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पढ़ा रहे देशभक्ति, जगा रहे हिंदू संस्कृति का संस्कार
भुवनेश्वर, शेषनाथ राय। यूं तो मुरली मनोहर शर्मा भाजपा के नेता हैं, लेकिन सेवा ही उनके जीवन का मूलमंत्र बन गया है। सियासत के अलावा विभिन्न सामाजिक, सांस्कृतिक, आध्यात्मिक संगठनों से जुड़े हैं। ओडिशा को पाश्चात्य संस्कृति से बचाने के लिए हर साल हिन्दू सेवा मेला का आयोजन करते हैं। सामूहिक वंदेमातरम कार्यक्रम के जरिए उन्होंने न सिर्फ लोगों का बल्कि सरकार का भी ध्यान अपनी ओर खींचा है।
आइएमसीटी फाउंडेशन नामक संगठन के जरिए हर साल तीन माह खुद अपने साथियों के साथ विभिन्न स्कूलों में जाकर बच्चों को देशभक्ति का पाठ पढ़ाते हैं। बच्चों के बीच हिंदू संस्कार जागृत करते हैं। उन्हें मातृ-पितृ पूजन, गौ पूजन, वृक्ष पूजन, जीव-जंतु पूजन का ज्ञान बांटते हैं।
ओडिशा प्रदेश के पिछड़े जिलों में से एक माने जाने वाले केंदुझर जिला में सन १९६६ में पिता ताराचन्द शर्मा एवं माता स्वर्गीय मोहिनी देवी के यहां जन्में मुरली मनोहर शर्मा के हृदय में बचपन से ही सेवा एवं संस्कार का भाव कूट-कूट कर भरा है। अपने बाल्य अवस्था में ही उन्होंने आपने सेवा भाव की मिशाल पेश कर दी। बाल्यावस्था में ही राष्ट्रीय स्वयं सेवक संघ से जुड़ गए। यही कारण है कि एक संभ्रांत परिवार में जन्में शर्मा स्कूल में पढ़ाई के समय ही एक नींबू एवं फल बेचने वाले के यहां नौकरी करने वाले दुखबंधु सेठी के साथ हाथ बंटाने के साथ उसके दुख-सुख में सहभागी बनकर रोजाना रात के समय दुकान बंद करने के समय उसके दुकान पहुंच जाते थे और फलों की टोकरी सिर पर रखकर उसके गोदाम पहुंचाते थे, जिसे लेकर वह चर्चा में आ गए थे। घर से मिले नाश्ता को दोस्तों में वितरित कर देना उनके सेवा मनोभाव को दर्शाने लगा था।
सन १९८३ में प्रथम श्रेणी से ग्रेजुएशन डिग्री हासिल करने के बाद पिताजी के साथ व्यवसाय से जुड़ गए। व्यवसाय के साथ सेवा भाव का उद्देश्य लेकर आपने अयोध्या में राममंदिर आंदोलन से प्रेरित होकर भाजपा से जुड़े। शर्मा ने बताया कि सेवा-संस्कार की प्रेरणा हमें अपने पिताजी से मिली है जो कि आज ७९ साल की उम्र में भी लोगों को मुफ्त में होम्योपैथी दवा के जरिए ५० साल से नियमित इलाज करते आ रहे हैं।
संपत्ति या पद-पदवी को समय का चक्र मानने वाले शर्मा की कार्यधारा ने न सिर्फ केंदुझर जिला, ओडिशा प्रांत बल्कि पूरे देश के लोगों से स्पर्श कर चुकी है। चाहे वह मातृभाषा ओड़िया को दफ्तर में प्रयोग के लिए आन्दोलन हो, नीलशैल जैसे विरल अनुष्ठान के जरिए मधुवर्णबोध के लिए आवाज बुलंद करनी हो, गैरकानूनी बांग्लादेशी अनुप्रवेश पर लगाम लगाना हो, पाइक अखाड़ा खेल को प्रोत्साहित करना हो, तूफान आपदा हो या मारवाड़ी युवा मंच के जरिए कृत्रिम अंग प्रत्यारोपण के जरिए लोगों की सेवा हो या फिर आइएमसीटी फाउंडेशन के जरिए बच्चों को राष्ट्रभक्ति का पाठ पढ़ाना हो या फिर हिन्दू आध्यात्म सेवा मेला के जरिए बच्चों को संस्कारी बनाने की मुहिम लगातार जारी रखे हुए हैं।
इसके साथ ही प्रकृति की सच्चाई से रू ब रू कराने के साथ बच्चों में देश भक्ति का पाठ पढ़ाना हो अपनी टीम के साथ निरंतर लगे रहते हैं। आपके सामूहिक वंदेमातराम गान कार्यक्रम से प्रेरित होकर प्रदेश सरकार भी सामूहिक वंदे उत्कल जननी गान जैसे कार्यक्रम को प्रोत्साहित करने के लिए आगे आई।
बांग्लादेशी अनुप्रवेशकारियों पर रोक लगाने के लिए कदम उठायी। शर्मा का मानना है कि आधुनिकता के नाम पर आज जिस कदर से हमारी प्राचीन भारतीय संस्कृति में पश्चिमी सभ्यता पैर पसार रही है, उसे हम सबको मिलकर युद्ध स्तर पर रोकना होगा। अन्यथा वह दिन दूर नहीं जिस संस्कृति में अपने माता-पिता को भगवान माना जाता है, वहीं के लोग अपने माता-पिता को वृद्धाश्रम में छोड़ने लगेंगे। अमेरिका जैसी सरकार पर आज माता-पिता की सेवा के लिए अरबों रुपये का भार पड़ रहा है, जिससे अपनी प्राचीन संस्कृति के चलते ही आज भारत बचा हुआ है। यही कारण है कि हम अपने बच्चों को संस्कारी, आध्यात्मिक ज्ञान, अपनी प्राचीन सभ्यता से अवगत कराने के लिए यह बीड़ा उठाया है।
जागरण एक्सक्लूसिव: सीपीसीबी के निर्देश के बावजूद पेट्रोल पंपों पर नहीं लग सका वेपर रिकवरी सिस्टम
महिला पुलिस अधिकारी के सामने छात्राएं बोलीं-कॉलेज के बाहर ही होती है छेड़छाड़
स्क के सामने खुला आम्रपाली का काला चिट्ठा, चपरासी-ड्राइवर के नाम पर हैं २३ कंपनियां
वेथर उपड़ते: दिल्ली-न्क्र को सता रही शीतलहर, जानिये- कब होगी बारिश
ई-कॉमर्स कंपनियों के कारोबार की हो जांच
बंगाल में ५ सभाएं करेंगे अमित शाह, सुप्रीम कोर्ट से रथयात्रा अनुमति नहीं मिलने पर तलाशा विकल्प | hindi |
जोया क्षेत्र में सोत नदी पर कब्जा कर बनाए गए मकान (फाइल फोटो) - फोटो : अमर उजाला
संभल के असमोली क्षेत्र के गांव मातीपुर में जिला प्रशासन ने नागरिकों का सहयोग लेकर सोत नदी को पुनर्जीवित करने की मुहिम का शनिवार को सुबह शुभारंभ कर दिया। शुभारंभ से पहले हवन पूजन के बाद नदी की खाली जमीन पर पौधे लगाए गए। साथ ही नदी की जमीन पर खुदाई कराई गई। इसके लिए खुद जिलाधिकारी अविनाश कृष्ण सिंह ने फावड़ा चलाया। उनके साथ भाजपा जिलाध्यक्ष राजेश सिंघल, पुलिस अधीक्षक यमुना प्रसाद ने भी नदी खुदाई के पावन काम में भागीदारी। राजनीतिक और सामाजिक क्षेत्र की कई हस्तियों ने शामिल रहीं।
आसपास के गांवों में रहने वाले लोगों ने भी सहयोग दिया। अमरोहा जिले से संभल जिले में प्रवेश करने वाली सोत नदी गुम गई है। नदी की जमीन पर अवैध कब्जे हो गए। सूखी नदी की जमीन पर हो रहे अवैध कब्जों को किसी ने नहीं रोका। अमर उजाला ने- कहां गए जल स्रोत शीर्षक खबरों की श्रृंखला से इन हालातों को जनता के साथ जिला प्रशासन और प्रदेश सरकार के सामने उठाया। प्रशासन ने इसे गंभीरता से लिया। खुद जिलाधिकारी अविनाश कृष्ण सिंह ने पूरे मामले को समझा और नदी को पुनर्जीवित करने का निर्णय लिया। उन्होंने बहजोई में जिलास्तरीय अधिकारियों के साथ बैठक की। मनरेगा में नदी की खुदाई कराने और खाली जमीन पर पौधे लगाने की कार्ययोजना बनाई।
खुदाई करके दोबारा जिंदा की जाएगी सोत
जिले में ४५-४५ किलोमीटर सोत नदी का हिस्सा है। इस पर लोगों ने कब्जा कर लिया। यह नदी कभी सदानीरा थी। अमर उजाला ने सोत की दुर्दशा के बारे में खबरें कीं तो जिला प्रशासन ने इसकी दशा सुधारने का फैसला किया। नदी की जमीन से अवैध कब्जे हटाए जा चुके हैं। अब नदी की खुदाई करके इसे पुनर्जीवित किया जाना है। इसके लिए शनिवार को नदी पूजन और हवन के साथ औपचारिक शुरुआत कर दी गई। इस मौके पर राजनीतिक नेताओं में नरेंद्र सिंह, हरेंद्र सिंह, विरलेश यादव, संजय सांख्यधर और अधिकारियों में मुख्य विकास अधिकारी उमेश कुमार त्यागी, अपर जिलाधिकारी लवकुश त्रिपाठी, संभल के एसडीएम दीपेंद्र यादव, चंदौसी के एसडीएम महेश दीक्षित, एआरटीओ छबि सिंह चौहान, डीएफओ डीके चतुर्वेदी ने भाग लिया। अधिकारियों ने जनता को पर्यावरण के खतरों के प्रति आगाह किया और भूजल स्रोतों को बचाने के लिए प्रेरित किया। अधिकारियों ने कहा कि यदि यही हाल रहा आने वाली पीढ़ी को गंभीर संकटों को सामना करना पड़ेगा।
वातावरण में बड़ी तेजी के साथ बदलाव हो रहा है। ग्लोबल वार्मिंग बढ़ रही है। जल स्रोत सूख रहे हैं। हमारे संभल जनपद में चालीस पचास वर्ष पहले सोत नदी बहा करती थी। लेकिन यह नदी धीरे-धीरे यह नदी मिट गई। नदी अब सिर्फ कागजों में रह गई। लेकिन हमारे अधिकारियों ने इस नदी को पुनर्जीवित करने का निर्णय लिया। जनता और जनप्रतिनिधियों से सहयोग लेकर नदी को पुनर्जीवित करने के काम की औपचारिक शुरूआत शनिवार को कर दी है। अब हर रोज कुछ न कुछ काम होगा और आने वाले दिनों में नदी बहेगी। नदी दोनों तरफ खाली जमीन पर छायादार पौधे लगाए जाएंगे। जो पर्यावरण को सुधारेंगे और आम लोगों को छाया देंगे। वहीं भूल जलस्तर भी बढ़ेगा।
अविनाश कृष्ण सिंह, जिलाधिकारी। संभल।
नजर आएगी हरियाली ही हरियाली
अमर उजाला ने कहां गए पर्यावरण पर आए संकट को शृंखलाबद्ध तरीके जनता के बीच रखा। कहां गए जल स्रोत शीर्षक से लगातार खबरें प्रकाशित की गईं। इसका संज्ञान जिला प्रशासन ने लिया। अब मनरेगा के तहत नदी की खुदाई होगी। वन विभाग खाली जमीन पर पौधे लगाएगा। अमर उजाला की पहल पर जिला प्रशासन ने अपना काम जारी रखा तो आने वाले वर्षों में जनपद संभल का नाम उन जनपदों में शामिल हो जाएगा जहां पर हरियाली ही हरियाली नजर आएगी। साथ ही सोत नदी भी पुनर्जीवित हो जाएगी।
-जिलाधिकारी ने फावड़ा चलाकर की अभियान की शुरुआत
-अमर उजाला ने सोत की दुर्दशा पर खबरें प्रकाशित कीं तो प्रशासन ने कसी कमर
नदी बहेगी, नदी के दोनों तरफ दिखेंगे छायादार वृक्ष
सोत नदी पर बने मकान (फाइल फोटो) - फोटो : अमर उजाला
सांसद आजम खां एवं राष्ट्रपति पदक विजेता पूर्व सीओ सिटी आलेहसन खां को भूमाफिया घोषित कर दिया गया है। दोनों का नाम एंटी भू-माफिया पोर्टल पर भी अपलोड कर दिया गया है।
एप्पल के शिखर पर पहुंचे मुरादाबाद के सबीह खान, खुशी से झूमा परिवार
यूपी: आठ वर्षीय बालिका के साथ हैवानित, अगवा कर सामूहिक दुष्कर्म
रामपुर में उत्पाती हाथियों ने एक शख्स की जान ली, एक घायल | hindi |
राष्ट्रीय अध्यक्ष पर लगाया ५ लाख में टिकट बेचने का आरोप
जौनपुर. भारतीय समाज पार्टी के कार्यकर्ताओं ने बृहस्पतिवार को अपनी ही पार्टी के खिलाफ बगावत कर दी। राष्ट्रीय अध्यक्ष ओम प्रकाश राजभर पर पैसे देकर टिकट बांटने का आरोप लगाया है। कार्यकर्ताओं ने उनके खिलाफ नारेबाजी करते हुए पार्टी का झंडा और टोपी जला दी।
भासपा और भाजपा का गठबंधन है। इस गठबंधन के तहत जौनपुर के शाहगंज विधानसभा सीट से भासपा को ही अपना प्रत्याशी उतारना है। काफी दिनों से कार्यकर्ता इस ओर निगाह लगाए बैठे थे। कई तो दावेदारी भी पेश कर रहे थे। इसी बीच मनीष सिंह और राणा अजीत सिंह का नाम तैरने लगा।
राणा अजीत सिंह व्यवसायी हैं। जबकि मनीष सिंह अधिवक्ता हैं। भासपा ने उम्मीदवारों की घोषणा की तो राणा अजीत सिंह को शाहगंज से अपना उम्मीदवार बनाया। घोषणा होते ही पाटी कार्यकर्ताओं ने हंगामा खड़ा कर दिया। आरोप लगाया कि अध्यक्ष ने ५ करोड़ रूपये लेकर राणा अजीत सिंह को टिकट बेचा है।
जमीनी कार्यकर्ताओं की अनदेखी की गई है। इसके विरोध में सभी एकजुट हो गए। जिला जेल के पास स्थित मनीष सिंह के आवास पर सभी ने ओमप्रकाश राजभर का पुतला बना कर उसमें आग लगा दी। उनके खिलाफ नारे लगाए जाने लगे। पार्टी का झंडा और टोपी आग के हवाले कर दी गई। साथ करीब ४० पदाधिकारियों व सदस्यों ने अपना इस्तीफा दे दिया।
बसपा भाईचारा सम्मेलन में अंसारी बंधुओं ने दिखाई ताकत
फर्जी पासपोर्ट बनवाने मामले में ८ के खिलाफ मुकदमा
जब ऐन शादी के वक्त पता चली दूल्हे की कमी तो लड़की के परिवार ने तुरंत लौटा दी बारात | hindi |
एक साल बाद भी मैट्रिक पास छात्राओं को नहीं मिली प्रोत्साहन राशि - अखंड भारत न्यूज
होम बिहार समस्तीपुर एक साल बाद भी मैट्रिक पास छात्राओं को नहीं मिली प्रोत्साहन राशि
दलसिंहसराय समस्तीपुर : शहर में स्थित बालिका उच्य विद्यालय दलसिंहसराय से २०१६ में मैट्रिक पास छात्राओं को अभी तक प्रोत्साहन राशि उपलब्ध नहीं करायी गयी है जबकि विभागीय स्तर पर स्कूल से छात्राओं को चेक देकर इसे बैंक में जमा करने को कहा गया था |अब बैंक की उदासीनता के कारण यह राशि छात्राओं को अभी तक नहीं मिल पायी है |
बता दें कि मैट्रिक प्रथम श्रेणी से पास सभी छात्राओं को सरकार द्वारा १० हजार की राशि प्रोत्साहन के तौर पर दी जाती है | स्कूल की छात्रा सपना कुमारी एवं दिव्या कुमारी समेत कई छात्राओं ने बताया कि स्कूल की ओर से उन्हें १० हजार का चेक देकर यह कहा गया था कि इसे स्थानीय पंजाब नेशनल बैंक में जमा कर दे, समस्तीपुर स्थित कॉरपोरेशन बैंक द्वारा इस राशि को भुगतान कर दिया जायेगा | सभी छात्राओं ने ऐसा ही किया लेकिन २०१६ में जमा किये गये चेक का कॉरपोरेशन बैंक ने अभी तक भुगतान नहीं किया है, इससे छात्राओं में काफी आक्रोश देखा जा रहा है | छात्राओं ने यह भी जानकारी दी है कि संबंधित पंजाब नेशनल बैंक से २० दिनों के बाद सभी छात्राओं का चेक यह कह कर लौटा दिया गया कि कॉरपोरेशन बैंक चेक नहीं भुना रहा है इसलिये आपका यह चेक आपको लौटाया जा रहा है |
प्रेवियस आर्टियलआबकारी अधिकारी बनकर पहुंचे बदमाश बलात्कारी नही निकले
नेक्स्ट आर्टियलछात्रों पर लाठी चार्ज के विरोध में छात्रों ने निकाला मार्च | hindi |
\begin{document}
\title{Introduction to Quantum Gate Set Tomography}
\tableofcontents
\renewcommand*{\thefootnote}{\arabic{footnote}}
\chapter{Introduction}
Progress in qubit technology requires accurate and reliable methods for qubit characterization. There are several reasons characterization is important. One is diagnostics. Qubit operations are susceptible to various types of errors due to imperfect control pulses, qubit-qubit couplings (crosstalk), and environmental noise. In order to improve qubit performance, it is necessary to identify the types and magnitudes of these errors and reduce them. Another reason is the desirability to have metrics of gate quality that are both platform independent and provide sufficient descriptive power to enable the assessment of qubit performance under a variety of conditions. Metrics are also necessary to ascertain if the requirements of quantum error correction (QEC) are being met, i.e., whether the gate error rates are below a suitable QEC threshold. Full characterization of quantum processes provides detailed information on gate errors as well as metrics of qubit quality such as gate fidelity.
Several methods of qubit characterization are currently available. In chronological order of their development, the main techniques are quantum state tomography (QST)~\cite{leibfried_experimental_1996}, quantum process tomography (QPT)~\cite{chuang_prescription_1997,poyatos_complete_1997}, randomized benchmarking (RB)~\cite{knill_randomized_2008,ryan_randomized_2009,magesan_scalable_2011,magesan_characterizing_2012}, and quantum gate set tomography (GST)~\cite{Merkel2013,RBK2013}. All of these tools, with the exception of GST, have been well studied and systematized, and have gained widespread acceptance and use in the quantum computing research community.
GST grew out of QPT, but is somewhat more demanding in terms of the number of experiments required as well as the post-processing. As we will see, obtaining a GST estimate involves solving a highly nonlinear optimization problem. In addition, the scaling with system size is polynomially worse than QPT because of the need to characterize multiple gates at once. Approximately 80 experiments are required for a single qubit and over 4,000 for 2 qubits to estimate a complete gate set, compared to 16 and 256 experiments respectively to reconstruct a single 1- or 2-qubit gate with QPT. Methods for streamlining the resource requirements for GST are under investigation~\cite{IBMpcomm,RBK-pcomm}. Nevertheless, single-qubit GST has been demonstrated by several groups~\cite{Merkel2013,RBK2013} and 2-qubit GST is widely believed to be achievable. Importantly, these groups have convincingly shown that GST outperforms QPT in situations relevant to fault-tolerant quantum information processing (QIP)~\cite{Merkel2013,RBK2013}. As a result, it seems clear that GST in either its present or some future form will supercede QPT as the only accurate method available to fully characterize qubits for fault-tolerant quantum computing. In this document, we present GST in the hopes that readers will be inspired to implement it with their qubits and explore the possibilities it offers.
GST arose from the observation that QPT is inaccurate in the presence of state preparation and measurement (SPAM) errors.
It will be useful to classify SPAM errors into two different types, which we will call {\em intrinsic} and {\em extrinsic}. Intrinsic SPAM errors are those that are inherent in the state preparation and measurement process. One example is an error initializing the $|0\rangle$ state due to thermal populations of excited states. Another is dark counts when attempting to measure, say, the $|1\rangle$ state. Extrinsic SPAM errors are those due to errors in the gates used to transform the initial state to the starting state (or set of states) for the experiment to be performed. In QPT, the starting states must form an informationally complete basis of the Hilbert-Schmidt space on which the gate being estimated acts. These are typically created by applying gates to a given initial state, usually the $|0\rangle$ state, and these gates themselves may be faulty.
Intrinsic SPAM errors are of particular relevance to fault-tolerant quantum computing, since it turns out that QEC requirements are much more stringent on gates than on SPAM. According to recent results from IBM~\cite{IBMpcomm}, a 50-fold increase in intrinsic SPAM error reduces the surface code threshold by only a factor of 3-4. Therefore QPT - the accuracy of which degrades with increasing SPAM - would not be able to determine if a qubit meets threshold requirements when the ratio of intrinsic SPAM to gate error is large.
This is not an issue for extrinsic SPAM errors, which go to zero as the errors on the gates go to zero. Nevertheless, extrinisic SPAM error interferes with diagnostics: as an example, QPT cannot distinguish an over-rotation error on a single gate from the same error on all gates (see Sec.~\ref{examples-subsection}). In addition, Merkel, et al. have found that, for a broad range of gate error -- including the thresholds of leading QEC code candidates -- the ratio of QPT estimation error to gate error {\em increases} as the gate error itself decreases~\cite{Merkel2013}. This makes QPT less reliable as gate quality improves.
Extrinsic SPAM error is also unsatisfactory from a theoretical point of view: QPT assumes the ability to perfectly prepare a complete set of states and measurements. In reality, these states and measurements are prepared using the same faulty gates that QPT attempts to characterize. One would like to have a characterization technique that takes account of SPAM gates self-consistently. We shall see that GST is able to resolve all of these issues.
Another approach to dealing with SPAM errors is provided by randomized benchmarking~\cite{knill_randomized_2008,ryan_randomized_2009,magesan_scalable_2011,magesan_characterizing_2012}. RB is based on the idea of twirling~\cite{bennett_mixed-state_1996,dankert_exact_2009} -- the gate being characterized is averaged in a such a way that the resulting process is depolarizing with the same average fidelity as the original gate. The depolarizing parameter of the averaged process is measured experimentally, and the result is related back to the average fidelity of the original gate. This technique is independent of the particular starting state of the experiment, and therefore is not affected by SPAM errors. However, RB has several shortcomings which make it unsatisfactory as a sole characterization technique for fault-tolerant QIP. For one thing, it is limited to Clifford gates\footnote{For recent work discussing generalizations of RB to non-Clifford gates, see Refs.~\cite{kimmel_robust_2014,dugas_characterizing_2015}.}, and so cannot be used to characterize a universal gate set for quantum computing. For another, RB provides only a single metric of gate quality, the average fidelity.\footnote{Recently, generalizations of RB have been proposed to measure leakage rates~\cite{wallman_robust_2015} and the coherence~\cite{wallman_estimating_2015} of errors. These advances have the promise to turn RB into a more widely applicable characterization tool.} This can be insufficient for determining the correct qubit error model to use for evaluating compatibility with QEC. Several groups have shown that qualitatively different errors can produce the same average gate fidelity, and in the case of coherent errors the depolarizing channel inferred from the RB gate fidelity underestimates the effect of the error~\cite{magesan_modeling_2013,gutierrez_approximation_2013,puzzuoli_tractable_2014}.
Finally, RB assumes the errors on subsequent gates are independent. This assumption fails in the presence of non-Markovian, or time-dependent noise. GST suffers from this assumption as well, but the long sequences used in RB make this a more pressing issue.
Despite these apparent shortcomings, RB has been used with great success by several groups to measure gate fidelities and to diagnose and correct errors~\cite{gaebler_randomized_2012,barends_superconducting_2014,kelly_optimal_2014,muhonen_quantifying_2014}. RB also has the advantage of scalability -- the resources required to implement RB (number of experiments, processing time) scale polynomially with the number of qubits being characterized~\cite{magesan_scalable_2011,magesan_characterizing_2012}. QPT and GST, on the other hand, scale exponentially with the number of qubits. As a result, these techniques will foreseeably be limited to addressing no more than 2-3 qubits at a time. In our view, GST and RB will end up complementing each other as elements of a larger characterization protocol for any future multi-qubit quantum computer. We will not discuss RB further in this document.
This document reviews GST and provides instructions and examples for workers who would like to implement GST to characterize their qubits. The goal is to provide a guide that is both practical and self-contained. For simplicity, consideration is restricted to a single qubit throughout. We begin in Chapter~\ref{math-chapter} with a review of the mathematical background common to all characterization techniques. This includes the representation of gates as quantum maps via the process matrix and Pauli transfer matrix, the superoperator formalism, and the Choi-Jamiolkowski representation, which allows the maximum likelihood estimation (MLE) problem to be formulated as a semidefinite program (SDP). Chapter~\ref{derivation-chapter} begins with a review of quantum state and process tomography, the characterization techniques that underlie GST. We then continue to GST itself, including a derivation of linear-inversion gate set tomography (LGST) following Ref.~\cite{RBK2013}. Although LGST is weaker than MLE since it does not in general provide physical estimates, it is useful both as a starting point for the nonlinear optimization associated with MLE, and also in its own right as a technique for getting some information about the gate set quickly and with little numerical effort. This is followed by a description of the MLE problem in both the process matrix~\cite{Chow2009SI} and Pauli transfer matrix~\cite{Chow2012SI} formulations. MLE is the standard approach for obtaining physical gate estimates from tomographic data. In Chapter~\ref{implementation-chapter} we continue with GST, presenting the detailed experimental protocol as well as numerical results implementing the ideas of Chapter~\ref{derivation-chapter}. The implementation utilizes simulated data with simplified but realistic errors. This data was designed to incorporate the important properties of actual data including coherent and incoherent errors and finite sampling noise. Using this simulated data, we compare the performance of QPT and GST.
\chapter{Mathematical preliminaries}\label{math-chapter}
In this chapter we review the mathematical background on quantum operations and establish notation that we will use later on.
\section{Quantum operations}\label{quantops}
Quantum operations are linear maps on density operators $\rho \rightarrow \Lambda(\rho)$, that take an arbitrary initial state $\rho \in \mathcal{H}$ in some Hilbert space $\mathcal{H}$ and output another state $\Lambda(\rho)\in\mathcal{H}$. Although the formalism we present is applicable to arbitrary ($d$-level, or qudit) quantum systems, we will restrict ourselves to systems of qubits, i.e. $\mathcal{H} = \mathcal{H}_n$, where $\mathcal{H}_n$ is some $2^n$-dimensional (n-qubit) Hilbert space. This is done in the interest of concreteness, and because most experiments involve qubit systems. Thus, for example, we will explicitly use the familiar qubit basis of Pauli operators rather than a more general qudit basis.
For such a map to describe a physical process, two basic requirements must be met: (1) for any initial state of the universe (system + environment), the final state after application of the map must have nonnegative probabilities for measuring the eigenstate of any observable (in other words, the density matrix of the universe is always positive semidefinite), and (2) total probability must be conserved.\footnote{Requirement (2) is relaxed in the presense of leakage errors (transitions outside the qubit subspace). In this case, the total probability must {\em not increase}. For an interesting discussion of non-trace preserving maps, see Ref.~\cite{bhandari_trace_preserving_2015}.} Requirement (1) is known as Complete Positivity (CP) and (2) is called Trace Preservation (TP), since the total probability of all eigenstates for the state $\rho$ is $\Tr\{\rho\}$. A physical map is then a CPTP map.
It turns out that a general CP map on just the system of interest (the environment having been traced out) can be written as~\cite{NielsenChuang}
\begin{equation}
\Lambda(\rho) = \sum_{i=1}^N K_i \rho K_i^\dag, \label{Kraus}
\end{equation}
for some $N\leq d^2$, where $d = 2^n$ is the Hilbert space dimension. This is called the {\em Kraus representation}. The $K_i$ are Kraus operators, and need not be unitary, hermitian, or invertible. The map is also TP if $\sum_i K_i^\dag K_i = I$, where $I$ denotes the identity.\footnote{To see this, we take the trace of Eq.~(\ref{Kraus}) and rearrange operators inside the trace: $\Tr\{\Lambda(\rho)\} = \Tr\{\sum_i K_i^\dag K_i \rho\}$. By trace preservation, $\Tr\{\Lambda(\rho)\} = \Tr\{\rho\}$. We therefore have $\Tr\{\sum_i K_i^\dag K_i \rho\} = \Tr\{\rho\}$ for any $\rho$, which can only be true if $\sum_i K_i^\dag K_i = I$.} This is called the {\em completeness condition}.
\subsection{Examples of quantum operations}
Eq. (\ref{Kraus}) contains unitary evolution as a special case: $N=1, K_1 \in SU(d)$. It also describes non-unitary processes, such as depolarization and amplitude damping, that can represent coupling to an environment (such as an external electromagnetic field). Below are some examples of the Kraus representation for a single qubit that describe familiar quantum processes.
\begin{enumerate}
\item {\em Depolarization} -- This operation replaces the input state with a completely mixed state with probability $p$, and does nothing with probability $1-p$. It describes a noise process where information is completely lost with probability $p$. For a single qubit, depolarization is given by the quantum map~\cite{NielsenChuang},
\begin{equation}
\Lambda_{dep}(\rho) = \left(1-\frac{3}{4}p\right)\rho + \frac{p}{4}\left(X\rho X + Y\rho Y + Z\rho Z\right). \label{example-depol}
\end{equation}
The Kraus operators are $\sqrt{1-3p/4}I,\sqrt{p}X/2,\sqrt{p}Y/2,\sqrt{p}Z/2$.\footnote{I, X, Y, Z are the single-qubit Pauli operators
\begin{eqnarray}
I = \left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right),\; X = \left(\begin{array}{cc} 0 &1\\ 1 &0\end{array}\right),\; Y =\left(\begin{array}{cc} 0 & -i\\ i &0\end{array}\right),\;
Z =\left(\begin{array}{cc} 1 & 0\\ 0 &-1\end{array}\right).\nonumber
\end{eqnarray}
}
\item {\em Dephasing} -- This process can arise when the energy splitting of a qubit fluctuates as a function of time due to coupling to the environment. Charge noise affecting a transmon qubit is of this type. Dephasing is represented by a phase-flip channel~\cite{NielsenChuang}, which describes the loss of phase information with probability $p$. This channel projects the state onto the $Z$-axis of the Bloch sphere with probability $p$, and does nothing with probability $1-p$:
\begin{equation}
\Lambda_{Z}(\rho) = \left(1-\frac{p}{2}\right)\rho + \frac{p}{2}Z\rho Z. \label{example-dephasing}
\end{equation}
Here the Kraus operators are $\sqrt{1-p/2}I,\sqrt{p/2}Z$.
\item {\em Spontaneous Emission / Amplitude Damping} -- This process describes energy loss from a quantum system, the standard example being spontaneous emission from an atom~\cite{NielsenChuang}. The Kraus operators are
\begin{eqnarray}
K_0 = \left(\begin{array}{cc} 1 & 0 \\ 0 & \sqrt{1-p} \end{array}\right),\; K_1 = \left(\begin{array}{cc} 0 &\sqrt{p}\\ 0 &0\end{array}\right).\label{amp-damp-kraus-ops}
\end{eqnarray}
$K_1$ changes $|1\rangle$ to $|0\rangle$, corresponding to the emission of a photon to the environment. $K_0$ leaves $|0\rangle$ unchanged but reduces the amplitude of $|1\rangle$, representing the change in the relative probabilities of the two states when no photon loss is observed. Note that the form of $K_0$ is fixed by $K_1$, the completeness condition, and the requirement that $K_0$ not change the amplitude of the $|0\rangle$ state.
\end{enumerate}
It is worth noting that at no point did we explicitly invoke the time dependence of quantum maps written in terms of Kraus operators. In fact, the time dependence is implicitly built in, through the parameter $p$ appearing in the examples above. Starting from Eq.~(\ref{Kraus}), and under the assumptions of local, Markovian noise, we can derive the Lindblad equation for the time evolution of the system density matrix, $\rho$. The Lindblad operators are related to the Kraus operators via $K_i = \sqrt{t} L_i$. (See for example, Ref.~\cite{Haroche-Raimond} Eq.~(4.62).) As a result, $p\propto t$, as we would expect from Fermi's golden rule for transitions induced by coupling to an environment. We do not worry about explicit time evolution in this document, and therefore use Eq.~(\ref{Kraus}) rather than the Lindblad equation, assuming implicitly that quantum operations take a finite time $t$.
For a given quantum operation, the Kraus operators are not unique~\cite{NielsenChuang}. They can be fixed by expressing them in a particular basis of $\mathcal{H}_n$. A typical choice is the Pauli basis, $\mathcal{P}^{\otimes n}$, where $\mathcal{P} = \{I,X,Y,Z\}$. We follow this convention here.
\subsection{Process matrix}\label{process-matrix-section}
The Pauli basis leads to a useful representation of a quantum map: the process matrix, or $\chi$-matrix. Expanding $K_i$ in terms of Pauli operators, we obtain $K_i = \sum_{j=1}^{d^2}a_{ij}P_j$, $P_j \in \mathcal{P}^{\otimes n}$. Inserting this expression into Eq. (\ref{Kraus}) gives
\begin{equation}
\Lambda(\rho) = \sum_{j,k=1}^{d^2} \chi_{jk}P_j \rho P_k, \label{chi}
\end{equation}
where $\chi_{jk} = \sum_i a_{ij}a_{ik}^*$. $\chi$ is a $d^2 \times d^2$ complex-valued matrix, $d = 2^n$. The $\chi$-matrix completely determines the map $\Lambda$. From its definition in terms of the $a$-coefficients, we find that $\chi$ is Hermitian and positive semidefinite.\footnote{Indeed, $\chi_{kj} = \sum_i a_{ik}a_{ij}^* = \chi_{jk}^*$, so $\chi$ is Hermitian. Also, for any $d^2$-dimensional complex vector $\vec{v}$, we have $\vec{v}^\dag \chi \vec{v} = \sum_{jk} v_j^* \chi_{jk} v_k = \sum_{ijk} (a_{ij}v_j^*)(a_{ik}^*v_k) = \sum_i |\sum_j a_{ij}v_j^*|^2 \geq 0$. Therefore $\chi$ is positive-semidefinite.} In addition, we have $\sum_i K_i^\dag K_i = \sum_{ijk}a_{ik}^*P_k a_{ij} P_j = \sum_{jk} (\sum_i a_{ij}a_{ik}^*)P_k P_j = \sum_{jk} \chi_{jk}P_kP_j$. Thus for a TP map, the completeness condition reads $\sum_{jk} \chi_{jk}P_kP_j = I$. Hence, the $\chi$-matrix for a physical (CPTP) map has $d^2 (d^2-1)$ free parameters. (A $d^2 \times d^2$ complex hermitian matrix has $d^4$ free parameters, and the completeness condition adds $d^2$ constraints.)
\subsubsection{Examples}
\begin{itemize}
\item[1.] {\em Depolarization} -- The depolarizing channel, Eq.~(\ref{example-depol}), is already written in the form of Eq.~(\ref{chi}). We can simply read off the coefficients of the process matrix,
\begin{eqnarray}
\chi = \left(\begin{array}{cccc} 1-3p/4 & 0 & 0 & 0\\ 0 & p/4 & 0 & 0\\ 0 & 0 & p/4 & 0\\ 0 & 0 & 0 & p/4 \end{array}\right).\nonumber
\end{eqnarray}
\item[2.] {\em Dephasing} -- As for depolarization, Eq.~(\ref{example-dephasing}) is already written in the form of Eq.~(\ref{chi}). The process matrix is,
\begin{eqnarray}
\chi = \left(\begin{array}{cccc} 1-p/2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & p/2 \end{array}\right).\nonumber
\end{eqnarray}
\item[3.] {\em Rotations} -- A rotation by angle $\theta$ about axis $\hat{n}$ applied to a state $\rho$ is given by $\Lambda_\theta(\rho) = e^{-i\theta \hat{n}\cdot\hat{\sigma}/2}\rho e^{i\theta \hat{n}\cdot\hat{\sigma}/2}$. For simplicity, let us choose the rotation axis to be the z-axis. Expanding $e^{-i\theta \hat{n}\cdot\hat{\sigma}/2} = \cos(\theta/2)-i\hat{n}\cdot\hat{\sigma}\sin(\theta/2)$, and setting $\hat{n} = \hat{z}$, we get
\begin{eqnarray}
\chi = \frac{1}{2}\left(\begin{array}{cccc} 1+\cos(\theta) & 0 & 0 & i\sin(\theta)\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ -i\sin(\theta) & 0 & 0 & 1-\cos(\theta) \end{array}\right).\nonumber
\end{eqnarray}
\item[4.] {\em Spontaneous Emission / Amplitude Damping} -- Expanding the Kraus operators for amplitude damping, Eq.~(\ref{amp-damp-kraus-ops}), in terms of the Paulis, we find
\begin{eqnarray}
\chi = \frac{1}{2}\left(\begin{array}{cccc} (1+\sqrt{1-p})^2 & 0 & 0 & p/2\\ 0 & p/2 & -ip/2 & 0\\ 0 & ip/2 & p/2 & 0\\ p/2 & 0 & 0 & (1-\sqrt{1-p})^2 \end{array}\right).\nonumber
\end{eqnarray}
\end{itemize}
\subsection{Pauli transfer matrix}\label{ptm-section}
Another useful representation of a quantum map is the Pauli transfer matrix (PTM):
\begin{equation}
(R_\Lambda)_{ij} = \frac{1}{d}\Tr\{P_i \Lambda(P_j)\}. \label{PTM}
\end{equation}
The PTM has several convenient properties. Since $\Lambda(\rho)$ is a density matrix, it can be expanded in the $\{P_i\}$ with real coefficients in the interval [-1,1]. (This is true for any Hermitian operator.) Inspection of the right-hand side of Eq.~(\ref{PTM}) then shows that all entries of the PTM must also be real and in the interval [-1,1]. Also, as we show in Section~\ref{superoperators}, the PTM of a composite map is simply the matrix product of the individual PTMs. This makes it easy to evaluate the result of multiple gates acting in succession. The process matrix does not have this property. We note that $R$ is generally not symmetric.
In the PTM representation, the trace preservation (TP) constraint has a particularly simple form. Since $\Tr\{P_j\}=\delta_{0j}$, we must have $\Tr\{\Lambda(P_j)\} = \delta_{0j}$. But $\Tr\{\Lambda(P_j)\} = \Tr\{P_0\Lambda(P_j)\} = (R_\Lambda)_{0j}$. Therefore the TP constraint is $(R_\Lambda)_{0j} = \delta_{0j}$. In other words, the first row of the PTM is one and all zeros.
Compete positivity on the other hand is not expressed as simply in the PTM representation as it is for the $\chi$-matrix. We discuss this more in Sec.~\ref{physicality-section}.
Another sometimes useful condition that is conveniently expressible via the PTM is {\em unitality}. A unital map is one that takes the identity to the identity. Physically, this is a map that does not increase the purity of a state, i.e. it does not `unmix' a mixed state. Many quantum operations of interest, including unitary maps, are unital. One interesting exception is amplitude damping, as we will see in the examples below. Mathematically, unitality is expressed as $\Lambda(I) = I$. Inserting this into the definition of the PTM, Eq.~(\ref{PTM}), we find $(R_\Lambda)_{i0} = \Tr\{P_i \Lambda(I)\} = \Tr\{P_i I\} = \delta_{i0}$. Therefore the PTM for a unital map must have one and all zeros in its first column.
The PTM and process matrix representations are equivalent, and one can express one in terms of the other. Substituting Eq. (\ref{chi}) into Eq. (\ref{PTM}) with $\rho = P_j$ gives $(R_\Lambda)_{ij} = \frac{1}{d}\sum_{kl}\chi_{kl}\Tr\{P_i P_k P_j P_l\}$. (Note that $\rho = P_j$ is not a physical density operator: $\Tr\{P_j\} \neq 1$. But $\Lambda(P_j)$ is still formally defined.) The inverse transformation (expressing $\chi$ in terms of $R$) cannot be written as a compact analytical expression but can be implemented straightforwardly for any particular PTM using symbolic or numerical software.
\subsubsection{Examples}
Using the above definitions, we can write down the PTMs for the examples in Sec.~\ref{process-matrix-section}. (See that section for definitions of $p$ and $\theta$.)
\begin{itemize}
\item[1.] {\em Depolarization}
\begin{eqnarray}
R = \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1-p & 0 & 0\\ 0 & 0 & 1-p & 0\\ 0 & 0 & 0 & 1-p \end{array}\right).\nonumber
\end{eqnarray}
\item[2.] {\em Dephasing}
\begin{eqnarray}
R = \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1- p & 0 & 0\\ 0 & 0 & 1 - p & 0\\ 0 & 0 & 0 & 1 \end{array}\right).\nonumber
\end{eqnarray}
\item[3.] {\em Rotation by $\theta$ about z-axis}
\begin{eqnarray}
R = \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & \cos(\theta) & -\sin(\theta) & 0\\ 0 & \sin(\theta) & \cos(\theta) & 0\\ 0 & 0 & 0 & 1 \end{array}\right).\nonumber
\end{eqnarray}
\item[4.] {\em Spontaneous Emission / Amplitude Damping}
\begin{eqnarray}
R = \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & \sqrt{1-p} & 0 & 0\\ 0 & 0 & \sqrt{1-p} & 0\\ p & 0 & 0 & 1-p \end{array}\right).\nonumber
\end{eqnarray}
\end{itemize}
These examples illustrate the convenient structure of the PTM. Note in particular examples 3 and 4. For rotations, the PTM reduces to block-diagonal form consisting of the 1-dimensional identity matrix and a $3 \times 3$ rotation matrix. This rotation matrix determines the transformation of the 1-qubit Bloch vector under the rotation map. For amplitude damping, the form of the PTM is much simpler than that of the corresponding $\chi$-matrix and makes the non-unitality apparent.
From the PTMs, we can see by inspection that depolarization, dephasing, and rotations are all trace preserving and unital. Amplitude damping, however, is trace preserving but not unital (the first column of the PTM has a non-zero entry beyond the first entry). Physically, amplitude damping corresponds to spontaneous emission from the $|1\rangle$ to the $|0\rangle$ state. Therefore, any mixed state tends towards the pure state $|0\rangle$ at long enough times. (This is in fact the procedure typically used experimentally to ``reset" a qubit to the $|0\rangle$ state -- waiting long enough for spontaneous emission to occur.)
We note in passing that invertible quantum maps form a group. It is therefore natural to look for group representations, since this allows quantum maps to be represented as matrices, and map composition as the product of representation matrices. Such group representations are called superoperators. We saw above that PTMs in fact have the properties we just described. Therefore PTMs form a superoperator group representation, based on the Pauli matrices. We study this representation below, and show how it can be used to simplify the description of quantum maps.
\section{Superoperator formalism}\label{superoperators}
Manipulations with quantum maps can be conveniently carried out using the superoperator formalism~\cite{KitaevBook}. In this formalism, density operators $\rho$ on a Hilbert space of dimension $d$ are represented as vectors $|\rho\rb$ in {\em Hilbert-Schmidt} space of dimension $d^2$. Quantum operations (linear maps on density operators) are represented as matrices of dimension $d^2 \times d^2$. The Hilbert-Schmidt inner product is defined as $\lb A| B\rb = \Tr\{A^\dag B\}/d$, where $A$, $B$ are density operators. As we shall see, defining the inner product in this way enables map composition to be represented as matrix multiplication. It also allows us to make contact with measurements. The expected value $p$ of a POVM $E$ for a state $\rho$ is $p = \Tr\{E\rho\}$.\footnote{POVM stands for Positive Operator-Valued Measure, and denotes a formalism for describing measurements. A POVM $E$ is also called a measurement operator and is used to represent the average outcome of measuring a state $\rho$, through the equation $p = \Tr\{E\rho\}$. POVMs are hermitian, positive semidefinite, and satisfy the completeness relation $\sum_i E_i = I$, where $i$ enumerates the possible experimental outcomes. In this document we are concerned primarily with the special case of projective measurements, for example $E=|0\rangle\langle 0|$, and use the POVM formalism simply as a matter of notational convenience. For this reason, we do not discuss POVMs in any detail, and refer the reader to Ref.~\cite{NielsenChuang}, Ch. 2.2.6 for a complete treatment.}
Although superoperators can be defined relative to any basis of $\mathcal{H}_n$, we continue to find it convenient to use the Pauli basis introduced above. From this point on, we use rescaled Pauli operators $P_i \rightarrow P_i/\sqrt{d}$. This way the basis is properly normalized, and we avoid having to write factors of $d$ everywhere.
We represent states $\rho$ as state vectors $|\rho\rb = \sum_k |k\rb\lb k|\rho\rb$, with components
\begin{eqnarray}
\lb k | \rho \rb &=& \Tr\{P_k \rho\}, \label{superop-component}\\
P_k &\in& \{I/\sqrt{2},X/\sqrt{2},Y/\sqrt{2},Z/\sqrt{2}\}^{\otimes n}. \label{PauliBasis}
\end{eqnarray}
This is simply a restatement of the completeness of the Pauli basis: $\rho = \sum_k P_k \Tr\{P_k \rho\}$ for any $\rho$, which implies that the operator $\sum_k P_k \Tr\{P_k \,\Cdot\,\}$ equals the identity. In superoperator notation, this is $\sum_k |k\rb\lb k| = I$, from which Eq.~(\ref{superop-component}) follows.
Quantum operations are represented as matrices,
\begin{eqnarray}
R_\Lambda &=& \sum_{jk} |j\rb\lb j| R_\Lambda |k\rb\lb k|, \\
\lb j| R_\Lambda |k\rb &=& \Tr\{P_j \Lambda (P_k)\}.\label{superop-PTM}
\end{eqnarray}
Note that $\lb j| R_\Lambda |k\rb$ is just the PTM, $(R_\Lambda)_{jk}$. (The $1/d$ factor was suppressed by our rescaling.)
From these definitions, it is straightforward to derive the following consequences:
\begin{eqnarray}
|\Lambda(\rho)\rb &=& R_\Lambda|\rho\rb,\label{superop-vec}\\
R_{\Lambda_2 \circ \Lambda_1} &=& R_{\Lambda_2} R_{\Lambda_1}.\label{superop-prod}
\end{eqnarray}
\begin{proof} To prove Eq.~(\ref{superop-vec}), we first use the completeness relation and then Eq.~(\ref{superop-component}),
\begin{equation}
|\Lambda(\rho)\rb = \sum_k |k\rb\lb k|\Lambda(\rho)\rb = \sum_k |k\rb \Tr\{P_k \Lambda(\rho)\}.\nonumber
\end{equation}
Next, we expand $\rho = \sum_j P_j \Tr\{P_j \rho\}$ and insert into the right-hand-side of the above equation, giving
\begin{equation}
|\Lambda(\rho)\rb = \sum_{jk} |k\rb \Tr\{P_k \Lambda(P_j)\}\Tr\{P_j \rho\}.\nonumber
\end{equation}
Substituting $\Tr\{P_k \Lambda(P_j)\} = \lb k | R_\Lambda | j\rb$ (see Eq.~(\ref{superop-PTM})) and $\Tr\{P_j \rho\} = \lb j|\rho\rb$ (Eq.~(\ref{superop-component}) again) we obtain
\begin{equation}
|\Lambda(\rho)\rb = \sum_{jk} |k\rb \lb k|R_\Lambda|j\rb\lb j|\rho\rb.\nonumber
\end{equation}
Finally, we eliminate the sums over $j, k$ using the completeness relation, thus proving Eq.~(\ref{superop-vec}).
To prove Eq.~(\ref{superop-prod}), we act with $R_{\Lambda_2 \circ \Lambda_1}$ on an arbitrary state $|\rho\rb$, and use Eq.~(\ref{superop-vec}).
\begin{equation}
R_{\Lambda_2 \circ \Lambda_1}|\rho\rb = |\Lambda_2\left(\Lambda_1(\rho)\right)\rb.\nonumber
\end{equation}
Using Eq.~(\ref{superop-vec}) again, we find
\begin{equation}
|\Lambda_2\left(\Lambda_1(\rho)\right)\rb = R_{\Lambda_2}|\Lambda_1(\rho)\rb = R_{\Lambda_2}R_{\Lambda_1}|\rho\rb.\nonumber
\end{equation}
This proves Eq.~(\ref{superop-prod}).
\end{proof}
In this way, the composition of quantum maps has been expressed as matrix multiplication. We now have a simple way to write down experimental outcomes. Consider the following experiment: prepare state $\rho$, perform quantum operation $\Lambda$, measure the POVM $E$. Repeat many times and calculate the average measured value, $m$. The expected value, $p = \mathrm{E}(m)$, of the measurement $m$ is the quantum expectation value, which can be written as $p=\Tr\{E\Lambda(\rho)\} = \lb E | R_\Lambda |\rho\rb$.
\section{Physicality constraints}\label{physicality-section}
As mentioned in Sec.~\ref{quantops}, a physical map must be completely positive and trace preserving (CPTP), which is equivalent to the following two conditions: (1) for any initial state of the universe, the final state after application of the map must have nonnegative probabilities for measuring the eigenstate of any observable (in other words, the density matrix of the universe is always positive semidefinite), and (2) total probability must be conserved. The mathematical statement of these requirements is different for the process matrix and PTM representations of a quantum map, and will be used later on to write down the constraints for the optimization problem for the gate set.
\subsection{Process matrix representation}
We have seen that hermiticity and positive semidefiniteness of the $\chi$ matrix follow from the Kraus representation, which (as we stated without proof) is a consequence of complete positivity. Also, we used the Kraus representation to show that trace preservation implies the completeness condition, $\sum_{ij} \chi_{ij}P_j P_i = I$. Since the process matrix is so central to our work, it is interesting to see how these constraints follow directly from the CPTP requirement and Eq.~(\ref{chi}), which is a general way of writing any quantum map. We now show this.
Since the CP condition must hold for any initial state, we consider initial states of the form $\rho = |\phi\rangle\langle\phi|\otimes I$, where $|\phi\rangle$ is a pure state of the system, or qubit, and $I$ refers to the ``rest of the universe''. We can neglect reference to the ``rest of the universe" since it will be traced out of all formulas in which the map $\Lambda$ acts only on the system. Therefore we omit the $\otimes I$ below.
The probability of observing the pure state $|\psi\rangle$ of the system after application of the map $\Lambda$ to the initial state $|\phi\rangle$ is
\begin{equation}
p_\psi = \Tr\{\Lambda(|\phi\rangle\langle\phi|)|\psi\rangle\langle \psi|\} = \langle \psi | \Lambda(|\phi\rangle\langle\phi|) |\psi\rangle.\label{pk-proof2}
\end{equation}
Using Eq.~(\ref{chi}), we obtain
\begin{equation}
\Lambda(|\phi\rangle\langle\phi|) = \sum_{ij}\chi_{ij}P_i|\phi\rangle\langle \phi| P_j.
\end{equation}
Inserting this into Eq.~(\ref{pk-proof2}) we get
\begin{equation}
p_\psi = \sum_{ij}\chi_{ij}\langle \psi|P_i|\phi\rangle\langle \phi| P_j|\psi \rangle.\label{pk2-proof2}
\end{equation}
Since $p_\psi$ is real, we find upon setting $p_\psi^* = p_\psi$ and interchanging the dummy indices $i,j$ in Eq.~(\ref{pk2-proof2}), that $\chi^\dag = \chi$.
Next, let us define a vector $\vec{v}$ with components $v_j = \langle \phi| P_j|\psi \rangle$. Then Eq.~(\ref{pk2-proof2}) becomes $p_\psi = \vec{v}^\dag \chi \vec{v}$. Since the probability $p_\psi \geq 0$ and $|\psi\rangle$, $|\phi\rangle$ are arbitrary, the matrix $\chi$ must be positive semidefinite.
Finally, the completeness condition is proved just like for the Kraus representation. Trace preservation means $\Tr\{\Lambda(\rho)\} = \sum_{ij}\chi_{ij}\Tr\{P_jP_i\rho\} = \Tr\{\rho\}$. Since this must hold for any $\rho$, the completeness condition follows.
Although we have only proved that these conditions on $\chi$ are necessary for CPTP (we only considered a restricted set of initial states), it turns out they are also sufficient. We use this in the formulation of the constraints for the optimization problem in Ch.~\ref{derivation-chapter}.
It is also clear now that the Kraus representation can be viewed as a special case of a general $\chi$-representation, where the process matrix is the identity. Indeed, the CP constraint implies that $\chi$ must have real and positive eigenvalues. Starting from Eq.~(\ref{chi}) for an arbitrary (not necessarily Pauli) basis $P_i$, we can diagonalize $\chi$ and absorb the (real and positive) eigenvalues into the definition of the $K_i$, resulting in Eq.~(\ref{Kraus}).
\subsection{PTM representation} \label{ptm-physicality}
As we saw above, the TP constraint in the PTM representation is $(R_\Lambda)_{0j} = \delta_{0j}$. The CP constraint, however, cannot be expressed in terms of the PTM directly. Instead, one typically uses the fact~\cite{magesan_gate_2011} that the Choi-Jamiolkowski (CJ) matrix~\cite{choi_completely_1975,jamiolkowski_linear_1972} associated with a CP map is positive semidefinite. The CJ matrix is a density matrix - like representation of a quantum map that lives in a tensor product of two Hilbert-Schmidt spaces. For a quantum map $\Lambda$ it can be written in terms of the PTM as~\cite{Merkel2013}
\begin{equation}
\rho_\Lambda = \frac{1}{d^2}\sum_{i,j=1}^{d^2} (R_\Lambda)_{ij} P_j^T \otimes P_i. \label{choi1}
\end{equation}
In terms of the CJ matrix, the CP constraint is $\rho_\Lambda \succcurlyeq 0$, where the curly inequality denotes positive-semidefiniteness. See Ref.~\cite{magesan_gate_2011} for further discussion.
\chapter{Derivation of GST}\label{derivation-chapter}
In this chapter, we derive GST and describe some of the relevant data analysis techniques. In the following chapter, we present the experimental and data analysis protocols in step-by-step fashion. The reader interested in getting right to the implementation may want to skip ahead to that chapter.
GST can be viewed as a self-consistent extension of quantum process tomography (QPT), which itself grew out of quantum state tomography (QST). We therefore begin with a brief review of QST and QPT.
\section{Quantum state tomography}\label{qst-section}
Quantum state tomography~\cite{leibfried_experimental_1996} attempts to characterize an unknown state $\rho$ by measuring its components $\lb k |\rho\rb$, usually in the Pauli basis. It can be more convenient to use a different basis of measurement operators $E_j$, $j=1,\ldots,d^2$ that span the Hilbert-Schmidt space $\mathcal{B}(\mathcal{H}_d)$.\footnote{We here use the notation $\mathcal{H}_d$ instead of $\mathcal{H}_n$ as in Ch.~\ref{math-chapter} to keep the discussion momentarily more general. For $n$ qubits, $d=2^n$.} Generally, one measures the $d^2$ {\em probabilities}
\begin{equation}
p_j = \lb E_j|\rho\rb = \sum_k \lb E_j|k\rb\lb k|\rho\rb.\label{p-qst}
\end{equation}
The matrix $A_{jk}=\lb E_j|k\rb$ is assumed known, since the $\lb E_j|$ were chosen in advance by the experimenter, and the $|k\rb$ are Pauli basis vectors.
The probability $p_j$ can be estimated as the sample average $m_j= \sum_{i=1}^N m_{ij}/N $ of $N$ single-shot measurements of $E_j$, where $m_{ij} = 0$ or 1 is the outcome of the $i$-th measurement. Since $p_j$ is the true probability, the expected value of $m_j$ is ${\rm E}(m_j) = \sum_{i=1}^N {\rm E}(m_{ij})/N = p_j$, and the variance is ${\rm Var}(m_j) = p_j(1-p_j)/N$. The sample average $m_j$ approaches the true value $p_j$ as $N\rightarrow \infty$.
Since $A_{jk}$ is known, Eq.~(\ref{p-qst}) can be solved by matrix inversion, $|\hat{\rho}\rb = A^{-1}|m\rb$, where $|m\rb$ is the vector of measurements $m_j$ and $\hat{\rho}$ is the linear inversion estimator for $\rho$. The only requirement is that $E_j$ be chosen such that $A$ is invertible.
Estimation is particularly simple if $|E_j\rb =|j\rb$. Then no matrix inversion is necessary and $|\hat{\rho}\rb = |m\rb$. (Technically $m_0$ should be set to $1/\sqrt{d}$, so that $\Tr\{P_0 \rho\}=1$.)
\section{Quantum process tomography}\label{qpt-section}
Quantum process tomography~\cite{chuang_prescription_1997,poyatos_complete_1997} is similar to QST, but now the goal is to characterize a gate $G$ rather than a state. This is done by measuring the process matrix $\chi$ or the PTM, $\lb k|G|l\rb$, via measurements of the $d^4$ probabilities
\begin{equation}
p_{ij} = \lb E_j|G|\rho_i\rb.\label{p-qpt}
\end{equation}
Here the $E_j$ are as before, and the set of vectors $|\rho_i\rb$ is a complete basis for the Hilbert-Schmidt space. Since we must have $\Tr\{\rho\} = 1$ and the Pauli matrices are traceless, we cannot choose $|\rho_i\rb = |i\rb$. Therefore, each $|\rho_i\rb$ must be chosen as some linear combination of Pauli basis vectors that can be inverted to give the PTM. Inserting complete sets of states in Eq.~(\ref{p-qpt}),
\begin{equation}
p_{ij} = \sum_{kl} \lb E_j|k\rb\lb k|G|l\rb \lb l|\rho_i\rb.\label{p-qst2}
\end{equation}
As before, the vectors $\lb E_j|$, $|\rho_i\rb$ are specified by the experimenter, and so the matrices $\lb E_j|k\rb$, $ \lb l|\rho_i\rb$ are assumed known. We can arrange the product of these two matrices into a single matrix, $S_{i+(j-1)d^2;k+(l-1)d^2} = \lb E_j|k\rb \lb l|\rho_i\rb$. Also, we can vectorize the PTM, defining $r_{k+(l-1)d^2} = G_{kl}$. Then, Eq.~(\ref{p-qst2}) becomes
\begin{equation}
\vec{p} = S\vec{r}.
\end{equation}
Given a vector of measurements $\vec{m}$ of the probabilities $\vec{p}$, this equation can be inverted to yield an estimate of the PTM:
\begin{equation}
\hat{\vec{r}} = S^{-1}\vec{m}.
\end{equation}
In the discussion so far, we have assumed S is full-rank. ($\{E_j\}$, $\{\rho_i\}$ each form a complete basis, and S is $d^4 \times d^4$.) The estimation can be improved by including an overcomplete ($>d^2$) set of states and measurements. Then $S$ has more rows than columns and cannot be inverted, but $S^\dag S$ can be. In this case, we obtain the ordinary least-squares estimate as
\begin{equation}
\hat{\vec{r}} = (S^\dag S)^{-1}S^\dag\vec{m}. \label{qpt-ols-estimate}
\end{equation}
In practice QST and QPT estimates are obtained using maximum likelihood estimation rather than linear inversion, since this allows physicality constraints to be put in. We discuss MLE in the context of GST in Sec.~\ref{mle-section} below.
\section{Gate set tomography}
We have seen that QST and QPT assume the initial states and final measurements are known. In fact, these states must be prepared using quantum gates which themselves may be faulty: $\lb E_j| = \lb E|F_j$, $|\rho_i\rb = F'_i|\rho\rb$, where $\lb E|$, $|\rho\rb$ are some particular starting state and measurement the experimenter is able to implement. This results in a self-consistency problem. If the state preparation and measurement (SPAM) gates $\{F_j\}$, $\{F'_i\}$ are sufficiently faulty, the QST and QPT estimates will be faulty as well. GST solves this problem by including the SPAM gates self-consistently in the gate set to be estimated.
The goal of GST is to completely characterize an unknown set of gates and states~\cite{RBK2013},
\begin{equation}
\mathcal{G} = \{|\rho\rb,\lb E|,G_0, ..., G_K\}, \label{gateset}
\end{equation}
where $|\rho\rb \in \mathcal{H}_n$ is some initial state, $\lb E| \in \mathcal{H}_n^*$ is a 2-outcome POVM, and each $G_k \in \mathcal{B}(\mathcal{H}_n)$ is an n-qubit quantum operation in the Hilbert-Schmidt space $\mathcal{B}$ of linear operators on $\mathcal{H}_n$. $\mathcal{G}$ is called the {\em gate set}. By {\em characterization} we mean a procedure to estimate the complete process matrices or PTMs for $\mathcal{G}$ based on experimental data. Sometimes we would like to focus only on the gates and not the states and measurements, and we write the gate set as $\mathcal{G}=\{G_0,\ldots,G_K\}$. The distinction should be clear from the context.
As in QPT, the information needed to reconstruct each gate $G_k$ is contained in measurements of $\lb E_i |G_k|\rho_j\rb$, the gate of interest sandwiched between a complete set of states and POVMs. The experimental requirements for GST are therefore similar to QPT: the ability to measure expectation values of the form $p=\Tr\{E G(\rho)\} = \lb E | R_G |\rho\rb$ for the gate set, Eq. (\ref{gateset}).
In QPT, the set $\{\lb E_i|, |\rho_j\rb\}$ is assumed given, and this leads to incorrect estimates when there are systematic errors on the gates preparing the initial and final states. For self-consistency, we must treat these SPAM gates on the same footing as the original gates $\{G_k\}$. This is done by introducing the SPAM gates\footnote{Reference~\cite{RBK2013} refers to these gates as {\em fiducial} gates, hence the letter $F$. We follow reference~\cite{Merkel2013} and refer to them as SPAM gates in order to make contact with the notion of SPAM introduced earlier.} explicitly: $|\rho_j\rb = F_j|\rho\rb$ and $\lb E_i| = \lb E |F_i$. The SPAM gates $\mathcal{F} = \{F_1,\ldots,F_N\}$ are composed of gates in the gate set $\mathcal{G}$, and therefore the minimal $\mathcal{G}$ must include sufficient gates to create a complete set of states and measurements. In contrast to QPT, it is not possible in GST to characterize one gate at a time. Instead, GST estimates every gate in the gate set simultaneously.\footnote{More accurately, there is a minimal set of gates that must be estimated all at once, which includes the gates required for SPAM. Additional gates may be added one at a time.}
Because the SPAM gates are included in the gate set, GST requires that only a single initial state $\rho$ be prepared and a single measurement $E$ implemented. This is close to the experimental reality, where $\rho$ is usually the ground state of a qubit or set of qubits, and $E$ is a measurement in the $Z$-basis.
\section{Linear inversion GST}\label{derivation-lgst}
We now derive a simple, closed-form algorithm for obtaining self-consistent gate estimates. This algorithm was introduced by Robin Blume-Kohout, et al.~\cite{RBK2013} and was inspired by the Gram matrix methods of Cyril Stark~\cite{stark_self-consistent_2014}. The limitation of linear-inversion GST (or LGST) is that it does not constrain the estimates to be physical. There is often an unphysical gate set that is a better fit to the data than any physical one. As a result, LGST by itself is insufficient to provide gate quality metrics.
Nevertheless, LGST provides a convenient method for diagnosing gate errors, and also gives a good starting point for the constrained maximum likelihood estimation (MLE) approaches we discuss later. Also, in cases where the LGST estimate happens to be physical, it is identical to the one found by MLE.
We begin by identifying a set of SPAM gate strings $\mathcal{F} = \{F_1,...,F_{d^2}\}$ that, when applied to our unknown fixed state $|\rho\rb$ and measurement $\lb E|$, produce a complete set of initial states $|\rho_k\rb = F_k |\rho\rb$, and final states $\lb E_k| = \lb E| F_k$.\footnote{Note that in defining $\lb E_k|$ we use $F_k$ and not $F_k^\dag$ as one might expect. This is important in the analysis that follows.} In terms of the Pauli basis, Eq. (\ref{PauliBasis}), the components of these states are
\begin{eqnarray}
|\rho_k\rb_i &=& \lb i|F_k|\rho\rb = \Tr\{P_i F_k(\rho)\}, \label{rho-component}\\
\lb E_k|_i &=& \lb E | F_k |i\rb = \Tr\{F_k^\dag(E) P_i\}. \label{E-component}
\end{eqnarray}
The SPAM gates must be composed of members from our gate set, $\mathcal{G}$.\footnote{If $\mathcal{G}$ is insufficient to produce a complete set of states and measurements in this way, we must add gates to $\mathcal{G}$ until this is possible. In practical applications, one is interested in characterizing a complete set of gates, so this is not a problem.} (For a single qubit, the completeness requirement means that $\{F_k |\rho\rb\}_{k=1}^{d^2}$ must span the Bloch sphere, and similarly for $\{\lb E | F_k\}$.) In general, SPAM gates have the form~\cite{RBK2013},
\begin{equation}
F_k = G_{f_{k1}}\circ G_{f_{k2}} \circ ... G_{f_{kL_k}}, \label{SPAMgates}
\end{equation}
where $\{f_{kl}\}$ are indices labeling the gates of $\mathcal{G} = \{G_0,...,G_K\}$, and $L_k$ is the length of the k-th SPAM gate string.
\subsection{Example gate sets}\label{example}
It is useful to have concrete examples of gate sets. Here we consider two of the simplest, both for a single qubit. We will continue to use both examples below. (We use the notation $A_\theta$ to denote a rotation by angle $\theta$ about axis $A$, for example $X_{\pi/2}$ is a $\pi/2$ rotation about the $X$-axis of the Bloch sphere.)
\begin{itemize}
\item[1.] $\mathcal{G} = \{\{\},X_{\pi/2},Y_{\pi/2}\} = \{G_0,G_1,G_2\}$. The symbol $\{\}$ denotes the ``null'' gate -- do nothing for no time. (We will always choose $G_0$ to be the null gate for reasons that will become clear later.) One choice of SPAM gates is $\mathcal{F} = \{\{\},X_{\pi/2},Y_{\pi/2},X_{\pi/2}\circ X_{\pi/2} \} = \{G_0,G_1,G_2,G_1\circ G_1\}$. It is easy to see that the set of states $F_k |\rho\rb$ (measurements $\lb E| F_k)$ spans the Bloch sphere for any pure state $|\rho\rb$ (measurement $\lb E|$).
\item[2.] $\mathcal{G} = \{\{\},X_{\pi/2},Y_{\pi/2},X_{\pi}\} = \{G_0,G_1,G_2,G_3\}$. Here the SPAM gates are $\mathcal{F} = \{\{\},X_{\pi/2},Y_{\pi/2},X_{\pi} \} = \mathcal{G}$. This includes one more gate in the gate set compared to Example 1, but now the SPAM gates do not contain products of gates from the gate set.
\end{itemize}
In Example 1, we chose $F_4 = X_{\pi/2}\circ X_{\pi/2}$ rather than $X_\pi$ so that we would not need to add the additional gate $X_\pi$ to the gate set $\mathcal{G}$. The choice $F_4 = G_1\circ G_1$ is more efficient from an experimental standpoint. This can be a consideration if the experimental resources required to implement GST for an additional gate (16 additional experiments) are significant. If this is not a consideration, it turns out to be a disadvantage for the purposes of data analysis to have multiples of the same gate within a single SPAM gate. This is because the order (the polynomial power in the optimization parameters) of the MLE objective function is proportional to the number of gates in the experiment. (This is not so much of an issue if an LGST analysis is sufficient.) Therefore, the choice of gate set depends on the constraints of the experimental implementation as well as the post-processing requirements.
\subsection{Gram matrix and A, B matrices}\label{gram-ab-section}
In GST, we work with expectation values,
\begin{equation}
p_{ijk} = \lb E | F_i G_k F_j |\rho\rb, \label{pijk-0}
\end{equation}
where $F_i,F_j \in \mathcal{F}$ and $G_k \in \mathcal{G}$. These quantities correspond to measurements that can (in principle) be carried out in the lab. Inserting a complete set of states on each side of $G_k$ in Eq.~(\ref{pijk-0}), we obtain
\begin{equation}
p_{ijk} = \sum_{rs}\lb E | F_i|r\rb\lb r| G_k |s\rb\lb s| F_j |\rho\rb \equiv \sum_{rs}A_{ir}(G_k)_{rs}B_{sj}. \label{pijk}
\end{equation}
This defines a set of Pauli transfer matrices, as follows. $(G_k)_{rs}=\lb r| G_k |s\rb$ is the $rs$-component of the PTM for the gate $G_k$. $A_{ir}=\lb E | F_i|r\rb$ and $B_{sj}=\lb s| F_j |\rho\rb$ are, respectively, the $r$-component of $\lb E_i|$ and the $s$-component of $|\rho_j\rb$ (See Eqs~(\ref{rho-component}),~(\ref{E-component})). It is useful to write $A$ and $B$ in component-free notation as
\begin{eqnarray}
A &=& \sum_i |i\rb\lb E|F_i, \\
B &=& \sum_j F_j|\rho\rb\lb j|.
\end{eqnarray}
One can easily verify that $A_{ir}=\lb E | F_i|r\rb$ and $B_{sj}=\lb s| F_j |\rho\rb$ as required. We then find that, according to Eq.~(\ref{pijk}), $p_{ijk} = (AG_kB)_{ij}$.
Experimentally measuring the values $p_{ijk}$ in Eq. (\ref{pijk}) amounts to measuring the ($ij$) components of the matrix
\begin{equation}
\tilde{G}_k = AG_k B.
\end{equation}
Since $G_0 = \{\}$ (the null gate), the $k=0$ experiment gives
\begin{equation}
g = \tilde{G}_0 = AB. \label{gram}
\end{equation}
The matrix $g$ is the Gram matrix of the $\{F_i\}$ in the Pauli basis.\footnote{The importance of the null gate is now clear. In order for Eq. (\ref{gram}) to hold, \{\} must be an {\em exact} identity. Anything else, e.g. an idle gate, would introduce an error term between $A$ and $B$.} We observe:
\begin{equation}
g^{-1}\tilde{G}_k = B^{-1}A^{-1}AG_kB = B^{-1}G_kB.\label{g-inv-G}
\end{equation}
Thus,
\begin{equation}
\hat{G}_k = g^{-1}\tilde{G}_k
\end{equation}
is an estimate of the gate set, up to a similarity transformation by the unobservable (see next section) matrix $B$.
It can happen that the SPAM gates that are implemented experimentally are not linearly independent within the bounds of sampling error. Then $\{|\rho_j\rb\}$ (and $\{\lb E_i|\}$) will not form a basis, and $g$ will have small or zero eigenvalues -- it will not be invertible. If this happens, the experimentalist must adjust the SPAM gates until $g$ is invertible. We will discuss this further in the next chapter.
\subsection{Gauge freedom}\label{gauge-freedom-section}
We now show that the matrix $B$ in Eq.~(\ref{g-inv-G}) is unobservable, as mentioned above. Recall that we assume no knowledge of $\mathcal{G}$, which includes $\lb E|$ and $|\rho\rb$. All experiments we consider are of the form $\lb E | G_{i_1} G_{i_2} ... G_{i_L}|\rho\rb$. (This includes measurements of $p_{ijk}$, since the $F_i$ are composed of elements of $\mathcal{G}$.) Transforming $\lb E | \rightarrow \lb E' | = \lb E| B$, $|\rho\rb \rightarrow |\rho'\rb = B^{-1}|\rho\rb$, $G_k \rightarrow G'_k = B^{-1}G_kB$, we find for a general expectation value: $\lb E' | G'_{i_1} G'_{i_2} ... G'_{i_L}|\rho'\rb = \lb E | G_{i_1} G_{i_2} ... G_{i_L}|\rho\rb$. Therefore, gates estimated by GST have a {\em gauge freedom} -- similarity transformation by a matrix, $B$. Any two sets of gates related to each other by a gauge transformation will describe a given set of measurements equally well (given that the states and measurements are transformed in the same way). They will be the same distance away from the actual gates (according to any distance metric) and, if they are physical, will have the same fidelity relative to the actual gate set.
Now consider the vectors
\begin{eqnarray}
|\tilde{\rho}\rb &\equiv& A|\rho\rb = \sum_i |i\rb\lb E|F_i|\rho\rb, \label{rhotilde}\\
\lb \tilde{E}| &\equiv& \lb E | B = \sum_j \lb E| F_j |\rho\rb \lb j|. \label{Etilde}
\end{eqnarray}
These vectors are componentwise identical, and the components are measurable quantities: $\lb i |\tilde{\rho}\rb = \lb \tilde{E} | i\rb = \lb E|F_i|\rho\rb$. They provide a way to construct a gate set consistent with our measurements, up to the gauge freedom $B$. Let
\begin{eqnarray}
|\hat{\rho}\rb &\equiv& g^{-1}|\tilde{\rho}\rb = B^{-1}|\rho\rb, \label{rhohat}\\
\lb\hat{E}| &\equiv& \lb\tilde{E}| = \lb E|B.\label{mhat}
\end{eqnarray}
As we saw before,
\begin{equation}
\hat{G}_k \equiv g^{-1}\tilde{G}_k = B^{-1}G_kB.\label{Ghat}
\end{equation}
The gate set $\hat{\mathcal{G}} = \{|\hat{\rho}\rb,\lb\hat{E}|,\{\hat{G}_k\}\}$ consists entirely of measurable quantities: $\lb E | F_i G_k F_j |\rho\rb$ for $g$ and $\tilde{G}_k$, $\lb E| F_i |\rho\rb$ for $|\tilde{\rho}\rb$ and $\lb\tilde{E}|$. We can therefore construct it from experimental data. By gauge freedom, measurements in $\hat{\mathcal{G}}$ are equivalent to those in $\mathcal{G}$:
\begin{equation}
\lb\hat{E}|\prod_k \hat{G}_{i_k}|\hat{\rho}\rb = \lb E|B\left(\prod_k B^{-1}G_{i_k}B\right)B^{-1}|\rho\rb = \lb E|\prod_k G_{i_k}|\rho\rb.
\end{equation}
Hence, $\hat{\mathcal{G}}$ is indistinguishable from the true gate set $\mathcal{G}$.
However, since $B$ is never the identity, $\hat{\mathcal{G}}$ is not equal to $\mathcal{G}$. Since $B$ cannot be measured, the best we can do is find an estimate of $B$ that brings us as close as possible to some ``target'' gate set. The target may be chosen arbitrarily in accordance with the gauge freedom. Choosing it to be the {\em intended} experimental gate set allows us to compare the actual gates to ideal ones.
\subsection{Gauge optimization}\label{gauge-opt-section}
As we just saw, the gauge matrix $B_{ij} = \lb i| F_j |\rho\rb = \Tr\{P_i F_j(\rho)\}$ cannot be the identity. (This would require $F_j(\rho) = P_j \forall j$, which is impossible for a physical state.) Therefore, the gate set $\hat{\mathcal{G}}$ estimated by LGST is necessarily different from the true gate set $\mathcal{G}$. Unfortunately, experiments have no access to the gauge matrix $B$. But, since experiments look the same regardless of the gauge, we are free to choose a gauge that suits our purposes. This choice makes no practical difference -- a quantum computation in any gauge is indistinguishable from the same computation in another gauge.
Experimental qubits these days are quite good -- randomized benchmarking gate fidelities in excess of 99\% are routinely reported~\cite{barends_superconducting_2014}, In this case we know a priori that the measured gates will differ from an ideal (or target) set of gates by some very small error. A reasonable protocol then is to select the gauge such that the estimated gate set is as close as possible to this target gate set. The remaining differences are attributed to systematic gate error and sampling error.
To define closeness of quantum gates, we choose a suitable matrix norm on superoperators. For our purposes, the trace norm is sufficient~\cite{RBK2013}. Given a target set of gates $\mathcal{T} = \{|\tau\rb,\lb \mu |, \{T_k\}\}$ and our LGST estimate $\hat{\mathcal{G}} = \{|\hat{\rho}\rb,\lb\hat{E}|,\{\hat{G}_k\}\}$, we find the matrix $\hat{B}^*$ that minimizes the RMS discrepancy (trace norm):
\begin{equation}
\hat{B}^* = \argmin_{\hat{B}} \sum_{k=1}^{K+1} \Tr\left\{\left(\hat{G}_k - \hat{B}^{-1} T_k \hat{B}\right)^T \left(\hat{G}_k - \hat{B}^{-1} T_k \hat{B}\right)\right\}. \label{gaugeopt}
\end{equation}
The index $k$ runs over the entire gate set, and we also include the ``gate" $G_{K+1} \equiv |\rho\rb\lb E|$. This allows the states to be fitted as well.
Our final LGST estimate is then
\begin{eqnarray}
\hat{G}_k^* &=& \hat{B}^* \hat{G}_k (\hat{B}^*)^{-1}, \label{lgst-final1}\\
|\hat{\rho}^*\rb &=& \hat{B}^* |\hat{\rho}\rb, \label{lgst-final2}\\
\lb\hat{E}^*| &=& \lb\hat{E}|(\hat{B}^*)^{-1}. \label{lgst-final3}
\end{eqnarray}
This is the closest gate set to the target that is consistent with experiments as well as the allowable gauge freedom.
\subsection{Discussion}\label{discussion-section}
A few points are worth mentioning. First, we note that $\det(\hat{B})$ can be fixed by a requirement on the normalization of $|\hat{\rho}^*\rb$. We do not do this, but rather let $\hat{B}$ vary over the entire group of real, invertible matrices, $GL(d^2,\mathcal{R})$, letting the optimizer find the (not necessarily physical) gate set consistent with the data. In addition, Eq. (\ref{gaugeopt}) contains a nonlinear objective function ($B$ and $B^{-1}$ each appear twice), and therefore has multiple minima. As a result, the gauge optimization problem requires a starting point to be specified. This may lead to some unexpected consequences, which we now describe.
Besides having multiple minima, the global minimum of the objective function, Eq. (\ref{gaugeopt}) is not unique. This indeterminacy is generic in quantum tomography, and appears in QST and QPT as well. In particular, we cannot distinguish $\lb U^\dag(E)| G |\rho\rb$ from $\lb E| G |U(\rho)\rb$ for any gate $G$, where $U$ is an arbitrary operation that commutes with $G$. (Depolarizing noise, for example.) The two sets $\{\lb U^\dag(E)| , |\rho\rb\}$, $\{\lb E|, |U(\rho)\rb\}$ are generally different. This indeterminacy can be recast as a type of gauge freedom, but in this case fixing the gauge provides no additional information about the true state and measurement. This has consequences for our analysis: an initialization error $|\mathcal{E}(\rho)\rb$ (e.g., a hot qubit) cannot be distinguished from a faulty measurement $\lb \mathcal{E}(E)|$ (e.g., dark counts).
Typically, the starting point for numerical optimization of Eq. (\ref{gaugeopt}) is taken to be the target gauge matrix, defined as $\hat{B}^{(0)}_{ij} = \lb i| S_j |\tau\rb$. ($S_j$ is the target for $F_j$, composed of gates in $\mathcal{T}$ in the same way that $F_j$ is composed of gates in $\mathcal{G}$.) This starting point depends on the target $|\tau\rb$ for $|\rho\rb$ but does not depend on the target $\lb \mu|$ for $\lb E|$. As a result, gauge optimization will always produce $|\hat{\rho}^*\rb \approx |\tau\rb$, and will attribute any initialization error to error in $\lb \hat{E}^*|$, regardless of whether the error was actually in $\lb E|$ or $|\rho\rb$. This must be kept in mind when interpreting the results of GST for estimating states and measurements. The gate estimates themselves are not affected by this particular gauge freedom.
\section{Maximum likelihood estimation}\label{mle-section}
Linear inversion typically does not produce estimates that are physical (it is not constrained to do so). In contrast to QPT (see Sec.~(\ref{qpt-section}), Eq.~(\ref{qpt-ols-estimate})), it is also incapable of working with overcomplete data, which could be used to improve the estimate. MLE solves both of these problems. (Another approach is to find the closest physical gate set to the LGST estimate, but this is not optimal~\cite{RBK2013} and also cannot be extended to overcomplete data.) Since the objective function for MLE is nonlinear, the linear inversion estimate is still useful as a starting point for the optimization algorithm.\footnote{Standardized optimization methods exist for convex (linear, quadratic) objective functions, which have a single minimum~\cite{BoydVandenberghe}. When the objective is nonlinear, with many local minima, there are no existing techniques that are guaranteed to find the global minimum. We must therefore use local optimization techniques and rely on a good starting point to put us close to the global minimum.}
Our goal is to estimate the true probabilities $p_{ijk}$ in Eq. (\ref{pijk-0}) based on a set of measurements $m_{ijk}$, subject to physicality constraints. MLE is the natural way to do this: the best estimate is found by fitting to experimental data a theoretical model of the probability of obtaining that data. The resulting estimates $\hat{p}_{ijk}$ are used to find the most likely set of gates $\hat{\mathcal{G}}$ that produced the data.
We begin by parameterizing the $4 \times 4$ estimate matrices $\hat{G}$, as well as the state and measurement estimates $\hat{\rho}$, $\hat{E}$, in terms of a vector of parameters, $\vec{t}$: $\hat{G}(\vec{t})$, $|\hat{\rho}(\vec{t})\rb$, $\hat{E}(\vec{t})$. (Although we have written the same vector $\vec{t}$ as the argument in each matrix, the different gates and states depend on non-overlapping subsets of parameters in $\vec{t}$.) There are several possibilities for the parameterization, which we discuss in detail later. The important point for now is that the parameterization is either linear or quadratic in $\vec{t}$: each matrix element of $G(\vec{t})$, $|\rho(\vec{t})\rb$, $E(\vec{t})$ is a polynomial of order 1 or 2 in the parameters $\{t_i\}$. There are $d^4=16$ parameters for each gate $G_k$, $k=1,\ldots,K$ ($k=0$ is not parameterized), and $d^2=4$ parameters each for $E$ and $\rho$.
A number of constraints reduce the total number of independent parameters, as we discuss later. The minimal case of GST on one qubit requires a gate set $\mathcal{G}$ with $K=2$. This gives $K d^4 + 2 d^2 = 40$ parameters. Equality constraints (trace preservation requirement) reduce the number of parameters by 4 per gate, and by 1 for $\rho$. Thus we are left with $31$ free parameters in the minimal instance of GST.
Putting everything together, the estimates $\hat{p}_{ijk}$ are written in terms of the parameter vector $\vec{t}$, as
\begin{equation}
\hat{p}_{ijk}(\vec{t}) = \lb \hat{E}(\vec{t})|\hat{F}_i(\vec{t})\hat{G}_k(\vec{t})\hat{F}_j(\vec{t}) |\hat{\rho}(\vec{t})\rb. \label{phat}
\end{equation}
Depending on the parameterization choice (discussed below), each of the gates and states in Eq.~(\ref{phat}) is either a linear or quadratic function of its parameters. The estimator $\hat{p}_{ijk}$ is therefore a homogeneous function of order 5 (linear parameterization) or 10 (quadratic parameterization).
MLE proceeds by finding the set of parameters $\vec{t}$ that minimizes an objective function. The objective, or likelihood function, is the probability distribution we assume produced the data. The most general likelihood function for the experiment we have described above is~\cite{RBK2013}
\begin{equation}
L(\hat{\mathcal{G}}) = \prod_{ijk} (\hat{p}_{ijk})^{m_{ijk}}(1-\hat{p}_{ijk})^{1-m_{ijk}}.
\end{equation}
Although several workers~\cite{RBK2013,Yudan} have successfully used this likelihood function with their experimental data, it is usually more convenient to use a simpler form. To this end, we invoke the central limit theorem to rewrite the likelihood as a normal distribution,
\begin{equation}
L(\hat{\mathcal{G}}) = \prod_{ijk}\exp\left[-\left(m_{ijk}-\hat{p}_{ijk}\right)^2/\sigma_{ijk}^2\right],
\end{equation}
where $\sigma^2 = p(1-p)/n$ is the sampling variance in the measurement $m$.\footnote{When performing numerical optimization, we typically approximate $p \approx m$ rather than $p \approx \hat{p}$ in the sampling variance, writing $\sigma^2 = m(1-m)/n$. This keeps the objective function from blowing up.}
Because the logarithm function is monotonic, maximizing the likelihood $L$ is equivalent to minimizing the negative log-likelihood $l = -\log L$. With a normal likelihood function, the problem reduces to weighted least-squares:
\begin{equation}
\mathrm{Minimize}:\;\;\; l(\hat{\mathcal{G}}) = \sum_{ijk}\left(m_{ijk}-\hat{p}_{ijk}(\vec{t})\right)^2/\sigma_{ijk}^2.
\end{equation}
Or, rewriting the estimators $\hat{p}_{ijk}$ in terms of the gate estimates, Eq. (\ref{phat}), the problem we need to solve is minimization over the parameters $\vec{t}$ of
\begin{equation}
l(\hat{\mathcal{G}}) = \sum_{ijk}\left(m_{ijk}-\lb \hat{E}(\vec{t})|\hat{F}_i(\vec{t})\hat{G}_k(\vec{t})\hat{F}_j(\vec{t}) |\hat{\rho}(\vec{t})\rb\right)^2/\sigma_{ijk}^2. \label{wlsq}
\end{equation}
As we saw above, the estimator $\hat{p}_{ijk}$ is of order 5 or 10 in the parameters. Therefore this objective function is 10-th to 20-th order in the parameters, making this a non-convex optimization problem.
It remains to express $\hat{p}_{ijk}$ in terms of the parameters $\vec{t}$, so that we have an explicit form for $ l(\hat{\mathcal{G}})$ in Eq. (\ref{wlsq}) in terms of $\vec{t}$. There are two parameterizations for gates that are commonly used: (1) The Pauli Process ($\chi$) Matrix representation and (2) the Pauli Transfer Matrix (PTM) representation. Each has its advantages, so we present both.
\subsection{Process matrix optimization problem}\label{process-matrix-optimization-section}
The process matrix $\chi_G$ for a gate $G$ is defined in terms of the gate's action on an arbitrary state $\rho$ to produce a new state $G(\rho)$ as in Section~\ref{process-matrix-section},
\begin{equation}
G(\rho) = \sum_{i,j=1}^{d^2} (\chi_G)_{ij} P_i \rho P_j.
\end{equation}
The gate action is expressed in terms of Pauli operators (see Section ~\ref{quantops}) $P_i$, $P_j$ acting on the state.
The $\chi$ matrix must be Hermitian positive semidefinite. This requirement follows from the Hermiticity of density matrices, $\rho^\dag = \rho$, $G(\rho)^\dag = G(\rho)$, and from the requirement of positive probabilities: $p_i = \Tr\{|i\rangle\langle i| \rho\} \geq 0$ for any pure state $|i\rangle$ and any density matrix $\rho$ (see Sec.~(\ref{physicality-section})). Any Hermitian positive semidefinite matrix can be written in terms of a Cholesky decomposition, $\chi = T^\dag T$, where $T$ is a lower-diagonal complex matrix with reals on the diagonal:
\begin{eqnarray}
T &=& \left(\begin{array}{cccc} t_1 & 0 & 0 & 0\\
t_5+i t_6 & t_2 & 0 & 0\\
t_{11}+i t_{12} & t_7+i t_8 & t_3 & 0\\
t_{15}+i t_{16} &t_{13}+i t_{14} & t_9+i t_{10} &t_4\end{array}\right). \label{T}
\end{eqnarray}
In addition, there will be a constraint on $\chi$ due to the requirement of trace preservation of the state after the application of the gate: $\Tr\{G(\rho)\} = \Tr\{\rho\}$. This leads to the completeness condition, $\sum_{ij}\chi_{ij}P_jP_i = I$, which is equivalent to the following four equations.
\begin{equation}
\sum_{ij}\chi_{ij}\Tr\{P_iP_kP_j\} = \delta_{k0},\;\;\; k=1,\ldots,4.
\end{equation}
This constraint reduces the number of free parameters for $\chi$ from 16 to 12. In practice, we leave the 16 parameters and introduce the constraint as an equality constraint in the numerical optimizer.
We now discuss parameterization of $E$ and $\rho$. Both are Hermitian positive semidefinite, so we may parameterize them via Cholesky decomposition in the same way as $\chi$. (Note only that they are $2 \times 2$ rather than $4 \times 4$ matrices, so that each will have 4 parameters.) The state $\rho$ has unit trace, and the measurement $E$ must be such that $I-E$ is also positive semidefinite. These conditions introduce additional constraints for optimization.
We can now write down an explicit form for the weighted least-squares objective function, Eq. (\ref{wlsq}), in terms of the parameter vector, $\vec{t}$. We expand the estimator $\hat{p}_{ijk}$ in terms of the $\chi$ matrices for its constituent gates using Eq. (\ref{chi}),
\begin{eqnarray}
\hat{p}_{ijk} &=& \lb \hat{E}(\vec{t})|\hat{F}_i(\vec{t})\hat{G}_k(\vec{t})\hat{F}_j(\vec{t}) |\hat{\rho}(\vec{t})\rb \nonumber \\
&=& \Tr\{\hat{E} \hat{F}_i(\hat{G}_k(\hat{F}_j(\rho)))\} \nonumber \\
&=& \sum_{mnrstu} (\chi_{F_i})_{tu}(\chi_{G_k})_{rs}(\chi_{F_j})_{mn} \Tr\{E P_t P_r P_m \rho P_n P_s P_u\}. \label{pchi}
\end{eqnarray}
Remember that each $\chi$ matrix as well as $\rho$ and $E$ is written as $T^\dag T$ in terms of its own set of parameters (the $\chi$s are $4 \times 4$ and $\rho$ and $E$ are $2\times 2$), and therefore $\hat{p}_{ijk}$ is generally a homogeneous function of order 10 in these parameters (it consists of 3 $\chi$-matrices plus $\rho$ and $E$). Also, it may be the case that some of the $F_i, F_j$ are composed of more than one gate $G \in \mathcal{G}$. In this case $\chi_{F_i}$ and $\chi_{F_j}$ can be decomposed further into process matrices for the gates in $F_i, F_j$, with corresponding additional Pauli matrix terms appearing inside the trace. This makes $\hat{p}_{ijk}$ a correspondingly higher order polynomial. For this reason we would like to avoid defining SPAM gates as combinations of gates in the gate set.
\subsubsection{Problem statement}
We can now state the optimization problem as follows.
\begin{eqnarray}
\mathrm{Minimize:}&&\;\;\; \sum_{ijk}\left(m_{ijk}-\sum_{mnrstu} (\chi_{F_i})_{tu}(\chi_{G_k})_{rs}(\chi_{F_j})_{mn} \Tr\{E P_t P_r P_m \rho P_n P_s P_u\}\right)^2/\sigma_{ijk}^2 \nonumber \\
\mathrm{Subject\;to:}&&\;\;\; \sum_{mn}(\chi_G)_{mn}\Tr\{P_m P_r P_n\} - \delta_{0r} = 0, \;\;\; \forall\; G \in \mathcal{G} \nonumber \\
&&\;\;\; \Tr\{\rho\} = 1, \nonumber \\
&&\;\;\; I - E \succcurlyeq 0 \nonumber
\end{eqnarray}
The wavy inequality denotes positive semidefiniteness. For each $G\in\mathcal{G}$ there is a $4\times 4$ matrix $\chi_G = T_G^\dag T_G$, where $T_G$ is a lower diagonal complex matrix with real entries on the diagonal, Eq. (\ref{T}). There are 16 free parameters $t_i$ for each $G \in \mathcal{G}$, and each matrix element of $\chi_G$ is a 2nd order polynomial in the $\{t_i\}$.
The matrices $\chi_F$ must be expressed in terms of matrices $\chi_G$. In the second example in Sec.~\ref{example} above, this is trivial since $\mathcal{F} = \mathcal{G}$. In the first example the only non-trivial case is $F_4 = G_1\circ G_1$. The simplest way to find $\chi_{F_4}$ is to calculate the PTM for $F_4$ by multiplying the PTMs for $G_1$, and then transforming back to the $\chi$ representation. This can be implemented with a simple numerical routine.
$E$ and $\rho$ are parameterized similarly to $\chi$ as $T^\dag T$, only they are $2 \times 2$ matrices. Each therefore contains 4 parameters.
Possible starting points are the target gate set or the LGST estimate (rather, the closest physical gate set to the LGST estimate). Typically LGST provides a better starting point than the target gate set.
\subsection{Pauli transfer matrix optimization problem}\label{ptm-optimization-section}
Next we discuss parameterization in terms of Pauli Transfer Matrices (PTMs). As we saw in Ch.~\ref{math-chapter}, the composition of gates is represented as matrix multiplication of PTMs. This avoids the cumbersome trace terms appearing in equations such as Eq. (\ref{pchi}). In addition, the PTM for each gate may be parameterized linearly rather than quadratically as we did for $\chi$. This reduces the order of the objective function by a factor of 2. The drawback is that the positivity constraint -- corresponding to positive-definiteness of $\chi$ that we imposed above via Cholesky decomposition -- has a more complicated structure in terms of PTMs. It is expressed by the requirement of positive semidefiniteness of the so-called Choi-Jamiolkowski (CJ) matrix representing the quantum map, see Sec.~\ref{ptm-physicality}. Imposing this type of constraint without Cholesky decomposition is difficult to do with standard nonlinear optimization techniques, but can be done using semidefinite programming (SDP)~\cite{Chow2012SI} if the objective function can be made quadratic.
The Pauli Transfer Matrix $R_G$ for a gate $G$ is defined in terms of the gate's action on Pauli matrices (see Section~\ref{ptm-section}),
\begin{equation}
(R_G)_{ij} = \Tr\{P_i G(P_j)\}. \label{PTM-R}
\end{equation}
The PTM contains the same information about the map as does $\chi$. It is a $d^2 \times d^2$ real-valued matrix with elements restricted to the interval $R_{ij}\in[-1,1]$, and is generally not symmetric. Like $\chi$, the PTM has 16 parameters. The trace-preserving condition is $R_{0j} = \delta_{0j}$. These 4 equations reduce the number of parameters to 12.
In the PTM representation, density matrices $\rho$ in Hilbert space are written as vectors in Hilbert-Schmidt space and denoted as $|\rho\rb$. The matrix elements (entries) of a general vector $|\rho\rb$ are defined as
\begin{equation}
\rho_i = \lb i | \rho\rb = \Tr\{P_i \rho\}. \label{rhovec}
\end{equation}
The left-hand term in the equality is the $i$-th component of the $4 \times 1$ vector $|\rho\rb$. The middle term is this same component written in Dirac notation. In the right-hand term, $\rho$ is the density matrix in standard $2 \times 2$ representation, and $P_i$ is a Pauli matrix.
Using these definitions, the state $G(\rho)$ after application of the gate $G$ can be written as a vector $|G(\rho)\rb$ in Hilbert-Schmidt space resulting from matrix multiplication by $R_G$:
\begin{equation}
|G(\rho)\rb = R_G|\rho\rb
\end{equation}
The standard density matrix form of $G(\rho)$ can be recovered from the vector $|G(\rho)\rb$ using the definition of matrix elements, Eq. (\ref{rhovec}),
\begin{equation}
\Tr\{P_i G(\rho)\} = \lb i | G(\rho)\rb,
\end{equation}
and the expansion of any density matrix in terms of Pauli matrices,
\begin{equation}
G(\rho) = \sum_{i=1}^4 \Tr\{P_i G(\rho)\} P_i.
\end{equation}
In Eq. (\ref{wlsq}), the weighted least-squares objective function has already been written in terms of PTMs. This was implicit in the superoperator notation we used to derive that equation. In terms of the $R$-matrix notation for PTMs, we rewrite Eq. (\ref{wlsq}) as
\begin{eqnarray}
l(\hat{\mathcal{G}}) = \sum_{ijk}\left(m_{ijk}-\lb \hat{E}(\vec{t})|\hat{R}_{F_i}(\vec{t})\hat{R}_{G_k}(\vec{t})\hat{R}_{F_j}(\vec{t}) |\hat{\rho}(\vec{t})\rb\right)^2/\sigma_{ijk}^2. \label{wlsq-R}
\end{eqnarray}
We can parameterize each matrix $\hat{R}$ linearly in terms of $\vec{t}$. This means each $R_{ij} = t_s$ for some index $s$. The term in double-brackets, $\lb\ldots\rb$, in Eq. (\ref{wlsq-R}) is a scalar given by applying the indicated matrix multiplications of $\hat{R}$-matrices to the vector $|\hat{\rho}\rb$, and then scalar multiplying by $\lb\hat{E}|$.
The state $|\hat{\rho}\rb$ and measurement $\lb \hat{E}|$ should also be parameterized linearly, and the constraints (positive semidefiniteness, hermiticity, trace-preservation of $\rho$) can be imposed during optimization.
Since each $\hat{R}$ is linear in the parameters, $\vec{t}$, the matrix product in Eq. (\ref{wlsq-R}) is 5th order in the parameters, $\vec{t}$. Therefore the objective function, Eq. (\ref{wlsq-R}) is a 10th-order polynomial in $\vec{t}$.
As mentioned earlier, the positivity constraint is expressed as the positive semidefiniteness of the Choi-Jamiolkowski matrix, Eq.~(\ref{choi1}).
\subsubsection{Problem statement}
\begin{eqnarray}
\mathrm{Minimize:}&&\;\;\; \sum_{ijk}\left(m_{ijk}-\lb \hat{E}(\vec{t})|\hat{R}_{F_i}(\vec{t})\hat{R}_{G_k}(\vec{t})\hat{R}_{F_j}(\vec{t}) |\hat{\rho}(\vec{t})\rb\right)^2/\sigma_{ijk}^2 \nonumber \\
\mathrm{Subject\;to:}&&\;\;\; \rho_G = \frac{1}{d^2}\sum_{i,j=1}^{d^2} (R_G)_{ij} P_j^T \otimes P_i \succcurlyeq 0, \;\;\; \forall\; G \in \mathcal{G} \nonumber \\
&&\;\;\; (R_G)_{0i} = \delta_{0i} \;\;\; \forall\; G \in \mathcal{G} \nonumber \\
&&\;\;\; (R_G)_{ij} \in [-1,1] \;\;\; \forall\;G,i,j \nonumber \\
&&\;\;\; \Tr\{\rho\} = 1, \nonumber \\
&&\;\;\; I - E \succcurlyeq 0 \nonumber
\end{eqnarray}
For each $G\in\mathcal{G}$ the $4\times 4$ matrix $R_G$ is parameterized linearly, $(R_G)_{ij} = t_s$ for some index $s$. $|\rho\rb$ is a $4\times 1$ vector and $\lb E|$ is a $1\times 4$ vector. $\rho$ and $E$ are parameterized linearly in such a way that they are hermitian and positive semidefinite. The components $\rho_i$ of $|\rho\rb$ are given as $\rho_i = \Tr\{P_i \rho\}$ and similarly for $\lb E|$.
The product in double brackets in the objective function is a scalar calculated by matrix multiplication of the Pauli transfer matrices for the gates as indicated. For the first example in Sec.~\ref{example}, the matrix $R_{F_4} = (R_{G_1})^2$. Note that this choice increases the order of the objective function in the parameters, $\vec{t}$, since $G_1$ appears twice in the same $F$-gate. As a result, the 2nd example gate set in Sec.~\ref{example} is a better choice.
This version of the optimization problem has the advantage over the $\chi$-matrix version in that the objective function is a 10th-order rather than a 20th-order polynomial in $\vec{t}$. This is still a highly nonlinear objective function. The tradeoff is the requirement of positive-semidefiniteness of the matrix $\rho_G$ for each $G$, which is not necessary in the $\chi$ matrix approach. This requirement naturally suggests a solution in terms of a semidefinite program (SDP). In Ref.~\cite{Merkel2013}, this is achieved by replacing the objective function by a quadratic approximation in order to recast the problem as a convex optimization problem, which is then solved by SDP. (Recall that convex optimization requires the objective function to be of order 2 or less in the optimization parameters.) This approximation amounts to linearizing the triple-gate-product in the estimator $\hat{p}_{ijk}$ about the target gate set, and eliminating the $t$-dependence of $\rho$ and $E$. The latter can be done either assuming the state and POVM are perfect, or by introducing imperfect ones by hand or from the LGST estimate.
Finally, we note that a different gate set, such as the linear inversion estimate, may be used as the starting point for nonlinear optimization, or as the point about which the gate set is linearized for convex optimization.
\chapter{Implementing GST}\label{implementation-chapter}
The data-processing challenge of GST boils down to that of solving a nonlinear optimization problem. We are presented with measurements $m_i$ of a set of probabilities $p_i = \lb E | G_i(\rho)\rb$, $i=0,...,K$. (Using a simplified notation with a single index $i$.) Based on these measurements alone, we would like to find the gate set $\mathcal{G}=\{|\rho\rb,\lb E|,\{G_i\}\}$ that produced the data. Practically speaking, we would like to find the best estimate $\hat{p}_i$ of the true probabilites $p_i$, given the experimentally measured values $m_i$, where the $\hat{p}_i$ are functions of the gate set (i.e., of the parameters used to define the gate set). This estimate should correspond to a physical (CPTP -- see Sec.~\ref{physicality-section}) set of gates.
Typically, each of the $K$ experiments is repeated a sufficiently large number of times $N$ that the central limit theorem applies. Therefore $m_i$ can be considered a sample from a Gaussian distribution.\footnote{More accurately, $m_i$ is a sample from a binomial distribution, which can be approximated as Gaussian when $p_i N$ and $(1-p_i)N$ are not too small. In a hypothetical ideal experiment with no intrinsic SPAM errors, it can happen that $p_i \approx 0$ or $1$ for some values of $i$ when the gate error is low. The Gaussian approximation then breaks down and we must use the full binomial objective function. Since real experiments are not perfect (noise affects initialization, readout, etc.) this should not be an issue.} Then, as we have seen, the problem of estimating $\hat{p}_i$ can be rephrased as maximum-likelihood estimation (MLE) with an objective function (log-likelihood) that is quadratic in the $\hat{p}_i$. Furthermore, if $G_i$ is a single gate (this is the case in QPT), the objective function is quadratic in the gate parameters, and thus convex. (A convex function has only a single local minimum, the global minimum, in its domain of definition.) In this case, the problem can be solved using standard convex optimization techniques~\cite{Chow2012SI,BoydVandenberghe}.
Computation of the QPT estimate is therefore a solved problem (aside from some technicalities, see Ref.~\cite{RBK2010}). For a linearly-parameterized gate set, it is straightforward (though possibly computationally intensive) to determine the most likely CPTP gates that produced the data, as well as the errors in the estimate. Unfortunately, QPT does not solve the correct problem. For self-consistency, we require state preparation and measurement (SPAM) gates to be included in the $\{G_i\}$, making the gate set at least 3rd order in the gate parameters. As a result the objective function is no longer convex (it has many local minima) and the problem is no longer solvable by standard convex optimization techniques. Instead one has to choose a combination of approximate and iterative methods that one hopes can locate the global optimum.
Our goal in this chapter is to illustrate the effectiveness of GST. We use the simplest optimization methods that get the job done. Thus, we implement full nonlinear optimization in Matlab, choosing reasonable settings for the optimization routines but making no effort to optimize these settings. We find this is sufficient to illustrate the main features of GST.
We begin by summarizing the steps required to implement GST, including how to gather and organize the data, and some tips on the analysis. We then present the results of GST for a single qubit using simulated data with simplified but realistic errors. We compare the performance of maximum likelihood GST (ML-GST) to that of QPT for varying levels of coherent error, incoherent error, and sampling noise. We corroborate the result of Ref.~\cite{Merkel2013} that coherent errors are poorly estimated by QPT near QEC thresholds while GST is accurate in this regime.
In Chapter~\ref{derivation-chapter}, we described two versions of ML-GST, a nonlinear optimization problem based on the process matrix (Sec.~\ref{process-matrix-optimization-section}) and a semidefinite program (SDP) based on the Pauli transfer matrix (Sec.~\ref{ptm-optimization-section}). The numerical results in the present chapter were obtained using the process matrix - based approach. Due to the high nonlinearity of the objective function in this approach, the LGST estimate was essential as a starting point for MLE and provided better estimates than using the target gate set as a starting point.
\section{Experimental protocol} \label{experimental-protocol}
Recalling the definitions from the last chapter, $\mathcal{G} = \{G_0,G_1,\ldots,G_K\}$ is the gate set, with $G_0$ denoting the ``null'' gate - do nothing for no time - a perfect identity. $\mathcal{F} = \{F_1,\ldots,F_N\}$ is the SPAM gate set. It is used to construct a complete basis of starting states and measurements from a given particular starting state $\rho$ and measurement operator $E$. Each gate $F_i \in \mathcal{F}$ is composed of gates in $\mathcal{G}$. For a single qubit, $d=2$, we must have $N\geq 4$. For simplicity, we can take $N=4$.
The experimental protocol is as follows.
\begin{enumerate}
\item[1.] Initialize the qubit to a particular state $|\rho\rb$. In most systems, the natural choice for $\rho$ is the ground state of the qubit, $\rho = |0\rangle\langle 0|$.
\item[2.] For a particular choice of $i,j \in \{1,\ldots,N\}$, $k\in\{0,\ldots,K\}$, apply the gate sequence $F_i \circ G_k \circ F_j$ to the qubit. Remember that the $F$ gates are composed of $G$s. So the gate sequence applied in this step is a sequence of gates $G \in \mathcal{G}$.
\item[3.] Measure the POVM $E$. $E$ is required to be a positive semidefinite Hermitian operator, such that $I-E$ is also positive semidefinite. ($I$ is the identity.) As is the case for $\rho$, the natural choice for $E$ in most systems is $E = |0\rangle\langle 0|$. Sometimes $E=|1\rangle\langle 1|$ is used.
\item[4.] Repeat steps 1-3 a large number of times, $n$. Typically, $n = $1,000 - 10,000. For the $r$-th repetition, record $n_r = 1$ if the measurement is success (i.e., the measured state is $|0\rangle\langle 0|$), and $n_r = 0$ if the measurement is failure (i.e., the measured state is not $|0\rangle\langle 0|$).
\item[5.] Average the results of step 4. The result, $m_{ijk} = \sum_{r=1}^n n_r/n$, is a measurement of the expectation value $p_{ijk} = \lb E|F_i G_k F_j |\rho\rb$. It is a random variable with mean $p_{ijk}$ and variance $p_{ijk} (1-p_{ijk}) / n$.
\item[6.] Repeat steps 1-5 for all $i,j \in \{1,\ldots,N\}$, $k\in\{0,\ldots,K\}$.
\item[7.] {\em Optional --} Repeat steps 1-5 to measure the expectation values $p_i = \lb E| F_i |\rho\rb$. Since typically $F_0 = G_0 = \{\}$, this data will already be contained in the measurements of $p_{ijk}$. However, it helps to have an independent measurement if possible.
\end{enumerate}
\section{Organizing and verifying the data}
The procedure above gives measurements of the following quantities.
\begin{eqnarray}
&\lb E| F_i \circ G_k \circ F_j|\rho\rb,&\nonumber \\
&\lb E| F_i \circ F_j|\rho\rb,&\nonumber \\
&\lb E| F_i |\rho\rb,&\nonumber
\end{eqnarray}
for $k=1,\ldots,K$ and $i,j=1,\ldots,N$.
The experimental data should then be organized into matrices, as follows (see Eqs.~(\ref{pijk}), (\ref{gram}), (\ref{rhotilde})-(\ref{Etilde})).
\begin{eqnarray}
(\tilde{G}_k)_{ij} & = & \lb E| F_i G_k F_j|\rho\rb, \label{exp2}\\
g_{ij} & = & \lb E| F_i F_j|\rho\rb, \label{exp1}\\
|\tilde{\rho}\rb_i & = & \lb E| F_i |\rho\rb = \lb \tilde{E}|_i, \label{exp3}
\end{eqnarray}
where in an abuse of notation we have written $\lb E|\ldots|\rho\rb$ to stand for the {\em measured} values of these quantities rather than the true values.
Once data is obtained, and before proceeding further, we must check that the Gram matrix, $g_{ij} = \lb E |F_i F_j|\rho\rb$, is nonsingular, so that it may be inverted to find the estimate as in Eqs. (\ref{rhohat})-(\ref{Ghat}). We want the smallest magnitude eigenvalue to be as large as possible. This is because the sampling error on the estimate scales roughly as the inverse of the smallest eigenvalue of the Gram matrix (multiplied by the sampling error in the data). A good rule of thumb is that no eigenvalue be less than 0.1 in absolute value. (Then for $N=2000$ samples the sampling error in the estimate is bounded at $5\%$.)
If the eigenvalues of the Gram matrix are very small, this indicates that the SPAM gates are only marginally linearly independent. Viewed as vectors, they are highly overlapping. In this case, the experimenter must go back and tweak the knobs of the experiment to make the SPAM gates more orthogonal. Then the Gram matrix must be measured again and its eigenvalues checked. This process is repeated until a suitable Gram matrix is obtained.\footnote{Typically experimentalists have other means at their disposal to ensure orthogonal, or nearly orthogonal, gate rotation axes. Thus it is not difficult to obtain an invertible Gram matrix in practice.}
\section{Linear inversion}
Once we have checked that the experimental Gram matrix is invertible, we apply its inverse to the data matrices in Eqs.~(\ref{exp2})-(\ref{exp3}), following the procedure in Section~\ref{derivation-lgst}. (Note that $g^{-1}$ is applied to all data matrices except the vector $\lb \tilde{E}|$.) We obtain the following estimates for the gates and states.
\begin{eqnarray}
|\hat{\rho}\rb &=& g^{-1}|\tilde{\rho}\rb, \label{lgst-est1}\\
\lb\hat{E}| &=& \lb\tilde{E}|,\label{lgst-est2}\\
\hat{G}_k &=& g^{-1}\tilde{G}_k.\label{lgst-est3}
\end{eqnarray}
Since $g \equiv \tilde{G}_0$, the LGST estimate of the null gate is exactly the identity, as it should be.
The gate set estimated in this way, $\hat{\mathcal{G}} = \{|\hat{\rho}\rb,\lb\hat{E}|,\{\hat{G}_k\}\}$, is in a different gauge from the actual gate set (Sec.~\ref{gram-ab-section} - \ref{gauge-opt-section}). To compensate, we transform to a more useful gauge. Since we do not know the actual gate set, only the intended (target) one, the most useful gauge is the one that brings the estimated gate set as close as possible (based on some distance metric) to the target. The difference between the final estimate and the target gate set is then a measure of the error in the actual gate set (how far away it is from what was intended).
The gauge transformation is found by solving the optimization problem defined in Eq.~\ref{gaugeopt}. The resulting gauge matrix, $\hat{B}^*$, is then applied to Eqs.~(\ref{lgst-est1}) - (\ref{lgst-est3}). The final LGST estimate is given by Eqs.~(\ref{lgst-final1}) - (\ref{lgst-final3}), which we reproduce here:
\begin{eqnarray}
\hat{G}_k^* &=& \hat{B}^* \hat{G}_k (\hat{B}^*)^{-1}, \\
|\hat{\rho}^*\rb &=& \hat{B}^* |\hat{\rho}\rb, \\
\lb\hat{E}^*| &=& \lb\hat{E}|(\hat{B}^*)^{-1}.
\end{eqnarray}
The linear inversion protocol is fairly easy to implement numerically and is useful for providing quick diagnostics without resorting to more computationally intensive estimation via constrained optimization. Since the LGST estimate is not generally physical,\footnote{There is no natural way to put physicality constraints into the LGST protocol. One option is to find the closest physical gate set to the LGST estimate, but this is suboptimal to MLE.} the information obtained is somewhat qualitative. Nevertheless, large enough errors can be easily detected with this approach.
More importantly, LGST is useful as a starting point for MLE. Since the starting point must be physical, while LGST is not, we should use the closest physical gate set to the LGST estimate. Such a gate set can be found in a similar way to the gauge optimization procedure described above, choosing a metric such as the trace norm as a measure of distance between gates.
\section{Estimation results and analysis for simulated experimental data}
In this section we present results of maximum likelihood - based GST (ML-GST) for several examples using simulated data, and compare to ML-QPT. The simulated errors are of three different types, representing coherent and incoherent gate errors as well as intrinsic SPAM errors. The MLE approach was discussed in Section~\ref{mle-section}.
Maximum likelihood estimation provides a convenient way to handle physicality constraints as well as overcomplete data (we will get to that later). Although it has been argued that MLE is suboptimal to Bayesian methods~\cite{RBK2010}, it is still the method of choice for GST and QPT~\cite{Chow2009SI, Chow2012, Merkel2013, RBK2013}. We shall see that MLE is highly accurate.
As a model system, we consider a single qubit with available gates consisting of rotations about two orthogonal axes, which we label X and Y. This is the case in many existing qubit implementations~\cite{barends_superconducting_2014,Chow2012}. Furthermore, we assume the qubit can be prepared in its ground state, $\rho = |0\rangle\langle 0|$ and measured in the $Z$-basis, distinguishing $E=|1\rangle\langle 1|$ and $I-E=|0\rangle\langle 0|$. This state and measurement may be faulty, and we will study the effect on estimation due to these errors.
The results in this section were obtained using gate set 2 in Sec.~\ref{example},
\begin{eqnarray}
\mathcal{G} &=& \{\{\},X_{\pi/2},Y_{\pi/2},X_{\pi}\},\\
\mathcal{F} &=& \mathcal{G}.
\end{eqnarray}
To illustrate the effect of systematic errors (all errors with the exclusion of statistical sampling noise are systematic errors), we use depolarizing noise as an example of incoherent environmental noise, and over-rotations in the Y-gate as an example of a coherent control error. Depolarizing noise is defined by the map
\begin{equation}
G_{dep}(\rho) = (1-3p)\rho + p(X\rho X + Y\rho Y + Z\rho Z) \label{depol-gate}
\end{equation}
For a 50~$\mathrm{ns}$ gate, a depolarizing parameter $p = 0.005$ corresponds to a decoherence time of 2.5~$\mathrm{\mu s}$. In our numerical experiments we apply $G_{dep}$ after every gate where depolarizing noise is required.
Over-rotation errors are obtained by applying the map,
\begin{equation}
G_{rot}(\rho) = \exp\left(-i \frac{\epsilon}{2}\hat{n}\cdot \vec{\sigma}\right) \rho \exp\left( i \frac{\epsilon}{2}\hat{n}\cdot \vec{\sigma} \right),
\end{equation}
after every gate that should have the error, in our case the Y-gate. The parameter $\epsilon$ is the angle of over-rotation, $\hat{n}=\hat{y}$ is the rotation axis, and $\vec{\sigma} = (X,Y,Z)$ is a vector of Pauli matrices.
In addition to these systematic errors, there will be noise due to finite sampling statistics. As an example, $N = 2000$ samples per experiment produces a sampling error in the data that is upper-bounded by $1/(4\sqrt{N}) = 0.005$.
\subsection{Systematic errors}\label{examples-subsection}
We first consider systematic errors only, no sampling error. This is useful for examining how well GST can do relative to QPT {\em in principle}. In practice, sampling error will contaminate both GST and QPT estimates, and efforts must be taken to reduce it. This can be done either by taking more samples (which can be impractical) or performing more independent experiments.
We calculate estimates using the process matrix formulation of the MLE problem, see Sec.~\ref{process-matrix-optimization-section}. This is a nonlinear optimization problem requiring a starting point to be specified. For the starting point, we use the LGST estimate for GST and the target gate set for QPT. More precisely, for GST we find the closest physical gate set to the LGST estimate (based on the trace norm) and use that for a starting point.
Since the number of samples is taken as infinite (corresponding to zero sampling error), we use a standard (non-weighted) least squares objective function,
\begin{equation}
l(\hat{\mathcal{G}}) = \sum_{i=1}^{N_{exp}} \left(m_i - \hat{p}_i\right)^2,
\end{equation}
where $N_{exp}$ is the number of experiments (16 for QPT, 84 for GST) and $m_i$ are the measured values of the true probabilities $p_i$, of which $\hat{p}_i$ are our estimates. Since the sampling error is zero, $m_i = p_i$.
The estimates in this section were generated using Matlab's built-in optimization function {\em fmincon}, running the active-set optimization algorithm. Optimization time varied between 2-4 minutes for each GST estimate on an Intel(R) Core(TM) i5-2500K (3.3GHz) processor. Optimizer settings were options.TolFun = $10^{-12}$, options.TolCon = $10^{-6}$, options.MaxFunEvals = 15000. Typically the objective function and constraint tolerances were satisfied before the maximum number of function evaluations was reached.
\subsubsection{Example 1: Over-rotation (coherent) error}
Fig.~\ref{est-vs-actual-error-theta-y} shows the estimation error of QPT and GST as a function of gate error for an over-rotation in the Y-gate, which is a type of coherent error. Gate error is defined as $\mathcal{E}_{gate} = 1-\bar{F}({\rm actual,ideal})$, where $\bar{F}({\rm actual, ideal})$ is the average fidelity of the actual gate relative to the ideal gate.\footnote{\label{average-fidelity-footnote}Average fidelity is defined as $\bar{F}(A,B) = \int d\rho \Tr\{\rho A^{-1}\circ B (\rho)\}$, where $A$ and $B$ are quantum operations, and the integral is over the uniform (Haar) measure on the space of density matrices. The average fidelity may be expressed in terms of Pauli transfer matrices $R_A, R_B$ as $\bar{F}(A,B) = (\Tr\{R_A^{-1}R_B\}+d)/[d(d+1)]$, where $d$ is the dimension of the Hilbert space. See Ref.~\cite{nielsen_simple_2002}.} Estimation error is defined similarly, as $\mathcal{E}_{est}=1-F({\rm estimated, actual})$. We are able to use fidelity as a metric because the estimates are constrained to be physical. (For unphysical gates, $0\leq \bar{F} \leq 1$ may not hold.)
\begin{figure}
\caption{Estimation error vs gate error for an over-rotation in $Y_{\pi/2}
\label{est-vs-actual-error-theta-y}
\end{figure}
\begin{figure}
\caption{Pauli transfer matrices for GST and QPT maximum likelihood estimates (2nd and 3rd columns). The actual gate set (1st column) contains an over-rotation error of $4\degree$ in the $Y_{\pi/2}
\label{ptm-theta-y}
\end{figure}
\begin{figure}
\caption{Pauli transfer matrices for GST and QPT maximum likelihood estimates (2nd and 3rd columns) for the same data as in Fig.~\ref{ptm-theta-y}
\label{ptm-diff-theta-y}
\end{figure}
We make the following observations: (1) QPT attributes error to all gates, even though only single gate ($Y_{\pi/2}$) is actually faulty. This is because the faulty gate is used in SPAM for all gates. (2) The GST estimation error is flat as a function of gate error, and represents the threshold for the optimizer.\footnote{\label{optimizer-threshold-footnote}We use a tolerance of $\epsilon = 10^{-12}$ on the (unweighted) least-squares objective function (see text), which corresponds to an estimation error of approximately $10^{-7}$. This can be derived by assuming an error in each entry of the PTM equal to the rms value of the residual in the objective function, $\sqrt{\epsilon/N_{exp}}$, where $N_{exp} \approx 10^2$ is the number of experiments. The estimation error is approximately equal to the error in the PTM elements, as can be verified using the trace formula in footnote~\ref{average-fidelity-footnote}, above.} (3) In the regime of gate error relevant to QEC thresholds ($\mathcal{E} = 10^{-3} - 10^{-2}$), (and given the numerical tolerances) GST is several orders of magnitude more accurate than QPT.
Fig.~\ref{ptm-theta-y} shows the PTM coefficients for the gate set at a representative gate error (infidelity) of $8.5 \times 10^{-4}$. Results are shown for the actual gates and for the GST and QPT estimates. Both GST and QPT are good at catching the error on the faulty ($Y_{\pi/2}$) gate, but QPT also attributes rotations to the other gates. The relative errors can also be seen in Fig.~\ref{ptm-diff-theta-y}, which shows the difference between the estimated and actual PTMs.
\subsubsection{Example 2: Depolarization (incoherent) error}
The depolarization map given by Eq.~(\ref{depol-gate}) is often used as a model of incoherent environmental noise. The depolarizing parameter $p$ is related to the gate error as $\mathcal{E} = 2p$.\footnote{This can be derived using the formula for average fidelity in footnote~\ref{average-fidelity-footnote}.} The estimation errors of QPT and GST as a function of gate error for depolarizing noise are plotted in Fig.~\ref{est-vs-actual-error-matnorm-depol}. As a measure of estimation error, we use the spectral norm (as implemented in Matlab with the function {\em norm.m}) of the difference between the estimated and actual gates. We use spectral norm rather than infidelity as in the previous example because the optimization routine was unable to satisfy the tolerance on the physicality constraints in the present case.\footnote{Technically, the diamond norm~\cite{KitaevBook} is a more correct metric for gate distance than the spectral norm. However, the spectral norm is easier to calculate. Since we are only interested in comparing the relative distance of QPT and GST estimates from the actual gate, and our simulated gates are {\em exactly} constrained to the single-qubit Hilbert space, the spectral norm is sufficient.} Although the constraint violation was small ($<10^{-4}$ deviation in each element the top row of the PTM), it was enough to invalidate fidelity as a metric.
As for the previous (over-rotation) example, we plot the PTMs for the estimated gates as well as the differences between estimated and actual PTMs for a representative gate error of $8.5 \times 10^{-4}$, the same as in the over-rotation example. At a given level of systematic noise, the estimation error of QPT is smaller for incoherent errors than coherent errors.
According to Fig.~\ref{est-vs-actual-error-matnorm-depol}, the estimation errors of QPT and GST are about the same.\footnote{Based on the previous analysis of coherent errors, we expect the GST estimate not to vary as a function of the depolarizing parameter, since depolarizing noise is treated self-consistently in the same way. We expect this to be achievable using a more robust optimization routine than the one we use.} However, Fig.~\ref{ptm-diff-depol} shows that GST does a better job of estimating the non-zero coefficients of the PTM. These coefficients are $1-p$, where $p$ is the depolarizing parameter. Therefore GST gives a better estimate of $p$ than does QPT. This is one illustration of the danger in relying on a single parameter as a metric of gate or estimation quality.
\begin{figure}
\caption{Estimation error vs gate error for depolarizing noise on all gates. Blue dots are ML-GST and green dots are ML-QPT. Estimation error is given by the spectral norm of the difference between the estimated and actual PTMs and gate error is the infidelity between the actual and ideal PTMs. The gate error is proportional to the depolarizing parameter. The vertical dashed line indicates the selected value of gate error ($E = 8.5 \times 10^{-4}
\label{est-vs-actual-error-matnorm-depol}
\end{figure}
\begin{figure}
\caption{Pauli transfer matrices for GST and QPT maximum likelihood estimates (2nd and 3rd columns). The actual gate set (1st column) contains a depolarizing gate error (infidelity) of $E(\mathrm{actual}
\label{ptm-depol}
\end{figure}
\begin{figure}
\caption{Pauli transfer matrices for GST and QPT maximum likelihood estimates (2nd and 3rd columns) for the same data as in Fig.~\ref{ptm-depol}
\label{ptm-diff-depol}
\end{figure}
\subsubsection{Example 3: Intrinsic SPAM error}
As an example of intrinsic SPAM error, we assume a faulty initial state $\rho$, produced by applying depolarizing noise of strength $p$, Eq.~(\ref{depol-gate}), to an ideal initial state. $\lb E|\rho\rb = 0$ in the ideal case, and $\lb E|\rho\rb = 2 p$ is a measure of the state error. This is the same value as would be obtained for the gate error had the depolarizing noise been applied to the gate instead of the state. Although depolarizing noise commutes with all gates (the $i,j = 2\ldots d^2$ sub-matrix is proportional to the identity), the example considered here is not equivalent to that in the last section. This is because in the present case there is a single depolarizing operation acting in each experiment, whereas the number of depolarizing operations in the last section varied between 1 and 3, depending on which opperators appeared in $\lb E| F_i G_k F_j|\rho\rb$.
As could be expected from the results on depolarizing noise in the last section, the error in the QPT estimate grows linearly with the error on $\rho$. In fact, the QPT estimation error is almost exactly equal to the initial state error, showing that QPT attributes the noise to the gates rather than the state. This makes sense, since QPT assumes ideal initial states, and is confirmed by looking at the difference between the estimated and actual PTMs as in Fig.~\ref{ptm-diff-depol-rho}. In contrast, GST is insensitive to the initial state error, and the GST estimation error is at the optimizer threshold (see footnote~\ref{optimizer-threshold-footnote}).
\begin{figure}
\caption{Estimation error vs gate error for depolarizing noise on the initial state. Blue dots are ML-GST and green dots are ML-QPT. Estimation error is given by the infidelity as in Fig.~\ref{est-vs-actual-error-theta-y}
\label{est-vs-actual-error-depol-rho}
\end{figure}
\begin{figure}
\caption{Pauli transfer matrices for GST and QPT maximum likelihood estimates (2nd and 3rd columns). The actual gate set (1st column) contains no error. The error is a depolarizing noise of strength $\lb E|\rho\rb = 8.5\times 10^{-4}
\label{ptm-depol-rho}
\end{figure}
\begin{figure}
\caption{Pauli transfer matrices for GST and QPT maximum likelihood estimates (2nd and 3rd columns) for the same data as in Fig.~\ref{ptm-depol-rho}
\label{ptm-diff-depol-rho}
\end{figure}
\subsection{Sampling noise}
In this section we repeat the first example from the last section, over-rotations in the Y-gate, but this time in the presence of sampling noise and intrinsic SPAM error. This situation is closer to real life, and illustrates what happens in general. We take $N_{Samples} = 10000$, $\lb E|\rho\rb = 0.01$, both of which could reasonably occur in an experiment. This value of $N_{Samples}$ corresponds to a sampling error of about $0.01$ per PTM entry, so both errors are about the same order of magnitude. We expect that, as we vary the strength of the systematic error, QPT will be swamped by both sampling and intrinsic SPAM error until the systematic gate error rises above the level (0.01) of these errors. We expect GST, on the other hand, to be insensitive to intrinsic SPAM but to also be unable to detect the systematic error unless it is larger than the sampling error.
The results turn out to be more subtle. Depending on the type of error, it may be detectable at a lower gate error than the sampling error. This is the case for over-rotations. Fig.~\ref{est-vs-actual-error-theta-y-samp} shows the estimation error vs gate error for our example. As expected, the estimation error is flat until the gate error exceeds 0.01, both for QPT and GST. QPT is limited by intrinsic SPAM while GST is not. Above 0.01, the estimation error behaves as in Fig.~\ref{est-vs-actual-error-theta-y} -- the QPT values increase while GST remains flat. However, the PTMs give a fuller picture. Figs.~\ref{ptm-depol-theta-y-samp} and~\ref{ptm-diff-depol-theta-y-samp} show the PTMs for a gate error near $10^{-3}$, well below the crossover point. GST is still able to find the over-rotation error. This is because, for a given gate error, the magnitude of non-zero PTM entries due to the over-rotation error is larger than the average magnitude of PTM entries for sampling noise.
The results in this section were generated by numerical optimization in Matlab, as in Sec.~\ref{examples-subsection}. Everything stated in that section regarding the optimization carries over to here, except that the objective function used was weighted least squares, and the objective function tolerance was set to options.TolFun = $10^{-6}$. The reason is that in the presence of sampling noise, the weighted-least squares objective function is equal to the chi-squared, which is of order $N_{exp}$. Without sampling noise, the unweighted least squares objective function is near zero, hence the smaller function tolerance in that case.
\begin{figure}
\caption{Estimation error vs gate error for over-rotation in the Y-gate, including sampling noise and intrinsic SPAM errors with parameters $N_{Samples}
\label{est-vs-actual-error-theta-y-samp}
\end{figure}
\begin{figure}
\caption{Pauli transfer matrices for GST and QPT maximum likelihood estimates (2nd and 3rd columns) of the gate set in column 1. The actual gate set shown in column 1 contains an over-rotation error of $4\degree$ in the Y-gate, corresponding to a gate error of $8.5\times 10^{-4}
\label{ptm-depol-theta-y-samp}
\end{figure}
\begin{figure}
\caption{Pauli transfer matrices for GST and QPT maximum likelihood estimates (2nd and 3rd columns) for the same data as in Fig.~\ref{ptm-depol-theta-y-samp}
\label{ptm-diff-depol-theta-y-samp}
\end{figure}
\chapter{Summary and outlook}
We have seen that gate set tomography is a robust and powerful tool for the full characterization of quantum gates. In the presence of SPAM errors, GST is accurate (within the limits of sampling error) while QPT typically overestimates the gate error. This discrepancy can be several orders of magnitude in the regime of gate error relevant to quantum error correction. GST is also capable of providing {\em qualitative} information about systematic gate errors -- via the estimated Pauli transfer matrices -- that is not accessible by QPT in the presence of SPAM error. Nevertheless, several topics remain the subject of current research. These include reduction of sampling error, treatment of non-Markovian noise, detection of leakage and extra dimensions, and tractable extension to multiple qubits.
Of these, the extension to multiple qubits seems the most clear-cut. For multiple qubits, the formalism of GST is the same but there is a formidable processing challenge because of the large amount of experimental data necessary (over 4,000 experiments for 2-qubit GST, as compared to $<100$ for 1-qubit). The nonlinear optimization routines we used to illustrate GST in Ch.~\ref{implementation-chapter} would likely take an unreasonable amount of time even for 2 qubits, assuming they run at all. The SDP method discussed in Ch.~\ref{derivation-chapter} has been reported to run slowly for 2 qubits as well~\cite{IBMpcomm}. One solution to this problem has been proposed by IBM, which is to use a type of semidefinite program used for compressed-sensing problems, called a first-order conic solver. Preliminary reports indicate that this runs much faster than standard SDP~\cite{IBMpcomm}.
Robin Blume-Kohout has proposed the inclusion of gate repetitions in the GST gate set in order to reduce sampling error and detect non-Markovian noise~\cite{RBK-pcomm}. Some initial evidence of the usefulness of this approach was presented in Ref.~\cite{RBK2013}. The detection of extra dimensions -- i.e. due to the system the leaving the qubit Hilbert space -- is also an active topic of research~\cite{RBK-pcomm,stark_compressibility_2014}.
One topic we have not discussed in detail but that deserves fuller attention is estimating the error in the GST estimates. This is important in order to determine the resource requirements (number of experiments, gate repetitions, etc.) for obtaining high-quality GST estimates within the tolerances required by QEC. It is a pressing question whether the required accuracy can be tractably obtained for two-qubit gates. E.g., for a CNOT gate below the surface code threshold -- probability of gate error $ \approx 10^{-3}$ -- we would like the error in the GST estimate of this quantity to be below roughly $10^{-4}$.
Part of the error in the maximum-likelihood estimate comes from sampling noise, part from any approximations used in the objective function, and part from the optimization routine (which caused the point-to-point variability in the plots in Ch.~\ref{implementation-chapter} of estimation vs gate error in the absence of sampling error). Neglecting errors due to the optimization itself, it is important to understand how statistical sampling errors propagate through the MLE procedure. Standard errors for MLE estimates (see, e.g. Ref.~\cite{statistics-book}, Ch.~14) are valid when these estimates can be shown to be asymptotically (limit of large number of experiments) normally distributed. In this case, an asymptotically valid estimator for the error can be written in terms of the covariance matrix of the estimated parameter vector $\hat{\vec{t}}$, which can be calculated from the MLE estimate and the objective function. However, the MLE estimates for GST may not be asymptotically normally distributed due to the constraint that $\vec{t}$ must describe a physical map. In fact, for very small errors - the regime we are interested in - the gates will be very close to ideal unitaries, which lie on the boundary of allowable maps. Therefore asymptotic normality cannot be assumed. Hence it is unclear how accurate this approach would be for our problem.
A commonly used~\cite{Chow2012SI} approach to estimating the error in the ML estimate is to resample from experimental data, known as bootstrapping. In Ch.~\ref{implementation-chapter}, we were able to plot estimation error vs gate error because we knew what the actual gates were. However, the goal of GST is to estimate an unknown set of gates. Operationally, the variance in the estimate can be found by repeating the same experiment many times, each time generating new data and a new best fit. Since the amount of work to do this can be impractical (and if it isn't one would prefer to include this additional data in a single, larger sample to produce a tighter estimate), resampling with replacement from the same data (say a set of single-shot measurements of size $N_{Samples}$) seems like a good alternative. Unfortunately, it is well known~\cite{blume-kohout_robust_2012} that bootstrapping is unreliable for biased estimators. As discussed in the previous paragraph, the MLE estimate is biased due to physicality constraints. Therefore, without a rigorous theory of error estimation for quantum tomography in the presence of sampling noise, it is impossible to evaluate the validity of bootstrapping for this problem.
A practical solution to both of these problems (validity of standard errors for MLE, validity of bootstrapping) may be possible via a Monte Carlo approach. One can numerically generate many sample data sets with a specified error model and a given level of sampling noise, as we have done in Ch.~\ref{implementation-chapter}. One can then perform GST on each sample data set and study the distribution of the resulting estimates. It is possible that general features, such as the amount of bias in the GST estimates for a given model of systematic errors, may be obtained in this way. This can then be used to determine under what conditions techniques such as bootstrapping are justified.
In conclusion, we have presented an overview of gate set tomography for a single qubit. It is hoped that this will be useful to practitioners aiming to implement full characterization of single as well as multi-qubit gates.
{}
\end{document} | math |
تھیٹرَس منٛز مُعٲینُک مُنٲسِب وقٕت چُھ اَہم | kashmiri |
/* ObjCryst++ Object-Oriented Crystallographic Library
(c) 2000-2002 Vincent Favre-Nicolin [email protected]
2000-2001 University of Geneva (Switzerland)
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., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*/
/*
* source file for Global Optimization Objects
*
*/
#include <iomanip>
#include "ObjCryst/RefinableObj/GlobalOptimObj.h"
#include "ObjCryst/ObjCryst/Crystal.h"
#include "ObjCryst/Quirks/VFNStreamFormat.h"
#include "ObjCryst/Quirks/VFNDebug.h"
#include "ObjCryst/Quirks/Chronometer.h"
#include "ObjCryst/ObjCryst/IO.h"
#include "ObjCryst/RefinableObj/LSQNumObj.h"
#include "ObjCryst/ObjCryst/Molecule.h"
#ifdef __WX__CRYST__
#include "ObjCryst/wxCryst/wxRefinableObj.h"
#undef GetClassName // Conflict from wxMSW headers ? (cygwin)
#endif
//For some reason, with wxWindows this must be placed after wx headers (Borland c++)
#include <fstream>
#include <sstream>
#include <stdio.h>
#include <boost/format.hpp>
namespace ObjCryst
{
void CompareWorlds(const CrystVector_long &idx,const CrystVector_long &swap, const RefinableObj &obj)
{
const long nb=swap.numElements();
const CrystVector_REAL *pv0=&(obj.GetParamSet(idx(swap(nb-1))));
for(long i=0;i<idx.numElements();++i)
{
REAL d=0.0;
const CrystVector_REAL *pv1=&(obj.GetParamSet(idx(swap(i))));
for(long j=0;j<pv0->numElements();++j) d += ((*pv0)(j)-(*pv1)(j))*((*pv0)(j)-(*pv1)(j));
cout<<"d("<<i<<")="<<sqrt(d)<<endl;
}
cout<<endl;
}
//#################################################################################
//
// OptimizationObj
//
//#################################################################################
ObjRegistry<OptimizationObj> gOptimizationObjRegistry("List of all Optimization objects");
OptimizationObj::OptimizationObj():
mName(""),mSaveFileName("GlobalOptim.save"),
mNbTrialPerRun(10000000),mNbTrial(0),mRun(0),mBestCost(-1),
mBestParSavedSetIndex(-1),
mContext(0),
mIsOptimizing(false),mStopAfterCycle(false),
mRefinedObjList("OptimizationObj: "+mName+" RefinableObj registry"),
mRecursiveRefinedObjList("OptimizationObj: "+mName+" recursive RefinableObj registry"),
mLastOptimTime(0)
{
VFN_DEBUG_ENTRY("OptimizationObj::OptimizationObj()",5)
// This must be done in a real class to avoid calling a pure virtual method
// if a graphical representation is automatically called upon registration.
// gOptimizationObjRegistry.Register(*this);
static bool need_initRandomSeed=true;
if(need_initRandomSeed==true)
{
srand(time(NULL));
need_initRandomSeed=false;
}
// We only copy parameters, so do not delete them !
mRefParList.SetDeleteRefParInDestructor(false);
VFN_DEBUG_EXIT("OptimizationObj::OptimizationObj()",5)
}
OptimizationObj::OptimizationObj(const string name):
mName(name),mSaveFileName("GlobalOptim.save"),
mNbTrialPerRun(10000000),mNbTrial(0),mRun(0),mBestCost(-1),
mBestParSavedSetIndex(-1),
mContext(0),
mIsOptimizing(false),mStopAfterCycle(false),
mRefinedObjList("OptimizationObj: "+mName+" RefinableObj registry"),
mRecursiveRefinedObjList("OptimizationObj: "+mName+" recursive RefinableObj registry"),
mLastOptimTime(0)
{
VFN_DEBUG_ENTRY("OptimizationObj::OptimizationObj()",5)
// This must be done in a real class to avoid calling a pure virtual method
// if a graphical representation is automatically called upon registration.
// gOptimizationObjRegistry.Register(*this);
static bool need_initRandomSeed=true;
if(need_initRandomSeed==true)
{
srand(time(NULL));
need_initRandomSeed=false;
}
// We only copy parameters, so do not delete them !
mRefParList.SetDeleteRefParInDestructor(false);
VFN_DEBUG_EXIT("OptimizationObj::OptimizationObj()",5)
}
OptimizationObj::OptimizationObj(const OptimizationObj &old):
mName(old.mName),mSaveFileName(old.mSaveFileName),
mNbTrialPerRun(old.mNbTrialPerRun),mNbTrial(old.mNbTrial),mRun(old.mRun),mBestCost(old.mBestCost),
mBestParSavedSetIndex(-1),
mContext(0),
mIsOptimizing(false),mStopAfterCycle(false),
mRefinedObjList("OptimizationObj: "+mName+" RefinableObj registry"),
mRecursiveRefinedObjList("OptimizationObj: "+mName+" recursive RefinableObj registry"),
mLastOptimTime(0)
{
VFN_DEBUG_ENTRY("OptimizationObj::OptimizationObj(&old)",5)
// This must be done in a real class to avoid calling a pure virtual method
// if a graphical representation is automatically called upon registration.
// gOptimizationObjRegistry.Register(*this);
static bool need_initRandomSeed=true;
if(need_initRandomSeed==true)
{
srand(time(NULL));
need_initRandomSeed=false;
}
// We only copy parameters, so do not delete them !
mRefParList.SetDeleteRefParInDestructor(false);
for(unsigned int i=0;i<old.mRefinedObjList.GetNb();i++)
this->AddRefinableObj(old.mRefinedObjList.GetObj(i));
VFN_DEBUG_EXIT("OptimizationObj::OptimizationObj(&old)",5)
}
OptimizationObj::~OptimizationObj()
{
VFN_DEBUG_ENTRY("OptimizationObj::~OptimizationObj()",5)
gOptimizationObjRegistry.DeRegister(*this);
VFN_DEBUG_EXIT("OptimizationObj::~OptimizationObj()",5)
}
void OptimizationObj::RandomizeStartingConfig()
{
VFN_DEBUG_ENTRY("OptimizationObj::RandomizeStartingConfig()",5)
this->PrepareRefParList();
for(int j=0;j<mRefParList.GetNbParNotFixed();j++)
{
if(true==mRefParList.GetParNotFixed(j).IsLimited())
{
const REAL min=mRefParList.GetParNotFixed(j).GetMin();
const REAL max=mRefParList.GetParNotFixed(j).GetMax();
mRefParList.GetParNotFixed(j).MutateTo(min+(max-min)*(rand()/(REAL)RAND_MAX) );
}
else if(true==mRefParList.GetParNotFixed(j).IsPeriodic())
mRefParList.GetParNotFixed(j).
Mutate(mRefParList.GetParNotFixed(j).GetPeriod()*rand()/(REAL)RAND_MAX);
}
//else cout << mRefParList.GetParNotFixed(j).Name() <<" Not limited :-(" <<endl;
VFN_DEBUG_EXIT("OptimizationObj::RandomizeStartingConfig()",5)
}
void OptimizationObj::FixAllPar()
{
VFN_DEBUG_ENTRY("OptimizationObj::FixAllPar()",5)
this->BuildRecursiveRefObjList();
for(int i=0;i<mRecursiveRefinedObjList.GetNb();i++)
mRecursiveRefinedObjList.GetObj(i).FixAllPar();
VFN_DEBUG_EXIT("OptimizationObj::FixAllPar():End",5)
}
void OptimizationObj::SetParIsFixed(const string& parName,const bool fix)
{
this->BuildRecursiveRefObjList();
for(int i=0;i<mRecursiveRefinedObjList.GetNb();i++)
mRecursiveRefinedObjList.GetObj(i).SetParIsFixed(parName,fix);
}
void OptimizationObj::SetParIsFixed(const RefParType *type,const bool fix)
{
this->BuildRecursiveRefObjList();
for(int i=0;i<mRecursiveRefinedObjList.GetNb();i++)
mRecursiveRefinedObjList.GetObj(i).SetParIsFixed(type,fix);
}
void OptimizationObj::UnFixAllPar()
{
this->BuildRecursiveRefObjList();
for(int i=0;i<mRecursiveRefinedObjList.GetNb();i++)
mRecursiveRefinedObjList.GetObj(i).UnFixAllPar();
}
void OptimizationObj::SetParIsUsed(const string& parName,const bool use)
{
this->BuildRecursiveRefObjList();
for(int i=0;i<mRecursiveRefinedObjList.GetNb();i++)
mRecursiveRefinedObjList.GetObj(i).SetParIsUsed(parName,use);
}
void OptimizationObj::SetParIsUsed(const RefParType *type,const bool use)
{
this->BuildRecursiveRefObjList();
for(int i=0;i<mRecursiveRefinedObjList.GetNb();i++)
mRecursiveRefinedObjList.GetObj(i).SetParIsUsed(type,use);
}
void OptimizationObj::SetLimitsRelative(const string &parName,
const REAL min, const REAL max)
{
this->BuildRecursiveRefObjList();
for(int i=0;i<mRecursiveRefinedObjList.GetNb();i++)
mRecursiveRefinedObjList.GetObj(i).SetLimitsRelative(parName,min,max);
}
void OptimizationObj::SetLimitsRelative(const RefParType *type,
const REAL min, const REAL max)
{
this->BuildRecursiveRefObjList();
for(int i=0;i<mRecursiveRefinedObjList.GetNb();i++)
mRecursiveRefinedObjList.GetObj(i).SetLimitsRelative(type,min,max);
}
void OptimizationObj::SetLimitsAbsolute(const string &parName,
const REAL min, const REAL max)
{
this->BuildRecursiveRefObjList();
for(int i=0;i<mRecursiveRefinedObjList.GetNb();i++)
mRecursiveRefinedObjList.GetObj(i).SetLimitsAbsolute(parName,min,max);
}
void OptimizationObj::SetLimitsAbsolute(const RefParType *type,
const REAL min, const REAL max)
{
this->BuildRecursiveRefObjList();
for(int i=0;i<mRecursiveRefinedObjList.GetNb();i++)
mRecursiveRefinedObjList.GetObj(i).SetLimitsAbsolute(type,min,max);
}
REAL OptimizationObj::GetLogLikelihood() const
{
TAU_PROFILE("OptimizationObj::GetLogLikelihood()","void ()",TAU_DEFAULT);
REAL cost =0.;
for(int i=0;i<mRecursiveRefinedObjList.GetNb();i++)
{
const REAL tmp=mRecursiveRefinedObjList.GetObj(i).GetLogLikelihood();
if(tmp!=0.)
{
LogLikelihoodStats* st=&((mvContextObjStats[mContext])
[&(mRecursiveRefinedObjList.GetObj(i))]);
st->mTotalLogLikelihood += tmp;
st->mTotalLogLikelihoodDeltaSq +=
(tmp-st->mLastLogLikelihood)*(tmp-st->mLastLogLikelihood);
st->mLastLogLikelihood=tmp;
}
cost += mvObjWeight[&(mRecursiveRefinedObjList.GetObj(i))].mWeight * tmp;
}
return cost;
}
void OptimizationObj::StopAfterCycle()
{
VFN_DEBUG_MESSAGE("OptimizationObj::StopAfterCycle()",5)
if(mIsOptimizing)
{
#ifdef __WX__CRYST__
wxMutexLocker lock(mMutexStopAfterCycle);
#endif
mStopAfterCycle=true;
}
}
void OptimizationObj::DisplayReport()
{
//:TODO: ask all objects to print their own report ?
}
void OptimizationObj::AddRefinableObj(RefinableObj &obj)
{
VFN_DEBUG_MESSAGE("OptimizationObj::AddRefinableObj():"<<obj.GetName(),5)
//in case some object has been modified, to avoid rebuilding the entire list
this->BuildRecursiveRefObjList();
mRefinedObjList.Register(obj);
RefObjRegisterRecursive(obj,mRecursiveRefinedObjList);
#ifdef __WX__CRYST__
if(0!=this->WXGet()) this->WXGet()->AddRefinedObject(obj);
#endif
}
RefinableObj& OptimizationObj::GetFullRefinableObj(const bool rebuild)
{
if(rebuild) this->PrepareRefParList();
return mRefParList;
}
const string& OptimizationObj::GetName()const { return mName;}
void OptimizationObj::SetName(const string& name) {mName=name;}
const string OptimizationObj::GetClassName()const { return "OptimizationObj";}
void OptimizationObj::Print()const {this->XMLOutput(cout);}
void OptimizationObj::RestoreBestConfiguration()
{
//:TODO: check list of refinableObj has not changed, and the list of
// RefPar has not changed in all sub-objects.
if(mBestParSavedSetIndex>0) mRefParList.RestoreParamSet(mBestParSavedSetIndex);
}
bool OptimizationObj::IsOptimizing()const{return mIsOptimizing;}
void OptimizationObj::TagNewBestConfig()
{
for(int i=0;i<mRecursiveRefinedObjList.GetNb();i++)
mRecursiveRefinedObjList.GetObj(i).TagNewBestConfig();
mMainTracker.AppendValues(mNbTrial);
}
REAL OptimizationObj::GetLastOptimElapsedTime()const
{
return mLastOptimTime;
}
MainTracker& OptimizationObj::GetMainTracker(){return mMainTracker;}
const MainTracker& OptimizationObj::GetMainTracker()const{return mMainTracker;}
RefObjOpt& OptimizationObj::GetXMLAutoSaveOption() {return mXMLAutoSave;}
const RefObjOpt& OptimizationObj::GetXMLAutoSaveOption()const {return mXMLAutoSave;}
const REAL& OptimizationObj::GetBestCost()const{return mBestCost;}
REAL& OptimizationObj::GetBestCost(){return mBestCost;}
void OptimizationObj::BeginOptimization(const bool allowApproximations, const bool enableRestraints)
{
mvContextObjStats.clear();
for(int i=0;i<mRefinedObjList.GetNb();i++)
{
mRefinedObjList.GetObj(i).BeginOptimization(allowApproximations,enableRestraints);
}
}
void OptimizationObj::EndOptimization()
{
for(int i=0;i<mRefinedObjList.GetNb();i++) mRefinedObjList.GetObj(i).EndOptimization();
}
long& OptimizationObj::NbTrialPerRun() {return mNbTrialPerRun;}
const long& OptimizationObj::NbTrialPerRun() const {return mNbTrialPerRun;}
long OptimizationObj::GetTrial() const {return mNbTrial;}
long OptimizationObj::GetRun() const {return mRun;}
unsigned int OptimizationObj::GetNbOption()const
{
return mOptionRegistry.GetNb();
}
ObjRegistry<RefObjOpt>& OptimizationObj::GetOptionList()
{
return mOptionRegistry;
}
RefObjOpt& OptimizationObj::GetOption(const unsigned int i)
{
VFN_DEBUG_MESSAGE("RefinableObj::GetOption()"<<i,3)
//:TODO: Check
return mOptionRegistry.GetObj(i);
}
RefObjOpt& OptimizationObj::GetOption(const string & name)
{
VFN_DEBUG_MESSAGE("OptimizationObj::GetOption()"<<name,3)
const long i=mOptionRegistry.Find(name);
if(i<0)
{
this->Print();
throw ObjCrystException("OptimizationObj::GetOption(): cannot find option: "+name+" in object:"+this->GetName());
}
return mOptionRegistry.GetObj(i);
}
const RefObjOpt& OptimizationObj::GetOption(const unsigned int i)const
{
VFN_DEBUG_MESSAGE("RefinableObj::GetOption()"<<i,3)
//:TODO: Check
return mOptionRegistry.GetObj(i);
}
const RefObjOpt& OptimizationObj::GetOption(const string & name)const
{
VFN_DEBUG_MESSAGE("OptimizationObj::GetOption()"<<name,3)
const long i=mOptionRegistry.Find(name);
if(i<0)
{
this->Print();
throw ObjCrystException("OptimizationObj::GetOption(): cannot find option: "+name+" in object:"+this->GetName());
}
return mOptionRegistry.GetObj(i);
}
const ObjRegistry<RefinableObj>& OptimizationObj::GetRefinedObjList() const
{
return mRefinedObjList;
}
unsigned int OptimizationObj::GetNbParamSet() const
{
return mvSavedParamSet.size();
}
long OptimizationObj::GetParamSetIndex(const unsigned int i) const
{
if(i>=mvSavedParamSet.size())
throw ObjCrystException("OptimizationObj::GetSavedParamSetIndex(i): i > nb saved param set");
return mvSavedParamSet[i].first;
}
long OptimizationObj::GetParamSetCost(const unsigned int i) const
{
if(i>=mvSavedParamSet.size())
throw ObjCrystException("OptimizationObj::GetSavedParamSetCost(i): i > nb saved param set");
return mvSavedParamSet[i].second;
}
void OptimizationObj::RestoreParamSet(const unsigned int i, const bool update_display)
{
mRefParList.RestoreParamSet(this->GetParamSetIndex(i));
if(update_display) this->UpdateDisplay();
}
void OptimizationObj::PrepareRefParList()
{
VFN_DEBUG_ENTRY("OptimizationObj::PrepareRefParList()",6)
this->BuildRecursiveRefObjList();
// As any parameter been added in the recursive list of objects ?
// or has any object been added/removed ?
RefinableObjClock clock;
GetRefParListClockRecursive(mRecursiveRefinedObjList,clock);
if( (clock>mRefParList.GetRefParListClock())
||(mRecursiveRefinedObjList.GetRegistryClock()>mRefParList.GetRefParListClock()) )
{
VFN_DEBUG_MESSAGE("OptimizationObj::PrepareRefParList():Rebuild list",6)
mRefParList.ResetParList();
mRefParList.EraseAllParamSet();
for(int i=0;i<mRecursiveRefinedObjList.GetNb();i++)
mRefParList.AddPar(mRecursiveRefinedObjList.GetObj(i),true);
mvSavedParamSet.clear();
mBestParSavedSetIndex=mRefParList.CreateParamSet("Best Configuration");
mvSavedParamSet.push_back(make_pair(mBestParSavedSetIndex,mBestCost));
mMainTracker.ClearTrackers();
REAL (OptimizationObj::*fl)() const;
fl=&OptimizationObj::GetLogLikelihood;
mMainTracker.AddTracker(new TrackerObject<OptimizationObj>
(this->GetName()+"::Overall LogLikelihood",*this,fl));
for(long i=0;i<mRecursiveRefinedObjList.GetNb();i++)
{
REAL (RefinableObj::*fp)() const;
fp=&RefinableObj::GetLogLikelihood;
mMainTracker.AddTracker(new TrackerObject<RefinableObj>
(mRecursiveRefinedObjList.GetObj(i).GetName()+"::LogLikelihood",mRecursiveRefinedObjList.GetObj(i),fp));
if(mRecursiveRefinedObjList.GetObj(i).GetClassName()=="Crystal")
{
REAL (Crystal::*fc)() const;
const Crystal *pCryst=dynamic_cast<const Crystal *>(&(mRecursiveRefinedObjList.GetObj(i)));
fc=&Crystal::GetBumpMergeCost;
mMainTracker.AddTracker(new TrackerObject<Crystal>
(pCryst->GetName()+"::BumpMergeCost",*pCryst,fc));
fc=&Crystal::GetBondValenceCost;
mMainTracker.AddTracker(new TrackerObject<Crystal>
(pCryst->GetName()+"::BondValenceCost",*pCryst,fc));
}
}
}
// Prepare for refinement, ie get the list of not fixed parameters,
// and prepare the objects...
mRefParList.PrepareForRefinement();
for(int i=0;i<mRecursiveRefinedObjList.GetNb();i++)
mRecursiveRefinedObjList.GetObj(i).PrepareForRefinement();
VFN_DEBUG_EXIT("OptimizationObj::PrepareRefParList()",6)
}
void OptimizationObj::InitOptions()
{
VFN_DEBUG_MESSAGE("OptimizationObj::InitOptions()",5)
static string xmlAutoSaveName;
static string xmlAutoSaveChoices[6];
static bool needInitNames=true;
if(true==needInitNames)
{
xmlAutoSaveName="Save Best Config Regularly";
xmlAutoSaveChoices[0]="No";
xmlAutoSaveChoices[1]="Every day";
xmlAutoSaveChoices[2]="Every hour";
xmlAutoSaveChoices[3]="Every 10mn";
xmlAutoSaveChoices[4]="Every new best config (a lot ! Not Recommended !)";
xmlAutoSaveChoices[5]="Every Run (Recommended)";
needInitNames=false;//Only once for the class
}
mXMLAutoSave.Init(6,&xmlAutoSaveName,xmlAutoSaveChoices);
this->AddOption(&mXMLAutoSave);
VFN_DEBUG_MESSAGE("OptimizationObj::InitOptions():End",5)
}
void OptimizationObj::UpdateDisplay() const
{
Chronometer chrono;
for(int i=0;i<mRefinedObjList.GetNb();i++)
mRefinedObjList.GetObj(i).UpdateDisplay();
mMainTracker.UpdateDisplay();
}
void OptimizationObj::BuildRecursiveRefObjList()
{
// First check if anything has changed (ie if a sub-object has been
// added or removed in the recursive refinable object list)
RefinableObjClock clock;
GetSubRefObjListClockRecursive(mRefinedObjList,clock);
if(clock>mRecursiveRefinedObjList.GetRegistryClock())
{
VFN_DEBUG_ENTRY("OptimizationObj::BuildRecursiveRefObjList()",5)
mRecursiveRefinedObjList.DeRegisterAll();
for(int i=0;i<mRefinedObjList.GetNb();i++)
RefObjRegisterRecursive(mRefinedObjList.GetObj(i),mRecursiveRefinedObjList);
VFN_DEBUG_EXIT("OptimizationObj::BuildRecursiveRefObjList()",5)
}
}
void OptimizationObj::AddOption(RefObjOpt *opt)
{
VFN_DEBUG_ENTRY("OptimizationObj::AddOption()",5)
mOptionRegistry.Register(*opt);
VFN_DEBUG_EXIT("OptimizationObj::AddOption()",5)
}
//#################################################################################
//
// MonteCarloObj
//
//#################################################################################
MonteCarloObj::MonteCarloObj():
OptimizationObj(""),
mCurrentCost(-1),
mTemperatureMax(1e6),mTemperatureMin(.001),mTemperatureGamma(1.0),
mMutationAmplitudeMax(8.),mMutationAmplitudeMin(.125),mMutationAmplitudeGamma(1.0),
mNbTrialRetry(0),mMinCostRetry(0)
#ifdef __WX__CRYST__
,mpWXCrystObj(0)
#endif
{
VFN_DEBUG_ENTRY("MonteCarloObj::MonteCarloObj()",5)
this->InitOptions();
mGlobalOptimType.SetChoice(GLOBAL_OPTIM_PARALLEL_TEMPERING);
mAnnealingScheduleTemp.SetChoice(ANNEALING_SMART);
mAnnealingScheduleMutation.SetChoice(ANNEALING_EXPONENTIAL);
mXMLAutoSave.SetChoice(5);//Save after each Run
mAutoLSQ.SetChoice(0);
gOptimizationObjRegistry.Register(*this);
VFN_DEBUG_EXIT("MonteCarloObj::MonteCarloObj()",5)
}
MonteCarloObj::MonteCarloObj(const string name):
OptimizationObj(name),
mCurrentCost(-1),
mTemperatureMax(1e6),mTemperatureMin(.001),mTemperatureGamma(1.0),
mMutationAmplitudeMax(8.),mMutationAmplitudeMin(.125),mMutationAmplitudeGamma(1.0),
mNbTrialRetry(0),mMinCostRetry(0)
#ifdef __WX__CRYST__
,mpWXCrystObj(0)
#endif
{
VFN_DEBUG_ENTRY("MonteCarloObj::MonteCarloObj()",5)
this->InitOptions();
mGlobalOptimType.SetChoice(GLOBAL_OPTIM_PARALLEL_TEMPERING);
mAnnealingScheduleTemp.SetChoice(ANNEALING_SMART);
mAnnealingScheduleMutation.SetChoice(ANNEALING_EXPONENTIAL);
mXMLAutoSave.SetChoice(5);//Save after each Run
mAutoLSQ.SetChoice(0);
gOptimizationObjRegistry.Register(*this);
VFN_DEBUG_EXIT("MonteCarloObj::MonteCarloObj()",5)
}
MonteCarloObj::MonteCarloObj(const MonteCarloObj &old):
OptimizationObj(old),
mCurrentCost(old.mCurrentCost),
mTemperatureMax(old.mTemperatureMax),mTemperatureMin(old.mTemperatureMin),
mTemperatureGamma(old.mTemperatureGamma),
mMutationAmplitudeMax(old.mMutationAmplitudeMax),mMutationAmplitudeMin(old.mMutationAmplitudeMin),
mMutationAmplitudeGamma(old.mMutationAmplitudeGamma),
mNbTrialRetry(old.mNbTrialRetry),mMinCostRetry(old.mMinCostRetry)
#ifdef __WX__CRYST__
,mpWXCrystObj(0)
#endif
{
VFN_DEBUG_ENTRY("MonteCarloObj::MonteCarloObj(&old)",5)
this->InitOptions();
for(unsigned int i=0;i<this->GetNbOption();i++)
this->GetOption(i).SetChoice(old.GetOption(i).GetChoice());
gOptimizationObjRegistry.Register(*this);
VFN_DEBUG_EXIT("MonteCarloObj::MonteCarloObj(&old)",5)
}
MonteCarloObj::MonteCarloObj(const bool internalUseOnly):
OptimizationObj(""),
mCurrentCost(-1),
mTemperatureMax(.03),mTemperatureMin(.003),mTemperatureGamma(1.0),
mMutationAmplitudeMax(16.),mMutationAmplitudeMin(.125),mMutationAmplitudeGamma(1.0),
mNbTrialRetry(0),mMinCostRetry(0)
#ifdef __WX__CRYST__
,mpWXCrystObj(0)
#endif
{
VFN_DEBUG_ENTRY("MonteCarloObj::MonteCarloObj(bool)",5)
this->InitOptions();
mGlobalOptimType.SetChoice(GLOBAL_OPTIM_PARALLEL_TEMPERING);
mAnnealingScheduleTemp.SetChoice(ANNEALING_SMART);
mAnnealingScheduleMutation.SetChoice(ANNEALING_EXPONENTIAL);
mXMLAutoSave.SetChoice(5);//Save after each Run
mAutoLSQ.SetChoice(0);
if(false==internalUseOnly) gOptimizationObjRegistry.Register(*this);
VFN_DEBUG_EXIT("MonteCarloObj::MonteCarloObj(bool)",5)
}
MonteCarloObj::~MonteCarloObj()
{
VFN_DEBUG_ENTRY("MonteCarloObj::~MonteCarloObj()",5)
gOptimizationObjRegistry.DeRegister(*this);
VFN_DEBUG_EXIT ("MonteCarloObj::~MonteCarloObj()",5)
}
void MonteCarloObj::SetAlgorithmSimulAnnealing(const AnnealingSchedule scheduleTemp,
const REAL tMax, const REAL tMin,
const AnnealingSchedule scheduleMutation,
const REAL mutMax, const REAL mutMin,
const long nbTrialRetry,const REAL minCostRetry)
{
VFN_DEBUG_MESSAGE("MonteCarloObj::SetAlgorithmSimulAnnealing()",5)
mGlobalOptimType.SetChoice(GLOBAL_OPTIM_SIMULATED_ANNEALING);
mTemperatureMax=tMax;
mTemperatureMin=tMin;
mAnnealingScheduleTemp.SetChoice(scheduleTemp);
mMutationAmplitudeMax=mutMax;
mMutationAmplitudeMin=mutMin;
mAnnealingScheduleMutation.SetChoice(scheduleMutation);
mNbTrialRetry=nbTrialRetry;
mMinCostRetry=minCostRetry;
VFN_DEBUG_MESSAGE("MonteCarloObj::SetAlgorithmSimulAnnealing():End",3)
}
void MonteCarloObj::SetAlgorithmParallTempering(const AnnealingSchedule scheduleTemp,
const REAL tMax, const REAL tMin,
const AnnealingSchedule scheduleMutation,
const REAL mutMax, const REAL mutMin)
{
VFN_DEBUG_MESSAGE("MonteCarloObj::SetAlgorithmParallTempering()",5)
mGlobalOptimType.SetChoice(GLOBAL_OPTIM_PARALLEL_TEMPERING);
mTemperatureMax=tMax;
mTemperatureMin=tMin;
mAnnealingScheduleTemp.SetChoice(scheduleTemp);
mMutationAmplitudeMax=mutMax;
mMutationAmplitudeMin=mutMin;
mAnnealingScheduleMutation.SetChoice(scheduleMutation);
//mNbTrialRetry=nbTrialRetry;
//mMinCostRetry=minCostRetry;
VFN_DEBUG_MESSAGE("MonteCarloObj::SetAlgorithmParallTempering():End",3)
}
void MonteCarloObj::Optimize(long &nbStep,const bool silent,const REAL finalcost,
const REAL maxTime)
{
//:TODO: Other algorithms !
TAU_PROFILE("MonteCarloObj::Optimize()","void (long)",TAU_DEFAULT);
VFN_DEBUG_ENTRY("MonteCarloObj::Optimize()",5)
this->BeginOptimization(true);
this->PrepareRefParList();
this->InitLSQ(false);
mIsOptimizing=true;
if(mTemperatureGamma<0.1) mTemperatureGamma= 0.1;
if(mTemperatureGamma>10.0)mTemperatureGamma=10.0;
if(mMutationAmplitudeGamma<0.1) mMutationAmplitudeGamma= 0.1;
if(mMutationAmplitudeGamma>10.0)mMutationAmplitudeGamma=10.0;
// prepare all objects
this->TagNewBestConfig();
mCurrentCost=this->GetLogLikelihood();
mBestCost=mCurrentCost;
mvObjWeight.clear();
mMainTracker.ClearValues();
Chronometer chrono;
chrono.start();
switch(mGlobalOptimType.GetChoice())
{
case GLOBAL_OPTIM_SIMULATED_ANNEALING:
{
this->RunSimulatedAnnealing(nbStep,silent,finalcost,maxTime);
break;
}//case GLOBAL_OPTIM_SIMULATED_ANNEALING
case GLOBAL_OPTIM_PARALLEL_TEMPERING:
{
this->RunParallelTempering(nbStep,silent,finalcost,maxTime);
break;
}//case GLOBAL_OPTIM_PARALLEL_TEMPERING
case GLOBAL_OPTIM_RANDOM_LSQ: //:TODO:
{
long cycles = 1;
this->RunRandomLSQMethod(cycles);
break;
}//case GLOBAL_OPTIM_GENETIC
}
mIsOptimizing=false;
#ifdef __WX__CRYST__
mMutexStopAfterCycle.Lock();
#endif
mStopAfterCycle=false;
#ifdef __WX__CRYST__
mMutexStopAfterCycle.Unlock();
#endif
mRefParList.RestoreParamSet(mBestParSavedSetIndex);
this->EndOptimization();
(*fpObjCrystInformUser)((boost::format("Finished Optimization, final cost=%12.2f (dt=%.1fs)") % this->GetLogLikelihood() % chrono.seconds()).str());
if(mSaveTrackedData.GetChoice()==1)
{
ofstream outTracker;
outTracker.imbue(std::locale::classic());
const string outTrackerName=this->GetName()+"-Tracker.dat";
outTracker.open(outTrackerName.c_str());
mMainTracker.SaveAll(outTracker);
outTracker.close();
}
for(vector<pair<long,REAL> >::iterator pos=mvSavedParamSet.begin();pos!=mvSavedParamSet.end();++pos)
if(pos->first==mBestParSavedSetIndex)
{
if( (pos->second>mBestCost)
||(pos->second<0))
{
pos->second=mBestCost;
break;
}
}
this->UpdateDisplay();
VFN_DEBUG_EXIT("MonteCarloObj::Optimize()",5)
}
void MonteCarloObj::MultiRunOptimize(long &nbCycle,long &nbStep,const bool silent,
const REAL finalcost,const REAL maxTime)
{
//:TODO: Other algorithms !
TAU_PROFILE("MonteCarloObj::MultiRunOptimize()","void (long)",TAU_DEFAULT);
VFN_DEBUG_ENTRY("MonteCarloObj::MultiRunOptimize()",5)
//Keep a copy of the total number of steps, and decrement nbStep
const long nbStep0=nbStep;
this->BeginOptimization(true);
this->PrepareRefParList();
this->InitLSQ(false);
mIsOptimizing=true;
if(mTemperatureGamma<0.1) mTemperatureGamma= 0.1;
if(mTemperatureGamma>10.0)mTemperatureGamma=10.0;
if(mMutationAmplitudeGamma<0.1) mMutationAmplitudeGamma= 0.1;
if(mMutationAmplitudeGamma>10.0)mMutationAmplitudeGamma=10.0;
// prepare all objects
mCurrentCost=this->GetLogLikelihood();
mBestCost=mCurrentCost;
this->TagNewBestConfig();
mvObjWeight.clear();
long nbTrialCumul=0;
const long nbCycle0=nbCycle;
Chronometer chrono;
mRun = 0;
while(nbCycle!=0)
{
if(!silent) cout <<"MonteCarloObj::MultiRunOptimize: Starting Run#"<<abs(nbCycle)<<endl;
nbStep=nbStep0;
for(int i=0;i<mRefinedObjList.GetNb();i++) mRefinedObjList.GetObj(i).RandomizeConfiguration();
mMainTracker.ClearValues();
chrono.start();
switch(mGlobalOptimType.GetChoice())
{
case GLOBAL_OPTIM_SIMULATED_ANNEALING:
{
try{this->RunSimulatedAnnealing(nbStep,silent,finalcost,maxTime);}
catch(...){cout<<"Unhandled exception in MonteCarloObj::MultiRunOptimize() ?"<<endl;}
break;
}
case GLOBAL_OPTIM_PARALLEL_TEMPERING:
{
try{this->RunParallelTempering(nbStep,silent,finalcost,maxTime);}
catch(...){cout<<"Unhandled exception in MonteCarloObj::MultiRunOptimize() ?"<<endl;}
break;
}
case GLOBAL_OPTIM_RANDOM_LSQ:
{
try{this->RunRandomLSQMethod(nbCycle);}
catch(...){cout<<"Unhandled exception in MonteCarloObj::RunRandomLSQMethod() ?"<<endl;}
//nbCycle=1;
break;
}
}
nbTrialCumul+=(nbStep0-nbStep);
if(finalcost>1)
(*fpObjCrystInformUser)((boost::format("Finished Run #%d, final cost=%12.2f, nbTrial=%d (dt=%.1fs), so far <nbTrial>=%d")
% (nbCycle0-nbCycle) % this->GetLogLikelihood() % (nbStep0-nbStep) % chrono.seconds() % (nbTrialCumul/(nbCycle0-nbCycle+1))).str());
else
(*fpObjCrystInformUser)((boost::format("Finished Run #%d, final cost=%12.2f, nbTrial=%d (dt=%.1fs)")
% (nbCycle0-nbCycle) % this->GetLogLikelihood() % (nbStep0-nbStep) % chrono.seconds()).str());
nbStep=nbStep0;
if(false==mStopAfterCycle) this->UpdateDisplay();
stringstream s;
s<<"Run #"<<abs(nbCycle);
mvSavedParamSet.push_back(make_pair(mRefParList.CreateParamSet(s.str()),mCurrentCost));
if(!silent) cout <<"MonteCarloObj::MultiRunOptimize: Finished Run#"
<<abs(nbCycle)<<", Run Best Cost:"<<mCurrentCost
<<", Overall Best Cost:"<<mBestCost<<endl;
if(mXMLAutoSave.GetChoice()==5)
{
string saveFileName=this->GetName();
time_t date=time(0);
char strDate[40];
strftime(strDate,sizeof(strDate),"%Y-%m-%d_%H-%M-%S",localtime(&date));//%Y-%m-%dT%H:%M:%S%Z
char costAsChar[30];
sprintf(costAsChar,"-Run#%ld-Cost-%f",abs(nbCycle),this->GetLogLikelihood());
saveFileName=saveFileName+(string)strDate+(string)costAsChar+(string)".xml";
XMLCrystFileSaveGlobal(saveFileName);
}
if(mSaveTrackedData.GetChoice()==1)
{
ofstream outTracker;
outTracker.imbue(std::locale::classic());
char runNum[40];
sprintf(runNum,"-Tracker-Run#%ld.dat",abs(nbCycle));
const string outTrackerName=this->GetName()+runNum;
outTracker.open(outTrackerName.c_str());
mMainTracker.SaveAll(outTracker);
outTracker.close();
}
nbCycle--;
#ifdef __WX__CRYST__
mMutexStopAfterCycle.Lock();
#endif
if(mStopAfterCycle)
{
#ifdef __WX__CRYST__
mMutexStopAfterCycle.Unlock();
#endif
break;
}
#ifdef __WX__CRYST__
mMutexStopAfterCycle.Unlock();
#endif
mRun++;
}
mIsOptimizing=false;
mRefParList.RestoreParamSet(mBestParSavedSetIndex);
for(vector<pair<long,REAL> >::iterator pos=mvSavedParamSet.begin();pos!=mvSavedParamSet.end();++pos)
if(pos->first==mBestParSavedSetIndex)
{
if( (pos->second>mBestCost)
||(pos->second<0))
{
pos->second=mBestCost;
break;
}
}
this->EndOptimization();
if(false==mStopAfterCycle) this->UpdateDisplay();
#ifdef __WX__CRYST__
mMutexStopAfterCycle.Lock();
#endif
mStopAfterCycle=false;
#ifdef __WX__CRYST__
mMutexStopAfterCycle.Unlock();
#endif
if(finalcost>1)
cout<<endl<<"Finished all runs, number of trials to reach cost="
<<finalcost<<" : <nbTrial>="<<nbTrialCumul/(nbCycle0-nbCycle)<<endl;
VFN_DEBUG_EXIT("MonteCarloObj::MultiRunOptimize()",5)
}
void MonteCarloObj::RunSimulatedAnnealing(long &nbStep,const bool silent,
const REAL finalcost,const REAL maxTime)
{
//Keep a copy of the total number of steps, and decrement nbStep
const long nbSteps=nbStep;
unsigned int accept;// 1 if last trial was accepted? 2 if new best config ? else 0
mNbTrial=0;
// time (in seconds) when last autoSave was made (if enabled)
unsigned long secondsWhenAutoSave=0;
if(!silent) cout << "Starting Simulated Annealing Optimization for"<<nbSteps<<" trials"<<endl;
if(!silent) this->DisplayReport();
REAL runBestCost;
mCurrentCost=this->GetLogLikelihood();
runBestCost=mCurrentCost;
const long lastParSavedSetIndex=mRefParList.CreateParamSet("MonteCarloObj:Last parameters (SA)");
const long runBestIndex=mRefParList.CreateParamSet("Best parameters for current run (SA)");
//Report each ... cycles
const int nbTryReport=3000;
// Keep record of the number of accepted moves
long nbAcceptedMoves=0;//since last report
long nbAcceptedMovesTemp=0;//since last temperature/mutation rate change
// Number of tries since best configuration found
long nbTriesSinceBest=0;
// Change temperature (and mutation) every...
const int nbTryPerTemp=300;
mTemperature=sqrt(mTemperatureMin*mTemperatureMax);
mMutationAmplitude=sqrt(mMutationAmplitudeMin*mMutationAmplitudeMax);
// Do we need to update the display ?
bool needUpdateDisplay=false;
Chronometer chrono;
chrono.start();
for(mNbTrial=1;mNbTrial<=nbSteps;)
{
if((mNbTrial % nbTryPerTemp) == 1)
{
VFN_DEBUG_MESSAGE("-> Updating temperature and mutation amplitude.",3)
// Temperature & displacements amplitude
switch(mAnnealingScheduleTemp.GetChoice())
{
case ANNEALING_BOLTZMANN:
mTemperature=
mTemperatureMin*log((REAL)nbSteps)/log((REAL)(mNbTrial+1));break;
case ANNEALING_CAUCHY:
mTemperature=mTemperatureMin*nbSteps/mNbTrial;break;
//case ANNEALING_QUENCHING:
case ANNEALING_EXPONENTIAL:
mTemperature=mTemperatureMax
*pow(mTemperatureMin/mTemperatureMax,
mNbTrial/(REAL)nbSteps);break;
case ANNEALING_GAMMA:
mTemperature=mTemperatureMax+(mTemperatureMin-mTemperatureMax)
*pow(mNbTrial/(REAL)nbSteps,mTemperatureGamma);break;
case ANNEALING_SMART:
{
if((nbAcceptedMovesTemp/(REAL)nbTryPerTemp)>0.30)
mTemperature/=1.5;
if((nbAcceptedMovesTemp/(REAL)nbTryPerTemp)<0.10)
mTemperature*=1.5;
if(mTemperature>mTemperatureMax) mTemperature=mTemperatureMax;
if(mTemperature<mTemperatureMin) mTemperature=mTemperatureMin;
nbAcceptedMovesTemp=0;
break;
}
default: mTemperature=mTemperatureMin;break;
}
switch(mAnnealingScheduleMutation.GetChoice())
{
case ANNEALING_BOLTZMANN:
mMutationAmplitude=
mMutationAmplitudeMin*log((REAL)nbSteps)/log((REAL)(mNbTrial+1));
break;
case ANNEALING_CAUCHY:
mMutationAmplitude=mMutationAmplitudeMin*nbSteps/mNbTrial;break;
//case ANNEALING_QUENCHING:
case ANNEALING_EXPONENTIAL:
mMutationAmplitude=mMutationAmplitudeMax
*pow(mMutationAmplitudeMin/mMutationAmplitudeMax,
mNbTrial/(REAL)nbSteps);break;
case ANNEALING_GAMMA:
mMutationAmplitude=mMutationAmplitudeMax+(mMutationAmplitudeMin-mMutationAmplitudeMax)
*pow(mNbTrial/(REAL)nbSteps,mMutationAmplitudeGamma);break;
case ANNEALING_SMART:
if((nbAcceptedMovesTemp/(REAL)nbTryPerTemp)>0.3) mMutationAmplitude*=2.;
if((nbAcceptedMovesTemp/(REAL)nbTryPerTemp)<0.1) mMutationAmplitude/=2.;
if(mMutationAmplitude>mMutationAmplitudeMax)
mMutationAmplitude=mMutationAmplitudeMax;
if(mMutationAmplitude<mMutationAmplitudeMin)
mMutationAmplitude=mMutationAmplitudeMax;
nbAcceptedMovesTemp=0;
break;
default: mMutationAmplitude=mMutationAmplitudeMin;break;
}
}
this->NewConfiguration();
accept=0;
REAL cost=this->GetLogLikelihood();
if(cost<mCurrentCost)
{
accept=1;
mCurrentCost=cost;
mRefParList.SaveParamSet(lastParSavedSetIndex);
if(mCurrentCost<runBestCost)
{
accept=2;
runBestCost=mCurrentCost;
this->TagNewBestConfig();
needUpdateDisplay=true;
mRefParList.SaveParamSet(runBestIndex);
if(runBestCost<mBestCost)
{
mBestCost=mCurrentCost;
mRefParList.SaveParamSet(mBestParSavedSetIndex);
if(!silent) cout << "Trial :" << mNbTrial
<< " Temp="<< mTemperature
<< " Mutation Ampl.: "<<mMutationAmplitude
<< " NEW OVERALL Best Cost="<<runBestCost<< endl;
}
else if(!silent) cout << "Trial :" << mNbTrial
<< " Temp="<< mTemperature
<< " Mutation Ampl.: "<<mMutationAmplitude
<< " NEW Run Best Cost="<<runBestCost<< endl;
nbTriesSinceBest=0;
}
nbAcceptedMoves++;
nbAcceptedMovesTemp++;
}
else
{
if( log((rand()+1)/(REAL)RAND_MAX) < (-(cost-mCurrentCost)/mTemperature) )
{
accept=1;
mCurrentCost=cost;
mRefParList.SaveParamSet(lastParSavedSetIndex);
nbAcceptedMoves++;
nbAcceptedMovesTemp++;
}
}
if(accept==0) mRefParList.RestoreParamSet(lastParSavedSetIndex);
if( (mNbTrial % nbTryReport) == 0)
{
if(!silent) cout <<"Trial :" << mNbTrial << " Temp="<< mTemperature;
if(!silent) cout <<" Mutation Ampl.: " <<mMutationAmplitude<< " Best Cost=" << runBestCost
<<" Current Cost=" << mCurrentCost
<<" Accepting "<<(int)((REAL)nbAcceptedMoves/nbTryReport*100)
<<"% moves" << endl;
nbAcceptedMoves=0;
#ifdef __WX__CRYST__
if(0!=mpWXCrystObj) mpWXCrystObj->UpdateDisplayNbTrial();
#endif
}
mNbTrial++;nbStep--;
#ifdef __WX__CRYST__
mMutexStopAfterCycle.Lock();
#endif
if((runBestCost<finalcost) || mStopAfterCycle ||( (maxTime>0)&&(chrono.seconds()>maxTime)))
{
#ifdef __WX__CRYST__
mMutexStopAfterCycle.Unlock();
#endif
if(!silent) cout << endl <<endl << "Refinement Stopped."<<endl;
break;
}
#ifdef __WX__CRYST__
mMutexStopAfterCycle.Unlock();
#endif
nbTriesSinceBest++;
if( ((mXMLAutoSave.GetChoice()==1)&&((chrono.seconds()-secondsWhenAutoSave)>86400))
||((mXMLAutoSave.GetChoice()==2)&&((chrono.seconds()-secondsWhenAutoSave)>3600))
||((mXMLAutoSave.GetChoice()==3)&&((chrono.seconds()-secondsWhenAutoSave)> 600))
||((mXMLAutoSave.GetChoice()==4)&&(accept==2)) )
{
secondsWhenAutoSave=(unsigned long)chrono.seconds();
string saveFileName=this->GetName();
time_t date=time(0);
char strDate[40];
strftime(strDate,sizeof(strDate),"%Y-%m-%d_%H-%M-%S",localtime(&date));//%Y-%m-%dT%H:%M:%S%Z
char costAsChar[30];
if(accept!=2) mRefParList.RestoreParamSet(mBestParSavedSetIndex);
sprintf(costAsChar,"-Cost-%f",this->GetLogLikelihood());
saveFileName=saveFileName+(string)strDate+(string)costAsChar+(string)".xml";
XMLCrystFileSaveGlobal(saveFileName);
if(accept!=2) mRefParList.RestoreParamSet(lastParSavedSetIndex);
}
if((mNbTrial%300==0)&&needUpdateDisplay)
{
this->UpdateDisplay();
needUpdateDisplay=false;
mRefParList.RestoreParamSet(lastParSavedSetIndex);
}
}
//cout<<"Beginning final LSQ refinement? ... ";
if(mAutoLSQ.GetChoice()>0)
{// LSQ
if(!silent) cout<<"Beginning final LSQ refinement"<<endl;
for(int i=0;i<mRefinedObjList.GetNb();i++) mRefinedObjList.GetObj(i).SetApproximationFlag(false);
mRefParList.RestoreParamSet(runBestIndex);
mCurrentCost=this->GetLogLikelihood();
try {mLSQ.Refine(-50,true,true,false,0.001);}
catch(const ObjCrystException &except){};
if(!silent) cout<<"LSQ cost: "<<mCurrentCost<<" -> "<<this->GetLogLikelihood()<<endl;
// Need to go back to optimization with approximations allowed (they are not during LSQ)
for(int i=0;i<mRefinedObjList.GetNb();i++) mRefinedObjList.GetObj(i).SetApproximationFlag(true);
REAL cost=this->GetLogLikelihood();
if(cost<mCurrentCost)
{
mCurrentCost=cost;
mRefParList.SaveParamSet(lastParSavedSetIndex);
if(mCurrentCost<runBestCost)
{
runBestCost=mCurrentCost;
mRefParList.SaveParamSet(runBestIndex);
if(runBestCost<mBestCost)
{
mBestCost=mCurrentCost;
mRefParList.SaveParamSet(mBestParSavedSetIndex);
if(!silent) cout << "LSQ : NEW OVERALL Best Cost="<<runBestCost<< endl;
}
else if(!silent) cout << " LSQ : NEW Run Best Cost="<<runBestCost<< endl;
}
}
if(!silent) cout<<"Finished LSQ refinement"<<endl;
}
mLastOptimTime=chrono.seconds();
//Restore Best values
mRefParList.RestoreParamSet(runBestIndex);
mRefParList.ClearParamSet(runBestIndex);
mRefParList.ClearParamSet(lastParSavedSetIndex);
mCurrentCost=this->GetLogLikelihood();
if(!silent) this->DisplayReport();
if(!silent) chrono.print();
}
/*
void MonteCarloObj::RunNondestructiveLSQRefinement( int nbCycle,bool useLevenbergMarquardt,
const bool silent, const bool callBeginEndOptimization,
const float minChi2var )
{
float bsigma=-1, bdelta=-1;
float asigma=-1, adelta=-1;
//set the sigma values lower - it makes the molecular model more stable for LSQ
for(int i=0;i<mRefinedObjList.GetNb();i++) {
if(mRefinedObjList.GetObj(i).GetClassName()=="Crystal") {
try {
Crystal * pCryst = dynamic_cast<Crystal *>(&(mRefinedObjList.GetObj(i)));
for(int s=0;s<pCryst->GetScattererRegistry().GetNb();s++) {
Molecule *pMol=dynamic_cast<Molecule*>(&(pCryst->GetScatt(s)));
if(pMol==NULL) continue; // not a Molecule
for(vector<MolBond*>::iterator pos = pMol->GetBondList().begin(); pos != pMol->GetBondList().end();++pos) {
bsigma = (*pos)->GetLengthSigma();
bdelta = (*pos)->GetLengthDelta();
(*pos)->SetLengthDelta(0.02);
(*pos)->SetLengthSigma(0.001);
}
for(vector<MolBondAngle*>::iterator pos=pMol->GetBondAngleList().begin();pos != pMol->GetBondAngleList().end();++pos)
{
asigma = (*pos)->GetAngleSigma();
adelta = (*pos)->GetAngleDelta();
(*pos)->SetAngleDelta(0.2*DEG2RAD);
(*pos)->SetAngleSigma(0.01*DEG2RAD);
}
}
} catch (const std::bad_cast& e) {
}
}
}
for(int i=0;i<mRefinedObjList.GetNb();i++) mRefinedObjList.GetObj(i).SetApproximationFlag(false);
try {
mLSQ.Refine(nbCycle,useLevenbergMarquardt,silent,callBeginEndOptimization,minChi2var);
}
catch(const ObjCrystException &except) {
};
for(int i=0;i<mRefinedObjList.GetNb();i++) mRefinedObjList.GetObj(i).SetApproximationFlag(true);
if(bsigma<0 || bdelta<0 || asigma<0 || adelta<0) return;
//restore the delta and sigma values
for(int i=0;i<mRefinedObjList.GetNb();i++) {
if(mRefinedObjList.GetObj(i).GetClassName()=="Crystal") {
try {
Crystal * pCryst = dynamic_cast<Crystal *>(&(mRefinedObjList.GetObj(i)));
for(int s=0;s<pCryst->GetScattererRegistry().GetNb();s++) {
Molecule *pMol=dynamic_cast<Molecule*>(&(pCryst->GetScatt(s)));
if(pMol==NULL) continue; // not a Molecule
for(vector<MolBond*>::iterator pos = pMol->GetBondList().begin(); pos != pMol->GetBondList().end();++pos) {
(*pos)->SetLengthDelta(bdelta);
(*pos)->SetLengthSigma(bsigma);
}
for(vector<MolBondAngle*>::iterator pos=pMol->GetBondAngleList().begin();pos != pMol->GetBondAngleList().end();++pos)
{
(*pos)->SetAngleDelta(adelta);
(*pos)->SetAngleSigma(asigma);
}
}
} catch (const std::bad_cast& e) {
}
}
}
}
*/
void MonteCarloObj::RunRandomLSQMethod(long &nbCycle)
{
//perform random move
mMutationAmplitude=mMutationAmplitudeMax;
float bsigma=-1, bdelta=-1;
float asigma=-1, adelta=-1;
//set the delta and sigma values - low values are good for LSQ!
for(int i=0;i<mRefinedObjList.GetNb();i++) {
if(mRefinedObjList.GetObj(i).GetClassName()=="Crystal") {
try {
Crystal * pCryst = dynamic_cast<Crystal *>(&(mRefinedObjList.GetObj(i)));
for(int s=0;s<pCryst->GetScattererRegistry().GetNb();s++)
{
Molecule *pMol=dynamic_cast<Molecule*>(&(pCryst->GetScatt(s)));
if(pMol==NULL) continue; // not a Molecule
for(vector<MolBond*>::iterator pos = pMol->GetBondList().begin(); pos != pMol->GetBondList().end();++pos) {
bsigma = (*pos)->GetLengthSigma();
bdelta = (*pos)->GetLengthDelta();
(*pos)->SetLengthDelta(0.02);
(*pos)->SetLengthSigma(0.001);
}
for(vector<MolBondAngle*>::iterator pos=pMol->GetBondAngleList().begin();pos != pMol->GetBondAngleList().end();++pos)
{
asigma = (*pos)->GetAngleSigma();
adelta = (*pos)->GetAngleDelta();
(*pos)->SetAngleDelta(0.2*DEG2RAD);
(*pos)->SetAngleSigma(0.01*DEG2RAD);
}
}
} catch (const std::bad_cast& e){
}
}
}
const long starting_point=mRefParList.CreateParamSet("MonteCarloObj:Last parameters (RANDOM-LSQ)");
mRefParList.SaveParamSet(starting_point);
mRun = 0;
while(nbCycle!=0) {
nbCycle--;
mRefParList.RestoreParamSet(starting_point);
//this->NewConfiguration();
for(int i=0;i<mRefinedObjList.GetNb();i++) mRefinedObjList.GetObj(i).RandomizeConfiguration();
this->UpdateDisplay();
//perform LSQ
for(int i=0;i<mRefinedObjList.GetNb();i++) mRefinedObjList.GetObj(i).SetApproximationFlag(false);
//mCurrentCost=this->GetLogLikelihood();
try {
mLSQ.Refine(20,true,true,false,0.001);
}
catch(const ObjCrystException &except) {
//cout<<"Something wrong?"<<endl;
};
for(int i=0;i<mRefinedObjList.GetNb();i++) mRefinedObjList.GetObj(i).SetApproximationFlag(true);
//cout<<"LSQ cost: "<<mCurrentCost<<" -> "<<this->GetLogLikelihood()<<endl;
REAL lsq_cost=this->GetLogLikelihood();
mCurrentCost = lsq_cost;
//mRefParList.SaveParamSet(lsqtParSavedSetIndex);
if(mCurrentCost<mBestCost)
{
mBestCost=mCurrentCost;
mRefParList.SaveParamSet(mBestParSavedSetIndex);
}
this->UpdateDisplay();
//save it to the file
string saveFileName=this->GetName();
time_t date=time(0);
char strDate[40];
strftime(strDate,sizeof(strDate),"%Y-%m-%d_%H-%M-%S",localtime(&date));//%Y-%m-%dT%H:%M:%S%Z
char costAsChar[30];
sprintf(costAsChar,"#Run%ld-Cost-%f",nbCycle, mCurrentCost);
saveFileName=saveFileName+(string)strDate+(string)costAsChar+(string)".xml";
XMLCrystFileSaveGlobal(saveFileName);
#ifdef __WX__CRYST__
mMutexStopAfterCycle.Lock();
#endif
if(mStopAfterCycle)
{
#ifdef __WX__CRYST__
mMutexStopAfterCycle.Unlock();
#endif
break;
}
#ifdef __WX__CRYST__
mMutexStopAfterCycle.Unlock();
#endif
mRun++;
}
if(bsigma<0 || bdelta<0 || asigma<0 || adelta<0) return;
//restore the delta and sigma values
for(int i=0;i<mRefinedObjList.GetNb();i++) {
if(mRefinedObjList.GetObj(i).GetClassName()=="Crystal") {
try {
Crystal * pCryst = dynamic_cast<Crystal *>(&(mRefinedObjList.GetObj(i)));
for(int s=0;s<pCryst->GetScattererRegistry().GetNb();s++)
{
Molecule *pMol=dynamic_cast<Molecule*>(&(pCryst->GetScatt(s)));
if(pMol==NULL) continue; // not a Molecule
for(vector<MolBond*>::iterator pos = pMol->GetBondList().begin(); pos != pMol->GetBondList().end();++pos) {
(*pos)->SetLengthDelta(bdelta);
(*pos)->SetLengthSigma(bsigma);
}
for(vector<MolBondAngle*>::iterator pos=pMol->GetBondAngleList().begin();pos != pMol->GetBondAngleList().end();++pos)
{
(*pos)->SetAngleDelta(adelta);
(*pos)->SetAngleSigma(asigma);
}
}
} catch (const std::bad_cast& e){
}
}
}
}
void MonteCarloObj::RunParallelTempering(long &nbStep,const bool silent,
const REAL finalcost,const REAL maxTime)
{
TAU_PROFILE("MonteCarloObj::RunParallelTempering()","void ()",TAU_DEFAULT);
TAU_PROFILE_TIMER(timer0a,"MonteCarloObj::RunParallelTempering() Begin 1","", TAU_FIELD);
TAU_PROFILE_TIMER(timer0b,"MonteCarloObj::RunParallelTempering() Begin 2","", TAU_FIELD);
TAU_PROFILE_TIMER(timer1,"MonteCarloObj::RunParallelTempering() New Config + LLK","", TAU_FIELD);
TAU_PROFILE_TIMER(timerN,"MonteCarloObj::RunParallelTempering() Finish","", TAU_FIELD);
TAU_PROFILE_START(timer0a);
//Keep a copy of the total number of steps, and decrement nbStep
const long nbSteps=nbStep;
unsigned int accept;// 1 if last trial was accepted? 2 if new best config ? else 0
mNbTrial=0;
// time (in seconds) when last autoSave was made (if enabled)
unsigned long secondsWhenAutoSave=0;
// Periodicity of the automatic LSQ refinements (if the option is set)
const unsigned int autoLSQPeriod=150000;
if(!silent) cout << "Starting Parallel Tempering Optimization"<<endl;
//Total number of parallel refinements,each is a 'World'. The most stable
// world must be i=nbWorld-1, and the most changing World (high mutation,
// high temperature) is i=0.
const long nbWorld=30;
CrystVector_long worldSwapIndex(nbWorld);
for(int i=0;i<nbWorld;++i) worldSwapIndex(i)=i;
// Number of successive trials for each World. At the end of these trials
// a swap is tried with the upper World (eg i-1). This number effectvely sets
// the rate of swapping.
const int nbTryPerWorld=10;
// Initialize the costs
mCurrentCost=this->GetLogLikelihood();
REAL runBestCost=mCurrentCost;
CrystVector_REAL currentCost(nbWorld);
currentCost=mCurrentCost;
// Init the different temperatures
CrystVector_REAL simAnnealTemp(nbWorld);
for(int i=0;i<nbWorld;i++)
{
switch(mAnnealingScheduleTemp.GetChoice())
{
case ANNEALING_BOLTZMANN:
simAnnealTemp(i)=
mTemperatureMin*log((REAL)nbWorld)/log((REAL)(i+2));break;
case ANNEALING_CAUCHY:
simAnnealTemp(i)=mTemperatureMin*nbWorld/(i+1);break;
//case ANNEALING_QUENCHING:
case ANNEALING_EXPONENTIAL:
simAnnealTemp(i)=mTemperatureMax
*pow(mTemperatureMin/mTemperatureMax,
i/(REAL)(nbWorld-1));break;
case ANNEALING_GAMMA:
simAnnealTemp(i)=mTemperatureMax+(mTemperatureMin-mTemperatureMax)
*pow(i/(REAL)(nbWorld-1),mTemperatureGamma);break;
case ANNEALING_SMART:
simAnnealTemp(i)=mCurrentCost/(100.+(REAL)i/(REAL)nbWorld*900.);break;
default:
simAnnealTemp(i)=mCurrentCost/(100.+(REAL)i/(REAL)nbWorld*900.);break;
}
}
//Init the different mutation rate parameters
CrystVector_REAL mutationAmplitude(nbWorld);
for(int i=0;i<nbWorld;i++)
{
switch(mAnnealingScheduleMutation.GetChoice())
{
case ANNEALING_BOLTZMANN:
mutationAmplitude(i)=
mMutationAmplitudeMin*log((REAL)(nbWorld-1))/log((REAL)(i+2));
break;
case ANNEALING_CAUCHY:
mutationAmplitude(i)=mMutationAmplitudeMin*(REAL)(nbWorld-1)/(i+1);break;
//case ANNEALING_QUENCHING:
case ANNEALING_EXPONENTIAL:
mutationAmplitude(i)=mMutationAmplitudeMax
*pow(mMutationAmplitudeMin/mMutationAmplitudeMax,
i/(REAL)(nbWorld-1));break;
case ANNEALING_GAMMA:
mutationAmplitude(i)=mMutationAmplitudeMax+(mMutationAmplitudeMin-mMutationAmplitudeMax)
*pow(i/(REAL)(nbWorld-1),mMutationAmplitudeGamma);break;
case ANNEALING_SMART:
mutationAmplitude(i)=sqrt(mMutationAmplitudeMin*mMutationAmplitudeMax);break;
default:
mutationAmplitude(i)=sqrt(mMutationAmplitudeMin*mMutationAmplitudeMax);break;
}
}
// Init the parameter sets for each World
// All Worlds start from the same (current) configuration.
CrystVector_long worldCurrentSetIndex(nbWorld);
for(int i=nbWorld-1;i>=0;i--)
{
if((i!=(nbWorld-1))&&(i%2==0))
for(int j=0;j<mRecursiveRefinedObjList.GetNb();j++)
mRecursiveRefinedObjList.GetObj(j).RandomizeConfiguration();
worldCurrentSetIndex(i)=mRefParList.CreateParamSet();
mRefParList.RestoreParamSet(worldCurrentSetIndex(nbWorld-1));
}
TAU_PROFILE_STOP(timer0a);
TAU_PROFILE_START(timer0b);
//mNbTrial=nbSteps;;
const long lastParSavedSetIndex=mRefParList.CreateParamSet("MonteCarloObj:Last parameters (PT)");
const long runBestIndex=mRefParList.CreateParamSet("Best parameters for current run (PT)");
CrystVector_REAL swapPar;
//Keep track of how many trials are accepted for each World
CrystVector_long worldNbAcceptedMoves(nbWorld);
worldNbAcceptedMoves=0;
//Do a report each... And check if mutation rate is OK (for annealing_smart)s
const int nbTrialsReport=3000;
// TEST : allow GENETIC mating of configurations
//Get gene groups list :TODO: check for missing groups
CrystVector_uint refParGeneGroupIndex(mRefParList.GetNbPar());
unsigned int first=1;
for(int i=0;i<mRecursiveRefinedObjList.GetNb();i++)
mRecursiveRefinedObjList.GetObj(i).GetGeneGroup(mRefParList,refParGeneGroupIndex,first);
#if 0
if(!silent)
for(int i=0;i<mRefParList.GetNbPar();i++)
{
cout << "Gene Group:"<<refParGeneGroupIndex(i)<<" :";
mRefParList.GetPar(i).Print();
}
#endif
// number of gene groups
// to select which gene groups are exchanged in the mating
//const unsigned int nbGeneGroup=refParGeneGroupIndex.max();
//CrystVector_int crossoverGroupIndex(nbGeneGroup);
//const long parSetOffspringA=mRefParList.CreateParamSet("Offspring A");
//const long parSetOffspringB=mRefParList.CreateParamSet("Offspring B");
// record the statistical distribution n=f(cost function) for each World
//CrystMatrix_REAL trialsDensity(100,nbWorld+1);
//trialsDensity=0;
//for(int i=0;i<100;i++) trialsDensity(i,0)=i/(float)100;
// Do we need to update the display ?
bool needUpdateDisplay=false;
//Do the refinement
bool makeReport=false;
Chronometer chrono;
chrono.start();
float lastUpdateDisplayTime=chrono.seconds();
TAU_PROFILE_STOP(timer0b);
for(;mNbTrial<nbSteps;)
{
for(int i=0;i<nbWorld;i++)
{
mContext=i;
//mRefParList.RestoreParamSet(worldCurrentSetIndex(i));
mMutationAmplitude=mutationAmplitude(i);
mTemperature=simAnnealTemp(i);
for(int j=0;j<nbTryPerWorld;j++)
{
//mRefParList.SaveParamSet(lastParSavedSetIndex);
TAU_PROFILE_START(timer1);
mRefParList.RestoreParamSet(worldCurrentSetIndex(i));
this->NewConfiguration();
accept=0;
REAL cost=this->GetLogLikelihood();
TAU_PROFILE_STOP(timer1);
//trialsDensity((long)(cost*100.),i+1)+=1;
if(cost<currentCost(i))
{
accept=1;
currentCost(i)=cost;
mRefParList.SaveParamSet(worldCurrentSetIndex(i));
if(cost<runBestCost)
{
accept=2;
runBestCost=currentCost(i);
this->TagNewBestConfig();
needUpdateDisplay=true;
mRefParList.SaveParamSet(runBestIndex);
if(runBestCost<mBestCost)
{
mBestCost=currentCost(i);
mRefParList.SaveParamSet(mBestParSavedSetIndex);
if(!silent) cout << "->Trial :" << mNbTrial
<< " World="<< worldSwapIndex(i)
<< " Temp="<< mTemperature
<< " Mutation Ampl.: "<<mMutationAmplitude
<< " NEW OVERALL Best Cost="<<mBestCost<< endl;
}
else if(!silent) cout << "->Trial :" << mNbTrial
<< " World="<< worldSwapIndex(i)
<< " Temp="<< mTemperature
<< " Mutation Ampl.: "<<mMutationAmplitude
<< " NEW RUN Best Cost="<<runBestCost<< endl;
if(!silent) this->DisplayReport();
}
worldNbAcceptedMoves(i)++;
}
else
{
if(log((rand()+1)/(REAL)RAND_MAX)<(-(cost-currentCost(i))/mTemperature) )
{
accept=1;
currentCost(i)=cost;
mRefParList.SaveParamSet(worldCurrentSetIndex(i));
worldNbAcceptedMoves(i)++;
}
}
//if(accept==1 && i==(nbWorld-1)){this->UpdateDisplay();}
if( ((mXMLAutoSave.GetChoice()==1)&&((chrono.seconds()-secondsWhenAutoSave)>86400))
||((mXMLAutoSave.GetChoice()==2)&&((chrono.seconds()-secondsWhenAutoSave)>3600))
||((mXMLAutoSave.GetChoice()==3)&&((chrono.seconds()-secondsWhenAutoSave)> 600))
||((mXMLAutoSave.GetChoice()==4)&&(accept==2)) )
{
secondsWhenAutoSave=(unsigned long)chrono.seconds();
string saveFileName=this->GetName();
time_t date=time(0);
char strDate[40];
strftime(strDate,sizeof(strDate),"%Y-%m-%d_%H-%M-%S",localtime(&date));//%Y-%m-%dT%H:%M:%S%Z
char costAsChar[30];
if(accept!=2) mRefParList.RestoreParamSet(mBestParSavedSetIndex);
sprintf(costAsChar,"-Cost-%f",this->GetLogLikelihood());
saveFileName=saveFileName+(string)strDate+(string)costAsChar+(string)".xml";
XMLCrystFileSaveGlobal(saveFileName);
//if(accept!=2) mRefParList.RestoreParamSet(lastParSavedSetIndex);
}
//if(accept==0) mRefParList.RestoreParamSet(lastParSavedSetIndex);
mNbTrial++;nbStep--;
if((mNbTrial%nbTrialsReport)==0) makeReport=true;
}//nbTryPerWorld trials
}//For each World
if(mAutoLSQ.GetChoice()==2)
if((mNbTrial%autoLSQPeriod)<(nbTryPerWorld*nbWorld))
{// Try a quick LSQ ?
for(int i=0;i<mRefinedObjList.GetNb();i++) mRefinedObjList.GetObj(i).SetApproximationFlag(false);
for(int i=nbWorld-5;i<nbWorld;i++)
{
#ifdef __WX__CRYST__
mMutexStopAfterCycle.Lock();
if(mStopAfterCycle)
{
mMutexStopAfterCycle.Unlock();
break;
}
mMutexStopAfterCycle.Unlock();
#endif
mRefParList.RestoreParamSet(worldCurrentSetIndex(i));
#if 0
// Report GoF values (Chi^2 / nbObs) values for all objects
for(map<RefinableObj*,unsigned int>::iterator pos=mLSQ.GetRefinedObjMap().begin();pos!=mLSQ.GetRefinedObjMap().end();++pos)
if(pos->first->GetNbLSQFunction()>0)
{
CrystVector_REAL tmp;
tmp =pos->first->GetLSQCalc(pos->second);
tmp-=pos->first->GetLSQObs (pos->second);
tmp*=tmp;
tmp*=pos->first->GetLSQWeight(pos->second);
cout<<pos->first->GetClassName()<<":"<<pos->first->GetName()<<": GoF="<<tmp.sum()/tmp.numElements();
}
cout<<endl;
#endif
const REAL cost0=this->GetLogLikelihood();// cannot use currentCost(i), approximations changed...
if(!silent) cout<<"LSQ: World="<<worldSwapIndex(i)<<": cost="<<cost0;
try {mLSQ.Refine(-30,true,true,false,0.001);}
catch(const ObjCrystException &except){};
#if 0
// Report GoF values (Chi^2 / nbObs) values for all objects
for(map<RefinableObj*,unsigned int>::iterator pos=mLSQ.GetRefinedObjMap().begin();pos!=mLSQ.GetRefinedObjMap().end();++pos)
if(pos->first->GetNbLSQFunction()>0)
{
CrystVector_REAL tmp;
tmp =pos->first->GetLSQCalc(pos->second);
tmp-=pos->first->GetLSQObs (pos->second);
tmp*=tmp;
tmp*=pos->first->GetLSQWeight(pos->second);
cout<<pos->first->GetClassName()<<":"<<pos->first->GetName()<<": GoF="<<tmp.sum()/tmp.numElements();
}
cout<<endl;
#endif
const REAL cost=this->GetLogLikelihood();
if(!silent) cout<<" -> "<<cost<<endl;
if(cost<cost0) mRefParList.SaveParamSet(worldCurrentSetIndex(i));
}
// Need to go back to optimization with approximations allowed (they are not during LSQ)
for(int i=0;i<mRefinedObjList.GetNb();i++) mRefinedObjList.GetObj(i).SetApproximationFlag(true);
// And recompute LLK - since they will be lower
for(int i=nbWorld-5;i<nbWorld;i++)
{
mRefParList.RestoreParamSet(worldCurrentSetIndex(i));
const REAL cost=this->GetLogLikelihood();
if(!silent) cout<<"LSQ2:"<<currentCost(i)<<"->"<<cost<<endl;
if(cost<currentCost(i))
{
const REAL oldcost=currentCost(i);
mRefParList.SaveParamSet(worldCurrentSetIndex(i));
currentCost(i)=cost;
if(cost<runBestCost)
{
runBestCost=currentCost(i);
this->TagNewBestConfig();
needUpdateDisplay=true;
mRefParList.SaveParamSet(runBestIndex);
if(runBestCost<mBestCost)
{
mBestCost=currentCost(i);
mRefParList.SaveParamSet(mBestParSavedSetIndex);
if(!silent) cout << "->Trial :" << mNbTrial
<< " World="<< worldSwapIndex(i)
<< " LSQ2: NEW OVERALL Best Cost="<<mBestCost<< endl;
}
else if(!silent) cout << "->Trial :" << mNbTrial
<< " World="<< worldSwapIndex(i)
<< " LSQ2: NEW RUN Best Cost="<<runBestCost<< endl;
if(!silent) this->DisplayReport();
}
// KLUDGE - after a successful LSQ, we will be close to a minimum,
// which will make most successive global optimization trials to
// be rejected, until the temperature is increased a lot - this
// is a problem as the temperature increases so much that the
// benefit of the LSQ is essentially negated.
// So we need to use a higher recorded cost, so that successive trials
// may be accepted
#if 0
mMutationAmplitude=mutationAmplitude(i);
for(unsigned int ii=0;ii<4;ii++) this->NewConfiguration(gpRefParTypeObjCryst,false);
currentCost(i)=(this->GetLogLikelihood()+cost)/2;
if(!silent) cout<<"LSQ3: #"<<worldSwapIndex(i)<<":"<<cost<<"->"<<currentCost(i)<<endl;
#else
currentCost(i)=oldcost;
#endif
}
}
}
//Try swapping worlds
for(int i=1;i<nbWorld;i++)
{
#if 0
mRefParList.RestoreParamSet(worldCurrentSetIndex(i));
mMutationAmplitude=mutationAmplitude(i);
cout<<i<<":"<<currentCost(i)<<":"<<this->GetLogLikelihood()<<endl;
#endif
#if 1
if( log((rand()+1)/(REAL)RAND_MAX)
< (-(currentCost(i-1)-currentCost(i))/simAnnealTemp(i)))
#else
// Compare World (i-1) and World (i) with the same amplitude,
// hence the same max likelihood error
mRefParList.RestoreParamSet(worldCurrentSetIndex(i-1));
mMutationAmplitude=mutationAmplitude(i);
if( log((rand()+1)/(REAL)RAND_MAX)
< (-(this->GetLogLikelihood()-currentCost(i))/simAnnealTemp(i)))
#endif
{
/*
if(i>2)
{
cout <<"->Swapping Worlds :" << i <<"(cost="<<currentCost(i)<<")"
<<" with "<< (i-1) <<"(cost="<< currentCost(i-1)<<")"<<endl;
}
*/
swapPar=mRefParList.GetParamSet(worldCurrentSetIndex(i));
mRefParList.GetParamSet(worldCurrentSetIndex(i))=
mRefParList.GetParamSet(worldCurrentSetIndex(i-1));
mRefParList.GetParamSet(worldCurrentSetIndex(i-1))=swapPar;
const REAL tmp=currentCost(i);
currentCost(i)=currentCost(i-1);
currentCost(i-1)=tmp;
const long tmpIndex=worldSwapIndex(i);
worldSwapIndex(i)=worldSwapIndex(i-1);
worldSwapIndex(i-1)=tmpIndex;
#if 0
// Compute correct costs in the case we use maximum likelihood
mRefParList.RestoreParamSet(worldCurrentSetIndex(i));
mMutationAmplitude=mutationAmplitude(i);
currentCost(i)=this->GetLogLikelihood();
mRefParList.RestoreParamSet(worldCurrentSetIndex(i-1));
mMutationAmplitude=mutationAmplitude(i-1);
currentCost(i-1)=this->GetLogLikelihood();
#endif
}
}
#if 0
//Try mating worlds- NEW !
TAU_PROFILE_TIMER(timer1,\
"MonteCarloObj::Optimize (Try mating Worlds)"\
,"", TAU_FIELD);
TAU_PROFILE_START(timer1);
if( (rand()/(REAL)RAND_MAX)<.1)
for(int k=nbWorld-1;k>nbWorld/2;k--)
for(int i=k-nbWorld/3;i<k;i++)
{
#if 0
// Random switching of gene groups
for(unsigned int j=0;j<nbGeneGroup;j++)
crossoverGroupIndex(j)= (int) floor(rand()/((REAL)RAND_MAX-1)*2);
for(int j=0;j<mRefParList.GetNbPar();j++)
{
if(0==crossoverGroupIndex(refParGeneGroupIndex(j)-1))
{
mRefParList.GetParamSet(parSetOffspringA)(j)=
mRefParList.GetParamSet(worldCurrentSetIndex(i))(j);
mRefParList.GetParamSet(parSetOffspringB)(j)=
mRefParList.GetParamSet(worldCurrentSetIndex(k))(j);
}
else
{
mRefParList.GetParamSet(parSetOffspringA)(j)=
mRefParList.GetParamSet(worldCurrentSetIndex(k))(j);
mRefParList.GetParamSet(parSetOffspringB)(j)=
mRefParList.GetParamSet(worldCurrentSetIndex(i))(j);
}
}
#endif
#if 1
// Switch gene groups in two parts
unsigned int crossoverPoint1=
(int)(1+floor(rand()/((REAL)RAND_MAX-1)*(nbGeneGroup)));
unsigned int crossoverPoint2=
(int)(1+floor(rand()/((REAL)RAND_MAX-1)*(nbGeneGroup)));
if(crossoverPoint2<crossoverPoint1)
{
int tmp=crossoverPoint1;
crossoverPoint1=crossoverPoint2;
crossoverPoint2=tmp;
}
if(crossoverPoint1==crossoverPoint2) crossoverPoint2+=1;
for(int j=0;j<mRefParList.GetNbPar();j++)
{
if((refParGeneGroupIndex(j)>crossoverPoint1)&&refParGeneGroupIndex(j)<crossoverPoint2)
{
mRefParList.GetParamSet(parSetOffspringA)(j)=
mRefParList.GetParamSet(worldCurrentSetIndex(i))(j);
mRefParList.GetParamSet(parSetOffspringB)(j)=
mRefParList.GetParamSet(worldCurrentSetIndex(k))(j);
}
else
{
mRefParList.GetParamSet(parSetOffspringA)(j)=
mRefParList.GetParamSet(worldCurrentSetIndex(k))(j);
mRefParList.GetParamSet(parSetOffspringB)(j)=
mRefParList.GetParamSet(worldCurrentSetIndex(i))(j);
}
}
#endif
// Try both offspring
for(int junk=0;junk<2;junk++)
{
if(junk==0) mRefParList.RestoreParamSet(parSetOffspringA);
else mRefParList.RestoreParamSet(parSetOffspringB);
REAL cost=this->GetLogLikelihood();
//if(log((rand()+1)/(REAL)RAND_MAX)
// < (-(cost-currentCost(k))/simAnnealTemp(k)))
if(cost<currentCost(k))
{
// Also exchange genes for higher-temperature World ?
//if(junk==0)
// mRefParList.GetParamSet(worldCurrentSetIndex(i))=
// mRefParList.GetParamSet(parSetOffspringB);
//else
// mRefParList.GetParamSet(worldCurrentSetIndex(i))=
// mRefParList.GetParamSet(parSetOffspringA);
currentCost(k)=cost;
mRefParList.SaveParamSet(worldCurrentSetIndex(k));
//worldNbAcceptedMoves(k)++;
if(!silent) cout << "Accepted mating :"<<k<<"(with"<<i<<")"
<<" (crossoverGene1="<< crossoverPoint1<<","
<<" crossoverGene2="<< crossoverPoint2<<")"
<<endl;
if(cost<runBestCost)
{
runBestCost=cost;
this->TagNewBestConfig();
needUpdateDisplay=true;
mRefParList.SaveParamSet(runBestIndex);
if(cost<mBestCost)
{
mBestCost=cost;
mRefParList.SaveParamSet(mBestParSavedSetIndex);
if(!silent) cout << "->Trial :" << mNbTrial
<< " World="<< worldSwapIndex(k)
<< " Temp="<< simAnnealTemp(k)
<< " Mutation Ampl.: "<<mMutationAmplitude
<< " NEW OVERALL Best Cost="<<mBestCost<< "(MATING !)"<<endl;
}
else if(!silent) cout << "->Trial :" << mNbTrial
<< " World="<< worldSwapIndex(k)
<< " Temp="<< simAnnealTemp(k)
<< " Mutation Ampl.: "<<mMutationAmplitude
<< " NEW RUN Best Cost="<<runBestCost<< "(MATING !)"<<endl;
bestConfigNb=mNbTrial;
if(!silent) this->DisplayReport();
//for(int i=0;i<mRefinedObjList.GetNb();i++)
// mRefinedObjList.GetObj(i).Print();
}
i=k;//Don't test other Worlds
break;
}
//mNbTrial++;nbStep--;
//if((mNbTrial%nbTrialsReport)==0) makeReport=true;
}
}
TAU_PROFILE_STOP(timer1);
#endif
if(true==makeReport)
{
makeReport=false;
worldNbAcceptedMoves*=nbWorld;
if(!silent)
{
#if 0
{// Experimental, dynamical weighting
REAL max=0.;
map<const RefinableObj*,REAL> ll,llvar;
map<const RefinableObj*,LogLikelihoodStats>::iterator pos;
for(pos=mvContextObjStats[0].begin();pos!=mvContextObjStats[0].end();++pos)
{
ll [pos->first]=0.;
llvar[pos->first]=0.;
}
for(int i=0;i<nbWorld;i++)
{
for(pos=mvContextObjStats[0].begin();pos!=mvContextObjStats[0].end();++pos)
{
ll [pos->first] += pos->second.mTotalLogLikelihood;
llvar[pos->first] += pos->second.mTotalLogLikelihoodDeltaSq;
}
}
for(pos=mvContextObjStats[0].begin();pos!=mvContextObjStats[0].end();++pos)
{
cout << pos->first->GetName()
<< " " << llvar[pos->first]
<< " " << mvObjWeight[pos->first].mWeight
<< " " << max<<endl;
llvar[pos->first] *= mvObjWeight[pos->first].mWeight;
if(llvar[pos->first]>max) max=llvar[pos->first];
}
map<const RefinableObj*,REAL>::iterator pos2;
for(pos2=llvar.begin();pos2!=llvar.end();++pos2)
{
const REAL d=pos2->second;
if(d<(max/mvObjWeight.size()/10.))
{
if(d<1) continue;
mvObjWeight[pos2->first].mWeight *=2;
}
}
REAL ll1=0;
REAL llt=0;
for(pos2=ll.begin();pos2!=ll.end();++pos2)
{
llt += pos2->second;
ll1 += pos2->second * mvObjWeight[pos2->first].mWeight;
}
map<const RefinableObj*,DynamicObjWeight>::iterator posw;
for(posw=mvObjWeight.begin();posw!=mvObjWeight.end();++posw)
{
posw->second.mWeight *= llt/ll1;
}
}
#endif //Experimental dynamical weighting
#if 1 //def __DEBUG__
for(int i=0;i<nbWorld;i++)
{
cout<<" World :"<<worldSwapIndex(i)<<":";
map<const RefinableObj*,LogLikelihoodStats>::iterator pos;
for(pos=mvContextObjStats[i].begin();pos!=mvContextObjStats[i].end();++pos)
{
cout << pos->first->GetName()
<< "(LLK="
<< pos->second.mLastLogLikelihood
//<< "(<LLK>="
//<< pos->second.mTotalLogLikelihood/nbTrialsReport
//<< ", <delta(LLK)^2>="
//<< pos->second.mTotalLogLikelihoodDeltaSq/nbTrialsReport
<< ", w="<<mvObjWeight[pos->first].mWeight
<<") ";
pos->second.mTotalLogLikelihood=0;
pos->second.mTotalLogLikelihoodDeltaSq=0;
}
cout << endl;
}
#endif
for(int i=0;i<nbWorld;i++)
{
//mRefParList.RestoreParamSet(worldCurrentSetIndex(i));
cout <<" World :" << worldSwapIndex(i)
<<" Temp.: " << simAnnealTemp(i)
<<" Mutation Ampl.: " << mutationAmplitude(i)
<<" Current Cost=" << currentCost(i)
<<" Accepting "
<< (int)((REAL)worldNbAcceptedMoves(i)/nbTrialsReport*100)
<<"% moves " <<endl;
// <<"% moves " << mRefParList.GetPar("Pboccup").GetValue()<<endl;
}
}
if(!silent) cout <<"Trial :" << mNbTrial << " Best Cost=" << runBestCost<< " ";
if(!silent) chrono.print();
//Change the mutation rate if necessary for each world
if(ANNEALING_SMART==mAnnealingScheduleMutation.GetChoice())
{
for(int i=0;i<nbWorld;i++)
{
if((worldNbAcceptedMoves(i)/(REAL)nbTrialsReport)>0.30)
mutationAmplitude(i)*=2.;
if((worldNbAcceptedMoves(i)/(REAL)nbTrialsReport)<0.10)
mutationAmplitude(i)/=2.;
if(mutationAmplitude(i)>mMutationAmplitudeMax)
mutationAmplitude(i)=mMutationAmplitudeMax;
if(mutationAmplitude(i)<mMutationAmplitudeMin)
mutationAmplitude(i)=mMutationAmplitudeMin;
}
}
if(ANNEALING_SMART==mAnnealingScheduleTemp.GetChoice())
{
for(int i=0;i<nbWorld;i++)
{
if((worldNbAcceptedMoves(i)/(REAL)nbTrialsReport)>0.30)
simAnnealTemp(i)/=1.5;
if((worldNbAcceptedMoves(i)/(REAL)nbTrialsReport)>0.80)
simAnnealTemp(i)/=1.5;
if((worldNbAcceptedMoves(i)/(REAL)nbTrialsReport)>0.95)
simAnnealTemp(i)/=1.5;
if((worldNbAcceptedMoves(i)/(REAL)nbTrialsReport)<0.10)
simAnnealTemp(i)*=1.5;
if((worldNbAcceptedMoves(i)/(REAL)nbTrialsReport)<0.04)
simAnnealTemp(i)*=1.5;
//if((worldNbAcceptedMoves(i)/(REAL)nbTrialsReport)<0.01)
// simAnnealTemp(i)*=1.5;
//cout<<"World#"<<i<<":"<<worldNbAcceptedMoves(i)<<":"<<nbTrialsReport<<endl;
//if(simAnnealTemp(i)>mTemperatureMax) simAnnealTemp(i)=mTemperatureMax;
//if(simAnnealTemp(i)<mTemperatureMin) simAnnealTemp(i)=mTemperatureMin;
}
}
worldNbAcceptedMoves=0;
//this->DisplayReport();
#ifdef __WX__CRYST__
if(0!=mpWXCrystObj) mpWXCrystObj->UpdateDisplayNbTrial();
#endif
}
if( (needUpdateDisplay&&(lastUpdateDisplayTime<(chrono.seconds()-1)))||(lastUpdateDisplayTime<(chrono.seconds()-10)))
{
mRefParList.RestoreParamSet(runBestIndex);
this->UpdateDisplay();
needUpdateDisplay=false;
lastUpdateDisplayTime=chrono.seconds();
}
#ifdef __WX__CRYST__
mMutexStopAfterCycle.Lock();
#endif
if((runBestCost<finalcost) || mStopAfterCycle ||( (maxTime>0)&&(chrono.seconds()>maxTime)))
{
#ifdef __WX__CRYST__
mMutexStopAfterCycle.Unlock();
#endif
if(!silent) cout << endl <<endl << "Refinement Stopped:"<<mBestCost<<endl;
break;
}
#ifdef __WX__CRYST__
mMutexStopAfterCycle.Unlock();
#endif
}//Trials
TAU_PROFILE_START(timerN);
if(mAutoLSQ.GetChoice()>0)
{// LSQ
if(!silent) cout<<"Beginning final LSQ refinement"<<endl;
for(int i=0;i<mRefinedObjList.GetNb();i++) mRefinedObjList.GetObj(i).SetApproximationFlag(false);
mRefParList.RestoreParamSet(runBestIndex);
mCurrentCost=this->GetLogLikelihood();
try {mLSQ.Refine(-50,true,true,false,0.001);}
catch(const ObjCrystException &except){};
if(!silent) cout<<"LSQ cost: "<<mCurrentCost<<" -> "<<this->GetLogLikelihood()<<endl;
// Need to go back to optimization with approximations allowed (they are not during LSQ)
for(int i=0;i<mRefinedObjList.GetNb();i++) mRefinedObjList.GetObj(i).SetApproximationFlag(true);
REAL cost=this->GetLogLikelihood();
if(cost<mCurrentCost)
{
mCurrentCost=cost;
mRefParList.SaveParamSet(lastParSavedSetIndex);
if(mCurrentCost<runBestCost)
{
runBestCost=mCurrentCost;
mRefParList.SaveParamSet(runBestIndex);
if(runBestCost<mBestCost)
{
mBestCost=mCurrentCost;
mRefParList.SaveParamSet(mBestParSavedSetIndex);
if(!silent) cout << "LSQ : NEW OVERALL Best Cost="<<runBestCost<< endl;
}
else if(!silent) cout << " LSQ : NEW Run Best Cost="<<runBestCost<< endl;
}
}
if(!silent) cout<<"Finished LSQ refinement"<<endl;
}
mLastOptimTime=chrono.seconds();
//Restore Best values
//mRefParList.Print();
if(!silent) this->DisplayReport();
mRefParList.RestoreParamSet(runBestIndex);
//for(int i=0;i<mRefinedObjList.GetNb();i++) mRefinedObjList.GetObj(i).Print();
mCurrentCost=this->GetLogLikelihood();
if(!silent) cout<<"Run Best Cost:"<<mCurrentCost<<endl;
if(!silent) chrono.print();
//Save density of states
//ofstream out("densityOfStates.txt");
//out << trialsDensity<<endl;
//out.close();
// Clear temporary param set
for(int i=0;i<nbWorld;i++)
{
mRefParList.ClearParamSet(worldCurrentSetIndex(i));
//mvSavedParamSet.push_back(make_pair(worldCurrentSetIndex(i),currentCost(i)));
}
mRefParList.ClearParamSet(lastParSavedSetIndex);
mRefParList.ClearParamSet(runBestIndex);
TAU_PROFILE_STOP(timerN);
}
void MonteCarloObj::XMLOutput(ostream &os,int indent)const
{
VFN_DEBUG_ENTRY("MonteCarloObj::XMLOutput():"<<this->GetName(),5)
for(int i=0;i<indent;i++) os << " " ;
XMLCrystTag tag("GlobalOptimObj");
tag.AddAttribute("Name",this->GetName());
tag.AddAttribute("NbTrialPerRun",(boost::format("%d")%(this->NbTrialPerRun())).str());
os <<tag<<endl;
indent++;
mGlobalOptimType.XMLOutput(os,indent);
os<<endl;
mAnnealingScheduleTemp.XMLOutput(os,indent);
os<<endl;
mXMLAutoSave.XMLOutput(os,indent);
os<<endl;
mAutoLSQ.XMLOutput(os,indent);
os<<endl;
{
XMLCrystTag tag2("TempMaxMin");
for(int i=0;i<indent;i++) os << " " ;
os<<tag2<<mTemperatureMax << " "<< mTemperatureMin;
tag2.SetIsEndTag(true);
os<<tag2<<endl;
}
mAnnealingScheduleMutation.XMLOutput(os,indent);
os<<endl;
mSaveTrackedData.XMLOutput(os,indent);
os<<endl;
{
XMLCrystTag tag2("MutationMaxMin");
for(int i=0;i<indent;i++) os << " " ;
os<<tag2<<mMutationAmplitudeMax << " "<< mMutationAmplitudeMin;
tag2.SetIsEndTag(true);
os<<tag2<<endl;
}
for(int j=0;j<mRefinedObjList.GetNb();j++)
{
XMLCrystTag tag2("RefinedObject",false,true);
tag2.AddAttribute("ObjectType",mRefinedObjList.GetObj(j).GetClassName());
tag2.AddAttribute("ObjectName",mRefinedObjList.GetObj(j).GetName());
for(int i=0;i<indent;i++) os << " " ;
os<<tag2<<endl;
}
indent--;
tag.SetIsEndTag(true);
for(int i=0;i<indent;i++) os << " " ;
os <<tag<<endl;
VFN_DEBUG_EXIT("MonteCarloObj::XMLOutput():"<<this->GetName(),5)
}
void MonteCarloObj::XMLInput(istream &is,const XMLCrystTag &tagg)
{
VFN_DEBUG_ENTRY("MonteCarloObj::XMLInput():"<<this->GetName(),5)
for(unsigned int i=0;i<tagg.GetNbAttribute();i++)
{
if("Name"==tagg.GetAttributeName(i)) this->SetName(tagg.GetAttributeValue(i));
if("NbTrialPerRun"==tagg.GetAttributeName(i))
{
stringstream ss(tagg.GetAttributeValue(i));
long v;
ss>>v;
this->NbTrialPerRun()=v;
}
}
while(true)
{
XMLCrystTag tag(is);
if(("GlobalOptimObj"==tag.GetName())&&tag.IsEndTag())
{
VFN_DEBUG_EXIT("MonteCarloObj::Exit():"<<this->GetName(),5)
this->UpdateDisplay();
return;
}
if("Option"==tag.GetName())
{
for(unsigned int i=0;i<tag.GetNbAttribute();i++)
if("Name"==tag.GetAttributeName(i))
{
if("Algorithm"==tag.GetAttributeValue(i))
{
mGlobalOptimType.XMLInput(is,tag);
break;
}
if("Temperature Schedule"==tag.GetAttributeValue(i))
{
mAnnealingScheduleTemp.XMLInput(is,tag);
break;
}
if("Displacement Amplitude Schedule"==tag.GetAttributeValue(i))
{
mAnnealingScheduleMutation.XMLInput(is,tag);
break;
}
if("Save Best Config Regularly"==tag.GetAttributeValue(i))
{
mXMLAutoSave.XMLInput(is,tag);
break;
}
if("Save Tracked Data"==tag.GetAttributeValue(i))
{
mSaveTrackedData.XMLInput(is,tag);
break;
}
if("Automatic Least Squares Refinement"==tag.GetAttributeValue(i))
{
mAutoLSQ.XMLInput(is,tag);
break;
}
}
continue;
}
if("TempMaxMin"==tag.GetName())
{
is>>mTemperatureMax>> mTemperatureMin;
if(false==tag.IsEmptyTag()) XMLCrystTag junk(is);//:KLUDGE: for first release
continue;
}
if("MutationMaxMin"==tag.GetName())
{
is>>mMutationAmplitudeMax>>mMutationAmplitudeMin;
if(false==tag.IsEmptyTag()) XMLCrystTag junk(is);//:KLUDGE: for first release
continue;
}
if("RefinedObject"==tag.GetName())
{
string name,type;
for(unsigned int i=0;i<tag.GetNbAttribute();i++)
{
if("ObjectName"==tag.GetAttributeName(i)) name=tag.GetAttributeValue(i);
if("ObjectType"==tag.GetAttributeName(i)) type=tag.GetAttributeValue(i);
}
RefinableObj* obj=& (gRefinableObjRegistry.GetObj(name,type));
this->AddRefinableObj(*obj);
continue;
}
}
}
const string MonteCarloObj::GetClassName()const { return "MonteCarloObj";}
LSQNumObj& MonteCarloObj::GetLSQObj() {return mLSQ;}
const LSQNumObj& MonteCarloObj::GetLSQObj() const{return mLSQ;}
void MonteCarloObj::NewConfiguration(const RefParType *type)
{
TAU_PROFILE("MonteCarloObj::NewConfiguration()","void ()",TAU_DEFAULT);
VFN_DEBUG_ENTRY("MonteCarloObj::NewConfiguration()",4)
for(int i=0;i<mRefinedObjList.GetNb();i++)
mRefinedObjList.GetObj(i).BeginGlobalOptRandomMove();
for(int i=0;i<mRefinedObjList.GetNb();i++)
mRefinedObjList.GetObj(i).GlobalOptRandomMove(mMutationAmplitude,type);
VFN_DEBUG_EXIT("MonteCarloObj::NewConfiguration()",4)
}
void MonteCarloObj::InitOptions()
{
VFN_DEBUG_MESSAGE("MonteCarloObj::InitOptions()",5)
this->OptimizationObj::InitOptions();
static string GlobalOptimTypeName;
static string GlobalOptimTypeChoices[2];//:TODO: Add Genetic Algorithm
static string AnnealingScheduleChoices[6];
static string AnnealingScheduleTempName;
static string AnnealingScheduleMutationName;
static string runAutoLSQName;
static string runAutoLSQChoices[3];
static string saveTrackedDataName;
static string saveTrackedDataChoices[2];
static bool needInitNames=true;
if(true==needInitNames)
{
GlobalOptimTypeName="Algorithm";
GlobalOptimTypeChoices[0]="Simulated Annealing";
GlobalOptimTypeChoices[1]="Parallel Tempering";
//GlobalOptimTypeChoices[2]="Random-LSQ";
AnnealingScheduleTempName="Temperature Schedule";
AnnealingScheduleMutationName="Displacement Amplitude Schedule";
AnnealingScheduleChoices[0]="Constant";
AnnealingScheduleChoices[1]="Boltzmann";
AnnealingScheduleChoices[2]="Cauchy";
AnnealingScheduleChoices[3]="Exponential";
AnnealingScheduleChoices[4]="Smart";
AnnealingScheduleChoices[5]="Gamma";
runAutoLSQName="Automatic Least Squares Refinement";
runAutoLSQChoices[0]="Never";
runAutoLSQChoices[1]="At the end of each run";
runAutoLSQChoices[2]="Every 150000 trials, and at the end of each run";
saveTrackedDataName="Save Tracked Data";
saveTrackedDataChoices[0]="No (recommended!)";
saveTrackedDataChoices[1]="Yes (for tests ONLY)";
needInitNames=false;//Only once for the class
}
mGlobalOptimType.Init(2,&GlobalOptimTypeName,GlobalOptimTypeChoices);
mAnnealingScheduleTemp.Init(6,&AnnealingScheduleTempName,AnnealingScheduleChoices);
mAnnealingScheduleMutation.Init(6,&AnnealingScheduleMutationName,AnnealingScheduleChoices);
mSaveTrackedData.Init(2,&saveTrackedDataName,saveTrackedDataChoices);
mAutoLSQ.Init(3,&runAutoLSQName,runAutoLSQChoices);
this->AddOption(&mGlobalOptimType);
this->AddOption(&mAnnealingScheduleTemp);
this->AddOption(&mAnnealingScheduleMutation);
this->AddOption(&mSaveTrackedData);
this->AddOption(&mAutoLSQ);
VFN_DEBUG_MESSAGE("MonteCarloObj::InitOptions():End",5)
}
void MonteCarloObj::InitLSQ(const bool useFullPowderPatternProfile)
{
mLSQ.SetRefinedObj(mRecursiveRefinedObjList.GetObj(0),0,true,true);
for(unsigned int i=1;i<mRefinedObjList.GetNb();++i)
mLSQ.SetRefinedObj(mRefinedObjList.GetObj(i),0,false,true);
if(!useFullPowderPatternProfile)
{// Use LSQ function #1 for powder patterns (integrated patterns - faster !)
for(map<RefinableObj*,unsigned int>::iterator pos=mLSQ.GetRefinedObjMap().begin();pos!=mLSQ.GetRefinedObjMap().end();++pos)
if(pos->first->GetClassName()=="PowderPattern") pos->second=1;
}
// Only refine structural parameters (excepting parameters already fixed) and scale factor
mLSQ.PrepareRefParList(true);
// Intensity corrections can be refined
std::list<RefinablePar*> vIntCorrPar;
for(int i=0; i<mLSQ.GetCompiledRefinedObj().GetNbPar();i++)
if(mLSQ.GetCompiledRefinedObj().GetPar(i).GetType()->IsDescendantFromOrSameAs(gpRefParTypeScattDataCorrInt) && mLSQ.GetCompiledRefinedObj().GetPar(i).IsFixed()==false)
vIntCorrPar.push_back(&mLSQ.GetCompiledRefinedObj().GetPar(i));
mLSQ.SetParIsFixed(gpRefParTypeScattData,true);
mLSQ.SetParIsFixed(gpRefParTypeScattDataScale,false);
for(std::list<RefinablePar*>::iterator pos=vIntCorrPar.begin();pos!=vIntCorrPar.end();pos++)
(*pos)->SetIsFixed(false);
mLSQ.SetParIsFixed(gpRefParTypeUnitCell,true);
mLSQ.SetParIsFixed(gpRefParTypeScattPow,true);
mLSQ.SetParIsFixed(gpRefParTypeRadiation,true);
}
void MonteCarloObj::UpdateDisplay() const
{
Chronometer chrono;
#ifdef __WX__CRYST__
if(0!=mpWXCrystObj) mpWXCrystObj->CrystUpdate(true,true);
#endif
this->OptimizationObj::UpdateDisplay();
}
#ifdef __WX__CRYST__
WXCrystObjBasic* MonteCarloObj::WXCreate(wxWindow *parent)
{
mpWXCrystObj=new WXMonteCarloObj (parent,this);
return mpWXCrystObj;
}
WXOptimizationObj* MonteCarloObj::WXGet()
{
return mpWXCrystObj;
}
void MonteCarloObj::WXDelete()
{
if(0!=mpWXCrystObj) delete mpWXCrystObj;
mpWXCrystObj=0;
}
void MonteCarloObj::WXNotifyDelete()
{
mpWXCrystObj=0;
}
#endif
}//namespace
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हाईकोर्ट के आदेश पर २४ तक रोकी गईं केजीबी की परीक्षाएं - अखंड भारत न्यूज
होम उत्तर प्रदेश हरदोई हाईकोर्ट के आदेश पर २४ तक रोकी गईं केजीबी की परीक्षाएं
हाईकोर्ट के आदेश पर २४ तक रोकी गईं केजीबी की परीक्षाएं
दरअसल जिले में २० कस्तूरबा गांधी आवासीय बालिका विद्यालय स्थापित हैं जिनमें सैकड़ों टीचर्स व अन्य का स्टाफ है। विगत ०१० वर्षों से इन विद्यालयों में शिक्षक, शिक्षिकाओं का स्वतः नवीनीकरण होता रहा है। किंतु इस बार प्रशासन के आदेश पर विभाग ने सभी संविदाकर्मियों की परीक्षा कराने का निर्णय लिया। परीक्षा उत्तीर्ण करने वाले ही कर्मी का नवीनीकरण किया जाना था।
किंतु इस परीक्षा का पाठ्यक्रम क्या होगा? इसकी कोई तैयारी प्रशासन ने नही की। ऐसे में संविदाकर्मी अजमंजस में हैं कि वह किस पाठ्यक्रम की तैयारी करें जिसका सवाल परीक्षा में पूछा जाएगा। द टेलीकास्ट वही दूसरा सवाल ये भी है कि २४ घण्टे की ड्यूटी करने वाले टीचर्स किस अवधि में परीक्षा की तैयारी करेंगे। बिना इसकी परवाह किये प्रशासन ने परीक्षा कराने का निर्णय ले लिया। इस निर्णय के खिलाफ संविदाकर्मियों ने उच्च न्यायालय का रुख किया जहां उन्हें फिलहाल राहत मिली है। जज ने सरकारी वकील अजय यादव से कहा है कि कस्तूरबा गांधी आवासीय बालिका विद्यालय की परीक्षा अब कोर्ट के अधीन हैं। इसलिए ये परीक्षाएं न कराई जाएं यदि कराई गयीं तो निरस्त कर दी जा सकती है। २४ अप्रैल को उच्च न्यायालय इस सम्बंध में अपना फैसला सुनाएगा।
प्रेवियस आर्टियलउत्तर प्रदेश लोकसेवा आयोग मे हुआ चयन,बना श्रम प्रवर्तन अधिकारी
नेक्स्ट आर्टियलनन्हे मुन्ने बच्चो ने दिया स्वच्छता का संदेश, डीएम ने फीता काट कर किया शुभारम्भ | hindi |
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تِم چھِ بنک منیجر | kashmiri |
مےٚ کٔر واریَہَن گٲنٹَن پوٗزا تہٕ پرٛارتھنا تہٕ پَتہٕ اوسُس گَرٕ کُن پکان تہٕ اَتہِ وُچھُم اَکھ جوٗگۍ پَنہٕ نِس أگنہٕ کۄنڈس برٛۄنٛہہ کَنہِ بہِتھ | kashmiri |
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\begin{document}
\title{\LARGE \bf
Approximate observability and back and forth observer\\
of a PDE model of crystallisation process
}
\thispagestyle{empty}
\pagestyle{empty}
\begin{abstract}
In this paper, we are interested in the estimation of Particle Size Distributions (PSDs) during
a batch crystallization process
in which particles of two different shapes coexist and evolve simultaneously.
The PSDs are estimated thanks to a measurement of an apparent Chord Length Distribution (CLD), a measure that we model for crystals of spheroidal shape.
Our main result is to prove the approximate observability of the infinite-dimensional system in any positive time.
Under this observability condition, we are able to apply a Back and Forth Nudging (BFN) algorithm to reconstruct the PSD.
\end{abstract}
\section{Introduction}
During a batch crystallization process, a critical issue is to monitor the Particle Size Distribution (PSD), which may affect the chemical-physical properties of the product.
In particular, multiple types of crystals may be evolving simultaneously in the reactor, with stable and meta-stable crystal formations.
In that case, estimating the PSD associated to each shape is an important task, but difficult to realize in practice.
Modern Process Analytical Technologies (PATs) offer a wide variety of approaches to extract PSD information from measurements, such as image processing based methods \cite{Benoit, GAO} for instance.
The use dynamical observers is a popular approach to the issue \cite{vissers2012model, porru2017monitoring, mesbah2011comparison, Uccheddu, lebaz, gruy:hal-01637703, BRIVADIS202017}.
A particular technique, on which we focus in this article,
is to use PATs giving access to the Chord Length Distribution (CLD) (such as the Focused Beam Reflectance Measurement or the BlazeMetrics\textsuperscript{\mbox{\scriptsize{\textregistered}}} technologies), and to reconstruct the PSD from the knowledge of the CLD \cite{worlitschek2005restoration, liu1998relationship, pandit2016chord, AGIMELEN2015629}.
When scanning across crystals, these sensors actually measure chords on the projection of the crystal on the plane that is orthogonal to the probe.
Hence, the measured CLD highly depends on the shapes that the crystals in the reactor may take.
And in many crystallization processes, several shapes can coexist due to polymorphism. Since the CLD sums up the contribution of each shape in one measurement, recovering the PSD associated to each shape only from the common CLD is a major challenge not yet tackled by the existing literature.
In the previous work \cite{brivadis:hal-03053999}, we proposed:
\textit{(i)} a model of the PSD-to-CLD relation for spheroid particles;
\textit{(ii)} a direct inversion method to instantly recover the PSD from the CLD when all particles have the same shape;
\textit{(iii)} a back and forth observer to reconstruct the PSD of several shapes from the
knowledge of their common CLD on a finite time interval and an evolution model of the process.
We were able to prove the convergence of the algorithm only in one case: when crystals have only two possible shapes (spheres and elongated spheroids of fixed eccentricity), each having a positive growth rate independent of the size (McCabe hypothesis).
Among the differences with the previous result, let us highlight the two main improvements of this paper:
\textit{(a)} we consider size-dependent growth rates;
\textit{(b)} we consider almost all possible combinations of two spheroid shapes with different eccentricities.
Then, we perform an observability analysis of the infinite-dimensional system, which is the main result of the paper.
\section{Evolution model and CLD}
A spheroid is a surface of revolution, obtained by rotating an ellipse along one of its axes of symmetry.
In particular, spheres are spheroids.
A spheroid is fully described by two scalar parameters: a radius $r$, characterizing its size and being the radius that is orthogonal to its rotation's axis, and an eccentricity $\eta$, characterizing its shape and being the ratio between the radius along its rotation's axis and $r$.
Consider a batch crystallisation process during which crystals of only two shapes appear: spheroids of eccentricities $\eta_1$ and $\eta_2$.
Let $\psi_1$ and $\psi_2$ be their corresponding PSDs. At any time $t\in[0, T]$ during the process,
$\int_{r_1}^{r_2}\psi_i(t, r)\mathrm{d} r$
is the number of crystals per unit of volume at time $t$
having the shape $\eta_i$ and a radius $r$ between $r_1$ and $r_2$.
Let $\rmin{i}$ be the minimal size at which crystals of shape $\eta_i$ appear
and $\rmax{i}$ be a maximal radius that no crystals can reach during the process.
At time $t=0$, assume that seed particles with PSD $\psi_{i, 0}$ for each shape $\eta_i$ lie in the reactor.
Denote by $G_i(t, r)$ the grow rate of crystals of shape $\eta_i$ and size $r$ at time $t$.
The usual (see, \emph{e.g.}, \cite{Mullin, Mersmann}) population balance equation leads to
\begin{equation}
\frac{\partial \psi_i}{\partial t}(t, r) + G_i(t, r) \frac{\partial \psi_i}{\partial r}(t, r) = 0
\end{equation}
Assume that $G_i$ is $C^1$.
Note that the growth rate may be positive or negative.
The boundary conditions are given by
\begin{align}
&\psi_i(t, \rmin{i}) = u_i(t)
\label{eq:rmin}
\\
&\psi_i(t, \rmax{i}) = 0
\label{eq:rmax}
\end{align}
where $u_i(t)$ denotes the appearance of particles of size $\rinf{i}$ and shape $\eta_i$ at time $t$.
Since $u_i$ is supposed to be unknown, it is part of the data to be estimated, with $\psi_{i, 0}$.
Note that the boundary conditions impose a relation between $u_i$ and $\psi_{i, 0}$ when $G_i(t, \rmin{i})<0$.
Set
\begin{align}
&\begin{aligned}
\rinf{i} = \min\Big\{&\rmin{i},
\rmin{i} - \max_{\tau\in[0, T]} \int_0^\tau G_i(t, \rmin{i})\mathrm{d} t\Big\}
\end{aligned}
\\
&\begin{aligned}
\rsup{i} = \max\Big\{&\rmax{i},
\rmax{i} - \min_{\tau\in[0, T]} \int_0^\tau G_i(t, \rmax{i})\mathrm{d} t\Big\}
\end{aligned}
\end{align}
In order to ensure the well-posedness of the evolution equation,
let us define $\psi_i(t, r)$ and $G_i(t, r)$ for $t\in[0, T]$ and $r\in[\rinf{i}, \rsup{i}]\setminus[\rmin{i}, \rmax{i}]$ by
\begin{align}
&\left\{\begin{aligned}
&\psi_i(t, r) = u_i(t+\tau),\\
&G_i(t, r) = G_i(t, \rmin{i}),
\end{aligned}\right.
&\forall r\in[\rinf{i}, \rmin{i}]
\label{eqtau}
\\
&\left\{\begin{aligned}
&\psi_i(t, r) = 0,\\
&G_i(t, r) = G_i(t, \rmax{i}),
\end{aligned}\right.
&\forall r\in[\rmax{i}, \rsup{i}]
\end{align}
where $\tau$ in \eqref{eqtau} is such that
\begin{equation}
\rmin{i} = r + \int_t^{t+\tau}G_i(s, \rmin{i})\mathrm{d} s
\end{equation}
Roughly speaking, $\psi_i(t, r)$ for $r<\rmin{i}$ represents crystals that did not yet appear at time $t$, but will appear later at some time $t+\tau$.
Then the evolution of the crystallization process can be modeled as
\begin{equation}
\left\{\begin{aligned}
&\frac{\partial \psi_i}{\partial t}(t, r) + G_i(t, r) \frac{\partial \psi_i}{\partial r}(t, r) = 0\\
&\psi_i(0, r) = \psi_{i, 0}(r)
\end{aligned}
\right.
\label{syst}
\end{equation}
where $i\in\{1, 2\}$, $t\in[0, T]$, $r\in[\rinf{i}, \rsup{i}]$
and
with the periodic boundary condition $\psi_i(t, \rinf{i}) = \psi_i(t, \rsup{i})$ since the boundary terms does not influence $\psi_i(t, r)$ for $r\in [\rmin{i}, \rmax{i}]$ and $t\leq T$.
The new initial condition $\psi_{i, 0}(r)$ contains both the information on the seed particles (for $r\in[\rmin{i}, \rmax{i}]$ and on all the crystals that will appear during the process (for $r\in[\rinf{i}, \rmin{i}]$).
Note that, contrary to \cite{brivadis:hal-03053999},
we do not assume that $\rmin{1} = \rmin{2}$, nor that $G_i(t, r)$ is independent of $r$. We rather make the assumption that $G_i$ has separate variables, that is,
\begin{equation}
G_i(t, r) = g_i f(t) h(r)
\end{equation}
for all $t\in[0, T]$ and all $r\in[\rmin{i}, \rmax{i}]$,
where $g_i$ is a constant (either positive or negative, depending on whether crystals of shape $\eta_i$ are appearing or disappearing), and $f$ and $h$ do not depend on $i$.
\begin{remark}
In modelling the growth rate $G_i$, it can be linked with individual crystal volume growth.
For $r$ the radius of an individual crystal, the volume of an individual crystal is $V=\frac{4}{3}\pi \eta r^3$. As a consequence, $\frac{d V}{d t }=4\pi \eta r^2 G_i$. This leads to choices such as $h(r)=1/r^2$ for linear volume growth, or $h(r)=1$ for volume growth proportional to the crystal surface (which corresponds to McCabe hypothesis).
\end{remark}
Denote by $L^2(\rinf{i}, \rsup{i})$ the set of square integrable real-valued functions over $(\rinf{i}, \rsup{i})$, and $H^p(\rinf{i}, \rsup{i})$ the usual real-valued Sobolev spaces for $p\in\mathbb{N}$.
\begin{theorem}[Well-posedness]\label{th:wp}
Assume that $f\in C^0([0, T]; \mathbb{R})$ has a finite number of zeros
and $h$ is Lipschitz over $[\rmin{i}, \rmax{i}]$ and has constant sign.
Then for all $\psi_{i, 0}\in L^2(\rinf{i}, \rsup{i})$,
there exists a unique solution $\psi_i\in C^0([0, T]; L^2(\rinf {i}, \rsup{i}))$ of the Cauchy problem \eqref{syst}.
\end{theorem}
\begin{proof}
Let $n$ be the number of zeros of $f$
and $([t_k, t_{k+1}])_{1\leq k\leq n}$ be intervals on which $f$ has constant sign, with $t_1 = 0$ and $t_n = T$.
Over each interval $[t_k, t_{k+1}]$, introduce the time reparametrization $\tilde{t} = \int_{t_k}^t |f(s)|\mathrm{d} s$.
Then $\psi$ is a solution of \eqref{syst} over $[t_k, t_{k+1}]$ if and only if $\tilde{\psi}(\tilde{t})= \psi(t)$ is a solution
\begin{equation}
\frac{\partial \tilde{\psi}_i}{\partial \tilde{t}}(\tilde{t}, r)
+ (\sign f) g_i h(r) \frac{\partial \tilde{\psi}_i}{\partial r}(\tilde{t}, r) = 0
\end{equation}
According to \cite[Appendix 1]{bastin2016stability},
this linear hyperbolic system with periodic boundary conditions admits a unique solution in $C^0([t_k, \int_{t_k}^{t_{k+1}} |f(s)|\mathrm{d} s]; L^2(\rinf{i}, \rsup{i}))$.
Reasoning by induction on each interval $[t_k, t_{k+1}]$, we find that there exists a unique $\psi_i\in C^0([0, T]; L^2(\rinf{i}, \rsup{i}))$ solution of \eqref{syst}.
\end{proof}
\begin{remark}
If $\psi_{i, 0}$ and $f$ are more regular, then the corresponding solutions of \eqref{syst} are also more regular. In particular, if
$\psi_{i, 0}\in H^2(\rinf{i}, \rsup{i})$ and
$f\in C^2([0, T], \mathbb{R})$,
then
$\psi_i\in C^0([0, T]; H^2(\rinf {i}, \rsup{i})) \cap C^1([0, T]; H^1(\rinf {i}, \rsup{i}))\cap C^2([0, T]; L^2(\rinf {i}, \rsup{i}))$.
This remark will be used in Theorem~\ref{th:main}.
\end{remark}
Now, let us recall the model of the accessible measurement, the CLD, denoted by $q$, given in \cite{brivadis:hal-03053999}.
For any $t\in[0, T]$, $\int_{\ell_1}^{\ell_2} q(t, \ell)\mathrm{d} \ell$ is the number of chords measured by the sensor at time $t$ with length $\ell$ between $\ell_1$ and $\ell_2$.
The cumulative CLD is given by $Q(t, \ell) = \int_0^\ell q(t, l)\mathrm{d} l$.
We model the PSD-to-CLD relation by
\begin{multline}\label{eqQ}
Q(t, \ell)
= \int_{\rmin{1}}^{\rmax{1}} k_1(\ell, r)\psi_1(t, r)\mathrm{d} r\\
+ \int_{\rmin{2}}^{\rmax{2}} k_2(\ell, r)\psi_2(t, r)\mathrm{d} r
\end{multline}
where $k_i$ is the kernel for the PSD-to-CLD of each crystal shape $i$.
As developed \cite{brivadis:hal-03053999}, the apparent shape of a crystal with respect to the sensor is that of an ellipse (by projection onto a plane). In that way, for a given ellipse in the plane, the probability that the measured chord-length is less than $\ell$ is $1- \sqrt{1-\alpha\ell^2/(4r^2)}$, with coefficient $\alpha>0$ depending on orientation and eccentricity of the ellipse (with maximum possible chord length $2r/\sqrt{\alpha}$). The apparent ellipse is linked to the shape of the crystal and the crystal's random orientation in the suspension, which follows a uniform distribution on the sphere given in spherical coordinates by the probability measure $\frac{\sin \theta}{4\pi} \mathrm{d} \phi\mathrm{d} \theta$ for $(\phi,\theta)\in[0,2\pi]\times [0,\pi]$.
The kernel $k_i$
is obtained by {total expectation over possible orientations:}
\begin{equation}\label{eq:ker}
k_i(\ell,r)
=
1-
\int_{\phi=0}^{2\pi}\int_{\theta=0}^\pi
\sqrt{1-\dfrac{\ell^2}{4r^2}\alpha_{\eta_i}(\phi,\theta)}\frac{\sin \theta}{4\pi} \mathrm{d} \theta\mathrm{d} \phi,
\end{equation}
with
\begin{equation}
\alpha_{\eta_i}(\phi,\theta)=\frac{\cos ^2\phi}{\cos^2\theta+\eta_i^2\sin^2 \theta }+\sin ^2\phi
\end{equation}
and the convention that $\sqrt{x}=0$ for $x<0$.
Expression \eqref{eqQ} comes from the law of total expectation, while kernels $k_i$ can be determined by a probabilistic analysis of the two sources of hazards in the measure of a chord on a spheroid crystal: the random orientation of the spheroid with respect to the probe, and the random chord measured by the sensor on the projection of the spheroid onto the plane that is orthogonal to the probe’s laser beam.
Note that in the particular case of spherical crystals (\emph{i.e.} $\eta=1$), expression \eqref{eq:ker} is simpler since $\alpha_1(\phi, \theta)=1$.
For a given shape $\eta$, the length of the largest chord possibly measured by the sensor on a crystal of size $r$ is $\ell^{\max} = 2r\max\{1, \eta\}$, since the direction of the largest diameter of a spheroid depends on whether $\eta>1$ or not.
Set $\ell^{\max} = 2\max\{\rmax{1}\max\{1, \eta_1\}, \rmax{2}\max\{1, \eta_2\}\}$.
Let $X_i = L^2(\rinf{i}, \rsup{i})$ be the function spaces of PSDs and $Y = L^2(0, \ell^{\max})$ the function space of CLD.
Define the operator $\mathcal{K}:X_1\times X_2\to Y$ that maps PSDs to their corresponding CLD:
\begin{equation}
\mathcal{K}(\psi_1, \psi_2) = \left(Q:\ell\mapsto
\sum_{i=1}^2 \int_{\rmin{i}}^{\rmax{i}} k_i(\ell, r)\psi_i(r)\mathrm{d} r\right).
\end{equation}
The estimation problem that we aim to solve in this paper is the following:
``From the knowledge of $Q(t) = \mathcal{K}(\psi_1(t), \psi_2(t))$ over $[0, T]$, where $(\psi_1, \psi_2)$ is a solution of \eqref{syst},
estimate $(\psi_{1, 0}, \psi_{2, 0})$.''
\section{Observability analysis}
First, we need to determine if the CLD $Q$ contains enough information to reconstruct the two PSDs $\psi_1$ and $\psi_2$.
In other words, we investigate the observability of the PDE \eqref{syst} with measured output $Q$.
Several observability notions exist on infinite-dimensional systems.
\begin{definition}[Observability]
Let $(\psi_1, \psi_2)$ be a solution of \eqref{syst} and $Q$ be the corresponding CLD.
Let $W(T) = \int_0^{T} \|Q(t)\|_Y^2 \mathrm{d} t$. System \eqref{syst} is said to be
\begin{itemize}
\item exactly observable if, for some $\kappa>0$,
$W(T) \geq \kappa(\|\psi_{1, 0}\|_{X_1}^2+\|\psi_{2, 0}\|_{X_2}^2)$ for all $\psi_{i, 0}\in X_i$;
\item approximately observable if
$W(T) >0 $ for all $(\psi_{1, 0}, \psi_{2, 0})\neq(0,0)$.
\end{itemize}
\end{definition}
These two notions are widely discussed in \cite{TW} for example.
Clearly, exact observability implies approximate observability, and they are equivalent on finite dimensional systems.
Unfortunately, the function $k_i(\ell, r)$ being bounded, the system is not exactly observable according to \cite[Proposition 6.3]{brivadis:hal-02529820}.
Therefore, we focus on approximate observability, as in \cite{Celle-etal.1989, xu1995observer, haine2014recovering}, and more recently in \cite{brivadis:hal-02529820}.
Let
$$
A(\eta)
=
\begin{cases}
1& \text{ if } \eta\geq 1,
\\
1/\eta^2& \text{ if } \eta< 1.
\end{cases}
$$
We prove approximate observability under the following geometric condition
\begin{equation}\label{eq:cond}
{(\rmin{1})}^2 A(\eta_2)\neq {(\rmin{2})}^2 A(\eta_1).
\end{equation}
\begin{theorem}\label{th:main}
Assume \eqref{eq:cond} holds.
Assume $f\in C^2([0, T]; \mathbb{R})$ has a finite number of zeros
and $h(r)=1/r^m$ for some $m\in\mathbb{N}$.
Let $\psi_{i, 0}\in H^2(\rinf{i}, \rsup{i})$ and
denote by $\psi_i$ for $i=1,2$ the corresponding
solution of \eqref{syst}, satisfying the condition \eqref{eq:rmax}.
If
$\mathcal{K}( \psi_1(t, \cdot), \psi_2(t, \cdot))= 0$ for all $t\in[0, T]$, then
$(\psi_1,\psi_2)=(0,0)$.
\end{theorem}
\begin{remark}
This statement generalizes the result of \cite{brivadis:hal-03053999},
which was limited to the case where $1 = \eta_1<\eta_2$, $\rmin{1}(0) = \rmin{2}(0)$, $\rmax{1}(0) = \rmax{2}(0)$, $g_1>0$, $g_2>0$ and $h(r) = 1$.
This case, not included in the statement of Theorem~\ref{th:main},
can be recovered thanks to technical comments found in Remark~\ref{rem:main}.
\end{remark}
We prove Theorem~\ref{th:main} in two steps. First the observability condition is translated into a sequential equality. Then we prove that this equality between sequences is actually asymptotically incompatible.
\noindent\textbf{Step 1}: From observability to sequence comparisons.
From \eqref{eq:ker}, we can derive
(from the power series expansion of $\ell\mapsto\sqrt{1-\ell^2}$)
a power series expansion at $0$ (with infinite convergence radius) of $\mathcal{K}_i$,
$$
\mathcal{K}_i(\psi_i)(\ell)=\sum_{n=1}^\infty a_n(\eta_i)b_n
\mathcal{F}_i^{2n}(\psi_i) \ell^{2n}
$$
with
$
a_n(\eta)=
\int_{\phi=0}^{2\pi}\int_{\theta=0}^\pi
\alpha_\eta^{n}(\phi,\theta) \frac{\sin \theta}{4\pi} \mathrm{d} \theta\mathrm{d} \phi$,
$b_n = \frac{1}{(n!)^2(1-2n)4^{2n}}$ and
$$
\mathcal{F}_i^n(\psi_i)
=
\int_{\rmin{i}}^{\rmax{i}}
\frac{\psi_i(r)}{r^{n}}\mathrm{d} r.
$$
In proving the approximate observability, we may as well assume
\begin{equation}\label{E:injectivity}
\mathcal{K}_1(\psi_1(t))=\mathcal{K}_2(\psi_2(t)),\qquad t\in[0,T].
\end{equation}
This is reduced to power series expansion comparisons. Term-wise, we have
\begin{equation}\label{E:id}
a_n(\eta_1) \mathcal{F}_1^{2n}(\psi_1)=a_n(\eta_2) \mathcal{F}_2^{2n}(\psi_2).
\end{equation}
If $f(t)\neq 0$ ($f$ is continuous and vanishes finitely many times), we differentiate \eqref{E:injectivity} with respect to time to obtain
$$
\frac{1}{f(t)}\partial_t
\left(
\frac{1}{f(t)}\partial_t
\left(
\mathcal{K}_1(\psi_1)
\right)
\right)
=
\frac{1}{f(t)}\partial_t
\left(
\frac{1}{f(t)}\partial_t
\left(
\mathcal{K}_2(\psi_2)
\right)
\right).
$$
Since
$
\frac{1}{f(t)}\partial_t
\left(
\frac{1}{f(t)}\partial_t
\psi_i
\right)=
g_i^2h(r)\partial_r(h(r)\partial_r\psi_i)
$
(for $i=1,2$),
we obtain
\begin{equation}\label{E:id_diff}
g_1^2 a_n(\eta_1)
\mathcal{F}_1^{2n}(h\partial_r(h\partial_r\psi_1))
=
g_2^2 a_n(\eta_2)
\mathcal{F}_2^{2n}(h\partial_r(h\partial_r\psi_2)),
\end{equation}
where, again,
$$
\mathcal{F}_i^{2n}(h\partial_r(h\partial_r\psi_i))
=
\int_{\rmin{i}}^{\rmax{i}}
\frac{h(r)\partial_r(h(r)\partial_r\psi_i(r))}{r^{2n}}\mathrm{d} r.
$$
\textit{Notation:} To unburden the notations, we will denote $r_i=\rmin{i}$ for the remaining of the section.
With $h(r)=1/ r^m$ and by integration by parts,
\begin{multline*}
\begin{aligned}
\mathcal{F}_i^{2n}(h\partial_r(h\partial_r\psi_i))
=&-\frac{\partial_r\psi_i(r_i)}{r_i^{2n+2m}}
\\
&+
(2n+m)
\mathcal{F}_i^{2n+2m+1}(\partial_r\psi_i)
\\
=&-\frac{\partial_r\psi_i(r_i)}{r_i^{2n+2m}}
-(2n+m)\frac{\psi_i(r_i)}{r_i^{2n+2m+1}}
\end{aligned}
\\
+(2n+m)(2n+2m+1)
\mathcal{F}_i^{2n+2m+2}(\psi_i)
\end{multline*}
That is, changing the variable $n$ to $ n-m$,
\begin{multline}\label{E:IPP}
\mathcal{F}^{2(n-m)}_i(h\partial_r(h\partial_r\psi_i))
=
-\frac{\partial_r\psi_i(r_i)}{r_i^{2n}}
-(2n-m)\frac{\psi_i(r_i)}{r_i^{2n+1}}
\\+
(2n-m)(2n+1)
\mathcal{F}^{2n+2}_i(\psi_i).
\end{multline}
Now we express \eqref{E:id_diff} in terms of \eqref{E:IPP}:
\begin{multline*}
\frac{g_2^2}{g_1^2}\frac{a_{n-m}(\eta_2)}{a_{n-m}(\eta_1)}\bigg[-\frac{\partial_r\psi_2(r_2)}{r_2^{2n}}
-(2n-m)\frac{\psi_2(r_2)}{r_2^{2n+1}}
\\+
(2n-m)(2n+1)
\mathcal{F}^{2n+2}_1(\psi_2)\bigg]
\qquad \\ \qquad
=-\frac{\partial_r\psi_1(r_1)}{r_1^{2n}}
-(2n-m)\frac{\psi_1(r_1)}{r_1^{2n+1}}
\\+
(2n-m)(2n+1)
\mathcal{F}^{2n+2}_1(\psi_1).
\end{multline*}
Finally, we switch $\mathcal{F}^{2n+2}_2(\psi_2)$ for $\mathcal{F}^{2n+2}_1(\psi_1)$ using \eqref{E:id}, leading to the sequence equality
\begin{equation}\label{E:final_eq}
U_n=V_n, \qquad \forall n\geq m+1, n\in \mathbb{N},
\end{equation}
where we have set
$$
U_n=\frac{g_2^2}{g_1^2}\frac{a_{n-m}(\eta_2)}{a_{n-m}(\eta_1)}\bigg(-\frac{\partial_r\psi_2(r_2)}{r_2^{2n}}
-(2n-m)\frac{\psi_2(r_2)}{r_2^{2n+1}}\bigg)
$$
\begin{multline*}
V_n= -\frac{\partial_r\psi_1(r_1)}{r_1^{2n}}
-(2n-m)\Bigg[\frac{\psi_1(r_1)}{r_1^{2n+1}} +
(2n+1)\times
\\
\Big(1-\frac{g_2^2}{g_1^2}\frac{a_{n-m}(\eta_2)}{a_{n+1}(\eta_2)}\frac{a_{n+1}(\eta_1)}{a_{n-m}(\eta_1)}\Big)
\mathcal{F}^{2n+2}_1(\psi_1)\Bigg].
\end{multline*}
\noindent
\textbf{Step 2:} Asymptotical identities.
To prove the approximate observability result, we prove that the asymptotics of both sides of \eqref{E:final_eq} are incompatible, imposing $(\psi_1,\psi_2)= (0,0)$.
To achieve the comparison, we need three identities. First, if $\psi_i(r_i)\neq 0$, then
\begin{equation}\label{E:id1}
\mathcal{F}^{2n}_i(\psi_i)\sim \frac{\psi_i(r_i)}{2nr_i^{2n-1}}.
\end{equation}
This can be obtained by comparing $\psi_i(r)$ to $\psi_i(r_{i})$ on any small interval $(r_{i},r_{i}+\delta)$ ($\delta>0$).
By integration by parts, we also get that if $\psi_i(r_i)=0$ but $\partial_r\psi_i(r_i)\neq 0$, then
\begin{equation}\label{E:id2}
\mathcal{F}^{2n}_i(\psi_i)\sim \frac{\partial_r \psi_i(r_i)}{4n^2r_i^{2n-2}}.
\end{equation}
Finally,
we have
$
\lim_{n\to +\infty}\frac{a_{n+1}(\eta)}{a_n(\eta)}
=
A(\eta).
$
This can be obtained thanks to the following remark. The function $(\phi,\theta)\mapsto\frac{\sin\theta}{4\pi}$ is the density of a probability measure $\mu$ on $(\phi,\theta)\in [0,2\pi]\times [0,\pi]$.
As such, $a_{n}(\eta)=\mathbb{E} \left(\alpha_\eta^{n}\right)$ where $\mathbb{E}$ denotes the expected value with respect to $\mu$.
In that respect,
$$
a_{n+1}(\eta)=\mathbb{E} \left(\alpha_\eta^{n+1}\right)\leq \|\alpha_\eta\|_\infty\mathbb{E} \left(\alpha_\eta^{n}\right)=\|\alpha\|_\infty a_{n}(\eta).
$$
On the other hand,
Jensen's inequality for $\mu$ yields
$$
a_{n+1}(\eta)
=
\mathbb{E} \left(\left(\alpha_\eta^{n}\right)^{\frac{n+1}{n}}\right)
\geq
\left(\mathbb{E} \left(\alpha_\eta^{n}\right)\right)^{1+\frac{1}{n}}
=
\left(a_{n}(\eta)\right)^{1+\frac{1}{n}}.
$$
Hence
$\left(a_{n}(\eta)\right)^{\frac{1}{n}}
\leq
\frac{a_{n+1}(\eta)}{a_{n}(\eta)}
\leq \|\alpha_\eta\|_\infty
$.
We obtain the result by noticing that both sides converge to $\|\alpha_\eta\|_\infty=A(\eta)$.
\noindent
\textbf{Step 3:} Proof of Theorem~\ref{th:main}.
\begin{proof}
First, let us analyse the influence of border terms.
Assuming either $\psi_2(r_2)\neq 0$ or $\partial_r\psi_2(r_2)\neq 0$, the quotient $U_{n+1}/U_n$, yields
\begin{multline*}
\frac{
\partial_r\psi_2(r_2)r_2
+(2n+2-m) \psi_2(r_2)
}{
\partial_r\psi_2(r_2)r_2^3
+(2n-m) \psi_2(r_2)r_2^2
}\times\\
\frac{a_{n-m+1}(\eta_2)}{a_{n-m}(\eta_2)}\frac{a_{n-m}(\eta_1)}{a_{n-m+1}(\eta_1)},
\end{multline*}
which has limit $\frac{A(\eta_2)}{r_2^2A(\eta_1)}$.
For the treatment of $V_n$, we use a natural generalization of the limit quotient of $a_n$:
$$
\frac{a_{n-m}(\eta_2)}{a_{n+1}(\eta_2)}\frac{a_{n+1}(\eta_1)}{a_{n-m}(\eta_1)}
\to
\frac{A(\eta_1)^{m+1}}{A(\eta_2)^{m+1}}.
$$
If $\psi_1(r_1)\neq 0$, then we deduce from \eqref{E:id1} that $\frac{V_n r_1^{2n}}{2n}$ has limit
$
-\frac{g_2^2}{g_1^2}\frac{A(\eta_1)^{m+1}}{A(\eta_2)^{m+1}}
\psi_1(r_1)$, which is incoherent with a limit quotient of $\frac{A(\eta_2)}{r_2^2 A(\eta_1)}\neq \frac{1}{r_1^2}$ by assumption \eqref{eq:cond}.
If $\psi_i(r_i)=0$ but $\partial_r\psi_i(r_i)\neq 0$, then
we deduce from \eqref{E:id1} that $\frac{V_n r_1^{2n}}{4n ^2}$ has limit
$
-\frac{g_2^2}{g_1^2}\frac{A(\eta_1)^{m+1}}{A(\eta_2)^{m+1}}
\partial_r\psi_1(r_1)$ which is again incoherent with a limit quotient of $\frac{A(\eta_2)}{r_2^2 A(\eta_1)}\neq \frac{1}{r_1^2}$.
Hence having $\psi_2(r_2)\neq 0$ or $\partial_r\psi_2(r_2)\neq 0$, is incoherent with having $\psi_1(r_1)\neq 0$ or $\partial_r\psi_1(r_1)\neq 0$. Now let's assume that $\psi_2(r_2)=\partial_r\psi_2(r_2)= 0$. Then if $\psi_1(r_1)\neq 0$, the $ \frac{V_nr_1^{2n}}{2n}$ has a non-zero limit despite being constantly zero, which is excluded. The same goes if $\partial \psi_1(r_1)\neq 0$ while $\psi_1(r_1)=0$.
The conclusion of this first step is that if there exists a pair $(\psi_1,\psi_2)$ of $C^2$ functions satisfying \eqref{E:final_eq}, they must satisfy
$$
\psi_1(r_1)=\partial_r\psi_1(r_1)=\psi_2(r_2)=\partial_r\psi_2(r_2)=0.
$$
We are now in a suitable position to conclude focusing on interior terms.
In that case, we are left with the equality
$$
\left(1-\frac{g_2^2}{g_1^2}\frac{a_{n-m}(\eta_2)}{a_{n+1}(\eta_2)}\frac{a_{n+1}(\eta_1)}{a_{n-m}(\eta_1)}\right)
\mathcal{F}^{2n+2}_1(\psi_1)=0.
$$
Naturally, $\mathcal{F}^{2n+2}_1(\psi_1)$ must have infinitely many non-zero terms, otherwise $\psi_1=0$ (the family $(1/r^{2n})_{n\geq n_0}$ is total on any bounded interval in $(a,b)$, $0<a<b$, for any $n_0$). But this would imply that
$$
\frac{a_{n-m}(\eta_2)}{a_{n+1}(\eta_2)}\frac{a_{n+1}(\eta_1)}{a_{n-m}(\eta_1)}=\frac{g_1^2}{g_2^2}.
$$
infinitely often, which is not true except if $\eta_1=\eta_2$ and $g_1=g_2$.
\end{proof}
\begin{remark}\label{rem:main}
In the case $r_1=r_2$, $\eta_1=1$ and $\eta_2>1$,
Theorem~\ref{th:wp} does not allow to answer but the approximate observability result still holds due the following observation.
In that case, $a_n(\eta_2)\to 0$ but $1>a_n(\eta_2)\geq 1/\sqrt{n}.$ Hence, in \eqref{E:final_eq}, $V_n\times r_1^{2n}$ is equivalent to an integer power in $n$, while $U_n\times r_1^{2n}$ cannot, because of the dominating term containing $a_{n-m}(\eta_2)$.
\end{remark}
\section{Observer and numerical simulations}
The observability analysis of the previous section guarantees the convergence of the state estimation by a BFN algorithm.
Recall that the goal is to estimate $\psi_{i, 0}$ from the measurement of the CLD $Q$ over $[0, T]$.
The BFN algorithm consists in applying iteratively of forward and backward Luenberger observers.
After each iteration of an observer over $[0, T]$, the final estimation obtained at $T$ is used as the initial condition of the next observer.
This strategy has been used in various contexts in recent decades \cite{auroux2005back, auroux2008nudging, auroux2012back}.
As shown in \cite{haine2014recovering} (which extended the results of \cite{ito2011time, Ramdani} which focused on exactly observable systems), the type of convergence depends on the observability properties of the system.
These results have been extended to the non-autonomous context (which is the case here since $G_i$ is time-varying) in \cite{brivadis:hal-02529820}
and applied to a crystallization process in \cite{brivadis:hal-03053999}.
In the context of this paper, the forward and backward observers are given by:
\begin{align}
&
\left\{\begin{aligned}
&\begin{aligned}
\frac{\partial \hat{\psi}_i^{2n}}{\partial t}(t, r) = &-G_i(t, r) \frac{\partial \hat{\psi}^{2n}_i}{\partial r}(t, r)\\
& - \mu\mathcal{K}^*(\mathcal{K} (\hat{\psi}^{2n}_1(t), \hat{\psi}^{2n}_2(t))- Q(t))
\end{aligned}
\\
&\hat{\psi}^{2n}_i(0, r) =
\begin{cases}
\hat{\psi}^{2n-1}_i(0, r) & \text{if } n\geq 1\\
\hat{\psi}_{i, 0}(r) & \text{otherwise}
\end{cases}
\end{aligned}\right.
\label{obs2}\\
&
\left\{\begin{aligned}
&\begin{aligned}
\frac{\partial \hat{\psi}^{2n+1}_i}{\partial t}(t, r) = &-G_i(t, r) \frac{\partial \hat{\psi}_i^{2n+1}}{\partial r}(t, r)\\
& + \mu\mathcal{K}^*(\mathcal{K} (\hat{\psi}^{2n+1}_1(t), \hat{\psi}^{2n+1}_2(t))- Q(t))
\end{aligned}
\\
&\hat{\psi}^{2n+1}_i(T, r) = \hat{\psi}^{2n}_i(T, r)
\end{aligned}\right.
\label{obs2b}
\end{align}
where $\hat{\psi}_i^{n}(t, r)$ represents the estimation of $\psi_i(t, r)$ obtained by the algorithm after $n$ iterations,
$Q(t) = \mathcal{K}(\psi_1(t), \psi_2(t))$ is the CLD at time $t$,
$\mu$ is a degree of freedom, called the observer gain,
and $\mathcal{K}^*$ is the adjoint of the operator $\mathcal{K}$:
\fonction{\mathcal{K}^*}{Y}{X_1\times X_2}{Q}{\left(r\mapsto
\int_{0}^{\ell^{\max}} k_i(\ell, r)Q(\ell)\mathrm{d} \ell\right)_{1\leq i\leq 2}.}
Note that \eqref{obs2} is the usual infinite-dimensional Luenberger observer of \eqref{syst}, while \eqref{obs2b} is a Luenberger observer of \eqref{syst} when reversed in time.
Then,
combining the observability analysis provided in Theorem~\ref{th:main} and the convergence result \cite[Theorem 4.2]{brivadis:hal-03053999},
we obtain the following result.
\begin{theorem}\label{th:conv}
Under the assumptions of Theorem~\ref{th:main}, for all $\mu>0$, all $t\in[0, T]$ and almost all $r\in[r_0, r_1]$,
\begin{equation}
\hat{\psi}^n(t, r)
\tol{n\to+\infty}\psi(t, r).
\end{equation}
\end{theorem}
We propose a numerical simulation of this algorithm.
System~\eqref{syst} and observer \eqref{obs2}-\eqref{obs2b} being transport equations, they are solved by the method of characteristics.
The characteristic equation is given by
\begin{equation}\label{carac}
\frac{\mathrm{d} \rho_i}{\mathrm{d} t} = G_i(t, \rho_i(t)).
\end{equation}
Along the solutions of this ODE,
$\psi_i$ and $\hat{\psi}_i^{n}$ satisfy
\begin{align}
&\frac{\mathrm{d}}{\mathrm{d} t}\psi_i(t, \rho_i(t)) = 0,
\label{ode1}\\
&\frac{\mathrm{d}}{\mathrm{d} t}\hat{\psi}^{n}_i(t, \rho_i(t)) = (-1)^{2n+1}
\mu\mathcal{K}^*(\mathcal{K} (\hat{\psi}^{n}_1(t), \hat{\psi}^{n}_2(t))- Q(t)).
\label{ode2}
\end{align}
We choose a spatial discretization of $[\rinf{i}, \rsup{i}]$ with space-step $\mathrm{d} x$, and integrate the characteristic equation \eqref{carac} over $[0, T]$ with time-step $\mathrm{d} t$ with initial conditions in this spatial discretization. Then, we integrate ODEs \eqref{ode1}-\eqref{ode2} along the characteristic curves with a first order Euler method.
Concerning the operator $\mathcal{K}$, integrals are computed with the rectangles methods.
We consider the set of parameters given in Table~\ref{tab:param}, and the observer gain $\mu=0.001$ (small enough to preserve stability of the numerical scheme).
Note that in this example, $G$ does not depends on $t$, but this actually does not affect the convergence properties, since it is always possible to use a time-reparametrization as it is done in the proof of Theorem~\ref{th:wp}.
Moreover, condition \eqref{eq:cond} is satisfied by the example.
\begin{table}[ht!]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
$\rmin{1}=\rmin{2} = 0.1$
&$\rmax{1}=\rmax{2} = 0.2$
&$T = 1$
\\
\hline
$g_1 = 0.1$
&$g_2 = 0.2$
&$f(t)h(r) = 1/r^2$
\\
\hline
$\eta_1 = 0.5$
&$\eta_2 = 2$
&$\mathrm{d} x=\mathrm{d} t = 0.01$
\\
\hline
\end{tabular}
\caption{Parameters of the numerical simulation.}
\label{tab:param}
\end{table}
The observer is initialized at $\hat\psi_{1, 0}=\hat\psi_{2, 0}=0$.
The initial conditions $\psi_{1, 0}$ and $\psi_{2, 0}$ are chosen as normal distributions centered at $r = 0.05$ and $r=0.15$, respectively.
Roughly speaking, crystals of shape $\eta_1$ will appear during $[0, T]$, but are not in the reactor at $t=0$, while crystals of shape $\eta_2$ are in the reactor at $t=0$ but disappear through the process.
The result of the simulation is presented in Figure~\ref{fig:simu} (numerical implementation can be found in repository \cite{git}).
After only $10$ iterations,
the locus of the maximum of the two PSDs is already well estimated.
In practice, this is the main information to be estimated.
After $1000$ iterations, the estimations are much more accurate.
Still, a peak at $r=0.15$ remains on $\hat{\psi}_{1, 0}$ while it is not in $\psi_{1, 0}$. This peak is due to the important contribution of $\psi_{2, 0}$ in the CLD at $r=0.15$. However, its amplitude decreases as the number of iterations increases, and eventually vanishes according to Theorem~\ref{th:conv}.
\begin{figure}
\caption{PSDs $\psi_1$ and $\psi_2$ at time $t=0$
and their estimations $\hat{\psi}
\label{fig:simu}
\end{figure}
\section{Conclusion}
In this paper,
we propose an observability analysis of a crystallization process.
We prove, under a geometric condition,
that two PSDs
of spheroid crystals of different shapes
are fully determined by their common CLD along the process.
Hence, the BFN algorithm is able to reconstruct the PSDs from the measurement of the CLD over a finite time interval,
by using iterations of forward and backward infinite-dimensional Luenberger observers. We provide a numerical simulation of the algorithm which suggest that possible applications of this method to experimental data could benefit from this theoretical study.
\end{document} | math |
बेतरतीब: स्टेशनरी शॉप...
पहले बात कहानी की करता हूँ, इस छोटे से कथानक और इसके कंटेंट के चित्रण ने विस्मित किया है। पहली बार मैं देखता रह गया कि किस प्रकार गाने के संगीत की तर्ज पर कहानी शुरू हुई और अपनी अभिव्यक्ति के ऐहसास छोड़ती हुई एक गाने की तरह ही समाप्त हो गयी। पहली विशेषता है कि इस कहानी को हर बार एक से अधिक बार पढ़ना होता है और सरलता से पढ़ा भी जाता है। न बड़े शब्दों का बोझ, न विन्यास की कसरत न ही पाठक को कदम कदम पर चौंका डालने वाला घटनाक्रम। यह अद्भुद है कि सीधे सरल शब्दों में सीधे सरल जीवन के भीतर पड़े मनोवैज्ञानिक आयाम को सरलता से चित्रित किया गया और पाठक उन्हीं भावनाओं में बहता रहा जिनसे कहानी के व्यक्तित्व के कपाट खुलते हैं। भावनाओं के मार्मिक बहाव में इतनी सादगी सच में यह विस्मित करने वाला लेखन है।
आँचल तुमने जिस गहराई से गद्य लेखन की अपनी शैली को विकसित किया है वह मुझे आश्चर्य में डालता है। कोई झिझक नहीं यह कहने में कि तुम बड़े लेखक के रूप में तेजी से विकसित हो रही हो, अपनी शैली और चित्रण पर पकड़ बनाये रखना। स्त्री लेखक होने का पूर्वाग्रह नहीं झलकता तुम में बल्कि वह तुम्हें अनुभव और अभिव्यक्ति की एडिशनल शक्ति दे रहा है। मै इस लेखन से प्रभावित हूँ और अब तुमसे इसी प्रकार कसी हुई सुंदर लेकिन लंबी कहानियों की अपेक्षा करता हूँ, वही अगला पड़ाव भी है। इस तेजी से गद्य की इतनी सुंदर विधा पकड़ने पर आँचल को दिल से बधाई ......:-) | hindi |
\begin{document}
\author{Jakub Kr\'asensk\'y}
\title{A cubic ring of integers with the smallest Pythagoras number}
\address{Charles University, Faculty of Mathematics and Physics, Department of Algebra,\newline
Sokolovsk\'{a}~83, 18600 Praha 8, Czech Republic}
\email{\hangindent=8em\hangafter=1 [email protected]}
\keywords{}
\thanks{The author aknowledges partial support by project PRIMUS/20/SCI/002 from Charles University, by Czech Science Foundation GA\v{C}R, grant 21-00420M, by projects UNCE/SCI/022 and GA UK No.\ 742120 from Charles University, and by SVV-2020-260589.}
\begin{abstract}
We prove that the ring of integers in the totally real cubic subfield $K^{(49)}$ of the cyclotomic field $\Q(\zeta_7)$ has Pythagoras number equal to $4$. This is the smallest possible value for a totally real number field of odd degree. Moreover, we determine which numbers are sums of integral squares in this field, and use this knowledge to construct a diagonal universal quadratic form in five variables.
\end{abstract}
\setcounter{tocdepth}{2} \maketitle
\section{Introduction}
In this note, we shall study sums of integral squares in the cubic field $K^{(49)} = \Q(\zeta_7+\zeta_7^{-1})$, which can be characterised e.g.\ as the unique number field of discriminant $49$ or as the maximal totally real subfield of the seventh cyclotomic field. In the main Theorem \ref{th:main}, we show that any sum of integral squares can in fact be written as a sum of four such squares -- in the terminology introduced below, the Pythagoras number of $\O_{K^{(49)}}$ is $4$. In Theorem~\ref{th:odd>3} we recall the generally known fact that this is the smallest possible value among totally real fields of odd degree. Moreover, with the information gained during the proof of the main theorem, it is fairly simple to fully characterise the numbers which are sums of integral squares in $K^{(49)}$ (Theorem \ref{th:characterisation}) and to use this to construct a diagonal universal quadratic form in five variables (Corollary \ref{co:universal}).
Let us start by summarising the known results. For standard definitions and notation, in particular regarding number fields, local fields and quadratic forms, we refer the reader to \cite{Mi} and \cite{OMeara}.
When $R$ is a commutative ring, $\sum R^2$ denotes the set of all elements of $R$ which can be written as a sum of squares, and for $\alpha \in \sum R^2$, its \emph{length} $\ell(\alpha)$ is the smallest integer $n$ such that $\alpha$ can be written as a sum of $n$ squares. The \emph{Pythagoras number} of $R$ is then defined as the largest length occurring in $\sum R^2$, i.e.
\[
\P(R) = \sup \bigl\{\ell(\alpha) : \alpha \in \textstyle{\sum} R^2\bigr\}.
\]
In this notation, Lagrange's four square theorem can be written as $\P(\Z)=4$. The Pythagoras number is primarily studied for fields \cite{Ho, TVY, BV, BL, BGV, Hu}, as this is arguably the easier case, but there are several results about the Pythagoras number of rings of integers $\O_K$ of a number field $K$ as well. In fact, if $K$ is not totally real, then $\P(\O_K) \leq 4$ \cite{Pf}. For totally real fields, the situation is dramatically different: By Scharlau \cite{Sch}, there are number fields $K$ with arbitrarily large $\P(\O_K)$. On the other hand, $\P(\O_K)$ is bounded by a constant depending only on the degree $d=[K : \Q]$, as shown by Kala and Yatsyna \cite{KY}. This constant is the so-called $g$-invariant $g_{\Z}(d)$, see e.g.\ \cite{KO}; thus for $1 \leq d \leq 5$, this bound is $d+3$, and for $d=6$, its value is $10$.
For a given number field $K$, it is fairly simple to obtain lower bounds for $\P(\O_K)$, since finding the length of any given $\alpha$ is only a computational task; the main content of several recent papers \cite{Ti, KRS, Kr} is finding good lower bounds for infinite families of fields. Obtaining tight upper bounds is much more difficult. The result of Kala and Yatsyna was generalised in \cite{KRS} to include $g$-invariants of any subfield of $K$, which in particular yielded $\P(\O_K) \leq 5$ for quadratic extensions of $\Q(\sqrt5)$; however, very few values of $g$-invariants of totally real orders are known, so this generalisation is of limited use.
Since cubic fields have no subfields besides $\Q$, the only easily available upper bound for their Pythagoras number is $\P(\O_K) \leq g_{\Z}(3) = 6$. Recently, Tinkov치 \cite{Ti} showed that this upper bound is attained in infinitely many totally real cubic fields $K$. More precisely, she studied the family of so-called \emph{simplest cubic fields} $\Q(\rho_a)$, introduced by Shanks \cite{Sh}, where $\rho_a$ is a root of the polynomial $x^3-ax^2-(a+3)x-1$, $a \geq -1$, and proved that $\P(\Z[\rho_a]) = 6$ for $a \geq 3$. (This order is the full ring of integers for a positive proportion of $a$.)
For $a=0,1,2$, Tinkov치 exhibits elements of length $5$, and for $a=-1$ she only gives $\ell(7)=4$, noting that using a computer program, she checked that no element with a reasonably small trace has bigger length. Therefore, it is natural to conjecture that $\P(\Z[\rho_{-1}])=4$. Our Theorem \ref{th:main} confirms this hypothesis, since $\Q(\rho_{-1}) = K^{(49)}$. Among totally real cubic fields, our result provides the first one where $\P(\O_K) \leq 4$, and by Tinkov치's result we know that it is the only simplest cubic field where the order $\Z[\rho_a]$ has Pythagoras number less than $5$.
In total, very few totally real fields $K$ with $\P(\O_K) \leq 4$ are known: For $K = \Q(\sqrt2), \Q(\sqrt3)$ and $\Q(\sqrt5)$ one has $\P(\O_K)=3$, while for $K = \Q, \Q(\sqrt6)$ and $\Q(\sqrt7)$ it is known that $\P(\O_K)=4$. For all other real quadratic fields, the maximal possible value $5 = g_{\Z}(2)$ is attained. The paper \cite{KRS} proves that there are at most seven real biquadratic fields with $\P(\O_K) \leq 4$, conjecturing that these seven fields indeed satisfy the inequality. This weak evidence may lead one to conjecture that in fact, there are only finitely many totally real $K$ with $\P(\O_K) \leq 4$.
We conclude this section by combining two well-known facts in order to show $\P(\O_K) \geq 4$ for every totally real number field of odd degree. The result is stated for general orders:
\begin{theorem}\label{th:odd>3}
Let $K$ be a totally real number field with $[K : \Q]$ odd. Then $\P(K) = 4$. If~$\O$ is an order in $K$, then $\P(\O) \geq 4$.
\end{theorem}
\begin{proof}
The fact that $\P(K)=4$ can be found in \cite[Ex.\ ~XI.5.9(2)]{La}. It is an application of the local-global principle, and the core of the argument is as follows: Since there is an odd number of real embeddings, by Hilbert's reciprocity law there exists at least one finite place where the form $x^2+y^2+z^2$ is anisotropic.
It is also well known that if $R$ is any integral domain and $F$ its field of fractions, then $\P(R) \geq \P(F)$. Indeed, if $\frac{\alpha}{\beta}$ is not a sum of $n$ squares in $F$, then the same holds for $\alpha\beta$ in $F$ and thus also in $R$. Now the proof is concluded since the field of fractions of $\O$ is $K$.
\end{proof}
\section{Preliminaries and results}
We have already defined the Pythagoras number, the set $\sum R^2$ and the length $\ell(\,\cdot\,)$. By $I_n$ we mean the quadratic form $x_1^2 + \cdots + x_n^2$. For brevity, we sometimes denote squares simply by $\square$. The symbol $h(\varphi)$ stands for the class number of the quadratic form $\varphi$. For its definition, as well as for other standard terminology and notation, we refer the reader to O'Meara's book \cite{OMeara}. The starting point of this paper is the following result, proven by analytical methods in \cite{Pe2}:
\begin{theorem}[Peters, 1977] \label{th:peters}
There are only six totally real number fields where $h(I_3)=1$; $K^{(49)}$ is one of them. (The other fields in question are $\Q$, $\Q(\sqrt2)$, $\Q(\sqrt5)$, $\Q(\sqrt{17})$ and the cubic field $K^{(148)}$ with discriminant $148$.)
\end{theorem}
From now on, we will leave general number fields and focus only on the field with discriminant $49$. Let $\zeta_7 = \mathrm{exp}(\frac{2\mathfrak{p}i\mathrm{i}}{7})$ be the primitive seventh root of unity and let $K^{(49)}$ stand for the totally real cubic field $\Q(\zeta_7+\zeta_7^{-1})$. Usually we denote its ring of integers $\Z[\zeta_7+\zeta_7^{-1}]$ simply by $\O$. Later, we will also use the element $\rho = \zeta_7+\zeta_7^{-1}$ which generates $\O$; its minimal polynomial is $x^3+x^2-2x-1$. Our main theorem is the following:
\begin{theorem} \label{th:main}
$\P\bigl(\Z[\zeta_7+\zeta_7^{-1}]\bigr)=4$.
\end{theorem}
There are many remarkable properties of $K^{(49)}$; for example, it has class number $1$, it is a Galois extension of $\Q$, and every other cubic Galois extension of $\Q$ has larger discriminant (in absolute value).
Thanks to Theorem \ref{th:peters}, sums of three integral squares in $\O$ satisfy the local-global principle; hence, our main tools are the completions $\O_\mathfrak{p}$ of the ring of integers $\O$ at places $\mathfrak{p}$. In particular, we will examine the only dyadic place of $\O$, corresponding to the prime ideal $(2)$. (The fact that $2$ is indeed a prime of $\O = \Z[\zeta_7+\zeta_7^{-1}]$ is easily checked using \cite[Th.\ ~3.41]{Mi}.)
In general, local conditions do not suffice to describe the set $\sum \O_K^2$; usually there are elements which are not a sum of squares (\enquote{globally}) despite being a sum of integral squares at all completions. Scharlau calls them \enquote{Ausnahmeelemente} (i.e.\ \emph{exceptional elements}) and examines them in depth in \cite{Sch2} -- for example, he shows that up to multiplication by squares of units, there are always only finitely many of them; on the other hand, he proves that in every cubic order there is at least one such element. As the discriminant of $K$ is odd, the only local condition for a sum of squares is total positivity \cite[Kap.\ ~0]{Sch2}. In our second result, we characterise the set of sums of squares, showing that up to multiplication by units there is only one exceptional element, namely $1+\rho+\rho^2$. (Remember that $\rho=\zeta_7+\zeta_7^{-1}$.)
\makeatletter
\newcommand{\mylabel}[2]{#2\def\@currentlabel{#2}\label{#1}}
\makeatother
\begin{theorem} \label{th:characterisation}
Let $\alpha \in \O$ be totally positive. Then the following statements are equivalent:
\begin{enumerate}
\item[\mylabel{a}{(1a)}] $\alpha \notin \sum \O^2$.
\item[\mylabel{b}{(1b)}] $\alpha$ is not a sum of four integral squares.
\item[\mylabel{c}{(2a)}] The norm of $\alpha$ is $7$.
\item[\mylabel{d}{(2b)}] $\alpha = u^2(1+\rho+\rho^2)$ for a unit $u \in \O$.
\item[\mylabel{e}{(2c)}] $\alpha$ is an indecomposable element and not a square.
\end{enumerate}
\end{theorem}
The proof of this result, as well as the therein contained notion of \emph{indecomposable elements}, is to be found in Section \ref{se:characterisation}. Note that while a full characterisation of $\sum \O_K^2$ is not usually known, for real quadratic fields, the problem is solved in \cite[Satz 2]{Pe}.
We can immediately state a simple corollary of our characterisation of $\sum \O^2$. Remember that a totally positive definite quadratic form is called \emph{universal} if it represents all totally positive integers. In \cite{KY}, it is proven that $K^{(49)}$ has the very rare property of admitting a universal quadratic form with rational integral coefficients (the form in question has four variables: $x^2+y^2+z^2+w^2+xw+yw+zw$). With our knowledge, one easily constructs a universal quadratic form in five variables:
\begin{corollary} \label{co:universal}
The totally positive definite quadratic form $x_1^2 + x_2^2 + x_3^2 + x_4^2 + (1+\rho+\rho^2)x_5^2$ is universal over $\O$.
\end{corollary}
While the form given in \cite{KY} has less variables, their form is \emph{non-classical}, i.e.\ the corresponding symmetric matrix contains non-integral entries. Our form is not only classical, but even diagonal. For monogenic simplest cubic fields, \cite[Th.\ 1.1]{KT} provides a construction of a diagonal universal quadratic form, which for $\O_{K^{(49)}}$ requires $12$ variables, so our form is much simpler. To the best of our knowledge, no classical universal form in five or less variables in a totally real field of odd degree has been previously known. (By a well-known argument involving Hilbert reciprocity, at least four variables are necessary for fields of odd degree \cite{EK}.)
\section{Sums of three squares}
It is well known \cite[102:5]{OMeara} that every quadratic form with class number $1$ satisfies the local-global principle. This means that a number is represented over $\O$ if and only if it is represented over $\O_{\mathfrak{p}}$ for all places $\mathfrak{p}$. While the sum of four squares $I_4$ does not have this property (we will later prove that $1+\rho+\rho^2$ is not a sum of any number of squares, while being a sum of four squares everywhere locally), we shall exploit that $I_3$ satisfies the local-global principle to fully determine which numbers are a sum of three integral squares in $K^{(49)}$:
\begin{proposition} \label{pr:localglobal}
In $\O = \Z[\zeta_7+\zeta_7^{-1}]$, the quadratic form $I_3$ represents a nonzero number $\alpha$ if and only if $\alpha$ satisfies both the following conditions:
\begin{enumerate}
\item $\alpha$ is totally positive;
\item at the dyadic place $\O_{(2)}$, $\alpha \neq -t^2$ for all $t \in \O_{(2)}$.
\end{enumerate}
\end{proposition}
The proof is given at the end of this section. To prepare for it, we need to examine the dyadic completion $\O_{(2)}$. We state our results more generally for the ring of integers $\mathfrak{O}$ in a dyadic local field $L$ with a few additional properties, since this generality makes the proofs clearer. We start with a useful observation:
\begin{observation}
Let $\mathfrak{O}$ be any commutative ring where $2$ is a prime element. Then the following statements are equivalent for $x,y \in \mathfrak{O}$:
\begin{enumerate}
\item $x \equiv y \mathfrak{p}mod2$; \label{it:1}
\item $x^2 \equiv y^2 \mathfrak{p}mod4$; \label{it:2}
\item $x^2 \equiv y^2 \mathfrak{p}mod2$. \label{it:3}
\end{enumerate}
Indeed, the implications (\ref{it:1}) $\Rightarrow$ (\ref{it:2}) $\Rightarrow$ (\ref{it:3}) are trivial. For (\ref{it:3}) $\Rightarrow$ (\ref{it:1}), one rewrites the condition (\ref{it:3}) as $2 \mid (x-y)^2$ and uses the fact that $2$ is a prime.
\end{observation}
This observation can further be exploited to get the following lemma:
\begin{lemma} \label{le:sumoftwosquares}
Let $\mathfrak{O}$ be the ring of integers of a dyadic local field $L$ where $2$ is a prime element and the degree $[L : \Q_2]$ is odd. Let $u,v \in \mathfrak{O}$ be units. Then:
\begin{enumerate}
\item $u^2 + v^2 \neq \square$;
\item there are no $y,z \in \O$ such that $u^2+y^2+z^2 \equiv 0 \mathfrak{p}mod4$.
\end{enumerate}
\end{lemma}
\begin{proof}
Assume that, contrary to the first statement, $u^2+v^2=w^2$ for some $w \in \mathfrak{O}$. Then $w^2 \equiv (u+v)^2 \mathfrak{p}mod2$, so the previous observation yields $w^2 \equiv (u+v)^2 \mathfrak{p}mod4$.
Plugging this into $u^2+v^2 = w^2$ gives $2uv \equiv 0 \mathfrak{p}mod4$. This is equivalent to $uv \equiv 0 \mathfrak{p}mod2$, which cannot happen since $u,v$ are units. Thus the first part is proven.
Similarly, if the second statement is violated, we write $z^2 \equiv (u+y)^2 \mathfrak{p}mod2$, so the above observation yields $z^2 \equiv (u+y)^2 \mathfrak{p}mod4$. Plugging in, we get $2u^2 + 2y^2 + 2uy \equiv 0 \mathfrak{p}mod4$. This is equivalent to $u^2 + uy + y^2 \equiv 0 \mathfrak{p}mod2$; as $u$ is a unit, it follows that there exists a root of $t^2+t+1$ modulo $2$, i.e.\ in $\mathfrak{O}/(2)$.
Since $2$ is the prime element, $\mathfrak{O}/(2)$ is the residue field. It is isomorphic to $\mathbb{F}_{2^d}$, where $d=[L:\Q_2]$ is by assumption an odd number. However, this is a contradiction -- no root of an irreducible quadratic polynomial over $\mathbb{F}_2$ can be contained in an extension of odd degree.
\end{proof}
We were building towards the following characterisation of sums of three squares in~$\O_{(2)}$ and more generally in the ring of integers in any unramified dyadic local field of odd degree:
\begin{lemma} \label{le:dyadicI3}
Let $L$ be any dyadic local field where $2$ is a prime element and the degree $[L : \Q_2]$ is odd. Then, over its ring of integers $\mathfrak{O}$, the form $I_3$ represents exactly those elements which are not equal to $-t^2$ for any $t \in \mathfrak{O}$.
\end{lemma}
\begin{proof}
Bear in mind that a square is integral if and only if the squared number is integral. Now, the proof has two parts. First we show that if a number in $\mathfrak{O}$ is a sum of three squares in $L$, then these squares are necessarily integral. Afterwards we use the known results about representations over a local field.
Let us examine whether a sum of three non-integral squares can belong to $\mathfrak{O}$. We use the fact that $2$ is the prime element, and multiply the equality by a suitable power of $2$, to reformulate the question as follows: Is it possible to find $x,y,z \in \mathfrak{O}$, with $x$ being a unit, so that $x^2+y^2+z^2 \equiv 0 \mathfrak{p}mod4$? By Lemma \ref{le:sumoftwosquares} (2), the answer is no.
We have just shown that if the sum of three squares is integral, then the squared numbers are integral as well (which, as a corollary, also yields that $0$ is not non-trivially represented, i.e.\ $I_3$ is anisotropic). Therefore, we now only have to examine the numbers represented by $I_3$ over the field $L$ instead of over the ring $\mathfrak{O}$. Since $I_3$ is anisotropic and its determinant is a square, \cite[63:21]{OMeara} yields that the only not-represented elements are minus squares.
\end{proof}
With this dyadic preparation, we are ready to characterise sums of three squares in $\O$.
\begin{proof}[Proof of Proposition \ref{pr:localglobal}]
By Theorem \ref{th:peters}, the form $I_3$ has class number $1$, so it satisfies the local-global principle. Thus, it suffices to check which numbers are represented locally.
In the real embeddings, every positive number is already a square, while negative numbers cannot be written as any number of squares. In a non-dyadic completion, every ternary unimodular form is universal \cite[92:1b]{OMeara}. Thus it remains to examine the only dyadic place $\O_{(2)}$. This was done in Lemma \ref{le:dyadicI3} -- this lemma applies since $(2)$ is inert, so $[K^{(49)}_{(2)} : \Q_2]$ is equal to $[K^{(49)} : \Q]$, which is odd.
\end{proof}
\section{The main proof}
Now we fully understand sums of three integral squares. To complete the proof that every sum of squares in $\Z[\zeta_7+\zeta_7^{-1}]$ is already a sum of four squares, one has to examine two types of problematic elements: Those which are equal to minus square of a unit in $\O_{(2)}$, and those which are minus square of a number divisible by $2$ in $\O_{(2)}$. To deal with the latter type, one simple trick suffices:
\begin{observation} \label{ob:2multiples}
Let $\alpha \in \O$ be totally positive and let it be equal to $-(2t)^2$ at the dyadic place. Then $\ell(\alpha)=4$, i.e.\ it is a sum of four but not of three squares.
Indeed, the number $\alpha/2$ is totally positive and it is clearly not equal to $-\square$ at the dyadic place, hence by Proposition \ref{pr:localglobal} we have $\alpha/2 = x^2 + y^2 + z^2$ in $\O$. Then $\alpha = (x+y)^2 + (x-y)^2 + z^2 + z^2$. On the other hand, by the same proposition, $\alpha$ is not a sum of three squares.
\end{observation}
As a side note, one could also use the universality of the quadratic form $x^2 + y^2 + z^2 + w^2 + xw + yw + zw$ to deduce that every totally positive number in $2\O$ is a sum of four squares, since $2(x^2+y^2+z^2+w^2+xw+yw+zw) = (x+y+w)^2+(x-y)^2+(z+w)^2+z^2$. Either way, combining Proposition \ref{pr:localglobal} with the above observation, we obtain the following neat statement, which is the basis of the proofs of Theorems \ref{th:main} and \ref{th:characterisation}:
\begin{corollary} \label{co:minusunitbad}
If a totally positive number in $\O$ is not a sum of four integral squares, then in $\O_{(2)}$ it is equal to $-u^2$ where $u$ is a unit.
\end{corollary}
Finally, we are ready to prove that the Pythagoras number of $\O$ is $4$:
\begin{proof}[Proof of Theorem \ref{th:main}]
Let $\alpha \in \sum \O^2$. Clearly, $\alpha$ is totally non-negative, so in most cases Corollary \ref{co:minusunitbad} applies and $\alpha$ is a sum of four squares. It remains to deal with the case when at the dyadic place, $\alpha = -u^2$ with $u$ a unit. In this case, the condition that $\alpha$ is totally positive is not enough for its being a sum of squares (compare Theorem \ref{th:characterisation}). However, we know that $\alpha = \sum x_i^2$ in $\O$ for some $x_i$. We shall show that if $x_j$ is a dyadic unit, then $\alpha - x_j^2$ is a sum of three squares in $\O$, implying that $\alpha$ is a sum of four squares. This will conclude the proof, since at least one of the summands must be a unit.
For the sake of contradiction, suppose that $\sum_{i \neq j} x_i^2$ is not a sum of three squares in $\O$. It is totally positive, so by Proposition \ref{pr:localglobal} it must be $-\square$ in $\O_{(2)}$. However, the equality $-u^2 = x_j^2 - \square$ in $\O_{(2)}$ is impossible by Lemma \ref{le:sumoftwosquares} (1) since $u$ a $x_j$ are units.
\end{proof}
\section{Characterisation of sums of squares} \label{se:characterisation}
In this section we are going to determine which numbers can be written as a sum of squares (by Theorem \ref{th:main} they can in fact be represented as a sum of \emph{four} squares). To achieve this, we shall need the following notion: A totally positive number in $\O$ is called \emph{indecomposable} if it cannot be written as a sum of two totally positive elements of $\O$.
The indecomposable elements are quite difficult to study; it is a significant success that \cite[Th.\ 1.2]{KT} fully characterises them for the order $\Z[\rho_a]$ in the simplest cubic fields. For our case, their result reads as follows (recall that $\rho = \zeta_7+\zeta_7^{-1}$):
\begin{lemma}[Kala, Tinkov치]
Up to multiplication by squares of units, the only indecomposables of $\O$ are $1$ and $1+\rho+\rho^2$.
\end{lemma}
We use this lemma to provide a very explicit description of the set $\sum \O^2$, namely: a totally positive $\alpha$ is an exceptional element if and only if it satisfies the simple condition \ref{c} or equivalently \ref{d}. However, note that the core of the proof does not use the indecomposable elements explicitly -- without the lemma, we would still have the equivalence of \ref{a} (and \ref{b}) with \ref{e}.
\setcounter{section}{2}
\setcounter{theorem}{2}
\begin{theorem}
Let $\alpha \in \O$ be totally positive. Then the following statements are equivalent:
\begin{enumerate}
\item[(1a)] $\alpha \notin \sum \O^2$.
\item[(1b)] $\alpha$ is not a sum of four integral squares.
\item[(2a)] The norm of $\alpha$ is $7$.
\item[(2b)] $\alpha = u^2(1+\rho+\rho^2)$ for a unit $u \in \O$.
\item[(2c)] $\alpha$ is an indecomposable element and not a square.
\end{enumerate}
\end{theorem}
\begin{proof}
The new input in this proof will be the equivalence between \ref{a} and \ref{e}. Most of the other implications are clear by now:
By Theorem \ref{th:main}, \ref{a} and \ref{b} are equivalent. Statements \ref{d} and \ref{e} are equivalent by the previous lemma. The equivalence of \ref{c} and \ref{d} is elementary: On the one hand, one easily checks (or looks up in \cite{KT}) that $1+\rho+\rho^2$ has norm $7$; on the other hand, since $(7)$ ramifies, any two (totally positive) elements of norm $7$ generate the same ideal, hence differ only by multiplication by a (totally positive) unit. It is well known \cite{Ti} that in simplest cubic fields, a unit is totally positive if and only if it is a square.
The implication from \ref{e} to \ref{a} is also immediate: An indecomposable element is either a square or not a sum of squares. Hence our task is to prove the converse: If $\alpha$ is decomposable, then $\alpha$ is a sum of squares. In the following, we will repeatedly exploit Corollary \ref{co:minusunitbad}: If a totally positive number is not a sum of squares, then in $\O_{(2)}$ it must be equal to $-u^2$ where $u$ is a unit.
Assume that $\alpha = \beta + \gamma$. If both $\beta$ and $\gamma$ are sums of squares, then so is $\alpha$. If neither of them is a sum of squares, then at the dyadic place, $\alpha$ is $-(u_1^2+u_2^2)$ where $u_1,u_2$ are units. However, by Lemma \ref{le:sumoftwosquares} (1) the sum of two unit squares is not a square in $\O_{(2)}$, hence $\alpha$ is a sum of squares in $\O$.
Hence we may assume that $\beta = \sum x_i^2$ and $\gamma$ is not a sum of squares. We also may assume that $x_1$ is a unit in $\O_{(2)}$; if not, we use the equality $(2x)^2=x^2+x^2+x^2+x^2$. Clearly it suffices to show that $\tilde{\gamma} = x_1^2 + \gamma$ is a sum of squares. And this is easy, since Lemma \ref{le:sumoftwosquares} (1) gives a contradiction if both $\gamma$ and $\tilde{\gamma}$ are minus squares of units in $\O_{(2)}$.
\end{proof}
\setcounter{section}{5}
\setcounter{theorem}{1}
The key idea behind the above proof is that while not every indecomposable element is a sum of squares, the sum of two indecomposables in $\O$ can always be rewritten as a sum of squares. With the just proven theorem, one immediately gets Corollary \ref{co:universal}: The quadratic form $x_1^2+x_2^2+x_3^2+x_4^2$ represents all totally positive elements except for those of the form $u^2(1+\rho+\rho^2)$, and these are represented by $(1+\rho+\rho^2)x_5^2$.
\section{Final notes}
The crucial ingredient of our proofs was the fact that the sum of three squares $I_3$ satisfies the local-global principle. The same arguments in fact prove that $\P(\O_K)=4$ for every totally real field $K$ of odd degree where $(2)$ is inert, \emph{provided that $I_3$ integrally represents all numbers which it integrally represents everywhere locally}, but the only straightforward way to prove this integral local-global principle is proving $h(I_3)=1$.
Therefore, the only totally real fields where it is reasonably simple to show that $\P(\O_K)$ is small are the six fields listed in Theorem \ref{th:peters} as the only fields with $h(I_3)=1$. Let us discuss them briefly:
For $\Q$, we have $\P(\Z)=4$ by Lagrange's four square theorem. For $\Q(\sqrt2)$ and $\Q(\sqrt5)$ it is well known \cite{Dz, Ma} that $\P(\O_K)=3$; the proofs are based on the local-global principle. For $\Q(\sqrt{17})$, the situation is different: $\P\bigl(\Z\bigl[\frac{1+\sqrt{17}}{2}\bigr]\bigr)=5$, which is the largest value allowed for quadratic orders by the bound $g_{\Z}(2)$; this is easily proven by checking that $7 + \bigl(\frac{1+\sqrt{17}}{2}\bigr)^2$ is not a sum of four squares. Actually, by \cite[Th.\ 5'']{CP}, there are many different elements of length $5$ in $\Z\bigl[\frac{1+\sqrt{17}}{2}\bigr]$: Any totally positive element of the norm $128\cdot 4^k$ has length $5$.
The only remaining field is $K^{(148)}$, the cubic field with discriminant $148$. There, the same strategy as in this paper might be applied -- first, one proves an analogy of Proposition~\ref{pr:localglobal}, characterising sums of three squares, and then hopefully exploits this result to prove $\P(\O_{K^{(148)}})=4$ or $\P(\O_{K^{(148)}})=5$. It should not be too difficult, but the situation is slightly complicated by the fact that $(2)$ ramifies, so the behaviour at the dyadic place is less simple than in our Lemma \ref{le:dyadicI3}. For example, total positivity is no longer the only local condition for a sum of squares.
\end{document} | math |
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<?php
/*
* This file is part of PHP CS Fixer.
*
* (c) Fabien Potencier <[email protected]>
* Dariusz Rumiński <[email protected]>
*
* This source file is subject to the MIT license that is bundled
* with this source code in the file LICENSE.
*/
namespace PhpCsFixer\Fixer\Phpdoc;
use PhpCsFixer\AbstractPhpdocTypesFixer;
/**
* @author Graham Campbell <[email protected]>
*/
final class PhpdocTypesFixer extends AbstractPhpdocTypesFixer
{
/**
* The types to process.
*
* @var string[]
*/
private static $types = array(
'array',
'bool',
'boolean',
'callable',
'double',
'false',
'float',
'int',
'integer',
'mixed',
'null',
'object',
'real',
'resource',
'self',
'static',
'string',
'true',
'void',
'$this',
);
/**
* {@inheritdoc}
*/
public function getDescription()
{
return 'The correct case must be used for standard PHP types in phpdoc.';
}
public function getPriority()
{
/*
* Should be run before all other docblock fixers apart from the
* phpdoc_to_comment and phpdoc_indent fixer to make sure all fixers
* apply correct indentation to new code they add. This should run
* before alignment of params is done since this fixer might change
* the type and thereby un-aligning the params. We also must run before
* the phpdoc_scalar_fixer so that it can make changes after us.
*/
return 16;
}
/**
* {@inheritdoc}
*/
protected function normalize($type)
{
$lower = strtolower($type);
if (in_array($lower, self::$types, true)) {
return $lower;
}
return $type;
}
}
| code |
افسوٗس افسوٗس بٔہ کیٛاہ وَنَے ژےٚ مےٚ چھُ خوف زِ ژےٚ تہِ ژٕھنی سُہ کھؠتھ | kashmiri |
On Tanabata’s book blog, In Spring it is the Dawn, she challenges her readers every month to do something Japanese. Each mini-challenge has guidelines and January’s was “try something Japanese that you haven’t tried before”, which I did. And it was most certainly an experience.
For my birthday earlier this month I booked a karaoke booth at a local Japanese restaurant. I love Japanese food, I love karaoke, as do several of my friends – what could possibly go wrong? The only real question was why I had never done this before.
There were some setbacks. A few karaoke-friendly friends couldn’t make it so I ended up with a group heavy on the “I’ll come but I probably won’t sing” side. On arrival, as we squeezed ourselves into a tiny room that could only possibly have seated the advertised occupancy of 20 if they were all model-thin, was boiling hot and had the music volume so loud we couldn’t hear each other across the table, I began to worry this wouldn’t be all it was cracked up to be. The hostess didn’t explain the computer properly and we appeared to have a songlist composed solely of Madonna, Britney, Mariah and Japanese acts we’d never heard of.
Thankfully, while I knocked back my first flask of warm sake and caught up with my friends over the as-always immensely tasty food there, some of my more computer-savvy friends worked out not only how to adjust the volume to an acceptable level but also that there was a huge long list of songs to choose from hidden in a sub-sub-menu. And we were off!
And it was a brilliant night. Sure the computer crashed a few times, wiping our carefully crafted playlist. We suspected that the karaoke tracks and videos were largely cheap knock-offs, with hilariously wrong lyrics and videos either from some tourist agency or a sort-of Japanese Pop Idol show. But everyone had a good time, everyone sang (sometimes all at once with harmonies and everything) and I laughed so much I cried.
I loved that we had to take our shoes off and that we sat at a table at floor level, something I’d only seen in films before. I loved that the most resistant of my friends let inhibitions go and belted out tunes wholeheartedly. If they had let us we could have carried on all through the night and they would have made a fortune out of our sake and Asahi consumption, but sadly they closed at 10.30pm.
It was a great way to spend an evening with friends and I shall definitely accept any opportunity to try it again.
I first discovered de Lint years ago and quickly fell in love with his world where fantasy and real-life middle America meet in stories that are both scarily dark and almost frothily light. It’s an amazing creation that this collection of short stories opens up beautifully.
Short stories are a perfect fit for de Lint. All his tales are set in the “realistic” city of Newford and the magical realm that many of its inhabitants travel to – some at will, others involuntarily or only in their sleep. The short story format allows all these different experiences to be depicted without laborious explanation.
The characters are mostly people who are struggling with life in some way and need an escape, or did when they were children. Many are or have been homeless. There are also a lot of artists and writers, presumably because their creativity and social circles tend to make them open-minded and curious about the world.
Not everyone in these stories goes to the magical world. Some have strange experiences that as a reader you put down to magic. Others are fully immersed, even to the point of drifting through the “real” world while living in the magical one. This is roughly the progression of the book. We see more and more of the magical world in later stories. In early ones, it’s more hints and whispers.
The stories are narrated in the first person, or alternate between first and third person, and de Lint’s writing allows you to quickly get to know each character, so that when they pop up later in someone else’s story it feels familiar and friendly.
The dark element comes from the reasons people have for escaping to another place, or wanting magic on their side. From chronic shyness to psychiatric problems to child abuse and the many reasons why a person might be homeless, the possibilities of magic are anchored heavily down to earth. There’s a strong sense of living inbetween, of the magic being a metaphor for other coping mechanisms.
The stories stand alone if you want to read them that way, which is good as most of them had been previously published in magazines or anthologies. Each has a strong storyline, a journey for its main character, a start, middle and end. They gain further dimensions by being in a collection but they don’t depend on it.
I do love being able to revisit characters from books I read years ago. One of the main linking characters is Jilly Peppercorn, an artist and star of my favourite de Lint novel, The Onion Girl. I didn’t realise when reading that book that she had featured in several previous Newford books. In fact, de Lint said in an interview that she is the “warm beating heart of the city” (can’t find the link right now).
One criticism I have of this book is that it’s occasionally trite. For all of the dark pasts and presents, most characters end their story in a better place and they almost always learn a life lesson. I suppose when dealing with depressing subjects it helps to have a lighter side.
In a similar vein, all of the narrators are such…good people. I know a lot of people struggle to read about a main character who’s bad or unpredictable, and it’s a nice idea that most people are good at heart, but I think there’s room in de Lint’s universe for a few more, if not evil, at least selfish or mischievous characters.
But they’re minor quibbles. I loved these stories. I think my favourites were “Bird bones and wood ash”, about a woman who is imbued with supernatural abilities by animal spirits and uses them to fight evil, literally donning a black bodysuit, gloves and hood, but it drains her and joining forces with a social worker almost ruins everything; and “Mr Truepenny’s Book Emporium and Gallery”, about a dreaming place invented by a child that is falling to rack and ruin because she never visits anymore.
I have been reminded how much I enjoy this blend of myth and reality – de Lint calls on all sorts of mythologies, from Native American to the Brothers Grimm to Shakespeare – and I will definitely have to look out for the Newford titles that are missing from my collection.
This was another book club pick, in fact this one was my choice, so I was pretty nervous before the meeting. I’d chosen it based on Némirovsky’s brilliant final work Suite Française but this was a much earlier novel of her’s, with no guarantee of the same brilliance. What if everyone hated it? Or was bored by it? What if it failed to generate any discussion?
I needn’t have worried. While this is a slim volume and not as good as Suite Française, in my opinion, it did have plenty for us to talk about.
David Golder is a Russian Jew who works endlessly on obscure international financial deals to maintain the fabulously wealthy lifestyle to which his wife and daughter are accustomed. However, while he lives in a Paris apartment, they live in a multimillion franc estate in Biarritz, accompanied by an endless stream of hangers-on.
Golder isn’t the most likeable character, but we meet him near the end of his life and the impression is given that it was a difficult life and that he worked incredibly hard for himself and his family. His wife and daughter seem to only care about money, only showing Golder affection immediately before asking for a handout and getting very aggressive when he honestly tells them that business is rough and he can’t afford it right now. Add to that his failing health and you have a very sad, lonely picture of a man.
Némirovsky toys with the reader a little regarding characters’ true selves. At first Golder’s daughter seems much nicer than her mother because that’s what Golder sees. Only later is her selfishness fully exposed. And with Golder it’s the reverse – at first all you see is obsessiveness about money and his scheming seems horrible but it becomes clear, as we discover more about him and especially when we learn about his past, that he has his reasons, that his family and business associates encourage him, maybe even force him, to be this person.
I was glad to discover I wasn’t the only one at book club weirded out by the way the narrative labels everyone as a Jew, in an insulting sounding way, even though the author herself was Jewish and indeed died because of it. It could be part of the characterisation of Golder, that he has an odd skewed view of Jewishness. Or it could just be the vernacular of the time.
There was a general feeling that the book is very bleak, there is no ray of hope, no good person to contrast everything else against. But despite that Némirovsky has an easy, fluid writing style that keeps you reading even though there’s no-one to like and a fairly uneventful story.
I can’t recommend this as highly as I had hoped but I will still be interested to read Némirovsky’s other novels if they continue to be translated into English.
It turns out that it’s a good idea to check back-ups have actually worked before relying on them. Somehow the result of my last site back-up is a directory full of empty folders. Not so helpful. So I had to reinstall an old version of my doctored-to-suit-me theme (which thankfully Tim had held onto long after I thought I had any need for it). At some point I’ll go back in and tweak it to how I like it.
Also, I hadn’t realised that the WordPress export file contains only posts and pages. It doesn’t include anything else that you’ve personalised like images, links, site name and description, user name…stuff like that. No doubt I’ll continue finding things I need to update for weeks to come. I also wouldn’t be surprised if there’s very clear instructions on all this on WordPress.org somewhere.
Life is still hectic and I’m getting very little reading done but if you ask nicely I might just blog about the fun and pains of my new discovery: hulaerobics. Yes, it is what it sounds like.
I now have a lot of new books, except I only physically have half of them so the photo doesn’t look as impressive as it might do. Stupid rubbish postal service. Not that I read fast enough to get through these before the end of the month.
…and if anything I have less reading time than last year, so this should be an interesting exercise in time management. Please don’t judge me if it takes months for my reviews of these titles to appear! | english |
मेट्रो सिटी में पेट्रोल-डीजल के रेट - मुंबई में पेट्रोल ७५.७५ रुपये और डीजल ६६.९९ रुपये, कोलकाता में पेट्रोल ७२.३१ रुपये और डीजल ६५.८२ रुपये, चेन्नई में पेट्रोल ७२.७८ रुपये और डीजल ६७.६० रुपये और गुरुग्राम में पेट्रोल ७०.४६ रुपये और डीजल ६३.३७ रुपये प्रति लीटर है.
अंतरराष्ट्रीय बाजार में पिछले दिनों कच्चे तेल की कीमत में बढ़ोतरी के बाद मंगलवार को हल्की नरमी दिखाई दी. पिछले दिनों क्रूड ऑयल में तेजी आने से तेल की कीमत में दिल्ली, कोलकाता और मुंबई में पेट्रोल और के दाम में १२ पैसे जबकि चेन्नई में १३ पैसे प्रति लीटर का इजाफा हुआ था. मंगलवार को डब्ल्यूटीआई क्रूड के रेट ५७.७७ डॉलर प्रति बैरल और ब्रेंट क्रूड ६४.१० डॉलर प्रति बैरल के स्तर पर पहुंच गया
पिछली खबर सुहाने मौसम में पति जय संग मस्ती करती दिखीं प्रेग्नेंट माही विज
अगली खबर दरवेश हत्याकांड: क्बी जांच की मांग वाली याचिका पर स्क ने सुनवाई से किया इंकार | hindi |
require 'flickraw'
module Ruboty
module FreeImage
module Actions
class Flickr < Ruboty::Actions::Base
def call
FlickRaw.api_key = ENV['FLICKR_API_KEY']
FlickRaw.shared_secret = ENV['FLICKR_API_SECRET']
count = message[:count] || 10
# CC BY, CC BY-SA
license = "4,5"
photos = flickr.photos.search(license: license, per_page: count, text: message[:keyword]).map do |photo|
{
image_url: FlickRaw.url_n(photo),
title: photo.title,
url: FlickRaw.url_photopage(photo)
}
end
photos_message = photos.map {|photo|
"#{photo[:title]}: #{photo[:url]}#{$/}#{photo[:image_url]}"
}.join($/ * 2)
message.reply(photos_message)
end
end
end
end
end
| code |
#region License
/*
* Copyright (C) 1999-2017 John Källén.
*
* 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, 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; see the file COPYING. If not, write to
* the Free Software Foundation, 675 Mass Ave, Cambridge, MA 02139, USA.
*/
#endregion
using Reko.Core.Machine;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Collections;
using Reko.Core;
using Reko.Core.Expressions;
namespace Reko.Arch.RiscV
{
public class RiscVDisassembler : DisassemblerBase<RiscVInstruction>
{
private static OpRec[] opRecs;
private static OpRec[] wideOpRecs;
private static OpRec[] compressed0;
private static OpRec[] compressed1;
private static OpRec[] compressed2;
private RiscVArchitecture arch;
private EndianImageReader rdr;
private Address addrInstr;
public RiscVDisassembler(RiscVArchitecture arch, EndianImageReader rdr)
{
this.arch = arch;
this.rdr = rdr;
}
public override RiscVInstruction DisassembleInstruction()
{
this.addrInstr = rdr.Address;
ushort hInstr;
if (!rdr.TryReadLeUInt16(out hInstr))
{
return null;
}
return opRecs[hInstr & 0x3].Decode(this, hInstr);
}
private RiscVInstruction DecodeCompressedOperands(Opcode opcode, string fmt, uint wInstr)
{
var ops = new List<MachineOperand>();
for (int i = 0; i < fmt.Length; ++i)
{
MachineOperand op;
switch (fmt[i++])
{
default: throw new InvalidOperationException(string.Format("Unsupported operand code {0}", fmt[i - 1]));
case ',': continue;
case 'd': op = GetRegister(wInstr, 7); break;
case 'I': op = GetImmediate(wInstr, 12, fmt[i++]); break;
}
ops.Add(op);
}
return BuildInstruction(opcode, ops);
}
private RiscVInstruction BuildInstruction(Opcode opcode, List<MachineOperand> ops)
{
var instr = new RiscVInstruction
{
Address = this.addrInstr,
opcode = opcode,
Length = (int)(this.rdr.Address - addrInstr)
};
if (ops.Count > 0)
{
instr.op1 = ops[0];
if (ops.Count > 1)
{
instr.op2 = ops[1];
if (ops.Count > 2)
{
instr.op3 = ops[2];
}
}
}
return instr;
}
private RiscVInstruction DecodeWideOperands(Opcode opcode, string fmt, uint wInstr)
{
var ops = new List<MachineOperand>();
for (int i = 0; i < fmt.Length; ++i)
{
MachineOperand op;
switch (fmt[i++])
{
default: throw new InvalidOperationException(string.Format("Unsupported operand code {0}", fmt[i - 1]));
case ',': continue;
case '1': op = GetRegister(wInstr, 15); break;
case '2': op = GetRegister(wInstr, 20); break;
case 'd': op = GetRegister(wInstr, 7); break;
case 'i': op = GetImmediate(wInstr, 20, 's'); break;
case 'B': op = GetBranchTarget(wInstr); break;
case 'F': op = GetFpuRegister(wInstr, fmt[i++]); break;
case 'J': op = GetJumpTarget(wInstr); break;
case 'I': op = GetImmediate(wInstr, 12, fmt[i++]); break;
case 'S': op = GetSImmediate(wInstr); break;
case 'L': // signed offset used in loads
op = GetImmediate(wInstr, 20, 's');
break;
case 'z': op = GetShiftAmount(wInstr, 5); break;
case 'Z': op = GetShiftAmount(wInstr, 6); break;
}
ops.Add(op);
}
return BuildInstruction(opcode, ops);
}
private RegisterOperand GetRegister(uint wInstr, int bitPos)
{
var reg = arch.GetRegister((int)(wInstr >> bitPos) & 0x1F);
return new RegisterOperand(reg);
}
private RegisterOperand GetFpuRegister(uint wInstr, char bitPos)
{
int pos;
switch (bitPos)
{
case '1': pos = 15; break;
case '2': pos = 20; break;
case 'd': pos = 7; break;
default: throw new InvalidOperationException();
}
var reg = arch.GetRegister(32 + ((int)(wInstr >> pos) & 0x1F));
return new RegisterOperand(reg);
}
private ImmediateOperand GetImmediate(uint wInstr, int bitPos, char sign)
{
if (sign == 's')
{
int s = ((int)wInstr) >> bitPos;
return ImmediateOperand.Int32(s);
}
else
{
uint u = wInstr >> bitPos;
return ImmediateOperand.Word32(u);
}
}
private ImmediateOperand GetShiftAmount(uint wInstr, int length)
{
return ImmediateOperand.UInt32(extract32(wInstr, 20, length));
}
private static bool bit(uint wInstr, int bitNo)
{
return (wInstr & (1u << bitNo)) != 0;
}
private static uint extract32(uint wInstr, int start, int length)
{
uint n = (wInstr >> start) & (~0U >> (32 - length));
return n;
}
private static ulong sextract64(ulong value, int start, int length)
{
long n = ((long)(value << (64 - length - start))) >> (64 - length);
return (ulong)n;
}
private AddressOperand GetBranchTarget(uint wInstr)
{
long offset = (long)
((extract32(wInstr, 8, 4) << 1)
| (extract32(wInstr, 25, 6) << 5)
| (extract32(wInstr, 7, 1) << 11)
| (sextract64(wInstr, 31, 1) << 12));
return AddressOperand.Create(addrInstr + offset);
}
private AddressOperand GetJumpTarget(uint wInstr)
{
long offset = (long)
((extract32(wInstr, 21, 10) << 1)
| (extract32(wInstr, 20, 1) << 11)
| (extract32(wInstr, 12, 8) << 12)
| (sextract64(wInstr, 31, 1) << 20));
return AddressOperand.Create(addrInstr + offset);
}
private ImmediateOperand GetSImmediate(uint wInstr)
{
var offset = (int)
(extract32(wInstr, 7, 5)
| (extract32(wInstr, 25, 7) << 5));
return ImmediateOperand.Int32(offset);
}
public abstract class OpRec
{
public abstract RiscVInstruction Decode(RiscVDisassembler dasm, uint hInstr);
}
public class COpRec : OpRec
{
private Opcode opcode;
private string fmt;
public COpRec(Opcode opcode, string fmt)
{
this.opcode = opcode;
this.fmt = fmt;
}
public override RiscVInstruction Decode(RiscVDisassembler dasm, uint wInstr)
{
return dasm.DecodeCompressedOperands(opcode, fmt, wInstr);
}
}
public class WOpRec : OpRec
{
private Opcode opcode;
private string fmt;
public WOpRec(Opcode opcode, string fmt)
{
this.opcode = opcode;
this.fmt = fmt;
}
public override RiscVInstruction Decode(RiscVDisassembler dasm, uint wInstr)
{
return dasm.DecodeWideOperands(opcode, fmt, wInstr);
}
}
public class FpuOpRec : OpRec
{
private string fmt;
private Opcode opcode;
public FpuOpRec(Opcode opcode, string fmt)
{
this.opcode = opcode;
this.fmt = fmt;
}
public override RiscVInstruction Decode(RiscVDisassembler dasm, uint wInstr)
{
return dasm.DecodeWideOperands(opcode, fmt, wInstr);
}
}
public class MaskOpRec : OpRec
{
private readonly int mask;
private readonly int shift;
private readonly OpRec[] subcodes;
public MaskOpRec(int shift, int mask, OpRec[] subcodes)
{
this.mask = mask;
this.shift = shift;
this.subcodes = subcodes;
}
public override RiscVInstruction Decode(RiscVDisassembler dasm, uint wInstr)
{
var slot = (wInstr >> shift) & mask;
return subcodes[slot].Decode(dasm, wInstr);
}
}
public class SparseMaskOpRec : OpRec
{
private readonly int mask;
private readonly int shift;
private readonly Dictionary<int, OpRec> subcodes;
public SparseMaskOpRec(int shift, int mask, Dictionary<int, OpRec> subcodes)
{
this.mask = mask;
this.shift = shift;
this.subcodes = subcodes;
}
public override RiscVInstruction Decode(RiscVDisassembler dasm, uint wInstr)
{
var slot = (int)((wInstr >> shift) & mask);
OpRec oprec;
if (!subcodes.TryGetValue(slot, out oprec))
{
return new RiscVInstruction
{
Address = dasm.addrInstr,
opcode = Opcode.invalid
};
}
return oprec.Decode(dasm, wInstr);
}
}
public class WideOpRec : OpRec
{
public WideOpRec()
{
}
public override RiscVInstruction Decode(RiscVDisassembler dasm, uint hInstr)
{
ushort hiword;
if (!dasm.rdr.TryReadUInt16(out hiword))
{
return new RiscVInstruction { opcode = Opcode.invalid, Address = dasm.addrInstr };
}
uint wInstr = (uint)hiword << 16;
wInstr |= hInstr;
var slot = (wInstr >> 2) & 0x1F;
return wideOpRecs[slot].Decode(dasm, wInstr);
}
}
public class ShiftOpRec : OpRec
{
private Opcode[] rl_ra;
private string fmt;
public ShiftOpRec(string fmt, params Opcode[] rl_ra)
{
this.rl_ra = rl_ra;
this.fmt = fmt;
}
public override RiscVInstruction Decode(RiscVDisassembler dasm, uint wInstr)
{
var opcode = rl_ra[bit(wInstr, 30) ? 1 : 0];
return dasm.DecodeWideOperands(opcode, fmt, wInstr);
}
}
static RiscVDisassembler()
{
var loads = new OpRec[]
{
new WOpRec(Opcode.lb, "d,1,Ls"),
new WOpRec(Opcode.lh, "d,1,Ls"),
new WOpRec(Opcode.lw, "d,1,Ls"),
new WOpRec(Opcode.ld, "d,1,Ls"),
new WOpRec(Opcode.lbu, "d,1,Ls"),
new WOpRec(Opcode.lhu, "d,1,Ls"),
new WOpRec(Opcode.lwu, "d,1,Ls"),
new WOpRec(Opcode.invalid, ""),
};
var fploads = new OpRec[]
{
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.flw, "Fd,1,Ls"),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
};
var stores = new OpRec[]
{
new WOpRec(Opcode.sb, "2,1,Ss"),
new WOpRec(Opcode.sh, "2,1,Ss"),
new WOpRec(Opcode.sw, "2,1,Ss"),
new WOpRec(Opcode.sd, "2,1,Ss"),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
};
var op = new OpRec[]
{
new ShiftOpRec( "d,1,2", Opcode.add, Opcode.sub),
new WOpRec(Opcode.sll, "d,1,2"),
new WOpRec(Opcode.slt, "d,1,2"),
new WOpRec(Opcode.sltu, "d,1,2"),
new WOpRec(Opcode.xor, "d,1,2"),
new ShiftOpRec("d,1,2", Opcode.srl, Opcode.sra),
new WOpRec(Opcode.or, "d,1,2"),
new WOpRec(Opcode.and, "d,1,2"),
};
var opimm = new OpRec[]
{
new WOpRec(Opcode.addi, "d,1,i"),
new ShiftOpRec("d,1,z", Opcode.slli, Opcode.invalid),
new WOpRec(Opcode.slti, "d,1,i"),
new WOpRec(Opcode.sltiu, "d,1,i"),
new WOpRec(Opcode.xori, "d,1,i"),
new ShiftOpRec("d,1,z", Opcode.srli, Opcode.srai),
new WOpRec(Opcode.ori, "d,1,i"),
new WOpRec(Opcode.andi, "d,1,i"),
};
var opimm32 = new OpRec[]
{
new WOpRec(Opcode.addiw, "d,1,i"),
new ShiftOpRec("d,1,Z", Opcode.slliw, Opcode.invalid),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new ShiftOpRec("d,1,Z", Opcode.srliw, Opcode.sraiw),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
};
var op32 = new OpRec[]
{
new ShiftOpRec("d,1,2", Opcode.addw, Opcode.subw),
new ShiftOpRec("d,1,2", Opcode.sllw, Opcode.invalid),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new ShiftOpRec("d,1,2", Opcode.srlw, Opcode.sraw),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
};
var opfp = new Dictionary<int, OpRec>
{
{ 0x00, new FpuOpRec(Opcode.fadd_s, "Fd,F1,F2") },
{ 0x01, new FpuOpRec(Opcode.fadd_d, "Fd,F1,F2") },
{ 0x21, new FpuOpRec(Opcode.fcvt_d_s, "Fd,F1") },
{ 0x50, new SparseMaskOpRec(12, 7, new Dictionary<int, OpRec>
{
{ 2, new WOpRec(Opcode.feq_s, "d,F1,F2") }
})
},
{ 0x71, new FpuOpRec(Opcode.fmv_d_x, "Fd,1") },
{ 0x78, new FpuOpRec(Opcode.fmv_s_x, "Fd,1") },
};
var branches = new OpRec[]
{
new WOpRec(Opcode.beq, "1,2,B"),
new WOpRec(Opcode.bne, "1,2,B"),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.blt, "1,2,B"),
new WOpRec(Opcode.bge, "1,2,B"),
new WOpRec(Opcode.bltu, "1,2,B"),
new WOpRec(Opcode.bgeu, "1,2,B"),
};
wideOpRecs = new OpRec[]
{
// 00
new MaskOpRec(12, 7, loads),
new MaskOpRec(12, 7, fploads),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new MaskOpRec(12, 7, opimm),
new WOpRec(Opcode.auipc, "d,Iu"),
new MaskOpRec(12, 7, opimm32),
new WOpRec(Opcode.invalid, ""),
new MaskOpRec(12, 7, stores),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new MaskOpRec(12, 7, op),
new WOpRec(Opcode.lui, "d,Iu"),
new MaskOpRec(12, 7, op32),
new WOpRec(Opcode.invalid, ""),
// 10
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new SparseMaskOpRec(25, 0x7F, opfp),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new MaskOpRec(12, 7, branches),
new WOpRec(Opcode.jalr, "d,1,i"),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.jal, "d,J"),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
new WOpRec(Opcode.invalid, ""),
};
compressed0 = new OpRec[]
{
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
};
compressed1 = new OpRec[]
{
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
};
compressed2 = new OpRec[]
{
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
new COpRec(Opcode.invalid, ""),
};
opRecs = new OpRec[]
{
new MaskOpRec(0x13, 7, compressed0),
new MaskOpRec(0x13, 7, compressed1),
new MaskOpRec(0x13, 7, compressed2),
new WideOpRec()
};
}
}
}
| code |
/*
* Tanaguru - Automated webpage assessment
* Copyright (C) 2008-2015 Tanaguru.org
*
* This program is free software: you can redistribute it and/or modify
* it under the terms of the GNU Affero General Public License as
* published by the Free Software Foundation, either version 3 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 Affero General Public License for more details.
*
* You should have received a copy of the GNU Affero General Public License
* along with this program. If not, see <http://www.gnu.org/licenses/>.
*
* Contact us by mail: tanaguru AT tanaguru DOT org
*/
package org.tanaguru.rules.accessiweb21;
import org.tanaguru.entity.audit.TestSolution;
import org.tanaguru.rules.accessiweb21.detection.AbstractTagDetectionPageRuleImplementation;
/**
* Implementation of the rule 11.8.3 of the referential Accessiweb 2.1.
* <br/>
* For more details about the implementation, refer to <a href="http://www.tanaguru.org/en/content/aw21-rule-11-8-3">the rule 11.8.3 design page.</a>
* @see <a href="http://www.braillenet.org/accessibilite/referentiel-aw21-en/index.php#test-11-8-3"> 11.8.3 rule specification</a>
*
* @author jkowalczyk
*/
public class Aw21Rule11083 extends AbstractTagDetectionPageRuleImplementation {
public static final String MESSAGE_CODE = "ManualCheckOnElements";
private static final String TAG_DETECTION_XPATH_EXPR ="//OPTGROUP[ancestor::SELECT and @label]";
/**
* Default constructor
*/
public Aw21Rule11083 () {
super();
setMessageCode(MESSAGE_CODE);
setSelectionExpression(TAG_DETECTION_XPATH_EXPR);
setDetectedSolution(TestSolution.NEED_MORE_INFO);
setNotDetectedSolution(TestSolution.NOT_APPLICABLE);
setIsRemarkCreatedOnDetection(true);
}
} | code |
/* dojox.mobile.RoundRectCategory */
.mblRoundRectCategory {
margin: 18px 0 0 20px;
padding: 0;
font-family: Helvetica;
font-size: 16px;
white-space: nowrap;
text-overflow: ellipsis;
overflow: hidden;
color: black;
}
| code |
import tensorflow as tf
from tensorflow.contrib import layers
from tensormate.graph import shape_info, graph_info, name_scope
from pprint import pprint
tf.logging.set_verbosity(tf.logging.INFO)
class ShapeInfoTest(tf.test.TestCase):
cached = True
@shape_info(cached=cached)
def fwd_graph(self, inputs, num_outputs):
return layers.conv2d(inputs, num_outputs=num_outputs, kernel_size=[3, 3])
def test(self):
input_shape = [10, 24, 24, 3]
inputs = tf.Variable(tf.zeros(shape=input_shape))
outputs = self.fwd_graph(inputs, num_outputs=16)
if ShapeInfoTest.cached:
pprint(self.fwd_graph.result)
input_name = "Variable:0"
output_name = "Conv/Relu:0"
self.assertAllEqual(self.fwd_graph.result[0][-1], input_shape)
output_expected_shape = input_shape.copy()
output_expected_shape[-1] = 16
self.assertAllEqual(self.fwd_graph.result[1][-1], output_expected_shape)
class NameScopeTest(tf.test.TestCase):
# @graph_info(cached=True)
@shape_info(cached=True)
@name_scope("my_scope")
def fwd_graph(self, inputs, num_outputs):
return layers.conv2d(inputs, num_outputs=num_outputs, kernel_size=[3, 3])
def test(self):
input_shape = [10, 24, 24, 3]
inputs = tf.Variable(tf.zeros(shape=input_shape))
outputs = self.fwd_graph(inputs, num_outputs=16)
print(self.fwd_graph.result)
class GraphInfoTest(tf.test.TestCase):
cached = True
@graph_info(cached=cached)
def fwd_graph(self, inputs, num_outputs, scope="test", reuse=False):
return layers.conv2d(inputs, num_outputs=num_outputs, kernel_size=[3, 3], scope=scope, reuse=reuse)
def test(self):
input_shape = [10, 24, 24, 3]
inputs = tf.Variable(tf.zeros(shape=input_shape))
inputs = tf.identity(inputs)
outputs = self.fwd_graph(inputs, num_outputs=16)
if GraphInfoTest.cached:
self.check_result("/home/guocong/git/github/tensormate/test1.html")
outputs_1 = self.fwd_graph(inputs, num_outputs=16, reuse=True)
if GraphInfoTest.cached:
self.check_result("/home/guocong/git/github/tensormate/test2.html")
self.assertEqual(self.fwd_graph.count, 2)
self.assertEqual(self.fwd_graph.__name__, self.fwd_graph.__wrapped__.__name__)
def check_result(self, output_file):
pprint(self.fwd_graph.result)
html = self.fwd_graph.viz_html_string
output_file = output_file
with open(output_file, "tw") as f:
f.write(html)
if __name__ == '__main__':
tf.test.main()
| code |
इस पुस्तक का नाम : प्राचीन भारतीय शासन-पद्धति है | इस पुस्तक के लेखक हैं : अनंत सदाशिव एल्टेकर | अनंत सदाशिव एल्टेकर की अन्य पुस्तकें पढने के लिए क्लिक करें : अनंत सदाशिव एल्टेकर | इस पुस्तक का कुल साइज ११६.९ म्ब है | पुस्तक में कुल 2९0 पृष्ठ हैं |नीचे प्राचीन भारतीय शासन-पद्धति का डाउनलोड लिंक दिया गया है जहाँ से आप इस पुस्तक को मुफ्त डाउनलोड कर सकते हैं | प्राचीन भारतीय शासन-पद्धति पुस्तक की श्रेणियां हैं : हिस्ट्री
मेरे ग्रन्थ अभीतक प्राय: पहले अंग्रेजी में प्रकाशित हए थे। पीछे उनका संस्करण मैंने अपनी मातृभाषा मराठी में प्रकाशित किया। मगर 'प्राचीन भारतीय शासन-पद्धति सर्वप्रथम हिंदी में ही प्रकाशित हो रही है। अनेक कठिनाइयों के कारण इसका अंग्रेजी संस्करण अभी तक प्रकाशित नहीं हो सका। मराठी संस्करण तैयार हो रहा है | hindi |
namespace Net.Sf.Dbdeploy.Database.SqlCmd
{
using System;
using System.Collections.Generic;
using System.Linq;
/// <summary>
/// Class that parses out connection string values.
/// </summary>
public static class ConnectionStringParser
{
/// <summary>
/// Parses the specified connection string into it's components.
/// </summary>
/// <param name="connectionString">The connection string.</param>
/// <returns>Connection string components.</returns>
public static ConnectionStringInfo Parse(string connectionString)
{
var info = new ConnectionStringInfo();
var entries = connectionString.Split(';').Where(s => !string.IsNullOrWhiteSpace(s));
foreach (var entry in entries)
{
var pair = entry.Split('=');
if (pair.Length == 2)
{
var name = pair[0].ToLowerInvariant().Trim();
var value = pair[1].Trim();
switch (name)
{
case "server":
case "data source":
case "datasource":
info.Server = value;
break;
case "database":
case "initial catalog":
info.Database = value;
break;
case "uid":
case "user":
case "user id":
info.UserId = value;
break;
case "pwd":
case "password":
info.Password = value;
break;
case "trusted_connection":
info.TrustedConnection = value.ToLowerInvariant() == "true";
break;
}
}
else
{
throw new FormatException("The connectionString does not have a correct matching of key value pairs.");
}
}
return info;
}
}
}
| code |
शिवपाल- घमंड की हार, उत्तराखंड में हरीश रावत हारे, यूपी मे ३०० सीट पर बीजेपी
पांच राज्यों के चुनाव परिणाम के रुझान आने शुरू हो गए हैं। शिवपाल यादव ने समाजवादी पार्टी काे विधानसभा चुनाव में पीछे हाेने पर अखिलेश यादव पर बड़ा हमला किया है। उन्हाेंने कहा कि यह हार समाजवादियाें का नहीं बल्कि घमंड का है। उन्हाेंने कहा कि नेता जी काे हटाकर उनका अाैर मेरा अपमान किया। यह उसी घमंड पर जनता ने जवाब दिया है।
इसके पहले सुबह उन्हाेंने कहा था कि हमारी जीत बड़ी होनी है। जनता इंतजार कर रही है। सरकार हमारी बन रही है। इटावा जसवंत नगर सीट से सपा प्रत्याशी वरिष्ठ सपा नेता शिवपाल सिंह यादव चुनाव के दौरान सपा में कलह की वजह से सबसे ज्यादा चर्चा में रहे।
गोवा से पहला परिणाम आ गया है। राज्य के पूर्व मुख्यमंत्री और कांग्रेस नेता प्रताप सिंह राणे पोरिम सीट से जीते
हरिद्वार ग्रामीण से उत्तराखंड के मुख्यमंत्री हरीश रावत हारे
मथुरा से बीजेपी के श्रीकांत शर्मा जीते
बागपत में नवें राउंड के बाद बीजेपी आगे
बाराबंकी में सपा के तीनों मंत्री बीजेपी प्रत्याशियों से पीछे
इटावा शिवपाल सिंह यादव ८वें राउंड में 1502८ मतों से आगे
हल्द्वानी ६ठे राउंड के बाद बीजेपी के जोगेन्द्र पाल 222६ वोट से आगे
लखनऊ पश्चिमी से बीजेपी के सुरेश श्रीवास्तव आगे
पटियाली के चौथे राउंड में सपा आगे
बहराइच ८वें राउंड में बीजेपी प्रत्याशी मुकुट बिहारी आगे
केदारनाथ से ८वें राउंड में कांग्रेस मनोज रावत आगे
गाजीपुर ९वें राउंड में बीजेपी से अलका राय आगे
मटेरा से नवें राउंड में बीजेपी प्रत्याशी याशर शाह आगे
हापुड़ ११वें राउंड में गढ़ से बीजेपी प्रत्याशी कमल मलिक आगे
बाराबंकी बीजेपी प्रत्याशी शरद अवस्थी से अरविंद सिंह गोप ११ हजार मतों से पीछे
कन्नौज सदर से छिबरामऊ से बीजेपी की अर्चना पाण्डेय आगे
बहराइच के नानपारा से आठवें चरण में बीजेपी की मधुरी वर्मा आगे
बहराइच के मटेरा से ११वें चरण सपा प्रत्याशी आगे
उत्तराखंड कालाढूंगी से बीजेपी प्रत्याशी बंसीधर भगत १८४३ वोटों से आगे
शाहजहांपुर तिलहर विधानसभा से जितिन प्रसाद १३वें राउंड में आगे
सरोजनीनगर में आठवें चरण में बीजेपी आगे
हल्द्वानी लालकुआं से ५वें राउंड में बीजेपी के नवीन दुम्का 97५6 वोट से आगे
लालकुआं सीट से ४ राउंड में बीजेपी के नवीन दुम्का आगे
यूपी से आया पहला नतीजा, मुस्लिम बहुल सीट देवबंद से जीती बीजेपी
सुल्तानपुर से बीजेपी के सूर्यभान सिंह आगे
पंजाब के रुझानः कांग्रेस ७०, आप २३, अकाली+बीजेपी २१, अन्य ३
उत्तराखंडः केदारनाथ सीट से ६ठे राउंड में कांग्रेस प्रत्याशी आगे
लखनऊः सरोजनीनगर सीट से ब्जप की स्वाति सिंह आगे, सपा प्रत्याशी अनुराग यादव से स्वाति सिंह आगे
बीजेपी की जीत पर प्रतिक्रिया देते हुए राजबब्बर ने कहा इस हार में गठबंधन की कोई खामी नहीं रही, बीजेपी ने चुनाव में धनबल का दुरुपयोग किया।
ईटीवी के साथ बातचीत में कांग्रेस प्रदेश अध्यक्ष राजबब्बर ने ईवीएम की विश्वसनीयता पर उठाए सवाल।
गोवा में सांत आद्रे सीट पर कांग्रेस के फ्रांसिस्को सिल्वेरा जीते। उन्होंने बीजेपी के रामराव सुर्या नाईक वाघ को हराया।
गोवा से आप के सीएम कैंडिडेट एल्विस गोम्स अपनी सीट पर तीसरे नंबर पर
बीजेपी उम्मीदवार स्वाति सिंह २० हजार मतों से आगे
यूपी के बख्शी का तालाब से बीजेपी के अविनाश त्रिवेदी आगे
मणिपुरः कांग्रेस-११, बीजेपी-९, एनसीपी- १ और अन्य ७
रायबरेली सदर से कांग्रेस की अदिति सिंह आगे। ऊंचाहार से सपा के मनोज पांडेय आगे। सलोन-कांग्रेस से सुरेश निर्मल आगे
प्रेवियस आर्टियलमणिपुर में मुख्यमंत्री ओकराम इबोबी सिंह ने इरोम शर्मिला को हराया
नेक्स्ट आर्टियलयूपी-उत्तराखंड में मोदी का रंग, पंजाब-गोवा-मणिपुर में कांग्रेस आगे | hindi |
२० लोगों पर पिस्तौल के बल पर लूट एवं मारपीट का आरोप, केस दर्ज
कैथल(सुखविंद्र): गांव नरड़ स्थित खेतों में घुसकर ३ लोगों से पिस्तौल के बल पर लूटपाट एवं मारपीट करने के आरोप में तितरम थाना पुलिस ने २० नामजद सहित २० अन्य अज्ञात लोगों पर विभिन्न धाराओं के तहत मामला दर्ज किया है। पुलिस को दी शिकायत में नरड़ निवासी गुरदेव सिंह ने बताया कि मैं कली राम व अजमेर सिंह के साथ खेत में बने कोठे में सो रहे थे। ७ फरवरी को सुबह ५ बजे हवा सिंह, गुरदेव, लाभ, गुड्डी, कुसुम, हवा सिंह की पत्नी, रामेश्वर की पत्नी, संदीप, गुरतेज, ईश्वर, रामफल, सामण, गुरदेव की पत्नी, संदीप की पत्नी, बलकार, गुलाब, अंग्रेज, ईश्वर की पत्नी व लगभग २० अन्य व्यक्ति हथियारों से लैस होकर आए। आरोपियों ने खेत में घुसकर मारपीट की व बंदूक के बल पर सैमसंग मोबाइल, कली राम के हाथ में पहनी हुई अंगूठी छीन ली।
आरोपी अन्य सामान भी लूटकर ले गए। आरोपियों ने खेत में मौजूद कली राम, अजमेर व गुरदेव को बंदूक दिखाकर खेतों से गमी के रास्ते भगा दिया और कहा कि इस बारे में किसी को कुछ बताया तो गोली मार देंगे। शिकायकत्र्ता एवं आरोपियों का बिजली कनैक्शन को लेकर पुराना विवाद है। ए.एस.आई. सतबीर सिंह ने बताया कि पुलिस ने आरोपियों पर मामला दर्ज कर जांच शुरू कर दी है।
१२ को प्रयागराज से कैथल पहुंचेंगे जूना अखाड़ा के साधु | hindi |
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| code |
/**
* Copyright © 2018 spring-data-dynamodb (https://github.com/boostchicken/spring-data-dynamodb)
*
* Licensed 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.socialsignin.spring.data.dynamodb.repository.support;
import org.springframework.data.repository.core.EntityInformation;
import java.util.Optional;
/**
* Encapsulates minimal information needed to load DynamoDB entities.
*
* As a minimum, provides access to hash-key related metadata.
*
* Implementing classes can elect to be either range-key aware or not. If a
* subclass is not range-key aware it should return null from getRangeKey(ID id)
* method, and return false from isRangeKeyAware and
* isCompositeHashAndRangeKeyProperty methods
*
* @author Michael Lavelle
* @author Sebastian Just
*/
public interface DynamoDBEntityInformation<T, ID>
extends
EntityInformation<T, ID>,
DynamoDBHashKeyExtractingEntityMetadata<T> {
default boolean isRangeKeyAware() {
return false;
}
boolean isCompositeHashAndRangeKeyProperty(String propertyName);
Object getHashKey(ID id);
default Object getRangeKey(ID id) {
return null;
}
Optional<String> getProjection();
Optional<Integer> getLimit();
}
| code |
package quickml.supervised.classifier.downsampling;
/**
* Created by ian on 4/23/14.
*/
public class DownsamplingUtils {
public static double correctProbability(final double dropProbability, final double uncorrectedProbability) {
return (1.0 - dropProbability)*uncorrectedProbability / (1.0 - dropProbability * uncorrectedProbability);
}
}
| code |
Green tea is a tea made from the leaves of the Camellia sinensis that has had little or no oxidation while being processed. Green tea originated in China four thousand years ago and was brought to Japan sometime in the 6th Century. Green Tea is hand or machine picked, dried to remove moisture and then shaped by rolling or twisting. Because Green tea is processed less than other varieties of tea, it has a higher EGCG level; EGCG is a catechin, which helps protect the body from harmful free radicals. Recently many of the long- standing and acclaimed medicinal benefits of tea have been scientifically supported by the medical and academic research, constantly making Green tea a more popular and trendy beverage. Green tea, much like coffee, has a distinctive taste based on the province or region it is grown in. Therefore if you think you do not like green tea try some different varieties, especially Teaopia green teas that often list the origin of the tea. I rate my teas on a scale from 0 to 10; 0 I will never let pass my lips again. 10 I like to always keep on hand, 7-9 I like to have around and enjoy often.
Some books just immediately impress you, either by their concept, the writing, or the overall skill in storytelling. This book does all three. Alexander Gordon Smith has started a series that looks to be amazing if this first book is any indication. It has been about 10 years since I got this excited about a series by the first book, and that time it was The Traveler by John Twelve Hawks.
The back story. At some point in the near future youth crime reaches epic heights. The populace responds strongly after the summer of slaughter, a period where youth murder rates soar. They create a new super max prison for young offenders, carved below the earth: Furnace. They say that below heaven is hell and below hell is furnace. There is now a zero tolerance policy on youth felons.
The story written in the first person narrative style, is the story of Alex Sawyer, a young criminal mostly responsible for break and enters and petty theft. Alex is framed for the murder of his criminal partner. Even though he is innocent, he is convicted and sent to Furnace. He arrives in Furnace, a terrifying dark hole carved from the earth, a place ruled by vicious gangs, and even more brutal guards. A place filled with horrifying creatures who come and steal boys from their cells in the night. A place with no hope and no future. Yet Alex struggles to maintain hope - that hope is escape, something believed to be impossible.
Alexander Gordon Smith does an amazing job of capturing the brutality of a prison environment, without going into too much gore. He tells a dark tale without becoming overly depressing. It is very well written.
This herbal tea is from the Rooibos (scientific name Aspalathus linearis) plant, a broom-like member of the legume family. This tea can be consumed in either a more native green state or a fermented red state. The tea is reddish brown if fermented, and greenish yellow if not. The green varieties tend to have a stronger and somewhat malty or nutty flavour compared to the red varieties. Rooibos is a naturally caffeine-free herbal, low in tannins and high in antioxidants. Rooibos is fairly new to North America, and becoming constantly more popular. It has been consumed in South Africa for generations. Research on Rooibos has been shown to aid in health problems such as insomnia, irritability and hypertension. Also since Rooibos contains high levels of antioxidants it also helps boost the immune system, destroy free radicals and slow the aging process. Rooibos is the good choice for introducing children to tea and for those with caffeine sensitivities. Distinctly Tea, a company based out of Stratford Ontario, lists on their website which teas are acceptable for children and what should be avoided. Teaopia in their tea guide (click on cover to right) give some indications but not nearly as explicit. Rooibos can be consumed straight, or is often combined with fruit or other botanicals to create flavoured varieties. I rate my teas on a scale from 0 to 10; 0 I will never try again. 10, I like to always keep on hand.
I have always been a tea drinker, but until recently I found coffee easier and more convenient. Brew up a pot and drink it all was my motto. I have always kept 15 -30 types of tea at home, a mix of bagged tea and loose leaf teas. Yet over the last 2 years I have dropped my coffee intake to 1 cup a day and upped my tea intake to upwards of 6 to 8 cups a day. A big part of the change is thanks to Teaopia, and their Tea Master. It is convenient and easy to use and reuse. Their tea is a loose enough cut to be re-steeped between 2 and 4 times depending on the tea. But back to the topic at hand, tea. If you do a Google search on the 'benefits of tea' you get 16,400,000 results.
The earliest recorded consumption of tea is from around 10,000 BC. The first book on how to buy and prepare tea was by Wang Bo and is dated at 59BC. Today tea comes in hundreds of varieties and flavors, with something to suit almost any pallet. Next to water, tea is the most widely consumed beverage around the word. And much of what is consumed in North American is not truly tea from its first definition. It is tea based on the second definition. According to Encyclopedia Britannica tea is: a beverage produced by steeping in freshly boiled water the young leaves and leaf buds of the tea plant, Camellia sinensis. At one point in history Britain ruled the world because it ruled the seas. One of the reasons it ruled the seas was to control the traffic of commodities from continent to continent. One way this was done was that the East India Company (aka East India Trading Company, English East India Company, and then the British East India Company) tried to monopolize the tea trade on a global scale. This led to the Tea Act in 1773 which in turn led to the Boston tea Party. But today tea is commonly considered almost any plant or botanicals steeped in hot water. Yet you might ask why would you drink tea?
Lose Weight, Stay Alert and Protect Your Heart: According to Health Canada's Natural Health Products Directorate (NHPD), tea helps support and maintain weight loss, increases alertness and helps protect against cardiovascular disease.
Fight Diseases: Tea is high in antioxidants, polyphenols, flavonoids and catechins including EGCG (Epigallocatechin gallate), which help protect the body from harmful free radicals and cellular damage.
Drink to your Health: After water, tea is the healthiest beverage you can consume according to a panel of nutrition expects published in the Journal of American Clinical Nutrition.
Boost your Memory: New scientific research suggests that drinking tea may lower the risk of developing neurodegenerative disorders such as dementia, Alzheimer's and Parkinson's disease.
Reduce Stress: The amino acid "L-theanine" found almost exclusively in tea is known to reduce stress and calm the mind, while keeping you more alert.
Age Gracefully: Beauty starts from the inside out. The antioxidants found in tea help slow down the aging process, making you look and feel great.
Lowers Cholesterol: Tea has been shown to reduce "bad" cholesterol (LDL).
Ease Arthritis: Tea has been shown to prevent and reduce the severity of rheumatoid arthritis.
Oral Health: Tea helps eliminate bad breath and is rich in fluoride, which strengthens teeth and protects tooth enamel.
Stay Hydrated: Drinking tea is a delicious way to help meet your recommended daily intake. Drinking tea may even be better than drinking water since it replenishes fluids and provides antioxidants.
Feel free to post comments and links to your favorite tea resources.
I have been planning on writing a series of posts about tea for a while now. I have done my research started writing but keep getting busy with other stuff. But Teaopia current has a cross promotion with Tim Burton's new Alice In Wonderland film. So I thought I would start the ball rolling. At work there are a number of us who love loose-leaf tea. I have a shelf full of it behind my desk. I also stock even more kinds at home. At work we almost have a tea party every day, and whenever one of us finds a particularly good new tea we share it around the office. So without further ado here are some photos of my teas at both work and home.
In each piece I will outline the benefits of that type of tea and highlight some of my favourites. So keep an eye out for a week of tea coming your way soon.
I have heard a lot of negative feedback on this workout. And after doing it I was a little surprised. I found it a very good workout and really challenging. I have a weak ankle from an injury about a decade back, and have small feet to begin with. Doing this workout pushed my balance and core stability to the limit. I found a good weight range about 1/3 to ½ my normal weight and just tried to do my best, as Tony always says.
You do three exercises on one leg then repeat on the other side, then three more and then the final two. Next is the bonus round.
This is a good workout that has some great variety and works very different muscles. I don't see myself doing this every week, but will put it in every 4th week during my recovery week at the end of each phase. It is worth giving it a try even if just to change things up from time to time.
To find out more about this series or other workouts in the collection follow the links below. I have created a random workout generator that uses 1 on 1's and P90X if your interested.
Volume 2, Disk 3: Patience "Hummingbird"
I am currently reading a really good book, the Art of Manliness by Brent and Kate McKay. I am enjoying it immensely. One of the first things I flipped to was a list of 100 Manly Books. I love books and I love lists so I decided to srat my comments about this book with it's book list. (The complete list of what I have read since October of 1995 is here.) Hat tips to James Hahn and Nod who both have mentioned the book and got me interested in it.
I have read 51 out of the 100, some I had to for school. Some of the authors I have read their complete works. A few of the books I have even read in numerous english translations. Post a comment and let em know how your did. Keep an eye out for more comments and thoughts as I read this book, I plan on doing a few posts about it as I process some of it's wisdom.
Out of the 16 books available in the Zipzer series, this is now my favourite for a few reasons. First, Hank really begins to accept his learning disability; he also chooses to seek extra help for it. Now true, there is a female interest that draws him to that option but when a young person chooses reading gym over Tae Kwon Do, you know something serious is up. Hank has his first crush and Zoe is in the reading gym also. Hank also tries to keep this change in after-school activities a secret from his dad. But best of all Hank learns how to describe his life with his learning challenges and how it feels to live and struggle with them day in and day out. In learning to admit he has a problem and really working on it, Hank learns how to communicate his struggles and feelings that have been pent up for years.
What is remarkable about this book, and the series as a whole, is the ability by the authors to capture the essence of struggling with a learning disability. I was diagnosed in grade 2 because of a very persistent teacher, but many people are now being diagnosed in college, university and even in the workforce. I know many people who were not diagnosed in school and feel the way Hank does, but they did not know why. These books, even though written for children, are excellent for anyone struggling with learning disabilities, no matter their age. For they will help them see it is not just them and in Hank will see a mirror of some of their own experiences. These books are wonderfully written and are also fun to read. I highly recommend them.
1. Niagara Falls, or Does It?
7. Help! Somebody Get me out of Fourth Grade!
8. Summer School? What Genius thought That Up?
13. Who Ordered This Baby? Definitely Not Me!
16. Dump Trucks and Dogsleds: I'm On My Way, Mom!
2. Mind If I Read Your Mind?
Back to your regularly scheduled programming!
Back to your regularly scheduled programming. Well at least sort of. I have migrated to Blogger custom domain, from FTP Publishing. I have created a new hidden blog, hacked the template, imported my old posts and gone through over 300 posts and put the pictures back in. Once I have finished the last 370 posts I will take down the temp back up. New posts should be appearing next week and then back to posting every 1 to 3 days.
"The only reason for being a professional writer is that you can't help it."
I have made a back up of this blog at srmcevoy.ca. That is the first in a few steps. Second I have created a hidden blogger blog with new templates and attempted to recreate the look and feel of this blog. I am about 99% satisfied with it (about 12 hours of design work). Next I will need to apply that template to this blog and redirect it to blogger hosting and test. If that goes well, there should be minimal disruption to this blog. What will take the longest is going through almost 700 posts and putting the book covers and other pictures back in. Thanks for your patience during the transition. New content is developed and should be reappearing soon. | english |
पीएम मोदी डरे हुए हैं, वह सो नहीं सकते : राहुल | उपुक्लाइव
पीएम मोदी डरे हुए हैं, वह सो नहीं सकते : राहुल
नयी दिल्ली। आलोक वर्मा को सीबीआई निदेशक पद से हटाए जाने के बाद कांग्रेस अध्यक्ष राहुल गांधी ने गुरुवार को प्रधानमंत्री नरेंद्र मोदी पर निशाना साधते हुए आरोप लगाया कि राफेल मामले का डर प्रधानमंत्री के दिमाग में घूम रहा है जिसके चलते वह सो भी नहीं सकते। उन्होंने ट्वीट करके कहा, मोदी जी, के दिमाग में डर घूम रहा है। वह सो नहीं सकते।
उन्होंने वायुसेना से ३० हजार करोड़ रुपये की चोरी की और अनिल अंबानी को दिए। उन्होंने कहा, सीबीआई प्रमुख आलोक वर्मा को दो बार हटाना स्पष्ट रूप से यह दिखाता है कि प्रधानमंत्री अब अपने ही झूठ से घिर चुके हैं। सत्यमेव जयते।
गौरतलब है कि राफेल मामले को लेकर गांधी प्रधानमंत्री पर निशाना साधने के साथ अनिल अंबानी पर भी आरोप लगाते हैं। अंबानी समूह उनके आरोपों को पहले ही खारिज कर चुका है। उधर, सीबीआई निदेशक को हटाए जाने के बाद कांग्रेस ने आरोप लगाया कि वर्मा को हटाने के इस कदम से फिर साबित हो गया है कि मोदी राफेल मामले की जांच से डरे हुए हैं।
पार्टी ने आधिकारिक ट्वीट में कहा, ''अपना पक्ष रखने का मौका दिए बिना आलोक वर्मा को हटाकर प्रधानमंत्री मोदी ने एक बार फिर दिखाया है कि वह राफेल मामले की किसी भी तरह की जांच से डरे हुए हैं, चाहे वह सीबीआई निदेशक द्वारा जांच हो या जेपीसी की जांच हो।
उपुक्लाइव: पीएम मोदी डरे हुए हैं, वह सो नहीं सकते : राहुल | hindi |
विडियो: ४७ साल के भारतीय गेंदबाज ने त-१० लीग में लगाई हैट्रिक, साथ में बनाया एक आैर रिकाॅर्ड
नई दिल्लीः टी१० लीग के दूसरे सीजन में भारत के ४७ साल के प्रवीण तांबे का कहर देखने को मिला। तांबे ने सिंधी टीम की ओर से खेलते हुए केरल नाइट्स के खिलाफ ५ विकेट झटके और वह इस लीग में ५ विकेट लेने वाले पहले गेंदबाज बन गए।
हैट्रिक भी लगाई
तांबे ने मैच के पहले ओवर में हैट्रिक लगाकर भी कोहराम मचा दिया। उन्होंने ओवर की दूसरी गेंद पर क्रिस गेल को आउट किया। इसके बाद चाैथी गेंद पर कप्तान इयोन मोर्गन, ५वीं गेंद पर किरोन पोलार्ड और आखिरी गेंद पर फेबियन एलीन का विकेट लेकर हैट्रिक पूरी कर ली।
इसके बाद मैच के तीसरे और अपने दूसरे ओवर में तांबे ने श्रीलंका के उपल थरंगा का विकेट लेकर अपना पांचवां विकेट भी ले लिया, जिससे केरल नाइट्स का स्कोर ३.१ ओवर के बाद केवल २१ रन पर छह विकेट हो गया था। इस तरह तांबे ने अपने २ ओवरों में कुल १५ रन देते हुए ५ विकेट झटक लिए।
टीम को मिली ९ विकेट से जीत
केरल के टाॅप ६ बल्लेबाज मिलकर कुल ५ रन ही बना सके, जिसमें उपल थरंगा के ४ आैर पॉल स्टर्लिंग का १ रन शामिल है। बाकी ४ बल्लेबाज ० पर आउट हुए, लेकिन बावजूद इसके केरल वेन पार्नेल (५९) और सोहेल तनवीर (२३) की बदाैलत सिंधी के सामने १०४ रनों का लक्ष्य रखने में कामयाब हो गया। जवाब में सिंधी ने ओपनर शेन वाॅटसन के नाबाद ५० आैर एंटोन डेवचिक के ४९ रनों की बदाैलत ९ विकेट से मैच जीत लिया। सामीुल्ला शेनवारी ३ रन पर नाबाद रहे।
विडियो: जडेजा के रन आउट होते ही धोनी को आया गुस्सा, बीच मैदान पर दिखाई नाराजगी
विडियो: धोनी ने मारा इप्ल २०१९ का सबसे लंबा छक्का, स्टेडियम के बाहर जा गिरी बॉल
इप्ल २०१९: मैच जीतने के बाद कप्तान विलियमसन ने इस खिलाड़ी को दिया जीत का श्रेय | hindi |
jsonp({"cep":"39404284","logradouro":"Rua Lagoa Caraj\u00e1s","bairro":"Interlagos","cidade":"Montes Claros","uf":"MG","estado":"Minas Gerais"});
| code |
होम > उत्पादों > डिजिटल सजावटी तस्वीर प्रिंटर
(डिजिटल सजावटी तस्वीर प्रिंटर के लिए कुल २४ उत्पादों)
थोक चीन से डिजिटल सजावटी तस्वीर प्रिंटर , लेकिन कम कीमत के अग्रणी निर्माताओं के रूप में सस्ते डिजिटल सजावटी तस्वीर प्रिंटर खोजने की आवश्यकता है। बस डिजिटल सजावटी तस्वीर प्रिंटर पर उच्च गुणवत्ता वाले ब्रांडों पा कारखाना उत्पादन, आप आप क्या चाहते हैं, बचत शुरू करते हैं और हमारे डिजिटल सजावटी तस्वीर प्रिंटर का पता लगाने के बारे में भी राय, आप में सबसे तेजी से उत्तर हम करूँगा कर सकते हैं। | hindi |
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| code |
रघुपतिप्रियभक्तं वातजातं नमामि!!
जो अतुल बलके धाम, सोने के पर्वत (सुमेरु) के समान कंतियुक्त शरीर वाले, दैत्यरूपी वन (को ध्वंश करने ) के लिये अग्निरूप, ज्ञानियों में अग्रगण्य, संपूर्ण गुणोंके निधान, वानरों के स्वामी, श्री रघुनाथजी के प्रिय भक्त हैं उन पवनपुत्र श्री हनुमान्जी को मैं प्रणाम करता हूँ ।
हनुमान जी की उत्पति के चार प्रमुख उद्देश्य
१) सत्यपथ नगरी के राजा केशरी और पटरानी अंजना के कोई संतान नहीं थी। इसलिए अंजना ने अपने पति की आज्ञा लेकर गंधमान पर्वत पर जा करके भगवान शिव की कठोरतपस्या की। बहुत दिनों की घोर तपस्या के बाद लोक-कल्याणकारी भगवान शिव जी प्रकट हुए और उन्होंने अंजना से वरदान माँगने को कहा। तब अंजना ने शिव जी के जैसा ही बली तथा तेजस्वी पुत्र का वरदान माँगा। तब महाराज शिव जी ने अंजना को वरदान दिया की उनके गर्भ से शिव जी हीं अवतार के रूप में जन्म लेंगेऐसा वरदन देकर शिव जी महाराज अंतर्ध्यान हो गये।
२) जलंधर नामक राक्षस का वध करने के लिये भगवान शिव को पवनदेव की शक्ति की आवश्यकता पड़ी। क्युंकि जलंधर बहुत ही तेज गति से आक्रमण करता था। भगवान शंकर के अनुरोध पर पवनदेव ने अपना पूरा बल, वेग भगवान शिव के साथ संलिष्ट कर दिया। उस वेग के सहयोग से ही भगवान शिव ने जलंधर का संहार किया। जलंधर के संहार के बाद शिव जी ने पवनदेव को उनकी शक्ति लौटा दी और उनसे वरदान माँगने को कहा। पवन देव ने भगवान शिव के अविचल भक्ति का वरदान माँगा। तब भगवान शिव ने पवनदेव से कहा कि इसके बदले में मैं तुम्हारा पुत्र होकर विश्व में कीर्ति फैलाऊँगा।
३) लंकाधिपति रावण के अत्याचार से समस्त संसार में हाहाकार मच गया। देवी-देवता भी उसकी शक्ति का लोहा मानते थे। पर्वताकार कुम्भकरण जब अपनी छ: महीने की निद्रा पूर्ण करके उठता था तो पृथ्वी के सम्स्त जीवों में हलचल मच जाती थी। रावण और कुम्भकरण भगवान शंकर के भक्त थे। रावण ने शिवजी को अपना शीश काटकर चढ़ाया था और ब्रह्मा जी का कठिन तप कर के यह वरदान प्राप्त किया था कि उसकी मृत्यु नर और वानर के अतिरिक्त किसी से भी न हो।
हम काहु के मरहि ना मारे।
वानर मनुज जाति दुउ टारे॥
इस बात को जानकर सभी देवताओं ने भगवान शिव की घोर तपस्या की। तब भगवान शिव ने उन्हें बताया कि वे वानर का अभिमान-शून्य रूप धारण करेंगे और श्री राम का अनुचर बनकर पृथ्वी पर अवतरित होंगे ।
४) कौतुकी नगर में नारद द्वारा श्राप पा चुकने पर भगवान विष्णु सदा ही चिंतित रहते थे। रावण का अत्याचार बहुत बढ़ा चुका था। अब यह आवश्यक था कि भगवान विष्णु अपने प्रमुख सहायकों के साथ जन्म लेते और उस घोर पराक्रमी और अत्याचारी का विनाश करते। तब भगवान विष्णु ने शिव जी की अराधना करना प्रारम्भ कर दिय। विष्णु जी की अराधना से प्रसन्न हो कर शिव जी प्रकट हुए । तब विष्णु जी बोले- प्रभो! आप हमारे अराध्यदेव हैं, आप ही के द्वारा दिया गया अमोघ सुदर्शन चक्र ही हमारा रक्षक है। अब आपको भी हमारे साथ पृथ्वी पर अवतरित होना है जिससे रावण का नाश हो सके। भगवान शिव ने बताया कि प्रभो ! मैं तो सदा ही आपकी सेवा के लिये तैयार हूँ। मैने भी देवी अंजना को वरदान दिया है कि उन्हीं के गर्भ से पुत्र रूप मे मैं उत्पन्न होऊँगा। वैसे तो रावण मेरा भक्त ही है किंतु अहंकार के रंग में रंगकर वह भले-बुरे का ज्ञान भूल चुका है।
अलग अलग देवताओं से मिले हुए वरदान
हनुमान जी अपने बाल काल में बहुत शरारती थे । एक बार सुर्य को देखकर उन्हें लगा की कोई फल है अत: उस फल को खाने की इच्छा से हनुमान जी सुर्य की ओर बढ़े और अपने मुख में भर लिया। तभी इंद्र ने घबराकर अपने वज्र से उसपर वार किया और वे मूर्छित हो गये। हनुमान को मूर्छित देखकर पवनदेव क्रोधित हो गये और पूरे संसार को वायुविहिन कर के एक गुफा में बैठ गये। चारों ओर हाहाकारमच गया। तब स्वयं ब्रह्मा जी ने आकर हनुमान को जीवित किया, और वायुदेव गुफा से बाहर निकले। उसी समय सभी देवताओं ने हनुमान जी को भिन्न-भिन्न वरदान दिए।
सूर्यदेव का वरदान
सूर्यदेव ने हनुमान को सर्वशक्तिमान बनने का वरदान दिया और अपने तेज का सौवा भाग प्रदान किया । सूर्य देव ने उन्हें पवनपुत्र को नौ विद्याओं का ज्ञान देकर एक अच्छा वक्ता और अद्भुत व्यक्तित्व का स्वामी भी बनाया।
यमराज का वरदान
यमराज ने हनुमान को अपने दंड से मुक्त होने का वरदान दिया।
कुबेर का वरदान
कुबेर ने अपने सभी अस्त्र-शस्त्रों के प्रभाव से हनुमान को मुक्त कर दिया था तथा अपनेगदा से भी उन्हें निर्भय कर दिया था।
भोलेनाथ का वरदान
भगवान शिव ने उन्हें किसी भी अस्त्र से मृत्यु नहीं होने का वरदान दिया था।
विश्वकर्मा का वरदान
विश्वकर्मा ने हनुमान को ऐसी शक्ति प्रदान की जिसकी वजह से विश्वकर्मा द्वारा निर्मित किसी भी अस्त्र से उनकी मृत्यु ना हो पाये, साथ ही उनको चिरंजीवी होने का वरदान भी प्रदान किया।
देवराज इन्द्र का वरदान
इन्द्र देव के द्वारा ही हनुमान की हनु ( ठुड्डी ) खंडित हुई थी, इसलिए इन्द्र ने ही उन्हें हनुमान नाम दिया था तथा उन्हें आशीर्वाद दिया था कि वज्र भी महावीर को चोट नहीं पहुंचा पाएगा।
वरुण देव का वरदान
वरुण देव ने हनुमान को दस लाख वर्ष तक जीवित रहने का वरदान दिया।
ब्रह्मा का वरदान
ब्रह्मा जी ने उन्हें हर प्रकार के ब्रह्मदंडों से मुक्त होने का और अपनी इच्छानुसार गति और वेश धारण करने का, तथा धर्मात्मा एवं परमज्ञानी होने का वरदान दिया।
हनुमानजी को सिंदूर चढ़ाने का कारण-
एक बार की बात है कि सीता जी अपनी मांग को सिंदूर से सजा रही थी । ठीक उसी समय हनुमानजी वहाँ पहुँचे । माता जानकी को सिंदूर लगाते देखकर हनुमान जी ने पूछा- हे माँ , आप सिंदूर अपनी मांग में क्यूँ भरती हैं ?
सीताजी ने उत्तर दिया कि हे पुत्र! इससे तुम्हारे स्वामी की आयु बढ़ती है । सिंदूर सौभाग्यवती स्त्रियों का एक आभूषण है। सिंदूर से स्वामी की आयु बढ़ेगी ऐसा सोचकर हनुमानजी ने माँ से सिंदूर मांगा और उसे पूरे शरीर पर पोतकर प्रसन्नता के साथ प्रभु श्री राम के पास जा पहुँचे । उन्हें सिंदूर से रंगा देखकर भगवान श्री राम ने हँसते हुए पूछा हे पवन पुत्र ! यह क्या किया है ? हनुमानजी ने गद्गद् स्वर में उत्तर दिया
हे नाथ ! माता सीता ने कहा था कि ऐसा करने से आपकी आयु बढ़ेगी। ऐसा सुनकर श्री राम ने हनुमानजी को हृदय से लगा लिया । | hindi |
function start(ms) {
ms.unlockUI();
ms.mapEffect("maplemap/enter/1020000");
} | code |
काशी में बनेगी कैटल कॉलोनी - इनेक्स्ट लाइव
काशी में बनेगी कैटल कॉलोनी
- नगर निगम ने कैटल कालोनी के लिए किया जगह का चयन
- प्रस्ताव तैयार कर शासन को भेजी जाएगी रिपोर्ट
मेरठ : हाईकोर्ट ने नगर निगम को सभी डेयरियों को तीन माह में शहर से बाहर करने के आदेश दिए थे. जिस पर अमल करते हुए नगर निगम ने कैटल कालोनी के लिए जगह चयन कर लिया है. काशी गांव में कैटल कॉलोनी बनाने की योजना बनाई है. नगर निगम इसके लिए प्रस्ताव तैयार कर रहा है. प्रस्ताव तैयार करने के बाद शासन को रिपोर्ट भेजी जाएगी.
पहले भी आए हैं निर्देश
इससे पहले भी हाईकोर्ट ने डेयरियों को बाहर भेजने के लिए आदेश दिए थे. लेकिन आदेश केवल फाइलों तक सिमटकर रह गए. ना तो नगर निगम ने डेयरी बाहर भेजने के लिए कुछ किया और न ही एमडीए ने कॉलोनी बनाने के लिए निगम को जमीन दी. अब दोबारा से इम्तियाज की शिकायत पर लखनऊ बेंच ने आदेश जारी किया है. जिस पर निगम ने इस बार कार्रवाई करते हुए जमीन चयनित की गई है.
एक बार हो चुकी है बैठक
महापौर ने डेयरियों को शहर से बाहर ले जाने के लिए दो माह पहले डेयरी संचालकों के साथ बैठक की थी. इस पर डेयरी संचालकों ने दूध बिकवाने की जिम्मेदारी नगर निगम पर डाल दी थी. इस पर मेयर ने पब्लिक से सहयोग करने की बात कही थी.
पशु पकड़ने की योजना नहीं
नगर निगम ने पशुओं को पकड़ने की कोई योजना तैयार नहीं की है. नगर निगम का कहना है कि निगम जब भी कोई अभियान चलाता है. पशुओं के लिए काम करने वाली संस्था इसका विरोध करती हैं. जिसके कारण अभियान को बीच में ही बंद करना पड़ता है. इसी कारण पशुओं को पकड़ने की योजना नहीं बनाई है.
डेयरी बाहर ले जाने से कोई दिक्कत नहीं है. लेकिन नगर निगम को कॉलोनी को तैयार करके देना होगा. यदि केवल जमीन देंगे तो किसी भी कीमत पर डेयरियों को बाहर नहीं ले जाया जाएगा. बाहर ले जाने से हमारे व्यापार पर खासा असर भी पड़ेगा.
महेंद्र उपाध्याय, अध्यक्ष डेयरी संचालक एसोसिएशन
कैटल कॉलोनी के लिए काशी गांव में जमीन चिंहित कर ली गई है. नगर निगम की वह जमीन है. प्रस्ताव तैयार करने के निर्देश दिए हैं. प्रस्ताव तैयार कर शासन को रिपोर्ट भेजी जाएगी.
डॉ. आरएस चौहान, नगर स्वास्थ्य अधिकारी, नगर निगम | hindi |
पुलिस की छवि ऐसी बने कि जनता सम्मान की दृष्टि से देखें ः रघुवर दास
| | २०१५-०८-१५ ०७:३१:२७.०
रांची, मुख्यमंत्री रघुवर दास ने कहा कि पुलिस की छवि और कार्य संस्कृति ऐसी हो कि जनता उन्हें सम्मान...
रांची, मुख्यमंत्री रघुवर दास ने कहा कि पुलिस की छवि और कार्य संस्कृति ऐसी हो कि जनता उन्हें सम्मान की दृष्टि से देखे। बदलते झारखण्ड में पुलिस को भी अपनी कार्य संस्कृति में बदलाव लाना है।
पुलिस पदाधिकारी, कर्मचारी शांति के वाहक हैं विकास के लिए शांति व्यवस्था आवश्यक है। देश की सीमा पर फौज और सशस्त्र बलों के जवान जोखिम में रहकर सीमाओं की चौकसी करते हैं। इसी प्रकार आन्तरिक सुरक्षा व्यवस्था के लिए हमारे पुलिस पदाधिकारी से लेकर आरक्षी तक दुर्गम क्षेत्रों में अपनी जान जोखिम में डाल कर राज्य के लोगों को सुरक्षित जीवन प्रदान करते हैं। भारत की संस्कृति में देश, राज्य और समाज की सुरक्षा को सर्वोपरि माना गया है।
श्री दास ने कहा कि यह हर्ष का विषय है कि पूरे देश में झारखण्ड पहला राज्य है, जहां पुलिस पदाधिकारियों, कर्मचारियों के लिए इस प्रकार की आवासन की व्यवस्था की गई है। मुख्यमंत्री आज स्थानीय डोरंडा स्थित जैप कम्पाउंड में पुलिस पदाधिकारियों व कर्मचारियों के लिए बने यू.एस. टावर एवं एल.एस. टावर का उद्धाटन कर रहे थे। मुख्यमंत्री ने इस कार्यक्रम के दौरान झारखण्ड पुलिस हाउसिंग कॉरपोरेशन द्वारा निर्मित १६ अन्य भवनों का ऑनलाईन उद्धाटन किया।
इस अवसर पर मुख्य सचिव राजीव गौबा, अपर मुख्य सचिव गृह विभाग एन.एन.पाण्डेय, पुलिस महानिदेशक डी.के.पाण्डेय सहित पुलिस मुख्यालय के वरीय पदाधिकारीगण उपस्थित थे। स्वागत भाषण झारखण्ड पुलिस हाउसिंग कॉरपोरेशन के अध्यक्ष राजीव कुमार ने किया वहीं धन्यवाद ज्ञापन झारखण्ड पुलिस हाउसिंग कॉरपोरेशन के प्रबंध निदेशक के.एस.मीणा ने किया। | hindi |
The purpose of the IAPSC is to establish and maintain the highest possible standards in the security consulting profession. For that purpose, the Association will provide opportunities for the professional enhancement of its members, and will promote greater awareness of the objective standards of its membership.
The IAPSC began to grow as an increasing number of consultants saw the opportunity to interact and network with their peers. This unique organization provided individual security consultants with the opportunity to exchange ideas within an interactive community.
Since our 1984 inception, IAPSC continues to hold annual meetings in charming U.S. locations. These annual meetings serve as opportunities for professional enrichment. Guest speakers, who are experts in various security-related fields, conduct seminars and give lectures to attendees. Occasionally, IAPSC members participate in meeting presentation in order to enlighten colleagues about their specific expertise. The variety of meeting locations also serves to broaden cultural experience for members and their spouses.
Several members have performed duties as Executive Director of IAPSC. Currently, the organization employs a professional management firm to handle its affairs, and the association is run by elected officers and a volunteer-elected Board of Directors.
The IAPSC is an organization of peers. No member is considered better than another, and frequent elections prevent any small group from dominating the association. The strict ethical standards and stringent membership requirements ensure that each member represents the highest professional standards. | english |
\begin{document}
\title{Atomic diffraction by light gratings with very short wavelengths}
\date{}
\author{P. Sancho \\ Centro de L\'aseres Pulsados, CLPU \\ E-37008, Salamanca, Spain}
\maketitle
\begin{abstract}
Lasers with wavelengths of the order of the atomic size are becoming
available. We explore the behavior of light-matter interactions in
this emergent field by considering the atomic Kapitza-Dirac effect.
We derive the diffraction patterns, which are in principle
experimentally testable. From a fundamental point of view, our
proposal provides an example of system where the periodicity of the
diffraction grating is comparable to the size of the diffracted
object.
\end{abstract}
\section{Introduction}
Lasers in the X-ray domain, with wavelengths of the order of the
atomic size, have been reported in the literature
\cite{Nat,xfe,All}. We must study the underlying physics in that unexplored
range of ultrashort wavelengths. In particular, we must analyze the
behavior of the light-matter interaction. In
order to carry out the analysis in a simple way, we shall consider a
well-known scheme, the Kapitza-Dirac effect, which can be described
with simple mathematical tools. In the Kapitza-Dirac effect a beam
of atoms or electrons is diffracted or scattered by a standing light
wave \cite{KD, Bat, Pri, Ba2,Ba3}. The effect can be extended to
two-particle systems \cite{Sa1,Sa2}.
First of all, we must derive the form of the light-atom interaction
in this regime. We shall show that the dynamics is ruled by a
lightshift potential in the dipole approximation. Being the light
wavelength comparable to the atomic size, different parts of the
atom feel different values of the field and the dipole approximation
provides an incomplete description of the problem. We must consider
higher multipole terms, in particular the quadrupole one. However,
because of the high frequency of the light field we must average the
interaction on time. The average of the higher permanent multipole
terms is zero, reducing the total interaction to the usual
lightshift potential in the dipole approximation. We shall also consider
the quadrupole induced by the electric field. The evaluation of
its actual contribution to the problem is difficult because of the
lack of reliable experimental data on quadrupole or higher order
polarizabilities. Nevertheless we shall show that if these effects
were of the same order of the lightshift potential they would not
modify the fundamental results of our paper.
In the standard atomic Kapitza-Dirac arrangement the wavelength of
the optical grating is close to an atomic transition, enhancing the
strength of the interaction. The very short wavelengths we consider
in this paper are fully detuned from the atomic transitions, making
much more weak the light-atom interaction, which is only related to
the atomic polarizability. However, considering a laser intensity of
the order of $10^{14}W/m^2$ (the values used in the observation of
the effect with electrons \cite{Ba2}) the strength of the light-atom
interaction is comparable to that in the on resonance case and we
can, in principle, observe the diffraction effects. We shall
evaluate the diffraction patterns, obtaining the same form of the
standard atomic Kapitza-Dirac arrangement (containing only even
order peaks).
The high frequency values of the light can give rise, depending on
the duration of the interaction, to the presence of large ionization
rates which would make more difficult the observation of the
diffraction effects. In these cases one must introduce some
procedure to remove the ionized atoms from the experiment.
In addition to provide an example of light-matter interaction in the
scale of very short wavelengths, our proposal is also interesting
from a fundamental point of view. There has been an increasing
interest in the study of quantum diffraction with large size
objects, for instance, with $C_{60}$ \cite{Ze1} and $C_{70}$
\cite{Ze2} molecules in solid nanostructures or structures made of
light \cite{Ze3}, and $Na_2$ molecules whose de Broglie wavelength
is smaller than their size \cite{Pr1}. However, there is yet a
fundamental question that has not been addressed, the limiting size
(relative to the spacing of the diffraction grating) of a quantum
object to observe diffraction. Our proposal can be the basis to
study quantum diffraction in the extreme regime where the
periodicity of the diffraction optical grating is similar to the
size of the diffracted system.
The plan of the paper is as follows. In Sect. 2 we present the
arrangement and evaluate the lightshift potential relevant for the
problem. In order to give a compact presentation of the main ideas
involved in the evaluation, some more technical aspects of the
treatment are discussed in Appendix 1. In Sect. 3 we derive the
diffraction patterns and consider how to eliminate the ionized
atoms. The values of the parameters involved in a realistic
experimental implementation of the arrangement are estimated in
Sect. 4. In the Discussion, we consider the potential impact of our
proposal for other physical problems, in particular, the exploration
of the limits of quantum diffraction with large size objects.
Finally, the possible effects associated with a large quadrupole
polarizability are presented in Appendix 2.
\section{The arrangement}
As usual in the Kapitza-Dirac effect our arrangement consists of a
standing light wave generated by a laser. A beam of atoms interacts
with that diffraction grating. Behind the grating we place
detectors. From the data collected at the detectors we can infer the
diffraction patterns (see Fig. 1).
Next, we consider the mathematical description of the interaction.
First of all, we note that we shall deal with high laser intensities.
Then we can resort to the semiclassical approximation, where the
electromagnetic fields can be treated classically. In Appendix 1
we estimate the intensity values for which we can safely use this
approximation in the framework of the Kapitza-Dirac effect.
\begin{figure}
\caption{The wavelength of the laser (green) is similar to the size
of the atom (blue). After the interaction we observe a diffraction
pattern.}
\end{figure}
The adiabatic condition plays an important role in this type of
problem. When it is fulfilled, the time dependence of the effective
optical potential experienced by the atoms is slow compared to the
internal evolution. The center of mass (CM) dynamics can be
decoupled from the internal one, and it can be described by an
optical potential, generating a phase shift as a function of
position \cite{Ada}. As we shall see later (Sect. 3), the adiabatic
condition is by far fulfilled in our case.
Once guaranteed adiabatic evolution, we can concentrate on the CM
dynamics. The phase shift of the CM wavefunction is given by the
dipole-type interaction between the electric field ${\bf E}$ and the
induced atomic dipole ${\bf d}=\alpha {\bf E}$, with $\alpha $ the
polarizability. We describe this interaction by the lightshift
potential \cite{Bat,Ze3}
\begin{equation}
U_{LS}({\bf R},t)=-\frac{1}{2} \alpha {\bf E}^2({\bf R},t)
\end{equation}
where ${\bf R}$ denotes the CM coordinate of the atom. Being the
CM timescale very different from the laser period, we must average
over that period. If we take for the electric field of the standing
wave the form
\begin{equation}
{\bf E}({\bf R},t)={\bf E}_0 \cos ({\bf k}_L \cdot {\bf R}) \cos (\omega _L t)
\end{equation}
with ${\bf k}_L$ and $\omega _L$ the wavevector and frequency of the
laser, the averaged lightshift potential takes the form
\begin{equation}
U({\bf R})= \frac{1}{T_L}\int _0^{T_L} U_{LS}dt =-\frac{1}{4} \alpha {\bf E}_0^2 \cos ^2({\bf k}_L \cdot {\bf R})
\end{equation}
with $T_L=2\pi /\omega _L$.
In the framework of the dipole approximation there is yet another
interaction channel (see Appendix 1). In addition to the
transitions between bound states that induce the atomic dipole,
there can be also transitions to the continuum. As a matter of fact,
due to the high energy of the involved photons we expect that
absorption will lead in most cases to ionization, and the ionization
rate in our problem can be large. This is not the case for the
values of the parameters we shall propose to observe the effect (see
Sect. 3). However, for other values of the parameters the ionization
rate could be large. In these cases, as we are only interested into
the diffraction properties of the non-ionized atoms, we must
consider an scheme to eliminate the ionized atoms. In the next
section we present such an scheme. Thus, due to the negligible value
of the ionization rate in some cases and to the possibility of
eliminate the ions in the rest of cases, we do not need to consider
this interaction channel.
Up to now we have restricted our considerations to the leading
dipole approximation. However, as remarked before, because the
optical wavelength is comparable to the atomic size, the dipole
approximation (which assumes no relevant variations of the field in
distances of the order of the atomic size) does not provide a
complete description of the problem. There are in the literature
several examples of how to go beyond this approximation in the
context of X-ray theory. For instance, quadrupole terms have been
used to study X-ray spectroscopy \cite{Ber}, numerical non-dipole
simulations of ionization by X-ray lasers have been presented in
\cite{Zho}, and the Bloch equations without the dipole approximation
have been derived in \cite{Zha}. In this paper we follow the
approach of considering higher multipole terms. Let us consider the
next term in the perturbative series, the permanent quadrupole term,
which has the form (see Appendix 1)
\begin{equation}
U_Q({\bf R},t)=\frac{1}{2} Q_{ij} \frac{\partial E_i}{\partial R_j}
\end{equation}
Using the form of the electric field we have that $U_Q \sim \sin
({\bf k}_L \cdot {\bf R}) \cos (\omega _L t)$. Now, when performing
the time average we have that the quadrupole interaction goes to
zero. Thus, although there is a quadrupole interaction associated
with the field variations along the atom size, its net effect
vanishes. As discussed in Appendix 1 a similar conclusion holds for
any higher order term. Thus, all the averaged multipole permanent
contributions vanish and the light-atom interaction can be described
via the dipole approximation. In Appendix 2 we shall discuss how
induced quadrupole multipoles could be present in the problem.
In conclusion, the analysis of this section shows that the light-atom interaction can be expressed in the form
\begin{equation}
U({\bf R})=U_0 \cos ^2 ({\bf k_L} \cdot {\bf R})
\end{equation}
with $U_0 = -\alpha {\bf E}_0^2/4$. This is the usual lightshift
potential used in the standard on resonance atomic Kapitza-Dirac effect.
\section{Predicted diffraction pattern}
We have shown in the previous section the existence of two
interaction channels in our arrangement. On the one hand, the large
energy of the photons in the range of frequencies considered can
lead, depending on the duration of the interaction, to a high
ionization rate. On the other hand, the atoms that are not ionized
generate a diffraction pattern. We are only interested into the
observation of these patterns. Then in the cases where the
ionization is large we must remove the ionized atoms (and the
electrons) in order to postselect the neutral ones.
One can easily design methods able to extract the ionized atoms and
electrons from the experiment. For instance, an electric field
perpendicular to the plane where the experiment takes place
(perpendicular to the longitudinal and transversal directions) would
deviate most electrons and ions from the detectors, whereas the
evolution of the neutral atoms would be almost unaffected.
After the removal of ions and electrons we can focus on the dynamics
of the neutral atoms interacting with the lightshift potential. The
state of the CM at time $t$ of the atoms with initial wavevector
$k_0$ (at time $t=0$) can be expressed as $e^{iU(X)t /\hbar}
e^{ik_0X}$ in the Raman-Nath approximation. In this approximation we
assume that the momentum of the atoms is large compared to that of
the photons. Then the kinetic energy remains approximately constant
and may be neglected. As it is well-known, this approximation is
valid in the diffraction regime \cite{Bat}. The coordinate $X$ is
that of the CM in the direction parallel to the granting. As usual
in the Kapitza-Dirac effect we only consider the one-dimensional
problem \cite{Bat}.
Using the relation $\exp (i\xi \cos \varphi)=\sum _n i^n J_n(\xi
)\exp (in\varphi )$, with $J_n$ the n-th order Bessel function, we
have
\begin{equation}
e^{iU(X)\tau /\hbar} e^{ik_0X}= e^{iU_0\tau /2\hbar} \sum _{n=-\infty}^{\infty } i^n J_n \left( \frac{U_0 \tau }{2\hbar } \right) e^{i(2nk_L + k_o)X}
\end{equation}
with $\tau $ the interaction time.
This pattern shows the standard form in Kapitza-Dirac diffraction
(that with on resonance light-atom interaction). Only even
diffraction orders are present. The intensity of the peaks is given
by $ J_n ( U_0 \tau /2\hbar )^2$.
\section{Experimental parameters}
We analyze in this section the values of the parameters of the
problem that would lead to observable effects in an experimental
realization of the above arrangement. We shall use a wavelength
$\lambda _L \approx 5 \times 10^{-10}m$. This wavelength differs in
several orders of magnitude of those associated with atomic
transitions, giving rise to a large detuning, which guarantees the
adiabatic evolution in the problem.
First of all, in order to be in the diffraction regime the
coefficient $U /\epsilon $, with $\epsilon =\hbar ^2k_L^2/2m$ the
recoil shift of the atom by absorption of a photon, must be larger
than unity \cite{Bat}. Taking the mass of the atom as ten to twenty times the
proton mass, we have $\epsilon \approx 10^{-4} eV$, and we
must take $U \approx 10^{-3}eV$.
From this value of the potential we can deduce the intensity of the
laser beam. We use the approximate relation $U \approx \alpha E_0^2$
and the definition of intensity $I=cE_0^2/8\pi $. The atomic
polarizability values are in the range of $10^{-29}m^3$ to
$10^{-31}m^3$ \cite{Sch}. Taking an atom in the high part of the
range, $\alpha \approx 10^{-29}m^3$, we would need a laser intensity
of the order of $10^{14}W/m^2$. This can seem a very high
value when compared with the intensities in the standard atomic
Kapitza-Dirac effect ($I \approx 10^7 W/m^2$). However, a value of
$5 \times 10^{14}W/m^2$ has been used to demonstrate the effect with
electrons \cite{Ba2}. The proposed value of $\lambda _L =5 \times
10^{-10}m$ has already been reached with beam intensities as high as
$10^{17}W/m^2$ \cite{Nat,xfe}. Note also that the proposed value, although
very high, is yet a long way from the threshold of non-linear
polarizabilities, around $I=10^{18}W/m^2$.
The condition of high visibility of the interference pattern, $U\tau
/\hbar \approx 1$ \cite{Bat}, implies a time of interaction of
around $10^{-12}s$. As the duration of the pulses in \cite{Nat,xfe}
is between $10^{-13}$ and $10^{-14}s$, a possibility to generate the
standing wave is using counter-propagating pulses slightly enlarged
on time (for instance, via dispersion of the ultrashort pulses).
Another possibility, which does not modify the pulse duration, is to
increase the velocity of the atoms. The smallest radios of the
focused spots for this range of wavelengths are around $1\mu m$
\cite{Nat}. Then $\tau \approx 10^{-12}s$ requires atom velocities
close to $10^6 ms^{-1}$, that is, an increase of three orders of
magnitude with respect to the usual values. These velocities could
be reached accelerating ions, which later would interact with free
electrons generating neutral atoms (the remaining ions should be
extracted from the beam by interaction with an electric field). In
this second scheme one cannot use counter-propagating laser beams,
which are shorter than $\tau$, and should generate the standing wave
with mirrors or other techniques.
We must also determine the fraction of atoms that are ionized during
the interaction. If a photon is absorbed the probability of
ionization is large in our arrangement. For $\lambda _L$ the energy
of one photon is $E_{ph}=\hbar \omega _L \approx 3 \times 10^2 eV$.
This energy is much larger than the ionization energy, giving rise
to a large probability of ionization in the case of photon
absorption. We estimate the number of non-ionized atoms, $N$, in the
usual way: $dN =-\Gamma N dt$ with $\Gamma $ the ionization rate,
which is assumed to be time-independent. This gives $N(\tau )=N_0
\exp (-\Gamma \tau)$ with $N_0$ the initial number of atoms in the
beam. On the other hand, for the range of intensities in our problem
the ionization rate can be expressed as $\Gamma =\sigma I/\hbar
\omega _L$ with $\sigma$ the photoionization cross section, which
depends on $\omega _L$. Values of $\sigma $ can be found in several
database. For instance for the Na atom and $\hbar \omega _L = 100eV$
we have $\sigma =5 \times 10^{-22} m^2$, which gives $\Gamma \tau
\approx 10^{-3}$. Due to the very short duration of the interaction
the fraction of ionized atoms is negligible. For the values of the
parameters here proposed it is not necessary to include the
procedure to remove the ionized atoms.
\section{Discussion}
Some coherent light sources are reaching the scale where the optical
wavelengths are comparable to the atomic sizes. Our work is one of
the first studies considering possible physical effects present in
this unexplored regime. Contrarily to a naive intuition, the
effective interaction does not depend on multipole terms beyond the
dipole one, and the evolution can be described via the usual
lightshift potential. We have shown the existence of diffraction
effects similar to those present when the light is on resonance with
atomic transitions. The main obstacle to carry out the experiment is
the short duration of the laser pulse. For other possible
experiments with much longer durations (for instance, Bragg's
scattering with very short wavelengths) the ionization rates can be
high and a procedure to eliminate the ionized atoms should be added
to the arrangement.
In addition to the interest of our proposal in atom optics, it could
also be relevant to other fields. We shall focus on three of them.
The first one concerns to fundamental physics, in particular, to the
understanding of the wave properties of quantum systems. Our scheme
provides the first example where the periodicity of the classical
diffraction grating is similar to the size of the quantum diffracted
object. This is an extreme scenario for quantum diffraction that
extends the research presented in \cite{Ze1,Ze2,Ze3,Pr1}. Our
analysis shows that in the limit of objects of the size of the grating
periodicity, diffraction survives.
Also, from a fundamental point of view, our proposal serves to test the
theory of light-matter interactions in the scale of very short
wavelengths. This knowledge would be necessary to study, for
instance, the dispersion of this type of light by atoms. It should
be also in the basis of techniques aimed to provide images of the
spatial structure of atoms, a process in principle possible with
light of the same wavelength of the size of the illuminated object.
Finally, from a more practical point of view, the diffraction of
atoms can be used to measure the polarizability, by
fitting the experimental data to the detection distributions.
This would be an alternative method of measurement of this atomic property.
{\bf Acknowledgments} I am grateful to Luis Plaja for discussions on
the problem. I acknowledge support from Spanish Ministerio de
Ciencia e Innovaci\'on through the research project FIS2009-09522.
{\bf Appendix 1}
In this Appendix we present some technical points that complement
the main developments in the paper.
{\it Semiclassical approximation.} We derive the values of the laser
intensity that guarantee the use of the semiclassical approximation.
The usual criterion for its validity is to have a large number of
photons in the interaction region. In standard Kapitza-Dirac
diffraction, where the approximation works well, the number of
photons per unit volume is $I/c\hbar \omega _L \approx 10^{18}
photons/m^3$, where we have used $\lambda _L =500 nm$ and $I=10^7
W/m^2$. The typical volume of the interaction region is $10^{-12}
m^3$ ($10 \mu m$ (aperture collimating the beam of atoms) $\times$
$1 mm$ (height of the laser beam) $\times$ $100 \mu m$ (width of the
laser beam)). The number of photons in this characteristic volume is
around $10^{6}$. For short wavelengths, the energy of each photon is
around $10^2 eV$ and, in order to have a similar number of photons
(assuming a similar characteristic volume), the intensity must be
above $10^{11} W/m^2$. Then in our proposal we can safely use the
semiclassical approximation.
{\it Multipole expansion.} For very short wavelengths different
parts of the atom can feel different electric fields. We must go
beyond the dipole approximation and consider higher multipole terms.
The light-atom interaction potential can be expressed in the
multipole form \cite{Lou}:
\begin{equation}
{\bf D}\cdot {\bf E}({\bf R},t) + \frac{1}{2} \sum _{i,j} Q_{ij} \left( \frac{\partial E_i}{\partial r_j} \right) ({\bf R},t) + \cdots
\label{eq:mul}
\end{equation}
where ${\bf D}$ and $Q_{ij}$ represent the dipole and $ij$-component
of the quadrupole momenta of the atom. They can be rewritten using
the second quantization of the atomic variables \cite{Lou}. If we
denote by $|h>$ and $|l>$ the eigenstates of the atom we have
$\hat{\bf D}=\sum _{l,h}{\bf d}_{lh}|l><h|$ with ${\bf D}_{lh}=
\sum _{e_n} <l|e_n {\bf r}_{e_n}|h>$, and $\hat{Q}_{ij}=\sum _{l,h} Q_{ij})_{lh}|l><h|$ with $(Q_{ij})_{lh}= \sum _{e_n} <l|e_n ( 3(r_{e_n})_i(r_{e_n})_j - \delta _{ij}{\bf r}_{e_n}^2)|h>$. The sum, $\sum _{e_n}$, is over all the electrons of the atom. The diagonal elements (expectation values) ${\bf D}_{ll}$ are null because the integrand has odd parity. The non-diagonal (transition) elements contribute to the atomic polarizability. In the static and
spherically symmetric case the polarizability has the form
\begin{equation}
\alpha =\frac{2}{3} \sum _{l \neq g} \frac{|<g|\sum _{e_n}{\bf r}_{e_n} |l>|^2}{{\cal E}_l-{\cal E}_g}
\end{equation}
with $g$ denoting the ground state, and ${\cal E}_l$ he energy of the state
\cite{Sch}.
The dipole term also describes the ionization of the atom. It is an
alternative channel associated with ground-continuum transitions,
with matrix elements in the form $<c|e{\bf r}|g>$, denoting $|c>$
states of the continuum.
The next term in Eq. (\ref{eq:mul}) represents the permanent
quadrupole effects. In general, the values of the permanent
quadrupole momenta can be expressed as $Q_{ij} \sim er_0^2$, with
$r_0$ Bohr's radius and a proportionality coefficient of the order
of unity \cite{Ita}. As signaled before, in order to be in the
diffraction regime the potential must be of the order of $U \approx
10^{-3}eV$. Then the quadrupole term must be similar, $U_Q \approx
10^{-3}eV$. Taking for the field the standard form we can express
the potential as $U_Q \approx er_0^2k_LE_0 \approx er_0E_0$. Using
this expression and the relation between $I$ and $E_0$ we have that
the intensity of the laser must be around $10^{8} W/m^2$ in order to
the quadrupolar effects to be relevant in the problem. This value is
smaller than the intensity required to have a non-negligible atomic
dipole induced by the polarizability. In Appendix 2 we discuss the
role of induced quadrupole terms in our problem.
Note that if the n-th order permanent multipole coefficient follows a relation
of the type $Q^n \sim er_0^n$ with a proportionality coefficient of
the order of unity, we have $U_{Q^n} \approx er_0^nk_L^{n-1}E_0
\approx er_0E_0$ and that multipole term also has to be taken into
account. However, as in the quadrupole case, its temporal dependence
remains in the form $\cos(\omega _L t)$, and after averaging it can
be neglected.
Finally, we recall the fact that the electric quadrupole and the
magnetic dipole energies have similar magnitudes. Thus, if the
electric quadrupole term is relevant for the problem, the magnetic
dipole one must also be taken into account. The magnetic dipole term
reads as $e{\bf D}_M \cdot {\bf B}({\bf R})$ with ${\bf B}({\bf R})$
the magnetic field at the atomic CM position and ${\bf D}_M$ the
magnetic dipole moment, which can be expressed as ${\bf
D}_M=-(1/2m)\sum _i {\bf l}_i$ with ${\bf l}_i$ the orbital angular
momentum of the i-th electron in the atom. From these expressions
and the form of the electromagnetic potential it is clear that the
temporal dependence of this term is also proportional to $\cos
(\omega _L t)$, and it will vanish after time averaging just as in
the electric quadrupole case.
{\bf Appendix 2} In this Appendix we analyze the atomic quadrupole
induced by the electric field or its gradient. When the
polarizability effects are taken into account the quadrupole momenta
can be expressed as
\begin{equation}
Q_{ij}=Q_{ij}^0 + \sum _k A_{kij}E_k + \sum _{kl} C_{ijkl} \frac{\partial E_k}{\partial x_l}
\end{equation}
where $ Q_{ij}^0$ are the permanent momenta, $A_{kij}$ is the
dipole-quadrupole polarizability and $C_{ijkl}$ the
quadrupole-quadrupole one. The first term in the r. h. s. of the
equation represents the quadrupole momenta in absence of external
electric fields, the second the momenta induced by an electric field
and the third these induced by the gradient of the field.
The order of magnitude of the coefficients is $A \approx
e^2r_0^3E_h^{-1}$ and $C \approx e^2r_0^4E_h^{-1}$ with $E_h = 4
\times 10^{-18} J$. Introducing numerical values we obtain for the
potentials associated with these two terms ($U_A
\approx AE_0k_LE_0$ and $U_C \approx C(E_0k_L)^2$): $U_A \approx U_C \approx (er_0E_0)^2 E_h^{-1} \approx 10^{-4} - 10^{-5} eV $, where we have used $r_0 k_L \approx 1$. We expect the potentials to be ten to one hundred times smaller than the lightshift potential. However, to extract precise conclusions of the importance of the induced terms we should know the actual values of the coefficients. Unfortunately, very few experimental data are available, and in many cases there are large uncertainties on their theoretical estimation (see, for instance, \cite{Por} for some recent work in this subject. Note that with the values suggested for Mg in this reference the induced potential would be comparable to the lightshift one).
In our particular problem, when the quadrupole polarizability is
taken into account the full potential can be written as
\begin{eqnarray}
U(X)=U_0 \cos ^2(k_LX) + U_A \cos ^3 (k_LX)\sin (k_L X) \nonumber
\\ + U_C \cos ^2 (k_LX)\sin ^2 (k_L X)
\end{eqnarray}
Note that the temporal dependence in all the terms of the r. h. s.
is $\cos ^2 (\omega _L t)$. All the terms remain after the time averaging.
The coefficient resulting from the integration is included in $U_0,
U_A, U_C$. Using simple trigonometric relations we obtain
\begin{eqnarray}
U(X)=\frac{U_0}{2}+\frac{U_C}{8} + \frac{U_0}{2} \cos (2k_LX)+
\frac{U_A}{4} \sin (2k_LX) \nonumber \\ + \frac{U_A}{8} \sin
(4k_LX)- \frac{U_C}{8} \cos (4k_LX)
\end{eqnarray}
The final detection pattern is
\begin{eqnarray}
e^{iU(X)\tau /\hbar} e^{ik_0X}= e^{i\left( \frac{U_0}{2} +
\frac{U_C}{8} \right) \tau /\hbar} \sum _{n,m,l,r=-\infty}^{\infty }
i^{n+r} J_n \left( \frac{U_0 \tau }{2\hbar } \right) \times
\nonumber \\ J_m \left( \frac{U_A \tau }{4\hbar } \right) J_l \left(
\frac{U_A \tau }{8\hbar } \right) J_r \left( -\frac{U_C \tau
}{8\hbar } \right) e^{i([2n+2m+4l+4r]k_L + k_o)X}
\end{eqnarray}
where we have used the relation $\exp (i\xi \sin \varphi)=\sum _n
J_n(\xi )\exp (in\varphi )$. The diffraction pattern shows the same
analytical form found for the lightshift potential, the peaks
correspond to even multiples of $k_L$. The difference lies in the
different intensities of the peaks, which in addition to $U_0$ now
also depend on $U_A$ and $U_C$. In conclusion, in presence of
quadrupole polarizability effects, we would obtain a diffraction
pattern similar to that associated with dipole polarizability.
Our main result remains valid, there are diffraction patterns in the
regime of very short wavelengths. In addition, the diffraction
pattern could be used to determine the $U_A$ and $U_C$ values and,
in consequence, the quadrupole polarizabilities, which are very difficult to
measure by other methods \cite{Por}.
\end{document} | math |
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سُے حصہٕ نظموومُت کٲشِرۍ پٲٹھۍ، یتھ | kashmiri |
use std::cell::{RefCell, RefMut};
use std::collections::HashMap;
use std::collections::hash_map::Entry::{Occupied, Vacant};
use buffer::*;
pub fn run() {
println!("********* HashMap<i64, RefCell<Buffer>> examples *********");
let mut fac = BufferFactory::new();
let mut buffers = BufferCollection::new();
let b1 = fac.new_empty_buffer();
let b2 = fac.new_empty_buffer();
buffers.insert(b1);
buffers.insert(b2);
let buffer_cell1 = buffers.get(0).unwrap();
let mut bor1 = buffer_cell1.borrow_mut();
let buffer_cell2 = buffers.get(1).unwrap();
let mut bor2 = buffer_cell2.borrow_mut();
bor1.set_title(String::from("phil"));
bor2.set_title(String::from("daniels"));
println!("bor1.Id = {}, bor1.title = {}", bor1.get_id(), bor1.title());
println!("bor2.Id = {}, bor2.title = {}", bor2.get_id(), bor2.title());
bor1.set_title(String::from("Philip"));
bor2.set_title(String::from("Daniels"));
println!("bor1.Id = {}, bor1.title = {}", bor1.get_id(), bor1.title());
println!("bor2.Id = {}, bor2.title = {}", bor2.get_id(), bor2.title());
}
| code |
$(document).ready(function() {
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$("#message2").hide();
$("#countdown").hide();
$("#photo").hide();
});
var socket = io.connect('http://raspbooth:8080');
socket.on('countdown', function (data) {
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socket.on('photo', function (data) {
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}
| code |
\begin{document}
\title{f Asymptotic Behaviour of Ergodic skip -2mm Integrals of
`Renormalizable' skip -2mm Parabolic Flows skip 6mm}
\thispagestyle{first} \setcounter{page}{317}
\begin{abstract}
\vskip 3mm
Ten years ago A. Zorich discovered, by computer experiments on interval
exchange transformations, some striking new power laws for the ergodic
integrals of generic non-exact Hamiltonian flows on higher genus surfaces.
In Zorich's later work and in a joint paper authored by M. Kontsevich,
Zorich and Kontsevich were able to explain conjecturally most of Zorich's
discoveries by relating them to the ergodic theory of Teichm\"uller flows
on moduli spaces of Abelian differentials.
In this article, we outline a generalization of the Kontsevich-Zorich
framework to a class of `renormalizable' flows on `pseudo-homogeneous'
spaces. We construct for such flows a `renormalization dynamics' on an
appropriate `moduli space', which generalizes the Teichm\"uller flow.
If a flow is renormalizable and the space of smooth functions is `stable',
in the sense that the Lie derivative operator on smooth functions has
closed range, the behaviour of ergodic integrals can be analyzed, at
least in principle, in terms of an Oseledec's decomposition for a
`renormalization cocycle' over the bundle of `basic currents' for the
orbit foliation of the flow.
This approach was suggested by the author's proof of the Kontsevich-Zorich
conjectures and it has since been applied, in collaboration with L. Flaminio,
to prove that the Zorich phenomenon generalizes to several classical examples
of volume preserving, uniquely ergodic, parabolic flows such as horocycle
flows and nilpotent flows on homogeneous $3$-manifolds.
\vskip 4.5mm
\noindent {\bf 2000 Mathematics Subject Classification:} 37C40, 37E35, 37A17,
37A25, 34D08, 43A85.
\noindent {\bf Keywords and Phrases:} Ergodic integrals, Renormalization
dynamics, Cohomological equations, Invariant distributions, Basic currents.
\end{abstract}
\vskip 12mm
\section{Introduction} \label{section 1}\setzero
\vskip-5mm
\hspace{5mm}
A fundamental problem in smooth ergodic theory is to establish quantitative
estimates on the asymptotics behaviour of ergodic integrals of smooth
functions. For several examples of {\it hyperbolic }flows, such as geodesic
flows on compact manifolds of negative curvature, the asymptotic behaviour
of ergodic integrals is described by the {\it Central Limit Theorem }(Y.
Sinai, M. Ratner). In these cases, the dynamical system can be described
as an approximation of a `random' stochastic process, like the outcomes of
flipping a coin. Non-hyperbolic systems as not as well understood, with the
important exception of toral flows. For generic non-singular area-preserving
flows on the $2$-torus logarithmic bounds on ergodic integrals of zero-average
functions of bounded variation can be derived by the {\it Denjoy-Koksma
inequality }and the theory of continued fractions. For a general ergodic
flow, ergodic integrals are bounded for all times for a special class of
functions: {\it coboundaries }with bounded `transfer' functions
(Gottschalk-Hedlund). In the hyperbolic examples and in the case of
generic toral flows, a smooth function is a coboundary if and only if it
has zero average with respect to all invariant measures.
In this article, we are interested in flows with {\it parabolic }behaviour.
Following A. Katok, a dynamical system is called parabolic if the rate of
divergence of nearby orbits is at most polynomial in time, while hyperbolic
systems are characterized by exponential divergence. Toral flows are a rather
special parabolic example, called {\it elliptic}, since there is no divergence
of orbits. It has been known for many years that typical examples of parabolic
flows, such as horocycle flows or generic nilpotent flows are {\it uniquely
ergodic}, but until recently not much was known on the asymptotic behaviour
of ergodic averages, with the exception of some polynomial bounds on the
speed of convergence in the horocycle case (M. Ratner, M. Burger), related
to the polynomial rate of mixing. We have been able to prove, in collaboration
with L. Flaminio, that for many examples of parabolic dynamics the behaviour
of ergodic averages is typically described as follows.
A smooth flow $\Phi^X$ on a finite dimensional manifold $M$ has {\it deviation
spectrum }$\{\lambda_1> ... > \lambda_i> ... >0\}$ with multiplicities $m_1,
...,m_i,...\in {\mathbb Z}^+$ if there exists a system $\{{\mathcal D}_{ij}
\,|\, i\in {\mathbb Z}^+\,,\,\,1\le j\le m_i\}$ of linearly independent
$X$-{\it invariant distributions }such that, for almost all $p\in M$, the
ergodic integrals of any smooth function $f\in C_0^{\infty}(M)$ have an
asymptotic expansion
\begin{equation}
\label{DS}
\int_0^T f\bigl(\Phi^X(t,p)\bigr) dt \,\,= \,\,\sum_{i\in \mathbb{N}}
\sum_{j=1}^{m_i} c_{ij}(p,T) \,{\mathcal D}_{ij}(f)\, T^{\lambda_i}\,\, +\,\,
R(p,T)(f)\,,
\end{equation}
where the real coefficients $c_{ij}(p,T)$ and the distributional remainder
$R(p,T)$ have, for almost all $p\in M$, a sub-polynomial behaviour, in the
sense that
\begin{equation}
\limsup_{T\to +\infty} \frac{ \log \sum_{j=1}^{m_i}|c_{ij}(p,T)|^2}{\log T}\,
=\,\limsup_{T\to +\infty} \frac{\log \|R(p,T)\|}{\log T}\,=\,0\,.
\end{equation}
The notion of a deviation spectrum first arose in the work of A. Zorich
and in his joint work with M. Kontsevich on non-exact Hamiltonian flows with
isolated saddle-like singularities on compact higher genus surfaces. Zorich
discovered in numerical experiments on interval exchange transformations an
unexpected new phenomenon \cite{Zone}. He found that, although a generic flow
on a surface of genus $g\ge 2$ is
uniquely ergodic (H. Masur, W. Veech), for large times the homology classes
of return orbits exhibit unbounded polynomial deviations with exponents
$\lambda_1>\lambda_2>...>\lambda_g>0$ from the line spanned in the homology
group by the {\it Schwartzmann's asymptotic cycle}. In his later work \cite
{Ztwo}, \cite{Zthree} and in joint work with M. Kontsevich \cite{K}, Zorich
was able to explain this phenomenon in terms of conjectures on the Lyapunov
exponents of the Teichm\"uller flow on moduli spaces of holomorphic
differentials on Riemann surfaces. Kontsevich and Zorich also conjectured
that Zorich's phenomenon is not merely topological, but it extends to ergodic
integrals of smooth functions. ``There is, presumably, an equivalent way of
describing the numbers $\lambda_i$. Namely, let $f$ be a smooth function ...
Then for a generic trajectory $p(t)$, we expect that the number $\int_0^T
f\bigl(p(t)\bigr) dt$ for large $T$ with high probability has size
$T^{\lambda_i+o(T)}$ for some $i\in \{1,...,g\}$. The exponent $\lambda_1$
appears for all functions with non-zero average value. The next exponent,
$\lambda_2$, should work for functions in a codimension $1$ subspace of
$C^{\infty}(S)$, etc.'' \cite{K}
Around the same time, we proved that for a generic non-exact Hamiltonian
flow $\Phi^X$ on a higher genus surface not all smooth zero average
functions are smooth coboundaries \cite{Fone}. In fact, we found that, in
contrast with the hyperbolic case and the elliptic case of toral flows,
there are $X$-invariant distributional obstructions, which are not signed
measures, to the existence of smooth solutions of the {\it cohomological
equation }$Xu=f$. This result suggested that Zorich's phenomenon should
be related to the presence of invariant distributions other than the (unique)
invariant probability measure. In fact, in \cite{Ftwo} we were able prove
the Kontsevich-Zorich conjectures that the deviation exponents are non-zero
and that generic non-exact Hamiltonian flows on higher genus surfaces have
a deviation spectrum. Recently, in collaboration with L. Flaminio, we have
proved that other classical parabolic examples, such as horocycle flows on
compact surfaces of constant negative curvature \cite{FFone} and generic
nilpotent flows on compact $3$-dimensional nilmanifolds \cite{FFtwo}, do
have a deviation spectrum, but of countable multiplicity, in contrast with
the case of flows on surfaces which have spectrum of finite multiplicity
equal to the genus.
We will outline below a general framework, derived mostly from \cite{Ftwo}
and successfully carried out in \cite{FFone}, \cite{FFtwo}, for proving that
a flow on a {\it pseudo-homogeneous space }has a deviation spectrum. Our
framework is based on the construction of an appropriate {\it renormalization
dynamics }on a moduli space of pseudo-homogeneous structures, which generalizes
the Teichm\"uller flow. A {\it renormalizable flow }for which the space of
smooth functions is {\it stable} (in the sense of A. Katok), has a deviation
spectrum determined by the Lyapunov exponents of a {\it renormalization
cocycle }over a bundle of {\it basic currents}. Pseudo-homogeneous spaces
are a generalization of homogeneous spaces. The motivating non-homogeneous
example is given by any punctured Riemann surface carrying a holomorphic
differential vanishing only at the punctures. It turns out that renormalizable
flows are necessarily parabolic. In fact, the class of renormalizable flows
encompasses all parabolic flows which are reasonably well-understood, while
not much is known for most non-renormalizable parabolic flows, such as generic
geodesic flows on flat surfaces with conical singularities. Our approach
unifies and generalizes several classical quantitative equidistribution
results such as the Zagier-Sarnak results for periodic horocycles on
non-compact hyperbolic surfaces of finite volume \cite{FFone} or number
theoretical results on the asymptotic behaviour of theta sums \cite{FFtwo}.
\section{Renormalizable flows} \label{section 2}
\setzero\vskip-5mm \hspace{5mm }
Let $\mathfrak g$ be a finite dimensional real Lie algebra. A $\mathfrak
g$-{\it structure }on a manifold $M$ is defined to be a homomorphism
$\tau$ from $\mathfrak g$ into the Lie algebra ${\mathcal V}(M)$ of all
smooth vector fields on $M$. This notion is well-known in the theory of
transformation groups (originated in the work of S. Lie) under the name of
`infinitesimal $G$-transformation group' (for a Lie group $G$ with $\mathfrak
g$ as Lie algebra). The second fundamental theorem of Lie states that any
infinitesimal $G$-transformation group $\tau$ on $M$ can be `integrated'
to yield an essentially unique local $G$-transformation group.
A $\mathfrak g$-structure $\tau$ will be called {\it faithful }if $\tau$
induces a linear isomorphism from ${\mathfrak g}$ onto $T_x M$, for all $x\in
M$. Let $\tau$ be a $\mathfrak g$-structure. For each element $X\in {\mathfrak
g}$, the vector field $X_{\tau}:=\tau(X)$ generates a (partially defined) flow
$\Phi^X_{\tau}$ on $M$. Let $E_t(X_{\tau})\subset M$ be the closure of the
complement of the domain of definition of the map $\Phi^X_{\tau}(t,\cdot)$
at time $t\in {\mathbb R}$. A faithful $\mathfrak g$-structure will be called
{\it pseudo-homogeneous }if for every $X\in \mathfrak g$ there exists $t>0$
such that $E_t(X_{\tau}) \cup E_{-t}(X_{\tau})$ has zero (Lebesgue) measure.
A manifold $M$ endowed with a pseudo-homogeneous $\mathfrak g$-structure will
be called a {\it pseudo-homogeneous }$\mathfrak g$-{\it space}. All
homogeneous spaces are pseudo-homogeneous.
Let ${\mathcal T}_{\mathfrak g}(M)$ be the space of all pseudo-homogeneous
$\mathfrak g$-structures on $M$. The automorphism group $\text{Aut}(\mathfrak
g)$ acts on ${\mathcal T}_{\mathfrak g}(M)$ by composition on the right. The
group $\text{Diff}(M)$ acts on ${\mathcal T}_{\mathfrak g}(M)$ by composition
on the left. The spaces
\begin{equation}
\text{T}_{\mathfrak g}(M):={\mathcal T}_{\mathfrak g}(M)/\text{Diff}_0(M)
\,\,,\,\,\,\,
{\mathcal M}_{\mathfrak g}(M):=\text{T}_{\mathfrak g}(M)/\Gamma(M) \,,
\end{equation}
where $\Gamma(M):=\text{Diff}^+(M)/\text{Diff}_0(M)$ is the {\it mapping
class group}, will be called respectively the {\it Teichm\"uller space }and
the {\it moduli space }of pseudo-homogeneous $\mathfrak g$-structures on $M$.
The group $\text{Aut}(\mathfrak g)$ acts on the Teichm\"uller space $\text{T}
_{\mathfrak g}(M)$ and on the moduli space ${\mathcal M}_{\mathfrak g}(M)$,
since in both cases the action of $\text{Aut}(\mathfrak g)$ on ${\mathcal
T}_{\mathfrak g}(M)$ passes to the quotient.
Let $\text{Aut}^{(1)}(\mathfrak g)$ be the subgroup of automorphisms with
determinant one. An element $X\in \mathfrak g$ will be called {\it a priori
renormalizable }if there exists a partially hyperbolic one-parameter subgroup
$\{G^X_t\}\subset \text{Aut}^{(1)}(\mathfrak g)$, $t\in {\mathbb R}$ ($t\in
{\mathbb Z}$), in general non-unique, with a single (simple) Lyapunov exponent
$\mu_X>0$ such that
\begin{equation}
\label{eq:RN}
G^X_t(X)=e^{t\mu_X}\,X \,\,.
\end{equation}
It follows from the definition that the subset of a priori renormalizable
elements of a Lie algebra $\mathfrak g$ is saturated with respect to the
action of $\text{Aut}(\mathfrak g)$. The subgroup $\{G^X_t\}$ acts on the
Teichm\"uller space and on the moduli space of pseudo-homogeneous $\mathfrak
g$-structures as a `renormalization dynamics' for the family of flows
$\Phi^X_{\tau}$ generated by the vector fields $\{X_{\tau}\,|\,\tau\in
{\mathcal T}_{\mathfrak g}(M)\}$ on $M$. It will be called a {\it generalized
Teichm\"uller flow (map)}. A flow $\Phi^X_{\tau}$ will be called {\it
renormalizable }if $\tau\in {\mathcal M}_{\mathfrak g}(M)$ is a recurrent
point for some generalized Teichm\"uller flow (map) $G_t^X$. If $\mu$ is
a probability $G^X_t$-invariant measure on the moduli space, then by
Poincar\'e recurrence the flow $\Phi^X_{\tau}$ is renormalizable for
$\mu$-almost all $\tau\in {\mathcal M}_{\mathfrak g}(M)$.
Let $R$ be an inner product on $\mathfrak g$. Every faithful $\mathfrak g$
structure $\tau$ induces a Riemannian metric $R_{\tau}$ of constant curvature
on $M$. Let $\omega_{\tau}$ be the volume form of $R_{\tau}$. The total volume
function $A:{\mathcal T}_{\mathfrak g}(M)\to {\mathbb R}^+ \cup \{+\infty\}$
is $\text{Diff}^+(M)$-invariant and $\text{Aut}^{(1)}(\mathfrak g)$-invariant.
Hence $A$ is well-defined as an $\text{Aut}^{(1)}(\mathfrak g)$-invariant
function on the Teichm\"uller space and on the moduli space. It follows that
the subspace of finite-volume $\mathfrak g$-structures has an $\text{Aut}^{(1)}
(\mathfrak g)$-invariant stratification by the level hypersurfaces of the total
volume function. Since different hypersurfaces are isomorphic up to a dilation,
when studying finite-volume spaces it is sufficient to consider the
hypersurface of volume-one $\mathfrak g$-structures:
\begin{equation}
\text{T}_{\mathfrak g}^{(1)}(M):=\text{T}_{\mathfrak g}(M)\cap A^{-1}(1)
\,\,,\,\,\,\,
{\mathcal M}_{\mathfrak g}^{(1)}(M):= {\mathcal M}_{\mathfrak g}(M)
\cap A^{-1}(1) \,.
\end{equation}
Let $\tau$ be a faithful $\mathfrak g$-structure and let $X\in\mathfrak g$. If
the linear map $ad_X$ on $\mathfrak g$ has zero trace, the flow $\Phi^X_{\tau}$
preserves the volume form $\omega_{\tau}$ and $X_{\tau}$ defines a symmetric
operator on $L^2(M,\omega_{\tau})$ with domain $C_0^{\infty}(M)$. If $\tau$
is pseudo-homogeneous, by E. Nelson's criterion \cite{N}, $X_{\tau}$ is
essentially skew-adjoint. It turns out that any a priori renormalizable
element $X\in \mathfrak g$ is {\it nilpotent}, in the sense that all
eigenvalues of the linear map $ad_X$ are equal to zero, hence the flow
$\Phi^X_{\tau}$ is volume preserving and parabolic. In all the examples
we have considered, the Lie algebra $\mathfrak g$ is {\it traceless}, in
the sense that for every element $X\in \mathfrak g$, the linear map $\text
{ad}_X$ has vanishing trace. In this case, any pseudo-homogeneous $\mathfrak
g$-structure induces a representation of the Lie algebra $\mathfrak g$ by
essentially skew-adjoint operators on the Hilbert space $L^2(M,\omega_{\tau})$
with common invariant domain $C_0^{\infty}(M)$.
\section{Examples} \label{section 3}
\setzero\vskip-5mm \hspace{5mm }
Homogeneous spaces provide a wide class of examples. Let $G$ be a finite
dimensional (non-compact) Lie group with Lie algebra $\mathfrak g$ and
let $M=G/\Gamma$ be a (compact) homogeneous space. The Teichm\"uller
space $\text{T}_G(M)\subset \text{T}_{\mathfrak g}(M)$ and the moduli space
${\mathcal M}_G(M)\subset {\mathcal M}_{\mathfrak g}(M)$ of all homogeneous
$G$-space structures on $M$ are respectively isomorphic to the Lie group
$\text{Aut}(G)$ and to the homogeneous space $\text{Aut}(G)/\text{Aut}(G,
\Gamma)$, where $\text{Aut}(G,\Gamma)<\text{Aut}(G)$ is the subgroup of
automorphisms which stabilize the lattice $\Gamma$. The Teichm\"uller
and moduli spaces $\text{T}^{(1)}_G(M)$ and ${\mathcal M}^{(1)}_G(M)$ of
homogeneous volume-one $G$-space structures are respectively isomorphic
to the subgroup $\text{Aut}^{(1)}(G)$ of orientation preserving, volume
preserving automorphisms and to the homogeneous space $\text{Aut}^{(1)}(G)
/\text{Aut}^{(1)}(G,\Gamma)$.
In the {\it Abelian }case ${\mathfrak g}={\mathbb R}^n$, any $X\in {\mathbb
R}^n$ is a priori renormalizable. In fact, the group $\text{Aut}^{(1)}
({\mathbb R}^n)=\text{SL}(n,{\mathbb R})$ acts transitively on ${\mathbb R}^n$
and $X_1=(1,0,...,0)$ is renormalized by the one-parameter group $G_t:=\text
{diag}(e^t,e^{-t/n},...,e^{-t/n})\subset \text{SL}(n,{\mathbb R})$.
Finite volume Abelian homogeneous spaces are diffeomorphic to $n$-dimensional
tori ${\mathbb T}^n$. The generalized Teichm\"uller flow $G_t$ on the moduli
space of all volume-one Abelian homogeneous structures on ${\mathbb T}^n$ is
a volume preserving Anosov flow on the finite-volume non-compact manifold
$\text{SL}(n,\mathbb R)/\text{SL}(n,\mathbb Z)$. Hence, in this case, almost
all homogeneous flows are renormalizable, by Poincar\'e recurrence theorem.
The dynamics of the flow $G_t$ has been investigated in depth by D. Kleinbock
and G. Margulis in connection with the theory of Diophantine approximations.
In the {\it semi-simple }case, let ${\mathfrak g}={\mathfrak s}{\mathfrak l}
(2,\mathbb R)$ be the unique $3$-dimensional simple Lie algebra. There is a
basis $\{H,H^{\perp},X\}$ with commutation relations $[X,H]=H$, $[X,H^{\perp}]
=-H^{\perp}$ and $[H,H^{\perp}]=2X$. The elements $H$, $H^{\perp}$ are
renormalized by the one-parameter group $G_t:=\text{diag}(e^t,e^{-t},1)\subset
\text{Aut}^{(1)}(\mathfrak g)$, while $X$ is not a priori renormalizable. The
unit tangent bundle of any hyperbolic surface $S$ can be identified to a
homogeneous $\mathfrak g$-space $M:=\text {PSL}(2,{\mathbb R})/\Gamma$. The
vector fields $H$, $H^{\perp}\in {\mathfrak g}$ generate the horocycle flows
and the vector field $X$ generates the geodesic flow on $S$. Since $G_t$ is a
group of {\it inner }automorphisms, it is in fact generated by the geodesic
vector field $X$, every point of the moduli space is fixed under $G_t$. Hence
horocycle flows are renormalizable on every homogeneous space $\text{PSL}
(2,{\mathbb R})/\Gamma$.
In the {\it nilpotent}, non-Abelian case, let $\mathfrak n$ be the Heisenberg
Lie algebra, spanned by elements $\{X,X^{\perp},Z\}$ such that $[X,X^{\perp}]
=Z$ and $Z$ is a generator of the one-dimensional center $Z_{\mathfrak n}$.
The element $X$ is renormalized by the one-parameter subgroup $G_t:=\text{diag}
(e^t,e^{-t},1)\subset \text{Aut}^{(1)}({\mathfrak n})$. Since the group
$\text{Aut}({\mathfrak n})$ acts transitively on ${\mathfrak n}\setminus
Z_{\mathfrak n}$, every $Y\in{\mathfrak n}\setminus Z_{\mathfrak n}$ is a
priori renormalizable, while the elements of the center are not. A compact
nilmanifold modeled over the Heisenberg group $N$ is a homogeneous space
$M=N/\Gamma$, where $\Gamma$ is a co-compact lattice. These spaces are
topologically circle bundles over ${\mathbb T}^2$ classified by their Euler
characteristic. The moduli space ${\mathcal M}^{(1)}_N(M)$ of volume-one
homogeneous $\mathfrak n$-structures on $M$ is a $5$-dimensional finite-volume
non-compact orbifold which fibers over the modular surface $\text{SL}(2,
{\mathbb R})/\text{SL}(2,{\mathbb Z})$ with fiber ${\mathbb T}^2$. The
generalized Teichm\"uller flow is an Anosov flow on ${\mathcal M}^{(1)}_N(M)$
\cite{FFtwo}.
The motivation for our definition of a pseudo-homogeneous space comes
from the theory of Riemann surfaces of higher genus. Any holomorphic
(Abelian) differential $h$ on a Riemann surface $S$ of genus $g\ge 2$,
vanishing at $Z_h\subset S$, induces a (non-unique) pseudo-homogeneous
${\mathbb R}^2$-structure on the open manifold $M_h:=S\setminus Z_h$
In fact, the frame $\{X,X^{\perp}\}$ of $TS|M_h$ uniquely determined by
the conditions
\begin{equation}
\frac{\sqrt{-1}}{2}\, \imath_X\,(h\wedge {\bar h})\,=\, \Im(h) \,\,\,\,,
\,\,\,\,
\frac{\sqrt{-1}}{2}\, \imath_{X^{\perp}} \,(h\wedge {\bar h})\,=\,-\Re(h)
\end{equation}
satisfies the Abelian commutation relation $[X,X^{\perp}]=0$ and the
homomorphism $\tau_h:{\mathbb R}^2\to {\mathcal V}(M_h)$ such that $\tau_h
(1,0)=X$, $\tau_h(0,1)=X^{\perp}$ is a pseudo-homogeneous ${\mathbb
R}^2$-structure on $M_h$. Let $Z\subset S$ be a given subset of cardinality
$\sigma\in{\mathbb N}$ and let $\kappa=(k_1,...,k_{\sigma})\in ({\mathbb
Z}^+)^{\sigma}$ with $\sum k_i=2g-2$. Let ${\mathcal H}_{\kappa}(S,Z)$ be
the space of Abelian differentials $h$ with $Z_h=Z$ and zeroes of
multiplicities $(k_1,...,k_{\sigma})$. The projection of the set
$\{\tau_h\in {\mathcal T}_{{\mathbb R}^2}(M)\,|\,h\in {\mathcal H}_{\kappa}
(S,Z)\}$ into the moduli space ${\mathcal M}_{{\mathbb R}^2}(M)$ of
pseudo-homogeneous ${\mathbb R}^2$-structures on $M:=S\setminus Z$ is
isomorphic to a stratum ${\mathcal H}({\kappa})$ of the moduli space of
Abelian differentials on $S$. The flow induced on ${\mathcal H}({\kappa})$
by the one-parameter group of automorphism $G_t=\text{diag}(e^t,e^{-t})
\subset \text{SL}(2,\mathbb R)$ coincides with the Teichm\"uller flow on
the stratum ${\mathcal H}({\kappa})$.
\section{Cohomological equations} \label{section 4}
\setzero\vskip-5mm \hspace{5mm }
Let $(\mathfrak g, R)$ be a finite dimensional Lie algebra endowed with an
inner product. Any pseudo-homogeneous $\mathfrak g$-structure $\tau$ on
a manifold $M$ induces a {\it Sobolev filtration }$\{\text{W}^s_{\tau}
(M)\}_{s\ge 0}$ on the space $W^0_{\tau}(M):=L^2(M,\omega_{\tau})$ of
square-integrable functions. Let $\triangle_{\tau}$ be the non-negative
{\it Laplace-Beltrami
operator }of the Riemannian metric $R_{\tau}$ on $M$. The Laplacian is densely
defined and symmetric on the Hilbert space $W^0_{\tau}(M)$ with domain
$C_0^{\infty}(M)$, but it is not in general essentially self-adjoint. In fact,
if $\mathfrak g$ is traceless, by a theorem of E. Nelson \cite{N}, $\triangle
_{\tau}$ is essentially self-adjoint if and only if the representation $\tau$
of the Lie algebra $\mathfrak g$ on $W^0_{\tau}(M)$ by essentially skew-adjoint
operators induces a unitary representation of a Lie group. Let then $\bar
\triangle_{\tau}$ be the {\it Friederichs extension }of $\triangle_{\tau}$.
The Sobolev space $W^s_{\tau}(M)$, $s>0$, is defined as the maximal domain
of the operator $(\text{I}+{\bar\triangle}_{\tau})^{s/2}$ endowed with the
norm
\begin{equation}
\|f\|_{s,\tau}:= \|(\text{I}+\bar{\triangle})^{s/2}f\|_{0,\tau}\,.
\end{equation}
The Sobolev spaces $W^{-s}_{\tau}(M)$ are defined as the duals of the Hilbert
spaces $W^s_{\tau}(M)$, for all $s > 0$. Let $C_B^0(M)$ be the space of
continous bounded functions on $M$. The pseudo-homogeneous space $(M,\tau)$
will be called of {\it bounded type }if there is a continous (Sobolev) embedding
$W^s_{\tau}(M)\subset C_B^0(M)$ for all $s>\text{dim}(M)/2$. The bounded-type
condition is essentially a geometric property of the pseudo-homogeneous
structure.
Let $X\in {\mathfrak g}$. Following A. Katok, the space $W^s_{\tau}(M)$ is
called $W^t_{\tau}(M)$-{\it stable }with respect to the flow $\Phi^X_{\tau}$
if the subspace
\begin{equation}
R^{s,t}(X_{\tau}):= \{ f\in W^s_{\tau}(M)\,|\, f=X_{\tau}u\,\,,\,\,\,\,
u\in W^t_{\tau}(M)\}
\end{equation}
is closed in $W^s_{\tau}(M)$. The flow $\Phi^X_{\tau}$ will be called {\it
tame }(of degree $\ell>0$) if $W^s_{\tau}(M)$ is $W^{s-\ell}_{\tau}(M)$-stable
with respect to $\Phi^X_{\tau}$ for all $s>\ell$. In all the examples of \S 3,
generic renormalizable flows are tame. In particular, it is well known that
generic toral flows are tame, horocycle flows and generic nilpotent flows on
$3$-dimensional compact nilmanifolds were proved tame of any degree $\ell>1$
in \cite{FFone}, \cite{FFtwo}, generic non-exact Hamiltonian flows on higher
genus surfaces were proved tame in \cite{Fone}. These results are based on the
appropriate harmonic analysis: in the homogeneous cases, the theory of unitary
representations for the Lie group $\text{SL}(2,{\mathbb R})$ \cite{FFone} and
the Heisenberg group \cite{FFtwo}; in the more difficult non-homogeneous case
of higher genus surfaces, the theory of boundary behaviour of holomorphic
functions on the unit disk plays a crucial role \cite{Ftwo}.
If the Sobolev space $W^s_{\tau}(M)$ is stable with respect to the flow
$\Phi^X_{\tau}$, the closed range $R^{s,t}(X_{\tau})$ of the operator
$X_{\tau}$ coincides with the distributional kernel ${\mathcal I}^s(X_{\tau})
\subset W^{-s}_{\tau}(M)$ of $X_{\tau}$, which is a space of $X_{\tau}$-{\it
invariant distributions}. Let $X$ be any smooth vector field on a manifold
$M$. A distribution ${\mathcal D}\in{\mathcal D}'(M)$ is called $X$-invariant
if $X{\mathcal D}=0$ in ${\mathcal D}'(M)$. Invariant distributions are in
bijective correspondence with (homogeneous) one-dimensional {\it basic
currents }for the orbit foliation ${\mathcal F}(X)$ of the flow $\Phi^X$. A
one-dimensional basic current $C$ for a foliation $\mathcal F$ on $M$ is a
continous linear functional on the space $\Omega^1_0(M)$ of smooth $1$-forms
with compact support such that, for all vector fields $Y$ tangent to
$\mathcal F$,
\begin{equation}
\label{eq:BC}
\imath_YC\,=\,{\mathcal L}_YC\,=\,0\,\, (\iff \,\, \imath_YC\,=\,dC\,=\,0)\,.
\end{equation}
It follows from the definitions that the one-dimensional current $C:=\imath_X
{\mathcal D}$ is basic for ${\mathcal F}(X)$ if and only if the distribution
$\mathcal D$ is $X$-invariant. Let ${\mathcal I}(X)$ be the space of all
$X$-invariant distributions and ${\mathcal B}(X)$ be the space of all
one-dimensional basic currents for the orbit foliation ${\mathcal F}(X)$. The
linear map $\imath_X:{\mathcal I}(X)\to {\mathcal B}(X)$ is bijective.
Let $(M,\tau)$ be a pseudo-homogeneous space. There is a
well-defined Hodge (star) operator and a space $\text{C}^0_{\tau}(M)$ of
square-integrable $1$-forms on $M$ associated with the metric $R_{\tau}$.
Since the Laplace operator $\triangle_{\tau}$ extends to $\text{C}^0_{\tau}
(M)$ with domain $\Omega^1_0(M)$, it is possible to define, as in the case
of functions, a Sobolev filtration $\{\text{C}^s_{\tau}(M)\}_{s\ge 0}$, on
the space $\text{C}^0_{\tau}(M)$. The Sobolev spaces $\text{C}^{-s}_{\tau}
(M)$ are defined as the duals of the Sobolev spaces $\text{C}^s(M)$, for
all $s>0$. Let ${\mathcal B}^s(X_{\tau}):={\mathcal B}(X_{\tau})\cap
\text{C}^{-s}_{\tau}(M)$ be the subspaces of basic currents of Sobolev
order $\le s$ for the orbit foliation ${\mathcal F}(X_{\tau})$. The
space ${\mathcal B}^s(X_{\tau})$ is the image of ${\mathcal I}^s(X_{\tau})
:={\mathcal I}(X_{\tau})\cap W^{-s}_{\tau}(M)$ under the bijective map
$\imath_X:{\mathcal I}(X)\to {\mathcal B}(X)$. In the case of minimal
toral flows the space ${\mathcal B}^s(X_{\tau})$ is one-dimensional for
all $s\ge 0$ (as all invariant distributions are scalar multiples of the
unique invariant probability measure). In the parabolic examples we
have studied, ${\mathcal B}^s(X_{\tau})$ has countable dimension,
as soon as $s>1/2$, for horocycle flows or generic nilpotent flows, while for
generic non-exact Hamiltonian flows on higher genus surfaces the dimension is
finite for all $s>0$ and grows linearly with respect to $s>0$. This finiteness
property seems to be an exceptional low dimensional feature.
\section{The renormalization cocycle} \label{section 5}
\setzero\vskip-5mm \hspace{5mm }
The Sobolev spaces $\text{C}^s_{\tau}(M)$ of one-dimensional currents form a
smooth infinite dimensional vector bundle over ${\mathcal T}_{\mathfrak g}(M)$.
Such bundles can be endowed with a flat connection with parallel transport
given locally by the identity maps $\text{C}^s_{\tau}(M)\to\text{C}^s_{\tau'}
(M)$, for any $\tau\approx\tau'\in {\mathcal T}_{\mathfrak g}(M)$. Since the
diffeomorphism group $\text{Diff}(M)$ acts on $\text{C}^s_{\tau}(M)$ by
push-forward, we can define (orbifold) vector bundles $\text{C}^s_{\mathfrak g}
(M)$ over the Teichm\"uller space $\text{T}_{\mathfrak g}(M)$ or the moduli
space ${\mathcal M}_{\mathfrak g}(M)$ of pseudo-homogeneous structures on $M$.
If $X\in \mathfrak g$ is a priori renormalizable, a generalized Teichm\"uller
flow (map) $G^X_t$ can be lifted by parallel transport to a `renormalization
cocycle' $R^X_t$ on the bundles of currents $\text{C}^s_{\mathfrak g}(M)$ over
the Teichm\"uller space or the moduli space. It follows from the definitions
that the sub-bundles ${\mathcal B}^s_{\mathfrak g}(X)\subset \text{C}^s_
{\mathfrak g}(M)$ with fibers the subspaces of basic currents ${\mathcal B}^s
(X_{\tau})\subset \text{C}^s_{\tau}(M)$ are $R^X_t$-invariant. It can be proved
that, for any $G_t$-ergodic probability measure $\mu$ on the moduli space, if
the flows $\Phi^X_{\tau}$ are tame of degree $\ell>0$ for $\mu$-almost all
$\tau\in{\mathcal M}_{\mathfrak g}(M)$, then the sub-bundles ${\mathcal B}^s
_{\mathfrak g}(X)$ are $\mu$-almost everywhere defined with closed (Hilbert)
fibers of constant rank, for all $s>\ell$.
In the examples considered, with the exception of flows on higher genus
surfaces, the Hilbert bundles of basic currents ${\mathcal B}^s_{\mathfrak g}
(X)$ are infinite dimensional, and to the author's best knowledge, available
Oseledec-type theorems for Hilbert bundles do not apply to the renormalization
cocycle. However, the cocycle has a well defined Lyapunov spectrum and an
Oseledec decomposition. We are therefore led to formulate the following
hypothesis:
\noindent $H_1(s)$. The renormalization cocyle $R^X_t$ on the bundle
${\mathcal B}^s_{\mathfrak g}(X)$ over the dynamical system $(G_t^X,\mu)$
has a Lyapunov spectrum $\{\nu_1> ...> \nu_k> ...>0>...\}$ and an Oseledec's
decomposition
\begin{equation}
\label{eq:OD}
{\mathcal B}^s_{\mathfrak g}(X)=E^s_{\mathfrak g}(\nu_1)\oplus ... \oplus
E^s_{\mathfrak g}(\nu_k)\oplus ... \oplus N^s_{\mathfrak g}\,\,,
\end{equation}
in which the components $E^s_{\mathfrak g}(\nu_k)$ correspond to the Lyapunov
exponents $\nu_k>0$, while the component $N^s_{\mathfrak g}$ has a non-positive
top Lyapunov exponent. Our result on the existence of a deviation spectrum
requires an additional technical hypothesis, verified in our examples.
\noindent $H_2(s)$. Let $\gamma_{\tau}^1(p)$ be the one-dimensional current
defined by the time $T=1$ orbit-segment of the flow $\Phi^X_{\tau}$ with
initial point $p\in M$. $(a)$ The essential supremum of the norm $\|\gamma
_{\tau}^1(p)\|_{\tau,s}$ over $p\in M$ is locally bounded for $\tau\in\text
{supp}(\mu)\subset {\mathcal M}_{\mathfrak g}(M)$; $(b)$ The orthogonal
projections of $\gamma_{\tau}^1(p)$ on all subspaces $E^s_{\tau}(\nu_k)
\subset \text{C}^{-s}_{\mathfrak g}(M)$ are non-zero for $\mu$-almost all
$\tau \in {\mathcal M}_{\mathfrak g}(M)$ and almost all $p\in M$.
{ Let }$X\in {\mathfrak g}$ { be a priori renormalizable and let
}$\mu$ { be a }$G_t^X$-{ invariant Borel probability measure on
}${\mathcal M} _{\mathfrak g}(M)$, { supported on a stratum of
bounded-type }$\mathfrak g$-{ structures. If the flow
}$\Phi_{\tau}^X$ { is tame of degree }$\ell
>0$ { and the hypoteses }$H_1(s)$, $H_2(s)$ { are verified for }$s>\ell+
\text{dim}(M)/2$, { for }$\mu$-{ almost }$\tau\in {\mathcal
M}_{\mathfrak g}$, { the flow }$\Phi_{\tau}^X$ { has a deviation
spectrum with deviation exponents}
\begin{equation}
\nu_1/\mu_X \,>\,...\,>\,\nu_k/\mu_X \,>\,...\,>\,0
\end{equation}
{ and multiplicities given by the decomposition }(\ref{eq:OD}) {
of the renormalization coycle.}
In the homogeneous examples, the Lyapunov spectrum of the renormalization
cocycle is computed explicitly in every irreducible unitary representation
of the structural Lie group. In the horocycle case, the existence of an
Oseledec's decomposition (\ref{eq:OD}) is equivalent to the statement that
the space of horocycle-invariant distributions is spanned by (generalized)
eigenvectors of the geodesic flow, well-known in the representation theory
of semi-simple Lie groups as {\it conical distributions }\cite{H}. In the
non-homogeneous case of higher genus surfaces, the Oseledec's theorem applies
since the bundles ${\mathcal B}^s_{\mathfrak g}(X)$ are finite dimensional.
We have found in all examples a surprising heuristic relation between the
Lyapunov exponents of the renormalization cocycle and the Sobolev regularity
of basic currents (or equivalently of invariant distributions): the subspaces
$E^s_{\tau}(\nu_k)$ are generated by basic currents of {\it Sobolev order }$1
-\nu_k/\mu_X\ge 0$. The Sobolev order of a one-dimensional current $C$ is
defined as the infimum of all $s>0$ such that $C\in \text{C}^{-s}_{\tau}(M)$.
In the special case of non-exact Hamiltonian flow on higher genus surfaces
the Lyapunov exponents of the renormalization cocycle are related to those of
the Teichm\"uller flow. In fact, let $S$ be compact orientable surface of
genus $g\ge 2$ and let ${\mathcal H}({\kappa})$ be a stratum of Abelian
differentials vanishing at $Z\subset S$. Let ${\mathcal B}_{\kappa}(X)
\subset {\mathcal B}_{{\mathbb R}^2}(X)$ be the measurable bundle of basic
currents over ${\mathcal H}({\kappa})\subset {\mathcal M}_{{\mathbb R}^2}
(S\setminus Z)$ and let $H^1_{\kappa}(S\setminus Z,\mathbb R)$ be the bundle
over ${\mathcal H}({\kappa})$ with fibers isomorphic to the real cohomology
$H^1(S\setminus Z,\mathbb R)$. Since basic currents are closed, there exists
a {\it cohomology map }$j_{\kappa}:{\mathcal B}_{\kappa}(X) \to H^1_{\kappa}
(S\setminus Z,{\mathbb R})$ such that, as proved in \cite{Ftwo}, the
restrictions $j_{\kappa}|{\cal B}^s_{\kappa}(X)$ are surjective for all
$s>>1$ and, for all $s\ge 1$, there are exact sequences
\begin{equation}
0\rightarrow \mathbb R \rightarrow {\mathcal B}_{\kappa}^{s-1}(X)\,\,
^{\underrightarrow{\,\,\,\, \delta_{\kappa}\,\,\,\,}}\,\, {\mathcal B}
_{\kappa}^s(X)\,\, ^{\underrightarrow{\,\,\,\, j_{\kappa} \,\,\,\,}}\,\,
H^1_{\kappa}(S\setminus Z,{\mathbb R})\,\,.
\end{equation}
The renormalization cocycle $R^X_t$ on ${\mathcal B}_{\kappa}^s(X)$ projects
for all $s>>1$ onto a cocycle on the cohomology bundle $H^1_{\kappa}(S\setminus
Z,{\mathbb R})$, introduced by M. Kontsevich and A. Zorich in order to explain
the {\it homological }asymptotic behaviour of orbits of the flow $\Phi_{\tau}^X$
for a generic $\tau\in {\cal H}(\kappa)\subset {\cal M}_{{\mathbb R}^2}(S
\setminus Z)$ \cite{K}. The Lyapunov exponents of the Kontsevich-Zorich cocycle
on $H^1_{\kappa}(S\setminus Z,{\mathbb R})$,
\begin{equation}
\label{eq:RKZE}
\lambda_1=1>\lambda_2\ge \dots \ge\lambda_g\ge \overbrace{0=\cdots=
0}^{\#Z-1} \ge -\lambda_g\ge \dots \ge -\lambda_2>-\lambda_1=-1 \,,
\end{equation}
are related the Lyapunov exponents of the Teich\"uller flow on
${\mathcal H}({\kappa})$ \cite{K}, \cite{Ftwo}. Since the bundle map
$\delta_{\kappa}$ shifts Lyapunov exponents by $-1$ and, as conjectured
in \cite{K} and proved in \cite{Ftwo}, the Kontsevich-Zorich exponents
$\lambda_1=1>\lambda_2\ge \dots \ge\lambda_g$ are non-zero, the strictly
positive exponents of the renormalization cocycle coincide with the
Kontsevich-Zorich exponents. This reduction explains why in the case of
non-exact Hamiltonian flows on surfaces the Lyapunov exponents of the
Teichm\"uller flow are related to the deviation exponents for the ergodic
averages of smooth functions.
\label{lastpage}
\end{document} | math |
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<p><strong>Transmutação de Nível 4</strong></p>
<p><strong>Tempo de Execução</strong>: Ação Padrão</p>
<p><strong>Alcance</strong>: 9 metros</p>
<p><strong>Alvo</strong>: até 5 humanoides</p>
<p><strong>Duração</strong>: 10 minutos</p>
<p><strong>Teste de Resistência</strong>: nenhum;</p>
<p>Esta magia dobra a altura de até 5 alvos, o que aumenta seu tamanho em uma categoria e fornece Força +2. O equipamento carregado pelo alvo também é afetado.
Aumentar pessoa anula reduzir pessoa.</p>
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In a bid to take On Samsung’s Rumoured Galaxy A8s. Huawei Was Rumoured to be Working on a Smartphone With A Similar Design. Touted To Be called The Nova 4, the Smartphone is said to come with a display hole for the selfie Camera.
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Do You Like Huawei Nova 4 Launch With In Screen Selfie Cameras Set To December 17 Post? Please LIke comment And Share My Improvement. And Share friend Social Media Just Example Facebook, Twitter And Instagram, Pinterest And Many Other Social Media platform. Thanks For Reading This Post. | english |
Want to add another hdd hard drive or ssd? Use this Lenovo thinkpad x61s hdd caddy to make your laptop have the second hard drive. Simply remove your CD-ROM/DVDRW drive and put the laptop caddy (with the added hard drive) in it's place, then there are two hard drives in laptop. | english |
आपने अकसर लोगों को कहते हुए सुना होगा कि आंवला सेहत के लिए बहुत लाभकारी है, लेकिन क्या आपको पता है कि यह आपकी सेहत से लेकर सौंदर्य तक हर पहलू में आपके काफी काम आ सकता है। आंवले का रस आपके शरीर को ऊर्जा देकर आपको पूरे दिन न सिर्फ तरोताजा रखता है, बल्कि इसमें मौजूद मिनरल्ज व विटामिन सी के कारण आप बहुत सी बीमारियों से भी बचे रहते हैं। जहां तक बात सौंदर्य लाभ की है, तो इसके सेवन से आप न सिर्फ खुद को लंबे समय तक जवां बनाए रख सकते हैं, बल्कि इससे आपके बालों को भी लाभ होता है। तो आइए जानते हैं आंवले के रस के ऐसे ही कुछ फायदों के बारे में।
कंट्रोल करे डायबिटीज-
शायद आपको पता न हो, लेकिन आंवले में गैलिक एसिड, गैलोटेनिन, एलैजिक एसिड और कोरिलैगिन जैसे तत्त्व पाए जाते हैं।
बढ़ाए गुड कोलेस्ट्रॉल-
आमतौर पर लोग कोलेस्ट्रॉल को शरीर के लिए हानिकारक ही मानते हैं, लेकिन शरीर में गुड कोलेस्ट्रॉल का होना भी बेहद आवश्यक है। कुछ शोध बताते हैं कि आंवले का रस न सिर्फ शरीर में गुड कोलेस्ट्रॉल की मात्रा में बढ़ोतरी करता है, बल्कि इसके नियमित सेवन से शरीर में मौजूद खराब कोलेस्ट्रॉल धीरे-धीरे कम होने लगता है। जिसके कारण आप कुछ ही समय में खुद को चुस्त व तंदुरुस्त पाते हैं। दरअसल, आंवले में मौजूद एमिनो एसिड व एंटीऑक्सीडेंट हृदय के कार्य करने की क्षमता को बेहतर बनाते हैं।
खांसी-जुकाम से छुटकारा-
कुछ लोगों को बदलते मौसम में खांसी-जुकाम की समस्या अकसर देखने को मिलती है। ऐसे लोगों के लिए आंवले का रस काफी मददगार साबित होता है। इसके लिए आप दो चम्मच आंवले के रस में दो चम्मच शहद मिलाकर रोजाना पिएं। आपको जल्द ही खांसी-जुकाम से राहत मिलेगी। वहीं अगर आपको मुंह में छालों की समस्या है, तो आप कुछ चम्मच आंवले का रस पानी में मिलाएं व उस पानी से कुल्ला करें। क्योंकि आंवले में विटामिन सी काफी प्रचुर मात्रा में पाया जाता है, जिससे यह आपकी इम्युनिटी, मैटाबॉलिज्म को बढ़ाता है व बैक्टीरियल इन्फेक्शन को कम करता है। इसलिए इसके नियमित सेवन से खांसी-जुकाम आदि होने की संभावना न के बराबर हो जाती है।
करे शरीर की सफाई-
आंवले का रस आपके शरीर में मौजूद सभी विषाक्त पदार्थों को बाहर निकालकर आपके शरीर की आंतरिक रूप से सफाई करता है। | hindi |
बता दें कि फ्लीपकर्ट का बिग बिलियन दए सले २९ सितंबर से शुरू होकर ४ अक्टूबर तक चलेगा. जिसमें फ्लिपकार्ट की ओर से कस्टमर्स को अलग- अलग प्रोडक्ट्स पर भारी डिस्काउंट और एक्सचेंज ऑफर्स दिया जाता है. फ्लिपकार्ट हर साल कुछ दिनों के लिए ही बिग बिलियन डे सेल लगाता है.
२९ सितंबर से ४ अक्टूबर तक जारी रहेगी सेल.
फ्लिपकार्ट स्मार्टफोन्स और टीवी पर दे रहा भारी छूट.
फ्लीपकर्ट बिग बिलियन डेस सले के दौरान कंपनी अपने हद टीवी मॉडलों से लेकर नए ऑफिशियल अंड्रॉयड ४क टीवी मॉडलों तक में भी छूट दी जा रही है. सेल के दौरान थॉमसन के 2४-इंच हद लेड टीवी को ५,९९९ रुपये में, ३२-इंच हद लेड टीवी को ६,९९९ रुपये और कंपनी के लेटेस्ट ऑफिशियल एंड्रॉयड टीवी ४क रेंज के ४3-इंच मॉडल को 2६,९९९ रुपये और ६५-इंच को ५५,९९९ रुपये में दिया जा रहा है.
इन मॉडलों पर मिलेगी छूट-
थॉमसन ब९ प्रो १०२कम (४०-इंच) फुल हद लेड स्मार्ट त्व को ग्राहक १६,९९९ रुपये की जगह १५,4९९ रुपये में मिल रहा है.
थॉमसन उड़९ १०२कम (४०-इंच) अल्ट्रा हद (४क) लेड स्मार्ट त्व को ग्राहक २०,४९९ रुपये की जगह १८,४९९ रुपये में मिल रहा है.
थॉमसन र९ ८०कम (३२-इंच) हद रेडी लेड त्व को ग्राहक ८,९९९ रुपये की जगह ६,९९९ रुपये में मिल रहा है.
थॉमसन ब९ प्रो ८०कम (३२-इंच) हद रेडी लेड स्मार्ट त्व को ग्राहक १०,९९९ रुपये की जगह ९,4९९ रुपये में मिल रहा है.
थॉमसन र९ ६०कम (२४-इंच) हद रेडी लेड त्व को ग्राहक ७,4९९ रुपये की जगह ५,९९९ रुपये में मिल रहा है.
इसी के साथ साथ स्मार्टफोन्स की रेंज पर भी भारी छूट दी जा रही है. बिग बिलियन डेस सले में आपको स्मार्टफोन्स पर मिलेंगे ऐसे ऑफर, जो आपको कहीं और नहीं मिल पाएंगे. यहां तक कि फ्लीपकर्ट के इस ऑफर जैसे डिस्काउंट और एक्सचेंज ऑफर्स किसी ऑफलाइन स्टोर या बाजारों की दुकानों में भी नहीं मिलेंगे.
हालांकि जब हम बात स्मार्टफोन्स की कर रहे हैं तो फ्लीपकर्ट की बिग बिलियन डे सेल में ४०००-७००० रुपए की कीमत में आने वाला रियामे च२ स्मार्टफोन अपने सेगमेंट में एक सुपरहिट डील दे रहा है. | hindi |
import mongoose from 'mongoose'
const Schema = mongoose.Schema;
const ThingSchema = new Schema({
active: {type: Boolean , default: false},
name: {type: String , required: true},
info: {type: String , required: true},
user : { type: Schema.ObjectId, ref: 'User', required: true }
}, {timestamps: true});
ThingSchema.set('toJSON', {
virtuals: false,
transform: (doc, ret, options) => {
delete ret.__v;
}
});
export default mongoose.model('Thing', ThingSchema) | code |
This page has been setup to provide information on how to start, organize, and become an advocacy group and/or a support group.
Where these groups are legitimate registered as business which are Nonprofit businesses/organizations.
even at the local level.
Please be patient as I collect and put up the information it may take some time.
If you have any information about organizing, managing, obtaining funding, or other information for Non profit groups please free to send it to me using the contact page.
and various Charities. Some examples of Not for Profits are publically owned (not private) Universities and Colleges (University of Toronto, McMaster University, York University, Sheridan College, and so on), as well as Hospitals and healthcare centres (Toronto General Hospital, Toronto SickKids Hospital, Credit Valley Hospital, and so on), World Wildlife Federation, MADD - Mothers Against Drunk Drivers, and many more.
Most people that have a cause, such as fighting for victims of workplace accidents. They try to make a difference, but the problem is they are unfamiliar with properly organizing to make a difference.
The first thing when you register as a Not For Profit is that people, the general public see that you are register and this gives your cause more credibility.
The opposition to your cause start to have more respect for your cause as it shows how committed and serious you are.
Once you are registered as a Not for Profit organization, you can legitimately apply for Government and non government grants, which is monies you do not pay back. With the money from the grants you will be able to get the much needed resources to further your cause. Hire staff, create a professional website, hire lawyers to make legal court challenges.
The one problem with registering and creating a Not for Profit Organization is having a physical Mailing address, well as telephone, fax, e-mail, and website. The reason to have all this is that it creates credibility and it makes it affordable for Not for Profits by not having a real physical location.
You can rent a Post office Box at a reasonable rates. This way it separates the people who are setting up and the actual organization. It legitimizes your cause by having a legitimate physical mailing address. You can obtain a post office box at any local post office or UPS store.
There is actually a service that offers a virtual phone number which is forwarded to a voice mail service or another number. There is a monthly fee for this service. That offers extensions automatic answering attendant and more the price ranges from $13 per month to $80- per month depending on the services needed.
Another service available is a virtual fax service where you are provided a fax number and whenever people send you a fax you receive it as an e-mail attachment. You can also send faxes as well by e-mail.
As well as being on social media such as Facebook, it is important to have a stable website with e-mail addresses. The first thing you need to do is pick a name, and a .ca or .com or whatever. for example fightwcb.ca There are many companies that offer website DNS hosting as well as website hosting, with e-mail hosting as well.
You now have a physical mailing address, phone number, fax and website with e-mail. This will now allow you to apply to register and become a legitimate not for profit business to further you cause at a very affordable cost, as opposed to a physical location.
If you would like more information about the requirements for setting up and operating a provincial not-for-profit corporation in Ontario, refer to the not-for-profit incorporator's handbook.
This allows your Not for Profit organization to issue tax receipt for donations made to the organization. It is a major revenue stream for Not for Profits as it encourages businesses and people to donate money as they get a tax receipt which they can claim on their tax returns. This has to be done through the Tax reporting agency for example in Canada it is Revenue Canada and in the United States it is the IRS.
These are various government and non government programs that offer funding to non for profit organizations. The best you can do is register your organization as a Not for profit. It will require a lot of work, but this is where the funding come sin to help you. It also legitimizes the cause. another way to raise funds for your organization is to become a tax recognize charity. Again it is a lot of work, but the rewards pay off in the end.
This is a centralized website for all provincial grants for not for profit organizations and other organizations. To apply for any grants in Ontario this is where to apply now.
The application deadline is February 3, 2017, but I am posting this anyways so people know about the program for future reference.
The Government of Canada Summer Jobs provides funding to not-for-profit organizations, public-sector employers and small businesses with 50 or fewer employees to create summer job opportunities for young people aged 15 to 30 years who are full-time students intending to return to their studies in the next school year.
Not-for-profit employers are eligible to receive funding for up to 100% of the provincial or territorial minimum hourly wage.
This is where you may get grants for publishing books or peridcal newsletters.
The Book and Periodical Council (BPC) is the umbrella organization for Canadian associations that are or whose members are primarily involved with the writing, editing, translating, publishing, producing, distributing, lending, marketing, reading and selling of written words. The BPC acts as a forum for its members to network, identify common goals and coordinate action to benefit the Canadian writing and publishing industry.
This is a charitable organization in Canada helping other charitable organizations. Including a funding grant program for not for profits.
This is a website design to help people raise money for many different reasons including not for profits.
Note they have different websites for different countries.
Google offers grants to advertise on Googles different sites to bring awareness of your cause or campaign. You can get up to $10,000 US in funding from them for advertising.
This website is dedicated for the improvement of women's lives. They offer numerous types of grants to help women.
GlobalGiving connects nonprofits, donors, and companies in nearly every country around the world. We help registered organizations access the funding, tools, training, and support they need to become more effective.
They provide a large list of organizations that can help Not for Profits, including funding programs.
The Community Forward Fund Assistance Corp. makes loans or arranges financing for Canadian nonprofits and charities. We invite you to learn more about us, whether you are a potential impact investor or a charity or nonprofit interested in finding out how your organization could benefit from a loan.
Shows you how to obtain funding for your Not for profit organization.
They also provide an extensive list as well.
Here learn how to find over 1,100 funding programs in Canada for Not for Profits.
The information here is presently only available for Canada, right now, but I will be adding more countries as time progresses. | english |
(including the new Voter ID law).
If you care about voting in Pennsylvania, this is the forum for you!
Join Three Rivers Community Foundation’s Building Social Change Committee and our voting rights allies Black Political Empowerment Project, Disability Rights Network, Just Harvest, ACLU, and more in a discussion about how to make your vote count on election day. We’ll also talk about what you need to do to make sure you meet the requirements of the new Voter ID law when you go to the polls.
The program will start at 3:00 PM, with a panel discussion on various parts of the voting process (preparing to register, registering, day of election issues, etc.). Speakers will include Allen Kukovich, Marybeth Kuznik, and others. At 4:00 we will move into breakout sessions that will take a more in-depth look at the issues the panel addresses.
The event ends at 5:00, but the room will be open for networking and further conversations until 7:00.
Feel free to let us know on Facebook if you’re attending, and invite your friends!
View the flyer for more details! | english |
package ch.zhaw.mapreduce.plugins.socket;
import java.util.concurrent.ConcurrentMap;
/**
* Der SocketResultCollector lebt auf dem Master und wartet auf Worker die ihm Resultate geben. Hier werden alle
* Resultate gesammelt.
*
* @author Reto Hablützel (rethab)
*
*/
public interface SocketResultCollector {
/**
* Speichert das Resultat von einem Agent
*
* @param res
* Resultat vom Agent
*/
void pushResult(SocketAgentResult res);
/**
* Ein ResultCollectorObserver (typischerweise SocketWorker) registriert sich beim SocketResultCollector und wir
* benachrichtigt, sobald das Resultat für einen bestimmten Task angekommen ist.
*
*/
SocketAgentResult registerObserver(String taskUuid, SocketResultObserver observer);
/**
* Liefert eine Referenz auf alle Resultat-Stati. Diese Methde ist für den Cleaner-Task gedacht.
*/
ConcurrentMap<String, ResultState> getResultStates();
}
| code |
Everybody loves to decorate their home in a nice fashion. Having a well decorated home has a number of advantages. The first and foremost among them is it creates a beautiful aesthetically pleasing appearance which soothes the eyes and give a stress free atmosphere where one can sit back and relax. In this article we will discuss about one of the products which play an important purpose in decorating a room.
Whenever someone enters a room the thing they notice is the wall. A nicely coloured wall with vibrant colours can change the very look of the room. People design their rooms according to their tastes and preferences and sometimes according to some theme. Furnishing is also done likewise so that they can also compliment the theme the decorator want to convey. Along with that canvas and wall arts also plays an important role to enhance the effect. A good looking canvas according to the theme can brighten up the wall and the entire room. So it is very important to choose the right kind of wall arts for a room to bring the theme to life.
Now we will site an example to make the idea clear. There are multiple kinds of theme one can choose depending upon innumerable ideas like sea beach theme. As the name of the theme suggest everything in the room should have something which reminds the viewers about some sea beach whenever they views the room. Beach canvas can be a useful tool in enhancing the idea of the theme. There are a number of stores both online and offline from where one can choose the best canvas on beach. There are websites where one can surf through the catalogue containing pictures of various beaches like rocky beach, tropical beach, beach sunrise sunset and many more.
Similarly fishing canvas depicting multiple moments related to fishing or fisherman can also add a completely different look to the entire room. A good quality canvas provides an accurate and realistic portraying of fishing scenes. These canvases are available in different shapes and sizes and are framed in different quality of material like wood, metal and many more. They can be kept anywhere whether it is a living room, bed room, or kitchen space. They light up the whole atmosphere of the room with their vibrant life like look.
A Marilyn Monroe wall art in a room can give a whole new looks to the entire space. Her larger than life persona looks very appealing when they are painted on a canvas and are an all-time favourite. When placed in home or in office space they create a look which is both elegant and vibrant. | english |
/*
* Copyright (C) 2007 ETH Zurich
*
* This file is part of Fosstrak (www.fosstrak.org).
*
* Fosstrak is free software; you can redistribute it and/or
* modify it under the terms of the GNU Lesser General Public
* License version 2.1, as published by the Free Software Foundation.
*
* Fosstrak 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
* Lesser General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public
* License along with Fosstrak; if not, write to the Free
* Software Foundation, Inc., 51 Franklin Street, Fifth Floor,
* Boston, MA 02110-1301 USA
*/
//
// This file was generated by the JavaTM Architecture for XML Binding(JAXB) Reference Implementation, v2.0-b26-ea3
// See <a href="http://java.sun.com/xml/jaxb">http://java.sun.com/xml/jaxb</a>
// Any modifications to this file will be lost upon recompilation of the source schema.
// Generated on: 2006.07.05 at 04:25:04 PM CEST
//
package org.fosstrak.reader.rprm.core.msg.notification;
import java.util.ArrayList;
import javax.xml.bind.JAXBElement;
import javax.xml.bind.annotation.XmlAccessType;
import javax.xml.bind.annotation.XmlAccessorType;
import javax.xml.bind.annotation.XmlElementRef;
import javax.xml.bind.annotation.XmlType;
import org.fosstrak.reader.rprm.core.msg.notification.HexStringListType;
/**
* <p>Java class for HexStringListType complex type.
*
* <p>The following schema fragment specifies the expected content contained within this class.
*
* <pre>
* <complexType name="HexStringListType">
* <complexContent>
* <restriction base="{http://www.w3.org/2001/XMLSchema}anyType">
* <sequence>
* <element name="list">
* <complexType>
* <complexContent>
* <restriction base="{http://www.w3.org/2001/XMLSchema}anyType">
* <sequence>
* <element name="value" type="{http://www.w3.org/2001/XMLSchema}hexBinary" maxOccurs="unbounded"/>
* </sequence>
* </restriction>
* </complexContent>
* </complexType>
* </element>
* </sequence>
* </restriction>
* </complexContent>
* </complexType>
* </pre>
*
*
*/
@XmlAccessorType(XmlAccessType.FIELD)
@XmlType(name = "HexStringListType", propOrder = {
"list"
})
public class HexStringListType {
protected org.fosstrak.reader.rprm.core.msg.notification.HexStringListType.List list;
/**
* Gets the value of the list property.
*
* @return
* possible object is
* {@link org.fosstrak.reader.rprm.core.msg.notification.HexStringListType.List }
*
*/
public org.fosstrak.reader.rprm.core.msg.notification.HexStringListType.List getList() {
return list;
}
/**
* Sets the value of the list property.
*
* @param value
* allowed object is
* {@link org.fosstrak.reader.rprm.core.msg.notification.HexStringListType.List }
*
*/
public void setList(org.fosstrak.reader.rprm.core.msg.notification.HexStringListType.List value) {
this.list = value;
}
/**
* <p>Java class for anonymous complex type.
*
* <p>The following schema fragment specifies the expected content contained within this class.
*
* <pre>
* <complexType>
* <complexContent>
* <restriction base="{http://www.w3.org/2001/XMLSchema}anyType">
* <sequence>
* <element name="value" type="{http://www.w3.org/2001/XMLSchema}hexBinary" maxOccurs="unbounded"/>
* </sequence>
* </restriction>
* </complexContent>
* </complexType>
* </pre>
*
*
*/
@XmlAccessorType(XmlAccessType.FIELD)
@XmlType(name = "", propOrder = {
"value"
})
public static class List {
@XmlElementRef(name = "value", type = JAXBElement.class)
protected java.util.List<JAXBElement<byte[]>> value;
/**
* Gets the value of the value property.
*
* <p>
* This accessor method returns a reference to the live list,
* not a snapshot. Therefore any modification you make to the
* returned list will be present inside the JAXB object.
* This is why there is not a <CODE>set</CODE> method for the value property.
*
* <p>
* For example, to add a new item, do as follows:
* <pre>
* getValue().add(newItem);
* </pre>
*
*
* <p>
* Objects of the following type(s) are allowed in the list
* {@link JAXBElement }{@code <}{@link byte[]}{@code >}
*
*
*/
public java.util.List<JAXBElement<byte[]>> getValue() {
if (value == null) {
value = new ArrayList<JAXBElement<byte[]>>();
}
return this.value;
}
}
}
| code |
\begin{document}
\title{Stampacchia's property, self-duality and orthogonality relations}
\author{Nikos Yannakakis}
\address{Department of Mathematics\\
National Technical University of Athens\\
Iroon Polytexneiou 9\\
15780 Zografou\\
Greece}
\email{[email protected]}
\subjclass[2000]{Primary 46C15; Secondary 47B99, 46B03}
\keywords{Variational inequality, complemented subspace, Hilbert space characterization, self-dual Banach space, positive operator, coercive operator, orthogonality relation, cosine of a linear operator, quadratic form, evolution triple.}
\date{}
\begin{abstract}
We show that if the conclusion of the well known Stampacchia Theorem, on variational inequalities, holds on a Banach space $X$, then $X$ is isomorphic to a Hilbert space. Motivated by this we obtain a relevant result concerning self-dual Banach spaces and investigate some connections between existing notions of orthogonality and self-duality. Moreover, we revisit the notion of the cosine of a linear operator and show that it can be used to characterize Hilbert space structure. Finally, we present some consequences of our results to quadratic forms and to evolution triples.
\end{abstract}
\maketitle
\section{Introduction}
Let $H$ be a real Hilbert space, $\|\cdot\|$ be its norm, $(\cdot\,,\cdot)$ its inner product and let $$a:H\times H\rightarrow \mathbb R$$
be a bounded bilinear form.
The well known Stampacchia Theorem (also called the Lions-Stampacchia Theorem, see \cite{Ernst}, \cite{Stampacchia1} and \cite{Stampacchia2}) states that if the above bilinear form is {\it coercive}, i.e. there exists $c>0$ such that
\begin{equation}
\label{coercive}
a(x,x)\geq c\| x\|^2,\text{ for all } x\in H,
\end{equation}
then for any nonempty, closed, convex subset $M$ of $H$ and $h\in H$, there exists a unique solution $x\in M$, of the variational inequality
\begin{equation}
\label{var}
a(x,z-x)\geq (h, z-x)\,,\text{ for all }z\in M.
\end{equation}
Our first aim, in this paper, is to investigate whether Stampacchia's Theorem can be generalized in the broader setting of an arbitrary Banach space $X$. As we will see, at least in its full generality, this is impossible since its conclusion implies that $X$ has to be isomorphic to a Hilbert space.
In the sequel we obtain a relevant result concerning self-dual Banach spaces, i.e. Banach spaces that are isomorphic to their dual spaces. Along the way we see that our approach brings out some connections between existing notions of orthogonality in general normed linear spaces and self-duality.
In the last section and motivated by the above, we revisit the cosine of a linear operator (a notion originally introduced by K. Gustafson in \cite{gus1}) and use it to obtain an additional Hilbert space characterization based on a result of J. R. Partington which can be found in \cite{Partington}.
Finally, we present some consequences of our results to quadratic forms and to evolution triples.
\section{Stampacchia's property}
\label{section1}
Let $X$ be a real Banach space, $X^\ast$ be its dual and $\langle \cdot\,,\cdot\rangle$ be their duality product. By $M^\bot$ we denote the annihilator of a subspace $M$ of $X$, i.e.
$$M^\bot=\left\{x^\ast\in X^\ast:\,<x^\ast,x>=0\,, \text{ for all }x\in M\right\}\,.$$
To obtain the natural analogue of the conclusion of Stampacchia's Theorem in this situation we need the following definition.
\begin{definition}
Let $X$ be a real Banach space. We say that $X$ has Stampacchia's property (property (S) for short), if there exists
a bounded, bilinear form
$$a:X\times X\rightarrow\mathbb{R}$$
such that if $M$ is any nonempty, convex, closed subset of $X$ and $x^\ast\in X^\ast$, then there exists
a unique $x\in M$ such that
$$a(x,z-x)\geq \langle x^\ast,z-x\rangle\,,\text{ for all } z\in M.$$
\end{definition}
Recall that a closed subspace $M$ of a Banach space $X$ is complemented in $X$, if there exists another closed subspace $N$ of $X$ such that $X$ is their direct sum, i.e.
$$M\cap N=\left\{\,0\right\}\text{ and }X=M+N\,.$$
Note that the existence of such a closed subspace $N$ is equivalent to the existence of a bounded linear projection from $X$ onto $M$.
Not all closed subspaces of an arbitrary Banach space are complemented. In fact we have the following well-known result by J. Lindenstrauss and L. Tzafriri \cite{LinTza}, which we will use in the sequel.
\begin{theorem}
\label{Lintza}
A Banach space $X$ is isomorphic to a Hilbert space if and only if all its closed subspaces are complemented.
\end{theorem}
To proceed with our task we need the following simple lemma.
\begin{lemma}
\label{complemented}
Let $X$ be a Banach space and $M$ be a closed subspace of $X$. If there exists another Banach space $Y$ and a bounded linear operator
$$S:M\rightarrow Y$$
$1-1$ and onto $Y$, which can be extended to the whole of $X$, then the closed subspace $M$ is complemented in $X$.
\end{lemma}
\begin{proof}
If
$$\hat{S}:X\rightarrow Y$$
denotes the extension of $S$ to the whole of $X$, then it is easy to see that the operator
$$S^{-1}\circ\hat{S}:X\rightarrow M$$
is the required bounded projection onto $M$.
\end{proof}
We can now show that property (S) characterizes Hilbert space structure.
\begin{theorem}
\label{theorem}
A real Banach space $X$ is isomorphic to a Hilbert space if and only if it has property (S).
\end{theorem}
\begin{proof}
The neccesity is obvious. We prove that property (S) is also sufficient.
To this end let $M$ be any closed subspace of $X$. We will show that $M$ is complemented. Since $M$ is a closed subspace of $X$, it is easy to see that property (S) in particular implies that for all
$x^\ast\in X^\ast$, there exists a unique $x\in M$ such that
\begin{equation}
\label{eq2}
a(x,z)=\langle x^\ast,z\rangle,\text{ for all }z\in M.
\end{equation}
Define the bounded linear operator
$$T:X\rightarrow X^\ast\,,$$
by
$$\langle Tx,z\rangle=a(x,z),\text{ for all }x, z\in X\,$$
and let
$$\pi:X^\ast\rightarrow X^\ast/M^\bot$$
be the natural quotient map.
Then the restriction of the operator $\pi \circ T$ on the subspace $M$ is $1-1$ and onto $X^\ast/M^\bot$.
To see this first note that if
$$(\pi \circ T)x=0$$
then $Tx\in M^\bot$, i.e. $\langle Tx,z\rangle=0$, for all $z\in M$. By the definition of $T$ this implies that $a(x,z)=0$, for all $z\in M$. But by hypothesis there is a unique $x\in M$ such that $a(x,z)=0$, for all $z\in M$, which by the boundedness of $a$ has to be $0$. Thus the restriction of the operator $\pi \circ T$ on the subspace $M$ is $1-1$.
To show that $\pi \circ T|_M$ is also onto, let $h=x^\ast+M^\bot$, for some $x^\ast\in X^\ast$. Then by (\ref{eq2}) we have that there exists a unique $x\in M$ such that $a(x,z)=\langle x^\ast,z\rangle$, for all $z\in M$. Hence again by the definition of $T$ we have that
$$\langle Tx,z\rangle=\langle x^\ast,z\rangle\,, \text{ for all }z\in M\,,$$
i.e. $Tx-x^\ast\in M^\bot$. Hence $(\pi\circ T)(x)=h$ and thus $\pi \circ T|_M$ is onto $X^\ast/M^\bot$.
Note now that by its definition the operator
$$S=\pi \circ T|_M$$
can be trivially extended to the whole of $X$ and thus by Lemma \ref{complemented} the closed subspace $M$ is a complemented subspace of $X$. Since $M$ was arbitrary we get by Theorem \ref{Lintza} that $X$ is isomorphic to a Hilbert space.
\end{proof}
\begin{remark}
A careful look in the above proof shows that if the Banach space $X$ has property (S) and $M$ is any closed subspace of $X$ then
$$X=M\oplus T^{-1}(M^\bot)\,,$$
where $T$ is the operator associated to the bilinear form $a(\cdot,\cdot)$.
\end{remark}
\begin{remark}
\label{remark}
A main hypothesis in Stampacchia's Theorem is the coercivity condition (\ref{coercive}). As it is well-known (see for example \cite{DY}, \cite{Lin}), such a hypothesis cannot hold in an arbitrary Banach space $X$ since if it did, then $X$ would have an equivalent Hilbertian norm induced by the inner product
$$(x,y)=\frac{1}{2}[a(x,y)+a(y,x)]$$
and thus would be isomorphic to a Hilbert space.
Hence our result implies that there can be no full generalization of Stampacchia's Theorem in an arbitrary Banach space even if one drops the coercivity condition (\ref{coercive}).
\end{remark}
\begin{remark}
\label{bounded}
Note that if we are restricted to bounded closed and convex subsets of a
Banach space $X$, then a generalization of Stampacchia's Theorem is possible by just assuming that the bilinear form is strictly positive i.e.
$$a(x,x)>0\,,\text{ for all }x\in X.$$
The proof is a straightforward application of a result due to Brezis \cite[Theorem 24]{Brezis1}, on pseudomonotone operators. It is easy to see that in this case the space $X$ need not be isomorphic to a Hilbert space.
\end{remark}
\begin{remark}
It seems appropriate to mention here a recent result by E. Ernst and M. Th\' era: if as in Remark \ref{bounded} we are restricted to bounded, closed and convex sets and moreover $X$ is a Hilbert space, then the pseudomonotonicity of the operator associated to the bilinear form $a(\cdot,\cdot)$, is a necessary and sufficient condition for the existence of a solution of the variational inequality (\ref{var}) (see \cite[Theorem 3.1]{Ernst}).
A similar result for unbounded sets has been obtained by A. Maugeri and F. Raciti in \cite{maugeri}.
\end{remark}
\section{Self-dual Banach spaces}
\label{self}
A self-dual Banach space is a Banach space isomorphic to its dual. It is well-known that Hilbert spaces are self-dual although they are far from being the only ones; if $Y$ is any reflexive Banach space then
$$X=Y\oplus Y^\ast$$
is self-dual.
We will now see that our approach in Section \ref{section1} can lead us to a result concerning self-dual Banach spaces.
The important observation is the fact that the operator $T$ associated to the bilinear form $a(\cdot,\cdot)$, in the proof of Theorem \ref{theorem}, is an isomorphism from $X$ onto $X^\ast$ and hence $X$ is a self-dual space.
Our result is the following.
\begin{proposition}
\label{th1}
Let $X$ be a real, self-dual, Banach space. If the isomorphism
$$T:X\rightarrow X^\ast$$
onto $X^\ast$ is such that for any closed subspace $M$ of $X$, the map
$\pi\circ T|_M$ is an isomorphism onto $X^\ast/M^\bot$, where $\pi$ is the natural quotient map from $X^\ast$ onto $X^\ast/M^\bot$, then $X$ is isomorphic to a Hilbert space.
\end{proposition}
\begin{proof}
We follow the proof of Theorem \ref{theorem}.
\end{proof}
\begin{remark}
Recalling that the quotient space $X^\ast/M^\bot$ is isomorphic to $M^\ast$ we can rephrase Proposition \ref{th1} as follows:
``Let $X$ be a self-dual space. If the isomorphism between $X$ and $X^\ast$ induces in a \textit{natural way} (through the natural quotient maps) isomorphisms between all closed subspaces of $X$ and their corresponding duals, then $X$ is isomorphic to a Hilbert space''.
\end{remark}
As one can easily see, a necessary and sufficient condition for $\pi\circ T|_M$ to be an isomorphism (not necessarily onto) from $M$ into $X^\ast/M^\bot$, is the existence of a positive constant $c$, such that whenever $x\in X$ and $x^\ast\in X^\ast$ are such that $\langle x^\ast,x\rangle=0$, we have that
\begin{equation}
\label{isom}
||Tx+x^\ast||\geq c||Tx||\,.
\end{equation}
In order to give some geometric intuition to condition (\ref{isom}) we recall the following definition.
\begin{definition}
Let $X$ be a normed space and $x\,,y\in X$. We say that $x$ is orthogonal, in the sense of Birkhoff-James, to $y$ if
$$||x+\lambda y||\geq ||x||, \text{ for all }\lambda\in \mathbb{R}.$$
\end{definition}
For more details about this notion of orthogonality the interested reader is referred to \cite{amir} and \cite{Istr}.
It is easy to see that if whenever $x\in X$ and $x^\ast\in X^\ast$ are such that $\langle x^\ast,x\rangle=0$, we have that
\begin{equation}
\nonumber
\label{bir}
Tx\;\bot\; x^\ast\,,
\end{equation}
in the sense of Birkhoff-James, then $T$ satisfies condition (\ref{isom}).
As a matter of fact Birkhoff-James orthogonality is not the only orthogonality relation that can be used to guarantee the validity of condition (\ref{isom}). To see this we recall that in \cite{Partington}, J. R. Partington has introduced the concept of \textit{boundedness} for an orthogonality relation in an arbitrary normed space as follows.
\begin{definition}
An orthogonality relation $\bot$ in a normed linear space is bounded if there exists $c>0$ such that if $x\bot y$ then
\begin{equation}
\nonumber
\label{bound}
||\lambda x+ y||\geq c||x||, \text{ whenever } |\lambda|\geq c.
\end{equation}
\end{definition}
Several well-known orthogonality relations (for example Birkhoff-James or Diminnie orthogonality, see \cite{diminnie} and \cite{Partington} for more details) are bounded.
\begin{definition}
\label{homog}
An orthogonality relation $\bot$ in a normed linear space is homogeneous if
\begin{equation}
\nonumber
x \bot y\;\text{ implies that }\;ax \bot by, \text{ for all }\;a,b\in\mathbb{R}.
\end{equation}
\end{definition}
\begin{remark}
\label{mil}
In \cite{Milicic} it was shown that if an orthogonality relation $\bot$ is homogeneous, then its boundedness is equivalent to the existence of $c>0$, such that $x\bot y$ implies
\begin{equation}
\nonumber
\label{bound2}
||x+ y||\geq c||x||.
\end{equation}
\end{remark}
Therefore if whenever $x\in X$ and $x^\ast\in X^\ast$ are such that $\langle x^\ast,x\rangle=0$, we have that
\begin{equation}
\nonumber
\label{general}
Tx\;\bot\; x^\ast,
\end{equation}
for a homogeneous and bounded orthogonality relation $\bot$, then $T$ satisfies (\ref{isom}).
To state our next result we need one more definition.
\begin{definition}
\label{nondeg}
An orthogonality relation $\bot$, in a normed linear space, is non-degenerate, if $x\bot x$ implies that $x=0$.
\end{definition}
We can now prove the following Hilbert space characterization.
\begin{theorem}
\label{orthogonal}
A real reflexive Banach space $X$ is isomorphic to a Hilbert space if and only if there exists an isomorphism
$$T:X\rightarrow X^\ast\,,$$
onto $X^\ast$, such that
\begin{equation}
\label{orth}
Tx\;\bot\; x^\ast, \text{ whenever } \langle x^\ast,x\rangle=0\,,
\end{equation}
for a non-degenerate, homogeneous and bounded orthogonality relation $\bot$ in $X^\ast$.
\end{theorem}
\begin{proof}
The necessity is obvious. To prove the sufficiency of our claim we will use Proposition \ref{th1}. To this end let $M$ be any closed subspace of $X$. By (\ref{orth}) and the discussion above, the operator $T$ satisfies condition (\ref{isom}) and hence $\pi\circ T|_M$ is an isomorphism. It remains to show that $\pi\circ T|_M$ is onto $X^\ast/M^\bot$.
Since $(\pi\circ T)(M)$ is closed it is enough to show that it is a dense subspace of $X^\ast/M^\bot$. Assume the contrary i.e.
$$(\pi\circ T)(M)\neq X^\ast/M^\bot\,.$$
Then by the Hahn-Banach Theorem there exists $0\neq f\in (X^\ast/M^\bot)^\ast$ such that
$$f(Tx)=0\,, \text{ for all } x\in M\,.$$
Since $X$ is reflexive so is $M$ and hence it is isometrically isomorphic to $(X^\ast/M^\bot)^\ast$. Therefore there exists $x\in M$, such that $$\langle Tx,x\rangle=f(Tx)=0$$
and thus again by (\ref{orth}) we get that $Tx\bot Tx$. Using the non-degeneracy of $\bot$ and the injectivity of $T$ we get that $x$ and consequently $f$ have to be 0, which is a contradiction.
Hence $\pi\circ T|_M$ is an isomorphism onto $X^\ast/M^\bot$ and by Proposition \ref{th1} the self-dual Banach space $X$ is isomorphic to a Hilbert space.
\end{proof}
\section{The cosine of a linear operator revisited}
A simple situation where condition (\ref{isom}) holds is when there exists $c>0$, such that the operator $T$ satisfies
\begin{equation}
\label{hes}
\langle Tx,x\rangle\geq c||Tx||^2,\text{ for all }x\in X\,.
\end{equation}
Recall the following well-known definition.
\begin{definition}
Let $X$ be a real Banach space. We say that the linear operator $$T:D(T)\subseteq X\rightarrow X^\ast$$
\begin{itemize}
\item[(i)] is positive, if $\langle Tx,x\rangle\geq 0$, for all $x\in D(T)$.
\item[(ii)] is strictly positive, if $\langle Tx,x\rangle>0$, for all $x\in D(T)$, with $x\neq 0$.
\item[(iii)] is coercive, if there exists $c>0$, such that
$$\langle Tx,x\rangle\geq c||x||^2\,,$$
for all $x\in D(T)$.
\item[(iv)] is symmetric, if $\langle Tx,y\rangle=\langle Ty,x\rangle$, for all $x\,,y\in D(T)$.
\end{itemize}
\end{definition}
Note that since in all our previous considerations (in Section \ref{self}), the operator $T$ was an isomorphism inequality (\ref{hes}) would imply that the operator $T$ was actually coercive.
In the general case though, operators satisfying (\ref{hes}) form a much larger class than that of coercive operators.
For example, see \cite{DY} and \cite{hess2} for more details, any positive, everywhere defined and symmetric operator $T$ satisfies (\ref{hes}).
On the other hand, unlike coercivity (see Remark \ref{remark}) condition (\ref{hes}) cannot guarantee on its own - i.e. when $T$ is no longer an isomorphism but just a continuous linear operator - the Hilbertian structure of $X$. Note that this is still the case even if $T$ has additional nice properties such as symmetry and positivity. It seems therefore quite natural that there may be some room between these two classes. To make things more precise we need the following definition.
\begin{definition}
\label{cosine}
Let $X$ be a real Banach space and let $$T:D(T)\subseteq X\rightarrow X^\ast$$ be a positive linear operator. The cosine of $T$ is defined as follows:
\begin{equation}
\label{cos}
\cos T=\inf\left\{\frac{\langle Tx,x\rangle}{||Tx||\,||x||}\;,\;\text{ for all } 0\neq x\in D(T) \text{, such that } Tx\neq 0\right\}\,.
\end{equation}
\end{definition}
Using expression (\ref{cos}) one can define the angle $\phi(T)$ of the linear operator $T$, which has an obvious geometric interpretation: it measures the maximum turning effect of $T$.
The above concepts were introduced, in the context of a complex Hilbert space, by K. Gustafson in \cite{gus1} and have attracted a lot of interest since then. We refer the interested reader to the book of K. Gustafson and D. Rao \cite{gus3}, for more details.
In order for the cosine of an operator to be a reliable tool, distinguishing between operators with different properties, it has to be positive for a large class of linear operators. As one can easily see this is the case for coercive everywhere defined - thus continuous - linear operators.
On the other hand things fail dramatically for unbounded linear operators: it was shown by K. Gustafson and B. Zwahlen in \cite{gus2} and by P. Hess (in a somewhat more general context) in \cite{hess}, that the cosine of an unbounded linear operator is always 0.
To return to our main theme note that if $\cos T>0$, then $T$ satisfies (\ref{hes}). Thus non-coercive operators with positive cosine form the aforementioned intermediate class, between (\ref{hes}) and coercivity. It turns out, as we shall see below, that if $X$ is not isomorphic to a Hilbert space then this class is quite small.
We need one more definition.
\begin{definition}[\cite{Partington}]
\label{properties}
An orthogonality relation $\bot$, in a normed linear space $X$ is
\begin{itemize}
\item[(i)] symmetric, if $x\bot y$ implies $y\bot x$.
\item[(ii)] right additive, if $x\bot y$ and $x\bot z$ implies $x\bot (y+z)$.
\item[(iii)] resolvable, if for any $x\,,y$ there exists $a\in\mathbb{R}$, such that $x\bot (ax+y)$.
\item[(iv)] continuous, if $x_n\rightarrow x$, $y_n\rightarrow y$ and $x_n\bot y_n$, then $x\bot y$.
\end{itemize}
\end{definition}
It should be noted that an orthogonality relation having all six properties of Definitions \ref{homog}, \ref{nondeg} and \ref{properties} (i.e. except boundedness) exists in any separable Banach space (see Theorem 3 in \cite{Partington}).
If boundedness is added things change drastically as the following result of J. R. Partington \cite{Partington} illustrates.
\begin{theorem}[\cite{Partington}, Theorem 4]
\label{part}
If $X$ is a Banach space and $\bot$ is an orthogonality relation in $X$, that is non-degenerate, symmetric, homogeneous, right additive, resolvable, continuous and bounded, then $X$ is isomorphic to a Hilbert space.
\end{theorem}
In the sequel we identify $T^\ast$ with the restriction on $X$ of the adjoint of the linear operator $T:X\rightarrow X^\ast$ (which is defined on the whole of $X^{\ast\ast}$).
Using Theorem \ref{part}, we can prove our main result for this section.
\begin{theorem}
\label{zero}
Let $X$ be a real Banach space, not isomorphic to a Hilbert space and
$$T:X\rightarrow X^\ast$$
a positive linear operator. If there exists $c>0$, such that
\begin{equation}
\label{ineq}
||T^\ast x||\leq c||Tx||\,,\text{ for all }x\in X\,,
\end{equation}
then $\cos T=0$.
\end{theorem}
\begin{proof}
Assume the contrary and let $\cos T=\delta>0$. The linear operator
$$S:X\rightarrow X^\ast$$
defined by
$$S=\frac{1}{2}(T+T^\ast)\,.$$
is strictly positive, everywhere defined and hence continuous. We define the following orthogonality relation in $X$:
$$x\bot y\,,\text{ if } \langle Sx,y\rangle =0\,.$$
It is easy to see that $\bot$ is non-degenerate, symmetric, homogeneous, right additive, resolvable and continuous. To see that $\bot$ is also bounded take $x\bot y$ with $x\neq 0$. Then
\begin{eqnarray*}
||x+y||&=&\sup_{x^\ast\neq 0}\frac{\langle x^\ast,x+y\rangle}{||x^\ast||}\\
&\geq &\frac{\langle Sx,x+y\rangle}{||Sx||}
=\frac{\langle Tx,x\rangle}{||Sx||}\\
&\geq &\frac{2\langle Tx,x\rangle}{||Tx||(1+c)}\\
&\geq &\frac{2\delta}{1+c}||x||\,,
\end{eqnarray*}
where the second inequality is justified by (\ref{ineq}).
Since $\bot$ is homogeneous, by Remark \ref{mil}, the orthogonality relation
$\bot$ is also bounded.
Hence by Theorem \ref{part} the Banach space $X$ is isomorphic to a Hilbert space, which is a contradiction.
Thus $\cos T=0$.
\end{proof}
\begin{remark}
The class of operators satisfying (\ref{ineq}) is quite large as it includes positive, everywhere defined, symmetric linear operators.
\end{remark}
Combining this last remark with Theorem \ref{zero} we can have the following simple Hilbert space characterization.
\begin{corollary}
\label{symmetric}
A real Banach space $X$ is isomorphic to a Hilbert space if and only if there exists
a positive and symmetric linear operator
$$T:X\rightarrow X^\ast$$
with $\cos T>0$.
\end{corollary}
It seems quite interesting that if $X$ is not isomorphic to a Hilbert space then an operator and its adjoint cannot have both positive cosines.
\begin{proposition}
Let $X$ be a real Banach space, not isomorphic to a Hilbert space and
$$A:X\rightarrow X^\ast$$
a positive linear operator with $\cos A>0$. Then $\cos A^\ast=0$.
\end{proposition}
\begin{proof}
Assume $\cos A=\delta>0$ and let $x\neq 0$. Then
\begin{eqnarray*}
||A^\ast x||&=&\sup_{y\neq 0}\frac{\langle A^\ast x,y\rangle}{||y||}\\
&\geq &\frac{\langle Ax,x\rangle}{||x||}\\
&\geq
&\delta||Ax||\,.
\end{eqnarray*}
If $T=A^\ast$, then $T$ is a positive linear operator that satisfies (\ref{ineq}). Thus by Theorem \ref{zero} we get that $\cos A^\ast=0$.
\end{proof}
\subsection{An application to quadratic forms}
Recall that a continuous quadratic form on a normed space $X$ is a function
$$q:X\rightarrow \mathbb{R}$$
for which there exists a bounded bilinear form
$$a:X\times X\rightarrow\mathbb{R}$$
such that
$$q(x)=\frac{1}{2}a(x,x)\,.$$
It is well known (see for example \cite{Kalton}) that there exists a one-to-one correspondence between continuous quadratic forms and symmetric linear operators
$$T:X\rightarrow X^\ast$$
through the formula
\begin{equation}
\label{quadratic}
q(x)=\frac{1}{2}\langle Tx,x\rangle\,.
\end{equation}
Moreover, each continuous quadratic form is everywhere Frechet differentiable and its derivative is equal to $2T$, where $T$ is the symmetric operator in (\ref{quadratic}).
Using Corollary \ref{symmetric}, we can have the following result.
\begin{proposition}
Let $X$ be a real Banach space, not isomorphic to a Hilbert space and
$$q:X\rightarrow \mathbb{R}$$
a continuous quadratic form. Then for any $\varepsilon>0$, there exists $x\in X$, such that
$$q(x)<\varepsilon ||q'(x))||\,||x||\,.$$
\end{proposition}
\begin{proof}
If $q(x)<0$, for some $x\in X$ we are done. If this is not the case then the symmetric linear operator $T$ that generates $q$, is positive and thus by Corollary \ref{symmetric}
$$\cos T=0\,.$$
Hence for any $\varepsilon>0$, there exists $x\neq 0$, such that
$$\frac{\langle Tx,x\rangle}{||Tx||\,||x||}<\varepsilon\,.$$
Since $q(x)=\displaystyle\frac{1}{2}\langle Tx,x\rangle$ and $q'=2T$ the result follows.
\end{proof}
\subsection{An application to evolution triples}
We end this paper with an application of Theorem \ref{zero} to evolution triples.
Recall that we say that a Banach space is continuously and densely embedded into another Banach space $Y$, if there exists an injective, bounded linear operator
$$i:X\rightarrow Y$$
such that $i(X)$ is dense in $Y$. We have the following Proposition.
\begin{proposition}
\label{evolution}
Let $X$ be a real reflexive Banach space that is continuously and densely embedded into a Hilbert space $H$ and assume that $X$ is not isomorphic to a Hilbert space. Then for any $\varepsilon>0$, there exists $x\in X$, such that
$$||i(x)||_H^2<\varepsilon ||i^\ast(i(x))||_{X^\ast}||x||_X\,,$$
where $i$ is the embedding operator from $X$ into $H$.
\end{proposition}
\begin{proof}
Since
$$i:X\rightarrow H$$
is the embedding operator from $X$ into $H$ and the embedding is continuous and dense, then (after identifying $H$ with its dual space $H^\ast$) the embedding
$$i^\ast:H\rightarrow X^\ast$$
is also continuous and dense (we say that $X$, $H$ and $X^\ast$ form an evolution or a Gelfand triple).
Let
$$T:X\rightarrow X^\ast$$
be defined by $T=i^\ast\circ i$. Then $T$ is a strictly positive, symmetric operator and by Corollary \ref{symmetric} we have that
$$\cos T=0\,.$$
Thus for any $\varepsilon>0$, there exists $x\in X$, such that
$$\frac{\langle Tx,x\rangle}{||Tx||_{X^\ast}||x||_X}<\varepsilon.$$
But $\langle Tx,x\rangle=(i(x),i(x))_H=||i(x)||_H^2$ and hence the result follows.
\end{proof}
A concrete example of the above situation is the following:
\begin{example}
Let $\Omega\subseteq \mathbb{R}^N$, open and bounded and assume $p>2$. Then for every $\varepsilon>0$, there exists $f\in L^p(\Omega)$, such that
$$||f||_{L^2(\Omega)}^2<\varepsilon ||f||_{L^q(\Omega)}||f||_{L^p(\Omega)}\,,$$
where $\displaystyle\frac{1}{p}+\frac{1}{q}=1$.
\end{example}
\begin{proof}
Let $i:L^p(\Omega)\rightarrow L^2(\Omega)$ be the identity operator and use the previous proposition.
\end{proof}
\begin{acknowledgment}
The author would like to thank Dr. D. Drivaliaris and Mr. M. Garagai for many fruitful discussions.
\end{acknowledgment}
\end{document} | math |
राज्य सरकार विश्वविद्यालय में पर्याप्त बुनियादी सुविधाएं उपलब्ध करवाने के प्रति बचनबद्ध मुख्यमंत्री जय राम ठाकुर - वेरिएंट - न्यूज मगजीन
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राज्य सरकार विश्वविद्यालय में पर्याप्त बुनियादी सुविधाएं उपलब्ध करवाने के प्रति बचनबद्ध मुख्यमंत्री जय राम ठाकुर
राज्य सरकार हिमाचल प्रदेश विश्वविद्यालय को केंद्रीय विश्वविद्यालय घोषित करने का मामला केन्द्र सरकार के साथ उठाएगी ताकि विश्वविद्यालय के विकास एवं विस्तार के लिए पर्याप्त धन उपलब्ध हो सके। मुख्यमंत्री जय राम ठाकुर ने प्रदेश की राजधानी में शिमला के समरहिल में विश्वविद्यालय के ५० वें स्थापना दिवस की अध्यक्षता करते हुए यह बात कही।
। उन्होंने कहा कि उन्होंने राज्य के लिए निवेश आकर्षित करने के लिए हाल ही में कुछ यूरोपीय देशों का दौरा किया है। उन्होंने कहा कि संभावित निवेशकों के साथ बैठकें बड़ी सफल रही हैं क्योंकि कई उद्यमियों ने राज्य में निवेश के लिए रुचि दिखाई है।
जय राम ठाकुर ने कहा कि विश्वविद्यालय परिसर में विशेष रूप से दृष्टिबाधित छात्रों के लिए पुस्तकालय की स्थापना विश्वविद्यालय की एक बड़ी पहल है। उन्होंने कहा कि राज्य सरकार दृष्टिबाधित छात्रों की सुविधा के लिए चयनित कॉलेजों और स्कूलों मे विशेष पुस्तकालय स्थापित करेगी। उन्होंने कहा कि राज्य सरकार विश्वविद्यालय परिसर में वाईफाई सुविधा सुनिश्चित करने के लिए हर संभव सहायता प्रदान करेगी।
मुख्यमंत्री ने कहा कि राज्य सरकार घणाहट्टी में नए परिसर की स्थापना के साथ-साथ विश्वविद्यालय की अन्य गतिविधियों के विस्तार के लिए हर संभव सहायता प्रदान करेगी। उन्होंने कहा कि राज्य सरकार यह सुनिश्चित करेगी कि विश्वविद्यालय की स्वायत्तता किसी भी कीमत पर बनी रहे। उन्होंने कहा कि राज्य के युवाओं में नशीली दवाओं के सेवन के प्रति बढ़ती लत पर अंकुश लगाने के लिए शिक्षकों को भी आगे आना चाहिए।
मुख्यमंत्री ने इस अवसर पर विश्वविद्यालय समाचार पत्र हिमशिखर, शोध पत्रिकाओं, स्मारिका और अन्य प्रकाशनों और पुस्तकों का भी विमोचन किया।
मुख्यमंत्री ने पद्मश्री डॉ. ओमेश भारती को इस अवसर पर रेबीज के प्रभावी उपचार प्रदान करने में योगदान के लिए सम्मानित किया।
मुख्यमंत्री ने इस अवसर पर शिक्षकों, छात्रों और कर्मचारियों को उनके उत्कृष्ट योगदान के लिए सम्मानित किया, जिसमें डॉ. सुषमा शर्मा, डॉ. सरस्वती भल्ला और डॉ. अनुराग शर्मा को उत्कृष्ट शिक्षक पुरस्कार से सम्मानित किया। उन्होंने प्रोफेसर राम प्रकाश शर्मा, डॉ. अनीता शर्मा और डॉ. अमरजीत सिंह को सर्वश्रेष्ठ शोधकर्ता का पुरस्कार प्रदान किया। उन्होंने सरवन कुमार, जमना दास और केवल कृष्ण को सर्वश्रेष्ठ कर्मचारी के पुरस्कार से सम्मानित किया। उन्होंने इस अवसर पर छात्रों मनीषा और दिव्या को भी सम्मानित किया। इससे पहले, मुख्यमंत्री ने विश्वविद्यालय में २.३८ करोड़ रुपये की लागत से निर्मित नॉन टीचिंग स्टाफ के आवासीय अटल भवन का उद्घाटन किया। उन्होंने २.५६ करोड़ रुपये की लागत से निर्मित किए जाने वाले चतुर्थ श्रेणी कर्मचारियों के आवासीय भवन की आधारशिला भी रखी। इसमें कर्मचारियों के लिए सोलह आवास होंगे।
मुख्यमंत्री ने १.५६ करोड़ रुपये की लागत से निर्मित किए गए विश्वविद्यालय मॉडल स्कूल (चरण-३) का उद्घाटन किया। उन्होंने २५ लाख रुपये की लागत से निर्मित दृष्टिबाधित और अन्य छात्रों के लिए पुस्तकालय का उद्घाटन भी किया। उन्होंने १.५० करोड़ रुपये की लागत से निर्मित हिमाचल प्रदेश विश्वविद्यालय और स्पेस टेली-इंफ्रा प्राइवेट लिमिटेड द्वारा संयुक्त रूप से स्थापित नेटवर्क का उद्घाटन भी किया।
शिक्षा मंत्री सुरेश भारद्वाज ने कहा कि हिमाचल प्रदेश विश्वविद्यालय ने मेहनती शिक्षकों, छात्रों और अन्य कर्मचारियों के कारण अपने लिए एक विशेष स्थान बनाया है। उन्होंने कहा कि १९७० में अपनी स्थापना के बाद से, विश्वविद्यालय ने कई उत्कृष्ट छात्र दिए हैं, जो विश्वविद्यालय और राज्य के लिए गौरवमयी साबित हुए हैं। उन्होंने कहा कि मुख्यमंत्री जय राम ठाकुर के कुशल और गतिशील नेतृत्व में राज्य सरकार राज्य के युवाओं को गुणवत्तापूर्ण शिक्षा प्रदान करने के लिए प्रतिबद्ध है। उन्होंने कहा कि राज्य सरकार ने छात्रों को गुणवत्तापूर्ण उच्च शिक्षा सुनिश्चित करने के लिए राज्य में उच्च शिक्षा परिषद की स्थापना की है। उन्होंने कहा कि केंद्र सरकार द्वारा देश में अनुसंधान गतिविधियों के लिए ४०० करोड़ रुपये रखे गए हैं।
हिमाचल प्रदेश विश्वविद्यालय के कुलपति प्रो. सिकंदर कुमार ने इस अवसर पर उपस्थित मुख्यमंत्री और अन्य गणमान्य व्यक्तियों का स्वागत करते हुए कहा कि हिमाचल प्रदेश विश्वविद्यालय के पचास वर्ष उपलब्धियों की दृष्टि से एक मी का पत्थर साबित हुए हैं। उन्होंने कहा कि इस विश्वविद्यालय से उत्तीर्ण छात्रों ने विभिन्न क्षेत्रों में उत्कृष्ट प्रदर्शन किया है। उन्होंने कहा कि इसका श्रेय विश्वविद्यालय के छात्रों, शिक्षकों और गैर-शिक्षण कर्मचारियों को जाता है। उन्होंने कहा कि विश्वविद्यालय इस साल जुलाई के अंत तक पूरी तरह से कम्प्यूटरीकृत हो जाएगा और बी.एड. की काउंसलिंग को ऑनलाईन किया जाएगा। उन्होंने कहा कि विश्वविद्यालय के परिसर में इस वर्ष के अंत तक वाईफाई की सुविधा प्रदान की जाएगी।
डीन ऑफ स्टडीज प्रो. अरविंद कालिया ने इस अवसर पर वर्ष २०१८-१९ के दौरान विश्वविद्यालय की उपलब्धियों का विस्तार से ब्यौरा दिया।
विधायक और एचपीयू के कार्यकारी परिषद के सदस्य सुभाष ठाकुर और राकेश जम्वाल, महापौर नगर निगम शिमला कुसुम सदरेट, राज्य समन्वयक कौशल विकास नवीन शर्मा, डीन, प्रोफेसर और विभागाध्यक्ष, शिक्षण और गैर शिक्षण स्टाफ के सदस्य और छात्र भी अन्यों सहित इस अवसर पर उपस्थित थे।
भारत वासियो के लिए गर्व की बात प्रधानमंत्री नरेंद्र मोदी दुनिया में छठे पसंदीदा...
रुड़की में खाना खाने के बाद ११ छात्राओं की तबीयत बिगड़ गंभीर
पावटा के विधायक ने इलाके में पौधा रोपण
हिमाचल से घूमकर लौट रहे पर्यटक की कार हादसे में मौत |
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कुमारस्वामी के पास संख्याबल घटकर ९८ हुआ; येदियुरप्पा ने...
अखिल भारतीय विद्यार्थी परिषद महाविद्यालय इकाई द्वारा धरना... | hindi |
/*
* SMP support for R-Mobile / SH-Mobile - r8a7779 portion
*
* Copyright (C) 2011 Renesas Solutions Corp.
* Copyright (C) 2011 Magnus Damm
*
* 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; version 2 of the License.
*
* 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.
*/
#include <linux/kernel.h>
#include <linux/init.h>
#include <linux/smp.h>
#include <linux/spinlock.h>
#include <linux/io.h>
#include <linux/delay.h>
#include <asm/cacheflush.h>
#include <asm/smp_plat.h>
#include <asm/smp_scu.h>
#include <asm/smp_twd.h>
#include "common.h"
#include "pm-rcar.h"
#include "r8a7779.h"
#define AVECR IOMEM(0xfe700040)
#define R8A7779_SCU_BASE 0xf0000000
static struct rcar_sysc_ch r8a7779_ch_cpu1 = {
.chan_offs = 0x40, /* PWRSR0 .. PWRER0 */
.chan_bit = 1, /* ARM1 */
.isr_bit = 1, /* ARM1 */
};
static struct rcar_sysc_ch r8a7779_ch_cpu2 = {
.chan_offs = 0x40, /* PWRSR0 .. PWRER0 */
.chan_bit = 2, /* ARM2 */
.isr_bit = 2, /* ARM2 */
};
static struct rcar_sysc_ch r8a7779_ch_cpu3 = {
.chan_offs = 0x40, /* PWRSR0 .. PWRER0 */
.chan_bit = 3, /* ARM3 */
.isr_bit = 3, /* ARM3 */
};
static struct rcar_sysc_ch *r8a7779_ch_cpu[4] = {
[1] = &r8a7779_ch_cpu1,
[2] = &r8a7779_ch_cpu2,
[3] = &r8a7779_ch_cpu3,
};
#if defined(CONFIG_HAVE_ARM_TWD) && !defined(CONFIG_ARCH_MULTIPLATFORM)
static DEFINE_TWD_LOCAL_TIMER(twd_local_timer, R8A7779_SCU_BASE + 0x600, 29);
void __init r8a7779_register_twd(void)
{
twd_local_timer_register(&twd_local_timer);
}
#endif
static int r8a7779_platform_cpu_kill(unsigned int cpu)
{
struct rcar_sysc_ch *ch = NULL;
int ret = -EIO;
cpu = cpu_logical_map(cpu);
if (cpu < ARRAY_SIZE(r8a7779_ch_cpu))
ch = r8a7779_ch_cpu[cpu];
if (ch)
ret = rcar_sysc_power_down(ch);
return ret ? ret : 1;
}
static int r8a7779_boot_secondary(unsigned int cpu, struct task_struct *idle)
{
struct rcar_sysc_ch *ch = NULL;
unsigned int lcpu = cpu_logical_map(cpu);
int ret;
if (lcpu < ARRAY_SIZE(r8a7779_ch_cpu))
ch = r8a7779_ch_cpu[lcpu];
if (ch)
ret = rcar_sysc_power_up(ch);
else
ret = -EIO;
return ret;
}
static void __init r8a7779_smp_prepare_cpus(unsigned int max_cpus)
{
/* Map the reset vector (in headsmp-scu.S, headsmp.S) */
__raw_writel(__pa(shmobile_boot_vector), AVECR);
shmobile_boot_fn = virt_to_phys(shmobile_boot_scu);
shmobile_boot_arg = (unsigned long)shmobile_scu_base;
/* setup r8a7779 specific SCU bits */
shmobile_scu_base = IOMEM(R8A7779_SCU_BASE);
shmobile_smp_scu_prepare_cpus(max_cpus);
r8a7779_pm_init();
/* power off secondary CPUs */
r8a7779_platform_cpu_kill(1);
r8a7779_platform_cpu_kill(2);
r8a7779_platform_cpu_kill(3);
}
#ifdef CONFIG_HOTPLUG_CPU
static int r8a7779_cpu_kill(unsigned int cpu)
{
if (shmobile_smp_scu_cpu_kill(cpu))
return r8a7779_platform_cpu_kill(cpu);
return 0;
}
static int r8a7779_cpu_disable(unsigned int cpu)
{
/* only CPU1->3 have power domains, do not allow hotplug of CPU0 */
return cpu == 0 ? -EPERM : 0;
}
#endif /* CONFIG_HOTPLUG_CPU */
struct smp_operations r8a7779_smp_ops __initdata = {
.smp_prepare_cpus = r8a7779_smp_prepare_cpus,
.smp_boot_secondary = r8a7779_boot_secondary,
#ifdef CONFIG_HOTPLUG_CPU
.cpu_disable = r8a7779_cpu_disable,
.cpu_die = shmobile_smp_scu_cpu_die,
.cpu_kill = r8a7779_cpu_kill,
#endif
};
| code |
\betagin{document}
\timestle[Compactifications of the Hitchin Moduli Space]{The Algebraic and Analytic
Compactifications\\ of the Hitchin Moduli Space}
\betagin{abstract}
Following the work of Mazzeo-Swoboda-Weiss-Witt \cite{mazzeo2016ends} and
Mochizuki \cite{Mochizukiasymptotic}, there is a map $\overline{\Xi}$ between the
algebraic compactification of the Dolbeault moduli space of $\mathrm{SL}C$
Higgs bundles on a smooth projective curve coming from the
$\mathbb{C}^\ast$ action, and the
analytic compactification of Hitchin's moduli space of solutions to the $\mathsf{SU}(2)$
self-duality equations on a Riemann surface obtained by adding solutions to the decoupled equations, known as ``limiting
configurations''. This map extends the classical Kobayashi-Hitchin correspondence. The main result of this paper is that $\overline{\Xi}$ fails to be continuous at the boundary over a certain subset of
the discriminant locus of the Hitchin fibration.
This suggests the possibility of a third, refined compactification
which dominates both.
\end{abstract}
\maketitle
\section{Introduction}
Let $\Sigmagma$ be a closed Riemann surface of genus $g\geq 2$.
The coarse Dolbeault moduli space of $\mathrm{SL}C$
semistable Higgs bundles on $\Sigmagma$, denoted by $\mathcal{M}D$,
and Hitchin's moduli space of solutions to the $\mathsf{SU}(2)$
self-duality equations on $\Sigmagma$, denoted by $\mathcal{M}H$,
have been extensively studied since their introduction over 35 years ago.
The Kobayashi-Hitchin correspondence, proved in \cite{hitchin1987self},
gives a homeomorphism between these two moduli spaces:
$$\Xi:\mathcal{M}D\isorightarrow \mathcal{M}H\ .$$
The space $\mathcal{M}D$ is naturally a quasiprojective variety
\cite{Nitsure:91, simpson1994moduli}, and similarly to monopole moduli
spaces, $\mathcal{M}H$ fails to be compact.
Recently, there has been interest from several directions on
natural compactifications of these two spaces.
A key feature on the Dolbeault side
is the existence of a $\mathbb{C}^\ast$ action with the Bia{\l}ynicki-Birula
property, and this may be used to define a completion
of $\mathcal{M}D$ as a projective variety
\cite{hausel1998compactification,de2018compactification,fan2022analytic}.
The ideal points are identified with
the $\mathbb{C}^*$ orbits in the complement of the nilpotent cone of $\mathcal{M}D$.
The Hitchin moduli space also admits a more recently introduced compactification, $\overline{\MM}_{\mathrm{Hit}}$,
based on the work of several authors (see \cite{mazzeo2016ends, Mochizukiasymptotic,
Taubes20133manifoldcompactness}).
The boundary of $\overline{\MM}_{\mathrm{Hit}}$ is given by gauge equivalence classes of \emph{limiting
configurations}. This compactification is relevant to many aspects of
Hitchin's moduli space. For more details, we refer the reader to
\cite{DumasNeitzke:19,mazzeo2012limiting,fredrickson2020exponential,fredrickson2022asymptotic,ott2020higgs,katzarkov2015harmonic,chen2022asymptotic},
and the references therein.
By the work of \cite{mazzeo2016ends,Mochizukiasymptotic}, there is a
natural extension
$$\overline{\Xi}:\overline{\MM}_{\mathrm{Dol}}\longrightarrow \overline{\MM}_{\mathrm{Hit}}$$
of the Kobayashi-Hitchin correspondence to the two compactifications
described above, and
it is of interest to study the geometry of this map.
This involves another key feature of Hitchin's moduli space; namely,
spectral curves.
Spectral curves and spectral data \cite{hitchin1992lie} play a central
role in the realization of the Dolbeault moduli space as an algebraically complete integrable
system $\mathcal{H}: \mathcal{M}D\to \mathcal{B}$.
In the case of $\mathrm{SL}C$, the base $\mathcal B$ is the space of
holomorphic quadratic differentials on $\Sigmagma$. Given
$q\in H^0(K^2)$, one obtains a (scheme theoretic) spectral curve $S_q$. This curve is
reduced if $q\neq 0$, irreducible if $q$ is not the square of an abelian
differential, and
smooth if $q$ has simple zeros. We let $\mathcal{B}^{\rm
reg}\mathfrak{su}bset \mathcal{B}$ denote the open cone of quadratic differentials
with simple zeros.
The ideal points of both compactifications $\overline{\MM}_{\mathrm{Dol}}$ and $\overline{\MM}_{\mathrm{Hit}}$ have
associated nonzero quadratic differentials, and therefore spectral curves.
We write $\overline{\MM}_{\mathrm{Dol}}r$ for the elements in $\overline{\MM}_{\mathrm{Dol}}$ with smooth spectral curves,
and $\overline{\MM}_{\mathrm{Dol}}s=\overline{\MM}_{\mathrm{Dol}}\setminus \overline{\MM}_{\mathrm{Dol}}r$ for those with singular spectral curves;
similarly for $\overline{\MM}_{\mathrm{Hit}}r$ and $\overline{\MM}_{\mathrm{Hit}}s$. We then have the following result.
\betagin{theorem}
The restriction of the compactified Kobayashi-Hitchin map
$\overline{\Xi}:\overline{\MM}_{\mathrm{Dol}}\to \overline{\MM}_{\mathrm{Hit}}$ to the locus with smooth associated spectral
curves
defines a homeomorphism $\overline{\MM}_{\mathrm{Dol}}^{\mathrm{reg}}\sigmameq \overline{\MM}_{\mathrm{Hit}}^{\mathrm{reg}}$.
On the singular spectral curve locus, however,
$\overline{\Xi}^{\sigmang}:\overline{\MM}_{\mathrm{Dol}}^{\sigmang}\to \overline{\MM}_{\mathrm{Hit}}^{\sigmang}$ is neither surjective nor injective.
\end{theorem}
It is convenient to analyze the behavior along rays in $\mathcal{B}$, where
the spectral curve is simply rescaled.
Let $q\neq 0$ be a quadratic differential and $\overline{\MM}_{\mathrm{Dol}}q$ (resp.\ $\overline{\MM}_{\mathrm{Hit}}q$) be the
points in $\overline{\MM}_{\mathrm{Dol}}$ (resp.\ $\overline{\MM}_{\mathrm{Hit}}$) with spectral curves $S_{tq}$, $t\in
\mathbb{C}^\ast$.
The restriction of $\overline{\Xi}$ gives us the map $\overline{\Xi}_q:\overline{\MM}_{\mathrm{Dol}}q\to \overline{\MM}_{\mathrm{Hit}}q$.
We shall study the continuous behavior of $\overline{\Xi}_{q}$ for points in the
fiber of $tq$ as $t\to \infty$.
For convenience, we set
$\mathcal{M}_{q^\ast}:=\overline{\MM}_{\mathrm{Dol}}q\cap \mathcal{M}D$.
When $q$ is irreducible, i.e.\ not a square, all elements in $\mathcal{M}_q$ are stable.
Via the Hitchin \cite{hitchin1987stable} and Beauville-Narasimhan-Ramanan (BNR) correspondence \cite{bnr1989spectral}, this reduces the
description of the fiber $\mathcal{H}^{-1}(q)$
to the characterization of rank $1$ torsion free sheaves on the integral
curve $S_q$.
In \cite{rego1980compactified}, parameter spaces for rank $1$ torsion free
sheaves on algebraic curves with Gorenstein singularities were studied in
the context of compactified Jacobians, and the crucial notion of a
parabolic module was introduced. This was extensively investigated by Cook in
\cite{cook1993local,cook1998compactified}, partially following ideas of
Bhosle \cite{Bhosle:92}. For simple plane curve singularities of the type appearing
in spectral curves, one makes use of the local classification of torsion free
modules of Greuel-Kn\"orrer \cite{Greuel1985}. These methods were applied
to study the Hitchin fibration by Gothen-Oliveira in
\cite{gothen2013singular} (see also \cite{kydonakis2022monodromy} for
recent study). In parallel, Horn \cite{horn2022semi} defines a stratification of
$\mathcal{M}_q=\bigcup_D\mathcal{M}_{q,D}$ by certain effective
divisors contained in the divisor of $q$ (see Section
\ref{sec:divisor-stratification},
and also \cite{hornna2022geometry} for
the more general situation).
Using the results from these references, we have a reinterpretation of Mochizuki's construction
\cite{Mochizukiasymptotic}. This leads to the following result.
\betagin{theorem}
Let $q\neq0$ be an irreducible quadratic differential.
\betagin{enumerate}
\item If $q$ has only zeros of odd order, then $\overline{\Xi}_q$ is
continuous.
\item If $q$ has at least one zero of even order, then for each $D\neq 0$
there exists an integer $n_D\geq 1$ so that for any Higgs bundle
$(\mathcal{F},\mathrm{ps}i)\in \mathcal{M}_{q,D}$, there exist $n_D$ sequences of
Higgs bundles $(\mathcal{E}_i^k,\varphi_i^k)$ with $k=1,\ldots, n_D$
satisfying
\betagin{itemize}
\item $\lim_{i\to \infty}(\mathcal{E}_i^k,\varphi_i^k)=(\mathcal{F},\mathrm{ps}i)$ for $k=1,\ldots,n_D$,
\item
and if we write $$\eta^k:=\lim_{i\to
\infty}\overline{\Xi}_q(\mathcal{E}_i^k,\varphi_i^k)\quad ,\quad \xi:=\lim_{i\to
\infty}\overline{\Xi}_q(\mathcal{F},t_i\mathrm{ps}i)\ ,$$ for some sequence
$t_i\in\mathbb{R}^+$, $t_i\to+\infty$, then $\xi,\eta^1,\ldots,\eta^{n_D}$ are $n_D+1$
different limiting configurations.
\end{itemize}
\end{enumerate}
\end{theorem}
When $q$ is reducible, the description of Higgs bundles in $\mathcal{M}_q$
becomes more complicated because, among other things, of the existence of
strictly semistable objects.
To understand this, we use the local descriptions of Gothen-Oliveira
and Mochizuki (see \cite{gothen2013singular,Mochizukiasymptotic}).
Our result, which focuses on the stable locus, is the following.
\betagin{theorem}
Suppose $q\neq 0$ is reducible. Let $\overline{\MM}_{\mathrm{Dol}}q^{\asta}$ denote the stable locus
of $\overline{\MM}_{\mathrm{Dol}}q$. If $g\geq 3$, then the restriction map
$\overline{\Xi}_q|_{\overline{\MM}_{\mathrm{Dol}}q^{\asta}}$ is discontinuous. However, if $g=2$, the map
$\overline{\Xi}_q|_{\overline{\MM}_{\mathrm{Dol}}q^{\asta}}$ is continuous.
\end{theorem}
We also note that there is recent work of Mochizuki and Szab\'{o}
\cite{mochizuki2023asymptotic} on the asymptotic behavior for families of
Higgs bundles in the higher rank case.
This paper is organized as follows:
in Section \ref{sec_background_Higgs_bundles}, we provide an overview of Higgs bundles and BNR correspondence.
In Section \ref{sec_filtered_bundles_and_compactness}, we delve into the concepts of
filtered bundles and their compactness properties.
Section \ref{sec_the_algebraic_and_analytic_compactifications} defines
the algebraic and analytic compactifications.
Section \ref{sec_parabolic_modules}
introduces parabolic modules and examines their connection to spectral curves.
The main results for Hitchin fibers with irreducible singular spectral curves are established
in Section \ref{sec_irreducible_singular_fiber}. In Section \ref{sec_reducible_singular_fiber},
the results for the reducible case are proven.
Finally, in Section \ref{sec_compactified_Kobayashi_Hitchin_map},
we construct the compactified Kobayashi-Hitchin map and prove the main results.
The Appendix, based on the work of Greuel-Kn\"orrer, calculates some invariants of rank $1$ torsion free
sheaves on the spectral curves we consider.
\textbf{Acknowledgements.} We extend our sincere gratitude to Takur\={o} Mochizuki for his valuable insights and stimulating
discussions during the BIRS conference in 2021.
The authors also wish to express their gratitude to a great many people for their interest and helpful comments.
Among them are Mark de Cataldo, Ron Donagi, Simon Donaldson, Laura
Fredrickson, Johannes Horn, Laura Schaposnik, Shizhang Li, Jie Liu, Tony
Pantev, Thomas Walpuski, Daxin Xu. S.H is supported by NSFC grant
No.12288201. R.W.'s research is supported by NSF grants DMS-1906403 and
DMS-2204346, and he thanks the Max Planck Institute for Mathematics in
Bonn for its hospitality and financial support.
The authors also gratefully acknowledge the support of the MSRI
under NSF grant DMS-1928930 during the program ``Analytic and Geometric Aspects of Gauge Theory'', Fall 2022.
\section{Background on Higgs bundles}
\label{sec_background_Higgs_bundles}
This section gives a very brief overview of
the Dolbeault and Hitchin moduli spaces, spectral curve descriptions,
and the nonabelian Hodge correspondence. For more details on these
topics, see \cite{hitchin1987self,
hitchin1987stable, Simpson1992, Wentworth2016}.
\mathfrak{su}bsection{Higgs bundles}
As in the Introduction, throughout this paper
$\Sigmagma$ will denote a closed Riemann surface of genus $g\geq 2$ with
structure sheaf $\mathcal{O}=\mathcal{O}_\Sigmagma$ and
canonical bundle $K=K_\Sigmagma$. Let $E\to \Sigmagma$ be a complex vector bundle.
A Higgs bundle consists of a pair $(\mathcal{E},\varphi)$, where
$\mathcal{E}$ is a holomorphic bundle and $\varphi\in
H^0(\mathrm{End}(\mathcal{E})\otimesimesmes K)$.
If $\rightarrownglek(\mathcal{E})=1$, then a Higgs field is just an abelian
differential $\omegaega$.
The pair $(\mathcal{E},\varphi)$ is called
an $\mathrm{SL}(2,\mathbb{C})$ Higgs bundle if $\mathrm{rank}(E)=2$,
$\det(\mathcal{E})$ has a fixed isomorphism with the trivial bundle, and $\mathrm{Tr}(\varphi)=0$. In this
paper we will focus mainly on $\mathrm{SL}(2,\mathbb{C})$ Higgs bundles,
but the rank $1$ case will also be important.
Let $(\mathcal{E},\varphi)$ be an $\mathrm{SL}(2,\mathbb{C})$ Higgs bundle.
A (proper) Higgs subbundle of $(\mathcal{E},\varphi)$ is a holomorphic line
bundle $\mathcal{L}\mathfrak{su}bset \mathcal{E}$ that is
$\varphi$-invariant, i.e.\ $\varphi:\mathcal{L}\to \mathcal{L}\otimesimesmes K$.
In this case, the restriction
$\varphi_{\mathcal{L}}:=\varphi\bigr|_{\mathcal L}$, makes
$(\mathcal{L},\varphi_{\mathcal{L}})$ a rank $1$ Higgs bundle.
Moreover, $\varphi$ induces a Higgs bundle structure on the quotient
$\mathcal{E}/\mathcal{L}$.
We say $(\mathcal{E},\varphi)$ is stable (resp.\ semistable) if for all Higgs
subbundles $\mathcal{L}$, $\deg \mathcal{L}<0$ (resp.\ $\deg \mathcal{L}\leq 0$). We say $(\mathcal{E},\varphi)$ is polystable
if $(\mathcal{E},\varphi)\sigmameq(\mathcal{L},\omega)\oplus (\mathcal{L}^{-1},-\omega)$, where $\mathcal{L}$ is a
degree zero holomorphic line bundle and $\omega \in H^0(K)$.
If $(\mathcal{E},\varphi)$ is strictly semistable, i.e.\ semistable but not stable,
the Seshadri filtration \cite{seshadri1967space} gives a unique Higgs
subbundle $0\mathfrak{su}bset (\mathcal{L},\omega)\mathfrak{su}bset (\mathcal{E},\varphi)$ with
$\deg(\mathcal{L})=\frac{1}{2}\deg(\mathcal{E})=0$. Write $(\mathcal{L}',\omega'):=(\mathcal{E},\varphi)/(\mathcal{L},\omega)$, then we have $\omega'=-\omega$ and $\mathcal{L}'=\mathcal{L}^{-1}$.
The associated graded bundle $\mathrm{Gr}(\mathcal{E},\varphi)=(\mathcal{L},\omega)\oplus (\mathcal{L}^{-1},-\omega)$ of this filtration
is a polystable $\mathrm{SL}C$ Higgs bundle. We say that $(\mathcal{E},\varphi)$ is
S-equivalent to $\mathrm{Gr}(\mathcal{E},\varphi)$.
Holomorphic bundles $\mathcal{E}$ with underlying $C^\infty$ bundle $E$
are in 1-1 correspondence with $\bar\partialrtial$-operators $\bar\partialrtial_E:
\Omega^0(E)\to \Omega^{0,1}(E)$. We use the
notation $\mathcal{E}:=(E,\bar{\partialrtial}_E)$.
Let $\mathcal{C}$ denote the space of pairs $ (\bar\partialrtial_E,\varphi)$,
$\bar\partialrtial_E\varphi=0$.
Let $\mathcal{C}^s$ and $\mathcal{C}^{ss}$ denote the subspaces of $\mathcal{C}$ where the Higgs
bundles are stable (resp.\ semistable).
The complex gauge transformation group $\mathcal{G}C:=\mathrm{Aut}(E)$ has a right action
on $\mathcal{C}$ by defining for $g\in\mathcal{G}C$,
$(\bar{\partial}_E,\varphi)g:=(g^{-1}\circ\bar{\partial}\circ g,g^{-1}\circ\varphi\circ g)$.
There is a quasiprojective scheme $\mathcal{M}D$ whose closed points are in 1-1
correspondence with polystable Higgs bundles constructed via (finite dimensional) Geometric
Invariant Theory (see \cite{Nitsure:91,
simpson1994moduli}). In \cite{Fan:22} it was shown that the infinite
dimensional quotient $\mathcal{C}^{\mathrm{ss}}\sslash\mathcal{G}C$,
where the double slash indicates that S-equivalent orbits are identified,
admits the structure of a complex analytic space that is
biholomorphic to the analytification $\mathcal{M}D^{\rm an}$ of $\mathcal{M}D$.
Henceforth, we shall work in the complex analytic category, identify the
algebro-geometric and gauge theoretic moduli spaces, and simply denote
them both by $\mathcal{M}D$.
We note that
the set of stable Higgs bundles modulo gauge transformations,
$\mathcal{M}D^s:=\mathcal{C}^{\mathrm{s}}\sslash\mathcal{G}C$, is a geometric quotient and an open subset of $\mathcal{M}D$.
Finally, notice that the a pair $(\mathcal{E},\varphi)$ is stable (resp.\
semistable) if and only if the same is true for
$(\mathcal{E},\lambdabda\varphi)$, $\lambdabda\in \mathbb{C}^\ast$. Hence,
$\mathcal{M}D$ admits an action of $\mathbb{C}^\ast$ that preserves $\mathcal{M}D^{s}$.
Though $\mathcal{M}D$ is only
quasiprojective, the $\mathbb{C}^\ast$ action satisfies the
Bia{\l}ynicki-Birula property:
\betagin{theorem} [{\cite{hitchin1987self,Simpson1992}}]
\label{thm:proper-action}
For any $[(\mathcal{E},\varphi)]\in \mathcal{M}D$,
$$
\lim_{\lambdabda\to 0}
\lambdabda\cdotot[(\mathcal{E},\varphi)]:=\lim_{\lambdabda\to 0}[(\mathcal{E},\lambdabda\varphi)]
$$
exists in $\mathcal{M}D$.
\end{theorem}
\mathfrak{su}bsection{Spectral curves and the Hitchin fibration}
The Hitchin map
is defined as
\betagin{equation*}
\mathcal{H}: \mathcal{M}D \longrightarrow H^0(K^2)\ , \ [(\mathcal{E},\varphi)] \mapsto \det(\varphi)\ ,
\end{equation*}
where $H^0(K^2)=:\mathcal{B}$ is known as the Hitchin base.
Hitchin \cite{hitchin1987self,hitchin1987stable} showed that $\mathcal{H}$ is a
proper map and a fibration by abelian varieties
over the open cone $\mathcal{B}^{\rm reg}\mathfrak{su}bset\mathcal{B}$
consisting of nonzero quadratic differentials with only simple zeros.
The discriminant
locus $\mathcal{B}^{\rm sing}:=\mathcal{B}\setminus \mathcal{B}^{\rm reg}$ consists of quadratic differentials that are
either identically zero or have at least one zero with multiplicity.
For $q\in \mathcal{B}$, let
$\mathcal{M}_q:=\mathcal{H}^{-1}(q)$. The ``most singular fiber''
$\mathcal{M}_0$ is called the \emph{nilpotent cone}.
Consider the total space $\mathrm{Tot}(K)$ of $K$,
along with its
projection $\pi: \mathrm{Tot}(K) \to \Sigmagma$. The pullback bundle $\pi^* K$ has
a tautological section, which we denote by $\lambdabda \in
H^0(\mathrm{Tot}(K),\pi^\ast K)$. Given any $q\neq 0 \in H^0(K^2)$,
the \emph{spectral curve} $S_q$ associated with $q$ is the
zero scheme of the section $\lambdabda^2 - \pi^\ast q \in H^0(\mathrm{Tot}(K),\pi^* K)$.
This is a reduced, but possibly reducible, projective algebraic curve.
The restriction of $\pi$ to $S_q$, also denoted by $\pi: S_q \to \Sigmagma$,
is a double covering branched along the zeros of $q$.
The spectral curve $S_q$ is smooth if and only if $q$ has only simple
zeros. It is reducible if and only if $q = -\omegaega \otimesimesmes \omegaega$ for some $\omegaega \in H^0(K)$.
We shall refer to such quadratic differentials as \emph{reducible}, and
\emph{irreducible} otherwise.
There is a noteworthy observation regarding irreducible spectral curves.
\betagin{proposition}[cf.\ {\cite{hitchin1987stable}}]
Let $(\mathcal{E},\varphi)$ be a Higgs bundle with $q = \det(\varphi)$, and suppose $q$ is irreducible. Then
$(\mathcal{E},\varphi)$ has no proper invariant subbundles. In particular, $(\mathcal{E},\varphi)$ is stable.
\end{proposition}
\betagin{proof}
Suppose $\mathcal{L}\mathfrak{su}bset\mathcal{E}$ is $\varphi$-invariant, and
let $\varphi_{\mathcal L}$ be the restriction.
Then
$$
\det\varphi=-\frac{1}{2}\mathrm{Tr}(\varphi^2)=-(\varphi_{\mathcal L})^2\ ,
$$
contradicting the assumption.
\end{proof}
Let us emphasize that being reducible is not the same as having only even zeros.
To see this,
suppose that $\mathrm{Div}(q)=2\lambdabdam'$. Then $K\sigmameq\mathcal{O}(\lambdabdam')\otimesimesmes \mathcal{I}$,
where $\mathcal{I}$ is a 2-torsion point in the Jacobian.
The spectral curve $S_q$ is reducible if and only if $\mathcal{I}$ is trivial.
\mathfrak{su}bsection{Rank 1 torsion free sheaves and the BNR correspondence}
In this subsection, we provide some
background on rank 1 torsion free sheaf theory over spectral curves in the
context of the Hitchin and BNR correspondence, as developed in \cite{hitchin1987stable,bnr1989spectral}.
Let $S$ be a reduced and irreducible complex projective curve and $\mathcal{O}_S$ its structure sheaf.
The moduli space of invertible sheaves on $S$ is denoted by $\mathrm{Pic}(S)$, and
$\mathrm{Pic}^d(S)\mathfrak{su}bset \mathrm{Pic}(S)$ is the degree $d$ component. If $\mathcal F$ is a
coherent analytic sheaf on $S$, we can define its cohomology groups
$H^i(X,\mathcal{F})$. Since $\dim S=1$, $H^i(X,\mathcal{F})=0$ for $i\geq 2$. The Euler characteristic
is defined as $\mathrm{ch}i(\mathcal{F})=\dim H^0(X,\mathcal{F})-\dim
H^1(X,\mathcal{F})$. The degree of a torsion free sheaf $\mathcal{F}$ is given by
$\deg(\mathcal{F})=\mathrm{ch}i(\mathcal{F})-\rightarrownglek(\mathcal{F})\mathrm{ch}i(\mathcal{O}_S)$.
If $\mathcal{F}$ is locally free, then
$\deg(\mathcal{F})$ coincides with the degree of the invertible sheaf $\det(\mathcal{F})$.
Let $\overline{\mathrm{Pic}}^d(S)$ be the moduli space of degree $d$ rank 1 torsion free
sheaves on $S$, and $\overline{\mathrm{Pic}}(S)=\mathrm{pr}od_{d\in \mathbb{Z}}\overline{\mathrm{Pic}}^d(S)$ \cite{cyril1979compactification}.
Then $\overline{\mathrm{Pic}}^d(S)$ is an irreducible projective scheme containing
$\mathrm{Pic}^d(S)$ as an open subscheme. When $S$
is smooth, we have $\overline{\mathrm{Pic}}^d(S)=\mathrm{Pic}^d(S)$. The relationship to Higgs
bundles is given by the following.
\betagin{theorem}[{\cite{hitchin1987stable,bnr1989spectral}}]
\label{thm_BNRcorrespondence}
Let $q\in H^0(K^2)$ be an irreducible quadratic differential with
spectral curve $S_q$. There is a bijective correspondence between
points in $\overline{\mathrm{Pic}}(S_q)$ and
isomorphism classes of rank 2 Higgs pairs $(\mathcal{E},\varphi)$ with
$\mathrm{Tr}(\varphi)=0$ and
$\det(\varphi)=q$. Explicitly:
if $\mathcal{L}\in \overline{\mathrm{Pic}}(S_q)$, then $\mathcal{E}:=\pi_{\ast}(\mathcal{L})$ is a rank 2 vector
bundle, and the homomorphism $\pi_{\ast}\mathcal{L}\to \pi_{\ast}\mathcal{L}\otimesimesmes K\cong
\pi_{\ast}(\mathcal{L}\otimesimesmes \pi^{\ast}K)$ given by multiplication by the
canonical section $\lambda$ defines the Higgs field $\varphi$.
\end{theorem}
This correspondence gives the very useful exact sequence
\betagin{equation} \label{eq_BNR}
\betagin{split}
0\to \mathcal{L}(-\Deltata)\to
\pi^{\ast}\mathcal{E}\xrightarrow{\pi^{\ast}\varphi-\lambda}\pi^{\ast}\mathcal{E}\otimesimesmes
\pi^{\ast}K\to \mathcal{L}\otimesimesmes \pi^{\ast}K\to 0\ ,
\end{split}
\end{equation}
where $\mathcal{O}_S(\Deltata):=K_S\otimesimesmes \pi^{\ast}K_{\Sigmagma}^{-1}$ is the ramification divisor.
This will be used below in Section
\ref{sec_irreducible_singular_fiber}.
Let $q$ be a quadratic differential with only simple zeros, and define a
divisor on $S$ by $\lambdabdam=\mathrm{Div}(\lambdabda)$.
Then $\lambdabdam$ is the ramification divisor of the map $\pi:S\to \Sigmagma$.
By the Riemann-Hurwitz formula, the genus of $S$ is $g(S)=4g-3$.
Furthermore, for any $\mathcal{L}\in \mathrm{Pic}(S)$, Riemann-Roch gives $\deg(\pi_{\ast}\mathcal{L})=\deg(\mathcal{L})-(2g-2).$
The $\mathrm{SL}C$ Higgs bundles are characterized by
\betagin{equation} \label{eq_definition_T}
\mathcal{T}:=\{\mathcal{L}\in\mathrm{Pic}^{2g-2}(S)\mid\det(\pi_{\ast}\mathcal{L})=\mathcal{O}_\Sigmagma\}.
\end{equation}
By the Hitchin-BNR correspondence (Theorem \ref{thm_BNRcorrespondence}), the map $\mathrm{ch}i_{BNR}:\mathcal{T}\to \mathcal{M}_q$ is a bijection.
The branched double cover $\pi:S\to \Sigmagma$
is given by an involution $\sigmagma:S\to S$.
We have the norm map
$\mathrm{Nm}_{S/\Sigmagma}:\mathrm{Jac}(S)\to \mathrm{Jac}(\Sigmagma)$, where $\mathrm{Nm}_{S/\Sigmagma}(\mathcal{O}_{S}(D)):=\mathcal{O}_{\Sigmagma}(\pi(D))$. The Prym variety is defined as
$$\mathrm{Prym}(S/\Sigmagma):=\ker(\mathrm{Nm}_{S/\Sigmagma})=\{\mathcal{L}\in\mathrm{Pic}(S)\mid\mathcal{L}\otimesimesmes
\sigmagma^{\ast}\mathcal{L}=\mathcal{O}_S\}\ .$$
Also, we have $\det(\pi_{\ast}\mathcal{L})\cong \mathrm{Nm}_{S/\Sigmagma}(\mathcal{L})\otimesimesmes K^{-1}$.
Thus, $\mathcal{T}$ can be expressed as
$$\mathcal{T}=\{\mathcal{L}\in\mathrm{Pic}^{2g-2}(S)\mid\mathrm{Nm}_{S/\Sigmagma}(\mathcal{L})\cong K\}\ .$$
Hence, $\mathcal{T}$ is a torsor over $\mathrm{Prym}(S,\Sigmagma)$.
Explicitly, by choosing
$\mathcal{L}_0\in \mathcal{T}$, we obtain an isomorphism $\mathcal{T}\isorightarrow
\mathrm{Prym}(S,\Sigmagma)$ given by $\mathcal{L}\to \mathcal{L}\otimesimesmes\mathcal{L}_0^{-1}$.
To summarize, we have the following:
\betagin{proposition}
Let $q$ be a quadratic differential with simple zeros. Then $\mathcal{M}_q\cong
\mathcal{T}\cong \mathrm{Prym}(S,\Sigmagma)$.
\end{proposition}
If $q\neq 0$ is irreducible but nongeneric, the spectral curve $S$ is
singular and irreducible. We may still define the set $\overline{\MT}\mathfrak{su}bset \overline{\mathrm{Pic}}^{2g-2}(S)$ as follows:
\betagin{equation*}
\overline{\MT}:=\{\mathcal{L}\in\overline{\mathrm{Pic}}^{2g-2}(S)\mid\det(\pi_{\ast}\mathcal{L})\cong \mathcal{O}_\Sigmagma\}\
,
\end{equation*}
We also set $\mathcal{T}:=\overline{\MT}\cap \mathrm{Pic}^{2g-2}$.
Then $\overline{\MT}$ is the natural compactification of $\mathcal{T}$ induced by the inclusion $\mathrm{Pic}^{2g-2}(S)\mathfrak{su}bset \overline{\mathrm{Pic}}(S)$. The BNR correspondence, as stated in Theorem \ref{thm_BNRcorrespondence}, implies that
$\mathrm{ch}i_{\mathrm{BNR}}:\overline{\MT}\to \mathcal{M}_q$ is an isomorphism.
\mathfrak{su}bsection{The Hitchin moduli space and the nonabelian Hodge
correspondence} \label{sec:nah}
We now recall the well-known nonabelian Hodge correspondence (NAH),
which relates the space of flat $\mathrm{SL}(2,\mathbb{C})$ connections, Higgs bundles, and solutions to the Hitchin equations. This result was developed in the work of Hitchin \cite{hitchin1987self}, Simpson \cite{Simpson1988Construction}, Corlette \cite{corlette1988flat}, and Donaldson \cite{donaldson1987twisted}.
As above, let $E$ be a trivial, smooth, rank 2 vector bundle over a Riemann surface
$\Sigmagma$, and let
$H_0$ be a fixed Hermitian metric on $E$.
We denote by $\mathfrak{sl}(E)$ (resp.\ $\mathfrak{su}(E)$) the bundle of
traceless (resp.\ traceless skew-hermitian) endomorphisms of $E$.
Let $A$ be a unitary (with respect to $H_0$)
connection on $E$ that induces the trivial connection on $\det E$,
and let $\phi \in \Omega^1(i\mathfrak{su}(E))$. We will sometimes also
refer to $\phi$ as a Higgs field.
The Hitchin equations for the pair $(A,\phi)$ are given by:
\betagin{equation} \label{eq_Hitchin_real}
\betagin{split}
F_{A} + \phi\wedgedge\phi= 0\ ,\ d_A \phi = d_A^\ast\phi= 0\ .
\end{split}
\end{equation}
If we split the Higgs field into type:
$\phi = \varphi + \varphi^{\daggergger}$, with $\varphi \in \Omega^{1,0}(\mathfrak{sl}(E))$, then
\eqref{eq_Hitchin_real} is equivalent to:
\betagin{equation} \label{eq_Hitchin}
\betagin{split}
F_{A} + [\varphi,\varphi^{\daggergger}]= 0\ ,\ \bar{\partialrtial}_A \varphi = 0\ .
\end{split}
\end{equation}
Notice that $(\bar\partialrtial_E,\varphi)$ then defines an $\mathrm{SL}(2,\mathbb{C})$ Higgs bundle.
The Hitchin moduli space, denoted by $\mathcal{M}H$, is the moduli space of solutions to the Hitchin equation, given by
$$\mathcal{M}H := \{(A,\phi) \mid (A,\phi)\;\mathrm{satisfies}\;\eqref{eq_Hitchin_real}\}/\mathcal{G},$$
where $\mathcal{G}$ is the gauge group of unitary automorphisms of $E$.
Recall that a flat connection $\mathcal{D}$ is called completely reducible if and
only if it is a direct sum
of irreducible flat connections. The NAH can be summarized as follows:
\betagin{theorem}[{\cite{hitchin1987self,simpson1990harmonic,corlette1988flat,donaldson1987twisted}}]
A Higgs bundle $(\mathcal{E},\varphi)$ is polystable if and only if there
exists a Hermitian metric $H$ such that the corresponding Chern
connection $A$ and Higgs field $\phi=\varphi+\varphi^{\daggergger}$ solve the Hitchin
equations \eqref{eq_Hitchin_real}. Moreover, the connection
$\mathcal{D}$ defined by $\mathcal{D}=\nablabla_A+\phi$ is a completely
reducible flat connection, and it is irreducible if and only if $(\mathcal{E},\varphi)$ is stable.
Conversely, a flat connection $\mathcal{D}$ is completely reducible if and
only if there exists a Hermitian metric $H$ on $E$ such that when we
express $\mathcal{D}=\nablabla_A+\varphi+\varphi^{\daggergger}$, we have
$\bar{\partialrtial}_{\mathcal{E}}\varphi=0$. Moreover, the corresponding
Higgs bundle
$(\mathcal{E},\varphi)$ is polystable, and it is stable if and only if $\mathcal{D}$ is irreducible.
\end{theorem}
The nonabelian Hodge correspondence defines the following
Kobayashi-Hitchin homeomorphism
\betagin{equation*}
\betagin{split}
\Xi:\mathcal{M}D\to \mathcal{M}H\ ,
\end{split}
\end{equation*}
which is a diffeomorphism to irreducible solutions of
\eqref{eq_Hitchin_real} when restricted to the stable locus.
Finally, we note that there is an
action of $S^1$ on $\mathcal{M}H$ defined by $(A,\phi) \to (A,e^{i\theta}\cdotot
\phi)$, where
$e^{i\theta}\cdotot\phi=e^{i\theta}\varphi+e^{-i\theta}\varphi^\daggergger$.
With respect to this and the $S^1\mathfrak{su}bset\mathbb{C}^\ast$ action on $\mathcal{M}D$,
the map $\Xi$ is $S^1$-equivariant.
\section{Filtered bundles and compactness} \label{sec_filtered_bundles_and_compactness}
Filtered (or parabolic) bundles
are described, for example, in \cite{simpson1990harmonic}.
They play a key role in the analytic compactification.
This section provides a brief overview of filtered line bundles and
demonstrates a compactness result.
\mathfrak{su}bsection{Filtered line bundles and nonabelian Hodge}
Let $Z$ be a finite collection of distinct points on a closed Riemann surface $\Sigmagma$, and
let $\Sigmagma' = \Sigmagma\setminus Z$. Viewing $\Sigmagma$ as a projective
algebraic curve, an algebraic line bundle $L$ over $\Sigmagma'$ is a
line bundle defined by regular transition functions on Zariski
open sets over $\Sigmagma'$. The sheaf of sections of $L$ can be extended
in infinitely many different ways over $Z$ to obtain
coherent (invertible) sheaves on $\Sigmagma$.
The sections of $L$ are then realized as
meromorphic sections of an extension, regular on $\Sigmagma'$.
A \emph{filtered line bundle} $\mathcal{F}_{\ast}(L)$ is an algebraic line bundle
$L$ with a collection of coherent extensions $L_{\alpha}$ across the punctures $
Z$ such that $L_{\alpha} \mathfrak{su}bset L_{\beta}$ for $\alpha \geq \beta$, for a fixed
sufficiently small $\epsilon$,
$L_{\alpha-\epsilon}=L_{\alpha}$, and $L_{\alpha} = L_{\alpha+1} \otimesimesmes
\mathcal{O}_\Sigmagma(Z)$.
Let $\mathrm{Gr}_{\alpha}= L_{\alpha+\epsilon}/L_{\alpha}$ denote the quotient (torsion)
sheaf. A value $\alphapha$ where $\mathrm{Gr}_{\alpha} \neq 0$ is called a jump.
Since we are considering line bundles, for each $p$ in
the support of $\mathrm{Gr}_{\alpha_p}$,
there is exactly
one jump $\alpha_p$ in the interval $[0,1)$. The collection of jumps $\{\alphapha_p\}$ fully determines the filtered
bundle structure. If we denote by
$\mathcal{L}:=L_{0}$, the degree of a filtered line bundle is defined as
$$\deg(\mathcal{F}_{\ast}(L)) = \deg(\mathcal{L}) + \mathfrak{su}m_{p\in Z}\alphapha_p
\ .$$
Alternatively,
a \emph{weighted line bundle} is a pair $(\mathcal{L},\mathrm{ch}i)$ where $\mathcal{L}\to \Sigmagma$
is a holomorphic line bundle and $\mathrm{ch}i:Z\to \mathbb{R}$ is a weight function.
The degree of a weighted bundle is defined as
$$\deg(\mathcal{L},\mathrm{ch}i) = \deg(\mathcal{L}) + \mathfrak{su}m_{p\in Z}\mathrm{ch}i_p\ .$$
The concepts of filtered line bundles and weighted line bundles are
nearly equivalent. Namely, given
a filtered line bundle $\mathcal{F}_{\ast}(L)$, we define $\mathcal{L}:=\mathcal{L}_{0}$ and
$\mathrm{ch}i_p=\alphapha_p$.
Conversely, given a weighted line bundle $(\mathcal{L},\mathrm{ch}i)$,
let $\alphapha_p=\mathrm{ch}i_p+n_p$, where $n_p\in \mathbb{Z}$ is the unique integer
so that $0\leq \mathrm{ch}i_p+n_p<1$. A filtered bundle $\mathcal{F}_{\ast}(L)$ is then determined by
setting $L_{0} =
\mathcal{L}(-\mathfrak{su}m_{p\in Z} n_p p)$ with jumps $\alphapha_p$.
Clearly,
$\deg(\mathcal{F}(L))=\deg(\mathcal{L},\mathrm{ch}i)$.
We shall use the notation $\mathcal{F}_{\ast}(\mathcal{L},\mathrm{ch}i)$ for the filtered
bundle associated to a weighted bundle $(\mathcal{L},\mathrm{ch}i)$ in this way.
Different weighted bundles can give rise to the same filtered bundle.
The following is a fact that will be frequently used in this paper.
If $D=\mathfrak{su}m_{x\in Z}d_xx$ is a divisor supported on $Z$,
let
$$
\mathrm{ch}i_D(x):=\betagin{cases}d_x\ ,&x\in Z\ ;\\ 0\ ,&x\in\Sigmagma\setminus Z\ .
\end{cases}
$$
Then for any weighted bundle $(\mathcal{L},\mathrm{ch}i)$ we have
$\mathcal{F}_\ast(\mathcal{L}(D),\mathrm{ch}i-\mathrm{ch}i_{D})=\mathcal{F}_\ast(\mathcal{L},\mathrm{ch}i)$.
Let $(\mathcal{L}_1,\mathrm{ch}i_1)$ and $(\mathcal{L}_2,\mathrm{ch}i_2)$ be two weighted lines bundles. We define the tensor product
$$(\mathcal{L}_1,\mathrm{ch}i_1)\otimesimesmes (\mathcal{L}_2,\mathrm{ch}i_2):=(\mathcal{L}_1\otimesimesmes \mathcal{L}_2,\mathrm{ch}i_1+\mathrm{ch}i_2)\
.$$
Then the degree is additive on tensor products. For filtered bundles, we
\emph{define}
\betagin{equation}\label{eqn:tensor}
\mathcal{F}_\ast(\mathcal{L}_1,\mathrm{ch}i_1)\otimesimesmes\mathcal{F}_\ast(\mathcal{L}_2,\mathrm{ch}i_2):=\mathcal{F}_\ast(\mathcal{L}_1\otimesimesmes
\mathcal{L}_2,\mathrm{ch}i_1+\mathrm{ch}i_2)\ .
\end{equation}
The degree is again additive for the tensor product of filtered bundles.
This agrees with the usual definition of tensor product for parabolic
bundles.
\mathfrak{su}bsection{Harmonic metrics for weighted line bundles}
\betagin{proposition}
\label{prop_NAHforfilteredlinebundle}
Let $(\mathcal{L},\mathrm{ch}i)$ be a degree 0 weighted bundle. Then there exists a Hermitian metric $h$
on $\mathcal{L}_{\Sigmagma'}$ such that:
\betagin{itemize}
\item [(i)] the Chern connection $A_h$ of $(\mathcal{L},h)$ is flat: $F_{A_h}=0$;
\item [(ii)] for $p\in Z$, and $(U_p,z)$ a holomorphic coordinate
centered at $p$, $|z|^{-2\mathrm{ch}i_p}h$ extends to a $\mathcal{C}^{\infty}$ Hermitian metric on $\mathcal{L}|_{U_p}$;
\item [(iii)] $h$ is uniquely determined up to a multiplication by a nonzero constant.
\end{itemize}
\end{proposition}
\betagin{proof}
We first choose a background Hermitian metric $h_0$ such that
$|z|^{-2\mathrm{ch}i_p}h_0$ defines a $\mathcal{C}^{\infty}$ Hermitian metric
defined on $U_p$. Let $A_{h_0}$ be the Chern connection, and $F_{A_0}$ the curvature.
Note that $F_{A_0}$ is smooth on $\Sigmagma$. By the Poincar\'e-Lelong
formula, we have
$\frac{\sqrt{-1}}{2\pi}\int_{\Sigmagma}F_{A_0}=\deg(\mathcal{L},\mathrm{ch}i)=0$.
Therefore, there exists a $\mathcal{C}^{\infty}$ function $\rho$ such
that $\Deltata\rho+\frac{\sqrt{-1}}{2\pi}\lambdabdambda F_{A_0}=0$. We define
$h=h_0e^{\rho}$. For the corresponding Chern connection $A_h$, we have
$F_{A_h}=0$, which implies (i). (ii) follows from the property for
$h_0$, since $\rho$ is a smooth function on $\Sigmagma$. As $\rho$ is well-defined up to a constant, $h$ is well-defined up to a constant, which implies (iii).
\end{proof}
The metric obtained above is called the \emph{harmonic metric}. For a weighted bundle $(\mathcal{L},\mathrm{ch}i)$, the holomorphic bundle $\mathcal{L}$ and the harmonic metric $h$ define a filtration as follows.
For $\epsilonsilon>0$ sufficiently small, let
$$L_{\alphapha}:=\{s\in\mathcal{L}(\ast Z) \mid |s|_h\leq Cr^{\alphapha-\epsilonsilon}\text{
for some $C$}\}\ . $$
Here, $r$ denotes the distance to $Z$ in any smooth conformal metric on
$\Sigmagma$. It is straightforward to check that this defines a filtered
bundle that matches $\mathcal{F}_\ast(\mathcal{L},\mathrm{ch}i)$ under the correspondence in the previous
section.
Even though the harmonic metric is only well-defined up to a constant, the
Chern connection $A=(\mathcal{L},h)$ is independent of this choice.
The $(1,0)$ part of $A$, denoted $\nablabla_h$, then defines
logarithmic connections $\nablabla_h:L_{\alphapha}\to
L_{\alphapha}\otimesimesmes K(Z).$
\mathfrak{su}bsection{Convergence of weighted line bundles}
In this subsection, we will consider the convergence of weighted line
bundles. The main result we prove here is a consequence of \cite[Theorem 1.8]{mochizuki2023asymptotic}. For the reader's convenience, we present a short proof here for our situation.
Let $(\Sigmagma_0,g_0)$ be a metrized Riemann surface (i.e.\ a Riemann surface
$\Sigmagma_0$
with conformal metric $g_0$). We view $\Sigmagma_0$ as given by an underlying
surface $C$ with almost complex structure $J_0$.
Consider a neighborhood $U_1$ of $J_0$ in the moduli space of holomorphic structures and a neighborhood $U_2$ of $g_0$ in the space of smooth metrics. We denote the product of these neighborhoods by $U=U_1\timesmes U_2$. We can define the fiber bundle $\mathrm{Pic}_U \to U$, where each fiber is the Picard group defined by the holomorphic structure.
Let $(\Sigmagma_t=(C,J_t),g_t)$ be a family of metrized Riemann surfaces that converge smoothly to $(\Sigmagma_0,g_0)$ as $t\to 0$. Let $Z_t\mathfrak{su}bset \Sigmagma_t$ be a collection of a finite number of points that converge to $Z_0$ in suitable symmetric products of $C$. For each $p\in Z_0$, we can write $Z_t=\cup_{p\in Z_0}Z_{t,p}$ such that all points in $Z_{t,p}$ converge to $p$. We define the convergence of weighted line bundles as follows:
\betagin{definition}
\label{def_convergence_parabolic_bundles}
A family of weighted line bundles $(\mathcal{L}_t,\mathrm{ch}i_t)$ over $\Sigmagma_t$ with weights $\mathrm{ch}i_t:Z_t\to \mathbb{R}$ converges to $(\mathcal{L}_0,\mathrm{ch}i_0)$ if
\betagin{itemize}
\item [(i)] $\mathcal{L}_t$ converges to $\mathcal{L}_0$ in $\mathrm{Pic}_U$,
\item [(ii)] for all $p\in Z_0$ and $t$ sufficiently small, $\mathfrak{su}m_{q\in Z_{t,p}}\mathrm{ch}i_t(q)=\mathrm{ch}i_0(p)$.
\end{itemize}
\end{definition}
A sequence of filtered bundles $\mathcal{F}_{\ast}(\mathcal{L}_t)$ converges to $\mathcal{F}_{\ast}(\mathcal{L}_0)$ if the corresponding weighted bundles converge. The following theorem provides insight into the compactness of a sequence of weighted line bundles:
\betagin{theorem}
\label{thm_convergence_family_harmonic_bundles}
Let $(\mathcal{L}_t,\mathrm{ch}i_t)$ be a sequence of weighted line bundles over
$\Sigmagma_t\setminus Z_t$, where $\deg(\mathcal{L}_t,\mathrm{ch}i_t)=0$, and let $h_t$ be
the corresponding harmonic metrics. If $Z_t$ converges to $Z_0$, we
write $Z_t=\cup_{p\in Z_0}Z_{t,p}$.
Then there exists a weighted line bundle $(\mathcal{L}_0,\mathrm{ch}i_0)$ over $Z_0$ with a harmonic metric $h_0$ such that:
\betagin{itemize}
\item [(i)] After being rescaled by $c_t>0$, $c_th_t$ converges to $h_0$ over $\Sigmagma_0\setminus Z_0$ in the $\mathcal{C}^{\infty}_{\mathrm{loc}}$ sense.
\item [(ii)] Let $\nabla_t$ be the unitary connection of $h_t$. Then on
$\Sigmagma_0\setminus Z_0$, $\lim_{t\to 0}\nabla_t=\nabla_0$ in $\mathcal{C}^{\infty}_{\mathrm{loc}}$.
\end{itemize}
\end{theorem}
\betagin{proof}
By the assumptions on weights,
$\deg(\mathcal{L}_t)$ is a fixed, $t$-independent constant. Let $\gamma_t=(J_t,g_t)$ be a path in $U$.
Then $\mathrm{Pic}_U|_{\gamma_t}$ is compact, and there exists an $\mathcal{L}_0\in \mathrm{Pic}(\Sigmagma_0)$ such that $\mathcal{L}_t$ converges to $\mathcal{L}_0$.
For $p\in Z_0$, define $\mathrm{ch}i_0(p)=\mathfrak{su}m_{q\in Z_{t,p}}\mathrm{ch}i_t(q)$, and thus
obtain a weighted line bundle $(\mathcal{L}_0,\mathrm{ch}i_0)$. We can choose a family
of approximate harmonic metrics $h_t^{\mathrm{app}}$, such that $|z|^{-2\mathrm{ch}i_p}h_t^{\mathrm{app}}$ extends to a smooth metric in a neighborhood of $p$ and $h_t^{\mathrm{app}}$ converges to $h_0^{\mathrm{app}}$ in $\mathcal{C}^{\infty}_{\mathrm{loc}}(\Sigmagma_0\setminus Z_0)$.
Moreover, we write $h_t=h_t^{\mathrm{app}}e^{s_t}$. After a suitable rescale of $h_t$, we can assume $\|s_t\|_{L^2}=1$.
Let $\rho_t:=\Deltata_th_t^{\mathrm{app}}$ be the curvature defined by the metric
$h^{\mathrm{app}}_t$. Then $s_t$ satisfies the equation $\Deltata_ts_t=\rho_t$
over $\Sigmagma$. As $\rho_t$ converges to $\rho_0\in\mathcal{C}^{\infty}_{\mathrm{loc}}(\Sigmagma\setminus Z_0)$, and $g_t$ is a family with bounded geometry, we obtain the estimate
$$
\|s_t\|_{\mathcal{C}^{k+2,\alpha}(\Sigmagma)}\leq
C_{k,\alpha}(\|\rho_t\|_{\mathcal{C}^{k,\alpha}(\Sigmagma)}+1)\ ,
$$
where $C_{k,\alpha}$ is a $t$-independent constant. Therefore, passing to a
subsequence, $s_t$ converges to $s_0$ in $\mathcal{C}^{\infty}(\Sigmagma)$, which
implies (i). The assertion (ii) follows from (i).
\end{proof}
\section{The algebraic and analytic compactifications}
\label{sec_the_algebraic_and_analytic_compactifications}
In this section, we introduce compactifications of the Dolbeault and
Hitchin moduli spaces.
\mathfrak{su}bsection{The algebraic compactification of the Dolbeault moduli space}
In this subsection, we present the algebraic method for compactifying the Dolbeault moduli space.
This technique is based on the $\mathbb{C}^{\ast}$ action on $\mathcal{M}D$, and was introduced in
\cite{simpson1996hodge, Schmitt:98, hausel1998compactification,de2018compactification,katzarkov2015harmonic}.
The gauge theoretic approach can be found in \cite{fan2022analytic}.
\betagin{theorem}[{\cite[Thm.\ 11.2]{simpson1996hodge},\cite{de2018compactification}}]
\label{thm_algebraiclemmasimpson}
Let $V$ be an algebraic variety with $\mathbb{C}^{\ast}$ action. Suppose
\betagin{itemize}
\item [(i)]the fixed point set of the $\mathbb{C}^{\ast}$ action is proper,
\item [(ii)]for every $t\in \mathbb{C}^{\ast},\;v\in V$, the limit $\displaystyle\lim_{t\to 0}t\cdotot v$ exists.
\end{itemize}
Then the space $\displaystyle U:=\{v\in V\mid \lim_{t\to\infty}t\cdotot
v\;\mathrm{does\;not\;exist}\}$ is open in $V$, and the quotient $U/\mathbb{C}^{\ast}$
is separated and proper.
\end{theorem}
We apply this to the Dolbeault moduli space. The first step is to note that
the possible isotropy subgroups are limited.
\betagin{lemma}{\cite[Thm.\ 6.2]{hausel1998compactification}}
Let $\xi=[(\mathcal{E},\varphi)]$ be a Higgs bundle equivalence class with $\mathcal{H}(\xi)\neq 0$. Then the stabilizer $\Gamma_{\xi}$
of $\xi$ for the $\mathbb{C}^{\ast}$ action is either trivial or $\mathbb{Z}/2$. The latter case holds if and only if
$(\mathcal{E},\varphi)$ and $(\mathcal{E},-\varphi)$ are complex gauge equivalent.
\end{lemma}
\betagin{proof}
For $t\in\Gamma_\xi$, $\mathcal{H}(t\cdotot \xi)=t^2\mathcal{H}(\xi)$, hence $t^2=1$ if $\mathcal{H}(\xi)\neq 0$.
\end{proof}
By this Lemma, the space
$(\mathcal{M}D\setminus \mathcal{H}^{-1}(0))/\mathbb{C}^{\ast}$ has an orbifold structure.
In passing, we note that the
fixed points of the $\mathbb{Z}/2$ action correspond to real representations under the nonabelian Hodge
correspondence \cite[Sec.\ 10]{hitchin1987self}.
By the properness of the Hitchin map $\mathcal{H}$ (see Theorem
\ref{thm:proper-action}),
it follows that $\displaystyle \lim_{t\to \infty}t\cdotot \xi$ exists if and only if $\mathcal{H}(\xi)=0$. Now define
\betagin{equation} \label{eqn:compactification}
\overline{\MM}_{\mathrm{Dol}} = \left\{(\mathcal{M}D \timesmes \mathbb{C}^{\ast}) \coprod (\mathcal{M}D \setminus \mathcal{H}^{-1}(0))\right\}/\mathbb{C}^{\ast}\ .
\end{equation}
The analytic topology on the disjoint union is generated by open sets
$U\timesmes W_1$ and $V\timesmes (W_2\cap\mathbb{C}^{\ast})\amalg V\cap(\mathcal{M}D \setminus \mathcal{H}^{-1}(0))$,
where
$U,V\mathfrak{su}bset\mathcal{M}D$,
$W_1,W_2\mathfrak{su}bset \mathbb{C}$ are
open, and $0\not\in W_1$, $0\in W_2$.
The topology on $\overline{\MM}_{\mathrm{Dol}}$ is then the quotient topology, and it is straightforward to see that with this topology,
it is compact.
Since $(\mathcal{M}D \timesmes \mathbb{C}^{\ast})/\mathbb{C}^{\ast}=\mathcal{M}D$, there is a natural inclusion
\betagin{equation*}
i\to\inftya : \mathcal{M}D \to \overline{\MM}_{\mathrm{Dol}},\;i\to\inftya(\xi) = [(\xi,1)]\ ,
\end{equation*}
where brackets denote the equivalence class under the $\mathbb{C}^{\ast}$ action. The boundary of $\overline{\MM}_{\mathrm{Dol}}$ is
$$\partialrtial \overline{\MM}_{\mathrm{Dol}} = \overline{\MM}_{\mathrm{Dol}} \setminus i\to\inftya(\mathcal{M}D)=(\mathcal{M}D \setminus
\mathcal{H}^{-1}(0))/\mathbb{C}^{\ast}\ .$$
There is a \emph{boundary map}
\betagin{equation*}
i\to\inftya_{\partialrtial} : \mathcal{M}D \setminus \mathcal{H}^{-1}(0) \longrightarrow \partialrtial
\overline{\MM}_{\mathrm{Dol}},\; \xi \mapsto [( \xi, 0)]\ ,
\end{equation*}
which is invariant under the $\mathbb{C}^{\ast}$ action, i.e., $i\to\inftya_{\partialrtial}(\lambdabda \xi) = i\to\inftya_{\partialrtial}(\xi)$
for $\lambdabda \in \mathbb{C}^{\ast}$.
The $\mathbb{C}^{\ast}$ action on $\mathcal{M}D$ covers the square of the action on $\mathcal{B}$.
Hence, it is natural to compactify $\mathcal{B}$ by projectivizing:
$$
\overline\mathcal{B} := \mathbb{P}(H^0(K^2)\oplus \mathbb{C}) \ .
$$
The inclusion is given, as usual, by
\betagin{equation*}
i\to\inftya_0:\mathcal{B}\to\phi_{z,\mathrm{mod}}B,\;i\to\inftya(q)=[q\times \{1\}]\ ,
\end{equation*}
where $q\times \{1\}\in H^0(K^2)\oplus \mathbb{C}$.
We also define $\partial\phi_{z,\mathrm{mod}}B=\phi_{z,\mathrm{mod}}B\setminus i\to\inftya_0(\mathcal{B})\sigmameq
\mathbb{P}(H^0(K^2))$, with boundary projection map
\betagin{equation*}
i\to\inftya_{0,\partial}:\mathcal{B}\setminus \{0\}\to \partial\phi_{z,\mathrm{mod}}B\ ,\ i\to\inftya_{0,\partial}(q)=[q\times
\{0\}]\ .
\end{equation*}
The Hitchin map $\mathcal{H}:\mathcal{M}D\to \mathcal{B}$ extends to $\overline{\MH}:\overline{\MM}_{\mathrm{Dol}}\to \phi_{z,\mathrm{mod}}B$, where
$\overline{\MH}|_{\mathcal{M}D}:=i\to\inftya_0\circ \mathcal{H}$, and for every $[(\mathcal{E},\varphi)]/\mathbb{C}^{\ast}\in \partial\overline{\MM}_{\mathrm{Dol}}$,
$$
\overline{\MH}([(\mathcal{E},\varphi)]/\mathbb{C}^{\ast}):=[(\mathcal{H}(\varphi), 0)]\mathfrak{su}bset\phi_{z,\mathrm{mod}}B\ .
$$
This is well defined, since $\det(\varphi) \neq 0$ if $[(\mathcal{E},\varphi)]/\mathbb{C}^{\ast}\in \partial
\overline{\MM}_{\mathrm{Dol}}$. Moreover,
$$
\betagin{tikzcd} \mathcal{M}D \arrow[r, "i\to\inftya"] \arrow[d, "\mathcal{H}"] & \overline{\MM}_{\mathrm{Dol}} \arrow[d,"\overline{\MH}"] \\ \mathcal{B} \arrow[r, "i\to\inftya_0"] & \phi_{z,\mathrm{mod}}B \end{tikzcd}
$$
commutes.
There is a good algebraic structure on this algebraic compactification:
\betagin{theorem}[{\cite{simpson1996hodge, Schmitt:98,
hausel1998compactification,de2018compactification,fan2022analytic}}]
The compactified space $\overline{\MM}_{\mathrm{Dol}}$ is a normal projective variety, and
$\partial\overline{\MM}_{\mathrm{Dol}}$ is a Cartier divisor of $\overline{\MM}_{\mathrm{Dol}}$.
\end{theorem}
The following characterization of sequential convergence is useful.
\betagin{proposition}
\label{prop_convergence_Dol_space}
Let $[(\mathcal{E}_i,\varphi_i)]\in\mathcal{M}D$ be a sequence of Higgs bundles, and write $q_i=\det(\varphi_i)$ and $r_i=\|q_i\|_{L^2}^{\frac12}$.
Suppose $\limsup r_i\to \infty$. Then up to subsequence:
\betagin{itemize}
\item [(i)] there exists a Higgs bundle $[(\widehat{\ME}_{\infty},\hat{\vp}_{\infty})]$ with $\tilde{q}_{\infty}=\det(\tilde{\vp}_{\infty})$ and
$\|\hat{q}_{\infty}\|_{L^2}=1$ such that $\lim_{i\to \infty}[(\mathcal{E}_i,r_i^{-1}\varphi_i)]=[(\widehat{\ME}_{\infty},\hat{\vp}_{\infty})]$ in $\mathcal{M}D$
and $\lim_{i\to \infty}r_i^{-1}q_i=\hat{q}_{\infty}$ in $H^0(K^2)$;
\item [(ii)]
\betagin{align*}
\lim_{i \to \infty}
i\to\inftya[(\mathcal{E}_i,\varphi_i)]&=i\to\inftya_{\partial}[(\widehat{\ME}_{\infty},\hat{\vp}_{\infty})]\ ,\
\mbox{on}\ \overline{\MM}_{\mathrm{Dol}}\ , \\
\lim_{i\to \infty}i\to\inftya_0(q_i)&=i\to\inftya_{0,\partial}(\hat{q}_{\infty})\ ,\ \mbox{on}\
\phi_{z,\mathrm{mod}}B\ .
\end{align*}
\end{itemize}
\end{proposition}
\betagin{proof}
The first point follows since the Hitchin map $\mathcal{H}$ is proper and $\mathcal{H}(r_i^{-1}\varphi_i)$ is
bounded. The second follows directly from the definition.
\end{proof}
\mathfrak{su}bsection{The analytic compactification of the Hitchin moduli space}
We next describe the compactification of the Hitchin moduli space, as
developed in \cite{mazzeo2012limiting,Mochizukiasymptotic,taubes2013compactness}.
\mathfrak{su}bsubsection{Decoupled Hitchin equations}
We begin by defining the decoupled Hitchin equations. Recall the notation from Section \ref{sec:nah}, let $E$ be a trivial, smooth, rank 2 vector bundle over a Riemann surface $\Sigmagma$, and let $H_0$ be a background Hermitian metric on $E$. Let $Z$ be a finite set of distinct points in $\Sigmagma$. For a smooth unitary connection $A$ on $E|_{\Sigmagma\setminus Z}$ and
smooth $\phi=\varphi+\varphi^{\dagger}\in \Omega^1(i\mathfrak{su}(E))|_{\Sigmagma\setminus Z}$, the
\emph{decoupled Hitchin equations} on $\Sigmagma\setminus Z$ are:
\betagin{equation} \label{eq_decoupled_Hitchin_equation}
\betagin{split}
F_A=0\ ,\ [\varphi,\varphi^{\dagger}]=0\ ,\ \bar{\partial}_A\varphi=0\ .
\end{split}
\end{equation}
Solutions to \eqref{eq_decoupled_Hitchin_equation} may be quite singular near $Z$, so we make the following restriction:
\betagin{definition}
A solution $(A,\phi)$ to the decoupled Hitchin equations over
$\Sigmagma\setminus Z$ is called \emph{admissible}
if $\phi\neq 0$, and $|\phi|$ extends to a continuous function on $\Sigmagma$ with $|\phi|^{-1}(0)=Z$.
\end{definition}
By a \emph{limiting configuration} we always mean an admissible solution to the decoupled Hitchin equations.
Clearly, $Z$ is determined by $(A,\phi)$. Admissibility guarantees that $\det(\varphi)$ extends to a holomorphic
quadratic differential $q=\det(\varphi)$ on $\Sigmagma$, with $Z=q^{-1}(0)$ the
zero locus. Hence, the spectral curve $S_q$
is well-defined. We emphasize that $Z$ may vary for different admissible solutions, but one always has
that $\# Z\leq 4g-4$.
The equivalence relation on limiting configurations is that $(A_1,\phi_1)\sigmam (A_2,\phi_2)$ if $Z_1=Z_2$ and
$(A_1,\phi_1)g=(A_2,\phi_2)$ for a smooth unitary gauge transformation $g$ on $\Sigmagma\setminus Z_1$.
The moduli space of decoupled Hitchin equations is then
\betagin{equation*}
\mathcal{M}H^{\mathrm{Lim}}=\{\text{admissible solutions to }\eqref{eq_decoupled_Hitchin_equation}\}/\sigmam\ .
\end{equation*}
We denote by $\mathcal{M}HQLC$ the elements in $\mathcal{M}H^{\mathrm{Lim}}$ with a fixed quadratic
differential $q$. In this case, the equivalence
relation is induced by the action of the unitary gauge group over $\Sigmagma\setminus Z$, $Z=q^{-1}(0)$.
There is a natural $\mathbb{C}^{\ast}$ action on the moduli space $\mathcal{M}H^{\mathrm{Lim}}$: given $(A,\phi=\varphi+\varphi^{\dagger})\in \mathcal{M}H^{\mathrm{Lim}}$
and $t\in \mathbb{C}^{\ast}$, we set $t\cdotot[(A,\phi)]=[(A,t\varphi+\bar{t}\varphi^{\dagger})]$, which is also a solution
to \eqref{eq_decoupled_Hitchin_equation}.
\mathfrak{su}bsubsection{Compactification of the Hitchin moduli space}
The following compactness result is due to Taubes \cite{Taubes20133manifoldcompactness}
and Mochizuki \cite{Mochizukiasymptotic} (see also \cite{he2020behavior}).
\betagin{proposition}
\label{prop_general_convergence_solutions}
Let $(A_{i},\varphi_i)$ be a sequence of
solutions to \eqref{eq_Hitchin_real}, with $q_i=\det(\varphi_i)\in
H^0(K^2)$. Then
\betagin{itemize}
\item [(i)] if $\limsup\|q_i\|_{L^2(\Sigmagma)}<\infty$, then there
is a subsequence (also denoted $\{i\}$),
a smooth solution $(A_{\infty},\phi_{\infty})$ to
\eqref{eq_Hitchin_real},
and a sequence $g_i$ of smooth unitary gauge transformations on
$\Sigmagma$,
such that $(A_i,\phi_i)g_i$ converges smoothly to
$(A_{\infty},\phi_{\infty})$ on $\Sigmagma$;
\item [(ii)] if $\lim\|q_i\|_{L^2(\Sigmagma)}=\infty$, then there
is a subsequence (also denoted $\{i\}$), and $q_{\infty}\in H^0(K^2)$
so that
$$\frac{q_i}{\Vert q_i\Vert_{L^2}}\longrightarrow
q_{\infty}$$ over $\Sigmagma$, and an admissible solution
$(A_\infty,\phi_{\infty}=\varphi_{\infty}+\varphi^{\dagger}_{\infty})$ to
\eqref{eq_decoupled_Hitchin_equation}, with
$Z_{\infty}:=q_{\infty}^{-1}(0)$, and smooth unitary gauge
transformations $g_i$ on $\Sigmagma\setminus Z_\infty$,
such that over any open set $ \Omega\Subset \Sigmagma\setminus Z_{\infty}$,
$(A_i)g_i\to A_\infty$, and
$$\frac{g_i^{-1}\phi_ig_i}{\Vert\phi\Vert_{L^2}}\longrightarrow
\phi_{\infty}$$
smoothly on $\Omega$.
\end{itemize}
\end{proposition}
The norm on $H^0(\Sigmagma,K^2)$ can be chosen arbitrarily, since it is a finite dimensional space.
There is also a compactness result for sequences of solutions in $\mathcal{M}H^{\mathrm{Lim}}$.
\betagin{proposition}
\label{prop_seq_compactification_limitingconfiguration}
Let $[(A_i,\phi_i=\varphi_i+\varphi_i^{\daggergger})]\in \mathcal{M}H^{\mathrm{Lim}}$ be a sequence
of admissible solutions to \eqref{eq_decoupled_Hitchin_equation},
and let $q_i=\det(\varphi_i)$ be the corresponding quadratic differentials.
Then after passing to a subsequence, there are $t_i\in\mathbb{C}^{\ast}$, a limiting
configuration $(A_{\infty},\phi_{\infty}=\varphi_{\infty}+\varphi_{\infty}^{\daggergger})$
with quadratic differential $q_{\infty}=\det(\varphi_{\infty})\neq 0$,
and a sequence $g_i$ of smooth gauge transformations on
$\Sigmagma\setminus Z_\infty$, such that:
\betagin{itemize}
\item [(i)] $t_i^2q_i$ converges smoothly to $q_{\infty}$,
\item [(ii)] over any open set $\Omega\Subset X\setminus Z_{\infty}$,
$(A_i,t_i\cdotot \phi_i)g_i$ converges smoothly to $(A_{\infty},\phi_{\infty})$.
\end{itemize}
\end{proposition}
\betagin{proof}
Write $q_i=\det(\varphi_i)\in H^0(K^2)$. Adjusting by $t_i$ if necessary, we
may assume $q_i$ converges to $q_{\infty}$ over $\Sigmagma$. Also, since $F_{A_i}=0$ over $\Sigmagma\setminus Z_i$
and $Z_i$ converges to $Z_{\infty}$, we can apply both Uhlenbeck
compactness and the classical bootstrapping method to obtain
$A_{\infty}$ such that up to gauge $A_i$ converges smoothly to $A_{\infty}$ over $\Sigmagma\setminus Z_{\infty}$.
Finally, the convergence of $\varphi_i$ follows by the bound on
$q_i$'s.
\end{proof}
\mathfrak{su}bsubsection{The topology on the compactified space}
We now carefully define the topology on the space $\mathcal{M}H \coprod
\mathcal{M}H^{\mathrm{Lim}}/\mathbb{C}^{\ast}$. Choose a metric in the conformal class on $\Sigmagma$. Let $W^{k,2}$ denote the Sobolev spaces on $\Sigmagma$ of
distributional sections with at least $k$ derivatives in $L^2$. For a finite set of points $Z \mathfrak{su}bset \Sigmagma$ (or indeed any
closed subset),
\betagin{equation*}
\betagin{split}
W^{k,2}_{\mathrm{loc}}(\Sigmagma\setminus Z):=\{f\mid f\in W^{k,2}(K),\;K\mathfrak{su}bset \Sigmagma\setminus Z,\;K\;\mathrm{compact}\}.
\end{split}
\end{equation*}
These definitions extend easily to the space of connections and $\Omega^1(i \mathfrak{su}(E))$ for a Hermitian vector bundle $(E,H_0)$
over $\Sigmagma$ with a fixed smooth background connection.
Let ${\omegaega_n}$ be a nested collection of open sets with $\omegaega_n\mathfrak{su}bset \overline{\omegaega_n} \mathfrak{su}bset \omegaega_{n+1}$,
with $\bigcup_{n}\omegaega_n= \Sigmagma\setminus Z$. We then define the seminorms $\|f\|_n:=\|f\|_{W^{k,2}(\omegaega_n)}$; in
terms of these, $W^{k,2}_{\mathrm{loc}}(\Sigmagma\setminus Z)$ a Fr\'echet space.
For any $q\in H^0(K^2)\setminus\{0\}$, set $Z_q:=q^{-1}(0)$, and consider the moduli space
\betagin{equation*}
\mathbb{M}_q=\{(A,\phi)\in
\mathcal{M}_{q^\ast}\cap W^{k,2}(\Sigmagma)\}\cup \{(A,\phi)\in\mathcal{M}HQLC\cap
W^{k,2}_{\mathrm{loc}}(\Sigmagma\setminus Z_q)\}/\mathbb{C}^{\ast}\ .
\end{equation*}
By classical bootstrapping of the gauge-theoretic elliptic equations, $\mathbb{M}_q$ is independent of $k\geq 2$.
Next define $\mathbb{M}:=\mathcal{M}_{0}\cup \bigcup_{q\in H^0(K^2)\setminus\{0\}}\mathbb{M}_q$. Its topology is generated by two types of open sets. For interior points
$\xi=[(A,\phi)]\in \mathcal{M}H\mathfrak{su}bset \mathbb{M}$ we use the open sets
\betagin{equation*}
\betagin{split}
V_{\xi,\epsilon}:=\{[(A',\phi')]\in \mathcal{M}H\mid\|A'-A\|_{W^{k,2}(\Sigmagma)}+\|\phi'-\phi\|_{W^{k,2}(\Sigmagma)}<\epsilon\}
\end{split}
\end{equation*}
from the topology of $\mathcal{M}H$. For any boundary point $\xi_0\in\mathcal{M}H^{\mathrm{Lim}}/\mathbb{C}^{\ast}$, choose a representative $(A_0,e^{i\theta}\phi_0)$ for some
constant $\theta$. Let $q=\det(\phi_0)$, and fix any open set $\omega\Subset \Sigmagma\setminus Z_q$. Then, setting $\mathcal{M}H^{\ast}=\mathcal{M}H\setminus \mathcal{H}^{-1}(0)$,
\betagin{equation*}
\betagin{split}
U_{\xi_0,\omega,\epsilon}:=&\{(A,\phi)\in\mathcal{M}H^{\ast}\mid
\|A-A_0\|_{W^{k,2}(\omega)}+\inf_{\theta\in S^1}\|\|\phi\|_{L^2}^{-\frac12}\phi-e^{i\theta}\phi_0\|_{W^{k,2}(\omega)}<\epsilon,\;\|\phi\|_{L^2}>\epsilon\}\\
&\bigcup \, \{(A,\phi)\in\mathcal{M}H^{\mathrm{Lim}}|
\|A-A_0\|_{W^{k,2}(\omega)}+\|\phi-\phi_0\|_{W^{k,2}(\omega)}<\epsilon\}
\end{split}
\end{equation*}
defines an open set around $\xi_0$. The sets $U_{\xi_0,\omega,\epsilon}$ and $V_{\xi,\epsilon}$ generate the topology on $\mathbb{M}$.
\betagin{theorem}
The space $\mathbb{M}$ is a Hausdorff and compact.
\end{theorem}
\betagin{proof}
The Hausdorff property follows from the definition of the topology. By Propositions \ref{prop_general_convergence_solutions} and
\ref{prop_seq_compactification_limitingconfiguration}, $\mathbb{M}$ is sequentially compact. Moreover, using this explicit base for
the topology $\mathbb{M}$ is first countable, and hence compact.
\end{proof}
We may now define the compactification of the Hitchin moduli space as the
closure $\overline{\MM}_{\mathrm{Hit}}\mathfrak{su}bset\mathbb{M}$;
we write $\partial\overline{\MM}_{\mathrm{Hit}}$ for the boundary of the closure, and $\overline{\MM}_{\mathrm{Hit}}q:=\overline{\MM}_{\mathrm{Hit}}\cap \mathbb{M}_q$ for the subset of elements
with a fixed quadratic differential.
The following result is described in \cite{mazzeo2016ends,ott2020higgs,mazzeo2019asymptotic}.
\betagin{theorem}[{\cite[Prop.\ 3.3]{mazzeo2019asymptotic}}]
\label{thm_simple_zero_bijective}
If $q$ has only simple zeros, then $\overline{\MM}_{\mathrm{Hit}}q=\mathbb{M}_q$.
\end{theorem}
In other words, the compactification of any slice where $q$ does not lie in
the discriminant locus is ``the obvious one''.
\section{Parabolic modules and stratification of BNR data}
\label{sec_parabolic_modules}
In this section, we review the notion of a parabolic module, as
described in \cite{rego1980compactified,cook1993local,cook1998compactified,
gothen2013singular}. This concept leads to a partial normalization of the
generalized Jacobian and Prym varieties of the spectral curve.
\mathfrak{su}bsection{Normalization of the spectral curve}
Let $q\neq 0$ be a quadratic differential with an irreducible, singular spectral curve $S=S_q$. The zeros of $q$ define a divisor
$\mathrm{Div}(q) = \mathfrak{su}m_{i=1}^{r_1} m_i p_i + \mathfrak{su}m_{j=1}^{r_2} n_j p_j'$, where the $m_i$ and $n_j$ are even and odd integers, respectively,
and hence $r_1$ and $r_2$ are the numbers of even and odd zeros, respectively, counted without
multiplicity. Write $Z_{\mathrm{even}} = \{p_1, \dots, p_{r_1}\}$, $Z_{\mathrm{odd}} = \{p_1', \dots, p_{r_2}'\}$, and $Z = Z_{\mathrm{even}} \cup Z_{\mathrm{odd}}$, so $\#Z=r = r_1 + r_2$.
The map $\pi: S \to \Sigmagma$ is a double covering branched along $Z$; hence, we may view $p_i$ and $p_i'$ as points in $S$.
For $x \in S$, let $\mathcal{O}_x$ be the local ring, $\mathcal{O}_x^\ast$ its group of units, and $R_x$ the completion.
We say that $S$ has an $A_n$ singularity at $x$ if $R_x \cong \mathbb{C}[[r,s]]/(r^2-s^{n+1})$, where $n \geq 1$. If $S$ has an $A_1$
singularity at $x$, we call it a \emph{nodal} singularity, and if $S$ has an $A_2$ singularity at $x$, we
call it a \emph{cusp} singularity.
Let $p:\widetilde{S} \to S$ be the normalization of $S$, and let $\tilde{\pi} := \pi \circ p$:
\[
\betagin{tikzcd}
\widetilde{S} \arrow{r}{p} \arrow[swap]{dr}{\tilde{\pi}} & S \arrow{d}{\pi} \\
& \Sigmagma
\end{tikzcd}
\]
For even zeros $p_i$ we write $p^{-1}(p_i) = \{\tilde{p}_i^{+}, \tilde{p}_i^{-}\}$, and for odd zeros $p_i'$ we write $p^{-1}(p_i') = \tilde{p}_i'$. Since
$\pi:S\to\Sigmagma$ is a branched double cover, the involution $\sigmagma$ on $S$ extends to an involution of $\widetilde{S}$ which we also denote
by $\sigmagma$. Note that $\sigmagma(\tilde{p}_i') = \tilde{p}_i'$ while $\sigmagma(\tilde{p}_i^\pm) = \tilde{p}_i^\mp$.
The ramification divisor
$\lambdabdam'=\frac{1}{2}\mathfrak{su}m_{i=1}^{r_1}m_ip_i+\mathfrak{su}m_{j=1}^{r_2}(n_j-1)p_j''$,
is a divisor on $S$, and there is an exact sequence:
\betagin{equation} \label{eq_normalization_exact_sequence}
0\longrightarrow \mathcal{O}_{S}\longrightarrow p_{\ast}\mathcal{O}_{\widetilde{S}}\longrightarrow \mathcal{O}_{\lambdabdam'}\longrightarrow 0\ .
\end{equation}
The genus of $\widetilde{S}$ is $g(\widetilde{S})=4g-3-\deg(\lambdabdam')=2g-1+r_2/2$.
\mathfrak{su}bsection{Jacobian under the pull-back of the normalization}
We now recall some facts about the Jacobian under the pull-back of the normalization, cf.\ \cite{gothen2013singular}. Let $x \in Z
\mathfrak{su}bset S$ be a singular point, i.e.\ either $x\inZ_{\mathrm{even}}$ or $x=p'_j$ with
$n_j\geq 3$. Let $\widetilde{\MO}_{x}$ be the
integral closure of $\mathcal{O}_x$. We take $V:=\mathrm{pr}od_{x\in
Z}\widetilde{\MO}_x^{\ast}/\mathcal{O}_x^{\ast}$. Then we have the following well-known short
exact sequence induced by the pull-back of the normalization.
\betagin{equation} \label{eq_normalization_Jacobian_fibration}
\betagin{split}
0\longrightarrow V\longrightarrow \mathrm{Jac}(S)\xrightarrow{p^{\ast}} \mathrm{Jac}(\widetilde{S})\longrightarrow 0\ .
\end{split}
\end{equation}
This will play an important role later on.
\mathfrak{su}bsubsection{Hitchin fiber}
Now we examine the locally free part $\mathcal{T}$ of the Hitchin fiber under
the pull-back. Here, $\mathcal{T}$ is defined to be the set of $L\in\mathrm{Pic}^{2g-2}(S)$ such that
$\det(\pi_{\ast}L)=\mathcal{O}_{\Sigmagma}$. For any $L\in\mathrm{Pic}(S)$, from \eqref{eq_normalization_exact_sequence} we
see that $\det(\tilde{\pi}_{\ast}p^{\ast}L)\cong \det(\pi_{\ast}L)\otimesimesmes \mathcal{O}_\Sigmagma(\lambdabdam')$. We define a new set, $\widetilde{\MT}$, as follows:
\betagin{equation*}
\widetilde{\MT}:=\{\widetilde{L}\in \mathrm{Pic}^{2g-2}(\widetilde{S})\mid\det(\tilde{\pi}_{\ast}L)\cong\mathcal{O}(\lambdabdam')\}.
\end{equation*}
It follows that $p^{\ast}$ maps $\mathcal{T}$ to $\widetilde{\MT}$. Furthermore, if $L_1,L_2\in \mathrm{Pic}(S)$ satisfy $p^{\ast}L_1\cong p^{\ast}L_2$, then we have $\pi_{\ast}L_1\cong \pi_{\ast}L_2$. This means that the fiber of $p^{\ast}:\mathrm{Jac}(S)\to \mathrm{Jac}(\widetilde{S})$ is the same as $p^{\ast}:\mathcal{T}\to \widetilde{\MT}$, resulting in the following exact sequence:
\betagin{equation} \label{eq_exactsequenceofT}
0\longrightarrow
V\longrightarrow\mathcal{T}\xrightarrow{\hat{s}pace{.4cm}p^{\ast}\ }\widetilde{\MT}\longrightarrow 0\ .
\end{equation}
\mathfrak{su}bsection{Torsion free sheaves}
\label{subsection_torsion_free_sheaf_integers}
Now we present Cook's parametrization of rank 1 torsion free sheaves on
curves with
Gorenstein singularities (see \cite[p.\ 40]{cook1998compactified} and also
\cite{cook1993local, rego1980compactified}). An explicit computation of the
invariants used in this paper is provided in Appendix \ref{appendixA}.
Let $x \in Z$ be a singular point of $S$, and let $L\to S$ be a rank 1 torsion free sheaf.
We again let $\mathcal{O}_x$ denote the local ring at $x$, $\widetilde{\MO}_x$ its integral closure,
$\mathcal{K}_x$ its fraction field,
and $\partialta_x = \dim_{\mathbb{C}}(\widetilde{\MO}_x/\mathcal{O}_x)$.
According to \cite[Lemma 1.1]{greuel1993moduli}, there exists a fractional
ideal $I_x$ that is isomorphic to $L_x$, uniquely defined up to multiplication
by a unit of $\widetilde{\MO}_x$, such that $\mathcal{O}_x \mathfrak{su}bset I_x \mathfrak{su}bset \widetilde{\MO}_x$.
We define $\ell_x := \dim_{\mathbb{C}} (I_x/\mathcal{O}_x)$ and $b_x:=\dim_{\mathbb{C}}(\mathrm{Tor}(I_x\otimesimesmes_{\mathcal{O}_x}\widetilde{\MO}_x))$.
Then, $\ell_x$ and $b_x$ are invariants of $L$.
Let $\mathcal{K}_x$ be the field of fractions of $\mathcal{O}_x$. The \emph{conductor} of
$I_x\mathfrak{su}bset \widetilde{\MO}_x$ is defined to be
$$C(I_x)=\{u\in\mathcal{K}_x\mid u\cdotot\widetilde{\MO}_x\mathfrak{su}bset I_x\}\ .$$
The singularity is characterized by the following dimensions:
\betagin{equation*}
\underbrace{C(\mathcal{O}_x)\mathfrak{su}bset\overbrace{C(I_x)\mathfrak{su}bset
\overbrace{\mathcal{O}_x\mathfrak{su}bset \underbrace{I_x\mathfrak{su}bset
\widetilde{\MO}_x}_{\partialta_x-\ell_x}}^{\partialta_x}}^{2\partialta_x-2\ell_x}}_{2\partialta_x}\ .
\end{equation*}
For $x = p_i \in Z_{\mathrm{even}}$, we have $\partialta_{p_i}=m_i/2$, and there are
two maximal ideals $\mathfrak{m}_{\pm}$ in $\widetilde{\MO}_x$ corresponding to the points
$\tilde{p}_i^{\pm}$. We let $(\widetilde{\MO}_{p_i}/C(I_{p_i}))_{\mathfrak{m}_{\pm}}$
be the localization by the ideals $\mathfrak{m}_{\pm}$, and define
$a_{\tilde{p}_i^{\pm}}:=\dim_{\mathbb{C}}(\widetilde{\MO}_{p_i}/C(I_{p_i}))_{\mathfrak{m}_{\pm}}$.
Moreover, we have
$\dim_{\mathbb{C}}(\widetilde{\MO}_{p_i}/C(\mathcal{O}_{p_i}))_{\mathfrak{m}_{\pm}}=m_i/2=\partialta_{p_i}$.
By Appendix \ref{appendixA},
$a_{\tilde{p}_i^{\pm}}=(m_i/2)-\ell_{p_i}$, and therefore $a_{\tilde{p}_i^+}+a_{\tilde{p}_i^-}=2\partialta_{p_i}-2\ell_{p_i}$, and also
$b_{p_i}=\ell_{p_i}$. Define
$$
V(L_{p_i})=\{(c_i^+,c_i^-)\mid c_{i}^{\pm}\in\mathbb{Z}_{\ge 0}\ ,\ c_i^++c_i^-=\ell_{p_i}\}\ .
$$
For $x=p_i' \in Z_{\mathrm{odd}}$, we have $\partialta_{p_i'}=(n_i-1)/2$, and the maximal ideal $\mathfrak{m}$ of $\widetilde{\MO}_x$ is unique.
Define $a_{\tilde{p}_i'}:=\dim_{\mathbb{C}}(\widetilde{\MO}_{p_i'}/C(I_{p_i'}))_{\mathfrak{m}}$. By Appendix \ref{appendixA},
we have $a_{\tilde{p}_i'}=2\partialta_{p_i'}-2\ell_{p_i'}$ and $b_{p_i'}=\ell_{p_i'}$. Moreover,
$\dim_{\mathbb{C}}(\widetilde{\MO}_{p_i'}/C(\mathcal{O}_{p_i'}))_{\mathfrak{m}}=n_i-1=2\partialta_{p_i'}$. In this case we set $V(L_{p_i'})=\{\ell_{p_i'}\}$.
Now consider modules compatible with $L_x$. Let $\eta:\widetilde{\MO}_x\to \widetilde{\MO}_x/C(\mathcal{O}_x)$ be the quotient map.
Define
$$
S(L_x):=\{\mathcal{O}_x\text{-submodules}\ F_x\mathfrak{su}bset \widetilde{\MO}_x/C(\mathcal{O}_x)\mid \dim_{\mathbb
C}(F_x)=\partialta\ ,\ \eta^{-1}(F_x)\cong L_x\}\ .
$$
Hence, if $J_x=\eta^{-1}(F_x)$ with $F_x\in S(L_x)$, there exists an ideal $\mathfrak{t}_x$ such that
$J_x=\mathfrak{t}_x\cdotot L_x$. For $x=p_i\in Z_{\mathrm{even}}$, we obtain two integers $c_{i}^{\pm}=\dim_{\mathbb{C}}(\widetilde{\MO}_x/(\mathfrak{t}_x\cdotot \widetilde{\MO}_x))_{\mathfrak{m}_{\pm}}$.
By \cite[Lemma 6]{cook1998compactified}, $(c_i^+,c_i^-)\in V(L_{p_i})$, for $x=p_i'\in Z_{\mathrm{odd}}$,
$\dim_{\mathbb{C}}(\widetilde{\MO}_x/(\mathfrak{t}_x\cdotot \widetilde{\MO}_x))=\ell_{p_i'}\in V(L_{p_i'})$,
and these only depend on $F_x$. Hence, there is a well-defined map:
\betagin{equation*}
\kappa_x:S(L_x)\longrightarrow V(L_x)\ :\ \betagin{cases} F_x\to
(c_i^+,c_i^-)&\mathrm{when}\;x=p_i,\;\\
F_x\to \ell_{p_i'}&\mathrm{when}\;x=p_i' \ . \end{cases}
\end{equation*}
\betagin{lemma}[{\cite[Lemma 6]{cook1998compactified}}]
For $x\in Z$, the components of $S(L_x)$ are parameterized by elements in $V(L_x)$.
\end{lemma}
Set $V(L):=\mathrm{pr}od_{x\in Z}V(L_x)$ and $S(L):=\mathrm{pr}od_{x\in Z}S(L_x)$. Write $N(L) := |V(L)|$ for the number of points in $V(L)$.
there is a map
\betagin{equation*}
\kappa:=\mathrm{pr}od_{x\in Z}\kappa_x: S(L)\longrightarrow V(L)\ .
\end{equation*}
For any $\mathbf{c}\in V(L)$, write $\mathbf{c}=(c_1^{\pm},\ldots,c_{r_1}^{\pm},\ell_{p_1'},\ldots,\ell_{p_{r_2}'})$. Associate to this the
divisor
\betagin{equation*}
D_{\mathbf{c}}=\mathfrak{su}m_{i=1}^{r_1}(c_i^+\tilde{p}_i^++c_i^-\tilde{p}_i^-)+\mathfrak{su}m_{i=1}^{r_2}\ell_{p_i'}\tilde{p}_i'
\end{equation*}
on $\widetilde{S}$.
Composing $\kappa$ with the map above, we define
\betagin{equation} \label{eq_divisor_map}
\varkappa: S(L)\longrightarrow \mathrm{Div}(\widetilde{S})\ :\ \mathrm{pr}od_{x\in Z}F_x \mapsto \mathbf{c}\mapsto D_{\mathbf{c}}\ .
\end{equation}
The following result is straightforward but important:
\betagin{proposition}
$L$ is locally free if and only if $\varkappa=0$ on $S(L)$.
\end{proposition}
\betagin{proof}
$L$ is locally free if and only if $\ell_x=0$ for $x\in Z$. The claim then follows directly from the definition of $D_{\mathbf{c}}$.
\end{proof}
\mathfrak{su}bsection{Parabolic modules}
In this subsection, we define the notion of a parabolic module, following \cite{rego1980compactified,cook1993local,cook1998compactified}.
First note that $\dim_{\mathbb{C}}(\widetilde{\MO}_x/C(\mathcal{O}_x))=2\partialta_x$.
Let $\mathrm{Gr}(\partialta_x,\widetilde{\MO}_x/C(\mathcal{O}_x))$ be the Grassmannian of $\partialta_x$ dimensional subspaces
of the vector space $\widetilde{\MO}_x/C(\mathcal{O}_x)$. Then $\widetilde{\MO}^{\ast}_x$ acts on $\mathrm{Gr}(\partialta_x,\widetilde{\MO}_x/C(\mathcal{O}_x))$
by multiplication, with fixed points corresponding to $\partialta_x$-dimensional $\mathcal{O}_x$ submodules of $\widetilde{\MO}_x/C(\mathcal{O}_x)$.
We write $\mathcal{P}(x)$ for the (reduced) variety of fixed points. This is a closed subvariety of $\mathrm{Gr}(\partialta_x,\widetilde{\MO}_x/C(\mathcal{O}_x))$.
Suppose $x$ is an $A_n$ singularity. For notational convenience, we write $\mathcal{P}(A_n):=\mathcal{P}(x)$. We have the following:
\betagin{proposition}[{\cite[Prop.\ 2]{cook1998compactified}}]
\label{prop_fiber_of_parabolic_module}
The following holds:
\betagin{itemize}
\item [(i)] $\mathcal{P}(A_n)$ is connected and depends only on $\partialta_x$.
Also, $\dim_{\mathbb{C}}\mathcal{P}(A_n)=n$, and we have isomorphisms $\mathcal{P}(A_{2n-1})\cong \mathcal{P}(A_{2n})$.
\item [(ii)] If $P(A_0)$ is defined to be a point, then the
inclusions $\mathcal{P}(A_0)\mathfrak{su}bset \mathcal{P}(A_2)\mathfrak{su}bset \cdotots \mathfrak{su}bset \mathcal{P}(A_{2n})$ give a cell decomposition of $\mathcal{P}(A_{2n})$.
\item [(iii)] The singular locus $\mathrm{Sing}(\mathcal{P}(A_{2n}))\cong
\mathcal{P}(A_{2n-4})$. In particular, it
has codimension $\geq 2$. Moreover, $\mathcal{P}(A_1)=\mathcal{P}(A_2)\cong
\mathbb{C}P^1$, and $\mathcal{P}(A_4)$ is a quadric cone.
\end{itemize}
\end{proposition}
Define $\mathcal{S}P(S)=\mathrm{pr}od_{x\in Z}\mathcal{P}(x)$. This only depends on the curve singularity of $S$.
Let $J\in\mathrm{Pic}(\widetilde{S})$. We have an
isomorphism $J_x\otimesimesmes \mathcal{O}_{\lambdabdam',x}\cong \widetilde{\MO}_x/C(\mathcal{O}_x)$ as
$\mathcal{O}_x$-modules. More explicitly, as vector spaces,
$$
J_{\tilde{p}_i^+}^{\oplus \frac{m_i}{2}}\oplus J_{\tilde{p}_i^-}^{\oplus \frac{m_i}{2}}\cong
\widetilde{\MO}_{p_i}/C(\mathcal{O}_{p_i'})\ , \ J_{\tilde{p}_i'}^{\oplus (n_i-1)}\cong \widetilde{\MO}_{p_i'}/C(\mathcal{O}_{p_i'})\ .
$$
\betagin{definition}
A parabolic module $\phi_{z,\mathrm{mod}}od$ consists of pairs $(J,v)$, where $J\in
\mathrm{Jac}(\widetilde{S})$ and $v=\mathrm{pr}od_{x\in Z} v_x$, with $v_x\in J_x\otimesimesmes \mathcal{O}_{\lambdabdam',x}$.
\end{definition}
By \cite[p.\ 41]{cook1998compactified}, $\phi_{z,\mathrm{mod}}od$ has a natural algebraic structure. Let $\mathrm{pr}:\phi_{z,\mathrm{mod}}od\to \mathrm{Jac}(\widetilde{S})$ be the projection
to the first component. Then $\mathrm{pr}$ defines a fibration of $\phi_{z,\mathrm{mod}}od$ with fiber $\mathcal{S}P(S)$.
Moreover, there is a finite morphism $\tau: \phi_{z,\mathrm{mod}}od\to \overline{\mathrm{Jac}}(S)$ defined by sending $(J,v) \to L$,
where $L$ is the kernel of the restriction map $p_{\ast}J\to (J\otimesimesmes \mathcal{O}_{\lambdabdam})/v$:
$$
0\longrightarrow L\longrightarrow p_{\ast}J\longrightarrow (J\otimesimesmes\mathcal{O}_{\lambdabdam})/v\longrightarrow 0\ .
$$
There is a diagram:
\betagin{equation*}
\betagin{tikzcd}
&\mathcal{S}P(S) \arrow[r, ] & \phi_{z,\mathrm{mod}}od \arrow[r, "\mathrm{pr}"] \arrow[d, "\tau"]
& \mathrm{Jac}(\widetilde{S}) \\
& &\overline{\mathrm{Jac}}(S) & &
\end{tikzcd}.
\end{equation*}
The map $\tau$ may be regarded as the compactification of the pull-back normalization map $p^{\ast}$ in \eqref{eq_normalization_Jacobian_fibration}.
\betagin{theorem}[{\cite[Thm.\ 1]{cook1998compactified}}]
\label{thm_parabolic_module_main_theorem}
For the map $\tau:\phi_{z,\mathrm{mod}}od\to \overline{\mathrm{Jac}}(S)$ defined above,
\betagin{itemize}
\item[(i)] $\tau$ is a finite morphism, where the fiber over $L$ consists of $N(L)$ points,
\item[(ii)] The restriction $\tau: \tau^{-1}\mathrm{Jac}(S)\to \mathrm{Jac}(S)$ is an isomorphism. Moreover, for $L\in\mathrm{Jac}(S)$, we have $\mathrm{pr}\circ \tau^{-1}(L)=p^{\ast}(L)$.
\item [(iii)] Suppose $\tau(J,v)=L$. For $x\in Z$, we have $v\in S(L)$.
Let $D_{v}=\varkappa(v)$ be the divisor defined in \eqref{eq_divisor_map}. Then
\betagin{equation*}
0\longrightarrow p^{\ast}L/\mathrm{Tor}(p^{\ast}L)\longrightarrow J\longrightarrow \mathcal{O}_{D_v}\longrightarrow 0\ .
\end{equation*}
In particular, $p^{\ast}L/\mathrm{Tor}(p^{\ast}L)=J(-D_v)$.
\end{itemize}
\end{theorem}
Suppose all of the zeros of the quadratic differential $q$ are odd.
Then for $L\in \overline{\mathrm{Jac}}(S)$, $N(L)=1$, and we can deduce the following.
\betagin{corollary}
If $q^{-1}(0)=\{p_1',\ldots, p_r'\}$ and all zeroes have odd multiplicity, then $\tau:\phi_{z,\mathrm{mod}}od\to \overline{\mathrm{Jac}}(S)$ is a bijection.
Moreover, for $L\in \overline{\mathrm{Jac}}(S)$ with $\tau(J,v)=L$, we have
$$
p^{\ast}L/\mathrm{Tor}(p^{\ast}L)=J(-\mathfrak{su}m \ell_{p_i'}\tilde{p}_i')\ .
$$
\end{corollary}
We will now present an example of a parabolic module.
\betagin{example}[{\cite[Ex.\ 2]{cook1998compactified}}]
Suppose $q$ contains $4g-2$ simple zeros and one zero $x$ of order $2$.
Then the spectral curve $S$ has one nodal singularity at $x$.
Denote $p:\widetilde{S}\to S$ the normalization, with $p^{-1}(x)=\{\tilde{x}_+,\tilde{x}_-\}$. Then
$\mathcal{S}P(S)=\mathbb{C}P^1$, and we obtain a fibration $\mathbb{C}P^1\to \phi_{z,\mathrm{mod}}od\to \mathrm{Jac}(\widetilde{S})$.
Let $L\in\overline{\mathrm{Jac}}(S)$. If we write $\widetilde{L}:=p^{\ast}L/\mathrm{Tor}(p^{\ast}L)$, then
$\tau^{-1}(L)={(\widetilde{L}\otimesimesmes \mathcal{O}(\tilde{x}_+),v_+),(\widetilde{L}\otimesimesmes \mathcal{O}(\tilde{x}_-),v_-)}$.
We can define two sections:
$$s_{\pm}:\overline{\mathrm{Jac}}(S)\longrightarrow \phi_{z,\mathrm{mod}}od,\;s_{\pm}:L\longrightarrow (\widetilde{L}\otimesimesmes
\mathcal{O}(\tilde{x}_{\pm}),v_{\pm})\ .$$
Then $\overline{\mathrm{Jac}}(S)$ is the quotient of $\phi_{z,\mathrm{mod}}od$ given by the identification
$$\overline{\mathrm{Jac}}(S)\cong \phi_{z,\mathrm{mod}}od/(s_+\sigmam \mathcal{O}(\tilde{x}_--\tilde{x}_+)s_-)\ .$$ In
particular, $\phi_{z,\mathrm{mod}}od$ is not a fibration over $\overline{\mathrm{Jac}}(S)$.
\end{example}
\betagin{proposition}
The singular set of $\phi_{z,\mathrm{mod}}od$ has codimension at least $2$. Moreover, if the spectral curve $S$ contains only cusp or nodal singularities, then $\phi_{z,\mathrm{mod}}od$ is smooth.
\end{proposition}
\betagin{proof}
As the singularities of $\phi_{z,\mathrm{mod}}od$ come from the space $\mathcal{S}P(S)$, the claim follows from Proposition \ref{prop_fiber_of_parabolic_module}.
\end{proof}
Since we focus on $\mathrm{SL}C$ Higgs bundles, we must consider the parabolic module compactification of the fibration
$$
0\longrightarrow V\longrightarrow \mathcal{P}\xrightarrow{\hat{s}pace{.3cm}p^{\ast}\ } \mathrm{Prym}(\widetilde{S}/\Sigmagma)\longrightarrow 0\ .
$$
Setting, $\widehat{\phi}Mod:=\tau^{-1}(\overline{\MP})$, then there is a diagram
from \cite[p.\ 17]{gothen2013singular}
\betagin{equation} \label{eq_parabolic_module_for_Prym}
\betagin{tikzcd}
&\mathcal{S}P(S) \arrow[r, ] & \widehat{\phi}Mod \arrow[r, "\mathrm{pr}"] \arrow[d, "\tau"]
& \mathrm{Prym}(\widetilde{S}/\Sigmagma) & \\
& &\overline{\MP} & &
\end{tikzcd}
\end{equation}
Theorem \ref{thm_parabolic_module_main_theorem} proves that $\mathrm{pr}\circ \tau^{-1}|_{\mathcal{P}}=p^{\ast}$.
\mathfrak{su}bsection{Stratifications of the BNR data}
\label{sec:divisor-stratification}
Parabolic modules define a stratification of $\overline{\MP}$ and $\overline{\MT}$. In the following, $\pi:S\to \Sigmagma$
is a branched double cover,
$\sigmagma$ the associated involution on $S$, and by $\sigmagma$ we also denote
its extension to an involution on the normalization $\widetilde{S}$ of $S$.
For a rank 1 torsion free sheaf $L\in\overline{\mathrm{Pic}}(S)$, consider the map
\betagin{equation*}
p^{\mathrm{st}r}_{\mathrm{tf}}:\overline{\mathrm{Pic}}(S)\longrightarrow \mathrm{Pic}(\widetilde{S})\ ,\ p^{\mathrm{st}r}_{\mathrm{tf}}(L):=p^{\ast}L/\mathrm{Tor}(p^{\ast}L)\ ,
\end{equation*}
i.e.\ the torsion free part of the pull-back to the normalization.
By \cite{rabinowitz1979monoidal}, $p^{\mathrm{st}r}_{\mathrm{tf}}(L)=p^{\ast}L$ at $x\in \widetilde{S}$ if and only if
$L$ is locally free at $p(x)\in S$.
\betagin{definition}[{\cite{horn2022semi}}]
An effective divisor $D\in \mathrm{Div}(\widetilde{S})$ is called a $\sigmagma$-divisor if
\betagin{itemize}
\item [(i)] $D\leq \lambdabdam$ and $\sigmagma^{\ast}D=D$;
\item [(ii)] and for any $x\in \mathrm{Fix}(\sigmagma)$, $D|_x=d_xx$, where $d_x\equiv 0\mathrm{mod}d 2$.
\end{itemize}
\end{definition}
The $\sigmagma$-divisors play an important role in describing the singular
Hitchin fibers.
\betagin{proposition}[{\cite{horn2022semi,Mochizukiasymptotic}}]
\label{prop_stratification_BNR_data}
Let $L\in \overline{\MP}$ and write $\widetilde{L}:=p^{\mathrm{st}r}_{\mathrm{tf}} L$. Then we have $\widetilde{L}\otimesimesmes \sigmagma^{\ast}\widetilde{L}=\mathcal{O}(\lambdabdam-D)$ for $D$ a $\sigmagma$-divisor.
\end{proposition}
For a $\sigmagma$-divisor $D$, define
\betagin{align}
\betagin{split}
\widetilde{\MT}_D&=\{J\in \mathrm{Pic}(\widetilde{S})\mid J\otimesimesmes
\sigmagma^{\ast}J=\mathcal{O}(\lambdabdam-D)\}\ ; \\
\widetilde{\MP}_D&=\{J\in\mathrm{Pic}(\widetilde{S})\mid J\otimesimesmes\sigmagma^{\ast}J=\mathcal{O}(-D)\}\ .
\end{split}
\end{align}
Then by \cite[Prop.\ 5.6]{horn2022semi}, $\widetilde{\MT}_D$ and $\widetilde{\MP}_D$
are abelian torsors over $\mathrm{Prym}(\widetilde{S}/\Sigmagma)$ with dimension $g(\widetilde{S})-g(S)=g-1+\frac12r_2$.
In addition, we define
\betagin{align}
\betagin{split}
\overline{\MT}_D&=\{L\in \overline{\MT}\mid p^{\mathrm{st}r}_{\mathrm{tf}} L\in \widetilde{\MT}_D\}\ ;\\
\overline{\MP}_D&=\{L\in\overline{\MP}\midp^{\mathrm{st}r}_{\mathrm{tf}} L\in \widetilde{\MP}_D\}\ .
\end{split}
\end{align}
Then the partial order on divisors defines a stratification of $\overline{\MT}$ (resp.\ $\overline{\MP}$)
by: $\cup_{D'\leq
D}\overline{\MT}_{D'}$ (resp.\ $\cup_{D'\leq D}\overline{\MP}_{D'}$). The top strata are
$\overline{\MT}_{D=0}$ (resp.\ $\overline{\MP}_{D=0}$), and these consist of the locally free
sheaves. From the definition, $\mathcal{T}=\overline{\MT}_{D=0}$ and $\mathcal{P}=\overline{\MP}_{D=0}$.
\betagin{theorem}[{\cite[Thm.\ 6.2]{horn2022semi}}]
\label{thm_stratification_fibration}
(i) Suppose $q$ contains at least one zero of odd order. Then for each
stratum indexed by $\sigmagma$-divisor $D$, if we let $n_{ss}$ be the
number of $p$ such that $D|_p=\lambdabdam|_p$, then there are holomorphic fiber bundles
\betagin{align}
\betagin{split}
(\mathbb{C}^{\ast})^{k_1}\times \mathbb{C}^{k_2}\longrightarrow
\overline{\MT}_D\xrightarrow{\hat{s}pace{.2cm}p^{\mathrm{st}r}_{\mathrm{tf}}\ }
\widetilde{\MT}_D\ ;\\
(\mathbb{C}^{\ast})^{k_1}\times \mathbb{C}^{k_2}\longrightarrow
\overline{\MP}_D\xrightarrow{\hat{s}pace{.2cm}p^{\mathrm{st}r}_{\mathrm{tf}}\ } \widetilde{\MP}_D\ ,
\end{split}
\end{align}
where $k_1=r_1-n_{ss}$, $k_2=2g-2-\frac12\deg(D)-r_1+n_{ss}-\frac {r_2}{2}$, and $r_1,r_2$ are the number of even
and odd zeros.
(ii)
Suppose $q$ is irreducible but all zeros are of even order. Then there exists an
analytic space $\overline{\MT}_D'$ and a double branched covering $p:\overline{\MT}_D\to
\overline{\MT}_D' $, with $\overline{\MT}_D'$ a holomorphic fibration
\betagin{equation*}
(\mathbb{C}^{\ast})^{k_1}\times \mathbb{C}^{k_2}\longrightarrow
\overline{\MT}_D'\xrightarrow{\hat{s}pace{.25cm}p^{\mathrm{st}r}_{\mathrm{tf}}\ } \widetilde{\MT}_D\ .
\end{equation*}
In particular, $\dim(\overline{\MP}_D)=\dim(\overline{\MT}_D)=3g-3-\frac12\deg(D).$
\end{theorem}
As explained in \cite{horn2022semi}, via the BNR correspondence the stratification above translates
into a stratification of the Hitchin fiber.
Let $\mathrm{ch}i_{\mathrm{BNR}}:\overline{\MT}\isorightarrow\mathcal{M}_q$ be the bijection in Theorem
\ref{thm_BNRcorrespondence}. Let $D$ be a $\sigmagma$-divisor. Define
$\mathcal{M}_{q,D}:=\mathrm{ch}i_{\mathrm{BNR}}(\overline{\MT}_D)$. Then the stratification of $\overline{\MT}$
induces a stratification on $\mathcal{M}_q=\bigcup_D\mathcal{M}_{q,D}$.
For each $\sigmagma$-divisor $D$, since $\sigmagma^{\ast}D=D$, we can write $D':=\frac{1}{2}\tilde{\pi}(D)$.
Then $D'$ is an effective divisor with $\mathfrak{su}pp D'\mathfrak{su}bset Z$. Moreover, for
$x\in q^{-1}(0)$, $D_x'\leq \frac12\lfloor \mathrm{ord}_x(q) \rfloor$.
Therefore, $\mathcal{M}_q$ may be regarded as also being stratified by divisors $D'$ defined over $\Sigmagma$.
\mathfrak{su}bsection{The structure of the parabolic module projection}
We now explain the relationship between the divisor $D_v$ in Theorem \ref{thm_parabolic_module_main_theorem} and the $\sigmagma$-divisor.
Given $L \in \overline{\MP}$, define
\betagin{align}
\betagin{split}
\mathcal{S}N_L&:=\{(J,v)\in\widehat{\phi}Mod\mid\tau(J,v)=L\}\ ;\\
\mathcal{S}D_L&:=\{D_v\mid(J,v)\in\mathcal{S}N_L\}\ ,
\end{split}
\end{align}
where $\mathcal{S}N_L$ is $\tau^{-1}(L)$, and $\mathcal{S}D_L$ is the collection of divisors $D_v$
such that $J(-D_v) = p^{\mathrm{st}r}_{\mathrm{tf}}(L)$. If $L$ is locally free, then $J = p^{\ast}
L$, and $\mathcal{S}D_L$ is empty. Moreover, if $\tau(J, v) = \tau(J', v)$, then $J' = J(D_{v'} - D_v)$.
The divisor $D_v$ satisfies the following symmetry property:
\betagin{proposition}
\label{prop_relationship_D_v_and_D}
Let $D$ be a $\sigmagma$-invariant divisor and $L \in \overline{\MP}_D$. For any $D_v \in \mathcal{S}D_L$, we have $D_v + \sigmagma^{\ast} D_v = D$.
\end{proposition}
\betagin{proof}
Let $\tau(J, v) = L$.
Then by Theorem \ref{thm_parabolic_module_main_theorem}, we have $\widetilde{L} =
J(-D_v)$, where $\widetilde{L} = p^{\mathrm{st}r}_{\mathrm{tf}}(L)$. As $L \in \overline{\MP}_D$ and $J \in
\mathrm{Prym}(\widetilde{S}/\Sigmagma)$, we have $\widetilde{L} \otimesimesmes \sigmagma^{\ast} \widetilde{L} = \mathcal{O}(-D)$ and $J \otimesimesmes \sigmagma^{\ast} J = \mathcal{O}_{\widetilde{S}}$, which implies $D_v + \sigmagma^{\ast} D_v = D$.
\end{proof}
As a consequence, we have the following:
\betagin{corollary}
Suppose $q$ has only zeroes of odd order. Then for $L \in \overline{\MP}_D$ and $D_v \in \mathcal{S}D_L$, we have $\sigmagma^{\ast} D_v = D_v$
and $D_v = \frac{1}{2}D$. In addition, $\tau: \widehat{\phi}Mod \to \overline{\MP}$ is a bijection.
\end{corollary}
\betagin{proof}
Since each zero has odd order $\mathfrak{su}pp(D_v) \mathfrak{su}bset \mathrm{Fix}(\sigmagma)$, which implies $D_v = \sigmagma^{\ast} D_v$.
By Proposition \ref{prop_relationship_D_v_and_D}, we must have $D_v = \frac{1}{2}D$.
\end{proof}
There are relationships between the integers appearing in the construction of the parabolic module:
\betagin{lemma}[\cite{Greuel1985}]
\label{lem_integers_in_parabolic_module}
Let $D=\mathfrak{su}m_{i=1}^{r_1} d_i(\tilde{p}_i^+ + \tilde{p}_i^-) + \mathfrak{su}m_{i=1}^{r_2} d_{i}'\tilde{p}_i'$ be a $\sigmagma$-divisor, and let $L\in \overline{\MP}_D$. Then we have
\betagin{itemize}
\item[(i)] $\ell_{p_i}=d_i$ and $\ell_{p_i'}=d_i'/2$;
\item[(ii)] $a_{\tilde{p}_i^+}=a_{\tilde{p}_i^-}=(m_i/2)-d_i$ and $a_{\tilde{p}_i'}=n_i-1-d_i'$.
\end{itemize}
\end{lemma}
\betagin{proof}
Since $L\in \overline{\MP}_D$, we have $\dim\mathrm{Tor}(p^{\ast}L_{p_i})=d_i$ and $\dim\mathrm{Tor}(p^{\ast}L_{p_i'})=d_i'/2$. The claim then follows
from Proposition \ref{prop_appendix_computation}.
\end{proof}
The elements in $\mathcal{S}D_L$ can be explicitly computed.
\betagin{proposition}
\label{prop_computation_NL}
Let $D=\mathfrak{su}m_{i=1}^{r_1} d_i(\tilde{p}_i^+ + \tilde{p}_i^-) + \mathfrak{su}m_{i=1}^{r_2} d_{i}'\tilde{p}_i'$ be a $\sigmagma$-divisor, and let $L\in \overline{\MP}_D$. Then
$N_L=\mathrm{pr}od_{i=1}^{r_1}(d_i+1)$. The number $N_L$ depends only on the $\sigmagma$-divisor $D$.
\end{proposition}
\betagin{proof}
By Lemma \ref{lem_integers_in_parabolic_module}, $V(L)$ can be rewritten as
\betagin{equation*}
\betagin{split}
V(L)=\{(c_1^{\pm},\ldots,c_{r_1}^{\pm},c'_{1}=l_{p_1'},\ldots,c'_{r_2}=l_{p'_{r_2}})\mid c_i^++c_i^-=d_i,c_i^{\pm}\in\mathbb{Z}_{\geq 0}\}\ ,
\end{split}
\end{equation*}
which implies the claim. The condition $D_v+\sigmagma^{\ast}D_v=D$ is automatically satisfied.
\end{proof}
If we define $n_L$ to be the number of $D_v\in\mathcal{S}D_L$ such that $\sigmagma^{\ast}D_v\neq D_v$, then we have the following:
\betagin{proposition}
\label{prop_computation_nL}
\betagin{itemize}
\item [(i)] $n_L$ is even;
\item [(ii)] if $L\in \overline{\MP}_D$ with
$$
D=\mathfrak{su}m_{i=1}^{r_1}d_i(\tilde{p}_i^++\tilde{p}_i^-)+\mathfrak{su}m_{i=1}^{r_2}d_{i}'\tilde{p}_i'\ ,$$
and if there exists $i_0\in\{1,\dots,r_1\}$ such that $d_{i_0}$ is not even, then $n_L=N_L$; otherwise, $n_L=N_L-1$.
\end{itemize}
\end{proposition}
\betagin{proof}
To prove (i), note that if $\sigmagma^{\ast}D_v\neq D_v$, then
$\sigmagma^{\ast}(\sigmagma^{\ast}D_v)\neq \sigmagma^{\ast}D_v$, which means that
$n_L$ is even. For (ii), by Proposition \ref{prop_computation_NL},
$D_v=\sigmagma^{\ast}D_v$ for $D_v\in\mathcal{S}D_L$ if and only if
$c_{i}^+=c_i^-=d_i/2$. Therefore, $n_L\neq N_L$ if and only if all $d_i$ are even, which implies (ii).
\end{proof}
\section{Irreducible singular fibers and the Mochizuki map}
\label{sec_irreducible_singular_fiber}
In this section, we provide a reinterpretation of the limiting configuration construction of a Higgs bundle
on an irreducible fiber, as introduced by Mochizuki in \cite{Mochizukiasymptotic} (see also \cite{horn2022semi}).
We also investigate the relationship between limiting configurations and the stratification.
\mathfrak{su}bsection{Abelianization of a Higgs bundle}
\label{subsection_abelianzation}
Let $q$ be a fixed irreducible quadratic differential with spectral curve $S$, with normalization $p:\widetilde{S}\to S$.
We define $\widetilde{K}:=\widetilde{\pi}^{\ast}K$ (but note that $\widetilde{K}\neq K_{\widetilde S}$) and
$\widetilde{q}:=\widetilde{\pi}^{\ast}q\in H^0(\widetilde{K}^2)$, where $\widetilde{\pi}$ is the double
branched covering of $\widetilde S\to\Sigmagma$ associated to the branching set $Z$ of $q$.
Choose a square root $\omegaega\in H^0(\widetilde{K})$ such that $\widetilde{q}=-\omegaega\otimesimesmes\omegaega$.
Let $\lambdabdambda:=\mathrm{Div}(\omegaega)$ and $\widetilde{Z}:=\mathrm{supp}(\lambdabdambda)$. We can then write
$$\lambdabdam=\mathfrak{su}m_{i=1}^{r_1}\frac{m_i}{2}(\tilde{p}_i^++\tilde{p}_i^-)+\mathfrak{su}m_{j=1}^{r_2}n_j\tilde{p}_j'\ .
$$
If $\sigmagma:\widetilde{S}\to \widetilde{S}$ is the involution, then $\sigmagma^{\ast}\omega=-\omega$.
Let $(\mathcal{E},\varphi)$ be a Higgs bundle with $\det\varphi=q$. Consider the pullback
$(\widetilde{\ME},\tilde{\vp}):=(\widetilde{\pi}^{\ast}\mathcal{E},\widetilde{\pi}^{\ast}\varphi)$. We have $\tilde{\vp}\in H^0(\mathrm{End}(\widetilde{\ME})\otimesimesmes\widetilde{K})$
and $\widetilde{q}=\det(\tilde{\vp})$. Since $\widetilde{q}=-\omegaega\otimesimesmes\omegaega$, $\pm\omegaega$ are well-defined eigenvalues
of $\tilde{\vp}$ over $\widetilde{S}$. Let $\timeslde{\lambdabda}$ be the canonical section of $\mathrm{Tot}(\widetilde{K})$.
The spectral curve for $(\widetilde{\ME},\tilde{\vp})$ is defined by the equation
$$
\widetilde{S}':=\{\tilde{\lam}^2-\tilde{q}=0\}.
$$
The set $\widetilde{S}'=\mathrm{Im}(\omegaega)\cup\mathrm{Im}(-\omegaega)\mathfrak{su}bset \mathrm{Tot}(\widetilde{K})$ decomposes into two irreducible pieces.
Having fixed a choice of $\omegaega$, the eigenvalues of $\tilde{\vp}$ are globally well-defined, and we can define the line bundle
$\widetilde{L}_+\mathfrak{su}bset \mathcal{E}$ as $\widetilde{L}_+:=\ker(\tilde{\vp}-\omegaega)$. Since $\sigmagma^{\ast}\omega=-\omega$, $\widetilde{L}_-=\sigmagma^{\ast}\widetilde{L}_+=\ker(\tilde{\vp}+\omegaega)$,
and there is an isomorphism $\widetilde{\ME}|_{\widetilde{S}\setminus \widetilde{Z}}\cong \widetilde{L}_+\oplus \widetilde{L}_-|_{\widetilde{S}\setminus \widetilde{Z}}$.
There is a local description of $(\widetilde{\ME},\tilde{\vp})$:
\betagin{lemma}[{\cite[Lemma 5.1, Thm.\ 5.3]{horn2022semi},\cite[Lemma
4.2]{Mochizukiasymptotic}}]
\label{lem_local_description}
Let $x\in \widetilde{Z}$ and write $\lambdabdambda|_x=m_xx$. Let $U$ be a holomorphic
coordinate neighborhood of $x$. Then there exists a frame $\mathfrak{e}\in H^0(U,\widetilde{K})$ such that, under a suitable trivialization of $\mathcal{E}|_U\cong U\timesmes \mathbb{C}^2$, we can write
\betagin{equation} \label{eq_local_description_Higgs_bundle}
\textrm{ }\tilde{\vp}=z^{d_x}\betagin{pmatrix}
0 & 1\\
z^{2m_x-2d_x} & 0
\end{pmatrix} \otimesimesmes \mathfrak{e}.
\end{equation}
Moreover, if we define $D:=\mathfrak{su}m_{x\in\widetilde{Z}}d_xx$, then $D$ is a $\sigmagma$-divisor.
\end{lemma}
\betagin{lemma}[{\cite[Sec.\ 4.1]{Mochizukiasymptotic}}]
\label{lem_exact_sequence_abelianization}
For the $\widetilde{L}_{\pm}$ defined above, we have $\widetilde{L}_+\otimesimesmes \widetilde{L}_-=\mathcal{O}(D-\lambdabdam)$. Moreover, if we denote $\widetilde{L}_0:=\widetilde{L}_+(\lambdabdam-D)$ and $\widetilde{L}_1:=\sigmagma^{\ast}\widetilde{L}_0$, then
$\widetilde{L}_+=\widetilde{\ME}\cap \widetilde{L}_0$, $\widetilde{L}_-=\widetilde{\ME}\cap \widetilde{L}_1$, and we have the exact sequences
\betagin{align*}
0\longrightarrow \widetilde{L}_+\longrightarrow \widetilde{\ME} \longrightarrow
\widetilde{L}_1\longrightarrow 0\ ;\\
0\longrightarrow\widetilde{L}_-\longrightarrow \widetilde{\ME} \longrightarrow
\widetilde{L}_0\longrightarrow 0\ .
\end{align*}
\end{lemma}
\betagin{proof}
The inclusion of $\widetilde{L}_{\pm}\to \mathcal{E}$ defines an exact sequence of sheaves
$$
0\longrightarrow \mathcal{O}(\widetilde{L}_+)\oplus \mathcal{O}(\widetilde{L}_-)\longrightarrow
\mathcal{O}(\mathcal{E})\longrightarrow \mathcal{T}\longrightarrow 0\ ,
$$ where $\mathcal{T}$ is a torsion sheaf with $\mathfrak{su}pp \mathcal{T}\mathfrak{su}bset \widetilde{Z}$. From the local description in
\eqref{eq_local_description_Higgs_bundle}, in the same trivialization, $\widetilde{L}_{\pm}$ are spanned by the bases $s_{\pm}=\betagin{pmatrix}
1 \\
\pm z^{m_x-d_x}
\end{pmatrix}.$ Therefore, as $\det(\mathcal{E})=\mathcal{O}_\Sigmagma$, we obtain $\widetilde{L}_+\otimesimesmes \widetilde{L}_-=\mathcal{O}(D-\lambdabdam)$.
Since $s_+,s_-$ are linear independent away from $z$, $\widetilde{\ME}/\widetilde{L}_+$ is
locally generated by the section $z^{d_x-m_x}s_-$. Therefore, $\widetilde{\ME}/\widetilde{L}_+\cong \widetilde{L}_-(\lambdabdam-D)=\widetilde{L}_1$. Using the involution, we obtain the other exact sequence.
\end{proof}
Therefore, if $\widetilde{L}\otimesimesmes \sigmagma^{\ast}\widetilde{L}=\mathcal{O}(D-\lambdabdam)$, we have
$\widetilde{L}_0=\widetilde{L}(\lambdabdam-D)\in \widetilde{\MT}_D$. In summary, the construction above leads us to consider the composition of the following maps:
\betagin{equation*}
\betagin{split}
\partialta &: \mathcal{M}_q \longrightarrow \mathrm{Pic}(\widetilde{S})\xrightarrow{\otimesimesmes \mathcal{O}(\lambdabdam-D)}\widetilde{\MT}_D\
;\\
& (\mathcal{E},\varphi)\mapsto \widetilde{L}_+\mapsto \widetilde{L}_+(\lambdabdam-D)\ ,
\end{split}
\end{equation*}
where the first map is given by taking the kernel of $(\tilde{\pi}^{\ast}\varphi-\omega)|_{\tilde{\pi}^{\ast}\mathcal{E}}$.
This construction is directly related to the torsion free pull-back. Recall that $\mathrm{ch}i_{\mathrm{BNR}}:\overline{\MT}\to \mathcal{M}_q$ is the BNR correspondence map,
and $p^{\mathrm{st}r}_{\mathrm{tf}}:\overline{\mathrm{Pic}}(S)\to\mathrm{Pic}(\widetilde{S})$ is the torsion free pull-back. Then we have
\betagin{proposition}
\label{prop_equivalent_torsionfree_pull_back}
$\partialta\circ\mathrm{ch}i_{\mathrm{BNR}}=p^{\mathrm{st}r}_{\mathrm{tf}}$. In particular, if $J\in \overline{\MT}_D$, then $\partialta\circ\mathrm{ch}i_{\mathrm{BNR}}(J)\in \widetilde{\MT}_D$.
\end{proposition}
\betagin{proof}
Let $J\in \overline{\MT}$, and write $(\mathcal{E},\varphi)=\mathrm{ch}i_{\mathrm{BNR}}(J)$, $(\widetilde{\ME},\tilde{\vp}):=\tilde{\pi}^{\ast}(\mathcal{E},\varphi)$.
Recall the BNR exact sequence on $S$ (see \eqref{eq_BNR}):
\betagin{equation*}
\betagin{split}
0\longrightarrow J(-\Deltata)\longrightarrow
\pi^{\ast}\mathcal{E}\xrightarrow{\pi^{\ast}\varphi-\lambda}\pi^{\ast}\mathcal{E}\otimesimesmes
\pi^{\ast}K\longrightarrow J\otimesimesmes
\pi^{\ast}K\longrightarrow 0\ .
\end{split}
\end{equation*}
As $p^{\ast}$ is right exact, we obtain
$$\widetilde{\ME}\xrightarrow{\hat{s}pace{.3cm}\tilde{\vp}-\tilde{\lam}\ }\widetilde{\ME}\otimesimesmes
\tilde{K}\longrightarrow p^{\ast}J\otimesimesmes \tilde{K}\longrightarrow
0\ . $$
Since the spectral curve is $\widetilde{S}'=\mathrm{Im}(\omega)\cup\mathrm{Im}(-\omega)$, we can consider the restriction to the component $\mathrm{Im}(\omega)$ and
write $\tilde{\lam}=\omega,\;\widetilde{L}_{\pm}:=\ker(\tilde{\vp}\mp\omega)$. We obtain an exact sequence
\betagin{equation*}
0\longrightarrow\widetilde{L}_+\longrightarrow\widetilde{\ME}\xrightarrow{\hat{s}pace{.25cm}\tilde{\vp}-\omega\
}\widetilde{\ME}\otimesimesmes \tilde{K}\longrightarrow p^{\ast}J\otimesimesmes \tilde{K}\longrightarrow 0\ ,
\end{equation*}
which breaks into short exact sequences
\betagin{align*}
0\longrightarrow \widetilde{L}_+\longrightarrow \widetilde{\ME}\longrightarrow
\mathrm{Im}(\tilde{\vp}-\omega)\longrightarrow 0\ ;\\
0\longrightarrow \mathrm{Im}(\tilde{\vp}-\omega)\longrightarrow \widetilde{\ME}\otimesimesmes
\tilde{K}\longrightarrow p^{\ast}J\otimesimesmes \tilde{K}\longrightarrow 0\ .
\end{align*}
Using the local trivialization in Lemma \ref{lem_local_description}, $\mathrm{Im}(\tilde{\vp}-\omega)$ is locally spanned by $\betagin{pmatrix}
z^{d_x}\\ -z^{m_x} \end{pmatrix}\mathfrak{e}$. From Lemma \ref{lem_exact_sequence_abelianization}, if we write
$\widetilde{L}_0:=\widetilde{L}_+(\lambdabdam-D)$ and $\widetilde{L}_1:=\sigmagma^{\ast}\widetilde{L}_0$, then
$$
\partialta\circ\mathrm{ch}i_{BNR}(J)=\widetilde{L}_+(\lambdabdam-D)\ .
$$
Moreover, there is an isomorphism $\mathrm{Im}(\tilde{\vp}-\omega)\cong \widetilde{L}_1$. Letting $\widetilde{L}_1'$ be the saturation of $\widetilde{L}_1$,
then we obtain the commutative diagram:
\betagin{equation*}
\betagin{tikzcd}
0\arrow[r, ] & \widetilde{L}_1 \arrow[r, ] \arrow[d, "i"] & \widetilde{\ME}\otimesimesmes\tilde{K} \arrow[r, ] \arrow[d, "\cong"] & p^{\ast}J\otimesimesmes \tilde{K} \arrow[r, ] \arrow[d, ] & 0 \\
0\arrow[r, ]& \widetilde{L}_1' \arrow[r, ]&\widetilde{\ME}\otimesimesmes \tilde{K} \arrow[r, ]&p^{\mathrm{st}r}_{\mathrm{tf}} J\otimesimesmes \tilde{K} \arrow[r, ] &0
\end{tikzcd}
\end{equation*}
where $i:\widetilde{L}_1\to \widetilde{L}_1'$ is the natural inclusion. Moreover, in the same
trivialization, $\widetilde{L}_1'$ is spanned by the section $\betagin{pmatrix}
1\\
-z^{m_x-d_x}
\end{pmatrix}\mathfrak{e}$. Therefore, $\widetilde{L}_1'\cong \widetilde{L}_-\otimesimesmes \tilde{K}$ and from
Lemma \ref{lem_exact_sequence_abelianization}, $p^{\mathrm{st}r}_{\mathrm{tf}} J=\partialta\circ\mathrm{ch}i_{\mathrm{BNR}}(J)$.
\end{proof}
If $(\mathcal{E},\varphi)$ is a Higgs
bundle with $(\mathcal{E},\varphi)=\mathrm{ch}i_{\mathrm{BNR}}(L)$, and $\widetilde{L}_0=\partialta\circ\mathrm{ch}i_{BNR}(L)$, then by Proposition \ref{prop_equivalent_torsionfree_pull_back}, $\widetilde{L}_0=p^{\mathrm{st}r}_{\mathrm{tf}}(L)$. We define a Higgs bundle $(\widetilde{\ME}_0,\tilde{\vp}_0)$
as follows
\betagin{equation*}
\betagin{split}
\widetilde{\ME}_0=\widetilde{L}_0\oplus \sigmagma^{\ast}\widetilde{L}_0,\;\tilde{\vp}_0=\betagin{pmatrix}
\omega & 0 \\
0 & -\omega
\end{pmatrix}.
\end{split}
\end{equation*}
Moreover, $\widetilde{\ME}$ is an $\mathcal{O}_{\widetilde{S}}$ submodule of $\widetilde{\ME}_0$ with a natural inclusion $i\to\inftya:\widetilde{\ME}\to \widetilde{\ME}_0$ satisfying the following:
\betagin{itemize}
\item [(i)] the induced morphism $\widetilde{\ME}\to \widetilde{L}_0,\;\widetilde{\ME}\to \sigmagma^{\ast}\widetilde{L}_0$ is surjective,
\item [(ii)] the restriction of $i\to\inftya|_{\widetilde{S}\setminus \widetilde{Z}}$ is an isomorphism,
\item [(iii)] $\tilde{\vp}_0\circ i\to\inftya=i\to\inftya\circ \tilde{\vp}$.
\end{itemize}
Following \cite[Sec.\ 4.1]{Mochizukiasymptotic},
we call $(\widetilde{\ME}_0,\tilde{\vp}_0)$ the \emph{abelianization of the Higgs bundle} $(\mathcal{E},\varphi)$.
\mathfrak{su}bsection{The construction of the algebraic Mochizuki map}
In this subsection, we define the algebraic Mochizuki map, as introduced in \cite{Mochizukiasymptotic}.
Recall that for any divisor $D=\mathfrak{su}m_{x\in Z}d_x x$, there is a canonical weight function
$$\mathrm{ch}i_D(x):=\betagin{cases}d_x &
x\in\mathfrak{su}pp D\ ;\\
0 &x\notin \mathfrak{su}pp D\ .
\end{cases}
$$
We also have the stratification
$\overline{\MT}=\cup_D\overline{\MT}_D$, for $\sigmagma$ divisors $D$. Let $\mathcal{S}F(\widetilde{S})$ be the space of all degree zero filtered
line bundles over $\widetilde{S}$. The \emph{algebraic Mochizuki map} $\Theta^{\mathrm{Moc}}$ is defined as
\betagin{equation*}
\Theta^{\mathrm{Moc}} : \overline{\MT}\longrightarrow \mathcal{S}F(\widetilde{S})\ ,\
L\mapsto \mathcal{F}_{\ast}(p^{\mathrm{st}r}_{\mathrm{tf}}(L),\tfrac{1}{2}\mathrm{ch}i_{D-\lambdabdam})\ .
\end{equation*}
\betagin{example}
When $q$ has only simple zeroes, this construction generalizes that of \cite{mazzeo2016ends} (see also \cite{fredrickson2018generic}).
In the case of a quadratic differential with simple zeros, the spectral curve $S$ is smooth, and every torsion free sheaf is locally free,
so that $\mathcal{T} = \overline{\MT}$. If $Z = \{p_1, \dots, p_{4g-4}\}$ are the branch points of $S$, and
$\lambdabdambda = \mathfrak{su}m_{i=1}^{4g-4} p_i$, then the weight function $\frac{1}{2}\mathrm{ch}i_{-\lambdabdambda}$ assigns a weight of $-\frac{1}{2}$
at each $p_i$. For $L\in \mathcal{T}$, $\Theta^{\mathrm{Moc}}(L)=\mathcal{F}_{\ast}(L,\frac12\mathrm{ch}i_{-\lambdabdam}).$
\end{example}
Below are some additional properties of $\Theta^{\mathrm{Moc}}$.
\betagin{proposition}
$\Theta^{\mathrm{Moc}}|_{\overline{\MT}_D}$ is a continuous map.
\end{proposition}
\betagin{proof}
This follows directly from the definition of $\Theta^{\mathrm{Moc}}$ and Theorem \ref{thm_convergence_family_harmonic_bundles}.
\end{proof}
From Theorem \ref{thm_stratification_fibration}, we know that for a
$\sigmagma$-divisor $D$, the preimage of the map $p^{\mathrm{st}r}_{\mathrm{tf}}:\overline{\MT}_D\to \widetilde{\MT}_D$ has
dimension $2g-2-\frac{1}{2}\deg(D)-r_2/2$, where $r_2$ is the number of odd
zeros of $q$. Even for the top stratum $D=0$, $p^{\mathrm{st}r}_{\mathrm{tf}}$ is not injective if
the spectral curve is not smooth. Indeed, if $L_1,L_2\in \overline{\MT}_D$ with $p^{\mathrm{st}r}_{\mathrm{tf}}(L_1)=p^{\mathrm{st}r}_{\mathrm{tf}}(L_2)$, then based on the construction
we have $\Theta^{\mathrm{Moc}}(L_1)=\Theta^{\mathrm{Moc}}(L_2)$. In summary, we have the following result:
\betagin{proposition}
\label{prop_simple_zero_injective}
If $q\in H^0(K^2)$ is irreducible, then $\Theta^{\mathrm{Moc}}$ is injective if and only if $q$ has simple zeros.
\end{proposition}
\mathfrak{su}bsection{Convergence of subsequences}
Fix a locally free $L_0\in \mathcal{T}$.
Using the isomorphism $\mathrm{ps}i_{L_0}:\overline{\MT}\to \overline{\MP}$ defined by $\mathrm{ps}i_{L_0}(L)=LL_0^{-1}$, we can extend the Mochizuki map $\Theta^{\mathrm{Moc}}$ to $\overline{\MP}$.
For $J\in \overline{\MP}_D$, we write $\widetilde{J}:=p^{\mathrm{st}r}_{\mathrm{tf}}(J)$ and choose the weight function $\frac{1}{2}\mathrm{ch}i_D$. We then define:
\betagin{equation*}
\Theta^{\mathrm{Moc}}_0:\overline{\MP}\longrightarrow
\mathcal{S}F(\widetilde{S})\ ,\ J\mapsto\mathcal{F}_{\ast}(\widetilde{J},\tfrac{1}{2}\mathrm{ch}i_D)\ .
\end{equation*}
\betagin{proposition}
\label{prop_algebraic_Mochizuki_map_on_PMod}
The map $\Theta^{\mathrm{Moc}}_0$ satisfies the following properties:
\betagin{itemize}
\item [(i)] Let $J\in \overline{\MP}$ and $L:=L_0J$, then
$$\Theta^{\mathrm{Moc}}_0(J)=\Theta^{\mathrm{Moc}}(L)\otimesimesmes \Theta^{\mathrm{Moc}}(L_0)^{-1}\ ,$$ where
$\otimesimesmes$ is the tensor product for filtered line bundles
\eqref{eqn:tensor}.
\item [(ii)] Suppose $L=\tau(I,v)$ with $(\mathcal{I},v)\in\widehat{\phi}Mod$ and $L\in \overline{\MP}_D$, then
\betagin{equation*}
\Theta^{\mathrm{Moc}}_0\circ\tau(I,v)=\mathcal{F}_{\ast}(I(-D_v),
\tfrac{1}{2}\mathrm{ch}i_{D_v+\sigmagma^{\ast}D_v})\ ,
\end{equation*}
where $D_v$ is the corresponding divisor defined in Theorem \ref{thm_parabolic_module_main_theorem}.
\item [(iii)] If $\sigmagma^{\ast}D_v=D_v$, then
$\Theta^{\mathrm{Moc}}_0\circ\tau(\mathcal{I},v)=\mathcal{F}_{\ast}(\mathcal{I},0)$, where $0$ means
all parabolic weights are zero.
\end{itemize}
\end{proposition}
\betagin{proof}
As $L_0$ is locally free, we have $p^{\mathrm{st}r}_{\mathrm{tf}} J=(p^{\ast}L_0)^{-1}\otimesimesmes p^{\mathrm{st}r}_{\mathrm{tf}} L$. By definition,
$$
\Theta^{\mathrm{Moc}}_0(J)=\mathcal{F}_{\ast}(p^{\mathrm{st}r}_{\mathrm{tf}} J,\tfrac{1}{2}\mathrm{ch}i_D)\ ,\
\Theta^{\mathrm{Moc}}(L)=\mathcal{F}_{\ast}(p^{\mathrm{st}r}_{\mathrm{tf}} L,\tfrac{1}{2}\mathrm{ch}i_{D-\lambdabdam})\ ,\ \Theta^{\mathrm{Moc}}(L_0)=
\mathcal{F}_{\ast}(p^{\mathrm{st}r}_{\mathrm{tf}} L_0,\tfrac{1}{2} \mathrm{ch}i_{-\lambdabdam})\ ,
$$
which implies (i). For (ii), by Theorem \ref{thm_parabolic_module_main_theorem}, $p^{\mathrm{st}r}_{\mathrm{tf}} L=I(-D_v)$, and from Proposition
\ref{prop_relationship_D_v_and_D}, we have $D=D_v+\sigmagma^{\ast}D_v$, which implies (ii). When $\sigmagma^{\ast}D_v=D_v$, we compute
$$
\mathcal{F}_{\ast}(I(-D_v)\ ,\
\tfrac{1}{2}\mathrm{ch}i_{D_v+\sigmagma^{\ast}D_v})=\mathcal{F}_{\ast}(I(-D_v)\ ,\
\mathrm{ch}i_{D_v})=\mathcal{F}_{\ast}(I,0)\ ,
$$
which implies (iii).
\end{proof}
We now give a criterion for the continuity of the map $\Theta^{\mathrm{Moc}}$. By
Proposition \ref{prop_algebraic_Mochizuki_map_on_PMod}, it is sufficient to study the map $\Theta^{\mathrm{Moc}}_0$.
Recall that for $L\in \overline{\MP}$, we have
$$
\mathcal{S}N_L:=\{(J,v)\in\widehat{\phi}Mod\mid \tau(J,v)=L\},\quad\mathcal{S}D_L:=\{D_v\mid (J,v)\in\mathcal{S}N_L\},
$$
and the number $n_L$ is defined to be the number of divisors $D_v\in \mathcal{S}D_L$ such that $\sigmagma^{\ast}D_v\neq D_v$.
\betagin{proposition}
Let $D$ be a $\sigmagma$-divisor, $L\in\overline{\MP}_D$, and assume that $\Theta^{\mathrm{Moc}}_0$ is continuous at $L$. Then, for $(J,v)\in \mathcal{S}N_L$ and
$D_v\in \mathcal{S}D_{L}$, we have $\sigmagma^{\ast}D_v=D_v$, i.e., $n_L=0$.
\end{proposition}
\betagin{proof}
As the top stratum $\mathcal{P}$ is dense in $\overline{\MP}$, there exists a family of $L_i\in \mathcal{P}$ such that $\lim_{i\to \infty}L_i=L$. Let $(J_i,v_i)\in \widehat{\phi}Mod$ be such that $\tau(J_i,v_i)=L_i$. Then, $\lim_{i\to\infty}(J_i,v_i)=(J_{\infty},v_{\infty})$, and $\tau(J_{\infty},v_{\infty})=L$. As $L_i$ is locally free, we have $D_{v_i}=0$. Moreover, by Theorem \ref{thm_parabolic_module_main_theorem}, we have $p^{\mathrm{st}r}_{\mathrm{tf}} L=J_{\infty}(-D_{v_\infty})$, and from Proposition \ref{prop_relationship_D_v_and_D}, we have $D=D_{v_{\infty}}+\sigmagma^{\ast}D_{v_{\infty}}$.
By Proposition \ref{prop_algebraic_Mochizuki_map_on_PMod}, we have
$$\Theta^{\mathrm{Moc}}_0(L_i)=\Theta^{\mathrm{Moc}}_0\circ\tau(J_i,v_i)=\mathcal{F}_{\ast}(J_i,0)\ ,$$
and we compute
$$\lim_{i\to\infty}\Theta^{\mathrm{Moc}}_0(L_i)=\mathcal{F}_{\ast}(J_{\infty},0)=\mathcal{F}_{\ast}(J_{\infty}(-D_{v_{\infty}}),\mathrm{ch}i_{D_{v_{\infty}}})
\ .$$
Moreover, by Proposition \ref{prop_algebraic_Mochizuki_map_on_PMod}, we have
$$
\Theta^{\mathrm{Moc}}_0(L)=\mathcal{F}_{\ast}(J_{\infty}(-D_{v_{\infty}}),\frac{1}{2}(\mathrm{ch}i_{D_{v_{\infty}}}+\mathrm{ch}i_{{\sigmagma^{\ast}D_{v_{\infty}}}})).$$
Since $\Theta^{\mathrm{Moc}}_0$
is continuous on $L$, we have $\lim_{i\to \infty}\Theta^{\mathrm{Moc}}_0(L_i)=\Theta^{\mathrm{Moc}}(L)$, which implies that $\mathrm{ch}i_{D_{v_{\infty}}}=\mathrm{ch}i_{\sigmagma^{\ast}D_{v_{\infty}}}$.
\end{proof}
By Proposition \ref{prop_computation_nL}, $n_L>0$ if and only if $q$ has at least one zero of even order. Hence, the following is immediate.
\betagin{corollary}
\label{cor_even_zero_not_continuous}
Suppose $q$ is irreducible and has a zero of even order, then $\Theta^{\mathrm{Moc}}_0$ is not continuous.
\end{corollary}
By contrast, we have the following.
\betagin{proposition}
If $q$ is irreducible with all zeroes of odd order, then $\Theta^{\mathrm{Moc}}_0$ is continuous.
\label{prop_odd_zero_continuous}
\end{proposition}
\betagin{proof}
Since all zeroes of $q$ are odd, for any $L\in \overline{\MP}$, we have $n_L=0$. Let $L_{\infty}\in \overline{\MP}$ be fixed and let $L_i\in \overline{\MP}$ be
any sequence such that $\lim_{i\to\infty}L_i=L_{\infty}$. Since $\tau:\widehat{\phi}Mod\to \overline{\MP}$ is bijective, we take $(J_i,v_i)\in\widehat{\phi}Mod$
with $\tau(J_i,v_i)=L_i$. Moreover, we assume $\lim_{i\to \infty}(J_i,v_i)=(J_{\infty},v_{\infty})$ with $\tau(J_{\infty},v_{\infty})=L_{\infty}$.
Since $q$ only contains odd zeros, it follows that $\mathfrak{su}pp D_v\mathfrak{su}bset \mathrm{Fix}(\sigmagma)$. By Proposition \ref{prop_algebraic_Mochizuki_map_on_PMod},
we have $\Theta^{\mathrm{Moc}}_0(L_i)=\mathcal{F}_{\ast}(J_i,0)$. Therefore, we have:
$$
\lim_{i\to \infty}\Theta^{\mathrm{Moc}}_0(L_i)=\lim_{i\to \infty}\mathcal{F}_{\ast}(J_i,0)=\mathcal{F}_{\ast}(J_{\infty},0)=\Theta^{\mathrm{Moc}}_0(L_{\infty}).
$$
This concludes the proof.
\end{proof}
\betagin{theorem}
\label{thm_algebraic_Mochizuki_map_continuous}
Suppose $q$ is irreducible. For the map $\Theta^{\mathrm{Moc}}:\mathcal{M}_q\rightarrow \mathcal{S}F(\widetilde{S})$, we have:
\betagin{itemize}
\item [(i)] $\Theta^{\mathrm{Moc}}$ is injective if and only if $q$ only has only simple zeros;
\item[(ii)] if $q$ has only zeroes of odd order, $\Theta^{\mathrm{Moc}}$ is continuous;
\item[(iii)] if $q$ contains a zero of even order, $\Theta^{\mathrm{Moc}}$ is not continuous.
\end{itemize}
\end{theorem}
\betagin{proof}
(i) follows from Proposition \ref{prop_simple_zero_injective}. (ii) follows from Proposition \ref{prop_odd_zero_continuous}. (iii) follows from
Corollary \ref{cor_even_zero_not_continuous}.
\end{proof}
\betagin{proposition}
\label{prop_number_of_limits_of_filtered_bundle}
Suppose $n_L> 0$. Then for $k=1,\ldots,n_L$, there exist sequences ${L_i^k}$ with $\lim_{i\to \infty}L_i^k=L$ such that if we denote
$\mathcal{F}_{\ast}^k:=\lim_{i\to \infty}\Theta^{\mathrm{Moc}}_0(L_i^k)$, $\mathcal{F}_{\ast}^0:=\Theta^{\mathrm{Moc}}_0(L)$, then $\mathcal{F}_{\ast}^{k_1}\neq \mathcal{F}_{\ast}^{k_2}$ for $k_1\neq k_2$.
Moreover, there exist $\{D_{1},\ldots,D_{n_L}\}\mathfrak{su}bset \mathcal{S}D_L$ such that $\mathcal{F}_{\ast}^{k}=\mathcal{F}_{\ast}(p^{\mathrm{st}r}_{\mathrm{tf}} L,\mathrm{ch}i_{D_k})$.
\end{proposition}
\betagin{proof}
By the definition of $n_L$, we can find $(J^k,v^k)$ with $\tau(J^k,v^k)=L$.
If we define $D_k:=D_{v^k}$, then $\sigmagma^{\ast}D_{k}\neq D_{k}$.
Moreover, by Theorem \ref{thm_parabolic_module_main_theorem}, we have
$p^{\mathrm{st}r}_{\mathrm{tf}} L=J^k(-D_k)$. As $\tau^{-1}(\mathcal{P})$ is dense in $\widehat{\phi}Mod$, for each
$(J^k,v^k)$, we can find a sequence $(J_{i}^k,v_{i}^k)\in \tau^{-1}(\mathcal{P})$ such that $\lim_{i\to \infty}(J_i^k,v_i^k)=(J^k,v^k)$ and we define $L_i^k:=\tau(J_{i}^k,v_{i}^k)$.
Since $L_i^k$ is locally free, $D_{v_i^k}=0$, and thus $\Theta^{\mathrm{Moc}}_0(L_i^k)=\mathcal{F}_{\ast}(J_i^k,0)$. We compute
\betagin{equation*}
\lim_{i\to\infty}\Theta^{\mathrm{Moc}}_0(L_i^k)=\mathcal{F}_{\ast}(J^k,0)=\mathcal{F}_{\ast}(p^{\mathrm{st}r}_{\mathrm{tf}} L,\mathrm{ch}i_{D_k})
\end{equation*}
and $\Theta^{\mathrm{Moc}}_0(L)=\mathcal{F}_{\ast}(p^{\mathrm{st}r}_{\mathrm{tf}} L,\frac12\mathrm{ch}i_{D})$. Based on our assumptions, we have $D_{k_1}\neq D_{k_2}$ for $k_1\neq k_2$ and $\sigmagma^{\ast}D_k\neq D_k$, which implies that $\mathrm{ch}i_{D_{k_1}}\neq \mathrm{ch}i_{D_{k_2}}$ for $k_1\neq k_2$ and $\mathrm{ch}i_{D_k}\neq \frac12\mathrm{ch}i_D$.
\end{proof}
We now present a computation for the case of a simple nodal curve.
\betagin{example}
Let $q$ be a quadratic differential with $2g-4$ simple zeros, and let $x$ be an even zero of $q$ of order two. Then $S$ has a singular point, which we also denote by $x$. Let $p:\widetilde{S}\to S$ be the normalization map and
$p^{-1}(x)=\{x_1,x_2\}$. Consider the $\sigmagma$-divisor $D=x_1+x_2$, and let $L\in \overline{\MP}_D$. Then $n_L=2$, and we can write $\mathcal{S}N_L={(J_1,v_1),(J_2,v_2)}$, where $D_{v_1}=x_1$ and $D_{v_2}=x_2$. Moreover, we have $p^{\mathrm{st}r}_{\mathrm{tf}} L=J_1\otimesimesmes \mathcal{O}(-x_1)=J_2\otimesimesmes \mathcal{O}(-x_2)$.
Let $(\alphapha,\betata)$ denote the parabolic weight that is equal to $\alphapha$ at $x_1$, $\betata$ at $x_2$, and $\frac12$ at all other zeros. Then the filtered bundles obtained in Proposition \ref{prop_number_of_limits_of_filtered_bundle} are
\betagin{equation*}
\mathcal{F}_{\ast}(p^{\mathrm{st}r}_{\mathrm{tf}} L, (1,0))\ ,\ \mathcal{F}_{\ast}(p^{\mathrm{st}r}_{\mathrm{tf}} L, (0,1))\ ,\
\mathcal{F}_{\ast}(p^{\mathrm{st}r}_{\mathrm{tf}} L, (\tfrac{1}{2},\tfrac{1}{2}))\ .
\end{equation*}
\end{example}
\mathfrak{su}bsection{Mochizuki's convergence theorem for irreducible fibers}
In this subsection, we recall Mochizuki's construction of the limiting configuration metric \cite[Section 4.2.1, 4.3.2]{Mochizukiasymptotic} and the convergence theorem.
\mathfrak{su}bsubsection{Limiting configuration metric}
Let $q$ be an irreducible quadratic differential and $(\mathcal{E},\varphi)\in
\mathcal{M}_q$ a Higgs bundle with $(\mathcal{E},\varphi)=\mathrm{ch}i_{\mathrm{BNR}}(L)$. We write
$\widetilde{L}_0=p^{\mathrm{st}r}_{\mathrm{tf}} L$ and $(\widetilde{\ME},\tilde{\vp}):=p^{\ast}(\mathcal{E},\varphi)$. Then the abelianization
of $(\mathcal{E},\varphi)$, which is a Higgs bundle over $\widetilde{S}$, can be written as
$\widetilde{\ME}_0=\widetilde{L}_0\oplus \sigmagma^{\ast}\widetilde{L}_0,\;\tilde{\vp}_0=\mathrm{diag}(\omega,\;-\omega)$.
The natural inclusion $i\to\inftya:(\widetilde{\ME},\tilde{\vp})\to (\widetilde{\ME}_0,\tilde{\vp}_0)$ is an isomorphism over $\widetilde{S}\setminus \widetilde{Z}$. Moreover, we write $D$ be the $\sigmagma$-divisor of $(\mathcal{E},\varphi)$.
From the construction of $\Theta^{\mathrm{Moc}}(L)$ and Proposition
\ref{prop_number_of_limits_of_filtered_bundle}, we have $n_L$ different
divisors $D_k$ for $k=1,\cdotots,n_L$ with $\sigmagma^{*}D_k\neq D_k$ and
$D_k+\sigmagma^{*}D_k=D$. Moreover, we can find $n_L+1$ different filtered
bundles with deg $0$. Define $$\mathcal{F}_{*,0}:=\Theta^{\mathrm{Moc}}(L)=\mathcal{F}_{*}(\widetilde{L}_0,\mathrm{ch}i_{\frac12(D-\lambdabdam)}),\; \mathcal{F}_{*,k}:=\mathcal{F}_{*}(\widetilde{L}_0,\mathrm{ch}i_{D_k-\frac12\lambdabdam)}),$$
which are all degree zero filtered bundles with different level of filtrations.
Now, we will introduce the construction in \cite[Section 4.2.1, 4.3.2]{Mochizukiasymptotic}. For $k=0,\cdotots,n_L$, we define $\tilde{h}_k$ to be the harmonic metric for the filtered bundle $\mathcal{F}_{*,k}$; this is well-defined up to a multiplicative constant. To fix this constant, assume that $\sigmagma^{\ast}\tilde{h}_k\otimesimesmes \tilde{h}_k=1$. This gives a unique choice of $\tilde{h}_0$. We then define the metric $\widetilde{H}_k=\mathrm{diag}(\tilde{h}_k,\;\sigmagma^{\ast}\tilde{h}_k)$ on $\widetilde{\ME}_0$, with $\det(\widetilde{H}_k)=1$. For the resulting harmonic bundle
$(\widetilde{\ME}_0,\varphi_0,\widetilde{H}_k)$, we define $\timeslde{\nabla}_k$ to be the unitary connection determined by $\widetilde{H}_k$. Since $\widetilde{H}_k$ is diagonal, over $\widetilde{S}\setminus \widetilde{Z}$, it follows that $F_{\timeslde{\nabla}_k}=0$, and we have $[\varphi_0,\varphi_0^{\dagger_{\widetilde{H}_k}}]=0$. Furthermore, as $i\to\inftya$ is an isomorphism on $\widetilde{S}\setminus \widetilde{Z}$, the metric $\widetilde{H}_k$ also defines a metric on $(\widetilde{\ME},\tilde{\vp})$ over $\widetilde{S}\setminus \widetilde{Z}$.
For any $\tilde{x}\in \widetilde{S}\setminus \widetilde{Z}$ with $x:=p(\tilde{x})$, we have isomorphisms
$$
(\widetilde{\ME}_0,\tilde{\vp}_0)|_{\sigmagma(\tilde{x})}\cong (\widetilde{\ME}_0,\tilde{\vp}_0)|_{\tilde{x}}\cong(\widetilde{\ME},\tilde{\vp})|_{\tilde{x}}\cong (\mathcal{E},\varphi)|_x.
$$
Therefore, $\widetilde{H}_k$ induces a metric $H^{\mathrm{Lim}}_k$ on $\Sigmagma\setminus Z$, and we may consider $H^{\mathrm{Lim}}_k$ as the
push-forward of $\tilde{h}_k$. In \cite[Theorem 5.2]{horn2022sltypesingularfibers}, the push-forward metric of $\Theta^{\mathrm{Moc}}(L)$ is explicitly written in local coordinates.
Recall the notation from Section \ref{sec:nah}, let $E$ be a trivial, smooth, rank 2 vector bundle over a Riemann surface $\Sigmagma$, and let $K$ be a background Hermitian metric on $E$. Over $\Sigmagma\setminus Z$, we write $\nabla_k^{\mathrm{Lim}}$ for the Chern connection defined by $H_k^{\mathrm{Lim}}$, which is unitary w.r.t. $H_0$ and $\phi^{\mathrm{Lim}}_k=\varphi_k^{\mathrm{Lim}}+\varphi_k^{\dagger_\mathrm{Lim}}$ be the corresponding Higgs field in the unitary gauge. They satisfy the decoupled Hitchin equations over $\Sigmagma\setminus Z$.
Thus from any Higgs bundle $(\mathcal{E},\varphi)$, we obtain $n_L+1$ limiting configurations $$(\nabla_k^{\mathrm{Lim}},\phi_k^{\mathrm{Lim}}=\varphi+\varphi_k^{\dagger_{\mathrm{Lim}}})\in \mathcal{M}H^{\mathrm{Lim}}.$$
The flat connection, which is defined over $\Sigmagma\setminus Z$, may be
understood by using the nonabelian Hodge correspondence for filtered vector
bundles \cite{simpson1990harmonic}. Given the filtered line bundles
$\mathcal{F}_{*,k}$, define filtered vector bundles $\widetilde{\ME}_{*,k}:=\mathcal{F}_{*,k}\oplus
\sigmagma^{*}\mathcal{F}_{*,k}$, which can be explicitly written as
\betagin{equation}
\betagin{split}
\widetilde{\ME}_{*,0}:&=\mathcal{F}_{*}(\widetilde{L}_0,\mathrm{ch}i_{\frac12(D-\lambdabdam)})\oplus
\mathcal{F}_{*}(\sigmagma^{*}\widetilde{L}_0,\mathrm{ch}i_{\frac12(D-\lambdabdam)})\ ;\\
\widetilde{\ME}_{*,k}:&=\mathcal{F}_{*}(\widetilde{L}_0,\mathrm{ch}i_{D_k-\frac12\lambdabdam})\oplus
\mathcal{F}_{*}(\sigmagma^{*}\widetilde{L}_0,\mathrm{ch}i_{\sigmagma^{*}D_k-\frac12\lambdabdam})\ , \
k\neq 0\ .
\end{split}
\end{equation}
These are polystable filtered vector bundles over $\widetilde{S}\setminus \widetilde{Z}$. As
for each $k=0,\cdotots,n_L$, $\sigmagma^{*}\widetilde{\ME}_{*,k}=\widetilde{\ME}_{*,k}$, the filtered
bundles $\widetilde{\ME}_{*,k}$ induce a filtered vector bundles $\mathcal{E}_{*,k}$ over
$\Sigmagma\setminus Z$. The flat connection $\nabla_k^{\mathrm{Lim}}$ will be the unique
harmonic unitary connection corresponding to the $\mathcal{E}_{*,k}$. Moreover, for
$0\leq k_1\neq k_2\leq n_L$, based on the definition of $D_{k_1}$ and
$D_{k_2}$, we can always find $\tilde{x}\in \widetilde{Z}_{\mathrm{even}}$, a preimage of
an even zero $x$ of $q$, such
that $\widetilde{\ME}_{*,k_1}$ and $\widetilde{\ME}_{*,k_2}$ have different filtered structures
near $\tilde{x}$. Since over even zeros, $\widetilde{S}\to \Sigmagma$ is not a branched
covering, we conclude that near $x$, $\mathcal{E}_{*,k_1}$ and $\mathcal{E}_{*,k_2}$ are
different filtered bundles. By \cite[Main theorem]{simpson1990harmonic}, the
harmonic connections $\nabla_{k_1}$ and $\nabla_{k_2}$ are not gauge equivalent.
We therefore conclude the following:
\betagin{proposition}
\label{prop_different_limiting_configurations}
For $0\leq k_1\neq k_2\leq n_L$, $(\nabla_{k_1}^{\mathrm{Lim}},\phi_{k_1}^{\mathrm{Lim}})$ and $(\nabla_{k_2}^{\mathrm{Lim}},\phi_{k_2}^{\mathrm{Lim}})$ are not gauge equivalent in $\mathcal{M}H^{\mathrm{Lim}}$.
\end{proposition}
This leads us to define the \emph{analytic Mochizuki map} $\UpsilonilonMoc$ as
\betagin{equation} \label{eq_analytic_moc_irreducible}
\UpsilonilonMoc:\mathcal{M}_q\longrightarrow \mathcal{M}HLC\ :\ [(\mathcal{E},\varphi)]\mapsto
[(\nabla_0^{\mathrm{Lim}},\phi_0^{\mathrm{Lim}})],
\end{equation}
which we recall is the limiting configuration defined by $\Theta^{\mathrm{Moc}}(L)$.
\mathfrak{su}bsubsection{The continuity of the limiting configurations} We now introduce the main result of
Mochizuki \cite{Mochizukiasymptotic}. Fix $(\mathcal{E},\varphi)=\mathrm{ch}i_{\mathrm{BNR}}(L)\in\mathcal{M}_q$. For any real parameter $t$,
$(\mathcal{E},t\varphi)$ is a stable Higgs bundle. By the Kobayashi-Hitchin correspondence, there exists a unique metric $H_t$
solving the Hitchin equation. Denote by $\nabla_t$ the unitary connection defined by $H_t$ and write
$\mathcal{D}_t=\nabla_t+t\phi_t$ for the full $\mathrm{SL}C$ flat connection. We then have:
\betagin{theorem}[{\cite{Mochizukiasymptotic}}]
\label{thm_moc_convergence_irreducible_case} The family $(\mathcal{E},t\varphi)$ has a unique limiting configuration
as limit for $t\to \infty$. Moreover, for any compact set $K\mathfrak{su}bset \Sigmagma\setminus Z$, let $d:=\min_{x\in K}|q|(x)$.
Then there exist $t$-independent constants $C_k$ and $C'_k$ such that
$$
|(\nabla_t,\phi_t)-\UpsilonilonMoc(\mathcal{E},\varphi)|_{\mathcal{C}^k}\leq C_ke^{-C'_ktd}\ .
$$
\end{theorem}
As the map $\UpsilonilonMoc$ is the composition of $\Theta^{\mathrm{Moc}} \circ \mathrm{ch}i_{\mathrm{BNR}}^{-1}$
with the nonabelian Hodge correspondence, the behavior of $\UpsilonilonMoc$ is the
same as $\Theta^{\mathrm{Moc}}$.
Recall the decomposition $\mathcal{M}_q = \bigcup \mathcal{M}_{q,D}$ from the end of
Section \ref{sec_parabolic_modules}.
By Theorem \ref{thm_algebraic_Mochizuki_map_continuous}, Proposition \ref{prop_number_of_limits_of_filtered_bundle} and Proposition \ref{prop_different_limiting_configurations}, we obtain:
\betagin{theorem}\label{thm_analytic_moc_irreducible_fiber}
Let $q$ be an irreducible quadratic differential. The map $\UpsilonilonMoc: \mathcal{M}_q \to \mathcal{M}HLC$ satisfies the following properties:
\betagin{itemize}
\item[(i)] if all the zeros of $q$ are odd, then $\UpsilonilonMoc$ is continuous;
\item[(ii)] if at least one zero of $q$ is even,
then for each $(\mathcal{E}, \varphi) \in \mathcal{M}_{q,D}$, there exists an integer $n_D$
that only depends on $D$, and $n_D$ sequences $\{(\mathcal{E}_i^k,
\varphi_i^k)\}$ for $k=1, \dots, n_D$, such that $\lim_{i \to
\infty}(\mathcal{E}_i^k, \varphi_i^k) = (\mathcal{E}_{\infty}, \varphi_{\infty})$, and
$$\lim_{i \to \infty} \UpsilonilonMoc(\mathcal{E}_i^{k_1}, \varphi_i^{k_1}) \neq
\lim_{i \to \infty} \UpsilonilonMoc(\mathcal{E}_i^{k_2}, \varphi_i^{k_2}) \neq
\UpsilonilonMoc(\mathcal{E}_{\infty}, \varphi_{\infty})\ , \ \text{for $k_1 \neq k_2$}\
.$$
\end{itemize}
\end{theorem}
\section{Reducible singular fiber and the Mochizuki map}
\label{sec_reducible_singular_fiber}
We now investigate the properties of the Hitchin fiber associated with a reducible quadratic differential, as discussed in
\cite{gothen2013singular}. Additionally, we will provide an overview of Mochizuki's technique for constructing limiting
configurations of Hitchin fibers for reducible quadratic differentials, as detailed in \cite{Mochizukiasymptotic}.
We also analyze the continuity of the Mochizuki map.
\mathfrak{su}bsection{Local description of a Higgs bundle}
\label{subsec_localdescription}
Write $q=-\omega\otimesimes\omega$ with $\omega\in H^0(K)$,
$\lambdabdam=\mathrm{Div}(\omegaega)$, $Z=\mathfrak{su}pp(\lambdabdam)$, and $\mathcal{M}_q=\mathcal{H}^{-1}(q)$. Compared to
the irreducible case, $\mathcal{M}_q$ contains strictly semistable Higgs bundles, so we let
$\mathcal{M}_q^{\rm st}$ denote the stable locus.
We point out that there is a sign ambiguity in the choice of $\omega$,
which actually plays an important role in the following.
\mathfrak{su}bsubsection{Local description}
Given a Higgs bundle $(\mathcal{E},\varphi)$ with $\det(\varphi)=q$, define line bundles
\betagin{equation} \label{eq_kernel_reducible}
L_{\pm}:=\ker(\varphi\pm\omega)\ .
\end{equation}
Then the inclusion maps $L_{\pm}\to \mathcal{E}$ are injective. Similarly, we may
define an abelianization of $(\mathcal{E},\varphi)$ by
$(\mathcal{E}_0=L_+\oplus L_-,\varphi_0=\mathrm{diag}(\omega,-\omega))$. We then have a natural inclusion
$i\to\inftya:\mathcal{E}_0\to \mathcal{E}$, which is
is an isomorphism on $\Sigmagma\setminus Z$, and $\varphi\circi\to\inftya=i\to\inftya\circ\varphi_0$.
It follows from \cite[Prop.\ 7.10]{gothen2013singular} that $L_{\pm}$ are
the only $\varphi$-invariant subbundles of $\mathcal{E}$.
If we write $d_{\pm}:=\deg(L_{\pm})$, then $(\mathcal{E},\varphi)$ is stable (resp.\ semistable)
if and only if $d_{\pm}<0\ (\text{resp.}\ \leq 0)$. As $\det(\mathcal{E})=\mathcal{O}$, the
map $\det(i\to\inftya):L_+\otimesimesmes L_-\to \mathcal{O}$ defines a divisor $D=\mathrm{Div}(\det(i\to\inftya))$ such
that $L_+\otimesimesmes L_-=\mathcal{O}(-D).$ Therefore, we obtain
\betagin{equation*}
d_++d_-+\deg D=0\ ,
\end{equation*}
and $0\leq D\leq \lambdabdam$. The Higgs bundle $(\mathcal{E},\varphi)$ is semistable if and
only if $-\deg D\leq d_+\leq 0$ and stable if the equalities are strict.
For the rest of this section, we always write $D=\mathfrak{su}m_{p\in Z}\ell_p p$.
\mathfrak{su}bsubsection{Semistable settings}
As the fiber $\mathcal{M}_q$ might contain strictly semistable Higgs bundles, we now explicitly enumerate all of the possible
$S$-equivalence classes. When $D=0$, then $L_-=L_+^{-1}$ and $\deg(L_+)=0$. The corresponding Higgs
bundle is polystable and can be explicitly written as
$$
\Bigl(L\oplus L^{-1}, \betagin{pmatrix}
\omega & 0\\
0 & -\omega
\end{pmatrix}\Bigr)\ ,
$$ where $L\in \mathrm{Jac}(\Sigmagma)$.
When $D\neq 0$, suppose $\deg(L_+)=-\deg(D)$. Then $L_-=L_+^{-1}(-D)$ and
$\deg(L_-)=0$. Under $S$-equivalence, the polystable Higgs bundle is
$$
\Bigl(L_+(D)\oplus L_+^{-1}(-D),\betagin{pmatrix}
\omega & 0\\
0 & -\omega
\end{pmatrix}\Bigr)\ ,
$$
where $L_+\in \mathrm{Pic}^{-\deg(D)}(\Sigmagma)$.
\mathfrak{su}bsection{Reducible spectral curves}
In this subsection, we introduce the algebraic data in \cite{gothen2013singular} which describes the singular fiber with
a reducible spectral curve. This plays a similar role to the parabolic modules. See \cite[Sec.\ 7.1]{gothen2013singular} for more details.
For any divisor $D$, and line bundle $L$, define the space
$$
H^0(D,L)=\bigoplus_{p\in \mathfrak{su}pp D}\mathcal{O}(L)_p/\sigmam\ ,
$$
where $s_1\sigmam s_2$ if and only if $\mathrm{ord}_p([s_1]-[s_2])\geq D_p$.
Let $L\in \mathrm{Pic}(\Sigmagma)$, define the following subspaces of $H^0(\lambdabdam,L^2K)$:
\betagin{equation*}
\betagin{split}
&\mathcal{V}( D,L):=\{s\in H^0(\lambdabdam,L^2K)\mid\mathrm{ord}_p(s)=\lambdabdam_p-D_p,\;\mathrm{if}\;D_p>0;\;s(p)=0,\;\mathrm{if}\;D_p=0\},\\
&\mathcal{W}( D,L)=\{s\in H^0(\lambdabdam,L^2K)\mid s|_{\mathfrak{su}pp(\lambdabdam-D)}=0\}\ .
\end{split}
\end{equation*}
One checks that $\mathcal{W}(D,L)=\cup_{D'\leq D}\mathcal{V}(D',L)$. Moreover, the space
$\mathcal{V}( D,L)$ is a linear subspace of $H^0(\lambdabdam,L^2K)$ with a hyperplane removed. In addition, $\mathbb{C}^{\ast}$ acts on $\mathcal{V}( D,L)$ by
multiplication, and $\dim(\mathcal{V}( D,L)/\mathbb{C}^{\ast})=\deg( D)-1$.
We define the fibrations
\betagin{equation*}
\betagin{split}
p_m:\mathcal{S}V( D,m)\longrightarrow \mathrm{Pic}^m(\Sigmagma),\;p_m:\mathcal{S}W( D,m)\longrightarrow \mathrm{Pic}^m(\Sigmagma)
\end{split}
\end{equation*}
such that for $L\in \mathrm{Pic}^m(\Sigmagma)$, the fibers are $\mathcal{V}(D,L)$ and $\mathcal{W}(D,L)$.
\mathfrak{su}bsubsection{Algebraic data from the extension}
The Higgs bundle $(\mathcal{E},\varphi)$ can be
understood in terms of an extension of exact sequence. As $\det(\mathcal{E})=\mathcal{O}$, we have the exact sequence
$$
0\longrightarrow L_+\longrightarrow \mathcal{E}\longrightarrow L_+^{-1}\longrightarrow 0\ .
$$
For each $p\in Z$, with $U\mathfrak{su}bset \Sigmagma$ a neighborhood of $p$,
$(\mathcal{E},\varphi)$ can be written as the following splitting of $\mathcal{C}^{\infty}$ bundles
\betagin{equation*}
\betagin{split}
\mathcal{E}=L_+\oplus_{\mathcal{C}^{\infty}} L_+^{-1}\ ,\ \bar{\partial}_{\mathcal{E}}=\betagin{pmatrix}
\bar{\pa}_{L_+} & b\\
0 & \bar{\pa}_{L_+^{-1}}
\end{pmatrix}\ ,\ \varphi=\betagin{pmatrix}
\omega & c\\
0 & -\omega
\end{pmatrix}\ .
\end{split}
\end{equation*}
We would like to consider the restriction of $\varphi+\omega\cdotot\mathrm{id}$ to $\lambdabdam$.
As $\omega|_{\lambdabdam}=0$, $\varphi+\omega\cdotot\mathrm{id}|_{L_+}=0$ and the image of
$\varphi+\omega\mathrm{id}\mathfrak{su}bset (L_+\oplus 0)\otimesimesmes K\mathfrak{su}bset \mathcal{E}\otimesimesmes K$. Therefore,
the restriction of $\varphi+\omega\cdotot \mathrm{id}$ to $\lambdabdam$ defines a holomorphic map $s :L_+^{-1}|_{\lambdabdam}\to L_+K|_{\lambdabdam},$ or equivalently a section $s\in H^0(\lambdabdam,L_+^2K).$ Moreover, by \cite[Lemma 7.12]{gothen2013singular}, $\mathrm{Div}(s)=\lambdabdam-D$. Therefore, given any $(\mathcal{E},\varphi)\in \mathcal{M}_q$, we obtain an $L\in\mathrm{Pic}^{m}(\Sigmagma)$ and an element in $\mathcal{V}(D,L)$. Moreover, the stability condition implies that $0\leq D\leq \lambdabdam$, we have $-\deg D\leq \deg L\leq 0.$
\mathfrak{su}bsubsection{Inverse construction}
The inverse of the construction above also holds; for further details, see \cite[Sec.\ 7]{gothen2013singular} and \cite[Sec.\ 5]{horn2022semi}.
Given $L\in \mathrm{Pic}^{m}(\Sigmagma)$ and $q\in \mathcal{V}(D,L)$, we define a Higgs bundle via extensions as follows. From $q,L$, we have a short exact sequence
of complexes of sheaves:
\betagin{equation*}
\betagin{tikzcd}
& C_1^{\ast}& C_2^{\ast}& C_3^{\ast}&\\
0\arrow[r, ] & L^2 \arrow[r, "="] \arrow[d, "\mathrm{id}"] & L^2 \arrow[r, "\mathrm{pr}"] \arrow[d, "c"] & 0 \arrow[r, ] \arrow[d, "0"] & 0 \\
0\arrow[r, ]& L^2 \arrow[r, "c"]&L^2K \arrow[r, "\mathrm{res}(\lambdabdam)"]&L^2K|_{\lambdabdam'} \arrow[r, ] & 0
\end{tikzcd},
\end{equation*}
where, for a section $s\in \Gamma(L^2)$, $c(s):=\sqrt{-1}\omegaega s$, and $\mathrm{res}(\lambdabdam)$ is the restriction map to the divisor $\lambdabdam$.
The long exact sequence in hypercohomology implies that $\mathrm{res}(\lambdabdam)$ induces an isomorphism
$$
\mathrm{res}(\lambdabdam):\mathbf{H}^1(C_2^{\ast})\cong
\mathbf{H}^1(C_3^{\ast})=H^0(\lambdabdam,L^2K)\ .
$$
Moreover, we have $\mathbf{H}^1(C_2^{\ast})\cong H^1(\Sigmagma,L^2)$, which parameterizes extensions
\betagin{equation*}
0\longrightarrow L\longrightarrow \mathcal{E}\longrightarrow L^{-1}\longrightarrow 0\ .
\end{equation*}
From $s\in H^0(\lambdabdam,L^2K)$ and $\mathcal{E}$ above, we can find a section
$c\in\Gamma(L^2K)$, and construct a Higgs bundle
\betagin{equation} \label{eq_Higgsbundle_construction_extension}
E=L\oplus_{\mathcal{C}^{\infty}} L^{-1},\;\bar{\partial}_E=\betagin{pmatrix}
\bar{\partial}_L & b\\
0 & \bar{\partial}_{L^{-1}}
\end{pmatrix},\;\varphi=\betagin{pmatrix}
\omega & c\\
0 & -\omega
\end{pmatrix},
\end{equation}
where $\bar{\partial}c=2b\omega$ for $b\in \Omega^{0,1}(L^2)$ and $c$ is an
extension of $q$.
For $0\leq D\leq \lambdabdam$ and $-\deg D\leq m\leq 0$, the construction above defines a map
\betagin{equation*}
\wp:\mathcal{S}V(D,m)\longrightarrow \mathcal{M}_q\ ,\ s\in\mathcal{V}(D,L)\mapsto(\mathcal{E},\varphi)\ ,
\end{equation*}
where $(\mathcal{E},\varphi)$ is the Higgs bundle constructed in \eqref{eq_Higgsbundle_construction_extension}.
When $ D=0$, $\mathcal{V}(\lambdabdam,L)=\{0\}$ and the image of $\wp:\mathcal{S}V(\lambdabdam,0)\to \mathcal{M}_q$ are the polystable Higgs bundles $\mathcal{E}=L\oplus L^{-1},\;\varphi=\mathrm{diag}(\omega,-\omega)$ such that $L^2\cong \mathcal{O}_{\Sigmagma}$.
\betagin{theorem}[{\cite[Thm.\ 7.7]{gothen2013singular}}]
\label{thm_reduciblefiberBNR}
For $ 0\leq D\leq \lambdabdam$ and $-\deg( D)\leq m_1\leq 0$ and the map $\wp:\mathcal{S}V(D,m_1)\to \mathcal{M}_q$, we have
\betagin{itemize}
\item [(i)] for $m_2=-\deg( D)-m_1$, we have $\wp(\mathcal{S}V(D,m_1))=\wp(\mathcal{S}V(D,m_2))$,
\item [(ii)] for the $\mathbb{C}^{\ast}$ action on $\mathcal{S}V(D,m_1)$ by multiplication, for $\xi\in \mathcal{S}V(D,m_1)$, $\wp(\mathbb{C}^{\ast} \xi)=\wp(\xi)$,
\item [(iii)] when $m_1\neq -\frac12\deg( D)$, $\wp:\mathcal{S}V(D,m_1)/\mathbb{C}^{\ast} \to \mathcal{M}_q$ is an isomorphism onto its image,
\item [(iv)] when $m_1=-\frac{1}{2}\deg( D)$, $\wp:\mathcal{S}V(D,m_1)/\mathbb{C}^{\ast} \to \mathcal{M}_q$ is a double branched covering, which branched along line bundles $L\in \mathrm{Pic}^{m_1}(\Sigmagma)$ such that $L^2\cong \mathcal{O}(- D)$,
\item [(v)] when $ D=0$, then $\wp:\mathcal{S}V(\lambdabdam,0)\to \mathcal{M}_q$ is a double branched covering, branched along $L\in \mathrm{Pic}^0(\Sigmagma)$ such that $L^2\cong\mathcal{O}.$
\end{itemize}
\end{theorem}
\betagin{example}
\label{ex_genus_two_stable.}
When $g=2$, for $q=-\omega\otimesimesmes \omega$, we can
write $\lambdabdam=p_1+p_2$ or $\lambdabdam=2p$. In either case, the
$\mathcal{M}_q^{\asta}=\wp(\mathcal{S}V(D,m))$ for $-\deg(D)<m<0$ and $0\leq D\leq
\lambdabdam$. Therefore, $m=-1,\;D=\lambdabdam$ and $\wp(\mathcal{S}V(\lambdabdam,-1))=\mathcal{M}_q^{\asta}$. Moreover, generically, the map $\wp:(\mathcal{S}V(\lambdabdam,-1))/\mathbb{C}^{\ast}\to \mathcal{M}_q^{\asta}$ is two-to-one.
\end{example}
\mathfrak{su}bsection{The stratification of the singular fiber}
We now present two stratifications of $\mathcal{M}_q$. Recall that from any Higgs bundle $(\mathcal{E},\varphi)$ we obtain two line bundles $L_{\pm}$ and a
divisor $D$. There are two different stratifications: one given by the divisor $D$ and the other by the degree of $L_+$.
\mathfrak{su}bsubsection{Divisor stratification} We first discuss the stratification
defined by the divisor. Indeed, using $D$, decompose into strata: $\mathcal{M}_q=\bigcup_{0\leq D\leq \lambdabdam}\mathcal{M}_{ D}$. As the definition of $L_{\pm}$ depends on the choice of the
square root, there is no natural map from $\mathcal{M}_{ D}$ to $\mathrm{Pic}(\Sigmagma)$. Consider the following space: $\mathcal{B}V_{ D}=\bigcup_{-\deg( D)\leq m\leq
0}\mathcal{S}V( D,m)$. This forms a fibration
$$
\tau:\mathcal{B}V_{ D}\longrightarrow \bigcup_{-\deg( D)\leq m\leq 0}\mathrm{Pic}^m(\Sigmagma)\ .
$$
Moreover, for $L\in \mathrm{Pic}^{m}(\Sigmagma)$, we have $\tau^{-1}(L)=\mathcal{S}V( D,L)$ and $\dim(\tau^{-1}(L)/\mathbb{C}^{\ast})=\deg( D)-1$.
By Theorem \ref{thm_reduciblefiberBNR}, $\wp|_{\mathcal{B}V_{ D}}:\mathcal{B}V_{ D}\to \mathcal{M}_{ D}$ is surjective.
Moreover, since $$\wp|_{\mathcal{S}V( D,m)}=\wp|_{\mathcal{S}V( D,-\deg( D)-m)}$$ generically,
$\wp|_{\mathcal{B}V_{ D}}$ is a two-to-one map.
In summary, we obtain the following map which characterizes the singular fiber.
$$
\wp: \mathcal{B}V=\bigcup_{0\leq D\leq \lambdabdam}\mathcal{B}V_{ D}\rightarrow \mathcal{M}_q=\bigcup_{0\leq D\leq \lambdabdam }\mathcal{M}_{ D}.
$$
The top stratum is given by $D=\lambdabdam$.
\mathfrak{su}bsubsection{Degree stratification}
We next introduce the stratification defined by degrees; this encodes how different divisor stratifications are pasted together.
For $-(2g-2)\leq m\leq 0$ and $L\in\mathrm{Pic}^m(\Sigmagma)$,
define $\mathcal{B}W(L):=\bigcup_{\deg D\geq -m}\mathcal{V}( D,L)$. This set is connected,
based on the definition and \cite[Lemma 7.14]{gothen2013singular}.
Moreover, if we define
$$\mathcal{B}W_m:=\bigcup_{-m\leq \deg D,\;0\leq D\leq \lambdabdam}\mathcal{S}V( D,m)\ ,\
\mathcal{B}W:=\bigcup_{-(2g-2)\leq m\leq 0}\mathcal{B}W_m\ ,$$ then we have $\wp(\mathcal{B}W)=\wp(\mathcal{B}V).$
We should also note that though $\mathcal{B}W_m\cap \mathcal{B}W_n=\emptyset$ for any
$m\neq n$, $\mathcal{B}W$ is connected. As $L_+,L_-$ are symmetric, by Theorem
\ref{thm_reduciblefiberBNR}, we have $\wp(\mathcal{S}V( D,m))=\wp(\mathcal{S}V( D,-\deg(
D)-m))$, which implies that for any integer $-(2g-2+m)\leq n\leq 0$,
$\wp\mathcal{B}W_m\cap \wp\mathcal{B}W_n\neq \emptyset$.
We now give an example of the degree stratification when $g=2$.
\betagin{example}
\label{example_genustwodegreestratification}
Suppose $\omega$ has only one zero with order 2. Then $\lambdabdam=2p$, and all
possible divisors are $D_2=2p,D_1=p,D_0=0$. The degree stratification
is
\betagin{equation*}
\betagin{split}
&\mathcal{B}W_{-2}=\mathcal{S}V(D_2,-2)\ ,\ \mathcal{B}W_{-1}=\mathcal{S}V(D_2,-1)\cup\mathcal{S}V(D_1,-1)\\
&\mathcal{B}W_0=\mathcal{S}V(D_0,0)\cup\mathcal{S}V(D_1,0)\cap \mathcal{S}V(D_2,0)\ .
\end{split}
\end{equation*}
The image of $\wp(\mathcal{S}V(D_2,-1))$ is stable, $\wp(\mathcal{S}V(D_0,0))$ is poly-stable and $\wp(\mathcal{S}W\setminus (\mathcal{S}V(D_2,-1)\cup\mathcal{S}V(D_0,0)))$ is semistable.
Moreover, we have $\wp(\mathcal{S}V(D_2,-2))=\wp(\mathcal{S}V(D_2,0))$, $\wp(\mathcal{S}V(D_1,-1))=\wp(\mathcal{S}V(D_1,0))$ and $\wp|_{\mathcal{S}V(D_2,-1)}$ is a branched covering. Moreover, we have $\wp(\mathcal{S}V(D_2,-1))\cap \wp(\mathcal{S}V(D_1,0))\neq 0$ and $\wp(\mathcal{S}V(D_2,-1))\cap \wp(\mathcal{S}V(D_0,0))=0$.
\end{example}
\mathfrak{su}bsection{Algebraic Mochizuki map}
Based on the study of the local rescaling properties of Higgs bundles,
Mochizuki introduced a weight for each $p \in Z$ in \cite[Sec.\ 3]{Mochizukiasymptotic}. To be more specific, let $c$ be a real number. For each $p \in Z$, the weight we consider is given by
$$\mathrm{ch}i_p(c) = \min\{\ell_p, (m_p + 1)c + \ell_p/2\}\ .$$
By utilizing the global geometry of a Higgs bundle, we can uniquely determine the constant $c$. We aim to choose the sign of $\omegaega$ such that $d_+ \leq d_-$.
\betagin{lemma}[{\cite[Lemma 4.3]{Mochizukiasymptotic}}]
\label{lemma_weightparabolic}
If $(\mathcal{E},\varphi)$ is stable, then there exists a unique constant $c>0$ such that
\betagin{equation*}
d_++\mathfrak{su}m_{p\in Z}\mathrm{ch}i_p(c)=0\ ,\ d_-+\mathfrak{su}m_{p\in
Z}(\ell_p-\mathrm{ch}i_p(c))=0\ .
\end{equation*}
\end{lemma}
\betagin{proof}
Since $(\mathcal{E},\varphi)$ is stable, we have $-\mathfrak{su}m \ell_p<d_{\pm}<0$. We define the function
\betagin{equation} \label{eq_mon_function_weight}
f(c) = d_++\mathfrak{su}m_p\mathrm{ch}i_p(c)\ ,
\end{equation}
which is strictly increasing. Moreover, for $c$ sufficiently large,
$\mathrm{ch}i_p(c)=\ell_p$, and therefore $f(c)=d_++\mathfrak{su}m_p\ell_p=-d_->0$.
Additionally, $f(0)=d_++\mathfrak{su}m_p (\ell_p/2)$. Since $d_+\leq d_-$ and
$d_++d_-+\mathfrak{su}m_p\ell_p=0$, we obtain $f(0)\leq 0$. The monotonicity of $f$ implies the existence of $c_0$ such that $f(c_0)=0$.
\end{proof}
From the construction, if $d_+\leq d_-$, two weighted bundles
$(L_+,\mathrm{ch}i_p(c_0))$ and $(L_{-},\ell_p-\mathrm{ch}i_p(c_0))$ are obtained with weights
$\mathrm{ch}i_p(c_0)$ and $\ell_p-\mathrm{ch}i_p(c_0)$ at each $p\in Z$, respectively. On the
other hand, if $d_+\geq d_-$, by symmetry, weighted bundles
$(L_+,\ell_p-\mathrm{ch}i_p(c_0))$ and $(L_-,\mathrm{ch}i_p(c_0))$ are obtained. When
$(\mathcal{E},\varphi)$ is strictly semistable, S-equivalent to
$(L,\omega)\oplus (L^{-1},-\omega)$, then we would like to consider the weighted bundles $(L,0)\oplus (L^{-1},0)$ with weight zero.
Next, we define the algebraic Mochizuki map. Let $\mathcal{S}F_{\pm}(\Sigmagma)$ be
the space of rank 1 degree zero filtered bundles on $\Sigmagma$, and let
$\mathcal{S}F_2(\Sigmagma):=\mathcal{S}F_+(\Sigmagma)\oplus \mathcal{S}F_-(\Sigmagma)$ be the direct sum.
Fix a choice of $\omega$. Then from any Higgs bundle $(\mathcal{E},\varphi)$, we obtain the subbundles $L_{\pm}$ with degree $d_{\pm}$ and define the algebraic Mochizuki map
\betagin{equation*}
\betagin{split}
\Theta^{\mathrm{Moc}}&:\mathcal{M}_q\longrightarrow \mathcal{S}F_2(\Sigmagma),\\
\Theta^{\mathrm{Moc}}&(\mathcal{E},\varphi):=\left\{\betagin{matrix}
\mathcal{F}_{\ast}(L_+,\mathrm{ch}i_p(c_0))\oplus \mathcal{F}_{\ast}(L_-,\ell_p-\mathrm{ch}i_p(c_0)),\;\mathrm{if}\;d_+\leq d_-\\
\mathcal{F}_{\ast}(L_+,\ell_p-\mathrm{ch}i_p(c_0))\oplus \mathcal{F}_{\ast}(L_-,\mathrm{ch}i_p(c_0)),\;\mathrm{if}\;d_-\leq d_+
\end{matrix}\right.,\;(\mathcal{E},\varphi)\;\mathrm{stable},\\
\Theta^{\mathrm{Moc}}&(\mathcal{E},\varphi):=\mathcal{F}_{\ast}(L,0)\oplus \mathcal{F}_{\ast}(L^{-1},0),\;(\mathcal{E},\varphi)\;\mathrm{semistable}.
\end{split}
\end{equation*}
We list some properties of this map.
\betagin{proposition}
\label{prop_reducible_continuous_on_strata}
For $\Theta^{\mathrm{Moc}}$, we have:
\betagin{itemize}
\item [(i)]for each $\mathcal{S}V( D,m)$ with $0\leq D\leq \lambdabdam$, $-\deg(
D)\leq m\leq 0$, $\Theta^{\mathrm{Moc}}|_{\wp(\mathcal{S}V( D,m))}$ is continuous,
\item [(ii)]for i=1,2 and $s_i\in \mathcal{B}V_D$ with $(\mathcal{E}_i,\varphi_i):=\wp(s_i)$, suppose $\tau(s_1)=\tau(s_2)$, then $\Theta^{\mathrm{Moc}}(\mathcal{E}_1,\varphi_1)=\Theta^{\mathrm{Moc}}(\mathcal{E}_2,\varphi_2)$. In particular, $\Theta^{\mathrm{Moc}}$ is not injective.
\end{itemize}
\end{proposition}
\betagin{proof}
The proof follows directly from the definition.
\end{proof}
\mathfrak{su}bsection{Exotic phenomena}
A Higgs bundle $(\mathcal{E},\varphi) \in \mathcal{M}_q$ is called ``exotic'' if the constant $c$ in Lemma \ref{lemma_weightparabolic} satisfies $c \neq 0$. This new behavior only appears in the Hitchin fiber with reducible spectral curve.
In this subsection, we aim to understand the exotic phenomenon of a Higgs bundle.
We first provide another expression for the weight in Lemma \ref{lemma_weightparabolic}.
\betagin{proposition}
\label{prop_choiceofconstant_c}
Suppose $-\deg D <d_+\leq -\frac12\deg D$ and let $Z_0:=\mathfrak{su}pp(D)$,
then the constant $c$ in Lemma \ref{lemma_weightparabolic} is given by
$$c=\frac{d_++\frac12\deg D}{\deg(\lambdabdam|_{Z_0})+|Z_0|}\ ,$$
where $\lambdabdam|_{Z_0}$ means the restriction of the divisor to $Z_0$, and $|Z_0|$ means the number of points in $Z_0$ without multiplicity.
\end{proposition}
\betagin{proof}
By the definition of $\mathrm{ch}i_p(c)$, if $p\in Z_1$, $\mathrm{ch}i_p(c)=0$ for any
$c\geq 0$. Therefore, the choice of $c$ is determined by the equation
$d_++\mathfrak{su}m_{p\in Z_0}\min\{\ell_p,(m_p+1)c+\frac{\ell_p}{2}\}=0$.
Define the function $F(c):=d_++\mathfrak{su}m_{p\in Z_0}((m_p+1)c+\ell_p/2)$,
and let $f(c)$ be the function defined in $\eqref{eq_mon_function_weight}$. Then $F(c)\geq f(c)$, $F(0)=0$ and for $c$ sufficiently large, $F(c)=f(c)$.
We compute
$$\mathfrak{su}m_{p\in
Z_0}(m_p+1)c=(\deg(\lambdabdam|_{Z_0})+|Z_0|))c\ ,\ \mathfrak{su}m_{p\in
Z_0}(\ell_p/2)=\frac{1}{2}\deg D\ .$$
Then for $$c_0=\frac{d_++\frac12\deg D}{\deg(\lambdabdam|_{Z_0})+|Z_0|}$$
we have $F(c_0)=0$. Therefore, $f(c_0)=0$, which determines the choice
of the constant $c$ in Lemma $\ref{lemma_weightparabolic}.$
\end{proof}
\betagin{corollary}
A Higgs bundle $(\mathcal{E},\varphi)$ is exotic if and only if its corresponding degrees $d_{\pm}$ satisfy $d_+\neq d_-$. Additionally, there are only a finite number of possible choices for $c$ over $\mathcal{M}_q$.
\end{corollary}
\betagin{proof}
The result follows directly from the formulas in Proposition \ref{prop_choiceofconstant_c}.
\end{proof}
We will compute some examples of possible weights in some special cases. First consider the generic case.
\betagin{example}
Suppose $\omega$ has only simple zeros, therefore $Z=\{p_1,\ldots,p_{2g-2}\}$ and $\lambdabdam=p_1+\cdotots+p_{2g-2}$. Given a stable Higgs bundle $(\mathcal{E},\varphi)$ with corresponding $L_+,L_-, D,d_{\pm}$, recall that from the stability condition, these degrees satisfy the followings:
\betagin{equation*}
d_++d_-+\deg D=0\ ,\ d_+\leq d_-\ ,\ d_+<0,\;d_-<0\ ,\ -\deg
D<d_+\leq -\frac{1}{2}\deg D\ .
\end{equation*}
If we write $Z_0:=\mathfrak{su}pp D$, then $\mathrm{ch}i_p=0$ for $p\in Z\setminus Z_0$
and $\mathrm{ch}i_p=2c+1/2$ for $p\in Z_0$. Moreover, we have $d_++\deg(
D)(2c+1/2)=0$. Therefore,
$c=-\frac{d_+}{2\deg( D)}+\frac14$, and two weighted bundles we obtained
are
\betagin{equation*}
(L_+,(-\frac{d_+}{\deg D}|_{Z_0},0|_{Z\setminus Z_0}))\ ,\ (L_-, (1+\frac{d_+}{\deg D}|_{Z_0},1|_{Z\setminus Z_0}))
\ .
\end{equation*}
\end{example}
Next, we consider the most nongeneric case.
\betagin{example}
Suppose $\omegaega$ only contains one zero. Write $\mathrm{Div}(\omega)=(2g-2)p$.
Then the possible divisors are $ D=\ell p$, for $0\leq \ell\leq 2g-2$.
Let $L_+$ be a line bundle with $d_+:=\deg(L_+)$ and $-\ell<d_+\leq
\ell/2$. Then the choice of $c$ is determined by the equation
\betagin{equation*}
\betagin{split}
d_++\min\{\ell,(2g-1)c+\ell/2\}=0\ .
\end{split}
\end{equation*}
As $d_+\neq -\ell$, we must have $c=-\frac{\ell}{2(2g-1)}$ and the corresponding
weighted bundles are
\betagin{equation*}
\betagin{split}
(L_+,-d_+),\;(L_+^{-1}\otimesimesmes \mathcal{O}(-\ell p),\ell-d_+)\ .
\end{split}
\end{equation*}
\end{example}
Finally, we explicitly compute the limiting configurations when $g=2$ for
the strictly semistable locus of the stratification in Example \ref{example_genustwodegreestratification}.
\betagin{example}
\label{example_limitingconfiguration_genus_2}
When $g=2$, we consider $\lambdabdam=2p$ and $D_i=i p$ for $i=0,1,2$. Let
$p_m:\mathcal{S}V(D,m)\to \mathrm{Pic}^m(\Sigmagma)$ be the projection. Over
$\wp(\mathcal{S}V(D_i,0))$, for $L\in \mathrm{Pic}^0(\Sigmagma)$, the S-equivalence class
for $\wp(p_0^{-1}(L))$ is $(L\oplus L^{-1},\betagin{pmatrix}
\omega & 0\\
0 & -\omega
\end{pmatrix})$ and the corresponding weighted bundles are $(L,0)\oplus (L^{-1},0)$.
Over $\wp(\mathcal{S}V(D_2,-1))$, for $L\in \mathrm{Pic}^{-1}(\Sigmagma)$, let
$(\mathcal{E},\varphi)=\wp(L,s\in \mathcal{V}(D_2,L))$, then
$$\Theta^{\mathrm{Moc}}(\mathcal{E},\varphi)=\mathcal{F}_{\ast}(L,1)\oplus
\mathcal{F}_{\ast}(L^{-1}(-D_2),1)=\mathcal{F}_{\ast}(L(p),0)\oplus
\mathcal{F}_{\ast}(L^{-1}(p-D_2),0)\ .$$
\end{example}
\betagin{comment}
Now, we will fully compute all the possible exotic weights when $g=3$ for stable.
\betagin{example}
When $g=3$, recall that the stable range we choose is $-\deg(D)<d_+\leq -\frac{1}{2}\deg(D)$.
When $\lambdabdam=2p_1+2p_2$, then
\betagin{itemize}
\item suppose $D=2p_1+p_2$, we have $d_+=2$, for any $L\in
\mathrm{Pic}^{-2}(\Sigmagma)$, the weights are $\mathrm{ch}i_{p_1}=\frac{13}{10}$
and $\mathrm{ch}i_{p_2}=\frac{7}{10}$. The weighted bundles are
$$
(L,(\frac{13}{10},\frac{7}{10}))\oplus (L^{-1}(- D),(\frac{7}{10},\frac{13}{10})).
$$
\item suppose $D=p_1+p_2$, as $d_+=d_-=-1=-\frac{\deg D}{2}$, we have $c=0$, which is nonexotic.
\end{itemize}
When $\lambdabdam=2p_1+p_2+p_3$, then
\betagin{itemize}
\item suppose $D=\lambdabdam$ and suppose $d_+=-2$, then this is a nonexotic case.
For any $L\in \mathrm{Pic}^{-3}(\Sigmagma)$, the weighted
bundles are
$$
(L,(\frac{10}{7},\frac{11}{14},\frac{11}{14}))\oplus (L^{-1}(-\lambdabdam),(\frac{4}{7},\frac{3}{14},\frac{3}{14})),
$$
\item suppose $D=p_1+p_2+p_3$, for $L\in \mathrm{Pic}^{-2}(\Sigmagma)$, the
weighted bundles are
$$
(L,(\frac57,\frac{9}{14},\frac{9}{14}))\oplus (L^{-1}(-D),(\frac27,\frac{5}{14},\frac{5}{14})),
$$
\item suppose $D=2p_1+p_2$, for $L\in \mathrm{Pic}^{-2}(\Sigmagma)$, the
weighted bundles are
$$
(L,(\frac{13}{10},\frac{7}{10},0))\oplus (L^{-1}(-D),(\frac{7}{10},\frac{13}{10},0),
$$
\item suppose $\deg D=2$, then $L\in \mathrm{Pic}^{-1}(\Sigmagma)$, this is nonexotic and $c=0$.
\end{itemize}
\end{example}
\end{comment}
\mathfrak{su}bsection{Discontinuous behavior}
In this subsection, we study the discontinuous behavior of $\Theta^{\mathrm{Moc}}$. Consider
a sequence of algebraic data $(L_i,q_i)\in \mathcal{B}W_m$, where $L_i\in\mathrm{Pic}^m$ and $q_i\in \mathcal{S}V(D,L_i)$.
We assume that $\lim_{i\to \infty}L_i=L_{\infty}$ in $\mathrm{Pic}^m$ and
$\lim_{i\to \infty}q_i=q_{\infty}\in \mathcal{S}V(D_{\infty},L)$, for
$D_{\infty}\neq D$. As the space $\bigcup_{\deg D'\geq -m}\mathcal{S}V(D',m)$ is
connected, we can always find such a sequence.
Let $L_+^i:=L_i$ and $L_-^i:=L_i^{-1}\otimesimesmes\mathcal{O}(-D)$. By Lemma \ref{lemma_weightparabolic}, the weight function,
which we denote by $\mathrm{ch}i_{\pm}$, is independent of $i$. In addition, we have
$$
\lim_{i\to\infty}\Theta^{\mathrm{Moc}}\circ
\wp(L_i,q_i)=\mathcal{F}_{\ast}(L_{\infty},\mathrm{ch}i_+)\oplus \mathcal{F}_{\ast}(L_{\infty}^{-1}(-
D),\mathrm{ch}i_-)\ .
$$
For $(L_{\infty},q_{\infty}\in \mathcal{S}V(D_{\infty},L))$, let
$\mathrm{ch}i^{\infty}_{\pm}$ be the corresponding weights. These depend on $D_{\infty}$ and $m$. Then $$
\Theta^{\mathrm{Moc}}\circ
\wp(L_{\infty},q_{\infty})=\mathcal{F}_{\ast}(L_{\infty},\mathrm{ch}i_+^{\infty})\oplus
\mathcal{F}_{\ast}(L_{\infty}^{-1}\otimesimesmes \mathcal{O}(- D_{\infty}),\mathrm{ch}i_-^{\infty})\ .
$$
Therefore, we obtain
\betagin{equation} \label{eq_limit_reducible_fiber}
\betagin{split}
&\lim_{i\to\infty}\Theta^{\mathrm{Moc}}\circ \wp(L_i,q_i)\\
=&\Theta^{\mathrm{Moc}}\circ \wp(L_{\infty},q_{\infty})\otimesimesmes
(\mathcal{F}_{\ast}(\mathcal{O},\mathrm{ch}i_+-\mathrm{ch}i_+^{\infty})\oplus \mathcal{F}_{\ast}(\mathcal{O}(
D_{\infty}- D),\mathrm{ch}i_--\mathrm{ch}i_-^{\infty}))\ .
\end{split}
\end{equation}
\betagin{proposition}
\label{prop_genus3_algebraic_Mochizuki}
When $g\geq 3$, there exists a sequence $(\mathcal{E}_i,\varphi_i)\in \mathcal{M}_q$ of stable Higgs bundles with stable limit $(\mathcal{E}_{\infty},\varphi_{\infty})=\lim_{i\to \infty}(\mathcal{E}_i,\varphi_i)$ such that
$$\lim_{i\to \infty}\Theta^{\mathrm{Moc}}(\mathcal{E}_i,\varphi_i)\neq
\Theta^{\mathrm{Moc}}(\mathcal{E}_{\infty},\varphi_{\infty})\ .$$
\end{proposition}
\betagin{proof}
Choose $ D=\lambdabdam$ and $d_+=-(g-1)$ with $L_i=L\in
\mathrm{Pic}^{d_+}(\Sigmagma)$, and study the degenerate behavior for a family
$q_i\in \mathcal{V}(\lambdabdam,L)$ which converges to $q_{\infty}\in \mathcal{S}V(
D_{\infty},L)$. Here, $D_{\infty}$ satisfies $D_{\infty}\leq D$ and $\deg(D_{\infty})=\deg(D)-1$.
As $q_i$ lies in the top stratum, we can always find such a family.
Take $(\mathcal{E}_i,\varphi_i)=\wp(L_i,q_i)$ and $(\mathcal{E}_{\infty},\varphi_{\infty})=\wp(L,q_{\infty})$.
When $g\geq 3$, we have $-\deg(D_{\infty})<d_+\leq -\frac12\deg(D_{\infty})$,
which implies $(\mathcal{E}_{\infty},\varphi_{\infty})$ is a stable Higgs bundle.
Write $D=\mathfrak{su}m_p\ell_p$. As $(\mathcal{E}_i,\varphi_i)$ is nonexotic,
the weights will be $\mathrm{ch}i_+(p)=\mathrm{ch}i_-(p)=\ell_p/2$. However, as
$\deg(D_{\infty})\neq 2d_+$, $(\mathcal{E}_{\infty},\varphi_{\infty})$ is exotic.
By Proposition \ref{prop_choiceofconstant_c}, if we write
$\mathrm{ch}i_{\pm}^{\infty}(p)$ for the weight functions with corresponding constant $c$, then $c>0$.
Therefore, for $p\neq p_0$, we have
$\mathrm{ch}i^{\infty}_+(p)=(m_p+1)c+m_p/2>m_p/2=\mathrm{ch}i_+(p)$.
By \eqref{eq_limit_reducible_fiber}, $\lim_{i\to
\infty}\Theta^{\mathrm{Moc}}(\mathcal{E}_i,\varphi_i)\neq \Theta^{\mathrm{Moc}}(\mathcal{E}_{\infty},\varphi_{\infty})$.
\end{proof}
When $g=2$, the stratification is simpler, and we have the following.
\betagin{proposition}
\label{prop_genus2_algebraic_Mochizuki}
When $g=2$, the following holds:
\betagin{itemize}
\item [(i)] Suppose $\lambdabdam=p_1+p_2$ for $p_1\neq p_2$, then $\Theta^{\mathrm{Moc}}|_{\mathcal{M}_q^{\asta}}$ is continuous.
Moreover, there exists a sequence of stable Higgs bundles $(\mathcal{E}_i,\varphi_i)\in \mathcal{M}_q$ where the
limit $(\mathcal{E}_{\infty},\varphi_{\infty})=\lim_{i\to \infty}(\mathcal{E}_i,\varphi_i)$ is semistable, and $\gamma(0)$ is also
semistable and furthermore
$$
\lim_{i\to \infty}\Theta^{\mathrm{Moc}}(\mathcal{E}_i,\varphi_i)\neq \Theta^{\mathrm{Moc}}(\mathcal{E}_{\infty},\varphi_{\infty})\ .
$$
\item [(ii)] Suppose $\lambdabdam=2p$. Then $\Theta^{\mathrm{Moc}}|_{\mathcal{M}_q^{\asta}}$ is continuous.
\end{itemize}
\end{proposition}
\betagin{proof}
For (i), suppose $\lambdabdam=p_1+p_2$, then by Example \ref{ex_genus_two_stable.}, we have $\mathcal{M}_q^{\asta}=\wp(\mathcal{S}V(\lambdabdam,-1))$.
By Proposition \ref{prop_reducible_continuous_on_strata}, $\Theta^{\mathrm{Moc}}_q|_{\mathcal{M}_q^{\asta}}$ is continuous.
However, for semistable elements other strata must be taken into consideration. Take $L\in \mathrm{Pic}^{-1}(\Sigmagma)$ and
$q_i\in \mathcal{V}(\lambdabdam,L)$ such that $q_i$ convergence to $q_{\infty}\in \mathcal{V}(p_1,L)$. We define $(\mathcal{E}_i,\varphi_i)=\wp(L,q_i)$ and
$(\mathcal{E}_{\infty},\varphi_{\infty})=\wp(L,q_{\infty})$. For each $i$,
$$
\Theta^{\mathrm{Moc}}(\mathcal{E}_i,\varphi_i)=\mathcal{F}_{\ast}(L,(\tfrac{1}{2},\tfrac{1}{2}))\oplus \mathcal{F}_{\ast}(L^{-1}(-\lambdabdam),(\tfrac{1}{2},\tfrac{1}{2})).
$$
Moreover, we have
$$
\Theta^{\mathrm{Moc}}(\mathcal{E}_{\infty},\varphi_{\infty})=\mathcal{F}_{\ast}(L(D),(0,0))\oplus \mathcal{F}_{\ast}(L^{-1}(-D),(0,0))\neq \lim_{i\to \infty}\Theta^{\mathrm{Moc}}(\mathcal{E}_i,\varphi_i).
$$
For (ii), by Example \ref{example_genustwodegreestratification},
$\wp(\mathcal{S}V(D_2,-1))=\mathcal{M}_q^{\asta}$ and by Proposition \ref{prop_reducible_continuous_on_strata}, $\Theta^{\mathrm{Moc}}_q|_{\mathcal{M}_q^{\asta}}$
is continuous. We now consider the behavior of the filtered bundle when crossing the divisors.
\end{proof}
\mathfrak{su}bsection{The analytic Mochizuki map and limiting configurations}
In this subsection, we construct the analytic Mochizuki map for the Hitchin fiber with a reducible spectral curve. We
also introduce the convergence theorem of Mochizuki as stated in \cite{Mochizukiasymptotic} and
examine the discontinuous behavior of the analytic Mochizuki map.
For $(\mathcal{E},\varphi)\in\mathcal{M}_q$, we can express the abelianization as $(\mathcal{E}_0,\varphi_0)=(L_+\oplus L_-,\betagin{pmatrix}
\omega & 0\\ 0 & -\omega
\end{pmatrix})$,
thus $\Theta^{\mathrm{Moc}}(\mathcal{E},\varphi)=\mathcal{F}_{\ast}(L_+,\mathrm{ch}i_+)\oplus \mathcal{L}_-(L_-,\mathrm{ch}i_-)\in \mathcal{F}_2(\Sigmagma)$. Via the nonabelian Hodge correspondence for filtered
bundles, we obtain two Hermitian metrics $h^{\mathrm{Lim}}_{\pm}$ with corresponding Chern connections $A_{h^{\mathrm{Lim}}_{\pm}}$. These metrics
satisfy the following proposition.
\betagin{proposition}[{\cite[Lemma 4.4]{mochizuki2003asymptoticdifferent}}]
\label{prop_limitingconfconstruct}
The metrics $h_{\pm}^{\mathrm{Lim}}$ over $L_{\pm}$ satisfy
\betagin{itemize}
\item [i)] $F_{A_{h^{\mathrm{Lim}}_{\pm}}}=0$ and $h_+^{\mathrm{Lim}}h_-^{\mathrm{Lim}}=1$,
\item [ii)] for every $p\in\Sigmagma$, there exists an open neighborhood $(U,z)$ with $P=\{z=0\}$ such that $|z|^{-2\mathrm{ch}i_p(c_0)}h_{+}^{\mathrm{Lim}}$ and $|z|^{2\mathrm{ch}i_p(c_0)+2l_P}h_-^{\mathrm{Lim}}$ extends smoothly to $L_{\pm}|_U$.
\end{itemize}
\end{proposition}
Now, $H^{\mathrm{Lim}}:=h_+^{\mathrm{Lim}} \oplus h_-^{\mathrm{Lim}}$ is a metric on $\mathcal{E}_0$ which induces a metric on $(\mathcal{E},\varphi)|_{\Sigmagma\setminus Z}$
because $(\mathcal{E},\varphi)|_{\Sigmagma\setminus Z} \cong (\mathcal{E}_0,\varphi_0)|_{\Sigmagma\setminus Z}$. Let $(A^{\mathrm{Lim}},\phi^{\mathrm{Lim}})$ be the Chern connection
defined by $(\mathcal{E},\varphi,H^{\mathrm{Lim}})$ over $\Sigmagma\setminus Z$. Then $(A^{\mathrm{Lim}},\phi^{\mathrm{Lim}})$ is a limiting configuration that satisfies
the decoupled Hitchin equations \eqref{eq_decoupled_Hitchin_equation}. The analytic Mochizuki map $\UpsilonilonMoc$ is defined as:
\betagin{equation} \label{eq_analytic_moc_reducible}
\UpsilonilonMoc: \mathcal{M}_q \longrightarrow \mathcal{M}HLC, \quad \UpsilonilonMoc(\mathcal{E},\varphi) = (A^{\mathrm{Lim}},\phi^{\mathrm{Lim}}).
\end{equation}
Note that $H^{\mathrm{Lim}}$ is not unique: for any constant $c$, the metric $c h_+^{\mathrm{Lim}} \oplus c^{-1} h_-^{\mathrm{Lim}}$ defines the same
Chern connection as $H^{\mathrm{Lim}}$. In any case, the map $\UpsilonilonMoc$ is well-defined.
Suppose $(\mathcal{E},\varphi)$ is an S-equivalence class of a semistable Higgs bundle. Let $H_t$ be the harmonic metric for $(\mathcal{E},t\varphi)$. For
each constant $C>0$, define $\mu_C$ to be the automorphism of $L_+ \oplus L_-$ given by $\mu_C = C\mathrm{id}_{L_+} \oplus C^{-1} \mathrm{id}_{L_-}$.
As $\mathcal{E} \cong L_+ \oplus L_-$ on $\Sigmagma\setminus Z$, $\mu_C^{\ast} H_t$ can be regarded as a metric on $\mathcal{E}|_{\Sigmagma\setminus Z}$.
Take any point $x \in \Sigmagma\setminus Z$ and a frame $e_x$ of $L_+|x$, and define:
\betagin{equation*}
C(x,t):=\biggl(\frac{h_{L_+}^{\mathrm{Lim}}(e_x,e_x)}{H_t(e_x,e_x)}\biggr)^{1/2}\ .
\end{equation*}
Writing $\nabla_t + t\phi_t$ as the corresponding flat connection of $(\mathcal{E},t\varphi)$ under the nonabelian Hodge correspondence, then
\betagin{theorem}[{\cite{Mochizukiasymptotic}}]
\label{thm_moc_convergence_reduciblecase}
On any compact subset $K$ of $\Sigmagma\setminus Z$,
$\mu^{\ast}_{C(x,t)}H_t$ converges smoothly to $H^{\mathrm{Lim}}$. Additionally, there exist $t$-independent constants $C_k$ and $C'_k$ such that
$$
|(\nabla_t,\phi_t)-\UpsilonilonMoc(\mathcal{E},\varphi)|_{\mathcal{C}^k}\leq C_ke^{-C'_kd}.
$$
\end{theorem}
Propositions \ref{prop_genus3_algebraic_Mochizuki} and \ref{prop_genus2_algebraic_Mochizuki} now give
\betagin{theorem}
\label{thm_analytic_moc_reducible_fiber}
When $g\geq 3$, $\UpsilonilonMoc|_{\mathcal{M}_q^{\asta}}$ is discontinuous, and when $g=2$, $\UpsilonilonMoc|_{\mathcal{M}_q^{\asta}}$ is continuous.
\end{theorem}
\section{The Compactified Kobayashi-Hitchin map}
\label{sec_compactified_Kobayashi_Hitchin_map}
In this section, we define a compactified version of the Kobayashi-Hitchin map and prove the main theorem of our paper.
The Kobayashi-Hitchin map $\Xi$ is a homeomorphism between the Dolbeault moduli space $\mathcal{M}D$ and the Hitchin
moduli space $\mathcal{M}H$. We wish to extend this to a map $\overline{\Xi}$ from the compactified Dolbeault moduli space $\overline{\MM}_{\mathrm{Dol}}$
to the compactification $\overline{\MM}_{\mathrm{Hit}}\mathfrak{su}bset \mathcal{M}H\cup\mathcal{M}HLC$ of the Hitchin moduli space, and to study the properties
of this extended map.
\mathfrak{su}bsection{The compactified Kobayashi-Hitchin map}
\label{subsec_constructionlimitngconfiguration}
We first summarize the results obtained above. By the construction in Section \ref{sec_the_algebraic_and_analytic_compactifications},
there is an identification $\partial\overline{\MM}_{\mathrm{Dol}}\cong (\mathcal{M}D\setminus \mathcal{H}^{-1}(0))/\mathbb{C}^{\ast}$. Moreover, through \eqref{eq_analytic_moc_irreducible} and
\eqref{eq_analytic_moc_reducible}, we have constructed the analytic Mochizuki map $\UpsilonilonMoc: \mathcal{M}D\setminus \mathcal{H}^{-1}(0)\to \mathcal{M}HLC$.
Writing $(A^{\mathrm{Lim}},\phi^{\mathrm{Lim}}=\varphi+\varphi^{\dagger_{\mathrm{Lim}}})=\UpsilonilonMoc(\mathcal{E},\varphi)$, then for $w\in \mathbb{C}^{\ast}$, we have
$$
\UpsilonilonMoc(\mathcal{E},w\varphi)=(A^{\mathrm{Lim}},\phi^{\mathrm{Lim}}=w\varphi+\bar{w}\varphi^{\dagger_{\mathrm{Lim}}})\ .
$$
Hence $\UpsilonilonMoc$ descends to a map $\partial \overline{\Xi}$ between $\mathbb{C}^{\ast}$ orbits.
\betagin{equation*}
\betagin{split}
\partial\overline{\Xi}:\partial\overline{\MM}_{\mathrm{Dol}}=(\mathcal{M}D\setminus \mathcal{H}^{-1}(0))/\mathbb{C}^{\ast}\longrightarrow \mathcal{M}H^{\mathrm{Lim}}/\mathbb{C}^{\ast}\ ,
\end{split}
\end{equation*}
Together with the initial Kobayashi-Hitchin map $\Xi:\mathcal{M}D\to \mathcal{M}H$, we obtain
\betagin{equation*}
\betagin{split}
\overline{\Xi}: \overline{\MM}_{\mathrm{Dol}}=\mathcal{M}D\cup\partial\overline{\MM}_{\mathrm{Dol}}\longrightarrow \mathcal{M}H\cup \mathcal{M}HLC/\mathbb{C}^{\ast}.
\end{split}
\end{equation*}
Theorems \ref{thm_moc_convergence_irreducible_case} and \ref{thm_moc_convergence_reduciblecase} show that for a Higgs bundle $(\mathcal{E},\varphi)\in
\mathcal{M}D\setminus \mathcal{H}^{-1}(0)$ and real $t$, $\lim_{t\to \infty}\Xi(\mathcal{E},t\varphi)=\partial\overline{\Xi}[(\mathcal{E},\varphi)/\mathbb{C}^{\ast}]$. Thus the image of $\overline{\Xi}$ lies in
$\overline{\MM}_{\mathrm{Hit}}$, the closure of $\mathcal{M}H$ in $\mathcal{M}H\cup\mathcal{M}D\setminus \mathcal{H}^{-1}(0)$.
There are natural extensions $\overline{\MH}_{\mathrm{Dol}}:\overline{\MM}_{\mathrm{Dol}}\to \phi_{z,\mathrm{mod}}B$ and $\overline{\MH}_{\mathrm{Hit}}:\overline{\MM}_{\mathrm{Hit}}\to \phi_{z,\mathrm{mod}}B$ such that
$\overline{\MH}_{\mathrm{Hit}}\circ \overline{\Xi}=\overline{\MH}_{\mathrm{Dol}}$.
In summary, there are commutative diagrams
\betagin{equation*}
\betagin{tikzcd} \mathcal{M}D \arrow[r, "\Xi"] \arrow[d, hook] & \mathcal{M}H \arrow[d,
hook] \\ \overline{\MM}_{\mathrm{Dol}} \arrow[r, "\overline{\Xi}"] & \overline{\MM}_{\mathrm{Hit}} \end{tikzcd}\ ,\ \betagin{tikzcd}
\overline{\MM}_{\mathrm{Dol}} \arrow[rd, "\overline{\MH}_{\mathrm{Dol}}",swap ] \arrow[r, "\overline{\Xi}"] & \overline{\MM}_{\mathrm{Hit}} \arrow[d, "\overline{\MH}_{\mathrm{Hit}}"] \\
& \phi_{z,\mathrm{mod}}B
\end{tikzcd}.
\end{equation*}
We now turn to the analysis of some properties of the compactified Kobayashi-Hitchin map. Define
$$
\phi_{z,\mathrm{mod}}B^{\mathrm{reg}}=\{[(q,w)]\in \phi_{z,\mathrm{mod}}B\mid q\neq 0\mathrm{\;has\;simple\;zeros}\}\ .
$$
This represents the compactified space of quadratic differentials with simple zeros. Let $\phi_{z,\mathrm{mod}}B^{\sigmang}=\phi_{z,\mathrm{mod}}B\setminus \phi_{z,\mathrm{mod}}B^{\mathrm{reg}}$.
Additionally, define the open sets $\overline{\MM}_{\mathrm{Dol}}^{\mathrm{reg}}=\overline{\MH}_{\mathrm{Dol}}^{-1}(\phi_{z,\mathrm{mod}}B^{\mathrm{reg}})$ and $\overline{\MM}_{\mathrm{Hit}}^{\mathrm{reg}}=\overline{\MH}_{\mathrm{Hit}}^{-1}(\phi_{z,\mathrm{mod}}B^{\mathrm{reg}})$
as the collections of elements with regular spectral curves. Now set $\overline{\MM}_{\mathrm{Dol}}^{\sigmang}=\overline{\MH}_{\mathrm{Dol}}^{-1}(\phi_{z,\mathrm{mod}}B^{\sigmang})$ and
$\overline{\MM}_{\mathrm{Hit}}^{\sigmang}=\overline{\MH}_{\mathrm{Hit}}^{-1}(\phi_{z,\mathrm{mod}}B^{\sigmang})$ to be the sets of singular
fibers. We can write $\overline{\Xi}=\overline{\Xi}^{\mathrm{reg}}\cup\overline{\Xi}^{\sigmang}$,
where
$$
\overline{\Xi}^{\mathrm{reg}}:\overline{\MM}_{\mathrm{Dol}}^{\mathrm{reg}}\longrightarrow \overline{\MM}_{\mathrm{Hit}}^{\mathrm{reg}},\quad \overline{\Xi}^{\sigmang}:\overline{\MM}_{\mathrm{Dol}}^{\sigmang}\longrightarrow \overline{\MM}_{\mathrm{Hit}}^{\sigmang}\ .
$$
\betagin{proposition}
The map $\overline{\Xi}^{\mathrm{reg}}:\overline{\MM}_{\mathrm{Dol}}^{\mathrm{reg}}\to \overline{\MM}_{\mathrm{Hit}}^{\mathrm{reg}}$ is bijective, whereas $\overline{\Xi}^{\sigmang}:\overline{\MM}_{\mathrm{Dol}}^{\sigmang}\to \overline{\MM}_{\mathrm{Hit}}^{\sigmang}$ is neither surjective nor injective.
\end{proposition}
\betagin{proof}
The bijectivity of $\overline{\Xi}^{\mathrm{reg}}$ is established by Theorem
\ref{thm_simple_zero_bijective}. The non-surjectivity and non-injectivity
of $\overline{\Xi}^{\sigmang}$ follow from Theorem \ref{thm_analytic_moc_irreducible_fiber} and Theorem \ref{thm_analytic_moc_reducible_fiber}.
\end{proof}
\mathfrak{su}bsection{Continuity properties of the compactified Kobayashi-Hitchin map}
In this subsection, we prove that the continuity of the compactified Kobayashi-Hitchin map is fully determined by the
continuity of the analytic Mochizuki map.
Let $(\mathcal{E}_{i},t_i\varphi_{i})$ be a sequence of Higgs bundles with real numbers
$t_i\to+\infty$, $\det(\varphi_i)=q_i$, $Z_i=q_i^{-1}(0)$, and $\|q_i\|_{L^2}=1$. We denote $\xi_i=[(\mathcal{E}_i,t_i\varphi_i)]\in \mathcal{M}D$.
By the compactness of $\overline{\MM}_{\mathrm{Dol}}$, after passing to a subsequence, we may
assume there is $\xi_{\infty}\in \partial\overline{\MM}_{\mathrm{Dol}}$ such that $\lim_{i\to\infty}\xi_i=\xi_{\infty}$. Since $\partial\overline{\MM}_{\mathrm{Dol}}\cong (\mathcal{M}D\setminus \mathcal{H}^{-1}(0))/\mathbb{C}^{\ast}$, we can select a representative $(\mathcal{E}_{\infty},\varphi_{\infty})$ of $\xi_{\infty}$. By Lemma \ref{prop_convergence_Dol_space}, we have that $(\mathcal{E}_i,\varphi_i)$ converges to $(\mathcal{E}_{\infty},\varphi_{\infty})$ in $\mathcal{M}D$, and $q_i$ converges to $q_{\infty}$. We write $Z_{\infty}=q_{\infty}^{-1}(0).$
By Proposition \ref{prop_general_convergence_solutions}, $\lim_{i\to
\infty}\overline{\Xi}(\mathcal{E}_i,t_i\varphi_i)$ exists. The following result
establishes the continuity of this map with respect to the analytic Mochizuki map $\UpsilonilonMoc$.
\betagin{proposition} \label{prop_topology_KHmap}
Under the previous convention, $\lim_{i\to \infty}\overline{\Xi}(\xi_i)=\overline{\Xi}(\xi_{\infty})$ if and only if $\lim_{i\to \infty}\UpsilonilonMoc(\mathcal{E}_i,\varphi_i)=\UpsilonilonMoc(\mathcal{E}_{\infty},\varphi_{\infty})$. In other words, $\overline{\Xi}$ is continuous at $\xi_\infty$ if and only if $\UpsilonilonMoc$ is continuous at $\xi_\infty$.
\end{proposition}
\betagin{proof}
Set
\betagin{align*}
\overline{\Xi}(\mathcal{E}_i,t_i\varphi_i)&=\Xi(\mathcal{E}_i,t_i\varphi_i)=A_i+t_i\phi_i\ ,\\
\UpsilonilonMoc(\mathcal{E}_i,\varphi_i)&=(A_i^{\mathrm{Lim}},\phi_i^{\mathrm{Lim}})\ ,\\
\UpsilonilonMoc(\mathcal{E}_{\infty},\varphi_{\infty})&=(A_{\infty}^{\mathrm{Lim}},\phi_{\infty}^{\mathrm{Lim}})\ .
\end{align*}
By Proposition \ref{prop_general_convergence_solutions},
there exists a limiting configuration $(A_{\infty},\phi_{\infty}):=\lim_{i\to \infty}(A_i,\phi_i)$ over $\Sigmagma\setminus Z_{\infty}$.
Let $K$ be any compact set in $\Sigmagma\setminus Z_{\infty}$. Note that by the convergence assumption, there exists $i_0$ such that $Z_i\cap
K=\emptyset$ for all $i\geq i_0$. Moreover, from Theorems \ref{thm_moc_convergence_irreducible_case} and \ref{thm_moc_convergence_reduciblecase},
for $d_i=\min_K|q_i|$, there exist $t$-independent constants $C,C'>0$ such that (up to gauge transformations)
\betagin{equation*}
|(A_i,\phi_i)-(A_i^{\mathrm{Lim}},\phi_i^{\mathrm{Lim}})|_{\mathcal{C}^k(K)}\leq Ce^{-C't_id_i}.
\end{equation*}
The convergence is uniform and exponential for fixed $K$. Therefore, over $K$, the size of
$|(A_{\infty},\phi_{\infty})-(A_{\infty}^{\mathrm{Lim}},\phi_{\infty}^{\mathrm{Lim}})|_{\mathcal{C}^k(K)}$ is the same as the size of
$|(A_{i}^{\mathrm{Lim}},\phi_{i}^{\mathrm{Lim}})-(A_{\infty}^{\mathrm{Lim}},\phi_{\infty}^{\mathrm{Lim}})|_{\mathcal{C}^k(K)}$. This proves the Proposition.
\end{proof}
\mathfrak{su}bsubsection{Continuity along rays}
We now investigate the behavior of the compactified Kobayashi-Hitchin map restricted to a singular fiber. Specifically,
fix $0\neq q\in H^0(K^2)$, and denote by $[q]$ the $\mathbb{C}^\ast$-orbit of $q\timesmes{1}$ in the compactified Hitchin base
$\phi_{z,\mathrm{mod}}B$. Define $\overline{\MM}_{\mathrm{Dol}}q:=\overline{\MH}^{-1}_{\mathrm{Dol}}([q])$, $\overline{\MM}_{\mathrm{Hit}}q:=\overline{\MH}^{-1}_{\mathrm{Hit}}([q])$. Then the restriction of $\overline{\Xi}$ on $\overline{\MM}_{\mathrm{Dol}}q$
defines a map $\overline{\Xi}_q:\overline{\MM}_{\mathrm{Dol}}q\to \overline{\MM}_{\mathrm{Hit}}q$.
\betagin{theorem}
Let $q$ be an irreducible quadratic differential.
\betagin{itemize}
\item [(i)] If $q$ contains only zeroes of odd order, then $\overline{\Xi}_q$ is continuous.
\item [(ii)] If $q$ contains a zero of even order, let $\mathcal{M}_q=\cup_D\mathcal{M}_{q,D}$ be the stratification defined earlier.
Then for each $D\neq 0$, there exists an integer $n_D>0$ such that for any Higgs bundle $(\mathcal{F},\mathrm{ps}i)\in \mathcal{M}_{q,D}$,
there exist $n_D$ sequences of Higgs bundles $(\mathcal{E}_i^k,\varphi_i^k)$ with $k=1,\ldots, n_D$ such that
\betagin{itemize}
\item [(a)] $\lim_{i\to \infty}(\mathcal{E}_i^k,\varphi_i^k)=(\mathcal{F},\mathrm{ps}i)$ for $k=1,\ldots,n_D$,
\item [(b)] if $\lim_{i\to \infty}t_i=\infty$, and we write
$$
\eta^k:=\lim_{i\to \infty}\overline{\Xi}_q(\mathcal{E}_i^k,\varphi_i^k),\;\xi:=\lim_{i\to \infty}\overline{\Xi}_q(\mathcal{F},t_i\mathrm{ps}i),
$$
then $\xi,\eta^1,\ldots,\eta^{n_D}$ are $n_D+1$ different limiting configurations.
\end{itemize}
\end{itemize}
\end{theorem}
\betagin{proof}
This follows from Theorem \ref{thm_analytic_moc_irreducible_fiber} and Proposition \ref{prop_topology_KHmap}.
\end{proof}
\betagin{theorem}
Suppose $q$ is reducible, and let $\overline{\MM}_{\mathrm{Dol}}q^{\asta}$ be the stable locus of $\overline{\MM}_{\mathrm{Dol}}q$. Then the restriction map
$\Theta^{\mathrm{Moc}}_q|_{\overline{\MM}_{\mathrm{Dol}}q^{\asta}}:\overline{\MM}_{\mathrm{Dol}}q^{\asta}\to \overline{\MM}_{\mathrm{Hit}}q$ is discontinuous when $g\geq 3$ and continuous when $g=2$.
\end{theorem}
\betagin{proof}
This follows from Propositions \ref{prop_genus3_algebraic_Mochizuki} and \ref{prop_topology_KHmap}.
\end{proof}
\mathfrak{su}bsubsection{Varying fiber}
With the conventions above, suppose $(\mathcal{E}_i,\varphi_i)$ converges to $(\mathcal{E}_{\infty},\varphi_{\infty})$ with
$q_{\infty}$ having only simple zeros, and $\xi_i=(\mathcal{E}_i,t_i\varphi_i)$ converges to $\xi_\infty$ on $\overline{\MM}_{\mathrm{Dol}}$. Since the condition
of having only simple zeros is open, the $q_i$ also have simple zeros for $i$ sufficiently large.
\betagin{proposition}[{cf.\ \cite[Thm.\ 2.12]{ott2020higgs}}]
\label{prop_homeomorphic_part_one}
Suppose $q_{\infty}$ has only simple zeros. Then, $\displaystyle\lim_{i\to\infty}\overline{\Xi}(\xi_i)=\overline{\Xi}(\xi_\infty)$. In particular, the map
$\overline{\Xi}^{\mathrm{reg}}:\overline{\MM}_{\mathrm{Dol}}^{\mathrm{reg}}\to \overline{\MM}_{\mathrm{Hit}}^{\mathrm{reg}}$ is continuous.
\end{proposition}
\betagin{proof}
Let $S_i$ denote the spectral curve of $(\mathcal{E}_i,\varphi_i)$ with branching locus $Z_i$. Also, let $L_i:=\mathrm{ch}i_{BNR}^{-1}(\mathcal{E}_i,\varphi_i)$ be the
eigenline bundles. By the construction in Section \ref{sec_irreducible_singular_fiber}, we have $\UpsilonilonMoc(\xi_i)=\mathcal{F}_{\ast}(L_i,\mathrm{ch}i_i)$, where
$\mathrm{ch}i_i=-\frac12\mathrm{ch}i_{Z_i}$. Our assumption implies that $\mathcal{F}_{\ast}(L_i,\mathrm{ch}i_i)$ converges to $\mathcal{F}_{\ast}(L_{\infty},\mathrm{ch}i_{\infty})$ in the sense
of Definition \ref{def_convergence_parabolic_bundles}. Thus, by Theorem \ref{thm_convergence_family_harmonic_bundles}, we obtain
the convergence of the limiting configurations: $\lim_{i\to \infty}\UpsilonilonMoc(\xi_i)=\UpsilonilonMoc(\xi_{\infty})$. The claim follows from
Proposition \ref{prop_topology_KHmap}.
\end{proof}
\betagin{theorem}
The map $\overline{\Xi}^{\mathrm{reg}}:\overline{\MM}_{\mathrm{Dol}}^{\mathrm{reg}}\to \overline{\MM}_{\mathrm{Hit}}^{\mathrm{reg}}$ is a homeomorphism.
\end{theorem}
\betagin{proof}
By Theorem \ref{thm_simple_zero_bijective}, $\overline{\Xi}^{\mathrm{reg}}$ is a bijection.
Moreover, by Proposition \ref{prop_homeomorphic_part_one},
$\overline{\Xi}^{\mathrm{reg}}$ is continuous. Finally, that $(\overline{\Xi}^{\mathrm{reg}})^{-1}$ is continuous follows directly from the construction in \cite{mazzeo2019asymptotic}.
\end{proof}
\mathrm{app}endix
\section{Classification of rank 1 torsion modules for $A_n$ singularities}
\label{appendixA}
In this appendix, we review the classification result for rank 1 torsion free modules at $A_n$ singularities, as given in \cite{Greuel1985}.
We compute the integer invariants defined in Subsection \ref{subsection_torsion_free_sheaf_integers}.
Let $S$ be the spectral curve of an $\mathrm{SL}C$ Higgs bundle, and $x$ a singular point with local defining equation given by
$r^2-s^{n+1}=0$; this is an $A_n$ singularity. Let $p:\widetilde{S}\to S$ be the normalization, where $p^{-1}(x)=\{\tilde{x}_+,\tilde{x}_-\}$ if $n$ is odd
and $p^{-1}(x)={\tilde{x}}$ if $n$ is even. We use $R$ to denote the completion
of the local ring $\mathcal{O}_x$, $K$ its
field of fractions, and $\widetilde{R}$ its normalization.
\mathfrak{su}bsection{$A_{2n}$ singularity}
The local equation is $r^2-s^{2n+1}=0$. The normalization induces a map between coordinate rings, and we can write
\betagin{equation*}
\mathrm{ps}i:\mathbb{C}[r,s]/(r^2-s^{2n+1})\longrightarrow \mathbb{C}[t],\quad \mathrm{ps}i(f(r,s))=f(t^{2n},t^2),
\end{equation*}
where $\widetilde{R}=\mathbb{C}[[t]]$ and $R=\mathbb{C}[[t^2,t^{2n+1}]]\mathfrak{su}bset \widetilde{R}$.
According to \cite[Anh.\ (1.1)]{Greuel1985}, any rank 1 torsion free $R$-module can be written as
\betagin{equation*}
M_k = R + R \cdotot t^k \mathfrak{su}bset \widetilde{R}, \quad k=1,3,\ldots,2n+1.
\end{equation*}
Here, $M_k$ is a fractional ideal that satisfies $R\mathfrak{su}bset M_k\mathfrak{su}bset \widetilde{R}$, with $M_1=\widetilde{R}$ and $M_{2n+1}=R$. We may express any $f \in M_k$ as $f = \mathfrak{su}m_{i=0}^{\frac{k-1}{2}} f_{2i}t^{2i}+\mathfrak{su}m_{i\geq k}f_it^i$, where $f_i\in \mathbb{C}$.
We are interested in the integers $\ell_x := \dim_{\mathbb{C}}(M_k/R)$, $a_{\tilde{x}}
:= \dim_{\mathbb{C}}(\widetilde{R}/C(M_k))$ and $b_x=\dim_{\mathbb{C}}(\mathrm{Tor}(M_k\otimesimesmes_R \widetilde{R}))$.
Thus, as a $\mathbb{C}$-vector space, $M_k/R$ is generated by $t^k, t^{k+2},
\ldots, t^{2n-1}$, implying that $\ell_x=\frac{2n+1-k}{2}$.
The conductor of $M_k$ is given by $C(M_k) = \{u\in K\mid u\cdotot \widetilde{R}\mathfrak{su}bset
M_k\}$. By the expression of $M_k$ and a straightforward computation, we
have $C(M_k) = (t^{k-1})$, where $(t^{k-1})$ is the ideal in $\widetilde{R}$
generated by $t^{k-1}$. Thus, $1,t,\ldots,t^{k-2}$ will form a basis for
$\widetilde{R}/C(M_k)$, and we have $a_{\tilde{x}}=k-1$. Therefore, we have
$a_{\tilde{x}}=2n-2\ell_x$.
For $i=0, 1,\ldots,\frac{2n-1-k}{2}$, we define $s_i=t^{k+2i}\otimesimesmes_R
1-1\otimesimesmes t^{k+2i}\in M_k\otimesimesmes_R\widetilde{R}$. As $k$ is odd, $t^{2n+1-k-2i}\in
R$ and $t^{2n+1-k-2i}s_i=t^{2n+1}\otimesimesmes_R 1-1\otimesimesmes_R t^{2n+1}=0$, where
the last equality is becasue $t^{2n+1}\in R$. Moreover, $\{s_1,\ldots,
s_{\frac{2n-1-k}{2}}\}$ form a basis of $\mathrm{Tor}(M_k\otimesimesmes_R\widetilde{R})$, thus
$b_x=\frac{2n+1-k}{2}=\ell_x$.
\mathfrak{su}bsection{$A_{2n-1}$ singularity}
The local equation is $r^2-s^{2n}=0$. The normalization induces a map
between the coordinate rings:
\betagin{equation*}
\mathrm{ps}i:\mathbb{C}[r,s]/(r^2-s^{2n})\longrightarrow \mathbb{C}[t]\oplus \mathbb{C}[t],\quad
\mathrm{ps}i(f(r,s))=(f(t^n,t),f(-t^n,t))\ ,
\end{equation*}
where $\widetilde{R}=\mathbb{C}[[t]]\oplus \mathbb{C}[[t]]$ and $R=\mathbb{C}[[(t,t),(t^n,-t^n)]]\cong \mathbb{C}[[(t,t),(t^n,0)]]$.
By \cite[Anh.\ (2.1)]{Greuel1985}, any rank 1 torsion free $R$-module can be written as:
\betagin{equation*}
M_k=R+R\cdotot (t^k,0)\mathfrak{su}bset \widetilde{R},\quad k=0,1,\ldots,n.
\end{equation*}
Then, $M_k$ is also a fractional ideal with $R\mathfrak{su}bset M_k\mathfrak{su}bset \widetilde{R}$. Moreover, $M_n=R$, and $M_0=\widetilde{R}$.
As $p^{-1}(x)=\{\tilde{x}_+,\tilde{x}_-\}$, $\widetilde{R}$ contains two maximal ideals,
$\mathfrak{m}_+=((t,1))$, $\mathfrak{m}_-=((1,t))$.
For $f\in M_k$, we can express $f$ as:
\betagin{equation*}
f=\mathfrak{su}m_{i=0}^{k-1}f_{ii}(t^i,t^i)+\mathfrak{su}m_{l\geq 0}f_{l0}(t^{k+l},0)+f_{0l}(0,t^{k+l}),
\end{equation*}
where $f_{ij}\in \mathbb{C}$. Therefore, $\ell_x=\dim_{\mathbb{C}}(M_k/R)=n-k$.
Moreover, using the expression, we can compute the conductor
$C(M_k)=((t^k,1))\cdotot((1,t^k))$, which implies $a_{\tilde{x}_{\pm}}=k$.
Similarly, for $i=k,\ldots,n-1$, we define
$s_i=(t^i,0)\otimesimesmes_R(1,1)-(1,1)\otimesimesmes_R (t^i,0)$, then $(t,t)^{n-i}\cdotot
s_i=0$ and $\{s_k,\ldots, s_{n-1}\}$ will be a basis for $\mathrm{Tor}(M_k\otimesimesmes_R
\widetilde{R})$ and $b_x=\ell_x$.
In summary, we have the following:
\betagin{proposition}
\label{prop_appendix_computation}
For the integers defined above, we have:
\betagin{itemize}
\item [(i)] for the $A_{2n}$ singularity, we have
$a_{\tilde{x}}=2n-2\ell_x$ and $b_x=\ell_x$,
\item [(ii)] for the $A_{2n-1}$ singularity, we have
$a_{\tilde{x}_{\pm}}=n-\ell_x$ and $b_x=\ell_x$.
\end{itemize}
\end{proposition}
\end{document} | math |
<?php
return [
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]; | code |
An immaculate three bedroom property to let situated in a convenient location for local shops and bus routes and with the Highcliffe School catchment area. Available from 8th July 2016. Sorry, no pets, DSS or smokers.
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White emulsion ceiling, ceiling light point, UPVC double glazed window overlooking rear garden aspect, radiator, brand new fitted carpet, range of fitted wardrobes to one wall, power points, TV aerial point.
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Admin Fees - £240. For up to two people. £60. For any further applicants and guarantors. Any fee lodged with Ross Nicholas & Company are non-refundable. Admin fees are only refunded if the Landlord withdraws the property from the market.
Please note that all deposits are lodged with The Deposit Protection Service (The DPS.) Further information can be found on their website www.depositprotection.com. | english |
Wow...Where oh where would I start. I am eternally grateful to my Lord and Savior Jesus Christ for carrying me through those moments in life when I thought I couldn't go on.
Life has been a roller coaster with many times up and many times down, but through it all I have Praised HIS name.
If you didn't know I had alot of losses last year. A Major Tragedy caused the loss of two family members, and all the other losses that followed were minor compared to that, But let's just say the domino effect left us in a bit of a transition and it is taken a while to readjust.
I Started this year off in a nice little apartment. EMPTY....Since all of My Belongings were stuck in a storage in Sunny Florida. God saw me through those difficult times and I feel like it had to do with the gratefulness that I felt in my heart. Every night when I filled that air mattress one more time I Praised the Lord for that wonderful King size fluffy bed that I owned! I had faith that the day would come that I would have it with me again. And I did, Thankfully...I mean there were a few losses, some things didn't come back, others broke, but My bed made it back to New England safely! Oh and the JOY I felt when I Gave HIM the Glory!!!! I am able to appreciate my bed that much more now whenever I get in it.
So even though the year is coming to an end and we are still in a little bit of transition trying to find a home where I can finally unpack the boxes I packed up last year. I would Say that I have alot to be thankful for this year.
I am thankful that GOD made Nature Beautiful!
I am thankful that I am able to decorate and put up a christmas tree this year.
I am thankful for the great LOVE and Understanding of my friends.
The meal was Turkey with stuffing and Traditional Puerto Rican pasteles and Arroz con gandules.
I am in the process of working on Some changes...Both on the Home and Work front....I'm going to take some time to really enjoy the holidays this year...I can't believe that the year is almost at it's end. However I am expecting to have the changes completed by the time my birthday rolls around in Mid January.
Just so your comfortable with the changes I promise not to change things here as much as I change my Hair color!!!
Thanking you in advance for your patience.
So the Autumn is my favorite time of year...I know some people think I'm crazy...Who likes the fall more than summer or spring? ME! YES I do!!!
I am especially mesmerized by all the lovely foliage This year, because I missed it soooo much last year....and I am a little more attentive this Year.
The Love radiates from them here....This is my favorite of the night!
Best Wishes Anupama and Niraj!!!
I'm finding that one of the best way a client can get the most out of their Photo Session is by ordering a uniquely customized Montage.
Montages are generally 16X20 which allows for the images to be large enough to really be enjoyed.... They can be for a specific occasion or simply just because. It's a great Senior Portrait option.
However, they can also be made in a 10X20 like the one below =) This was my niece's 18th Birthday Present...She is Ecstatic!!!
Schedule your Session Today, and you too can order your very own completely customizable Montage.
Who doesn't like FREE? I know alot of people that like free....LOL!
So I thought I would share it. I love the way the filters make my ordinary images...EXTRODINARY!!!
Or in the words of one my Friends...."WOW" Thanks Evelyn for your encouragement!
Love the leaves......and yes I'm the girl that put up an autumn tree up and brought real leaves from outside to put on the tree =) but Really...I am NOT crazy I just Love Autumn!!
Ok...so maybe I was getting my hair done and getting my picture taken tooo... I'm alllowed....Right!?
I have recently started reading this book by Bruce Wilkinson. It's titled The Dream Giver. It's part of a book club of sorts. The women's church group....Women of Wisdom are reading it and I have joined in.
I don't know yet what everyone else's response is to the book, only a few comments on Facebook here and there. There will be a meeting to discuss the book, that should be fun.
I find it amazing that I feel like this book is speaking directly to ME... The book is about a person named Ordinary who lives in Familiar is given a dream and decides to pursue it. Oh but to pursue our dreams we Must Step out of our Comfort Zone. | english |
Photos of sweethearts and girls with nice-looking and wet..
Photos of sweethearts and girls with nice-looking and wet giant asses. Those tight gazoo so sexually excited they wanna clap, to knead, to cling to the bubble gazoo by face. | english |
पात्रा भारतीय रिजर्व बैंक में चौथे डिप्टी गवर्नर के रूप में पदभार संभालेंगे. वह सभी महत्वपूर्ण मौद्रिक नीति समिति में भी शामिल होंगे, जो ब्याज दर पर निर्णय लेती है.
माइकल पात्रा को भारतीय रिजर्व बैंक (रबी) का नया डिप्टी गवर्नर नियुक्त किया गया है. रबी के मौजूदा कार्यकारी निदेशक पात्रा को यह जिम्मेदारी मंगलवार को सौंपी गई. यह पद करीब छह महीने पहले विरल आचार्य के इस्तीफे के बाद से खाली पड़ा हुआ था. कैबिनेट की नियुक्ति समिति की ओर से जारी एक बयान के अनुसार, पात्रा को तीन साल के लिए नियुक्त किया गया है. पात्रा ने आचार्य का स्थान लिया, जिन्होंने पिछले साल २३ जुलाई को पद छोड़ दिया था.
पात्रा भारतीय रिजर्व बैंक में चौथे डिप्टी गवर्नर के रूप में पदभार संभालेंगे. वह संभवत: आचार्य द्वारा संचालित मौद्रिक नीति का कार्यभार संभालेंगे. वह सभी महत्वपूर्ण मौद्रिक नीति समिति में भी शामिल होंगे, जो ब्याज दर पर निर्णय लेती है.
पात्रा उन उम्मीदवारों में से एक हैं, जिनका वित्त मंत्रालय की समिति ने इंटरव्यू लिया था. समिति में बैंकिंग और वित्त सचिव राजीव कुमार शामिल थे. समझा जाता है कि प्रधानमंत्री कार्यालय ने भी पात्रा के नाम पर मुहर लगाई है.
इस पद पर केंद्रीय बैंक के बाहर के अर्थशास्त्रियों का चयन होता रहा है. आचार्य से पहले इस पद पर उर्जित पटेल थे, जिन्हें बाद में गवर्नर बना दिया गया था. उर्जित भी इस्तीफा दे चुके हैं.
वर्ष २०१७ में रबी के साथ करियर शुरू करने वाले माइकल पात्रा की मौद्रिक नीति को लेकर सोच रबी गवर्नर शक्तिकांत दास से मेल खाती है. दास के दिसंबर २०१८ में पद संभालने के बाद से रेपो रेट में लगातार तीन बार हुई कटौती में पात्रा ने हमेशा पक्ष में मतदान किया है.
गवर्नर की तरह उनका भी मानना है कि अर्थव्यवस्था को गति देने के लिए नरम मौद्रिक नीति के साथ राजकोषीय स्तर पर मदद मिलना बहुत जरूरी है. | hindi |
\begin{document}
\title{Bayesian Image Analysis in Fourier Space}
\begin{abstract}
Bayesian image analysis has played a large role over the last 40+~years in solving problems in image noise-reduction, de-blurring, feature enhancement, and object detection. However, these problems can be complex and lead to computational difficulties, due to the modeled interdependence between spatial locations. The Bayesian image analysis in Fourier space (BIFS) approach proposed here reformulates the conventional Bayesian image analysis paradigm as a large set of independent (but heterogeneous) processes over Fourier space. The original high-dimensional estimation problem in image space is thereby broken down into (trivially parallelizable) independent one-dimensional problems in Fourier space. The BIFS approach leads to easy model specification with fast and direct computation, a wide range of possible prior characteristics, easy modeling of isotropy into the prior, and models that are effectively invariant to changes in image resolution.
\textbf{Keywords:} Bayesian image analysis, Fourier space, Image priors, k\nobreakdash-space, Markov random fields, Statistical image analysis.
\end{abstract}
\section{Introduction\label{intro}}
Bayesian image analysis models provide a solution for improving image quality in image reconstruction/ enhancement problems by incorporating \emph{a~priori} expectations of image characteristics along with a model for image noise, i.e., for the image degradation process~\citep{winkler1995image,guyon1995random,li2009markov}.
However, conventional Bayesian image analysis models, defined in the space of conventional images (hereafter referred to as ``image space'') can be limited in practice because they can be difficult to specify and implement (requiring problem-specific code) and they can be slow to compute estimates for.
Furthermore, Markov random field (MRF) model priors in conventional Bayesian image analysis (as commonly used for the type of problem discussed here) are not invariant to changes in image resolution (i.e., model parameters and MRF neighborhood size need to change when pixel dimensions change in order to retain the same spatial characteristics; this can be a problem when images are collected across multiple sites with different acquisition parameters or hardware) and are difficult to specify with isotropic autocovariance (i.e., with direction-invariant covariance).
Our approach to overcoming the difficulties and limitations of the conventional Bayesian image analysis paradigm, is to move the problem to the Fourier domain and reformulate in terms of spatial frequencies: \textit{Bayesian image analysis in Fourier space} (BIFS). Spatially correlated prior distributions (priors) that are difficult to model and compute in conventional image space, can be successfully modeled via a set of independent priors across locations in Fourier space.
A prior is specified for the signal at each Fourier space location (i.e. at each spatial frequency) and \emph{Parameter functions} are specified to define the values of the parameters in the prior distribution across Fourier space locations, i.e. we specify probability density functions (pdfs) over Fourier space that are conditionally independent given known values of parameter functions. The original high-dimensional problem in image space is thereby broken down into a set of one-dimensional problems, leading to easier specification and implementation, and faster computation that is further trivially parallelizable. The fast computation coupled with trivial parallelization has the potential to open up Bayesian image analysis to big data imaging problems. Furthermore, the BIFS approach carries with it numerous useful properties, including easy specification of isotropy and consistency in priors across differing image resolutions for the same field of view.
Note that the BIFS approach is distinct from shrinkage prior methods that have been used previously in Fourier, wavelet or other basis set prior specifications (e.g. ~\citet{olshausen1996emergence,levin2007user}). BIFS does not seek to smooth by generating a sparse representation in the transformed space through simple thresholding with the hope that it provides \emph{a~priori} desired spatial characteristics. Rather, the goal of BIFS is to fully specify the prior distribution over Fourier space in the same spirit as Markov random field or other spatial priors as specified in Bayesian image analysis, i.e., to fully represent as faithfully as possible \emph{a~priori} expected characteristics of the true/optimal image.
\subsection{Bayesian image analysis}
The general image analysis problem can be described as follows: Consider observed image data, $y$, that have been degraded by some "noise" process. The goal is to get an optimal estimate of the undegraded (and generally unobserved) version of the image ($x$); this process of estimating the undegraded image will be referred to as \emph{reconstruction}. The objective of the Bayesian image analysis approach is to attempt to optimally reconstruct $x$ based on observing $y$ given knowledge of the image noise/degradation process and prior knowledge about properties that $x$ should have. The optimization being performed with respect to minimizing some loss function of the posterior. Sometimes, interest may lie not in reconstructing the true image itself, but in generating a version of the image that may enhance certain features or properties (e.g. in cancer detection it would be useful to enhance tumors in an image to make it easier for the radiologist to detect and delineate them). In this case, the optimal $x$ that we wish to construct will be an enhanced version of the true (undegraded) image with the prior designed to emphasize desirable characteristics of an enhanced image.
\subsection{Conventional Bayesian image analysis:}
Consider $x$ to be a true or idealized image (e.g., noise-free or with enhanced features) that we wish to recover from a sub-optimal image dataset $y$. (Note that we are using the common shorthand notation of not explicitly distinguishing the random variables and the corresponding image realizations \citep{besag1989digital,besag1991bayesian}, i.e., we use lower case $x$ and $y$.)
The Bayesian image analysis paradigm incorporates \emph{a~priori} desired spatial characteristics of the reconstructed image via a prior distribution (``the prior'') for the ``true'' image $x$: $\pi(x)$; and the noise degradation process via the likelihood: $\pi(y|x)$. The prior and likelihood are combined via Bayes' Theorem to give the posterior: $\pi(x|y) \propto \pi(y|x) \pi(x)$ from which an estimate of $x$ can be extracted, e.g., the ubiquitous \emph{maximum a posteriori} (MAP) solution obtained by determining the image associated with the mode of the joint posterior distribution.
\subsubsection{Markov random field (MRF) priors:}
The most common choice for the prior in conventional Bayesian image analysis is a Markov random field (MRF) model~\citep{besag1974spatial,geman1984stochastic,besag1986statistical,besag1989digital,besag1991bayesian,winkler1995image,guyon1995random,li2009markov,swain2013efficient,sonka2014image}. MRF priors are used for imposing expected \emph{contextual information} to an image such as spatial smoothness, textural information (small-scale pattern repetition), edge configurations (patterns of locations of boundaries with large intensity differences) etc. MRF methods provide improvement over deterministic filtering methods by \emph{probabilistically} interacting with the data to smooth, clean, or enhance images, by appropriately weighting information from the data (via the likelihood) with the MRF prior to form the posterior probability distribution (the posterior)~\citep{besag1991bayesian}.
The definition of an MRF over a set of locations $S$ is given via a conditional specification of each pixel (or voxel in 3D) intensity $x_s$ at location $s$, given the set of neighboring pixels $\partial s$; where "neighboring pixels" are defined in terms of being close to each other in space. Specifically, $\pi(x_s|x_{-s})=\pi(x_s|x_{\partial s})$, i.e. if the neighboring pixel values are known then remaining pixels add no further knowledge about the conditional distribution of the intensity at $s$. The \emph{full conditional posterior} for $x_s$ given the data $y$ and the values of $x$ at all sites other than $s$, denoted by $x_{-s}$, can therefore be written as
\begin{equation}
\pi(x_s|y,x_{-s}) \propto \pi(y_s|x_s) \pi(x_s|x_{\partial s}), \label{conventional}
\end{equation}
(making the common assumption that the noise degradation process is independent across pixels). Note that the full conditional posterior at a pixel depends on its set of neighbors (even if that neighborhood is small) and therefore the joint distribution over all pixels is highly interdependent (the covariance matrix will generally be dense). Note also that the Hammersley-Clifford Theorem~\citep{besag1974spatial} allows an alternate (and equivalent) joint specification of MRFs in terms of the product of Gibbs measures over cliques (sets of inter-connected neighbors); these include the highly used set of intrinsic pairwise difference priors \citep{besag1989digital}. However, the difficulties of dealing with a highly inter-dependent process remain.
\subsubsection{Other conventional Bayesian image analysis priors}
Other ``higher level'' Bayesian image analysis models exist that use priors to describe characteristics of objects in images through their geometrical specification~\citep{grenander1993general,baddeley1993stochastic,rue1999bayesian,grenander2000asymptotic,ritter2002bayesian,sheikh2005bayesian}. However, these models have been used less frequently than MRF-based models, most likely because they often need highly problem-specific computational approaches, and are more difficult to specify, implement and compute.
\subsection{Basis set representation methods for image analysis}
There is considerable literature on methods for representing processes in terms of basis set representations, see e.g., the field of functional analysis~\citep{james2005functional,ramsay2002applied,morris2014functional}. Both Fourier and wavelet space basis sets can be used for functional analysis (including approaches with a Bayesian emphasis)~\citep{chipman1997adaptive,abramovich1998wavelet,leporini2001bayesian,johnstone2005empirical,ray2006functional,nadarajah2007bkf,christmas2014bayesian}. However, these basis function approaches have mostly focused on using either simple priors based on L1/L2 regularization functions, and/or coefficient thresholding in the transformed space of the basis set; the aim being to shrink or get rid of the majority of coefficients.
Fourier/wavelet basis set shrinkage/thresholding based methods have seen multiple applications to the field of image analysis (including from a Bayesian perspective through L1/L2 regularization). In particular, methods for image processing have been developed using Fourier, wavelet and other bases sets~\citep{olshausen1996emergence,donoho1999combined,buccigrossi1999image,figueiredo1999bayesian,donoho2002beamlets,Chang2000adaptive,portilla2003image,candes2004new,levin2007image,levin2007user,pavlicova2008detecting,vijay2012image,li2014bayesian}. By using an appropriate basis set, the sparse representation in that space is expected to generate processed images with certain characteristics. For example, sparse Fourier or wavelet representations can lead to noise reduction back in image space. (The intuition here is that removed coefficients with a small contribution to the overall signal are considered to be more likely dominated by noise.) However, in contrast to the Bayesian image analysis in the Fourier space approach that we develop here, there is no explicit \emph{a~priori} model for expected structure in the true image, other than that the true image might be well represented by a small subset of the basis functions. The BIFS paradigm provides a comprehensive approach to characterizing image priors by modeling specific priors at all Fourier space locations.
Fourier representations of Gaussian Markov random fields (GMRFs) have been used in order to generate fast simulations when the GMRF neighborhood structure can be represented by a block-circulant matrix, i.e. such that the GMRF can be considered as wrapped on a 2D torus:~see~Ch.~2.6 of~\citet{rue2005gaussian}. We will explore the relationship between this Fourier space representation and a special case of the Bayesian image analysis in Fourier space approach we are proposing in Section~\ref{MRImatch}.
Finally, there are a couple of interesting Fourier space-based Bayesian image analysis approaches that are of interest. \citet{baskaran1999bayesian} define a prior for the modulus (alternatively referred to as the magnitude) of the signal in Fourier space to infer the signal. It is motivated by a specific problem in X-ray crystallography where only the modulus of the Fourier transform can be measured, but not the argument (often referred to as the phase by physicists and engineers), and uses prior information over a known part of the signal which is spherically symmetric. \citet{staib1992boundary} use a different idea of generating deformable models for finding the boundaries of 2D objects in images based on elliptic Fourier decompositions.
\subsection{The Bayesian Image analysis in Fourier space approach}
In this paper, we define a complete framework for specifying a wide range of spatial priors for continuous-valued images as models in Fourier space. The methodological benefit of working with Bayesian image analysis in Fourier space (BIFS) is that it provides the ability to model a range of stationary spatially correlated processes in conventional image space as independent processes across spatial frequencies in Fourier space.
The range of advantages afforded by transforming the Bayesian image analysis problem into Fourier space include:
\begin{description}
\item[a)] \emph{easy model specification:} expected spatial characteristics in images are modeled as \emph{a~priori} expectations of the contribution of spatial frequencies to the BIFS reconstruction (e.g.\, smooth reconstructions require higher signal at lower spatial frequencies).
\item[b)] \emph{fast and easy computation:} specifying independence over Fourier space through BIFS means that optimization is based on a large set of low-dimensional problem (as opposed to a single high-dimensional problem).
\item[c)] \emph{modular structure:} allows for relatively straightforward changes in prior model.
\item[d)] \emph{resolution invariance in model specification:} BIFS allows for simple, generic specification of the prior, independent of image resolution.
\item[e)] \emph{straightforward specification of isotropic models:} BIFS allows for consistent specification of the prior in different directions, even when pixel (or voxel) dimensions are not isotropic.
\item[f)] \emph{ability to determine fast (and if desired, spatially isotropic) approximations to Bayesian MRF priors:} BIFS priors can be generated to mimic the behavior of many traditional prior models, providing users experienced with using MRFs with fast (non-iterative) counter-parts that can be implemented via BIFS representations.
\end{description}
\section{BIFS Modeling Framework}
Consider $x$ to be the true (or idealized/enhanced) image that we wish to recover from a degraded or sub-optimal image dataset $y$.
Instead of the conventional Bayesian image analysis approach of generating prior and likelihood models for the true image $x$ based on image data $y$ directly in terms of pixel values, we formulate the models via their discrete Fourier transform representations: $\mathcal{F} x$ and $\mathcal{F} y$. Using Bayes' Theorem, the posterior, $\pi(\mathcal{F} x | \mathcal{F} y)$, is then,
\begin{equation}
\pi(\mathcal{F} x | \mathcal{F} y) \propto \pi(\mathcal{F} y | \mathcal{F} x) \pi(\mathcal{F} x) \enspace.
\end{equation}
The key aspect of the BIFS formulation that leads to its useful properties of easy specification and computational speed, is that we specify both the prior and likelihood (and therefore the posterior) to consist of a set of independent processes over Fourier space locations. In order to induce spatial correlation in image space, the parameters of the prior distributions are specified so as to change in a systematic fashion over Fourier space;
independent (but heterogeneous) processes in Fourier space are thereby transformed into spatially correlated processes in image space~\citep{zeger1985exploring,lange1997non,peligrad2006central}. Heuristically, the realized signal at each position in Fourier space corresponds to a spatially correlated process in image space (at one particular spatial frequency). In general therefore, linear combinations of these spatially correlated signals (such as that given by the discrete Fourier transform) will also lead to a correlated process in image space. This independence-based specification over Fourier space can be contrasted with the conventional Bayesian image analysis approach of using Markov random field (MRF) priors for imposing spatial correlation properties, where the Markovian neighborhood structures are used to induce correlation patterns across pixels via joint or conditional distributional specifications \citep{besag1974spatial,geman1984stochastic,besag1989digital}. In fact, as we discuss in Section \ref{MRImatch} in certain instances MRF models exactly correspond to uncorrelated processes in Fourier space.
When specifying a spatially correlated prior in image space via a set of independent processes across Fourier space, the full conditional posterior at a Fourier space location
$k= (k_x,k_y) \in [-\pi, \pi)^2$, or for volumetric data $(k_x,k_y,k_z) \in [-\pi, \pi)^3$, now only depends on the prior and likelihood at that same Fourier space location $k$, i.e.,
\begin{equation}
\pi(\mathcal{F} x_k | \mathcal{F} y) = \pi(\mathcal{F} x| \mathcal{F} y_k) \propto \pi(\mathcal{F} y_k| \mathcal{F} x_k) \pi(\mathcal{F} x_k) \enspace,
\end{equation}
where we use $\mathcal{F} x_k$ as shorthand for $(\mathcal{F} x)_k$. The joint posterior density for the image is then
\begin{equation}
\pi(\mathcal{F} x| \mathcal{F} y) \propto \prod_{k \in K} \pi(\mathcal{F} y_k| \mathcal{F} x_k) \pi(\mathcal{F} x_k) \enspace,
\end{equation}
where $K$ is the set of all Fourier space point locations in the (discrete) Fourier transformed image. Note that for our purposes we index Fourier space along direction $v \in \{x,y,z\}$ by $\{-N_v/2,\ldots,0,1,\ldots,N_v/2 - 1\}$, rather than the common alternative of, $\{0,\ldots,N_v-1$\}, as it leads to a more convenient formulation for specifying BIFS prior models, i.e. such that they are centered at the zero frequency position of Fourier space. Furthermore, in order to account for the fact that (most) images are in practice real-valued, the Fourier transform must be conjugate (Hermitian) symmetric on the plane (or volume if 3D). A real-valued image output is ensured by considering a realization of the posterior distribution to be determined by the half-plane (half volume), the other half being conjugate symmetric to the first (see \citet{liang2000principles}, pp. 31 and 322). Therefore, for real-valued images, the BIFS posterior is only evaluated over half of Fourier space (and points on the line $x=0$ if taking half-plane in the $y$-direction or conversely $y=0$ for half-plane in the $x$-direction) and the remainder is obtained by conjugate reflection.
In defining priors as a process over Fourier space we are restricting the space of possible priors to generally stationary processes, similar to MRFs with neighborhood structure wrapped on the torus. (The ``generally'' qualifier is because of the very specific exception that the BIFS priors can be specified as non-stationary with respect to identification of the overall mean by placing an improper uniform prior at k-space point $(0,0)$ for the modulus, i.e., leading to models with the same property as the intrinsic pairwise MRF priors. In practice, for image analysis problems where the goal is to enhance features, this restriction to stationarity is minimal except toward the edges of an image; and the effects at the edges can be mitigated by expanding the field of view of the data (e.g. by setting pixel values in the expanded edges to the overall image mean or to an expanded neighborhood mean). Furthermore, in many medical imaging applications the area of interest is far from the edges of the field of view and much of the boundary corresponds to regions outside of the body and therefore the intensity levels are flat toward the edges.
\subsection{The BIFS prior} \label{methods}
A two-step process is used to specify the BIFS prior distribution over Fourier space.
First, the distributional form of the prior for the signal intensity at each Fourier space location is specified, i.e., $\pi(\mathcal{F} x_k)$.
Second, the parameters of each of the priors are specified at each Fourier space location using \emph{parameter functions}.
In order to specify the parameter values across all Fourier space locations simultaneously, we specify a parameter function over Fourier space that identifies the value of each parameter at each Fourier space location. Specifically, for some parameter $\alpha_k$ of $\pi(\mathcal{F} x_k)$ we choose the parameter function $f_{\alpha}$ such that the $\alpha_k = f_{\alpha}(k)$. For most problems in practice it is desirable to choose a spatially isotropic prior, which can be induced by specifying $\alpha_k = f_{\alpha}(|k|)$, where $|k| = \sqrt{k_x^2 + k_y^2}$ in 2D or $\sqrt{k_x^2+k_y^2+k_z^2}$ in 3D, i.e., such that $f$ only depends on the distance from the origin of Fourier space. In the remainder of this paper, description is given in terms of 2D analysis, but notational extension and application to 3D volumetric imaging is straightforward.
\subsection{The Parameter Functions:}
In order to control spatial characteristics of the prior and likelihood we develop the concept of a \emph{parameter function}. The parameters of the independent priors over Fourier space are specified according to parameter functions that describe the pattern of parameter values over Fourier space (see illustrations of Figure~\ref{parfn}); the parameter function traces out the values of parameters for the prior over Fourier space. In general, the parameter function is multivariate, with one dimension for each parameter of the prior distribution used at each Fourier space location, e.g. Figure~\ref{parfn} illustrates parameter functions for distributions with two parameters (scale and location). Note that for some models it proves more convenient to specify parameter functions for transformations of the original distribution parameters; e.g. for the gamma distribution the parameter functions might be defined for the mean and variance which can then be transformed to the shape and scale (or rate) parameters.
Separate and independent priors, and associated parameter functions, are specified for each of the modulus and argument of the complex value at each Fourier space location. Working with the modulus and argument provides a more convenient framework for incorporating prior information at specific Fourier space locations (i.e., specific spatial frequencies) than working with the real and imaginary components. The convenience arises because prior information (e.g., expected characteristics of smoothness, edges, or features of interest) can be directly specified via the modulus of the process, with the argument being treated independently of the modulus. However, real and imaginary components are more difficult to specify since signal can shift between them through the translation of objects in the corresponding image space; for example, a rigid movement of an object in an otherwise constant intensity image will cause shifts between real and imaginary components (by the Fourier transform shift Theorem) whereas in the modulus/argument specification it will only change the argument of the signal.
\begin{figure}
\caption{Schematic of parameter function set up for the signal modulus of the signal where the distribution at each Fourier space location requires specification of a location parameter, $\mu$, and scale parameter, $\sigma$. Panel~(\subref{fig:fig_a}
\label{parfn}
\end{figure}
\subsection{Priors and parameter functions for signal modulus}
\subsubsection*{Modulus prior form:}
Any continuous distribution can be used to represent the prior at each point in Fourier space. It is also possible to allow the distribution itself to differ in different regions of Fourier space, though we do not pursue that here except to allow for a different distributional form at the $k = (k_x,k_y) = (0,0)$ spatial frequency (corresponding to the mean intensity). However, there is an advantage to specifying priors that only have mass for non-negative real numbers; if priors are chosen that can take negative values then a switch from positive to negative corresponds to a $\pi$ discontinuous change in the argument which would typically lead to an overall representation of prior beliefs that is unrealistic. In practice, we find that the choice for the form of the parameter function (discussed below) has a larger influence on the spatial characteristics of the prior than the choice of prior distribution model. For the purpose of computational expedience it therefore often make sense to choose conjugate priors to the likelihood for the modulus when available. For example, we could use a Gaussian (normal) prior for the location parameter of a lognormal likelihood with known scale parameter for the noise process; specification of noise parameters will be discussed in Section~\ref{constvar}.
\subsubsection*{Modulus parameter function forms:}
The parameter functions for the modulus are specified as a set of 2D functions over Fourier space (in terms of $k_x$ and $k_y$): often one for each of location (e.g. mean) and scale (e.g. standard deviation, sd), or/and any other parameters of the prior.
The center of Fourier space, i.e., the $k = (k_x,k_y) = (0,0)$ frequency, is the prior for the overall image intensity mean. At this location, it is reasonable to choose a different pdf than at all other locations. Indeed, the prior at $k = (k_x,k_y) = (0,0)$ can be modeled as improper, e.g. uniform on the real line, leading to an intrinsic, non-stationary prior for the image \citep{kunsch1987intrinsic,besag1991bayesian}.
In general, useful priors across Fourier space are generated by allowing the parameter function for the location parameter to decrease with increasing distance from the center of Fourier space. In our experience, the functional form of the descent has a major impact on the properties of the prior model; much more so than the form of the pdf chosen for each Fourier space location. In addition, specific ranges of frequencies can be accentuated by increasing the parameter function for the location parameter over those frequencies, leading to enhanced spatial frequency bands as a function of distance from the origin.
When considering how to define the modulus parameter function for the scale parameter, we find that setting it to be proportional to that for the mean is often a good strategy though the method allows for different strategies wherever warranted.
Other parameter functions (i.e. parameter functions that do not represent location or scale) may be intuitively more difficult to specify. We do not consider any such functions here, as we find priors with one or two parameters at each Fourier space location to be satisfactory for problems we have encountered.
One situation where a parameter function for something other than the location and scale parameters might be useful would be when mixture distribution priors are appropriate. In particular, a mixture of two distributions could be considered where one of the distributions consists of a probability mass specified at zero. The variation in probability mass for zero at each Fourier space location can itself be specified by a parameter function. For example, one might want to specify increased probability of exact zeros the further away a point is from the origin of Fourier space. Such a prior/parameter function combination would have the effect of encouraging more sparsity at Fourier space locations further from the origin. If high enough mass were specified for zeros across Fourier space this could lead to sparsely represented reconstructions that could in turn be useful when storage space is an issue. The end result would provide lasso style shrinkage in combination with prior information about spatial properties.
\subsection{Priors and parameter functions for signal argument}
Generally, we have limited prior knowledge about the argument of the signal; the argument is related to the relative positioning of objects in the image; moving (shifting) objects around in an image will change the argument of related frequencies. We therefore specify the prior for the argument to reflect \emph{a~priori} ignorance by using an \emph{i.i.d.\,} uniform distribution on the circle at all Fourier space locations, i.e., we define flat parameter functions representing $-\pi$ and $\pi$ as the parameters for $U(a,b)$ distributed arguments at all Fourier space locations. A notable exception to using uniform priors for the argument occurs when the prior is being built empirically from a database of images; we discuss this scenario in our final (simulation) example of Section~\ref{simstudy}.
\subsection{BIFS likelihood}
Similar to the prior, the BIFS likelihood is modeled separately for the modulus and argument of the signal at each Fourier space location. Because we model based on independence across Fourier space points, a range of different noise structures (specified in Fourier space) can readily be incorporated into the likelihood $\pi(\mathcal{F} y_k| \mathcal{F} x_k)$.
The parameter(s) of the likelihood needs to be provided or estimated for the BIFS algorithm. A straightforward approach to this estimation is to extrapolate any areas of the original image that are known to consist only of noise to an image of equal size to the original image, and then Fourier transform to estimate the corresponding noise distribution in Fourier space.
In practice, not knowing the noise standard deviation is not a major impediment to proceeding with Bayesian image analysis. The noise in the image and the precision in the prior trade off with one another in the Bayesian paradigm and therefore the parameter functions of the prior can be adjusted to produce a desired effect \emph{a posteriori}. This \emph{ad hoc} approach is common to much Markov random field prior modeling in Bayesian image analysis; the appropriate setting of hyper-parameter values is a difficult problem in general and often they are left as parameters to be tuned by the user \citep{sorbye2014scaling}.
\subsection{Posterior estimation}
Posterior estimation in conventional Bayesian image analysis tends to focus on MAP estimation (i.e., minimizing a $0-1$ loss function) primarily because it is usually the most computationally tractable. In the BIFS formulation the MAP estimate can be efficiently obtained by independently maximizing the posterior distribution at each Fourier space location, i.e, $x_{\mathrm{\tiny MAP}} = \mathcal{F}^{-1}(\mathcal{F} x_{\mathrm{\tiny MAP}})$ where $\mathcal{F} x_{\mathrm{\tiny MAP}} = \{\mathcal{F} x_{k, \mathrm{\tiny MAP}}, k = 1, \ldots, K \} $ and $ \mathcal{F} x_{k, \mathrm{\tiny MAP}} = \max_{\mathcal{F} x_k} \left\{ \pi(\mathcal{F} x_k | \mathcal{F} y) \right\} = \max_{\mathcal{F} x_k} \left\{ \pi(\mathcal{F} x_k | \mathcal{F} y_k) \right\} $. This contrasts with conventional Bayesian image analysis, where even the most computationally convenient MAP estimates typically require iterative computation methods such as conjugate gradients or expectation-maximization. Beyond the MAP estimate, it is straightforward to simulate from the posterior of BIFS models to get mean estimates such as minimum mean squares estimate (MMSE) estimates \citep{winkler1995image} or other summaries of samples from the posterior. The independence of posterior distributions over Fourier space implies that simple Monte Carlo simulation is all that is required to obtain posterior samples. This is in contrast to conventional Bayesian image analysis, where some form of Markov chain Monte Carlo (MCMC) simulation is typically needed: leading to issues of chain convergence and mixing that need to be dealt with \citep{gilks1996markov}. The general computational approach to implementing BIFS follows that of Algorithm~\ref{bifsalg}.
Note that, when an uninformative uniform prior is used for the argument, and the likelihood is symmetric about the observed argument in the data, the corresponding maximum of the posterior at that Fourier space point is simply the argument of the Fourier transformed data at that point. Therefore, under these conditions, the exact form of the likelihood is unimportant for the MAP estimate. This leads to added simplicity for obtaining the MAP image estimate and is particularly beneficial when working with Gaussian noise in image space where the corresponding distribution for the argument in Fourier space is difficult to work with given the lack of analytical solution (See Section~\ref{ricelik}). \label{likarg}
\begin{algorithm}
\caption{General BIFS implementation} \label{bifsalg}
\begin{algorithmic}
\State Fast Fourier transform (FFT) image data, $y$, into Fourier space, $\mathcal{F} y$
\State Specify noise distribution/likelihood in Fourier space $\pi(\mathcal{F} y_k | \mathcal{F} x_k)$
\State Specify prior distribution form $\pi(\mathcal{F} x_k)$
\State Specify parameter functions for each of modulus and argument of the signal
\ForEach{$k \in K$, }
\State Obtain $\pi(\mathcal{F} x | \mathcal{F} y) \propto \pi(\mathcal{F} y | \mathcal{F} x) \pi(\mathcal{F} x)$
\State Generate posterior estimates/summaries/simulations at each $k$ via MAP or Monte Carlo
\EndFor
\State Inverse FFT posterior estimates/summaries/simulations back to image space.
\end{algorithmic}
\end{algorithm}
\subsection{Modeling Gaussian \emph{i.i.d.\,} noise in image space} \label{ricelik}
A common model for the noise in images is to assume that the image intensities are contaminated by \emph{i.i.d.\,}~Gaussian noise. Although \emph{i.i.d.\,} Gaussian noise in image space transforms to complex Gaussian noise in Fourier space with independent real and imaginary components, when the real and imaginary components are transformed to modulus and argument, the associated errors are not Gaussian distributed.
The corresponding likelihood model for the modulus is the Rician distribution which takes the following form \citep{rice1944mathematical,rice1945mathematical,gudbjartsson1995rician,rowe2004complex,miolane2017template}:
\begin{equation}
\pi(r|\rho,\sigma) = \frac{r}{\sigma^2} \exp \left(- \frac{r^2 + \rho^2}{2 \sigma^2} \right) I_0 \left( \frac{r \rho}{\sigma^2} \right); \qquad r, \rho, \sigma \ge 0 \label{ricelikform}
\end{equation}
where $I_0(z)$ is the modified Bessel function of the first kind with order zero, $\sigma$ is the standard deviation of the real and imaginary Gaussian noise components in Fourier space (\emph{i.i.d.\,} real and imaginary parts), $r$ is the observed modulus of the signal, i.e., $\operatorname{Mod} (\mathcal{F} y_k)$ in the BIFS formulation, and $\rho$ is the noise-free modulus, $\operatorname{Mod} (\mathcal{F} x_k)$.
Note that in the Rician likelihood, the standard deviation of each of the real and imaginary components is the $\sigma$ parameter of the Rician distribution. Therefore, by obtaining an estimate of the noise level in image space, the $\sigma$ parameter over Fourier space can itself be estimated by dividing the estimated standard deviation of the noise in image space by $4$. \label{constvar}
The corresponding likelihood for the argument takes the form \citep{gudbjartsson1995rician,rowe2004complex}:
\begin{equation}
\pi(\psi|\rho,\theta,\sigma) = \frac{\exp \left(-\frac{\rho^2}{2 \sigma^2}\right)}{2 \pi}\left[1 + \frac{\rho}{\sigma} \cos(\psi - \theta) \exp \left(\frac{\rho^2 \cos^2 (\psi - \theta)}{2 \sigma^2} \right) \int_{r = - \infty}^{\frac{\rho \cos (\psi - \theta)}{\sigma}} \exp \left( - \frac{z^2}{2} \right) \, dz \right] \label{ModPhase}
\end{equation}
where $\psi \in [-\pi, \pi)$ is the observed argument of the signal, $\operatorname{Arg} (\mathcal{F} y_k)$, and $\theta \in [-\pi, \pi)$ is the noise-free argument, $\operatorname{Arg} (\mathcal{F} x_k)$.
\subsubsection*{Posterior estimation for \emph{i.i.d.\,} Gaussian noise in image space}
\textbf{Modulus MAP estimate:} If one is interested in the MAP image estimate (i.e. the image associated with minimizing the $0-1$ loss function of the posterior) then the mode of the posterior for the modulus at each Fourier space location needs to be estimated. A first approach might be to consider a direct off-the-shelf optimization, but we have found this to be problematic. The problem is that this is a non-trivial optimization (at least in the Rician case) because at many Fourier space locations the prior and likelihood can be highly discordant, i.e. the modes of the prior and the likelihood can be very far apart with very little density in between for both distributions. This discordance is not entirely surprising and exists for the same reason that typical conventional image analysis priors are not full representations of prior beliefs for an image, but instead typically only represent expected local characteristics such as smoothness of the image \citep{besag1993toward,green1990bayesian}. In practice, we experience direct numerical optimization to break down in extremely discordant cases. \label{discord}
We therefore propose the following approach to MAP estimation with the Rician likelihood (we drop the $k$ subscript for location in Fourier space to aid clarity). Take the Rician likelihood given in Equation (5) and multiply it by the prior for the modulus $\pi(\rho)$. The posterior is $\pi(\rho|r,\sigma) \propto \pi(r|\rho,\sigma) \pi(\rho)$ and we can take logs to simplify, drop constant terms, and find the maximum. Finally, take the second derivative to check that it is concave by bounding the derivative of the Bessel function.
For example, if we assume an exponential prior for $\rho$ i.e. $\pi(\rho) \propto \exp \left(- \frac{\rho}{m} \right)$, take logs and simplify we get
\begin{equation*}
\log \pi(\rho | r, \sigma) = c + \log \left( \frac{r}{\sigma^2} \right) - \frac{r^2+\rho^2}{2\sigma^2} + \log \left[I_0 \left( \frac{r \rho}{\sigma^2} \right) \right] -\frac{\rho}{m}
\end{equation*}
where $c$ is a constant term. Now differentiate w.r.t. $\rho$ and set to $0$
\begin{equation*}
- \frac{\rho}{\sigma^2} - \frac{1}{m} + \frac{r I_1 \left( \frac{r \rho}{\sigma^2} \right)}{\sigma^2 I_0 \left( \frac{r \rho}{\sigma^2} \right)} = 0
\end{equation*}
where $I_0(z)$ is the modified Bessel function of the first kind with order one, and define
\begin{equation*}
b(\rho) = \frac{I_1 \left( \frac{r \rho}{\sigma^2} \right)}{I_0 \left( \frac{r \rho}{\sigma^2} \right)}
\end{equation*}
to get
\begin{equation} \label{exprho}
\rho = r b(\rho) - \frac{\sigma^2}{m}
\end{equation}
The posterior estimate of $\rho$ can then be estimated quickly through iteration. Start with $\rho_0=r$ and then iterate Equation~\ref{exprho} repeatedly as $\rho_{n+1} = r b(\rho_n) - \frac{\sigma^2}{m}$ to compute the MAP estimate of $\rho$. In practice this requires only a few iterations to get high accuracy. Note that the unconstrained maximum of the function may occur at negative values of $\rho$. In that case the posterior maximum for $\rho$ is set to $0$ because the function is monotonically decreasing to the right of the maximum. A similar iterative approach can be applied when considering other prior distributions for $\rho$.
\textbf{Argument MAP estimate:} Note that for a noise process with a uniformly random argument on the circle (as for \emph{i.i.d.\,} noise in real and imaginary components), the likelihood at a Fourier space point has highest density at the argument of the data at that Fourier space location, i.e. $\operatorname{Arg}(\mathcal{F} y_k)$. Therefore, since we adopt a uniform prior for the argument on the circle, then the argument corresponding to the maximum of the posterior is also $\operatorname{Arg}(\mathcal{F} y_k)$. This property will always be true when using the uniform prior on the circle for the Argument provided the likelihood for the argument is symmetric about its maximum; this condition will be necessarily true for all noise processes with uniformly random argument.
\subsection{Example 1 -- Smoothing/denoising} \label{denoise}
Figure~\ref{mandrillpics} shows the BIFS MAP reconstruction results of a grayscale test image of a Mandrill monkey face using the exponential prior, Rician likelihood, and a parameter function for the exponential distribution mean of the form $f_{\mu}(|k|; a, b) = a/|k|^b$ (inverse exponentiated distance) at all locations except for the origin, $k=(0,0)$, where the prior was an improper uniform distribution on the real line. The top-left panel~(a) shows the original noise-free image and the top-middle panel~(b) shows the same image with added Gaussian noise (zero mean with SD $\approx$ one third of the dynamic range of the original image). The noisy image in panel~(b) is used as the input degraded image into the BIFS models. The remaining panels show BIFS MAP reconstructions for different values of $b$, namely (c)~$b=1.5$, (d)~$b=1.75$, (e)~$b=2$, and (f)~$b=2.5$. The parameter value for $a$ was chosen based on matching the power of the parameter function (sum of square magnitude) to that in the observed data over all Fourier space points other than at $k=(0,0)$. The image intensities in each panel are linearly re-scaled to use the full dynamic range. Re-scaling is appropriate in situations where image features are of interest, but not if quantification of intensities is of interest.
\begin{figure}
\caption{\hspace*{0.28 cm}
\label{man_a}
\caption{\hspace*{0.28 cm}
\label{man_b}
\caption{\hspace*{0.28 cm}
\label{man_c}
\caption{\hspace*{0.28 cm}
\label{man_d}
\caption{\hspace*{0.28 cm}
\label{man_e}
\caption{\hspace*{0.28 cm}
\label{man_f}
\caption{Mandrill pics, (a) original image; (b) Gaussian noise degraded image; (c) BIFS MAP with $b=1.5$; (d) $b=1.75$; (e) $b=2$; (f) $b=2.5$. \label{mandrillpics}
\label{mandrillpics}
\end{figure}
In examining the MAP reconstructions of Panels~(c)~to~(e) it is clear that the overall level of smoothness increases (and noise suppressed) as the value of $b$ increases. This is to be expected since higher $b$ implies faster decay of the parameter function for the signal modulus with increasing distance from the origin. However, as a price for higher noise suppression, finer level features are lost. For example, by the $b=2.5$ the whiskers and even nostrils of the Mandrill are smoothed away.
\textbf{Robustness to heavy-tailed noise:} In order to examine whether the BIFS reconstructions were robust to noise distributions with heavier tails, the BIFS procedure was repeated for data with added noise generated from Student $t$-distributions controlled to have the same SD but with different degrees-of-freedom (d.f.). Figure \ref{Tmandrillpics} displays the results of the reconstructions using the same priors as above with $b = 2$ and with d.f. of (a)~10~d.f.; (b)~5~d.f.; and (c)~3~d.f. The reconstructions are visually quite similar to each other with slight differences only showing up with the very heavy-tailed 3~d.f. reconstruction. Results from other values of $b$ were also similarly robust.
\begin{figure}
\caption{\hspace*{0.28 cm}
\label{tman_10}
\caption{\hspace*{0.28 cm}
\label{tman_5}
\caption{\hspace*{0.28 cm}
\label{tman_3}
\caption{Mandrill pics reconstructed under wrong noise distribution (Student-$t$) all with $b=2$, (a) 10 d.f.; (b) 5 d.f.; (c) 3 d.f. \label{Tmandrillpics}
\label{Tmandrillpics}
\end{figure}
\subsection{Example 2 -- Frequency enhancement}
The example in Figure~\ref{moon1} displays a range of BIFS reconstructions for a grayscale test image of a surface patch on the moon. The reconstructions are again focused on using an exponential prior with Rician likelihood for the parameter function of the signal modulus. Panel~(a) shows the original image and Panel~(b) shows the \emph{i.i.d.\,} additive Gaussian noise degraded image serving as the image to apply reconstruction. Panel~(c) displays the BIFS reconstruction based on applying the denoising prior parameter function used in Example~1 with $b=2$. Panels~(d) through~(f) show frequency selective priors for which prior weight is only given to Fourier space locations within a specific range of distances from the origin (\emph{frequency selective torus parameter function}); these priors are also smoothed by an isotropic Gaussian spatial kernel with SD of 1.5 Fourier space pixels in each of the $k_x$ and $k_y$ directions. For Panel~(d) distances of 1 to 5 pixels from the origin are given non-zero mass; in Panel~(e) 10.01 to 15 pixels; and Panel~(f) 15.01 to 60 pixels. Panels~(g) through (i) show corresponding reconstructions where the parameter function is a weighted average of the torus parameter function directly above and the denoising parameter function of panel~(C); weighted at 90\% torus and 10\% denoising. In all of the examples, the level of each of the parameter functions is adjusted to approximately match total power to that observed in the image data.
It is clear that the different torus parameter functions are providing results as expected in terms of focusing in on specific frequency windows. However, when mixed with other parameter functions such as the denoising prior useful compromises between the parameter function forms can be achieved. The addition of the denoising component in Panels~(g) through~(i) softens the harsh restriction to the specific frequency ranges. For example, while in Panel~(d) the reconstruction is highly blurry, the reconstruction of Panel~(g) allows enough higher resolution information to make the image less obviously blurry to view while still enhancing low frequency features. For the high range of frequencies of Panel~(f) it is clear that the addition of the denoising component in Panel~(i) is critical to be able to even understand the context of the image. The effect of combining the high-frequency selective prior with the denoising prior is to enhance smaller features. In particular, notice that this prior is able to identify the small crater that is pointed to by the top arrow in Panel~(i) which is missed by the other priors with less high frequency information. Notice also that even though it is observable in panel~(f) it is not easily distinguishable as a different type of object than the white spot at the middle of the dip in the large crater at the bottom indicated by the bottom arrow.
Clearly, the design of these parameter functions is critical to the properties of the prior in a similar way to how the clique penalties of Markov random field priors operate. However, it is difficult to see how clique functions could be defined to produce priors with the properties of the frequency selective priors or their mixtures with denoising priors.
\begin{figure}
\caption{\hspace*{0.28 cm}
\label{moon_a}
\caption{\hspace*{0.28 cm}
\label{moon_b}
\caption{\hspace*{0.28 cm}
\label{moon_c}
\caption{\hspace*{0.28 cm}
\label{moon_d}
\caption{\hspace*{0.28 cm}
\label{moon_e}
\caption{\hspace*{0.28 cm}
\label{moon_f}
\caption{\hspace*{0.28 cm}
\label{moon_g}
\caption{\hspace*{0.28 cm}
\label{moon_h}
\caption{\hspace*{0.28 cm}
\label{moon_i}
\caption{Moon pics, (a) original image; (b) Gaussian noise degraded image; (c) BIFS MAP estimate using denoising prior from Section~\protect\ref{denoise}
\label{moon1}
\end{figure}
There is clearly considerable potential for designing parameter function / prior combinations for BIFS that can produce a range of image processing characteristics that are not readily achievable with MRF priors.
\subsection{Example 3 -- edge detection}
The family of parameter functions described in the previous example can also prove useful for edge detection in images as illustrated in this example using a Gaussian noise contaminated version of the standard grayscale pirate test image. The sequence of images Figure~\ref{pirate1} (d) to (f) shows how the removal of low frequency information isolates edge information. The upper limit on frequencies is chosen as a trade off between capturing the highest frequency edge information vs. potentially masking the edge information if there is too much high frequency noise, Figure~\ref{pirate1} (g) to (i).
\begin{figure}
\caption{\hspace*{0.28 cm}
\label{pirate_a}
\caption{\hspace*{0.28 cm}
\label{pirate_b}
\caption{\hspace*{0.28 cm}
\label{pirate_c}
\caption{\hspace*{0.28 cm}
\label{pirate_d}
\caption{\hspace*{0.28 cm}
\label{pirate_e}
\caption{\hspace*{0.28 cm}
\label{pirate_f}
\caption{\hspace*{0.28 cm}
\label{pirate_g}
\caption{\hspace*{0.28 cm}
\label{pirate_h}
\caption{\hspace*{0.28 cm}
\label{pirate_i}
\caption{Pirate pics, (a) original image; (b) Gaussian noise degraded image; (c) BIFS denoising $b=2$; (d) BIFS 10.01 - 50; (e) 15.01 - 50; (f) 20.01 - 60; (g) 20.01 - 30; (h) 20.01 - 100; (i) 20.01 - 200. \label{pirate1}
\label{pirate1}
\end{figure}
\section{BIFS Properties}
There are multiple properties of the BIFS formulation that prove advantageous relative to conventional MRF-priors and other Bayesian image analysis models:
\subsection{Computational Speed:} \label{compspeed}
The independence property of the BIFS formulation generally leads to improved computational efficiency over MRF-based or other conventional Bayesian image analysis models that incorporate spatial correlation structures into the priors in image space.
For MAP estimation, Bayesian image analysis using MRF priors requires high-dimensional iterative optimization algorithms such as iterated conditional modes (ICM) \citep{besag1989digital}, conjugate gradients\\ \citep{hestenes1969multiplier}, or simulated annealing \citep{geman1984stochastic}. Iterative processes are necessary when working in the space of the images because of the inter-dependence between pixels. In contrast, BIFS simply requires independent posterior mode estimation at each Fourier space location, and the inverse discrete Fourier transform of this set of Fourier space local posterior modes will provide the global posterior mode image. This independent-over-Fourier-space optimization approach also scales well with respect to increasing image size, increasing dimensionality (e.g. to 3D), or increasing prior model complexity. The level of computational complexity is basically $n g(k)$, where $g(k)$ is the complexity of the optimization at each Fourier space point. Therefore, for increased image size or dimension, the computation time is scaled by the proportional increase increase in the number of Fourier space points, whereas for increased complexity at each Fourier space point to $h(k)$ the total computation is simply scaled by $\frac{h(k)}{g(k)}$.
Computational improvements can similarly be obtained if one wishes to perform posterior mean or other estimation and credible interval generation. (Note that one has to be careful interpreting credible intervals for Bayesian image analysis models since the prior models are at best only rough approximations about characteristics of prior beliefs.) Markov random field prior models generally require high-dimensional Markov chain Monte Carlo (MCMC) posterior simulations to obtain the MMSE image estimate as the posterior mean. However, BIFS only requires at most low-dimensional MCMC to be performed at each Fourier space location. A single realization of the posterior parameter set from each Fourier space location can be inverse Fourier transformed to produce an independent realization of an image from the posterior distribution, thereby avoiding the difficulties of dealing with potentially slow mixing chains that often occur in Bayesian image analysis modeling when updating at the pixel-level.
In addition to the algorithmic speed up for MAP and MMSE estimation, the independence property of the Fourier space approach also allows for trivial parallelization because sampling for each Fourier space location can be performed independently. Therefore, acceleration of a factor close to the number of processors available can be achieved up to having as many processors as Fourier space points.
Finally, another aspect in which it is computationally more efficient to work in the BIFS framework comes about because it is much easier to try different prior models defined in Fourier space. In particular, one can very easily change the parameter functions for the prior, which in our experience has the biggest effect on the properties of the prior overall. Changing the form of the prior distribution at each Fourier space location takes more coding effort but is still much less work than writing code for different MRF-prior-based posterior reconstructions; trying different MRF priors (beyond simply changing parameter values) is typically a non-trivial task.
Overall, the computational speed afforded by BIFS, in addition to the potential for massive parallelization and flexibility of parameter functions, has great potential for the advancement of Bayesian image analysis and opens it up to play a larger role in Big Data imaging problems.
\subsection{Resolution invariance}
Specifying the prior in Fourier space leads to easy translation of the priors for changes in image resolution. When increasing resolution in image space by factors of two, the central region of the Fourier transform at higher resolution corresponds very closely to that of the complete Fourier transform at lower resolution; they correspond to the same spatial frequencies within the field of view. The lower resolution image is very close to a band-limited version of the image at higher resolution. Note that this is only approximate as the increased resolution can slightly alter the magnitude of the measured Fourier components whose frequency is significantly lower than that of the resolution. However, this effect can be bounded by the ratio of the maximum resolution to that of the wavelength of the measured frequency so is typically quite small. This small change in magnitude will remain small in the posterior estimate when the parameter function is continuous.
When the resolution increases but not by a factor of two, the same approach of matching parameter functions over the frequency range can still be applied. However, there will no longer be a direct match of points over the lower frequencies and therefore there will be a less perfect match of the prior distributions overall.
The above described BIFS approach to matching over different resolutions contrasts with MRF models, for which in order to retain the spatial properties of the prior at lower frequencies, an increase in resolution would require careful manipulation of neighborhood structure and prior parameters to match spatial auto-covariance structures between the different resolution images~\citep{rue2002fitting}.
\subsection{Isotropy} \label{isotropy}
In order to specify a \textit{maximally isotropic} BIFS prior, all that is required is for the prior to be specified in such a way that the distribution at each Fourier space location only depends on the distance from the center of Fourier space. This can be achieved by defining the parameter functions for the prior completely in terms of distance from the origin in Fourier space, i.e., $\pi(\mathcal{F} x_k) = g(|k|)$, and not the orientation with respect to the center of Fourier space. (The "maximally" qualifier is needed because the prior will be isotropic up to the maximal level afforded by the discrete -- and anisotropic -- specification of Fourier space on a regular, i.e. square, grid.) The relative ease with which isotropy is specified can be contrasted with that of MRF-based priors where local neighborhood characteristics need to be carefully manipulated by increasing neighborhood size and adjusting parameter values to induce approximate spatial isotropy~\citep{rue2002fitting}. For MRFs, pairwise interaction parameters for diagonal neighbors need to be specified differently to horizontal/vertical ones, requiring something of a balancing process to lead to an approximately isotropic process overall. However, for BIFS all that is required to achieve "maximal" isotropy is that the parameter functions are specified such that they only depend on the distance from the origin of Fourier space. Note however, that even for BIFS priors it may still be possible to adjust the parameter function to lead to greater isotropy in practice by tweaking the parameter function to undo anisotropy effects due to Fourier space discretization, though it is not obvious how one might go about achieving this.
Although easy to specify as such, isotropy is clearly not a requirement of the BIFS prior formulation. Anisotropy can be induced by allowing the parameter functions for the prior to vary in different ways with distance from the origin of Fourier space along different directions.
\section{Approximating Markov random fields with BIFS} \label{MRFmatch}
Given that MRF priors have become something of a standard in Bayesian image analysis, it would be useful to generate BIFS models to try and match these commonly used priors. We propose an approach to specifying a prior that is close to an MRF of interest but also potentially "maximally isotropic". The goal is to create BIFS models that are approximate matches, but that have 1)~increased computational ease and speed (by specifying as independent over Fourier space locations); 2)~increased isotropy if desired (i.e., reduced directional preferences); and 3)~resolution invariance.
To find a formulation in Fourier space that approximately matches a corresponding MRF model we propose taking the steps described in Algorithm \ref{simMRFalg}.
\begin{algorithm}
\caption{Simulating MRF models with BIFS} \label{simMRFalg}
\begin{algorithmic}[1]
\State Simulate a set of images from the MRF prior distribution and take the FFT of each image
\State Determine a model for the prior probability distribution of the modulus to be used and independently estimate its' parameters at each Fourier space location
\State Determine an appropriate parameter function form over Fourier space for each of the parameters from the chosen prior based on estimates from the simulated data. (If maximally isotropic approximations to the prior are required then the parameter functions need to be chosen subject to the constraint that it only depends only on distance from the origin of Fourier space)
\State Estimate the coefficients of the parameter function by fitting to the marginally estimated parameters in Fourier space, e.g. via least squares
\end{algorithmic}
\end{algorithm}
Note that it is possible to take this approach when estimating a BIFS approximation to any Bayesian image analysis prior model, though the potential to match higher-level priors (e.g. ones that directly model geometric properties of objects in the image) is likely to be much less of a good approximation.
\subsection{Example 4 - Matching Gaussian MRFs} \label{MRImatch}
We here consider using the above ideas to approximate the simple first-order pairwise difference intrinsic Gaussian MRF (IG-MRF)
\begin{equation*}
\pi(x) \propto \exp \left\{ - \frac{\kappa}{2} \sum_{i \sim j} \left(x_i-x_j \right)^2 \right\}
\end{equation*}
as described in \cite{besag1989digital}. The sum over $i \sim j$ is over all unordered pairs of pixels such that $i$ and $j$ are vertically or horizontally adjacent neighbors in the image. The model is called \textit{intrinsic} because the overall mean is not defined and therefore the prior is improper with respect to the overall mean level.
We used the R-INLA package to simulate 1,000 realizations of a first-order IG-MRF with $\kappa = 1.0$ and first-order neighborhood structure wrapped on a torus. At each Fourier space location we adopted a prior distribution for the modulus such that the square of the modulus is distributed as exponential. This choice of prior falls in line with the theoretical results in \citep{rue2005gaussian}, Ch~2.6 for the specific case of simulating a Gaussian Markov random field with neighborhood structure wrapped on a torus (i.e. with block-circulant precision matrix).
Specifically, an IG-MRF has a precision matrix $\bm{Q}$ that is block circulant, in which case we can show using the analysis in \citep{rue2005gaussian} that the priors in Fourrier space are such that the power at each frequency pair is exponentially distributed. The proof is straightforward but notationally complex so we present the 1 dimensional proof to provide intuition.
The key idea is that if $\bm{Q}$ is a circulant matrix then we can decompose it as
$\bm{Q}=\bm{F\Lambda F}^H$, where $\bm{F}$ is the (discrete) Fourier transform (DFT) matrix, $\bm{F}^H$ is the Hermitian (i.e. conjugate transpose of $\bm{F}$), and $\bm{\Lambda}$ is a diagonal matrix of eigenvalues of $\bm{Q}$. Now we can compute,
\begin{equation*}
\pi(\bm{x}) \propto \exp \left( -\bm{x}^T \bm{Qx} \right)
= \exp \left( -\bm{x}^T \bm{F\Lambda F}^H \bm{x} \right)
= \exp \left( -\bm{F}^T \bm{x} \bm{\Lambda F}^H \bm{x} \right)
= \exp \left( -\bm{F} \bm{x} \bm{\Lambda F}^H \bm{x} \right) \\
\end{equation*}
Now let $f$ be the DFT of $\bm{x}$, $f^\dagger$ the inverse DFT (IDFT) of $\bm{x}$, and $p_k$ the power at frequency $k$, so
\begin{equation*}
\pi(\bm{x}) \propto \exp \left( -f \bm{\Lambda} f^\dagger \right)
= \exp \left( \sum_k - \lambda_k f_k f^\dagger_k \right)
=\exp \left( \sum_k - \lambda_k p_k \right)
=\prod_k \exp \left( -\lambda_k p_k \right)
\end{equation*}
This final term is a product of functions so the distributions at each frequency are independent and each has an exponential distribution.
\begin{equation*}
\pi(p,\theta) \propto \prod_k \exp \left( - \lambda_k p_k \right).
\end{equation*}
To summarize, this shows that for GMRFs with neighborhood structure specified on the torus, the power spectrum (the square of the signal modulus) is made up of independent exponential random variables.
We need to obtain the posterior maximum for the prior where the modulus square follows an exponential distribution coupled with Rician noise, analogous to that of Equation~\ref{exprho} for when the exponential prior was used directly for the modulus.
Assume that $X \sim \textrm{Exp}(1/m)$ for each point in Fourier space and that $P = \sqrt{X}$. Then for any $\rho \ge 0$;
\begin{equation*}
\Pr(P \le \rho) = \Pr(\sqrt{X} \le \rho) = \Pr(X \le \rho^2) = 1 - \exp \left(- \frac{\rho^2}{m} \right) \;\;\;\; \rho \ge 0
\end{equation*}
differentiating, we get the pdf for the prior of $\rho$:
\begin{equation*}
\pi(\rho | m) = \frac{2 \rho}{m} \exp \left( - \frac{\rho^2}{m} \right) \;\;\; \rho \ge 0
\end{equation*}
now applying Bayes' Theorem and taking logs to simplify
\begin{equation*}
\log \pi(\rho | r, \sigma, m) = c - \frac{\rho^2}{2\sigma^2} + \log \left(I_0 \left( \frac{r \rho}{\sigma^2} \right) \right) + \log \left( \frac{2 \rho}{m} \right) -\frac{\rho^2}{m}
\end{equation*}
differentiate w.r.t. $\rho$
\begin{equation*}
\dfrac{\mathrm{d} \log \pi(\rho | r, \sigma, m)}{\mathrm{d} \rho} = - \frac{\rho}{\sigma^2} + \frac{r I_1 \left( \frac{r \rho}{\sigma^2} \right)}{\sigma^2 I_0 \left( \frac{r \rho}{\sigma^2} \right)} + \frac{1}{\rho} - \frac{2 \rho}{m}
\end{equation*}
then set equal to zero and solve to get the positive solution for $\rho$ of
\begin{equation}
\rho = \frac{rm b(\rho) + \sqrt{\left(b(\rho) r m \right)^2 + 8 \sigma^4 m + 4 (\sigma m)^2}}{4 \sigma^2 + 2 m}.
\end{equation}
However, instead of exactly matching the MRF using the eigenvalues to model the IG-MRF in Fourier space as in \citep{rue2005gaussian}, we instead fit an isotropic parameter function to the mean of the modulus at each Fourier space location using least squares. The form of the parameter function used was $$f(|k|; \textbf{a}) = \frac{a_0}{a_1 + a_2 |k| + a_3 |k|^2 + a_4|k|^3}$$ (chosen based on a trial-and-error approach to get a good least squares fit -- See Figure~\ref{fig:GMRFmodFit}).
\begin{figure}
\caption{Parameter model fit for modulus as a function of distance from the origin of Fourier space}
\label{fig:GMRFmodFit}
\end{figure}
Simulations from the BIFS prior were well-matched to their MRF counterpart. Figure~\ref{simMatch} shows example realizations from each of the BIFS and direct MRF simulations in panels~(a) and~(b) respectively and they clearly exhibit similar properties. Panel~(c) shows the respective estimated autocovariance functions (ACFs) as a function of distance from 1,000 simulations of each random field. The spread observed in the estimated ACFs at longer distances is due to the anisotropy of the processes induced by the rectangular lattice and for the MRF because of the anisotropic representation of the neighborhood structure; hence the slightly narrower band for the BIFS simulations (see Section~\ref{isotropy}).
\begin{figure}
\caption{\hspace*{0.28 cm}
\label{gsim_a}
\caption{\hspace*{0.28 cm}
\label{gsim_b}
\caption{\hspace*{0.28 cm}
\label{acf_c}
\caption{Example simulations for the first order intrinsic GMRF using a direct approach in R-INLA in panel~(a) and the BIFS approximation with isotropic parameter function in panel~(b). The estimated ACF as a function of distance for each of the simulated models is given in panel~(c).}
\label{simMatch}
\end{figure}
In order to compare posterior estimates between the two approaches we examine a simulated dataset generated using a version of the Montreal Neurological Institute (MNI) brain \citep{cocosco5online} that had been segmented into gray matter (GM), white matter (WM), and cerebro-spinal fluid (CSF)/outside brain at 128 by 128 resolution. GM is the outer ribbon (the cortex) around the brain where neuronal activity occurs, WM is made up of the connective strands that enable different regions of the cortex to communicate with each other, and CSF is fluid in the brain. The signal was generated with intensity 20.0 in GM, 10.0 in WM and 0.0 elsewhere and is displayed in Panel~(a) of Figure~\ref{MRImatchFig}. Gaussian noise (\emph{i.i.d.\,}) with SD of 2.5 was added to generate a degraded image as displayed in Panel~(b).
The image was reconstructed as a MAP estimate from the degraded data using each of the IG-MRF prior (with conjugate gradients optimization) and BIFS ``maximally isotropic'' equivalent prior approximation. Comparing the IG-MRF and corresponding BIFS reconstructions in Panels~(c) and~(d), respectively, indicates that the MRF and BIFS are indistinguishable visually. This is confirmed when looking at the residual maps in panels~(e) and~(f) which are also indistinguishable. Note that Panels~(a) to~(d) are normalized to be on the same dynamic scale (i.e. such that the same value corresponds to each gray-scale shade). Similarly, Panels~(e) and~(f) are matched to a dynamic scale over the range of the residuals in both images. Of note is that the residuals for both MRFs and BIFS carry considerable residual structure. This is simply a reflection of the nature of the priors in only representing local characteristics as discussed in Section \ref{discord}
\begin{figure}
\caption{\hspace*{0.28 cm}
\label{gmrf_a}
\caption{\hspace*{0.28 cm}
\label{gmrf_b}
\caption{\hspace*{0.28 cm}
\label{gmrf_c}
\caption{\hspace*{0.28 cm}
\label{gmrf_d}
\caption{\hspace*{0.28 cm}
\label{gmrf_e}
\caption{\hspace*{0.28 cm}
\label{gmrf_f}
\caption{BIFS match to IG-MRF prior for perfusion MRI simulation study. (a) simulated brain signal image; (b) image of the same data with added Gaussian noise (c) IG-MRF MAP reconstruction from the noisy data (using conjugate gradients optimization); (d) BIFS MAP reconstruction based on maximally isotropic approximation to IG-MRF; (e) IG-MRF MAP residuals (relative to true signal of Panel~(a)); (f) BIFS MAP residuals. Panels~(a) to~(d) are normalized to be on the same dynamic scale. Similarly, Panels~(e) and~(f) are matched to a dynamic scale over the range of the residuals in both images.}
\label{MRImatchFig}
\end{figure}
The level of similarity between the two reconstructions is further emphasized in Table~\ref{cfMRF}. The table gives a comparison of mean signal estimates in each tissue type along with overall predictive accuracy based on root-mean-square-error (RMSE) over all pixels in the image. The results are very similar comparing IG-MRF and BIFS, indicating that BIFS is doing a good job of approximating the results of IG-MRF. Both models shrink estimates of mean tissue levels toward neighboring tissue types due to the overall smoothing effect of the Gaussian pairwise difference prior. This bias effect is largest in GM where the tissue region is narrow and therefore the conditional distributions within GM regions often include neighbors from other tissue types, which for GM will lead to biasing down the pixel estimates. In fact, this smoothing bias is strong enough to increase the RMSE for the Bayesian reconstructions compared with just using the noisy data. This emphasizes the dangers of blindly using Bayesian image analysis when the goal is estimation.
The results are very slightly better for BIFS than ICM-GMRF in terms of having slightly smaller differences from the true values, though one needs to be careful not to over-interpret this very small difference. The different results are simply due to using slightly different models and one could easily find datasets or/and models that fit better with respect to RMSE in either the MRF or BIFS frameworks. However, it should be noted again, that as mentioned in Section~\ref{compspeed} it is much easier to try different models in the BIFS framework e.g. via simply changing the form of the parameter function than it is to try different MRF priors.
\begin{table}[!ht]
\begin{center}
\caption{Comparison between estimates of mean signal in each tissue type and overall RMSE of reconstructions.}
\label{cfMRF}
\begin{tabular}{l|r|r|r|r}
& \textbf{True} & \textbf{True + noise} & \textbf{IG-MRF} & \textbf{BIFS}\\
\hline
GM & 20.0 & 19.92 & 13.48 & 13.53\\
WM & 10.0 & 9.99 & 11.10 & 11.08\\
CSF/out & 0.0 & -0.01 & 0.54 & 0.54\\
\hline
RMSE & 0.0 & 2.47 & 2.74 & 2.71
\end{tabular}
\end{center}
\end{table}
\section{The data-driven BIFS Prior (DD-BIFS)}
The standard process of generating the BIFS prior distribution described in Section \ref{methods} is based on choosing a pair of distributions to be applied as priors at each location in Fourier space (one for the modulus and the other for the argument of the complex value signal) and a set of parameter functions to define how the parameters of the distributions vary over Fourier space. In contrast, for the data-driven approach the parameters at each Fourier space location are estimated empirically from a database of transformed images. The database of images would be of high-quality and have the characteristics that are required to represent the prior specification.
To estimate the parameters, all of the images in the database are first Fourier transformed, the data at each location in Fourier space are extracted (i.e. for all images), and the distribution parameters for that Fourier space location are estimated from that data. These parameter estimates are then used to define the parameters for the prior at each Fourier space location, i.e. they form the basis of the parameter functions. (Note that when the database is not large it may be more beneficial to fit parameter functions to the empirical data rather than use estimates generated separately at each Fourier space location.)
The implementation of DD-BIFS modeling follows the steps of Algorithm~\ref{DDBIFS}.
\begin{algorithm}
\caption{DD-BIFS modeling} \label{DDBIFS}
\begin{algorithmic}[1]
\State Fast Fourier transform (FFT) all images in the database that are to be used to build the DD-BIFS prior
\State Choose the distributional form of the prior at each location in Fourier space
\State Estimate the parameters of the prior at each location in Fourier space using the data from that Fourier space location across the subjects in the database
\State Scale the sets of parameters over Fourier space to adjust the influence of the prior -- this is the DD-BIFS prior
\State Define the likelihood in Fourier space
\State FFT the dataset to be reconstructed from image space into Fourier space
\State Combine the DD-BIFS prior and likelihood for the image at each Fourier space location via Bayes' Theorem to generate the DD-BIFS posterior
\State Generate the Fourier space MAP estimate by maximizing the posterior at each Fourier space location
\State Inverse FFT the Fourier space MAP estimate back to image space and display
\end{algorithmic}
\end{algorithm}
\subsection{Example 5 - Data-driven prior simulation study} \label{simstudy}
To illustrate the DD-BIFS approach we simulated 10,000 256$\times$256 images containing ellipsoid objects. The number of objects was modeled as a Poisson process; the objects were simulated as randomly positioned 2D Gaussian probability density functions (resembling bumps) with random intensity, and standard deviation on each axis, and correlation between the standard deviations on each axis distributed uniformly between~-1 and~0, i.e., so that the process was not isotropic.
We generated an additional realization (separate to the 10,000 used to build the prior) displayed in Figure~\ref{gal_a} and contaminated it with added Gaussian noise (Figure~\ref{gal_b}). We then performed DD-BIFS with MAP estimation for this single new realization from the process.
For this reconstruction we used a deliberately miss-specified prior and likelihood below to allow a simple illustration using conjugate prior forms. At each Fourier space location $k$ we adopt a truncated Gaussian prior for the modulus: $\operatorname{Mod}(\mathcal{F} x_k) \sim TN(\mu_k, \tau_k^2, 0, \infty)$, with $\mu_k \ge 0$; a Uniform prior on the circle for the argument: $\operatorname{Arg}(\mathcal{F} x_k) \sim U(0,2 \pi)$, a Gaussian noise model for the modulus $\operatorname{Mod}(\epsilon_k) \sim N(0,\sigma^2)$, and a Uniform noise model for the argument $\operatorname{Arg}(\epsilon_k) \sim U(0,2 \pi)$,
where $\epsilon_k$ is the complex noise treated as independent across Fourier space locations $k$.
(Note this model does not correspond to the simulated Gaussian noise in image space, which would require the Rician likelihood used previously.) The values of $\mu_k$ and $\sigma_k$ at each Fourier space location are estimated using the approach outlined in Algorithm \ref{DDBIFS}.
The global posterior mode is then obtained by generating the posterior mean based on conjugate Bayes for the corresponding non-truncated Gaussian prior and likelihood, which is equivalent to the posterior mode and hence the posterior mode of the truncated prior version, at each Fourier space location \citep{gelman2014bayesian} with
\begin{equation*}
x_{k,\mathrm{\tiny MAP}} = \frac{ \left( \frac{m \mu_k}{\tau_k^2} + \frac{y_k}{\sigma^2} \right) } { \left( \frac{m}{\tau_k^2} + \frac{1}{\sigma^2} \right) }
\end{equation*}
Note that the value of $m$ in the prior is specified by the user and can be considered to represent how many observations we want the weight of the prior to count for in the posterior.
Panel~(c) of Figure~\ref{bumps} shows an IG-MRF reconstruction of the noisy data. It clearly denoises the image but at the expense of smoothing the objects in the image. Panels~(d) through (f) shows the a DD-BIFS reconstruction where the database prior is given weight equivalent to 0.1, 1.0 and 10.0 observations respectively. Note that in general for these priors the bumps that are elongated have their length better preserved than in the Gaussian prior case. As the number of observations that the DD-BIFS prior represents increases the features of the true signal begin to diminish and the noise level of the reconstruction is reduced. This makes sense because in the limit we would effectively be obtaining the MAP estimate based on the prior alone which is an average over 10,000 simulations. Note that this high-level capturing of the bump features occurs despite the independence specification in Fourier space; the BIFS formulation is able to capture the anisotropic characteristics of these features through the empirical parameter function. Note also that the DD-BIFS prior could itself be modified by changing the form of the parameter function. For example, if one wanted to diminish the anisotropic features of the background signal the prior could be taken as a function of both the database prior and a denoising parameter function.
\begin{figure}
\caption{\hspace*{0.0 cm}
\label{gal_a}
\caption{\hspace*{0.0 cm}
\label{gal_b}
\caption{\hspace*{0.0 cm}
\label{gal_c}
\caption{\hspace*{0.0 cm}
\label{gal_d}
\caption{\hspace*{0.0 cm}
\label{gal_e}
\caption{\hspace*{0.0 cm}
\label{gal_f}
\caption{Simulation study and reconstruction of anisotropic bump patterns. Panel~(a)~new process realization; (b)~realization with added noise; (c)~first order IG-MRF reconstruction; (d)~DD-BIFS prior equivalent to 0.1~observations (e)~DD-BIFS prior 1~observation; (f)~DD-BIFS for 10~observations. All panels except for Panel~(b) are normalized to be on the same dynamic scale. \label{bumps}
\label{bumps}
\end{figure}
\section{Discussion and Conclusion}
The BIFS modeling framework provides a new family of Bayesian image analysis models with the capacity to a)~enhance images beyond conventional standard Bayesian image analysis methods; b)~allow straightforward specification and implementation across a wide range of imaging research applications; and c)~enable fast and high-throughput processing. These benefits, along with the inherent properties of resolution invariance and isotropy, make BIFS a powerful tool for the image analysis practitioner.
A particular strength of the BIFS approach is the ease with which one is able to try different prior models. Experimenting with different priors might be considered problematic in other fields, but in Bayesian image analysis we typically know in advance that any model we can specify will be wrong and at best can approximately capture some characteristics from the image; if we simulate from Bayesian image analysis priors we would expect to be waiting an extremely long time before we see a realization that would be representative of an object of interest (e.g. a brain or a car). It is therefore often of benefit to try different models until finding a prior that has the desired impact on the posterior. This approach would have additional legitimacy if one reserves images purely for the purpose of formulating a preferred prior before analyzing the images of interest.
The BIFS framework has much potential for future work to expand the foundations presented here: BIFS could be applied to spatio-temporal modeling, multi-image analysis, multi-modal medical imaging, color images, 3D images, other spatial basis spaces such as wavelets, multifractal modeling, non-continuous valued MRFs with hidden latent models, etc. We hope that other statisticians, engineers, and computer scientists with an interest in image analysis will begin to explore these potential areas.
\section*{Code}
All code used in this manuscript is available at: https://github.com/ucsf-deb/BIFSpaper1
\end{document} | math |
begin{document}
\title{The $1$-Eigenspace for matrices in $\mathbb{G}L_2(\mathbb{Z}_\emphll)$}
\alphauthor{Davide~Lombardo and Antonella~Perucca}
begin{abstract}
Fix some prime number $\emphll$ and consider an open subgroup $G$ either of $\mathbb{G}L_2(\mathbb{Z}_{\emphll})$ or of the normalizer of a Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_{\emphll})$. The elements of $G$ act on $(\mathbb{Z}/\emphll^n \mathbb{Z})^2$ for every $n\gammaeqslant 1$ and also on the direct limit, and we call $1$-Eigenspace the group of fixed points. We partition $G$ by considering the possible group structures for the $1$-Eigenspace and show how to evaluate with a finite procedure the Haar measure of all sets in the partition. The results apply to all elliptic curves defined over a number field, where we consider the image of the $\emphll$-adic representation and the Galois action on the torsion points of order a power of $\emphll$.
\emphnd{abstract}
\alphaddress[]{Dipartimento di Matematica, Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, Italy}
\emphmail[]{[email protected]}
\alphaddress[]{Universit\"at Regensburg, Universit\"atsstra{\ss}e 31,
93053 Regensburg, Germany}
\emphmail[]{[email protected]}
\keywords{Haar measure, general linear group, Cartan subgroup, $\emphll$-adic representation, elliptic curve}
\subjclass[2010]{28C10, 16S50, 11G05, 11F80}
\maketitle
\section{Introduction}
Fix a prime number $\emphll$, and let $G$ be an open subgroup of either $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ or the normalizer of a (possibly ramified) Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$. This general framework can be applied to elliptic curves defined over a number field, where $G$ is the image of the $\emphll$-adic representation.
We identify an element of $G$ with an automorphism of the direct limit in $n$ of $(\mathbb{Z}/\emphll^n \mathbb{Z})^2$: for elliptic curves this means considering the Galois action on the group of torsion points whose order is a power of $\emphll$.
We equip $G$ with its Haar measure, normalized so as to assign volume one to $G$, and we compute the measure of subsets of $G$ of arithmetic interest.
For $M \in G$, we call \textit{$1$-Eigenspace of $M$} the subgroup of fixed points of $M$ for its action on the direct limit $\varinjlim_n (\mathbb{Z}/\emphll^n\mathbb{Z})^2$. This leads to partitioning $G$ into subsets according to the group structure of the 1-Eigenspace.
More specifically, the matrices whose $1$-Eigenspace is an infinite group form a subset of $G$ that has Haar measure zero, so we only investigate the possible finite group structures. For all integers $a,b\gammaeqslant 0$ we consider the set
begin{equation*}
\mathcal M_{a,b}:=\{M\in G: \, \ker (M-I) \simeq \mathbb{Z}/{\emphll^a}\mathbb{Z}\times \mathbb{Z}/{\emphll^{a+b}}\mathbb{Z} \}
\emphnd{equation*}
and its Haar measure in $G$, which is well-defined for each pair $(a, b)$ and that we call $\mu_{a,b}$. The aim of this paper is to show that the whole countable family $\mu_{a,b}$ can be effectively computed:
begin{thm}\label{main-thm}
Fix a prime number $\emphll$ and an open subgroup $G$ of either $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ or the normalizer of a (possibly ramified) Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$. It is possible to compute the whole family $\{\mu_{a,b}\}$ for $(a,b)\in \mathbb N^2$ with a finite procedure. More precisely, we can partition $\mathbb N^2$ in finitely many subsets $S$ (as in Definition \ref{admissible} and explicitly computable) such that the following holds: there is some (explicitly computable) rational number $c_S\gammaeqslant 0$ such that for every $(a,b)\in S$ we have
$$\mu_{a,b}= c_S\cdot \emphll^{-(\dim(G) a+b)}$$
where the dimension of $G$ is either $4$ or $2$, according to whether $G$ is open in $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ or in the
normalizer of a Cartan subgroup. The sets $S$ and the constants $c_S$ may depend on $\emphll$ and $G$.
\emphnd{thm}
Some explicit results are as follows:
begin{thm}\label{thm-countGL2}
For $\mathbb{G}L_2(\mathbb Z_{\emphll})$, we have:
$$\mu_{a,b}= \left\{
begin{array}{llll}
\dfrac{\emphll^3-2\emphll^2-\emphll+3}{(\emphll-1)^2 \cdot (\emphll+1)} & \text{if $a=0, b=0$}
\\
\dfrac{\emphll^2-\emphll-1}{\emphll(\emphll-1)}\cdot \emphll^{-b} & \text{if $a=0, b>0$}
\\
\emphll^{-4a} & \text{if $a>0, b=0$}
\\
(\emphll + 1)\cdot \emphll^{-4a-b-1} & \text{if $a>0, b>0$ .} \\
\emphnd{array}\right.$$
\emphnd{thm}
begin{thm}\label{thm-countsplitnonsplitCartan}
For a Cartan subgroup of $\mathbb{G}L_2(\mathbb Z_{\emphll})$ which is either split or nonsplit (see Definition \ref{Cartan}) we respectively have:
$$\mu_{a,b} = \left\{
begin{array}{lllll}
\dfrac{(\emphll-2)^2}{(\emphll-1)^2} & \text{if $a=0, b=0$}
\\
\dfrac{2(\emphll-2)}{\emphll-1} \cdot \emphll^{-b} & \text{if $a=0, b>0$}
\\
\emphll^{-2a} & \text{if $a>0, b=0$} \\
2\cdot \emphll^{-2a-b} & \text{if $a>0, b>0$} \\
\emphnd{array}\right.
\mathfrak{q}quad
\mu_{a,b} = \left\{
begin{array}{lllll}
\dfrac{\emphll^2-2}{\emphll^2-1} & \text{if $a=0, b=0$}
\\
\emphll^{-2a} & \text{if $a>0, b=0$}
\\
0 & \text{if $b>0$} \,. \\
\emphnd{array}\right.
$$
\emphnd{thm}
begin{thm}\label{thm-countnormalizerCartan}
For the normalizer of a split or nonsplit Cartan subgroup $C$ of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ we have
$$\mu_{a,b} =\frac{1}{2} \cdot \mu_{a,b}^{C} + \frac{1}{2} \cdot \mu^*_{a,b}$$
where $\mu_{a,b}^C$ is the Haar measure in $C$ of $\mathcal M_{a,b}\cap C$ (which can be read off Theorem \ref{thm-countsplitnonsplitCartan}) and where we set
$$\mu^*_{a,b}=\left\{
begin{array}{lllll}
\dfrac{\emphll-2}{\emphll-1} & \text{if $a=0, b=0$}
\\
\emphll^{-b} & \text{if $a=0, b>0$}
\\
0 & \text{if $a>0$ .}\\
\emphnd{array}\right.$$
\emphnd{thm}
The Haar measure $\mu_{a,b}$ can computed as the limit in $n$ of the ratio $\# \mathcal{M}_{a,b}(n) / \# G(n)$, where for a subset $X$ of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ the symbol $X(n)$ denotes the image of $X$ in $\mathbb{G}L_2(\mathbb{Z}/\emphll^n\mathbb{Z})$.
For fixed $a$ and $b$, the quantity $\# \mathcal{M}_{a,b}(n) / \# G(n)$ stabilizes for $n$ sufficiently large by the higher-dimensional version of Hensel's Lemma. However, since we cannot fix a single value of $n$ which is good for every pair $(a,b)$, we need technical results about counting the number of lifts of any given matrix in $\mathbb{G}L_2(\mathbb{Z}/\emphll^n\mathbb{Z})$ to $\mathbb{G}L_2(\mathbb{Z}/\emphll^{n+1}\mathbb{Z})$.
The structure of the paper is as follows. In Section \ref{sect:PreliminariesCartan} we define Cartan subgroups of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ in full generality and prove a classification result which might be of independent interest. In Section \ref{sec-1ES} we prove general results about the group structure of the $1$-Eigenspace and set the notation for the subsequent sections. These contain further results, in particular Theorem \ref{thm-lifts} (about the reductions of $\mathcal M_{a,b}$) and the two technical results Theorems \ref{thm:GeneralLift} and \ref{thm:LiftsNormalizer}.
Finally, the last section is devoted to the proof of Theorems 1 to 4. In \cite{LombardoPeruccaRedEC} we apply the results of this paper to solve a problem about elliptic curves:
begin{rem}
Let $E$ be an elliptic curve defined over a number field $K$. If $\emphll$ is a prime number and $E[\emphll^\infty]$ is the group of $\overline{K}$-points on $E$ of order a power of $\emphll$, we have general results and a computational strategy for:
begin{itemize}
\item classifying the elements in the image of the $\emphll$-adic representation according to the group structure of the fixed points in $E[\emphll^\infty]$;
\item computing the density of reductions such that the $\emphll$-part of the group of local points has some prescribed group structure, for the whole family of possible group structures.
\emphnd{itemize}
\emphnd{rem}
\section{Cartan subgroups of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$}\label{sect:PreliminariesCartan}
\subsection{General definition of Cartan subgroups}
Classical references are \cite[Chapter 4]{MR1102012} and \cite[Section 2]{MR0387283}.
Let $\emphll$ be a prime number and $F$ be a reduced $\mathbb{Q}_\emphll$-algebra of degree 2 with ring of integers $\mathcal{O}_F$.
Concretely, $F$ is either a quadratic extension of $\mathbb{Q}_\emphll$, or the ring $\mathbb{Q}_\emphll^2$ (in the latter case we define the $\emphll$-adic valuation as the minimum of those of the two coordinates and by $\mathcal{O}_F$ we mean the valuation ring $\mathbb{Z}_\emphll^2$).
Let furthermore $R$ be a $\mathbb{Z}_\emphll$-order in $F$, by which we mean a subring of $F$ (containing $1$) which is a finitely generated $\mathbb{Z}_\emphll$-module and satisfies $\mathbb{Q}_\emphll R=F$ (i.e. $R$ spans $F$ over $\mathbb{Q}_\emphll$).
The \emphmph{Cartan subgroup $C$ of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ associated with $R$} is the group of units of $R$: the embedding $R^\times \hookrightarrow \mathbb{G}L_2(\mathbb{Z}_\emphll)$ is given by fixing a $\mathbb{Z}_\emphll$-basis of $R$ and considering the left multiplication action of $R^\times$. The Cartan subgroup is only well-defined up to conjugation in $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ because of the choice of the basis. Writing $C_R:=\operatorname{Res}_{R/\mathbb{Z}_\emphll}(\mathbb{G}_m)$, where $\operatorname{Res}$ is the Weil restriction of scalars, we have $C =C_R (\mathbb Z_{\emphll})$, provided that the Weil restriction is computed using the same $\mathbb{Z}_\emphll$-basis for $R$.
Equivalently, a Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ can be described as follows: there exists a maximal torus $\mathcal{T}$ of $\mathbb{G}L_{2,\mathbb{Z}_\emphll}$, flat over $\mathbb{Z}_\emphll$, such that $C=\mathcal{T}(\mathbb{Z}_\emphll)$.
begin{defi}\label{Cartan}
We shall say that the Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ associated with $R$ is:
begin{itemize}
\item \emphmph{maximal}, if $\emphll$ does not divide the index of $R$ in $\mathcal{O}_F$;
\item \emphmph{split}, if it is maximal and furthermore $\emphll$ is split in $F$;
\item \emphmph{nonsplit}, if it is maximal and furthermore $\emphll$ is inert in $F$;
\item \emphmph{ramified}, if it is neither split nor nonsplit.
\emphnd{itemize}
\emphnd{defi}
Notice in particular that \emphmph{unramified} means the same as \emphmph{either split or nonsplit}. Thus a Cartan subgroup is either split, nonsplit or ramified: a Cartan subgroup can be ramified because it is not maximal ($\emphll$ divides $[\mathcal{O}_F:R]$), or because $\emphll$ ramifies in $F$.
Note that we always understand `maximal' in the sense of the above definition (in particular, even if a Cartan subgroup is not maximal, it is still the group of $\mathbb Z_{\emphll}$-points of a maximal subtorus of $\mathbb{G}L_2$). A proper subgroup of a Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ is not a Cartan subgroup in our terminology.
begin{rem}
A strict inclusion of quadratic rings $R \hookrightarrow S$ over $\mathbb{Z}_\emphll$ does not induce an inclusion of Cartan subgroups according to our definition. This is because the multiplication action of $R^\times$ on $R$ (resp.~of $S^\times$ on~$S$) is represented with respect to a $\mathbb{Z}_\emphll$-basis of $R$ (resp.~$S$), and the base-change matrix relating a basis of $R$ with a basis of $S$ is not $\emphll$-integral. More concretely, write $S=\mathbb{Z}_\emphll[\omega]$ and $R=\mathbb{Z}_\emphll[\emphll^k\omega]$ for some $k> 0$. Suppose for simplicity that $\emphll \neq 2$ and $\omega^2=d \in \mathbb{Z}_\emphll$, and consider the bases $\{1,\omega\}$ and $\{1,\emphll^k\omega\}$ of $S, R$ respectively. An element $a+b \emphll^k \omega$ (where $a,b\in \mathbb{Z}_\emphll$) corresponds to
$$
begin{pmatrix}
a & b \emphll^{k} d \\ b \emphll^k & a
\emphnd{pmatrix}\in C_S(\mathbb{Z}_\emphll) \mathfrak{q}quad \text{and} \mathfrak{q}quad begin{pmatrix}
a & b \emphll^{2k} d \\ b & a
\emphnd{pmatrix} \in C_R(\mathbb{Z}_\emphll)\,.$$
One can check that for $b \neq 0$ there is no $\mathbb{Z}_\emphll$-integral change of basis relating these two matrices, and a similar conclusion holds for any choice of $\mathbb{Z}_\emphll$-bases of $R, S$.
\emphnd{rem}
For a maximal Cartan subgroup we have $R = \mathcal{O}_F$ and for a split Cartan subgroup we have $R \cong \mathbb{Z}_\emphll^2$ and hence $C \cong (\mathbb{Z}_\emphll^\times)^2$.
\subsection{A classification for quadratic rings}
It is apparent from the previous discussion that classifying the Cartan subgroups of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ up to conjugacy is equivalent to classifying the \emphmph{quadratic rings} over $\mathbb{Z}_\emphll$ (i.e.\@ the orders in integral quadratic $\mathbb{Q}_\emphll$-algebras) up to a $\mathbb{Z}_\emphll$-linear ring isomorphism.
A Cartan subgroup is maximal if and only if the corresponding quadratic ring $R$ is the maximal order; a maximal Cartan subgroup is unramified if and only if the corresponding $\mathbb{Q}_\emphll$-algebra is \'etale i.e.\@ it is either $\mathbb{Q}_\emphll^2$ (in the split case) or the unique unramified quadratic extension of $\mathbb{Q}_\emphll$ (in the nonsplit case). We have an \'etale $\mathbb{Q}_\emphll$-algebra if and only if the $\emphll$-adic valuation on $R$, normalized so that $v_\emphll(\emphll)=1$, takes integer values.
begin{thm}{(Classification of quadratic rings)}\label{quadratic-rings}
If $R$ is a quadratic ring over $\mathbb{Z}_\emphll$ then there exist a $\mathbb{Z}_\emphll$-basis $(1,\omega)$ of $R$ and parameters $(c,d)$ in $\mathbb{Z}_\emphll$ satisfying $\omega^2=c\omega+d$ and such that one of the following holds: $c=0$ (and hence $d\neq 0$); $\emphll=2$, $c=1$, and $d$ is either zero or odd.
\emphnd{thm}
begin{proof}
Let $(1,\omega_0)$ be a $\mathbb{Z}_\emphll$-basis of $R$ and write $\omega_0^2=c_0\omega_0+d_0$ for some $c_0,d_0 \in \mathbb{Z}_\emphll$. If $\emphll$ is odd or $c_0$ is even, we set $\omega=\omega_0-c_0/2$ and have parameters $(0,d_0+{c_0^2}/{4})$. If $\emphll=2$ and $c_0$ is odd, we set $\omega=\omega_0-{(c_0-1)}/{2}$ and $d_1=d_0+{(c_0^2-1)}/{4}$. If $d_1$ is odd, we are done because we have $\omega^2=\omega+d_1$. If $d_1$ is even, the quadratic equation
$\omega^2=\omega+d_1$ has solutions in $\mathbb{Q}_2$ because its discriminant $1-4d_1 \emphquiv 1 \mathfrak{p}mod 8$ is a square. Thus $R$ is an order in $\mathbb{Q}^2_2$ and hence it is of the form $\mathbb{Z}_2 (1,1) \oplus \mathbb{Z}_2 (0,beta)$ for some $beta \in \mathbb{Z}_2$. If $beta$ is odd, we have $R=\mathbb{Z}^2_2$ so we set $\omega=(0,1)$ and have parameters $(1,0)$. If $beta$ is even, we set $\omega=(-{beta}/{2},{beta}/{2})$ and have parameters $(0,{beta^2}/{4})$. \emphnd{proof}
\subsection{Parameters for a Cartan subgroup}\label{subsec-classification}
We call the parameters $(c,d)$ as in Theorem \ref{quadratic-rings}
\emphmph{parameters for the Cartan subgroup} of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ corresponding to $R$: they are in general not uniquely determined.
begin{rem}\label{rmk:IntegralParameters}
Since $\mathbb{Z}$ is dense in $\mathbb{Z}_\emphll$ we may assume that the parameters $(c,d)$ are integers. Indeed, one can prove that the isomorphism class of the ring $\mathbb{Z}_\emphll[x]/(x^2-cx-d)$ is a locally constant function of $(c,d) \in \mathbb{Z}_\emphll^2$ (this property is closely related to Krasner's Lemma \cite[Tag 0BU9]{stacks-project}). We also give a direct argument.
Consider first a Cartan subgroup $C$ with parameters $(0,d)$.
If $u$ is an $\emphll$-adic unit, $(0,u^2d)$ are also parameters for $C$. Thus $C$ depends on $d$ only through its class in $(\mathbb{Z}_\emphll \setminus \{0\})/\mathbb{Z}_\emphll^{\times 2}$ (quotient as multiplicative monoids): this is isomorphic to $\mathbb{N} \times \mathbb{Z}_\emphll^\times/\mathbb{Z}_\emphll^{\times 2}$, where the first factor is the valuation and the second factor is finite (indeed $\mathbb{Z}_\emphll^\times/\mathbb{Z}_\emphll^{\times 2} \cong \mathbb{F}_\emphll^\times/\mathbb{F}_\emphll^{\times 2}$ if $\emphll$ is odd, and $\mathbb{Z}_2^\times / \mathbb{Z}_2^{\times 2} \cong (\mathbb{Z}/8\mathbb{Z})^\times$).
With powers of $\emphll$ we can realize every integral valuation (recall that $d$ is an element of $\mathbb{Z}_{\emphll}$), and the integers coprime to $\emphll$ represent all elements of $\mathbb{Z}_\emphll^\times/\mathbb{Z}_\emphll^{\times 2}$, thus there is an integer representative.
Now suppose $\emphll=2$ and consider a Cartan subgroup with parameters $(1,d)$ where $d$ is odd: in Proposition \ref{prop:ConditionsCD2} we show that the quadratic ring is $\mathbb{Z}_2[\zeta_6]$ and hence we can take as parameters $(1,-1)$.
\emphnd{rem}
begin{prop}[Classification of Cartan subgroups for $\emphll$ odd]\label{prop:ConditionsCDodd}
Suppose that $\emphll$ is odd, and consider a Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ with parameters $(0,d)$. It is
maximal if and only if $v_\emphll(d)\leqslant 1$. It is unramified if and only if $\emphll \nmid d$: it is then split if $d$ is a square in $\mathbb{Z}_\emphll^\times$, and nonsplit otherwise.
\emphnd{prop}
begin{proof}
If $v_\emphll(d)=1$ then $v_\emphll(\omega)=1/2$ and hence $F$ is a ramified extension of $\mathbb{Q}_\emphll$. If $v_\emphll(d) \gammaeqslant 2$ then $C$ is not maximal because $(\omega/ \emphll)^{2}={d}/ \emphll^{2} \in \mathbb{Z}_\emphll$ and hence ${\omega}/{\emphll}$ is in $\mathbb{Z}_{\emphll}$. If $v_\emphll(d) \leqslant 1$ then $R$ is a maximal order. Indeed, let $R'$ be an order in $F$ containing $R$ and choose a $\mathbb{Z}_\emphll$-basis $(1,\omega_1)$ of $R'$ satisfying $\omega_1^2=d_1 \in \mathbb{Z}_\emphll$: writing $\omega=a \omega_1+b$ for some $a,b \in \mathbb{Z}_\emphll$, we have $$d=\omega^2=(a^2 d_1+b^2)\cdot 1+(2ab)\cdot \omega_1$$ which implies $b=0$, thus $v_\emphll(d)=2v_\emphll(a)+v_\emphll(d_1)$ and hence $v_\emphll(a)=0$ and $R'=R$.
Now suppose $v_\emphll(d)=0$.
If $d$ is not a square, then $F=\mathbb{Q}_\emphll(\sqrt{d})$ is an unramified extension of $\mathbb{Q}_\emphll$ while if $d$ is a square the map $a + b \omega \mapsto (a + b \sqrt{d},a - b \sqrt{d})$ identifies $R$ and $\mathbb{Z}_\emphll^2$.
\emphnd{proof}
begin{prop}[Classification of Cartan subgroups for $\emphll=2$]\label{prop:ConditionsCD2}
Suppose that $\emphll=2$, and consider a Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ with parameters $(c,d)$.
It is unramified if and only if $c=1$: it is then split for $d=0$ and nonsplit for $d$ odd. It is maximal and ramified if and only if $c=0$ and either $v_2(d)=1$ or $v_2(d)=0$ and $d \emphquiv 3 \mathfrak{p}mod 4$.
\emphnd{prop}
begin{proof}
We keep the notation of the previous proof. If $c=1$ and $d=0$ then $a+b\omega \mapsto (a,a+b)$ is an isomorphism between $R$ and $\mathbb{Z}_2^2$, so that $C$ is split. If $c=1$ and $d$ is odd then up to isomorphism we may suppose $\omega^2=\omega+d$. This equation is separable over $\mathbb{Z}/2\mathbb{Z}$, so $R$ is contained in the unique unramified quadratic extension of $\mathbb{Q}_2$, which is $\mathbb{Q}_2(\zeta_6)$. Since $2$ is inert, we will have shown that $C$ is nonsplit once we prove $R=\mathbb{Z}_2[\zeta_6]$. To show $\zeta_6\in R$, we write $\omega=a+b\zeta_6$ (with $a,b \in \mathbb{Z}_2$) and prove that $b$ is a unit: the equation $\omega^2=\omega+d$ gives $$(a^2-b^2)+\zeta_6(b^2+2ab)=(a+d)+\zeta_6b,$$ so $a+d$ has the same parity as $a^2-b^2$ and we deduce that $b$ is odd.
Conversely, if $C$ is unramified then no $\mathbb{Z}_2$-basis $(1,\omega)$ of $R$ satisfies $\omega^2 \in \mathbb{Z}_2$ and we must have $c=1$.
Since $t^2=t+d$ has no solutions in $\mathbb{Z}_2^2$ for $d$ odd while it has solutions in $\mathbb{Q}_2$ if $d=0$, we deduce that $d=0$ (resp. $d$ is odd) for a split (resp. nonsplit) Cartan subgroup.
If $c=1$ we have seen that $C$ is maximal, so suppose $c=0$ and hence $\omega^2=d \in \mathbb{Z}_2$. Analogously to the previous remark we have $v_2(d) \leqslant 1$ if $C$ is maximal.
By Remark \ref{rmk:IntegralParameters} we only need to consider those $d$ in $(\mathbb{Z}_2 \setminus\{0\})/ \mathbb{Z}_2^{\times 2} \cong (\mathbb{N} \times \mathbb{Z}_2^\times)/ \mathbb{Z}_2^{\times 2}$ with valuation $0$ or $1$, namely
$d=1,3,5,7$ and $d=2, 6, 10, 14$. We may conclude because it is known whether $\mathbb{Z}_2[\sqrt{d}]$ has index $1$ or $2$ in the ring of integers of $\mathbb{Q}_2(\sqrt{d})$, where for $d=1$ we set $\sqrt{d}=(1,-1)\in \mathbb{Z}_2^2$.
\emphnd{proof}
\subsection{A concrete description of Cartan subgroups}
Let $(c,d)$ be parameters as in Theorem \ref{quadratic-rings}; we can then give a precise description for $C_R:=\operatorname{Res}_{R/\mathbb{Z}_\emphll}(\mathbb{G}_m) \subset \mathbb{G}L_{2, \mathbb{Z}_\emphll}$ as follows. For every $\mathbb{Z}_\emphll$-algebra $A$, the $A$-points of $C_R$ are the subgroup of $\mathbb{G}L_{2, \mathbb{Z}_\emphll}(A)$ given by:
begin{equation*}
C_R(A)=\left\{ begin{pmatrix}
x & dy \\ y & x+yc
\emphnd{pmatrix} : x,y \in A,\; \det begin{pmatrix}
x & dy \\ y & x+yc
\emphnd{pmatrix} \in A^\times \right\}\,.
\emphnd{equation*}
In particular the Cartan subgroup $C=C_R(\mathbb{Z}_\emphll) $ is the set
begin{equation}\label{nfaa}
C=\left\{ begin{pmatrix}
x & dy \\ y & x+yc
\emphnd{pmatrix} : x,y \in \mathbb{Z}_{\emphll},\; v_{\emphll}(x(x+yc)-dy^2)=0 \right\}\,.
\emphnd{equation}
begin{rem}[Diagonal model for a split Cartan subgroup]\label{rmk:DiagonalModelWorks}
For the parameters $(c,d)$ of a split Cartan subgroup $C$ we have shown: if $\emphll$ is odd, we have $c=0$ and $d$ is a square in $\mathbb{Z}_\emphll^\times$; if $\emphll=2$, we have $(c,d)=(1,0)$. We deduce the existence of an isomorphism between $C$ and the group of diagonal matrices in $\mathbb{G}L_2(\mathbb Z_{\emphll})$:
begin{equation}\label{eq:SplitCartan}
\left\{ begin{pmatrix}
X& 0 \\ 0 & Y
\emphnd{pmatrix} :\, X,Y \in \mathbb Z_{\emphll}^\times
\right\} \, .
\emphnd{equation}
We can define such an isomorphism for $\emphll$ odd and for $\emphll=2$ respectively as:
begin{equation}\label{phil}
\varphi_{\emphll} : begin{pmatrix}
x & dy \\ y & x
\emphnd{pmatrix} \mapsto begin{pmatrix}
x-y\sqrt{d} & 0 \\ 0 & x+y \sqrt{d}
\emphnd{pmatrix}
\mathfrak{q}quad
\varphi_2 : begin{pmatrix}
x & 0 \\ y & x+y
\emphnd{pmatrix} \mapsto begin{pmatrix}
x & 0 \\ 0 & x+ y
\emphnd{pmatrix}\,.
\emphnd{equation}
We have $\det(\varphi_\emphll(M)-I)=\det(M-I)$ and for any $n\gammaeqslant 1$ we have $\varphi_\emphll(M)-I\emphquiv 0 bmod (\emphll^n)$ if and only if $M-I\emphquiv 0 bmod (\emphll^n)$.
\emphnd{rem}
{bf Notation.} For a subset $X$ of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ we denote by $X(n)$ the image of $X$ in $\mathbb{G}L_2(\mathbb{Z}/\emphll^n\mathbb{Z})$.
begin{lem}\label{lemma:CardinalityCartans}
If $C$ is a Cartan subgroup of $\mathbb{G}L_2(\mathbb Z_{\emphll})$ we have
$$\# C(1)= \left\{
begin{array}{ll}
(\emphll-1)^2 & \text{if $C$ is split}\\
(\emphll-1)\cdot (\emphll+1) & \text{if $C$ is nonsplit}\\
(\emphll-1)\cdot \emphll & \text{if $C$ is ramified}\\
\emphnd{array}\right.$$
and for any $n\gammaeqslant 1$ we have $\# C(n)=\# C(1) \cdot \emphll^{2n-2}$.
\emphnd{lem}
begin{proof}
The assertion for
$n=1$ is a straightforward computation, while for $n>1$ it follows from the (higher-dimensional version of) Hensel's Lemma \cite[Proposition 7.8]{Nekovar} because the Zariski closure of $C$ in $\mathbb{G}L_{2,\mathbb Z_{\emphll}}$ is smooth of relative dimension $2$.
\emphnd{proof}
\subsection{Normalizers of Cartan subgroups}
begin{lem}\label{lem-Norm}
A Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ has index 2 in its normalizer. If $C$ is as in \emphqref{nfaa}, its normalizer $N$ in $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ is the disjoint union of $C$ and
$$C':=begin{pmatrix}
1 & c \\ 0 & -1
\emphnd{pmatrix} \cdot C\,.$$
We have instead $C':=begin{pmatrix}
0 & 1 \\ 1 & 0
\emphnd{pmatrix}\cdot C$ for a split Cartan subgroup as in \emphqref{eq:SplitCartan}.
\emphnd{lem}
begin{proof}
An easy verification shows $C'\subset N$. If $A\in N$, there exist $x, y \in \mathbb{Z}_\emphll$ such that we have
begin{equation*}A begin{pmatrix}
0 & d \\ 1 & c
\emphnd{pmatrix} A^{-1} = begin{pmatrix}
x & yd \\ y & x+yc
\emphnd{pmatrix}.
\emphnd{equation*}
If $c=0$, by comparing traces we find $x=0$ and hence by comparing determinants we have $(x,y) = (0,\mathfrak{p}m 1)$. If $\emphll=2$ and $c=1$, by comparing traces we find $y=1-2x$ and hence by comparing determinants we have $-x^2+x=0$, so $(x,y)$ is either $(0,1)$ or $(1,-1)$. We compute
begin{equation*}A begin{pmatrix}
0 & d \\ 1 & c
\emphnd{pmatrix} = begin{pmatrix}
x & yd \\ y & x+yc
\emphnd{pmatrix} A
\emphnd{equation*}
for any explicit value of $(x,y)$ as above, finding in each case $A \in C \cup C'$.
The last assertion about a split Cartan subgroup is well-known and easy to prove.
\emphnd{proof}
begin{rem}
If one considers a Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ as the $\mathbb{Z}_\emphll$-valued points of a maximal torus of $\mathbb{G}L_2$, the previous lemma also follows from the fact that any maximal torus in $\mathbb{G}L_2$ has index 2 in its normalizer (the Weyl group of $\mathbb{G}L_2$ is $\mathbb{Z}/2\mathbb{Z}$).\emphnd{rem}
begin{lem}\label{lemma:NormalizerAIsZero}
If $C$ is as in \emphqref{nfaa} and $N$ is its normalizer then we have
begin{equation}\label{eqcomplement}
N\setminus C =Big\{begin{pmatrix}
z & -dw+cz \\ w & -z
\emphnd{pmatrix} \mathfrak{q}uad \text{
with $z,w\in \mathbb{Z}_{\emphll}$\; and\; $v_{\emphll}(-z^2+dw^2-czw)=0$Big\}\,.}
\emphnd{equation}
Consider $M \in N \setminus C$. If $\emphll$ is odd, we have $M \not \emphquiv I \mathfrak{p}mod{\emphll}$; if $\emphll=2$, we have $M \not \emphquiv I \mathfrak{p}mod{4}$, and if $C$ is unramified we also have $M \not \emphquiv I \mathfrak{p}mod{2}$.
\emphnd{lem}
begin{proof}
The first assertion follows from the previous lemma and \emphqref{nfaa}.
Since $M$ has trace zero, we have $M \not \emphquiv I \mathfrak{p}mod{\emphll}$ for $\emphll$ odd and $M \not \emphquiv I \mathfrak{p}mod{4}$ for $\emphll=2$. If $\emphll=2$ and $C$ is unramified we know $c=1$ thus $M \emphquiv I \mathfrak{p}mod{2}$ is impossible.
\emphnd{proof}
begin{rem}\label{nor}
By comparing \emphqref{nfaa} and \emphqref{eqcomplement}, we see that the sets $C(n)$ and $(N\setminus C) (n)$ are disjoint for $n\gammaeqslant 2$ (if $\emphll$ is odd or $C$ is unramified, for $n\gammaeqslant 1$). By Lemma \ref{lem-Norm} we then have $\# N(n)=2 \cdot \# C(n)$.
\emphnd{rem}
\subsection{The tangent space of a Cartan subgroup}\label{subsec-Lie}
Let $G$ be an open subgroup of either $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ or of the normalizer of a Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$. Then $G$ is an $\emphll$-adic manifold, and there is a well-defined notion of a tangent space $T_{I}G$ at the identity. This is a $\mathbb{Q}_\emphll$-vector subspace of $\operatorname{Mat}_2(\mathbb{Q}_\emphll)$, of dimension equal to the dimension of $G$ as a manifold (in particular, if $G=\mathbb{G}L_2(\mathbb{Z}_\emphll)$ we have $T_{I}G =\operatorname{Mat}_2(\mathbb{Q}_\emphll)$). For our lifting questions, however, we are more interested in the `mod-$\emphll$' tangent space, which can be defined either as the reduction modulo $\emphll$ of the intersection of $T_{I}G$ with $\operatorname{Mat}_2(\mathbb{Z}_\emphll)$, or as the tangent space to the modulo-$\emphll$ fiber of the Zariski closure of $G$ in $\mathbb{G}L_{2,\mathbb{Z}_\emphll}$. More concretely, the next two definitions describe the tangent space explicitly:
begin{defi} If $C$ is as in \emphqref{nfaa}, its tangent space is
\[
\mathbb{T}:=Big\{begin{pmatrix}
x & dy \\ y & x+cy
\emphnd{pmatrix} \, : \, x,y \in \mathbb{Z}/\emphll \mathbb{Z} Big\}
\]
where $(c,d)$ are here the reductions modulo $\emphll$ of the parameters of $C$.
Write $\mathbb{T}^\times=C(1)$ for the subset of $\mathbb{T}$ consisting of the invertible matrices.
\emphnd{defi}
We clearly have $\#\mathbb{T}=\emphll^2$ and by Lemma \ref{lemma:CardinalityCartans} we also know $\#\mathbb{T}^\times$. So we get:
begin{equation}\label{eqprop:XB}
begin{tabular}{|c|c|c|c|}
\hline
{Type of $C$} &$\#\mathbb{T}$ & $\#\mathbb{T}^\times$ & $\#\mathbb{T}-\#\mathbb{T}^\times-1$ \\ \hline
split & $\emphll^2$ & $(\emphll-1)^2$ & $2(\emphll-1)$ \\ \hline
nonsplit & $\emphll^2$ & $\emphll^2-1$ & $0$ \\ \hline
ramified & $\emphll^2$ & $\emphll(\emphll-1)$ & $\emphll-1$
\\ \hline
\emphnd{tabular}
\emphnd{equation}
We define the tangent space of an open subgroup of the normalizer of $C$ as the tangent space of $C$. We also define the tangent space of an open subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ as follows:
begin{defi}\label{LieGL}
Let $G$ be an open subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$. The tangent space of $G$ is $\mathbb{T}:=\operatorname{Mat}_2(\mathbb{Z}/\emphll \mathbb{Z})$ and we write $\mathbb{T}^\times=\mathbb{G}L_2(\mathbb{Z}/\emphll \mathbb{Z})$.
\emphnd{defi}
For $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ we have $\#\mathbb{T}=\emphll^4$, $\#\mathbb{T^\times}=\emphll (\emphll-1)^2(\emphll+1)$ and $\#\mathbb{T}-\#\mathbb{T}^\times-1=(\emphll+1)(\emphll^2-1)$.
\section{The group structure of the $1$-Eigenspace}\label{sec-1ES}
\subsection{The level} Let $G'$ be either $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ or the normalizer of a Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$. Let $G$ be an open subgroup of $G'$ of finite index $[G':G]$. Call $G'(n)$ and $G(n)$ the reductions of $G'$ and $G$ modulo $\emphll^n$, that is their respective images in $\mathbb{G}L_2(\mathbb{Z}/\emphll^n\mathbb{Z})$.
If $n$ is the smallest positive integer such that we have $[G'(n):G(n)]=[G':G]$, we define the \emphmph{level} $n_0$ of $G$ as
\[
n_0 = begin{cases} begin{array}{ll}
\max\{n,2\} & \text{if } \emphll = 2 \text{ and $G'$ is the normalizer of a ramified Cartan} \\ n & \text{otherwise.} \emphnd{array} \emphnd{cases}
\]
begin{rem}\label{rem:Level}
It is easy to check that all our statements involving the notion of \emphmph{level} remain true if $n_0$ is replaced by any larger integer.
\emphnd{rem}
All matrices in $G'$ that are congruent to the identity modulo $\emphll^{n_0}$ belong to $G$. In other words, $G$ is the inverse image of $G{(n_0)}$ for the reduction map $G' \to G'(n_0)$.
Consequently we have
begin{equation}\label{n0-forH}
[G'(n):G(n)]=[G':G]\mathfrak{q}quad \text{for every $n\gammaeqslant n_0$\,.}
\emphnd{equation}
The dimension of $G'$ is $4$ if $G'=\mathbb{G}L_2(\mathbb{Z}_\emphll)$ and is $2$ otherwise, and we have
begin{equation}\label{dimG}
[G(n+1):G(n)]=[G'(n+1):G'(n)]=\emphll^{\dim G'}\mathfrak{q}quad \text{for every $n\gammaeqslant n_0$\,.}
\emphnd{equation}
begin{rem}\label{kerT}
Let $G$ be an open subgroup of either $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ or the normalizer of a Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$. Let $n_0$ be the level of $G$.
For any $n\gammaeqslant n_0$ the map $M \mapsto \emphll^{-n}(M-I)$ identifies the tangent space of $G$ with the kernel of $G(n+1) \to G(n)$. This is immediate for $\mathbb{G}L_2(\mathbb{Z}_\emphll)$, and for Cartan subgroups it follows from \emphqref{nfaa}. The assertion also holds for normalizers of Cartan subgroups because by Lemma \ref{lemma:NormalizerAIsZero} all matrices reducing to the identity in $G(n)$ are contained in the Cartan subgroup.
\emphnd{rem}
\subsection{The $1$-Eigenspace}
We identify an element of $\mathbb{G}L_2(\mathbb Z_{\emphll})$ with an automorphism of the direct limit in $n$ of $(\mathbb{Z}/\emphll^n \mathbb{Z})^2$. For all integers $a,b\gammaeqslant 0$, if $X\subseteq \mathbb{G}L_2(\mathbb Z_{\emphll})$ we define
begin{equation}
\mathcal M_{a,b}(X):=\{M\in X: \, \ker (M-I) \simeq \mathbb{Z}/{\emphll^a}\mathbb{Z}\times \mathbb{Z}/{\emphll^{a+b}}\mathbb{Z}\}
\emphnd{equation}
and call $\mathcal M_{a,b}(X;n)$ the reduction of $\mathcal M_{a,b}(X)$ modulo $\emphll^n$. To ease notation, we write $\mathcal M_{a,b}:=\mathcal M_{a,b}(G)$ and $\mathcal M_{a,b}(n):=\mathcal M_{a,b}(G;n)$.
We consider the normalized Haar measure on $G$ and call $\mu_{a,b}$ the measure of the set $\mathcal M_{a,b}$. Since $\mathcal M_{a,b}(n)$ is a subset of $G(n)$, we may consider its measure $$\mu_{a,b}(n):=\#\mathcal M_{a,b}(n)/\#G(n)\,.$$
The sets $\mathcal{M}_{a,b}$ are pairwise disjoint, but the same is not necessarily true for the sets $\mathcal{M}_{a,b}(n)$.
The sequence $\mu_{a,b}(n)$ is constant for $n>a+b$: this shows that $\mu_{a,b}$ is well-defined and that we have $\mu_{a,b}=\mu_{a,b}(n)$ for every $n>a+b$.
begin{rem}\label{vuoto}
We clearly have $\mathcal M_{a,b}=G\cap \mathcal M_{a,b}(G')$. Moreover, we have
$$\mathcal M_{a,b}=\emphmptyset \mathfrak{q}quad \Lambdaeftrightarrow\mathfrak{q}quad G(n_0)\cap \mathcal M_{a,b}(G'; n_0)=\emphmptyset\,.$$
Indeed we know $\mathcal M_{a,b}(n_0)\subseteq G(n_0)\cap \mathcal M_{a,b}(G';n_0)$ so if the latter is empty so is $\mathcal{M}_{a,b}$.
Conversely, matrices in $\mathcal M_{a,b}(G')$ whose reduction modulo $\emphll^{n_0}$ lies in $G(n_0)$ are in $\mathcal M_{a,b}$ because $G$ is the inverse image in $G'$ of $G(n_0)$.
\emphnd{rem}
\subsection{Additional notation}
We write $\det_{\emphll}$ for the \emphmph{$\emphll$-adic valuation of the determinant}. If $M$ is in $\operatorname{Mat}_2(\mathbb{Z}/\emphll^n\mathbb{Z})$, then $\det(M)$ is well-defined modulo $\emphll^n$ so we can write $\det_{\emphll}(M)\gammaeqslant n$ if the determinant is zero modulo $\emphll^n$. Notice that the matrices in $\operatorname{Mat}_2(\mathbb{Z}_{\emphll})$ that are zero modulo $\emphll^a$ for some $a\gammaeqslant 0$ and with a given reduction modulo $\emphll^n$ for some $n>a$ have a determinant which is well-defined modulo $\emphll^{a+n}$.
More generally, if $p$ is a polynomial with integer coefficients and $z_1, z_2$ are in $\mathbb{Z}/\emphll^n\mathbb{Z}$ then we write $v_\emphll(p(z_1,z_2))$ for the minimum of $v_\emphll(p(Z_1,Z_2))$ over all lifts $Z_1, Z_2$ of $z_1, z_2$ to $\mathbb{Z}_\emphll$. For example, if $z \emphquiv \emphll^{t} \mathfrak{p}mod{\emphll^n}$ with $t<n$ then we have $v_\emphll(z^2)=2t$ because all lifts $Z$ of $z$ satisfy $v_\emphll(Z^2)=2t$.
\subsection{Conditions related to the group structure of the $1$-Eigenspace}
begin{lem}\label{condi-det}
The set $\mathcal M_{a,b}$ consists of the matrices $M\in G$ that satisfy
begin{equation}\label{conditions-Mab}
M-I \emphquiv 0\!\!\!\! \mathfrak{p}mod{\emphll^a},\; M-I \not \emphquiv 0 \!\!\!\!\mathfrak{p}mod{\emphll^{a+1}}\mathfrak{q}uad\text{and}\mathfrak{q}uad \mathrm{det}_{\emphll}(M-I)=2a+b\,.
\emphnd{equation}
For every $n>a+b$ the set $\mathcal M_{a,b}$ is the preimage of $\mathcal M_{a,b}(n)$ in $G$, and $\mathcal{M}_{a,b}(n)$ consists of the matrices $M\in G(n)$ satisfying \emphqref{conditions-Mab}.
\emphnd{lem}
begin{proof}
The necessity of \emphqref{conditions-Mab} follows from the fact that for $A\in \operatorname{Mat}_2(\mathbb{Z}_\emphll)$ the order of the kernel of $A$ (considered as acting on the direct limit $\varinjlim_n (\mathbb{Z}/\emphll^n\mathbb{Z})^2$) equals $\emphll^{\det_\emphll A}$, that is, there are $\emphll^{\det_\emphll A}$ points $x$ in $\varinjlim_n (\mathbb{Z}/\emphll^n\mathbb{Z})^2$ such that $Ax=0$.
Now suppose that $M\in G$ satisfies \emphqref{conditions-Mab}, and write $M=I+\emphll^a A$ for some $A\in \operatorname{Mat}_2(\mathbb{Z}_{\emphll})$ which is non-zero modulo $\emphll$. We have $\det_\emphll(A)=b$. Since $A$ is nonzero modulo $\emphll$, the kernel of $A$ is cyclic. Thus $\ker(A)\simeq \mathbb{Z}/\emphll^b\mathbb{Z}$ and hence $\ker(M-I)\simeq \mathbb{Z}/\emphll^a \mathbb{Z} \times \mathbb{Z}/\emphll^{a+b}\mathbb{Z}$. If $n>a+b$, \emphqref{conditions-Mab} holds for the matrices in $\mathcal{M}_{a,b}(n)$ and their preimages in $G$.
\emphnd{proof}
By Remark \ref{rmk:DiagonalModelWorks} and Lemma \ref{condi-det}, the maps in \emphqref{phil} preserve $\mathcal{M}_{a,b}$ and $\mathcal{M}_{a,b}(n)$, thus for a split Cartan subgroup we can indifferently use the general model \emphqref{nfaa} or the diagonal model \emphqref{eq:SplitCartan}.
\subsection{Existence of the Haar measure}
A fundamental tool in dealing with Haar measures on profinite groups is the following simple lemma:
begin{lem}\label{lemma:Haar}{\cite[Lemma 18.1.1]{MR2445111}}
Let $\mathcal G_1$ be a profinite group equipped with its normalized Haar measure, and let $\mathcal G_2$ be an open normal subgroup of $\mathcal G_1$. Call $\mathfrak{p}i$ the natural projection $\mathcal G_1 \to \mathcal G_1/\mathcal G_2$. For any subset $S$ of the finite group $\mathcal G_1/\mathcal G_2$, the set $\mathfrak{p}i^{-1}(S)$ is measurable in $\mathcal G_1$, and its Haar measure is $\#S/\#(\mathcal G_1/\mathcal G_2)$.
\emphnd{lem}
begin{lem}\label{lemma:EverythingHasWellDefinedAB}
For all integers $a,b\gammaeqslant 0$ the set $\mathcal{M}_{a,b}$ is measurable in $G$ and we have $\mu_{a,b}=\mu_{a,b}(n)$ whenever $n>a+b$. In particular we have $\mu_{a,b}=0$ if and only if $\mathcal{M}_{a,b}=\emphmptyset$. The set $bigcup_{a,b \in \mathbb N} \mathcal{M}_{a,b}$ is measurable in $G$, and its complement has measure zero.
\emphnd{lem}
begin{proof}
For the first assertion apply Lemma \ref{lemma:Haar} to $G$, $\ker(G \to G(n))$ and $\mathcal{M}_{a,b}(n)$, noticing that $\mathcal{M}_{a,b}$ is the preimage of
$\mathcal{M}_{a,b}(n)$ in $G$ by Lemma \ref{condi-det}. The set $\mathcal M :=bigcup_{a,b \in \mathbb N} \mathcal{M}_{a,b}$ is measurable because it is a countable union of measurable sets. We now prove $\mu(G\setminus \mathcal M)=0$. Fix $n_0$ as in \emphqref{n0-forH} and for $n\gammaeqslant n_0$ call $\mathfrak{p}i_n: G\rightarrow G(n)$ the reduction modulo $\emphll^n$. We have $G\setminus \mathcal M \subseteq \mathfrak{p}i_n^{-1}\left( \mathfrak{p}i_n\left(G\setminus \mathcal M \right) \right)$, so by Lemma \ref{lemma:Haar} it suffices to show that
begin{equation}\label{10}
\mu(\mathfrak{p}i_n\left(G\setminus \mathcal M \right))=\frac{\# \mathfrak{p}i_n\left( G\setminus \mathcal M \right)}{\#G(n)}
\emphnd{equation}
tends to 0 as $n$ tends to infinity. By \emphqref{dimG} we know that $\#G(n)$ is a constant times $\emphll^{n \dim G'}$. Let $G'_{\infty}$ be the closed $\emphll$-adic analytic subvariety of $G'$ defined by $\det(M-I)=0$.
We have $G\setminus \mathcal M\subseteq G'_{\infty}$ because for any $M\in G$ with $\det(M-I) \neq 0$ there exists $n$ such that $M \not \emphquiv I \mathfrak{p}mod{\emphll^n}$ and $\det_\emphll(M-I) \leqslant n$, whence $M \in \mathcal{M}$. Thus the numerator in \emphqref{10} is at most $\# \mathfrak{p}i_n(G'_{\infty})$, which by \cite[Theorem 4]{MR656627} is at most a constant times $\emphll^{n \dim(G'_{\infty})}=\emphll^{n(\dim G' -1)}$.\emphnd{proof}
\subsection{The complement of a Cartan subgroup in its normalizer}\label{star}
Fix a Cartan subgroup $C$ of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ and denote by $N$ its normalizer. If $G$ is an open subgroup of $N$, set
\[
\mathcal{M}^*_{a,b}:=(N \setminus C) \cap \mathcal{M}_{a,b}\,.
\]
We denote by $\mathcal{M}^*_{a,b}(n)$ the reduction of $\mathcal{M}^*_{a,b}$ modulo $\emphll^n$, that is its image in $G(n)$.
If $G$ is not contained in $C$, the sets $G \cap C$ and $G \cap (N \setminus C)$ are measurable and have measure $1/2$ in $G$ because of Lemma \ref{lemma:Haar} applied to the canonical projection $G \to G/(G \cap C) \cong \mathbb{Z}/2\mathbb{Z}$. In particular we have
\[
\mu(\mathcal{M}_{a,b}) = \mu\left(\mathcal{M}_{a,b} \cap C\right) + \mu\left( \mathcal{M}^*_{a,b} \right).
\]
Since $\mu_N\left( \mathcal{M}_{a,b} \cap C \right)=1/2 \cdot \mu_C\left( \mathcal{M}_{a,b} \cap C \right)$, to determine $\mu_{a,b}$ we are reduced to computing $\mu( \mathcal{M}^*_{a,b})$ and studying $G\cap C$, which is open in the Cartan subgroup $C$.
begin{prop}\label{complement}
We have $\mathcal{M}^*_{a,b} = \emphmptyset$ for $a > 1$ (if $\emphll$ is odd or $C$ is unramified, for $a>0$).
\emphnd{prop}
begin{proof} This is a consequence of Lemma \ref{lemma:NormalizerAIsZero}.
\emphnd{proof}
\section{First results on the cardinality of $\mathcal{M}_{a,b}(n)$}
begin{thm}\label{thm:GeneralLift}
Let $G'$ be either $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ or the normalizer of an unramified Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$. Let $G$ be an open subgroup of $G'$ of level $n_0$.
Call ${\mathcal{H}_{a,b}(n)}$ the set of matrices $M$ in $G(n)$ satisfying the following conditions:
begin{enumerate}
\item if $a>0$, $M \emphquiv I \mathfrak{p}mod{\emphll^a}$; if $n>a$, $M \not \emphquiv I \mathfrak{p}mod{\emphll^{a+1}}$;
\item if $a<n \leqslant a+b$, $\det_\emphll(M-I) \gammaeqslant a+n$; if $n>a+b$, $\det_\emphll(M-I)=2a+b$.
\emphnd{enumerate}
For every integer $n\gammaeqslant 1$ define
\[
f(n)=\left\{begin{array}{ll}
1 & \text{if $n < a$} \\
\#\mathbb{T}^\times & \text{if $n=a, b=0$} \\
\#\mathbb{T}-\#\mathbb{T}^\times-1 & \text{if $n=a, b>0$}\\
\#\mathbb{T}\cdot \emphll^{-1} & \text{if $a<n<a+b$} \\
\#\mathbb{T}\cdot (1-\emphll^{-1}) & \text{if $n=a+b$, $b>0$} \\
\#\mathbb{T} & \text{if $n>a+b$}\,. \\
\emphnd{array}\right.
\]
Then the following holds:
begin{enumerate}
\item[(i)] For every $n\gammaeqslant n_0$ we have ${\# \mathcal{H}_{a,b}(n+1)}=f(n)\cdot {\#\mathcal{H}_{a,b}(n)}$. For each $M \in\mathcal{H}_{a,b}(n)$ the number of matrices ${M'} \in\mathcal{H}_{a,b}(n+1)$ such that ${M'} \emphquiv M \mathfrak{p}mod{\emphll^n}$ equals $f(n)$.
\item[(ii)] If $\mathcal{M}_{a,b}\neq \emphmptyset$ and $n \gammaeqslant n_0$, or if $n>a+b$, we have $\mathcal{M}_{a,b}(n)=\mathcal{H}_{a,b}(n)$.
\item[(iii)] For $a\gammaeqslant n_0$ we have $\mathcal{M}_{a,b}= \emphmptyset$ if and only if $b>0$ and $\#\mathbb{T}-\#\mathbb{T}^\times-1=0$.
\emphnd{enumerate}
\emphnd{thm}
begin{proof} We first prove (i). Since $n\gammaeqslant n_0$, all lifts to $G'(n+1)$ of matrices in ${\mathcal{H}_{a,b}(n)}$ are in $G(n+1)$. If $n>a+b$ then clearly every lift to $G(n+1)$ of a matrix in $\mathcal{H}_{a,b}(n)$ belongs to $\mathcal{H}_{a,b}(n+1)$. If $n<a$, the sets ${\mathcal{H}_{a,b}(n)}$ and ${\mathcal{H}_{a,b}(n+1)}$ contain only the identity and we are done.
Suppose $n=a$: the only matrix in ${\mathcal{H}_{a,b}(a)}$ is the identity, so we apply Remark \ref{kerT}. For $b=0$ we count the matrices of the form $I+\emphll^a T$ with $T \in \mathbb{T}$ and $\det_{\emphll}(T)= 0$. For $b>0$ we count those $M\in \mathcal{H}_{a,b}(a+1)$ that are congruent to the identity modulo $\emphll^{a}$ but not modulo $\emphll^{a+1}$ and such that $\det_\emphll(M-I) \gammaeqslant 2a+1$: this means $M=I+\emphll^a T$, where $T \in \mathbb{T}$ with $\det_\emphll T \neq 0$, and excluding $T=0$.
Now consider the case $a<n<a+b$. Let $M \in\mathcal{H}_{a,b}(n)$ and fix some lift $L$ to $G(n+1)$.
The lifts of $M$ are those matrices of the form $M' =L+\emphll^n T$ with $T \in \mathbb{T}$, unless $G'$ is the normalizer of a Cartan subgroup $C$ and $M\notin C(n)$, for which by Lemma \ref{lemma:NormalizerAIsZero} we have $a=0$ and $T\in \mathbb T_1$, where
begin{equation}\label{defT}
\mathbb T_1:=Big\{begin{pmatrix}
z & -dw+cz \\ w & -z
\emphnd{pmatrix}\mathfrak{q}uad \text{where $z,w\in \mathbb{Z}/\emphll\mathbb{Z}$}Big\}\,.
\emphnd{equation}
Write $L-I=\emphll^a N$ for some $N\in \operatorname{Mat}_2(\mathbb{Z}/\emphll^{n+1-a}\mathbb{Z})$. Since $b>n-a$, we have $\det(N)=\emphll^{n-a} z$ for some $z \in \mathbb{Z}/\emphll\mathbb{Z}$.
Setting $(N bmod \emphll)=(n_{ij})$ and $T=(t_{ij})$, we get the following congruence modulo $\emphll^{n+1-a}$:
\[
\det (M'-I ) \emphquiv \emphll^{2a}\cdot \det (N+\emphll^{n-a}T)
\emphquiv
\emphll^{n+a} \left(z + n_{11}t_{22}+ n_{22}t_{11}-n_{21}t_{12}-n_{12}t_{21} \right).
\]
So the condition for $M'$ to be in ${\mathcal{H}_{a,b}(n+1)}$ is
begin{equation}\label{eq:Lifts1}
z + n_{22}t_{11}+n_{11}t_{22}-n_{21}t_{12}-n_{12}t_{21}= 0.
\emphnd{equation}
We conclude by checking that this equation defines an affine subspace of codimension 1 in $\mathbb{T}$ (resp. in $\mathbb{T}_1$, if $M \not \in C(n)$). The equation is nontrivial because at least one of the $n_{ij}$ is nonzero, and this remark suffices for $\mathbb{G}L_2(\mathbb{Z}_{\emphll})$. If $G'$ is the normalizer of a Cartan subgroup, we also have to check that \emphqref{eq:Lifts1} is independent from the equations defining $\mathbb{T}$ (resp. $\mathbb{T}_1$), which are
\[
begin{cases}
t_{22}=t_{11}+ct_{21} \\
t_{12}=dt_{21}.
\emphnd{cases}
\mathfrak{q}quad\text{resp.}\mathfrak{q}quad
begin{cases} t_{22}=-t_{11} \\ t_{12}=-dt_{21}+ct_{11}. \emphnd{cases}
\]
For the elements of $\mathbb{T}$, noticing that $(N bmod \emphll )$ depends only on $(M bmod \emphll^{a+1})$ and it is in $\mathbb T\setminus \{0\}$,
we can rewrite \emphqref{eq:Lifts1} as
begin{equation*}
z + (2n_{11}+cn_{21})t_{11}+(n_{11}c-2dn_{21})t_{21}=0
\emphnd{equation*}
and we can easily check by Proposition \ref{prop:ConditionsCD2} that $2n_{11}+cn_{21}$ or $n_{11}c-2dn_{21}$ is nonzero.
For the elements of $\mathbb{T}_1$, we again conclude by Proposition \ref{prop:ConditionsCD2} because $a=0$ and we have $(N+I bmod \emphll )\in \mathbb{T}_1\setminus\{0\}$, thus \emphqref{eq:Lifts1} becomes
begin{equation}\label{eq:LiftsNormalizer}
z - (2(n_{11}+1)+c n_{21})t_{11}+ (2dn_{21}-c(n_{11}+1))t_{21}=0.
\emphnd{equation}
If $n=a+b$ and $b>0$, we can reason as in the previous case. Now the condition for $M'$ to be in ${\mathcal{H}_{a,b}(a+b+1)}$ is that \emphqref{eq:Lifts1} is \textit{not} satisfied: we conclude because that equation has $\emphll^{-1}\cdot \#\mathbb{T}$ solutions.
We now prove (ii). The assertion for $n>a+b$ is the content of Lemma \ref{condi-det}, so in particular we know $\mathcal{M}_{a,b}(a+b+1)=\mathcal{H}_{a,b}(a+b+1)$ and we may suppose $n \leqslant a+b$. We clearly have $\mathcal{M}_{a,b}(n) \subseteq\mathcal{H}_{a,b}(n)$ and are left to prove the other inclusion.
The assumption $\mathcal{M}_{a,b}\neq \emphmptyset$ implies that for all $x\gammaeqslant 1$ the sets $\mathcal{M}_{a,b}(x)$ and $\mathcal{H}_{a,b}(x)$ are non-empty and hence by (i) we know $f(x)\neq 0$ for all $x \gammaeqslant n_0$.
Thus for any $M \in\mathcal{H}_{a,b}(n)$ there is some ${M'} \in {\mathcal{H}_{a,b}(a+b+1)}$ satisfying ${M'} \emphquiv M \mathfrak{p}mod{\emphll^n}$, and we deduce $M\in \mathcal{M}_{a,b}(n)$.
Finally, we prove (iii). The condition $f(a)=0$ is equivalent to $b>0$ and $\#\mathbb{T}-\#\mathbb{T}^\times-1=0$. By (i), if $f(a)=0$ then $\mathcal{H}_{a,b}(a+1)$ is empty and hence also $\mathcal{M}_{a,b}(a+1)$ and $\mathcal{M}_{a,b}$ are empty. If $f(a)\neq 0$ then we have $f(x)\neq 0$ for all $x\gammaeqslant 1$. Since $\mathcal{H}_{a,b}(a)$ contains the identity, we deduce that $\mathcal{H}_{a,b}(a+b+1)=\mathcal{M}_{a,b}(a+b+1)$ is nonempty, and hence $\mathcal{M}_{a,b}\neq \emphmptyset$.
\emphnd{proof}
\section{The number of lifts for the reductions of matrices}
\subsection{Main result}
We study the lifts of a matrix $M\in \mathcal{M}_{a,b}(n)$ to $\mathcal{M}_{a,b}(n+1)$, namely the matrices in $\mathcal{M}_{a,b}(n+1)$ which are congruent to $M$ modulo $\emphll^n$.
begin{thm}\label{thm-lifts}
Let $G$ be an open subgroup of either $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ or the normalizer $N$ of a Cartan subgroup $C$ of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$. Let $n_0$ be the level of $G$.
For $n\gammaeqslant n_0$ the number of lifts of a matrix $M\in \mathcal{M}_{a,b}(n)$ to $\mathcal{M}_{a,b}(n+1)$ is independent of $M$ in the first case, while in the second case it depends at most on whether $M$ belongs to either $C(n)$ or $(N\setminus C)(n)$.
\emphnd{thm}
begin{proof}[Proof of Theorem \ref{thm-lifts}]
If $G$ is open in $\operatorname{GL}_2(\mathbb{Z}_\emphll)$ or if $C$ is unramified, the number of lifts of $M$ to $G(n+1)$ is independent of $M$ by Theorem \ref{thm:GeneralLift} (i).
If $C$ is ramified the assertion follows from Theorems \ref{thm:LiftsRamifiedCartan} and \ref{thm:LiftsNormalizer}.
\emphnd{proof}
begin{exa}
The number of lifts may indeed depend on the coset of $N/C$. Suppose that $\emphll$ is odd and consider the Cartan subgroup $C$ of $\operatorname{GL}_2(\mathbb{Z}_\emphll)$ with parameters $(0,\emphll)$. If $G$ is the normalizer of $C$ then the matrices
\[
begin{pmatrix}
1 & \emphll \\ 1 & 1
\emphnd{pmatrix}
\mathfrak{q}quad\text{and}\mathfrak{q}quad
begin{pmatrix}
1 & -\emphll \\ 1 & -1
\emphnd{pmatrix}
\]
are in $\mathcal M_{0,1}$ and their reductions modulo $\emphll$ have respectively $\emphll^2$ and $\emphll^2-\emphll$ lifts to $\mathcal M_{0,1}(2)$. Indeed, their lifts to $G(2)$ are of the form
$$L=begin{pmatrix}
1+\emphll u & \emphll \\ 1+\emphll v & 1+\emphll u
\emphnd{pmatrix}\mathfrak{q}quad \text{and}\mathfrak{q}quad
L'=begin{pmatrix}
1+\emphll u & -\emphll \\ 1+\emphll v & -1-\emphll u
\emphnd{pmatrix}
$$
respectively, where $u,v\in \mathbb{Z}/\emphll\mathbb{Z}$: we have
$\det_\emphll (L-I)=1$ for every $u,v$ while $\det_\emphll (L'-I)=1$ holds if and only if $2u-1 \not \emphquiv 0 \mathfrak{p}mod \emphll$.
\emphnd{exa}
\subsection{Ramified Cartan subgroups}
begin{thm}\label{thm:LiftsRamifiedCartan}
Let $G$ be open in a ramified Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$.
Let $n_0$ be the level of $G$. For all $a,b\gammaeqslant 0$ and for all $n\gammaeqslant n_0$ the number of lifts of a matrix $M\in \mathcal{M}_{a,b}(n)$ to $\mathcal{M}_{a,b}(n+1)$ is independent of $M$.
\emphnd{thm}
begin{proof}
For $n \leqslant a$ the set $\mathcal{M}_{a,b}(n)$ consists at most of the identity matrix, so suppose $n>a$. Let $(0,d)$ be the parameters for the Cartan subgroup (for convenience, we do not use a different notation for $d$ and its reductions modulo powers of $\emphll$). The matrices in $\mathcal{M}_{a,b}(n)$ are of the form
begin{equation}\label{emmmmmm}
M=I+\emphll^abegin{pmatrix}
x & dy \\ y & x
\emphnd{pmatrix}
\emphnd{equation}
where $x,y\in \mathbb{Z}/\emphll^{n-a}\mathbb{Z}$ are not both divisible by $\emphll$ and have lifts $X,Y\in \mathbb{Z}_\emphll$ satisfying $v_\emphll(X^2-dY^2)=b$.
\emphmph{If all matrices in $\mathcal{M}_{a,b}(n)$ satisfy $x \emphquiv 0 \mathfrak{p}mod{\emphll^{n-a}}$ then they all have the same number of lifts to $\mathcal{M}_{a,b}(n+1)$.} Since $y$ is a unit, for any $M_1, M_2 \in \mathcal{M}_{a,b}(n)$ there is an obvious bijection between the lifts of $M_1-I$ and of $M_2-I$ given by rescaling by a suitable unit.
\emphmph{If some $M_0\in \mathcal{M}_{a,b}(n)$ satisfies $x_0 \not \emphquiv 0 \mathfrak{p}mod{\emphll^{n-a}}$ and either $v_\emphll(x_0^2)\neq v_\emphll(d)$ or $v_\emphll(x_0^2)=v_\emphll(d)=v_\emphll(x_0^2-dy_0^2)$ then every matrix in $\mathcal{M}_{a,b}(n)$ has $\emphll^2$ lifts to $\mathcal{M}_{a,b}(n+1)$.}
It suffices to show that for $M\in \mathcal{M}_{a,b}(n)$ all lifts $X,Y$ of $x,y$ to $\mathbb{Z}_{\emphll}$ satisfy $v_\emphll(X^2-dY^2)=b$ because this implies that all lifts of $M$ to $G$ belong to $\mathcal M_{a,b}$.
If $v_\emphll(x_0^2)< v_\emphll(d)$ then for any $X_0,Y_0$ lifting $x_0,y_0$ we have $v_\emphll(X_0^2-dY_0^2)=v_\emphll(X_0^2)=v_\emphll(x_0^2)$, so this number is independent of the lift and it is equal to $b$. In particular, we have $b<v_\emphll(d)$ and $b<2(n-a)$. For $M\in \mathcal{M}_{a,b}(n)$ there exist lifts $X,Y \in \mathbb{Z}_\emphll$ of $x,y$ that satisfy $v_\emphll(X^2-dY^2)=b$. We deduce $v_\emphll(X^2)=b$ and hence $v_\emphll(x^2)\leqslant b$: since $v_\emphll(x^2)< v_\emphll(d)$ and $x \not \emphquiv 0 \mathfrak{p}mod{\emphll^{n-a}}$ we can reason as for $M_0$ and we conclude.
If $v_\emphll(x_0^2)> v_\emphll(d)$ then $y_0$ must be a unit, so we have $v_\emphll(d)=v_\emphll(x_0^2-dy_0^2)$, and the same holds for all lifts $X_0,Y_0$. In particular, we have $b=v_\emphll(d)<2(n-a)$.
If $v_\emphll(x_0^2)=v_\emphll(d)=v_\emphll(x_0^2-dy_0^2)$, we write $x_0=\emphll^k u_0$ and $d=\emphll^{2k} \delta$, where $u_0, \delta$ are units and $k<n-a$. Then $u_0^2-\delta y_0^2$ is a unit and hence $v_\emphll(U_0^2-\delta Y_0^2)=0$ for all lifts $U_0, Y_0$ of $u_0, y_0$. We deduce $v_\emphll(X_0^2-dY_0^2)= v_\emphll(d)$ for all lifts $X_0,Y_0$ of $x_0,y_0$ and again we have $b = v_\emphll(d)<2(n-a)$.
So suppose $b = v_\emphll(d)<2(n-a)$. For $M\in \mathcal{M}_{a,b}(n)$ there are lifts $X,Y$ of $x,y$ satisfying $v_\emphll(X^2-dY^2)=b$ and hence $v_\emphll(X^2)\gammaeqslant v_\emphll(d)$ and $v_\emphll(x^2-dy^2)\leqslant v_\emphll(d)$.
If $x \not \emphquiv 0 \mathfrak{p}mod{\emphll^{n-a}}$ then either $v_\emphll(x^2) \neq v_\emphll(d)$ or we have $v_\emphll(x^2)= v_\emphll(d)$ and $v_\emphll(x^2-dy^2)=v_\emphll(d)$, so we can reason as for $M_0$. If $x \emphquiv 0 \mathfrak{p}mod{\emphll^{n-a}}$ then $v_\emphll(x^2)>v_\emphll(d)$ and $y$ is a unit: we deduce $v_\emphll(X^2-dY^2)=b$ for all lifts $X,Y$.
\emphmph{If some $M_0 \in \mathcal{M}_{a,b}(n)$ satisfies $x_0 \not \emphquiv 0 \mathfrak{p}mod{\emphll^{n-a}}$, $v_\emphll(x_0^2)=v_\emphll(d)$ and $v_\emphll(x_0^2-dy_0^2)>v_\emphll(d)$, then no $M\in \mathcal{M}_{a,b}(n)$ has $x \emphquiv 0 \mathfrak{p}mod{\emphll^{n-a}}$.} From $v_\emphll(x_0^2-dy_0^2)>v_\emphll(d)$ we deduce $b>v_\emphll(d)$. Supposing that such an $M$ exists, let $X,Y$ be lifts of $x, y$ to $\mathbb{Z}_\emphll$ such that $v_\emphll(X^2-dY^2)=b$.
Since $y$ must be a unit, $v_\emphll(X^2-dY^2)>v_\emphll(d)$ implies $v_\emphll(X^2)=v_\emphll(dY^2)=v_\emphll(d)$. We deduce $v_\emphll(x_0)=v_\emphll(X) \gammaeqslant v_\emphll(x)$, which contradicts $v_\emphll(x_0) < n-a \leqslant v_\emphll(x)$.
\emphmph{Finally, if all $M \in \mathcal{M}_{a,b}(n)$ satisfy $x \not \emphquiv 0 \mathfrak{p}mod{\emphll^{n-a}}$, $v_\emphll(x^2)=v_\emphll(d)$ and $v_\emphll(x^2-dy^2)>v_\emphll(d)$ then the number of lifts of $M$ to $\mathcal{M}_{a,b}(n+1)$ only depends on $G,d,n,a,b$.} The hypotheses imply $v_\emphll(x^2)=v_\emphll(dy^2)$ and hence $y$ is a unit, otherwise neither $x$ nor $y$ would be units. We can write
$d=\emphll^{2k} \delta$, $x=\emphll^{k} u$ and $X = \emphll^k U$ where $\delta,u,U$ are units.
We are counting the reductions modulo $\emphll^{n-a+1}$ of the pairs $(X,Y)\in \mathbb{Z}_\emphll^2$ that satisfy:
begin{equation}\label{syss}
begin{cases}
U \emphquiv u \mathfrak{p}mod{\emphll^{n-a-k}} \\
Y \emphquiv y \mathfrak{p}mod{\emphll^{n-a}} \\
v_\emphll(U^2- \delta Y^2)=b-2k.
\emphnd{cases}
\emphnd{equation}
{Consider the case where $\emphll$ is odd.} If $b-2k\leqslant n-a-k$, the third condition of \emphqref{syss} is a consequence of the first two because it only depends on $U,Y$ through $u,y$ (since by assumption it holds for {some} lifts, it then holds for {all} lifts). So $M$ has $\emphll^2$ lifts to $\mathcal{M}_{a,b}(n+1)$. Now suppose that $b-2k> n-a-k$.
We know that $\delta$ is a square in $\mathbb{Z}_\emphll^\times$ because $\emphll \mid u^2-\delta y^2$ and $\emphll \nmid y$. Since $\emphll$ is odd, we may assume without loss of generality that $u-\sqrt{\delta}y\emphquiv 0 \mathfrak{p}mod{\emphll}$ and $u+\sqrt{\delta}y\not\emphquiv 0 \mathfrak{p}mod{\emphll}$.
We may then rewrite the third condition of \emphqref{syss} as
begin{equation}\label{syss2}
U-\sqrt{\delta}\, Y \emphquiv 0 \mathfrak{p}mod{\emphll^{b-2k}}, \mathfrak{q}quad U-\sqrt{\delta}\, Y \not \emphquiv 0 \mathfrak{p}mod{\emphll^{b-2k+1}}.
\emphnd{equation}
If we choose $(Y bmod {\emphll^{n-a+1}})$ arbitrarily among the lifts of $y$, \emphqref{syss2} uniquely determines the value of $(U bmod {\emphll^{n-a-k+1}})$, so $M$ has $\emphll$ lifts to $\mathcal{M}_{a,b}(n+1)$.
Now consider the case $\emphll=2$. If $b-2k \leqslant n-a-k+1$ there are $4$ lifts for $M$ to $\mathcal{M}_{a,b}(n+1)$ because again the third condition of \emphqref{syss} is a consequence of the first two: notice that $(u bmod 2^{n-a-k})$ determines $(u^2 bmod 2^{n-a-k+1})$, and likewise for $y$. Suppose instead that $b-2k > n-a-k+1$. If $\delta$ is a square in $\mathbb{Z}_2^\times$ we can proceed as for $\emphll$ odd, where we may suppose $v_2(U-\sqrt{\delta}Y)=b-2k-1$ and $v_2(U+\sqrt{\delta}Y)=1$ because $U-\sqrt{\delta}Y$ and $U+\sqrt{\delta}Y$ are even and not both divisible by 4. Thus $M$ has 2 lifts to $\mathcal{M}_{a,b}(n+1)$. Finally, suppose that $\delta$ is not a square in $\mathbb{Z}_2^\times$, i.e. $\delta \not \emphquiv 1 \mathfrak{p}mod 8$.
For all $X,Y\in \mathbb{Z}_2$ lifting $x,y$ we know that $Y$ is odd, and we have $$v_2(X^2-dY^2)=2k+v_2(U^2-\delta Y^2)=2k+ begin{cases} 1, \text{ if }\delta \emphquiv 3 \mathfrak{p}mod 4 \\ 2, \text{ if } \delta \emphquiv 5 \mathfrak{p}mod 8\,. \emphnd{cases}$$
Since $v_2(X^2-dY^2)$ is independent of $X,Y$, the matrix $M$ has $4$ lifts to $\mathcal{M}_{a,b}(n+1)$.
\emphnd{proof}
\subsection{Normalizers of ramified Cartan subgroups}
Recall from Proposition \ref{complement} that $\mathcal{M}^*_{a,b}=\emphmptyset$ if $\emphll$ is odd and $a>0$, or if $\emphll=2$ and $a>1$.
begin{thm}\label{thm:LiftsNormalizer}
Let $G$ be open in the normalizer of a ramified Cartan subgroup $C$ of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$.
Let $n_0$ be the level of $G$. Assume $a=0$ if $\emphll$ is odd, and $a\in \{0,1\}$ if $\emphll=2$. Let $n\gammaeqslant 1$.
If $\emphll$ is odd, define $\mathcal{N}_{a,b}(n)$ as the subset of $G(n) \setminus C(n)$ consisting of those matrices $M$ that satisfy the following conditions:
begin{itemize}
\item[$bullet$] $\det_\emphll(M-I) \gammaeqslant n$, if $n \leqslant b$; $\det_\emphll(M-I)=b$, if $n>b$.
\emphnd{itemize}
If $\emphll=2$, define $\mathcal{N}_{a,b}(n)$ as the subset of $G(n) \setminus C(n)$ consisting of those matrices $M$ that satisfy the following conditions:
begin{itemize}
\item[$bullet$] $M \emphquiv I \mathfrak{p}mod{2^a}$, $M \not \emphquiv I \mathfrak{p}mod{2^{a+1}}$;
\item[$bullet$] $\det_2(M-I) \gammaeqslant n+1$, if $n<2a+b$; $\det_2(M-I) = 2a+b$, if $n \gammaeqslant 2a+b$.
\emphnd{itemize}
Define for $\emphll$ odd and $\emphll=2$ respectively:
\[
f(n)=\left\{begin{array}{ll}
\emphll & \text{if $n<b$} \\
\emphll(\emphll-1) & \text{if $n=b$} \\
\emphll^2 & \text{if $n>b$}
\emphnd{array}\right.
\mathfrak{q}quad
f(n) = \left\{begin{array}{ll}
2 & \text{if $ n < 2a+b$} \\
4 & \text{if $n \gammaeqslant 2a+b$\,.}
\emphnd{array}\right.
\]
begin{enumerate}
\item[(i)] For every $n \gammaeqslant n_0$ we have $\#\mathcal{N}_{a,b}(n+1)=f(n)\cdot \#\mathcal{N}_{a,b}(n)$. More precisely, for every matrix in $\mathcal{N}_{a,b}(n)$
the number of lifts to $\mathcal{N}_{a,b}(n+1)$ is $f(n)$.
\item[(ii)] If $n \gammaeqslant n_0$ or if $n>a+b$ we have $\mathcal{M}^*_{a,b}(n)=\mathcal{N}_{a,b}(n)$.
\emphnd{enumerate}
\emphnd{thm}
begin{proof}
We first prove (i). The parameters for $C$ are $(0,d)$, where $\emphll \mid d$ if $\emphll$ is odd, and by Lemma \ref{lemma:NormalizerAIsZero} any matrix in $G \setminus C$ is of the form
begin{equation}\label{nieuw}
M=begin{pmatrix}
x & d y \\ -y & -x
\emphnd{pmatrix}\,.
\emphnd{equation}
\textit{The case $\emphll$ odd ($n \gammaeqslant n_0$ and $a=0$).} If $b<n$, every lift of a matrix in $\mathcal{N}_{0,b}(n)$ to $G(n+1)$ is in $\mathcal{N}_{0,b}(n+1)$.
If $b\gammaeqslant n>0$ we proceed as for Theorem \ref{thm:GeneralLift}, noticing two facts: by Proposition \ref{complement} no matrix in $G\setminus C$ is congruent to the identity modulo $\emphll$; the coefficient of $t_{11}$ in \emphqref{eq:LiftsNormalizer} is nonzero because $\mathrm{det}_\emphll(M-I)>0$ gives $x^2 \emphquiv 1 \mathfrak{p}mod{\emphll}$, and we have $n_{11}+1\emphquiv x \mathfrak{p}mod{\emphll}$.
\textit{The case $\emphll=2$ ($n \gammaeqslant n_0$ and $a\in \{0,1\}$).} Remark that $(M bmod 2^n)$ determines $\det(M-I)$ modulo $2^{n+1}$. In particular, $\mathcal{N}_{a,b}(n)$ is well-defined.
Fix $M\in \mathcal{N}_{a,b}(n)$, and let $L$ be a lift of $M$ to $G(n+1)$.
Since $n\gammaeqslant 2$, we know $L \emphquiv I \mathfrak{p}mod{2^a}$ and $L \not \emphquiv I \mathfrak{p}mod{2^{a+1}}$. If $n \gammaeqslant 2a+b$, by the above remark all $4$ lifts of $M$ to $G(n+1)$ are in $\mathcal{N}_{a,b}(n+1)$. If $n < 2a+b$, we have $\det_2(M-I)\gammaeqslant n+1$ and hence $\det_2(L-I)\gammaeqslant n+1$: writing any lift of $M$ in the form $L'=L+2^n T$ with $T$ as in \emphqref{defT}, we are left to verify $\det_2(L'-I)\gammaeqslant n+2$ for $n+1<2a+b$ and $\det_2(L'-I)= n+1$ for $n+1=2a+b$. We thus study the inequality $\mathrm{det}_2\left( (L-I) + 2^n T\right) \gammaeqslant n+2$
and an explicit verification (by Lemma \ref{lemma:NormalizerAIsZero} and because $2^{n+2}\mid 2^{2n}$) shows that there are precisely two lifts in $\mathcal{N}_{a,b}(n+1)$ as claimed.
We can prove (i)$\Rightarrow$(ii) as for Theorem \ref{thm:GeneralLift}: we clearly have $f(n)\neq 0$ for all $n\gammaeqslant 1$, and we have $\mathcal{N}_{a,b}(2a+b+1)=\mathcal{M}^*_{a,b}(2a+b+1)$ because the defining conditions hold for a matrix if and only if they hold for its lifts to $G$.
\emphnd{proof}
\section{Measures related to the $1$-Eigenspace}
\subsection{The case of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ and unramified Cartan subgroups}
begin{prop}\label{prop:FinitelyManyCab}
Suppose that $G$ is open either in $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ or in the normalizer of an unramified Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$. Suppose $\mathcal{M}_{a,b}\neq \emphmptyset$. We have
$\mathcal{M}_{a,b}(n_0)=\{I\}$ if $n_0 \leqslant a$ and
$\mathcal{M}_{a,b}(n_0)=\mathcal{M}_{a,n_0-a}(n_0)$ if $a<n_0 \leqslant a+b$, and in particular we have:
$$\mu_{a,b}(n_0)=\left\{
begin{array}{ll}
\#G(n_0)^{-1} & \text{if $n_0 \leqslant a $}\\
\mu_{a,n_0-a}(n_0) & \text{if $a<n_0 \leqslant a+b$\,.}\\
\emphnd{array}
\right. $$
\emphnd{prop}
begin{proof}
For $n_0 \leqslant a$ the set $\mathcal{M}_{a,b}(n_0)$ contains at most the identity and it is non-empty by the assumption on $\mathcal{M}_{a,b}$. Now suppose $a<n_0 \leqslant a+b$. We claim that $\mathcal{M}_{a,n_0-a} \neq \emphmptyset$: the statement then follows from Theorem \ref{thm:GeneralLift} (ii) because by definition $\mathcal{H}_{a,b}(n_0)=\mathcal{H}_{a,n_0-a}(n_0)$.
We prove the claim by making use of Theorem \ref{thm:GeneralLift}. The assumption $\mathcal{M}_{a,b}\neq \emphmptyset$ implies that the set $\mathcal{M}_{a,b}(n_0)=\mathcal{H}_{a,b}(n_0)=\mathcal{H}_{a,n_0-a}(n_0)$ is nonempty. Since $n_0>a$ we have $f(n_0)\neq 0$ and hence $\mathcal{H}_{a,n_0-a}(n_0+1)$ is nonempty. This set equals $\mathcal{M}_{a,n_0-a}(n_0+1)$ because $n_0+1>a+(n_0-a)$.
\emphnd{proof}
begin{prop}\label{prop:GeneralCount}
Let $G'$ be either $\mathbb{G}L_2(\mathbb{Z}_\emphll)$, an unramified Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$, or the normalizer of an unramified Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$. If $G$ is open in $G'$, we have:
\[
\mu_{a,b} = \mu_{a,b}(n_0) \cdot \left\{ begin{array}{ll}
\#\mathbb{T}^{-(a+1-n_0)}\cdot \#\mathbb{T}^\times & \text{if $n_0 \leqslant a, b=0$} \\
\#\mathbb{T}^{-(a+1-n_0)} \cdot (\#\mathbb{T}-\#\mathbb{T}^\times-1)\cdot \emphll^{-b}(\emphll-1) & \text{if $n_0 \leqslant a, b>0$} \\
\emphll^{-(a+b+1-n_0)} (\emphll-1) & \text{if $a < n_0 \leqslant a+b$} \\
1 & \text{if $n_0>a+b$}\,.
\emphnd{array}\right.
\]
We also have \[
\mu_{a,b} = [G':G]\cdot \#\mathbb{T}^{-a} \cdot \varepsilon \cdot \left\{ begin{array}{ll}
1 & \text{if $n_0 \leqslant a, b=0$} \\
\displaystyle \frac{\#\mathbb{T}-\#\mathbb{T}^\times-1}{\#\mathbb{T}^\times} \cdot \emphll^{-b} (\emphll-1) & \text{if $n_0 \leqslant a, b>0$} \\
\emphnd{array}\right.
\]
where $\varepsilon=\frac{1}{2}$ if $G'$ is the normalizer of a Cartan subgroup and $\varepsilon=1$ otherwise.
\emphnd{prop}
begin{proof}
To prove the first assertion we may suppose $\mathcal{M}_{a,b}\neq \emphmptyset$, because otherwise $\mu_{a,b}=\mu_{a,b}(n_0)=0$. The formula for $n_0>a+b$ has been proven in Lemma \ref{lemma:EverythingHasWellDefinedAB}. We have
begin{equation*}
\#\mathcal{M}_{a,b}(a+b+1)=\#\mathcal{M}_{a,b}(n_0) \mathfrak{p}rod_{j=n_0}^{a+b} \frac{\#\mathcal{M}_{a,b}(j+1)}{\#\mathcal{M}_{a,b}(j)}
\emphnd{equation*}
and by definition of $n_0$ we know $\#G(a+b+1)=\#G(n_0) \cdot \#\mathbb{T}^{a+b+1-n_0}$. We then obtain
begin{equation*}
\mu_{a,b} = \mu_{a,b}(n_0) \cdot \mathfrak{p}rod_{j=n_0}^{a+b} \#\mathbb{T}^{-1}\cdot \frac{\#\mathcal{M}_{a,b}(j+1)}{\#\mathcal{M}_{a,b}(j)}
\emphnd{equation*}
and the formulas for $n_0 \leqslant a+b$ can easily be deduced from Theorem \ref{thm:GeneralLift}.
We now turn to the second assertion.
By Theorem \ref{thm:GeneralLift} (iii), $b=0$ implies $\mathcal M_{a,b}\neq \emphmptyset$ while $b>0$ and $\mathcal M_{a,b}= \emphmptyset$ imply $\#\mathbb{T}-\#\mathbb{T}^\times-1=0$. In the latter case the formula for $\mu_{a,b}$ clearly holds, so we can assume $\mathcal M_{a,b}\neq \emphmptyset$ and hence
$\mu_{a,b}(n_0)=\#G(n_0)^{-1}$ by Proposition \ref{prop:FinitelyManyCab}.
By Remark \ref{nor} and Lemma \ref{lemma:CardinalityCartans} (respectively, by Definition \ref{LieGL})
we know that $\#G'(1)=\varepsilon^{-1}\cdot \#\mathbb{T}^\times$ and $\#G'(n_0)=\#G'(1) \cdot \#\mathbb{T}^{n_0-1}$. We conclude because we have
\[
\#G(n_0)^{-1}=[G':G]\cdot (\#G'(n_0))^{-1}=[G':G] \cdot \varepsilon \cdot (\#\mathbb{T}^\times)^{-1} \cdot \#\mathbb{T}^{1-n_0}.
\]
\emphnd{proof}
begin{exa}
Let $G$ be the inverse image in $\mathbb{G}L_2(\mathbb{Z}_2)$ of
$$
G(2)=\langle
begin{pmatrix}
3 & 3 \\ 0 & 1
\emphnd{pmatrix}, begin{pmatrix}
1 & 1 \\ 3 & 0
\emphnd{pmatrix}
\rangle \subset \mathbb{G}L_2(\mathbb{Z}/4\mathbb{Z}).
$$
Since $G$ has index $8$ and level $2$ in $\mathbb{G}L_2(\mathbb{Z}_2)$, by Proposition \ref{prop:FinitelyManyCab} we get $\mu_{a,b}(2)=1/12$ if $a\gammaeqslant 2$ and $\mu_{a,b}(2)=\mu_{a,2-a}(2)$ if $a=0,1$ and $a+b\gammaeqslant 2$. A direct computation gives $\mu_{0,0}(2)=1/3$, $\mu_{1,0}(2)=1/12$, $\mu_{0,2}(2)=1/2$ and
$\mu_{0,1}(2)=\mu_{1,1}(2)=0$. So by Proposition \ref{prop:GeneralCount} we have:
\[
\mu_{a,b}=\left\{begin{array}{lllll}
0 & \textit{ if $ a\in \{0,1\}, b=1$} \\
{1}/{3} & \textit{ if $a=b=0$} \\
{1}/{12} & \textit{ if $a=1, b=0$} \\
2^{-b} & \textit{ if $a=0, b \gammaeqslant 2$} \\
8 \cdot 2^{-4a} & \textit{ if $a \gammaeqslant 2, b=0$} \\
12 \cdot 2^{-4a-b} & \textit{ if $a \gammaeqslant 2, b>0$}\,.
\emphnd{array}\right.
\]
\emphnd{exa}
begin{lem}\label{finitea}
Suppose that $G$ is open in a Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$. Let $n_0$ be the level of $G$. For all $a \gammaeqslant n_0$ we have $\mu_{a,b}=\emphll^{-2(a-n_0)}\mu_{n_0,b}$.
\emphnd{lem}
begin{proof}
We prove that for all $a\gammaeqslant n_0$ we have $\mu(\mathcal{M}_{a+1,b})=\emphll^{-2} \mu(\mathcal{M}_{a,b})$. We claim that the map
\[
begin{array}{cccc}
\mathfrak{p}hi: & \mathcal{M}_{a,b}(a+b+2) & \to & \mathcal{M}_{a+1,b}(a+b+2) \\
& M & \mapsto & I+\emphll(M-I)
\emphnd{array}
\]
is well-defined, surjective and $\emphll^2$-to-$1$, so we have:
\[
\mu(\mathcal{M}_{a+1,b})=\frac{\#\mathcal{M}_{a+1,b}(a+b+2) }{\#G(a+b+2)} = \frac{\emphll^{-2} \#\mathcal{M}_{a,b}(a+b+2)}{\#G(a+b+2)}=\emphll^{-2} \mu(\mathcal{M}_{a,b})\,.
\]
We are left to prove the claim. Since $a\gammaeqslant n_0$, all matrices in the Cartan subgroup that are congruent to the identity modulo $\emphll^a$ are in $G$ thus we may suppose that $G$ is the Cartan subgroup.
A matrix $M\in G(a+b+2)$ is in the domain of $\mathfrak{p}hi$ if and only if the conditions in \emphqref{conditions-Mab} hold, and these imply that $\mathfrak{p}hi(M)$ is in $\mathcal{M}_{a+1,b}(a+b+2)$ by \emphqref{nfaa} and because we have:
$$\mathfrak{p}hi(M) \emphquiv I \mathfrak{p}mod{\emphll^{a+1}}\mathfrak{q}quad \mathfrak{p}hi(M )\not \emphquiv I \mathfrak{p}mod{\emphll^{a+2}}\mathfrak{q}quad \mathrm{det}_{\emphll}(\mathfrak{p}hi(M)-I)=2(a+1)+b.$$ If $N$ is in the codomain of $\mathfrak{p}hi$ then $I+\emphll^{-1}(N-I)$ is well-defined modulo $\emphll^{a+b+1}$: by Lemma \ref{condi-det} this matrix belongs to $\mathcal{M}_{a,b}(a+b+1)$ and if $M$ is any lift of it to $\mathcal{M}_{a,b}(a+b+2)$ we have $\mathfrak{p}hi(M)=N$.
This proves that $\mathfrak{p}hi$ is surjective (we may suppose that domain and codomain are nonempty, otherwise they must both be empty and the statement holds trivially). The set of preimages of $N$ consists of the matrices in $\mathcal{M}_{a,b}(a+b+2)$ congruent to $M$ modulo $\emphll^{a+b+1}$, thus there are $\emphll^2$ such preimages by Theorem \ref{thm:GeneralLift} (i)-(ii).
\emphnd{proof}
begin{rem}\label{rem:emptyness}
For every $a,b\gammaeqslant 0$ we have $\mathcal M_{a,b}(\mathbb{G}L_2(\mathbb Z_\emphll))\neq \emphmptyset$ because this set contains
\[\mathfrak{q}quad begin{pmatrix} 2 & 1 \\ 1 & 1 \emphnd{pmatrix} \mathfrak{q}uad\text{for $a=b=0$ ,\; and}\mathfrak{q}quad
begin{pmatrix} 1 & \emphll^{a+b}\\ \emphll^a & 1 \emphnd{pmatrix}\mathfrak{q}quad\text{otherwise}.\]
If $C$ is a split Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$, we have $\mathcal M_{a,b}(C)\neq \emphmptyset$ for every $a,b\gammaeqslant 0$, with the exception of $\emphll=2$ and $a=0$:
considering the diagonal model, if $\emphll \neq 2$ or if $a \gammaeqslant 1$ the set $\mathcal M_{a,b}(C)$ contains
$
begin{pmatrix}
1+ \emphll^{a}& 0\\
0 & 1+ \emphll^{a+b}
\emphnd{pmatrix};
$
however, for $\emphll=2$ every diagonal invertible matrix is congruent to the identity modulo $2$.
If $C$ is a nonsplit Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$, we have $\mathcal M_{a,b}(C)= \emphmptyset$ for every $b>0$.
Indeed, if $M\in \mathcal M_{a,b}(C)$ then for $\emphll$ odd (resp. $\emphll=2$) we have
$$\emphll^{-a}(M-I)= begin{pmatrix} z & dw \\ w & z \emphnd{pmatrix}\mathfrak{q}quad \text{resp.}\mathfrak{q}quad 2^{-a}(M-I)= begin{pmatrix} z & dw \\ w & z+w \emphnd{pmatrix} $$
for some $z,w\in \mathbb{Z}_{\emphll}$, and by Propositions \ref{prop:ConditionsCDodd} and \ref{prop:ConditionsCD2} these matrices are invertible unless $z$ and $w$ are zero modulo $\emphll$.
\emphnd{rem}
\subsection{Ramified Cartan subgroups}
begin{lem}\label{lem:ReductionCartan}
Suppose that $G$ is open in a Cartan subgroup $C$ of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ with parameters $(0,d)$.
Write $d=m \emphll^{v}$ with $\emphll\nmid m$.
begin{enumerate}
\item[\textit{(i)}] If $v$ is odd, we have $\mu_{a,b}=0$ for every $b>v$.
\item[\textit{(ii)}] If $v$ is even and $m$ is not a square in $\mathbb{Z}_\emphll^\times$, we have $\mu_{a,b}=0$ for every $b>v+2$ (if $\emphll$ is odd we have $\mu_{a,b}=0$ for $b>v$).
\item[\textit{(iii)}] If $v$ is even and $m$ is a square in $\mathbb{Z}_\emphll^\times$, consider the Cartan subgroup $C'$ of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ with parameters $(0,1)$. There exists a closed subgroup $G_1$ of $C'$ such that the following holds: there is an explicit isomorphism between $G$ and $G_1$; the level of $G_1$ does not exceed the level of $G$ by more than ${v}/{2}$;
for all $b>v$ the sets $\mathcal{M}_{a, b}(G)$ and
$\mathcal{M}_{a+{v}/{2}, b-v}(G_1)$ have the same Haar measure in $G$ and $G_1$ respectively.
\emphnd{enumerate}
\emphnd{lem}
begin{proof}
Fix $n>a+b+1$. By \emphqref{nfaa} we can write any matrix in $ \mathcal M_{a,b}(n)$ as
$$M=I+\emphll^a begin{pmatrix}
x & dy \\ y & x
\emphnd{pmatrix}
$$
where $x,y\in \mathbb{Z}/\emphll^{n-a}\mathbb{Z}$ satisfy $v_\emphll(x^2-dy^2)=b$ and we have $v_\emphll(x)=0$ or $v_\emphll(y)=0$.
\emphmph{Proof of (i):} We have $v_\emphll(x^2)\neq v_\emphll(dy^2)$ and hence
$v_\emphll(x^2-dy^2)\leqslant v$, which implies that $\mathcal{M}_{a,b}(n)$ is empty for $b>v$.
\emphmph{Proof of (ii):} For $b>v$ we have $v_\emphll(x^2)\gammaeqslant v$ and we can write $b=v + v_\emphll(x_1^2-my^2)$, where $x=\emphll^{v/2} x_1$. We must have $x_1^2 \emphquiv my^2 \mathfrak{p}mod {\emphll}$, which is impossible for $\emphll$ odd because $m$ is not a square modulo $\emphll$.
If $\emphll=2$ and $b>v+2$ we should similarly have $x_1^2 \emphquiv my^2 \mathfrak{p}mod {8}$, which is impossible because $m$ is not a square modulo $8$.
\emphmph{Proof of (iii):} Since $d$ is a square in $\mathbb{Z}_\emphll$, we may fix a square root $\sqrt{d}$ of it. Define $G_1$ to be the image of
\[
begin{array}{cccc}
\mathfrak{p}hi: & G & \to & C'
\\ & I+\emphll^a begin{pmatrix}
x & d y \\ y & x
\emphnd{pmatrix} & \mapsto & I+\emphll^a begin{pmatrix}
x & y \sqrt{d} \\ y \sqrt{d} & x
\emphnd{pmatrix}\,.
\emphnd{array}
\]
Embedding $G$ in $\mathbb{G}L_2(\mathbb Q_{\emphll})$, the map $\mathfrak{p}hi$ can be identified with the conjugation by $begin{pmatrix}
1 & 0 \\ 0 & \sqrt{d}
\emphnd{pmatrix}
$, thus $\mathfrak{p}hi$ is a continuous group isomorphism between $G$ and $G'$, and for every $M\in G$ we have $\det \mathfrak{p}hi(M)=\det(M)$ and $\det (\mathfrak{p}hi(M)-I)=\det(M-I)$.
Let $n_0$ denote the level of $G$. The level of $G_1$ is at most $n_0+v/2$ because any matrix in $C'$ which is congruent to the identity modulo $\emphll^{n_0+v/2}$ is the image via $\mathfrak{p}hi$ of a matrix in $C$ that is congruent to the identity modulo $\emphll^{n_0}$:
$$I+\emphll^{n_0} begin{pmatrix} \emphll^{v/2} z & y d \\ y & \emphll^{v/2} z \emphnd{pmatrix} \stackrel{\mathfrak{p}hi}{\longrightarrow} I+\emphll^{n_0+v/2} begin{pmatrix} z & y\sqrt{m} \\ y\sqrt{m} & z \emphnd{pmatrix}\,. $$
If $b>v$, we have $v_\emphll(x^2)\gammaeqslant v$ and a straightforward verification shows that $\mathfrak{p}hi$ induces a bijection from $\mathcal{M}_{a,b}(G)$ to $\mathcal{M}_{a+v/2,b-v}(G_1)$, and we conclude because a continuous isomorphism of profinite groups preserves the normalized Haar measure, see \cite[Proposition 18.2.2]{MR2445111}.\emphnd{proof}
begin{lem}\label{lem:FakeSplitCartan}
If $G$ is open in the Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_2)$ with parameters $(0,1)$ the sets $\mathcal{M}_{a,1}$ and $\mathcal{M}_{a,2}$ are empty. Moreover, there exists an open subgroup $G_1$ of the subgroup of diagonal matrices in $\mathbb{G}L_2(\mathbb{Z}_2)$ such that the following holds: there is an explicit isomorphism between $G$ and $G_1$; the level of $G_1$ does not exceed the level of $G$ by more than 1; for all $b>2$ the sets $\mathcal{M}_{a,b}(G)$ and $\mathcal{M}_{a+1,b-2}(G_1)$ have the same Haar measure in $G$ and $G_1$ respectively.
\emphnd{lem}
begin{proof}
We can write any matrix in $\mathcal{M}_{a,b}(G)$ as
begin{equation}\label{proofM}
M=I + 2^a begin{pmatrix}
x & y \\ y & x
\emphnd{pmatrix}
\emphnd{equation}
where at least one between $x$ and $y$ is a $2$-adic unit. Working modulo $8$, we see that $b=v_2(x^2-y^2)$ cannot be $1$ or $2$.
We sketch the rest of the proof, which mimics Lemma \ref{lem:ReductionCartan} (iii). We define a map $\mathfrak{p}hi$ from $G$ to $\mathbb{G}L_2(\mathbb{Z}_2)$, denoting $G_1$ its image:
begin{equation}\label{proofM'}
\mathfrak{p}hi(M)=I + 2^a begin{pmatrix}
x+y & 0 \\ 0 & x-y
\emphnd{pmatrix}.
\emphnd{equation}
We clearly have $\det_2(\mathfrak{p}hi(M)-I)=\det_2(M-I)$. If $b>2$, then $x+y$ and $x-y$ must be even and not both divisible by $4$, and it follows that $\mathfrak{p}hi(M)\in \mathcal{M}_{a+1,b-2}(G_1)$.
\emphnd{proof}
begin{lem}\label{lem:MabN0IsIndependentOfB}
Let $G$ be open in a Cartan subgroup $C$ of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$ with parameters $(0,d)$, where $d$ is a square in $\mathbb{Z}_\emphll$.
For any fixed value of $a$, the set $\mathcal{M}_{a,b}(n_0)$ does not depend on $b$ provided that $b\gammaeqslant b_0$, where $b_0:=\max\{1+v_\emphll(4d), n_0-a+v_\emphll(2d)\}$.
\emphnd{lem}
begin{proof}
Let $b \gammaeqslant b_0>0$ and consider a matrix in $\mathcal{M}_{a,b}$:
$M=I+\emphll^a begin{pmatrix}
x & dy \\ y & x
\emphnd{pmatrix}$.
It suffices to show that for every $b' \gammaeqslant b_0$ there is $M' \in \mathcal{M}_{a,b'}$ that is congruent to $M$ modulo $\emphll^{n_0}$.
We have $v_\emphll(x^2-d y^2)\gammaeqslant b>v_\emphll(d)$ and at least one among $x$ and $y$ is a unit. One checks easily that $y$ cannot be divisible by $\emphll$, so we have $v_\emphll(x^2)= v_\emphll(d)$ and we can define $y''=x/\sqrt{d}$. Given two units in $\mathbb Z_\emphll$, either their sum or their difference has valuation $v_\emphll(2)$, so up to replacing $\sqrt{d}$ by $-\sqrt{d}$ we get $v_\emphll(x-\sqrt{d} y)\gammaeqslant b-v_\emphll(d)/2-v_\emphll(2)\gammaeqslant n_0-a+v_\emphll(d)/2$ and hence $y'' \emphquiv y \mathfrak{p}mod{\emphll^{n_0-a}}$. Defining $B:=b'-v_\emphll(d)/2-v_\emphll(2)\gammaeqslant n_0-a$, the matrix
$$M'=I+\emphll^a begin{pmatrix}
x+\emphll^B & dy'' \\ y'' & x +\emphll^B
\emphnd{pmatrix}$$ is congruent to $M$ modulo $\emphll^{n_0}$ and we have $\det(M'-I)=\emphll^{2a}(2x\emphll^B+\emphll^{2B})$.
Since $B>v_\emphll(d)/2+v_\emphll(2)=v_\emphll(x)+v_\emphll(2)$ we have $\det_\emphll(M'-I)=2a+b'$ and hence $M'\in \mathcal{M}_{a,b'}$.
\emphnd{proof}
\subsection{Normalizers of Cartan subgroups}
Recall the notation from Section \ref{star}.
begin{thm}\label{thm:CountCPrime}
Let $G$ be open in the normalizer of a Cartan subgroup of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$. Let $n_0$ be the level of $G$.
begin{enumerate}
\item[(i)] If $\emphll$ is odd or $C$ is unramified, we have:
$$\mu(\mathcal M^*_{a,b})= \left\{
begin{array}{lllll}
0 & \text{if $a>0$} \\
\mu(\mathcal M^*_{0,b}(n_0)) & \text{if $a=0, b < n_0$} \\
\mu(\mathcal M^*_{0,n_0}(n_0)) \cdot (\emphll-1) \cdot \emphll^{n_0 -b- 1} & \text{if $a=0, b \gammaeqslant n_0$\,.}
\emphnd{array}\right.$$
\item[(ii)] If $\emphll=2$ and $C$ is ramified, we have:
$$\mu(\mathcal M^*_{a,b})= \left\{
begin{array}{lllll}
0 & \text{if $a> 1$} \\
\mu(\mathcal M^*_{a,b}(n_0)) & \text{if $a\leqslant 1$ and $2a+b \leqslant n_0$} \\
\mu(\mathcal M^*_{a,n_0+1}(n_0)) \cdot 2^{n_0-2a-b} & \text{if $a\leqslant 1$ and $2a+b > n_0$\,.}
\emphnd{array}\right.$$
\emphnd{enumerate}
\emphnd{thm}
begin{proof}
For the first assertion, by Proposition \ref{complement} we have $\mu(\mathcal M^*_{a,b})=0$ for $a>0$.
For $b<n_0$ we clearly have $\mu(\mathcal M^*_{0,b})=\mu(\mathcal M^*_{0,b}(n_0))$.
If $b \gammaeqslant n_0$, Theorem \ref{thm:LiftsNormalizer} (ii) implies
\[
\mathcal{M}^*_{0,b}(n_0) = \{ M \in (G \setminus C)(n_0) bigm\vert \mathrm{det}_\emphll(M-I) \gammaeqslant n_0 \}=\mathcal{M}_{0,n_0}^*(n_0)
\]
so in particular we have $\mu(\mathcal M^*_{0,b}(n_0))=\mu(\mathcal M^*_{0,n_0}(n_0))$. We conclude by Theorem \ref{thm:GeneralLift} and \ref{thm:LiftsNormalizer} respectively, by the same argument used to prove Proposition \ref{prop:GeneralCount}.
The second assertion follows analogously from Proposition \ref{complement} and Theorem \ref{thm:LiftsNormalizer}. Indeed, if $2a+b > n_0$ then $\mathcal{M}^*_{a,b}(n_0)$ is independent of $b$ by Theorem \ref{thm:LiftsNormalizer} (ii).
\emphnd{proof}
begin{cor}\label{cor:AsymptoticGrowthCartan}
Let $G$ be open in the normalizer $N$ of a Cartan subgroup $C$ of $\mathbb{G}L_2(\mathbb{Z}_\emphll)$.
For $a \in \{0,1\}$ there exist (effectively computable) rational numbers $c_1(a), c_2(a), c_3(a)$ such that
\[
\mu(\mathcal{M}_{a,b}) = c_1(a) \emphll^{-b}, \mathfrak{q}quad \mu( \mathcal{M}_{a,b} \cap (N\setminus C)) = c_2(a) \emphll^{-b}, \mathfrak{q}quad \mu( \mathcal{M}_{a,b} \cap C) = c_3(a) \emphll^{-b}
\]
hold for all sufficiently large $b$ (and the bound is effective). The rational constants $c_i(a)$ may depend on $\emphll$ and $G$, as well as on $a$.
\emphnd{cor}
begin{proof}
The assertion for $\mathcal{M}_{a,b}$ follows from the other two, and the assertion for $\mathcal{M}_{a,b} \cap (N\setminus C)$ holds by Theorem \ref{thm:CountCPrime}.
Now consider $\mathcal{M}_{a,b} \cap C$. Because of Lemmas \ref{lem:ReductionCartan} and \ref{lem:FakeSplitCartan}, we only need to consider the case when $C$ is a split Cartan subgroup. We apply Proposition \ref{prop:FinitelyManyCab} (in view of Remark \ref{rem:emptyness}) to deduce that $\mu_{a,b}(n_0)$ is constant for $b\gammaeqslant n_0$ and then apply Proposition \ref{prop:GeneralCount}.
\emphnd{proof}
\section{The results of the Introduction}
\subsection{Proof of Theorem \ref{main-thm}}
begin{defi}\label{admissible}
We call a subset of $\mathbb N^2$ \emphmph{admissible} if it is the product of two subsets of $\mathbb N$, each of which is either finite or consists of all integers greater than some given one. The family of finite unions of admissible sets is closed w.r.t. intersection, union and complement.
\emphnd{defi}
We describe a general computational strategy to determine $\mu_{a,b}$ for all $a,b\gammaeqslant 0$.
Depending on the input data (i.e.\@ a finite amount of information about the group $G$), we can choose which of the previous results must be applied, and we can compute the finitely many rational parameters that appear in the statements. After a case distinction, we have formulas for all measures $\mu_{a,b}$ that depend only on $a$, $b$, and finitely many known constants. As it can be seen from the explicit description below, the cases give a partition of $\mathbb N^2$ into finitely many admissible subsets and on each of them the formula for $\mu_{a,b}$ is as requested.
We first need to express the relevant properties of $G$ in terms of finitely many parameters:
begin{enumerate}
\item The group $G$ is open in $G'$, which is either $\mathbb{G}L_2(\mathbb{Z}_\emphll)$,
a Cartan subgroup, or the normalizer of a Cartan subgroup. We describe a Cartan subgroup with the integer parameters $(c,d)$ of Section \ref{subsec-classification}, which also determine whether this is split, nonsplit or ramified. The cardinality of the tangent space $\mathbb T$ and of its subset $\mathbb T^{\times}$ is known, see Section \ref{subsec-Lie}.
\item We fix an integer $n_0 \gammaeqslant 1$ such that $G$ is the inverse image in $G'$ of $G(n_0)$ for the reduction modulo $\emphll^{n_0}$. If $\emphll=2$ and $G'$ is (the normalizer of) a ramified Cartan subgroup, we take $n_0 \gammaeqslant 2$ ($n_0$ is not necessarily the level of $G$, see Remark \ref{rem:Level}).
\item We need to know the finite group $G(n_0)$ explicitly. From this we extract various data, including the order of $G(n_0)$, the index $[G'(n_0): G(n_0)]=[G':G]$, and the following information: for each of the finitely many pairs $(a,b)$ such that $a<n_0$ and $b \leqslant n_0-a$, we need to know the counting measure $\mu_{a,b}(n_0)$ and whether the set $G(n_0)\cap \mathcal M_{a,b}(G'; n_0)$ is empty or not. For (normalizers of) ramified Cartan subgroups we may also need finitely many other quantities which can all be read off $G(n_0)$, see the description below.
\emphnd{enumerate}
We make repeated use of the following remark: suppose that for $(a,b)$ in some admissible set $S=A \times B$ with $A$ finite we have $\mu_{a,b}=c(a) \emphll^{-b}$, where $c(a)$ is a rational number depending on $a$. Then $S$ is the finite union of the sets $S_a=\{a\} \times B$, and by choosing the constant $c'(a)$ appropriately we have $\mu_{a,b} = c'(a) \emphll^{-a \dim(G)-b}$ for all $(a,b) \in S_a$.
\emphmph{If $G'=\mathbb{G}L_2(\mathbb{Z}_\emphll)$:}
We can compute the values $\mu_{a,b}$ for all pairs $(a,b)$ such that $\mathcal M_{a,b}\neq \emphmptyset$ by Propositions \ref{prop:FinitelyManyCab} and \ref{prop:GeneralCount}.
Up to refining the partition, we can ensure that $\mu_{a,b}$ is a constant multiple of $\emphll^{-4a-b}$ on every set of the partition.
We are left to determine the pairs $(a,b)$ such that $\mathcal M_{a,b}= \emphmptyset$ (and hence $\mu_{a,b}=0$) and show that they form an admissible subset of $\mathbb N^2$.
By Remark \ref{rem:emptyness} we know $\mathcal{M}_{a,b}(G') \neq \emphmptyset$, and by Remark \ref{vuoto} we just need to know whether $G(n_0)\cap \mathcal M_{a,b}(G'; n_0)$ is empty. By Proposition \ref{prop:FinitelyManyCab} (applied to $G'$) there are only finitely many distinct sets of the form $\mathcal M_{a,b}(G'; n_0)$ to consider and it is a finite computation to determine those that intersect $G(n_0)$ trivially.
\emphmph{If $G'$ is a nonsplit Cartan subgroup:}
By Lemma \ref{finitea} and Remark \ref{rem:emptyness} we reduce to the case $a\leqslant n_0$ and $b=0$. Thus by Proposition \ref{prop:GeneralCount} we only need to evaluate $\mu_{a,0}(n_0)$ for $a \leqslant n_0$. Since we only have finitely many values of $a$ to consider, up to refining the partition we find that $\mu_{a,b}$ is a constant multiple of $\emphll^{-2a-b}$ on every set of the partition.
\emphmph{If $G'$ is a split Cartan subgroup:}
By Lemma \ref{finitea} we reduce to the case $a\leqslant n_0$, so fix one of those finitely many values for $a$. By Proposition \ref{prop:GeneralCount} it suffices to evaluate $\mu_{a,b}(n_0)$ for all $b\gammaeqslant 0$. If $\mathcal M_{a,b}\neq \emphmptyset$, by Proposition \ref{prop:FinitelyManyCab} we only need to consider the finitely many cases for which $b\leqslant n_0$.
We are left to determine the pairs $(a,b)$ such that $\mathcal M_{a,b}= \emphmptyset$ (and hence $\mu_{a,b}=0$) and show that they form an admissible subset of $\mathbb N^2$. By Remark \ref{rem:emptyness} the set $\mathcal{M}_{a,b}(G')$ is empty (and hence $\mathcal M_{a,b}= \emphmptyset$) if and only if $\emphll=2$ and $a=0$. In the remaining cases we have $\mathcal{M}_{a,b}(G') \neq \emphmptyset$ and $a\leqslant n_0$, so we may reason as for $G'=\mathbb{G}L_2(\mathbb{Z}_\emphll)$.
Up to refining the partition, we find that $\mu_{a,b}$ is a constant multiple of $\emphll^{-2a-b}$ on every set of the partition.
\emphmph{If $G'$ is a ramified Cartan subgroup:}
By Lemma \ref{finitea} we reduce to the case $a\leqslant n_0$, so fix one of these finitely many values for $a$. By Propositions \ref{prop:ConditionsCDodd} and \ref{prop:ConditionsCD2}, the parameters for $C$ are $(0,d)$ and we can apply Lemma \ref{lem:ReductionCartan}.
If we are in cases (i)-(ii) of this lemma, we only need to consider the finitely many values $b\leqslant v_\emphll (d)+2$. The measure $\mu_{a,b}$ for a single pair $(a,b)$ can be computed explicitly as $\mu_{a,b}(a+b+1)$. Notice that the group $G(a+b+1)$ and hence its subset $\mathcal{M}_{a,b}(a+b+1)$ can be determined from the knowledge of $G'$ and $G(n_0)$.
Now suppose that we are in case (iii) of Lemma \ref{lem:ReductionCartan}. Recalling that $a\leqslant n_0$ is fixed, we may compute the finitely many measures $\mu_{a,b}$ where $b\leqslant v_\emphll (d)$. For $b> v_\emphll (d)$ we reduce to a similar problem for an unramified Cartan subgroup: if $\emphll$ is odd, the Cartan subgroup with parameters $(0,1)$ is unramified; if $\emphll=2$ we further apply Lemma \ref{lem:FakeSplitCartan}.
Once more, this gives a partition as requested.
\emphmph{The case when $G'$ is the normalizer of a Cartan subgroup:}
As shown in Section \ref{star}, to reduce to the case when $G'$ is a Cartan subgroup it suffices to compute the measures $\mu(\mathcal M^*_{a,b})$ for all $a,b\gammaeqslant 0$. We achieve this by Theorem \ref{thm:CountCPrime}: it suffices to compute $\mu(\mathcal M^*_{a,b}(n_0))$ for finitely many pairs $(a,b)$.
The measures in the Cartan subgroup and those related to its complement in the normalizer add up to an expression of the desired form, because they can both be written as $\emphll^{-2a-b}$ times a constant.
\subsection{The special case where $G$ has index $1$}
begin{proof}[Proof of Theorem \ref{thm-countGL2}]
Consider Definition \ref{LieGL} and Proposition \ref{prop:GeneralCount}. The cases with $a>0$ are clear because $n_0=1 \leqslant a$. If $a=b=0$, we have $\mu_{0,0}=\mu_{0,0}(1)=\#\mathcal{M}_{0,0}(1)/\#\mathbb{G}L_2(\mathbb{Z}/\emphll\mathbb{Z})$ so it suffices to prove
$\#\mathcal{M}_{0,0}(1)=\emphll(\emphll^3-2\emphll^2-\emphll+3)\,.$
Equivalently, we have to show that there are $\#\mathbb{G}L_2(\mathbb{Z}/\emphll\mathbb{Z})-\#\mathcal{M}_{0,0}(1)=\emphll^3-2\emphll$ matrices in $\mathbb{G}L_2(\mathbb{Z}/\emphll\mathbb{Z})$ that have $1$ as an eigenvalue. This is done e.g. in the course of \cite[Proof of Theorem 5.5]{MR2640290} (see also \cite[Section 4]{MR1995144}), but for the convenience of the reader we sketch the computation. Matrices admitting 1 as an eigenvalue are the identity and those that are conjugate to one of the following:
$$J_1=begin{pmatrix} 1& 1\\ 0& 1\emphnd{pmatrix}\mathfrak{q}quad J_{\lambda}=begin{pmatrix} 1& 0\\ 0& \lambda \emphnd{pmatrix}\mathfrak{q}quad \lambda \neq 0,1\,.$$
Since the centralizer of $J_1$ has size $\emphll(\emphll-1)$ while that of $J_\lambda$ has size $(\emphll-1)^2$,
we may conclude by computing the size of the conjugacy classes as the index of the centralizer.
If $a=0$ and $b>0$, we have to evaluate $(\emphll-1)\cdot \emphll^{-b}\cdot \mu_{0,b}(1)$ for $b>0$. By Proposition \ref{prop:FinitelyManyCab} and Remark \ref{rem:emptyness} we have
$\mu_{0,b}(1)=\mu_{0,1}(1)=\#\mathcal{M}_{0,1}(1)/\#\mathbb{G}L_2(\mathbb{Z}/\emphll\mathbb{Z})$ so it suffices to prove
$$\#\mathcal{M}_{0,1}(1)=(\emphll^2-\emphll-1)(\emphll+1)\,.$$
We may conclude by noticing that $\mathcal{M}_{0,1}(1)$ consists of the $\emphll^3-2\emphll$ matrices in $\#\mathbb{G}L_2(\mathbb{Z}/\emphll\mathbb{Z})$ that have $1$ as an eigenvalue, with the exception of the identity matrix.
\emphnd{proof}
begin{proof}[Proof of Theorem \ref{thm-countsplitnonsplitCartan}]
We can take $n_0=1$ so the cases with $a\gammaeqslant 1$ follow immediately from Proposition \ref{prop:GeneralCount} and \emphqref{eqprop:XB}. Now suppose $a=0$.
Consider a split Cartan subgroup with the diagonal model. For $b=0$, in order to evaluate $\mu_{0,0}(1)$ we count those diagonal matrices $\operatorname{diag}(x,y)$ such that $xy$ and $(x-1)(y-1)$ are in $(\mathbb{Z}/\emphll\mathbb{Z})^\times$: this means $x, y \not \emphquiv 0, 1 \mathfrak{p}mod \emphll$, hence there are $(\emphll-2)^2$ choices, out of $(\emphll-1)^2$ total elements.
For $b>0$ we have $\mu_{0,b}(1)=\mu_{0,1}(1)$ by Proposition \ref{prop:FinitelyManyCab} and Remark \ref{rem:emptyness}. There are $2(\emphll-2)$ diagonal matrices in $\mathbb{G}L_2(\mathbb{Z}/\emphll\mathbb{Z})$ such that exactly one of the two diagonal entries is congruent to 1 modulo $\emphll$, so we get by Proposition \ref{prop:GeneralCount}:
\[
\mu_{0,b}=\mu_{0,b}(1) \emphll^{-b} (\emphll-1) = \frac{2(\emphll-2)}{(\emphll-1)^2} (\emphll-1) \emphll^{-b}\,.
\]
Now consider the nonsplit case. By Remark \ref{rem:emptyness} we know $\mathcal{M}_{a,b}=\emphmptyset$ for $b>0$. For $b=0$ we need to evaluate $\mu_{0,0}$: by Lemma \ref{lemma:EverythingHasWellDefinedAB} and by the previous case $a \gammaeqslant 1$ we have
\[
1 = \sum_{a,b \gammaeqslant 0} \mu_{a,b} = \sum_{a \gammaeqslant 0} \mu_{a,0}= \mu_{0,0} + \sum_{a \gammaeqslant 1} \emphll^{-2a}\,.
\]
\emphnd{proof}
begin{proof}[Proof of Theorem \ref{thm-countnormalizerCartan}]
Let $C$ be the Cartan subgroup and let $C'$ be as in Lemma \ref{lem-Norm}. Fixing some $n> a+b$ we get
$$\mu_{a,b}=\mu_{a,b}(n)=\frac{\#\mathcal{M}_{a,b}(n)}{\#(C\cup C')(n)}=\frac{\#(\mathcal{M}_{a,b}(n)\cap C(n))+\#(\mathcal{M}_{a,b}(n)\cap C'(n))}{2\cdot \#C(n)}\,.$$
By definition we have $\mu^C_{a,b}=\#(\mathcal{M}_{a,b}(n)\cap C(n))/\#C(n)$, so it suffices to show $\mu^*_{a,b}=\#(\mathcal{M}_{a,b}(n)\cap C'(n))/\#C(n)$.
If $a>0$ then no matrix in $C'(n)$ is in $\mathcal{M}_{a,b}(n)$ by Lemma \ref{lemma:NormalizerAIsZero}. For $a=0$ we are left to prove that for $n= b+1$ we have $\#\mathcal{M}_{0,b}(n)\cap C'(n)=\mu^*_{0,b} \cdot \#C(n)$. By Lemma \ref{lemma:NormalizerAIsZero} the elements of $C'(b+1)$ are those matrices of the form
begin{equation}\label{form}
M=begin{pmatrix}
\alphalpha & -d beta + c \alphalpha \\ beta & -\alphalpha
\emphnd{pmatrix} \mathfrak{q}quad \alphalpha, beta \in \mathbb{Z}/\emphll^{b+1}\mathbb{Z}
\emphnd{equation}
where $c, d$ are here the reductions modulo $\emphll^{b+1}$ of the parameters of $C$.
Thus we need to count the pairs $(\alphalpha, beta) \in (\mathbb{Z}/\emphll^{b+1}\mathbb{Z})^2$ satisfying
begin{equation}\label{proof1}
\mathrm{det}_\emphll (M-I)=v_\emphll(1-\alphalpha^2+dbeta^2 -c \alphalpha beta)=b\,.
\emphnd{equation}
We also need $\mathrm{det}_\emphll (M)=v_\emphll(-\alphalpha^2+dbeta^2 -c \alphalpha beta)=0$, which for $b>0$ follows from \emphqref{proof1}.
The count for the split case will give $(\emphll-1)(\emphll-2)$ for $b=0$ and $(\emphll-1)^2\emphll^b$ for $b>0$. The count for the nonsplit case will give $(\emphll+1)(\emphll-2)$ for $b=0$ and $(\emphll^2-1)\emphll^{b}$ for $b>0$. We then conclude by Lemma \ref{lemma:CardinalityCartans} because $\# C(b+1)$ equals $(\emphll-1)^2\emphll^{2b}$ and $(\emphll^2-1)\emphll^{2b}$ for the split and the nonsplit case respectively.
One can easily check that the affine curve $\mathcal{D}: 1-x^2+y(dy-cx)=0$ (defined over $\mathbb{Z}_\emphll$) is smooth over $\mathbb{Z}/\emphll\mathbb{Z}$. We have
$\#\mathcal{D}(\mathbb{Z}/\emphll\mathbb{Z})=\emphll \mathfrak{p}m 1$, where the sign is $-$ (resp.~$+$) if $C$ is split (resp.~nonsplit). Indeed, $\mathcal{D}$ can be identified over $\mathbb{Z}/\emphll\mathbb{Z}$ with the open subscheme of $\{Z^2-X^2+Y(dY-cX)=0\} \cong \mathbb{P}^1$ given by $Z \neq 0$, and by Propositions \ref{prop:ConditionsCDodd} and \ref{prop:ConditionsCD2} there are two (resp. zero) $\mathbb{Z}/\emphll\mathbb{Z}$-points with $Z=0$ if $C$ is split (resp. nonsplit).
\emphmph{The case $b=0$.} There are precisely $\emphll^2-(\emphll \mathfrak{p}m 1)$ pairs $(\alphalpha, beta)\in (\mathbb{Z}/\emphll \mathbb{Z})^2$ that do {not} correspond to points in $\mathcal{D}(\mathbb{Z}/\emphll\mathbb{Z})$. Since we only want invertible matrices, we need to exclude those pairs such that $-\alphalpha^2+dbeta^2-c\alphalphabeta =0$.
By Propositions \ref{prop:ConditionsCDodd} and \ref{prop:ConditionsCD2} this equation has $2\emphll-1$ solutions if $C$ is split and has only the trivial solution $\alphalpha= beta=0$ if $C$ is nonsplit.
\emphmph{The case $b>0$.} As $\mathcal{D}$ is smooth over $\mathbb{F}_\emphll$, by (the higher-dimensional version of) Hensel's Lemma \cite[Proposition 7.8]{Nekovar} we have $\#\mathcal{D}(\mathbb{Z}/\emphll^b\mathbb{Z})=\emphll^{b-1}\cdot \#\mathcal{D}(\mathbb{Z}/\emphll\mathbb{Z})$. A pair $(\alphalpha, beta)\in (\mathbb{Z}/\emphll^{b+1}\mathbb{Z})^2$ as in \emphqref{proof1} reduces to a point in $\mathcal{D}(\mathbb{Z}/\emphll^b\mathbb{Z})$, so it suffices to prove that there are precisely $\emphll^2-\emphll$ pairs $(\alphalpha, beta)$ as in \emphqref{proof1} that lie over some fixed $(\overline{\alphalpha}, \overline{beta}) \in \mathcal{D}(\mathbb{Z}/\emphll^b\mathbb{Z})$. There are $\emphll^2$ lifts of $(\overline{\alphalpha}, \overline{beta})$ to $(\mathbb{Z}/\emphll^{b+1}\mathbb{Z})^2$ and we must avoid those in $\mathcal{D}(\mathbb{Z}/\emphll^{b+1}\mathbb{Z})$, which are exactly $\emphll$ again by Hensel's Lemma.
\emphnd{proof}
bibliographystyle{abbrv}
bibliography{biblio}
\end{document} | math |
\begin{document}
\title{Resource analysis for quantum-aided Byzantine agreement with the four-qubit singlet state}
\author{Zolt\'an Guba}
\affiliation{Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary}
\author{Istv\'an Finta}
\affiliation{Nokia Bell Labs}
\affiliation{\'Obuda University}
\author{\'Akos Budai}
\affiliation{Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary}
\affiliation{Wigner Research Centre for Physics, H-1525 Budapest, P.O.Box 49, Hungary}
\affiliation{Nokia Bell Labs}
\author{L\'or\'ant Farkas}
\affiliation{Nokia Bell Labs}
\author{Zolt\'an Zimbor\'as}
\affiliation{Wigner Research Centre for Physics, H-1525 Budapest, P.O.Box 49, Hungary}
\affiliation{E\"otv\"os University, Budapest, Hungary}
\author{Andr\'as P\'alyi}
\affiliation{Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary}
\maketitle
\begin{abstract}
In distributed computing, a Byzantine fault is a condition
where a component behaves inconsistently, showing different symptoms to
different components of the system.
Consensus among the correct components can be reached by appropriately crafted communication protocols even in the presence of byzantine faults.
Quantum-aided protocols built upon distributed entangled quantum states are worth considering, as they are more resilient than traditional ones.
Based on earlier ideas, here we establish a parameter-dependent
family of quantum-aided weak broadcast protocols.
We compute upper bounds on the failure probability of the protocol, and define and illustrate a procedure that minimizes the quantum resource requirements.
Following earlier work demonstrating the suitability of noisy intermediate scale quantum (NISQ) devices for the study of quantum networks, we experimentally create our resource quantum state on publicly available quantum computers.
Our work highlights important engineering aspects of the future deployment of quantum communication protocols with multi-qubit entangled states.
\end{abstract}
\tableofcontents
\section{Introduction}
In distributed computing, a Byzantine fault is a condition
where a component behaves inconsistently, showing different symptoms to
different components of the system.
To reach consensus among the correctly functioning components,
communication protocols resilient to such faults must be used.
An early example of such protocols was developed by Pease et al. \cite{<pease>}.
That protocol provides consensus if $t<n/3$, where $n$ is the number of components in
the distributed system, and $t$ is the maximum number of components exhibiting Byzantine fault.
For example, if the maximum number of faulty components is 1, then at least 4 components
are needed to reach consensus.
We will refer to this property of the protocol as \emph{resilience}.
The efficiency of early protocols \cite{<pease>,<lamport>} has since
been improved \cite{<garaymoses>,<pbft>,<raft>,<paxos>}.
Quantum-aided `weak broadcast' protocols developed in the past two decades,
built upon distributed entangled quantum states \cite{fitzi_proceedings}
such as the four-qubit singlet state
\cite{cabello,cabstateprep}, are worth considering, as they offer a higher resilience:
$t$ faulty components are tolerated as long as $t < n/2$.
For example, to be resilient against $t=1$ faulty component, having $n=3$ components
is sufficient.
In this work, we build on those earlier ideas, and
introduce a parameter-dependent family of quantum-aided weak broadcast protocols that relies on an entangled four-qubit singlet state \cite{cabello}. Our main contributions are as follows.
(1) \emph{Proof.} We prove the weak broadcast functionality of the protocol in a certain range of protocol parameters. The proof relies on considering the failure probability $p_f(m)$ as the function of the number $m$ of resource states, and upper bounding this failure probability exponentially, that is, proving $p_f \leq e^{-bm}$ with a certain $b>0$ as $m \to \infty$.
(2) \emph{Tight upper bound for failure probability.} We compute a tight upper bound to the failure probability of the protocol, as function of the number of quantum states used to broadcast a single classical bit.
(3) \emph{Resource optimisation.} We use the tight upper bound to perform a resource optimisation, i.e., compute the answer to these questions: (i) For a given allowed failure probability, how to set the protocol parameters to minimise the number of resource states? (ii) What is the number of required resources states?
(4) \emph{Preparation of the four-qubit singlet state.} Following earlier work demonstrating the suitability of noisy intermediate-scale quantum (NISQ) devices
for the study of quantum networks \cite{<pathumsoot>}, we experimentally create our resource quantum
state on the public quantum-computer prototypes of IBM and IonQ, and characterise its state fidelity.
In a spirit similar to a recent study \cite{Taherkhani_2017}
of an alternative quantum-aided protocol \cite{<benor>},
our analysis illustrates multiple engineering aspects of future deployment of such protocols in practice.
The rest of the paper is structured as follows.
In Sec.~\ref{sec:weakbroadcastprotocol},
we define a two-parameter weak broadcast protocol
based on the four-qubit singlet state.
In Sec.~\ref{sec:securityproof}, we prove the functionality of the protocol.
In Sec.~\ref{sec:analysis}, we outline how to compute the tight upper bound for the failure probability, and discuss the resource optimisation protocol.
In Sec.~\ref{sec:experiment},
we construct two four-qubit quantum circuits that can be used to generate the
singlet state, and implement those circuits on IBM's quantum computer
devices.
We discuss our results and follow-up ideas, and draw conclusions, in
Sec.~\ref{sec:discussion}.
\section{A two-parameter Weak Broadcast protocol}
\label{sec:weakbroadcastprotocol}
Multi-party communication protocols exist for various functionalities.
Two examples are broadcast and weak broadcast \cite{fitzi_phd}. An important building block for more complex protocols
is a weak broadcast protocol that is resilient
up to one faulty component in the presence of 3 components.
Following Ref.~\cite{fitzi_phd}, we will denote such protocols as WBC(3,1).
A key result of Ref.~\cite{fitzi_phd} is that WBC(3,1) can be used as a
primitive, to efficiently build a broadcast protocol with $n\geq 3$ components,
resilient against any $t<n/2$ faulty components.
This motivates us to define and study a protocol achieving WBC(3,1).
In a distributed system with 3 components, the WBC(3,1) protocol has the following goal: the sender ($S$) wants to distribute a single bit $x_S \in \{0,1\}$ of classical information to the two receivers $R_0$ and $R_1$, in such a way that there is a consensus about the bit value even if one of the three components is an active adversary.
More formally, in a WBC(3,1) protocol, the output of the receivers can take a value $\perp$ (often called `abort') besides 0 and 1, and the protocol should fulfill the conditions of `Validity' and `Consistency'.
Validity means that if the Sender $S$ is correct (i.e., exactly follows the prescribed protocol), then the output bit values of all correct components are equal to $x_S$.
Consistency means that if any correct component has an output $y \in \{0,1\}$, then all other correct components should have an output from $\{y,\perp\}$.
(See a more detailed description in Appendix \ref{app:broadcast}.)
The two-parameter WBC(3,1) protocol we define here is derived from similar protocols
in earlier works \cite{fitzi_proceedings,cabello,cabstateprep}.
It prescribes communication between three components ($S$, $R_0$, $R_1$) to achieve consensus in the sense of WBC(3,1) described above.
The quantum resource of this protocol is a four-qubit singlet state \cite{cabello}:
\begin{align}
\label{eq:cabello}
\begin{split}
\ket{\psi} =\frac{1}{2 \sqrt{3}} (2\ket{0011}-\ket{0101}- \ket{0110}-
\\
\ket{1010}-\ket{1001}+2\ket{1100}).
\end{split}
\end{align}
In fact, we assume that to achieve WBC(3,1) with a single data bit,
the parties have distributed $m \in \mathbb{Z}^+$ four-qubit singlet states
through public quantum channels such that the first two qubits are distributed to $S$, the third qubit is distributed to $R_0$,
and the last qubit is distributed to $R_1$, as shown in Fig.~\ref{fig:Cabstate_distribution}a.
In rough terms, the more four-qubit singlet states are used to broadcast one bit, the higher the probability of success; hence the integer $m$ is a key quantity in assessing the performance of the protocol.
\begin{figure}
\caption{Using a four-qubit singlet state for
weak broadcast.
(a) Yellow circles indicate the four qubits of the singlet state $\ket{\psi}
\label{fig:Cabstate_distribution}
\end{figure}
Having $m$ singlet states distributed, the components perform measurements in the
computational basis, each obtaining its own random list of bit values:
$S$ obtains $m$ bit-pairs, while $R_0$ and $R_1$ obtain $m$ bits each.
An example for such a collection of random correlated bitstrings
for $m=12$ is shown in Fig.$~\ref{fig:Cabstate_distribution}$b,
and the probability density function
of the six possible outcomes of four-bit strings, derived straightforwardly from Eq.~\eqref{eq:cabello},
is shown in Fig.$~\ref{fig:Cabstate_distribution}$c.
We will refer to the six possible four-long bitstring outcomes
as \emph{Elementary Events}, and we will refer to
a particular instance of this collection of random data, as shown in Fig.~\ref{fig:Cabstate_distribution}b, as an \emph{Event}.
In what follows, we will use $Q_j[\alpha]$ to denote the
measurement result of component $j$ in its row $\alpha$,
where $j \in (S,R_0,R_1) \equiv (S,0,1)$
and
$\alpha \in (1,2,\dots, m) \equiv (\text{a},\text{b} \dots)$.
For example, in Fig.~\ref{fig:Cabstate_distribution}c, we see $Q_S[\text{d}] = 10$, $Q_0[\text{e}] = 1$, etc.
The Weak Broadcast protocol presented below
has two real-valued parameters:
$0 < \mu < 1/3$ and $1/2 < \lambda < 1$.
We anticipate that the protocol
works well if $\mu$ and $\lambda$ are
close to their upper bound.
The Weak Broadcast protocol we propose is as follows.
In all steps, the classical channels are assumed to be
authenticated, and all the channels,
the source of quantum states, and the measurements,
are assumed to be perfect.
\begin{enumerate}
\item
\emph{Invocation Phase.}
$S$ sends the data bit ($x_S$) to be broadcasted to the other components.
We denote the bits of the components $R_0$ and $R_1$ received
as $x_0 = x_S$ and $x_1 = x_S$.
Now the Sender measures all its qubits ($2$ qubits in each distributed state) in the computational basis.
For each qubit pair of $S$, if both measurements corresponding to
state index $\alpha$
(see Fig.~\ref{fig:Cabstate_distribution}b)
yielded $x_S$, then
$S$ adds index $\alpha$ to its \emph{check set} $\sigma_S$.
After assembling $\sigma_S$,
this check set is sent to both receivers.
At this point, both receivers hold a bit value $x_j$ and a set $\sigma_j$ from the Sender.
The Sender also sets its output to $y_S = x_S$.
In the example of Fig.~\ref{fig:Cabstate_distribution}b,
and assuming $x_S = 0$, the check set is $\sigma_S = \{\mathrm{b},\mathrm{e},\mathrm{f},\mathrm{i}\}$.
\item
\emph{Check Phase.}
Now both $R_0$ and $R_1$ check the consistency of their data received from $S$. For this, $R_j$ measures all of its qubits in the check set $\sigma_j$, and if all the results differ from $x_j$
(\emph{Consistency Condition}),
and also the check set is large enough
\emph{(Length Condition)},
then it accepts the message $x_j$.
For $R_0$, this implies that it chooses its
output to be $y_0 = x_0$.
For $R_1$, it chooses the value of an intermediate
variable $\tilde{y}_1 = x_1$.
The check set is large enough if the number
of its elements is at least
$T \equiv \lceil \mu \cdot m \rceil$, where $0<\mu<1/3$.
If $R_0$ finds that any of the two conditions is violated, then it sets its output to `abort', $y_0 = \perp$.
If $R_1$ finds that any of the two conditions is violated, then it sets its intermediate value to `abort', $\tilde{y}_1 = \perp$.
\item
\emph{Cross-calling Phase.}
$R_0$ sends to $R_1$ its output value $y_0$
and the check set $\sigma_0$ it received from $S$.
$R_1$ receives these as $y_{01}$ and $\rho_{01}$,
respectively.
\item
\emph{Cross-check Phase.}
$R_1$ evaluates the following three conditions
and if all of them are true, then it outputs the value
received from $R_0$, that is, $y_1 = y_{01}$;
otherwise it outputs its intermediate value, $y_1 = \tilde{y}_1$.
The three conditions:
(i) Confusion Condition: $y_{01}$ is different from
$\tilde{y}_1$, and none of them is $\perp$.
(ii) Length Condition:
The size of the check set $\rho_{01}$ is at least $T \equiv \lceil \mu \cdot m \rceil$,
similarly to the Length Condition of the Check Phase above.
(iii) Consistency Condition:
For a large fraction of the indices in $\rho_{01}$,
$R_1$ measures the bit value opposite to $y_{01}$.
The tolerance parameter defining this fraction is $\lambda$,
assumed to fulfill $1/2<\lambda <1$;
that is, this condition is true if the number of indices in $\rho_{01}$
where $R_1$ measures the bit opposite to $y_{01}$ is greater or equal to
$\lambda T + |\rho_{01}| - T$.
Note that this Consistency Condition is less stringent than the
one in the Check Phase.
\label{lst:fitzicabello_text}
\end{enumerate}
We also provide a more compact and more formal
definition of the protocol:
\begin{enumerate}
\item $S \rightarrow R_{0}$, $R_{1}$: $x_{S}, \sigma_{S}=\{\alpha \in\{1, \ldots, m\}: Q_{S}[\alpha]=x_{S} x_{S}\}, R_{j}$ receive $\left\{x_{j}, \sigma_{j}\right\}$
$S: y_{S}=x_{S}$
\item $R_{0}:$ if $(| \sigma_{0} | \geq T)$ $\wedge$ $( \{\alpha \in \sigma_{0}: Q_{0}[\alpha]=x_{0}\}=\emptyset)$ then $y_{0}=x_{0}$ else $y_{0}=\perp$ fi
$R_{1}:$ if $(| \sigma_{1} | \geq T)$ $\wedge$ $( \{\alpha \in \sigma_{1}: Q_{1}[\alpha]=x_{1}\}=\emptyset)$ then $\tilde{y}_{1}=x_{1}$ else $\tilde{y}_{1}=\perp$ fi
\item $R_{0} \rightarrow R_{1}: y_{0}, \rho_{0}= \sigma_0 ; R_{1}:$ receives $ (y_{01}, \rho_{01} )$
\item $R_1$: if $ ( \perp \neq y_{01} \neq y_{1} \neq \perp ) \wedge ( |\rho_{01} | \geq T ) \wedge$
$\wedge ( |\{\alpha \in \rho_{01} : Q_{1}[\alpha]=1-y_{01} \}| \geq \lambda T + |\rho_{01} | - T)$
then $y_{1}:=y_{01}$ else $y_1 = \tilde{y}_1$ fi
\captionof{enumcnt}{Formal description of the Weak Broadcast protocol.}
\label{lst:fitzicabello}
\end{enumerate}
In what follows, we will use the label `Weak Broadcast protocol' or simply `Weak Broadcast'
to denote the above protocol.
The description above shows how the correct components behave.
Our goal is to show that this
protocol is secure for at most one faulty component in a certain range of the values of the parameters $\mu$ and $\lambda$.
Roughly speaking (see below for the theorem),
by security \cite{fitzi_phd} we mean that the \emph{failure probability} of the protocol, that is, the probability of not achieving weak broadcast,
converges to zero as the number $m$ of four-qubit singlet states is increased to infinity.
Even though the Weak Broadcast protocol itself
follows straightforwardly from earlier works
\cite{cabello,cabstateprep}, the security theorem
and its proof has not been published before, to our knowledge.
We point out an important property of the above Weak Broadcast protocol, which is shared with similar earlier protocols \cite{fitzi_phd,fitzi_proceedings,<fitzi_2001>,cabstateprep}. There is an asymmetry between the two Receivers $R_0$ and $R_1$. In contrast to $R_0$, $R_1$ never sends messages, thus it has no impact on the output of the other components. In particular, if $R_1$ is the faulty component,
then the other two components will reach consensus and hence weak broadcast is guaranteed.
Therefore, from now on we do not consider
the case when $R_1$ is faulty; we restrict our attention
to three other adversary configurations, which we will refer to as 'no faulty', `S faulty', and `$R_0$ faulty'.
\section{Security of Weak Broadcast}
\label{sec:securityproof}
The Weak Broadcast protocol defined in Sec.~\ref{sec:weakbroadcastprotocol}
is based on Refs.~
\cite{fitzi_proceedings,fitzi_phd,cabello,cabstateprep}.
Reference \cite{fitzi_phd} provides a security proof of the
weak broadcast protocol proposed therein.
That protocol uses entangled three-qutrit states.
In this work, we apply elements from there to construct
a security proof for the above-defined Weak Broadcast
protocol, which is based on four-qubit singlet states.
Preliminaries and the theorem are presented in this section;
for the proof we refer to Appendix \ref{app:securityproof}.
We start by specifying our framework.
We assume that there is a complete network of pairwise authenticated
classical channels
among the three components, $S$, $R_0$ and $R_1$.
We also assume that this classical network is synchronous.
In particular,
(1) before the protocol, the components have already agreed on a common point in time when the protocol is to be started,
and
(2) all components operate according to a global clock and every message
sent during a clock cycle
(our protocol consists of 2 clock cycles)
will arrive to the receiver by the end of the cycle.
Recall also that we consider three \emph{adversary configurations}:
`no faulty', `$S$ faulty', and `$R_0$ faulty'.
Here, \emph{adversary} or \emph{faulty component} means
an \emph{active adversary}, which might act as a properly functioning
component, but it might try to undermine the protocol by sending
confusing information to the other components.
In fact, we assume a conscious adversary
following a rational strategy.
We also assume that the adversary has only local information.
In other words, each random bit generated
from the four-qubit singlet states by a local measurement,
see Fig.~\ref{fig:Cabstate_distribution}b, is known only for
the component that has performed the measurement.
We also assume that there is a quantum source that distributes the
four-qubit singlet states to the components as shown in
Fig.~\ref{fig:Cabstate_distribution}.
This distribution happens before the measurements required for the
communication steps of the Weak Broadcast protocol are performed.
After these preliminaries, we state the theorem
claiming the security of Weak Broadcast:
{\bf Theorem:} For any parameter pairs $(\mu,\lambda)$
fulfilling
\begin{subequations}
\label{eq:conditionsoftheorem}
\begin{eqnarray}
\label{eq:conditionmu}
2/9 &<& \mu < 1/3, \mbox{ and} \\
\frac{2+9\mu}{18\mu} &<& \lambda < 1,
\label{eq:conditionlambda}
\end{eqnarray}
\end{subequations}
the failure probability $p_f(m)$ of the Weak Broadcast protocol, defined as Protocol \ref{lst:fitzicabello} above, converges to zero as $m \to \infty$.
In particular, in all three adversary configurations, the failure probability
is upper-bounded as
\begin{equation}
p_f(m) \leq a e^{-b m},
\end{equation}
with $a,b>0$.
The proof is found in Appendix \ref{app:securityproof}.
The parameter range where our proof guarantees security is shown in Fig.~\ref{fig:R0FaultyCompleteCondition}.
In the spirit of Refs.~\cite{fitzi_phd,fitzi_proceedings}, we make a remark on the complexity of the protocol:
Let $\mathcal{R}_\mathrm{q}$ and $\mathcal{B}_\mathrm{q}$ be the round and bit complexities of distributing and measuring a four-qubit singlet state. Then, the Weak Broadcast protocol with a failure probability of at most $0 < \epsilon \ll 1$ requires a round complexity of $\mathcal{R} = \mathcal{R}_q+2$, and a bit complexitiy of $\mathcal{B} = \mathcal{O}(\log(1/\epsilon)) \mathcal{B}_q$.
\begin{figure}
\caption{
Parameter region where the Weak Broadcast
protocol (Protocol \ref{lst:fitzicabello}
\label{fig:R0FaultyCompleteCondition}
\end{figure}
A natural question is if using secure classical channels
instead of authenticated ones brings any advantage.
Here we reason that the answer is no.
If the Sender is the faulty component,
then even if it can eavesdrop on the communication between
$R_0$ and $R_1$, that has no effect on the outcome of the
protocol.
Furthermore, if $R_0$ is the faulty component, then
$S$ is correct, hence $S$ sends the very same information to
both receivers.
Therefore, $R_0$ eavesdropping on the communication between
$S$ and $R_1$ does not gain extra information for $R_0$.
These imply that authenticated classical
channels where eavesdropping is possible
are just as useful in this case as secure channels.
\section{Resource optimization of the Weak Broadcast protocol}
\label{sec:analysis}
In this section, our first goal is to calculate a tight upper bound for the failure probability $p_f(m)$ of the Weak Broadcast protocol.
This tight upper bound will be used to estimate the resource requirements of the protocol.
In particular, we will estimate the number $m$
of singlet states that are required to achieve
weak broadcast with a small failure probability, say, 5\%.
This analysis is similar in spirit to Ref.~ \cite{Taherkhani_2017}, which was carried out
for a different quantum-aided multi-party communication
protocol \cite{<benor>}.
First, we derive the tight upper bound of the failure probability of the Weak Broadcast protocol as function of $\mu$ and $\lambda$.
We obtain analytical results, and cross-check those against Monte-Carlo simulations.
Then, we optimize the protocol in the $\mu$-$\lambda$ parameter plane.
That is, we identify
the parameter values that require the least resources, i.e., the smallest number of four-qubit singlet states such that the failure probability is guaranteed to be below the desired failure probability threshold.
\subsection{Tight upper bound for the failure probability -- analytical results}
\label{anexpressions}
We analyse the resource requirement of the Weak Broadcast protocol is as follows.
We can fix the parameter values $\mu$ and $\lambda$, and the \emph{target failure threshold} $p_{f,t}$;
e.g., $\mu =0.272$, $\lambda = 0.94$, and $p_{f,t} = 0.05$.
The target failure threshold $p_{t,f} = 0.05$ means that the user can tolerate at most 5\% failure probability.
Then, the resource requirement of the protocol is characterized by the minimal number $m_\text{min}(\mu,\lambda,p_{f,t}) \in \mathbb{Z}^+$ of four-qubit singlet states required to suppress the failure probability $p_f$ below the threshold $p_{f,t}$.
A natural way to compute $m_\text{min}$ would be to identify the optimal adversary strategies for the $S$ faulty and $R_0$ faulty configurations, i.e. those strategies that maximize the failure probability.
We do not complete this task here.
Instead, we identify optimal \emph{incomplete} strategies, which are incomplete in the sense that they provide a description of the adversary's behavior for certain Events but not for others.
Using these optimal incomplete strategies, we can compute tight upper bounds on the failure probabilities $p_f(m)$, and can use those upper bounds to compute an upper bound $m_{\text{min},\uparrow}$ of the minimal resource requirement.
For the no faulty configuration, we derive the exact result $p_f^\text{(nf)}$ for the failure probability shown as Eq.~\eqref{eq:pfnofaulty}.
The optimal incomplete adversary strategies for the $S$ faulty and $R_0$ faulty configurations are described in App.~\ref{app:adversarystrategies} and App.~\ref{app:proofsofoptimality}.
The corresponding failure-probability upper bounds $p_{f,\uparrow}^{(S)}$ and
$p_{f,\uparrow}^{(R)}$ are derived in App.~\ref{app:failureprobabilities}, and are shown in Eq.~\eqref{eq:ps_upper} and Eq.~\eqref{equation:pr0_upper}, respectively.
To evaluate these analytical formulas, we carry out numerical summation.
The results, for a fixed value of protocol parameters, are shown in the three panels of Fig.~\ref{fig:theoryandsimulation}, corresponding to the
three adversary configurations. See caption for parameter values.
All three panels show that the failure probability $p_f$ exhibits a
decreasing trend as we increase the number of
four-qubit singlet states, as expected.
From the data, we can read off the minimum number of states
required to suppress the upper bound of the failure probability below $p_{f,t} = 0.05$:
these are 143, 246 and 280, respectively,
for the three adversary configurations.
Our upper bound $m_{\text{min},\uparrow}$ of the minimal resource requirement of this protocol for $p_{f,t} = 0.05$ is the maximum of those
three integers, that is, $m_{\text{min},\uparrow} = 280$.
\begin{figure}
\caption{Failure probability of the Weak Broadcast protocol for the three adversary configurations.
(a) `No faulty' configuration. Exact results (green circles) and Monte-Carlo results (red crosses) for the failure probability.
(b)`$S$ faulty configuration.
(c) `$R_0$ faulty' configuration.
In (b) and (c), the failure-probability upper bound is shown.
Parameters (all panels): $\mu = 0.272$ and $\lambda = 0.94$.
Simulation results are obtained from $N=10,000$ random Events for each $m$.
(Fig.~\ref{fig:Cabstate_distribution}
\label{fig:theoryandsimulation}
\end{figure}
\subsection{Tight upper bound for the failure probability - Monte-Carlo simulations}
\label{sec:montecarlo}
To test our analytical calculations described in the previous subsection, here we use Monte-Carlo simulations to compute the failure probability $p^{\text{(nf)}}_f$ and the failure-probability upper bounds $p^{(S)}_{f,\uparrow}$ and $p^{(R)}_{f,\uparrow}$.
In these simulations, we randomly generate, on a classical computer, the lists of correlated 4-bit strings measured by the components (i.e., the Events, see Fig.~\ref{fig:Cabstate_distribution}b).
For the no faulty configuration, we evaluate for each random Event if the protocol leads to failure or weak broadcast.
Then, we express the failure probability as the ratio $N_f/N$ of the number $N_f$ of failures and the number $N$ of random Events.
The resulting failure probability as function of $m$ is shown as the red crosses in Fig.~\ref{fig:theoryandsimulation}a.
Data corresponds to $N=10,000$ random Events.
For the $S$ faulty and $R_0$ faulty adversary configurations, our simulation corresponds to the upper-bound calculation described in the previous subsection.
For example, for the $S$ faulty configuration, we use the optimal incomplete strategy described in App.~\ref{app:adversarystrategies}, and do the following steps.
(1) We generate a random Event.
(2) We check if the random Event is in the domain of the optimal incomplete strategy.
(3a) If the Event is not in the domain, then we assume failure.
(3b) If the Event is in the domain, then we evaluate the protocol such that $S$ is acting according to its adversary strategy; this leads to either failure or weak broadcast.
(4) After repeating these steps for $N$ different random Events, we express the failure-probability upper bound as the ratio $N_f/N$ of the number $N_f$ of failures and the number $N$ of random Events.
These results are shown as the red crosses in Fig.~\ref{fig:theoryandsimulation}b and c, for the $S$ faulty and $R_0$ faulty configurations, respectively.
Data corresponds to $N=10,000$ random Events.
In Fig.~\ref{fig:theoryandsimulation}a, b, and c, the results of the Monte-Carlo simulations (red crosses) are in excellent agreement with the exact results (green circles).
We interpret the small differences between the Monte-Carlo result and the exact result as statistical fluctuations of the Monte-Carlo result.
For example, estimating the failure probability $p=0.2$ from $N=10,000$ samples has a statistical error of
$\sqrt{p(1-p)/N} \approx 0.4$ \%.
\subsection{Optimization in the parameter space}
\label{sec:optimization}
In the previous subsections, we computed the resource requirements
of the Weak Broadcast protocol for fixed values of the parameters $\mu$ and $\lambda$,
and for a fixed value of the target failure threshold $p_{f,t}$.
A natural next step is the optimisation of the protocol: finding the optimal parameter values $\lambda$ and $\mu$, which minimize the resource requirements, such that it is guaranteed that the failure probability is kept below the target failure threshold $p_{f,t}$.
Even though we do not know the failure probabilities of the best adversary strategies, we do know tight upper bounds for the failure probability, using which we can carry out the optimization.
In particular, we will minimize the function $m_{\text{min},\uparrow}(\mu,\lambda,p_{f,t})$
in the $(\mu,\lambda)$ parameter plane, with the specific value of the target failure threshold $p_{f,t} = 0.05$.
We expect that future deployment of such communication protocols will require similar optimization procedures.
We perform this optimization in two steps, as follows.
We use the analytical formulas discussed in section \ref{anexpressions}.
For a rough optimization in the $(\mu,\lambda)$ parameter space,
we consider the rectangle in Fig.~\ref{fig:parspace}a, and the grid of
parameter pairs shown there.
For each point (box) of the grid, we evaluate the failure-probability upper bound $p_{\uparrow}(m)$ for
$m=290,300$, and color the box as blue or green, respectively, according to the minimal $m$ within this set where $p_{f,\uparrow}<p_{f,t} = 0.05$.
This procedure results in the pattern in Fig.~\ref{fig:parspace}a,
which suggest a finer optimization focusing on the area
of the dark blue rectangle of Fig.~\ref{fig:parspace}a.
Note that grey boxes correspond to parameter values $\mu$, $\lambda$ lying outside the green zone of Fig.~\ref{fig:R0FaultyCompleteCondition}.
The results of this finer optimization are shown in
Fig.~\ref{fig:parspace}b.
The procedure is analogous to the rough optimization,
with the difference that here the set of $m$ values is
$m = 270, 271, \dots, 300$.
Figure ~\ref{fig:parspace}b reveals a `sweet rectangle', a rectangle
of points where using 280 four-qubit singlet states is sufficient
to achieve weak broadcast with at most 5 \% error.
We emphasize that for an error threshold different from 5 \%, this optimization
must be repeated, and might yield different optimal values for the parameters
$\mu$ and $\lambda$.
To summarize, in this section we have outlined and implemented
resource optimization procedures for the Weak Broadcast protocol.
This illustrates the engineering aspects of the deployment
of future quantum-aided distributed systems.
\begin{figure}
\caption{Optimization of the Weak Broadcast
protocol in the $(\mu,\lambda)$ parameter space,
for a fixed target failure threshold.
Both panels indicate the resource requirement, i.e.,
number of four-qubit singlet states,
to achieve failure probability below the
target failure threshold $p_\text{th}
\label{fig:parspace}
\end{figure}
\section{Four-qubit singlet state on IBM Q and IonQ devices}
\label{sec:experiment}
An important goal in the field of quantum communication is
to advance the hardware technology and thereby
to enable the practical deployment
of quantum-aided multi-party communication \cite{cabstateprep,Wehner,Pompili}.
With this goal in mind, we studied the preparation of the resource
states of the Weak Broadcast protocol, i.e., four-qubit singlet states,
on NISQ hardware \cite{preskill2018quantum}, namely using superconducting qubits of IBM Q \cite{ibm} and trapped ion qubits of IonQ \cite{monroe2021ionq}.
Both superconducting and trapped ion qubits have already been shown to be promising candidates
for implementing quantum networks \cite{<pathumsoot>}.
Quantum network functionalities using
superconducting qubits housed in spatially
separated cryogenic systems have
been experimentally demonstrated recently \cite{Magnard}, while scalable remote entanglement between trapped ions has been achieved via photon and phonon interactions \cite{hucul2015modular}, raising the hope that entangled states prepared
on-chip on quantum processors can be used
as resources in quantum networks.
In the rest of this section we identify two quantum circuits that
prepare the four-qubit singlet state.
To characterize the quality of our experimental state preparation, we use two quantities.
The first one is the \emph{classical fidelity} $F_c$, that is the fidelity between (i) the
probability distribution $P_\text{exp}$ of bitstrings
obtained experimentally by doing computational-basis
measurements,
and (ii) the ideal probability distribution
$P_\text{id}$
of measured bitstrings, following from
Eq.~\eqref{eq:cabello} and
shown in Fig.~\ref{fig:Cabstate_distribution}c:
\begin{equation}
\label{eq:classicalfidelity}
F_c = \left( \sum_{s = (0000)}^{(1111)}
\sqrt{P_\text{exp}(s) P_\text{id}(s)} \right) ^2.
\end{equation}
The second figure of merit for state preparation
is the \emph{quantum state fidelity} $F_q$.
We obtain this by performing quantum state
tomography at the end of the circuit,
reconstructing the density matrix $\rho_\text{exp}$
from the experimental data,
and evaluating the fidelity between this
state and the ideal
four-qubit singlet state
$\rho_\text{id} = \ket{\psi}\bra{\psi}$:
\begin{equation}
\label{eq:quantumfidelity}
F_q = \left( \mathrm{Tr}\sqrt{ \rho_\text{id}^{1/2} \rho_\text{exp}
\rho_\text{id}^{1/2}} \right) ^2.
\end{equation}
\subsection{Circuit A}
First, we need to identify a quantum circuit that
prepares the four-qubit singlet state.
We rely on
CNOT as the two-qubit gate.
On today's noisy devices, CNOT is `expensive' in the sense
that its gate error is greater than the single-qubit
gate errors.
Therefore, a second target for our circuit search
is to minimize the number of CNOTs.
A third aspect is related to the incomplete connectivity
of IBM Q devices: in this subsection we will pay special attention to devices that host a chain of four qubits with linear connectivity.
On linear qubit chains, we performed our measurements
using Circuit A, shown in Fig.~\ref{fig:bryancirc},
which is based on the results of Ref.~\cite{bryan}.
The construction of Ref.~\cite{bryan} provides a circuit
with 18 CNOTs.
This we have further simplified, partly by hand, partly by
Qiskit's transpiler, to a circuit containing 14 CNOTs. In the circuit we use general single-particle $U$ gates, with the parametrization
\begin{equation}
U(\theta,\phi,\lambda) =
\left(\begin{array}{cc}
\cos(\theta/2) & - e^{i \lambda} \sin(\theta/2)
\\
e^{i\phi} \sin(\theta/2) & e^{i(\lambda + \phi)} \cos(\theta/2)
\end{array} \right).
\end{equation}
\begin{figure}
\caption{Circuit A, a 14-CNOT circuit
that prepares the four-qubit singlet state
of Eq.~\eqref{eq:cabello}
\label{fig:bryancirc}
\end{figure}
We have run Circuit A in all four-qubit linear chains
in all IBM Q devices with Quantum Volume 32 (available in 2021 April).
This yields 32 qubit configurations,
and the corresponding
results are
listed in Table \ref{tab:classicaltable}.
In Table \ref{tab:classicaltable},
we characterize the quality of state
preparation using classical and quantum fidelities introduced in Eqs. \ref{eq:classicalfidelity} and \ref{eq:quantumfidelity}.
Without readout error mitigation, our best experiment was the
one using the qubit chain 0-1-2-3 on \verbsantiago, yielding $F_c = 0.8510$. When we correct for the readout errors
using the built-in readout error mitigation
procedure of Qiskit (see Refs.~\cite{chen2019detector,maciejewski2020mitigation} for theoretical details), then the best result
is found on the same qubit chain, with
$F_c = 0.9021$.
For reference, the classical fidelity
between a uniform bitstring distribution
and the ideal bitstring distribution
is $1/\sqrt{3} \approx 0.58$.
Hence we conclude that the bitstring
distribution
provided by IBM Q hardware is much closer to the
ideal one than to random noise.
The data in Table \ref{tab:classicaltable}
show that the best-performing arrangement is
the 0-1-2-3 qubit chain on \verbsantiago.
Without readout error mitigation, the
fidelity is $F_q = 0.7435$,
whereas readout error mitigation
improves this to $F_q^\text{mitig} = 0.8116$.
As a reference, the quantum state fidelity
of the ideal four-qubit singlet state
and the fully mixed four-qubit state is
$\approx 0.0625$.
\subsection{Circuit B}
Another four-qubit circuit that prepares
the four-qubit singlet state $\ket{\psi}$, is Circuit B depicted in Fig.~\ref{fig:circB}, which includes only 5 CNOT gates.
We found Circuit B using a combination
of random circuit search and gradient descent:
We assembled random circuits using CNOTs and
single-qubit gates from the set
$\{I,X,T,S,H\}$; searching through approximately $10^7$ such
instances, we selected the one whose
output state had the maximal overlap with the
four-qubit singlet state.
Then, for the selected instance, we replaced all the one-qubit gates with generic three-parameter single-qubit gates, and performed gradient descent in this multi-dimensional parameter space, with the goal of maximizing the fidelity between the output state
of the circuit and the four-qubit singlet state.
\begin{figure}
\caption{Circuit B, a 5-CNOT circuit that
prepares the four-qubit singlet state
of Eq.~\eqref{eq:cabello}
\label{fig:circB}
\end{figure}
Although the decrease in the number of CNOTs with respect to Circuit A is spectacular (5 instead of 14), Circuit B did not produce better results than Circuit A in our state-preparation experiments with the IBM Q quantum processors. The reason is that Circuit B does not suit the linear qubit chain arrangements of the QV32 backends of IBM Q, hence it can only be run without qubit rearrangement on lower-quality hardware (e.g,. \verbmelbourne), which has the loop-type connectivity of Circuit B.
However, we also implemented Circuit B on IonQ’s trapped ion quantum computer, which has full connectivity. Beside the better connectivity, these devices have also smaller readout and gate error rates. The experiments were run through
the AWS Braket cloud service \cite{gonzalez2021cloud}. As we expected, the results were better; we achieved classical fidelities $\geq 95\%$ in all bases and a quantum fidelity (without readout error mitigation) of 88.5\%, both being considerably higher than the fidelities obtained using the IBM Q quantum processors.
To conclude, we have identified two circuits, Circuit A and
Circuit B, that can be used to prepare the four-qubit
singlet state on IBM Q and IonQ hardware.
According to our best data, the circuit on IBM Q
produces the desired bitstring probability
distribution with classical fidelity
$F_c^\text{mitig} = 0.9002$,
and it produces the desired state
with quantum state fidelity $F_q^\text{mitig} = 0.8116$. Correspondingly, our single run of the experiment on IonQ achieved classical fidelities $\geq 95\%$ in all bases and a quantum fidelity without readout error mitigation of 88.5\%.
\section{Conclusion and outlook}
\label{sec:discussion}
In this work, we introduced a two-parameter
family of quantum-aided weak broadcast protocols,
and proved their security in a certain range of its parameters.
We computed a tight upper bound $m_{\text{min},\uparrow}$ on the number of resource quantum states required to suppress the failure probability of the protocol below
a given target failure probability threshold.
We optimized the protocol, that is, we located the parameter range where the resource requirements, as indicated by $m_{\text{min},\uparrow}$, are minimal.
We have experimentally implemented the preparation
and characterization of our four-qubit
resource state on real qubits: in the superconducting quantum processors of IBM and the ion-trap quantum processors of IonQ.
Our work illustrates the engineering aspects of future
deployments of such protocols in practice.
The methods and results of this work can be utilized in
future research toward multi-party quantum communication in distributed systems.
Let us note that throughout the theoretical part of this work, we assumed that the four-qubit singlet state $\ket{\psi}$ of Eq.~\eqref{eq:cabello} is prepared and distributed through public quantum channels to the three components of the distributed system. The role of these entangled states is to provide random bits that are correlated among the different components. The same functionality cannot be substituted by a source of correlated classical random bits, distributed through public channels. In this case, the adversary can eavesdrop on the classical channel transmitting the random bits from the source to the components, and hence the criterion that each component knows only its own measurement outcomes can be violated. In the quantum case, considered in this work, the eavesdropping can be detected by sacrificing a fraction of the four-qubit singlet state in a distribute-and-test procedure of the protocol, similar to the procedures described in preceding work \cite{<fitzi_2001>,cabello,cabstateprep}. We leave it for future work to elaborate the quantitative details of such a distribute-and-test procedure.
The above observation points toward further
important follow-up tasks to this work.
For example, an attack against the protocol
could be based on the adversary gaining control
over the source, or over the measurement
devices generating the correlated random bits.
A further difficulty is posed by physical errors
of the source, the quantum channels, and the measurement
devices, as highlighted by our experiments
presented in Sec.~\ref{sec:experiment}.
We anticipate that the effect of these imperfections
on the protocol will be studied in a framework similar
to that of device-independent quantum key
distribution \cite{Acin,Xu_2020}.
Some of these issues have already been
discussed \cite{fitzi_proceedings,cabello,cabstateprep},
but a quantitative study is, to our knowledge,
still missing.
\acknowledgments
We acknowledge useful discussions with T. Coopmans,
M. Farkas, M. Fitzi, B. T. Gard, T. Kriv\'achy, and A. Pereszl\'enyi. This research was supported by the Ministry of Innovation and Technology and the National Research, Development and Innovation Office (NKFIH) within the Quantum Information National Laboratory of Hungary, the Quantum Technology National Excellence Program (Project No. 2017-1.2.1-NKP-2017-00001) and the Thematic Excellence Programmes TKP2020-NKA-06 and TKP2020 IES (Grant No. BME-IE-NAT), under the auspices of the Ministry for Innovation and Technology, and by NKFIH through the OTKA Grants FK 124723, FK 132146, K 124351.
Furthermore, we acknowledge the use of the IBM Q for this work. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Q team. We acknowledge the access to advanced services provided by the IBM Quantum Researchers Program.
\printbibliography
\onecolumn
\appendix
\section{Broadcast, weak broadcast, and their truth tables}
\label{app:broadcast}
In Sec.~\ref{sec:weakbroadcastprotocol}, we introduced
the broadcast and the weak broadcast
functionalities of multi-party communication protocols.
Here, we recall their definitions \cite{fitzi_phd}.
Furthermore, we show
their tabular representations (`truth tables'),
for the case of $n=3$ componenents and at most
$t=1$ faulty component.
\begin{mydef}{(broadcast)}
\label{broadcast}
A protocol among \textit{n} components such that one distinct component \textit{S} (the Sender) holds an input value $x_S \in \{ 0,1 \} $, and all other components (the receivers) eventually decide on an output value in $\{ 0,1 \}$ is said to achieve \textbf{broadcast} if the protocol guarantees the following conditions:
\begin{itemize}[label = {}]
\item \textit{Validity}: if the Sender is correct then all correct components decide on $y = x_S$.
\item \textit{Consistency}: all correct components decide on the same output.
\end{itemize}
\end{mydef}
The truth table of the broadcast
functionality is shown in Table \ref{tab:broadcast},
for the special case when the number of components
is $n=3$, and resilience is expected up to $t=1$
faulty component.
\begin{table}[t]
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$S$ & $R_0$ & $R_1$ & no faulty & $S$ faulty & $R_0$ faulty \\ \hline
0 & 0 & 0 & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; \\ \hline
0 & 0 & 1 & $\times$ & $\times$ & $\times$ \\ \hline
0 & 1 & 0 & $\times$ & $\times$ & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; \\ \hline
0 & 1 & 1 & $\times$ & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & $\times$ \\ \hline
1 & 0 & 0 & $\times$ & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & $\times$ \\ \hline
1 & 0 & 1 & $\times$ & $\times$ & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; \\ \hline
1 & 1 & 0 & $\times$ & $\times$ & $\times$ \\ \hline
1 & 1 & 1 & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; \\ \hline
\end{tabular}
\caption{Truth table of broadcast with
$n=3$ components, resilient up to $t=1$ faulty component.
Column 1 is the input and output bit of the sender
($S$), column 2 (3) is the output bit of receiver $R_0$
($R_1$).
Columns 4, 5, 6 indicate whether the list
of output bits and the faulty configuration
together satisfies the conditions of
broadcast or not.
The faulty configuration `$R_1$ faulty' is not shown as it is
analogous to $R_0$ faulty.
\label{tab:broadcast}}
\end{table}
In the definition of the second functionality, weak broadcast,
the conditions of broadcast are relaxed to
some extent, and in addition to 0 and 1,
a third output value called `abort' ($\perp$) is
also allowed.
As discussed in the main text, the
weak broadcast functionality is important because
it can be used as a building block to achieve
broadcast \cite{fitzi_phd}.
\begin{mydef}{(weak broadcast)}
\label{def:wb}
A protocol among $n$ components such that one distinct component $S$ (the Sender) holds an input value $x_S \in \{0, 1\} $ and all other components eventually decide on an output value in $\{0,1, \perp \} $ is said to achieve \textbf{weak broadcast} if the protocol guarantees the following conditions:
\begin{itemize}[label={}]
\item \textit{Validity}: if the Sender is correct then all correct components decide on $y = x_S$.
\item \textit{Consistency}: if any correct component decides on an output $y \in \{0, 1\} $ then all correct components decide on a value in $ \{ y, \perp \}$; that is, either they choose the value $y$ or choose abort.
\end{itemize}
\end{mydef}
The truth table of the weak broadcast
functionality is shown in Table \ref{tab:broadcast},
for the special case when the number of components
is $n=3$, and resilience is expected up to $t=1$
faulty component.
In the main text, this special case
is denoted by WBC(3,1).
\begin{table}[t]
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$S$ & $R_0$ & $R_1$ & no faulty & $S$ faulty & $R_0$ faulty \\ \hline
0 & 0 & 0 & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; \\ \hline
0 & 0 & 1 & $\times$ & $\times$ & $\times$ \\ \hline
0 & 0 & $\perp$ & $\times$ & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & $\times$ \\ \hline
0 & 1 & 0 & $\times$ & $\times$ & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; \\ \hline
0 & 1 & 1 & $\times$ & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & $\times$ \\ \hline
0 & 1 & $\perp$ & $\times$ & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & $\times$ \\ \hline
0 & $\perp$ & 0 & $\times$ & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; \\ \hline
0 & $\perp$ & 1 & $\times$ & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & $\times$ \\ \hline
0 & $\perp$ & $\perp$ & $\times$ & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & $\times$ \\ \hline
1 & 0 & 0 & $\times$ & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & $\times$ \\ \hline
1 & 0 & 1 & $\times$ & $\times$ & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; \\ \hline
1 & 0 & $\perp$ & $\times$ & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & $\times$ \\ \hline
1 & 1 & 0 & $\times$ & $\times$ & $\times$ \\ \hline
1 & 1 & 1 & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; \\ \hline
1 & 1 & $\perp$ & $\times$ & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & $\times$ \\ \hline
1 & $\perp$ & 0 & $\times$ & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & $\times$ \\ \hline
1 & $\perp$ & 1 & $\times$ & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; \\ \hline
1 & $\perp$ & $\perp$ & $\times$ & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & $\times$ \\ \hline
\end{tabular}
\caption{Truth table of WBC(3,1), that is,
weak broadcast with
$n=3$ components, resilient up to $t=1$ faulty component.
Column 1 is the input and output bit of the sender
($S$), column 2 (3) is the output value of receiver $R_0$
($R_1$).
Columns 4, 5, 6 indicate whether the list
of output values and the faulty configuration
together satisfies the conditions of
weak broadcast or not.
The faulty configuration `$R_1$ faulty' is not shown as it is
analogous to $R_0$ faulty.
\label{tab:weakbroadcast}}
\end{table}
\section{Adversary strategies against the Weak Broadcast protocol}
\label{app:adversarystrategies}
In this section, we define adversary strategies, and propose an $S$ faulty strategy and an $R_0$ faulty strategy.
We prove that the strategies proposed here are optimal, i.e., they maximise the failure probability, in App.~\ref{app:proofsofoptimality}.
As a preliminary for this section, recall that in Sec.~\ref{sec:weakbroadcastprotocol}, we defined
the Elementary Events as the possible correlated random outcomes
of the measurement of the four-qubit singlet state $\ket{\psi}$
of Eq.~\eqref{eq:cabello}.
There are six such outcomes, 0011, 1100, 1010, 0101, 1001, and 0110.
It follows from Eq.~\eqref{eq:cabello} that
the probabilities of 0011 and 1100 are $1/3$,
whereas the probabilities of the remaining four bitstrings are $1/12$.
Furthermore, we have denoted the number of four-qubit singlet states, available for the weak broadcast of a single bit from $S$, as $m$.
In Sec.~\ref{sec:weakbroadcastprotocol}, we defined
an Event as the random configuration of the $m$ bitstrings measured on the $m$ distributed four-qubit singlet states by
the three components.
Fig.~\ref{fig:sfaultyexamples}a shows an example of an Event for $m=12$
(actually, this is the same Event as in Fig.~\ref{fig:Cabstate_distribution}b).
The first column is the index of the singlet state, and columns
2-4 show the qubit measurement outcomes (Elementary Events) obtained by the three components.
Note that the number of possible different Events is $6^m$, and for a given $m$, it is straightforward to calculate the occurence probability of each possible Event. For example, the Event in Fig.~\ref{fig:sfaultyexamples} occurs with probability $(1/3)^8 (1/12)^4$.
\subsection{$S$ faulty}
\label{app:sfaultystrategy}
For concreteness, and without the loss of generality, here we assume that the faulty $S$ is a conscious adversary, which wants to reach failure of Weak Broadcast such that receiver $R_0$ outputs $y_0 = 0$ and receiver $R_1$ outputs $y_1 = 1$.
Hence, the message sent by $S$ to $R_0$ is $x_0 = 0$, whereas the message sent by $S$ to $R_1$ is $x_1 = 1$.
\begin{figure}
\caption{
An example for a Sender ($S$)
adversary strategy, which causes
failure of Weak Broadcast in this
particular random instance (Event).
(a) Random bits resulting from the
measurement of $m=12$ four-qubit singlet states.
Boldface (italic) denotes the information possessed
by (hidden from) $S$.
(b) Flowchart of the communication
and decision events in case of a faulty $S$
and correct $R_0$ and $R_1$, for
the Event in (a) and the
adversary strategy
$\zeta=(3,1,0;0,0,\ell_3)$.
For this random instance,
the strategy $\mathcal{S}
\label{fig:sfaultyexamples}
\end{figure}
Let us first formalize what we mean by an adversary strategy (of $S$).
To this end, we introduce two natural classifications of the Events; a finer, \emph{global} classification, and a coarser, \emph{local} one.
Here, the term `local' refers to the property that the classification is based only on the information available to the adversary sender $S$.
We define the \emph{global count list} of an Event as the list of the six non-negative integers that count the frequencies of the six different measurement outcomes of the Event. The set of the global count lists will be denoted as GCL.
That is, an element of GCL is
\begin{equation}
\label{eq:globalcountlist}
g = (g_1, g_2, g_3, g_4, g_5, g_6) =
(m_{0011}, m_{0101}, m_{0110}, m_{1001}, m_{1010}, m_{1100}).
\end{equation}
Note that for a certain Event, the adversary $S$ does not know the corresponding global count list $g$, as $S$ knows only the first two bit values of each row of the Event.
Clearly, for any $g \in \text{GCL}$ it holds that $\sum_{i=1}^6 g_i = m$.
Furthermore, we will denote the counting function that associates an element of GCL to an Event as
\begin{equation}
G: \text{Events} \rightarrow \text{GCL}.
\end{equation}
Note that this function is not injective, since in general, multiple Events are mapped to a single global count list $g \in \text{GCL}$.
In fact, for any $g \in \text{GCL}$, the number of Events that are mapped to $g$ via $G$ is
\begin{equation}
|G^{-1}(g)| = \binom{m}{g_1,g_2,g_3,g_4,g_5,g_6},
\end{equation}
where the bracket on the right hand side denotes the multinomial coefficient.
An adversary strategy of the sender $S$ describes how $S$ uses its own local information to act during the protocol.
More precisely, the strategy is a procedure, according to which the adversary $S$ selects the check sets $\sigma_0$ and $\sigma_1$, to be sent to $R_0$ and $R_1$, respectively, in the Invocation Phase.
Locally, the sender $S$ cannot distinguish between the outcomes $0101$ and $0110$, as $S$ knows only the first two bits of these bitstrings.
Similarly, $S$ cannot distinguish between the outcomes $1010$ and $1001$.
These imply that the basis of an adversary strategy of $S$ is provided by the four non-negative integers $(g_1,g_2+g_3,g_4+g_5,g_6) =
(m_{0011},m_{0101}+m_{0110},m_{1001}+m_{1010},m_{1100})$.
For our purposes, it is sufficient to use an even coarser local classification, which does not distinguish between the local bit pairs $01$ and $10$ of the sender $S$.
This coarser classification is defined by the map
\begin{equation}
\mathcal{L}_S: \text{GCL} \to \text{LCL}_S, \,
g \mapsto
\ell \equiv
(\ell_1,\ell_2,\ell_3) \equiv
(g_1,g_2+g_3+g_4+g_5,g_6).
\end{equation}
Here, $\text{LCL}_S$ is the set of \emph{local count lists} for the $S$ faulty scenario, which is implicitly defined as the range of $\mathcal{L}_S$, containing triplets of non-negative integers, such that $\ell_1 + \ell_2 + \ell_3 = m$.
The term \emph{local count list} hence refers to the integer triplets $\ell \in \text{LCL}_S$.
Because of the sum rule $\ell_1 + \ell_2 + \ell_3 = m$, a local count list $\ell$ is represented by two of its elements, e.g., $\ell_1$ and $\ell_3$, and hence the set $\text{LCL}_S$ can be visualised in a two-dimensional plot. This is illustrated for the case $m=12$ in Fig.~\ref{fig:sfaulty_strategydomain}, where each non-grey square (i.e., orange and blue squares) represents a local count list.
\begin{figure}
\caption{Local count lists and the domain of an incomplete strategy of an adversary Sender ($S$ faulty configuration).
Each non-grey square depicts a local count list in case of $m=12$.
Orange region depicts the domain $D(\zeta)$ of the optimal incomplete adversary strategy $\zeta$, for parameter values $\mu = 0.272$, and $\lambda = 0.94$.
The strategy $\zeta$ is defined in Eq.~\eqref{eq:sstrategy}
\label{fig:sfaulty_strategydomain}
\end{figure}
The adversary strategy of $S$ describes how the behavior of $S$ is derived from the local information $S$ has.
For our purposes, it is sufficent to regard the local count list $\ell$ associated to the Event as the local information possessed by $S$.
We define an adversary strategy of $S$ as a function $\zeta_S$ that maps the local count list of $S$ to a list of 6 non-negative integers,
\begin{equation}
\label{eq:sstrategydef}
\zeta_S: \text{LCL}_S \to \left(\mathbb{Z}_0^+\right)^6, \
\ell \mapsto (k^{(0)}_{0011},k^{(0)}_\text{mixed},k^{(0)}_{1100};
k^{(1)}_{0011},k^{(1)}_\text{mixed},k^{(1)}_{1100}).
\end{equation}
Here, $k^{(0)}_{0011} \leq \ell_1$ indicates how many of the 0011 indices does $S$ include in the check set $\sigma_0$ sent to $R_0$ in the Invocation phase.
Similarly, $k^{(0)}_\text{mixed} \leq \ell_2$ indicates the number of mixed-outcome (01XX and 10XX) indices to be included in $\sigma_0$, and $k^{(1)}_\text{1100} \leq \ell_3$ indicates the number of 1100 indices to be included in $\sigma_1$, etc.
Our purpose in this work is to analyse
the failure probability of certain strategies.
For this purpose, it is sufficient to define a strategy by the numbers above: how many
indices from each class are included in each check set.
It is \emph{not} necessary to specify \emph{which} indices are included from each class, because the failure probability is independent of the specific choice.
Nevertheless, in the examples considered in this work, we assume that $S$ includes the `smallest' indices from each class, in alphabetical order of the indices (see Fig.~\ref{fig:sfaultyexamples}).
We call a strategy, as defined in Eq.~\eqref{eq:sstrategydef}, a \emph{complete strategy}, if its domain is the complete $\text{LCL}_S$; otherwise, we call it an \emph{incomplete strategy}.
For a complete strategy, the failure probability $p_f$ is a well-defined real number $p_f \in [0,1]$.
From now on, we will deal with incomplete strategies, and hence we will not be able to compute the specific failure probability; however, we can derive a relatively tight upper bound for the failure probability.
In what follows, we will often consider sequences of strategies: such a strategy sequence is a function that specifies a strategy for each $m$.
A natural notation for a strategy in such a sequence $\zeta_{S,m}$; note that we will often suppress the $m$, and call such a sequence of strategies simply as a strategy.
In general, for different strategies in a given sequence, the domain and the failure probability (or the failure-probability upper bound, for incomplete strategies) are different.
In this work, we focus on a sequence of optimal incomplete adversary strategies of $S$.
This is given as
\begin{equation}
\label{eq:sstrategy}
\zeta_S(\ell_1, \ell_2, \ell_3) =
\left(
T-Q, Q, 0; 0, 0, \ell_3
\right),
\end{equation}
where
\begin{eqnarray}
\label{eq:Tdef}
T&=&\ceil{\mu m},
\\
\label{eq:Qdef}
Q &=& T- \ceil{\lambda T} +1,
\end{eqnarray}
with $\ceil{.}$ denoting the ceiling function.
E.g., for $\mu = 0.26$, $\lambda = 0.94$, and $m=1200$, these are $T=312$ and $Q=19$.
Recall that $T$, originally defined in the Weak Broadcast protocol in Sec.~\ref{sec:weakbroadcastprotocol}, is the minimal check set length that satisfies the Length Conditions of the Weak Broadcast protocol.
Integer $Q$ is the minimal number of inconsistent indices in $\rho_{01}$ that implies the violation of the Consistency Condition of the Cross-check Phase, in the case when the length of the check set is $|\rho_{01}| = T$.
Note that Eq.~\eqref{eq:sstrategy} contains two implicit restrictions on the domain of $\zeta_S$.
First, the adversary $S$ can include $T-Q$ indices of 0011 outcomes in its check set $\sigma_0$ only if the number $\ell_1$ of 0011 outcomes is sufficient, that is, if
\begin{equation}
\label{eq:cond1}
T-Q \leq \ell_1.
\end{equation}
Second, the adversary $S$ can include $Q$ indices of mixed outcomes in its check set $\sigma_0$ only if the number $\ell_2$ of mixed outcomes is sufficient, that is, if
\begin{equation}
\label{eq:cond2}
Q \leq \ell_2.
\end{equation}
Besides these two restrictions, we further restrict the domain of $\zeta$ by requiring
\begin{equation}
\label{eq:cond3}
T \leq \ell_3.
\end{equation}
The above 3 conditions define the domain $D(\zeta)$ of the strategy $\zeta$.
With this extra condition, the incomplete strategy $\zeta$ of Eq.~\eqref{eq:sstrategy} is optimal in its domain $D(\zeta)$, which will be proven in App.~\ref{app:sfaultyoptimality}.
The domain $D(\zeta)$ is shown as the orange region in Fig.~\ref{fig:sfaulty_strategydomain}, and the conditions Eqs.~\eqref{eq:cond1}, \eqref{eq:cond2}, and \eqref{eq:cond3} are depicted as the blue, black, and red lines in Fig.~\ref{fig:sfaulty_strategydomain}, respectively.
How can the strategy $\zeta$ in Eq.~\eqref{eq:sstrategy} lead to failure? This is exemplified by a specific instance of the Weak Broadcast protocol in Fig.~\ref{fig:sfaultyexamples}, for $m=12$.
To be specific, we consider a Weak Broadcast protocol with parameter values $\mu =0.26$ and $\lambda = 0.94$.
The adversary strategy derived from the optimal strategy of Eq.~\eqref{eq:sstrategy} using these values of $\mu$, $\lambda$ and $m$ can be written as
\begin{equation}
\label{eq:sstrategyexample}
\zeta = (3,1,0;0,0,\ell_3),
\end{equation}
since $T = 4$ and $Q=1$.
The table in Fig.~\ref{fig:sfaultyexamples}a shows an Event for $m=12$.
The local information of $S$, i.e., the information $S$ possesses, is denoted by boldface characters in Fig.~\ref{fig:sfaultyexamples}a:
$S$ knows the indices of the four-qubit singlet states
used in the protocol, it knows its own
measurement outcomes (bit-pairs), and it also knows
the measurement outcomes of $R_0$ and $R_1$ for those rows where
$S$ measured 00 or 11.
The information $S$ does not possess is denoted by italic characters in Fig.~\ref{fig:sfaultyexamples}a.
Based on the local information $S$ has, it classifies the indices $\alpha \in \{1,\dots,m\} \equiv \{\mathrm{a}, \mathrm{b}, \dots, \mathrm{l}\}$
of the Event into the three classes defined above: 0011, mixed, 1100.
In the example of Fig.~\ref{fig:sfaultyexamples}a, these
classes can be identified as the corresponding check sets:
$0011 = \{\mathrm{b,e,f,i}\}$,
$\text{mixed} = \{\mathrm{d,g,j,l}\}$,
and $1100 = \{\mathrm{a,c,h,k}\}$.
This implies $m_{0011} = m_\text{mixed} = m_{1100} = 4$, i.e,
$\ell_1 = \ell_2 = \ell_3 = 4$.
This local count list $\ell = (4,4,4)$ is in the domain, defined by Eqs.~\eqref{eq:cond1}, \eqref{eq:cond2}, \eqref{eq:cond3}, of the incomplete strategy $\zeta$ in Eq.~\eqref{eq:sstrategyexample}.
Hence $S$ can follow the incomplete strategy $\zeta_S$, which results in the communication and decision steps depicted in Fig.~\ref{fig:sfaultyexamples}b.
There, the receivers $R_0$ and $R_1$ follow the Weak Broadcast protocol, and their output bit values are $y_0 = 0$ and $y_1 = 0$, respectively, implying failure of the Weak Broadcast protocol.
The above example sheds light on why the strategy $\zeta_S$ in Eq.~\eqref{eq:sstrategy} is efficient (in fact, optimal in its domain) in achieving failure.
In more general terms, its efficiency is reasoned as follows.
According to this strategy, the adversary sender $S$ tries to send a check set $\sigma_0$ such that $R_0$ finds it convincingly long (that is, the Length Condition of the Check Phase is satisfied) and
consistent (that is, the Consistency Condition of the Check Phase is satisfied);
but when sent over to $R_1$ as $\rho_{01}$, then $R_1$ finds it inconsistent
(Consistency Condition of the Cross-Check Phase is violated), resulting
in failure via $y_0 = 0$ and $y_1 = 1$.
This is achieved by the strategy $\zeta_S$ of Eq.~\eqref{eq:sstrategy}, if \emph{each} of the $Q$ rows chosen from the mixed class by $S$ contain bit 1 at $R_0$ and bit 0 at $R_1$.
If at least a single one of these rows contains bit 0 at $R_0$ and bit 1 at $R_1$, then $R_0$ finds the check set $\sigma_0$ inconsistent with the data bit $x_0 = 0$ and hence produces an output $y_0 = \perp$, leading to Weak Broadcast (see Table \ref{tab:weakbroadcast},
row 8 or row 17).
\subsection{$R_0$ faulty}
\label{app:rfaultystrategy}
For concreteness, and without the loss of generality, here we assume that the correct sender $S$ sends the message $x_S = x_0 = x_1 = 0$ to the receivers.
Furthermore, we assume that the faulty $R_0$ is a conscious adversary, which wants to reach failure of Weak Broadcast such that the correct $R_1$ has an output bit value $y_1 =1$ that conflicts with the output bit value of $x_S = 0$ of $S$.
We start by formalizing what we mean by an adversary strategy of $R_0$.
Similarly to the $S$ faulty configuration, our starting point is the set GCL of global count lists, see Eq.~\eqref{eq:globalcountlist}.
The adversary strategy of the receiver $R_0$ describes how $R_0$ uses its own local information to act during the protocol.
Then, locally, the receiver $R_0$ can distinguish between three types of outcomes.
First, if the third bit of the outcome, i.e., the bit $R_0$ has measured, carries the value `1', and the index of the outcome is sent from $S$ to $R_0$ in the check set $\sigma_0$, then $R_0$ can identify the outcome as 0011.
Second, if the third bit of the outcome carries the value `1', and the index of the outcome is not included in $\sigma_0$, then $R_0$ identifies the outcome as `either 0110 or 1010'.
In any other case, i.e., if the third bit of the outcome carries the value of `0', then $R_0$ identifies the outcome as `either 0101, or 1001, or 1100'.
It is natural to denote the frequency of each type of outcome in a given Events as $m_{0011}$, $m_\text{XX10}$, and $m_\text{XX0X}$.
This threefold classification of the Events according to $R_0$'s local information is therefore defined by the map
\begin{equation}
\mathcal{L}_R\, : \,
\text{GCL} \to \text{LCL}_R,
g \mapsto \ell \equiv (\ell_1,\ell_2,\ell_3)
\equiv
(m_{0011}, m_\text{XX10}, m_\text{XX0X})
\equiv
(g_1,g_3+g_5,g_2+g_4 + g_6).
\end{equation}
Here, $\text{LCL}_R$ is the set of local count lists for the $R_0$ faulty scenario, which is implicitly defined as the range of $\mathcal{L}_R$, containing triplets of non-negative integers, such that $\ell_1 + \ell_2 + \ell_3 = m$.
The term \emph{local count list} hence refers to the integer triplets $\ell \in \text{LCL}_R$.
Because of the sum rule $\ell_1 + \ell_2 + \ell_3 = m$, a local count list $\ell$ is represented by two of its elements, e.g., $\ell_1$ and $\ell_2$, and hence the set $\text{LCL}_R$ can be visualised in a two-dimensional plot.
This is illustrated for the case $m=12$ in Fig.~\ref{fig:r0faulty_stratdom}, where each non-grey square represents a local count list.
\begin{figure}
\caption{Local count lists and the domain of an incomplete optimal strategy of an adversary receiver $R_0$
($R_0$ faulty configuration).
Each non-grey square depicts a local count list in the case of $m=12$.
The union of orange, blue and pink regions depicts the domain $D(\zeta)$ of the optimal incomplete strategy $\zeta$, for parameter values $\mu = 0.272$ and $\lambda = 0.94$.
The strategy $\zeta$ is defined in Eq.~\eqref{eq:r0faultydef}
\label{fig:r0faulty_stratdom}
\end{figure}
The adversary strategy of $R_0$ describes the procedure to assemble the check set $\rho_{01}$, based on the local information $R_0$ has.
Hence, we define an adversary strategy of $R_0$ as a function $\zeta_R$ that maps the local count list of $R_0$ to a list of 3 non-negative integers:
\begin{equation}
\label{eq:r0strategy}
\zeta_R: \text{LCL}_R \to \left(\mathbb{Z}_0^+\right)^3,
\ell \mapsto \left(
k_{0011},k_\text{XX10}, k_\text{XX0X}
\right).
\end{equation}
Here, $k_{0011} \leq \ell_1$ indicates how many of the 0011 indices does $R_0$ include in the check set $\rho_{01}$ sent to $R_1$ in the Cross-calling Phase.
Similarly, $k_\text{XX10} \leq \ell_2$ indicates the number of XX10 indices to be included in $\rho_{01}$, and $k_\text{XX0X} \leq \ell_3$ indicates the number of XX0X indices to be included in $\rho_{01}$.
Similarly to the case of the $S$ adversary strategy discussed above, here it is also sufficient to define a strategy by the `$k$' numbers above in Eq.~\eqref{eq:r0strategy}:
how many indices from each class are included in each check set.
It is \emph{not} necessary to specify \emph{which} indices are included from each class, because the failure probability is independent of the specific choice.
Nevertheless, in the examples considered in this work, we assume that $R_0$ includes the `smallest' indices from each class, in alphabetical order of the indices (see Fig.~\ref{fig:r0faultyexamples}).
\begin{figure}
\caption{An example for a Receiver ($R_0$) adversary strategy,
which causes failure of Weak Broadcast in this particular random instance (Event).
(a) Random bits resulting from the measurement of $m=12$ four-qubit
singlet states.
Boldface (italic) denotes the information possessed by (hidden from)
$R_0$.
(b) Flowchart of the communication and decision steps in case of a
faulty $R_0$ and correct $S$ and $R_1$, for the Event in (a) and
the adversary strategy $\mathcal{S}
\label{fig:r0faultyexamples}
\end{figure}
We call a strategy, as defined in Eq.~\eqref{eq:r0strategy}, a complete strategy, if its domain is the complete $\text{LCL}_R$; otherwise, we call it an incomplete strategy.
From now on, we deal with incomplete strategies, and hence we will not be able to compute the specific failure probability; however, we will derive an upper bound for the failure probability.
We consider the incomplete strategy defined as
\begin{equation}
\label{eq:r0faultydef}
\zeta_R(\ell_1,\ell_2,\ell_3) = (0,\ell_2,n_\text{min}),
\end{equation}
where $n_\text{min}$ is the smallest non-negative integer such that $\ell_2+n_\text{min} \geq T$.
That is,
\begin{equation}
\label{eq:nmindef}
n_\text{min} =
\begin{cases}
T-\ell_2 & \text{if}~ \ell_2 < T \\
0 & \mbox{if }\ell_2 \geq T.
\end{cases}
\end{equation}
The above definition of $\zeta_R$ contains an implicit assumption on the domain of $\zeta_R$:
the strategy can be followed only if the value of $n_\text{min}$, as defined in Eq.~\eqref{eq:nmindef}, is not greater than $\ell_3$.
More precisely, if $\ell_2 < T$, then $T-\ell_2 \leq \ell_3$ must hold.
Formulating this in terms of $\ell_1$ and $\ell_2$:
if $\ell_2 < T$, then $\ell_1 \leq m-T$ must hold.
As $\ell_2 \geq T$ implies $\ell_1 \leq m-T$, we conclude that the domain of the above strategy is defined by
\begin{equation}
\label{eq:r0faultydomain}
\ell_1 \leq m-T.
\end{equation}
In Fig.~\ref{fig:r0faulty_stratdom}, this domain $D(\zeta_R)$ is illustrated for a specific choice of parameters (see caption).
There, the domain $D(\zeta_R)$ of the strategy is shown as the union of the orange, blue and pink regions, whereas the green region is the complement $\bar{D}(\zeta_R)$ of the domain.
Furthermore, the condition \eqref{eq:r0faultydomain} is depicted as the red line.
We claim that the incomplete strategy $\zeta_R$ is optimal on its domain.
This will be proven in App.~\ref{app:r0faultyoptimality}.
How can the strategy $\zeta_R$ of Eq.~\eqref{eq:r0faultydef} lead to failure? This is exemplified by a specific instance of the Weak Broadcast protocol in Fig.~\ref{fig:r0faultyexamples}, for $m=12$.
To be specific, we consider a Weak Broadcast protocol with parameter values $\mu =0.26$ and $\lambda = 0.94$.
The table in Fig.~\ref{fig:r0faultyexamples}a shows an Event for $m=12$.
The local information possessed by $R_0$ is denoted by boldface characters in Fig.~\ref{fig:r0faultyexamples}a (cf.~the second paragraph of this section); the information hidden from $R_0$ is denoted as italic.
Based on the local information $R_0$ has, it classifies the indices $\alpha \in \{1,\dots,m\} \equiv \{\mathrm{a}, \mathrm{b}, \dots, \mathrm{l}\}$
of the Event into the three classes defined above: 0011, XX10, XX0X.
In the example of Fig.~\ref{fig:r0faultyexamples}a, these
classes can be identified as the corresponding check sets:
$0011 = \{\mathrm{b,e,f,i}\}$,
$\text{XX10} = \{\mathrm{d,g}\}$,
and $\text{XX0X} = \{\mathrm{a,c,h,j,k,l}\}$.
This implies $m_{0011} \equiv \ell_1 = 4$,
$m_\text{XX10} \equiv \ell_2 = 2$, and $m_\text{XX0X} \equiv \ell_3 = 6$.
This local count list $\ell = (4,2,6)$ is in the domain $D(\zeta_R)$ of the incomplete strategy $\zeta_R$; in fact, it is in the orange region of Fig.~\ref{fig:r0faulty_stratdom}.
Hence $R_0$ can follow the incomplete strategy $\zeta_R$, which results in the communication and decision steps depicted in Fig.~\ref{fig:r0faultyexamples}b.
There, the sender $S$ and the receiver $R_1$ follow the Weak Broadcast protocol, and their output bit values are $x_S = 0$ and $y_1 = 1$, respectively, implying failure of the Weak Broadcast protocol.
The above example sheds light on why the strategy $\zeta_R$ in Eq.~\eqref{eq:r0strategy} is efficient (in fact, optimal in its domain) in achieving failure.
In more general terms, its efficiency is reasoned as follows.
According to this strategy, the adversary receiver $R_0$ sends a false data bit value ($y_{01} = 1 \neq 0 = x_S$) and tries to `back it up' by a check set $\rho_{01}$ that is convincingly long (that is, the Length Condition of the Cross-Check Phase is satisfied) and consistent (that is, to achieve that the Consistency Condition of the Cross-Check Phase is satisfied).
This is achieved by the strategy $\zeta_R$ of Eq.~\eqref{eq:r0strategy}, if the number of consistent indices is large enough; hence it is reasonable for $R_0$ to include \emph{all} indices that are certainly consistent with the bit value $y_{01}=1$ (red indices in Fig.~\ref{fig:r0faultyexamples}), and include the minimal amount of indices which are only potentially consistent with the bit value $y_{01} = 1$ (these are the two blue indices in Fig.~\ref{fig:r0faultyexamples}b), such that the Length Condition is satisfied.
\section{Failure probabilities}
\label{app:failureprobabilities}
In this section, we compute the exact failure probability for the `no faulty' adversary configuration, and compute tight upper bounds of the failure probabilities for the adversary configurations `$S$ faulty' and `$R_0$ faulty'.
These results allow future users of the protocol to allocate resources sparingly.
In fact, these results are analysed, and are used to optimize the protocol, in the main text.
\subsection{No faulty}
In this subsection, we consider the no faulty configuration.
Without the loss of generality, assume that the data bit sent by $S$ is $x_S = 0$.
Then, we have the following exact result for the failure probability $p_f$, which we denote as $p_f^{(\textrm{nf})}$:
\begin{equation}
\label{eq:pfnofaulty}
p_f^{(\mathrm{nf})}
= \sum_{m_{0011}=0}^{T-1} \binom{m}{m_{0011}} \left( \frac{1}{3} \right) ^{m_{0011}} \left( \frac{2}{3} \right)^{m-m_{0011}}.
\end{equation}
Recall the definition $T = \ceil{\mu m}$, introduced in the Weak Broadcast protocol, Sec.~\ref{sec:weakbroadcastprotocol}.
Recall also that $T$ is the minimal check set length that satisfies the Length Conditions.
To derive this result, it is sufficient to note that the protocol leads to failure only if the Length Condition of the Check Phase evaluates as False, that is, if the number of 0011 outcomes is less than the minimal check set length that satisfies the Length Conditions; i.e. if the following condition holds:
\begin{equation}
\label{eq:nofaultycondition}
m_{0011} < T.
\end{equation}
Since the Elementary Event 0011 is generated with a probability of $1/3$, the probability of
Eq.~\eqref{eq:nofaultycondition} holding true can be expressed by the binomial formula Eq.~\eqref{eq:pfnofaulty}.
This concludes the proof.
Note that the analysis generalises straightforwardly to the case when the message of the sender $S$ is $x_S = 1$, and yields the same result for $p_f^{(nf)}$.
\subsection{$S$ faulty}
\label{app:sfaultyfailureprobability}
Here, we provide an upper bound $p^{(S)}_{f, \uparrow}$ for the failure probability in the $S$ faulty configuration.
To express this upper bound,
recall that $T = \ceil{\mu m}$
is the minimal check set length that satisfies the Length Conditions, which was introduced in Eq.~\eqref{eq:Tdef}.
Also recall that the integer $Q$ was defined in Eq.~\eqref{eq:Qdef}. Here we use its exact definition, that is,
$Q = T- \ceil{T \lambda }+1$.
With these notations, the upper bound $p^{(S)}_{f, \uparrow}$ reads
\begin{eqnarray}
\label{eq:ps_upper}
p_{f,\uparrow}^{(S)} &=& p_{f,\downarrow}^{(S)}+
\left( 1- \sum_{\ell_3=T}^{m-T} \; \sum_{\ell_1=T-Q}^{m-Q-\ell_3} \binom{m}{\ell_3,\ell_1,m-\ell_1-\ell_3} \left (\frac{1}{3} \right )^{m} \right),
\end{eqnarray}
with
\begin{eqnarray}
\label{equation:ps_lower}
p_{f,\downarrow}^{(S)} &=& \sum_{\ell_3=T}^{m-T} \; \sum_{\ell_1=T-Q}^{m-Q-\ell_3} \binom{m}{\ell_3,\ell_1,m-\ell_1-\ell_3} \left (\frac{1}{3} \right )^{m} 2^{-Q}.
\end{eqnarray}
In the derivation of this upper bound, first we express the upper bound formally for a general incomplete strategy, and then we apply it to the $S$ faulty strategy defined above.
Assume that we have identified an incomplete strategy $\zeta$ that is optimal in its domain $D(\zeta) \subset \text{LCL}_S$.
An upper bound of the failure probability of this strategy can be expressed as
\begin{equation}
\label{eq:psuppersimple}
p^{(S)}_{f,\uparrow} =
\sum_{\ell \in D(\zeta)}
P_\zeta(\text{failure} | \ell)
P(\ell) +
\sum_{\ell \in \bar{D}(\zeta)}
P(\ell).
\end{equation}
Here, $P(\ell)$ is the probability that a random Event produces the local count list $\ell$.
Furthermore, $P_\zeta(\text{failure}| \ell)$ is the conditional probability of failure, given the random Event produces the local count list $\ell$.
Finally, $\bar{D}(\zeta)$ is the complement of $D(\zeta)$, that is, $\bar{D}(\zeta) = \text{LCL}_S \setminus D(\zeta)$.
In words, the formula \eqref{eq:psuppersimple} for the upper bound is interpreted as follows: the random Event either produces a local count list $\ell$ that is in the domain of the strategy $\zeta$, and then we compute the corresponding failure probability exactly (first sum); or, the Event produces a local count list $\ell$ that is outside the domain of the strategy $\zeta$, and then we assume a worst-case scenario, i.e., that failure happens with unit probability (second sum).
Using the normalization condition
\begin{equation}
1 = \sum_{\ell \in \text{LCL}_S}
P(\ell)
= \sum_{\ell \in D(\zeta)} P(\ell)
+ \sum_{\ell \in \bar{D}(\zeta)} P(\ell),
\end{equation}
we see that Eq.~\eqref{eq:psuppersimple} can be reformulated as
\begin{equation}
\label{eq:puppergeneral2}
p^{(S)}_{f,\uparrow} =
\sum_{\ell \in D(\zeta)}
P_\zeta(\text{failure} | \ell)
P(\ell) +
\left(
1-
\sum_{\ell \in D(\zeta)}
P(\ell)
\right).
\end{equation}
Note that these considerations suggest that for a given incomplete strategy, a lower bound $p^{(S)}_{f,\downarrow}$ of failure probability can also be expressed, by assuming a best-case scenario, i.e., zero failure probability, for Events that produce local count lists that are outside of the domain of the strategy.
Formally, this lower bound is expressed as
\begin{equation}
\label{eq:pslowerwithell}
p^{(S)}_{f,\downarrow} =
\sum_{\ell \in D(\zeta)}
P_\zeta(\text{failure} | \ell)
P(\ell).
\end{equation}
To evaluate the upper bound $p^{(S)}_{f,\uparrow}$ of the failure probability for this optimal partial strategy $\zeta$ via Eq.~\eqref{eq:puppergeneral2}, we first express
\begin{equation}
\label{eq:Pell}
P(\ell) =
\binom{m}{\ell_1, m-\ell_1-\ell_3, \ell_3} \left (\frac{1}{3} \right )^{m}.
\end{equation}
Recall that $\ell = (\ell_1,\ell_2,\ell_3)$ is determined by two of its components, e.g., $\ell_1$ and $\ell_3$, because $\ell_1 + \ell_2 + \ell_3 = m$.
Equation \eqref{eq:Pell} is a straighforward consequence of the fact that the probability of a 0011 outcome, the probability of a mixed outcome (0101 or 0110 or 1001 or 1010), and the probability of a 1100 outcome, are all equal to $1/3$.
Finally, we express the failure probability of $\zeta$ on its domain, which is
\begin{equation}
\label{eq:conditional}
P_\zeta(\text{failure} | \ell) = (1/2)^Q, \mbox{ for all $\ell \in D(\zeta)$}.
\end{equation}
This follows from the fact that $S$ includes $Q$ mixed indices in its check set $\sigma_0$, and this leads to failure only if all the $Q$ corresponding mixed outcomes contain a `1' at $R_0$.
Inserting Eqs.~\eqref{eq:Pell} and \eqref{eq:conditional} into Eq.~\eqref{eq:puppergeneral2}, we obtain the result Eq.~\eqref{equation:ps_lower}, which concludes our derivation.
Note also that the first term of Eq.~\eqref{equation:ps_lower} is in fact the failure-probability lower bound of our specific strategy $\zeta$, in line with the general expression in Eq.~\eqref{eq:pslowerwithell}.
\subsection{$R_0$ faulty}
\label{app:rfaultyfailureprobability}
Here, we provide an upper bound $p^{(R)}_{f,\uparrow}$ for the failure probability in the $R_0$ faulty configuration.
Again, recall that $T = \ceil{\mu m}$ (original definition in Eq.~\eqref{eq:Tdef}) and
$Q = T- \ceil{T \lambda }+1$ (original definition in Eq.~\eqref{eq:Qdef}.
With these notations, the failure-probability upper bound reads:
\begin{equation}
\label{equation:pr0_upper}
\begin{aligned}
p_{f,\uparrow}^{(R)} = p_{f,\downarrow}^{(R)} + \sum_{\ell_1 = m- T+1}^{m} \binom{m}{\ell_1} \left (\frac{1}{3} \right )^{\ell_1} \left (\frac{2}{3} \right )^{m-\ell_1},
\end{aligned}
\end{equation}
where
\begin{equation}
\label{equation:pr0_lower}
\begin{aligned}
p_{f,\downarrow}^{(R)} &=\sum_{\ell_1=T}^{m-T} \sum_{\ell_2=0}^{T-Q} \binom{m}{\ell_1,\ell_2,\ell_3} \left(\frac{1}{3} \right)^{\ell_1} \left(\frac{1}{6} \right)^{\ell_2} \left(\frac{1}{2} \right)^{\ell_3} \sum_{k = T- Q + 1 - \ell_2}^{T-\ell_2} \binom{T-\ell_2}{k} \left(\frac{2}{3} \right)^k \left(\frac{1}{3} \right)^{T-\ell_2-k} + \\
&+ \sum_{\ell_1=T}^{m-T} \sum_{\ell_2=T-Q+1}^{m-\ell_1} \binom{m}{\ell_1,\ell_2,\ell_3} \left(\frac{1}{3} \right)^{\ell_1} \left(\frac{1}{6} \right)^{\ell_2} \left(\frac{1}{2} \right)^{\ell_3}+\\
&+ \sum_{\ell_1=0}^{T-1} \binom{m}{\ell_1} \left( \frac{1}{3} \right) ^{\ell_1} \left( \frac{2}{3} \right)^{m-\ell_1}.
\end{aligned}
\end{equation}
The result \eqref{equation:pr0_upper} is valid both for the $x_S = 0$ and the $x_S=1$ scenario.
In what follows, we derive this result. The derivation follows the 3-step scheme outlined in the $S$ faulty case below Eq.~\eqref{equation:ps_lower}.
Assume that we have identified an incomplete strategy $\zeta$ that is optimal on its domain $D(\zeta) \subset \text{LCL}_R$.
An upper bound of the failure probability of this strategy can be expressed as shown in Eqs.~\eqref{eq:psuppersimple}.
A lower bound can also be expressed as Eq.~\eqref{eq:pslowerwithell}.
For our optimal incomplete strategy $\zeta$, the formula for the failure-probability upper bound, Eq.~\eqref{eq:psuppersimple}, can be structured further, according to the three regions (orange, pink, blue) in Fig.~\ref{fig:r0faulty_stratdom}:
\begin{equation}
p^{(R)}_{f,\uparrow} =
\sum_{\ell \in \text{orange}}
P_\zeta(\text{failure}|\ell) P(\ell)
+
\sum_{\ell \in \text{pink}}
P_\zeta(\text{failure}|\ell) P(\ell)
+
\sum_{\ell \in \text{blue}}
P_\zeta(\text{failure}|\ell) P(\ell)
+
\sum_{\ell \in \bar{D}(\ell)}
P(\ell).
\end{equation}
A simplification is allowed by the observation that for local count lists ($\ell$) in the pink and blue regions, the failure probability is $P_\ell(\text{failure}|\ell) = 1$,
hence
\begin{equation}
\label{eq:pfcolorsimplified}
p^{(R)}_{f,\uparrow} =
\sum_{\ell \in \text{orange}}
P_\zeta(\text{failure}|\ell) P(\ell)
+
\sum_{\ell \in \text{pink}}
P(\ell)
+
\sum_{\ell \in \text{blue}}
P(\ell)
+
\sum_{\ell \in \bar{D}(\ell)}
P(\ell).
\end{equation}
In the pink region, failure is guaranteed, in fact, for any $R_0$ adversary strategy, since $\ell_1 < T$ implies that the check list $\sigma_0 = \sigma_1$ sent by $S$ violates the Length Condition of the Check Phase.
In the blue region, however, failure is guaranteed because the adversary $R_0$ was lucky to find enough outcomes XX10, that is, $\ell_2 \geq T-Q+1$, such that its check set $\rho_{01}$ automatically satisfies the Consistency Condition of the Cross-check Phase.
The last three terms of Eq.~\eqref{eq:pfcolorsimplified} can be readily expressed by binomial and trinomial distributions.
The blue, pink, and green regions correspond to, respectively, to the second term of Eq.~\eqref{equation:pr0_lower}, to the third term of Eq.~\eqref{equation:pr0_lower}, and to the second term of Eq.~\eqref{equation:pr0_upper}.
To complete the derivation of the failure-probability upper bound, we express $P_\zeta(\text{failure}| \ell)$ for $\ell \in \text{orange}$:
\begin{equation}
\label{eq:orangefailure}
P_\zeta(\text{failure} | \ell) = \sum_{k = T-Q+1-\ell_2}^{T-\ell_2} \binom{T-\ell_2}{k} \left(\frac{2}{3} \right)^k \left(\frac{1}{3} \right)^{T-\ell_2-k}, \mbox{ for $\ell \in$ orange}.
\end{equation}
To interpret this formula, let us first recall that in the orange region of $D(\zeta)$, it holds that $n_\text{min} = T-\ell_2$.
Furthermore, note also that the probability that the number of XX10 outcomes in the check set $\rho_{01}$ of length $n_\text{min} = T- \ell_2$ is $k$ is given by the binomial distribution.
Hence, Eq.~\eqref{eq:orangefailure} sums up the probabilities where $k$ is large enough that $\rho_{01}$ satisfies the Consistency Condition of the Cross-check Phase.
Combining Eq.~\eqref{eq:orangefailure} with the trinomial distribution expressing the probability of the local count lists of the orange region, we obtain the first term of Eq.~\eqref{equation:pr0_lower}. This concludes the derivation of the failure-probability upper bound in the case of an adversary $R_0$.
\section{Proofs of optimality}
\label{app:proofsofoptimality}
\subsection{$S$ faulty}
\label{app:sfaultyoptimality}
Here, we show that the $S$ faulty adversary strategy defined above is an optimal incomplete strategy, i.e., it is optimal on its domain.
Consider therefore an arbitrary strategy
\begin{eqnarray}
\mathcal{\zeta}_B = \left(
k^{(0)}_{0011},
k^{(0)}_\text{mixed},
k^{(0)}_{1100};
k^{(1)}_{0011},
k^{(1)}_\text{mixed},
k^{(1)}_{1100}
\right).
\end{eqnarray}
It is straightforward to see that the strategy
\begin{eqnarray}
\mathcal{\zeta}'_B = \left(
k^{(0)}_{0011},
k^{(0)}_\text{mixed},
0;
0,
0,
m_{1100}
\right),
\end{eqnarray}
derived from $\mathcal{S}_B$,
has a higher or equal failure probability than the initial Bad Strategy $\mathcal{S}_B$:
$p_f(\mathcal{S}_B) \leq p_f(\mathcal{S}'_B)$.
The reason is as follows.
On the one hand, deviating from
$\left(
k^{(1)}_{0011},
k^{(1)}_\text{mixed},
k^{(1)}_{1100}
\right) = (0,0,m_{1100})$
either decreases the failure probability, or does not change that.
In words: from the viewpoint of an adversary $S$,
it is useless to deviate
from the correct communication
toward $R_1$.
On the other hand, $k^{(0)}_{1100} > 0$ leads to weak broadcast, implying $p_f = 0$.
To analyze the failure probability of
the `improved' strategy $\mathcal{\zeta}'_B$
further, we introduce
$Q' = k^{(0)}_\text{mixed}$ and
$T' = k^{(0)}_{0011} + Q'$.
We address the question:
are any strategies defined by $(T',Q') \neq (T,Q)$ better than the
strategy the strategy $\zeta$?
No, as we argue now.
\begin{figure}
\caption{Optimal, suboptimal, and dysfunctional strategies.
(a) Adversary strategies of a faulty sender $S$.
(b) Adversary strategies of a faulty receiver $R_0$.
Filled green circle represents the incomplete optimal adversary strategy $\zeta_S$ in (a), $\zeta_R$ in (b).
Red, purple, and blue circles represent dysfunctional strategies.
Opaque green circles represent suboptimal adversary strategies.
Arrows depict increasing failure probability.}
\label{fig:StrategyFailureFlow}
\end{figure}
(i)
If $T' < T$, then the strategy $\mathcal{\zeta}'_B$ is dysfunctional, $p_f = 0$, since
there are not enough indices in $\sigma_0$ and $\sigma_1$ to convince the receivers;
that is, the Length Condition of the Check Phase evaluates is violated, implying weak broadcast.
These strategies are depicted as red circles in Fig.~\ref{fig:StrategyFailureFlow}a.
(ii) If $T' \geq T$ and $Q' < Q$, then the strategy $\mathcal{\zeta}'_B$ is dysfunctional, since this excludes the possibility that $\rho_{01}$ contains enough inconsistent indices that compromise $R_0$.
These strategies are depicted as purple circles in Fig.~\ref{fig:StrategyFailureFlow}.
(iii) The failure probability of the strategy $\zeta'_B$ along the `line' $(T',Q') = (T+n,Q+n)$ decreases as $n\geq 0$ is increased.
I.e., along this line, depicted as the line of diagonal arrows in Fig.~\ref{fig:StrategyFailureFlow}a, the filled green circle depicting
$n=0$ represents the optimal strategy.
The reason is that by increasing $T'$ from $T$ to $T+n$, the number of mixed outcomes in $\sigma_0$ for which $R_0$ needs to measure a `1' to cause failure is increased to $Q+n$, as demanded by the Consistency Condition of the Cross-Check Phase.
This reduces the corresponding failure probability from $(1/2)^Q$ to
$(1/2)^{Q+n}$.
This trend is depicted in Fig.~\ref{fig:StrategyFailureFlow}a by the diagonal arrows.
(iv) The strategy $\zeta'_B$ is dysfunctional `below the line' defined in (iii), see the blue circles in Fig.~\ref{fig:StrategyFailureFlow}a.
There, even if all $Q'$ mixed outcomes included in $\sigma_0$ have `1' at $R_0$, that is insufficient to violate the Consistency Condition of the Cross-Check phase.
(v) The failure probability of the strategies $\zeta'_B$ `above the line' defined in (iii), that is, strategies with $(T',Q') = (T+n,Q+n')$ (where $n\geq0$ and $n' \geq n$), decrease as $n'$ is increased.
Indeed, as $n'$ is increased, the check set size remains unchanged ($T+n$), but the number of `guessed' indices ($Q+n'$) is increased, and hence the failure probability $(1/2)^{Q+n'}$ decreases.
This trend is depicted in Fig.~\ref{fig:StrategyFailureFlow}a by the vertical arrows.
This concludes the proof that the strategy $\zeta = (T-Q,Q,0;0,0,m_{1100})$ is indeed optimal on its domain.
\subsection{$R_0$ faulty}
\label{app:r0faultyoptimality}
Here, we prove that the $R_0$ adversary strategy
$\mathcal{\zeta} = (0,m_\text{XX10},n_\text{min})$
defined above is optimal on its domain.
(i) Increasing the first strategy parameter $k_{0011}$ from zero decreases the failure probability,
as it lowers the chance of the Consistency Condition of the Cross-check Phase evaluating as True.
(ii) Consider the alternative strategies $\zeta' = (0,k_\text{XX10},k_\text{XX0X})$, with $k_\text{XX10} = m_\text{XX10} - n$ and $k_\text{XX0X} = n_\text{min} + n'$, with $n, n' \geq 0$.
The alternative strategies are dysfunctional ($p_f = 0$) `below the line' of the $(k_\text{XX10},k_\text{XX0X})$ plane defined by $n = n'$, i.e., for $n > n'$.
This is because for such a strategy, the check set size $|\rho_{01}|$ is below the minimum size $T$ required to satisfy the Length Conditions.
These dysfunctional strategies are shown as the red circles in Fig.~\ref{fig:StrategyFailureFlow}b.
(iii) The alternative strategies along the line $n=n'$, shown in Fig.~\ref{fig:StrategyFailureFlow}b as the line of diagonal arrows, have a decreasing failure probability as $n$ is increased.
Along this line, the check set size $|\rho_{01}|$ is fixed to the minimum size $T$, but the number $n_\text{min} + n'$ of guessed indices increases, and hence the probability of satisfying the Consistency Condition of the Cross-check Phase decreases.
This trend is depicted in Fig.~\ref{fig:StrategyFailureFlow}b by the diagonal arrows.
(iv) Consider the alternative strategies `above the line' of the $(k_\text{XX10},k_\text{XX0X})$ plane defined by $n = n'$, i.e., above the line of diagonal arrows in Fig.~\ref{fig:StrategyFailureFlow}b.
These are the strategies of the form $\zeta = (0, m_\text{XX10}-n,n_\text{min}+n')$ with $n' \geq n$.
For a fixed $n$, the failure probability of these strategies decreases as $n'$ is increased.
This is because increasing $n'$ by 1 increases the check set size $|\rho_{01}|$ by 1, and hence the right hand side of the Consistency Condition of the Check Phase by 1, and at the same time, increases the minimum number of lucky indices needed to satisfy that Consistency Condition, leading to a decrease of the failure probability.
This trend is depicted in Fig.~\ref{fig:StrategyFailureFlow}b by the horizontal arrows.
This concludes the proof that the strategy $\zeta = (0,m_\text{XX10},n_\text{min})$ is indeed optimal on its domain
\section{Security proof}
\label{app:securityproof}
In this Appendix, we derive asymptotic ($m \to \infty$) upper bounds for the failure probability for generic adversary strategies.
\subsection{No faulty}
Without the loss of generality, we assume that $S$ sends the message $x_S =0$.
The protocol will achieve consensus, $y_0 = y_1 = x_S = 0$, if the number of 0011 outcomes in the event is at least $T$, cf. the exact failure-probability result of Eq.~\eqref{eq:pfnofaulty}.
The probability of achieving consensus, i.e., that a random event contains the outcome 0011 with frequency $m_{0011}$ is
\begin{eqnarray}
P(m_{0011}) = P\left( \left(\sum_{j=1}^m X_j\right) = m_{0011}\right),
\end{eqnarray}
where $X_j$ are binary (Bernoulli) random variables, taking
values 0 or 1, with probability $p=1/3$ for the latter.
This implies that
\begin{eqnarray}
p_f(m) =
P\left( \left(\sum_{j=1}^{m} X_j \right) \leq \mu m \right)
\end{eqnarray}
Recall that $\mu < 1/3$; hence, the failure probability $p_f$ can be upper bounded by the Chernoff bound as
\begin{eqnarray}
p_f(m) \leq
\exp\left(
- \frac 1 2 \frac m 3 (1-3\mu)^2
\right).
\end{eqnarray}
That is, in the absence of an adversary, the failure probability of the Weak Broadcast protocol is subject to an upper bound that decreases exponentially as $m$ increases.
Recall that the general form of the Chernoff bound used to derive this result is
\begin{equation}
\label{eq:chernoff1}
P(X \leq (1-\delta) \bar{X}) \leq e^{- \bar{X} \delta^2/2},
\mbox{ for }0 \leq \delta,
\end{equation}
where $\bar{X}$ is the expectation value of the sum variable $X$.
\subsection{$S$ faulty}
The security proof for the $S$ faulty adversary configuration will be based on the tight failure-probability upper bound calculation presented in App.~\ref{app:sfaultyfailureprobability}.
Our goal is here is to derive an exponentially decaying upper bound for the failure probability.
To this end, we restrict the domain of the $S$ faulty adversary strategy $\zeta$, defined in \ref{app:sfaultyfailureprobability}.
The restricted strategy is denoted as $\zeta_r$.
In particular, we restrict the domain of $\zeta$ for `regular' or `typical' local count lists, where $\ell_1$ and $\ell_3$ are approximately equal to their expected values $m/3$.
Formally, we define $D(\zeta_r)$ as formed by those $(\ell_1,\ell_3)$ pairs where
\begin{equation}
\label{eq:srestriction}
\mu m < \ell_1, \ell_3 < 2m/3 - \mu m.
\end{equation}
When is $\zeta_r$ indeed a restriction of $\zeta$?
Conditions \eqref{eq:cond1} and \eqref{eq:cond3} are automatically satisfied if Eq.~\eqref{eq:srestriction} holds.
However, Eq.~\eqref{eq:cond2} demands $Q \leq \ell_2$; this is also guaranteed if $\lambda \geq 1/2$.
Since $D(\zeta_r) \subset D(\zeta)$, and $D(\zeta)$ is optimal on its domain, the restricted strategy $\zeta_r$ is also optimal on its domain.
In the spirit of Eq.~\eqref{eq:psuppersimple}, the failure-probability upper bound can be expressed using the restricted strategy $\zeta_r$ as follows:
\begin{equation}
\label{eq:p_s_securityproof}
p_{f}(m) \leq
\sum_{\ell \in D(\zeta_r)}
P_{\zeta_r}(\text{failure} | \ell)
P(\ell) +
\sum_{\ell \in \bar{D}(\zeta_r)}
P(\ell).
\end{equation}
It is straightforward to exponentially upper bound the first term using the fact that
\begin{equation}
P_{\zeta_r}(\text{failure}|\ell) = 2^{-Q} \approx 2^{-(1-\lambda)\mu m},
\end{equation}
which is independent of $\ell$.
Therefore, we have
\begin{equation}
\sum_{\ell \in D(\zeta_r)}
P_{\zeta_r}(\text{failure} | \ell)
P(\ell) \approx
2^{-(1-\lambda)\mu m}
\sum_{\ell \in D(\zeta_r)}
P(\ell)
\leq
2^{-(1-\lambda)\mu m},
\end{equation}
where we used $\sum_{\ell \in D(\zeta_r)} P(\ell) \leq 1$.
To exponentially upper bound the second term of Eq.~\eqref{eq:p_s_securityproof}, we express it as
\begin{eqnarray}
\sum_{\ell \in \bar{D}(\zeta_r)}
P(\ell) &=&
P(
(\ell_1 \leq \mu m) \mbox{ OR }
(\ell_1 \geq 2m/3 - \mu m) \mbox{ OR }
(\ell_3 \leq \mu m/3) \mbox{ OR }
(\ell_3 \geq 2m/3 - \mu m)
) \nonumber
\\
&=&
P(\ell_1 \leq \mu m) +
P(\ell_1 \geq 2m/3 - \mu m) +
P(\ell_3 \leq \mu m/3) +
P(\ell_3 \geq 2m/3 - \mu m)
\\
&\leq&
4 e^{-\frac 1 3 \frac m 3 (1-3\mu)^2}.
\end{eqnarray}
In the last step, we used another Chernoff bound (cf.~Eq.~\eqref{eq:chernoff1}):
\begin{equation}
\label{eq:chernoff2}
P(|X - \bar{X}| \geq \delta \cdot \bar{X} ) \leq 2 e^{- \bar{X} \delta^2/3},
\mbox{ for }0 \leq \delta.
\end{equation}
\subsection{$R_0$ faulty}
Here, we describe the security proof for the $R_0$ faulty adversary configuration. This proof is based on the tight failure-probability upper bound calculation presented in App.~\ref{app:rfaultyfailureprobability}.
We restrict the domain of the $R_0$ faulty optimal incomplete adversary strategy $\zeta$, defined in \ref{app:rfaultystrategy}.
The restricted strategy is denoted as $\zeta_r$.
In particular, we restrict the domain of $\zeta$ for `regular' local count lists, where $\ell_1$ and $\ell_2$ are close to their expected values, $m/3$ and $m/6$, respectively.
Formally, we define the domain $D(\zeta_r)$ of the restricted strategy as formed by those $(\ell_1,\ell_2)$ pairs which satisfy
\begin{eqnarray}
\label{eq:rrestriction}
\mu m &< \ell_1 <& 2m/3 - \mu m, \\
\mu m / 2 & < \ell_2 < & 2m/6 - \mu m / 2.
\end{eqnarray}
Note that $\zeta_r$ is indeed a restriction of $\zeta$, since the upper bound in Eq.~\eqref{eq:rrestriction} is a stricter condition than \eqref{eq:r0faultydomain}.
Furthermore, since $D(\zeta_r) \subset D(\zeta)$, and $\zeta$ is optimal on its domain, the restricted strategy $\zeta_r$ is also optimal on its domain.
In the spirit of Eq.~\eqref{eq:psuppersimple}, the failure-probability upper bound can be expressed using the restricted strategy $\zeta_r$ as follows:
\begin{equation}
\label{eq:p_r_securityproof}
p_{f}(m) \leq
\sum_{\ell \in D(\zeta_r)}
P_{\zeta_r}(\text{failure} | \ell)
P(\ell) +
\sum_{\ell \in \bar{D}(\zeta_r)}
P(\ell).
\end{equation}
To exponentially upper bound the second term of Eq.~\eqref{eq:p_r_securityproof}, we express it as
\begin{eqnarray}
\sum_{\ell \in \bar{D}(\zeta_r)}
P(\ell) &=&
P(
(\ell_1 \leq \mu m) \mbox{ OR }
(\ell_1 \geq 2m/3 - \mu m) \mbox{ OR }
(\ell_2 \leq \mu m/6) \mbox{ OR }
(\ell_2 \geq 2m/6 - \mu m/2)
) \nonumber
\\
&=&
P(\ell_1 \leq \mu m) +
P(\ell_1 \geq 2m/3 - \mu m) +
P(\ell_2 \leq \mu m/6) +
P(\ell_2 \geq 2m/6 - \mu m/2)
\\
&\leq&
2 e^{-\frac 1 3 \frac m 3 (1-3\mu)^2}
+
2 e^{-\frac 1 3 \frac m 6 (1-3\mu)^2}.
\end{eqnarray}
In the last step, we used the Chernoff bound in Eq.~\eqref{eq:chernoff2}.
Finally, we provide an upper bound for the first term of Eq.~\eqref{eq:p_r_securityproof}.
Within the restricted strategy domain $D(\zeta_r)$, the adversary $R_0$ has the greatest chance to cause failure if the number of `safe' indices, $\ell_2$, is maximal, $\ell_2 = \ell_{2,\text{max}} \equiv \lfloor 2m/6 - \mu m/2 \rfloor$.
For a local count list containing $\ell_{2,\text{max}}$, Eq.~\eqref{eq:orangefailure} specifies $P_\zeta(\text{failure}|\ell_1,\ell_{2,\text{max}})$ as a sum of the upper part of a binomial distribution, between $k = T - Q + 1 - \ell_{2,\text{max}}$ to $k=T-\ell_{2,\text{max}}$.
In fact, the quantity $P_{\zeta_r}(\text{failure}|\ell_1,\ell_{2,\text{max}})$
is independent of $\ell_1$.
The Chernoff bound
\begin{equation}
\label{eq:chernoff3}
P(X \geq (1+\delta) \bar{X}) \leq e^{- \bar{X} \delta^2/3},
\mbox{ for }0 \leq \delta,
\end{equation}
can be used to upper bound this sum, if the the latter is an upper tail sum, that is, the lower end of the $k$ sum is above the expectation value $\bar{X} = \frac 2 3 (T-\ell_{2,\text{max}})$ of $k$:
\begin{equation}
{T-Q+1 - \ell_{2,\text{max}}} > \frac{2}{3} \left(T-\ell_{2,\text{max}}\right).
\end{equation}
The latter condition is readily transformed as
\begin{equation}
\label{eq:lambdacondition}
\lambda > \frac{2+9\mu}{18\mu},
\end{equation}
where we disregarded ceiling and floor functions for simplicity.
Note that since $\lambda < 1$, the condition of Eq.~\eqref{eq:lambdacondition} can be satisfied only if
\begin{equation}
\label{eq:mucondition}
\mu > 2/9\approx 0.222.
\end{equation}
If the condition \eqref{eq:lambdacondition} holds, then the Chernoff bound of Eq.~\eqref{eq:chernoff3} is applied with the substitution
\begin{equation}
\delta = \frac{2+ 9 \mu - 18 \lambda \mu}{6-27 \mu},
\end{equation}
yielding
\begin{eqnarray}
\sum_{\ell \in D(\zeta_r)} P_{\zeta_r}(\text{failure}|\ell) P(\ell)
&\leq&
\sum_{\ell \in D(\zeta_r)} P_{\zeta_r}(\text{failure}|\ell_1,\ell_{2,\text{max}}) P(\ell)
\\
&=&
P_{\zeta_r}(\text{failure}|\ell_{2,\text{max}})
\sum_{\ell \in D(\zeta_r)} P(\ell)
\\
&\leq&
P_{\zeta_r}(\text{failure}|\ell_{2,\text{max}})
\\
&\leq&
e^{-\bar{X} \delta^2/3},
\end{eqnarray}
with
\begin{equation}
\label{eq:chernoff3applied}
\bar{X} = \frac 2 3 (T-\ell_{2,\text{max}}) \approx \left(\frac 3 2 \mu - \frac 1 3 \right) m.
\end{equation}
Note that Eq.~\eqref{eq:mucondition} implies that the coefficient of $m$ on the right hand side is positive, i.e., the upper bound in Eq.~\eqref{eq:chernoff3applied} is exponentially decreasing with increasing $m$.
\section{Preparation and benchmarking of the four-qubit singlet
state on IBM Q}
This appendix exhibits the data we obtained by conducting
experiments on IBM Q quantum computer prototypes.
We aimed these experiments to prepare the
four-qubit singlet state $\ket{\psi}$ of Eq.~\eqref{eq:cabello}
using Circuit A of Fig.~\ref{fig:bryancirc},
and to quantify how close the experiment resembles
the preparation of an ideal four-qubit singlet state.
Further details are in the table caption.
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
backend & qubits & $F_c$ & $F_c^\text{mitig}$ & date(c) & $F_q$ & $F_q^\text{mitig}$ & date(q) \\ \hline
\multirow{8}{*}{\verbcasablanca} & 0-1-3-5 & 0.3234 & 0.3115 & 2021.04.26 & 0.1812 & 0.2071 & 2021.04.28 \\ \cline{2-8}
& 2-1-3-5 & 0.3109 & 0.3033 & 2021.04.26 & 0.1963 & 0.2166 & 2021.04.28 \\ \cline{2-8}
& 1-3-5-4 & 0.3409 & 0.1259 & 2021.04.26 & 0.3182 & 0.3436 & 2021.04.28 \\ \cline{2-8}
& 1-3-5-6 & 0.3066 & 0.1182 & 2021.04.26 & 0.1118 & 0.1105 & 2021.04.28 \\ \cline{2-8}
& 5-3-1-0 & 0.3061 & 0.1566 & 2021.04.26 & 0.5006 & 0.5527 & 2021.04.28 \\ \cline{2-8}
& 5-3-1-2 & 0.3130 & 0.1587 & 2021.04.26 & 0.3726 & 0.4141 & 2021.04.28 \\ \cline{2-8}
& 4-5-3-1 & 0.3399 & 0.3011 & 2021.04.26 & 0.1701 & 0.1818 & 2021.04.28 \\ \cline{2-8}
& 6-5-3-1 & 0.3532 & 0.2047 & 2021.04.26 & 0.1301 & 0.1324 & 2021.04.28 \\ \hline
\multirow{4}{*}{\verbsantiago} & 0-1-2-3 & 0.8510 & 0.9021 & 2021.04.25 & 0.7435 & 0.8116 & 2021.04.26 \\ \cline{2-8}
& 1-2-3-4 & 0.8501 & 0.9002 & 2021.04.25 & 0.7221 & 0.8011 & 2021.04.26 \\ \cline{2-8}
& 3-2-1-0 & 0.7595 & 0.8604 & 2021.04.25 & 0.5431 & 0.5939 & 2021.04.26 \\ \cline{2-8}
& 4-3-2-1 & 0.8332 & 0.8795 & 2021.04.25 & 0.6659 & 0.7803 & 2021.04.26 \\ \hline
\multirow{4}{*}{\verbathens} & 0-1-2-3 & 0.7836 & 0.8370 & 2021.04.23 & 0.6632 & 0.7311 & 2021.04.26 \\ \cline{2-8}
& 1-2-3-4 & 0.8050 & 0.8608 & 2021.04.23 & 0.6652 & 0.7868 & 2021.04.26 \\ \cline{2-8}
& 3-2-1-0 & 0.7686 & 0.8828 & 2021.04.23 & 0.5599 & 0.6283 & 2021.04.26 \\ \cline{2-8}
& 4-3-2-1 & 0.8025 & 0.8573 & 2021.04.23 & 0.6847 & 0.7811 & 2021.04.26 \\ \hline
\multirow{4}{*}{\verbbogota} & 0-1-2-3 & 0.7566 & 0.8840 & 2021.04.26 & 0.4710 & 0.6022 & 2021.04.26 \\ \cline{2-8}
& 1-2-3-4 & 0.7322 & 0.8813 & 2021.04.26 & 0.5278 & 0.7802 & 2021.04.27 \\ \cline{2-8}
& 3-2-1-0 & 0.6904 & 0.7909 & 2021.04.26 & 0.4998 & 0.6512 & 2021.04.27 \\ \cline{2-8}
& 4-3-2-1 & 0.7295 & 0.8352 & 2021.04.26 & 0.4003 & 0.6269 & 2021.04.27 \\ \hline
\multirow{4}{*}{\verbrome} & 0-1-2-3 & 0.6347 & 0.7103 & 2021.04.26 & 0.3893 & 0.4727 & 2021.04.26 \\ \cline{2-8}
& 1-2-3-4 & 0.7106 & 0.7896 & 2021.04.26 & 0.5587 & 0.6827 & 2021.04.26 \\ \cline{2-8}
& 3-2-1-0 & 0.5531 & 0.6039 & 2021.04.26 & 0.1537 & 0.1734 & 2021.04.26 \\ \cline{2-8}
& 4-3-2-1 & 0.6424 & 0.7083 & 2021.04.26 & 0.3775 & 0.4475 & 2021.04.26 \\ \hline
\end{tabular}
\caption{Preparation and benchmarking of the four-qubit singlet
state on IBM Q prototype quantum computers.
We prepared the state $\ket{\psi}$ of Eq.~\eqref{eq:cabello}
using Circuit A shown in Fig.~\ref{fig:bryancirc}.
The first column indicates the quantum computer (`backend') used;
the second column identifies the
qubits of Circuit A with the qubits of the backend.
The third (fourth) column, $F_c$ ($F_c^\text{mitig}$), shows
the classical fidelity, Eq.~\eqref{eq:classicalfidelity},
of the ideal distribution (Fig.~\ref{fig:Cabstate_distribution}c)
and the distribution generated by the backend,
without (with) readout error mitigation.
The third (fourth) column, $F_c$ ($F_c^\text{mitig}$), shows
the classical fidelity, Eq.~\eqref{eq:classicalfidelity},
of the ideal distribution (Fig.~\ref{fig:Cabstate_distribution}c)
and the distribution generated by the backend,
without (with) readout error mitigation.
The sixth (seventh) column,
$F_q$ ($F_q^\text{mitig}$), shows
the quantum state fidelity, Eq.~\eqref{eq:quantumfidelity},
of $\ket{\psi}$
and the output density matrix of
Circuit A, as inferred from quantum state tomography
without (with) readout error mitigation.
Measurement dates are indicated in the fourth and eighth columns.
\label{tab:classicaltable}}
\end{table}
\end{document} | math |
# manter-colaborador
Projeto para manter um colaborador.
Inicio: 02/01/2017.
| code |
# Ligusticum maireii M.Hiroe SPECIES
#### Status
ACCEPTED
#### According to
International Plant Names Index
#### Published in
null
#### Original name
null
### Remarks
null | code |
/*
* Copyright (C) 2009, 2010 Jayway AB
*
* Licensed 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 com.jayway.maven.plugins.lab;
import static org.junit.Assert.assertEquals;
import static org.junit.Assert.assertFalse;
import static org.junit.Assert.assertTrue;
import static org.junit.Assert.fail;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.StringWriter;
import org.junit.Test;
import com.jayway.maven.plugins.lab.LabCreator;
import com.jayway.maven.plugins.lab.Version;
import com.jayway.maven.plugins.lab.VersionedContents;
public class LabCreatorTest {
public static String newline = System.getProperty("line.separator");
@Test
public void correctContentsOfVersion() throws IOException {
String name = "VersionTest.txt";
VersionedContents versionedContents = new LabCreator().labify(new InputStreamReader(LabCreatorTest.class.getClassLoader().getResourceAsStream(name)), name);
checkCorrectContentsOfVersion(versionedContents);
}
@Test
public void correctContentsOfVersionWithConstants() throws IOException {
String name = "VersionTestWithConstants.txt";
VersionedContents versionedContents = new LabCreator("noll", "ETT", "TVÅ", "tre").labify(new InputStreamReader(LabCreatorTest.class.getClassLoader().getResourceAsStream(name)), name);
checkCorrectContentsOfVersion(versionedContents);
}
private void checkCorrectContentsOfVersion(VersionedContents versionedContents) {
String contents1 = getContents(versionedContents, 1);
assertEquals("VERSION1" + newline, contents1);
String contents2 = getContents(versionedContents, 2);
assertEquals("VERSION1" + newline + "VERSION2" + newline, contents2);
String contents3 = getContents(versionedContents, 3);
assertEquals("---" + newline + "VERSION1" + newline + "VERSION2" + newline + "---" + newline, contents3);
}
private String getContents(VersionedContents versionedContents, int step) {
StringWriter writer = new StringWriter();
versionedContents.writeVersion(writer, step);
return writer.getBuffer().toString();
}
@Test
public void nonJavaFilesShouldWork() throws IOException {
String name = "NonJavaFile.properties";
VersionedContents versionedContents = new LabCreator().labify(new InputStreamReader(LabCreatorTest.class.getClassLoader().getResourceAsStream(name)), name);
assertEquals(new Version(2), versionedContents.getMaxVersion());
assertEquals(new Version(0), versionedContents.getLeastVersion());
}
@Test
public void normalLab() throws IOException {
String name = "Dummy.java.txt";
VersionedContents versionedContents = new LabCreator().labify(new InputStreamReader(LabCreatorTest.class.getClassLoader().getResourceAsStream(name)), name);
assertEquals(new Version(3), versionedContents.getMaxVersion());
assertEquals(new Version(1), versionedContents.getLeastVersion());
}
@Test
public void incorrectlyNestedTagsShouldGenerateException() throws IOException {
String name = "Evil.java.txt";
try {
new LabCreator().labify(new InputStreamReader(LabCreatorTest.class.getClassLoader().getResourceAsStream(name)), name);
fail("Expected exception!");
} catch (IllegalArgumentException e) {
// ok!
}
}
@Test
public void firstStepsMightBeEmpty() throws IOException {
String name = "StartingEmpty.java.txt";
VersionedContents versionedContents = new LabCreator().labify(new InputStreamReader(LabCreatorTest.class.getClassLoader().getResourceAsStream(name)), name);
assertEquals(new Version(3), versionedContents.getLeastVersion());
assertFalse(versionedContents.hasContents(2));
assertTrue(versionedContents.hasContents(3));
assertEquals(new Version(7), versionedContents.getMaxVersion());
}
@Test
public void correctContentsOfVersionThisOnly() throws IOException {
String name = "VersionTestThisOnly.txt";
VersionedContents versionedContents = new LabCreator().labify(new InputStreamReader(LabCreatorTest.class.getClassLoader().getResourceAsStream(name)), name);
checkCorrectContetsOfVersionThisOnly(versionedContents);
}
private void checkCorrectContetsOfVersionThisOnly(VersionedContents versionedContents) {
String contents1 = getContents(versionedContents, 1);
assertEquals("Checking version 1: ", "VERSION1" + newline + "VERSION1 that should be left" + newline, contents1);
String contents2 = getContents(versionedContents, 2);
assertEquals("Checking version 2: ", "VERSION2" + newline + "VERSION1 that should be left" + newline, contents2);
String contents3 = getContents(versionedContents, 3);
assertEquals("Checking version 3: ", "---" + newline + "VERSION2" + newline + "---" + newline + "VERSION1 that should be left" + newline, contents3);
}
@Test
public void correctContentsOfAnotherVersionThisOnly() throws IOException {
String name = "VersionTestThisOnlyWithConstants.txt";
VersionedContents versionedContents = new LabCreator("Zero", "One", "Two").labify(new InputStreamReader(LabCreatorTest.class.getClassLoader().getResourceAsStream(name)), name);
checkCorrectContetsOfVersionThisOnlyWithConstants(versionedContents);
}
private void checkCorrectContetsOfVersionThisOnlyWithConstants(VersionedContents versionedContents) {
String contents0 = getContents(versionedContents, 0);
assertEquals("Checking version 0: ",
"html" + newline +
" head" + newline +
" title" + newline +
" some initial stuff ie Zero" + newline +
" some more initial stuff" + newline +
" head" + newline +
" body" + newline +
" Zero" + newline +
" div" + newline +
" body" + newline +
"html" + newline, contents0);
String contents1 = getContents(versionedContents, 1);
assertEquals("Checking version 1: ",
"html" + newline +
" head" + newline +
" title" + newline +
" some initial stuff ie Zero" + newline +
" One to keep" + newline +
" some more initial stuff" + newline +
" head" + newline +
" body" + newline +
" One" + newline +
" div" + newline +
" body" + newline +
"html" + newline, contents1);
String contents2 = getContents(versionedContents, 2);
assertEquals("Checking version 2: ",
"html" + newline +
" head" + newline +
" title" + newline +
" Two to keep" + newline +
" some initial stuff ie Zero" + newline +
" One to keep" + newline +
" Two to keep" + newline +
" some more initial stuff" + newline +
" head" + newline +
" body" + newline +
" Two" + newline +
" div" + newline +
" body" + newline +
"html" + newline, contents2);
}
@Test
public void correctContentsOfUpToVersion() throws IOException {
String name = "VersionTestUpToVersion.txt";
VersionedContents versionedContents = new LabCreator().labify(new InputStreamReader(LabCreatorTest.class.getClassLoader().getResourceAsStream(name)), name);
checkCorrectContetsOfUpToVersion(versionedContents);
}
private void checkCorrectContetsOfUpToVersion(VersionedContents versionedContents) {
String contents1 = getContents(versionedContents, 1);
assertEquals("Checking version 1: ", "@Ignore" + newline + "VERSION1 that should be left" + newline, contents1);
String contents2 = getContents(versionedContents, 2);
assertEquals("Checking version 2: ", "@Ignore" + newline + "VERSION2" + newline + "VERSION1 that should be left" + newline, contents2);
String contents3 = getContents(versionedContents, 3);
assertEquals("Checking version 3: ", "---" + newline + "VERSION2" + newline + "---" + newline + "VERSION1 that should be left" + newline, contents3);
String contents4 = getContents(versionedContents, 4);
assertEquals("Checking version 4: ", "---" + newline + "@Ignore" + newline + "VERSION2" + newline + "---" + newline + "VERSION1 that should be left" + newline, contents4);
}
}
| code |
७वां वेतन आयोग : दशहरे से पहले इन २ लाख कर्मचारियों की डबल लॉटरी लगी, केंद्र के बराबर पाएंगे सैलरी
नई दिल्ली। ७वां वेतन आयोग, त्रिपुरा सरकार ने दशहरे से पहले अपने दो लाख कर्मचारियों को जबरदस्त तोहफा दिया है। राज्य सरकार ने १ अक्टूबर 20१8 से इन कर्मचारियों को ७वां वेतन आयोग देने का ऐलान किया है। सरकार का दावा है कि राज्य कर्मचारियों के वेतन में केंद्रीय कर्मचारियों की सैलरी के बराबर बढ़ोतरी की गई है। इससे इन कर्मचारियों की खुशी दोहरी हो गई है। क्योंकि जिन राज्यों में नया वेतनमान लागू हुआ है वहां के कर्मचारियों की शिकायत है कि राज्य और केंद्र में एक ही स्तर पर काम कर रहे अफसर की तनख्वाह में करीब ५ हजार रुपए का अंतर है।
त्रिपुरा के मुख्यमंत्री बिपलव देव ने बताया कि संशोधित वेतनमान असम के पूर्व मुख्य सचिव पीपी वर्मा की अध्यक्षता वाली समिति की सिफारिशों के आधार पर लागू हुआ है। फाइनेंशियल एक्सप्रेस की खबर के मुताबिक समिति ने बीते हफ्ते अपनी रिपोर्ट सरकार को सौंपी थी। इसके बाद इसे कैबिनेट ने मंजूरी दे दी। देव ने कहा कि ढाई दशक तक राज्य पर वाम सरकार का शासन रहा। इसमें राज्य की हालत काफी खस्ता हो गई है लेकिन बीजेपी ने चुनाव से पहले कर्मचारियों को नया वेतनमान देने का वादा किया था और उस वादे को अब निभाया है।
नए वेतनमान के मुताबिक राज्य में एंट्री लेवल के कर्मचारी की सैलरी १८००० रुपए हो गई है। यह सैलरी ग्रुप सी लेवल के कर्मचारी की है जबकि ग्रुप डी के कर्मी की सैलरी १६००० रुपए कर दी गई है। जिन कर्मचारियों की तनख्वाह फिक्स्ड है उन्हें नियमित कर्मचारियों के आधार पर लाभ मिलेगा। वहीं पेंशनरों की न्यूनतम पेंशन ८००० रुपए प्रति माह कर दी गई है, जो अधिकतम १,०७,४५० प्रति माह होगी।
७वें वेतन आयोग के लागू होने के बाद केंद्रीय कर्मचारियों को पे बैंड या पे स्केल की बजाय पे मेट्रिक्स के आधार पर सैलरी मिलती है। पे मेट्रिक्स में लेवल पर न्यूनतम पे १८ हजार रुपए है। वहीं लेवल १८ पर यह ढाई लाख रुपए है। वित्त मंत्रालय के एक अधिकारी के मुताबिक केंद्रीय कर्मचारी पे मेट्रिक्स लेवल के आधार पर सैलरी पा रहे हैं। बेस फिटमेंट फैक्टरी २.5७ गुणा है। आगे के लेवल पर यह बढ़ता जाता है। | hindi |
Download HP Photosmart C6280 Driver below this descriptions for Windows, Mac, and Linux. ThweHP C6280 Photosmart All In One Printer offers print efficiently with six individual inks and more. In addition, its save money and paper using the included duplexer to print on both sides of the paper. Even, it also support share on an existing home computer network by using Ethernet interfaces.
HP Photosmart C6280 lets you print documents and photos without swapping out paper by using an automated photo tray. In addition, you can also create high quality photo scans with 4800 dpi resolution and 48 bit color using a 2.4 inch display. In addition, its lets you get lab-quality photo and document print results in 6 ink color. even, it make you easily print photos without a PC by using memory card slots and HP Photosmart Express. HP C6280 ensure you to get photos print quality using HP Auto Sense which it enhance your photos and remove red eye with the HP Red eye Removal button.
Download HP Photosmart C6280 Driver and Software from HP Support Downloads.
Select a method to identify printer model (if prompted), and then follow the instructions to the download HP Photosmart C6280 Drivers.
Use the HP Download and Install Assistant for a guided HP Photosmart C6280 Driver installation and download. You also can selecting Download only to manually options or run the driver file through your internet browser. | english |
एग सैंडविच बनाने की रेसिपी हिंदी में - वेब रफ़्तार
होम खाना खज़ाना नॉन वेज रेसिपी एग सैंडविच बनाने की रेसिपी हिंदी में
एग सैंडविच बनाने की रेसिपी
नमस्कार दोस्तों वेब रफ़्तार में आपका स्वागत हें| दोस्तों आज हम आपको एग सैंडविच बनाने की रेसिपी के बारें में जानकारी देने जा रहे हें| दोस्तों हर घर में माँ को टेंशन रहता हे की सुबह अपने बच्चो को नाश्ते में क्या दू| जिसे बच्चे प्यार से पसंद कर के खाए| जि हा तो अब चिंता छोड़ दीजिए और एग सैंडविच तरी कीजिए| एग सैंडविच सेहत के लिए पौष्टिक और स्वादिष्ट होता हें| दोस्तों एग सैंडविच आप घर पर आसानी से और कम समय में बना सकते हें| इसलिए आज हम आपको एग सैंडविच बनाने की रेसिपी बता रहे हें| हमे आशा हे की आपको यह रेसिपी बेहद पसंद आएगी|तो आईये पहले जानते हे इसकी सामग्री के बारें में और फिर जानेंगे इसकी विधि के बारें में|
एग सैंडविच बनाने की सामग्री:
ब्रेड के स्लाइसेस कटे हुए
डिश बोउल तवा कढाई
एग सैंडविच बनाने की रेसिपी:
तो आईये अब जानते हें इसकी रेसिपी के बारें में|
दोस्तों सबसे पहले ते गैस पर एक छोटा पातेला रखे और इसमें थोडा पानी गरम करने के लिए डाल दें| इसके बाद २ अंडे इस पानी में उबलने के लिए डाल दें| अंडे अच्छी तरह उबल ने के बाद गैस बंद कर दीजिए|
अब इन अन्डो को पानी से बहार निकाल लीजिए और एक बोउल में रख दीजिए| ठंडा होने पर अन्डो के छिलके निकाल दीजिए| अन्डो के ४ समान भाग कर लीजिए| तवे पर इन्हें भी थोडा गरम कर लीजिए|
अब गैस पर तवा रख दीजिए और गरम होने पर थोडासा मक्खन लगाए और तवे पर गरम होने के लिए रख दीजिए| ब्रेड थोडा टोस्ट होने के बाद दुसरे बाजु से भी मक्खन लगाए आयर हल्का टोस्ट कर लीजिए| २ मिनट पकाने के बाद तवे से निकाल लीजिए| ब्रेड टोस्ट तैयार हो गया हें|
अब ब्रेड स्लाइसेस पर चीज लगाए| इसके बाद एक स्लाइस पे लाल मिर्च पावडर और दुसरे स्लाइस पे काली मिर्च पावडर डाले और नमक स्वादानुसार छिडके|
अब अंडे के टुकड़े को ब्रेड स्लाइसेस के बिच में रखे और टोस्ट को बोहोत नाजुकता से २ भागो में काट लीजिए|
आपका गरमा गरम एग सैंडविच खाने के लिए तैयार हें|
वास्तु शास्त्र में कपूर के चमत्कारी फायदे और उपाय
अंडा ब्रेड रेसिपी
ब्रेड सैंडविच रेसिपी
कोल्हापुरी बैंगन मसाला रेसिपी हिंदी में जानकारी
बंगाली फिश फ्राई बनाने की रेसिपी इन हिंदी
लजीज मटन मसाला बनाने की रेसिपी इन हिंदी | hindi |
You may already know that at you can purchase an exclusive range of skincare products from our clinics that are not available on the high street.
Lifting and protective anti-aging day lotion. It contains the exclusive absorbable transdermal filler Phosphatrans®, which delivers the active ingredients deeper than ever, allowing a much greater absorption and efficacy. A polyactive, moisturising, anti-fatigue and anti-free radicals formula, for an outstanding youthful skin.
Restorative and regenerative anti-wrinkle night cream. It contains the exclusive absorbable transdermal filler Hydroxytrans®, which delivers the active ingredients deeper than ever, allowing a much greater absorption and efficacy. A polyactive, nutritious, and firming care formula that stimulates cellular activity, for an outstanding youthful skin.
Skin-enhancing instant beauty flash serum. It contains the exclusive absorbable transdermal filler Phosphatrans®, which delivers the active ingredients deeper than ever, allowing a much greater absorption and efficacy. A concentrated anti-aging essence with tensor peptides that smooths expression lines, revitalizing and energizing the skin, for an outstanding youthful skin.
Cool anti-fatigue, anti-dark circles and anti-puffiness gel. It contains the exclusive absorbable transdermal filler Hyalutrans®, which delivers the active ingredients deeper than ever, allowing a much greater absorption and efficacy. A polyactive anti-aging formula that smooths expression lines, draining the eye area and increasing micro-circulation, for an outstanding youthful skin.
If you would like to order any of the products available from the Juvilis skin care range please email [email protected] with the products you would like to order. The perfect Christmas gift and the ultimate treat for your skin. | english |
مشرقی ٹایِم زونَس منٛز کیٛاہ ٹایِم چھُ وٕنۍ کیٚنَس | kashmiri |
/*
* Copyright (c) 2006-2007 Hyperic, Inc.
* Copyright (c) 2010 VMware, Inc.
*
* Licensed 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.hyperic.sigar.cmd;
import java.io.File;
import java.io.FileFilter;
import java.util.HashMap;
import java.util.Iterator;
import java.util.Map;
import java.lang.reflect.Method;
import java.lang.reflect.InvocationTargetException;
import java.net.URLClassLoader;
import java.net.URL;
import org.hyperic.sigar.Sigar;
import org.hyperic.sigar.SigarLoader;
public class Runner {
private static HashMap wantedJars = new HashMap();
private static final String JAR_EXT = ".jar";
static {
wantedJars.put("junit", Boolean.FALSE);
wantedJars.put("log4j", Boolean.FALSE);
}
private static void printMissingJars() {
for (Iterator it = wantedJars.entrySet().iterator();
it.hasNext();)
{
Map.Entry entry = (Map.Entry)it.next();
String jar = (String)entry.getKey();
if (wantedJars.get(jar) == Boolean.FALSE) {
System.out.println("Unable to locate: " + jar + JAR_EXT);
}
}
}
private static boolean missingJars() {
for (Iterator it = wantedJars.entrySet().iterator();
it.hasNext();)
{
Map.Entry entry = (Map.Entry)it.next();
String jar = (String)entry.getKey();
if (wantedJars.get(jar) == Boolean.FALSE) {
return true;
}
}
return false;
}
public static URL[] getLibJars(String dir) throws Exception {
File[] jars = new File(dir).listFiles(new FileFilter() {
public boolean accept(File file) {
String name = file.getName();
int jarIx = name.indexOf(JAR_EXT);
if (jarIx == -1) {
return false;
}
int ix = name.indexOf('-');
if (ix != -1) {
name = name.substring(0, ix); //versioned .jar
}
else {
name = name.substring(0, jarIx);
}
if (wantedJars.get(name) != null) {
wantedJars.put(name, Boolean.TRUE);
return true;
}
else {
return false;
}
}
});
if (jars == null) {
return new URL[0];
}
URL[] urls = new URL[jars.length];
for (int i=0; i<jars.length; i++) {
URL url =
new URL("jar", null,
"file:" + jars[i].getAbsolutePath() + "!/");
urls[i] = url;
}
return urls;
}
private static void addURLs(URL[] jars) throws Exception {
URLClassLoader loader =
(URLClassLoader)Thread.currentThread().getContextClassLoader();
//bypass protected access.
Method addURL =
URLClassLoader.class.getDeclaredMethod("addURL",
new Class[] {
URL.class
});
addURL.setAccessible(true); //pound sand.
for (int i=0; i<jars.length; i++) {
addURL.invoke(loader, new Object[] { jars[i] });
}
}
private static boolean addJarDir(String dir) throws Exception {
URL[] jars = getLibJars(dir);
addURLs(jars);
return !missingJars();
}
private static String getenv(String key) {
try {
return System.getenv("ANT_HOME"); //check for junit.jar
} catch (Error e) {
/*1.4*/
Sigar sigar = new Sigar();
try {
return sigar.getProcEnv("$$", "ANT_HOME");
} catch (Exception se) {
return null;
}
finally { sigar.close(); }
}
}
public static void main(String[] args) throws Exception {
if (args.length < 1) {
args = new String[] { "Shell" };
}
else {
//e.g. convert
// "ifconfig", "eth0"
// to:
// "Shell", "ifconfig", "eth0"
if (Character.isLowerCase(args[0].charAt(0))) {
String[] nargs = new String[args.length + 1];
System.arraycopy(args, 0, nargs, 1, args.length);
nargs[0] = "Shell";
args = nargs;
}
}
String name = args[0];
String[] pargs = new String[args.length - 1];
System.arraycopy(args, 1, pargs, 0, args.length-1);
String sigarLib = SigarLoader.getLocation();
String[] dirs = { sigarLib, "lib", "." };
for (int i=0; i<dirs.length; i++) {
if (addJarDir(dirs[i])) {
break;
}
}
if (missingJars()) {
File[] subdirs = new File(".").listFiles(new FileFilter() {
public boolean accept(File file) {
return file.isDirectory();
}
});
for (int i=0; i<subdirs.length; i++) {
File lib = new File(subdirs[i], "lib");
if (lib.exists()) {
if (addJarDir(lib.getAbsolutePath())) {
break;
}
}
}
if (missingJars()) {
String home = getenv("ANT_HOME"); //check for junit.jar
if (home != null) {
addJarDir(home + "/lib");
}
}
}
Class cmd = null;
String[] packages = {
"org.hyperic.sigar.cmd.",
"org.hyperic.sigar.test.",
"org.hyperic.sigar.",
"org.hyperic.sigar.win32.",
"org.hyperic.sigar.jmx.",
};
for (int i=0; i<packages.length; i++) {
try {
cmd = Class.forName(packages[i] + name);
break;
} catch (ClassNotFoundException e) {}
}
if (cmd == null) {
System.out.println("Unknown command: " + args[0]);
return;
}
Method main = cmd.getMethod("main",
new Class[] {
String[].class
});
try {
main.invoke(null, new Object[] { pargs });
} catch (InvocationTargetException e) {
Throwable t = e.getTargetException();
if (t instanceof NoClassDefFoundError) {
System.out.println("Class Not Found: " +
t.getMessage());
printMissingJars();
}
else {
t.printStackTrace();
}
}
}
}
| code |
1. A tweet from US president and subsequent signing of a decree declaring recognition of sovereign rights of Israel over the Golden Heights captured in 1967 during six day Arab – Israeli war has global strategic implications. On the very outset it has challenged the sanctity of UN charter, besides setting a precedence of changing the international boundaries where it suits the big powers for their geo political interests.
2. The Russia annexed Crimea in 2014 for their strategic politico-military interests in Ukraine theatre, and now US has chosen to indulge in supporting Israel in legitimizing their occupation of Syrian territory. It surely has an inimical import for regional peace and political equilibrium in the West Asia. Syria and Iran are the obvious targets in this diplomatic exercise who continue to defy the US dictates. China has also done it somewhat similar in South China Sea for their territorial expansion in violation of laws of sea to which they are signatories. It is a case where in the global conscience keepers have indulged in an obvious opportunist unilateral heady power play.
3. The world seem to be wearing around once again to early twentieth century environment which saw carving out artificial boundaries disturbing local political structures as it suited the European colonial powers. As a result it brought unprecedented socio-political upheavals which continue even today impacting the global peace. Almost all the conflict zones in Africa, West and South Asia have their genesis rooted in political intrusions into their ethnic, religious and cultural milieu of ancient lands ensconced in traditional feudal mindset.
4. In that, the military intervention has been central to political manipulations for controlling material & mineral resources through regime changes, ethnic and religious divide as a proven western world policy. Iran –Iraq, Iran- Saudi Arab, Arab- Israel, Syria, Lebanon, Yemen, Turkey- Kurds, Afghanistan, India-Pakistan, African landscape, Korea, Taiwan and many more, all have common fabric of the colonial era legacies.
5. The Syria, since capture of Golan Heights, has been talking of evicting Israel from their illegal occupation of their sovereign territories. The Iranian nuclear ambitions are known to be anti Israel as well as against Sunni fraternity led by Saudi Arab. Hence, an omnipresent military threat exists for the Israel from Syria –Iran combine. In that, the Golan Heights provide depth to Israel for their defence from the Northern Syrian flank. Both Israel as well as Saudi Arab are close US allies in the West Asia who are expected to take care of US interests once they move out of the region. Accordingly, their security interests have to be taken care of by the US in appropriate manner.
6. Apropos, the US intensions of regime change in Syria and installation of a more pliable political dispensation seem to be aiming at reducing the Syrian threat to Israel, besides geo politics of oil and gas. However, Iran is supporting Syrian president with Shiite credentials and pushes her anti Israeli and anti sectarian rival Saudi Arab agenda. The Iran is also keen for her economic expansion through Syria which happens to be against known US interests. Therefore, the US grand game seems to make Syria as well as Iran incapable of imposing any worthwhile military threat in the region. To do that, they created a political divide on sectarian lines amongst Syrians and provided military support to the rebel groups.
7. The US backed Syrian rebels had achieved substantive progress and were ready for the final push towards Damascus by mid 2015. However, the Russian intervention in Syria in sep 2015 changed the strategic dynamics giving a much needed boost to Syrian government forces and their Iranian allies. In the mean time, the emergence of ISIS in the Syrian landscape became a common threat to both US as well Russia which brought them together to fight them jointly, besides initiating process for reconciliation amongst the Syrian society. Having stabilizing the situation to reasonable level, the US indicated their intensions of moving out from the Asian landscape.
8. However, there were security concerns in the region consequent to likely US draw down as and when it so happens. In that, Iran with her nuclear credentials is considered to be a threat likely to indulge in coercive politics to occupy regional strategic space. The president Trump reviewed the US policy on Iranian nuclear deal brokered by the president Obama alongside France, Germany, UK and China in 2015. The US decided to pull out of the deal with intentions of rolling back Iranian nuclear as well as their missile programmes.
9. No sooner the US decision was announced, the military ante was increased by Iran by firing rockets on the Israeli positions on the Golan Heights on 10 May 2018. It conveyed a political message that Iran would continue to follow their resolve, since Islamic revolution, to obliterate Israel as a national entity. Israel, in retaliation, conducted counter air raids on Iranian installations on the Golan Heights and subsequently inside Damascus also. The threat quotient to Israel has certainly increased with such military exchanges.
10. Besides above, the UN sponsored talks for reconciliation amongst the Syrians have not been progressing as per US expectations with government forces gaining ground. In the given circumstances, the dilution of the political intransigence of the Syrian government is contingent on change in the force balance in favour of the Syrian rebels. To do that, the Golan Heights fit into this scheme, albeit it would need to change the existing politico-military equation of higher threat level for dislocation of Syrian forces. The current political declaration by the US has done that job.
11. With that, the Israel has been empowered legitimizing their offensive responses in case of infringement of their sovereign rights over the Golan Heights. The message is clear that whosoever dares to temper with her (new) territorial status, should be prepared for its consequences. While Israel is already in occupation of 2/3rd of the plateau since last 52 years and their law prevails since 1981, it makes no difference in the actual ground position at all. However, the mere declaration of Israeli sovereignty over Golan Heights raises the hackles and conveys provocative political intentions. With this strategic shift, the politico-military paradigms have changed forever in the region to the benefit of Israel.
12. Syria in the given situation have option to either accept it as a fait accompli, or alternately launch military operations to evict Israel from the occupied Syrian territory. Since the later does not seem to be feasible given the Syrian commitments to save their turf in main land against the rebels, the former is likely to acquire permanency over period of time. The Iran with crippling US economic sanctions is unlikely to be in a position to initiate something substantial which would have a major impact on Israeli status in Golan Heights. Moreover, unlike earlier times there are no violent overtures from the Arab world on such a sensitive politico- religious issue except demonstrations to maintain sanctity of territorial integrity of Syria. Hence, it appears to be a political checkmate on the face of it.
13. The media is abuzz with immediate aim of this US political intrusion as an electoral boost to the sitting Israeli prime minister Netanyahu in the forth coming elections. Whereas, such a deliberate declaration certainly goes beyond this seemingly innocuous purpose in the bigger geo-political grand game. It has also opened up a debate, if this precedence would not be followed in the case of other conflict zones wherever interests of the big powers lie? Is rest of the West Asia, though contextually different in each case, would be impacted by this new normal as established by the US? Is there any possibility of such political intrusions in the South Asia also in some form? All these, and many more questions need to be looked at by the world community for the possible fallouts of this new political trend.
14. It would be in order for the India to take cognizance of such contingencies which may be imposed upon us with little or no warning at all. There would certainly be a shift in strategic paradigms with US plans of draw down from the Asian landscape. India would surely be impacted being a major stake holder in the region. We better be aware of its possible manifestations in the back drop of big powers trespassing the established international political sanctities and protocols.
Disclaimer:- Views expressed are of the author and do not necessarily reflect the views of CENJOWS. | english |
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<ul>
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<ul>
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<ul>
<li><a href="namespace-Gedmo.IpTraceable.Mapping.html">Mapping<span></span></a>
<ul>
<li><a href="namespace-Gedmo.IpTraceable.Mapping.Driver.html">Driver</a>
</li>
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<ul>
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</ul></li></ul></li>
<li><a href="namespace-Gedmo.IpTraceable.Traits.html">Traits</a>
</li>
</ul></li>
<li><a href="namespace-Gedmo.Loggable.html">Loggable<span></span></a>
<ul>
<li><a href="namespace-Gedmo.Loggable.Document.html">Document<span></span></a>
<ul>
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</ul></li>
<li><a href="namespace-Gedmo.Loggable.Entity.html">Entity<span></span></a>
<ul>
<li><a href="namespace-Gedmo.Loggable.Entity.MappedSuperclass.html">MappedSuperclass</a>
</li>
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</li>
</ul></li>
<li><a href="namespace-Gedmo.Loggable.Mapping.html">Mapping<span></span></a>
<ul>
<li><a href="namespace-Gedmo.Loggable.Mapping.Driver.html">Driver</a>
</li>
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<ul>
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</li>
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<li><a href="namespace-Gedmo.Mapping.html">Mapping<span></span></a>
<ul>
<li><a href="namespace-Gedmo.Mapping.Annotation.html">Annotation</a>
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<ul>
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<ul>
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<ul>
<li><a href="namespace-Gedmo.Mapping.Mock.Extension.Encoder.Mapping.Driver.html">Driver</a>
</li>
<li><a href="namespace-Gedmo.Mapping.Mock.Extension.Encoder.Mapping.Event.html">Event<span></span></a>
<ul>
<li><a href="namespace-Gedmo.Mapping.Mock.Extension.Encoder.Mapping.Event.Adapter.html">Adapter</a>
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<li><a href="namespace-Gedmo.Mapping.Mock.Mapping.html">Mapping<span></span></a>
<ul>
<li><a href="namespace-Gedmo.Mapping.Mock.Mapping.Event.html">Event<span></span></a>
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<ul>
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<li><a href="namespace-Gedmo.References.html">References<span></span></a>
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<ul>
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<li><a href="namespace-Gedmo.Sluggable.Handler.html">Handler</a>
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<ul>
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</li>
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<ul>
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<ul>
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<ul>
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<li><a href="namespace-Gedmo.Timestampable.html">Timestampable<span></span></a>
<ul>
<li><a href="namespace-Gedmo.Timestampable.Mapping.html">Mapping<span></span></a>
<ul>
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</li>
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<ul>
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<ul>
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<ul>
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<ul>
<li><a href="namespace-Gedmo.Translatable.Mapping.Driver.html">Driver</a>
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<li><a href="namespace-Gedmo.Translatable.Mapping.Event.Adapter.html">Adapter</a>
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<ul>
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<li><a href="namespace-Gedmo.Tree.Entity.MappedSuperclass.html">MappedSuperclass</a>
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<li><a href="namespace-Gedmo.Tree.Mapping.html">Mapping<span></span></a>
<ul>
<li><a href="namespace-Gedmo.Tree.Mapping.Driver.html">Driver</a>
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<ul>
<li><a href="namespace-Gedmo.Tree.Mapping.Event.Adapter.html">Adapter</a>
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<li><a href="namespace-Gedmo.Uploadable.html">Uploadable<span></span></a>
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<li><a href="namespace-Gedmo.Uploadable.Event.html">Event</a>
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<li><a href="namespace-Gedmo.Uploadable.Mapping.Driver.html">Driver</a>
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<li><a href="namespace-Incenteev.html">Incenteev<span></span></a>
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<li><a href="namespace-IpTraceable.Fixture.html">Fixture<span></span></a>
<ul>
<li><a href="namespace-IpTraceable.Fixture.Document.html">Document</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-JMS.html">JMS<span></span></a>
<ul>
<li><a href="namespace-JMS.Parser.html">Parser<span></span></a>
<ul>
<li><a href="namespace-JMS.Parser.Tests.html">Tests</a>
</li>
</ul></li>
<li><a href="namespace-JMS.Serializer.html">Serializer<span></span></a>
<ul>
<li><a href="namespace-JMS.Serializer.Annotation.html">Annotation</a>
</li>
<li><a href="namespace-JMS.Serializer.Builder.html">Builder</a>
</li>
<li><a href="namespace-JMS.Serializer.Construction.html">Construction</a>
</li>
<li><a href="namespace-JMS.Serializer.EventDispatcher.html">EventDispatcher<span></span></a>
<ul>
<li><a href="namespace-JMS.Serializer.EventDispatcher.Subscriber.html">Subscriber</a>
</li>
</ul></li>
<li><a href="namespace-JMS.Serializer.Exception.html">Exception</a>
</li>
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</li>
<li><a href="namespace-JMS.Serializer.Handler.html">Handler</a>
</li>
<li><a href="namespace-JMS.Serializer.Metadata.html">Metadata<span></span></a>
<ul>
<li><a href="namespace-JMS.Serializer.Metadata.Driver.html">Driver</a>
</li>
</ul></li>
<li><a href="namespace-JMS.Serializer.Naming.html">Naming</a>
</li>
<li><a href="namespace-JMS.Serializer.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-JMS.Serializer.Tests.Exclusion.html">Exclusion</a>
</li>
<li><a href="namespace-JMS.Serializer.Tests.Fixtures.html">Fixtures<span></span></a>
<ul>
<li><a href="namespace-JMS.Serializer.Tests.Fixtures.Discriminator.html">Discriminator</a>
</li>
<li><a href="namespace-JMS.Serializer.Tests.Fixtures.Doctrine.html">Doctrine</a>
</li>
<li><a href="namespace-JMS.Serializer.Tests.Fixtures.DoctrinePHPCR.html">DoctrinePHPCR</a>
</li>
</ul></li>
<li><a href="namespace-JMS.Serializer.Tests.Handler.html">Handler</a>
</li>
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<ul>
<li><a href="namespace-JMS.Serializer.Tests.Metadata.Driver.html">Driver</a>
</li>
</ul></li>
<li><a href="namespace-JMS.Serializer.Tests.Serializer.html">Serializer<span></span></a>
<ul>
<li><a href="namespace-JMS.Serializer.Tests.Serializer.EventDispatcher.html">EventDispatcher<span></span></a>
<ul>
<li><a href="namespace-JMS.Serializer.Tests.Serializer.EventDispatcher.Subscriber.html">Subscriber</a>
</li>
</ul></li>
<li><a href="namespace-JMS.Serializer.Tests.Serializer.Naming.html">Naming</a>
</li>
</ul></li>
<li><a href="namespace-JMS.Serializer.Tests.Twig.html">Twig</a>
</li>
</ul></li>
<li><a href="namespace-JMS.Serializer.Twig.html">Twig</a>
</li>
<li><a href="namespace-JMS.Serializer.Util.html">Util</a>
</li>
</ul></li>
<li><a href="namespace-JMS.SerializerBundle.html">SerializerBundle<span></span></a>
<ul>
<li><a href="namespace-JMS.SerializerBundle.DependencyInjection.html">DependencyInjection<span></span></a>
<ul>
<li><a href="namespace-JMS.SerializerBundle.DependencyInjection.Compiler.html">Compiler</a>
</li>
</ul></li>
<li><a href="namespace-JMS.SerializerBundle.Serializer.html">Serializer</a>
</li>
<li><a href="namespace-JMS.SerializerBundle.Templating.html">Templating</a>
</li>
<li><a href="namespace-JMS.SerializerBundle.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-JMS.SerializerBundle.Tests.DependencyInjection.html">DependencyInjection<span></span></a>
<ul>
<li><a href="namespace-JMS.SerializerBundle.Tests.DependencyInjection.Fixture.html">Fixture</a>
</li>
</ul></li></ul></li></ul></li>
<li><a href="namespace-JMS.TranslationBundle.html">TranslationBundle<span></span></a>
<ul>
<li><a href="namespace-JMS.TranslationBundle.Annotation.html">Annotation</a>
</li>
<li><a href="namespace-JMS.TranslationBundle.Command.html">Command</a>
</li>
<li><a href="namespace-JMS.TranslationBundle.Controller.html">Controller</a>
</li>
<li><a href="namespace-JMS.TranslationBundle.DependencyInjection.html">DependencyInjection<span></span></a>
<ul>
<li><a href="namespace-JMS.TranslationBundle.DependencyInjection.Compiler.html">Compiler</a>
</li>
</ul></li>
<li><a href="namespace-JMS.TranslationBundle.Exception.html">Exception</a>
</li>
<li><a href="namespace-JMS.TranslationBundle.Logger.html">Logger</a>
</li>
<li><a href="namespace-JMS.TranslationBundle.Model.html">Model</a>
</li>
<li><a href="namespace-JMS.TranslationBundle.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-JMS.TranslationBundle.Tests.Functional.html">Functional<span></span></a>
<ul>
<li><a href="namespace-JMS.TranslationBundle.Tests.Functional.TestBundle.html">TestBundle<span></span></a>
<ul>
<li><a href="namespace-JMS.TranslationBundle.Tests.Functional.TestBundle.Controller.html">Controller</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-JMS.TranslationBundle.Tests.Model.html">Model</a>
</li>
<li><a href="namespace-JMS.TranslationBundle.Tests.Translation.html">Translation<span></span></a>
<ul>
<li><a href="namespace-JMS.TranslationBundle.Tests.Translation.Comparison.html">Comparison</a>
</li>
<li><a href="namespace-JMS.TranslationBundle.Tests.Translation.Dumper.html">Dumper</a>
</li>
<li><a href="namespace-JMS.TranslationBundle.Tests.Translation.Extractor.html">Extractor<span></span></a>
<ul>
<li><a href="namespace-JMS.TranslationBundle.Tests.Translation.Extractor.File.html">File<span></span></a>
<ul>
<li><a href="namespace-JMS.TranslationBundle.Tests.Translation.Extractor.File.Fixture.html">Fixture</a>
</li>
</ul></li>
<li><a href="namespace-JMS.TranslationBundle.Tests.Translation.Extractor.Fixture.html">Fixture<span></span></a>
<ul>
<li><a href="namespace-JMS.TranslationBundle.Tests.Translation.Extractor.Fixture.SimpleTest.html">SimpleTest<span></span></a>
<ul>
<li><a href="namespace-JMS.TranslationBundle.Tests.Translation.Extractor.Fixture.SimpleTest.Controller.html">Controller</a>
</li>
<li><a href="namespace-JMS.TranslationBundle.Tests.Translation.Extractor.Fixture.SimpleTest.Form.html">Form</a>
</li>
</ul></li></ul></li></ul></li>
<li><a href="namespace-JMS.TranslationBundle.Tests.Translation.Loader.html">Loader<span></span></a>
<ul>
<li><a href="namespace-JMS.TranslationBundle.Tests.Translation.Loader.Symfony.html">Symfony</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-JMS.TranslationBundle.Tests.Twig.html">Twig</a>
</li>
</ul></li>
<li><a href="namespace-JMS.TranslationBundle.Translation.html">Translation<span></span></a>
<ul>
<li><a href="namespace-JMS.TranslationBundle.Translation.Comparison.html">Comparison</a>
</li>
<li><a href="namespace-JMS.TranslationBundle.Translation.Dumper.html">Dumper</a>
</li>
<li><a href="namespace-JMS.TranslationBundle.Translation.Extractor.html">Extractor<span></span></a>
<ul>
<li><a href="namespace-JMS.TranslationBundle.Translation.Extractor.File.html">File</a>
</li>
</ul></li>
<li><a href="namespace-JMS.TranslationBundle.Translation.Loader.html">Loader<span></span></a>
<ul>
<li><a href="namespace-JMS.TranslationBundle.Translation.Loader.Symfony.html">Symfony</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-JMS.TranslationBundle.Twig.html">Twig</a>
</li>
<li><a href="namespace-JMS.TranslationBundle.Util.html">Util</a>
</li>
</ul></li></ul></li>
<li class="active"><a href="namespace-Knp.html">Knp<span></span></a>
<ul>
<li class="active"><a href="namespace-Knp.Bundle.html">Bundle<span></span></a>
<ul>
<li class="active"><a href="namespace-Knp.Bundle.MenuBundle.html">MenuBundle<span></span></a>
<ul>
<li><a href="namespace-Knp.Bundle.MenuBundle.DependencyInjection.html">DependencyInjection<span></span></a>
<ul>
<li><a href="namespace-Knp.Bundle.MenuBundle.DependencyInjection.Compiler.html">Compiler</a>
</li>
</ul></li>
<li><a href="namespace-Knp.Bundle.MenuBundle.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Knp.Bundle.MenuBundle.Provider.html">Provider</a>
</li>
<li><a href="namespace-Knp.Bundle.MenuBundle.Renderer.html">Renderer</a>
</li>
<li><a href="namespace-Knp.Bundle.MenuBundle.Templating.html">Templating<span></span></a>
<ul>
<li><a href="namespace-Knp.Bundle.MenuBundle.Templating.Helper.html">Helper</a>
</li>
</ul></li>
<li class="active"><a href="namespace-Knp.Bundle.MenuBundle.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Knp.Bundle.MenuBundle.Tests.DependencyInjection.html">DependencyInjection<span></span></a>
<ul>
<li><a href="namespace-Knp.Bundle.MenuBundle.Tests.DependencyInjection.Compiler.html">Compiler</a>
</li>
</ul></li>
<li><a href="namespace-Knp.Bundle.MenuBundle.Tests.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Knp.Bundle.MenuBundle.Tests.Provider.html">Provider</a>
</li>
<li><a href="namespace-Knp.Bundle.MenuBundle.Tests.Renderer.html">Renderer</a>
</li>
<li class="active"><a href="namespace-Knp.Bundle.MenuBundle.Tests.Stubs.html">Stubs<span></span></a>
<ul>
<li><a href="namespace-Knp.Bundle.MenuBundle.Tests.Stubs.Child.html">Child<span></span></a>
<ul>
<li><a href="namespace-Knp.Bundle.MenuBundle.Tests.Stubs.Child.Menu.html">Menu</a>
</li>
</ul></li>
<li class="active"><a href="namespace-Knp.Bundle.MenuBundle.Tests.Stubs.Menu.html">Menu</a>
</li>
</ul></li>
<li><a href="namespace-Knp.Bundle.MenuBundle.Tests.Templating.html">Templating</a>
</li>
</ul></li></ul></li></ul></li>
<li><a href="namespace-Knp.Menu.html">Menu<span></span></a>
<ul>
<li><a href="namespace-Knp.Menu.Factory.html">Factory</a>
</li>
<li><a href="namespace-Knp.Menu.Integration.html">Integration<span></span></a>
<ul>
<li><a href="namespace-Knp.Menu.Integration.Silex.html">Silex</a>
</li>
<li><a href="namespace-Knp.Menu.Integration.Symfony.html">Symfony</a>
</li>
</ul></li>
<li><a href="namespace-Knp.Menu.Iterator.html">Iterator</a>
</li>
<li><a href="namespace-Knp.Menu.Loader.html">Loader</a>
</li>
<li><a href="namespace-Knp.Menu.Matcher.html">Matcher<span></span></a>
<ul>
<li><a href="namespace-Knp.Menu.Matcher.Voter.html">Voter</a>
</li>
</ul></li>
<li><a href="namespace-Knp.Menu.Provider.html">Provider</a>
</li>
<li><a href="namespace-Knp.Menu.Renderer.html">Renderer</a>
</li>
<li><a href="namespace-Knp.Menu.Silex.html">Silex<span></span></a>
<ul>
<li><a href="namespace-Knp.Menu.Silex.Voter.html">Voter</a>
</li>
</ul></li>
<li><a href="namespace-Knp.Menu.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Knp.Menu.Tests.Factory.html">Factory</a>
</li>
<li><a href="namespace-Knp.Menu.Tests.Integration.html">Integration<span></span></a>
<ul>
<li><a href="namespace-Knp.Menu.Tests.Integration.Silex.html">Silex</a>
</li>
</ul></li>
<li><a href="namespace-Knp.Menu.Tests.Iterator.html">Iterator</a>
</li>
<li><a href="namespace-Knp.Menu.Tests.Loader.html">Loader</a>
</li>
<li><a href="namespace-Knp.Menu.Tests.Matcher.html">Matcher<span></span></a>
<ul>
<li><a href="namespace-Knp.Menu.Tests.Matcher.Voter.html">Voter</a>
</li>
</ul></li>
<li><a href="namespace-Knp.Menu.Tests.Provider.html">Provider</a>
</li>
<li><a href="namespace-Knp.Menu.Tests.Renderer.html">Renderer</a>
</li>
<li><a href="namespace-Knp.Menu.Tests.Silex.html">Silex</a>
</li>
<li><a href="namespace-Knp.Menu.Tests.Twig.html">Twig</a>
</li>
<li><a href="namespace-Knp.Menu.Tests.Util.html">Util</a>
</li>
</ul></li>
<li><a href="namespace-Knp.Menu.Twig.html">Twig</a>
</li>
<li><a href="namespace-Knp.Menu.Util.html">Util</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Loggable.html">Loggable<span></span></a>
<ul>
<li><a href="namespace-Loggable.Fixture.html">Fixture<span></span></a>
<ul>
<li><a href="namespace-Loggable.Fixture.Document.html">Document<span></span></a>
<ul>
<li><a href="namespace-Loggable.Fixture.Document.Log.html">Log</a>
</li>
</ul></li>
<li><a href="namespace-Loggable.Fixture.Entity.html">Entity<span></span></a>
<ul>
<li><a href="namespace-Loggable.Fixture.Entity.Log.html">Log</a>
</li>
</ul></li></ul></li></ul></li>
<li><a href="namespace-Lunetics.html">Lunetics<span></span></a>
<ul>
<li><a href="namespace-Lunetics.LocaleBundle.html">LocaleBundle<span></span></a>
<ul>
<li><a href="namespace-Lunetics.LocaleBundle.Controller.html">Controller</a>
</li>
<li><a href="namespace-Lunetics.LocaleBundle.Cookie.html">Cookie</a>
</li>
<li><a href="namespace-Lunetics.LocaleBundle.DependencyInjection.html">DependencyInjection<span></span></a>
<ul>
<li><a href="namespace-Lunetics.LocaleBundle.DependencyInjection.Compiler.html">Compiler</a>
</li>
</ul></li>
<li><a href="namespace-Lunetics.LocaleBundle.Event.html">Event</a>
</li>
<li><a href="namespace-Lunetics.LocaleBundle.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Lunetics.LocaleBundle.Form.html">Form<span></span></a>
<ul>
<li><a href="namespace-Lunetics.LocaleBundle.Form.Extension.html">Extension<span></span></a>
<ul>
<li><a href="namespace-Lunetics.LocaleBundle.Form.Extension.ChoiceList.html">ChoiceList</a>
</li>
<li><a href="namespace-Lunetics.LocaleBundle.Form.Extension.Type.html">Type</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Lunetics.LocaleBundle.LocaleGuesser.html">LocaleGuesser</a>
</li>
<li><a href="namespace-Lunetics.LocaleBundle.LocaleInformation.html">LocaleInformation</a>
</li>
<li><a href="namespace-Lunetics.LocaleBundle.Matcher.html">Matcher</a>
</li>
<li><a href="namespace-Lunetics.LocaleBundle.Session.html">Session</a>
</li>
<li><a href="namespace-Lunetics.LocaleBundle.Switcher.html">Switcher</a>
</li>
<li><a href="namespace-Lunetics.LocaleBundle.Templating.html">Templating<span></span></a>
<ul>
<li><a href="namespace-Lunetics.LocaleBundle.Templating.Helper.html">Helper</a>
</li>
</ul></li>
<li><a href="namespace-Lunetics.LocaleBundle.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Lunetics.LocaleBundle.Tests.Controller.html">Controller</a>
</li>
<li><a href="namespace-Lunetics.LocaleBundle.Tests.DependencyInjection.html">DependencyInjection<span></span></a>
<ul>
<li><a href="namespace-Lunetics.LocaleBundle.Tests.DependencyInjection.Compiler.html">Compiler</a>
</li>
</ul></li>
<li><a href="namespace-Lunetics.LocaleBundle.Tests.Event.html">Event</a>
</li>
<li><a href="namespace-Lunetics.LocaleBundle.Tests.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Lunetics.LocaleBundle.Tests.Form.html">Form<span></span></a>
<ul>
<li><a href="namespace-Lunetics.LocaleBundle.Tests.Form.Extension.html">Extension<span></span></a>
<ul>
<li><a href="namespace-Lunetics.LocaleBundle.Tests.Form.Extension.ChoiceList.html">ChoiceList</a>
</li>
<li><a href="namespace-Lunetics.LocaleBundle.Tests.Form.Extension.Type.html">Type</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Lunetics.LocaleBundle.Tests.LocaleGuesser.html">LocaleGuesser</a>
</li>
<li><a href="namespace-Lunetics.LocaleBundle.Tests.LocaleInformation.html">LocaleInformation</a>
</li>
<li><a href="namespace-Lunetics.LocaleBundle.Tests.Templating.html">Templating<span></span></a>
<ul>
<li><a href="namespace-Lunetics.LocaleBundle.Tests.Templating.Helper.html">Helper</a>
</li>
</ul></li>
<li><a href="namespace-Lunetics.LocaleBundle.Tests.Twig.html">Twig<span></span></a>
<ul>
<li><a href="namespace-Lunetics.LocaleBundle.Tests.Twig.Extension.html">Extension</a>
</li>
</ul></li>
<li><a href="namespace-Lunetics.LocaleBundle.Tests.Validator.html">Validator</a>
</li>
</ul></li>
<li><a href="namespace-Lunetics.LocaleBundle.Twig.html">Twig<span></span></a>
<ul>
<li><a href="namespace-Lunetics.LocaleBundle.Twig.Extension.html">Extension</a>
</li>
</ul></li>
<li><a href="namespace-Lunetics.LocaleBundle.Validator.html">Validator</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Mapping.html">Mapping<span></span></a>
<ul>
<li><a href="namespace-Mapping.Fixture.html">Fixture<span></span></a>
<ul>
<li><a href="namespace-Mapping.Fixture.Compatibility.html">Compatibility</a>
</li>
<li><a href="namespace-Mapping.Fixture.Document.html">Document</a>
</li>
<li><a href="namespace-Mapping.Fixture.Unmapped.html">Unmapped</a>
</li>
<li><a href="namespace-Mapping.Fixture.Xml.html">Xml</a>
</li>
<li><a href="namespace-Mapping.Fixture.Yaml.html">Yaml</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Metadata.html">Metadata<span></span></a>
<ul>
<li><a href="namespace-Metadata.Cache.html">Cache</a>
</li>
<li><a href="namespace-Metadata.Driver.html">Driver</a>
</li>
<li><a href="namespace-Metadata.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Metadata.Tests.Cache.html">Cache</a>
</li>
<li><a href="namespace-Metadata.Tests.Driver.html">Driver<span></span></a>
<ul>
<li><a href="namespace-Metadata.Tests.Driver.Fixture.html">Fixture<span></span></a>
<ul>
<li><a href="namespace-Metadata.Tests.Driver.Fixture.A.html">A</a>
</li>
<li><a href="namespace-Metadata.Tests.Driver.Fixture.B.html">B</a>
</li>
<li><a href="namespace-Metadata.Tests.Driver.Fixture.C.html">C<span></span></a>
<ul>
<li><a href="namespace-Metadata.Tests.Driver.Fixture.C.SubDir.html">SubDir</a>
</li>
</ul></li>
<li><a href="namespace-Metadata.Tests.Driver.Fixture.T.html">T</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Metadata.Tests.Fixtures.html">Fixtures<span></span></a>
<ul>
<li><a href="namespace-Metadata.Tests.Fixtures.ComplexHierarchy.html">ComplexHierarchy</a>
</li>
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<li><a href="namespace-Monolog.html">Monolog<span></span></a>
<ul>
<li><a href="namespace-Monolog.Formatter.html">Formatter</a>
</li>
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<ul>
<li><a href="namespace-Monolog.Handler.FingersCrossed.html">FingersCrossed</a>
</li>
<li><a href="namespace-Monolog.Handler.SyslogUdp.html">SyslogUdp</a>
</li>
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<li><a href="namespace-Monolog.Processor.html">Processor</a>
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<li><a href="namespace-MyProject.html">MyProject<span></span></a>
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<ul>
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<ul>
<li><a href="namespace-MyProject.Proxies.__CG__.Doctrine.html">Doctrine<span></span></a>
<ul>
<li><a href="namespace-MyProject.Proxies.__CG__.Doctrine.Tests.html">Tests<span></span></a>
<ul>
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<li><a href="namespace-MyProject.Proxies.__CG__.Doctrine.Tests.Common.Util.html">Util</a>
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<li><a href="namespace-MyProject.Proxies.__CG__.OtherProject.html">OtherProject<span></span></a>
<ul>
<li><a href="namespace-MyProject.Proxies.__CG__.OtherProject.Proxies.html">Proxies<span></span></a>
<ul>
<li><a href="namespace-MyProject.Proxies.__CG__.OtherProject.Proxies.__CG__.html">__CG__<span></span></a>
<ul>
<li><a href="namespace-MyProject.Proxies.__CG__.OtherProject.Proxies.__CG__.Doctrine.html">Doctrine<span></span></a>
<ul>
<li><a href="namespace-MyProject.Proxies.__CG__.OtherProject.Proxies.__CG__.Doctrine.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-MyProject.Proxies.__CG__.OtherProject.Proxies.__CG__.Doctrine.Tests.Common.html">Common<span></span></a>
<ul>
<li><a href="namespace-MyProject.Proxies.__CG__.OtherProject.Proxies.__CG__.Doctrine.Tests.Common.Util.html">Util</a>
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<li><a href="namespace-NamespaceCollision.html">NamespaceCollision<span></span></a>
<ul>
<li><a href="namespace-NamespaceCollision.A.html">A<span></span></a>
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<ul>
<li><a href="namespace-Pequiven.SEIPBundle.html">SEIPBundle<span></span></a>
<ul>
<li><a href="namespace-Pequiven.SEIPBundle.Controller.html">Controller</a>
</li>
<li><a href="namespace-Pequiven.SEIPBundle.DataFixtures.html">DataFixtures</a>
</li>
<li><a href="namespace-Pequiven.SEIPBundle.DependencyInjection.html">DependencyInjection</a>
</li>
<li><a href="namespace-Pequiven.SEIPBundle.Entity.html">Entity</a>
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<li><a href="namespace-Symfony.Bundle.AsseticBundle.Twig.html">Twig</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.html">FrameworkBundle<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.CacheWarmer.html">CacheWarmer</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Command.html">Command</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Console.html">Console<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Console.Descriptor.html">Descriptor</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Console.Helper.html">Helper</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Controller.html">Controller</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.DataCollector.html">DataCollector</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.DependencyInjection.html">DependencyInjection<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.DependencyInjection.Compiler.html">Compiler</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Fragment.html">Fragment</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.HttpCache.html">HttpCache</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Routing.html">Routing</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Templating.html">Templating<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Templating.Asset.html">Asset</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Templating.Helper.html">Helper</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Templating.Loader.html">Loader</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Test.html">Test</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.CacheWarmer.html">CacheWarmer</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.Console.html">Console<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.Console.Descriptor.html">Descriptor</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.Controller.html">Controller</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.DependencyInjection.html">DependencyInjection<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.DependencyInjection.Compiler.html">Compiler</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.Fixtures.html">Fixtures<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.Fixtures.BaseBundle.html">BaseBundle</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.Fragment.html">Fragment</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.Functional.html">Functional<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.Functional.app.html">app</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.Functional.Bundle.html">Bundle<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.Functional.Bundle.TestBundle.html">TestBundle<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.Functional.Bundle.TestBundle.Controller.html">Controller</a>
</li>
</ul></li></ul></li></ul></li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.Routing.html">Routing</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.Templating.html">Templating<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.Templating.Helper.html">Helper<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.Templating.Helper.Fixtures.html">Fixtures</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.Templating.Loader.html">Loader</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.Translation.html">Translation</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Tests.Validator.html">Validator</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Translation.html">Translation</a>
</li>
<li><a href="namespace-Symfony.Bundle.FrameworkBundle.Validator.html">Validator</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.MonologBundle.html">MonologBundle<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.MonologBundle.DependencyInjection.html">DependencyInjection<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.MonologBundle.DependencyInjection.Compiler.html">Compiler</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.MonologBundle.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.MonologBundle.Tests.DependencyInjection.html">DependencyInjection<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.MonologBundle.Tests.DependencyInjection.Compiler.html">Compiler</a>
</li>
</ul></li></ul></li></ul></li>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.html">SecurityBundle<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Command.html">Command</a>
</li>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.DataCollector.html">DataCollector</a>
</li>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.DependencyInjection.html">DependencyInjection<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.DependencyInjection.Compiler.html">Compiler</a>
</li>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.DependencyInjection.Security.html">Security<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.DependencyInjection.Security.Factory.html">Factory</a>
</li>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.DependencyInjection.Security.UserProvider.html">UserProvider</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Security.html">Security</a>
</li>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Templating.html">Templating<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Templating.Helper.html">Helper</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Tests.DependencyInjection.html">DependencyInjection<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Tests.DependencyInjection.Security.html">Security<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Tests.DependencyInjection.Security.Factory.html">Factory</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Tests.Functional.html">Functional<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Tests.Functional.app.html">app</a>
</li>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Tests.Functional.Bundle.html">Bundle<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Tests.Functional.Bundle.CsrfFormLoginBundle.html">CsrfFormLoginBundle<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Tests.Functional.Bundle.CsrfFormLoginBundle.Controller.html">Controller</a>
</li>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Tests.Functional.Bundle.CsrfFormLoginBundle.Form.html">Form</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Tests.Functional.Bundle.FormLoginBundle.html">FormLoginBundle<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Tests.Functional.Bundle.FormLoginBundle.Controller.html">Controller</a>
</li>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Tests.Functional.Bundle.FormLoginBundle.DependencyInjection.html">DependencyInjection</a>
</li>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Tests.Functional.Bundle.FormLoginBundle.Security.html">Security</a>
</li>
</ul></li></ul></li></ul></li></ul></li>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Twig.html">Twig<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.SecurityBundle.Twig.Extension.html">Extension</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Bundle.SwiftmailerBundle.html">SwiftmailerBundle<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.SwiftmailerBundle.Command.html">Command</a>
</li>
<li><a href="namespace-Symfony.Bundle.SwiftmailerBundle.DataCollector.html">DataCollector</a>
</li>
<li><a href="namespace-Symfony.Bundle.SwiftmailerBundle.DependencyInjection.html">DependencyInjection<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.SwiftmailerBundle.DependencyInjection.Compiler.html">Compiler</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.SwiftmailerBundle.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Symfony.Bundle.SwiftmailerBundle.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.SwiftmailerBundle.Tests.DependencyInjection.html">DependencyInjection</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Bundle.TwigBundle.html">TwigBundle<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.TwigBundle.CacheWarmer.html">CacheWarmer</a>
</li>
<li><a href="namespace-Symfony.Bundle.TwigBundle.Command.html">Command</a>
</li>
<li><a href="namespace-Symfony.Bundle.TwigBundle.Controller.html">Controller</a>
</li>
<li><a href="namespace-Symfony.Bundle.TwigBundle.Debug.html">Debug</a>
</li>
<li><a href="namespace-Symfony.Bundle.TwigBundle.DependencyInjection.html">DependencyInjection<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.TwigBundle.DependencyInjection.Compiler.html">Compiler</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.TwigBundle.Extension.html">Extension</a>
</li>
<li><a href="namespace-Symfony.Bundle.TwigBundle.Loader.html">Loader</a>
</li>
<li><a href="namespace-Symfony.Bundle.TwigBundle.Node.html">Node</a>
</li>
<li><a href="namespace-Symfony.Bundle.TwigBundle.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.TwigBundle.Tests.Controller.html">Controller</a>
</li>
<li><a href="namespace-Symfony.Bundle.TwigBundle.Tests.DependencyInjection.html">DependencyInjection<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.TwigBundle.Tests.DependencyInjection.Compiler.html">Compiler</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.TwigBundle.Tests.Loader.html">Loader</a>
</li>
<li><a href="namespace-Symfony.Bundle.TwigBundle.Tests.TokenParser.html">TokenParser</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.TwigBundle.TokenParser.html">TokenParser</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Bundle.WebProfilerBundle.html">WebProfilerBundle<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.WebProfilerBundle.Controller.html">Controller</a>
</li>
<li><a href="namespace-Symfony.Bundle.WebProfilerBundle.DependencyInjection.html">DependencyInjection</a>
</li>
<li><a href="namespace-Symfony.Bundle.WebProfilerBundle.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Symfony.Bundle.WebProfilerBundle.Profiler.html">Profiler</a>
</li>
<li><a href="namespace-Symfony.Bundle.WebProfilerBundle.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Bundle.WebProfilerBundle.Tests.Controller.html">Controller</a>
</li>
<li><a href="namespace-Symfony.Bundle.WebProfilerBundle.Tests.DependencyInjection.html">DependencyInjection</a>
</li>
<li><a href="namespace-Symfony.Bundle.WebProfilerBundle.Tests.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Symfony.Bundle.WebProfilerBundle.Tests.Profiler.html">Profiler</a>
</li>
</ul></li></ul></li></ul></li>
<li><a href="namespace-Symfony.Component.html">Component<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.BrowserKit.html">BrowserKit<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.BrowserKit.Tests.html">Tests</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.ClassLoader.html">ClassLoader<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.ClassLoader.Tests.html">Tests</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Config.html">Config<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Config.Definition.html">Definition<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Config.Definition.Builder.html">Builder</a>
</li>
<li><a href="namespace-Symfony.Component.Config.Definition.Dumper.html">Dumper</a>
</li>
<li><a href="namespace-Symfony.Component.Config.Definition.Exception.html">Exception</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Config.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.Config.Loader.html">Loader</a>
</li>
<li><a href="namespace-Symfony.Component.Config.Resource.html">Resource</a>
</li>
<li><a href="namespace-Symfony.Component.Config.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Config.Tests.Definition.html">Definition<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Config.Tests.Definition.Builder.html">Builder</a>
</li>
<li><a href="namespace-Symfony.Component.Config.Tests.Definition.Dumper.html">Dumper</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Config.Tests.Fixtures.html">Fixtures<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Config.Tests.Fixtures.Configuration.html">Configuration</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Config.Tests.Loader.html">Loader</a>
</li>
<li><a href="namespace-Symfony.Component.Config.Tests.Resource.html">Resource</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Config.Util.html">Util</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Console.html">Console<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Console.Command.html">Command</a>
</li>
<li><a href="namespace-Symfony.Component.Console.Descriptor.html">Descriptor</a>
</li>
<li><a href="namespace-Symfony.Component.Console.Event.html">Event</a>
</li>
<li><a href="namespace-Symfony.Component.Console.Formatter.html">Formatter</a>
</li>
<li><a href="namespace-Symfony.Component.Console.Helper.html">Helper</a>
</li>
<li><a href="namespace-Symfony.Component.Console.Input.html">Input</a>
</li>
<li><a href="namespace-Symfony.Component.Console.Output.html">Output</a>
</li>
<li><a href="namespace-Symfony.Component.Console.Tester.html">Tester</a>
</li>
<li><a href="namespace-Symfony.Component.Console.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Console.Tests.Command.html">Command</a>
</li>
<li><a href="namespace-Symfony.Component.Console.Tests.Descriptor.html">Descriptor</a>
</li>
<li><a href="namespace-Symfony.Component.Console.Tests.Fixtures.html">Fixtures</a>
</li>
<li><a href="namespace-Symfony.Component.Console.Tests.Formatter.html">Formatter</a>
</li>
<li><a href="namespace-Symfony.Component.Console.Tests.Helper.html">Helper</a>
</li>
<li><a href="namespace-Symfony.Component.Console.Tests.Input.html">Input</a>
</li>
<li><a href="namespace-Symfony.Component.Console.Tests.Output.html">Output</a>
</li>
<li><a href="namespace-Symfony.Component.Console.Tests.Tester.html">Tester</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Component.CssSelector.html">CssSelector<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.CssSelector.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.CssSelector.Node.html">Node</a>
</li>
<li><a href="namespace-Symfony.Component.CssSelector.Parser.html">Parser<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.CssSelector.Parser.Handler.html">Handler</a>
</li>
<li><a href="namespace-Symfony.Component.CssSelector.Parser.Shortcut.html">Shortcut</a>
</li>
<li><a href="namespace-Symfony.Component.CssSelector.Parser.Tokenizer.html">Tokenizer</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.CssSelector.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.CssSelector.Tests.Node.html">Node</a>
</li>
<li><a href="namespace-Symfony.Component.CssSelector.Tests.Parser.html">Parser<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.CssSelector.Tests.Parser.Handler.html">Handler</a>
</li>
<li><a href="namespace-Symfony.Component.CssSelector.Tests.Parser.Shortcut.html">Shortcut</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.CssSelector.Tests.XPath.html">XPath</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.CssSelector.XPath.html">XPath<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.CssSelector.XPath.Extension.html">Extension</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Component.Debug.html">Debug<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Debug.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.Debug.FatalErrorHandler.html">FatalErrorHandler</a>
</li>
<li><a href="namespace-Symfony.Component.Debug.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Debug.Tests.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.Debug.Tests.FatalErrorHandler.html">FatalErrorHandler</a>
</li>
<li><a href="namespace-Symfony.Component.Debug.Tests.Fixtures.html">Fixtures</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Component.DependencyInjection.html">DependencyInjection<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.DependencyInjection.Compiler.html">Compiler</a>
</li>
<li><a href="namespace-Symfony.Component.DependencyInjection.Dump.html">Dump</a>
</li>
<li><a href="namespace-Symfony.Component.DependencyInjection.Dumper.html">Dumper</a>
</li>
<li><a href="namespace-Symfony.Component.DependencyInjection.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.DependencyInjection.Extension.html">Extension</a>
</li>
<li><a href="namespace-Symfony.Component.DependencyInjection.LazyProxy.html">LazyProxy<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.DependencyInjection.LazyProxy.Instantiator.html">Instantiator</a>
</li>
<li><a href="namespace-Symfony.Component.DependencyInjection.LazyProxy.PhpDumper.html">PhpDumper</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.DependencyInjection.Loader.html">Loader</a>
</li>
<li><a href="namespace-Symfony.Component.DependencyInjection.ParameterBag.html">ParameterBag</a>
</li>
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<ul>
<li><a href="namespace-Symfony.Component.DependencyInjection.Tests.Compiler.html">Compiler</a>
</li>
<li><a href="namespace-Symfony.Component.DependencyInjection.Tests.Dumper.html">Dumper</a>
</li>
<li><a href="namespace-Symfony.Component.DependencyInjection.Tests.Extension.html">Extension</a>
</li>
<li><a href="namespace-Symfony.Component.DependencyInjection.Tests.LazyProxy.html">LazyProxy<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.DependencyInjection.Tests.LazyProxy.Instantiator.html">Instantiator</a>
</li>
<li><a href="namespace-Symfony.Component.DependencyInjection.Tests.LazyProxy.PhpDumper.html">PhpDumper</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.DependencyInjection.Tests.Loader.html">Loader</a>
</li>
<li><a href="namespace-Symfony.Component.DependencyInjection.Tests.ParameterBag.html">ParameterBag</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Component.DomCrawler.html">DomCrawler<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.DomCrawler.Field.html">Field</a>
</li>
<li><a href="namespace-Symfony.Component.DomCrawler.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.DomCrawler.Tests.Field.html">Field</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Component.EventDispatcher.html">EventDispatcher<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.EventDispatcher.Debug.html">Debug</a>
</li>
<li><a href="namespace-Symfony.Component.EventDispatcher.Tests.html">Tests</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.ExpressionLanguage.html">ExpressionLanguage<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.ExpressionLanguage.Node.html">Node</a>
</li>
<li><a href="namespace-Symfony.Component.ExpressionLanguage.ParserCache.html">ParserCache</a>
</li>
<li><a href="namespace-Symfony.Component.ExpressionLanguage.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.ExpressionLanguage.Tests.Node.html">Node</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Component.Filesystem.html">Filesystem<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Filesystem.Exception.html">Exception</a>
</li>
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<li><a href="namespace-Symfony.Component.Finder.html">Finder<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Finder.Adapter.html">Adapter</a>
</li>
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</li>
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</li>
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</li>
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</li>
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</li>
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<ul>
<li><a href="namespace-Symfony.Component.Finder.Tests.Comparator.html">Comparator</a>
</li>
<li><a href="namespace-Symfony.Component.Finder.Tests.Expression.html">Expression</a>
</li>
<li><a href="namespace-Symfony.Component.Finder.Tests.FakeAdapter.html">FakeAdapter</a>
</li>
<li><a href="namespace-Symfony.Component.Finder.Tests.Iterator.html">Iterator</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Component.Form.html">Form<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Form.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Extension.html">Extension<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Form.Extension.Core.html">Core<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Form.Extension.Core.ChoiceList.html">ChoiceList</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Extension.Core.DataMapper.html">DataMapper</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Extension.Core.DataTransformer.html">DataTransformer</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Extension.Core.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Extension.Core.Type.html">Type</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Extension.Core.View.html">View</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Form.Extension.Csrf.html">Csrf<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Form.Extension.Csrf.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Extension.Csrf.Type.html">Type</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Form.Extension.DataCollector.html">DataCollector<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Form.Extension.DataCollector.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Extension.DataCollector.Proxy.html">Proxy</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Extension.DataCollector.Type.html">Type</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Form.Extension.DependencyInjection.html">DependencyInjection</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Extension.HttpFoundation.html">HttpFoundation<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Form.Extension.HttpFoundation.Type.html">Type</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Form.Extension.Templating.html">Templating</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Extension.Validator.html">Validator<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Form.Extension.Validator.Constraints.html">Constraints</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Extension.Validator.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Extension.Validator.Type.html">Type</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Extension.Validator.Util.html">Util</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Extension.Validator.ViolationMapper.html">ViolationMapper</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Component.Form.Guess.html">Guess</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Test.html">Test</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.html">Extension<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.Core.html">Core<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.Core.ChoiceList.html">ChoiceList</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.Core.DataMapper.html">DataMapper</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.Core.DataTransformer.html">DataTransformer</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.Core.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.Core.Type.html">Type</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.Csrf.html">Csrf<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.Csrf.CsrfProvider.html">CsrfProvider</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.Csrf.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.Csrf.Type.html">Type</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.DataCollector.html">DataCollector<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.DataCollector.Type.html">Type</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.HttpFoundation.html">HttpFoundation<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.HttpFoundation.EventListener.html">EventListener</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.Validator.html">Validator<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.Validator.Constraints.html">Constraints</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.Validator.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.Validator.Type.html">Type</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.Validator.Util.html">Util</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Tests.Extension.Validator.ViolationMapper.html">ViolationMapper</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Component.Form.Tests.Fixtures.html">Fixtures</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Tests.Guess.html">Guess</a>
</li>
<li><a href="namespace-Symfony.Component.Form.Tests.Util.html">Util</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Form.Util.html">Util</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.HttpFoundation.html">HttpFoundation<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.HttpFoundation.File.html">File<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.HttpFoundation.File.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.HttpFoundation.File.MimeType.html">MimeType</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.HttpFoundation.Session.html">Session<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.HttpFoundation.Session.Attribute.html">Attribute</a>
</li>
<li><a href="namespace-Symfony.Component.HttpFoundation.Session.Flash.html">Flash</a>
</li>
<li><a href="namespace-Symfony.Component.HttpFoundation.Session.Storage.html">Storage<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.HttpFoundation.Session.Storage.Handler.html">Handler</a>
</li>
<li><a href="namespace-Symfony.Component.HttpFoundation.Session.Storage.Proxy.html">Proxy</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Component.HttpFoundation.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.HttpFoundation.Tests.File.html">File<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.HttpFoundation.Tests.File.MimeType.html">MimeType</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.HttpFoundation.Tests.Session.html">Session<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.HttpFoundation.Tests.Session.Attribute.html">Attribute</a>
</li>
<li><a href="namespace-Symfony.Component.HttpFoundation.Tests.Session.Flash.html">Flash</a>
</li>
<li><a href="namespace-Symfony.Component.HttpFoundation.Tests.Session.Storage.html">Storage<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.HttpFoundation.Tests.Session.Storage.Handler.html">Handler</a>
</li>
<li><a href="namespace-Symfony.Component.HttpFoundation.Tests.Session.Storage.Proxy.html">Proxy</a>
</li>
</ul></li></ul></li></ul></li></ul></li>
<li><a href="namespace-Symfony.Component.HttpKernel.html">HttpKernel<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.HttpKernel.Bundle.html">Bundle</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.CacheClearer.html">CacheClearer</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.CacheWarmer.html">CacheWarmer</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Config.html">Config</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Controller.html">Controller</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.DataCollector.html">DataCollector<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.HttpKernel.DataCollector.Util.html">Util</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.HttpKernel.Debug.html">Debug</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.DependencyInjection.html">DependencyInjection</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Event.html">Event</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Fragment.html">Fragment</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.HttpCache.html">HttpCache</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Log.html">Log</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Profiler.html">Profiler</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.Bundle.html">Bundle</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.CacheClearer.html">CacheClearer</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.CacheWarmer.html">CacheWarmer</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.Config.html">Config</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.Controller.html">Controller</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.DataCollector.html">DataCollector</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.Debug.html">Debug</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.DependencyInjection.html">DependencyInjection</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.Fixtures.html">Fixtures<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.Fixtures.ExtensionAbsentBundle.html">ExtensionAbsentBundle</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.Fixtures.ExtensionLoadedBundle.html">ExtensionLoadedBundle<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.Fixtures.ExtensionLoadedBundle.DependencyInjection.html">DependencyInjection</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.Fixtures.ExtensionPresentBundle.html">ExtensionPresentBundle<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.Fixtures.ExtensionPresentBundle.Command.html">Command</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.Fixtures.ExtensionPresentBundle.DependencyInjection.html">DependencyInjection</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.Fragment.html">Fragment</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.HttpCache.html">HttpCache</a>
</li>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.Profiler.html">Profiler<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.HttpKernel.Tests.Profiler.Mock.html">Mock</a>
</li>
</ul></li></ul></li></ul></li>
<li><a href="namespace-Symfony.Component.Icu.html">Icu<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Icu.Tests.html">Tests</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Intl.html">Intl<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Intl.Collator.html">Collator</a>
</li>
<li><a href="namespace-Symfony.Component.Intl.DateFormatter.html">DateFormatter<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Intl.DateFormatter.DateFormat.html">DateFormat</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Intl.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.Intl.Globals.html">Globals</a>
</li>
<li><a href="namespace-Symfony.Component.Intl.Locale.html">Locale</a>
</li>
<li><a href="namespace-Symfony.Component.Intl.NumberFormatter.html">NumberFormatter</a>
</li>
<li><a href="namespace-Symfony.Component.Intl.ResourceBundle.html">ResourceBundle<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Intl.ResourceBundle.Compiler.html">Compiler</a>
</li>
<li><a href="namespace-Symfony.Component.Intl.ResourceBundle.Reader.html">Reader</a>
</li>
<li><a href="namespace-Symfony.Component.Intl.ResourceBundle.Transformer.html">Transformer<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Intl.ResourceBundle.Transformer.Rule.html">Rule</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Intl.ResourceBundle.Util.html">Util</a>
</li>
<li><a href="namespace-Symfony.Component.Intl.ResourceBundle.Writer.html">Writer</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Intl.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Intl.Tests.Collator.html">Collator<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Intl.Tests.Collator.Verification.html">Verification</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Intl.Tests.DateFormatter.html">DateFormatter<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Intl.Tests.DateFormatter.Verification.html">Verification</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Intl.Tests.Globals.html">Globals<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Intl.Tests.Globals.Verification.html">Verification</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Intl.Tests.Locale.html">Locale<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Intl.Tests.Locale.Verification.html">Verification</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Intl.Tests.NumberFormatter.html">NumberFormatter<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Intl.Tests.NumberFormatter.Verification.html">Verification</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Intl.Tests.ResourceBundle.html">ResourceBundle<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Intl.Tests.ResourceBundle.Reader.html">Reader</a>
</li>
<li><a href="namespace-Symfony.Component.Intl.Tests.ResourceBundle.Util.html">Util</a>
</li>
<li><a href="namespace-Symfony.Component.Intl.Tests.ResourceBundle.Writer.html">Writer</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Intl.Tests.Util.html">Util</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Intl.Util.html">Util</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Locale.html">Locale<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Locale.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Locale.Tests.Stub.html">Stub</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Component.OptionsResolver.html">OptionsResolver<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.OptionsResolver.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.OptionsResolver.Tests.html">Tests</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Process.html">Process<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Process.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.Process.Tests.html">Tests</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.PropertyAccess.html">PropertyAccess<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.PropertyAccess.Exception.html">Exception</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Routing.html">Routing<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Routing.Annotation.html">Annotation</a>
</li>
<li><a href="namespace-Symfony.Component.Routing.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.Routing.Generator.html">Generator<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Routing.Generator.Dumper.html">Dumper</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Routing.Loader.html">Loader</a>
</li>
<li><a href="namespace-Symfony.Component.Routing.Matcher.html">Matcher<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Routing.Matcher.Dumper.html">Dumper</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Routing.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Routing.Tests.Annotation.html">Annotation</a>
</li>
<li><a href="namespace-Symfony.Component.Routing.Tests.Fixtures.html">Fixtures<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Routing.Tests.Fixtures.AnnotatedClasses.html">AnnotatedClasses</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Routing.Tests.Generator.html">Generator<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Routing.Tests.Generator.Dumper.html">Dumper</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Routing.Tests.Loader.html">Loader</a>
</li>
<li><a href="namespace-Symfony.Component.Routing.Tests.Matcher.html">Matcher<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Routing.Tests.Matcher.Dumper.html">Dumper</a>
</li>
</ul></li></ul></li></ul></li>
<li><a href="namespace-Symfony.Component.Security.html">Security<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Acl.html">Acl<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Acl.Dbal.html">Dbal</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Acl.Domain.html">Domain</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Acl.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Acl.Model.html">Model</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Acl.Permission.html">Permission</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Acl.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Acl.Tests.Dbal.html">Dbal</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Acl.Tests.Domain.html">Domain</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Acl.Tests.Permission.html">Permission</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Acl.Tests.Voter.html">Voter</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Security.Acl.Voter.html">Voter</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Security.Core.html">Core<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Core.Authentication.html">Authentication<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Core.Authentication.Provider.html">Provider</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Core.Authentication.RememberMe.html">RememberMe</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Core.Authentication.Token.html">Token</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Security.Core.Authorization.html">Authorization<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Core.Authorization.Voter.html">Voter</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Security.Core.Encoder.html">Encoder</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Core.Event.html">Event</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Core.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Core.Role.html">Role</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Core.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Core.Tests.Authentication.html">Authentication<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Core.Tests.Authentication.Provider.html">Provider</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Core.Tests.Authentication.RememberMe.html">RememberMe</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Core.Tests.Authentication.Token.html">Token</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Security.Core.Tests.Authorization.html">Authorization<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Core.Tests.Authorization.Voter.html">Voter</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Security.Core.Tests.Encoder.html">Encoder</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Core.Tests.Role.html">Role</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Core.Tests.User.html">User</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Core.Tests.Util.html">Util</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Core.Tests.Validator.html">Validator<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Core.Tests.Validator.Constraints.html">Constraints</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Component.Security.Core.User.html">User</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Core.Util.html">Util</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Core.Validator.html">Validator<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Core.Validator.Constraints.html">Constraints</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Component.Security.Csrf.html">Csrf<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Csrf.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Csrf.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Csrf.Tests.TokenGenerator.html">TokenGenerator</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Csrf.Tests.TokenStorage.html">TokenStorage</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Security.Csrf.TokenGenerator.html">TokenGenerator</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Csrf.TokenStorage.html">TokenStorage</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Security.Http.html">Http<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Http.Authentication.html">Authentication</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Http.Authorization.html">Authorization</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Http.EntryPoint.html">EntryPoint</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Http.Event.html">Event</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Http.Firewall.html">Firewall</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Http.Logout.html">Logout</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Http.RememberMe.html">RememberMe</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Http.Session.html">Session</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Http.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Http.Tests.Authentication.html">Authentication</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Http.Tests.EntryPoint.html">EntryPoint</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Http.Tests.Firewall.html">Firewall</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Http.Tests.Logout.html">Logout</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Http.Tests.RememberMe.html">RememberMe</a>
</li>
<li><a href="namespace-Symfony.Component.Security.Http.Tests.Session.html">Session</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Component.Security.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Tests.Core.html">Core<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Tests.Core.Authentication.html">Authentication<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Tests.Core.Authentication.Token.html">Token</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Security.Tests.Core.User.html">User</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Security.Tests.Http.html">Http<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Security.Tests.Http.Firewall.html">Firewall</a>
</li>
</ul></li></ul></li></ul></li>
<li><a href="namespace-Symfony.Component.Serializer.html">Serializer<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Serializer.Encoder.html">Encoder</a>
</li>
<li><a href="namespace-Symfony.Component.Serializer.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.Serializer.Normalizer.html">Normalizer</a>
</li>
<li><a href="namespace-Symfony.Component.Serializer.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Serializer.Tests.Encoder.html">Encoder</a>
</li>
<li><a href="namespace-Symfony.Component.Serializer.Tests.Fixtures.html">Fixtures</a>
</li>
<li><a href="namespace-Symfony.Component.Serializer.Tests.Normalizer.html">Normalizer</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Component.Stopwatch.html">Stopwatch</a>
</li>
<li><a href="namespace-Symfony.Component.Templating.html">Templating<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Templating.Asset.html">Asset</a>
</li>
<li><a href="namespace-Symfony.Component.Templating.Helper.html">Helper</a>
</li>
<li><a href="namespace-Symfony.Component.Templating.Loader.html">Loader</a>
</li>
<li><a href="namespace-Symfony.Component.Templating.Storage.html">Storage</a>
</li>
<li><a href="namespace-Symfony.Component.Templating.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Templating.Tests.Fixtures.html">Fixtures</a>
</li>
<li><a href="namespace-Symfony.Component.Templating.Tests.Helper.html">Helper</a>
</li>
<li><a href="namespace-Symfony.Component.Templating.Tests.Loader.html">Loader</a>
</li>
<li><a href="namespace-Symfony.Component.Templating.Tests.Storage.html">Storage</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Symfony.Component.Translation.html">Translation<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Translation.Catalogue.html">Catalogue</a>
</li>
<li><a href="namespace-Symfony.Component.Translation.Dumper.html">Dumper</a>
</li>
<li><a href="namespace-Symfony.Component.Translation.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.Translation.Extractor.html">Extractor</a>
</li>
<li><a href="namespace-Symfony.Component.Translation.Loader.html">Loader</a>
</li>
<li><a href="namespace-Symfony.Component.Translation.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Translation.Tests.Catalogue.html">Catalogue</a>
</li>
<li><a href="namespace-Symfony.Component.Translation.Tests.Dumper.html">Dumper</a>
</li>
<li><a href="namespace-Symfony.Component.Translation.Tests.Loader.html">Loader</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Translation.Writer.html">Writer</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Validator.html">Validator<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Validator.Constraints.html">Constraints</a>
</li>
<li><a href="namespace-Symfony.Component.Validator.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.Validator.Mapping.html">Mapping<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Validator.Mapping.Cache.html">Cache</a>
</li>
<li><a href="namespace-Symfony.Component.Validator.Mapping.Loader.html">Loader</a>
</li>
</ul></li>
<li><a href="namespace-Symfony.Component.Validator.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Validator.Tests.Constraints.html">Constraints</a>
</li>
<li><a href="namespace-Symfony.Component.Validator.Tests.Fixtures.html">Fixtures</a>
</li>
<li><a href="namespace-Symfony.Component.Validator.Tests.Mapping.html">Mapping<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Validator.Tests.Mapping.Cache.html">Cache</a>
</li>
<li><a href="namespace-Symfony.Component.Validator.Tests.Mapping.Loader.html">Loader</a>
</li>
</ul></li></ul></li></ul></li>
<li><a href="namespace-Symfony.Component.Yaml.html">Yaml<span></span></a>
<ul>
<li><a href="namespace-Symfony.Component.Yaml.Exception.html">Exception</a>
</li>
<li><a href="namespace-Symfony.Component.Yaml.Tests.html">Tests</a>
</li>
</ul></li></ul></li></ul></li>
<li><a href="namespace-Tecnocreaciones.html">Tecnocreaciones<span></span></a>
<ul>
<li><a href="namespace-Tecnocreaciones.Bundle.html">Bundle<span></span></a>
<ul>
<li><a href="namespace-Tecnocreaciones.Bundle.AjaxFOSUserBundle.html">AjaxFOSUserBundle<span></span></a>
<ul>
<li><a href="namespace-Tecnocreaciones.Bundle.AjaxFOSUserBundle.Controller.html">Controller</a>
</li>
<li><a href="namespace-Tecnocreaciones.Bundle.AjaxFOSUserBundle.DependencyInjection.html">DependencyInjection<span></span></a>
<ul>
<li><a href="namespace-Tecnocreaciones.Bundle.AjaxFOSUserBundle.DependencyInjection.Compiler.html">Compiler</a>
</li>
</ul></li>
<li><a href="namespace-Tecnocreaciones.Bundle.AjaxFOSUserBundle.Event.html">Event</a>
</li>
<li><a href="namespace-Tecnocreaciones.Bundle.AjaxFOSUserBundle.EventListener.html">EventListener</a>
</li>
<li><a href="namespace-Tecnocreaciones.Bundle.AjaxFOSUserBundle.Handler.html">Handler</a>
</li>
<li><a href="namespace-Tecnocreaciones.Bundle.AjaxFOSUserBundle.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Tecnocreaciones.Bundle.AjaxFOSUserBundle.Tests.Controller.html">Controller</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Tecnocreaciones.Bundle.InstallBundle.html">InstallBundle<span></span></a>
<ul>
<li><a href="namespace-Tecnocreaciones.Bundle.InstallBundle.Command.html">Command</a>
</li>
<li><a href="namespace-Tecnocreaciones.Bundle.InstallBundle.DependencyInjection.html">DependencyInjection</a>
</li>
<li><a href="namespace-Tecnocreaciones.Bundle.InstallBundle.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Tecnocreaciones.Bundle.InstallBundle.Tests.Controller.html">Controller</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Tecnocreaciones.Bundle.TemplateBundle.html">TemplateBundle<span></span></a>
<ul>
<li><a href="namespace-Tecnocreaciones.Bundle.TemplateBundle.Controller.html">Controller</a>
</li>
<li><a href="namespace-Tecnocreaciones.Bundle.TemplateBundle.DependencyInjection.html">DependencyInjection</a>
</li>
<li><a href="namespace-Tecnocreaciones.Bundle.TemplateBundle.Menu.html">Menu<span></span></a>
<ul>
<li><a href="namespace-Tecnocreaciones.Bundle.TemplateBundle.Menu.Template.html">Template<span></span></a>
<ul>
<li><a href="namespace-Tecnocreaciones.Bundle.TemplateBundle.Menu.Template.Developer.html">Developer</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Tecnocreaciones.Bundle.TemplateBundle.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Tecnocreaciones.Bundle.TemplateBundle.Tests.Controller.html">Controller</a>
</li>
</ul></li></ul></li></ul></li>
<li><a href="namespace-Tecnocreaciones.Vzla.html">Vzla<span></span></a>
<ul>
<li><a href="namespace-Tecnocreaciones.Vzla.GovernmentBundle.html">GovernmentBundle<span></span></a>
<ul>
<li><a href="namespace-Tecnocreaciones.Vzla.GovernmentBundle.DependencyInjection.html">DependencyInjection</a>
</li>
<li><a href="namespace-Tecnocreaciones.Vzla.GovernmentBundle.Form.html">Form<span></span></a>
<ul>
<li><a href="namespace-Tecnocreaciones.Vzla.GovernmentBundle.Form.Type.html">Type</a>
</li>
</ul></li>
<li><a href="namespace-Tecnocreaciones.Vzla.GovernmentBundle.Menu.html">Menu<span></span></a>
<ul>
<li><a href="namespace-Tecnocreaciones.Vzla.GovernmentBundle.Menu.Template.html">Template<span></span></a>
<ul>
<li><a href="namespace-Tecnocreaciones.Vzla.GovernmentBundle.Menu.Template.Developer.html">Developer</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Tecnocreaciones.Vzla.GovernmentBundle.Model.html">Model</a>
</li>
<li><a href="namespace-Tecnocreaciones.Vzla.GovernmentBundle.Tests.html">Tests<span></span></a>
<ul>
<li><a href="namespace-Tecnocreaciones.Vzla.GovernmentBundle.Tests.Controller.html">Controller</a>
</li>
</ul></li>
<li><a href="namespace-Tecnocreaciones.Vzla.GovernmentBundle.Twig.html">Twig<span></span></a>
<ul>
<li><a href="namespace-Tecnocreaciones.Vzla.GovernmentBundle.Twig.Extension.html">Extension</a>
</li>
</ul></li></ul></li></ul></li></ul></li>
<li><a href="namespace-TestBundle.html">TestBundle<span></span></a>
<ul>
<li><a href="namespace-TestBundle.Fabpot.html">Fabpot<span></span></a>
<ul>
<li><a href="namespace-TestBundle.Fabpot.FooBundle.html">FooBundle<span></span></a>
<ul>
<li><a href="namespace-TestBundle.Fabpot.FooBundle.Controller.html">Controller</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-TestBundle.FooBundle.html">FooBundle<span></span></a>
<ul>
<li><a href="namespace-TestBundle.FooBundle.Controller.html">Controller<span></span></a>
<ul>
<li><a href="namespace-TestBundle.FooBundle.Controller.Sub.html">Sub</a>
</li>
<li><a href="namespace-TestBundle.FooBundle.Controller.Test.html">Test</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-TestBundle.Sensio.html">Sensio<span></span></a>
<ul>
<li><a href="namespace-TestBundle.Sensio.Cms.html">Cms<span></span></a>
<ul>
<li><a href="namespace-TestBundle.Sensio.Cms.FooBundle.html">FooBundle<span></span></a>
<ul>
<li><a href="namespace-TestBundle.Sensio.Cms.FooBundle.Controller.html">Controller</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-TestBundle.Sensio.FooBundle.html">FooBundle<span></span></a>
<ul>
<li><a href="namespace-TestBundle.Sensio.FooBundle.Controller.html">Controller</a>
</li>
</ul></li></ul></li></ul></li>
<li><a href="namespace-TestFixtures.html">TestFixtures</a>
</li>
<li><a href="namespace-Timestampable.html">Timestampable<span></span></a>
<ul>
<li><a href="namespace-Timestampable.Fixture.html">Fixture<span></span></a>
<ul>
<li><a href="namespace-Timestampable.Fixture.Document.html">Document</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Tool.html">Tool</a>
</li>
<li><a href="namespace-Translatable.html">Translatable<span></span></a>
<ul>
<li><a href="namespace-Translatable.Fixture.html">Fixture<span></span></a>
<ul>
<li><a href="namespace-Translatable.Fixture.Document.html">Document<span></span></a>
<ul>
<li><a href="namespace-Translatable.Fixture.Document.Personal.html">Personal</a>
</li>
</ul></li>
<li><a href="namespace-Translatable.Fixture.Issue114.html">Issue114</a>
</li>
<li><a href="namespace-Translatable.Fixture.Issue138.html">Issue138</a>
</li>
<li><a href="namespace-Translatable.Fixture.Issue165.html">Issue165</a>
</li>
<li><a href="namespace-Translatable.Fixture.Issue173.html">Issue173</a>
</li>
<li><a href="namespace-Translatable.Fixture.Issue75.html">Issue75</a>
</li>
<li><a href="namespace-Translatable.Fixture.Issue922.html">Issue922</a>
</li>
<li><a href="namespace-Translatable.Fixture.Personal.html">Personal</a>
</li>
<li><a href="namespace-Translatable.Fixture.Template.html">Template</a>
</li>
<li><a href="namespace-Translatable.Fixture.Type.html">Type</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Translator.html">Translator<span></span></a>
<ul>
<li><a href="namespace-Translator.Fixture.html">Fixture</a>
</li>
</ul></li>
<li><a href="namespace-Tree.html">Tree<span></span></a>
<ul>
<li><a href="namespace-Tree.Fixture.html">Fixture<span></span></a>
<ul>
<li><a href="namespace-Tree.Fixture.Closure.html">Closure</a>
</li>
<li><a href="namespace-Tree.Fixture.Document.html">Document</a>
</li>
<li><a href="namespace-Tree.Fixture.Genealogy.html">Genealogy</a>
</li>
<li><a href="namespace-Tree.Fixture.Mock.html">Mock</a>
</li>
<li><a href="namespace-Tree.Fixture.Repository.html">Repository</a>
</li>
<li><a href="namespace-Tree.Fixture.Transport.html">Transport</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Uploadable.html">Uploadable<span></span></a>
<ul>
<li><a href="namespace-Uploadable.Fixture.html">Fixture<span></span></a>
<ul>
<li><a href="namespace-Uploadable.Fixture.Entity.html">Entity</a>
</li>
</ul></li></ul></li>
<li><a href="namespace-Wrapper.html">Wrapper<span></span></a>
<ul>
<li><a href="namespace-Wrapper.Fixture.html">Fixture<span></span></a>
<ul>
<li><a href="namespace-Wrapper.Fixture.Document.html">Document</a>
</li>
<li><a href="namespace-Wrapper.Fixture.Entity.html">Entity</a>
</li>
</ul></li></ul></li>
</ul>
</div>
<hr />
<div id="elements">
<h3>Classes</h3>
<ul>
<li class="active"><a href="class-Knp.Bundle.MenuBundle.Tests.Stubs.Menu.Builder.html">Builder</a></li>
<li><a href="class-Knp.Bundle.MenuBundle.Tests.Stubs.Menu.ContainerAwareBuilder.html">ContainerAwareBuilder</a></li>
</ul>
</div>
</div>
</div>
<div id="splitter"></div>
<div id="right">
<div id="rightInner">
<form id="search">
<input type="hidden" name="cx" value="" />
<input type="hidden" name="ie" value="UTF-8" />
<input type="text" name="q" class="text" />
<input type="submit" value="Search" />
</form>
<div id="navigation">
<ul>
<li>
<a href="index.html" title="Overview"><span>Overview</span></a>
</li>
<li>
<a href="namespace-Knp.Bundle.MenuBundle.Tests.Stubs.Menu.html" title="Summary of Knp\Bundle\MenuBundle\Tests\Stubs\Menu"><span>Namespace</span></a>
</li>
<li class="active">
<span>Class</span> </li>
</ul>
<ul>
<li>
<a href="tree.html" title="Tree view of classes, interfaces, traits and exceptions"><span>Tree</span></a>
</li>
</ul>
<ul>
</ul>
</div>
<div id="content" class="class">
<h1>Class Builder</h1>
<div class="info">
<b>Namespace:</b> <a href="namespace-Knp.html">Knp</a>\<a href="namespace-Knp.Bundle.html">Bundle</a>\<a href="namespace-Knp.Bundle.MenuBundle.html">MenuBundle</a>\<a href="namespace-Knp.Bundle.MenuBundle.Tests.html">Tests</a>\<a href="namespace-Knp.Bundle.MenuBundle.Tests.Stubs.html">Stubs</a>\<a href="namespace-Knp.Bundle.MenuBundle.Tests.Stubs.Menu.html">Menu</a><br />
<b>Located at</b> <a href="source-class-Knp.Bundle.MenuBundle.Tests.Stubs.Menu.Builder.html#7-18" title="Go to source code">vendor/knplabs/knp-menu-bundle/Knp/Bundle/MenuBundle/Tests/Stubs/Menu/Builder.php</a><br />
</div>
<table class="summary" id="methods">
<caption>Methods summary</caption>
<tr data-order="mainMenu" id="_mainMenu">
<td class="attributes"><code>
public
</code>
</td>
<td class="name"><div>
<a class="anchor" href="#_mainMenu">#</a>
<code><a href="source-class-Knp.Bundle.MenuBundle.Tests.Stubs.Menu.Builder.html#9-12" title="Go to source code">mainMenu</a>( <span><code><a href="class-Knp.Menu.FactoryInterface.html">Knp\Menu\FactoryInterface</a></code> <var>$factory</var></span> )</code>
<div class="description short">
</div>
<div class="description detailed hidden">
</div>
</div></td>
</tr>
<tr data-order="invalidMethod" id="_invalidMethod">
<td class="attributes"><code>
public
</code>
</td>
<td class="name"><div>
<a class="anchor" href="#_invalidMethod">#</a>
<code><a href="source-class-Knp.Bundle.MenuBundle.Tests.Stubs.Menu.Builder.html#14-17" title="Go to source code">invalidMethod</a>( <span><code><a href="class-Knp.Menu.FactoryInterface.html">Knp\Menu\FactoryInterface</a></code> <var>$factory</var></span> )</code>
<div class="description short">
</div>
<div class="description detailed hidden">
</div>
</div></td>
</tr>
</table>
</div>
<div id="footer">
seip API documentation generated by <a href="http://apigen.org">ApiGen 2.8.0</a>
</div>
</div>
</div>
</body>
</html>
| code |
#!/bin/bash
README=($(git diff --cached --name-only | grep -Ei '^README\.[R]?md$'))
MSG="use 'git commit --no-verify' to override this check"
if [[ ${#README[@]} == 0 ]]; then
exit 0
fi
if [[ README.Rmd -nt README.md ]]; then
echo -e "README.md is out of date; please re-knit README.Rmd\n$MSG"
exit 1
elif [[ ${#README[@]} -lt 2 ]]; then
echo -e "README.Rmd and README.md should be both staged\n$MSG"
exit 1
fi
| code |
Are you worried about your MBA admission? Not sure which Business schools to apply to? Do not worry, for we have carefully indexed and ranked MBA programs in top MBA colleges in Telangana. Whether you are aiming for your MBA in finance, marketing, HR, IT or in any other specialization; most of these B-Schools give the best business schools in the world a run for their money.
Telangana is one of the more popular destinations for management students in India. Hitbullseye ranking of top MBA Colleges in Telangana helps you decide which institutions you should be focusing on for your MBA admissions. | english |
Billed as the Unofficial Underground Boston Pub of Dallas, McSwiggan's Irish Pub is located in the heart of Austin Ranch. Amid the impressive surroundings of Irish decor and sports memorabilia that graces each wall, is a full service bar featuring more than 50 different beers from all over the world. Also famous for their traditional pub fare, McSwiggan's is perfect any time of day. | english |
The 2019 Hyundai Sonata. This 4 door, 5 passenger sedan stands out among competitors in its class! It features an automatic transmission, front-wheel drive, and a 2.4 liter 4 cylinder engine. All of the following features are included: delay-off headlights, 1-touch window functionality, heated seats, turn signal indicator mirrors, blind spot sensor, an overhead console, and air conditioning. Audio features include an AM/FM radio, steering wheel mounted audio controls, and 6 speakers, providing excellent sound throughout the cabin. Passenger security is always assured thanks to various safety features, such as: dual front impact airbags, front side impact airbags, traction control, brake assist, a security system, an emergency communication system, and 4 wheel disc brakes with ABS. This car was designed with safety in mind, allowing you to drive with even greater assurance. Our sales staff will help you find the vehicle that you've been searching for. We'd be happy to answer any questions that you may have. Call now to schedule a test drive. | english |
کلس چھےٚ مشکل رٲچھۍ | kashmiri |
The Rohwer outpost . (McGehee, AR), Apr. 14 1943. https://0-www.loc.gov.oasys.lib.oxy.edu/item/sn84025150/1943-04-14/ed-1/.
(1943, April 14) The Rohwer outpost . Retrieved from the Library of Congress, https://0-www.loc.gov.oasys.lib.oxy.edu/item/sn84025150/1943-04-14/ed-1/.
The Rohwer outpost . (McGehee, AR) 14 Apr. 1943. Retrieved from the Library of Congress, www.loc.gov/item/sn84025150/1943-04-14/ed-1/. | english |
- 'जब मैंने अपने बेटे को देखा तो वो मिर्च पाउडर से नहाए था, उसके गले में ब्रा बंधी थी'
तिरुवंतपुरम: केरल के कासरगोड जिले में कक्षा ११ में पढ़ने वाले लड़के को केवल शक के चलते बर्बरता पूर्वक मारा गया. लड़के पर आरोप था कि उसने महिलाओं के अंडरवियर चुराए हैं. लड़के का आरोप है कि बस इसी वजह से एक पड़ोसी ने उसको बांध दिया और चेहरे पर मिर्ची पाउडर लगाया. लड़के ने इस आरोप का खंडन करते हुए बताया कि वो अपने पड़ोसी के घर कृष्णा फल लेने गया था.
एक पुलिस अधिकारी ने बताया, यह घटना बेलूर गांव के अटेंगनम के पास सोमवार शाम ५.३० बजे हुई. अपने पड़ोसी के अंडरवियर चोरी करने के आरोप पर लड़के पर हमला किया गया था, लेकिन लड़के ने कहा कि वह कृष्णा फल लेने के लिए उनके घर गया था. हमें मामले की जांच करनी होगी. पड़ोसी ने पुलिस को बताया कि उनके पास कपड़े चुराने का वीडियो है. लड़का अब कोडंगल के वेलारिककुंडु तालुक अस्पताल में भर्ती है. डॉक्टरों ने उन्हें कान्हांगड़ के एक ईएनटी विशेषज्ञ के पास भेज दिया.
लड़के की मां ने बताया, लड़के ने उनको बताया था कि वह अपने पड़ोसी के घर में फल लेने के लिए जा रहा है. उसके लगभग ३० मिनट के बाद, मैंने अपने लड़के को दर्द से कराहते हुए सुना तब मैं भागकर पड़ोसी के घर गई. पड़ोसी का घर हमारे घर के ऊपर ही है.
मां ने कहा, जब मैं वहां पहुंची तो मैंने देखा कि मेरे लड़का पूरी तरह से मिर्च पाउडर से नहाया हुआ था और उसकी गर्दन के चारो ओर ब्रा बंधी थी. पड़ोसी उसे जान से मारना चाहता था. उसने लड़के को बचाया और उसे घर वापस लाईं. पड़ोसी ने पुलिस को बताया कि लड़का था दिसंबर से इनर वियर चुरा रहा है और उनके पास इसका वीडियो है. लड़के की मां दिहाड़ी मजदूर है. मां ने उससे वीडियो दिखाने की बात कही. उसने वीडियो में कुछ भी नहीं दिखा. वीडियो में साफ दिख रहा था कि लड़का कृष्णा फल के बगीचे में था. उसने फल भी नहीं लिया था और जब वह पकड़ा गया था और तब वह लौट रहा था. | hindi |
Architectural accents exude charisma including intricate ceiling millwork, master carpenter level trim and molding, 12' and 14' ceilings, and natural stone and concrete counters. An energy ââ¬â efficient Geothermal HVAC system eliminates the need for condenser replacement and the interruption of start-up noise. Convenience begins first floor with a luscious master suite including master bath stone flooring and an exterior full-sized claw tub. Each of three additional bedrooms, one incorporating bunk beds for four campers, includes a private bath in an en suite configuration. A private sitting area and screened porch ensure guests ultimate comfort. Take the elevator or the stairs to second floor living that is designed for gathering with a large kitchen complete with Thermador and Subzero appliances and full-sized pantry with wine storage. Dining is either in a cozy, sunny alcove for four or overlooking living at a table for eight. Get-togethers tend to flow from relaxed seating, fireside to the full-sized pool table serviced by its own wet bar with wine fridge. Wind-downs lead straight to the airy screened porch with lake views and its own gas fireplace. The 2nd floor office is all business with a built-in desk and bookcases and its own commanding, wooded view from a private balcony. Movie night is just around the corner in the private media viewing room on the 3rd floor. A fourth-floor light filled sitting/reading/work room overlooks enchanting Western Lake and entertaining Seaside. A separate carriage house over a two-car garage includes its own kitchen and sleeping/sitting area and a very private, porch with views all its own. Designer furnishings remain for a move-in ready experience in this ideally located home within walking distance to dining, shopping, and beach club in one-of-a-kind Watercolor. | english |
Mondo is Italian for world, and Deli Di Mondo’s shelves groan with an assortment of boutique comestibles from around the globe including olive oils, unusual teas and exotic sauces.
There’s also great range of petrochemical-free shampoos and body lotions, 40/50s china and glassware at the cafe/gourmet deli.
In fact there’s so much to look at lunch was on the table before my eyes had made it halfway down the funky cushions and pouffes aisle.
The cafe food presented a similar problem and I had agonised between the cauliflower fritter with minted yoghurt and a Mediterranean tart.
The tart ($13.80 with salad) won out—just—with its glorious combination of colours; bright red capsicum and tomato and white fetta.
A feast for the eyes, it was also a winner in the taste stakes, the sharpness of the cheese combining perfectly with the filling and the sweet capsicum and tomato topping.
The accompanying salads were also delicious, including an Asian-style coleslaw with shredded coconut and sultanas drizzled in a sweet vinegar dressing, and a puy and chick pea one.
The other half of the D’Anger duo had the tuna pattie ($13.80), and it was a massive ball of potato and tuna subtly laced with herbs and spices.
Having gulped down a deliciously fresh-squeezed orange and mango juice ($6.90) it was time for tea and cake and there was no going past the apple and butterscotch or lemon syrup ones ($4.50).
It was hard to say which was the better.
Both were moist and flavoursome, with the apple cake rich with a tooth-achingly sweet butterscotch melange running through it like a motherload, and the syrup cake sharp and sweet.
My brother and his wife put me onto di Mondo after a memorable breakfast.
They were so impressed by the big brekkie’s lashings of bacon, sausage, egg and hash browns he was texting photos before his coffee had cooled.
Owners Sherry and John have been on Hislop Road, Attadale for a couple of years but only recently added the cafe extension.
All but the occasional apple pie is made daily on the premises.
Sherry sources the eclectic deli provisions as a foil to the generic blandness of chain stores.
“I’m so scared we are in trouble of losing our identity,” she says.
She hopes to source more regional hand-made goods in coming months and is happy to hear from local artisans. | english |
NASCAR fans have gotten a small taste of action during the start of Daytona Speedweeks, but now things are really about to pick up.
Over the next four days, there will be some form of competitive racing across NASCAR's top three series each day. That begins with Thursday night's Monster Energy Cup Series Gander RV Duels at Daytona, the qualifying races that will set the starting order for Sunday's Daytona 500.
There will also be the Gander Outdoors Truck Series race on Friday night, followed by the Xfinity Series race on Saturday afternoon. NASCAR's opening week will conclude with the Great American Race on Sunday afternoon.
So far during Daytona Speedweeks, William Byron and Alex Bowman secured the top two starting spots for the Daytona 500 during qualifying this past Sunday afternoon. Then, later in the day, Jimmie Johnson won the rain-shortened Advance Auto Parts Clash.
But there's plenty more action for fans to catch this weekend.
Perhaps one of the most interesting things about this year's Daytona 500 is that it will be the final Cup Series race to feature restrictor plates.
After Sunday, the cars' engines will no longer have restrictor plates at the two superspeedway tracks, Daytona and Talladega. This will certainly change how these races unfold in future years.
But for now, fans can expect one more afternoon filled with pack racing in which drivers will try to avoid major wrecks that could collect up to half the field.
Over the past nine years, there have been nine different Daytona 500 winners, and six of them will be in this year's field—Jamie McMurray (2010 winner), Jimmie Johnson (2013), Joey Logano (2015), Denny Hamlin (2016), Kurt Busch (2017) and Austin Dillon (2018).
Dillon notched his first career Cup Series victory in last year's Great American Race, but he was winless in the remaining 35 races of the season.
If the early parts of Daytona Speedweeks are any indication, then the four Hendrick Motorsports cars could be strong contenders to win the Daytona 500. Not only did Johnson win the Advance Auto Parts Clash, but he and his three teammates (Byron, Bowman and Chase Elliott) had the four fastest lap times during the final round of qualifying.
Logano may have the most momentum entering the 2019 season, as he won last year's Cup Series championship, the first of his career, by winning last year's season finale at Homestead-Miami Speedway.
Logano, who is entering his 11th full-time Cup Series season, finished in the top five in five of the last seven races of 2018. | english |
यू अरे हेरे: होम २०१८ ऑगस्ट एम्स ऋषिकेश ने ६६८ नर्सिंग ऑफिसर, ऑफिस असिस्टेंट, तकनीशियन, सेक्रेटरी, प्रोग्रामर और विभिन्न रिक्ति के लिए आवेदन आमंत्रित किया है। अंतिम तिथि: १४ सितंबर २०१८
ऑल इंडिया इंस्टीट्यूट ऑफ मेडिकल साइंसेज (एम्स ऋषिकेश भर्ती २०१८) ऋषिकेश ने ६६८ विभिन्न पदों के लिए आवेदन आमंत्रित किया है। अगर आप इस एम्स ऋषिकेश भर्ती २०१८ के इच्छुक हैं तो आप आवेदन कर सकते हैं। पद इस प्रकार हैं ६६८ नर्सिंग ऑफिसर, ऑफिस असिस्टेंट, तकनीशियन, सेक्रेटरी, प्रोग्रामर और विभिन्न रिक्ति
वेतनमान: र्स. ९३००- ३४,८००/-
ग्रेड वेतन: ८५०० /
१. नर्सिंग ऑफिसर उम्मीदवार: ई) बीएससी नर्सिंग (४ साल का कोर्स) मान्यता प्राप्त संस्थान / विश्वविद्यालय इंडियन नर्सिंग काउंसिल से या बीएससी (पोस्ट सर्टिफिकेट) या बीएससी जैसे समकक्ष। एक भारतीय नर्सिंग
ई) नर्स और मिडवाइफ स्टेट / भारतीय नर्सिंग काउंसिल के रूप में पंजीकृत।
२. ऑफिस असिस्टेंट उम्मीदवार: ई) मान्यता प्राप्त विश्वविद्यालय या समकक्ष की डिग्री।
ई) कंप्यूटर में प्रवीणता।
आवेदन शुल्क: सामान्य / पिछड़ा वर्ग के संबंधित उम्मीदवारों के लिए ३०००/- डेबिट कार्ड, क्रेडिट कार्ड, नेट बैंकिंग के माध्यम से परीक्षा शुल्क का भुगतान करें। और अनुसूचित जाति / अनुसूचित जनजाति / पीडब्ल्यूडी / महिला उम्मीदवारों के लिए कोई शुल्क नहीं
आवेदन पत्र शुरू करने की तारीख -२७ अगस्त २०१८
आवेदन पत्र जमा करने की अंतिम तिथि- १४ सितंबर २०१८
थे पोस्ट एम्स ऋषिकेश ने ६६८ नर्सिंग ऑफिसर, ऑफिस असिस्टेंट, तकनीशियन, सेक्रेटरी, प्रोग्रामर और विभिन्न रिक्ति के लिए आवेदन आमंत्रित किया है। अंतिम तिथि: १४ सितंबर २०१८ एपेअरड फर्स्ट ऑन फ्री जॉब अलर्ट हिंदी. | hindi |
र्स के भैयाजी ने बताया देश को किस पैमाने पर बांटा जाए
रायपुर। विजयदशमी उत्सव पर राष्ट्रीय स्वयंसेवक संघ के सरकार्यवाह सुरेश भैयाजी जोशी ने कहा कि देश को हिंदू-मुस्लिम के पैमाने पर नहीं, देशभक्त और देशद्रोही के पैमाने पर बांटने की जरूरत है। राजधानी के स्वामी विवेकानंद स्पोर्ट्स कांप्लेक्स में रविवार को स्वयंसेवकों को संबोधित करते हुए भैयाजी ने कहा कि जो देश के विरोध में बोलेगा, वह देशद्रोही है। इसको राजनीतिक दृष्टि से नहीं देखना चाहिए। डेढ़ घंटे के भाषण में वे रोहिंग्या से लेकर देश की आंतरिक सुरक्षा, गौहत्या, जैविक खेती, जलसंवर्धन और स्वच्छता अभियान पर भी बोले। रोहिंग्या मुसलमानों के मुद्दे पर भैयाजी ने कहा कि म्यांमार से रोहिंग्या आए तो वहां हिंदुओं की हत्या हुई।
म्यांमार सरकार ने उनको खदेड़ा। अब कोई म्यांमार से चलकर कश्मीर पहुंच जाए और अपने अधिकारों की मांग करे, तो उसे कितना जायज माना जाएगा। रोहिंग्या को बाहर किया जाना चाहिए। उन्होंने इस मामले में केंद्र सरकार और सुप्रीम कोर्ट के रुख का स्वागत किया। उन्होंने कहा कि चुनाव तो कभी-कभी आते हैं। देश, समाज और आर्थिक हित के मुद्दे आते हैं, तो देशहित में सबको साथ देना चाहिए। भारत को कोई बांट नहीं सकता है। ना भाषा ना भौगोलिक सीमा के आधार पर। हमारी सेना सीमा पर मुस्तैदी से खड़ी है, उनके हाथ में आधुनिक हथियार हैं। कुछ राजनेता उनका मनोबल तोड़ने की कोशिश करते हैं। सेना को मजबूत करने की जरूरत है।
गौमाता आर्थिक विकास का आधार, इसलिए गौरक्षा जरूरी
भैयाजी ने कहा कि गौमाता आर्थिक विकास का आधार है, इसलिए देश में गौरक्षा जरूरी है। गौरक्षा को धर्म के आधार पर जोड़कर विवाद करना दुखद है। गौरक्षा किसी संप्रदाय के खिलाफ नहीं है। देश में बदलाव आया है। इस्लाम का एक वर्ग गौरक्षा की मांग कर रहा है। कुछ गलत लोगों के कारण गौरक्षा के क्षेत्र में सही काम करने वालों को प्रताड़ित नहीं करना चाहिए।
सामान खरीदने से पहले देखे, किस देश का बना है
उन्होंने कहा- भारत के उद्योगों के सामने संकट है। चीन के पटाखे से लेकर भगवान तक बाजार में उपलब्ध हैं। हमें ये देखना चाहिए कि किस देश का बना हुआ समान है। देश के आर्थिक तंत्र में सरकार ने कुछ नए प्रयोग किए हैं। अब छोटे उद्योगों के बारे में भी बड़े कदम उठाने की जरूरत है।
फसल के मूल्य निर्धारण के लिए हो वैज्ञानिक व्यवस्था
किसानों की आत्महत्या रोकने के लिए किसानों को उनकी फसल का उचित मूल्य देने की जरूरत है। फसल के मूल्य निर्धारण के लिए वैज्ञानिक व्यवस्था होनी चाहिए। किसान खेती के लिए कर्ज ले, लेकिन कर्ज लौटने की स्थिति में पहुंच जाए, ऐसी व्यवस्था करनी होगी। कर्जमाफी किसानों की आत्महत्या को रोकने का उपाय नहीं है। नीति निर्धारकों को जैविक खेती के विस्तार के बारे में सोचना चाहिए। सबको एक पौधा लगाना चाहिए, जिससे बारिश के लिए वातावरण बन सके। | hindi |
Piano Instrumental Music for Prayer, Meditation, Deep Healing, Soaking, Worship, Study, Rest, Reflection & Relaxation. May the peace of God which surpasses all understanding guard your heart and mind as you listen. This was played in the key of C#. | english |
أمِس نفرس تہِ کَتہِ آسہِ یژھہٕ کَتھہٕ وہمہٕ گمانس منٛز | kashmiri |
/*
* Xen event channels
*
* Xen models interrupts with abstract event channels. Because each
* domain gets 1024 event channels, but NR_IRQ is not that large, we
* must dynamically map irqs<->event channels. The event channels
* interface with the rest of the kernel by defining a xen interrupt
* chip. When an event is received, it is mapped to an irq and sent
* through the normal interrupt processing path.
*
* There are four kinds of events which can be mapped to an event
* channel:
*
* 1. Inter-domain notifications. This includes all the virtual
* device events, since they're driven by front-ends in another domain
* (typically dom0).
* 2. VIRQs, typically used for timers. These are per-cpu events.
* 3. IPIs.
* 4. PIRQs - Hardware interrupts.
*
* Jeremy Fitzhardinge <[email protected]>, XenSource Inc, 2007
*/
#define pr_fmt(fmt) "xen:" KBUILD_MODNAME ": " fmt
#include <linux/linkage.h>
#include <linux/interrupt.h>
#include <linux/irq.h>
#include <linux/module.h>
#include <linux/string.h>
#include <linux/bootmem.h>
#include <linux/slab.h>
#include <linux/irqnr.h>
#include <linux/pci.h>
#ifdef CONFIG_X86
#include <asm/desc.h>
#include <asm/ptrace.h>
#include <asm/irq.h>
#include <asm/idle.h>
#include <asm/io_apic.h>
#include <asm/xen/pci.h>
#include <xen/page.h>
#endif
#include <asm/sync_bitops.h>
#include <asm/xen/hypercall.h>
#include <asm/xen/hypervisor.h>
#include <xen/xen.h>
#include <xen/hvm.h>
#include <xen/xen-ops.h>
#include <xen/events.h>
#include <xen/interface/xen.h>
#include <xen/interface/event_channel.h>
#include <xen/interface/hvm/hvm_op.h>
#include <xen/interface/hvm/params.h>
#include <xen/interface/physdev.h>
#include <xen/interface/sched.h>
#include <xen/interface/vcpu.h>
#include <asm/hw_irq.h>
#include "events_internal.h"
const struct evtchn_ops *evtchn_ops;
/*
* This lock protects updates to the following mapping and reference-count
* arrays. The lock does not need to be acquired to read the mapping tables.
*/
static DEFINE_MUTEX(irq_mapping_update_lock);
static LIST_HEAD(xen_irq_list_head);
/* IRQ <-> VIRQ mapping. */
static DEFINE_PER_CPU(int [NR_VIRQS], virq_to_irq) = {[0 ... NR_VIRQS-1] = -1};
/* IRQ <-> IPI mapping */
static DEFINE_PER_CPU(int [XEN_NR_IPIS], ipi_to_irq) = {[0 ... XEN_NR_IPIS-1] = -1};
int **evtchn_to_irq;
#ifdef CONFIG_X86
static unsigned long *pirq_eoi_map;
#endif
static bool (*pirq_needs_eoi)(unsigned irq);
#define EVTCHN_ROW(e) (e / (PAGE_SIZE/sizeof(**evtchn_to_irq)))
#define EVTCHN_COL(e) (e % (PAGE_SIZE/sizeof(**evtchn_to_irq)))
#define EVTCHN_PER_ROW (PAGE_SIZE / sizeof(**evtchn_to_irq))
/* Xen will never allocate port zero for any purpose. */
#define VALID_EVTCHN(chn) ((chn) != 0)
static struct irq_chip xen_dynamic_chip;
static struct irq_chip xen_percpu_chip;
static struct irq_chip xen_pirq_chip;
static void enable_dynirq(struct irq_data *data);
static void disable_dynirq(struct irq_data *data);
static void clear_evtchn_to_irq_row(unsigned row)
{
unsigned col;
for (col = 0; col < EVTCHN_PER_ROW; col++)
evtchn_to_irq[row][col] = -1;
}
static void clear_evtchn_to_irq_all(void)
{
unsigned row;
for (row = 0; row < EVTCHN_ROW(xen_evtchn_max_channels()); row++) {
if (evtchn_to_irq[row] == NULL)
continue;
clear_evtchn_to_irq_row(row);
}
}
static int set_evtchn_to_irq(unsigned evtchn, unsigned irq)
{
unsigned row;
unsigned col;
if (evtchn >= xen_evtchn_max_channels())
return -EINVAL;
row = EVTCHN_ROW(evtchn);
col = EVTCHN_COL(evtchn);
if (evtchn_to_irq[row] == NULL) {
/* Unallocated irq entries return -1 anyway */
if (irq == -1)
return 0;
evtchn_to_irq[row] = (int *)get_zeroed_page(GFP_KERNEL);
if (evtchn_to_irq[row] == NULL)
return -ENOMEM;
clear_evtchn_to_irq_row(row);
}
evtchn_to_irq[EVTCHN_ROW(evtchn)][EVTCHN_COL(evtchn)] = irq;
return 0;
}
int get_evtchn_to_irq(unsigned evtchn)
{
if (evtchn >= xen_evtchn_max_channels())
return -1;
if (evtchn_to_irq[EVTCHN_ROW(evtchn)] == NULL)
return -1;
return evtchn_to_irq[EVTCHN_ROW(evtchn)][EVTCHN_COL(evtchn)];
}
/* Get info for IRQ */
struct irq_info *info_for_irq(unsigned irq)
{
return irq_get_handler_data(irq);
}
/* Constructors for packed IRQ information. */
static int xen_irq_info_common_setup(struct irq_info *info,
unsigned irq,
enum xen_irq_type type,
unsigned evtchn,
unsigned short cpu)
{
int ret;
BUG_ON(info->type != IRQT_UNBOUND && info->type != type);
info->type = type;
info->irq = irq;
info->evtchn = evtchn;
info->cpu = cpu;
ret = set_evtchn_to_irq(evtchn, irq);
if (ret < 0)
return ret;
irq_clear_status_flags(irq, IRQ_NOREQUEST|IRQ_NOAUTOEN);
return xen_evtchn_port_setup(info);
}
static int xen_irq_info_evtchn_setup(unsigned irq,
unsigned evtchn)
{
struct irq_info *info = info_for_irq(irq);
return xen_irq_info_common_setup(info, irq, IRQT_EVTCHN, evtchn, 0);
}
static int xen_irq_info_ipi_setup(unsigned cpu,
unsigned irq,
unsigned evtchn,
enum ipi_vector ipi)
{
struct irq_info *info = info_for_irq(irq);
info->u.ipi = ipi;
per_cpu(ipi_to_irq, cpu)[ipi] = irq;
return xen_irq_info_common_setup(info, irq, IRQT_IPI, evtchn, 0);
}
static int xen_irq_info_virq_setup(unsigned cpu,
unsigned irq,
unsigned evtchn,
unsigned virq)
{
struct irq_info *info = info_for_irq(irq);
info->u.virq = virq;
per_cpu(virq_to_irq, cpu)[virq] = irq;
return xen_irq_info_common_setup(info, irq, IRQT_VIRQ, evtchn, 0);
}
static int xen_irq_info_pirq_setup(unsigned irq,
unsigned evtchn,
unsigned pirq,
unsigned gsi,
uint16_t domid,
unsigned char flags)
{
struct irq_info *info = info_for_irq(irq);
info->u.pirq.pirq = pirq;
info->u.pirq.gsi = gsi;
info->u.pirq.domid = domid;
info->u.pirq.flags = flags;
return xen_irq_info_common_setup(info, irq, IRQT_PIRQ, evtchn, 0);
}
static void xen_irq_info_cleanup(struct irq_info *info)
{
set_evtchn_to_irq(info->evtchn, -1);
info->evtchn = 0;
}
/*
* Accessors for packed IRQ information.
*/
unsigned int evtchn_from_irq(unsigned irq)
{
if (unlikely(WARN(irq >= nr_irqs, "Invalid irq %d!\n", irq)))
return 0;
return info_for_irq(irq)->evtchn;
}
unsigned irq_from_evtchn(unsigned int evtchn)
{
return get_evtchn_to_irq(evtchn);
}
EXPORT_SYMBOL_GPL(irq_from_evtchn);
int irq_from_virq(unsigned int cpu, unsigned int virq)
{
return per_cpu(virq_to_irq, cpu)[virq];
}
static enum ipi_vector ipi_from_irq(unsigned irq)
{
struct irq_info *info = info_for_irq(irq);
BUG_ON(info == NULL);
BUG_ON(info->type != IRQT_IPI);
return info->u.ipi;
}
static unsigned virq_from_irq(unsigned irq)
{
struct irq_info *info = info_for_irq(irq);
BUG_ON(info == NULL);
BUG_ON(info->type != IRQT_VIRQ);
return info->u.virq;
}
static unsigned pirq_from_irq(unsigned irq)
{
struct irq_info *info = info_for_irq(irq);
BUG_ON(info == NULL);
BUG_ON(info->type != IRQT_PIRQ);
return info->u.pirq.pirq;
}
static enum xen_irq_type type_from_irq(unsigned irq)
{
return info_for_irq(irq)->type;
}
unsigned cpu_from_irq(unsigned irq)
{
return info_for_irq(irq)->cpu;
}
unsigned int cpu_from_evtchn(unsigned int evtchn)
{
int irq = get_evtchn_to_irq(evtchn);
unsigned ret = 0;
if (irq != -1)
ret = cpu_from_irq(irq);
return ret;
}
#ifdef CONFIG_X86
static bool pirq_check_eoi_map(unsigned irq)
{
return test_bit(pirq_from_irq(irq), pirq_eoi_map);
}
#endif
static bool pirq_needs_eoi_flag(unsigned irq)
{
struct irq_info *info = info_for_irq(irq);
BUG_ON(info->type != IRQT_PIRQ);
return info->u.pirq.flags & PIRQ_NEEDS_EOI;
}
static void bind_evtchn_to_cpu(unsigned int chn, unsigned int cpu)
{
int irq = get_evtchn_to_irq(chn);
struct irq_info *info = info_for_irq(irq);
BUG_ON(irq == -1);
#ifdef CONFIG_SMP
cpumask_copy(irq_get_affinity_mask(irq), cpumask_of(cpu));
#endif
xen_evtchn_port_bind_to_cpu(info, cpu);
info->cpu = cpu;
}
static void xen_evtchn_mask_all(void)
{
unsigned int evtchn;
for (evtchn = 0; evtchn < xen_evtchn_nr_channels(); evtchn++)
mask_evtchn(evtchn);
}
/**
* notify_remote_via_irq - send event to remote end of event channel via irq
* @irq: irq of event channel to send event to
*
* Unlike notify_remote_via_evtchn(), this is safe to use across
* save/restore. Notifications on a broken connection are silently
* dropped.
*/
void notify_remote_via_irq(int irq)
{
int evtchn = evtchn_from_irq(irq);
if (VALID_EVTCHN(evtchn))
notify_remote_via_evtchn(evtchn);
}
EXPORT_SYMBOL_GPL(notify_remote_via_irq);
static void xen_irq_init(unsigned irq)
{
struct irq_info *info;
#ifdef CONFIG_SMP
/* By default all event channels notify CPU#0. */
cpumask_copy(irq_get_affinity_mask(irq), cpumask_of(0));
#endif
info = kzalloc(sizeof(*info), GFP_KERNEL);
if (info == NULL)
panic("Unable to allocate metadata for IRQ%d\n", irq);
info->type = IRQT_UNBOUND;
info->refcnt = -1;
irq_set_handler_data(irq, info);
list_add_tail(&info->list, &xen_irq_list_head);
}
static int __must_check xen_allocate_irqs_dynamic(int nvec)
{
int i, irq = irq_alloc_descs(-1, 0, nvec, -1);
if (irq >= 0) {
for (i = 0; i < nvec; i++)
xen_irq_init(irq + i);
}
return irq;
}
static inline int __must_check xen_allocate_irq_dynamic(void)
{
return xen_allocate_irqs_dynamic(1);
}
static int __must_check xen_allocate_irq_gsi(unsigned gsi)
{
int irq;
/*
* A PV guest has no concept of a GSI (since it has no ACPI
* nor access to/knowledge of the physical APICs). Therefore
* all IRQs are dynamically allocated from the entire IRQ
* space.
*/
if (xen_pv_domain() && !xen_initial_domain())
return xen_allocate_irq_dynamic();
/* Legacy IRQ descriptors are already allocated by the arch. */
if (gsi < NR_IRQS_LEGACY)
irq = gsi;
else
irq = irq_alloc_desc_at(gsi, -1);
xen_irq_init(irq);
return irq;
}
static void xen_free_irq(unsigned irq)
{
struct irq_info *info = irq_get_handler_data(irq);
if (WARN_ON(!info))
return;
list_del(&info->list);
irq_set_handler_data(irq, NULL);
WARN_ON(info->refcnt > 0);
kfree(info);
/* Legacy IRQ descriptors are managed by the arch. */
if (irq < NR_IRQS_LEGACY)
return;
irq_free_desc(irq);
}
static void xen_evtchn_close(unsigned int port)
{
struct evtchn_close close;
close.port = port;
if (HYPERVISOR_event_channel_op(EVTCHNOP_close, &close) != 0)
BUG();
}
static void pirq_query_unmask(int irq)
{
struct physdev_irq_status_query irq_status;
struct irq_info *info = info_for_irq(irq);
BUG_ON(info->type != IRQT_PIRQ);
irq_status.irq = pirq_from_irq(irq);
if (HYPERVISOR_physdev_op(PHYSDEVOP_irq_status_query, &irq_status))
irq_status.flags = 0;
info->u.pirq.flags &= ~PIRQ_NEEDS_EOI;
if (irq_status.flags & XENIRQSTAT_needs_eoi)
info->u.pirq.flags |= PIRQ_NEEDS_EOI;
}
static void eoi_pirq(struct irq_data *data)
{
int evtchn = evtchn_from_irq(data->irq);
struct physdev_eoi eoi = { .irq = pirq_from_irq(data->irq) };
int rc = 0;
irq_move_irq(data);
if (VALID_EVTCHN(evtchn))
clear_evtchn(evtchn);
if (pirq_needs_eoi(data->irq)) {
rc = HYPERVISOR_physdev_op(PHYSDEVOP_eoi, &eoi);
WARN_ON(rc);
}
}
static void mask_ack_pirq(struct irq_data *data)
{
disable_dynirq(data);
eoi_pirq(data);
}
static unsigned int __startup_pirq(unsigned int irq)
{
struct evtchn_bind_pirq bind_pirq;
struct irq_info *info = info_for_irq(irq);
int evtchn = evtchn_from_irq(irq);
int rc;
BUG_ON(info->type != IRQT_PIRQ);
if (VALID_EVTCHN(evtchn))
goto out;
bind_pirq.pirq = pirq_from_irq(irq);
/* NB. We are happy to share unless we are probing. */
bind_pirq.flags = info->u.pirq.flags & PIRQ_SHAREABLE ?
BIND_PIRQ__WILL_SHARE : 0;
rc = HYPERVISOR_event_channel_op(EVTCHNOP_bind_pirq, &bind_pirq);
if (rc != 0) {
pr_warn("Failed to obtain physical IRQ %d\n", irq);
return 0;
}
evtchn = bind_pirq.port;
pirq_query_unmask(irq);
rc = set_evtchn_to_irq(evtchn, irq);
if (rc)
goto err;
info->evtchn = evtchn;
bind_evtchn_to_cpu(evtchn, 0);
rc = xen_evtchn_port_setup(info);
if (rc)
goto err;
out:
unmask_evtchn(evtchn);
eoi_pirq(irq_get_irq_data(irq));
return 0;
err:
pr_err("irq%d: Failed to set port to irq mapping (%d)\n", irq, rc);
xen_evtchn_close(evtchn);
return 0;
}
static unsigned int startup_pirq(struct irq_data *data)
{
return __startup_pirq(data->irq);
}
static void shutdown_pirq(struct irq_data *data)
{
unsigned int irq = data->irq;
struct irq_info *info = info_for_irq(irq);
unsigned evtchn = evtchn_from_irq(irq);
BUG_ON(info->type != IRQT_PIRQ);
if (!VALID_EVTCHN(evtchn))
return;
mask_evtchn(evtchn);
xen_evtchn_close(evtchn);
xen_irq_info_cleanup(info);
}
static void enable_pirq(struct irq_data *data)
{
startup_pirq(data);
}
static void disable_pirq(struct irq_data *data)
{
disable_dynirq(data);
}
int xen_irq_from_gsi(unsigned gsi)
{
struct irq_info *info;
list_for_each_entry(info, &xen_irq_list_head, list) {
if (info->type != IRQT_PIRQ)
continue;
if (info->u.pirq.gsi == gsi)
return info->irq;
}
return -1;
}
EXPORT_SYMBOL_GPL(xen_irq_from_gsi);
static void __unbind_from_irq(unsigned int irq)
{
int evtchn = evtchn_from_irq(irq);
struct irq_info *info = irq_get_handler_data(irq);
if (info->refcnt > 0) {
info->refcnt--;
if (info->refcnt != 0)
return;
}
if (VALID_EVTCHN(evtchn)) {
unsigned int cpu = cpu_from_irq(irq);
xen_evtchn_close(evtchn);
switch (type_from_irq(irq)) {
case IRQT_VIRQ:
per_cpu(virq_to_irq, cpu)[virq_from_irq(irq)] = -1;
break;
case IRQT_IPI:
per_cpu(ipi_to_irq, cpu)[ipi_from_irq(irq)] = -1;
break;
default:
break;
}
xen_irq_info_cleanup(info);
}
BUG_ON(info_for_irq(irq)->type == IRQT_UNBOUND);
xen_free_irq(irq);
}
/*
* Do not make any assumptions regarding the relationship between the
* IRQ number returned here and the Xen pirq argument.
*
* Note: We don't assign an event channel until the irq actually started
* up. Return an existing irq if we've already got one for the gsi.
*
* Shareable implies level triggered, not shareable implies edge
* triggered here.
*/
int xen_bind_pirq_gsi_to_irq(unsigned gsi,
unsigned pirq, int shareable, char *name)
{
int irq = -1;
struct physdev_irq irq_op;
int ret;
mutex_lock(&irq_mapping_update_lock);
irq = xen_irq_from_gsi(gsi);
if (irq != -1) {
pr_info("%s: returning irq %d for gsi %u\n",
__func__, irq, gsi);
goto out;
}
irq = xen_allocate_irq_gsi(gsi);
if (irq < 0)
goto out;
irq_op.irq = irq;
irq_op.vector = 0;
/* Only the privileged domain can do this. For non-priv, the pcifront
* driver provides a PCI bus that does the call to do exactly
* this in the priv domain. */
if (xen_initial_domain() &&
HYPERVISOR_physdev_op(PHYSDEVOP_alloc_irq_vector, &irq_op)) {
xen_free_irq(irq);
irq = -ENOSPC;
goto out;
}
ret = xen_irq_info_pirq_setup(irq, 0, pirq, gsi, DOMID_SELF,
shareable ? PIRQ_SHAREABLE : 0);
if (ret < 0) {
__unbind_from_irq(irq);
irq = ret;
goto out;
}
pirq_query_unmask(irq);
/* We try to use the handler with the appropriate semantic for the
* type of interrupt: if the interrupt is an edge triggered
* interrupt we use handle_edge_irq.
*
* On the other hand if the interrupt is level triggered we use
* handle_fasteoi_irq like the native code does for this kind of
* interrupts.
*
* Depending on the Xen version, pirq_needs_eoi might return true
* not only for level triggered interrupts but for edge triggered
* interrupts too. In any case Xen always honors the eoi mechanism,
* not injecting any more pirqs of the same kind if the first one
* hasn't received an eoi yet. Therefore using the fasteoi handler
* is the right choice either way.
*/
if (shareable)
irq_set_chip_and_handler_name(irq, &xen_pirq_chip,
handle_fasteoi_irq, name);
else
irq_set_chip_and_handler_name(irq, &xen_pirq_chip,
handle_edge_irq, name);
out:
mutex_unlock(&irq_mapping_update_lock);
return irq;
}
#ifdef CONFIG_PCI_MSI
int xen_allocate_pirq_msi(struct pci_dev *dev, struct msi_desc *msidesc)
{
int rc;
struct physdev_get_free_pirq op_get_free_pirq;
op_get_free_pirq.type = MAP_PIRQ_TYPE_MSI;
rc = HYPERVISOR_physdev_op(PHYSDEVOP_get_free_pirq, &op_get_free_pirq);
WARN_ONCE(rc == -ENOSYS,
"hypervisor does not support the PHYSDEVOP_get_free_pirq interface\n");
return rc ? -1 : op_get_free_pirq.pirq;
}
int xen_bind_pirq_msi_to_irq(struct pci_dev *dev, struct msi_desc *msidesc,
int pirq, int nvec, const char *name, domid_t domid)
{
int i, irq, ret;
mutex_lock(&irq_mapping_update_lock);
irq = xen_allocate_irqs_dynamic(nvec);
if (irq < 0)
goto out;
for (i = 0; i < nvec; i++) {
irq_set_chip_and_handler_name(irq + i, &xen_pirq_chip, handle_edge_irq, name);
ret = xen_irq_info_pirq_setup(irq + i, 0, pirq + i, 0, domid,
i == 0 ? 0 : PIRQ_MSI_GROUP);
if (ret < 0)
goto error_irq;
}
ret = irq_set_msi_desc(irq, msidesc);
if (ret < 0)
goto error_irq;
out:
mutex_unlock(&irq_mapping_update_lock);
return irq;
error_irq:
for (; i >= 0; i--)
__unbind_from_irq(irq + i);
mutex_unlock(&irq_mapping_update_lock);
return ret;
}
#endif
int xen_destroy_irq(int irq)
{
struct physdev_unmap_pirq unmap_irq;
struct irq_info *info = info_for_irq(irq);
int rc = -ENOENT;
mutex_lock(&irq_mapping_update_lock);
/*
* If trying to remove a vector in a MSI group different
* than the first one skip the PIRQ unmap unless this vector
* is the first one in the group.
*/
if (xen_initial_domain() && !(info->u.pirq.flags & PIRQ_MSI_GROUP)) {
unmap_irq.pirq = info->u.pirq.pirq;
unmap_irq.domid = info->u.pirq.domid;
rc = HYPERVISOR_physdev_op(PHYSDEVOP_unmap_pirq, &unmap_irq);
/* If another domain quits without making the pci_disable_msix
* call, the Xen hypervisor takes care of freeing the PIRQs
* (free_domain_pirqs).
*/
if ((rc == -ESRCH && info->u.pirq.domid != DOMID_SELF))
pr_info("domain %d does not have %d anymore\n",
info->u.pirq.domid, info->u.pirq.pirq);
else if (rc) {
pr_warn("unmap irq failed %d\n", rc);
goto out;
}
}
xen_free_irq(irq);
out:
mutex_unlock(&irq_mapping_update_lock);
return rc;
}
int xen_irq_from_pirq(unsigned pirq)
{
int irq;
struct irq_info *info;
mutex_lock(&irq_mapping_update_lock);
list_for_each_entry(info, &xen_irq_list_head, list) {
if (info->type != IRQT_PIRQ)
continue;
irq = info->irq;
if (info->u.pirq.pirq == pirq)
goto out;
}
irq = -1;
out:
mutex_unlock(&irq_mapping_update_lock);
return irq;
}
int xen_pirq_from_irq(unsigned irq)
{
return pirq_from_irq(irq);
}
EXPORT_SYMBOL_GPL(xen_pirq_from_irq);
int bind_evtchn_to_irq(unsigned int evtchn)
{
int irq;
int ret;
if (evtchn >= xen_evtchn_max_channels())
return -ENOMEM;
mutex_lock(&irq_mapping_update_lock);
irq = get_evtchn_to_irq(evtchn);
if (irq == -1) {
irq = xen_allocate_irq_dynamic();
if (irq < 0)
goto out;
irq_set_chip_and_handler_name(irq, &xen_dynamic_chip,
handle_edge_irq, "event");
ret = xen_irq_info_evtchn_setup(irq, evtchn);
if (ret < 0) {
__unbind_from_irq(irq);
irq = ret;
goto out;
}
/* New interdomain events are bound to VCPU 0. */
bind_evtchn_to_cpu(evtchn, 0);
} else {
struct irq_info *info = info_for_irq(irq);
WARN_ON(info == NULL || info->type != IRQT_EVTCHN);
}
out:
mutex_unlock(&irq_mapping_update_lock);
return irq;
}
EXPORT_SYMBOL_GPL(bind_evtchn_to_irq);
static int bind_ipi_to_irq(unsigned int ipi, unsigned int cpu)
{
struct evtchn_bind_ipi bind_ipi;
int evtchn, irq;
int ret;
mutex_lock(&irq_mapping_update_lock);
irq = per_cpu(ipi_to_irq, cpu)[ipi];
if (irq == -1) {
irq = xen_allocate_irq_dynamic();
if (irq < 0)
goto out;
irq_set_chip_and_handler_name(irq, &xen_percpu_chip,
handle_percpu_irq, "ipi");
bind_ipi.vcpu = cpu;
if (HYPERVISOR_event_channel_op(EVTCHNOP_bind_ipi,
&bind_ipi) != 0)
BUG();
evtchn = bind_ipi.port;
ret = xen_irq_info_ipi_setup(cpu, irq, evtchn, ipi);
if (ret < 0) {
__unbind_from_irq(irq);
irq = ret;
goto out;
}
bind_evtchn_to_cpu(evtchn, cpu);
} else {
struct irq_info *info = info_for_irq(irq);
WARN_ON(info == NULL || info->type != IRQT_IPI);
}
out:
mutex_unlock(&irq_mapping_update_lock);
return irq;
}
int bind_interdomain_evtchn_to_irq(unsigned int remote_domain,
unsigned int remote_port)
{
struct evtchn_bind_interdomain bind_interdomain;
int err;
bind_interdomain.remote_dom = remote_domain;
bind_interdomain.remote_port = remote_port;
err = HYPERVISOR_event_channel_op(EVTCHNOP_bind_interdomain,
&bind_interdomain);
return err ? : bind_evtchn_to_irq(bind_interdomain.local_port);
}
EXPORT_SYMBOL_GPL(bind_interdomain_evtchn_to_irq);
static int find_virq(unsigned int virq, unsigned int cpu)
{
struct evtchn_status status;
int port, rc = -ENOENT;
memset(&status, 0, sizeof(status));
for (port = 0; port < xen_evtchn_max_channels(); port++) {
status.dom = DOMID_SELF;
status.port = port;
rc = HYPERVISOR_event_channel_op(EVTCHNOP_status, &status);
if (rc < 0)
continue;
if (status.status != EVTCHNSTAT_virq)
continue;
if (status.u.virq == virq && status.vcpu == cpu) {
rc = port;
break;
}
}
return rc;
}
/**
* xen_evtchn_nr_channels - number of usable event channel ports
*
* This may be less than the maximum supported by the current
* hypervisor ABI. Use xen_evtchn_max_channels() for the maximum
* supported.
*/
unsigned xen_evtchn_nr_channels(void)
{
return evtchn_ops->nr_channels();
}
EXPORT_SYMBOL_GPL(xen_evtchn_nr_channels);
int bind_virq_to_irq(unsigned int virq, unsigned int cpu, bool percpu)
{
struct evtchn_bind_virq bind_virq;
int evtchn, irq, ret;
mutex_lock(&irq_mapping_update_lock);
irq = per_cpu(virq_to_irq, cpu)[virq];
if (irq == -1) {
irq = xen_allocate_irq_dynamic();
if (irq < 0)
goto out;
if (percpu)
irq_set_chip_and_handler_name(irq, &xen_percpu_chip,
handle_percpu_irq, "virq");
else
irq_set_chip_and_handler_name(irq, &xen_dynamic_chip,
handle_edge_irq, "virq");
bind_virq.virq = virq;
bind_virq.vcpu = cpu;
ret = HYPERVISOR_event_channel_op(EVTCHNOP_bind_virq,
&bind_virq);
if (ret == 0)
evtchn = bind_virq.port;
else {
if (ret == -EEXIST)
ret = find_virq(virq, cpu);
BUG_ON(ret < 0);
evtchn = ret;
}
ret = xen_irq_info_virq_setup(cpu, irq, evtchn, virq);
if (ret < 0) {
__unbind_from_irq(irq);
irq = ret;
goto out;
}
bind_evtchn_to_cpu(evtchn, cpu);
} else {
struct irq_info *info = info_for_irq(irq);
WARN_ON(info == NULL || info->type != IRQT_VIRQ);
}
out:
mutex_unlock(&irq_mapping_update_lock);
return irq;
}
static void unbind_from_irq(unsigned int irq)
{
mutex_lock(&irq_mapping_update_lock);
__unbind_from_irq(irq);
mutex_unlock(&irq_mapping_update_lock);
}
int bind_evtchn_to_irqhandler(unsigned int evtchn,
irq_handler_t handler,
unsigned long irqflags,
const char *devname, void *dev_id)
{
int irq, retval;
irq = bind_evtchn_to_irq(evtchn);
if (irq < 0)
return irq;
retval = request_irq(irq, handler, irqflags, devname, dev_id);
if (retval != 0) {
unbind_from_irq(irq);
return retval;
}
return irq;
}
EXPORT_SYMBOL_GPL(bind_evtchn_to_irqhandler);
int bind_interdomain_evtchn_to_irqhandler(unsigned int remote_domain,
unsigned int remote_port,
irq_handler_t handler,
unsigned long irqflags,
const char *devname,
void *dev_id)
{
int irq, retval;
irq = bind_interdomain_evtchn_to_irq(remote_domain, remote_port);
if (irq < 0)
return irq;
retval = request_irq(irq, handler, irqflags, devname, dev_id);
if (retval != 0) {
unbind_from_irq(irq);
return retval;
}
return irq;
}
EXPORT_SYMBOL_GPL(bind_interdomain_evtchn_to_irqhandler);
int bind_virq_to_irqhandler(unsigned int virq, unsigned int cpu,
irq_handler_t handler,
unsigned long irqflags, const char *devname, void *dev_id)
{
int irq, retval;
irq = bind_virq_to_irq(virq, cpu, irqflags & IRQF_PERCPU);
if (irq < 0)
return irq;
retval = request_irq(irq, handler, irqflags, devname, dev_id);
if (retval != 0) {
unbind_from_irq(irq);
return retval;
}
return irq;
}
EXPORT_SYMBOL_GPL(bind_virq_to_irqhandler);
int bind_ipi_to_irqhandler(enum ipi_vector ipi,
unsigned int cpu,
irq_handler_t handler,
unsigned long irqflags,
const char *devname,
void *dev_id)
{
int irq, retval;
irq = bind_ipi_to_irq(ipi, cpu);
if (irq < 0)
return irq;
irqflags |= IRQF_NO_SUSPEND | IRQF_FORCE_RESUME | IRQF_EARLY_RESUME;
retval = request_irq(irq, handler, irqflags, devname, dev_id);
if (retval != 0) {
unbind_from_irq(irq);
return retval;
}
return irq;
}
void unbind_from_irqhandler(unsigned int irq, void *dev_id)
{
struct irq_info *info = irq_get_handler_data(irq);
if (WARN_ON(!info))
return;
free_irq(irq, dev_id);
unbind_from_irq(irq);
}
EXPORT_SYMBOL_GPL(unbind_from_irqhandler);
/**
* xen_set_irq_priority() - set an event channel priority.
* @irq:irq bound to an event channel.
* @priority: priority between XEN_IRQ_PRIORITY_MAX and XEN_IRQ_PRIORITY_MIN.
*/
int xen_set_irq_priority(unsigned irq, unsigned priority)
{
struct evtchn_set_priority set_priority;
set_priority.port = evtchn_from_irq(irq);
set_priority.priority = priority;
return HYPERVISOR_event_channel_op(EVTCHNOP_set_priority,
&set_priority);
}
EXPORT_SYMBOL_GPL(xen_set_irq_priority);
int evtchn_make_refcounted(unsigned int evtchn)
{
int irq = get_evtchn_to_irq(evtchn);
struct irq_info *info;
if (irq == -1)
return -ENOENT;
info = irq_get_handler_data(irq);
if (!info)
return -ENOENT;
WARN_ON(info->refcnt != -1);
info->refcnt = 1;
return 0;
}
EXPORT_SYMBOL_GPL(evtchn_make_refcounted);
int evtchn_get(unsigned int evtchn)
{
int irq;
struct irq_info *info;
int err = -ENOENT;
if (evtchn >= xen_evtchn_max_channels())
return -EINVAL;
mutex_lock(&irq_mapping_update_lock);
irq = get_evtchn_to_irq(evtchn);
if (irq == -1)
goto done;
info = irq_get_handler_data(irq);
if (!info)
goto done;
err = -EINVAL;
if (info->refcnt <= 0)
goto done;
info->refcnt++;
err = 0;
done:
mutex_unlock(&irq_mapping_update_lock);
return err;
}
EXPORT_SYMBOL_GPL(evtchn_get);
void evtchn_put(unsigned int evtchn)
{
int irq = get_evtchn_to_irq(evtchn);
if (WARN_ON(irq == -1))
return;
unbind_from_irq(irq);
}
EXPORT_SYMBOL_GPL(evtchn_put);
void xen_send_IPI_one(unsigned int cpu, enum ipi_vector vector)
{
int irq;
#ifdef CONFIG_X86
if (unlikely(vector == XEN_NMI_VECTOR)) {
int rc = HYPERVISOR_vcpu_op(VCPUOP_send_nmi, cpu, NULL);
if (rc < 0)
printk(KERN_WARNING "Sending nmi to CPU%d failed (rc:%d)\n", cpu, rc);
return;
}
#endif
irq = per_cpu(ipi_to_irq, cpu)[vector];
BUG_ON(irq < 0);
notify_remote_via_irq(irq);
}
static DEFINE_PER_CPU(unsigned, xed_nesting_count);
static void __xen_evtchn_do_upcall(void)
{
struct vcpu_info *vcpu_info = __this_cpu_read(xen_vcpu);
int cpu = get_cpu();
unsigned count;
do {
vcpu_info->evtchn_upcall_pending = 0;
if (__this_cpu_inc_return(xed_nesting_count) - 1)
goto out;
xen_evtchn_handle_events(cpu);
BUG_ON(!irqs_disabled());
count = __this_cpu_read(xed_nesting_count);
__this_cpu_write(xed_nesting_count, 0);
} while (count != 1 || vcpu_info->evtchn_upcall_pending);
out:
put_cpu();
}
void xen_evtchn_do_upcall(struct pt_regs *regs)
{
struct pt_regs *old_regs = set_irq_regs(regs);
irq_enter();
#ifdef CONFIG_X86
exit_idle();
inc_irq_stat(irq_hv_callback_count);
#endif
__xen_evtchn_do_upcall();
irq_exit();
set_irq_regs(old_regs);
}
void xen_hvm_evtchn_do_upcall(void)
{
__xen_evtchn_do_upcall();
}
EXPORT_SYMBOL_GPL(xen_hvm_evtchn_do_upcall);
/* Rebind a new event channel to an existing irq. */
void rebind_evtchn_irq(int evtchn, int irq)
{
struct irq_info *info = info_for_irq(irq);
if (WARN_ON(!info))
return;
/* Make sure the irq is masked, since the new event channel
will also be masked. */
disable_irq(irq);
mutex_lock(&irq_mapping_update_lock);
/* After resume the irq<->evtchn mappings are all cleared out */
BUG_ON(get_evtchn_to_irq(evtchn) != -1);
/* Expect irq to have been bound before,
so there should be a proper type */
BUG_ON(info->type == IRQT_UNBOUND);
(void)xen_irq_info_evtchn_setup(irq, evtchn);
mutex_unlock(&irq_mapping_update_lock);
bind_evtchn_to_cpu(evtchn, info->cpu);
/* This will be deferred until interrupt is processed */
irq_set_affinity(irq, cpumask_of(info->cpu));
/* Unmask the event channel. */
enable_irq(irq);
}
/* Rebind an evtchn so that it gets delivered to a specific cpu */
static int rebind_irq_to_cpu(unsigned irq, unsigned tcpu)
{
struct evtchn_bind_vcpu bind_vcpu;
int evtchn = evtchn_from_irq(irq);
int masked;
if (!VALID_EVTCHN(evtchn))
return -1;
if (!xen_support_evtchn_rebind())
return -1;
/* Send future instances of this interrupt to other vcpu. */
bind_vcpu.port = evtchn;
bind_vcpu.vcpu = tcpu;
/*
* Mask the event while changing the VCPU binding to prevent
* it being delivered on an unexpected VCPU.
*/
masked = test_and_set_mask(evtchn);
/*
* If this fails, it usually just indicates that we're dealing with a
* virq or IPI channel, which don't actually need to be rebound. Ignore
* it, but don't do the xenlinux-level rebind in that case.
*/
if (HYPERVISOR_event_channel_op(EVTCHNOP_bind_vcpu, &bind_vcpu) >= 0)
bind_evtchn_to_cpu(evtchn, tcpu);
if (!masked)
unmask_evtchn(evtchn);
return 0;
}
static int set_affinity_irq(struct irq_data *data, const struct cpumask *dest,
bool force)
{
unsigned tcpu = cpumask_first_and(dest, cpu_online_mask);
return rebind_irq_to_cpu(data->irq, tcpu);
}
static void enable_dynirq(struct irq_data *data)
{
int evtchn = evtchn_from_irq(data->irq);
if (VALID_EVTCHN(evtchn))
unmask_evtchn(evtchn);
}
static void disable_dynirq(struct irq_data *data)
{
int evtchn = evtchn_from_irq(data->irq);
if (VALID_EVTCHN(evtchn))
mask_evtchn(evtchn);
}
static void ack_dynirq(struct irq_data *data)
{
int evtchn = evtchn_from_irq(data->irq);
irq_move_irq(data);
if (VALID_EVTCHN(evtchn))
clear_evtchn(evtchn);
}
static void mask_ack_dynirq(struct irq_data *data)
{
disable_dynirq(data);
ack_dynirq(data);
}
static int retrigger_dynirq(struct irq_data *data)
{
unsigned int evtchn = evtchn_from_irq(data->irq);
int masked;
if (!VALID_EVTCHN(evtchn))
return 0;
masked = test_and_set_mask(evtchn);
set_evtchn(evtchn);
if (!masked)
unmask_evtchn(evtchn);
return 1;
}
static void restore_pirqs(void)
{
int pirq, rc, irq, gsi;
struct physdev_map_pirq map_irq;
struct irq_info *info;
list_for_each_entry(info, &xen_irq_list_head, list) {
if (info->type != IRQT_PIRQ)
continue;
pirq = info->u.pirq.pirq;
gsi = info->u.pirq.gsi;
irq = info->irq;
/* save/restore of PT devices doesn't work, so at this point the
* only devices present are GSI based emulated devices */
if (!gsi)
continue;
map_irq.domid = DOMID_SELF;
map_irq.type = MAP_PIRQ_TYPE_GSI;
map_irq.index = gsi;
map_irq.pirq = pirq;
rc = HYPERVISOR_physdev_op(PHYSDEVOP_map_pirq, &map_irq);
if (rc) {
pr_warn("xen map irq failed gsi=%d irq=%d pirq=%d rc=%d\n",
gsi, irq, pirq, rc);
xen_free_irq(irq);
continue;
}
printk(KERN_DEBUG "xen: --> irq=%d, pirq=%d\n", irq, map_irq.pirq);
__startup_pirq(irq);
}
}
static void restore_cpu_virqs(unsigned int cpu)
{
struct evtchn_bind_virq bind_virq;
int virq, irq, evtchn;
for (virq = 0; virq < NR_VIRQS; virq++) {
if ((irq = per_cpu(virq_to_irq, cpu)[virq]) == -1)
continue;
BUG_ON(virq_from_irq(irq) != virq);
/* Get a new binding from Xen. */
bind_virq.virq = virq;
bind_virq.vcpu = cpu;
if (HYPERVISOR_event_channel_op(EVTCHNOP_bind_virq,
&bind_virq) != 0)
BUG();
evtchn = bind_virq.port;
/* Record the new mapping. */
(void)xen_irq_info_virq_setup(cpu, irq, evtchn, virq);
bind_evtchn_to_cpu(evtchn, cpu);
}
}
static void restore_cpu_ipis(unsigned int cpu)
{
struct evtchn_bind_ipi bind_ipi;
int ipi, irq, evtchn;
for (ipi = 0; ipi < XEN_NR_IPIS; ipi++) {
if ((irq = per_cpu(ipi_to_irq, cpu)[ipi]) == -1)
continue;
BUG_ON(ipi_from_irq(irq) != ipi);
/* Get a new binding from Xen. */
bind_ipi.vcpu = cpu;
if (HYPERVISOR_event_channel_op(EVTCHNOP_bind_ipi,
&bind_ipi) != 0)
BUG();
evtchn = bind_ipi.port;
/* Record the new mapping. */
(void)xen_irq_info_ipi_setup(cpu, irq, evtchn, ipi);
bind_evtchn_to_cpu(evtchn, cpu);
}
}
/* Clear an irq's pending state, in preparation for polling on it */
void xen_clear_irq_pending(int irq)
{
int evtchn = evtchn_from_irq(irq);
if (VALID_EVTCHN(evtchn))
clear_evtchn(evtchn);
}
EXPORT_SYMBOL(xen_clear_irq_pending);
void xen_set_irq_pending(int irq)
{
int evtchn = evtchn_from_irq(irq);
if (VALID_EVTCHN(evtchn))
set_evtchn(evtchn);
}
bool xen_test_irq_pending(int irq)
{
int evtchn = evtchn_from_irq(irq);
bool ret = false;
if (VALID_EVTCHN(evtchn))
ret = test_evtchn(evtchn);
return ret;
}
/* Poll waiting for an irq to become pending with timeout. In the usual case,
* the irq will be disabled so it won't deliver an interrupt. */
void xen_poll_irq_timeout(int irq, u64 timeout)
{
evtchn_port_t evtchn = evtchn_from_irq(irq);
if (VALID_EVTCHN(evtchn)) {
struct sched_poll poll;
poll.nr_ports = 1;
poll.timeout = timeout;
set_xen_guest_handle(poll.ports, &evtchn);
if (HYPERVISOR_sched_op(SCHEDOP_poll, &poll) != 0)
BUG();
}
}
EXPORT_SYMBOL(xen_poll_irq_timeout);
/* Poll waiting for an irq to become pending. In the usual case, the
* irq will be disabled so it won't deliver an interrupt. */
void xen_poll_irq(int irq)
{
xen_poll_irq_timeout(irq, 0 /* no timeout */);
}
/* Check whether the IRQ line is shared with other guests. */
int xen_test_irq_shared(int irq)
{
struct irq_info *info = info_for_irq(irq);
struct physdev_irq_status_query irq_status;
if (WARN_ON(!info))
return -ENOENT;
irq_status.irq = info->u.pirq.pirq;
if (HYPERVISOR_physdev_op(PHYSDEVOP_irq_status_query, &irq_status))
return 0;
return !(irq_status.flags & XENIRQSTAT_shared);
}
EXPORT_SYMBOL_GPL(xen_test_irq_shared);
void xen_irq_resume(void)
{
unsigned int cpu;
struct irq_info *info;
/* New event-channel space is not 'live' yet. */
xen_evtchn_mask_all();
xen_evtchn_resume();
/* No IRQ <-> event-channel mappings. */
list_for_each_entry(info, &xen_irq_list_head, list)
info->evtchn = 0; /* zap event-channel binding */
clear_evtchn_to_irq_all();
for_each_possible_cpu(cpu) {
restore_cpu_virqs(cpu);
restore_cpu_ipis(cpu);
}
restore_pirqs();
}
static struct irq_chip xen_dynamic_chip __read_mostly = {
.name = "xen-dyn",
.irq_disable = disable_dynirq,
.irq_mask = disable_dynirq,
.irq_unmask = enable_dynirq,
.irq_ack = ack_dynirq,
.irq_mask_ack = mask_ack_dynirq,
.irq_set_affinity = set_affinity_irq,
.irq_retrigger = retrigger_dynirq,
};
static struct irq_chip xen_pirq_chip __read_mostly = {
.name = "xen-pirq",
.irq_startup = startup_pirq,
.irq_shutdown = shutdown_pirq,
.irq_enable = enable_pirq,
.irq_disable = disable_pirq,
.irq_mask = disable_dynirq,
.irq_unmask = enable_dynirq,
.irq_ack = eoi_pirq,
.irq_eoi = eoi_pirq,
.irq_mask_ack = mask_ack_pirq,
.irq_set_affinity = set_affinity_irq,
.irq_retrigger = retrigger_dynirq,
};
static struct irq_chip xen_percpu_chip __read_mostly = {
.name = "xen-percpu",
.irq_disable = disable_dynirq,
.irq_mask = disable_dynirq,
.irq_unmask = enable_dynirq,
.irq_ack = ack_dynirq,
};
int xen_set_callback_via(uint64_t via)
{
struct xen_hvm_param a;
a.domid = DOMID_SELF;
a.index = HVM_PARAM_CALLBACK_IRQ;
a.value = via;
return HYPERVISOR_hvm_op(HVMOP_set_param, &a);
}
EXPORT_SYMBOL_GPL(xen_set_callback_via);
#ifdef CONFIG_XEN_PVHVM
/* Vector callbacks are better than PCI interrupts to receive event
* channel notifications because we can receive vector callbacks on any
* vcpu and we don't need PCI support or APIC interactions. */
void xen_callback_vector(void)
{
int rc;
uint64_t callback_via;
if (xen_have_vector_callback) {
callback_via = HVM_CALLBACK_VECTOR(HYPERVISOR_CALLBACK_VECTOR);
rc = xen_set_callback_via(callback_via);
if (rc) {
pr_err("Request for Xen HVM callback vector failed\n");
xen_have_vector_callback = 0;
return;
}
pr_info("Xen HVM callback vector for event delivery is enabled\n");
/* in the restore case the vector has already been allocated */
if (!test_bit(HYPERVISOR_CALLBACK_VECTOR, used_vectors))
alloc_intr_gate(HYPERVISOR_CALLBACK_VECTOR,
xen_hvm_callback_vector);
}
}
#else
void xen_callback_vector(void) {}
#endif
#undef MODULE_PARAM_PREFIX
#define MODULE_PARAM_PREFIX "xen."
static bool fifo_events = true;
module_param(fifo_events, bool, 0);
void __init xen_init_IRQ(void)
{
int ret = -EINVAL;
if (fifo_events)
ret = xen_evtchn_fifo_init();
if (ret < 0)
xen_evtchn_2l_init();
evtchn_to_irq = kcalloc(EVTCHN_ROW(xen_evtchn_max_channels()),
sizeof(*evtchn_to_irq), GFP_KERNEL);
BUG_ON(!evtchn_to_irq);
/* No event channels are 'live' right now. */
xen_evtchn_mask_all();
pirq_needs_eoi = pirq_needs_eoi_flag;
#ifdef CONFIG_X86
if (xen_pv_domain()) {
irq_ctx_init(smp_processor_id());
if (xen_initial_domain())
pci_xen_initial_domain();
}
if (xen_feature(XENFEAT_hvm_callback_vector))
xen_callback_vector();
if (xen_hvm_domain()) {
native_init_IRQ();
/* pci_xen_hvm_init must be called after native_init_IRQ so that
* __acpi_register_gsi can point at the right function */
pci_xen_hvm_init();
} else {
int rc;
struct physdev_pirq_eoi_gmfn eoi_gmfn;
pirq_eoi_map = (void *)__get_free_page(GFP_KERNEL|__GFP_ZERO);
eoi_gmfn.gmfn = virt_to_mfn(pirq_eoi_map);
rc = HYPERVISOR_physdev_op(PHYSDEVOP_pirq_eoi_gmfn_v2, &eoi_gmfn);
/* TODO: No PVH support for PIRQ EOI */
if (rc != 0) {
free_page((unsigned long) pirq_eoi_map);
pirq_eoi_map = NULL;
} else
pirq_needs_eoi = pirq_check_eoi_map;
}
#endif
}
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