{proof}
\begin{corol}
Let \(m\in \ensuremath{\mathbb{N}}\). In the case \(m<n\) we have
\[
AK(m)_{**}(C_\eta\wedge \ensuremath{\mathbb{X}}_n)\cong0
\]
and in the case \(m=n\) we have:
\[AK(n)_{**}(C_\eta\wedge \ensuremath{\mathbb{X}}_n)\cong AK(n)_{**}(C_\eta)\underset{AK(n)_{**}}{\otimes} AK(n)_{**}(\ensuremath{\mathbb{X}}_n)\neq 0\]
\end{corol}
\begin{proof}
By the preceding lemma the finite cell spectrum \(C_\eta\) has free \(AK(m)\)-homology and thus satisfies the requirements of the Künneth formula \cite[Remark 8.7]{DI2}.\\
Application of the Künneth formula yields:
\[AK(m)_{**}(C_\eta \wedge \ensuremath{\mathbb{X}}_n)\cong AK(m)_{**}(C_\eta)\underset{AK(m)_{**}}{\otimes} AK(m)_{**}(\ensuremath{\mathbb{X}}_n)\]
If \(m<n\) the factor \(AK(m)_{**}(X)=0\) vanishes by \ref{XnFacts}. This implies the first part of the statement. If \(m=n\) the result contains \[AK(n)_{**}(\ensuremath{\mathbb{X}}_n)\underset{AK(n)_{**}}{\otimes}AK(n)_{**}=AK(n)_{**}(\ensuremath{\mathbb{X}}_n)\neq 0\]
as a direct summand, so \(AK(n)_{**}(C_\eta\wedge \ensuremath{\mathbb{X}}_n)\) cannot vanish.
\end{proof}
\begin{prop}
The spectrum \(\ensuremath{\mathbb{X}}_{n+1}\) is contained in the intersection of thick ideals \(\text{thickid}(C_\eta)\cap \mathcal{C}_{AK(n)}\), but not in \(\text{thickid}(C_\eta)\cap \mathcal{C}_{AK(n+1)}\). In particular, these intersections are nonzero and distinct for all \(n\in \ensuremath{\mathbb{N}}\).
\end{prop}
\begin{proof}
Clearly \(C_\eta \wedge \ensuremath{\mathbb{X}}_{n+1}\) is in the thick ideal generated by \(C_\eta\). The preceding corollary tells us on the one hand that \(C_\eta \wedge \ensuremath{\mathbb{X}}_{n+1} \in \mathcal{C}_{AK(n)}\), and on the other hand that \(C_\eta \wedge \ensuremath{\mathbb{X}}_{n+1} \notin \mathcal{C}_{AK(n+1)}\).
\end{proof}
\section{A counterexample to a statement about thick subcategories in \cite{JOA}}
In this section we construct a counterexample to the inclusion \[\text{thickid}(c\mathcal{C}_2) \subset \mathcal{C}_{AK(1)}\] claimed in \cite[Chapter 9, last section]{JOA}, based on an error in \cite[Proposition 8.7.3]{JOA}.\\
Let \(l\) be an odd prime, and consider the topological mod-\(l\) Moore spectrum \(S/l \in \mathcal{SH}\). We can easily compute its \(K(1)\)-homology:
\begin{lemma}
\(K(1)_*(S/l)\cong K(1)_*\oplus K(1)_{*-1}\)
\end{lemma}
\begin{proof}
The Moore spectrum is defined via the cofiber sequence \(S\overset{\cdot l}\rightarrow S \rightarrow S/l\) and the map induced by \(l\) is trivial in \(K(1)\)-homology. Therefore the long exact sequence in \(K(1)\)-homology induced by this cofiber sequence splits up into short exact sequences, and these short exact sequences split because all graded \(K(1)\)-modules are free.
\end{proof}
\ \\
In \cite{ADA} Adams proved the existence of a non-nilpotent self map
\[v: \Sigma^{2l-2}S/l\rightarrow S/l\]
on the Moore spectrum which induces an isomorphism in \(K(1)\)-homology; namely multiplication by the invertible element \(v_1^{top}\). Consequently, the \(K(1)\)-homology of the cone \(C_v\) vanishes: \(K(1)_*(C_v)=0\), or equivalently \(C_v \in \mathcal{C}_2\).\\
\\
Applying the constant simplicial presheaf functor \(c\) to the construction gives us the cofiber sequence \[\Sigma^{2l-2,0}S/l\overset{cv}\longrightarrow S/l\rightarrow C_{cv}\] in \(\mathcal{SH}_\ensuremath{\mathbb{C}}\). The cone \(C_{cv}\) of \(cv\) is equivalent to \(c(C_v)\) because \(c\) is a triangulated functor, and the Moore spectrum is mapped to the Moore spectrum (\(cS/l=S/l\) because \(cl=l\).) We can compute the \(AK(1)\)-homology of the mod-\(l\)-Moore spectrum using the same argument as in the topological case:
\[AK(1)_{**}(S/l)\cong AK(1)_{**}\oplus AK(1)_{*-1,*}\]
However, the algebraic Morava K-theory of \(C_{cv}\) does not vanish:
\begin{lemma}
\label{NonVanishingMorava}
\(AK(1)(C_{cv}))\cong AK(1)_{**}(S/l)/(\tau^{l-1})\neq 0\)
\end{lemma}
\begin{proof}
The cofiber sequence \(S/l\overset{cv}\rightarrow S/l \rightarrow C_{cv}\) induces a long exact sequence in \(AK(1)\)-homology:
\[
...\rightarrow AK(1)_{p+(2l-2),q}(S/l) \overset{AK(1)_{**}(cv)}\longrightarrow AK(1)_{pq}(S/l)\rightarrow AK(1)_{pq}(C_{cv})\rightarrow ...
\]
The map \(AK(1)_{**}(cv)\) must be given by multiplication with \(\tau^{l-1}v_1\), because Betti realization maps \(AK(1)_{**}(cv)\) to multiplication with \(v_1^{top}\) and there is only one map realizing to this in the appropiate bidegree. This map is injective but, unlike the topological case, no longer an isomorphism. Hence the long exact sequence splits into short exact sequences
\[0 \rightarrow AK(1)_{pq}(cS/l) \overset{\cdot \tau^{l-1}v_1}\longrightarrow AK(1)_{pq}(cS/l)\rightarrow AK(1)_{pq}(C_{cv}) \rightarrow 0\]
and because \(v_1\) is invertible, the last term is isomorphic to \(AK(1)_{**}(S/l)/(\tau^{l-1})\).
\end{proof}
\ \\
Because \(\mathcal{C}_{AK(1)}\) was defined by the vanishing of \(AK(1)\)-homology and \(AK(1)_{**}(C_{cv})\neq 0\) does not vanish, we have \(C_{cv}\notin \mathcal{C}_{AK(1)}\). On the other hand, we have shown that \(C_v\in \mathcal{C}_2\). Because \(R(C_{cv})=C_v\), this implies \(C_{cv}\in R^{-1}(\mathcal{C}_2)\). Therefore we can conclude the following corollary from the preceding lemma:
\begin{corol}
The inclusion \[\mathcal{C}_{AK(1)}\subsetneq R^{-1}(\mathcal{C}_2)\] is proper.
\end{corol}
\ \\
Furthermore we have \(C_{cv}=cC_v\in \text{thickid}(c\mathcal{C}_2)\). Therefore \(cC_v\) is our desired counterexample and proves:
\begin{prop}
\(\text{thickid}(c\mathcal{C}_2) \not\subset \mathcal{C}_{AK(1)}\)
\end{prop}
\begin{rem}The mistake on which the incorrect assertion is based occurs in \cite[Proposition 8.7.3]{JOA}. This proposition states that for a finite topological CW spectrum \(Y\), \(AK(n)^{**}(cY)=0\) if and only if \(K(n)_{*}(Y)=0\). In the proof of this proposition Joachimi shows that the differentials in the motivic Atiyah-Hirzebruch spectral sequence are determined by the differentials of the topological Atiyah-Hirzebruch spectral sequence, and that the \(E_2\)-page of the motivic spectral sequence is given by adjoining a generator \(\tau\) to each entry in the topological spectral sequence, where all entries are generated in motivic weight 0. The problem that now occurs is that the differentials in the motivic spectral sequence do not preserve the weight, but lower it. Hence a nontrivial differential can generate \(\tau\)-primary torsion in the spectral sequence. The above example shows that this in fact happens.
\end{rem}
\ \\
This argument can in fact be made for any topological spectrum \(X\in \mathcal{C}_{n+1}\setminus\mathcal{C}_{n+2}\). Any such spectrum has nontrivial \(
K(n)\)-homology and a self map \(v: \Sigma^m X\rightarrow X\) that induces multiplication by some power of \(v_n^{top}\). We know by \ref{taurealization} that the map \(AK(n)_{**}(cX)\rightarrow K(n)_*(X)\) induced by Betti realization is surjective and its kernel is exactly the \(\tau\)-primary torsion elements. In particular we know that \(AK(n)_{**}(cX)\neq 0\) , and the self map provides us with a motivic map \(cv\). This map induces multiplication by the same power of \(\tau^{l-1}v_n\) in \(AK(n)\)-homology - up to a possible error term, which has to be \(\tau\)-primary torsion. We can eliminate this error term by taking sufficiently large \(l\)-fold powers of this map. We end up with a \(v_n^{top}\)-self map \(v'\) of \(X\) whose image \(cv'\) under the constant simplicial presheaf funtor \(c\) induces multiplication by some power of \(\tau^{l-1}v_n\) in \(AK(n)\)-homology. In particular, its cone has nonvanishing \(AK(n)\)-homology by the same argument as for our earlier counterexample and thus proves:
\begin{prop}
\(\text{thickid}(c\mathcal{C}_{n+1}) \not\subset \mathcal{C}_{AK(n)}\)
\end{prop}
Just as before, this also proves:
\begin{corol} The inclusion \[\mathcal{C}_{AK(n)}\subsetneq R^{-1}(\mathcal{C}_{n+1})\]
is proper.
\end{corol}
\appendix
\ \\
\textit{Sven-Torben Stahn\\
Fachgruppe Mathematik und Informatik\\
Bergische Universität Wuppertal\\}
SvenTorbenStahn@gmail.com
\end{document} | math |
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